LMFDB - Embedded newform 1840.2.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1840,2,Mod(1,1840)] 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("1840.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,2,0,-1,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.6924739719$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{3} +1.00000 q^{5} +0.618034 q^{7} -0.381966 q^{9} +2.85410 q^{11} -7.09017 q^{13} -1.61803 q^{15} +6.09017 q^{17} -1.85410 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +5.47214 q^{27} -9.23607 q^{29} -9.09017 q^{31} -4.61803 q^{33} +0.618034 q^{35} +6.47214 q^{37} +11.4721 q^{39} +3.32624 q^{41} -0.381966 q^{45} +3.70820 q^{47} -6.61803 q^{49} -9.85410 q^{51} +0.472136 q^{53} +2.85410 q^{55} +3.00000 q^{57} -1.70820 q^{59} -9.32624 q^{61} -0.236068 q^{63} -7.09017 q^{65} -14.4721 q^{67} +1.61803 q^{69} +4.09017 q^{71} +3.23607 q^{73} -1.61803 q^{75} +1.76393 q^{77} -1.52786 q^{79} -7.70820 q^{81} +6.94427 q^{83} +6.09017 q^{85} +14.9443 q^{87} -10.4721 q^{89} -4.38197 q^{91} +14.7082 q^{93} -1.85410 q^{95} +12.3820 q^{97} -1.09017 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{5} - q^{7} - 3 q^{9} - q^{11} - 3 q^{13} - q^{15} + q^{17} + 3 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 14 q^{29} - 7 q^{31} - 7 q^{33} - q^{35} + 4 q^{37} + 14 q^{39}+ \cdots + 9 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^5 - q^7 - 3 * q^9 - q^11 - 3 * q^13 - q^15 + q^17 + 3 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 - 14 * q^29 - 7 * q^31 - 7 * q^33 - q^35 + 4 * q^37 + 14 * q^39 - 9 * q^41 - 3 * q^45 - 6 * q^47 - 11 * q^49 - 13 * q^51 - 8 * q^53 - q^55 + 6 * q^57 + 10 * q^59 - 3 * q^61 + 4 * q^63 - 3 * q^65 - 20 * q^67 + q^69 - 3 * q^71 + 2 * q^73 - q^75 + 8 * q^77 - 12 * q^79 - 2 * q^81 - 4 * q^83 + q^85 + 12 * q^87 - 12 * q^89 - 11 * q^91 + 16 * q^93 + 3 * q^95 + 27 * 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
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.618034	0.233595	0.116797	−	0.993156i	$-0.462737\pi$
$7$	0.618034	0.233595	0.116797	+	0.993156i	$0.462737\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	2.85410	0.860544	0.430272	−	0.902699i	$-0.358418\pi$
$11$	2.85410	0.860544	0.430272	+	0.902699i	$0.358418\pi$
$12$	0	0
$13$	−7.09017	−1.96646	−0.983230	−	0.182372i	$-0.941623\pi$
$13$	−7.09017	−1.96646	−0.983230	+	0.182372i	$0.941623\pi$
$14$	0	0
$15$	−1.61803	−0.417775
$16$	0	0
$17$	6.09017	1.47708	0.738542	−	0.674208i	$-0.235515\pi$
$17$	6.09017	1.47708	0.738542	+	0.674208i	$0.235515\pi$
$18$	0	0
$19$	−1.85410	−0.425360	−0.212680	−	0.977122i	$-0.568219\pi$
$19$	−1.85410	−0.425360	−0.212680	+	0.977122i	$0.568219\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	−9.23607	−1.71509	−0.857547	−	0.514405i	$-0.828013\pi$
$29$	−9.23607	−1.71509	−0.857547	+	0.514405i	$0.828013\pi$
$30$	0	0
$31$	−9.09017	−1.63264	−0.816321	−	0.577598i	$-0.803990\pi$
$31$	−9.09017	−1.63264	−0.816321	+	0.577598i	$0.803990\pi$
$32$	0	0
$33$	−4.61803	−0.803897
$34$	0	0
$35$	0.618034	0.104467
$36$	0	0
$37$	6.47214	1.06401	0.532006	−	0.846740i	$-0.321438\pi$
