LMFDB - Embedded newform 7803.2.a.ce.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7803,2,Mod(1,7803)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7803.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7803, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7803 = 3^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7803.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,0,0,24,0,0,-16,0,0,-16,0,0,-8,0,0,24,0,0,-8,0,0,-56,0,0,16, 0,0,0,0,0,-16,0,0,0,0,0,-72,0,0,-72,0,0,-8,0,0,-40,0,0,24,0,0,-32,0,0, -40,0,0,-72,0,0,-48,0,0,48,0,0,-32,0,0,24,0,0,-88] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(73)] == traces)

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

Self dual:	yes
Analytic conductor:	$62.3072686972$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 459)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	7803.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61878 q^{2} +0.620447 q^{4} -0.812904 q^{5} -2.09960 q^{7} +2.23319 q^{8} +1.31591 q^{10} -1.37936 q^{11} +5.60334 q^{13} +3.39879 q^{14} -4.85594 q^{16} +0.00715563 q^{19} -0.504364 q^{20} +2.23288 q^{22} +6.20479 q^{23} -4.33919 q^{25} -9.07057 q^{26} -1.30269 q^{28} +0.436138 q^{29} -4.24842 q^{31} +3.39431 q^{32} +1.70677 q^{35} +0.923561 q^{37} -0.0115834 q^{38} -1.81537 q^{40} -6.05356 q^{41} -1.17902 q^{43} -0.855820 q^{44} -10.0442 q^{46} -5.43379 q^{47} -2.59168 q^{49} +7.02419 q^{50} +3.47658 q^{52} +12.1209 q^{53} +1.12129 q^{55} -4.68881 q^{56} -0.706011 q^{58} -7.91642 q^{59} -9.76442 q^{61} +6.87726 q^{62} +4.21724 q^{64} -4.55498 q^{65} +10.4241 q^{67} -2.76289 q^{70} +10.9333 q^{71} -9.64990 q^{73} -1.49504 q^{74} +0.00443969 q^{76} +2.89611 q^{77} -5.43506 q^{79} +3.94741 q^{80} +9.79938 q^{82} +1.28033 q^{83} +1.90858 q^{86} -3.08038 q^{88} +0.603172 q^{89} -11.7648 q^{91} +3.84974 q^{92} +8.79610 q^{94} -0.00581684 q^{95} +5.30194 q^{97} +4.19536 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{4} - 16 q^{7} - 16 q^{10} - 8 q^{13} + 24 q^{16} - 8 q^{19} - 56 q^{22} + 16 q^{25} - 16 q^{31} - 72 q^{37} - 72 q^{40} - 8 q^{43} - 40 q^{46} + 24 q^{49} - 32 q^{52} - 40 q^{55} - 72 q^{58}+ \cdots - 112 q^{97}+O(q^{100}) $ 24 * q + 24 * q^4 - 16 * q^7 - 16 * q^10 - 8 * q^13 + 24 * q^16 - 8 * q^19 - 56 * q^22 + 16 * q^25 - 16 * q^31 - 72 * q^37 - 72 * q^40 - 8 * q^43 - 40 * q^46 + 24 * q^49 - 32 * q^52 - 40 * q^55 - 72 * q^58 - 48 * q^61 + 48 * q^64 - 32 * q^67 + 24 * q^70 - 88 * q^73 + 8 * q^76 - 8 * q^82 - 144 * q^88 - 56 * q^91 + 72 * q^94 - 112 * q^97

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 $

$2$	−1.61878	−1.14465	−0.572325	−	0.820027i	$-0.693958\pi$
$2$	−1.61878	−1.14465	−0.572325	+	0.820027i	$0.693958\pi$
$3$	0	0
$3$	0	0
$4$	0.620447	0.310224
$5$	−0.812904	−0.363542	−0.181771	−	0.983341i	$-0.558183\pi$
$5$	−0.812904	−0.363542	−0.181771	+	0.983341i	$0.558183\pi$
$6$	0	0
$7$	−2.09960	−0.793574	−0.396787	−	0.917911i	$-0.629875\pi$
$7$	−2.09960	−0.793574	−0.396787	+	0.917911i	$0.629875\pi$
$8$	2.23319	0.789553
$9$	0	0
$10$	1.31591	0.416128
$11$	−1.37936	−0.415893	−0.207946	−	0.978140i	$-0.566678\pi$
$11$	−1.37936	−0.415893	−0.207946	+	0.978140i	$0.566678\pi$
$12$	0	0
$13$	5.60334	1.55409	0.777043	−	0.629447i	$-0.216718\pi$
$13$	5.60334	1.55409	0.777043	+	0.629447i	$0.216718\pi$
$14$	3.39879	0.908365
$15$	0	0
$16$	−4.85594	−1.21398
$17$	0	0
$17$	0	0
$18$	0	0
$19$	0.00715563	0.00164162	0.000820808	−	1.00000i	$-0.499739\pi$
$19$	0.00715563	0.00164162	0.000820808		1.00000i	$0.499739\pi$
$20$	−0.504364	−0.112779
$21$	0	0
$22$	2.23288	0.476052
$23$	6.20479	1.29379	0.646894	−	0.762580i	$-0.276068\pi$
$23$	6.20479	1.29379	0.646894	+	0.762580i	$0.276068\pi$
$24$	0	0
$25$	−4.33919	−0.867837
$26$	−9.07057	−1.77889
$27$	0	0
$28$	−1.30269	−0.246185
$29$	0.436138	0.0809888	0.0404944	−	0.999180i	$-0.487107\pi$
$29$	0.436138	0.0809888	0.0404944	+	0.999180i	$0.487107\pi$
$30$	0	0
$31$	−4.24842	−0.763039	−0.381520	−	0.924361i	$-0.624599\pi$
$31$	−4.24842	−0.763039	−0.381520	+	0.924361i	$0.624599\pi$
$32$	3.39431	0.600035
$33$	0	0
$34$	0	0
$35$	1.70677	0.288497
$36$	0	0
$37$	0.923561	0.151833	0.0759163	−	0.997114i	$-0.475812\pi$
$37$	0.923561	0.151833	0.0759163	+	0.997114i	$0.475812\pi$
$38$	−0.0115834	−0.00187907
$39$	0	0
$40$	−1.81537	−0.287035
$41$	−6.05356	−0.945407	−0.472704	−	0.881221i	$-0.656722\pi$
$41$	−6.05356	−0.945407	−0.472704	+	0.881221i	$0.656722\pi$
$42$	0	0
$43$	−1.17902	−0.179799	−0.0898996	−	0.995951i	$-0.528655\pi$
$43$	−1.17902	−0.179799	−0.0898996	+	0.995951i	$0.528655\pi$
$44$	−0.855820	−0.129020
$45$	0	0
$46$	−10.0442	−1.48093
$47$	−5.43379	−0.792599	−0.396300	−	0.918121i	$-0.629706\pi$
$47$	−5.43379	−0.792599	−0.396300	+	0.918121i	$0.629706\pi$
$48$	0	0
$49$	−2.59168	−0.370240
$50$	7.02419	0.993370
$51$	0	0
$52$	3.47658	0.482114
$53$	12.1209	1.66494	0.832468	−	0.554073i	$-0.186927\pi$
$53$	12.1209	1.66494	0.832468	+	0.554073i	$0.186927\pi$
$54$	0	0
$55$	1.12129	0.151194
$56$	−4.68881	−0.626569
$57$	0	0
$58$	−0.706011	−0.0927038
