LMFDB - Embedded newform 108.6.i.a.13.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,6,Mod(13,108)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 8]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("108.13");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	108.i (of order $9$, degree $6$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.3214525398$
Analytic rank:	$0$
Dimension:	$90$
Relative dimension:	$15$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			13.13
Character	$\chi$	$=$	108.13
Dual form			108.6.i.a.25.13

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(13.5505 - 7.70608i) q^{3} +(23.4643 + 19.6889i) q^{5} +(-110.864 - 40.3512i) q^{7} +(124.233 - 208.843i) q^{9} +O(q^{10})$	\(q+(13.5505 - 7.70608i) q^{3} +(23.4643 + 19.6889i) q^{5} +(-110.864 - 40.3512i) q^{7} +(124.233 - 208.843i) q^{9} +(379.679 - 318.588i) q^{11} +(30.4951 - 172.946i) q^{13} +(469.678 + 85.9766i) q^{15} +(588.286 - 1018.94i) q^{17} +(450.271 + 779.893i) q^{19} +(-1813.21 + 307.548i) q^{21} +(-546.936 + 199.068i) q^{23} +(-379.729 - 2153.55i) q^{25} +(74.0541 - 3787.27i) q^{27} +(-200.718 - 1138.33i) q^{29} +(4221.86 - 1536.63i) q^{31} +(2689.77 - 7242.87i) q^{33} +(-1806.88 - 3129.61i) q^{35} +(-2211.22 + 3829.94i) q^{37} +(-919.514 - 2578.51i) q^{39} +(-1169.17 + 6630.67i) q^{41} +(15982.3 - 13410.7i) q^{43} +(7026.92 - 2454.35i) q^{45} +(5769.67 + 2099.99i) q^{47} +(-2212.30 - 1856.34i) q^{49} +(119.527 - 18340.6i) q^{51} -12237.5 q^{53} +15181.6 q^{55} +(12111.3 + 7098.11i) q^{57} +(-18769.2 - 15749.2i) q^{59} +(-2036.37 - 741.177i) q^{61} +(-22200.0 + 18140.2i) q^{63} +(4120.67 - 3457.65i) q^{65} +(-6908.37 + 39179.3i) q^{67} +(-5877.22 + 6912.22i) q^{69} +(-34866.1 + 60389.9i) q^{71} +(15482.6 + 26816.7i) q^{73} +(-21741.0 - 26255.5i) q^{75} +(-54948.1 + 19999.5i) q^{77} +(13735.2 + 77896.4i) q^{79} +(-28181.6 - 51890.1i) q^{81} +(20136.4 + 114199. i) q^{83} +(33865.6 - 12326.1i) q^{85} +(-11491.9 - 13878.1i) q^{87} +(14646.8 + 25369.0i) q^{89} +(-10359.4 + 17943.0i) q^{91} +(45366.9 - 53356.1i) q^{93} +(-4789.92 + 27165.0i) q^{95} +(56340.4 - 47275.2i) q^{97} +(-19366.4 - 118872. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 90 q - 87 q^{5} + 330 q^{9}+O(q^{10}) $ 90 * q - 87 * q^5 + 330 * q^9	$ 90 q - 87 q^{5} + 330 q^{9} - 1257 q^{11} + 531 q^{15} - 3468 q^{17} + 12894 q^{21} + 8106 q^{23} + 4959 q^{25} - 17415 q^{27} + 3468 q^{29} - 6651 q^{31} + 33624 q^{33} - 8229 q^{35} - 10545 q^{39} + 68673 q^{41} + 9459 q^{43} - 53469 q^{45} - 57087 q^{47} - 5490 q^{49} + 42831 q^{51} - 4146 q^{53} + 24624 q^{57} + 5388 q^{59} + 70110 q^{61} - 98115 q^{63} - 172425 q^{65} - 15039 q^{67} + 251037 q^{69} + 67812 q^{71} - 27009 q^{73} - 75273 q^{75} + 23991 q^{77} - 216180 q^{79} + 177822 q^{81} - 76725 q^{83} - 53100 q^{85} - 201483 q^{87} - 98814 q^{89} - 90999 q^{91} + 21765 q^{93} - 143490 q^{95} - 71739 q^{97} + 13635 q^{99}+O(q^{100}) $ 90 * q - 87 * q^5 + 330 * q^9 - 1257 * q^11 + 531 * q^15 - 3468 * q^17 + 12894 * q^21 + 8106 * q^23 + 4959 * q^25 - 17415 * q^27 + 3468 * q^29 - 6651 * q^31 + 33624 * q^33 - 8229 * q^35 - 10545 * q^39 + 68673 * q^41 + 9459 * q^43 - 53469 * q^45 - 57087 * q^47 - 5490 * q^49 + 42831 * q^51 - 4146 * q^53 + 24624 * q^57 + 5388 * q^59 + 70110 * q^61 - 98115 * q^63 - 172425 * q^65 - 15039 * q^67 + 251037 * q^69 + 67812 * q^71 - 27009 * q^73 - 75273 * q^75 + 23991 * q^77 - 216180 * q^79 + 177822 * q^81 - 76725 * q^83 - 53100 * q^85 - 201483 * q^87 - 98814 * q^89 - 90999 * q^91 + 21765 * q^93 - 143490 * q^95 - 71739 * q^97 + 13635 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{4}{9}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	13.5505	−	7.70608i	0.869265	−	0.494346i
$3$	13.5505	−	7.70608i	0.869265	−	0.494346i
$4$		0			0
$5$	23.4643	+	19.6889i	0.419743	+	0.352206i	0.828065	−	0.560632i	$-0.189442\pi$
$5$	23.4643	+	19.6889i	0.419743	+	0.352206i	−0.408323	+	0.912838i	$0.633886\pi$
$6$		0			0
$7$	−110.864	−	40.3512i	−0.855156	−	0.311251i	−0.123015	−	0.992405i	$-0.539256\pi$
$7$	−110.864	−	40.3512i	−0.855156	−	0.311251i	−0.732141	+	0.681153i	$0.761479\pi$
$8$		0			0
$9$	124.233	−	208.843i	0.511245	−	0.859435i
$10$		0			0
$11$	379.679	−	318.588i	0.946094	−	0.793867i	−0.0325411	−	0.999470i	$-0.510360\pi$
$11$	379.679	−	318.588i	0.946094	−	0.793867i	0.978635	+	0.205603i	$0.0659155\pi$
$12$		0			0
$13$	30.4951	−	172.946i	0.0500462	−	0.283826i	−0.949506	−	0.313749i	$-0.898415\pi$
$13$	30.4951	−	172.946i	0.0500462	−	0.283826i	0.999552	+	0.0299226i	$0.00952608\pi$
$14$		0			0
$15$	469.678	+	85.9766i	0.538979	+	0.0986625i
$16$		0			0
$17$	588.286	−	1018.94i	0.493704	−	0.855120i	−0.506270	−	0.862375i	$-0.668976\pi$
$17$	588.286	−	1018.94i	0.493704	−	0.855120i	0.999974	+	0.00725521i	$0.00230942\pi$
$18$		0			0
$19$	450.271	+	779.893i	0.286148	+	0.495622i	0.972887	−	0.231282i	$-0.0742919\pi$
$19$	450.271	+	779.893i	0.286148	+	0.495622i	−0.686739	+	0.726904i	$0.740959\pi$
$20$		0			0
$21$	−1813.21	+	307.548i	−0.897224	+	0.152183i
$22$		0			0
$23$	−546.936	+	199.068i	−0.215584	+	0.0784663i	−0.447555	−	0.894257i	$-0.647705\pi$
$23$	−546.936	+	199.068i	−0.215584	+	0.0784663i	0.231970	+	0.972723i	$0.425483\pi$
$24$		0			0
$25$	−379.729	−	2153.55i	−0.121513	−	0.689136i
$26$		0			0
$27$	74.0541	−	3787.27i	0.0195497	−	0.999809i
$28$		0			0
$29$	−200.718	−	1138.33i	−0.0443190	−	0.251346i	0.954597	−	0.297901i	$-0.0962867\pi$
$29$	−200.718	−	1138.33i	−0.0443190	−	0.251346i	−0.998916	+	0.0465556i	$0.985176\pi$
$30$		0			0
$31$	4221.86	−	1536.63i	0.789041	−	0.287187i	0.0841036	−	0.996457i	$-0.473197\pi$
$31$	4221.86	−	1536.63i	0.789041	−	0.287187i	0.704937	+	0.709270i	$0.250975\pi$
$32$		0			0
$33$	2689.77	−	7242.87i	0.429962	−	1.15778i
$34$		0			0
$35$	−1806.88	−	3129.61i	−0.249321	−	0.431837i
$36$		0			0
$37$	−2211.22	+	3829.94i	−0.265538	+	0.459926i	−0.967704	−	0.252087i	$-0.918883\pi$
$37$	−2211.22	+	3829.94i	−0.265538	+	0.459926i	0.702166	+	0.712013i	$0.252216\pi$
$38$		0			0
$39$	−919.514	−	2578.51i	−0.0968048	−	0.271461i
$40$		0			0
$41$	−1169.17	+	6630.67i	−0.108622	+	0.616024i	0.881090	+	0.472948i	$0.156810\pi$
$41$	−1169.17	+	6630.67i	−0.108622	+	0.616024i	−0.989712	+	0.143076i	$0.954301\pi$
$42$		0			0
$43$	15982.3	−	13410.7i	1.31816	−	1.10606i	0.331464	−	0.943468i	$-0.392458\pi$
$43$	15982.3	−	13410.7i	1.31816	−	1.10606i	0.986693	−	0.162596i	$-0.0519869\pi$
$44$		0			0
$45$	7026.92	−	2454.35i	0.517289	−	0.180678i
$46$		0			0
$47$	5769.67	+	2099.99i	0.380983	+	0.138667i	0.525411	−	0.850849i	$-0.323912\pi$
$47$	5769.67	+	2099.99i	0.380983	+	0.138667i	−0.144427	+	0.989515i	$0.546134\pi$
$48$		0			0
$49$	−2212.30	−	1856.34i	−0.131630	−	0.110450i
$50$		0			0
$51$	119.527	−	18340.6i	0.00643487	−	0.987386i
$52$		0			0
$53$	−12237.5			−0.598416			−0.299208	−	0.954188i	$-0.596723\pi$
$53$	−12237.5			−0.598416			−0.299208	+	0.954188i	$0.596723\pi$
$54$		0			0
$55$	15181.6			0.676721
$56$		0			0
$57$	12111.3	+	7098.11i	0.493747	+	0.289371i
$58$		0			0
$59$	−18769.2	−	15749.2i	−0.701964	−	0.589018i	0.220367	−	0.975417i	$-0.429274\pi$
$59$	−18769.2	−	15749.2i	−0.701964	−	0.589018i	−0.922332	+	0.386399i	$0.873719\pi$
$60$		0			0