$37$	6.47214	1.06401	0.532006	+	0.846740i	$0.321438\pi$
$38$	0	0
$39$	11.4721	1.83701
$40$	0	0
$41$	3.32624	0.519471	0.259736	−	0.965680i	$-0.416365\pi$
$41$	3.32624	0.519471	0.259736	+	0.965680i	$0.416365\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−0.381966	−0.0569401
$46$	0	0
$47$	3.70820	0.540897	0.270449	−	0.962734i	$-0.412828\pi$
$47$	3.70820	0.540897	0.270449	+	0.962734i	$0.412828\pi$
$48$	0	0
$49$	−6.61803	−0.945433
$50$	0	0
$51$	−9.85410	−1.37985
$52$	0	0
$53$	0.472136	0.0648529	0.0324264	−	0.999474i	$-0.489677\pi$
$53$	0.472136	0.0648529	0.0324264	+	0.999474i	$0.489677\pi$
$54$	0	0
$55$	2.85410	0.384847
$56$	0	0
$57$	3.00000	0.397360
$58$	0	0
$59$	−1.70820	−0.222389	−0.111195	−	0.993799i	$-0.535468\pi$
$59$	−1.70820	−0.222389	−0.111195	+	0.993799i	$0.535468\pi$
$60$	0	0
$61$	−9.32624	−1.19410	−0.597051	−	0.802203i	$-0.703661\pi$
$61$	−9.32624	−1.19410	−0.597051	+	0.802203i	$0.703661\pi$
$62$	0	0
$63$	−0.236068	−0.0297418
$64$	0	0
$65$	−7.09017	−0.879427
$66$	0	0
$67$	−14.4721	−1.76805	−0.884026	−	0.467437i	$-0.845177\pi$
$67$	−14.4721	−1.76805	−0.884026	+	0.467437i	$0.845177\pi$
$68$	0	0
$69$	1.61803	0.194788
$70$	0	0
$71$	4.09017	0.485414	0.242707	−	0.970100i	$-0.421965\pi$
$71$	4.09017	0.485414	0.242707	+	0.970100i	$0.421965\pi$
$72$	0	0
$73$	3.23607	0.378753	0.189377	−	0.981905i	$-0.439353\pi$
$73$	3.23607	0.378753	0.189377	+	0.981905i	$0.439353\pi$
$74$	0	0
$75$	−1.61803	−0.186834
$76$	0	0
$77$	1.76393	0.201019
$78$	0	0
$79$	−1.52786	−0.171898	−0.0859491	−	0.996300i	$-0.527392\pi$
$79$	−1.52786	−0.171898	−0.0859491	+	0.996300i	$0.527392\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	6.94427	0.762233	0.381116	−	0.924527i	$-0.375540\pi$
$83$	6.94427	0.762233	0.381116	+	0.924527i	$0.375540\pi$
$84$	0	0
$85$	6.09017	0.660572
$86$	0	0
$87$	14.9443	1.60219
$88$	0	0
$89$	−10.4721	−1.11004	−0.555022	−	0.831836i	$-0.687290\pi$
$89$	−10.4721	−1.11004	−0.555022	+	0.831836i	$0.687290\pi$
$90$	0	0
$91$	−4.38197	−0.459355
$92$	0	0
$93$	14.7082	1.52517
$94$	0	0
$95$	−1.85410	−0.190227
$96$	0	0
$97$	12.3820	1.25720	0.628599	−	0.777730i	$-0.283629\pi$
$97$	12.3820	1.25720	0.628599	+	0.777730i	$0.283629\pi$
$98$	0	0
$99$	−1.09017	−0.109566
$100$	0	0
$101$	−0.291796	−0.0290348	−0.0145174	−	0.999895i	$-0.504621\pi$
$101$	−0.291796	−0.0290348	−0.0145174	+	0.999895i	$0.504621\pi$
$102$	0	0
$103$	−16.5623	−1.63193	−0.815966	−	0.578100i	$-0.803795\pi$
$103$	−16.5623	−1.63193	−0.815966	+	0.578100i	$0.803795\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−18.1803	−1.75756	−0.878780	−	0.477227i	$-0.841642\pi$
$107$	−18.1803	−1.75756	−0.878780	+	0.477227i	$0.841642\pi$
$108$	0	0
$109$	−11.5623	−1.10747	−0.553734	−	0.832694i	$-0.686798\pi$
$109$	−11.5623	−1.10747	−0.553734	+	0.832694i	$0.686798\pi$
$110$	0	0
$111$	−10.4721	−0.993971
$112$	0	0
$113$	1.05573	0.0993145	0.0496573	−	0.998766i	$-0.484187\pi$
$113$	1.05573	0.0993145	0.0496573	+	0.998766i	$0.484187\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	2.70820	0.250374
$118$	0	0
$119$	3.76393	0.345039
$120$	0	0
$121$	−2.85410	−0.259464
$122$	0	0
$123$	−5.38197	−0.485276