$59$	−7.91642	−1.03063	−0.515315	−	0.857001i	$-0.672325\pi$
$59$	−7.91642	−1.03063	−0.515315	+	0.857001i	$0.672325\pi$
$60$	0	0
$61$	−9.76442	−1.25021	−0.625103	−	0.780542i	$-0.714943\pi$
$61$	−9.76442	−1.25021	−0.625103	+	0.780542i	$0.714943\pi$
$62$	6.87726	0.873413
$63$	0	0
$64$	4.21724	0.527155
$65$	−4.55498	−0.564975
$66$	0	0
$67$	10.4241	1.27351	0.636753	−	0.771068i	$-0.280277\pi$
$67$	10.4241	1.27351	0.636753	+	0.771068i	$0.280277\pi$
$68$	0	0
$69$	0	0
$70$	−2.76289	−0.330228
$71$	10.9333	1.29754	0.648769	−	0.760985i	$-0.275284\pi$
$71$	10.9333	1.29754	0.648769	+	0.760985i	$0.275284\pi$
$72$	0	0
$73$	−9.64990	−1.12943	−0.564717	−	0.825284i	$-0.691015\pi$
$73$	−9.64990	−1.12943	−0.564717	+	0.825284i	$0.691015\pi$
$74$	−1.49504	−0.173795
$75$	0	0
$76$	0.00443969	0.000509268 0
$77$	2.89611	0.330042
$78$	0	0
$79$	−5.43506	−0.611491	−0.305746	−	0.952113i	$-0.598906\pi$
$79$	−5.43506	−0.611491	−0.305746	+	0.952113i	$0.598906\pi$
$80$	3.94741	0.441334
$81$	0	0
$82$	9.79938	1.08216
$83$	1.28033	0.140535	0.0702675	−	0.997528i	$-0.477615\pi$
$83$	1.28033	0.140535	0.0702675	+	0.997528i	$0.477615\pi$
$84$	0	0
$85$	0	0
$86$	1.90858	0.205807
$87$	0	0
$88$	−3.08038	−0.328369
$89$	0.603172	0.0639361	0.0319681	−	0.999489i	$-0.489823\pi$
$89$	0.603172	0.0639361	0.0319681	+	0.999489i	$0.489823\pi$
$90$	0	0
$91$	−11.7648	−1.23328
$92$	3.84974	0.401364
$93$	0	0
$94$	8.79610	0.907249
$95$	−0.00581684	−0.000596796 0
$96$	0	0
$97$	5.30194	0.538331	0.269165	−	0.963094i	$-0.413252\pi$
$97$	5.30194	0.538331	0.269165	+	0.963094i	$0.413252\pi$
$98$	4.19536	0.423795
$99$	0	0
$100$	−2.69224	−0.269224
$101$	18.8690	1.87753	0.938767	−	0.344552i	$-0.111969\pi$
$101$	18.8690	1.87753	0.938767	+	0.344552i	$0.111969\pi$
$102$	0	0
$103$	15.7661	1.55348	0.776740	−	0.629821i	$-0.216872\pi$
$103$	15.7661	1.55348	0.776740	+	0.629821i	$0.216872\pi$
$104$	12.5133	1.22703
$105$	0	0
$106$	−19.6211	−1.90577
$107$	−2.14449	−0.207315	−0.103658	−	0.994613i	$-0.533055\pi$
$107$	−2.14449	−0.207315	−0.103658	+	0.994613i	$0.533055\pi$
$108$	0	0
$109$	−2.38886	−0.228811	−0.114405	−	0.993434i	$-0.536496\pi$
$109$	−2.38886	−0.228811	−0.114405	+	0.993434i	$0.536496\pi$
$110$	−1.81512	−0.173065
$111$	0	0
$112$	10.1955	0.963387
$113$	−11.8064	−1.11066	−0.555328	−	0.831631i	$-0.687407\pi$
$113$	−11.8064	−1.11066	−0.555328	+	0.831631i	$0.687407\pi$
$114$	0	0
$115$	−5.04390	−0.470346
$116$	0.270601	0.0251246
$117$	0	0
$118$	12.8149	1.17971
$119$	0	0
$120$	0	0
$121$	−9.09736	−0.827033
$122$	15.8064	1.43105
$123$	0	0
$124$	−2.63592	−0.236713
$125$	7.59186	0.679037
$126$	0	0
$127$	0.556912	0.0494179	0.0247090	−	0.999695i	$-0.492134\pi$
$127$	0.556912	0.0494179	0.0247090	+	0.999695i	$0.492134\pi$
$128$	−13.6154	−1.20344
$129$	0	0
$130$	7.37350	0.646699
$131$	14.8327	1.29594	0.647969	−	0.761667i	$-0.275619\pi$
$131$	14.8327	1.29594	0.647969	+	0.761667i	$0.275619\pi$
$132$	0	0
$133$	−0.0150240	−0.00130274
$134$	−16.8743	−1.45772
$135$	0	0
$136$	0	0
$137$	−9.06382	−0.774374	−0.387187	−	0.922001i	$-0.626553\pi$
$137$	−9.06382	−0.774374	−0.387187	+	0.922001i	$0.626553\pi$
$138$	0	0
$139$	−13.7396	−1.16537	−0.582687	−	0.812697i	$-0.697999\pi$
$139$	−13.7396	−1.16537	−0.582687	+	0.812697i	$0.697999\pi$
$140$	1.05896	0.0894986
$141$	0	0
$142$	−17.6985	−1.48523
$143$	−7.72903	−0.646334
$144$	0	0
$145$	−0.354538	−0.0294428
$146$	15.6211	1.29281
$147$	0	0
$148$	0.573021	0.0471020
$149$	10.4965	0.859904	0.429952	−	0.902852i	$-0.358531\pi$
$149$	10.4965	0.859904	0.429952	+	0.902852i	$0.358531\pi$
$150$	0	0
$151$	−7.94556	−0.646600	−0.323300	−	0.946296i	$-0.604792\pi$
$151$	−7.94556	−0.646600	−0.323300	+	0.946296i	$0.604792\pi$
$152$	0.0159799	0.00129614
$153$	0	0
$154$	−4.68816	−0.377782
$155$	3.45356	0.277397
$156$	0	0
$157$	7.65841	0.611208	0.305604	−	0.952159i	$-0.401142\pi$
$157$	7.65841	0.611208	0.305604	+	0.952159i	$0.401142\pi$
$158$	8.79816	0.699944
$159$	0	0
$160$	−2.75925	−0.218138
$161$	−13.0276	−1.02672
$162$	0	0
$163$	7.88871	0.617891	0.308946	−	0.951080i	$-0.400024\pi$
$163$	7.88871	0.617891	0.308946	+	0.951080i	$0.400024\pi$
$164$	−3.75591	−0.293288
$165$	0	0
$166$	−2.07258	−0.160863
$167$	20.9901	1.62426	0.812131	−	0.583475i	$-0.198307\pi$
$167$	20.9901	1.62426	0.812131	+	0.583475i	$0.198307\pi$
$168$	0	0
$169$	18.3974	1.41519
$170$	0	0
$171$	0	0
$172$	−0.731521	−0.0557779
$173$	22.0238	1.67444	0.837219	−	0.546868i	$-0.184180\pi$
$173$	22.0238	1.67444	0.837219	+	0.546868i	$0.184180\pi$
$174$	0	0
$175$	9.11056	0.688693
$176$	6.69809	0.504888
$177$	0	0
$178$	−0.976403	−0.0731845
$179$	−10.8811	−0.813292	−0.406646	−	0.913586i	$-0.633302\pi$
$179$	−10.8811	−0.813292	−0.406646	+	0.913586i	$0.633302\pi$
$180$	0	0
$181$	−9.07488	−0.674531	−0.337265	−	0.941410i	$-0.609502\pi$
$181$	−9.07488	−0.674531	−0.337265	+	0.941410i	$0.609502\pi$
$182$	19.0446	1.41168
$183$	0	0
$184$	13.8565	1.02151
$185$	−0.750767	−0.0551975
$186$	0	0
$187$	0	0
$188$	−3.37138	−0.245883
$189$	0	0
$190$	0.00941619	0.000683122 0
$191$	8.60933	0.622949	0.311475	−	0.950254i	$-0.399177\pi$