$61$	−2036.37	−	741.177i	−0.0700699	−	0.0255034i	0.306747	−	0.951791i	$-0.400759\pi$
$61$	−2036.37	−	741.177i	−0.0700699	−	0.0255034i	−0.376817	+	0.926288i	$0.622982\pi$
$62$		0			0
$63$	−22200.0	+	18140.2i	−0.704695	+	0.575826i
$64$		0			0
$65$	4120.67	−	3457.65i	0.120972	−	0.101507i
$66$		0			0
$67$	−6908.37	+	39179.3i	−0.188013	+	1.06628i	0.734009	+	0.679140i	$0.237647\pi$
$67$	−6908.37	+	39179.3i	−0.188013	+	1.06628i	−0.922022	+	0.387137i	$0.873464\pi$
$68$		0			0
$69$	−5877.22	+	6912.22i	−0.148611	+	0.174781i
$70$		0			0
$71$	−34866.1	+	60389.9i	−0.820839	+	1.42173i	0.0842201	+	0.996447i	$0.473160\pi$
$71$	−34866.1	+	60389.9i	−0.820839	+	1.42173i	−0.905059	+	0.425287i	$0.860173\pi$
$72$		0			0
$73$	15482.6	+	26816.7i	0.340046	+	0.588977i	0.984441	−	0.175716i	$-0.0562240\pi$
$73$	15482.6	+	26816.7i	0.340046	+	0.588977i	−0.644395	+	0.764693i	$0.722891\pi$
$74$		0			0
$75$	−21741.0	−	26255.5i	−0.446299	−	0.538973i
$76$		0			0
$77$	−54948.1	+	19999.5i	−1.05615	+	0.384408i
$78$		0			0
$79$	13735.2	+	77896.4i	0.247610	+	1.40427i	0.814352	+	0.580372i	$0.197093\pi$
$79$	13735.2	+	77896.4i	0.247610	+	1.40427i	−0.566741	+	0.823896i	$0.691796\pi$
$80$		0			0
$81$	−28181.6	−	51890.1i	−0.477257	−	0.878764i
$82$		0			0
$83$	20136.4	+	114199.i	0.320838	+	1.81956i	0.537439	+	0.843303i	$0.319392\pi$
$83$	20136.4	+	114199.i	0.320838	+	1.81956i	−0.216601	+	0.976260i	$0.569497\pi$
$84$		0			0
$85$	33865.6	−	12326.1i	0.508407	−	0.185045i
$86$		0			0
$87$	−11491.9	−	13878.1i	−0.162777	−	0.196577i
$88$		0			0
$89$	14646.8	+	25369.0i	0.196005	+	0.339491i	0.947230	−	0.320556i	$-0.103870\pi$
$89$	14646.8	+	25369.0i	0.196005	+	0.339491i	−0.751224	+	0.660047i	$0.770536\pi$
$90$		0			0
$91$	−10359.4	+	17943.0i	−0.131139	+	0.227139i
$92$		0			0
$93$	45366.9	−	53356.1i	0.543916	−	0.639701i
$94$		0			0
$95$	−4789.92	+	27165.0i	−0.0544527	+	0.308817i
$96$		0			0
$97$	56340.4	−	47275.2i	0.607982	−	0.510158i	−0.286018	−	0.958224i	$-0.592332\pi$
$97$	56340.4	−	47275.2i	0.607982	−	0.510158i	0.894000	+	0.448067i	$0.147887\pi$
$98$		0			0
$99$	−19366.4	−	118872.i	−0.198592	−	1.21897i
$100$		0			0
$101$	−173238.	−	63053.6i	−1.68982	−	0.615044i	−0.695218	−	0.718799i	$-0.744692\pi$
$101$	−173238.	−	63053.6i	−1.68982	−	0.615044i	−0.994603	+	0.103755i	$0.966914\pi$
$102$		0			0
$103$	135083.	+	113348.i	1.25461	+	1.05274i	0.996235	+	0.0866994i	$0.0276320\pi$
$103$	135083.	+	113348.i	1.25461	+	1.05274i	0.258377	+	0.966044i	$0.416812\pi$
$104$		0			0
$105$	−48601.1	−	28483.8i	−0.430203	−	0.252130i
$106$		0			0
$107$	−3792.67			−0.0320248			−0.0160124	−	0.999872i	$-0.505097\pi$
$107$	−3792.67			−0.0320248			−0.0160124	+	0.999872i	$0.505097\pi$
$108$		0			0
$109$	133286.			1.07453			0.537264	−	0.843414i	$-0.319458\pi$
$109$	133286.			1.07453			0.537264	+	0.843414i	$0.319458\pi$
$110$		0			0
$111$	−449.270	+	68937.5i	−0.00346099	+	0.531065i
$112$		0			0
$113$	15648.2	+	13130.4i	0.115284	+	0.0967345i	0.698607	−	0.715505i	$-0.253804\pi$
$113$	15648.2	+	13130.4i	0.115284	+	0.0967345i	−0.583323	+	0.812240i	$0.698248\pi$
$114$		0			0
$115$	−16752.9	−	6097.57i	−0.118126	−	0.0429944i
$116$		0			0
$117$	−32330.1	−	27854.2i	−0.218344	−	0.188116i
$118$		0			0
$119$	−106335.	+	89225.9i	−0.688351	+	0.577595i
$120$		0			0
$121$	14691.2	−	83318.0i	0.0912209	−	0.517339i
$122$		0			0
$123$	35253.7	+	98858.6i	0.210108	+	0.589185i
$124$		0			0
$125$	81351.1	−	140904.i	0.465681	−	0.806583i
$126$		0			0
$127$	13990.5	+	24232.3i	0.0769705	+	0.133317i	0.901941	−	0.431858i	$-0.142142\pi$
$127$	13990.5	+	24232.3i	0.0769705	+	0.133317i	−0.824971	+	0.565175i	$0.808809\pi$
$128$		0			0
$129$	113224.	−	304882.i	0.599050	−	1.61309i
$130$		0			0
$131$	−150971.	+	54948.9i	−0.768626	+	0.279757i	−0.696421	−	0.717633i	$-0.745226\pi$
$131$	−150971.	+	54948.9i	−0.768626	+	0.279757i	−0.0722042	+	0.997390i	$0.523003\pi$
$132$		0			0
$133$	−18449.3	−	104631.i	−0.0904378	−	0.512898i
$134$		0			0
$135$	76304.9	−	87407.7i	0.360344	−	0.412777i
$136$		0			0
$137$	18621.7	+	105609.i	0.0847652	+	0.480727i	0.997407	+	0.0719670i	$0.0229276\pi$
$137$	18621.7	+	105609.i	0.0847652	+	0.480727i	−0.912642	+	0.408760i	$0.865961\pi$
$138$		0			0
$139$	−109574.	+	39881.7i	−0.481029	+	0.175080i	−0.571142	−	0.820851i	$-0.693499\pi$
$139$	−109574.	+	39881.7i	−0.481029	+	0.175080i	0.0901130	+	0.995932i	$0.471277\pi$
$140$		0			0
$141$	94364.6	−	16005.6i	0.399725	−	0.0677993i
$142$		0			0
$143$	−43520.3	−	75379.4i	−0.177972	−	0.308257i
$144$		0			0
$145$	17702.7	−	30662.0i	0.0699229	−	0.121110i
$146$		0			0
$147$	−44282.9	−	8106.18i	−0.169022	−	0.0309402i
$148$		0			0
$149$	3336.53	−	18922.4i	0.0123120	−	0.0698250i	−0.978033	−	0.208452i	$-0.933158\pi$
$149$	3336.53	−	18922.4i	0.0123120	−	0.0698250i	0.990345	+	0.138627i	$0.0442688\pi$
$150$		0			0
$151$	73065.6	−	61309.3i	0.260778	−	0.218819i	−0.503019	−	0.864275i	$-0.667777\pi$
$151$	73065.6	−	61309.3i	0.260778	−	0.218819i	0.763797	+	0.645457i	$0.223333\pi$
$152$		0			0
$153$	−139714.	−	249445.i	−0.482516	−	0.861482i
$154$		0			0
$155$	129318.	+	47067.8i	0.432343	+	0.157360i
$156$		0			0
$157$	−270136.	−	226671.i	−0.874650	−	0.733918i	0.0904223	−	0.995904i	$-0.471178\pi$
$157$	−270136.	−	226671.i	−0.874650	−	0.733918i	−0.965072	+	0.261985i	$0.915623\pi$
$158$		0			0
$159$	−165825.	+	94303.3i	−0.520183	+	0.295824i
$160$		0			0
$161$	68668.2			0.208781
$162$		0			0
$163$	653167.			1.92555			0.962776	−	0.270299i	$-0.0871227\pi$
$163$	653167.			1.92555			0.962776	+	0.270299i	$0.0871227\pi$
$164$		0			0
$165$	205718.	−	116990.i	0.588250	−	0.334534i
$166$		0			0
$167$	−227439.	−	190844.i	−0.631065	−	0.529527i	0.270194	−	0.962806i	$-0.412912\pi$
$167$	−227439.	−	190844.i	−0.631065	−	0.529527i	−0.901260	+	0.433279i	$0.857356\pi$
$168$		0			0
$169$	319921.	+	116442.i	0.861640	+	0.313611i
$170$		0			0
$171$	218813.	+	2852.16i	0.572247	+	0.00745906i
$172$		0			0
$173$	308448.	−	258818.i	0.783549	−	0.657476i	−0.160591	−	0.987021i	$-0.551340\pi$
$173$	308448.	−	258818.i	0.783549	−	0.657476i	0.944140	+	0.329545i	$0.106895\pi$
$174$		0			0
$175$	−44800.1	+	254074.i	−0.110582	+	0.627140i
$176$		0			0
$177$	−375697.	−	68772.9i	−0.901372	−	0.165000i
$178$		0			0
$179$	133815.	−	231774.i	0.312155	−	0.540669i	−0.666673	−	0.745350i	$-0.732282\pi$
$179$	133815.	−	231774.i	0.312155	−	0.540669i	0.978829	+	0.204681i	$0.0656157\pi$
$180$		0			0
$181$	72760.7	+	126025.i	0.165082	+	0.285931i	0.936684	−	0.350174i	$-0.113878\pi$
$181$	72760.7	+	126025.i	0.165082	+	0.285931i	−0.771602	+	0.636105i	$0.780544\pi$
$182$		0			0
$183$	−33305.4	+	5649.09i	−0.0735168	+	0.0124696i
$184$		0			0
$185$	−127292.	+	46330.5i	−0.273446	+	0.0995263i
$186$		0			0
$187$	−101263.	−	574291.i	−0.211762	−	1.20096i
$188$		0			0
$189$	−161031.	+	416884.i	−0.327910	+	0.848908i
$190$		0			0
$191$	135364.	+	767687.i	0.268485	+	1.52265i	0.758925	+	0.651178i	$0.225725\pi$
$191$	135364.	+	767687.i	0.268485	+	1.52265i	−0.490441	+	0.871475i	$0.663164\pi$
$192$		0			0
$193$	−864409.	+	314619.i	−1.67042	+	0.607984i	−0.991949	−	0.126641i	$-0.959580\pi$
$193$	−864409.	+	314619.i	−1.67042	+	0.607984i	−0.678473	+	0.734625i	$0.737358\pi$
$194$		0			0
$195$	29192.2	−	78607.2i	0.0549769	−	0.148039i
$196$		0			0
$197$	−142186.	−	246273.i	−0.261030	−	0.452117i	0.705486	−	0.708724i	$-0.250729\pi$