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.1803	−1.43577	−0.717886	−	0.696160i	$-0.754890\pi$
$127$	−16.1803	−1.43577	−0.717886	+	0.696160i	$0.754890\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.94427	−0.257242	−0.128621	−	0.991694i	$-0.541055\pi$
$131$	−2.94427	−0.257242	−0.128621	+	0.991694i	$0.541055\pi$
$132$	0	0
$133$	−1.14590	−0.0993620
$134$	0	0
$135$	5.47214	0.470966
$136$	0	0
$137$	−10.3262	−0.882230	−0.441115	−	0.897451i	$-0.645417\pi$
$137$	−10.3262	−0.882230	−0.441115	+	0.897451i	$0.645417\pi$
$138$	0	0
$139$	−12.7639	−1.08262	−0.541311	−	0.840822i	$-0.682072\pi$
$139$	−12.7639	−1.08262	−0.541311	+	0.840822i	$0.682072\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	−20.2361	−1.69223
$144$	0	0
$145$	−9.23607	−0.767014
$146$	0	0
$147$	10.7082	0.883198
$148$	0	0
$149$	−7.85410	−0.643433	−0.321717	−	0.946836i	$-0.604260\pi$
$149$	−7.85410	−0.643433	−0.321717	+	0.946836i	$0.604260\pi$
$150$	0	0
$151$	2.56231	0.208517	0.104259	−	0.994550i	$-0.466753\pi$
$151$	2.56231	0.208517	0.104259	+	0.994550i	$0.466753\pi$
$152$	0	0
$153$	−2.32624	−0.188065
$154$	0	0
$155$	−9.09017	−0.730140
$156$	0	0
$157$	−3.70820	−0.295947	−0.147973	−	0.988991i	$-0.547275\pi$
$157$	−3.70820	−0.295947	−0.147973	+	0.988991i	$0.547275\pi$
$158$	0	0
$159$	−0.763932	−0.0605838
$160$	0	0
$161$	−0.618034	−0.0487079
$162$	0	0
$163$	−1.38197	−0.108244	−0.0541220	−	0.998534i	$-0.517236\pi$
$163$	−1.38197	−0.108244	−0.0541220	+	0.998534i	$0.517236\pi$
$164$	0	0
$165$	−4.61803	−0.359513
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	37.2705	2.86696
$170$	0	0
$171$	0.708204	0.0541577
$172$	0	0
$173$	−1.43769	−0.109306	−0.0546529	−	0.998505i	$-0.517405\pi$
$173$	−1.43769	−0.109306	−0.0546529	+	0.998505i	$0.517405\pi$
$174$	0	0
$175$	0.618034	0.0467190
$176$	0	0
$177$	2.76393	0.207750
$178$	0	0
$179$	−2.18034	−0.162966	−0.0814831	−	0.996675i	$-0.525966\pi$
$179$	−2.18034	−0.162966	−0.0814831	+	0.996675i	$0.525966\pi$
$180$	0	0
$181$	−12.1459	−0.902797	−0.451399	−	0.892322i	$-0.649075\pi$
$181$	−12.1459	−0.902797	−0.451399	+	0.892322i	$0.649075\pi$
$182$	0	0
$183$	15.0902	1.11550
$184$	0	0
$185$	6.47214	0.475841
$186$	0	0
$187$	17.3820	1.27110
$188$	0	0
$189$	3.38197	0.246002
$190$	0	0
$191$	13.7082	0.991891	0.495945	−	0.868354i	$-0.334822\pi$
$191$	13.7082	0.991891	0.495945	+	0.868354i	$0.334822\pi$
$192$	0	0
$193$	0.763932	0.0549890	0.0274945	−	0.999622i	$-0.491247\pi$
$193$	0.763932	0.0549890	0.0274945	+	0.999622i	$0.491247\pi$
$194$	0	0
$195$	11.4721	0.821537
$196$	0	0
$197$	−22.5623	−1.60750	−0.803749	−	0.594969i	$-0.797164\pi$
$197$	−22.5623	−1.60750	−0.803749	+	0.594969i	$0.797164\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	23.4164	1.65167
$202$	0	0
$203$	−5.70820	−0.400637
$204$	0	0
$205$	3.32624	0.232315
$206$	0	0
$207$	0.381966	0.0265485
$208$	0	0
$209$	−5.29180	−0.366041
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	0	0
$213$	−6.61803	−0.453460
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.61803	−0.381377
$218$	0	0
$219$	−5.23607	−0.353821
$220$	0	0
$221$	−43.1803	−2.90462
$222$	0	0
$223$	20.9443	1.40253	0.701266	−	0.712900i	$-0.252618\pi$