$191$	8.60933	0.622949	0.311475	+	0.950254i	$0.399177\pi$
$192$	0	0
$193$	−8.93418	−0.643096	−0.321548	−	0.946893i	$-0.604203\pi$
$193$	−8.93418	−0.643096	−0.321548	+	0.946893i	$0.604203\pi$
$194$	−8.58268	−0.616200
$195$	0	0
$196$	−1.60800	−0.114857
$197$	12.2661	0.873924	0.436962	−	0.899480i	$-0.356054\pi$
$197$	12.2661	0.873924	0.436962	+	0.899480i	$0.356054\pi$
$198$	0	0
$199$	−22.5931	−1.60159	−0.800793	−	0.598942i	$-0.795588\pi$
$199$	−22.5931	−1.60159	−0.800793	+	0.598942i	$0.795588\pi$
$200$	−9.69024	−0.685203
$201$	0	0
$202$	−30.5447	−2.14912
$203$	−0.915715	−0.0642706
$204$	0	0
$205$	4.92096	0.343695
$206$	−25.5218	−1.77819
$207$	0	0
$208$	−27.2095	−1.88664
$209$	−0.00987020	−0.000682736 0
$210$	0	0
$211$	−11.4966	−0.791460	−0.395730	−	0.918367i	$-0.629508\pi$
$211$	−11.4966	−0.791460	−0.395730	+	0.918367i	$0.629508\pi$
$212$	7.52038	0.516502
$213$	0	0
$214$	3.47145	0.237303
$215$	0.958432	0.0653645
$216$	0	0
$217$	8.91999	0.605528
$218$	3.86703	0.261908
$219$	0	0
$220$	0.695700	0.0469041
$221$	0	0
$222$	0	0
$223$	16.0279	1.07331	0.536655	−	0.843802i	$-0.319688\pi$
$223$	16.0279	1.07331	0.536655	+	0.843802i	$0.319688\pi$
$224$	−7.12669	−0.476172
$225$	0	0
$226$	19.1120	1.27131
$227$	−5.78973	−0.384278	−0.192139	−	0.981368i	$-0.561542\pi$
$227$	−5.78973	−0.384278	−0.192139	+	0.981368i	$0.561542\pi$
$228$	0	0
$229$	−11.7447	−0.776110	−0.388055	−	0.921636i	$-0.626853\pi$
$229$	−11.7447	−0.776110	−0.388055	+	0.921636i	$0.626853\pi$
$230$	8.16496	0.538382
$231$	0	0
$232$	0.973980	0.0639449
$233$	−14.4563	−0.947066	−0.473533	−	0.880776i	$-0.657022\pi$
$233$	−14.4563	−0.947066	−0.473533	+	0.880776i	$0.657022\pi$
$234$	0	0
$235$	4.41715	0.288143
$236$	−4.91172	−0.319726
$237$	0	0
$238$	0	0
$239$	29.6210	1.91602	0.958012	−	0.286727i	$-0.0925673\pi$
$239$	29.6210	1.91602	0.958012	+	0.286727i	$0.0925673\pi$
$240$	0	0
$241$	−27.2211	−1.75346	−0.876732	−	0.480979i	$-0.840281\pi$
$241$	−27.2211	−1.75346	−0.876732	+	0.480979i	$0.840281\pi$
$242$	14.7266	0.946663
$243$	0	0
$244$	−6.05831	−0.387843
$245$	2.10679	0.134598
$246$	0	0
$247$	0.0400955	0.00255121
$248$	−9.48754	−0.602460
$249$	0	0
$250$	−12.2896	−0.777259
$251$	−22.5761	−1.42499	−0.712495	−	0.701677i	$-0.752435\pi$
$251$	−22.5761	−1.42499	−0.712495	+	0.701677i	$0.752435\pi$
$252$	0	0
$253$	−8.55865	−0.538077
$254$	−0.901517	−0.0565662
$255$	0	0
$256$	13.6059	0.850366
$257$	29.8412	1.86144	0.930721	−	0.365731i	$-0.119181\pi$
$257$	29.8412	1.86144	0.930721	+	0.365731i	$0.119181\pi$
$258$	0	0
$259$	−1.93911	−0.120490
$260$	−2.82612	−0.175269
$261$	0	0
$262$	−24.0108	−1.48340
$263$	−28.8696	−1.78018	−0.890088	−	0.455788i	$-0.849357\pi$
$263$	−28.8696	−1.78018	−0.890088	+	0.455788i	$0.849357\pi$
$264$	0	0
$265$	−9.85314	−0.605273
$266$	0.0243205	0.00149119
$267$	0	0
$268$	6.46760	0.395071
$269$	−8.75721	−0.533937	−0.266968	−	0.963705i	$-0.586022\pi$
$269$	−8.75721	−0.533937	−0.266968	+	0.963705i	$0.586022\pi$
$270$	0	0
$271$	−22.6956	−1.37866	−0.689329	−	0.724448i	$-0.742095\pi$
$271$	−22.6956	−1.37866	−0.689329	+	0.724448i	$0.742095\pi$
$272$	0	0
$273$	0	0
$274$	14.6723	0.886387
$275$	5.98530	0.360927
$276$	0	0
$277$	13.3862	0.804297	0.402148	−	0.915575i	$-0.368264\pi$
$277$	13.3862	0.804297	0.402148	+	0.915575i	$0.368264\pi$
$278$	22.2413	1.33395
$279$	0	0
$280$	3.81155	0.227784
$281$	−21.9082	−1.30693	−0.653466	−	0.756956i	$-0.726686\pi$
$281$	−21.9082	−1.30693	−0.653466	+	0.756956i	$0.726686\pi$
$282$	0	0
$283$	8.54095	0.507707	0.253853	−	0.967243i	$-0.418302\pi$
$283$	8.54095	0.507707	0.253853	+	0.967243i	$0.418302\pi$
$284$	6.78351	0.402527
$285$	0	0
$286$	12.5116	0.739826
$287$	12.7101	0.750251
$288$	0	0
$289$	0	0
$290$	0.573919	0.0337017
$291$	0	0
$292$	−5.98725	−0.350377
$293$	−7.30183	−0.426577	−0.213289	−	0.976989i	$-0.568417\pi$
$293$	−7.30183	−0.426577	−0.213289	+	0.976989i	$0.568417\pi$
$294$	0	0
$295$	6.43529	0.374677
$296$	2.06249	0.119880
$297$	0	0
$298$	−16.9915	−0.984289
$299$	34.7676	2.01066
$300$	0	0
$301$	2.47547	0.142684
$302$	12.8621	0.740131
$303$	0	0
$304$	−0.0347473	−0.00199290
$305$	7.93754	0.454502
$306$	0	0
$307$	11.4855	0.655512	0.327756	−	0.944762i	$-0.393708\pi$
$307$	11.4855	0.655512	0.327756	+	0.944762i	$0.393708\pi$
$308$	1.79688	0.102387
$309$	0	0
$310$	−5.59055	−0.317522
$311$	−22.9552	−1.30167	−0.650834	−	0.759220i	$-0.725580\pi$
$311$	−22.9552	−1.30167	−0.650834	+	0.759220i	$0.725580\pi$
$312$	0	0
$313$	−34.5357	−1.95207	−0.976036	−	0.217610i	$-0.930174\pi$
$313$	−34.5357	−1.95207	−0.976036	+	0.217610i	$0.930174\pi$
$314$	−12.3973	−0.699619
$315$	0	0
$316$	−3.37216	−0.189699
$317$	32.1677	1.80672	0.903360	−	0.428883i	$-0.141093\pi$
$317$	32.1677	1.80672	0.903360	+	0.428883i	$0.141093\pi$
$318$	0	0
$319$	−0.601592	−0.0336827
$320$	−3.42821	−0.191643
$321$	0	0
$322$	21.0888	1.17523
$323$	0	0
$324$	0	0
$325$	−24.3139	−1.34869
$326$	−12.7701	−0.707269
$327$	0	0
$328$	−13.5188	−0.746449
$329$	11.4088	0.628986
$330$	0	0
$331$	17.5835	0.966476	0.483238	−	0.875489i	$-0.339461\pi$
$331$	17.5835	0.966476	0.483238	+	0.875489i	$0.339461\pi$
