LMFDB - Embedded newform 920.2.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [920,2,Mod(1,920)] 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("920.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14510 q^{3} -1.00000 q^{5} -1.14510 q^{7} +1.60147 q^{9} +5.89167 q^{11} -4.89167 q^{13} +2.14510 q^{15} -5.89167 q^{17} -2.34803 q^{19} +2.45636 q^{21} +1.00000 q^{23} +1.00000 q^{25} +3.00000 q^{27} +3.74657 q^{29} +5.68874 q^{31} -12.6382 q^{33} +1.14510 q^{35} +4.00000 q^{37} +10.4931 q^{39} -1.05783 q^{41} +11.4931 q^{43} -1.60147 q^{45} +7.74657 q^{47} -5.68874 q^{49} +12.6382 q^{51} +12.9863 q^{53} -5.89167 q^{55} +5.03677 q^{57} +0.797069 q^{59} +13.8917 q^{61} -1.83384 q^{63} +4.89167 q^{65} -15.5667 q^{67} -2.14510 q^{69} +2.94217 q^{71} +6.32698 q^{73} -2.14510 q^{75} -6.74657 q^{77} -11.2397 q^{81} -0.912726 q^{83} +5.89167 q^{85} -8.03677 q^{87} -15.4931 q^{89} +5.60147 q^{91} -12.2029 q^{93} +2.34803 q^{95} +13.3848 q^{97} +9.43531 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{17} + 3 q^{19} + 12 q^{21} + 3 q^{23} + 3 q^{25} + 9 q^{27} + 3 q^{29} + 6 q^{31} - 15 q^{33} - 3 q^{35} + 12 q^{37} + 15 q^{39} - 6 q^{41} + 18 q^{43}+ \cdots + 9 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^17 + 3 * q^19 + 12 * q^21 + 3 * q^23 + 3 * q^25 + 9 * q^27 + 3 * q^29 + 6 * q^31 - 15 * q^33 - 3 * q^35 + 12 * q^37 + 15 * q^39 - 6 * q^41 + 18 * q^43 - 3 * q^45 + 15 * q^47 - 6 * q^49 + 15 * q^51 + 6 * q^53 - 3 * q^55 - 6 * q^57 + 6 * q^59 + 27 * q^61 + 12 * q^63 + 12 * q^67 + 6 * q^71 - 15 * q^73 - 12 * q^77 - 9 * q^81 - 12 * q^83 + 3 * q^85 - 3 * q^87 - 30 * q^89 + 15 * q^91 - 33 * q^93 - 3 * q^95 + 9 * q^97 + 9 * q^99

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
$3$	−2.14510	−1.23848	−0.619238	−	0.785204i	$-0.712558\pi$
$3$	−2.14510	−1.23848	−0.619238	+	0.785204i	$0.712558\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.14510	−0.432808	−0.216404	−	0.976304i	$-0.569433\pi$
$7$	−1.14510	−0.432808	−0.216404	+	0.976304i	$0.569433\pi$
$8$	0	0
$9$	1.60147	0.533822
$10$	0	0
$11$	5.89167	1.77641	0.888203	−	0.459452i	$-0.151954\pi$
$11$	5.89167	1.77641	0.888203	+	0.459452i	$0.151954\pi$
$12$	0	0
$13$	−4.89167	−1.35671	−0.678353	−	0.734736i	$-0.737306\pi$
$13$	−4.89167	−1.35671	−0.678353	+	0.734736i	$0.737306\pi$
$14$	0	0
$15$	2.14510	0.553863
$16$	0	0
$17$	−5.89167	−1.42894	−0.714470	−	0.699666i	$-0.753332\pi$
$17$	−5.89167	−1.42894	−0.714470	+	0.699666i	$0.753332\pi$
$18$	0	0
$19$	−2.34803	−0.538676	−0.269338	−	0.963046i	$-0.586805\pi$
$19$	−2.34803	−0.538676	−0.269338	+	0.963046i	$0.586805\pi$
$20$	0	0
$21$	2.45636	0.536022
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.00000	0.577350
$28$	0	0
$29$	3.74657	0.695720	0.347860	−	0.937546i	$-0.386908\pi$
$29$	3.74657	0.695720	0.347860	+	0.937546i	$0.386908\pi$
$30$	0	0
$31$	5.68874	1.02173	0.510864	−	0.859662i	$-0.329326\pi$
$31$	5.68874	1.02173	0.510864	+	0.859662i	$0.329326\pi$
$32$	0	0
$33$	−12.6382	−2.20004
$34$	0	0
$35$	1.14510	0.193558
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	10.4931	1.68025
$40$	0	0
$41$	−1.05783	−0.165205	−0.0826025	−	0.996583i	$-0.526323\pi$
$41$	−1.05783	−0.165205	−0.0826025	+	0.996583i	$0.526323\pi$
$42$	0	0
$43$	11.4931	1.75269	0.876343	−	0.481687i	$-0.159976\pi$
$43$	11.4931	1.75269	0.876343	+	0.481687i	$0.159976\pi$
$44$	0	0
$45$	−1.60147	−0.238732
$46$	0	0
$47$	7.74657	1.12995	0.564977	−	0.825107i	$-0.308885\pi$
$47$	7.74657	1.12995	0.564977	+	0.825107i	$0.308885\pi$
$48$	0	0
$49$	−5.68874	−0.812677
$50$	0	0
$51$	12.6382	1.76971
$52$	0	0
$53$	12.9863	1.78380	0.891901	−	0.452231i	$-0.149372\pi$
$53$	12.9863	1.78380	0.891901	+	0.452231i	$0.149372\pi$
$54$	0	0
$55$	−5.89167	−0.794433
$56$	0	0
$57$	5.03677	0.667137
$58$	0	0
$59$	0.797069	0.103770	0.0518848	−	0.998653i	$-0.483477\pi$
$59$	0.797069	0.103770	0.0518848	+	0.998653i	$0.483477\pi$
$60$	0	0
$61$	13.8917	1.77865	0.889323	−	0.457279i	$-0.151176\pi$
$61$	13.8917	1.77865	0.889323	+	0.457279i	$0.151176\pi$
$62$	0	0
$63$	−1.83384	−0.231042
$64$	0	0
$65$	4.89167	0.606737
$66$	0	0
$67$	−15.5667	−1.90177	−0.950887	−	0.309540i	$-0.899825\pi$
$67$	−15.5667	−1.90177	−0.950887	+	0.309540i	$0.899825\pi$
$68$	0	0
$69$	−2.14510	−0.258240
$70$	0	0
$71$	2.94217	0.349172	0.174586	−	0.984642i	$-0.444141\pi$
$71$	2.94217	0.349172	0.174586	+	0.984642i	$0.444141\pi$
$72$	0	0
$73$	6.32698	0.740517	0.370258	−	0.928929i	$-0.379269\pi$
$73$	6.32698	0.740517	0.370258	+	0.928929i	$0.379269\pi$
$74$	0	0
$75$	−2.14510	−0.247695
$76$	0	0
$77$	−6.74657	−0.768843