$223$	20.9443	1.40253	0.701266	+	0.712900i	$0.252618\pi$
$224$	0	0
$225$	−0.381966	−0.0254644
$226$	0	0
$227$	18.7639	1.24541	0.622703	−	0.782458i	$-0.286035\pi$
$227$	18.7639	1.24541	0.622703	+	0.782458i	$0.286035\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−2.85410	−0.187786
$232$	0	0
$233$	6.29180	0.412189	0.206095	−	0.978532i	$-0.433925\pi$
$233$	6.29180	0.412189	0.206095	+	0.978532i	$0.433925\pi$
$234$	0	0
$235$	3.70820	0.241897
$236$	0	0
$237$	2.47214	0.160582
$238$	0	0
$239$	20.3607	1.31702	0.658511	−	0.752571i	$-0.271186\pi$
$239$	20.3607	1.31702	0.658511	+	0.752571i	$0.271186\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	−6.61803	−0.422811
$246$	0	0
$247$	13.1459	0.836453
$248$	0	0
$249$	−11.2361	−0.712057
$250$	0	0
$251$	−6.14590	−0.387926	−0.193963	−	0.981009i	$-0.562134\pi$
$251$	−6.14590	−0.387926	−0.193963	+	0.981009i	$0.562134\pi$
$252$	0	0
$253$	−2.85410	−0.179436
$254$	0	0
$255$	−9.85410	−0.617088
$256$	0	0
$257$	−7.81966	−0.487777	−0.243888	−	0.969803i	$-0.578423\pi$
$257$	−7.81966	−0.487777	−0.243888	+	0.969803i	$0.578423\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	3.52786	0.218369
$262$	0	0
$263$	−20.7426	−1.27905	−0.639523	−	0.768772i	$-0.720868\pi$
$263$	−20.7426	−1.27905	−0.639523	+	0.768772i	$0.720868\pi$
$264$	0	0
$265$	0.472136	0.0290031
$266$	0	0
$267$	16.9443	1.03697
$268$	0	0
$269$	−14.1803	−0.864591	−0.432295	−	0.901732i	$-0.642296\pi$
$269$	−14.1803	−0.864591	−0.432295	+	0.901732i	$0.642296\pi$
$270$	0	0
$271$	30.3262	1.84219	0.921094	−	0.389341i	$-0.127297\pi$
$271$	30.3262	1.84219	0.921094	+	0.389341i	$0.127297\pi$
$272$	0	0
$273$	7.09017	0.429117
$274$	0	0
$275$	2.85410	0.172109
$276$	0	0
$277$	29.4164	1.76746	0.883730	−	0.467996i	$-0.155024\pi$
$277$	29.4164	1.76746	0.883730	+	0.467996i	$0.155024\pi$
$278$	0	0
$279$	3.47214	0.207871
$280$	0	0
$281$	22.7639	1.35798	0.678991	−	0.734146i	$-0.262417\pi$
$281$	22.7639	1.35798	0.678991	+	0.734146i	$0.262417\pi$
$282$	0	0
$283$	26.9443	1.60167	0.800835	−	0.598885i	$-0.204389\pi$
$283$	26.9443	1.60167	0.800835	+	0.598885i	$0.204389\pi$
$284$	0	0
$285$	3.00000	0.177705
$286$	0	0
$287$	2.05573	0.121346
$288$	0	0
$289$	20.0902	1.18177
$290$	0	0
$291$	−20.0344	−1.17444
$292$	0	0
$293$	−19.8885	−1.16190	−0.580951	−	0.813939i	$-0.697319\pi$
$293$	−19.8885	−1.16190	−0.580951	+	0.813939i	$0.697319\pi$
$294$	0	0
$295$	−1.70820	−0.0994555
$296$	0	0
$297$	15.6180	0.906250
$298$	0	0
$299$	7.09017	0.410035
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0.472136	0.0271235
$304$	0	0
$305$	−9.32624	−0.534019
$306$	0	0
$307$	28.4508	1.62378	0.811888	−	0.583813i	$-0.198440\pi$
$307$	28.4508	1.62378	0.811888	+	0.583813i	$0.198440\pi$
$308$	0	0
$309$	26.7984	1.52451
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	12.7984	0.723407	0.361703	−	0.932293i	$-0.382195\pi$
$313$	12.7984	0.723407	0.361703	+	0.932293i	$0.382195\pi$
$314$	0	0
$315$	−0.236068	−0.0133009
$316$	0	0
$317$	−11.0902	−0.622886	−0.311443	−	0.950265i	$-0.600812\pi$
$317$	−11.0902	−0.622886	−0.311443	+	0.950265i	$0.600812\pi$
$318$	0	0
$319$	−26.3607	−1.47591
