LMFDB - Embedded newform 392.6.i.g.361.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,6,Mod(177,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, 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("392.177");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	392.i (of order $3$, degree $2$, not 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:	$62.8704573667$
Analytic rank:	$1$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-59})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 14x^{2} - 15x + 225 $ x^4 - x^3 - 14x^2 - 15x + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.2
Root			$3.57603 - 1.48727i$ of defining polynomial
Character	$\chi$	$=$	392.361
Dual form			392.6.i.g.177.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.152067 - 0.263388i) q^{3} +(17.7603 + 30.7618i) q^{5} +(121.454 + 210.364i) q^{9} +O(q^{10})$	\(q+(0.152067 - 0.263388i) q^{3} +(17.7603 + 30.7618i) q^{5} +(121.454 + 210.364i) q^{9} +(-282.912 + 490.019i) q^{11} -983.594 q^{13} +10.8031 q^{15} +(100.088 - 173.357i) q^{17} +(414.374 + 717.716i) q^{19} +(-2217.71 - 3841.19i) q^{23} +(931.641 - 1613.65i) q^{25} +147.781 q^{27} -3717.74 q^{29} +(496.231 - 859.498i) q^{31} +(86.0435 + 149.032i) q^{33} +(4179.84 + 7239.70i) q^{37} +(-149.572 + 259.067i) q^{39} +13473.0 q^{41} +298.798 q^{43} +(-4314.12 + 7472.27i) q^{45} +(-9368.26 - 16226.3i) q^{47} +(-30.4401 - 52.7238i) q^{51} +(-8018.18 + 13887.9i) q^{53} -20098.5 q^{55} +252.051 q^{57} +(6374.58 - 11041.1i) q^{59} +(-17488.0 - 30290.1i) q^{61} +(-17469.0 - 30257.1i) q^{65} +(-5989.44 + 10374.0i) q^{67} -1348.97 q^{69} -12924.9 q^{71} +(40588.6 - 70301.5i) q^{73} +(-283.344 - 490.767i) q^{75} +(-23499.4 - 40702.2i) q^{79} +(-29490.8 + 51079.5i) q^{81} +111544. q^{83} +7110.36 q^{85} +(-565.347 + 979.210i) q^{87} +(-17363.4 - 30074.3i) q^{89} +(-150.921 - 261.403i) q^{93} +(-14718.8 + 25493.8i) q^{95} -92655.6 q^{97} -137443. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 26 q^{3} - 62 q^{5} - 206 q^{9}+O(q^{10}) $ 4 * q - 26 * q^3 - 62 * q^5 - 206 * q^9	$ 4 q - 26 q^{3} - 62 q^{5} - 206 q^{9} - 972 q^{11} - 156 q^{13} + 5152 q^{15} + 560 q^{17} + 2642 q^{19} - 2272 q^{23} - 4522 q^{25} + 11128 q^{27} - 15616 q^{29} + 5444 q^{31} - 10512 q^{33} - 576 q^{37} - 24120 q^{39} + 33776 q^{41} - 16792 q^{43} - 52406 q^{45} - 4532 q^{47} + 9404 q^{51} - 1420 q^{53} + 39024 q^{55} - 94888 q^{57} + 34146 q^{59} + 19106 q^{61} - 123252 q^{65} - 56952 q^{67} - 116512 q^{69} - 14448 q^{71} + 128828 q^{73} - 168526 q^{75} - 52808 q^{79} - 92366 q^{81} + 168972 q^{83} - 55960 q^{85} + 106460 q^{87} - 130972 q^{89} + 116792 q^{93} + 147392 q^{95} - 389248 q^{97} + 89784 q^{99}+O(q^{100}) $ 4 * q - 26 * q^3 - 62 * q^5 - 206 * q^9 - 972 * q^11 - 156 * q^13 + 5152 * q^15 + 560 * q^17 + 2642 * q^19 - 2272 * q^23 - 4522 * q^25 + 11128 * q^27 - 15616 * q^29 + 5444 * q^31 - 10512 * q^33 - 576 * q^37 - 24120 * q^39 + 33776 * q^41 - 16792 * q^43 - 52406 * q^45 - 4532 * q^47 + 9404 * q^51 - 1420 * q^53 + 39024 * q^55 - 94888 * q^57 + 34146 * q^59 + 19106 * q^61 - 123252 * q^65 - 56952 * q^67 - 116512 * q^69 - 14448 * q^71 + 128828 * q^73 - 168526 * q^75 - 52808 * q^79 - 92366 * q^81 + 168972 * q^83 - 55960 * q^85 + 106460 * q^87 - 130972 * q^89 + 116792 * q^93 + 147392 * q^95 - 389248 * q^97 + 89784 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	0.152067	−	0.263388i	0.00975512	−	0.0168964i	−0.861107	−	0.508424i	$-0.830228\pi$
$3$	0.152067	−	0.263388i	0.00975512	−	0.0168964i	0.870862	+	0.491528i	$0.163561\pi$
$4$		0			0
$5$	17.7603	+	30.7618i	0.317707	+	0.550284i	0.980009	−	0.198953i	$-0.0637540\pi$
$5$	17.7603	+	30.7618i	0.317707	+	0.550284i	−0.662303	+	0.749236i	$0.730421\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	121.454	+	210.364i	0.499810	+	0.865696i
$10$		0			0
$11$	−282.912	+	490.019i	−0.704969	+	1.22104i	0.261733	+	0.965140i	$0.415706\pi$
$11$	−282.912	+	490.019i	−0.704969	+	1.22104i	−0.966703	+	0.255903i	$0.917627\pi$
$12$		0			0
$13$	−983.594			−1.61420			−0.807100	−	0.590415i	$-0.798964\pi$
$13$	−983.594			−1.61420			−0.807100	+	0.590415i	$0.798964\pi$
$14$		0			0
$15$	10.8031			0.0123971
$16$		0			0
$17$	100.088	−	173.357i	0.0839959	−	0.145485i	−0.820967	−	0.570976i	$-0.806565\pi$
$17$	100.088	−	173.357i	0.0839959	−	0.145485i	0.904963	+	0.425490i	$0.139898\pi$
$18$		0			0
$19$	414.374	+	717.716i	0.263335	+	0.456109i	0.967126	−	0.254298i	$-0.0818443\pi$
$19$	414.374	+	717.716i	0.263335	+	0.456109i	−0.703791	+	0.710407i	$0.748511\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2217.71	−	3841.19i	−0.874149	−	1.51407i	−0.857666	−	0.514207i	$-0.828086\pi$
$23$	−2217.71	−	3841.19i	−0.874149	−	1.51407i	−0.0164834	−	0.999864i	$-0.505247\pi$
$24$		0			0
$25$	931.641	−	1613.65i	0.298125	−	0.516368i
$26$		0			0
$27$	147.781			0.0390131
$28$		0			0
$29$	−3717.74			−0.820889			−0.410444	−	0.911886i	$-0.634626\pi$
$29$	−3717.74			−0.820889			−0.410444	+	0.911886i	$0.634626\pi$
$30$		0			0
$31$	496.231	−	859.498i	0.0927427	−	0.160635i	−0.815922	−	0.578163i	$-0.803770\pi$
$31$	496.231	−	859.498i	0.0927427	−	0.160635i	0.908664	+	0.417527i	$0.137103\pi$
$32$		0			0
$33$	86.0435	+	149.032i	0.0137541	+	0.0238229i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4179.84	+	7239.70i	0.501945	+	0.869393i	0.999997	+	0.00224683i	$0.000715188\pi$