$197$	−142186.	−	246273.i	−0.261030	−	0.452117i	−0.966516	+	0.256607i	$0.917395\pi$
$198$		0			0
$199$	−299147.	+	518138.i	−0.535490	+	0.927497i	0.463649	+	0.886019i	$0.346540\pi$
$199$	−299147.	+	518138.i	−0.535490	+	0.927497i	−0.999139	+	0.0414777i	$0.986793\pi$
$200$		0			0
$201$	208307.	+	584136.i	0.363675	+	1.01982i
$202$		0			0
$203$	−23680.5	+	134299.i	−0.0403320	+	0.228734i
$204$		0			0
$205$	−157984.	+	132565.i	−0.262561	+	0.220314i
$206$		0			0
$207$	−26373.3	+	138954.i	−0.0427797	+	0.225396i
$208$		0			0
$209$	419423.	+	152657.i	0.664181	+	0.241742i
$210$		0			0
$211$	−396361.	−	332586.i	−0.612893	−	0.514278i	0.282668	−	0.959218i	$-0.408781\pi$
$211$	−396361.	−	332586.i	−0.612893	−	0.514278i	−0.895560	+	0.444940i	$0.853225\pi$
$212$		0			0
$213$	−7084.02	+	1.08699e6i	−0.0106987	+	1.64164i
$214$		0			0
$215$	639055.			0.942849
$216$		0			0
$217$	−530057.			−0.764141
$218$		0			0
$219$	416450.	+	244070.i	0.586749	+	0.343877i
$220$		0			0
$221$	−158282.	−	132815.i	−0.217998	−	0.182922i
$222$		0			0
$223$	−718009.	−	261334.i	−0.966870	−	0.351912i	−0.190148	−	0.981755i	$-0.560897\pi$
$223$	−718009.	−	261334.i	−0.966870	−	0.351912i	−0.776722	+	0.629844i	$0.783119\pi$
$224$		0			0
$225$	−496928.	−	188237.i	−0.654391	−	0.247885i
$226$		0			0
$227$	−270626.	+	227082.i	−0.348582	+	0.292495i	−0.800220	−	0.599706i	$-0.795284\pi$
$227$	−270626.	+	227082.i	−0.348582	+	0.292495i	0.451638	+	0.892201i	$0.350840\pi$
$228$		0			0
$229$	−153056.	+	868024.i	−0.192869	+	1.09381i	0.722552	+	0.691316i	$0.242969\pi$
$229$	−153056.	+	868024.i	−0.192869	+	1.09381i	−0.915421	+	0.402497i	$0.868142\pi$
$230$		0			0
$231$	−590457.	+	694438.i	−0.728045	+	0.856256i
$232$		0			0
$233$	651067.	−	1.12768e6i	0.785663	−	1.36081i	−0.142940	−	0.989731i	$-0.545656\pi$
$233$	651067.	−	1.12768e6i	0.785663	−	1.36081i	0.928602	−	0.371076i	$-0.121011\pi$
$234$		0			0
$235$	94034.9	+	162873.i	0.111076	+	0.192389i
$236$		0			0
$237$	786396.	+	949691.i	0.909432	+	1.09828i
$238$		0			0
$239$	−704867.	+	256551.i	−0.798202	+	0.290522i	−0.708741	−	0.705469i	$-0.750737\pi$
$239$	−704867.	+	256551.i	−0.798202	+	0.290522i	−0.0894607	+	0.995990i	$0.528514\pi$
$240$		0			0
$241$	−306811.	−	1.74001e6i	−0.340274	−	1.92979i	−0.367181	−	0.930149i	$-0.619677\pi$
$241$	−306811.	−	1.74001e6i	−0.340274	−	1.92979i	0.0269075	−	0.999638i	$-0.491434\pi$
$242$		0			0
$243$	−781744.	−	485968.i	−0.849276	−	0.527949i
$244$		0			0
$245$	−15360.8	−	87115.5i	−0.0163493	−	0.0927215i
$246$		0			0
$247$	148611.	−	54089.8i	0.154991	−	0.0564122i
$248$		0			0
$249$	1.15289e6	+	1.39228e6i	1.17839	+	1.42308i
$250$		0			0
$251$	95464.9	+	165350.i	0.0956444	+	0.165661i	0.909877	−	0.414877i	$-0.136176\pi$
$251$	95464.9	+	165350.i	0.0956444	+	0.165661i	−0.814233	+	0.580538i	$0.802842\pi$
$252$		0			0
$253$	−144239.	+	249829.i	−0.141671	+	0.245382i
$254$		0			0
$255$	363910.	−	427996.i	0.350464	−	0.412182i
$256$		0			0
$257$	240768.	−	1.36546e6i	0.227387	−	1.28958i	−0.630682	−	0.776041i	$-0.717225\pi$
$257$	240768.	−	1.36546e6i	0.227387	−	1.28958i	0.858069	−	0.513534i	$-0.171664\pi$
$258$		0			0
$259$	399687.	−	335377.i	0.370229	−	0.310659i
$260$		0			0
$261$	−262667.	−	99498.7i	−0.238673	−	0.0904099i
$262$		0			0
$263$	1.16455e6	+	423860.i	1.03817	+	0.377862i	0.804185	−	0.594379i	$-0.202602\pi$
$263$	1.16455e6	+	423860.i	1.03817	+	0.377862i	0.233982	+	0.972241i	$0.424824\pi$
$264$		0			0
$265$	−287145.	−	240943.i	−0.251181	−	0.210766i
$266$		0			0
$267$	393968.	+	230894.i	0.338207	+	0.198214i
$268$		0			0
$269$	1.44877e6			1.22073			0.610364	−	0.792121i	$-0.291023\pi$
$269$	1.44877e6			1.22073			0.610364	+	0.792121i	$0.291023\pi$
$270$		0			0
$271$	1.46988e6			1.21579			0.607895	−	0.794018i	$-0.292014\pi$
$271$	1.46988e6			1.21579			0.607895	+	0.794018i	$0.292014\pi$
$272$		0			0
$273$	−2104.80	+	322967.i	−0.00170924	+	0.262272i
$274$		0			0
$275$	−830271.	−	696680.i	−0.662046	−	0.555522i
$276$		0			0
$277$	−1.18834e6	−	432519.i	−0.930550	−	0.338693i	−0.168123	−	0.985766i	$-0.553770\pi$
$277$	−1.18834e6	−	432519.i	−0.930550	−	0.338693i	−0.762428	+	0.647073i	$0.775993\pi$
$278$		0			0
$279$	203578.	−	1.07260e6i	0.156574	−	0.824952i
$280$		0			0
$281$	−284459.	+	238689.i	−0.214908	+	0.180330i	−0.743887	−	0.668306i	$-0.767020\pi$
$281$	−284459.	+	238689.i	−0.214908	+	0.180330i	0.528978	+	0.848635i	$0.322575\pi$
$282$		0			0
$283$	40598.0	−	230243.i	0.0301328	−	0.170891i	−0.966028	−	0.258439i	$-0.916792\pi$
$283$	40598.0	−	230243.i	0.0301328	−	0.170891i	0.996160	+	0.0875476i	$0.0279030\pi$
$284$		0			0
$285$	144430.	+	405011.i	0.105328	+	0.295362i
$286$		0			0
$287$	397174.	−	687925.i	0.284627	−	0.492988i
$288$		0			0
$289$	17767.3	+	30773.8i	0.0125134	+	0.0216739i
$290$		0			0
$291$	399135.	−	1.07477e6i	0.276304	−	0.744016i
$292$		0			0
$293$	23222.4	−	8452.27i	0.0158030	−	0.00575181i	−0.334107	−	0.942535i	$-0.608435\pi$
$293$	23222.4	−	8452.27i	0.0158030	−	0.00575181i	0.349910	+	0.936783i	$0.386212\pi$
$294$		0			0
$295$	−130321.	−	739089.i	−0.0871888	−	0.494472i
$296$		0			0
$297$	−1.17846e6	−	1.46154e6i	−0.775220	−	0.961433i
$298$		0			0
$299$	17749.3	+	100661.i	0.0114816	+	0.0651154i
$300$		0			0
$301$	−2.31299e6	+	841861.i	−1.47149	+	0.535580i
$302$		0			0
$303$	−2.83336e6	+	480581.i	−1.77295	+	0.300718i
$304$		0			0
$305$	−33189.0	−	57485.1i	−0.0204289	−	0.0353839i
$306$		0			0
$307$	351319.	−	608502.i	0.212743	−	0.368482i	−0.739829	−	0.672795i	$-0.765094\pi$
$307$	351319.	−	608502.i	0.212743	−	0.368482i	0.952572	+	0.304313i	$0.0984270\pi$
$308$		0			0
$309$	2.70392e6	+	494965.i	1.61101	+	0.294902i
$310$		0			0
$311$	306250.	−	1.73683e6i	0.179546	−	1.01825i	−0.753219	−	0.657769i	$-0.771500\pi$
$311$	306250.	−	1.73683e6i	0.179546	−	1.01825i	0.932765	−	0.360485i	$-0.117389\pi$
$312$		0			0
$313$	1.02155e6	−	857185.i	0.589387	−	0.494554i	−0.298628	−	0.954370i	$-0.596529\pi$
$313$	1.02155e6	−	857185.i	0.589387	−	0.494554i	0.888014	+	0.459816i	$0.152084\pi$
$314$		0			0
$315$	−878068.	−	11445.4i	−0.498600	−	0.00649910i
$316$		0			0
$317$	1.83479e6	+	667807.i	1.02550	+	0.373253i	0.799367	−	0.600843i	$-0.205168\pi$
$317$	1.83479e6	+	667807.i	1.02550	+	0.373253i	0.226136	+	0.974096i	$0.427391\pi$
$318$		0			0
$319$	−438865.	−	368252.i	−0.241465	−	0.202613i
$320$		0			0
$321$	−51392.6	+	29226.7i	−0.0278380	+	0.0158313i
$322$		0			0
$323$	1.05955e6			0.565088
$324$		0			0
$325$	−384028.			−0.201676
$326$		0			0
$327$	1.80609e6	−	1.02711e6i	0.934050	−	0.531188i
$328$		0			0
$329$	−554911.	−	465626.i	−0.282640	−	0.237163i
$330$		0			0
$331$	95903.7	+	34906.1i	0.0481133	+	0.0175118i	0.365965	−	0.930629i	$-0.380739\pi$
$331$	95903.7	+	34906.1i	0.0481133	+	0.0175118i	−0.317851	+	0.948141i	$0.602961\pi$
$332$		0			0
$333$	525150.	+	937600.i	0.259521	+	0.463348i
$334$		0			0
$335$	−933498.	+	783298.i	−0.454466	+	0.381342i
$336$		0			0
$337$	−173587.	+	984458.i	−0.0832609	+	0.472196i	0.914457	+	0.404682i	$0.132618\pi$
$337$	−173587.	+	984458.i	−0.0832609	+	0.472196i	−0.997718	+	0.0675140i	$0.978493\pi$
$338$		0			0
$339$	313225.	+	57337.2i	0.148032	+	0.0270980i
$340$		0			0
$341$	1.11340e6	−	1.92846e6i	0.518518	−	0.898100i
$342$		0			0
$343$	1.16180e6	+	2.01229e6i	0.533205	+	0.923539i
