LMFDB - Embedded newform 576.7.h.a.161.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,7,Mod(161,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("576.161");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	576.h (of order $2$, degree $1$, 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:	$132.511152165$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 964 x^{14} - 6608 x^{13} + 404616 x^{12} - 2342156 x^{11} + 95125730 x^{10} + \cdots + 39\!\cdots\!00 $ x^16 - 8x^15 + 964x^14 - 6608x^13 + 404616x^12 - 2342156x^11 + 95125730x^10 - 454315424x^9 + 13646460877x^8 - 51885446412x^7 + 1232155738966x^6 - 3516750797184x^5 + 70625158468274x^4 - 135448702737480x^3 + 2383004402377884x^2 - 2315857662932040*x + 39335549337519300
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{48}\cdot 3^{36} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.16
Root			$4.07458 + 13.6726i$ of defining polynomial
Character	$\chi$	$=$	576.161
Dual form			576.7.h.a.161.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+168.989 q^{5} +13.9908 q^{7} +O(q^{10})$	$q+168.989 q^{5} +13.9908 q^{7} -1500.86 q^{11} +744.941i q^{13} -2963.54i q^{17} -6311.61i q^{19} +9997.60i q^{23} +12932.2 q^{25} +20393.1 q^{29} -39212.3 q^{31} +2364.29 q^{35} +89287.8i q^{37} +20087.0i q^{41} +55492.9i q^{43} +108404. i q^{47} -117453. q^{49} +143755. q^{53} -253629. q^{55} +11065.7 q^{59} +175980. i q^{61} +125887. i q^{65} -42074.2i q^{67} +103977. i q^{71} +539416. q^{73} -20998.3 q^{77} +479208. q^{79} -485444. q^{83} -500805. i q^{85} -310114. i q^{89} +10422.3i q^{91} -1.06659e6i q^{95} -199839. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 19888 q^{25} + 388784 q^{49} + 465728 q^{73} + 6555136 q^{97}+O(q^{100}) $ 16 * q - 19888 * q^25 + 388784 * q^49 + 465728 * q^73 + 6555136 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	168.989	1.35191	0.675955	−	0.736943i	$-0.263731\pi$
$5$	168.989	1.35191	0.675955	+	0.736943i	$0.263731\pi$
$6$	0	0
$7$	13.9908	0.0407895	0.0203948	−	0.999792i	$-0.493508\pi$
$7$	13.9908	0.0407895	0.0203948	+	0.999792i	$0.493508\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1500.86	−1.12762	−0.563809	−	0.825905i	$-0.690665\pi$
$11$	−1500.86	−1.12762	−0.563809	+	0.825905i	$0.690665\pi$
$12$	0	0
$13$	744.941i	0.339072i	0.985524	+	0.169536i	$0.0542269\pi$
$13$	744.941i	0.339072i	−0.985524	+	0.169536i	$0.945773\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2963.54i	− 0.603203i	−0.953434	−	0.301602i	$-0.902479\pi$
$17$	− 2963.54i	− 0.603203i	0.953434	−	0.301602i	$-0.0975213\pi$
$18$	0	0
$19$	− 6311.61i	− 0.920194i	−0.887869	−	0.460097i	$-0.847815\pi$
$19$	− 6311.61i	− 0.920194i	0.887869	−	0.460097i	$-0.152185\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9997.60i	0.821698i	0.911703	+	0.410849i	$0.134768\pi$
$23$	9997.60i	0.821698i	−0.911703	+	0.410849i	$0.865232\pi$
$24$	0	0
$25$	12932.2	0.827662
$26$	0	0
$27$	0	0
$28$	0	0
$29$	20393.1	0.836161	0.418081	−	0.908410i	$-0.362703\pi$
$29$	20393.1	0.836161	0.418081	+	0.908410i	$0.362703\pi$
$30$	0	0
$31$	−39212.3	−1.31625	−0.658123	−	0.752910i	$-0.728650\pi$
$31$	−39212.3	−1.31625	−0.658123	+	0.752910i	$0.728650\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2364.29	0.0551438
$36$	0	0
$37$	89287.8i	1.76273i	0.472432	+	0.881367i	$0.343376\pi$
$37$	89287.8i	1.76273i	−0.472432	+	0.881367i	$0.656624\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	20087.0i	0.291450i	0.989325	+	0.145725i	$0.0465515\pi$
$41$	20087.0i	0.291450i	−0.989325	+	0.145725i	$0.953448\pi$
$42$	0	0
$43$	55492.9i	0.697962i	0.937130	+	0.348981i	$0.113472\pi$
$43$	55492.9i	0.697962i	−0.937130	+	0.348981i	$0.886528\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	108404.i	1.04412i	0.852909	+	0.522060i	$0.174836\pi$
$47$	108404.i	1.04412i	−0.852909	+	0.522060i	$0.825164\pi$
$48$	0	0
$49$	−117453.	−0.998336
$50$	0	0
$51$	0	0
$52$	0	0
$53$	143755.	0.965596	0.482798	−	0.875732i	$-0.339621\pi$
$53$	143755.	0.965596	0.482798	+	0.875732i	$0.339621\pi$
$54$	0	0
$55$	−253629.	−1.52444
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11065.7	0.0538792	0.0269396	−	0.999637i	$-0.491424\pi$
$59$	11065.7	0.0538792	0.0269396	+	0.999637i	$0.491424\pi$
$60$	0	0
$61$	175980.i	0.775305i	0.921806	+	0.387653i	$0.126714\pi$
$61$	175980.i	0.775305i	−0.921806	+	0.387653i	$0.873286\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	125887.i	0.458395i
$66$	0	0
$67$	− 42074.2i	− 0.139892i	−0.997551	−	0.0699458i	$-0.977717\pi$
$67$	− 42074.2i	− 0.139892i	0.997551	−	0.0699458i	$-0.0222826\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	103977.i	0.290511i	0.989394	+	0.145256i	$0.0464005\pi$
$71$	103977.i	0.290511i	−0.989394	+	0.145256i	$0.953600\pi$
$72$	0	0
