LMFDB - Embedded newform 252.6.k.f.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,6,Mod(37,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, degree $2$, 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:	$40.4167225929$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 $ x^8 - 2x^7 + 703x^6 + 2770x^5 + 427565x^4 + 718170x^3 + 42175732x^2 - 40929504*x + 3559792896
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$-11.2416 + 19.4709i$ of defining polynomial
Character	$\chi$	$=$	252.109
Dual form			252.6.k.f.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-46.4128 - 80.3893i) q^{5} +(-118.369 + 52.8745i) q^{7} +O(q^{10})$	$q+(-46.4128 - 80.3893i) q^{5} +(-118.369 + 52.8745i) q^{7} +(70.3812 - 121.904i) q^{11} -1111.24 q^{13} +(27.4435 - 47.5335i) q^{17} +(855.929 + 1482.51i) q^{19} +(-1643.95 - 2847.41i) q^{23} +(-2745.79 + 4755.85i) q^{25} +3790.72 q^{29} +(2423.66 - 4197.90i) q^{31} +(9744.39 + 7061.57i) q^{35} +(5683.35 + 9843.86i) q^{37} +10385.6 q^{41} +7137.16 q^{43} +(-8207.53 - 14215.9i) q^{47} +(11215.6 - 12517.4i) q^{49} +(-10487.2 + 18164.3i) q^{53} -13066.3 q^{55} +(-18106.0 + 31360.5i) q^{59} +(-2474.35 - 4285.70i) q^{61} +(51575.9 + 89332.0i) q^{65} +(11482.8 - 19888.8i) q^{67} +26341.8 q^{71} +(-27693.7 + 47966.9i) q^{73} +(-1885.36 + 18151.0i) q^{77} +(24978.3 + 43263.7i) q^{79} -44858.9 q^{83} -5094.91 q^{85} +(63972.5 + 110804. i) q^{89} +(131537. - 58756.4i) q^{91} +(79452.1 - 137615. i) q^{95} -65685.9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 42 q^{7}+O(q^{10}) $ 8 * q - 42 * q^7	$ 8 q - 42 q^{7} + 462 q^{11} - 1204 q^{13} - 228 q^{17} + 358 q^{19} + 2148 q^{23} - 5454 q^{25} + 11064 q^{29} + 830 q^{31} - 7692 q^{35} - 3914 q^{37} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} + 41876 q^{49} - 22164 q^{53} + 7784 q^{55} - 32886 q^{59} + 83732 q^{61} + 93192 q^{65} - 80034 q^{67} - 89544 q^{71} - 22470 q^{73} - 138732 q^{77} - 75286 q^{79} + 34836 q^{83} + 278504 q^{85} - 28944 q^{89} - 12058 q^{91} + 144120 q^{95} - 433356 q^{97}+O(q^{100}) $ 8 * q - 42 * q^7 + 462 * q^11 - 1204 * q^13 - 228 * q^17 + 358 * q^19 + 2148 * q^23 - 5454 * q^25 + 11064 * q^29 + 830 * q^31 - 7692 * q^35 - 3914 * q^37 + 16632 * q^41 - 29036 * q^43 - 41700 * q^47 + 41876 * q^49 - 22164 * q^53 + 7784 * q^55 - 32886 * q^59 + 83732 * q^61 + 93192 * q^65 - 80034 * q^67 - 89544 * q^71 - 22470 * q^73 - 138732 * q^77 - 75286 * q^79 + 34836 * q^83 + 278504 * q^85 - 28944 * q^89 - 12058 * q^91 + 144120 * q^95 - 433356 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−46.4128	−	80.3893i	−0.830257	−	1.43805i	−0.897834	−	0.440333i	$-0.854860\pi$
$5$	−46.4128	−	80.3893i	−0.830257	−	1.43805i	0.0675775	−	0.997714i	$-0.478473\pi$
$6$		0			0
$7$	−118.369	+	52.8745i	−0.913049	+	0.407851i
$7$	−118.369	+	52.8745i	−0.913049	+	0.407851i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	70.3812	−	121.904i	0.175378	−	0.303763i	−0.764914	−	0.644132i	$-0.777219\pi$
$11$	70.3812	−	121.904i	0.175378	−	0.303763i	0.940292	+	0.340369i	$0.110552\pi$
$12$		0			0
$13$	−1111.24			−1.82369			−0.911844	−	0.410537i	$-0.865341\pi$
$13$	−1111.24			−1.82369			−0.911844	+	0.410537i	$0.865341\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	27.4435	−	47.5335i	0.0230312	−	0.0398912i	−0.854280	−	0.519813i	$-0.826002\pi$
$17$	27.4435	−	47.5335i	0.0230312	−	0.0398912i	0.877311	+	0.479922i	$0.159335\pi$
$18$		0			0
$19$	855.929	+	1482.51i	0.543943	+	0.942138i	0.998673	+	0.0515079i	$0.0164027\pi$
$19$	855.929	+	1482.51i	0.543943	+	0.942138i	−0.454729	+	0.890630i	$0.650264\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1643.95	−	2847.41i	−0.647993	−	1.12236i	−0.983602	−	0.180355i	$-0.942275\pi$
$23$	−1643.95	−	2847.41i	−0.647993	−	1.12236i	0.335609	−	0.942001i	$-0.391058\pi$
$24$		0			0
$25$	−2745.79	+	4755.85i	−0.878653	+	1.52187i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3790.72			0.837003			0.418501	−	0.908216i	$-0.362555\pi$
$29$	3790.72			0.837003			0.418501	+	0.908216i	$0.362555\pi$
$30$		0			0
$31$	2423.66	−	4197.90i	0.452968	−	0.784563i	−0.545601	−	0.838045i	$-0.683699\pi$
$31$	2423.66	−	4197.90i	0.452968	−	0.784563i	0.998569	+	0.0534817i	$0.0170319\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9744.39	+	7061.57i	1.34457	+	0.974386i
$36$		0			0
$37$	5683.35	+	9843.86i	0.682496	+	1.18212i	0.974217	+	0.225615i	$0.0724391\pi$
$37$	5683.35	+	9843.86i	0.682496	+	1.18212i	−0.291720	+	0.956504i	$0.594228\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10385.6			0.964881			0.482440	−	0.875929i	$-0.339751\pi$
$41$	10385.6			0.964881			0.482440	+	0.875929i	$0.339751\pi$
$42$		0			0
$43$	7137.16			0.588646			0.294323	−	0.955706i	$-0.404906\pi$
$43$	7137.16			0.588646			0.294323	+	0.955706i	$0.404906\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8207.53	−	14215.9i	−0.541961	−	0.938704i	−0.998791	−	0.0491499i	$-0.984349\pi$