$320$	0	0
$321$	29.4164	1.64186
$322$	0	0
$323$	−11.2918	−0.628292
$324$	0	0
$325$	−7.09017	−0.393292
$326$	0	0
$327$	18.7082	1.03457
$328$	0	0
$329$	2.29180	0.126351
$330$	0	0
$331$	−19.2361	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$331$	−19.2361	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$332$	0	0
$333$	−2.47214	−0.135472
$334$	0	0
$335$	−14.4721	−0.790697
$336$	0	0
$337$	13.6738	0.744857	0.372429	−	0.928061i	$-0.378525\pi$
$337$	13.6738	0.744857	0.372429	+	0.928061i	$0.378525\pi$
$338$	0	0
$339$	−1.70820	−0.0927769
$340$	0	0
$341$	−25.9443	−1.40496
$342$	0	0
$343$	−8.41641	−0.454443
$344$	0	0
$345$	1.61803	0.0871120
$346$	0	0
$347$	6.38197	0.342602	0.171301	−	0.985219i	$-0.445203\pi$
$347$	6.38197	0.342602	0.171301	+	0.985219i	$0.445203\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−38.7984	−2.07090
$352$	0	0
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	4.09017	0.217084
$356$	0	0
$357$	−6.09017	−0.322326
$358$	0	0
$359$	−26.3607	−1.39126	−0.695632	−	0.718399i	$-0.744875\pi$
$359$	−26.3607	−1.39126	−0.695632	+	0.718399i	$0.744875\pi$
$360$	0	0
$361$	−15.5623	−0.819069
$362$	0	0
$363$	4.61803	0.242384
$364$	0	0
$365$	3.23607	0.169384
$366$	0	0
$367$	−6.47214	−0.337843	−0.168921	−	0.985630i	$-0.554028\pi$
$367$	−6.47214	−0.337843	−0.168921	+	0.985630i	$0.554028\pi$
$368$	0	0
$369$	−1.27051	−0.0661401
$370$	0	0
$371$	0.291796	0.0151493
$372$	0	0
$373$	20.1803	1.04490	0.522449	−	0.852670i	$-0.325018\pi$
$373$	20.1803	1.04490	0.522449	+	0.852670i	$0.325018\pi$
$374$	0	0
$375$	−1.61803	−0.0835549
$376$	0	0
$377$	65.4853	3.37266
$378$	0	0
$379$	22.4508	1.15322	0.576611	−	0.817019i	$-0.304375\pi$
$379$	22.4508	1.15322	0.576611	+	0.817019i	$0.304375\pi$
$380$	0	0
$381$	26.1803	1.34126
$382$	0	0
$383$	17.8885	0.914062	0.457031	−	0.889451i	$-0.348913\pi$
$383$	17.8885	0.914062	0.457031	+	0.889451i	$0.348913\pi$
$384$	0	0
$385$	1.76393	0.0898983
$386$	0	0
$387$	0	0
$388$	0	0
$389$	21.3262	1.08128	0.540642	−	0.841253i	$-0.318182\pi$
$389$	21.3262	1.08128	0.540642	+	0.841253i	$0.318182\pi$
$390$	0	0
$391$	−6.09017	−0.307993
$392$	0	0
$393$	4.76393	0.240309
$394$	0	0
$395$	−1.52786	−0.0768752
$396$	0	0
$397$	7.32624	0.367693	0.183847	−	0.982955i	$-0.441145\pi$
$397$	7.32624	0.367693	0.183847	+	0.982955i	$0.441145\pi$
$398$	0	0
$399$	1.85410	0.0928212
$400$	0	0
$401$	−1.70820	−0.0853036	−0.0426518	−	0.999090i	$-0.513581\pi$
$401$	−1.70820	−0.0853036	−0.0426518	+	0.999090i	$0.513581\pi$
$402$	0	0
$403$	64.4508	3.21053
$404$	0	0
$405$	−7.70820	−0.383024
$406$	0	0
$407$	18.4721	0.915630
$408$	0	0
$409$	−30.2148	−1.49402	−0.747012	−	0.664810i	$-0.768512\pi$
$409$	−30.2148	−1.49402	−0.747012	+	0.664810i	$0.768512\pi$
$410$	0	0
$411$	16.7082	0.824155
$412$	0	0
$413$	−1.05573	−0.0519490
$414$	0	0
$415$	6.94427	0.340881
$416$	0	0
$417$	20.6525	1.01136
$418$	0	0
$419$	−14.4721	−0.707010	−0.353505	−	0.935433i	$-0.615010\pi$
$419$	−14.4721	−0.707010	−0.353505	+	0.935433i	$0.615010\pi$
$420$	0	0
$421$	13.7426	0.669776	0.334888	−	0.942258i	$-0.391302\pi$
$421$	13.7426	0.669776	0.334888	+	0.942258i	$0.391302\pi$
$422$	0	0