$37$	4179.84	+	7239.70i	0.501945	+	0.869393i	−0.498053	+	0.867147i	$0.665951\pi$
$38$		0			0
$39$	−149.572	+	259.067i	−0.0157467	+	0.0272741i
$40$		0			0
$41$	13473.0			1.25171			0.625855	−	0.779940i	$-0.284750\pi$
$41$	13473.0			1.25171			0.625855	+	0.779940i	$0.284750\pi$
$42$		0			0
$43$	298.798			0.0246437			0.0123218	−	0.999924i	$-0.496078\pi$
$43$	298.798			0.0246437			0.0123218	+	0.999924i	$0.496078\pi$
$44$		0			0
$45$	−4314.12	+	7472.27i	−0.317586	+	0.550074i
$46$		0			0
$47$	−9368.26	−	16226.3i	−0.618606	−	1.07146i	−0.989740	−	0.142878i	$-0.954364\pi$
$47$	−9368.26	−	16226.3i	−0.618606	−	1.07146i	0.371134	−	0.928579i	$-0.378969\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−30.4401	−	52.7238i	−0.00163878	−	0.00283845i
$52$		0			0
$53$	−8018.18	+	13887.9i	−0.392090	+	0.679121i	−0.992725	−	0.120403i	$-0.961581\pi$
$53$	−8018.18	+	13887.9i	−0.392090	+	0.679121i	0.600635	+	0.799524i	$0.294915\pi$
$54$		0			0
$55$	−20098.5			−0.895894
$56$		0			0
$57$	252.051			0.0102754
$58$		0			0
$59$	6374.58	−	11041.1i	0.238408	−	0.412935i	−0.721849	−	0.692050i	$-0.756708\pi$
$59$	6374.58	−	11041.1i	0.238408	−	0.412935i	0.960258	+	0.279115i	$0.0900410\pi$
$60$		0			0
$61$	−17488.0	−	30290.1i	−0.601748	−	1.04226i	−0.992556	−	0.121786i	$-0.961138\pi$
$61$	−17488.0	−	30290.1i	−0.601748	−	1.04226i	0.390808	−	0.920472i	$-0.372196\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−17469.0	−	30257.1i	−0.512842	−	0.888268i
$66$		0			0
$67$	−5989.44	+	10374.0i	−0.163004	+	0.282332i	−0.935945	−	0.352147i	$-0.885452\pi$
$67$	−5989.44	+	10374.0i	−0.163004	+	0.282332i	0.772940	+	0.634479i	$0.218785\pi$
$68$		0			0
$69$	−1348.97			−0.0341097
$70$		0			0
$71$	−12924.9			−0.304285			−0.152143	−	0.988359i	$-0.548617\pi$
$71$	−12924.9			−0.304285			−0.152143	+	0.988359i	$0.548617\pi$
$72$		0			0
$73$	40588.6	−	70301.5i	0.891450	−	1.54404i	0.0533123	−	0.998578i	$-0.483022\pi$
$73$	40588.6	−	70301.5i	0.891450	−	1.54404i	0.838138	−	0.545459i	$-0.183645\pi$
$74$		0			0
$75$	−283.344	−	490.767i	−0.00581649	−	0.0100745i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−23499.4	−	40702.2i	−0.423632	−	0.733753i	0.572659	−	0.819793i	$-0.305912\pi$
$79$	−23499.4	−	40702.2i	−0.423632	−	0.733753i	−0.996292	+	0.0860408i	$0.972578\pi$
$80$		0			0
$81$	−29490.8	+	51079.5i	−0.499429	+	0.865037i
$82$		0			0
$83$	111544.			1.77726			0.888632	−	0.458621i	$-0.151656\pi$
$83$	111544.			1.77726			0.888632	+	0.458621i	$0.151656\pi$
$84$		0			0
$85$	7110.36			0.106744
$86$		0			0
$87$	−565.347	+	979.210i	−0.00800787	+	0.0138700i
$88$		0			0
$89$	−17363.4	−	30074.3i	−0.232359	−	0.402458i	0.726143	−	0.687544i	$-0.241311\pi$
$89$	−17363.4	−	30074.3i	−0.232359	−	0.402458i	−0.958502	+	0.285086i	$0.907978\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−150.921	−	261.403i	−0.00180943	−	0.00313403i
$94$		0			0
$95$	−14718.8	+	25493.8i	−0.167326	+	0.289818i
$96$		0			0
$97$	−92655.6			−0.999867			−0.499933	−	0.866064i	$-0.666642\pi$
$97$	−92655.6			−0.999867			−0.499933	+	0.866064i	$0.666642\pi$
$98$		0			0
$99$	−137443.			−1.40940
$100$		0			0
$101$	−77382.6	+	134031.i	−0.754814	+	1.30738i	0.190653	+	0.981657i	$0.438939\pi$
$101$	−77382.6	+	134031.i	−0.754814	+	1.30738i	−0.945467	+	0.325718i	$0.894394\pi$
$102$		0			0
$103$	−98572.8	−	170733.i	−0.915512	−	1.58571i	−0.806149	−	0.591712i	$-0.798452\pi$
$103$	−98572.8	−	170733.i	−0.915512	−	1.58571i	−0.109363	−	0.994002i	$-0.534881\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−31950.6	−	55340.1i	−0.269786	−	0.467284i	0.699020	−	0.715102i	$-0.253620\pi$
$107$	−31950.6	−	55340.1i	−0.269786	−	0.467284i	−0.968807	+	0.247818i	$0.920286\pi$
$108$		0			0
$109$	−33455.2	+	57946.1i	−0.269710	+	0.467152i	−0.968787	−	0.247895i	$-0.920261\pi$
$109$	−33455.2	+	57946.1i	−0.269710	+	0.467152i	0.699077	+	0.715047i	$0.253595\pi$
$110$		0			0
$111$	2542.47			0.0195861
$112$		0			0
$113$	−100524.			−0.740580			−0.370290	−	0.928916i	$-0.620742\pi$
$113$	−100524.			−0.740580			−0.370290	+	0.928916i	$0.620742\pi$
$114$		0			0
$115$	78774.6	−	136442.i	0.555446	−	0.962061i
$116$		0			0
$117$	−119461.	−	206913.i	−0.806793	−	1.39741i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−79553.4	−	137790.i	−0.493964	−	0.855570i
$122$		0			0
$123$	2048.80	−	3548.62i	0.0122106	−	0.0211493i
$124$		0			0
$125$	177187.			1.01428
$126$		0			0
$127$	−155009.			−0.852800			−0.426400	−	0.904535i	$-0.640218\pi$
$127$	−155009.			−0.852800			−0.426400	+	0.904535i	$0.640218\pi$
$128$		0			0
$129$	45.4373	−	78.6998i	0.000240402	−	0.000416389i
$130$		0			0
$131$	−35198.4	−	60965.4i	−0.179203	−	0.310388i	0.762405	−	0.647100i	$-0.224018\pi$
$131$	−35198.4	−	60965.4i	−0.179203	−	0.310388i	−0.941608	+	0.336712i	$0.890685\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2624.65	+	4546.02i	0.0123947	+	0.0214683i
$136$		0			0
$137$	−87865.9	+	152188.i	−0.399962	+	0.692755i	−0.993721	−	0.111888i	$-0.964310\pi$
$137$	−87865.9	+	152188.i	−0.399962	+	0.692755i	0.593758	+	0.804643i	$0.297643\pi$
$138$		0			0
$139$	−44017.9			−0.193238			−0.0966189	−	0.995321i	$-0.530803\pi$
$139$	−44017.9			−0.193238			−0.0966189	+	0.995321i	$0.530803\pi$
$140$		0			0
$141$	−5698.43			−0.0241383