$344$		0			0
$345$	−273999.	+	46474.4i	−0.123937	+	0.0210216i
$346$		0			0
$347$	−3.70125e6	+	1.34714e6i	−1.65015	+	0.600607i	−0.988771	−	0.149442i	$-0.952252\pi$
$347$	−3.70125e6	+	1.34714e6i	−1.65015	+	0.600607i	−0.661383	+	0.750049i	$0.730030\pi$
$348$		0			0
$349$	214292.	+	1.21531e6i	0.0941766	+	0.534102i	0.994997	+	0.0999091i	$0.0318552\pi$
$349$	214292.	+	1.21531e6i	0.0941766	+	0.534102i	−0.900820	+	0.434193i	$0.857034\pi$
$350$		0			0
$351$	−652736.	−	128301.i	−0.282794	−	0.0555854i
$352$		0			0
$353$	324504.	+	1.84036e6i	0.138607	+	0.786077i	0.972280	+	0.233819i	$0.0751224\pi$
$353$	324504.	+	1.84036e6i	0.138607	+	0.786077i	−0.833674	+	0.552257i	$0.813767\pi$
$354$		0			0
$355$	−2.00712e6	+	730532.i	−0.845284	+	0.307658i
$356$		0			0
$357$	−753315.	+	2.02848e6i	−0.312828	+	0.842367i
$358$		0			0
$359$	−1.74184e6	−	3.01696e6i	−0.713301	−	1.23547i	−0.963611	−	0.267307i	$-0.913866\pi$
$359$	−1.74184e6	−	3.01696e6i	−0.713301	−	1.23547i	0.250311	−	0.968166i	$-0.419467\pi$
$360$		0			0
$361$	832561.	−	1.44204e6i	0.336239	−	0.582383i
$362$		0			0
$363$	−442982.	−	1.24221e6i	−0.176449	−	0.494800i
$364$		0			0
$365$	−164702.	+	934073.i	−0.0647094	+	0.366985i
$366$		0			0
$367$	3.60909e6	−	3.02839e6i	1.39873	−	1.17367i	0.437065	−	0.899430i	$-0.356018\pi$
$367$	3.60909e6	−	3.02839e6i	1.39873	−	1.17367i	0.961661	−	0.274241i	$-0.0884266\pi$
$368$		0			0
$369$	1.23952e6	+	1.06792e6i	0.473900	+	0.408293i
$370$		0			0
$371$	1.35670e6	+	493798.i	0.511740	+	0.186258i
$372$		0			0
$373$	291243.	+	244382.i	0.108389	+	0.0909489i	0.695371	−	0.718651i	$-0.255240\pi$
$373$	291243.	+	244382.i	0.108389	+	0.0909489i	−0.586983	+	0.809599i	$0.699684\pi$
$374$		0			0
$375$	16528.8	−	2.53622e6i	0.00606963	−	0.931342i
$376$		0			0
$377$	−202990.			−0.0735566
$378$		0			0
$379$	4.89796e6			1.75153			0.875764	−	0.482740i	$-0.160358\pi$
$379$	4.89796e6			1.75153			0.875764	+	0.482740i	$0.160358\pi$
$380$		0			0
$381$	376314.	+	220548.i	0.132812	+	0.0778377i
$382$		0			0
$383$	1.36011e6	+	1.14127e6i	0.473780	+	0.397549i	0.848171	−	0.529722i	$-0.177704\pi$
$383$	1.36011e6	+	1.14127e6i	0.473780	+	0.397549i	−0.374391	+	0.927271i	$0.622148\pi$
$384$		0			0
$385$	−1.68309e6	−	612594.i	−0.578702	−	0.210630i
$386$		0			0
$387$	−815212.	−	5.00382e6i	−0.276690	−	1.69834i
$388$		0			0
$389$	−3.31866e6	+	2.78468e6i	−1.11196	+	0.933043i	−0.998171	−	0.0604619i	$-0.980743\pi$
$389$	−3.31866e6	+	2.78468e6i	−1.11196	+	0.933043i	−0.113787	+	0.993505i	$0.536298\pi$
$390$		0			0
$391$	−118916.	+	674405.i	−0.0393367	+	0.223089i
$392$		0			0
$393$	−1.62229e6	+	1.90798e6i	−0.529843	+	0.623150i
$394$		0			0
$395$	−1.21141e6	+	2.09822e6i	−0.390659	+	0.676641i
$396$		0			0
$397$	2.88147e6	+	4.99086e6i	0.917568	+	1.58927i	0.803097	+	0.595848i	$0.203184\pi$
$397$	2.88147e6	+	4.99086e6i	0.917568	+	1.58927i	0.114471	+	0.993427i	$0.463483\pi$
$398$		0			0
$399$	−1.05629e6	−	1.27563e6i	−0.332163	−	0.401137i
$400$		0			0
$401$	−5.51169e6	+	2.00609e6i	−1.71169	+	0.623003i	−0.997069	−	0.0765128i	$-0.975621\pi$
$401$	−5.51169e6	+	2.00609e6i	−1.71169	+	0.623003i	−0.714617	+	0.699516i	$0.753399\pi$
$402$		0			0
$403$	−137009.	−	777014.i	−0.0420228	−	0.238323i
$404$		0			0
$405$	360398.	−	1.77243e6i	0.109181	−	0.536947i
$406$		0			0
$407$	380622.	+	2.15861e6i	0.113896	+	0.645935i
$408$		0			0
$409$	−1.07825e6	+	392452.i	−0.318722	+	0.116005i	−0.496427	−	0.868078i	$-0.665355\pi$
$409$	−1.07825e6	+	392452.i	−0.318722	+	0.116005i	0.177705	+	0.984084i	$0.443133\pi$
$410$		0			0
$411$	1.06616e6	+	1.28755e6i	0.311329	+	0.375976i
$412$		0			0
$413$	1.44533e6	+	2.50338e6i	0.416957	+	0.722190i
$414$		0			0
$415$	−1.77597e6	+	3.07607e6i	−0.506191	+	0.876749i
$416$		0			0
$417$	−1.17745e6	+	1.38481e6i	−0.331592	+	0.389986i
$418$		0			0
$419$	−1.01869e6	+	5.77726e6i	−0.283469	+	1.60763i	0.427234	+	0.904141i	$0.359488\pi$
$419$	−1.01869e6	+	5.77726e6i	−0.283469	+	1.60763i	−0.710703	+	0.703492i	$0.751623\pi$
$420$		0			0
$421$	−2.03140e6	+	1.70454e6i	−0.558585	+	0.468709i	−0.877836	−	0.478962i	$-0.841013\pi$
$421$	−2.03140e6	+	1.70454e6i	−0.558585	+	0.468709i	0.319250	+	0.947670i	$0.396569\pi$
$422$		0			0
$423$	1.15535e6	−	944066.i	0.313951	−	0.256538i
$424$		0			0
$425$	−2.41773e6	−	879982.i	−0.649285	−	0.236321i
$426$		0			0
$427$	195852.	+	164340.i	0.0519828	+	0.0436187i
$428$		0			0
$429$	−1.17060e6	−	686058.i	−0.307090	−	0.179977i
$430$		0			0
$431$	−2.95480e6			−0.766188			−0.383094	−	0.923709i	$-0.625142\pi$
$431$	−2.95480e6			−0.766188			−0.383094	+	0.923709i	$0.625142\pi$
$432$		0			0
$433$	−528654.			−0.135504			−0.0677520	−	0.997702i	$-0.521583\pi$
$433$	−528654.			−0.135504			−0.0677520	+	0.997702i	$0.521583\pi$
$434$		0			0
$435$	3596.80	−	551904.i	0.000911366	−	0.139843i
$436$		0			0
$437$	−401522.	−	336917.i	−0.100579	−	0.0843954i
$438$		0			0
$439$	5.09922e6	+	1.85597e6i	1.26282	+	0.459630i	0.884716	−	0.466131i	$-0.154352\pi$
$439$	5.09922e6	+	1.85597e6i	1.26282	+	0.459630i	0.378108	+	0.925761i	$0.376575\pi$
$440$		0			0
$441$	−662522.	+	231405.i	−0.162220	+	0.0566599i
$442$		0			0
$443$	−1.11654e6	+	936886.i	−0.270311	+	0.226818i	−0.767859	−	0.640618i	$-0.778678\pi$
$443$	−1.11654e6	+	936886.i	−0.270311	+	0.226818i	0.497548	+	0.867436i	$0.334234\pi$
$444$		0			0
$445$	−155811.	+	883647.i	−0.0372990	+	0.211533i
$446$		0			0
$447$	−100606.	−	282120.i	−0.0238153	−	0.0667829i
$448$		0			0
$449$	2.86213e6	−	4.95735e6i	0.669998	−	1.16047i	−0.307906	−	0.951417i	$-0.599628\pi$
$449$	2.86213e6	−	4.95735e6i	0.669998	−	1.16047i	0.977904	−	0.209054i	$-0.0670384\pi$
$450$		0			0
$451$	1.66855e6	+	2.89001e6i	0.386275	+	0.669048i
$452$		0			0
$453$	517621.	−	1.39382e6i	0.118513	−	0.319126i
$454$		0			0
$455$	−596354.	+	217055.i	−0.135044	+	0.0491521i
$456$		0			0
$457$	−107286.	−	608449.i	−0.0240299	−	0.136280i	0.970433	−	0.241372i	$-0.0775975\pi$
$457$	−107286.	−	608449.i	−0.0240299	−	0.136280i	−0.994463	+	0.105092i	$0.966486\pi$
$458$		0			0
$459$	−3.81544e6	−	2.30346e6i	−0.845305	−	0.510327i
$460$		0			0
$461$	255365.	+	1.44825e6i	0.0559641	+	0.317388i	0.999920	−	0.0126847i	$-0.00403777\pi$
$461$	255365.	+	1.44825e6i	0.0559641	+	0.317388i	−0.943955	+	0.330073i	$0.892927\pi$
$462$		0			0
$463$	4.82890e6	−	1.75757e6i	1.04688	−	0.381032i	0.239393	−	0.970923i	$-0.423051\pi$
$463$	4.82890e6	−	1.75757e6i	1.04688	−	0.381032i	0.807483	+	0.589891i	$0.200829\pi$
$464$		0			0
$465$	2.11503e6	−	358741.i	0.453611	−	0.0769393i
$466$		0			0
$467$	473479.	+	820090.i	0.100464	+	0.174008i	0.911876	−	0.410466i	$-0.134634\pi$
$467$	473479.	+	820090.i	0.100464	+	0.174008i	−0.811412	+	0.584474i	$0.801301\pi$
$468$		0			0
$469$	2.34682e6	−	4.06481e6i	0.492661	−	0.853313i
$470$		0			0
$471$	−5.40724e6	−	989818.i	−1.12311	−	0.205591i
$472$		0			0
$473$	1.79563e6	−	1.01835e7i	0.369032	−	2.09288i
$474$		0			0
$475$	1.50856e6	−	1.26583e6i	0.306780	−	0.257419i
$476$		0			0
$477$	−1.52030e6	+	2.55572e6i	−0.305937	+	0.514300i
$478$		0			0
$479$	−4.31394e6	−	1.57015e6i	−0.859084	−	0.312681i	−0.125345	−	0.992113i	$-0.540004\pi$
$479$	−4.31394e6	−	1.57015e6i	−0.859084	−	0.312681i	−0.733738	+	0.679432i	$0.762226\pi$
$480$		0			0
$481$	594943.	+	499216.i	0.117250	+	0.0983843i
$482$		0			0