$423$	−1.41641	−0.0688681
$424$	0	0
$425$	6.09017	0.295417
$426$	0	0
$427$	−5.76393	−0.278936
$428$	0	0
$429$	32.7426	1.58083
$430$	0	0
$431$	−3.34752	−0.161245	−0.0806223	−	0.996745i	$-0.525691\pi$
$431$	−3.34752	−0.161245	−0.0806223	+	0.996745i	$0.525691\pi$
$432$	0	0
$433$	8.50658	0.408800	0.204400	−	0.978887i	$-0.434476\pi$
$433$	8.50658	0.408800	0.204400	+	0.978887i	$0.434476\pi$
$434$	0	0
$435$	14.9443	0.716523
$436$	0	0
$437$	1.85410	0.0886937
$438$	0	0
$439$	13.3820	0.638686	0.319343	−	0.947639i	$-0.396538\pi$
$439$	13.3820	0.638686	0.319343	+	0.947639i	$0.396538\pi$
$440$	0	0
$441$	2.52786	0.120374
$442$	0	0
$443$	−25.0902	−1.19207	−0.596035	−	0.802958i	$-0.703258\pi$
$443$	−25.0902	−1.19207	−0.596035	+	0.802958i	$0.703258\pi$
$444$	0	0
$445$	−10.4721	−0.496427
$446$	0	0
$447$	12.7082	0.601077
$448$	0	0
$449$	−1.56231	−0.0737298	−0.0368649	−	0.999320i	$-0.511737\pi$
$449$	−1.56231	−0.0737298	−0.0368649	+	0.999320i	$0.511737\pi$
$450$	0	0
$451$	9.49342	0.447028
$452$	0	0
$453$	−4.14590	−0.194791
$454$	0	0
$455$	−4.38197	−0.205430
$456$	0	0
$457$	−37.7771	−1.76714	−0.883569	−	0.468301i	$-0.844866\pi$
$457$	−37.7771	−1.76714	−0.883569	+	0.468301i	$0.844866\pi$
$458$	0	0
$459$	33.3262	1.55554
$460$	0	0
$461$	−39.2361	−1.82741	−0.913703	−	0.406383i	$-0.866790\pi$
$461$	−39.2361	−1.82741	−0.913703	+	0.406383i	$0.866790\pi$
$462$	0	0
$463$	−2.00000	−0.0929479	−0.0464739	−	0.998920i	$-0.514798\pi$
$463$	−2.00000	−0.0929479	−0.0464739	+	0.998920i	$0.514798\pi$
$464$	0	0
$465$	14.7082	0.682077
$466$	0	0
$467$	17.1246	0.792433	0.396216	−	0.918157i	$-0.370323\pi$
$467$	17.1246	0.792433	0.396216	+	0.918157i	$0.370323\pi$
$468$	0	0
$469$	−8.94427	−0.413008
$470$	0	0
$471$	6.00000	0.276465
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.85410	−0.0850720
$476$	0	0
$477$	−0.180340	−0.00825720
$478$	0	0
$479$	31.8885	1.45702	0.728512	−	0.685033i	$-0.240212\pi$
$479$	31.8885	1.45702	0.728512	+	0.685033i	$0.240212\pi$
$480$	0	0
$481$	−45.8885	−2.09234
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	12.3820	0.562236
$486$	0	0
$487$	19.8197	0.898115	0.449057	−	0.893503i	$-0.351760\pi$
$487$	19.8197	0.898115	0.449057	+	0.893503i	$0.351760\pi$
$488$	0	0
$489$	2.23607	0.101118
$490$	0	0
$491$	−6.18034	−0.278915	−0.139457	−	0.990228i	$-0.544536\pi$
$491$	−6.18034	−0.278915	−0.139457	+	0.990228i	$0.544536\pi$
$492$	0	0
$493$	−56.2492	−2.53334
$494$	0	0
$495$	−1.09017	−0.0489995
$496$	0	0
$497$	2.52786	0.113390
$498$	0	0
$499$	−12.3607	−0.553340	−0.276670	−	0.960965i	$-0.589231\pi$
$499$	−12.3607	−0.553340	−0.276670	+	0.960965i	$0.589231\pi$
$500$	0	0
$501$	−12.9443	−0.578307
$502$	0	0
$503$	−36.3262	−1.61971	−0.809853	−	0.586632i	$-0.800453\pi$
$503$	−36.3262	−1.61971	−0.809853	+	0.586632i	$0.800453\pi$
$504$	0	0
$505$	−0.291796	−0.0129848
$506$	0	0
$507$	−60.3050	−2.67824
$508$	0	0
$509$	36.6525	1.62459	0.812296	−	0.583245i	$-0.198217\pi$
$509$	36.6525	1.62459	0.812296	+	0.583245i	$0.198217\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	−10.1459	−0.447952
$514$	0	0
$515$	−16.5623	−0.729822
$516$	0	0
$517$	10.5836	0.465466
$518$	0	0