$142$		0			0
$143$	278271.	−	481979.i	1.13796	−	1.97101i
$144$		0			0
$145$	−66028.4	−	114364.i	−0.260802	−	0.451722i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	60568.1	+	104907.i	0.223500	+	0.387114i	0.955868	−	0.293795i	$-0.0949183\pi$
$149$	60568.1	+	104907.i	0.223500	+	0.387114i	−0.732368	+	0.680909i	$0.761585\pi$
$150$		0			0
$151$	−199231.	+	345079.i	−0.711075	+	1.23162i	0.253379	+	0.967367i	$0.418458\pi$
$151$	−199231.	+	345079.i	−0.711075	+	1.23162i	−0.964454	+	0.264251i	$0.914875\pi$
$152$		0			0
$153$	48624.1			0.167928
$154$		0			0
$155$	35252.9			0.117860
$156$		0			0
$157$	−68369.3	+	118419.i	−0.221367	+	0.383418i	−0.955223	−	0.295886i	$-0.904385\pi$
$157$	−68369.3	+	118419.i	−0.221367	+	0.383418i	0.733857	+	0.679304i	$0.237718\pi$
$158$		0			0
$159$	2438.61	+	4223.79i	0.00764978	+	0.0132498i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−321205.	−	556343.i	−0.946919	−	1.64011i	−0.751863	−	0.659319i	$-0.770845\pi$
$163$	−321205.	−	556343.i	−0.946919	−	1.64011i	−0.195056	−	0.980792i	$-0.562489\pi$
$164$		0			0
$165$	−3056.32	+	5293.71i	−0.00873955	+	0.0151374i
$166$		0			0
$167$	290841.			0.806984			0.403492	−	0.914983i	$-0.367796\pi$
$167$	290841.			0.806984			0.403492	+	0.914983i	$0.367796\pi$
$168$		0			0
$169$	596163.			1.60564
$170$		0			0
$171$	−100654.	+	174339.i	−0.263234	+	0.455935i
$172$		0			0
$173$	−281117.	−	486909.i	−0.714121	−	1.23689i	−0.963298	−	0.268436i	$-0.913493\pi$
$173$	−281117.	−	486909.i	−0.714121	−	1.23689i	0.249177	−	0.968458i	$-0.419840\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1938.73	−	3357.98i	−0.00465141	−	0.00805647i
$178$		0			0
$179$	156084.	−	270345.i	0.364103	−	0.630646i	−0.624528	−	0.781002i	$-0.714709\pi$
$179$	156084.	−	270345.i	0.364103	−	0.630646i	0.988632	+	0.150356i	$0.0480421\pi$
$180$		0			0
$181$	−254232.			−0.576811			−0.288405	−	0.957508i	$-0.593125\pi$
$181$	−254232.			−0.576811			−0.288405	+	0.957508i	$0.593125\pi$
$182$		0			0
$183$	−10637.4			−0.0234805
$184$		0			0
$185$	−148471.	+	257159.i	−0.318942	+	0.552424i
$186$		0			0
$187$	56632.0	+	98089.6i	0.118429	+	0.205125i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	94529.8	+	163730.i	0.187493	+	0.324748i	0.944414	−	0.328759i	$-0.106630\pi$
$191$	94529.8	+	163730.i	0.187493	+	0.324748i	−0.756921	+	0.653507i	$0.773297\pi$
$192$		0			0
$193$	−330213.	+	571946.i	−0.638118	+	1.10525i	0.347727	+	0.937596i	$0.386953\pi$
$193$	−330213.	+	571946.i	−0.638118	+	1.10525i	−0.985845	+	0.167657i	$0.946380\pi$
$194$		0			0
$195$	−10625.8			−0.0200113
$196$		0			0
$197$	−786619.			−1.44410			−0.722052	−	0.691838i	$-0.756801\pi$
$197$	−786619.			−1.44410			−0.722052	+	0.691838i	$0.756801\pi$
$198$		0			0
$199$	180265.	−	312228.i	0.322685	−	0.558907i	−0.658356	−	0.752707i	$-0.728748\pi$
$199$	180265.	−	312228.i	0.322685	−	0.558907i	0.981041	+	0.193800i	$0.0620812\pi$
$200$		0			0
$201$	1821.60	+	3155.10i	0.00318025	+	0.00550836i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	239284.	+	414453.i	0.397676	+	0.688795i
$206$		0			0
$207$	538699.	−	933054.i	0.873817	−	1.51349i
$208$		0			0
$209$	−468926.			−0.742571
$210$		0			0
$211$	−1.13491e6			−1.75491			−0.877455	−	0.479658i	$-0.840761\pi$
$211$	−1.13491e6			−1.75491			−0.877455	+	0.479658i	$0.840761\pi$
$212$		0			0
$213$	−1965.45	+	3404.27i	−0.00296834	+	0.00514132i
$214$		0			0
$215$	5306.74	+	9191.55i	0.00782946	+	0.0135610i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−12344.4	−	21381.1i	−0.0173924	−	0.0301245i
$220$		0			0
$221$	−98445.5	+	170513.i	−0.135586	+	0.234842i
$222$		0			0
$223$	783934.			1.05564			0.527822	−	0.849355i	$-0.323009\pi$
$223$	783934.			1.05564			0.527822	+	0.849355i	$0.323009\pi$
$224$		0			0
$225$	452605.			0.596023
$226$		0			0
$227$	300377.	−	520268.i	0.386903	−	0.670136i	−0.605128	−	0.796128i	$-0.706878\pi$
$227$	300377.	−	520268.i	0.386903	−	0.670136i	0.992031	+	0.125992i	$0.0402115\pi$
$228$		0			0
$229$	739959.	+	1.28165e6i	0.932435	+	1.61503i	0.779145	+	0.626844i	$0.215654\pi$
$229$	739959.	+	1.28165e6i	0.932435	+	1.61503i	0.153291	+	0.988181i	$0.451013\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	521749.	+	903696.i	0.629610	+	1.09052i	0.987630	+	0.156803i	$0.0501188\pi$
$233$	521749.	+	903696.i	0.629610	+	1.09052i	−0.358020	+	0.933714i	$0.616548\pi$
$234$		0			0
$235$	332767.	−	576369.i	0.393070	−	0.680818i
$236$		0			0
$237$	−14294.0			−0.0165303
$238$		0			0
$239$	262948.			0.297767			0.148883	−	0.988855i	$-0.452432\pi$
$239$	262948.			0.297767			0.148883	+	0.988855i	$0.452432\pi$
$240$		0			0
$241$	−899728.	+	1.55837e6i	−0.997857	+	1.72834i	−0.442266	+	0.896884i	$0.645825\pi$
$241$	−899728.	+	1.55837e6i	−0.997857	+	1.72834i	−0.555591	+	0.831456i	$0.687508\pi$
$242$		0			0
$243$	26924.6	+	46634.8i	0.0292505	+	0.0506634i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−407575.	−	705941.i	−0.425075	−	0.736251i
$248$		0			0
$249$	16962.2	−	29379.5i	0.0173374	−	0.0300293i
$250$		0			0
$251$	−1.06631e6			−1.06831			−0.534156	−	0.845386i	$-0.679370\pi$
$251$	−1.06631e6			−1.06831			−0.534156	+	0.845386i	$0.679370\pi$
$252$		0			0
$253$	2.50967e6			2.46499
$254$		0			0
$255$	1081.25	−	1872.79i	0.00104130	−	0.00180359i
$256$		0			0
$257$	337589.	+	584721.i	0.318827	+	0.552225i	0.980244	−	0.197794i	$-0.0633777\pi$