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−11.2397	−1.24886
$82$	0	0
$83$	−0.912726	−0.100185	−0.0500923	−	0.998745i	$-0.515952\pi$
$83$	−0.912726	−0.100185	−0.0500923	+	0.998745i	$0.515952\pi$
$84$	0	0
$85$	5.89167	0.639041
$86$	0	0
$87$	−8.03677	−0.861633
$88$	0	0
$89$	−15.4931	−1.64227	−0.821135	−	0.570735i	$-0.806659\pi$
$89$	−15.4931	−1.64227	−0.821135	+	0.570735i	$0.806659\pi$
$90$	0	0
$91$	5.60147	0.587193
$92$	0	0
$93$	−12.2029	−1.26539
$94$	0	0
$95$	2.34803	0.240903
$96$	0	0
$97$	13.3848	1.35902	0.679511	−	0.733666i	$-0.262192\pi$
$97$	13.3848	1.35902	0.679511	+	0.733666i	$0.262192\pi$
$98$	0	0
$99$	9.43531	0.948284
$100$	0	0
$101$	2.79707	0.278319	0.139159	−	0.990270i	$-0.455560\pi$
$101$	2.79707	0.278319	0.139159	+	0.990270i	$0.455560\pi$
$102$	0	0
$103$	−9.89167	−0.974655	−0.487328	−	0.873219i	$-0.662028\pi$
$103$	−9.89167	−0.974655	−0.487328	+	0.873219i	$0.662028\pi$
$104$	0	0
$105$	−2.45636	−0.239716
$106$	0	0
$107$	12.6961	1.22738	0.613688	−	0.789549i	$-0.289685\pi$
$107$	12.6961	1.22738	0.613688	+	0.789549i	$0.289685\pi$
$108$	0	0
$109$	3.65197	0.349795	0.174897	−	0.984587i	$-0.444041\pi$
$109$	3.65197	0.349795	0.174897	+	0.984587i	$0.444041\pi$
$110$	0	0
$111$	−8.58041	−0.814417
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−7.83384	−0.724239
$118$	0	0
$119$	6.74657	0.618457
$120$	0	0
$121$	23.7118	2.15562
$122$	0	0
$123$	2.26915	0.204602
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.1524	1.07835	0.539177	−	0.842193i	$-0.318735\pi$
$127$	12.1524	1.07835	0.539177	+	0.842193i	$0.318735\pi$
$128$	0	0
$129$	−24.6540	−2.17066
$130$	0	0
$131$	−6.94950	−0.607181	−0.303590	−	0.952803i	$-0.598185\pi$
$131$	−6.94950	−0.607181	−0.303590	+	0.952803i	$0.598185\pi$
$132$	0	0
$133$	2.68874	0.233143
$134$	0	0
$135$	−3.00000	−0.258199
$136$	0	0
$137$	−2.44904	−0.209235	−0.104618	−	0.994513i	$-0.533362\pi$
$137$	−2.44904	−0.209235	−0.104618	+	0.994513i	$0.533362\pi$
$138$	0	0
$139$	9.23970	0.783702	0.391851	−	0.920029i	$-0.371835\pi$
$139$	9.23970	0.783702	0.391851	+	0.920029i	$0.371835\pi$
$140$	0	0
$141$	−16.6172	−1.39942
$142$	0	0
$143$	−28.8201	−2.41006
$144$	0	0
$145$	−3.74657	−0.311136
$146$	0	0
$147$	12.2029	1.00648
$148$	0	0
$149$	−2.05783	−0.168584	−0.0842919	−	0.996441i	$-0.526863\pi$
$149$	−2.05783	−0.168584	−0.0842919	+	0.996441i	$0.526863\pi$
$150$	0	0
$151$	9.34803	0.760732	0.380366	−	0.924836i	$-0.375798\pi$
$151$	9.34803	0.760732	0.380366	+	0.924836i	$0.375798\pi$
$152$	0	0
$153$	−9.43531	−0.762799
$154$	0	0
$155$	−5.68874	−0.456931
$156$	0	0
$157$	−3.60879	−0.288013	−0.144007	−	0.989577i	$-0.545999\pi$
$157$	−3.60879	−0.288013	−0.144007	+	0.989577i	$0.545999\pi$
$158$	0	0
$159$	−27.8569	−2.20920
$160$	0	0
$161$	−1.14510	−0.0902467
$162$	0	0
$163$	21.2554	1.66485	0.832427	−	0.554135i	$-0.186951\pi$
$163$	21.2554	1.66485	0.832427	+	0.554135i	$0.186951\pi$
$164$	0	0
$165$	12.6382	0.983886
$166$	0	0
$167$	−10.9863	−0.850143	−0.425072	−	0.905160i	$-0.639751\pi$
$167$	−10.9863	−0.850143	−0.425072	+	0.905160i	$0.639751\pi$
$168$	0	0
$169$	10.9284	0.840650
$170$	0	0
$171$	−3.76030	−0.287557
$172$	0	0
$173$	−6.28288	−0.477678	−0.238839	−	0.971059i	$-0.576767\pi$
$173$	−6.28288	−0.477678	−0.238839	+	0.971059i	$0.576767\pi$
$174$	0	0
$175$	−1.14510	−0.0865616
$176$	0	0
$177$	−1.70979	−0.128516
$178$	0	0
$179$	10.1524	0.758828	0.379414	−	0.925227i	$-0.376126\pi$
$179$	10.1524	0.758828	0.379414	+	0.925227i	$0.376126\pi$
$180$	0	0
$181$	11.1451	0.828409	0.414204	−	0.910184i	$-0.364060\pi$
$181$	11.1451	0.828409	0.414204	+	0.910184i	$0.364060\pi$
$182$	0	0
$183$	−29.7991	−2.20281
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−34.7118	−2.53838
$188$	0	0
$189$	−3.43531	−0.249882
$190$	0	0
$191$	−14.6961	−1.06337	−0.531685	−	0.846942i	$-0.678441\pi$
$191$	−14.6961	−1.06337	−0.531685	+	0.846942i	$0.678441\pi$
$192$	0	0
$193$	−26.4005	−1.90035	−0.950176	−	0.311715i	$-0.899097\pi$
$193$	−26.4005	−1.90035	−0.950176	+	0.311715i	$0.899097\pi$
$194$	0	0
$195$	−10.4931	−0.751429
$196$	0	0
$197$	−4.26076	−0.303567	−0.151783	−	0.988414i	$-0.548502\pi$
$197$	−4.26076	−0.303567	−0.151783	+	0.988414i	$0.548502\pi$
$198$	0	0
$199$	18.0735	1.28120	0.640600	−	0.767875i	$-0.278686\pi$
$199$	18.0735	1.28120	0.640600	+	0.767875i	$0.278686\pi$
$200$	0	0
$201$	33.3921	2.35530
$202$	0	0
$203$	−4.29021	−0.301113
$204$	0	0
$205$	1.05783	0.0738819
$206$	0	0
$207$	1.60147	0.111310
$208$	0	0
$209$	−13.8338	−0.956907
$210$	0	0
$211$	−11.8990	−0.819161	−0.409580	−	0.912274i	$-0.634325\pi$
$211$	−11.8990	−0.819161	−0.409580	+	0.912274i	$0.634325\pi$
$212$	0	0
$213$	−6.31126	−0.432440