$47$	−8207.53	−	14215.9i	−0.541961	−	0.938704i	0.456831	−	0.889554i	$-0.348985\pi$
$48$		0			0
$49$	11215.6	−	12517.4i	0.667315	−	0.744775i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10487.2	+	18164.3i	−0.512824	+	0.888238i	0.487065	+	0.873366i	$0.338067\pi$
$53$	−10487.2	+	18164.3i	−0.512824	+	0.888238i	−0.999889	+	0.0148720i	$0.995266\pi$
$54$		0			0
$55$	−13066.3			−0.582435
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−18106.0	+	31360.5i	−0.677161	+	1.17288i	0.298672	+	0.954356i	$0.403456\pi$
$59$	−18106.0	+	31360.5i	−0.677161	+	1.17288i	−0.975832	+	0.218521i	$0.929877\pi$
$60$		0			0
$61$	−2474.35	−	4285.70i	−0.0851405	−	0.147468i	0.820311	−	0.571918i	$-0.193801\pi$
$61$	−2474.35	−	4285.70i	−0.0851405	−	0.147468i	−0.905451	+	0.424451i	$0.860467\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	51575.9	+	89332.0i	1.51413	+	2.62255i
$66$		0			0
$67$	11482.8	−	19888.8i	0.312507	−	0.541279i	−0.666397	−	0.745597i	$-0.732164\pi$
$67$	11482.8	−	19888.8i	0.312507	−	0.541279i	0.978905	+	0.204318i	$0.0654978\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	26341.8			0.620154			0.310077	−	0.950711i	$-0.399645\pi$
$71$	26341.8			0.620154			0.310077	+	0.950711i	$0.399645\pi$
$72$		0			0
$73$	−27693.7	+	47966.9i	−0.608238	+	1.05350i	0.383293	+	0.923627i	$0.374790\pi$
$73$	−27693.7	+	47966.9i	−0.608238	+	1.05350i	−0.991531	+	0.129872i	$0.958543\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1885.36	+	18151.0i	−0.0362383	+	0.348879i
$78$		0			0
$79$	24978.3	+	43263.7i	0.450293	+	0.779930i	0.998404	−	0.0564752i	$-0.0179862\pi$
$79$	24978.3	+	43263.7i	0.450293	+	0.779930i	−0.548111	+	0.836406i	$0.684653\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−44858.9			−0.714749			−0.357374	−	0.933961i	$-0.616328\pi$
$83$	−44858.9			−0.714749			−0.357374	+	0.933961i	$0.616328\pi$
$84$		0			0
$85$	−5094.91			−0.0764873
$86$		0			0
$87$		0			0
$88$		0			0
$89$	63972.5	+	110804.i	0.856087	+	1.48279i	0.875633	+	0.482978i	$0.160445\pi$
$89$	63972.5	+	110804.i	0.856087	+	1.48279i	−0.0195454	+	0.999809i	$0.506222\pi$
$90$		0			0
$91$	131537.	−	58756.4i	1.66512	−	0.743793i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	79452.1	−	137615.i	0.903226	−	1.56443i
$96$		0			0
$97$	−65685.9			−0.708831			−0.354415	−	0.935088i	$-0.615320\pi$
$97$	−65685.9			−0.708831			−0.354415	+	0.935088i	$0.615320\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	87891.4	−	152232.i	0.857320	−	1.48492i	−0.0171565	−	0.999853i	$-0.505461\pi$
$101$	87891.4	−	152232.i	0.857320	−	1.48492i	0.874476	−	0.485068i	$-0.161205\pi$
$102$		0			0
$103$	77054.7	+	133463.i	0.715659	+	1.23956i	0.962705	+	0.270554i	$0.0872068\pi$
$103$	77054.7	+	133463.i	0.715659	+	1.23956i	−0.247046	+	0.969004i	$0.579460\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	91681.0	+	158796.i	0.774141	+	1.34085i	0.935276	+	0.353919i	$0.115151\pi$
$107$	91681.0	+	158796.i	0.774141	+	1.34085i	−0.161135	+	0.986932i	$0.551515\pi$
$108$		0			0
$109$	67322.3	−	116606.i	0.542741	−	0.940055i	−0.456004	−	0.889978i	$-0.650720\pi$
$109$	67322.3	−	116606.i	0.542741	−	0.940055i	0.998745	−	0.0500775i	$-0.0159468\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−176955.			−1.30367			−0.651833	−	0.758362i	$-0.726000\pi$
$113$	−176955.			−1.30367			−0.651833	+	0.758362i	$0.726000\pi$
$114$		0			0
$115$	−152601.	+	264313.i	−1.07600	+	1.86369i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−735.153	+	7077.57i	−0.00475894	+	0.0458159i
$120$		0			0
$121$	70618.5	+	122315.i	0.438485	+	0.759479i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	219679.			1.25752
$126$		0			0
$127$	144432.			0.794608			0.397304	−	0.917687i	$-0.369946\pi$
$127$	144432.			0.794608			0.397304	+	0.917687i	$0.369946\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−118754.	−	205688.i	−0.604602	−	1.04720i	−0.992114	−	0.125337i	$-0.959999\pi$
$131$	−118754.	−	205688.i	−0.604602	−	1.04720i	0.387512	−	0.921865i	$-0.373335\pi$
$132$		0			0
$133$	−179703.	−	130227.i	−0.880899	−	0.638370i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−48101.6	+	83314.5i	−0.218957	+	0.379244i	−0.954489	−	0.298245i	$-0.903599\pi$
$137$	−48101.6	+	83314.5i	−0.218957	+	0.379244i	0.735533	+	0.677489i	$0.236932\pi$
$138$		0			0
$139$	−391373.			−1.71812			−0.859060	−	0.511874i	$-0.828951\pi$
$139$	−391373.			−1.71812			−0.859060	+	0.511874i	$0.828951\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−78210.6	+	135465.i	−0.319835	+	0.553970i
$144$		0			0
$145$	−175938.	−	304733.i	−0.694927	−	1.20365i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−40339.4	−	69869.9i	−0.148855	−	0.257825i	0.781949	−	0.623342i	$-0.214225\pi$
$149$	−40339.4	−	69869.9i	−0.148855	−	0.257825i	−0.930805	+	0.365517i	$0.880892\pi$
$150$		0			0
$151$	−47841.3	+	82863.6i	−0.170750	+	0.295748i	−0.938682	−	0.344783i	$-0.887952\pi$