$332$	0.794380	0.0435973
$333$	0	0
$334$	−33.9783	−1.85921
$335$	−8.47379	−0.462973
$336$	0	0
$337$	−26.4074	−1.43850	−0.719252	−	0.694749i	$-0.755515\pi$
$337$	−26.4074	−1.43850	−0.719252	+	0.694749i	$0.755515\pi$
$338$	−29.7814	−1.61989
$339$	0	0
$340$	0	0
$341$	5.86011	0.317343
$342$	0	0
$343$	20.1387	1.08739
$344$	−2.63298	−0.141961
$345$	0	0
$346$	−35.6517	−1.91664
$347$	−21.7865	−1.16956	−0.584781	−	0.811191i	$-0.698820\pi$
$347$	−21.7865	−1.16956	−0.584781	+	0.811191i	$0.698820\pi$
$348$	0	0
$349$	−19.6719	−1.05301	−0.526505	−	0.850172i	$-0.676498\pi$
$349$	−19.6719	−1.05301	−0.526505	+	0.850172i	$0.676498\pi$
$350$	−14.7480	−0.788313
$351$	0	0
$352$	−4.68198	−0.249550
$353$	−0.396746	−0.0211166	−0.0105583	−	0.999944i	$-0.503361\pi$
$353$	−0.396746	−0.0211166	−0.0105583	+	0.999944i	$0.503361\pi$
$354$	0	0
$355$	−8.88769	−0.471709
$356$	0.374236	0.0198345
$357$	0	0
$358$	17.6141	0.930935
$359$	19.0564	1.00576	0.502878	−	0.864357i	$-0.332274\pi$
$359$	19.0564	1.00576	0.502878	+	0.864357i	$0.332274\pi$
$360$	0	0
$361$	−18.9999	−0.999997
$362$	14.6902	0.772101
$363$	0	0
$364$	−7.29942	−0.382593
$365$	7.84444	0.410597
$366$	0	0
$367$	32.0678	1.67393	0.836963	−	0.547260i	$-0.184329\pi$
$367$	32.0678	1.67393	0.836963	+	0.547260i	$0.184329\pi$
$368$	−30.1301	−1.57064
$369$	0	0
$370$	1.21533	0.0631818
$371$	−25.4491	−1.32125
$372$	0	0
$373$	−5.54108	−0.286906	−0.143453	−	0.989657i	$-0.545821\pi$
$373$	−5.54108	−0.286906	−0.143453	+	0.989657i	$0.545821\pi$
$374$	0	0
$375$	0	0
$376$	−12.1347	−0.625799
$377$	2.44383	0.125864
$378$	0	0
$379$	12.3097	0.632308	0.316154	−	0.948708i	$-0.397608\pi$
$379$	12.3097	0.632308	0.316154	+	0.948708i	$0.397608\pi$
$380$	−0.00360904	−0.000185140 0
$381$	0	0
$382$	−13.9366	−0.713059
$383$	8.72976	0.446070	0.223035	−	0.974810i	$-0.428404\pi$
$383$	8.72976	0.446070	0.223035	+	0.974810i	$0.428404\pi$
$384$	0	0
$385$	−2.35426	−0.119984
$386$	14.4625	0.736120
$387$	0	0
$388$	3.28958	0.167003
$389$	−0.863655	−0.0437891	−0.0218945	−	0.999760i	$-0.506970\pi$
$389$	−0.863655	−0.0437891	−0.0218945	+	0.999760i	$0.506970\pi$
$390$	0	0
$391$	0	0
$392$	−5.78772	−0.292324
$393$	0	0
$394$	−19.8561	−1.00034
$395$	4.41818	0.222303
$396$	0	0
$397$	−14.8100	−0.743292	−0.371646	−	0.928375i	$-0.621206\pi$
$397$	−14.8100	−0.743292	−0.371646	+	0.928375i	$0.621206\pi$
$398$	36.5733	1.83325
$399$	0	0
$400$	21.0708	1.05354
$401$	−25.6795	−1.28238	−0.641188	−	0.767384i	$-0.721558\pi$
$401$	−25.6795	−1.28238	−0.641188	+	0.767384i	$0.721558\pi$
$402$	0	0
$403$	−23.8054	−1.18583
$404$	11.7072	0.582455
$405$	0	0
$406$	1.48234	0.0735674
$407$	−1.27392	−0.0631461
$408$	0	0
$409$	33.0421	1.63383	0.816913	−	0.576761i	$-0.195684\pi$
$409$	33.0421	1.63383	0.816913	+	0.576761i	$0.195684\pi$
$410$	−7.96595	−0.393410
$411$	0	0
$412$	9.78203	0.481926
$413$	16.6213	0.817882
$414$	0	0
$415$	−1.04079	−0.0510903
$416$	19.0195	0.932507
$417$	0	0
$418$	0.0159777	0.000781494 0
$419$	−3.02327	−0.147697	−0.0738483	−	0.997269i	$-0.523528\pi$
$419$	−3.02327	−0.147697	−0.0738483	+	0.997269i	$0.523528\pi$
$420$	0	0
$421$	7.79933	0.380116	0.190058	−	0.981773i	$-0.439132\pi$
$421$	7.79933	0.380116	0.190058	+	0.981773i	$0.439132\pi$
$422$	18.6105	0.905945
$423$	0	0
$424$	27.0683	1.31455
$425$	0	0
$426$	0	0
$427$	20.5014	0.992131
$428$	−1.33054	−0.0643141
$429$	0	0
$430$	−1.55149	−0.0748195
$431$	−35.1509	−1.69316	−0.846579	−	0.532263i	$-0.821342\pi$
$431$	−35.1509	−1.69316	−0.846579	+	0.532263i	$0.821342\pi$
$432$	0	0
$433$	−8.45280	−0.406216	−0.203108	−	0.979156i	$-0.565104\pi$
$433$	−8.45280	−0.406216	−0.203108	+	0.979156i	$0.565104\pi$
$434$	−14.4395	−0.693118
$435$	0	0
$436$	−1.48216	−0.0709825
$437$	0.0443992	0.00212390
$438$	0	0
$439$	3.78223	0.180516	0.0902580	−	0.995918i	$-0.471231\pi$
$439$	3.78223	0.180516	0.0902580	+	0.995918i	$0.471231\pi$
$440$	2.50405	0.119376
$441$	0	0
$442$	0	0
$443$	−36.1780	−1.71887	−0.859433	−	0.511248i	$-0.829183\pi$
$443$	−36.1780	−1.71887	−0.859433	+	0.511248i	$0.829183\pi$
$444$	0	0
$445$	−0.490321	−0.0232435
$446$	−25.9457	−1.22856
$447$	0	0
$448$	−8.85451	−0.418336
$449$	15.1134	0.713243	0.356622	−	0.934249i	$-0.383929\pi$
$449$	15.1134	0.713243	0.356622	+	0.934249i	$0.383929\pi$
$450$	0	0
$451$	8.35004	0.393188
$452$	−7.32527	−0.344552
$453$	0	0
$454$	9.37230	0.439864
$455$	9.56363	0.448350
$456$	0	0
$457$	−34.7220	−1.62423	−0.812113	−	0.583501i	$-0.801682\pi$
$457$	−34.7220	−1.62423	−0.812113	+	0.583501i	$0.801682\pi$
$458$	19.0120	0.888374
$459$	0	0
$460$	−3.12947	−0.145912
$461$	−23.1121	−1.07644	−0.538218	−	0.842806i	$-0.680902\pi$
$461$	−23.1121	−1.07644	−0.538218	+	0.842806i	$0.680902\pi$
$462$	0	0
$463$	−17.1158	−0.795437	−0.397719	−	0.917507i	$-0.630198\pi$
$463$	−17.1158	−0.795437	−0.397719	+	0.917507i	$0.630198\pi$
$464$	−2.11786	−0.0983192
$465$	0	0
$466$	23.4016	1.08406
$467$	17.0663	0.789736	0.394868	−	0.918738i	$-0.370790\pi$
$467$	17.0663	0.789736	0.394868	+	0.918738i	$0.370790\pi$
$468$	0	0
$469$	−21.8864	−1.01062
$470$	−7.15039	−0.329823