$214$	0	0
$215$	−11.4931	−0.783825
$216$	0	0
$217$	−6.51419	−0.442212
$218$	0	0
$219$	−13.5720	−0.917112
$220$	0	0
$221$	28.8201	1.93865
$222$	0	0
$223$	4.58041	0.306727	0.153363	−	0.988170i	$-0.450989\pi$
$223$	4.58041	0.306727	0.153363	+	0.988170i	$0.450989\pi$
$224$	0	0
$225$	1.60147	0.106764
$226$	0	0
$227$	2.21666	0.147125	0.0735624	−	0.997291i	$-0.476563\pi$
$227$	2.21666	0.147125	0.0735624	+	0.997291i	$0.476563\pi$
$228$	0	0
$229$	−19.1608	−1.26618	−0.633091	−	0.774077i	$-0.718214\pi$
$229$	−19.1608	−1.26618	−0.633091	+	0.774077i	$0.718214\pi$
$230$	0	0
$231$	14.4721	0.952193
$232$	0	0
$233$	−21.7466	−1.42467	−0.712333	−	0.701842i	$-0.752361\pi$
$233$	−21.7466	−1.42467	−0.712333	+	0.701842i	$0.752361\pi$
$234$	0	0
$235$	−7.74657	−0.505330
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20.9495	1.35511	0.677555	−	0.735472i	$-0.263039\pi$
$239$	20.9495	1.35511	0.677555	+	0.735472i	$0.263039\pi$
$240$	0	0
$241$	24.0735	1.55071	0.775357	−	0.631523i	$-0.217570\pi$
$241$	24.0735	1.55071	0.775357	+	0.631523i	$0.217570\pi$
$242$	0	0
$243$	15.1103	0.969328
$244$	0	0
$245$	5.68874	0.363440
$246$	0	0
$247$	11.4858	0.730825
$248$	0	0
$249$	1.95789	0.124076
$250$	0	0
$251$	−7.82651	−0.494005	−0.247003	−	0.969015i	$-0.579446\pi$
$251$	−7.82651	−0.494005	−0.247003	+	0.969015i	$0.579446\pi$
$252$	0	0
$253$	5.89167	0.370406
$254$	0	0
$255$	−12.6382	−0.791437
$256$	0	0
$257$	9.82012	0.612562	0.306281	−	0.951941i	$-0.400915\pi$
$257$	9.82012	0.612562	0.306281	+	0.951941i	$0.400915\pi$
$258$	0	0
$259$	−4.58041	−0.284613
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−14.6887	−0.905746	−0.452873	−	0.891575i	$-0.649601\pi$
$263$	−14.6887	−0.905746	−0.452873	+	0.891575i	$0.649601\pi$
$264$	0	0
$265$	−12.9863	−0.797740
$266$	0	0
$267$	33.2344	2.03391
$268$	0	0
$269$	−0.253432	−0.0154520	−0.00772600	−	0.999970i	$-0.502459\pi$
$269$	−0.253432	−0.0154520	−0.00772600	+	0.999970i	$0.502459\pi$
$270$	0	0
$271$	−19.6099	−1.19121	−0.595607	−	0.803276i	$-0.703088\pi$
$271$	−19.6099	−1.19121	−0.595607	+	0.803276i	$0.703088\pi$
$272$	0	0
$273$	−12.0157	−0.727224
$274$	0	0
$275$	5.89167	0.355281
$276$	0	0
$277$	29.1976	1.75431	0.877157	−	0.480204i	$-0.159437\pi$
$277$	29.1976	1.75431	0.877157	+	0.480204i	$0.159437\pi$
$278$	0	0
$279$	9.11032	0.545421
$280$	0	0
$281$	−27.1755	−1.62115	−0.810577	−	0.585633i	$-0.800846\pi$
$281$	−27.1755	−1.62115	−0.810577	+	0.585633i	$0.800846\pi$
$282$	0	0
$283$	18.1471	1.07873	0.539366	−	0.842071i	$-0.318664\pi$
$283$	18.1471	1.07873	0.539366	+	0.842071i	$0.318664\pi$
$284$	0	0
$285$	−5.03677	−0.298353
$286$	0	0
$287$	1.21132	0.0715021
$288$	0	0
$289$	17.7118	1.04187
$290$	0	0
$291$	−28.7118	−1.68311
$292$	0	0
$293$	0.681412	0.0398085	0.0199043	−	0.999802i	$-0.493664\pi$
$293$	0.681412	0.0398085	0.0199043	+	0.999802i	$0.493664\pi$
$294$	0	0
$295$	−0.797069	−0.0464071
$296$	0	0
$297$	17.6750	1.02561
$298$	0	0
$299$	−4.89167	−0.282893
$300$	0	0
$301$	−13.1608	−0.758577
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−13.8917	−0.795435
$306$	0	0
$307$	26.5877	1.51744	0.758721	−	0.651415i	$-0.225824\pi$
$307$	26.5877	1.51744	0.758721	+	0.651415i	$0.225824\pi$
$308$	0	0
$309$	21.2186	1.20709
$310$	0	0
$311$	13.5299	0.767211	0.383605	−	0.923497i	$-0.374682\pi$
$311$	13.5299	0.767211	0.383605	+	0.923497i	$0.374682\pi$
$312$	0	0
$313$	19.8412	1.12149	0.560745	−	0.827989i	$-0.310515\pi$
$313$	19.8412	1.12149	0.560745	+	0.827989i	$0.310515\pi$
$314$	0	0
$315$	1.83384	0.103325
$316$	0	0
$317$	3.55096	0.199442	0.0997210	−	0.995015i	$-0.468205\pi$
$317$	3.55096	0.199442	0.0997210	+	0.995015i	$0.468205\pi$
$318$	0	0
$319$	22.0735	1.23588
$320$	0	0
$321$	−27.2344	−1.52007
$322$	0	0
$323$	13.8338	0.769736
$324$	0	0
$325$	−4.89167	−0.271341
$326$	0	0
$327$	−7.83384	−0.433212
$328$	0	0
$329$	−8.87062	−0.489053
$330$	0	0
$331$	−13.3133	−0.731763	−0.365881	−	0.930662i	$-0.619232\pi$
$331$	−13.3133	−0.731763	−0.365881	+	0.930662i	$0.619232\pi$
$332$	0	0
$333$	6.40586	0.351039
$334$	0	0
$335$	15.5667	0.850499
$336$	0	0
$337$	−20.4951	−1.11644	−0.558220	−	0.829693i	$-0.688516\pi$
$337$	−20.4951	−1.11644	−0.558220	+	0.829693i	$0.688516\pi$
$338$	0	0
$339$	21.4510	1.16506
$340$	0	0
$341$	33.5162	1.81500
$342$	0	0
$343$	14.5299	0.784541
$344$	0	0
$345$	2.14510	0.115488
$346$	0	0
$347$	−7.60147	−0.408068	−0.204034	−	0.978964i	$-0.565405\pi$
$347$	−7.60147	−0.408068	−0.204034	+	0.978964i	$0.565405\pi$
$348$	0	0
$349$	5.55736	0.297479	0.148739	−	0.988876i	$-0.452478\pi$
$349$	5.55736	0.297479	0.148739	+	0.988876i	$0.452478\pi$
$350$	0	0