$519$	2.32624	0.102111
$520$	0	0
$521$	−15.5279	−0.680288	−0.340144	−	0.940373i	$-0.610476\pi$
$521$	−15.5279	−0.680288	−0.340144	+	0.940373i	$0.610476\pi$
$522$	0	0
$523$	26.0000	1.13690	0.568450	−	0.822718i	$-0.307543\pi$
$523$	26.0000	1.13690	0.568450	+	0.822718i	$0.307543\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	−55.3607	−2.41155
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.652476	0.0283150
$532$	0	0
$533$	−23.5836	−1.02152
$534$	0	0
$535$	−18.1803	−0.786005
$536$	0	0
$537$	3.52786	0.152239
$538$	0	0
$539$	−18.8885	−0.813587
$540$	0	0
$541$	22.8328	0.981659	0.490830	−	0.871256i	$-0.336694\pi$
$541$	22.8328	0.981659	0.490830	+	0.871256i	$0.336694\pi$
$542$	0	0
$543$	19.6525	0.843368
$544$	0	0
$545$	−11.5623	−0.495275
$546$	0	0
$547$	−27.9230	−1.19390	−0.596950	−	0.802278i	$-0.703621\pi$
$547$	−27.9230	−1.19390	−0.596950	+	0.802278i	$0.703621\pi$
$548$	0	0
$549$	3.56231	0.152036
$550$	0	0
$551$	17.1246	0.729533
$552$	0	0
$553$	−0.944272	−0.0401545
$554$	0	0
$555$	−10.4721	−0.444517
$556$	0	0
$557$	−22.8328	−0.967457	−0.483729	−	0.875218i	$-0.660718\pi$
$557$	−22.8328	−0.967457	−0.483729	+	0.875218i	$0.660718\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−28.1246	−1.18742
$562$	0	0
$563$	13.8885	0.585332	0.292666	−	0.956215i	$-0.405458\pi$
$563$	13.8885	0.585332	0.292666	+	0.956215i	$0.405458\pi$
$564$	0	0
$565$	1.05573	0.0444148
$566$	0	0
$567$	−4.76393	−0.200066
$568$	0	0
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	−15.9787	−0.668688	−0.334344	−	0.942451i	$-0.608515\pi$
$571$	−15.9787	−0.668688	−0.334344	+	0.942451i	$0.608515\pi$
$572$	0	0
$573$	−22.1803	−0.926597
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−3.52786	−0.146867	−0.0734335	−	0.997300i	$-0.523396\pi$
$577$	−3.52786	−0.146867	−0.0734335	+	0.997300i	$0.523396\pi$
$578$	0	0
$579$	−1.23607	−0.0513692
$580$	0	0
$581$	4.29180	0.178054
$582$	0	0
$583$	1.34752	0.0558087
$584$	0	0
$585$	2.70820	0.111970
$586$	0	0
$587$	−13.6180	−0.562076	−0.281038	−	0.959697i	$-0.590679\pi$
$587$	−13.6180	−0.562076	−0.281038	+	0.959697i	$0.590679\pi$
$588$	0	0
$589$	16.8541	0.694461
$590$	0	0
$591$	36.5066	1.50168
$592$	0	0
$593$	39.2361	1.61123	0.805616	−	0.592438i	$-0.201834\pi$
$593$	39.2361	1.61123	0.805616	+	0.592438i	$0.201834\pi$
$594$	0	0
$595$	3.76393	0.154306
$596$	0	0
$597$	3.23607	0.132443
$598$	0	0
$599$	18.3820	0.751067	0.375533	−	0.926809i	$-0.377460\pi$
$599$	18.3820	0.751067	0.375533	+	0.926809i	$0.377460\pi$
$600$	0	0
$601$	−33.2705	−1.35713	−0.678566	−	0.734539i	$-0.737398\pi$
$601$	−33.2705	−1.35713	−0.678566	+	0.734539i	$0.737398\pi$
$602$	0	0
$603$	5.52786	0.225112
$604$	0	0
$605$	−2.85410	−0.116036
$606$	0	0
$607$	−26.4721	−1.07447	−0.537235	−	0.843432i	$-0.680531\pi$
$607$	−26.4721	−1.07447	−0.537235	+	0.843432i	$0.680531\pi$
$608$	0	0
$609$	9.23607	0.374264
$610$	0	0
$611$	−26.2918	−1.06365
$612$	0	0
$613$	19.3050	0.779720	0.389860	−	0.920874i	$-0.372523\pi$
$613$	19.3050	0.779720	0.389860	+	0.920874i	$0.372523\pi$
$614$	0	0
$615$	−5.38197	−0.217022
$616$	0	0
$617$	34.0902	1.37242	0.686209	−	0.727404i	$-0.259273\pi$