$73$	539416.	1.38661	0.693307	−	0.720643i	$-0.256153\pi$
$73$	539416.	1.38661	0.693307	+	0.720643i	$0.256153\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−20998.3	−0.0459951
$78$	0	0
$79$	479208.	0.971946	0.485973	−	0.873974i	$-0.338465\pi$
$79$	479208.	0.971946	0.485973	+	0.873974i	$0.338465\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−485444.	−0.848995	−0.424498	−	0.905429i	$-0.639549\pi$
$83$	−485444.	−0.848995	−0.424498	+	0.905429i	$0.639549\pi$
$84$	0	0
$85$	− 500805.i	− 0.815477i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 310114.i	− 0.439898i	−0.975511	−	0.219949i	$-0.929411\pi$
$89$	− 310114.i	− 0.439898i	0.975511	−	0.219949i	$-0.0705891\pi$
$90$	0	0
$91$	10422.3i	0.0138306i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 1.06659e6i	− 1.24402i
$96$	0	0
$97$	−199839.	−0.218960	−0.109480	−	0.993989i	$-0.534919\pi$
$97$	−199839.	−0.218960	−0.109480	+	0.993989i	$0.534919\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.20189e6	−1.16654	−0.583270	−	0.812279i	$-0.698227\pi$
$101$	−1.20189e6	−1.16654	−0.583270	+	0.812279i	$0.698227\pi$
$102$	0	0
$103$	416228.	0.380907	0.190454	−	0.981696i	$-0.439004\pi$
$103$	416228.	0.380907	0.190454	+	0.981696i	$0.439004\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−664061.	−0.542071	−0.271036	−	0.962569i	$-0.587366\pi$
$107$	−664061.	−0.542071	−0.271036	+	0.962569i	$0.587366\pi$
$108$	0	0
$109$	− 1.42279e6i	− 1.09865i	−0.835608	−	0.549326i	$-0.814885\pi$
$109$	− 1.42279e6i	− 1.09865i	0.835608	−	0.549326i	$-0.185115\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.58232e6i	1.78968i	0.446390	+	0.894838i	$0.352709\pi$
$113$	2.58232e6i	1.78968i	−0.446390	+	0.894838i	$0.647291\pi$
$114$	0	0
$115$	1.68948e6i	1.11086i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 41462.3i	− 0.0246044i
$120$	0	0
$121$	481022.	0.271524
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−455049.	−0.232985
$126$	0	0
$127$	3.13676e6	1.53134	0.765668	−	0.643237i	$-0.222409\pi$
$127$	3.13676e6	1.53134	0.765668	+	0.643237i	$0.222409\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−418255.	−0.186049	−0.0930245	−	0.995664i	$-0.529653\pi$
$131$	−418255.	−0.186049	−0.0930245	+	0.995664i	$0.529653\pi$
$132$	0	0
$133$	− 88304.6i	− 0.0375343i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.05135e6i	0.797772i	0.917000	+	0.398886i	$0.130603\pi$
$137$	2.05135e6i	0.797772i	−0.917000	+	0.398886i	$0.869397\pi$
$138$	0	0
$139$	2.00434e6i	0.746322i	0.927767	+	0.373161i	$0.121726\pi$
$139$	2.00434e6i	0.746322i	−0.927767	+	0.373161i	$0.878274\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.11805e6i	− 0.382344i
$144$	0	0
$145$	3.44621e6	1.13042
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.32337e6	0.702361	0.351180	−	0.936308i	$-0.385780\pi$
$149$	2.32337e6	0.702361	0.351180	+	0.936308i	$0.385780\pi$
$150$	0	0
$151$	−6.44651e6	−1.87238	−0.936190	−	0.351493i	$-0.885674\pi$
$151$	−6.44651e6	−1.87238	−0.936190	+	0.351493i	$0.885674\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.62644e6	−1.77945
$156$	0	0
$157$	2.70983e6i	0.700233i	0.936706	+	0.350117i	$0.113858\pi$
$157$	2.70983e6i	0.700233i	−0.936706	+	0.350117i	$0.886142\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	139875.i	0.0335167i
$162$	0	0
$163$	940013.i	0.217056i	0.994093	+	0.108528i	$0.0346136\pi$
$163$	940013.i	0.217056i	−0.994093	+	0.108528i	$0.965386\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.67355e6i	1.86229i	0.364647	+	0.931146i	$0.381190\pi$
$167$	8.67355e6i	1.86229i	−0.364647	+	0.931146i	$0.618810\pi$
$168$	0	0
$169$	4.27187e6	0.885030
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.50876e6	0.291395	0.145697	−	0.989329i	$-0.453457\pi$
$173$	1.50876e6	0.291395	0.145697	+	0.989329i	$0.453457\pi$
$174$	0	0
$175$	180932.	0.0337600
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.07397e7	−1.87255	−0.936277	−	0.351262i	$-0.885753\pi$
$179$	−1.07397e7	−1.87255	−0.936277	+	0.351262i	$0.885753\pi$
$180$	0	0
$181$	6.45716e6i	1.08894i	0.838779	+	0.544472i	$0.183270\pi$
$181$	6.45716e6i	1.08894i	−0.838779	+	0.544472i	$0.816730\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.50886e7i	2.38306i
$186$	0	0
$187$	4.44786e6i	0.680183i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 7.54231e6i	− 1.08244i	−0.840881	−	0.541220i	$-0.817962\pi$
$191$	− 7.54231e6i	− 1.08244i	0.840881	−	0.541220i	$-0.182038\pi$
$192$	0	0
$193$	8.76476e6	1.21918	0.609591	−	0.792716i	$-0.291334\pi$
$193$	8.76476e6	1.21918	0.609591	+	0.792716i	$0.291334\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.83274e6	0.501315	0.250657	−	0.968076i	$-0.419353\pi$
$197$	3.83274e6	0.501315	0.250657	+	0.968076i	$0.419353\pi$
$198$	0	0
$199$	−1.16120e7	−1.47349	−0.736744	−	0.676171i	$-0.763638\pi$