$471$	0	0
$472$	−17.6789	−0.813737
$473$	1.62630	0.0747772
$474$	0	0
$475$	−0.0310496	−0.00142466
$476$	0	0
$477$	0	0
$478$	−47.9499	−2.19318
$479$	−24.3984	−1.11479	−0.557397	−	0.830246i	$-0.688200\pi$
$479$	−24.3984	−1.11479	−0.557397	+	0.830246i	$0.688200\pi$
$480$	0	0
$481$	5.17503	0.235961
$482$	44.0649	2.00710
$483$	0	0
$484$	−5.64443	−0.256565
$485$	−4.30997	−0.195706
$486$	0	0
$487$	−17.2386	−0.781154	−0.390577	−	0.920570i	$-0.627724\pi$
$487$	−17.2386	−0.781154	−0.390577	+	0.920570i	$0.627724\pi$
$488$	−21.8058	−0.987103
$489$	0	0
$490$	−3.41042	−0.154067
$491$	11.6234	0.524558	0.262279	−	0.964992i	$-0.415526\pi$
$491$	11.6234	0.524558	0.262279	+	0.964992i	$0.415526\pi$
$492$	0	0
$493$	0	0
$494$	−0.0649057	−0.00292025
$495$	0	0
$496$	20.6301	0.926318
$497$	−22.9555	−1.02969
$498$	0	0
$499$	19.9655	0.893778	0.446889	−	0.894589i	$-0.352532\pi$
$499$	19.9655	0.893778	0.446889	+	0.894589i	$0.352532\pi$
$500$	4.71035	0.210653
$501$	0	0
$502$	36.5457	1.63111
$503$	11.3550	0.506295	0.253148	−	0.967428i	$-0.418534\pi$
$503$	11.3550	0.506295	0.253148	+	0.967428i	$0.418534\pi$
$504$	0	0
$505$	−15.3387	−0.682562
$506$	13.8546	0.615910
$507$	0	0
$508$	0.345534	0.0153306
$509$	−9.82575	−0.435519	−0.217759	−	0.976002i	$-0.569875\pi$
$509$	−9.82575	−0.435519	−0.217759	+	0.976002i	$0.569875\pi$
$510$	0	0
$511$	20.2609	0.896290
$512$	5.20592	0.230071
$513$	0	0
$514$	−48.3063	−2.13070
$515$	−12.8163	−0.564755
$516$	0	0
$517$	7.49515	0.329636
$518$	3.13899	0.137919
$519$	0	0
$520$	−10.1721	−0.446078
$521$	−16.6105	−0.727717	−0.363859	−	0.931454i	$-0.618541\pi$
$521$	−16.6105	−0.727717	−0.363859	+	0.931454i	$0.618541\pi$
$522$	0	0
$523$	13.4702	0.589012	0.294506	−	0.955650i	$-0.404845\pi$
$523$	13.4702	0.589012	0.294506	+	0.955650i	$0.404845\pi$
$524$	9.20289	0.402030
$525$	0	0
$526$	46.7335	2.03768
$527$	0	0
$528$	0	0
$529$	15.4994	0.673888
$530$	15.9501	0.692826
$531$	0	0
$532$	−0.00932158	−0.000404142 0
$533$	−33.9202	−1.46925
$534$	0	0
$535$	1.74326	0.0753678
$536$	23.2790	1.00550
$537$	0	0
$538$	14.1760	0.611171
$539$	3.57486	0.153980
$540$	0	0
$541$	24.3360	1.04629	0.523143	−	0.852245i	$-0.324759\pi$
$541$	24.3360	1.04629	0.523143	+	0.852245i	$0.324759\pi$
$542$	36.7391	1.57808
$543$	0	0
$544$	0	0
$545$	1.94191	0.0831823
$546$	0	0
$547$	23.9143	1.02250	0.511250	−	0.859432i	$-0.329183\pi$
$547$	23.9143	1.02250	0.511250	+	0.859432i	$0.329183\pi$
$548$	−5.62362	−0.240229
$549$	0	0
$550$	−9.68889	−0.413136
$551$	0.00312084	0.000132952 0
$552$	0	0
$553$	11.4114	0.485264
$554$	−21.6692	−0.920638
$555$	0	0
$556$	−8.52467	−0.361526
$557$	−0.00668957	−0.000283446 0	−0.000141723	−	1.00000i	$-0.500045\pi$
$557$	−0.00668957	−0.000283446 0	−0.000141723		1.00000i	$0.500045\pi$
$558$	0	0
$559$	−6.60646	−0.279424
$560$	−8.28799	−0.350231
$561$	0	0
$562$	35.4645	1.49598
$563$	9.81215	0.413533	0.206766	−	0.978390i	$-0.433706\pi$
$563$	9.81215	0.413533	0.206766	+	0.978390i	$0.433706\pi$
$564$	0	0
$565$	9.59750	0.403770
$566$	−13.8259	−0.581146
$567$	0	0
$568$	24.4161	1.02448
$569$	−14.8826	−0.623910	−0.311955	−	0.950097i	$-0.600984\pi$
$569$	−14.8826	−0.623910	−0.311955	+	0.950097i	$0.600984\pi$
$570$	0	0
$571$	18.3999	0.770012	0.385006	−	0.922914i	$-0.374199\pi$
$571$	18.3999	0.770012	0.385006	+	0.922914i	$0.374199\pi$
$572$	−4.79545	−0.200508
$573$	0	0
$574$	−20.5748	−0.858774
$575$	−26.9237	−1.12280
$576$	0	0
$577$	−17.1344	−0.713316	−0.356658	−	0.934235i	$-0.616084\pi$
$577$	−17.1344	−0.713316	−0.356658	+	0.934235i	$0.616084\pi$
$578$	0	0
$579$	0	0
$580$	−0.219972	−0.00913385
$581$	−2.68819	−0.111525
$582$	0	0
$583$	−16.7191	−0.692435
$584$	−21.5501	−0.891748
$585$	0	0
$586$	11.8200	0.488282
$587$	4.31079	0.177925	0.0889626	−	0.996035i	$-0.471645\pi$
$587$	4.31079	0.177925	0.0889626	+	0.996035i	$0.471645\pi$
$588$	0	0
$589$	−0.0304002	−0.00125262
$590$	−10.4173	−0.428874
$591$	0	0
$592$	−4.48476	−0.184322
$593$	18.9392	0.777741	0.388871	−	0.921292i	$-0.372865\pi$
$593$	18.9392	0.777741	0.388871	+	0.921292i	$0.372865\pi$
$594$	0	0
$595$	0	0
$596$	6.51250	0.266762
$597$	0	0
$598$	−56.2810	−2.30150
$599$	1.82078	0.0743950	0.0371975	−	0.999308i	$-0.488157\pi$
$599$	1.82078	0.0743950	0.0371975	+	0.999308i	$0.488157\pi$
$600$	0	0
$601$	−28.8295	−1.17598	−0.587990	−	0.808869i	$-0.700080\pi$
$601$	−28.8295	−1.17598	−0.587990	+	0.808869i	$0.700080\pi$
$602$	−4.00725	−0.163323
$603$	0	0
$604$	−4.92980	−0.200591
$605$	7.39528	0.300661
$606$	0	0
$607$	0.229266	0.00930562	0.00465281	−	0.999989i	$-0.498519\pi$
$607$	0.229266	0.00930562	0.00465281	+	0.999989i	$0.498519\pi$
$608$	0.0242885	0.000985027 0
$609$	0	0
$610$	−12.8491	−0.520246
$611$	−30.4474	−1.23177
$612$	0	0
$613$	−34.9225	−1.41051	−0.705253	−	0.708956i	$-0.749166\pi$
$613$	−34.9225	−1.41051	−0.705253	+	0.708956i	$0.749166\pi$
$614$	−18.5925	−0.750332
$615$	0	0
$616$	6.46756	0.260585
$617$	−3.02767	−0.121889	−0.0609447	−	0.998141i	$-0.519411\pi$
$617$	−3.02767	−0.121889	−0.0609447	+	0.998141i	$0.519411\pi$
$618$	0	0
$619$	−4.39550	−0.176670	−0.0883350	−	0.996091i	$-0.528155\pi$