$483$	930489.	−	529163.i	0.181486	−	0.103210i
$484$		0			0
$485$	2.25279e6			0.434877
$486$		0			0
$487$	−2.30746e6			−0.440871			−0.220436	−	0.975402i	$-0.570748\pi$
$487$	−2.30746e6			−0.440871			−0.220436	+	0.975402i	$0.570748\pi$
$488$		0			0
$489$	8.85075e6	−	5.03336e6i	1.67382	−	0.951888i
$490$		0			0
$491$	1.74143e6	+	1.46123e6i	0.325988	+	0.273537i	0.791063	−	0.611735i	$-0.209528\pi$
$491$	1.74143e6	+	1.46123e6i	0.325988	+	0.273537i	−0.465075	+	0.885271i	$0.653973\pi$
$492$		0			0
$493$	−1.27797e6	−	465142.i	−0.236811	−	0.0861922i
$494$		0			0
$495$	1.88604e6	−	3.17056e6i	0.345970	−	0.581598i
$496$		0			0
$497$	6.30220e6	−	5.28818e6i	1.14446	−	0.960318i
$498$		0			0
$499$	−987238.	+	5.59891e6i	−0.177489	+	1.00659i	0.757743	+	0.652553i	$0.226302\pi$
$499$	−987238.	+	5.59891e6i	−0.177489	+	1.00659i	−0.935232	+	0.354035i	$0.884809\pi$
$500$		0			0
$501$	−4.55258e6	−	833370.i	−0.810333	−	0.148335i
$502$		0			0
$503$	−3.30169e6	+	5.71870e6i	−0.581858	+	1.00781i	0.413401	+	0.910549i	$0.364341\pi$
$503$	−3.30169e6	+	5.71870e6i	−0.581858	+	1.00781i	−0.995259	+	0.0972584i	$0.968993\pi$
$504$		0			0
$505$	−2.82347e6	−	4.89039e6i	−0.492668	−	0.853325i
$506$		0			0
$507$	5.23240e6	−	887493.i	0.904026	−	0.153336i
$508$		0			0
$509$	1.06463e6	−	387492.i	0.182139	−	0.0662931i	−0.249341	−	0.968416i	$-0.580214\pi$
$509$	1.06463e6	−	387492.i	0.182139	−	0.0662931i	0.431480	+	0.902123i	$0.357992\pi$
$510$		0			0
$511$	−634381.	−	3.59775e6i	−0.107473	−	0.609507i
$512$		0			0
$513$	2.98701e6	−	1.64754e6i	0.501122	−	0.276404i
$514$		0			0
$515$	937935.	+	5.31929e6i	0.155831	+	0.883763i
$516$		0			0
$517$	2.85965e6	−	1.04083e6i	0.470529	−	0.171259i
$518$		0			0
$519$	2.18515e6	−	5.88404e6i	0.356092	−	0.958865i
$520$		0			0
$521$	−351923.	−	609549.i	−0.0568007	−	0.0983816i	0.836227	−	0.548384i	$-0.184757\pi$
$521$	−351923.	−	609549.i	−0.0568007	−	0.0983816i	−0.893028	+	0.450002i	$0.851423\pi$
$522$		0			0
$523$	3.37091e6	−	5.83858e6i	0.538881	−	0.933369i	−0.460084	−	0.887875i	$-0.652181\pi$
$523$	3.37091e6	−	5.83858e6i	0.538881	−	0.933369i	0.998965	−	0.0454934i	$-0.0144860\pi$
$524$		0			0
$525$	1.35085e6	+	3.78806e6i	0.213899	+	0.599817i
$526$		0			0
$527$	917924.	−	5.20581e6i	0.143973	−	0.816510i
$528$		0			0
$529$	−4.67101e6	+	3.91945e6i	−0.725725	+	0.608955i
$530$		0			0
$531$	−5.62085e6	+	1.96324e6i	−0.865099	+	0.302160i
$532$		0			0
$533$	1.11110e6	+	404406.i	0.169408	+	0.0616594i
$534$		0			0
$535$	−88992.5	−	74673.6i	−0.0134422	−	0.0112793i
$536$		0			0
$537$	27188.2	−	4.17184e6i	0.00406859	−	0.624297i
$538$		0			0
$539$	−1.43137e6			−0.212217
$540$		0			0
$541$	−3.50446e6			−0.514788			−0.257394	−	0.966307i	$-0.582864\pi$
$541$	−3.50446e6			−0.514788			−0.257394	+	0.966307i	$0.582864\pi$
$542$		0			0
$543$	1.95711e6	+	1.14701e6i	0.284849	+	0.166942i
$544$		0			0
$545$	3.12746e6	+	2.62425e6i	0.451025	+	0.378455i
$546$		0			0
$547$	8.17676e6	+	2.97610e6i	1.16846	+	0.425284i	0.852112	−	0.523360i	$-0.175322\pi$
$547$	8.17676e6	+	2.97610e6i	1.16846	+	0.425284i	0.316346	+	0.948644i	$0.397544\pi$
$548$		0			0
$549$	−407772.	+	333202.i	−0.0577414	+	0.0471821i
$550$		0			0
$551$	797394.	−	669093.i	0.111891	−	0.0938875i
$552$		0			0
$553$	1.62047e6	−	9.19014e6i	0.225335	−	1.27794i
$554$		0			0
$555$	−1.36785e6	+	1.60873e6i	−0.188497	+	0.221692i
$556$		0			0
$557$	−2.67883e6	+	4.63986e6i	−0.365853	+	0.633676i	−0.988913	−	0.148498i	$-0.952556\pi$
$557$	−2.67883e6	+	4.63986e6i	−0.365853	+	0.633676i	0.623060	+	0.782174i	$0.285889\pi$
$558$		0			0
$559$	−1.83195e6	−	3.17303e6i	−0.247961	−	0.429482i
$560$		0			0
$561$	−5.79770e6	−	7.00160e6i	−0.777766	−	0.939269i
$562$		0			0
$563$	−1.15894e7	+	4.21819e6i	−1.54095	+	0.560861i	−0.966274	−	0.257518i	$-0.917095\pi$
$563$	−1.15894e7	+	4.21819e6i	−1.54095	+	0.560861i	−0.574679	+	0.818379i	$0.694873\pi$
$564$		0			0
$565$	108651.	+	616191.i	0.0143190	+	0.0812072i
$566$		0			0
$567$	1.03049e6	+	6.88991e6i	0.134613	+	0.900027i
$568$		0			0
$569$	−2.06543e6	−	1.17137e7i	−0.267442	−	1.51674i	−0.761989	−	0.647590i	$-0.775777\pi$
$569$	−2.06543e6	−	1.17137e7i	−0.267442	−	1.51674i	0.494546	−	0.869151i	$-0.335334\pi$
$570$		0			0
$571$	356546.	−	129772.i	0.0457642	−	0.0166568i	−0.319037	−	0.947742i	$-0.603359\pi$
$571$	356546.	−	129772.i	0.0457642	−	0.0166568i	0.364801	+	0.931086i	$0.381137\pi$
$572$		0			0
$573$	7.75011e6	+	9.35942e6i	0.986101	+	1.19087i
$574$		0			0
$575$	636391.	+	1.10226e6i	0.0802703	+	0.139032i
$576$		0			0
$577$	−2.49136e6	+	4.31517e6i	−0.311528	+	0.539583i	−0.978693	−	0.205327i	$-0.934174\pi$
$577$	−2.49136e6	+	4.31517e6i	−0.311528	+	0.539583i	0.667165	+	0.744910i	$0.267507\pi$
$578$		0			0
$579$	−9.28870e6	+	1.09245e7i	−1.15149	+	1.35427i
$580$		0			0
$581$	2.37567e6	−	1.34731e7i	0.291975	−	1.65587i
$582$		0			0
$583$	−4.64632e6	+	3.89873e6i	−0.566158	+	0.475063i
$584$		0			0
$585$	−210184.	−	1.29012e6i	−0.0253928	−	0.155863i
$586$		0			0
$587$	2.73020e6	+	993711.i	0.327039	+	0.119032i	0.500321	−	0.865840i	$-0.333215\pi$
$587$	2.73020e6	+	993711.i	0.327039	+	0.119032i	−0.173283	+	0.984872i	$0.555437\pi$
$588$		0			0
$589$	3.09939e6	+	2.60070e6i	0.368119	+	0.308888i
$590$		0			0
$591$	−3.82449e6	−	2.24143e6i	−0.450406	−	0.263971i
$592$		0			0
$593$	−2.44021e6			−0.284964			−0.142482	−	0.989797i	$-0.545508\pi$
$593$	−2.44021e6			−0.284964			−0.142482	+	0.989797i	$0.545508\pi$
$594$		0			0
$595$	−4.25185e6			−0.492363
$596$		0			0
$597$	−60780.0	+	9.32628e6i	−0.00697951	+	1.07096i
$598$		0			0
$599$	1.09896e7	+	9.22138e6i	1.25146	+	1.05010i	0.996539	+	0.0831325i	$0.0264925\pi$
$599$	1.09896e7	+	9.22138e6i	1.25146	+	1.05010i	0.254917	+	0.966963i	$0.417952\pi$
$600$		0			0
$601$	−1.08665e7	−	3.95507e6i	−1.22716	−	0.446650i	−0.354537	−	0.935042i	$-0.615361\pi$
$601$	−1.08665e7	−	3.95507e6i	−1.22716	−	0.446650i	−0.872624	+	0.488392i	$0.837584\pi$
$602$		0			0
$603$	7.32407e6	+	6.31011e6i	0.820274	+	0.706713i
$604$		0			0
$605$	1.98516e6	−	1.66575e6i	0.220499	−	0.185021i
$606$		0			0
$607$	765482.	−	4.34126e6i	0.0843263	−	0.478238i	−0.913174	−	0.407571i	$-0.866376\pi$
$607$	765482.	−	4.34126e6i	0.0843263	−	0.478238i	0.997500	−	0.0706675i	$-0.0225129\pi$
$608$		0			0
$609$	714034.	+	2.00230e6i	0.0780145	+	0.218769i
$610$		0			0
$611$	539131.	−	933803.i	0.0584240	−	0.101193i
$612$		0			0
$613$	−4.28769e6	−	7.42649e6i	−0.460863	−	0.798238i	0.538141	−	0.842855i	$-0.319127\pi$
$613$	−4.28769e6	−	7.42649e6i	−0.460863	−	0.798238i	−0.999004	+	0.0446167i	$0.985793\pi$
$614$		0			0
$615$	−1.11921e6	+	3.01376e6i	−0.119323	+	0.321307i
$616$		0			0
$617$	9.12363e6	−	3.32073e6i	0.964839	−	0.351173i	0.188911	−	0.981994i	$-0.439504\pi$
$617$	9.12363e6	−	3.32073e6i	0.964839	−	0.351173i	0.775928	+	0.630822i	$0.217282\pi$
$618$		0			0
$619$	745358.	+	4.22714e6i	0.0781877	+	0.443424i	0.998620	+	0.0525219i	$0.0167259\pi$
$619$	745358.	+	4.22714e6i	0.0781877	+	0.443424i	−0.920432	+	0.390902i	$0.872163\pi$
$620$		0			0
$621$	713423.	+	2.08614e6i	0.0742367	+	0.217077i
$622$		0			0
$623$	−600134.	−	3.40353e6i	−0.0619481	−	0.351325i
$624$		0			0
$625$	−1.73844e6	+	632740.i	−0.178016	+	0.0647926i
$626$		0			0
$627$	6.85978e6	−	1.16352e6i	0.696854	−	0.118197i
$628$		0			0
$629$	2.60166e6	+	4.50620e6i	0.262194	+	0.454134i