$257$	337589.	+	584721.i	0.318827	+	0.552225i	−0.661416	+	0.750019i	$0.730044\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−451534.	−	782079.i	−0.410288	−	0.710640i
$262$		0			0
$263$	−150968.	+	261485.i	−0.134585	+	0.233108i	−0.925439	−	0.378897i	$-0.876303\pi$
$263$	−150968.	+	261485.i	−0.134585	+	0.233108i	0.790854	+	0.612005i	$0.209637\pi$
$264$		0			0
$265$	−569622.			−0.498279
$266$		0			0
$267$	−10561.6			−0.00906678
$268$		0			0
$269$	−408714.	+	707914.i	−0.344381	+	0.596485i	−0.985241	−	0.171172i	$-0.945244\pi$
$269$	−408714.	+	707914.i	−0.344381	+	0.596485i	0.640860	+	0.767658i	$0.278578\pi$
$270$		0			0
$271$	448310.	+	776495.i	0.370813	+	0.642267i	0.989691	−	0.143220i	$-0.0457458\pi$
$271$	448310.	+	776495.i	0.370813	+	0.642267i	−0.618878	+	0.785487i	$0.712412\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	527146.	+	913043.i	0.420338	+	0.728047i
$276$		0			0
$277$	454810.	−	787754.i	0.356148	−	0.616866i	−0.631166	−	0.775648i	$-0.717423\pi$
$277$	454810.	−	787754.i	0.356148	−	0.616866i	0.987314	+	0.158782i	$0.0507566\pi$
$278$		0			0
$279$	241077.			0.185415
$280$		0			0
$281$	762090.			0.575758			0.287879	−	0.957667i	$-0.407050\pi$
$281$	762090.			0.575758			0.287879	+	0.957667i	$0.407050\pi$
$282$		0			0
$283$	465530.	−	806322.i	0.345527	−	0.598470i	−0.639923	−	0.768439i	$-0.721034\pi$
$283$	465530.	−	806322.i	0.345527	−	0.598470i	0.985449	+	0.169969i	$0.0543669\pi$
$284$		0			0
$285$	4476.51	+	7753.54i	0.00326458	+	0.00565441i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	689893.	+	1.19493e6i	0.485889	+	0.841585i
$290$		0			0
$291$	−14089.9	+	24404.4i	−0.00975382	+	0.0168941i
$292$		0			0
$293$	−204188.			−0.138951			−0.0694755	−	0.997584i	$-0.522133\pi$
$293$	−204188.			−0.138951			−0.0694755	+	0.997584i	$0.522133\pi$
$294$		0			0
$295$	452859.			0.302976
$296$		0			0
$297$	−41809.2	+	72415.6i	−0.0275030	+	0.0476366i
$298$		0			0
$299$	2.18133e6	+	3.77817e6i	1.41105	+	2.44401i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	23534.7	+	40763.3i	0.0147266	+	0.0255072i
$304$		0			0
$305$	621184.	−	1.07592e6i	0.382359	−	0.662265i
$306$		0			0
$307$	1.00786e6			0.610314			0.305157	−	0.952302i	$-0.401291\pi$
$307$	1.00786e6			0.610314			0.305157	+	0.952302i	$0.401291\pi$
$308$		0			0
$309$	−59958.8			−0.0357238
$310$		0			0
$311$	888681.	−	1.53924e6i	0.521009	−	0.902413i	−0.478693	−	0.877982i	$-0.658889\pi$
$311$	888681.	−	1.53924e6i	0.521009	−	0.902413i	0.999702	−	0.0244310i	$-0.00777741\pi$
$312$		0			0
$313$	−865502.	−	1.49909e6i	−0.499352	−	0.864904i	0.500647	−	0.865651i	$-0.333095\pi$
$313$	−865502.	−	1.49909e6i	−0.499352	−	0.864904i	−1.00000		0.000747608i	$0.999762\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.11036e6	+	1.92320e6i	0.620605	+	1.07492i	0.989373	+	0.145398i	$0.0464463\pi$
$317$	1.11036e6	+	1.92320e6i	0.620605	+	1.07492i	−0.368768	+	0.929521i	$0.620220\pi$
$318$		0			0
$319$	1.05180e6	−	1.82176e6i	0.578701	−	1.00234i
$320$		0			0
$321$	−19434.6			−0.0105272
$322$		0			0
$323$	165895.			0.0884761
$324$		0			0
$325$	−916356.	+	1.58718e6i	−0.481233	+	0.833521i
$326$		0			0
$327$	10174.9	+	17623.4i	0.00526212	+	0.00911425i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.22609e6	+	2.12365e6i	0.615109	+	1.06540i	0.990365	+	0.138479i	$0.0442214\pi$
$331$	1.22609e6	+	2.12365e6i	0.615109	+	1.06540i	−0.375256	+	0.926921i	$0.622445\pi$
$332$		0			0
$333$	−1.01532e6	+	1.75858e6i	−0.501753	+	0.869063i
$334$		0			0
$335$	−425498.			−0.207150
$336$		0			0
$337$	−2.06932e6			−0.992549			−0.496275	−	0.868166i	$-0.665299\pi$
$337$	−2.06932e6			−0.992549			−0.496275	+	0.868166i	$0.665299\pi$
$338$		0			0
$339$	−15286.4	+	26476.7i	−0.00722445	+	0.0125131i
$340$		0			0
$341$	280780.	+	486325.i	0.130762	+	0.226486i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−23958.1	−	41496.7i	−0.0108369	−	0.0187700i
$346$		0			0
$347$	1.10969e6	−	1.92205e6i	0.494743	−	0.856920i	−0.505238	−	0.862980i	$-0.668595\pi$
$347$	1.10969e6	−	1.92205e6i	0.494743	−	0.856920i	0.999982	+	0.00605935i	$0.00192876\pi$
$348$		0			0
$349$	−2.92580e6			−1.28582			−0.642911	−	0.765941i	$-0.722273\pi$
$349$	−2.92580e6			−1.28582			−0.642911	+	0.765941i	$0.722273\pi$
$350$		0			0
$351$	−145357.			−0.0629749
$352$		0			0
$353$	−1.51660e6	+	2.62682e6i	−0.647788	+	1.12200i	0.335862	+	0.941911i	$0.390972\pi$
$353$	−1.51660e6	+	2.62682e6i	−0.647788	+	1.12200i	−0.983650	+	0.180091i	$0.942361\pi$
$354$		0			0
$355$	−229550.	−	397593.i	−0.0966735	−	0.167443i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.83298e6	−	3.17482e6i	−0.750623	−	1.30012i	−0.947521	−	0.319693i	$-0.896420\pi$
$359$	−1.83298e6	−	3.17482e6i	−0.750623	−	1.30012i	0.196898	−	0.980424i	$-0.436913\pi$
$360$		0			0
$361$	894639.	−	1.54956e6i	0.361310	−	0.625807i
$362$		0			0
$363$	−48389.9			−0.0192747
$364$		0			0
$365$	2.88347e6			1.13288
$366$		0			0
$367$	−1.73677e6	+	3.00818e6i	−0.673098	+	1.16584i	0.303923	+	0.952697i	$0.401703\pi$
$367$	−1.73677e6	+	3.00818e6i	−0.673098	+	1.16584i	−0.977021	+	0.213143i	$0.931630\pi$
$368$		0			0
$369$	1.63634e6	+	2.83423e6i	0.625616	+	1.08360i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−290715.	−	503534.i	−0.108192	−	0.187394i	0.806846	−	0.590762i	$-0.201173\pi$