$151$	−47841.3	+	82863.6i	−0.170750	+	0.295748i	0.767932	+	0.640531i	$0.221286\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−449955.			−1.50432
$156$		0			0
$157$	97812.7	−	169417.i	0.316699	−	0.548538i	−0.663098	−	0.748532i	$-0.730759\pi$
$157$	97812.7	−	169417.i	0.316699	−	0.548538i	0.979797	+	0.199994i	$0.0640923\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	345149.	+	250123.i	1.04940	+	0.760481i
$162$		0			0
$163$	−18822.8	−	32602.1i	−0.0554902	−	0.0961118i	0.836946	−	0.547286i	$-0.184339\pi$
$163$	−18822.8	−	32602.1i	−0.0554902	−	0.0961118i	−0.892436	+	0.451174i	$0.851005\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−646778.			−1.79458			−0.897292	−	0.441437i	$-0.854469\pi$
$167$	−646778.			−1.79458			−0.897292	+	0.441437i	$0.854469\pi$
$168$		0			0
$169$	863568.			2.32584
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−193454.	−	335072.i	−0.491431	−	0.851184i	0.508520	−	0.861050i	$-0.330193\pi$
$173$	−193454.	−	335072.i	−0.491431	−	0.851184i	−0.999951	+	0.00986616i	$0.996859\pi$
$174$		0			0
$175$	73554.0	−	708129.i	0.181556	−	1.74790i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	13705.3	−	23738.2i	0.0319709	−	0.0553752i	−0.849597	−	0.527432i	$-0.823155\pi$
$179$	13705.3	−	23738.2i	0.0319709	−	0.0553752i	0.881568	+	0.472057i	$0.156488\pi$
$180$		0			0
$181$	−196155.			−0.445044			−0.222522	−	0.974928i	$-0.571429\pi$
$181$	−196155.			−0.445044			−0.222522	+	0.974928i	$0.571429\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	527560.	−	913761.i	1.13329	−	1.96292i
$186$		0			0
$187$	−3863.01	−	6690.93i	−0.00807833	−	0.0139921i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	133491.	+	231213.i	0.264770	+	0.458595i	0.967503	−	0.252858i	$-0.0813706\pi$
$191$	133491.	+	231213.i	0.264770	+	0.458595i	−0.702733	+	0.711453i	$0.748037\pi$
$192$		0			0
$193$	50681.9	−	87783.6i	0.0979398	−	0.169637i	−0.812892	−	0.582415i	$-0.802108\pi$
$193$	50681.9	−	87783.6i	0.0979398	−	0.169637i	0.910832	+	0.412778i	$0.135441\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−362366.			−0.665245			−0.332623	−	0.943060i	$-0.607934\pi$
$197$	−362366.			−0.665245			−0.332623	+	0.943060i	$0.607934\pi$
$198$		0			0
$199$	−112706.	+	195213.i	−0.201751	+	0.349443i	−0.949093	−	0.314997i	$-0.897997\pi$
$199$	−112706.	+	195213.i	−0.201751	+	0.349443i	0.747342	+	0.664440i	$0.231330\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−448705.	+	200433.i	−0.764224	+	0.341372i
$204$		0			0
$205$	−482026.	−	834894.i	−0.801099	−	1.38754i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	240965.			0.381583
$210$		0			0
$211$	−327801.			−0.506878			−0.253439	−	0.967351i	$-0.581562\pi$
$211$	−327801.			−0.506878			−0.253439	+	0.967351i	$0.581562\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−331255.	−	573751.i	−0.488727	−	0.846501i
$216$		0			0
$217$	−64924.7	+	625053.i	−0.0935968	+	0.901088i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−30496.4	+	52821.2i	−0.0420017	+	0.0727492i
$222$		0			0
$223$	109690.			0.147708			0.0738538	−	0.997269i	$-0.476470\pi$
$223$	109690.			0.147708			0.0738538	+	0.997269i	$0.476470\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	608733.	−	1.05436e6i	0.784083	−	1.35807i	−0.145463	−	0.989364i	$-0.546467\pi$
$227$	608733.	−	1.05436e6i	0.784083	−	1.35807i	0.929545	−	0.368708i	$-0.120200\pi$
$228$		0			0
$229$	668013.	+	1.15703e6i	0.841776	+	1.45800i	0.888392	+	0.459086i	$0.151823\pi$
$229$	668013.	+	1.15703e6i	0.841776	+	1.45800i	−0.0466161	+	0.998913i	$0.514844\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	23394.2	+	40520.0i	0.0282305	+	0.0488967i	0.879795	−	0.475352i	$-0.157679\pi$
$233$	23394.2	+	40520.0i	0.0282305	+	0.0488967i	−0.851565	+	0.524249i	$0.824346\pi$
$234$		0			0
$235$	−761868.	+	1.31959e6i	−0.899933	+	1.55873i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−86350.2			−0.0977841			−0.0488921	−	0.998804i	$-0.515569\pi$
$239$	−86350.2			−0.0977841			−0.0488921	+	0.998804i	$0.515569\pi$
$240$		0			0
$241$	−412649.	+	714730.i	−0.457655	+	0.792682i	−0.998837	−	0.0482236i	$-0.984644\pi$
$241$	−412649.	+	714730.i	−0.457655	+	0.792682i	0.541181	+	0.840906i	$0.317977\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.52681e6	−	320642.i	−1.62507	−	0.341276i
$246$		0			0
$247$	−951145.	−	1.64743e6i	−0.991983	−	1.71817i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	130188.			0.130432			0.0652162	−	0.997871i	$-0.479226\pi$
$251$	130188.			0.130432			0.0652162	+	0.997871i	$0.479226\pi$
$252$		0			0
$253$	−462814.			−0.454574
$254$		0			0
$255$		0			0
$256$		0			0
$257$	51395.1	+	89018.9i	0.0485388	+	0.0840716i	0.889274	−	0.457375i	$-0.151210\pi$
$257$	51395.1	+	89018.9i	0.0485388	+	0.0840716i	−0.840735	+	0.541446i	$0.817877\pi$
$258$		0			0
$259$	−1.19322e6	−	864706.i	−1.10528	−	0.800975i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−17750.4	+	30744.5i	−0.0158241	+	0.0274081i	−0.873829	−	0.486233i	$-0.838370\pi$