$619$	−4.39550	−0.176670	−0.0883350	+	0.996091i	$0.528155\pi$
$620$	2.14275	0.0860549
$621$	0	0
$622$	37.1593	1.48995
$623$	−1.26642	−0.0507381
$624$	0	0
$625$	15.5245	0.620979
$626$	55.9056	2.23444
$627$	0	0
$628$	4.75164	0.189611
$629$	0	0
$630$	0	0
$631$	9.86746	0.392817	0.196409	−	0.980522i	$-0.437072\pi$
$631$	9.86746	0.392817	0.196409	+	0.980522i	$0.437072\pi$
$632$	−12.1375	−0.482805
$633$	0	0
$634$	−52.0725	−2.06806
$635$	−0.452716	−0.0179655
$636$	0	0
$637$	−14.5221	−0.575385
$638$	0.973844	0.0385549
$639$	0	0
$640$	11.0680	0.437502
$641$	−24.1140	−0.952446	−0.476223	−	0.879325i	$-0.657994\pi$
$641$	−24.1140	−0.952446	−0.476223	+	0.879325i	$0.657994\pi$
$642$	0	0
$643$	−0.484669	−0.0191135	−0.00955674	−	0.999954i	$-0.503042\pi$
$643$	−0.484669	−0.0191135	−0.00955674	+	0.999954i	$0.503042\pi$
$644$	−8.08292	−0.318512
$645$	0	0
$646$	0	0
$647$	−11.3102	−0.444651	−0.222326	−	0.974972i	$-0.571365\pi$
$647$	−11.3102	−0.444651	−0.222326	+	0.974972i	$0.571365\pi$
$648$	0	0
$649$	10.9196	0.428632
$650$	39.3589	1.54378
$651$	0	0
$652$	4.89453	0.191684
$653$	−43.0687	−1.68541	−0.842705	−	0.538376i	$-0.819038\pi$
$653$	−43.0687	−1.68541	−0.842705	+	0.538376i	$0.819038\pi$
$654$	0	0
$655$	−12.0575	−0.471127
$656$	29.3957	1.14771
$657$	0	0
$658$	−18.4683	−0.719969
$659$	−48.4534	−1.88748	−0.943738	−	0.330695i	$-0.892717\pi$
$659$	−48.4534	−1.88748	−0.943738	+	0.330695i	$0.892717\pi$
$660$	0	0
$661$	42.4555	1.65133	0.825663	−	0.564164i	$-0.190801\pi$
$661$	42.4555	1.65133	0.825663	+	0.564164i	$0.190801\pi$
$662$	−28.4638	−1.10628
$663$	0	0
$664$	2.85923	0.110960
$665$	0.0122130	0.000473602 0
$666$	0	0
$667$	2.70615	0.104782
$668$	13.0232	0.503884
$669$	0	0
$670$	13.7172	0.529941
$671$	13.4687	0.519952
$672$	0	0
$673$	7.75060	0.298764	0.149382	−	0.988780i	$-0.452272\pi$
$673$	7.75060	0.298764	0.149382	+	0.988780i	$0.452272\pi$
$674$	42.7478	1.64658
$675$	0	0
$676$	11.4146	0.439024
$677$	−40.9960	−1.57560	−0.787802	−	0.615929i	$-0.788781\pi$
$677$	−40.9960	−1.57560	−0.787802	+	0.615929i	$0.788781\pi$
$678$	0	0
$679$	−11.1320	−0.427205
$680$	0	0
$681$	0	0
$682$	−9.48622	−0.363246
$683$	−36.7584	−1.40652	−0.703260	−	0.710932i	$-0.748273\pi$
$683$	−36.7584	−1.40652	−0.703260	+	0.710932i	$0.748273\pi$
$684$	0	0
$685$	7.36801	0.281517
$686$	−32.6001	−1.24468
$687$	0	0
$688$	5.72526	0.218273
$689$	67.9176	2.58745
$690$	0	0
$691$	9.91773	0.377288	0.188644	−	0.982046i	$-0.439591\pi$
$691$	9.91773	0.377288	0.188644	+	0.982046i	$0.439591\pi$
$692$	13.6646	0.519450
$693$	0	0
$694$	35.2676	1.33874
$695$	11.1689	0.423662
$696$	0	0
$697$	0	0
$698$	31.8444	1.20533
$699$	0	0
$700$	5.65262	0.213649
$701$	−18.8994	−0.713819	−0.356910	−	0.934139i	$-0.616170\pi$
$701$	−18.8994	−0.713819	−0.356910	+	0.934139i	$0.616170\pi$
$702$	0	0
$703$	0.00660867	0.000249251 0
$704$	−5.81709	−0.219240
$705$	0	0
$706$	0.642244	0.0241712
$707$	−39.6173	−1.48996
$708$	0	0
$709$	16.9622	0.637030	0.318515	−	0.947918i	$-0.396816\pi$
$709$	16.9622	0.637030	0.318515	+	0.947918i	$0.396816\pi$
$710$	14.3872	0.539942
$711$	0	0
$712$	1.34700	0.0504809
$713$	−26.3606	−0.987211
$714$	0	0
$715$	6.28296	0.234969
$716$	−6.75115	−0.252302
$717$	0	0
$718$	−30.8481	−1.15124
$719$	−1.11134	−0.0414459	−0.0207230	−	0.999785i	$-0.506597\pi$
$719$	−1.11134	−0.0414459	−0.0207230	+	0.999785i	$0.506597\pi$
$720$	0	0
$721$	−33.1025	−1.23280
$722$	30.7567	1.14465
$723$	0	0
$724$	−5.63048	−0.209255
$725$	−1.89248	−0.0702851
$726$	0	0
$727$	31.4813	1.16758	0.583788	−	0.811906i	$-0.301570\pi$
$727$	31.4813	1.16758	0.583788	+	0.811906i	$0.301570\pi$
$728$	−26.2730	−0.973742
$729$	0	0
$730$	−12.6984	−0.469989
$731$	0	0
$732$	0	0
$733$	16.1073	0.594935	0.297467	−	0.954732i	$-0.403858\pi$
$733$	16.1073	0.594935	0.297467	+	0.954732i	$0.403858\pi$
$734$	−51.9107	−1.91606
$735$	0	0
$736$	21.0610	0.776318
$737$	−14.3786	−0.529642
$738$	0	0
$739$	28.8022	1.05951	0.529753	−	0.848152i	$-0.322285\pi$
$739$	28.8022	1.05951	0.529753	+	0.848152i	$0.322285\pi$
$740$	−0.465811	−0.0171236
$741$	0	0
$742$	41.1964	1.51237
$743$	−44.6548	−1.63823	−0.819113	−	0.573632i	$-0.805534\pi$
$743$	−44.6548	−1.63823	−0.819113	+	0.573632i	$0.805534\pi$
$744$	0	0
$745$	−8.53262	−0.312611
$746$	8.96979	0.328407
$747$	0	0
$748$	0	0
$749$	4.50256	0.164520
$750$	0	0
$751$	−38.1637	−1.39261	−0.696306	−	0.717745i	$-0.745174\pi$
$751$	−38.1637	−1.39261	−0.696306	+	0.717745i	$0.745174\pi$
$752$	26.3861	0.962203
$753$	0	0
$754$	−3.95602	−0.144070
$755$	6.45898	0.235066
$756$	0	0
$757$	−21.7159	−0.789278	−0.394639	−	0.918836i	$-0.629130\pi$
$757$	−21.7159	−0.789278	−0.394639	+	0.918836i	$0.629130\pi$
$758$	−19.9267	−0.723772
$759$	0	0
$760$	−0.0129901	−0.000471202 0
$761$	−42.8323	−1.55267	−0.776334	−	0.630321i	$-0.782923\pi$
$761$	−42.8323	−1.55267	−0.776334	+	0.630321i	$0.782923\pi$
$762$	0	0
$763$	5.01564	0.181578
$764$	5.34163	0.193254
$765$	0	0
$766$	−14.1316	−0.510594
$767$	−44.3584	−1.60169
$768$	0	0
$769$	16.6231	0.599443	0.299721	−	0.954027i	$-0.403106\pi$