$373$	−290715.	−	503534.i	−0.108192	−	0.187394i	−0.915038	+	0.403368i	$0.867840\pi$
$374$		0			0
$375$	26944.4	−	46669.0i	0.00989441	−	0.0171376i
$376$		0			0
$377$	3.65675e6			1.32508
$378$		0			0
$379$	1.83859e6			0.657486			0.328743	−	0.944419i	$-0.393375\pi$
$379$	1.83859e6			0.657486			0.328743	+	0.944419i	$0.393375\pi$
$380$		0			0
$381$	−23571.8	+	40827.5i	−0.00831917	+	0.0144092i
$382$		0			0
$383$	69566.4	+	120493.i	0.0242328	+	0.0419724i	0.877887	−	0.478867i	$-0.158952\pi$
$383$	69566.4	+	120493.i	0.0242328	+	0.0419724i	−0.853655	+	0.520839i	$0.825619\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	36290.1	+	62856.3i	0.0123172	+	0.0213339i
$388$		0			0
$389$	−1.10992e6	+	1.92245e6i	−0.371894	+	0.644139i	−0.989857	−	0.142068i	$-0.954625\pi$
$389$	−1.10992e6	+	1.92245e6i	−0.371894	+	0.644139i	0.617963	+	0.786207i	$0.287958\pi$
$390$		0			0
$391$	−887862.			−0.293700
$392$		0			0
$393$	−21410.1			−0.00699258
$394$		0			0
$395$	834715.	−	1.44577e6i	0.269182	−	0.466236i
$396$		0			0
$397$	252999.	+	438207.i	0.0805643	+	0.139541i	0.903492	−	0.428604i	$-0.140994\pi$
$397$	252999.	+	438207.i	0.0805643	+	0.139541i	−0.822928	+	0.568145i	$0.807661\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.03331e6	−	1.78975e6i	−0.320901	−	0.555817i	0.659773	−	0.751465i	$-0.270652\pi$
$401$	−1.03331e6	−	1.78975e6i	−0.320901	−	0.555817i	−0.980674	+	0.195648i	$0.937319\pi$
$402$		0			0
$403$	−488090.	+	845396.i	−0.149705	+	0.259297i
$404$		0			0
$405$	−2.09507e6			−0.634688
$406$		0			0
$407$	−4.73012e6			−1.41542
$408$		0			0
$409$	2.98356e6	−	5.16767e6i	0.881913	−	1.52752i	0.0327024	−	0.999465i	$-0.489589\pi$
$409$	2.98356e6	−	5.16767e6i	0.881913	−	1.52752i	0.849211	−	0.528054i	$-0.177078\pi$
$410$		0			0
$411$	26723.1	+	46285.7i	0.00780337	+	0.0135158i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1.98106e6	+	3.43130e6i	0.564648	+	0.978000i
$416$		0			0
$417$	−6693.68	+	11593.8i	−0.00188506	+	0.00326502i
$418$		0			0
$419$	569081.			0.158358			0.0791788	−	0.996860i	$-0.474770\pi$
$419$	569081.			0.158358			0.0791788	+	0.996860i	$0.474770\pi$
$420$		0			0
$421$	−171377.			−0.0471247			−0.0235623	−	0.999722i	$-0.507501\pi$
$421$	−171377.			−0.0471247			−0.0235623	+	0.999722i	$0.507501\pi$
$422$		0			0
$423$	2.27562e6	−	3.94149e6i	0.618371	−	1.07105i
$424$		0			0
$425$	−186491.	−	323013.i	−0.0500826	−	0.0867455i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−84631.8	−	146587.i	−0.0222019	−	0.0384548i
$430$		0			0
$431$	−2.37965e6	+	4.12168e6i	−0.617049	+	1.06876i	0.372972	+	0.927843i	$0.378339\pi$
$431$	−2.37965e6	+	4.12168e6i	−0.617049	+	1.06876i	−0.990021	+	0.140918i	$0.954995\pi$
$432$		0			0
$433$	−623369.			−0.159781			−0.0798906	−	0.996804i	$-0.525457\pi$
$433$	−623369.			−0.159781			−0.0798906	+	0.996804i	$0.525457\pi$
$434$		0			0
$435$	−40163.0			−0.0101766
$436$		0			0
$437$	1.83792e6	−	3.18338e6i	0.460388	−	0.797415i
$438$		0			0
$439$	−2.62289e6	−	4.54298e6i	−0.649559	−	1.12507i	−0.983228	−	0.182379i	$-0.941620\pi$
$439$	−2.62289e6	−	4.54298e6i	−0.649559	−	1.12507i	0.333669	−	0.942690i	$-0.391713\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.82463e6	−	4.89241e6i	−0.683837	−	1.18444i	−0.973801	−	0.227403i	$-0.926977\pi$
$443$	−2.82463e6	−	4.89241e6i	−0.683837	−	1.18444i	0.289964	−	0.957038i	$-0.406357\pi$
$444$		0			0
$445$	616760.	−	1.06826e6i	0.147644	−	0.255727i
$446$		0			0
$447$	36841.7			0.00872110
$448$		0			0
$449$	812156.			0.190118			0.0950591	−	0.995472i	$-0.469696\pi$
$449$	812156.			0.190118			0.0950591	+	0.995472i	$0.469696\pi$
$450$		0			0
$451$	−3.81167e6	+	6.60200e6i	−0.882417	+	1.52839i
$452$		0			0
$453$	60593.2	+	104951.i	0.0138733	+	0.0240292i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	463460.	+	802736.i	0.103806	+	0.179797i	0.913250	−	0.407400i	$-0.133565\pi$
$457$	463460.	+	802736.i	0.103806	+	0.179797i	−0.809444	+	0.587197i	$0.800231\pi$
$458$		0			0
$459$	14791.1	−	25618.9i	0.00327694	−	0.00567582i
$460$		0			0
$461$	3.67043e6			0.804386			0.402193	−	0.915555i	$-0.368248\pi$
$461$	3.67043e6			0.804386			0.402193	+	0.915555i	$0.368248\pi$
$462$		0			0
$463$	−2.70462e6			−0.586345			−0.293173	−	0.956060i	$-0.594711\pi$
$463$	−2.70462e6			−0.586345			−0.293173	+	0.956060i	$0.594711\pi$
$464$		0			0
$465$	5360.82	−	9285.21i	0.00114974	−	0.00199140i
$466$		0			0
$467$	−4.12215e6	−	7.13978e6i	−0.874645	−	1.51493i	−0.857140	−	0.515083i	$-0.827761\pi$
$467$	−4.12215e6	−	7.13978e6i	−0.874645	−	1.51493i	−0.0175052	−	0.999847i	$-0.505572\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	20793.5	+	36015.4i	0.00431892	+	0.00748059i
$472$		0			0
$473$	−84533.5	+	146416.i	−0.0173731	+	0.0300910i
$474$		0			0
$475$	1.54419e6			0.314027
$476$		0			0
$477$	−3.89535e6			−0.783882
$478$		0			0
$479$	−1.27501e6	+	2.20839e6i	−0.253908	+	0.439781i	−0.964598	−	0.263724i	$-0.915049\pi$
$479$	−1.27501e6	+	2.20839e6i	−0.253908	+	0.439781i	0.710690	+	0.703505i	$0.248383\pi$
$480$		0			0
$481$	−4.11127e6	−	7.12092e6i	−0.810239	−	1.40337i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.64559e6	−	2.85025e6i	−0.317664	−	0.550211i
$486$		0			0
$487$	−3.83408e6	+	6.64081e6i	−0.732552	+	1.26882i	0.223238	+	0.974764i	$0.428337\pi$
$487$	−3.83408e6	+	6.64081e6i	−0.732552	+	1.26882i	−0.955789	+	0.294053i	$0.904996\pi$