$199$	−1.16120e7	−1.47349	−0.736744	+	0.676171i	$0.763638\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	285317.	0.0341066
$204$	0	0
$205$	3.39449e6i	0.394015i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.47285e6i	1.03763i
$210$	0	0
$211$	1.50589e7i	1.60305i	0.597962	+	0.801524i	$0.295977\pi$
$211$	1.50589e7i	1.60305i	−0.597962	+	0.801524i	$0.704023\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.37768e6i	0.943583i
$216$	0	0
$217$	−548612.	−0.0536891
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.20766e6	0.204529
$222$	0	0
$223$	1.46388e7	1.32005	0.660027	−	0.751242i	$-0.270545\pi$
$223$	1.46388e7	1.32005	0.660027	+	0.751242i	$0.270545\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.60373e7	−1.37105	−0.685526	−	0.728048i	$-0.740428\pi$
$227$	−1.60373e7	−1.37105	−0.685526	+	0.728048i	$0.740428\pi$
$228$	0	0
$229$	− 1.16093e7i	− 0.966721i	−0.875421	−	0.483360i	$-0.839416\pi$
$229$	− 1.16093e7i	− 0.966721i	0.875421	−	0.483360i	$-0.160584\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.35696e7i	1.07275i	0.843980	+	0.536375i	$0.180207\pi$
$233$	1.35696e7i	1.07275i	−0.843980	+	0.536375i	$0.819793\pi$
$234$	0	0
$235$	1.83190e7i	1.41156i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 4.26081e6i	− 0.312103i	−0.987749	−	0.156052i	$-0.950123\pi$
$239$	− 4.26081e6i	− 0.312103i	0.987749	−	0.156052i	$-0.0498766\pi$
$240$	0	0
$241$	1.40563e7	1.00420	0.502101	−	0.864809i	$-0.332561\pi$
$241$	1.40563e7	1.00420	0.502101	+	0.864809i	$0.332561\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.98483e7	−1.34966
$246$	0	0
$247$	4.70178e6	0.312012
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.96750e7	−1.87659	−0.938296	−	0.345833i	$-0.887596\pi$
$251$	−2.96750e7	−1.87659	−0.938296	+	0.345833i	$0.887596\pi$
$252$	0	0
$253$	− 1.50050e7i	− 0.926562i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.40873e7i	1.41902i	0.704696	+	0.709509i	$0.251083\pi$
$257$	2.40873e7i	1.41902i	−0.704696	+	0.709509i	$0.748917\pi$
$258$	0	0
$259$	1.24921e6i	0.0719011i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.72669e7i	− 1.49888i	−0.662070	−	0.749442i	$-0.730322\pi$
$263$	− 2.72669e7i	− 1.49888i	0.662070	−	0.749442i	$-0.269678\pi$
$264$	0	0
$265$	2.42930e7	1.30540
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.01871e6	−0.206457	−0.103229	−	0.994658i	$-0.532917\pi$
$269$	−4.01871e6	−0.206457	−0.103229	+	0.994658i	$0.532917\pi$
$270$	0	0
$271$	2.31382e7	1.16258	0.581288	−	0.813698i	$-0.302549\pi$
$271$	2.31382e7	1.16258	0.581288	+	0.813698i	$0.302549\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.94095e7	−0.933288
$276$	0	0
$277$	− 6.24314e6i	− 0.293740i	−0.989156	−	0.146870i	$-0.953080\pi$
$277$	− 6.24314e6i	− 0.293740i	0.989156	−	0.146870i	$-0.0469200\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.15373e6i	0.232275i	0.993233	+	0.116137i	$0.0370513\pi$
$281$	5.15373e6i	0.232275i	−0.993233	+	0.116137i	$0.962949\pi$
$282$	0	0
$283$	− 1.21978e7i	− 0.538174i	−0.963116	−	0.269087i	$-0.913278\pi$
$283$	− 1.21978e7i	− 0.538174i	0.963116	−	0.269087i	$-0.0867218\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	281034.i	0.0118881i
$288$	0	0
$289$	1.53550e7	0.636146
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.58367e6	0.381003	0.190502	−	0.981687i	$-0.438989\pi$
$293$	9.58367e6	0.381003	0.190502	+	0.981687i	$0.438989\pi$
$294$	0	0
$295$	1.86997e6	0.0728399
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.44761e6	−0.278614
$300$	0	0
$301$	776391.i	0.0284696i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.97386e7i	1.04814i
$306$	0	0
$307$	3.65639e7i	1.26368i	0.775098	+	0.631841i	$0.217700\pi$
$307$	3.65639e7i	1.26368i	−0.775098	+	0.631841i	$0.782300\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.14969e7i	0.714653i	0.933980	+	0.357327i	$0.116312\pi$
$311$	2.14969e7i	0.714653i	−0.933980	+	0.357327i	$0.883688\pi$
$312$	0	0
$313$	2.15571e7	0.703004	0.351502	−	0.936187i	$-0.385671\pi$
$313$	2.15571e7	0.703004	0.351502	+	0.936187i	$0.385671\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.84431e7	−1.20681	−0.603407	−	0.797433i	$-0.706190\pi$
$317$	−3.84431e7	−1.20681	−0.603407	+	0.797433i	$0.706190\pi$
$318$	0	0
$319$	−3.06073e7	−0.942871
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.87047e7	−0.555064
$324$	0	0
$325$	9.63374e6i	0.280637i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.51665e6i	0.0425892i
$330$	0	0
$331$	4.00607e7i	1.10467i	0.833621	+	0.552337i	$0.186264\pi$
$331$	4.00607e7i	1.10467i	−0.833621	+	0.552337i	$0.813736\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 7.11007e6i	− 0.189121i
$336$	0	0
$337$	−3.98363e7	−1.04085	−0.520426	−	0.853907i	$-0.674227\pi$