$630$		0			0
$631$	−7.11512e6	+	1.23237e7i	−0.711391	+	1.23217i	0.252944	+	0.967481i	$0.418601\pi$
$631$	−7.11512e6	+	1.23237e7i	−0.711391	+	1.23217i	−0.964335	+	0.264685i	$0.914732\pi$
$632$		0			0
$633$	−7.93383e6	−	1.45232e6i	−0.786998	−	0.144063i
$634$		0			0
$635$	−148829.	+	844052.i	−0.0146472	+	0.0830682i
$636$		0			0
$637$	−388511.	+	326000.i	−0.0379363	+	0.0318323i
$638$		0			0
$639$	8.28048e6	+	1.47839e7i	0.802239	+	1.43231i
$640$		0			0
$641$	−1.06365e7	−	3.87137e6i	−1.02248	−	0.372151i	−0.224266	−	0.974528i	$-0.571998\pi$
$641$	−1.06365e7	−	3.87137e6i	−1.02248	−	0.372151i	−0.798212	+	0.602377i	$0.794221\pi$
$642$		0			0
$643$	−4.75522e6	−	3.99010e6i	−0.453568	−	0.380589i	0.387190	−	0.922000i	$-0.373446\pi$
$643$	−4.75522e6	−	3.99010e6i	−0.453568	−	0.380589i	−0.840758	+	0.541411i	$0.817890\pi$
$644$		0			0
$645$	8.65952e6	−	4.92461e6i	0.819586	−	0.466093i
$646$		0			0
$647$	1.03138e7			0.968632			0.484316	−	0.874893i	$-0.339068\pi$
$647$	1.03138e7			0.968632			0.484316	+	0.874893i	$0.339068\pi$
$648$		0			0
$649$	−1.21438e7			−1.13173
$650$		0			0
$651$	−7.18254e6	+	4.08467e6i	−0.664241	+	0.377750i
$652$		0			0
$653$	1.61184e7	+	1.35250e7i	1.47924	+	1.24123i	0.906989	+	0.421154i	$0.138375\pi$
$653$	1.61184e7	+	1.35250e7i	1.47924	+	1.24123i	0.572253	+	0.820077i	$0.306070\pi$
$654$		0			0
$655$	−4.62431e6	−	1.68311e6i	−0.421157	−	0.153289i
$656$		0			0
$657$	7.52392e6	+	98072.1i	0.680035	+	0.00886405i
$658$		0			0
$659$	1.36971e7	−	1.14932e7i	1.22861	−	1.03093i	0.230286	−	0.973123i	$-0.426034\pi$
$659$	1.36971e7	−	1.14932e7i	1.22861	−	1.03093i	0.998328	−	0.0578059i	$-0.0184105\pi$
$660$		0			0
$661$	608613.	−	3.45161e6i	0.0541798	−	0.307269i	−0.945660	−	0.325157i	$-0.894583\pi$
$661$	608613.	−	3.45161e6i	0.0541798	−	0.307269i	0.999840	+	0.0178877i	$0.00569412\pi$
$662$		0			0
$663$	−3.16829e6	−	579969.i	−0.279924	−	0.0512414i
$664$		0			0
$665$	1.62717e6	−	2.81834e6i	0.142685	−	0.247138i
$666$		0			0
$667$	336385.	+	582635.i	0.0292767	+	0.0507086i
$668$		0			0
$669$	−1.17433e7	+	1.99183e6i	−1.01443	+	0.172063i
$670$		0			0
$671$	−1.00930e6	+	367353.i	−0.0865390	+	0.0314976i
$672$		0			0
$673$	894318.	+	5.07193e6i	0.0761122	+	0.431654i	0.998923	+	0.0464002i	$0.0147749\pi$
$673$	894318.	+	5.07193e6i	0.0761122	+	0.431654i	−0.922811	+	0.385254i	$0.874114\pi$
$674$		0			0
$675$	−8.18420e6	+	1.27866e6i	−0.691380	+	0.108018i
$676$		0			0
$677$	637960.	+	3.61805e6i	0.0534961	+	0.303391i	0.999802	−	0.0198801i	$-0.00632845\pi$
$677$	637960.	+	3.61805e6i	0.0534961	+	0.303391i	−0.946306	+	0.323271i	$0.895217\pi$
$678$		0			0
$679$	−8.15374e6	+	2.96772e6i	−0.678707	+	0.247029i
$680$		0			0
$681$	−1.91721e6	+	5.16255e6i	−0.158417	+	0.426576i
$682$		0			0
$683$	−8.69351e6	−	1.50576e7i	−0.713088	−	1.23511i	−0.963692	−	0.267015i	$-0.913963\pi$
$683$	−8.69351e6	−	1.50576e7i	−0.713088	−	1.23511i	0.250604	−	0.968090i	$-0.419371\pi$
$684$		0			0
$685$	−1.64238e6	+	2.84468e6i	−0.133735	+	0.231637i
$686$		0			0
$687$	4.61508e6	+	1.29416e7i	0.373068	+	1.04616i
$688$		0			0
$689$	−373184.	+	2.11643e6i	−0.0299485	+	0.169846i
$690$		0			0
$691$	−4.19867e6	+	3.52310e6i	−0.334515	+	0.280692i	−0.794537	−	0.607216i	$-0.792286\pi$
$691$	−4.19867e6	+	3.52310e6i	−0.334515	+	0.280692i	0.460021	+	0.887908i	$0.347842\pi$
$692$		0			0
$693$	−2.64960e6	+	1.39601e7i	−0.209579	+	1.10422i
$694$		0			0
$695$	−3.35631e6	−	1.22160e6i	−0.263573	−	0.0959326i
$696$		0			0
$697$	6.06846e6	+	5.09204e6i	0.473148	+	0.397018i
$698$		0			0
$699$	132282.	−	2.02978e7i	0.0102402	−	1.57129i
$700$		0			0
$701$	−2.23129e7			−1.71499			−0.857494	−	0.514493i	$-0.827980\pi$
$701$	−2.23129e7			−1.71499			−0.857494	+	0.514493i	$0.827980\pi$
$702$		0			0
$703$	−3.98259e6			−0.303933
$704$		0			0
$705$	2.52934e6	+	1.48237e6i	0.191661	+	0.112327i
$706$		0			0
$707$	1.66616e7	+	1.39808e7i	1.25363	+	1.05192i
$708$		0			0
$709$	7.66978e6	+	2.79157e6i	0.573017	+	0.208561i	0.612243	−	0.790669i	$-0.290267\pi$
$709$	7.66978e6	+	2.79157e6i	0.573017	+	0.208561i	−0.0392267	+	0.999230i	$0.512489\pi$
$710$		0			0
$711$	1.79745e7	+	6.80877e6i	1.33347	+	0.505120i
$712$		0			0
$713$	−2.00319e6	+	1.68088e6i	−0.147570	+	0.123826i
$714$		0			0
$715$	462963.	−	2.62559e6i	0.0338673	−	0.192071i
$716$		0			0
$717$	−7.57431e6	+	8.90816e6i	−0.550231	+	0.647128i
$718$		0			0
$719$	−1.93036e6	+	3.34348e6i	−0.139257	+	0.241200i	−0.927215	−	0.374528i	$-0.877805\pi$
$719$	−1.93036e6	+	3.34348e6i	−0.139257	+	0.241200i	0.787959	+	0.615728i	$0.211138\pi$
$720$		0			0
$721$	−1.04021e7	−	1.80170e7i	−0.745221	−	1.29076i
$722$		0			0
$723$	−1.75661e7	−	2.12137e7i	−1.24977	−	1.50928i
$724$		0			0
$725$	−2.37522e6	+	864511.i	−0.167826	+	0.0610837i
$726$		0			0
$727$	2.77579e6	+	1.57423e7i	0.194783	+	1.10467i	0.912728	+	0.408568i	$0.133972\pi$
$727$	2.77579e6	+	1.57423e7i	0.194783	+	1.10467i	−0.717945	+	0.696100i	$0.754917\pi$
$728$		0			0
$729$	−1.43379e7	−	560926.i	−0.999236	−	0.0390919i
$730$		0			0
$731$	−4.26258e6	−	2.41743e7i	−0.295039	−	1.67325i
$732$		0			0
$733$	1.82932e7	−	6.65817e6i	1.25756	−	0.457715i	0.374611	−	0.927182i	$-0.377776\pi$
$733$	1.82932e7	−	6.65817e6i	1.25756	−	0.457715i	0.882950	+	0.469468i	$0.155554\pi$
$734$		0			0
$735$	−879466.	−	1.06209e6i	−0.0600483	−	0.0725174i
$736$		0			0
$737$	9.85910e6	+	1.70765e7i	0.668604	+	1.15806i
$738$		0			0
$739$	1.07120e7	−	1.85537e7i	0.721536	−	1.24974i	−0.238848	−	0.971057i	$-0.576770\pi$
$739$	1.07120e7	−	1.85537e7i	0.721536	−	1.24974i	0.960384	−	0.278680i	$-0.0898970\pi$
$740$		0			0
$741$	1.59693e6	−	1.87815e6i	0.106841	−	0.125656i
$742$		0			0
$743$	3.75633e6	−	2.13032e7i	0.249627	−	1.41571i	−0.559869	−	0.828581i	$-0.689149\pi$
$743$	3.75633e6	−	2.13032e7i	0.249627	−	1.41571i	0.809496	−	0.587125i	$-0.199740\pi$
$744$		0			0
$745$	450851.	−	378309.i	0.0297607	−	0.0249722i
$746$		0			0
$747$	2.63512e7	+	9.98190e6i	1.72782	+	0.654503i
$748$		0			0
$749$	420471.	+	153039.i	0.0273862	+	0.00996775i
$750$		0			0
$751$	−1.20029e7	−	1.00717e7i	−0.776583	−	0.651631i	0.165803	−	0.986159i	$-0.446979\pi$
$751$	−1.20029e7	−	1.00717e7i	−0.776583	−	0.651631i	−0.942386	+	0.334528i	$0.891423\pi$
$752$		0			0
$753$	2.56780e6	+	1.50492e6i	0.165034	+	0.0967219i
$754$		0			0
$755$	2.92155e6			0.186529
$756$		0			0
$757$	1.00375e7			0.636626			0.318313	−	0.947986i	$-0.396884\pi$
$757$	1.00375e7			0.636626			0.318313	+	0.947986i	$0.396884\pi$
$758$		0			0
$759$	−29306.2	+	4.49684e6i	−0.00184652	+	0.283336i
$760$		0			0
$761$	1.29642e6	+	1.08783e6i	0.0811492	+	0.0680923i	0.682460	−	0.730923i	$-0.260910\pi$
$761$	1.29642e6	+	1.08783e6i	0.0811492	+	0.0680923i	−0.601311	+	0.799015i	$0.705355\pi$
$762$		0			0
$763$	−1.47766e7	−	5.37824e6i	−0.918889	−	0.334448i
$764$		0			0
$765$	1.63300e6	−	8.60388e6i	0.100886	−	0.531546i
$766$		0			0
$767$	−3.29613e6	+	2.76578e6i	−0.202310	+	0.169758i
$768$		0			0
$769$	2.48161e6	−	1.40739e7i	0.151328	−	0.858222i	−0.810739	−	0.585408i	$-0.800935\pi$
$769$	2.48161e6	−	1.40739e7i	0.151328	−	0.858222i	0.962067	−	0.272814i	$-0.0879544\pi$
$770$		0			0
$771$	−7.25984e6	−	2.03581e7i	−0.439836	−	1.23339i
$772$		0			0
$773$	−1.05749e7	+	1.83163e7i	−0.636545	+	1.10253i	0.349640	+	0.936884i	$0.386304\pi$