$351$	−14.6750	−0.783294
$352$	0	0
$353$	1.96323	0.104492	0.0522460	−	0.998634i	$-0.483362\pi$
$353$	1.96323	0.104492	0.0522460	+	0.998634i	$0.483362\pi$
$354$	0	0
$355$	−2.94217	−0.156154
$356$	0	0
$357$	−14.4721	−0.765944
$358$	0	0
$359$	10.5069	0.554531	0.277266	−	0.960793i	$-0.410572\pi$
$359$	10.5069	0.554531	0.277266	+	0.960793i	$0.410572\pi$
$360$	0	0
$361$	−13.4867	−0.709828
$362$	0	0
$363$	−50.8642	−2.66968
$364$	0	0
$365$	−6.32698	−0.331169
$366$	0	0
$367$	33.7412	1.76128	0.880639	−	0.473788i	$-0.157114\pi$
$367$	33.7412	1.76128	0.880639	+	0.473788i	$0.157114\pi$
$368$	0	0
$369$	−1.69408	−0.0881901
$370$	0	0
$371$	−14.8706	−0.772044
$372$	0	0
$373$	−17.4510	−0.903580	−0.451790	−	0.892124i	$-0.649214\pi$
$373$	−17.4510	−0.903580	−0.451790	+	0.892124i	$0.649214\pi$
$374$	0	0
$375$	2.14510	0.110773
$376$	0	0
$377$	−18.3270	−0.943887
$378$	0	0
$379$	−33.4584	−1.71864	−0.859320	−	0.511438i	$-0.829113\pi$
$379$	−33.4584	−1.71864	−0.859320	+	0.511438i	$0.829113\pi$
$380$	0	0
$381$	−26.0682	−1.33551
$382$	0	0
$383$	13.0873	0.668728	0.334364	−	0.942444i	$-0.391478\pi$
$383$	13.0873	0.668728	0.334364	+	0.942444i	$0.391478\pi$
$384$	0	0
$385$	6.74657	0.343837
$386$	0	0
$387$	18.4059	0.935623
$388$	0	0
$389$	14.1083	0.715321	0.357660	−	0.933852i	$-0.383575\pi$
$389$	14.1083	0.715321	0.357660	+	0.933852i	$0.383575\pi$
$390$	0	0
$391$	−5.89167	−0.297955
$392$	0	0
$393$	14.9074	0.751978
$394$	0	0
$395$	0	0
$396$	0	0
$397$	16.6245	0.834360	0.417180	−	0.908824i	$-0.363018\pi$
$397$	16.6245	0.834360	0.417180	+	0.908824i	$0.363018\pi$
$398$	0	0
$399$	−5.76762	−0.288742
$400$	0	0
$401$	22.2628	1.11175	0.555874	−	0.831266i	$-0.312384\pi$
$401$	22.2628	1.11175	0.555874	+	0.831266i	$0.312384\pi$
$402$	0	0
$403$	−27.8274	−1.38618
$404$	0	0
$405$	11.2397	0.558505
$406$	0	0
$407$	23.5667	1.16816
$408$	0	0
$409$	15.6887	0.775758	0.387879	−	0.921710i	$-0.373208\pi$
$409$	15.6887	0.775758	0.387879	+	0.921710i	$0.373208\pi$
$410$	0	0
$411$	5.25343	0.259133
$412$	0	0
$413$	−0.912726	−0.0449123
$414$	0	0
$415$	0.912726	0.0448039
$416$	0	0
$417$	−19.8201	−0.970595
$418$	0	0
$419$	−5.66769	−0.276885	−0.138442	−	0.990371i	$-0.544210\pi$
$419$	−5.66769	−0.276885	−0.138442	+	0.990371i	$0.544210\pi$
$420$	0	0
$421$	15.0716	0.734543	0.367271	−	0.930114i	$-0.380292\pi$
$421$	15.0716	0.734543	0.367271	+	0.930114i	$0.380292\pi$
$422$	0	0
$423$	12.4059	0.603194
$424$	0	0
$425$	−5.89167	−0.285788
$426$	0	0
$427$	−15.9074	−0.769813
$428$	0	0
$429$	61.8221	2.98480
$430$	0	0
$431$	29.3500	1.41374	0.706870	−	0.707343i	$-0.250106\pi$
$431$	29.3500	1.41374	0.706870	+	0.707343i	$0.250106\pi$
$432$	0	0
$433$	29.6245	1.42366	0.711832	−	0.702350i	$-0.247866\pi$
$433$	29.6245	1.42366	0.711832	+	0.702350i	$0.247866\pi$
$434$	0	0
$435$	8.03677	0.385334
$436$	0	0
$437$	−2.34803	−0.112322
$438$	0	0
$439$	−20.7907	−0.992285	−0.496142	−	0.868241i	$-0.665251\pi$
$439$	−20.7907	−0.992285	−0.496142	+	0.868241i	$0.665251\pi$
$440$	0	0
$441$	−9.11032	−0.433825
$442$	0	0
$443$	21.2975	1.01188	0.505938	−	0.862570i	$-0.331146\pi$
$443$	21.2975	1.01188	0.505938	+	0.862570i	$0.331146\pi$
$444$	0	0
$445$	15.4931	0.734445
$446$	0	0
$447$	4.41425	0.208787
$448$	0	0
$449$	−19.3197	−0.911751	−0.455875	−	0.890044i	$-0.650674\pi$
$449$	−19.3197	−0.911751	−0.455875	+	0.890044i	$0.650674\pi$
$450$	0	0
$451$	−6.23238	−0.293471
$452$	0	0
$453$	−20.0525	−0.942148
$454$	0	0
$455$	−5.60147	−0.262601
$456$	0	0
$457$	0.912726	0.0426955	0.0213478	−	0.999772i	$-0.493204\pi$
$457$	0.912726	0.0426955	0.0213478	+	0.999772i	$0.493204\pi$
$458$	0	0
$459$	−17.6750	−0.824999
$460$	0	0
$461$	2.83384	0.131985	0.0659926	−	0.997820i	$-0.478979\pi$
$461$	2.83384	0.131985	0.0659926	+	0.997820i	$0.478979\pi$
$462$	0	0
$463$	−17.2618	−0.802225	−0.401112	−	0.916029i	$-0.631376\pi$
$463$	−17.2618	−0.802225	−0.401112	+	0.916029i	$0.631376\pi$
$464$	0	0
$465$	12.2029	0.565897
$466$	0	0
$467$	−21.4510	−0.992635	−0.496318	−	0.868141i	$-0.665315\pi$
$467$	−21.4510	−0.992635	−0.496318	+	0.868141i	$0.665315\pi$
$468$	0	0
$469$	17.8255	0.823103
$470$	0	0
$471$	7.74123	0.356697
$472$	0	0
$473$	67.7138	3.11348
$474$	0	0
$475$	−2.34803	−0.107735
$476$	0	0
$477$	20.7971	0.952232
$478$	0	0
$479$	9.49314	0.433752	0.216876	−	0.976199i	$-0.430413\pi$
$479$	9.49314	0.433752	0.216876	+	0.976199i	$0.430413\pi$
$480$	0	0
$481$	−19.5667	−0.892164
$482$	0	0
$483$	2.45636	0.111768
$484$	0	0
$485$	−13.3848	−0.607773
$486$	0	0
$487$	38.3270	1.73676	0.868381	−	0.495898i	$-0.165161\pi$
$487$	38.3270	1.73676	0.868381	+	0.495898i	$0.165161\pi$
$488$	0	0
$489$	−45.5951	−2.06188