$337$	−3.98363e7	−1.04085	−0.520426	+	0.853907i	$0.674227\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.88522e7	1.48422
$342$	0	0
$343$	−3.28927e6	−0.0815112
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.80254e6	−0.0910092	−0.0455046	−	0.998964i	$-0.514490\pi$
$347$	−3.80254e6	−0.0910092	−0.0455046	+	0.998964i	$0.514490\pi$
$348$	0	0
$349$	2.37735e7i	0.559263i	0.960107	+	0.279632i	$0.0902123\pi$
$349$	2.37735e7i	0.559263i	−0.960107	+	0.279632i	$0.909788\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.78223e7i	0.632512i	0.948674	+	0.316256i	$0.102426\pi$
$353$	2.78223e7i	0.632512i	−0.948674	+	0.316256i	$0.897574\pi$
$354$	0	0
$355$	1.75710e7i	0.392745i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.62940e7i	0.568293i	0.958781	+	0.284147i	$0.0917103\pi$
$359$	2.62940e7i	0.568293i	−0.958781	+	0.284147i	$0.908290\pi$
$360$	0	0
$361$	7.20942e6	0.153242
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.11553e7	1.87458
$366$	0	0
$367$	−6.54421e7	−1.32391	−0.661956	−	0.749542i	$-0.730274\pi$
$367$	−6.54421e7	−1.32391	−0.661956	+	0.749542i	$0.730274\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.01125e6	0.0393862
$372$	0	0
$373$	6.34853e7i	1.22334i	0.791114	+	0.611669i	$0.209502\pi$
$373$	6.34853e7i	1.22334i	−0.791114	+	0.611669i	$0.790498\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.51917e7i	0.283519i
$378$	0	0
$379$	− 8.56684e7i	− 1.57363i	−0.617188	−	0.786816i	$-0.711728\pi$
$379$	− 8.56684e7i	− 1.57363i	0.617188	−	0.786816i	$-0.288272\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.73160e7i	1.73216i	0.499906	+	0.866080i	$0.333368\pi$
$383$	9.73160e7i	1.73216i	−0.499906	+	0.866080i	$0.666632\pi$
$384$	0	0
$385$	−3.54847e6	−0.0621812
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.00954e8	−1.71504	−0.857519	−	0.514452i	$-0.827995\pi$
$389$	−1.00954e8	−1.71504	−0.857519	+	0.514452i	$0.827995\pi$
$390$	0	0
$391$	2.96283e7	0.495651
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.09807e7	1.31398
$396$	0	0
$397$	2.94102e7i	0.470031i	0.971992	+	0.235015i	$0.0755140\pi$
$397$	2.94102e7i	0.470031i	−0.971992	+	0.235015i	$0.924486\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 9.81649e7i	− 1.52238i	−0.648529	−	0.761190i	$-0.724616\pi$
$401$	− 9.81649e7i	− 1.52238i	0.648529	−	0.761190i	$-0.275384\pi$
$402$	0	0
$403$	− 2.92108e7i	− 0.446302i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.34009e8i	− 1.98769i
$408$	0	0
$409$	2.36974e7	0.346363	0.173181	−	0.984890i	$-0.444595\pi$
$409$	2.36974e7	0.346363	0.173181	+	0.984890i	$0.444595\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	154818.	0.00219771
$414$	0	0
$415$	−8.20347e7	−1.14777
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−8.91039e7	−1.21131	−0.605654	−	0.795728i	$-0.707089\pi$
$419$	−8.91039e7	−1.21131	−0.605654	+	0.795728i	$0.707089\pi$
$420$	0	0
$421$	− 8.70306e7i	− 1.16634i	−0.812350	−	0.583170i	$-0.801812\pi$
$421$	− 8.70306e7i	− 1.16634i	0.812350	−	0.583170i	$-0.198188\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 3.83251e7i	− 0.499249i
$426$	0	0
$427$	2.46210e6i	0.0316244i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.98013e7i	0.497125i	0.968616	+	0.248563i	$0.0799582\pi$
$431$	3.98013e7i	0.497125i	−0.968616	+	0.248563i	$0.920042\pi$
$432$	0	0
$433$	2.08244e7	0.256512	0.128256	−	0.991741i	$-0.459062\pi$
$433$	2.08244e7	0.256512	0.128256	+	0.991741i	$0.459062\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.31010e7	0.756122
$438$	0	0
$439$	1.38090e8	1.63218	0.816090	−	0.577926i	$-0.196138\pi$
$439$	1.38090e8	1.63218	0.816090	+	0.577926i	$0.196138\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.08013e7	1.04443	0.522217	−	0.852813i	$-0.325105\pi$
$443$	9.08013e7	1.04443	0.522217	+	0.852813i	$0.325105\pi$
$444$	0	0
$445$	− 5.24058e7i	− 0.594702i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 8.73485e7i	− 0.964976i	−0.875903	−	0.482488i	$-0.839733\pi$
$449$	− 8.73485e7i	− 0.964976i	0.875903	−	0.482488i	$-0.160267\pi$
$450$	0	0
$451$	− 3.01479e7i	− 0.328645i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.76126e6i	0.0186977i
$456$	0	0
$457$	−1.75726e8	−1.84114	−0.920570	−	0.390578i	$-0.872275\pi$
$457$	−1.75726e8	−1.84114	−0.920570	+	0.390578i	$0.872275\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.37345e8	−1.40187	−0.700936	−	0.713224i	$-0.747234\pi$
$461$	−1.37345e8	−1.40187	−0.700936	+	0.713224i	$0.747234\pi$
$462$	0	0
$463$	3.30418e7	0.332905	0.166452	−	0.986049i	$-0.446769\pi$
$463$	3.30418e7	0.332905	0.166452	+	0.986049i	$0.446769\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.48968e8	1.46266	0.731330	−	0.682024i	$-0.238900\pi$
$467$	1.48968e8	1.46266	0.731330	+	0.682024i	$0.238900\pi$
$468$	0	0
$469$	− 588652.i	− 0.00570611i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 8.32871e7i	− 0.787036i