$263$	−17750.4	+	30744.5i	−0.0158241	+	0.0274081i	0.858005	+	0.513641i	$0.171704\pi$
$264$		0			0
$265$	1.94695e6			1.70310
$266$		0			0
$267$		0			0
$268$		0			0
$269$	523171.	−	906158.i	0.440821	−	0.763525i	−0.556929	−	0.830560i	$-0.688021\pi$
$269$	523171.	−	906158.i	0.440821	−	0.763525i	0.997751	+	0.0670350i	$0.0213539\pi$
$270$		0			0
$271$	−88639.2	−	153528.i	−0.0733167	−	0.126988i	0.827036	−	0.562148i	$-0.190025\pi$
$271$	−88639.2	−	153528.i	−0.0733167	−	0.126988i	−0.900353	+	0.435160i	$0.856692\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	386504.	+	669445.i	0.308193	+	0.533805i
$276$		0			0
$277$	−72793.0	+	126081.i	−0.0570020	+	0.0987304i	−0.893118	−	0.449822i	$-0.851487\pi$
$277$	−72793.0	+	126081.i	−0.0570020	+	0.0987304i	0.836116	+	0.548552i	$0.184821\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.03073e6			0.778714			0.389357	−	0.921087i	$-0.372697\pi$
$281$	1.03073e6			0.778714			0.389357	+	0.921087i	$0.372697\pi$
$282$		0			0
$283$	558631.	−	967577.i	0.414628	−	0.718157i	−0.580761	−	0.814074i	$-0.697245\pi$
$283$	558631.	−	967577.i	0.414628	−	0.718157i	0.995389	+	0.0959167i	$0.0305783\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.22934e6	+	549136.i	−0.880983	+	0.393527i
$288$		0			0
$289$	708422.	+	1.22702e6i	0.498939	+	0.864188i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.02032e6			1.37484			0.687418	−	0.726262i	$-0.258744\pi$
$293$	2.02032e6			1.37484			0.687418	+	0.726262i	$0.258744\pi$
$294$		0			0
$295$	3.36139e6			2.24887
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.82683e6	+	3.16417e6i	1.18174	+	2.04683i
$300$		0			0
$301$	−844820.	+	377374.i	−0.537462	+	0.240080i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−229683.	+	397822.i	−0.141377	+	0.244872i
$306$		0			0
$307$	−535150.			−0.324063			−0.162031	−	0.986786i	$-0.551805\pi$
$307$	−535150.			−0.324063			−0.162031	+	0.986786i	$0.551805\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.15009e6	+	1.99202e6i	−0.674268	+	1.16787i	0.302415	+	0.953176i	$0.402207\pi$
$311$	−1.15009e6	+	1.99202e6i	−0.674268	+	1.16787i	−0.976682	+	0.214689i	$0.931126\pi$
$312$		0			0
$313$	1.37863e6	+	2.38786e6i	0.795404	+	1.37768i	0.922582	+	0.385801i	$0.126075\pi$
$313$	1.37863e6	+	2.38786e6i	0.795404	+	1.37768i	−0.127178	+	0.991880i	$0.540592\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.27404e6	+	2.20671e6i	0.712092	+	1.23338i	0.964070	+	0.265647i	$0.0855856\pi$
$317$	1.27404e6	+	2.20671e6i	0.712092	+	1.23338i	−0.251978	+	0.967733i	$0.581081\pi$
$318$		0			0
$319$	266795.	−	462103.i	0.146792	−	0.254251i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	93958.7			0.0501107
$324$		0			0
$325$	3.05124e6	−	5.28490e6i	1.60239	−	2.77542i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.72318e6	+	1.24875e6i	0.877688	+	0.636043i
$330$		0			0
$331$	244667.	+	423776.i	0.122746	+	0.212602i	0.920849	−	0.389918i	$-0.127497\pi$
$331$	244667.	+	423776.i	0.122746	+	0.212602i	−0.798104	+	0.602520i	$0.794163\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.13179e6			−1.03785
$336$		0			0
$337$	−1.47140e6			−0.705757			−0.352878	−	0.935669i	$-0.614797\pi$
$337$	−1.47140e6			−0.705757			−0.352878	+	0.935669i	$0.614797\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−341160.	−	590907.i	−0.158881	−	0.275190i
$342$		0			0
$343$	−665725.	+	2.07470e6i	−0.305534	+	0.952181i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.20963e6	+	2.09513e6i	−0.539296	+	0.934088i	0.459646	+	0.888102i	$0.347976\pi$
$347$	−1.20963e6	+	2.09513e6i	−0.539296	+	0.934088i	−0.998942	+	0.0459859i	$0.985357\pi$
$348$		0			0
$349$	−2.58571e6			−1.13636			−0.568180	−	0.822905i	$-0.692352\pi$
$349$	−2.58571e6			−1.13636			−0.568180	+	0.822905i	$0.692352\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	502198.	−	869832.i	0.214505	−	0.371534i	−0.738614	−	0.674128i	$-0.764519\pi$
$353$	502198.	−	869832.i	0.214505	−	0.371534i	0.953119	+	0.302594i	$0.0978528\pi$
$354$		0			0
$355$	−1.22260e6	−	2.11760e6i	−0.514887	−	0.891811i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.13383e6	−	1.96384e6i	−0.464313	−	0.804213i	0.534858	−	0.844942i	$-0.320365\pi$
$359$	−1.13383e6	−	1.96384e6i	−0.464313	−	0.804213i	−0.999170	+	0.0407292i	$0.987032\pi$
$360$		0			0
$361$	−227180.	+	393487.i	−0.0917490	+	0.158914i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.14136e6			2.01998
$366$		0			0
$367$	−2.39312e6	+	4.14500e6i	−0.927467	+	1.60642i	−0.139923	+	0.990162i	$0.544685\pi$
$367$	−2.39312e6	+	4.14500e6i	−0.927467	+	1.60642i	−0.787545	+	0.616258i	$0.788648\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	280929.	−	2.70460e6i	0.105965	−	1.02016i
$372$		0			0
$373$	2.31351e6	+	4.00711e6i	0.860991	+	1.49128i	0.870973	+	0.491331i	$0.163489\pi$
$373$	2.31351e6	+	4.00711e6i	0.860991	+	1.49128i	−0.00998166	+	0.999950i	$0.503177\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.21241e6			−1.52643