$490$	0	0
$491$	−13.9947	−0.631570	−0.315785	−	0.948831i	$-0.602268\pi$
$491$	−13.9947	−0.631570	−0.315785	+	0.948831i	$0.602268\pi$
$492$	0	0
$493$	−22.0735	−0.994143
$494$	0	0
$495$	−9.43531	−0.424086
$496$	0	0
$497$	−3.36909	−0.151124
$498$	0	0
$499$	−0.0642277	−0.00287522	−0.00143761	−	0.999999i	$-0.500458\pi$
$499$	−0.0642277	−0.00287522	−0.00143761	+	0.999999i	$0.500458\pi$
$500$	0	0
$501$	23.5667	1.05288
$502$	0	0
$503$	−24.2785	−1.08252	−0.541262	−	0.840854i	$-0.682053\pi$
$503$	−24.2785	−1.08252	−0.541262	+	0.840854i	$0.682053\pi$
$504$	0	0
$505$	−2.79707	−0.124468
$506$	0	0
$507$	−23.4426	−1.04112
$508$	0	0
$509$	−2.65929	−0.117871	−0.0589356	−	0.998262i	$-0.518771\pi$
$509$	−2.65929	−0.117871	−0.0589356	+	0.998262i	$0.518771\pi$
$510$	0	0
$511$	−7.24504	−0.320502
$512$	0	0
$513$	−7.04410	−0.311005
$514$	0	0
$515$	9.89167	0.435879
$516$	0	0
$517$	45.6402	2.00726
$518$	0	0
$519$	13.4774	0.591593
$520$	0	0
$521$	−17.0598	−0.747404	−0.373702	−	0.927549i	$-0.621912\pi$
$521$	−17.0598	−0.747404	−0.373702	+	0.927549i	$0.621912\pi$
$522$	0	0
$523$	−17.5667	−0.768137	−0.384069	−	0.923305i	$-0.625477\pi$
$523$	−17.5667	−0.768137	−0.384069	+	0.923305i	$0.625477\pi$
$524$	0	0
$525$	2.45636	0.107204
$526$	0	0
$527$	−33.5162	−1.45999
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	1.27648	0.0553944
$532$	0	0
$533$	5.17455	0.224135
$534$	0	0
$535$	−12.6961	−0.548899
$536$	0	0
$537$	−21.7780	−0.939790
$538$	0	0
$539$	−33.5162	−1.44364
$540$	0	0
$541$	−8.36909	−0.359815	−0.179908	−	0.983684i	$-0.557580\pi$
$541$	−8.36909	−0.359815	−0.179908	+	0.983684i	$0.557580\pi$
$542$	0	0
$543$	−23.9074	−1.02596
$544$	0	0
$545$	−3.65197	−0.156433
$546$	0	0
$547$	−16.7486	−0.716117	−0.358058	−	0.933699i	$-0.616561\pi$
$547$	−16.7486	−0.716117	−0.358058	+	0.933699i	$0.616561\pi$
$548$	0	0
$549$	22.2470	0.949480
$550$	0	0
$551$	−8.79707	−0.374768
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.58041	0.364218
$556$	0	0
$557$	−7.26182	−0.307693	−0.153847	−	0.988095i	$-0.549166\pi$
$557$	−7.26182	−0.307693	−0.153847	+	0.988095i	$0.549166\pi$
$558$	0	0
$559$	−56.2206	−2.37788
$560$	0	0
$561$	74.4603	3.14372
$562$	0	0
$563$	34.5530	1.45623	0.728117	−	0.685453i	$-0.240396\pi$
$563$	34.5530	1.45623	0.728117	+	0.685453i	$0.240396\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	0	0
$567$	12.8706	0.540515
$568$	0	0
$569$	34.7275	1.45585	0.727926	−	0.685655i	$-0.240484\pi$
$569$	34.7275	1.45585	0.727926	+	0.685655i	$0.240484\pi$
$570$	0	0
$571$	14.7034	0.615318	0.307659	−	0.951497i	$-0.400454\pi$
$571$	14.7034	0.615318	0.307659	+	0.951497i	$0.400454\pi$
$572$	0	0
$573$	31.5246	1.31696
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−6.13777	−0.255519	−0.127759	−	0.991805i	$-0.540779\pi$
$577$	−6.13777	−0.255519	−0.127759	+	0.991805i	$0.540779\pi$
$578$	0	0
$579$	56.6318	2.35354
$580$	0	0
$581$	1.04516	0.0433607
$582$	0	0
$583$	76.5108	3.16876
$584$	0	0
$585$	7.83384	0.323890
$586$	0	0
$587$	−1.33338	−0.0550344	−0.0275172	−	0.999621i	$-0.508760\pi$
$587$	−1.33338	−0.0550344	−0.0275172	+	0.999621i	$0.508760\pi$
$588$	0	0
$589$	−13.3574	−0.550380
$590$	0	0
$591$	9.13977	0.375960
$592$	0	0
$593$	1.88434	0.0773807	0.0386903	−	0.999251i	$-0.487681\pi$
$593$	1.88434	0.0773807	0.0386903	+	0.999251i	$0.487681\pi$
$594$	0	0
$595$	−6.74657	−0.276582
$596$	0	0
$597$	−38.7696	−1.58673
$598$	0	0
$599$	−43.7632	−1.78812	−0.894058	−	0.447951i	$-0.852154\pi$
$599$	−43.7632	−1.78812	−0.894058	+	0.447951i	$0.852154\pi$
$600$	0	0
$601$	4.50046	0.183578	0.0917889	−	0.995778i	$-0.470742\pi$
$601$	4.50046	0.183578	0.0917889	+	0.995778i	$0.470742\pi$
$602$	0	0
$603$	−24.9295	−1.01521
$604$	0	0
$605$	−23.7118	−0.964021
$606$	0	0
$607$	6.17455	0.250617	0.125309	−	0.992118i	$-0.460008\pi$
$607$	6.17455	0.250617	0.125309	+	0.992118i	$0.460008\pi$
$608$	0	0
$609$	9.20293	0.372922
$610$	0	0
$611$	−37.8937	−1.53301
$612$	0	0
$613$	−9.72445	−0.392767	−0.196383	−	0.980527i	$-0.562920\pi$
$613$	−9.72445	−0.392767	−0.196383	+	0.980527i	$0.562920\pi$
$614$	0	0
$615$	−2.26915	−0.0915010
$616$	0	0
$617$	−0.730849	−0.0294229	−0.0147114	−	0.999892i	$-0.504683\pi$
$617$	−0.730849	−0.0294229	−0.0147114	+	0.999892i	$0.504683\pi$
$618$	0	0
$619$	11.1598	0.448549	0.224274	−	0.974526i	$-0.427999\pi$
$619$	11.1598	0.448549	0.224274	+	0.974526i	$0.427999\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	17.7412	0.710787
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	29.6750	1.18511
$628$	0	0
$629$	−23.5667	−0.939665
$630$	0	0
$631$	−4.87062	−0.193896	−0.0969481	−	0.995289i	$-0.530908\pi$