$773$	−1.05749e7	+	1.83163e7i	−0.636545	+	1.10253i	−0.986186	+	0.165644i	$0.947030\pi$
$774$		0			0
$775$	−4.91237e6	−	8.50848e6i	−0.293790	−	0.508859i
$776$		0			0
$777$	2.83152e6	−	7.62456e6i	0.168255	−	0.453067i
$778$		0			0
$779$	−5.69765e6	+	2.07378e6i	−0.336397	+	0.122439i
$780$		0			0
$781$	6.00159e6	+	3.40367e7i	0.352078	+	1.99673i
$782$		0			0
$783$	−4.32601e6	+	675874.i	−0.252164	+	0.0393968i
$784$		0			0
$785$	−1.87566e6	−	1.06374e7i	−0.108637	−	0.616114i
$786$		0			0
$787$	1.89920e7	−	6.91254e6i	1.09304	−	0.397833i	0.268292	−	0.963338i	$-0.413541\pi$
$787$	1.89920e7	−	6.91254e6i	1.09304	−	0.397833i	0.824745	+	0.565505i	$0.191319\pi$
$788$		0			0
$789$	1.90465e7	−	3.23057e6i	1.08924	−	0.184751i
$790$		0			0
$791$	−1.20499e6	−	2.08711e6i	−0.0684768	−	0.118605i
$792$		0			0
$793$	−190283.	+	329580.i	−0.0107453	+	0.0186113i
$794$		0			0
$795$	−5.74769e6	−	1.05214e6i	−0.322534	−	0.0590413i
$796$		0			0
$797$	5.68528e6	−	3.22428e7i	0.317034	−	1.79799i	−0.243545	−	0.969890i	$-0.578310\pi$
$797$	5.68528e6	−	3.22428e7i	0.317034	−	1.79799i	0.560579	−	0.828101i	$-0.310579\pi$
$798$		0			0
$799$	5.53398e6	−	4.64356e6i	0.306669	−	0.257326i
$800$		0			0
$801$	7.11775e6	+	92777.7i	0.391978	+	0.00510931i
$802$		0			0
$803$	1.44219e7	+	5.24915e6i	0.789286	+	0.287276i
$804$		0			0
$805$	1.61125e6	+	1.35200e6i	0.0876343	+	0.0735339i
$806$		0			0
$807$	1.96316e7	−	1.11644e7i	1.06114	−	0.603462i
$808$		0			0
$809$	851912.			0.0457639			0.0228820	−	0.999738i	$-0.492716\pi$
$809$	851912.			0.0457639			0.0228820	+	0.999738i	$0.492716\pi$
$810$		0			0
$811$	−9.08561e6			−0.485067			−0.242534	−	0.970143i	$-0.577978\pi$
$811$	−9.08561e6			−0.485067			−0.242534	+	0.970143i	$0.577978\pi$
$812$		0			0
$813$	1.99176e7	−	1.13270e7i	1.05684	−	0.601020i
$814$		0			0
$815$	1.53261e7	+	1.28602e7i	0.808237	+	0.678191i
$816$		0			0
$817$	1.76553e7	+	6.42599e6i	0.925377	+	0.336810i
$818$		0			0
$819$	2.46029e6	+	4.39259e6i	0.128167	+	0.228829i
$820$		0			0
$821$	−1.08040e7	+	9.06561e6i	−0.559404	+	0.469396i	−0.878111	−	0.478458i	$-0.841196\pi$
$821$	−1.08040e7	+	9.06561e6i	−0.559404	+	0.469396i	0.318706	+	0.947853i	$0.396752\pi$
$822$		0			0
$823$	8575.64	−	48634.9i	0.000441333	−	0.00250293i	−0.984586	−	0.174900i	$-0.944040\pi$
$823$	8575.64	−	48634.9i	0.000441333	−	0.00250293i	0.985028	+	0.172397i	$0.0551511\pi$
$824$		0			0
$825$	−1.66193e7	−	3.04223e6i	−0.850113	−	0.155617i
$826$		0			0
$827$	824850.	−	1.42868e6i	0.0419383	−	0.0726393i	−0.844294	−	0.535880i	$-0.819980\pi$
$827$	824850.	−	1.42868e6i	0.0419383	−	0.0726393i	0.886233	+	0.463240i	$0.153313\pi$
$828$		0			0
$829$	−1.53301e7	−	2.65524e7i	−0.774743	−	1.34189i	−0.934939	−	0.354809i	$-0.884546\pi$
$829$	−1.53301e7	−	2.65524e7i	−0.774743	−	1.34189i	0.160196	−	0.987085i	$-0.448787\pi$
$830$		0			0
$831$	−1.94356e7	+	3.29657e6i	−0.976327	+	0.165600i
$832$		0			0
$833$	−3.19297e6	+	1.16214e6i	−0.159434	+	0.0580293i
$834$		0			0
$835$	−1.57920e6	−	8.95607e6i	−0.0783826	−	0.444530i
$836$		0			0
$837$	−5.50699e6	−	1.61031e7i	−0.271707	−	0.794505i
$838$		0			0
$839$	−5.11432e6	−	2.90048e7i	−0.250832	−	1.42254i	−0.806548	−	0.591168i	$-0.798667\pi$
$839$	−5.11432e6	−	2.90048e7i	−0.250832	−	1.42254i	0.555716	−	0.831372i	$-0.312444\pi$
$840$		0			0
$841$	1.80187e7	−	6.55826e6i	0.878482	−	0.319741i
$842$		0			0
$843$	−2.01520e6	+	5.42642e6i	−0.0976674	+	0.262993i
$844$		0			0
$845$	5.21412e6	+	9.03112e6i	0.251211	+	0.435111i
$846$		0			0
$847$	−4.99071e6	+	8.64416e6i	−0.239031	+	0.414013i
$848$		0			0
$849$	−1.22415e6	−	3.43276e6i	−0.0582860	−	0.163446i
$850$		0			0
$851$	446974.	−	2.53492e6i	0.0211572	−	0.119989i
$852$		0			0
$853$	−2.30502e7	+	1.93414e7i	−1.08468	+	0.910155i	−0.996301	−	0.0859318i	$-0.972613\pi$
$853$	−2.30502e7	+	1.93414e7i	−1.08468	+	0.910155i	−0.0883795	+	0.996087i	$0.528169\pi$
$854$		0			0
$855$	5.07815e6	+	4.37512e6i	0.237569	+	0.204680i
$856$		0			0
$857$	−2.44658e7	−	8.90480e6i	−1.13791	−	0.414164i	−0.296749	−	0.954955i	$-0.595903\pi$
$857$	−2.44658e7	−	8.90480e6i	−1.13791	−	0.414164i	−0.841157	+	0.540791i	$0.818125\pi$
$858$		0			0
$859$	1.88176e7	+	1.57898e7i	0.870124	+	0.730121i	0.964124	−	0.265451i	$-0.0855209\pi$
$859$	1.88176e7	+	1.57898e7i	0.870124	+	0.730121i	−0.0939999	+	0.995572i	$0.529965\pi$
$860$		0			0
$861$	80696.9	−	1.23824e7i	0.00370979	−	0.569242i
$862$		0			0
$863$	−2.91737e7			−1.33341			−0.666706	−	0.745321i	$-0.732296\pi$
$863$	−2.91737e7			−1.33341			−0.666706	+	0.745321i	$0.732296\pi$
$864$		0			0
$865$	1.23334e7			0.560456
$866$		0			0
$867$	477901.	+	280085.i	0.0215919	+	0.0126544i
$868$		0			0
$869$	3.00319e7	+	2.51997e7i	1.34906	+	1.13200i
$870$		0			0
$871$	6.56524e6	+	2.38955e6i	0.293228	+	0.106726i
$872$		0			0
$873$	−2.87377e6	−	1.76394e7i	−0.127620	−	0.783337i
$874$		0			0
$875$	−1.47046e7	+	1.23386e7i	−0.649280	+	0.544811i
$876$		0			0
$877$	−3.42198e6	+	1.94070e7i	−0.150238	+	0.852041i	0.812774	+	0.582580i	$0.197957\pi$
$877$	−3.42198e6	+	1.94070e7i	−0.150238	+	0.852041i	−0.963011	+	0.269461i	$0.913154\pi$
$878$		0			0
$879$	249542.	−	293487.i	0.0108936	−	0.0128120i
$880$		0			0
$881$	−1.98152e7	+	3.43210e7i	−0.860121	+	1.48977i	0.0116898	+	0.999932i	$0.496279\pi$
$881$	−1.98152e7	+	3.43210e7i	−0.860121	+	1.48977i	−0.871811	+	0.489842i	$0.837054\pi$
$882$		0			0
$883$	1.50221e7	+	2.60190e7i	0.648377	+	1.12302i	0.983511	+	0.180851i	$0.0578852\pi$
$883$	1.50221e7	+	2.60190e7i	0.648377	+	1.12302i	−0.335134	+	0.942171i	$0.608781\pi$
$884$		0			0
$885$	−7.46140e6	−	9.01077e6i	−0.320230	−	0.386726i
$886$		0			0
$887$	−3.33248e7	+	1.21293e7i	−1.42220	+	0.517637i	−0.934684	−	0.355480i	$-0.884317\pi$
$887$	−3.33248e7	+	1.21293e7i	−1.42220	+	0.517637i	−0.487511	+	0.873117i	$0.662095\pi$
$888$		0			0
$889$	−573243.	−	3.25102e6i	−0.0243268	−	0.137964i
$890$		0			0
$891$	−2.72315e7	−	1.07233e7i	−1.14915	−	0.452514i
$892$		0			0
$893$	960150.	+	5.44528e6i	0.0402912	+	0.228503i
$894$		0			0
$895$	7.70324e6	−	2.80375e6i	0.321452	−	0.116999i
$896$		0			0
$897$	1.01622e6	+	1.22723e6i	0.0421701	+	0.0509267i
$898$		0			0
$899$	−2.59659e6	−	4.49742e6i	−0.107153	−	0.185594i
$900$		0			0
$901$	−7.19916e6	+	1.24693e7i	−0.295440	+	0.511718i
$902$		0			0
$903$	−2.48548e7	+	2.92318e7i	−1.01436	+	1.19299i
$904$		0			0
$905$	−774018.	+	4.38968e6i	−0.0314145	+	0.178160i
$906$		0			0
$907$	1.69926e7	−	1.42585e7i	0.685869	−	0.575512i	−0.231846	−	0.972753i	$-0.574476\pi$
$907$	1.69926e7	−	1.42585e7i	0.685869	−	0.575512i	0.917715	+	0.397240i	$0.130032\pi$
$908$		0			0
$909$	−3.46901e7	+	2.83463e7i	−1.39250	+	1.13785i
$910$		0			0
$911$	1.15656e7	+	4.20952e6i	0.461712	+	0.168049i	0.562394	−	0.826869i	$-0.309880\pi$
$911$	1.15656e7	+	4.20952e6i	0.461712	+	0.168049i	−0.100682	+	0.994919i	$0.532103\pi$
$912$		0			0
$913$	4.40278e7	+	3.69437e7i	1.74803	+	1.46678i
$914$		0			0
$915$	−892713.	−	523195.i	−0.0352500	−	0.0206591i
$916$		0			0
$917$	1.89545e7			0.744370
$918$		0			0
$919$	−1.69957e7			−0.663818			−0.331909	−	0.943311i	$-0.607693\pi$
$919$	−1.69957e7			−0.663818			−0.331909	+	0.943311i	$0.607693\pi$
$920$		0			0
$921$	71380.2	−	1.09528e7i	0.00277286	−	0.425477i
$922$		0			0
$923$	9.38096e6	+	7.87156e6i	0.362446	+	0.304128i
$924$		0			0
$925$	9.08763e6	+	3.30763e6i	0.349218	+	0.127105i