$378$		0			0
$379$	−497760.			−0.178001			−0.0890004	−	0.996032i	$-0.528367\pi$
$379$	−497760.			−0.178001			−0.0890004	+	0.996032i	$0.528367\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	693745.	+	1.20160e6i	0.241659	+	0.418566i	0.961187	−	0.275898i	$-0.0889751\pi$
$383$	693745.	+	1.20160e6i	0.241659	+	0.418566i	−0.719528	+	0.694463i	$0.755642\pi$
$384$		0			0
$385$	1.54665e6	−	690877.i	0.531791	−	0.237547i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−293028.	+	507539.i	−0.0981826	+	0.170057i	−0.910932	−	0.412556i	$-0.864636\pi$
$389$	−293028.	+	507539.i	−0.0981826	+	0.170057i	0.812750	+	0.582613i	$0.197970\pi$
$390$		0			0
$391$	−180463.			−0.0596962
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.31862e6	−	4.01598e6i	0.747718	−	1.29509i
$396$		0			0
$397$	−1.52668e6	−	2.64429e6i	−0.486151	−	0.842039i	0.513722	−	0.857957i	$-0.328266\pi$
$397$	−1.52668e6	−	2.64429e6i	−0.486151	−	0.842039i	−0.999873	+	0.0159181i	$0.994933\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.76158e6	+	4.78320e6i	0.857624	+	1.48545i	0.874189	+	0.485585i	$0.161393\pi$
$401$	2.76158e6	+	4.78320e6i	0.857624	+	1.48545i	−0.0165657	+	0.999863i	$0.505273\pi$
$402$		0			0
$403$	−2.69327e6	+	4.66489e6i	−0.826072	+	1.43080i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.60000e6			0.478779
$408$		0			0
$409$	1.67464e6	−	2.90055e6i	0.495008	−	0.857379i	−0.504975	−	0.863134i	$-0.668498\pi$
$409$	1.67464e6	−	2.90055e6i	0.495008	−	0.857379i	0.999983	+	0.00575475i	$0.00183180\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	485021.	−	4.66946e6i	0.139922	−	1.34707i
$414$		0			0
$415$	2.08203e6	+	3.60618e6i	0.593425	+	1.02784i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.97012e6			−0.826493			−0.413247	−	0.910619i	$-0.635605\pi$
$419$	−2.97012e6			−0.826493			−0.413247	+	0.910619i	$0.635605\pi$
$420$		0			0
$421$	−5.41478e6			−1.48894			−0.744468	−	0.667658i	$-0.767297\pi$
$421$	−5.41478e6			−1.48894			−0.744468	+	0.667658i	$0.767297\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	150708.	+	261034.i	0.0404729	+	0.0701011i
$426$		0			0
$427$	519491.	+	376465.i	0.137882	+	0.0999205i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.52975e6	+	4.38165e6i	−0.655970	+	1.13617i	0.325679	+	0.945480i	$0.394407\pi$
$431$	−2.52975e6	+	4.38165e6i	−0.655970	+	1.13617i	−0.981650	+	0.190694i	$0.938926\pi$
$432$		0			0
$433$	3.84174e6			0.984711			0.492355	−	0.870394i	$-0.336136\pi$
$433$	3.84174e6			0.984711			0.492355	+	0.870394i	$0.336136\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.81422e6	−	4.87437e6i	0.704943	−	1.22100i
$438$		0			0
$439$	762859.	+	1.32131e6i	0.188922	+	0.327223i	0.944891	−	0.327385i	$-0.106167\pi$
$439$	762859.	+	1.32131e6i	0.188922	+	0.327223i	−0.755969	+	0.654607i	$0.772834\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−735068.	−	1.27317e6i	−0.177958	−	0.308233i	0.763223	−	0.646135i	$-0.223616\pi$
$443$	−735068.	−	1.27317e6i	−0.177958	−	0.308233i	−0.941181	+	0.337903i	$0.890283\pi$
$444$		0			0
$445$	5.93828e6	−	1.02854e7i	1.42154	−	2.46219i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.05071e6			1.41641			0.708207	−	0.706004i	$-0.249504\pi$
$449$	6.05071e6			1.41641			0.708207	+	0.706004i	$0.249504\pi$
$450$		0			0
$451$	730954.	−	1.26605e6i	0.169219	−	0.293095i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.08284e7	−	7.84712e6i	−2.45208	−	1.77698i
$456$		0			0
$457$	2.87710e6	+	4.98328e6i	0.644413	+	1.11616i	0.984437	+	0.175739i	$0.0562315\pi$
$457$	2.87710e6	+	4.98328e6i	0.644413	+	1.11616i	−0.340024	+	0.940417i	$0.610435\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.83684e6			−0.621703			−0.310851	−	0.950458i	$-0.600614\pi$
$461$	−2.83684e6			−0.621703			−0.310851	+	0.950458i	$0.600614\pi$
$462$		0			0
$463$	5.19089e6			1.12535			0.562677	−	0.826677i	$-0.309771\pi$
$463$	5.19089e6			1.12535			0.562677	+	0.826677i	$0.309771\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	544520.	+	943136.i	0.115537	+	0.200116i	0.917994	−	0.396594i	$-0.129808\pi$
$467$	544520.	+	943136.i	0.115537	+	0.200116i	−0.802457	+	0.596710i	$0.796474\pi$
$468$		0			0
$469$	−307600.	+	2.96137e6i	−0.0645734	+	0.621670i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	502322.	−	870047.i	0.103235	−	0.178809i
$474$		0			0
$475$	−9.40081e6			−1.91175
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.84705e6	+	4.93124e6i	−0.566966	+	0.982013i	0.429898	+	0.902877i	$0.358549\pi$
$479$	−2.84705e6	+	4.93124e6i	−0.566966	+	0.982013i	−0.996864	+	0.0791359i	$0.974784\pi$
$480$		0			0
$481$	−6.31559e6	−	1.09389e7i	−1.24466	−	2.15582i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.04866e6	+	5.28044e6i	0.588512	+	1.01933i
$486$		0			0
$487$	2.05713e6	−	3.56305e6i	0.393042	−	0.680768i	−0.599807	−	0.800144i	$-0.704756\pi$