$474$	0	0
$475$	− 8.16232e7i	− 0.761610i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 1.47748e8i	− 1.34436i	−0.740389	−	0.672178i	$-0.765359\pi$
$479$	− 1.47748e8i	− 1.34436i	0.740389	−	0.672178i	$-0.234641\pi$
$480$	0	0
$481$	−6.65141e7	−0.597693
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.37705e7	−0.296014
$486$	0	0
$487$	1.43723e8	1.24434	0.622170	−	0.782882i	$-0.286251\pi$
$487$	1.43723e8	1.24434	0.622170	+	0.782882i	$0.286251\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.74135e8	1.47110	0.735550	−	0.677471i	$-0.236924\pi$
$491$	1.74135e8	1.47110	0.735550	+	0.677471i	$0.236924\pi$
$492$	0	0
$493$	− 6.04358e7i	− 0.504375i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.45473e6i	0.0118498i
$498$	0	0
$499$	− 2.57033e7i	− 0.206865i	−0.994636	−	0.103433i	$-0.967017\pi$
$499$	− 2.57033e7i	− 0.206865i	0.994636	−	0.103433i	$-0.0329826\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 6.86336e6i	− 0.0539303i	−0.999636	−	0.0269651i	$-0.991416\pi$
$503$	− 6.86336e6i	− 0.0539303i	0.999636	−	0.0269651i	$-0.00858431\pi$
$504$	0	0
$505$	−2.03105e8	−1.57706
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.26759e6	0.0247784	0.0123892	−	0.999923i	$-0.496056\pi$
$509$	3.26759e6	0.0247784	0.0123892	+	0.999923i	$0.496056\pi$
$510$	0	0
$511$	7.54687e6	0.0565593
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7.03378e7	0.514953
$516$	0	0
$517$	− 1.62699e8i	− 1.17737i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 7.16898e7i	− 0.506926i	−0.967345	−	0.253463i	$-0.918430\pi$
$521$	− 7.16898e7i	− 0.506926i	0.967345	−	0.253463i	$-0.0815695\pi$
$522$	0	0
$523$	− 1.78584e8i	− 1.24835i	−0.781284	−	0.624176i	$-0.785435\pi$
$523$	− 1.78584e8i	− 1.24835i	0.781284	−	0.624176i	$-0.214565\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.16207e8i	0.793964i
$528$	0	0
$529$	4.80840e7	0.324813
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.49637e7	−0.0988226
$534$	0	0
$535$	−1.12219e8	−0.732832
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.76281e8	1.12574
$540$	0	0
$541$	− 5.04327e7i	− 0.318508i	−0.987238	−	0.159254i	$-0.949091\pi$
$541$	− 5.04327e7i	− 0.318508i	0.987238	−	0.159254i	$-0.0509089\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 2.40435e8i	− 1.48528i
$546$	0	0
$547$	5.13884e7i	0.313981i	0.987600	+	0.156990i	$0.0501791\pi$
$547$	5.13884e7i	0.313981i	−0.987600	+	0.156990i	$0.949821\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1.28714e8i	− 0.769431i
$552$	0	0
$553$	6.70450e6	0.0396453
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.14558e8	0.662920	0.331460	−	0.943469i	$-0.392459\pi$
$557$	1.14558e8	0.662920	0.331460	+	0.943469i	$0.392459\pi$
$558$	0	0
$559$	−4.13389e7	−0.236659
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.54440e7	−0.478803	−0.239401	−	0.970921i	$-0.576951\pi$
$563$	−8.54440e7	−0.478803	−0.239401	+	0.970921i	$0.576951\pi$
$564$	0	0
$565$	4.36383e8i	2.41948i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 3.26748e8i	− 1.77368i	−0.462074	−	0.886841i	$-0.652895\pi$
$569$	− 3.26748e8i	− 1.77368i	0.462074	−	0.886841i	$-0.347105\pi$
$570$	0	0
$571$	− 2.77811e8i	− 1.49225i	−0.665806	−	0.746125i	$-0.731912\pi$
$571$	− 2.77811e8i	− 1.49225i	0.665806	−	0.746125i	$-0.268088\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.29291e8i	0.680088i
$576$	0	0
$577$	−1.83022e8	−0.952744	−0.476372	−	0.879244i	$-0.658048\pi$
$577$	−1.83022e8	−0.952744	−0.476372	+	0.879244i	$0.658048\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.79176e6	−0.0346301
$582$	0	0
$583$	−2.15756e8	−1.08882
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.57537e8	1.27328	0.636641	−	0.771160i	$-0.280323\pi$
$587$	2.57537e8	1.27328	0.636641	+	0.771160i	$0.280323\pi$
$588$	0	0
$589$	2.47493e8i	1.21120i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1.96060e8i	− 0.940210i	−0.882610	−	0.470105i	$-0.844216\pi$
$593$	− 1.96060e8i	− 0.940210i	0.882610	−	0.470105i	$-0.155784\pi$
$594$	0	0
$595$	− 7.00667e6i	− 0.0332629i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.79045e8i	0.833072i	0.909119	+	0.416536i	$0.136756\pi$
$599$	1.79045e8i	0.833072i	−0.909119	+	0.416536i	$0.863244\pi$
$600$	0	0
$601$	1.01945e8	0.469617	0.234808	−	0.972042i	$-0.424554\pi$
$601$	1.01945e8	0.469617	0.234808	+	0.972042i	$0.424554\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.12873e7	0.367077
$606$	0	0
$607$	5.68302e7	0.254105	0.127052	−	0.991896i	$-0.459448\pi$
$607$	5.68302e7	0.254105	0.127052	+	0.991896i	$0.459448\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.07542e7	−0.354031
$612$	0	0
$613$	− 1.94135e8i	− 0.842797i	−0.906876	−	0.421399i	$-0.861539\pi$