$926$		0			0
$927$	4.04538e7	−	1.41296e7i	1.54618	−	0.540047i
$928$		0			0
$929$	−2.36670e7	+	1.98590e7i	−0.899713	+	0.754948i	−0.970134	−	0.242569i	$-0.922010\pi$
$929$	−2.36670e7	+	1.98590e7i	−0.899713	+	0.754948i	0.0704217	+	0.997517i	$0.477566\pi$
$930$		0			0
$931$	451611.	−	2.56121e6i	0.0170762	−	0.0968437i
$932$		0			0
$933$	−9.23431e6	−	2.58949e7i	−0.347297	−	0.973891i
$934$		0			0
$935$	8.93110e6	−	1.54691e7i	0.334100	−	0.578678i
$936$		0			0
$937$	1.31815e7	+	2.28311e7i	0.490475	+	0.849528i	0.999940	−	0.0109637i	$-0.00348994\pi$
$937$	1.31815e7	+	2.28311e7i	0.490475	+	0.849528i	−0.509465	+	0.860491i	$0.670157\pi$
$938$		0			0
$939$	7.23703e6	−	1.94875e7i	0.267853	−	0.721260i
$940$		0			0
$941$	−4.58147e7	+	1.66752e7i	−1.68667	+	0.613899i	−0.994200	−	0.107544i	$-0.965701\pi$
$941$	−4.58147e7	+	1.66752e7i	−1.68667	+	0.613899i	−0.692474	+	0.721443i	$0.743479\pi$
$942$		0			0
$943$	−680498.	−	3.85930e6i	−0.0249200	−	0.141328i
$944$		0			0
$945$	−1.19865e7	+	6.61138e6i	−0.436628	+	0.240831i
$946$		0			0
$947$	5.95173e6	+	3.37539e7i	0.215659	+	1.22306i	0.879759	+	0.475420i	$0.157704\pi$
$947$	5.95173e6	+	3.37539e7i	0.215659	+	1.22306i	−0.664099	+	0.747644i	$0.731185\pi$
$948$		0			0
$949$	5.11000e6	−	1.85989e6i	0.184185	−	0.0670380i
$950$		0			0
$951$	3.00085e7	−	5.08988e6i	1.07595	−	0.182497i
$952$		0			0
$953$	1.31238e7	+	2.27310e7i	0.468087	+	0.810750i	0.999335	−	0.0364662i	$-0.0116101\pi$
$953$	1.31238e7	+	2.27310e7i	0.468087	+	0.810750i	−0.531248	+	0.847216i	$0.678277\pi$
$954$		0			0
$955$	−1.19387e7	+	2.06784e7i	−0.423593	+	0.733684i
$956$		0			0
$957$	−8.78463e6	−	1.60806e6i	−0.310058	−	0.0567576i
$958$		0			0
$959$	2.19697e6	−	1.24596e7i	0.0771396	−	0.437480i
$960$		0			0
$961$	−6.46834e6	+	5.42758e6i	−0.225936	+	0.189582i
$962$		0			0
$963$	−471173.	+	792072.i	−0.0163725	+	0.0275232i
$964$		0			0
$965$	−2.64773e7	−	9.63695e6i	−0.915283	−	0.333136i
$966$		0			0
$967$	−3.87064e7	−	3.24785e7i	−1.33112	−	1.11694i	−0.983815	−	0.179187i	$-0.942653\pi$
$967$	−3.87064e7	−	3.24785e7i	−1.33112	−	1.11694i	−0.347302	−	0.937753i	$-0.612902\pi$
$968$		0			0
$969$	1.43575e7	−	8.16501e6i	0.491212	−	0.279349i
$970$		0			0
$971$	−4.80793e7			−1.63648			−0.818239	−	0.574878i	$-0.805050\pi$
$971$	−4.80793e7			−1.63648			−0.818239	+	0.574878i	$0.805050\pi$
$972$		0			0
$973$	1.37571e7			0.465849
$974$		0			0
$975$	−5.20378e6	+	2.95935e6i	−0.175310	+	0.0996978i
$976$		0			0
$977$	1.28202e7	+	1.07574e7i	0.429693	+	0.360555i	0.831836	−	0.555022i	$-0.187290\pi$
$977$	1.28202e7	+	1.07574e7i	0.429693	+	0.360555i	−0.402143	+	0.915577i	$0.631734\pi$
$978$		0			0
$979$	1.36434e7	+	4.96577e6i	0.454951	+	0.165589i
$980$		0			0
$981$	1.65584e7	−	2.78358e7i	0.549347	−	0.923487i
$982$		0			0
$983$	4.48200e7	−	3.76084e7i	1.47941	−	1.24137i	0.572594	−	0.819839i	$-0.305937\pi$
$983$	4.48200e7	−	3.76084e7i	1.47941	−	1.24137i	0.906814	−	0.421532i	$-0.138507\pi$
$984$		0			0
$985$	1.51255e6	−	8.57811e6i	0.0496729	−	0.281709i
$986$		0			0
$987$	−1.11075e7	−	2.03327e6i	−0.362930	−	0.0664359i
$988$		0			0
$989$	−6.07162e6	+	1.05164e7i	−0.197385	+	0.341881i
$990$		0			0
$991$	−5.06100e6	−	8.76591e6i	−0.163701	−	0.283539i	0.772492	−	0.635024i	$-0.219010\pi$
$991$	−5.06100e6	−	8.76591e6i	−0.163701	−	0.283539i	−0.936193	+	0.351485i	$0.885677\pi$
$992$		0			0
$993$	1.56853e6	−	266047.i	0.0504801	−	0.00856219i
$994$		0			0
$995$	−1.72208e7	+	6.26787e6i	−0.551438	+	0.200707i
$996$		0			0
$997$	−985950.	−	5.59160e6i	−0.0314136	−	0.178155i	0.965064	−	0.262014i	$-0.0843866\pi$
$997$	−985950.	−	5.59160e6i	−0.0314136	−	0.178155i	−0.996478	+	0.0838588i	$0.973276\pi$
$998$		0			0
$999$	1.43413e7	+	8.65810e6i	0.454647	+	0.274479i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.i.a.13.13	✓	90
3.2	odd	2		324.6.i.a.253.6		90
27.2	odd	18		324.6.i.a.73.6		90
27.25	even	9	inner	108.6.i.a.25.13	yes	90

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.i.a.13.13	✓	90	1.1	even	1	trivial
108.6.i.a.25.13	yes	90	27.25	even	9	inner
324.6.i.a.73.6		90	27.2	odd	18
324.6.i.a.253.6		90	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	108.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(13.5505 - 7.70608i) q^{3} +(23.4643 + 19.6889i) q^{5} +(-110.864 - 40.3512i) q^{7} +(124.233 - 208.843i) q^{9} +O(q^{10})\)	\(q+(13.5505 - 7.70608i) q^{3} +(23.4643 + 19.6889i) q^{5} +(-110.864 - 40.3512i) q^{7} +(124.233 - 208.843i) q^{9} +(379.679 - 318.588i) q^{11} +(30.4951 - 172.946i) q^{13} +(469.678 + 85.9766i) q^{15} +(588.286 - 1018.94i) q^{17} +(450.271 + 779.893i) q^{19} +(-1813.21 + 307.548i) q^{21} +(-546.936 + 199.068i) q^{23} +(-379.729 - 2153.55i) q^{25} +(74.0541 - 3787.27i) q^{27} +(-200.718 - 1138.33i) q^{29} +(4221.86 - 1536.63i) q^{31} +(2689.77 - 7242.87i) q^{33} +(-1806.88 - 3129.61i) q^{35} +(-2211.22 + 3829.94i) q^{37} +(-919.514 - 2578.51i) q^{39} +(-1169.17 + 6630.67i) q^{41} +(15982.3 - 13410.7i) q^{43} +(7026.92 - 2454.35i) q^{45} +(5769.67 + 2099.99i) q^{47} +(-2212.30 - 1856.34i) q^{49} +(119.527 - 18340.6i) q^{51} -12237.5 q^{53} +15181.6 q^{55} +(12111.3 + 7098.11i) q^{57} +(-18769.2 - 15749.2i) q^{59} +(-2036.37 - 741.177i) q^{61} +(-22200.0 + 18140.2i) q^{63} +(4120.67 - 3457.65i) q^{65} +(-6908.37 + 39179.3i) q^{67} +(-5877.22 + 6912.22i) q^{69} +(-34866.1 + 60389.9i) q^{71} +(15482.6 + 26816.7i) q^{73} +(-21741.0 - 26255.5i) q^{75} +(-54948.1 + 19999.5i) q^{77} +(13735.2 + 77896.4i) q^{79} +(-28181.6 - 51890.1i) q^{81} +(20136.4 + 114199. i) q^{83} +(33865.6 - 12326.1i) q^{85} +(-11491.9 - 13878.1i) q^{87} +(14646.8 + 25369.0i) q^{89} +(-10359.4 + 17943.0i) q^{91} +(45366.9 - 53356.1i) q^{93} +(-4789.92 + 27165.0i) q^{95} +(56340.4 - 47275.2i) q^{97} +(-19366.4 - 118872. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 87 q^{5} + 330 q^{9}+O(q^{10}) \) 90 * q - 87 * q^5 + 330 * q^9	\( 90 q - 87 q^{5} + 330 q^{9} - 1257 q^{11} + 531 q^{15} - 3468 q^{17} + 12894 q^{21} + 8106 q^{23} + 4959 q^{25} - 17415 q^{27} + 3468 q^{29} - 6651 q^{31} + 33624 q^{33} - 8229 q^{35} - 10545 q^{39} + 68673 q^{41} + 9459 q^{43} - 53469 q^{45} - 57087 q^{47} - 5490 q^{49} + 42831 q^{51} - 4146 q^{53} + 24624 q^{57} + 5388 q^{59} + 70110 q^{61} - 98115 q^{63} - 172425 q^{65} - 15039 q^{67} + 251037 q^{69} + 67812 q^{71} - 27009 q^{73} - 75273 q^{75} + 23991 q^{77} - 216180 q^{79} + 177822 q^{81} - 76725 q^{83} - 53100 q^{85} - 201483 q^{87} - 98814 q^{89} - 90999 q^{91} + 21765 q^{93} - 143490 q^{95} - 71739 q^{97} + 13635 q^{99}+O(q^{100}) \) 90 * q - 87 * q^5 + 330 * q^9 - 1257 * q^11 + 531 * q^15 - 3468 * q^17 + 12894 * q^21 + 8106 * q^23 + 4959 * q^25 - 17415 * q^27 + 3468 * q^29 - 6651 * q^31 + 33624 * q^33 - 8229 * q^35 - 10545 * q^39 + 68673 * q^41 + 9459 * q^43 - 53469 * q^45 - 57087 * q^47 - 5490 * q^49 + 42831 * q^51 - 4146 * q^53 + 24624 * q^57 + 5388 * q^59 + 70110 * q^61 - 98115 * q^63 - 172425 * q^65 - 15039 * q^67 + 251037 * q^69 + 67812 * q^71 - 27009 * q^73 - 75273 * q^75 + 23991 * q^77 - 216180 * q^79 + 177822 * q^81 - 76725 * q^83 - 53100 * q^85 - 201483 * q^87 - 98814 * q^89 - 90999 * q^91 + 21765 * q^93 - 143490 * q^95 - 71739 * q^97 + 13635 * q^99

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