$487$	2.05713e6	−	3.56305e6i	0.393042	−	0.680768i	0.992849	+	0.119376i	$0.0380895\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	609530.			0.114101			0.0570507	−	0.998371i	$-0.481830\pi$
$491$	609530.			0.114101			0.0570507	+	0.998371i	$0.481830\pi$
$492$		0			0
$493$	104031.	−	180186.i	0.0192772	−	0.0333891i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.11806e6	+	1.39281e6i	−0.566231	+	0.252930i
$498$		0			0
$499$	−497896.	−	862382.i	−0.0895133	−	0.155042i	0.817792	−	0.575514i	$-0.195198\pi$
$499$	−497896.	−	862382.i	−0.0895133	−	0.155042i	−0.907306	+	0.420472i	$0.861864\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.02730e7			−1.81042			−0.905209	−	0.424966i	$-0.860286\pi$
$503$	−1.02730e7			−1.81042			−0.905209	+	0.424966i	$0.860286\pi$
$504$		0			0
$505$	−1.63171e7			−2.84718
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.39213e6	−	2.41125e6i	−0.238170	−	0.412522i	0.722020	−	0.691873i	$-0.243214\pi$
$509$	−1.39213e6	−	2.41125e6i	−0.238170	−	0.412522i	−0.960189	+	0.279351i	$0.909881\pi$
$510$		0			0
$511$	741856.	−	7.14210e6i	0.125680	−	1.20997i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.15264e6	−	1.23887e7i	1.18836	−	2.05830i
$516$		0			0
$517$	−2.31062e6			−0.380192
$518$		0			0
$519$		0			0
$520$		0			0
$521$	522678.	−	905306.i	0.0843607	−	0.146117i	−0.820758	−	0.571276i	$-0.806449\pi$
$521$	522678.	−	905306.i	0.0843607	−	0.146117i	0.905119	+	0.425159i	$0.139782\pi$
$522$		0			0
$523$	1.81578e6	+	3.14503e6i	0.290275	+	0.502771i	0.973875	−	0.227086i	$-0.0729199\pi$
$523$	1.81578e6	+	3.14503e6i	0.290275	+	0.502771i	−0.683600	+	0.729857i	$0.739587\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−133027.	−	230410.i	−0.0208648	−	0.0361389i
$528$		0			0
$529$	−2.18700e6	+	3.78800e6i	−0.339789	+	0.588532i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.15410e7			−1.75964
$534$		0			0
$535$	8.51034e6	−	1.47403e7i	1.28547	−	2.22650i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−736558.	−	2.24821e6i	−0.109203	−	0.333323i
$540$		0			0
$541$	4.72442e6	+	8.18294e6i	0.693994	+	1.20203i	0.970519	+	0.241025i	$0.0774837\pi$
$541$	4.72442e6	+	8.18294e6i	0.693994	+	1.20203i	−0.276525	+	0.961007i	$0.589183\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.24985e7			−1.80246
$546$		0			0
$547$	2.69014e6			0.384421			0.192210	−	0.981354i	$-0.438434\pi$
$547$	2.69014e6			0.384421			0.192210	+	0.981354i	$0.438434\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.24459e6	+	5.61979e6i	0.455282	+	0.788572i
$552$		0			0
$553$	−5.24421e6	−	3.80038e6i	−0.729235	−	0.528462i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.44490e6	+	7.69880e6i	−0.607050	+	1.05144i	0.384674	+	0.923052i	$0.374314\pi$
$557$	−4.44490e6	+	7.69880e6i	−0.607050	+	1.05144i	−0.991724	+	0.128389i	$0.959020\pi$
$558$		0			0
$559$	−7.93112e6			−1.07351
$560$		0			0
$561$		0			0
$562$		0			0
$563$	629963.	−	1.09113e6i	0.0837615	−	0.145079i	−0.821101	−	0.570782i	$-0.806640\pi$
$563$	629963.	−	1.09113e6i	0.0837615	−	0.145079i	0.904863	+	0.425703i	$0.139973\pi$
$564$		0			0
$565$	8.21297e6	+	1.42253e7i	1.08238	+	1.87473i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.91019e6	−	5.04059e6i	−0.376826	−	0.652681i	0.613773	−	0.789483i	$-0.289651\pi$
$569$	−2.91019e6	−	5.04059e6i	−0.376826	−	0.652681i	−0.990598	+	0.136801i	$0.956318\pi$
$570$		0			0
$571$	−2.32980e6	+	4.03534e6i	−0.299040	+	0.517952i	−0.975917	−	0.218144i	$-0.930000\pi$
$571$	−2.32980e6	+	4.03534e6i	−0.299040	+	0.517952i	0.676877	+	0.736096i	$0.263333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.80558e7			2.27744
$576$		0			0
$577$	762391.	−	1.32050e6i	0.0953319	−	0.165120i	−0.814415	−	0.580283i	$-0.802942\pi$
$577$	762391.	−	1.32050e6i	0.0953319	−	0.165120i	0.909747	+	0.415163i	$0.136275\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.30992e6	−	2.37189e6i	0.652600	−	0.291511i
$582$		0			0
$583$	1.47620e6	+	2.55685e6i	0.179876	+	0.311554i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.50087e7			1.79783			0.898914	−	0.438124i	$-0.144357\pi$
$587$	1.50087e7			1.79783			0.898914	+	0.438124i	$0.144357\pi$
$588$		0			0
$589$	8.29792e6			0.985556
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.53973e6	−	6.13099e6i	−0.413364	−	0.715968i	0.581891	−	0.813267i	$-0.302313\pi$
$593$	−3.53973e6	−	6.13099e6i	−0.413364	−	0.715968i	−0.995255	+	0.0972987i	$0.968980\pi$
$594$		0			0
$595$	603081.	−	269391.i	0.0698366	−	0.0311954i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.08761e6	+	1.88379e6i	−0.123853	+	0.214519i	−0.921284	−	0.388891i	$-0.872858\pi$
$599$	−1.08761e6	+	1.88379e6i	−0.123853	+	0.214519i	0.797431	+	0.603410i	$0.206192\pi$
$600$		0			0
$601$	2.69785e6			0.304671			0.152336	−	0.988329i	$-0.451321\pi$
$601$	2.69785e6			0.304671			0.152336	+	0.988329i	$0.451321\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.55520e6	−	1.13539e7i	0.728111	−	1.26112i
$606$		0			0