$631$	−4.87062	−0.193896	−0.0969481	+	0.995289i	$0.530908\pi$
$632$	0	0
$633$	25.5246	1.01451
$634$	0	0
$635$	−12.1524	−0.482254
$636$	0	0
$637$	27.8274	1.10256
$638$	0	0
$639$	4.71179	0.186395
$640$	0	0
$641$	32.6540	1.28975	0.644877	−	0.764286i	$-0.276909\pi$
$641$	32.6540	1.28975	0.644877	+	0.764286i	$0.276909\pi$
$642$	0	0
$643$	47.9579	1.89127	0.945637	−	0.325223i	$-0.105439\pi$
$643$	47.9579	1.89127	0.945637	+	0.325223i	$0.105439\pi$
$644$	0	0
$645$	24.6540	0.970749
$646$	0	0
$647$	12.1103	0.476106	0.238053	−	0.971252i	$-0.423491\pi$
$647$	12.1103	0.476106	0.238053	+	0.971252i	$0.423491\pi$
$648$	0	0
$649$	4.69607	0.184337
$650$	0	0
$651$	13.9736	0.547669
$652$	0	0
$653$	−43.2133	−1.69107	−0.845534	−	0.533922i	$-0.820718\pi$
$653$	−43.2133	−1.69107	−0.845534	+	0.533922i	$0.820718\pi$
$654$	0	0
$655$	6.94950	0.271539
$656$	0	0
$657$	10.1324	0.395304
$658$	0	0
$659$	20.4333	0.795969	0.397984	−	0.917392i	$-0.369710\pi$
$659$	20.4333	0.795969	0.397984	+	0.917392i	$0.369710\pi$
$660$	0	0
$661$	48.1398	1.87242	0.936210	−	0.351441i	$-0.114308\pi$
$661$	48.1398	1.87242	0.936210	+	0.351441i	$0.114308\pi$
$662$	0	0
$663$	−61.8221	−2.40097
$664$	0	0
$665$	−2.68874	−0.104265
$666$	0	0
$667$	3.74657	0.145068
$668$	0	0
$669$	−9.82545	−0.379874
$670$	0	0
$671$	81.8452	3.15960
$672$	0	0
$673$	16.5436	0.637710	0.318855	−	0.947803i	$-0.396702\pi$
$673$	16.5436	0.637710	0.318855	+	0.947803i	$0.396702\pi$
$674$	0	0
$675$	3.00000	0.115470
$676$	0	0
$677$	−48.9863	−1.88270	−0.941348	−	0.337438i	$-0.890440\pi$
$677$	−48.9863	−1.88270	−0.941348	+	0.337438i	$0.890440\pi$
$678$	0	0
$679$	−15.3270	−0.588195
$680$	0	0
$681$	−4.75496	−0.182210
$682$	0	0
$683$	−21.8779	−0.837136	−0.418568	−	0.908185i	$-0.637468\pi$
$683$	−21.8779	−0.837136	−0.418568	+	0.908185i	$0.637468\pi$
$684$	0	0
$685$	2.44904	0.0935728
$686$	0	0
$687$	41.1019	1.56814
$688$	0	0
$689$	−63.5246	−2.42009
$690$	0	0
$691$	−3.30393	−0.125688	−0.0628438	−	0.998023i	$-0.520017\pi$
$691$	−3.30393	−0.125688	−0.0628438	+	0.998023i	$0.520017\pi$
$692$	0	0
$693$	−10.8044	−0.410425
$694$	0	0
$695$	−9.23970	−0.350482
$696$	0	0
$697$	6.23238	0.236068
$698$	0	0
$699$	46.6486	1.76441
$700$	0	0
$701$	−4.46369	−0.168591	−0.0842956	−	0.996441i	$-0.526864\pi$
$701$	−4.46369	−0.168591	−0.0842956	+	0.996441i	$0.526864\pi$
$702$	0	0
$703$	−9.39214	−0.354231
$704$	0	0
$705$	16.6172	0.625839
$706$	0	0
$707$	−3.20293	−0.120459
$708$	0	0
$709$	−2.39853	−0.0900789	−0.0450394	−	0.998985i	$-0.514341\pi$
$709$	−2.39853	−0.0900789	−0.0450394	+	0.998985i	$0.514341\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.68874	0.213045
$714$	0	0
$715$	28.8201	1.07781
$716$	0	0
$717$	−44.9388	−1.67827
$718$	0	0
$719$	12.0725	0.450228	0.225114	−	0.974332i	$-0.427725\pi$
$719$	12.0725	0.450228	0.225114	+	0.974332i	$0.427725\pi$
$720$	0	0
$721$	11.3270	0.421839
$722$	0	0
$723$	−51.6402	−1.92052
$724$	0	0
$725$	3.74657	0.139144
$726$	0	0
$727$	11.4269	0.423801	0.211900	−	0.977291i	$-0.432035\pi$
$727$	11.4269	0.423801	0.211900	+	0.977291i	$0.432035\pi$
$728$	0	0
$729$	1.30592	0.0483676
$730$	0	0
$731$	−67.7138	−2.50448
$732$	0	0
$733$	−3.08727	−0.114031	−0.0570155	−	0.998373i	$-0.518158\pi$
$733$	−3.08727	−0.114031	−0.0570155	+	0.998373i	$0.518158\pi$
$734$	0	0
$735$	−12.2029	−0.450112
$736$	0	0
$737$	−91.7138	−3.37832
$738$	0	0
$739$	−8.36909	−0.307862	−0.153931	−	0.988082i	$-0.549193\pi$
$739$	−8.36909	−0.307862	−0.153931	+	0.988082i	$0.549193\pi$
$740$	0	0
$741$	−24.6382	−0.905108
$742$	0	0
$743$	15.1598	0.556158	0.278079	−	0.960558i	$-0.410302\pi$
$743$	15.1598	0.556158	0.278079	+	0.960558i	$0.410302\pi$
$744$	0	0
$745$	2.05783	0.0753930
$746$	0	0
$747$	−1.46170	−0.0534808
$748$	0	0
$749$	−14.5383	−0.531218
$750$	0	0
$751$	−33.6823	−1.22909	−0.614543	−	0.788883i	$-0.710660\pi$
$751$	−33.6823	−1.22909	−0.614543	+	0.788883i	$0.710660\pi$
$752$	0	0
$753$	16.7887	0.611813
$754$	0	0
$755$	−9.34803	−0.340210
$756$	0	0
$757$	−29.1608	−1.05987	−0.529934	−	0.848039i	$-0.677783\pi$
$757$	−29.1608	−1.05987	−0.529934	+	0.848039i	$0.677783\pi$
$758$	0	0
$759$	−12.6382	−0.458739
$760$	0	0
$761$	−6.91912	−0.250818	−0.125409	−	0.992105i	$-0.540024\pi$
$761$	−6.91912	−0.250818	−0.125409	+	0.992105i	$0.540024\pi$
$762$	0	0
$763$	−4.18188	−0.151394
$764$	0	0
$765$	9.43531	0.341134
$766$	0	0
$767$	−3.89900	−0.140785
$768$	0	0
$769$	−14.3491	−0.517442	−0.258721	−	0.965952i	$-0.583301\pi$
$769$	−14.3491	−0.517442	−0.258721	+	0.965952i	$0.583301\pi$
$770$	0	0
$771$	−21.0652	−0.758643
$772$	0	0
$773$	50.6265	1.82091	0.910454	−	0.413609i	$-0.135732\pi$