$769$	16.6231	0.599443	0.299721	+	0.954027i	$0.403106\pi$
$770$	3.81102	0.137340
$771$	0	0
$772$	−5.54319	−0.199504
$773$	6.93237	0.249340	0.124670	−	0.992198i	$-0.460213\pi$
$773$	6.93237	0.249340	0.124670	+	0.992198i	$0.460213\pi$
$774$	0	0
$775$	18.4347	0.662194
$776$	11.8403	0.425041
$777$	0	0
$778$	1.39807	0.0501232
$779$	−0.0433171	−0.00155199
$780$	0	0
$781$	−15.0809	−0.539637
$782$	0	0
$783$	0	0
$784$	12.5850	0.449466
$785$	−6.22555	−0.222199
$786$	0	0
$787$	30.1387	1.07433	0.537164	−	0.843478i	$-0.319496\pi$
$787$	30.1387	1.07433	0.537164	+	0.843478i	$0.319496\pi$
$788$	7.61047	0.271112
$789$	0	0
$790$	−7.15206	−0.254459
$791$	24.7888	0.881388
$792$	0	0
$793$	−54.7134	−1.94293
$794$	23.9741	0.850809
$795$	0	0
$796$	−14.0178	−0.496849
$797$	0.412813	0.0146226	0.00731128	−	0.999973i	$-0.497673\pi$
$797$	0.412813	0.0146226	0.00731128	+	0.999973i	$0.497673\pi$
$798$	0	0
$799$	0	0
$800$	−14.7286	−0.520733
$801$	0	0
$802$	41.5695	1.46787
$803$	13.3107	0.469724
$804$	0	0
$805$	10.5902	0.373254
$806$	38.5356	1.35736
$807$	0	0
$808$	42.1381	1.48241
$809$	−18.8299	−0.662023	−0.331011	−	0.943627i	$-0.607390\pi$
$809$	−18.8299	−0.662023	−0.331011	+	0.943627i	$0.607390\pi$
$810$	0	0
$811$	−54.4668	−1.91259	−0.956294	−	0.292408i	$-0.905544\pi$
$811$	−54.4668	−1.91259	−0.956294	+	0.292408i	$0.905544\pi$
$812$	−0.568153	−0.0199383
$813$	0	0
$814$	2.06220	0.0722802
$815$	−6.41276	−0.224629
$816$	0	0
$817$	−0.00843665	−0.000295161 0
$818$	−53.4878	−1.87016
$819$	0	0
$820$	3.05320	0.106622
$821$	−5.72330	−0.199744	−0.0998722	−	0.995000i	$-0.531843\pi$
$821$	−5.72330	−0.199744	−0.0998722	+	0.995000i	$0.531843\pi$
$822$	0	0
$823$	−38.8966	−1.35585	−0.677925	−	0.735131i	$-0.737121\pi$
$823$	−38.8966	−1.35585	−0.677925	+	0.735131i	$0.737121\pi$
$824$	35.2087	1.22655
$825$	0	0
$826$	−26.9063	−0.936188
$827$	37.3432	1.29855	0.649275	−	0.760553i	$-0.275072\pi$
$827$	37.3432	1.29855	0.649275	+	0.760553i	$0.275072\pi$
$828$	0	0
$829$	14.6224	0.507855	0.253928	−	0.967223i	$-0.418277\pi$
$829$	14.6224	0.507855	0.253928	+	0.967223i	$0.418277\pi$
$830$	1.68481	0.0584805
$831$	0	0
$832$	23.6306	0.819244
$833$	0	0
$834$	0	0
$835$	−17.0629	−0.590487
$836$	−0.00612394	−0.000211801 0
$837$	0	0
$838$	4.89401	0.169061
$839$	8.93026	0.308307	0.154153	−	0.988047i	$-0.450735\pi$
$839$	8.93026	0.308307	0.154153	+	0.988047i	$0.450735\pi$
$840$	0	0
$841$	−28.8098	−0.993441
$842$	−12.6254	−0.435100
$843$	0	0
$844$	−7.13304	−0.245529
$845$	−14.9553	−0.514479
$846$	0	0
$847$	19.1008	0.656312
$848$	−58.8584	−2.02121
$849$	0	0
$850$	0	0
$851$	5.73051	0.196439
$852$	0	0
$853$	22.9810	0.786853	0.393426	−	0.919356i	$-0.371290\pi$
$853$	22.9810	0.786853	0.393426	+	0.919356i	$0.371290\pi$
$854$	−33.1872	−1.13564
$855$	0	0
$856$	−4.78905	−0.163686
$857$	9.41958	0.321767	0.160883	−	0.986973i	$-0.448566\pi$
$857$	9.41958	0.321767	0.160883	+	0.986973i	$0.448566\pi$
$858$	0	0
$859$	−31.8111	−1.08538	−0.542690	−	0.839933i	$-0.682594\pi$
$859$	−31.8111	−1.08538	−0.542690	+	0.839933i	$0.682594\pi$
$860$	0.594656	0.0202776
$861$	0	0
$862$	56.9015	1.93807
$863$	28.4968	0.970041	0.485020	−	0.874503i	$-0.338812\pi$
$863$	28.4968	0.970041	0.485020	+	0.874503i	$0.338812\pi$
$864$	0	0
$865$	−17.9032	−0.608728
$866$	13.6832	0.464975
$867$	0	0
$868$	5.53438	0.187849
$869$	7.49690	0.254315
$870$	0	0
$871$	58.4098	1.97914
$872$	−5.33477	−0.180658
$873$	0	0
$874$	−0.0718725	−0.00243112
$875$	−15.9399	−0.538866
$876$	0	0
$877$	−53.5156	−1.80709	−0.903546	−	0.428491i	$-0.859045\pi$
$877$	−53.5156	−1.80709	−0.903546	+	0.428491i	$0.859045\pi$
$878$	−6.12260	−0.206628
$879$	0	0
$880$	−5.44491	−0.183548
$881$	33.8902	1.14179	0.570895	−	0.821023i	$-0.306596\pi$
$881$	33.8902	1.14179	0.570895	+	0.821023i	$0.306596\pi$
$882$	0	0
$883$	2.09301	0.0704353	0.0352177	−	0.999380i	$-0.488788\pi$
$883$	2.09301	0.0704353	0.0352177	+	0.999380i	$0.488788\pi$
$884$	0	0
$885$	0	0
$886$	58.5641	1.96750
$887$	10.1669	0.341371	0.170686	−	0.985326i	$-0.445402\pi$
$887$	10.1669	0.341371	0.170686	+	0.985326i	$0.445402\pi$
$888$	0	0
$889$	−1.16929	−0.0392168
$890$	0.793722	0.0266056
$891$	0	0
$892$	9.94448	0.332966
$893$	−0.0388822	−0.00130114
$894$	0	0
$895$	8.84529	0.295666
$896$	28.5869	0.955021
$897$	0	0
$898$	−24.4652	−0.816414
$899$	−1.85290	−0.0617976
$900$	0	0
$901$	0	0
$902$	−13.5169	−0.450063
$903$	0	0
$904$	−26.3660	−0.876921
$905$	7.37701	0.245220
$906$	0	0
$907$	16.2158	0.538437	0.269218	−	0.963079i	$-0.413235\pi$
$907$	16.2158	0.538437	0.269218	+	0.963079i	$0.413235\pi$
$908$	−3.59222	−0.119212
$909$	0	0
$910$	−15.4814	−0.513204
$911$	7.88531	0.261252	0.130626	−	0.991432i	$-0.458301\pi$
$911$	7.88531	0.261252	0.130626	+	0.991432i	$0.458301\pi$
$912$	0	0
$913$	−1.76604	−0.0584475
$914$	56.2072	1.85917
$915$	0	0
$916$	−7.28695	−0.240768
$917$	−31.1427	−1.02842
$918$	0	0
$919$	−55.1843	−1.82036	−0.910181	−	0.414210i	$-0.864058\pi$
$919$	−55.1843	−1.82036	−0.910181	+	0.414210i	$0.864058\pi$
$920$	−11.2640	−0.371363
$921$	0	0
$922$	37.4133	1.23214
$923$	61.2627	2.01649
$924$	0	0