$613$	− 1.94135e8i	− 0.842797i	0.906876	−	0.421399i	$-0.138461\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.21008e8i	0.515180i	0.966254	+	0.257590i	$0.0829284\pi$
$617$	1.21008e8i	0.515180i	−0.966254	+	0.257590i	$0.917072\pi$
$618$	0	0
$619$	− 3.38818e8i	− 1.42855i	−0.699867	−	0.714273i	$-0.746757\pi$
$619$	− 3.38818e8i	− 1.42855i	0.699867	−	0.714273i	$-0.253243\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 4.33875e6i	− 0.0179432i
$624$	0	0
$625$	−2.78964e8	−1.14264
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.64608e8	1.06329
$630$	0	0
$631$	−2.31097e8	−0.919827	−0.459913	−	0.887964i	$-0.652120\pi$
$631$	−2.31097e8	−0.919827	−0.459913	+	0.887964i	$0.652120\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.30078e8	2.07023
$636$	0	0
$637$	− 8.74957e7i	− 0.338508i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.38832e8i	0.527125i	0.964642	+	0.263563i	$0.0848976\pi$
$641$	1.38832e8i	0.527125i	−0.964642	+	0.263563i	$0.915102\pi$
$642$	0	0
$643$	1.50268e8i	0.565242i	0.959232	+	0.282621i	$0.0912039\pi$
$643$	1.50268e8i	0.565242i	−0.959232	+	0.282621i	$0.908796\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.22339e8i	0.820922i	0.911878	+	0.410461i	$0.134632\pi$
$647$	2.22339e8i	0.820922i	−0.911878	+	0.410461i	$0.865368\pi$
$648$	0	0
$649$	−1.66080e7	−0.0607552
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.97454e7	0.322309	0.161155	−	0.986929i	$-0.448478\pi$
$653$	8.97454e7	0.322309	0.161155	+	0.986929i	$0.448478\pi$
$654$	0	0
$655$	−7.06804e7	−0.251521
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2.46917e8	−0.862770	−0.431385	−	0.902168i	$-0.641975\pi$
$659$	−2.46917e8	−0.862770	−0.431385	+	0.902168i	$0.641975\pi$
$660$	0	0
$661$	1.72712e8i	0.598024i	0.954249	+	0.299012i	$0.0966571\pi$
$661$	1.72712e8i	0.598024i	−0.954249	+	0.299012i	$0.903343\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.49225e7i	− 0.0507430i
$666$	0	0
$667$	2.03882e8i	0.687072i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 2.64121e8i	− 0.874249i
$672$	0	0
$673$	−1.66236e8	−0.545356	−0.272678	−	0.962105i	$-0.587909\pi$
$673$	−1.66236e8	−0.545356	−0.272678	+	0.962105i	$0.587909\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.38587e8	−1.09120	−0.545601	−	0.838045i	$-0.683698\pi$
$677$	−3.38587e8	−1.09120	−0.545601	+	0.838045i	$0.683698\pi$
$678$	0	0
$679$	−2.79591e6	−0.00893127
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.36588e7	0.0428698	0.0214349	−	0.999770i	$-0.493177\pi$
$683$	1.36588e7	0.0428698	0.0214349	+	0.999770i	$0.493177\pi$
$684$	0	0
$685$	3.46656e8i	1.07852i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.07089e8i	0.327406i
$690$	0	0
$691$	2.87568e8i	0.871577i	0.900049	+	0.435789i	$0.143530\pi$
$691$	2.87568e8i	0.871577i	−0.900049	+	0.435789i	$0.856470\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.38711e8i	1.00896i
$696$	0	0
$697$	5.95287e7	0.175804
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.04539e8	−0.884075	−0.442037	−	0.896997i	$-0.645744\pi$
$701$	−3.04539e8	−0.884075	−0.442037	+	0.896997i	$0.645744\pi$
$702$	0	0
$703$	5.63550e8	1.62206
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.68154e7	−0.0475826
$708$	0	0
$709$	− 5.94975e8i	− 1.66940i	−0.550706	−	0.834699i	$-0.685642\pi$
$709$	− 5.94975e8i	− 1.66940i	0.550706	−	0.834699i	$-0.314358\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 3.92029e8i	− 1.08156i
$714$	0	0
$715$	− 1.88938e8i	− 0.516894i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 6.96990e8i	− 1.87517i	−0.347760	−	0.937583i	$-0.613058\pi$
$719$	− 6.96990e8i	− 1.87517i	0.347760	−	0.937583i	$-0.386942\pi$
$720$	0	0
$721$	5.82336e6	0.0155370
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.63729e8	0.692059
$726$	0	0
$727$	7.97034e7	0.207431	0.103716	−	0.994607i	$-0.466927\pi$
$727$	7.97034e7	0.207431	0.103716	+	0.994607i	$0.466927\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.64455e8	0.421013
$732$	0	0
$733$	− 5.90402e8i	− 1.49912i	−0.661937	−	0.749559i	$-0.730266\pi$
$733$	− 5.90402e8i	− 1.49912i	0.661937	−	0.749559i	$-0.269734\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.31475e7i	0.157744i
$738$	0	0
$739$	− 5.48328e7i	− 0.135865i	−0.997690	−	0.0679324i	$-0.978360\pi$
$739$	− 5.48328e7i	− 0.135865i	0.997690	−	0.0679324i	$-0.0216402\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 2.92105e8i	− 0.712153i	−0.934457	−	0.356076i	$-0.884114\pi$
$743$	− 2.92105e8i	− 0.712153i	0.934457	−	0.356076i	$-0.115886\pi$
$744$	0	0
$745$	3.92624e8	0.949529
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.29075e6	−0.0221108
$750$	0	0
$751$	4.91813e8	1.16113	0.580564	−	0.814215i	$-0.302832\pi$
$751$	4.91813e8	1.16113	0.580564	+	0.814215i	$0.302832\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.08939e9	−2.53129
$756$	0	0
$757$	1.26268e8i	0.291076i	0.989353	+	0.145538i	$0.0464912\pi$