$773$	50.6265	1.82091	0.910454	+	0.413609i	$0.135732\pi$
$774$	0	0
$775$	5.68874	0.204346
$776$	0	0
$777$	9.82545	0.352486
$778$	0	0
$779$	2.48382	0.0889920
$780$	0	0
$781$	17.3343	0.620270
$782$	0	0
$783$	11.2397	0.401674
$784$	0	0
$785$	3.60879	0.128803
$786$	0	0
$787$	20.9442	0.746579	0.373289	−	0.927715i	$-0.378230\pi$
$787$	20.9442	0.746579	0.373289	+	0.927715i	$0.378230\pi$
$788$	0	0
$789$	31.5089	1.12174
$790$	0	0
$791$	11.4510	0.407152
$792$	0	0
$793$	−67.9535	−2.41310
$794$	0	0
$795$	27.8569	0.987982
$796$	0	0
$797$	19.0009	0.673047	0.336524	−	0.941675i	$-0.390749\pi$
$797$	19.0009	0.673047	0.336524	+	0.941675i	$0.390749\pi$
$798$	0	0
$799$	−45.6402	−1.61464
$800$	0	0
$801$	−24.8117	−0.876679
$802$	0	0
$803$	37.2765	1.31546
$804$	0	0
$805$	1.14510	0.0403596
$806$	0	0
$807$	0.543637	0.0191369
$808$	0	0
$809$	18.4300	0.647963	0.323982	−	0.946063i	$-0.394978\pi$
$809$	18.4300	0.647963	0.323982	+	0.946063i	$0.394978\pi$
$810$	0	0
$811$	−21.1387	−0.742280	−0.371140	−	0.928577i	$-0.621033\pi$
$811$	−21.1387	−0.742280	−0.371140	+	0.928577i	$0.621033\pi$
$812$	0	0
$813$	42.0652	1.47529
$814$	0	0
$815$	−21.2554	−0.744545
$816$	0	0
$817$	−26.9863	−0.944130
$818$	0	0
$819$	8.97055	0.313457
$820$	0	0
$821$	−16.7550	−0.584752	−0.292376	−	0.956303i	$-0.594446\pi$
$821$	−16.7550	−0.584752	−0.292376	+	0.956303i	$0.594446\pi$
$822$	0	0
$823$	24.9220	0.868728	0.434364	−	0.900737i	$-0.356973\pi$
$823$	24.9220	0.868728	0.434364	+	0.900737i	$0.356973\pi$
$824$	0	0
$825$	−12.6382	−0.440007
$826$	0	0
$827$	−6.62252	−0.230288	−0.115144	−	0.993349i	$-0.536733\pi$
$827$	−6.62252	−0.230288	−0.115144	+	0.993349i	$0.536733\pi$
$828$	0	0
$829$	52.7001	1.83035	0.915174	−	0.403058i	$-0.132053\pi$
$829$	52.7001	1.83035	0.915174	+	0.403058i	$0.132053\pi$
$830$	0	0
$831$	−62.6318	−2.17267
$832$	0	0
$833$	33.5162	1.16127
$834$	0	0
$835$	10.9863	0.380196
$836$	0	0
$837$	17.0662	0.589895
$838$	0	0
$839$	42.5951	1.47054	0.735272	−	0.677772i	$-0.237054\pi$
$839$	42.5951	1.47054	0.735272	+	0.677772i	$0.237054\pi$
$840$	0	0
$841$	−14.9632	−0.515973
$842$	0	0
$843$	58.2942	2.00776
$844$	0	0
$845$	−10.9284	−0.375950
$846$	0	0
$847$	−27.1524	−0.932969
$848$	0	0
$849$	−38.9274	−1.33598
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	−57.7255	−1.97648	−0.988242	−	0.152898i	$-0.951139\pi$
$853$	−57.7255	−1.97648	−0.988242	+	0.152898i	$0.951139\pi$
$854$	0	0
$855$	3.76030	0.128599
$856$	0	0
$857$	−55.3721	−1.89148	−0.945738	−	0.324930i	$-0.894659\pi$
$857$	−55.3721	−1.89148	−0.945738	+	0.324930i	$0.894659\pi$
$858$	0	0
$859$	−14.3544	−0.489767	−0.244883	−	0.969553i	$-0.578750\pi$
$859$	−14.3544	−0.489767	−0.244883	+	0.969553i	$0.578750\pi$
$860$	0	0
$861$	−2.59841	−0.0885536
$862$	0	0
$863$	15.9211	0.541961	0.270981	−	0.962585i	$-0.412652\pi$
$863$	15.9211	0.541961	0.270981	+	0.962585i	$0.412652\pi$
$864$	0	0
$865$	6.28288	0.213624
$866$	0	0
$867$	−37.9936	−1.29033
$868$	0	0
$869$	0	0
$870$	0	0
$871$	76.1471	2.58015
$872$	0	0
$873$	21.4353	0.725475
$874$	0	0
$875$	1.14510	0.0387115
$876$	0	0
$877$	−30.2975	−1.02308	−0.511538	−	0.859261i	$-0.670924\pi$
$877$	−30.2975	−1.02308	−0.511538	+	0.859261i	$0.670924\pi$
$878$	0	0
$879$	−1.46170	−0.0493019
$880$	0	0
$881$	−11.1883	−0.376943	−0.188471	−	0.982079i	$-0.560353\pi$
$881$	−11.1883	−0.376943	−0.188471	+	0.982079i	$0.560353\pi$
$882$	0	0
$883$	25.2882	0.851016	0.425508	−	0.904955i	$-0.360095\pi$
$883$	25.2882	0.851016	0.425508	+	0.904955i	$0.360095\pi$
$884$	0	0
$885$	1.70979	0.0574741
$886$	0	0
$887$	−32.2849	−1.08402	−0.542010	−	0.840372i	$-0.682336\pi$
$887$	−32.2849	−1.08402	−0.542010	+	0.840372i	$0.682336\pi$
$888$	0	0
$889$	−13.9158	−0.466720
$890$	0	0
$891$	−66.2206	−2.21847
$892$	0	0
$893$	−18.1892	−0.608679
$894$	0	0
$895$	−10.1524	−0.339358
$896$	0	0
$897$	10.4931	0.350356
$898$	0	0
$899$	21.3133	0.710837
$900$	0	0
$901$	−76.5108	−2.54895
$902$	0	0
$903$	28.2313	0.939479
$904$	0	0
$905$	−11.1451	−0.370476
$906$	0	0
$907$	−1.47848	−0.0490922	−0.0245461	−	0.999699i	$-0.507814\pi$
$907$	−1.47848	−0.0490922	−0.0245461	+	0.999699i	$0.507814\pi$
$908$	0	0
$909$	4.47941	0.148573
$910$	0	0
$911$	5.70979	0.189174	0.0945870	−	0.995517i	$-0.469847\pi$
$911$	5.70979	0.189174	0.0945870	+	0.995517i	$0.469847\pi$
$912$	0	0
$913$	−5.37748	−0.177969
$914$	0	0
$915$	29.7991	0.985127
$916$	0	0
$917$	7.95789	0.262793
$918$	0	0
$919$	22.6265	0.746379	0.373190	−	0.927755i	$-0.378264\pi$
$919$	22.6265	0.746379	0.373190	+	0.927755i	$0.378264\pi$
$920$	0	0
$921$	−57.0334	−1.87932
$922$	0	0
$923$	−14.3921	−0.473723