$925$	−4.00751	−0.131766
$926$	27.7067	0.910497
$927$	0	0
$928$	1.48039	0.0485961
$929$	0.749896	0.0246033	0.0123016	−	0.999924i	$-0.496084\pi$
$929$	0.749896	0.0246033	0.0123016	+	0.999924i	$0.496084\pi$
$930$	0	0
$931$	−0.0185451	−0.000607792 0
$932$	−8.96939	−0.293802
$933$	0	0
$934$	−27.6266	−0.903971
$935$	0	0
$936$	0	0
$937$	−46.3906	−1.51551	−0.757757	−	0.652537i	$-0.773705\pi$
$937$	−46.3906	−1.51551	−0.757757	+	0.652537i	$0.773705\pi$
$938$	35.4293	1.15681
$939$	0	0
$940$	2.74061	0.0893887
$941$	−15.7019	−0.511866	−0.255933	−	0.966694i	$-0.582383\pi$
$941$	−15.7019	−0.511866	−0.255933	+	0.966694i	$0.582383\pi$
$942$	0	0
$943$	−37.5611	−1.22316
$944$	38.4417	1.25117
$945$	0	0
$946$	−2.63262	−0.0855937
$947$	−38.7968	−1.26073	−0.630363	−	0.776300i	$-0.717094\pi$
$947$	−38.7968	−1.26073	−0.630363	+	0.776300i	$0.717094\pi$
$948$	0	0
$949$	−54.0717	−1.75524
$950$	0.0502625	0.00163073
$951$	0	0
$952$	0	0
$953$	−17.7642	−0.575439	−0.287719	−	0.957715i	$-0.592897\pi$
$953$	−17.7642	−0.575439	−0.287719	+	0.957715i	$0.592897\pi$
$954$	0	0
$955$	−6.99856	−0.226468
$956$	18.3783	0.594396
$957$	0	0
$958$	39.4957	1.27605
$959$	19.0304	0.614523
$960$	0	0
$961$	−12.9509	−0.417771
$962$	−8.37723	−0.270093
$963$	0	0
$964$	−16.8892	−0.543966
$965$	7.26263	0.233792
$966$	0	0
$967$	−47.2166	−1.51838	−0.759191	−	0.650867i	$-0.774405\pi$
$967$	−47.2166	−1.51838	−0.759191	+	0.650867i	$0.774405\pi$
$968$	−20.3162	−0.652986
$969$	0	0
$970$	6.97689	0.224015
$971$	2.48379	0.0797085	0.0398542	−	0.999206i	$-0.487311\pi$
$971$	2.48379	0.0797085	0.0398542	+	0.999206i	$0.487311\pi$
$972$	0	0
$973$	28.8476	0.924811
$974$	27.9054	0.894148
$975$	0	0
$976$	47.4154	1.51773
$977$	2.64911	0.0847525	0.0423762	−	0.999102i	$-0.486507\pi$
$977$	2.64911	0.0847525	0.0423762	+	0.999102i	$0.486507\pi$
$978$	0	0
$979$	−0.831992	−0.0265906
$980$	1.30715	0.0417554
$981$	0	0
$982$	−18.8158	−0.600435
$983$	−25.1160	−0.801075	−0.400538	−	0.916280i	$-0.631177\pi$
$983$	−25.1160	−0.801075	−0.400538	+	0.916280i	$0.631177\pi$
$984$	0	0
$985$	−9.97117	−0.317708
$986$	0	0
$987$	0	0
$988$	0.0248771	0.000791446 0
$989$	−7.31558	−0.232622
$990$	0	0
$991$	9.88780	0.314096	0.157048	−	0.987591i	$-0.449802\pi$
$991$	9.88780	0.314096	0.157048	+	0.987591i	$0.449802\pi$
$992$	−14.4205	−0.457850
$993$	0	0
$994$	37.1598	1.17864
$995$	18.3661	0.582243
$996$	0	0
$997$	−30.3886	−0.962416	−0.481208	−	0.876606i	$-0.659802\pi$
$997$	−30.3886	−0.962416	−0.481208	+	0.876606i	$0.659802\pi$
$998$	−32.3197	−1.02306
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7803.2.a.ce.1.7		24
3.2	odd	2	inner	7803.2.a.ce.1.18		24
17.5	odd	16		459.2.l.b.433.9	yes	48
17.7	odd	16		459.2.l.b.406.9	yes	48
17.16	even	2		7803.2.a.cf.1.7		24
51.5	even	16		459.2.l.b.433.4	yes	48
51.41	even	16		459.2.l.b.406.4	✓	48
51.50	odd	2		7803.2.a.cf.1.18		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.l.b.406.4	✓	48	51.41	even	16
459.2.l.b.406.9	yes	48	17.7	odd	16
459.2.l.b.433.4	yes	48	51.5	even	16
459.2.l.b.433.9	yes	48	17.5	odd	16
7803.2.a.ce.1.7		24	1.1	even	1	trivial
7803.2.a.ce.1.18		24	3.2	odd	2	inner
7803.2.a.cf.1.7		24	17.16	even	2
7803.2.a.cf.1.18		24	51.50	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7803 = 3^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7803.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61878 q^{2} +0.620447 q^{4} -0.812904 q^{5} -2.09960 q^{7} +2.23319 q^{8} +1.31591 q^{10} -1.37936 q^{11} +5.60334 q^{13} +3.39879 q^{14} -4.85594 q^{16} +0.00715563 q^{19} -0.504364 q^{20} +2.23288 q^{22} +6.20479 q^{23} -4.33919 q^{25} -9.07057 q^{26} -1.30269 q^{28} +0.436138 q^{29} -4.24842 q^{31} +3.39431 q^{32} +1.70677 q^{35} +0.923561 q^{37} -0.0115834 q^{38} -1.81537 q^{40} -6.05356 q^{41} -1.17902 q^{43} -0.855820 q^{44} -10.0442 q^{46} -5.43379 q^{47} -2.59168 q^{49} +7.02419 q^{50} +3.47658 q^{52} +12.1209 q^{53} +1.12129 q^{55} -4.68881 q^{56} -0.706011 q^{58} -7.91642 q^{59} -9.76442 q^{61} +6.87726 q^{62} +4.21724 q^{64} -4.55498 q^{65} +10.4241 q^{67} -2.76289 q^{70} +10.9333 q^{71} -9.64990 q^{73} -1.49504 q^{74} +0.00443969 q^{76} +2.89611 q^{77} -5.43506 q^{79} +3.94741 q^{80} +9.79938 q^{82} +1.28033 q^{83} +1.90858 q^{86} -3.08038 q^{88} +0.603172 q^{89} -11.7648 q^{91} +3.84974 q^{92} +8.79610 q^{94} -0.00581684 q^{95} +5.30194 q^{97} +4.19536 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{4} - 16 q^{7} - 16 q^{10} - 8 q^{13} + 24 q^{16} - 8 q^{19} - 56 q^{22} + 16 q^{25} - 16 q^{31} - 72 q^{37} - 72 q^{40} - 8 q^{43} - 40 q^{46} + 24 q^{49} - 32 q^{52} - 40 q^{55} - 72 q^{58}+ \cdots - 112 q^{97}+O(q^{100}) \) 24 * q + 24 * q^4 - 16 * q^7 - 16 * q^10 - 8 * q^13 + 24 * q^16 - 8 * q^19 - 56 * q^22 + 16 * q^25 - 16 * q^31 - 72 * q^37 - 72 * q^40 - 8 * q^43 - 40 * q^46 + 24 * q^49 - 32 * q^52 - 40 * q^55 - 72 * q^58 - 48 * q^61 + 48 * q^64 - 32 * q^67 + 24 * q^70 - 88 * q^73 + 8 * q^76 - 8 * q^82 - 144 * q^88 - 56 * q^91 + 72 * q^94 - 112 * q^97

Introduction

L-functions

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