$607$	−2.49417e6	−	4.32003e6i	−0.274760	−	0.475899i	0.695314	−	0.718706i	$-0.255265\pi$
$607$	−2.49417e6	−	4.32003e6i	−0.274760	−	0.475899i	−0.970075	+	0.242807i	$0.921932\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.12056e6	+	1.57973e7i	0.988367	+	1.71190i
$612$		0			0
$613$	−556820.	+	964441.i	−0.0598500	+	0.103663i	−0.894398	−	0.447272i	$-0.852396\pi$
$613$	−556820.	+	964441.i	−0.0598500	+	0.103663i	0.834548	+	0.550935i	$0.185729\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.14757e6			0.544364			0.272182	−	0.962246i	$-0.412255\pi$
$617$	5.14757e6			0.544364			0.272182	+	0.962246i	$0.412255\pi$
$618$		0			0
$619$	−121588.	+	210597.i	−0.0127545	+	0.0220915i	−0.872332	−	0.488914i	$-0.837393\pi$
$619$	−121588.	+	210597.i	−0.0127545	+	0.0220915i	0.859578	+	0.511005i	$0.170727\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.34311e7	−	9.73322e6i	−1.38641	−	1.00470i
$624$		0			0
$625$	−1.61533e6	−	2.79783e6i	−0.165410	−	0.286498i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	623884.			0.0628749
$630$		0			0
$631$	9.94255e6			0.994087			0.497044	−	0.867726i	$-0.334419\pi$
$631$	9.94255e6			0.994087			0.497044	+	0.867726i	$0.334419\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.70347e6	−	1.16108e7i	−0.659729	−	1.14268i
$636$		0			0
$637$	−1.24632e7	+	1.39099e7i	−1.21697	+	1.35824i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.77767e6	+	1.00072e7i	−0.555402	+	0.961985i	0.442470	+	0.896783i	$0.354102\pi$
$641$	−5.77767e6	+	1.00072e7i	−0.555402	+	0.961985i	−0.997872	+	0.0652017i	$0.979231\pi$
$642$		0			0
$643$	1.35139e6			0.128900			0.0644500	−	0.997921i	$-0.479471\pi$
$643$	1.35139e6			0.128900			0.0644500	+	0.997921i	$0.479471\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.77678e6	−	1.52018e7i	0.824279	−	1.42769i	−0.0781895	−	0.996939i	$-0.524914\pi$
$647$	8.77678e6	−	1.52018e7i	0.824279	−	1.42769i	0.902469	−	0.430755i	$-0.141753\pi$
$648$		0			0
$649$	2.54864e6	+	4.41437e6i	0.237518	+	0.411393i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.99936e6	−	6.92710e6i	−0.367035	−	0.635724i	0.622065	−	0.782965i	$-0.286294\pi$
$653$	−3.99936e6	−	6.92710e6i	−0.367035	−	0.635724i	−0.989101	+	0.147242i	$0.952961\pi$
$654$		0			0
$655$	−1.10234e7	+	1.90931e7i	−1.00395	+	1.73889i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−229419.			−0.0205786			−0.0102893	−	0.999947i	$-0.503275\pi$
$659$	−229419.			−0.0205786			−0.0102893	+	0.999947i	$0.503275\pi$
$660$		0			0
$661$	1.64279e6	−	2.84539e6i	0.146244	−	0.253302i	−0.783593	−	0.621275i	$-0.786615\pi$
$661$	1.64279e6	−	2.84539e6i	0.146244	−	0.253302i	0.929836	+	0.367973i	$0.119948\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.12835e6	+	2.04904e7i	−0.186634	+	1.79678i
$666$		0			0
$667$	−6.23177e6	−	1.07937e7i	−0.542372	−	0.939416i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−696590.			−0.0597271
$672$		0			0
$673$	8.22188e6			0.699734			0.349867	−	0.936799i	$-0.386227\pi$
$673$	8.22188e6			0.699734			0.349867	+	0.936799i	$0.386227\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.07702e7	+	1.86546e7i	0.903138	+	1.56428i	0.823397	+	0.567465i	$0.192076\pi$
$677$	1.07702e7	+	1.86546e7i	0.903138	+	1.56428i	0.0797405	+	0.996816i	$0.474591\pi$
$678$		0			0
$679$	7.77519e6	−	3.47311e6i	0.647197	−	0.289097i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.08706e6	+	1.40072e7i	−0.663344	+	1.14895i	0.316387	+	0.948630i	$0.397530\pi$
$683$	−8.08706e6	+	1.40072e7i	−0.663344	+	1.14895i	−0.979731	+	0.200316i	$0.935803\pi$
$684$		0			0
$685$	8.93012e6			0.727162
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.16538e7	−	2.01850e7i	0.935231	−	1.61987i
$690$		0			0
$691$	−5.32318e6	−	9.22002e6i	−0.424108	−	0.734576i	0.572229	−	0.820094i	$-0.306079\pi$
$691$	−5.32318e6	−	9.22002e6i	−0.424108	−	0.734576i	−0.996337	+	0.0855179i	$0.972746\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.81647e7	+	3.14622e7i	1.42648	+	2.47074i
$696$		0			0
$697$	285018.	−	493666.i	0.0222224	−	0.0384903i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.55461e6			0.350071			0.175035	−	0.984562i	$-0.443996\pi$
$701$	4.55461e6			0.350071			0.175035	+	0.984562i	$0.443996\pi$
$702$		0			0
$703$	−9.72909e6	+	1.68513e7i	−0.742479	+	1.28601i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.35442e6	+	2.26668e7i	−0.177148	+	1.70546i
$708$		0			0
$709$	−6.55174e6	−	1.13479e7i	−0.489487	−	0.847816i	0.510440	−	0.859913i	$-0.329483\pi$
$709$	−6.55174e6	−	1.13479e7i	−0.489487	−	0.847816i	−0.999927	+	0.0120971i	$0.996149\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.59375e7			−1.17408
$714$		0			0
$715$	1.45199e7			1.06218
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.95942e6	+	1.55182e7i	0.646335	+	1.11949i	0.983991	+	0.178216i	$0.0570325\pi$
$719$	8.95942e6	+	1.55182e7i	0.646335	+	1.11949i	−0.337656	+	0.941269i	$0.609634\pi$
$720$		0			0