$488$		0			0
$489$	−195379.			−0.0369492
$490$		0			0
$491$	9.33268e6			1.74704			0.873519	−	0.486790i	$-0.161832\pi$
$491$	9.33268e6			1.74704			0.873519	+	0.486790i	$0.161832\pi$
$492$		0			0
$493$	−372100.	+	644496.i	−0.0689513	+	0.119427i
$494$		0			0
$495$	−2.44104e6	−	4.22800e6i	−0.447776	−	0.775571i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	595327.	+	1.03114e6i	0.107030	+	0.185381i	0.914566	−	0.404437i	$-0.132533\pi$
$499$	595327.	+	1.03114e6i	0.107030	+	0.185381i	−0.807536	+	0.589818i	$0.799199\pi$
$500$		0			0
$501$	44227.5	−	76604.2i	0.00787223	−	0.0136351i
$502$		0			0
$503$	−2.34213e6			−0.412753			−0.206377	−	0.978473i	$-0.566167\pi$
$503$	−2.34213e6			−0.412753			−0.206377	+	0.978473i	$0.566167\pi$
$504$		0			0
$505$	−5.49736e6			−0.959237
$506$		0			0
$507$	90657.0	−	157022.i	0.0156632	−	0.0271295i
$508$		0			0
$509$	−3.34884e6	−	5.80035e6i	−0.572927	−	0.992339i	−0.996263	−	0.0863663i	$-0.972474\pi$
$509$	−3.34884e6	−	5.80035e6i	−0.572927	−	0.992339i	0.423336	−	0.905973i	$-0.360859\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	61236.7	+	106065.i	0.0102735	+	0.0177942i
$514$		0			0
$515$	3.50137e6	−	6.06456e6i	0.581729	−	1.00758i
$516$		0			0
$517$	1.06016e7			1.74439
$518$		0			0
$519$	−170995.			−0.0278654
$520$		0			0
$521$	2.90863e6	−	5.03790e6i	0.469456	−	0.813121i	−0.529934	−	0.848039i	$-0.677783\pi$
$521$	2.90863e6	−	5.03790e6i	0.469456	−	0.813121i	0.999390	+	0.0349174i	$0.0111168\pi$
$522$		0			0
$523$	5.12853e6	+	8.88287e6i	0.819858	+	1.42003i	0.905787	+	0.423734i	$0.139281\pi$
$523$	5.12853e6	+	8.88287e6i	0.819858	+	1.42003i	−0.0859292	+	0.996301i	$0.527386\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−99333.2	−	172050.i	−0.0155800	−	0.0269854i
$528$		0			0
$529$	−6.61833e6	+	1.14633e7i	−1.02827	+	1.78102i
$530$		0			0
$531$	3.09687e6			0.476635
$532$		0			0
$533$	−1.32519e7			−2.02051
$534$		0			0
$535$	1.13491e6	−	1.96572e6i	0.171426	−	0.296918i
$536$		0			0
$537$	−47470.4	−	82221.2i	−0.00710375	−	0.0123041i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2.63300e6	−	4.56050e6i	−0.386775	−	0.669914i	0.605239	−	0.796044i	$-0.293078\pi$
$541$	−2.63300e6	−	4.56050e6i	−0.386775	−	0.669914i	−0.992014	+	0.126130i	$0.959744\pi$
$542$		0			0
$543$	−38660.3	+	66961.7i	−0.00562686	+	0.00974601i
$544$		0			0
$545$	−2.37670e6			−0.342755
$546$		0			0
$547$	4.89900e6			0.700066			0.350033	−	0.936737i	$-0.386170\pi$
$547$	4.89900e6			0.700066			0.350033	+	0.936737i	$0.386170\pi$
$548$		0			0
$549$	4.24796e6	−	7.35768e6i	0.601519	−	1.04186i
$550$		0			0
$551$	−1.54053e6	−	2.66828e6i	−0.216168	−	0.374415i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	45155.1	+	78211.0i	0.00622264	+	0.0107779i
$556$		0			0
$557$	4.58973e6	−	7.94965e6i	0.626829	−	1.08570i	−0.361355	−	0.932428i	$-0.617686\pi$
$557$	4.58973e6	−	7.94965e6i	0.626829	−	1.08570i	0.988184	−	0.153272i	$-0.0489809\pi$
$558$		0			0
$559$	−293895.			−0.0397799
$560$		0			0
$561$	34447.5			0.00462116
$562$		0			0
$563$	−124736.	+	216049.i	−0.0165852	+	0.0287264i	−0.874199	−	0.485568i	$-0.838613\pi$
$563$	−124736.	+	216049.i	−0.0165852	+	0.0287264i	0.857614	+	0.514294i	$0.171946\pi$
$564$		0			0
$565$	−1.78533e6	−	3.09229e6i	−0.235287	−	0.407529i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.57490e6	−	1.13881e7i	−0.851351	−	1.47458i	−0.879989	−	0.474993i	$-0.842450\pi$
$569$	−6.57490e6	−	1.13881e7i	−0.851351	−	1.47458i	0.0286383	−	0.999590i	$-0.490883\pi$
$570$		0			0
$571$	−7.55189e6	+	1.30803e7i	−0.969316	+	1.67891i	−0.271774	+	0.962361i	$0.587610\pi$
$571$	−7.55189e6	+	1.30803e7i	−0.969316	+	1.67891i	−0.697542	+	0.716544i	$0.745723\pi$
$572$		0			0
$573$	57499.6			0.00731608
$574$		0			0
$575$	−8.26445e6			−1.04242
$576$		0			0
$577$	−363911.	+	630312.i	−0.0455046	+	0.0788163i	−0.887881	−	0.460074i	$-0.847823\pi$
$577$	−363911.	+	630312.i	−0.0455046	+	0.0788163i	0.842376	+	0.538890i	$0.181156\pi$
$578$		0			0
$579$	100429.	+	173949.i	0.0124498	+	0.0215638i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−4.53689e6	−	7.85812e6i	−0.552823	−	0.957518i
$584$		0			0
$585$	4.24334e6	−	7.34968e6i	0.512647	−	0.887930i
$586$		0			0
$587$	−1.24256e7			−1.48841			−0.744207	−	0.667949i	$-0.767172\pi$
$587$	−1.24256e7			−1.48841			−0.744207	+	0.667949i	$0.767172\pi$
$588$		0			0
$589$	822500.			0.0976895
$590$		0			0
$591$	−119619.	+	207186.i	−0.0140874	+	0.0244001i
$592$		0			0
$593$	3.08990e6	+	5.35187e6i	0.360835	+	0.624984i	0.988098	−	0.153823i	$-0.0491586\pi$
$593$	3.08990e6	+	5.35187e6i	0.360835	+	0.624984i	−0.627264	+	0.778807i	$0.715825\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−54824.9	−	94959.5i	−0.00629567	−	0.0109044i
$598$		0			0
$599$	−6.53016e6	+	1.13106e7i	−0.743630	+	1.28800i	0.207202	+	0.978298i	$0.433564\pi$
$599$	−6.53016e6	+	1.13106e7i	−0.743630	+	1.28800i	−0.950832	+	0.309706i	$0.899769\pi$
$600$		0			0
$601$	−9.01833e6			−1.01845			−0.509225	−	0.860633i	$-0.670068\pi$
$601$	−9.01833e6			−1.01845			−0.509225	+	0.860633i	$0.670068\pi$
$602$		0			0
$603$	−2.90976e6			−0.325884
$604$		0			0
$605$	2.82579e6	−	4.89441e6i	0.313871	−	0.543641i
$606$		0			0
$607$	−6.84174e6	−	1.18502e7i	−0.753694	−	1.30544i	−0.946021	−	0.324105i	$-0.894937\pi$