$757$	1.26268e8i	0.291076i	−0.989353	+	0.145538i	$0.953509\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.59472e8i	1.26947i	0.772728	+	0.634737i	$0.218892\pi$
$761$	5.59472e8i	1.26947i	−0.772728	+	0.634737i	$0.781108\pi$
$762$	0	0
$763$	− 1.99059e7i	− 0.0448135i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.24326e6i	0.0182689i
$768$	0	0
$769$	7.51300e8	1.65209	0.826046	−	0.563603i	$-0.190585\pi$
$769$	7.51300e8	1.65209	0.826046	+	0.563603i	$0.190585\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−7.22477e8	−1.56417	−0.782087	−	0.623169i	$-0.785845\pi$
$773$	−7.22477e8	−1.56417	−0.782087	+	0.623169i	$0.785845\pi$
$774$	0	0
$775$	−5.07102e8	−1.08941
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.26782e8	0.268191
$780$	0	0
$781$	− 1.56055e8i	− 0.327586i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.57931e8i	0.946653i
$786$	0	0
$787$	4.69415e8i	0.963014i	0.876442	+	0.481507i	$0.159910\pi$
$787$	4.69415e8i	0.963014i	−0.876442	+	0.481507i	$0.840090\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.61288e7i	0.0730001i
$792$	0	0
$793$	−1.31094e8	−0.262884
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.15029e8	−1.01732	−0.508659	−	0.860968i	$-0.669859\pi$
$797$	−5.15029e8	−1.01732	−0.508659	+	0.860968i	$0.669859\pi$
$798$	0	0
$799$	3.21258e8	0.629816
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8.09588e8	−1.56357
$804$	0	0
$805$	2.36372e7i	0.0453116i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.78752e8i	0.715335i	0.933849	+	0.357668i	$0.116428\pi$
$809$	3.78752e8i	0.715335i	−0.933849	+	0.357668i	$0.883572\pi$
$810$	0	0
$811$	− 8.01790e8i	− 1.50314i	−0.659656	−	0.751568i	$-0.729298\pi$
$811$	− 8.01790e8i	− 1.50314i	0.659656	−	0.751568i	$-0.270702\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.58852e8i	0.293440i
$816$	0	0
$817$	3.50250e8	0.642261
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.04505e9	1.88845	0.944226	−	0.329298i	$-0.106812\pi$
$821$	1.04505e9	1.88845	0.944226	+	0.329298i	$0.106812\pi$
$822$	0	0
$823$	5.24543e7	0.0940982	0.0470491	−	0.998893i	$-0.485018\pi$
$823$	5.24543e7	0.0940982	0.0470491	+	0.998893i	$0.485018\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.55035e8	0.627703	0.313851	−	0.949472i	$-0.398381\pi$
$827$	3.55035e8	0.627703	0.313851	+	0.949472i	$0.398381\pi$
$828$	0	0
$829$	− 3.06153e8i	− 0.537372i	−0.963228	−	0.268686i	$-0.913411\pi$
$829$	− 3.06153e8i	− 0.537372i	0.963228	−	0.268686i	$-0.0865894\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.48077e8i	0.602200i
$834$	0	0
$835$	1.46573e9i	2.51765i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.16944e8i	0.875301i	0.899145	+	0.437650i	$0.144189\pi$
$839$	5.16944e8i	0.875301i	−0.899145	+	0.437650i	$0.855811\pi$
$840$	0	0
$841$	−1.78943e8	−0.300834
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.21899e8	1.19648
$846$	0	0
$847$	6.72989e6	0.0110754
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.92663e8	−1.44844
$852$	0	0
$853$	7.67664e8i	1.23687i	0.785836	+	0.618435i	$0.212233\pi$
$853$	7.67664e8i	1.23687i	−0.785836	+	0.618435i	$0.787767\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.58878e8i	0.729045i	0.931195	+	0.364523i	$0.118768\pi$
$857$	4.58878e8i	0.729045i	−0.931195	+	0.364523i	$0.881232\pi$
$858$	0	0
$859$	7.79347e8i	1.22957i	0.788697	+	0.614783i	$0.210756\pi$
$859$	7.79347e8i	1.22957i	−0.788697	+	0.614783i	$0.789244\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1.69372e8i	− 0.263518i	−0.991282	−	0.131759i	$-0.957937\pi$
$863$	− 1.69372e8i	− 0.263518i	0.991282	−	0.131759i	$-0.0420625\pi$
$864$	0	0
$865$	2.54963e8	0.393940
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7.19224e8	−1.09599
$870$	0	0
$871$	3.13428e7	0.0474333
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.36650e6	−0.00950335
$876$	0	0
$877$	6.85516e8i	1.01629i	0.861271	+	0.508146i	$0.169669\pi$
$877$	6.85516e8i	1.01629i	−0.861271	+	0.508146i	$0.830331\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8.99510e7i	0.131546i	0.997835	+	0.0657731i	$0.0209513\pi$
$881$	8.99510e7i	0.131546i	−0.997835	+	0.0657731i	$0.979049\pi$
$882$	0	0
$883$	7.65510e8i	1.11191i	0.831213	+	0.555954i	$0.187647\pi$
$883$	7.65510e8i	1.11191i	−0.831213	+	0.555954i	$0.812353\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 6.43357e8i	− 0.921894i	−0.887428	−	0.460947i	$-0.847510\pi$
$887$	− 6.43357e8i	− 0.921894i	0.887428	−	0.460947i	$-0.152490\pi$
$888$	0	0
$889$	4.38858e7	0.0624625
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.84202e8	0.960793
$894$	0	0
$895$	−1.81489e9	−2.53153
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.99662e8	−1.10059
$900$	0	0
$901$	− 4.26023e8i	− 0.582450i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.09119e9i	1.47216i