$721$	−1.61777e7	−	1.17236e7i	−1.15899	−	0.839894i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.04085e7	+	1.80281e7i	−0.735435	+	1.27381i
$726$		0			0
$727$	1.41466e7			0.992693			0.496347	−	0.868125i	$-0.334675\pi$
$727$	1.41466e7			0.992693			0.496347	+	0.868125i	$0.334675\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	195868.	−	339254.i	0.0135572	−	0.0234818i
$732$		0			0
$733$	6.22241e6	+	1.07775e7i	0.427758	+	0.740899i	0.996674	−	0.0814969i	$-0.0259701\pi$
$733$	6.22241e6	+	1.07775e7i	0.427758	+	0.740899i	−0.568915	+	0.822396i	$0.692637\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.61634e6	−	2.79959e6i	−0.109614	−	0.189857i
$738$		0			0
$739$	883322.	−	1.52996e6i	0.0594988	−	0.103055i	−0.834742	−	0.550642i	$-0.814383\pi$
$739$	883322.	−	1.52996e6i	0.0594988	−	0.103055i	0.894241	+	0.447587i	$0.147716\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.73250e6			−0.380953			−0.190477	−	0.981692i	$-0.561003\pi$
$743$	−5.73250e6			−0.380953			−0.190477	+	0.981692i	$0.561003\pi$
$744$		0			0
$745$	−3.74453e6	+	6.48571e6i	−0.247176	+	0.428122i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.92485e7	−	1.39490e7i	−1.25370	−	0.908528i
$750$		0			0
$751$	−1.31566e7	−	2.27879e7i	−0.851224	−	1.47436i	−0.880104	−	0.474780i	$-0.842528\pi$
$751$	−1.31566e7	−	2.27879e7i	−0.851224	−	1.47436i	0.0288804	−	0.999583i	$-0.490806\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.88179e6			0.567066
$756$		0			0
$757$	1.63104e7			1.03449			0.517245	−	0.855838i	$-0.326958\pi$
$757$	1.63104e7			1.03449			0.517245	+	0.855838i	$0.326958\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.42080e6	+	1.45853e7i	0.527098	+	0.912961i	0.999501	+	0.0315785i	$0.0100534\pi$
$761$	8.42080e6	+	1.45853e7i	0.527098	+	0.912961i	−0.472403	+	0.881383i	$0.656613\pi$
$762$		0			0
$763$	−1.80342e6	+	1.73622e7i	−0.112147	+	1.07967i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.01201e7	−	3.48491e7i	1.23493	−	2.13896i
$768$		0			0
$769$	1.19890e7			0.731084			0.365542	−	0.930795i	$-0.380884\pi$
$769$	1.19890e7			0.731084			0.365542	+	0.930795i	$0.380884\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8.48868e6	−	1.47028e7i	0.510965	−	0.885018i	−0.488954	−	0.872310i	$-0.662621\pi$
$773$	8.48868e6	−	1.47028e7i	0.510965	−	0.885018i	0.999919	−	0.0127081i	$-0.00404522\pi$
$774$		0			0
$775$	1.33097e7	+	2.30531e7i	0.796003	+	1.37872i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.88937e6	+	1.53968e7i	0.524841	+	0.909050i
$780$		0			0
$781$	1.85397e6	−	3.21117e6i	0.108761	−	0.188380i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.81590e7			−1.05177
$786$		0			0
$787$	1.59750e7	−	2.76696e7i	0.919400	−	1.59245i	0.119072	−	0.992886i	$-0.462008\pi$
$787$	1.59750e7	−	2.76696e7i	0.919400	−	1.59245i	0.800328	−	0.599563i	$-0.204659\pi$
$788$		0			0
$789$		0			0
$790$		0

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-46.4128 - 80.3893i) q^{5} +(-118.369 + 52.8745i) q^{7} +O(q^{10})\)	\(q+(-46.4128 - 80.3893i) q^{5} +(-118.369 + 52.8745i) q^{7} +(70.3812 - 121.904i) q^{11} -1111.24 q^{13} +(27.4435 - 47.5335i) q^{17} +(855.929 + 1482.51i) q^{19} +(-1643.95 - 2847.41i) q^{23} +(-2745.79 + 4755.85i) q^{25} +3790.72 q^{29} +(2423.66 - 4197.90i) q^{31} +(9744.39 + 7061.57i) q^{35} +(5683.35 + 9843.86i) q^{37} +10385.6 q^{41} +7137.16 q^{43} +(-8207.53 - 14215.9i) q^{47} +(11215.6 - 12517.4i) q^{49} +(-10487.2 + 18164.3i) q^{53} -13066.3 q^{55} +(-18106.0 + 31360.5i) q^{59} +(-2474.35 - 4285.70i) q^{61} +(51575.9 + 89332.0i) q^{65} +(11482.8 - 19888.8i) q^{67} +26341.8 q^{71} +(-27693.7 + 47966.9i) q^{73} +(-1885.36 + 18151.0i) q^{77} +(24978.3 + 43263.7i) q^{79} -44858.9 q^{83} -5094.91 q^{85} +(63972.5 + 110804. i) q^{89} +(131537. - 58756.4i) q^{91} +(79452.1 - 137615. i) q^{95} -65685.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 42 q^{7}+O(q^{10}) \) 8 * q - 42 * q^7	\( 8 q - 42 q^{7} + 462 q^{11} - 1204 q^{13} - 228 q^{17} + 358 q^{19} + 2148 q^{23} - 5454 q^{25} + 11064 q^{29} + 830 q^{31} - 7692 q^{35} - 3914 q^{37} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} + 41876 q^{49} - 22164 q^{53} + 7784 q^{55} - 32886 q^{59} + 83732 q^{61} + 93192 q^{65} - 80034 q^{67} - 89544 q^{71} - 22470 q^{73} - 138732 q^{77} - 75286 q^{79} + 34836 q^{83} + 278504 q^{85} - 28944 q^{89} - 12058 q^{91} + 144120 q^{95} - 433356 q^{97}+O(q^{100}) \) 8 * q - 42 * q^7 + 462 * q^11 - 1204 * q^13 - 228 * q^17 + 358 * q^19 + 2148 * q^23 - 5454 * q^25 + 11064 * q^29 + 830 * q^31 - 7692 * q^35 - 3914 * q^37 + 16632 * q^41 - 29036 * q^43 - 41700 * q^47 + 41876 * q^49 - 22164 * q^53 + 7784 * q^55 - 32886 * q^59 + 83732 * q^61 + 93192 * q^65 - 80034 * q^67 - 89544 * q^71 - 22470 * q^73 - 138732 * q^77 - 75286 * q^79 + 34836 * q^83 + 278504 * q^85 - 28944 * q^89 - 12058 * q^91 + 144120 * q^95 - 433356 * q^97

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