$607$	−6.84174e6	−	1.18502e7i	−0.753694	−	1.30544i	0.192327	−	0.981331i	$-0.438397\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.21456e6	+	1.59601e7i	0.998554	+	1.72955i
$612$		0			0
$613$	2.11633e6	−	3.66559e6i	0.227474	−	0.393997i	−0.729585	−	0.683891i	$-0.760287\pi$
$613$	2.11633e6	−	3.66559e6i	0.227474	−	0.393997i	0.957059	+	0.289894i	$0.0936199\pi$
$614$		0			0
$615$	145549.			0.0155175
$616$		0			0
$617$	3.32584e6			0.351713			0.175856	−	0.984416i	$-0.443731\pi$
$617$	3.32584e6			0.351713			0.175856	+	0.984416i	$0.443731\pi$
$618$		0			0
$619$	−5.99934e6	+	1.03912e7i	−0.629328	+	1.09003i	0.358359	+	0.933584i	$0.383336\pi$
$619$	−5.99934e6	+	1.03912e7i	−0.629328	+	1.09003i	−0.987687	+	0.156444i	$0.949997\pi$
$620$		0			0
$621$	−327737.	−	567656.i	−0.0341033	−	0.0590686i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	235525.	+	407942.i	0.0241178	+	0.0417732i
$626$		0			0
$627$	−71308.3	+	123510.i	−0.00724388	+	0.0125468i
$628$		0			0
$629$	1.67340e6			0.168645
$630$		0			0
$631$	9.99962e6			0.999793			0.499896	−	0.866085i	$-0.333371\pi$
$631$	9.99962e6			0.999793			0.499896	+	0.866085i	$0.333371\pi$
$632$		0			0
$633$	−172583.	+	298922.i	−0.0171194	+	0.0296516i
$634$		0			0
$635$	−2.75301e6	−	4.76835e6i	−0.270940	−	0.469282i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−1.56978e6	−	2.71893e6i	−0.152085	−	0.263419i
$640$		0			0
$641$	1.58312e6	−	2.74204e6i	0.152184	−	0.263590i	−0.779846	−	0.625971i	$-0.784703\pi$
$641$	1.58312e6	−	2.74204e6i	0.152184	−	0.263590i	0.932030	+	0.362381i	$0.118036\pi$
$642$		0			0
$643$	−950578.			−0.0906693			−0.0453346	−	0.998972i	$-0.514435\pi$
$643$	−950578.			−0.0906693			−0.0453346	+	0.998972i	$0.514435\pi$
$644$		0			0
$645$	3227.93			0.000305510
$646$		0			0
$647$	−5.73353e6	+	9.93076e6i	−0.538470	+	0.932657i	0.460517	+	0.887651i	$0.347664\pi$
$647$	−5.73353e6	+	9.93076e6i	−0.538470	+	0.932657i	−0.998987	+	0.0450059i	$0.985669\pi$
$648$		0			0
$649$	3.60689e6	+	6.24732e6i	0.336141	+	0.582214i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.40468e6	+	2.43297e6i	0.128912	+	0.223282i	0.923255	−	0.384187i	$-0.125518\pi$
$653$	1.40468e6	+	2.43297e6i	0.128912	+	0.223282i	−0.794343	+	0.607469i	$0.792185\pi$
$654$		0			0
$655$	1.25027e6	−	2.16553e6i	0.113868	−	0.197225i
$656$		0			0
$657$	1.97186e7			1.78222
$658$		0			0
$659$	−4.23794e6			−0.380138			−0.190069	−	0.981771i	$-0.560871\pi$
$659$	−4.23794e6			−0.380138			−0.190069	+	0.981771i	$0.560871\pi$
$660$		0			0
$661$	854514.	−	1.48006e6i	0.0760703	−	0.131758i	−0.825481	−	0.564430i	$-0.809096\pi$
$661$	854514.	−	1.48006e6i	0.0760703	−	0.131758i	0.901551	+	0.432672i	$0.142429\pi$
$662$		0			0
$663$	29940.7	+	51858.8i	0.00264532	+	0.00458183i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	8.24488e6	+	1.42806e7i	0.717579	+	1.24288i
$668$		0			0
$669$	119211.	−	206479.i	0.0102979	−	0.0178365i
$670$		0			0
$671$	1.97903e7			1.69686
$672$		0			0
$673$	1.11371e7			0.947836			0.473918	−	0.880569i	$-0.342839\pi$
$673$	1.11371e7			0.947836			0.473918	+	0.880569i	$0.342839\pi$
$674$		0			0
$675$	137679.	−	238467.i	0.0116308	−	0.0201451i
$676$		0			0
$677$	4.04936e6	+	7.01370e6i	0.339559	+	0.588133i	0.984350	−	0.176226i	$-0.0563889\pi$
$677$	4.04936e6	+	7.01370e6i	0.339559	+	0.588133i	−0.644791	+	0.764359i	$0.723056\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−91355.1	−	158232.i	−0.00754857	−	0.0130745i
$682$		0			0
$683$	7.49319e6	−	1.29786e7i	0.614632	−	1.06457i	−0.375817	−	0.926694i	$-0.622638\pi$
$683$	7.49319e6	−	1.29786e7i	0.614632	−	1.06457i	0.990449	−	0.137880i	$-0.0440287\pi$
$684$		0			0
$685$	−6.24211e6			−0.508283
$686$		0			0
$687$	450094.			0.0363841
$688$		0			0
$689$	7.88663e6	−	1.36600e7i	0.632912	−	1.09624i
$690$		0			0
$691$	1.09285e7	+	1.89286e7i	0.870691	+	1.50808i	0.861284	+	0.508125i	$0.169661\pi$
$691$	1.09285e7	+	1.89286e7i	0.870691	+	1.50808i	0.00940700	+	0.999956i	$0.497006\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−781772.	−	1.35407e6i	−0.0613929	−	0.106336i
$696$		0			0
$697$	1.34848e6	−	2.33563e6i	0.105138	−	0.182105i
$698$		0			0
$699$	317364.			0.0245677
$700$		0			0
$701$	−1.21819e7			−0.936307			−0.468154	−	0.883647i	$-0.655081\pi$
$701$	−1.21819e7			−0.936307			−0.468154	+	0.883647i	$0.655081\pi$
$702$		0			0
$703$	−3.46403e6	+	5.99988e6i	−0.264359	+	0.457883i
$704$		0			0
$705$	−101206.	−	175294.i	−0.00766890	−	0.0132829i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	4.01520e6	+	6.95453e6i	0.299980	+	0.519580i	0.976131	−	0.217183i	$-0.0696868\pi$
$709$	4.01520e6	+	6.95453e6i	0.299980	+	0.519580i	−0.676151	+	0.736763i	$0.736353\pi$
$710$		0			0
$711$	5.70818e6	−	9.88686e6i	0.423471	−	0.733473i
$712$		0			0
$713$	−4.40199e6			−0.324284
$714$		0			0
$715$	1.97687e7			1.44615
$716$		0			0
$717$	39985.9	−	69257.6i	0.00290475	−	0.00503117i
$718$		0			0
$719$	−8.83404e6	−	1.53010e7i	−0.637290	−	1.10382i	−0.986025	−	0.166598i	$-0.946722\pi$
$719$	−8.83404e6	−	1.53010e7i	−0.637290	−	1.10382i	0.348735	−	0.937221i	$-0.386611\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	273638.	+	473956.i	0.0194684	+	0.0337203i
$724$		0			0
$725$	−3.46360e6	+	5.99913e6i	−0.244728	+	0.423880i
$726$		0			0
$727$	−1.71905e7			−1.20629			−0.603146	−	0.797631i	$-0.706086\pi$
$727$	−1.71905e7			−1.20629			−0.603146	+	0.797631i	$0.706086\pi$