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	−15.8412	−0.520292
$928$	0	0
$929$	−50.0829	−1.64317	−0.821583	−	0.570089i	$-0.806909\pi$
$929$	−50.0829	−1.64317	−0.821583	+	0.570089i	$0.806909\pi$
$930$	0	0
$931$	13.3574	0.437770
$932$	0	0
$933$	−29.0230	−0.950172
$934$	0	0
$935$	34.7118	1.13520
$936$	0	0
$937$	9.98428	0.326172	0.163086	−	0.986612i	$-0.447855\pi$
$937$	9.98428	0.326172	0.163086	+	0.986612i	$0.447855\pi$
$938$	0	0
$939$	−42.5613	−1.38894
$940$	0	0
$941$	−36.0809	−1.17620	−0.588101	−	0.808787i	$-0.700124\pi$
$941$	−36.0809	−1.17620	−0.588101	+	0.808787i	$0.700124\pi$
$942$	0	0
$943$	−1.05783	−0.0344476
$944$	0	0
$945$	3.43531	0.111751
$946$	0	0
$947$	−32.4730	−1.05523	−0.527616	−	0.849483i	$-0.676914\pi$
$947$	−32.4730	−1.05523	−0.527616	+	0.849483i	$0.676914\pi$
$948$	0	0
$949$	−30.9495	−1.00466
$950$	0	0
$951$	−7.61718	−0.247004
$952$	0	0
$953$	−39.3427	−1.27443	−0.637217	−	0.770684i	$-0.719915\pi$
$953$	−39.3427	−1.27443	−0.637217	+	0.770684i	$0.719915\pi$
$954$	0	0
$955$	14.6961	0.475554
$956$	0	0
$957$	−47.3500	−1.53061
$958$	0	0
$959$	2.80440	0.0905587
$960$	0	0
$961$	1.36176	0.0439278
$962$	0	0
$963$	20.3323	0.655200
$964$	0	0
$965$	26.4005	0.849863
$966$	0	0
$967$	23.5720	0.758025	0.379013	−	0.925392i	$-0.376264\pi$
$967$	23.5720	0.758025	0.379013	+	0.925392i	$0.376264\pi$
$968$	0	0
$969$	−29.6750	−0.953299
$970$	0	0
$971$	12.9368	0.415163	0.207581	−	0.978218i	$-0.433441\pi$
$971$	12.9368	0.415163	0.207581	+	0.978218i	$0.433441\pi$
$972$	0	0
$973$	−10.5804	−0.339192
$974$	0	0
$975$	10.4931	0.336049
$976$	0	0
$977$	−22.7118	−0.726614	−0.363307	−	0.931669i	$-0.618352\pi$
$977$	−22.7118	−0.726614	−0.363307	+	0.931669i	$0.618352\pi$
$978$	0	0
$979$	−91.2805	−2.91734
$980$	0	0
$981$	5.84850	0.186728
$982$	0	0
$983$	26.1819	0.835072	0.417536	−	0.908660i	$-0.362894\pi$
$983$	26.1819	0.835072	0.417536	+	0.908660i	$0.362894\pi$
$984$	0	0
$985$	4.26076	0.135759
$986$	0	0
$987$	19.0284	0.605680
$988$	0	0
$989$	11.4931	0.365460
$990$	0	0
$991$	−56.3143	−1.78888	−0.894442	−	0.447185i	$-0.852427\pi$
$991$	−56.3143	−1.78888	−0.894442	+	0.447185i	$0.852427\pi$
$992$	0	0
$993$	28.5583	0.906270
$994$	0	0
$995$	−18.0735	−0.572970
$996$	0	0
$997$	−3.74123	−0.118486	−0.0592430	−	0.998244i	$-0.518869\pi$
$997$	−3.74123	−0.118486	−0.0592430	+	0.998244i	$0.518869\pi$
$998$	0	0
$999$	12.0000	0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.a.g.1.1	✓	3
3.2	odd	2		8280.2.a.bo.1.1		3
4.3	odd	2		1840.2.a.t.1.3		3
5.2	odd	4		4600.2.e.r.4049.5		6
5.3	odd	4		4600.2.e.r.4049.2		6
5.4	even	2		4600.2.a.y.1.3		3
8.3	odd	2		7360.2.a.ca.1.1		3
8.5	even	2		7360.2.a.cb.1.3		3
20.19	odd	2		9200.2.a.cd.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.g.1.1	✓	3	1.1	even	1	trivial
1840.2.a.t.1.3		3	4.3	odd	2
4600.2.a.y.1.3		3	5.4	even	2
4600.2.e.r.4049.2		6	5.3	odd	4
4600.2.e.r.4049.5		6	5.2	odd	4
7360.2.a.ca.1.1		3	8.3	odd	2
7360.2.a.cb.1.3		3	8.5	even	2
8280.2.a.bo.1.1		3	3.2	odd	2
9200.2.a.cd.1.1		3	20.19	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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q-2.14510 q^{3} -1.00000 q^{5} -1.14510 q^{7} +1.60147 q^{9} +5.89167 q^{11} -4.89167 q^{13} +2.14510 q^{15} -5.89167 q^{17} -2.34803 q^{19} +2.45636 q^{21} +1.00000 q^{23} +1.00000 q^{25} +3.00000 q^{27} +3.74657 q^{29} +5.68874 q^{31} -12.6382 q^{33} +1.14510 q^{35} +4.00000 q^{37} +10.4931 q^{39} -1.05783 q^{41} +11.4931 q^{43} -1.60147 q^{45} +7.74657 q^{47} -5.68874 q^{49} +12.6382 q^{51} +12.9863 q^{53} -5.89167 q^{55} +5.03677 q^{57} +0.797069 q^{59} +13.8917 q^{61} -1.83384 q^{63} +4.89167 q^{65} -15.5667 q^{67} -2.14510 q^{69} +2.94217 q^{71} +6.32698 q^{73} -2.14510 q^{75} -6.74657 q^{77} -11.2397 q^{81} -0.912726 q^{83} +5.89167 q^{85} -8.03677 q^{87} -15.4931 q^{89} +5.60147 q^{91} -12.2029 q^{93} +2.34803 q^{95} +13.3848 q^{97} +9.43531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{17} + 3 q^{19} + 12 q^{21} + 3 q^{23} + 3 q^{25} + 9 q^{27} + 3 q^{29} + 6 q^{31} - 15 q^{33} - 3 q^{35} + 12 q^{37} + 15 q^{39} - 6 q^{41} + 18 q^{43}+ \cdots + 9 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^17 + 3 * q^19 + 12 * q^21 + 3 * q^23 + 3 * q^25 + 9 * q^27 + 3 * q^29 + 6 * q^31 - 15 * q^33 - 3 * q^35 + 12 * q^37 + 15 * q^39 - 6 * q^41 + 18 * q^43 - 3 * q^45 + 15 * q^47 - 6 * q^49 + 15 * q^51 + 6 * q^53 - 3 * q^55 - 6 * q^57 + 6 * q^59 + 27 * q^61 + 12 * q^63 + 12 * q^67 + 6 * q^71 - 15 * q^73 - 12 * q^77 - 9 * q^81 - 12 * q^83 + 3 * q^85 - 3 * q^87 - 30 * q^89 + 15 * q^91 - 33 * q^93 - 3 * q^95 + 9 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more