$906$	0	0
$907$	− 2.38011e8i	− 0.318988i	−0.987199	−	0.159494i	$-0.949014\pi$
$907$	− 2.38011e8i	− 0.318988i	0.987199	−	0.159494i	$-0.0509863\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.29557e9i	1.71358i	0.515664	+	0.856791i	$0.327545\pi$
$911$	1.29557e9i	1.71358i	−0.515664	+	0.856791i	$0.672455\pi$
$912$	0	0
$913$	7.28584e8	0.957343
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.85173e6	−0.00758885
$918$	0	0
$919$	−1.83739e8	−0.236731	−0.118366	−	0.992970i	$-0.537766\pi$
$919$	−1.83739e8	−0.236731	−0.118366	+	0.992970i	$0.537766\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7.74568e7	−0.0985042
$924$	0	0
$925$	1.15469e9i	1.45895i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.72117e8i	1.08775i	0.839167	+	0.543873i	$0.183043\pi$
$929$	8.72117e8i	1.08775i	−0.839167	+	0.543873i	$0.816957\pi$
$930$	0	0
$931$	7.41319e8i	0.918663i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.51638e8i	0.919547i
$936$	0	0
$937$	7.39899e8	0.899402	0.449701	−	0.893179i	$-0.351531\pi$
$937$	7.39899e8	0.899402	0.449701	+	0.893179i	$0.351531\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5.86767e8	−0.704201	−0.352101	−	0.935962i	$-0.614532\pi$
$941$	−5.86767e8	−0.704201	−0.352101	+	0.935962i	$0.614532\pi$
$942$	0	0
$943$	−2.00822e8	−0.239484
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.35408e9	−1.59439	−0.797195	−	0.603722i	$-0.793684\pi$
$947$	−1.35408e9	−1.59439	−0.797195	+	0.603722i	$0.793684\pi$
$948$	0	0
$949$	4.01833e8i	0.470161i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8.06231e8i	0.931495i	0.884918	+	0.465748i	$0.154215\pi$
$953$	8.06231e8i	0.931495i	−0.884918	+	0.465748i	$0.845785\pi$
$954$	0	0
$955$	− 1.27457e9i	− 1.46336i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.87001e7i	0.0325408i
$960$	0	0
$961$	6.50102e8	0.732506
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.48115e9	1.64822
$966$	0	0
$967$	6.61478e8	0.731536	0.365768	−	0.930706i	$-0.380806\pi$
$967$	6.61478e8	0.731536	0.365768	+	0.930706i	$0.380806\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.01203e8	0.438234	0.219117	−	0.975699i	$-0.429682\pi$
$971$	4.01203e8	0.438234	0.219117	+	0.975699i	$0.429682\pi$
$972$	0	0
$973$	2.80423e7i	0.0304422i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 7.93550e8i	− 0.850924i	−0.904976	−	0.425462i	$-0.860112\pi$
$977$	− 7.93550e8i	− 0.850924i	0.904976	−	0.425462i	$-0.139888\pi$
$978$	0	0
$979$	4.65438e8i	0.496037i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 6.47001e8i	− 0.681153i	−0.940217	−	0.340576i	$-0.889378\pi$
$983$	− 6.47001e8i	− 0.681153i	0.940217	−	0.340576i	$-0.110622\pi$
$984$	0	0
$985$	6.47690e8	0.677733
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.54796e8	−0.573514
$990$	0	0
$991$	2.18163e8	0.224161	0.112081	−	0.993699i	$-0.464248\pi$
$991$	2.18163e8	0.224161	0.112081	+	0.993699i	$0.464248\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.96229e9	−1.99203
$996$	0	0
$997$	− 1.47959e9i	− 1.49299i	−0.665393	−	0.746494i	$-0.731736\pi$
$997$	− 1.47959e9i	− 1.49299i	0.665393	−	0.746494i	$-0.268264\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.7.h.a.161.16	yes	16
3.2	odd	2	inner	576.7.h.a.161.4	yes	16
4.3	odd	2	inner	576.7.h.a.161.14	yes	16
8.3	odd	2	inner	576.7.h.a.161.1	✓	16
8.5	even	2	inner	576.7.h.a.161.3	yes	16
12.11	even	2	inner	576.7.h.a.161.2	yes	16
24.5	odd	2	inner	576.7.h.a.161.15	yes	16
24.11	even	2	inner	576.7.h.a.161.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.7.h.a.161.1	✓	16	8.3	odd	2	inner
576.7.h.a.161.2	yes	16	12.11	even	2	inner
576.7.h.a.161.3	yes	16	8.5	even	2	inner
576.7.h.a.161.4	yes	16	3.2	odd	2	inner
576.7.h.a.161.13	yes	16	24.11	even	2	inner
576.7.h.a.161.14	yes	16	4.3	odd	2	inner
576.7.h.a.161.15	yes	16	24.5	odd	2	inner
576.7.h.a.161.16	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+168.989 q^{5} +13.9908 q^{7} +O(q^{10})\)	\(q+168.989 q^{5} +13.9908 q^{7} -1500.86 q^{11} +744.941i q^{13} -2963.54i q^{17} -6311.61i q^{19} +9997.60i q^{23} +12932.2 q^{25} +20393.1 q^{29} -39212.3 q^{31} +2364.29 q^{35} +89287.8i q^{37} +20087.0i q^{41} +55492.9i q^{43} +108404. i q^{47} -117453. q^{49} +143755. q^{53} -253629. q^{55} +11065.7 q^{59} +175980. i q^{61} +125887. i q^{65} -42074.2i q^{67} +103977. i q^{71} +539416. q^{73} -20998.3 q^{77} +479208. q^{79} -485444. q^{83} -500805. i q^{85} -310114. i q^{89} +10422.3i q^{91} -1.06659e6i q^{95} -199839. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 19888 q^{25} + 388784 q^{49} + 465728 q^{73} + 6555136 q^{97}+O(q^{100}) \) 16 * q - 19888 * q^25 + 388784 * q^49 + 465728 * q^73 + 6555136 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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