$728$		0			0
$729$	−1.43161e7			−0.997717
$730$		0			0
$731$	29905.9	−	51798.6i	0.00206997	−	0.00358529i
$732$		0			0
$733$	4.65110e6	+	8.05593e6i	0.319739	+	0.553804i	0.980433	−	0.196851i	$-0.0630715\pi$
$733$	4.65110e6	+	8.05593e6i	0.319739	+	0.553804i	−0.660695	+	0.750655i	$0.729738\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.38897e6	−	5.86987e6i	−0.229826	−	0.398070i
$738$		0			0
$739$	6.48126e6	−	1.12259e7i	0.436564	−	0.756151i	−0.560858	−	0.827912i	$-0.689528\pi$
$739$	6.48126e6	−	1.12259e7i	0.436564	−	0.756151i	0.997422	+	0.0717608i	$0.0228618\pi$
$740$		0			0
$741$	−247915.			−0.0165866
$742$		0			0
$743$	1.45466e7			0.966693			0.483347	−	0.875429i	$-0.339421\pi$
$743$	1.45466e7			0.966693			0.483347	+	0.875429i	$0.339421\pi$
$744$		0			0
$745$	−2.15142e6	+	3.72637e6i	−0.142015	+	0.245977i
$746$		0			0
$747$	1.35475e7	+	2.34649e7i	0.888294	+	1.53857i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−7.24749e6	−	1.25530e7i	−0.468908	−	0.812173i	0.530460	−	0.847710i	$-0.322019\pi$
$751$	−7.24749e6	−	1.25530e7i	−0.468908	−	0.812173i	−0.999368	+	0.0355371i	$0.988686\pi$
$752$		0			0
$753$	−162150.	+	280853.i	−0.0104215	+	0.0180506i
$754$		0			0
$755$	−1.41537e7			−0.903653
$756$		0			0
$757$	2.49809e7			1.58441			0.792207	−	0.610252i	$-0.208932\pi$
$757$	2.49809e7			1.58441			0.792207	+	0.610252i	$0.208932\pi$
$758$		0			0
$759$	381639.	−	661019.i	0.0240463	−	0.0416495i
$760$		0			0
$761$	7.82969e6	+	1.35614e7i	0.490098	+	0.848874i	0.999935	−	0.0113966i	$-0.00362773\pi$
$761$	7.82969e6	+	1.35614e7i	0.490098	+	0.848874i	−0.509837	+	0.860271i	$0.670294\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	863580.	+	1.49576e6i	0.0533518	+	0.0924080i
$766$		0			0
$767$	−6.26999e6	+	1.08599e7i	−0.384839	+	0.666560i
$768$		0			0
$769$	−1.30420e7			−0.795292			−0.397646	−	0.917539i	$-0.630173\pi$
$769$	−1.30420e7			−0.795292			−0.397646	+	0.917539i	$0.630173\pi$
$770$		0			0
$771$	205345.			0.0124408
$772$		0			0
$773$	−7.30422e6	+	1.26513e7i	−0.439668	+	0.761527i	−0.997664	−	0.0683164i	$-0.978237\pi$
$773$	−7.30422e6	+	1.26513e7i	−0.439668	+	0.761527i	0.557996	+	0.829844i	$0.311571\pi$
$774$		0			0
$775$	−924619.	−	1.60149e6i	−0.0552979	−	0.0957787i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.58284e6	+	9.66976e6i	0.329618	+	0.570916i
$780$		0			0
$781$	3.65661e6	−	6.33344e6i	0.214512	−	0.371546i
$782$		0			0
$783$	−549413.			−0.0320254
$784$		0			0
$785$	−4.85705e6			&m

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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	392.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.152067 - 0.263388i) q^{3} +(17.7603 + 30.7618i) q^{5} +(121.454 + 210.364i) q^{9} +O(q^{10})\)	\(q+(0.152067 - 0.263388i) q^{3} +(17.7603 + 30.7618i) q^{5} +(121.454 + 210.364i) q^{9} +(-282.912 + 490.019i) q^{11} -983.594 q^{13} +10.8031 q^{15} +(100.088 - 173.357i) q^{17} +(414.374 + 717.716i) q^{19} +(-2217.71 - 3841.19i) q^{23} +(931.641 - 1613.65i) q^{25} +147.781 q^{27} -3717.74 q^{29} +(496.231 - 859.498i) q^{31} +(86.0435 + 149.032i) q^{33} +(4179.84 + 7239.70i) q^{37} +(-149.572 + 259.067i) q^{39} +13473.0 q^{41} +298.798 q^{43} +(-4314.12 + 7472.27i) q^{45} +(-9368.26 - 16226.3i) q^{47} +(-30.4401 - 52.7238i) q^{51} +(-8018.18 + 13887.9i) q^{53} -20098.5 q^{55} +252.051 q^{57} +(6374.58 - 11041.1i) q^{59} +(-17488.0 - 30290.1i) q^{61} +(-17469.0 - 30257.1i) q^{65} +(-5989.44 + 10374.0i) q^{67} -1348.97 q^{69} -12924.9 q^{71} +(40588.6 - 70301.5i) q^{73} +(-283.344 - 490.767i) q^{75} +(-23499.4 - 40702.2i) q^{79} +(-29490.8 + 51079.5i) q^{81} +111544. q^{83} +7110.36 q^{85} +(-565.347 + 979.210i) q^{87} +(-17363.4 - 30074.3i) q^{89} +(-150.921 - 261.403i) q^{93} +(-14718.8 + 25493.8i) q^{95} -92655.6 q^{97} -137443. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 26 q^{3} - 62 q^{5} - 206 q^{9}+O(q^{10}) \) 4 * q - 26 * q^3 - 62 * q^5 - 206 * q^9	\( 4 q - 26 q^{3} - 62 q^{5} - 206 q^{9} - 972 q^{11} - 156 q^{13} + 5152 q^{15} + 560 q^{17} + 2642 q^{19} - 2272 q^{23} - 4522 q^{25} + 11128 q^{27} - 15616 q^{29} + 5444 q^{31} - 10512 q^{33} - 576 q^{37} - 24120 q^{39} + 33776 q^{41} - 16792 q^{43} - 52406 q^{45} - 4532 q^{47} + 9404 q^{51} - 1420 q^{53} + 39024 q^{55} - 94888 q^{57} + 34146 q^{59} + 19106 q^{61} - 123252 q^{65} - 56952 q^{67} - 116512 q^{69} - 14448 q^{71} + 128828 q^{73} - 168526 q^{75} - 52808 q^{79} - 92366 q^{81} + 168972 q^{83} - 55960 q^{85} + 106460 q^{87} - 130972 q^{89} + 116792 q^{93} + 147392 q^{95} - 389248 q^{97} + 89784 q^{99}+O(q^{100}) \) 4 * q - 26 * q^3 - 62 * q^5 - 206 * q^9 - 972 * q^11 - 156 * q^13 + 5152 * q^15 + 560 * q^17 + 2642 * q^19 - 2272 * q^23 - 4522 * q^25 + 11128 * q^27 - 15616 * q^29 + 5444 * q^31 - 10512 * q^33 - 576 * q^37 - 24120 * q^39 + 33776 * q^41 - 16792 * q^43 - 52406 * q^45 - 4532 * q^47 + 9404 * q^51 - 1420 * q^53 + 39024 * q^55 - 94888 * q^57 + 34146 * q^59 + 19106 * q^61 - 123252 * q^65 - 56952 * q^67 - 116512 * q^69 - 14448 * q^71 + 128828 * q^73 - 168526 * q^75 - 52808 * q^79 - 92366 * q^81 + 168972 * q^83 - 55960 * q^85 + 106460 * q^87 - 130972 * q^89 + 116792 * q^93 + 147392 * q^95 - 389248 * q^97 + 89784 * q^99

Introduction

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