$617$	34.0902	1.37242	0.686209	+	0.727404i	$0.259273\pi$
$618$	0	0
$619$	2.79837	0.112476	0.0562381	−	0.998417i	$-0.482089\pi$
$619$	2.79837	0.112476	0.0562381	+	0.998417i	$0.482089\pi$
$620$	0	0
$621$	−5.47214	−0.219589
$622$	0	0
$623$	−6.47214	−0.259301
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.56231	0.341946
$628$	0	0
$629$	39.4164	1.57164
$630$	0	0
$631$	−42.0689	−1.67474	−0.837368	−	0.546640i	$-0.815907\pi$
$631$	−42.0689	−1.67474	−0.837368	+	0.546640i	$0.815907\pi$
$632$	0	0
$633$	22.6525	0.900355
$634$	0	0
$635$	−16.1803	−0.642097
$636$	0	0
$637$	46.9230	1.85916
$638$	0	0
$639$	−1.56231	−0.0618039
$640$	0	0
$641$	0.360680	0.0142460	0.00712300	−	0.999975i	$-0.497733\pi$
$641$	0.360680	0.0142460	0.00712300	+	0.999975i	$0.497733\pi$
$642$	0	0
$643$	8.29180	0.326997	0.163498	−	0.986544i	$-0.447722\pi$
$643$	8.29180	0.326997	0.163498	+	0.986544i	$0.447722\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.2492	1.42510	0.712552	−	0.701619i	$-0.247539\pi$
$647$	36.2492	1.42510	0.712552	+	0.701619i	$0.247539\pi$
$648$	0	0
$649$	−4.87539	−0.191376
$650$	0	0
$651$	9.09017	0.356272
$652$	0	0
$653$	−8.03444	−0.314412	−0.157206	−	0.987566i	$-0.550249\pi$
$653$	−8.03444	−0.314412	−0.157206	+	0.987566i	$0.550249\pi$
$654$	0	0
$655$	−2.94427	−0.115042
$656$	0	0
$657$	−1.23607	−0.0482236
$658$	0	0
$659$	46.2492	1.80161	0.900807	−	0.434220i	$-0.142976\pi$
$659$	46.2492	1.80161	0.900807	+	0.434220i	$0.142976\pi$
$660$	0	0
$661$	18.6738	0.726325	0.363163	−	0.931726i	$-0.381697\pi$
$661$	18.6738	0.726325	0.363163	+	0.931726i	$0.381697\pi$
$662$	0	0
$663$	69.8673	2.71342
$664$	0	0
$665$	−1.14590	−0.0444360
$666$	0	0
$667$	9.23607	0.357622
$668$	0	0
$669$	−33.8885	−1.31021
$670$	0	0
$671$	−26.6180	−1.02758
$672$	0	0
$673$	−10.9443	−0.421871	−0.210935	−	0.977500i	$-0.567651\pi$
$673$	−10.9443	−0.421871	−0.210935	+	0.977500i	$0.567651\pi$
$674$	0	0
$675$	5.47214	0.210623
$676$	0	0
$677$	−50.9443	−1.95795	−0.978974	−	0.203986i	$-0.934610\pi$
$677$	−50.9443	−1.95795	−0.978974	+	0.203986i	$0.934610\pi$
$678$	0	0
$679$	7.65248	0.293675
$680$	0	0
$681$	−30.3607	−1.16342
$682$	0	0
$683$	31.5623	1.20770	0.603849	−	0.797099i	$-0.293633\pi$
$683$	31.5623	1.20770	0.603849	+	0.797099i	$0.293633\pi$
$684$	0	0
$685$	−10.3262	−0.394545
$686$	0	0
$687$	−16.1803	−0.617318
$688$	0	0
$689$	−3.34752	−0.127531
$690$	0	0
$691$	29.2361	1.11219	0.556096	−	0.831118i	$-0.312299\pi$
$691$	29.2361	1.11219	0.556096	+	0.831118i	$0.312299\pi$
$692$	0	0
$693$	−0.673762	−0.0255941
$694$	0	0
$695$	−12.7639	−0.484164
$696$	0	0
$697$	20.2574	0.767302
$698$	0	0
$699$	−10.1803	−0.385056
$700$	0	0
$701$	43.3394	1.63691	0.818453	−	0.574573i	$-0.194832\pi$
$701$	43.3394	1.63691	0.818453	+	0.574573i	$0.194832\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	−6.00000	−0.225973
$706$	0	0
$707$	−0.180340	−0.00678238
$708$	0	0
$709$	−26.0902	−0.979837	−0.489918	−	0.871768i	$-0.662973\pi$
$709$	−26.0902	−0.979837	−0.489918	+	0.871768i	$0.662973\pi$
$710$	0	0
$711$	0.583592	0.0218864
$712$	0	0
$713$	9.09017	0.340430
$714$	0	0
$715$	−20.2361	−0.756786
$716$	0	0
$717$	−32.9443	−1.23033
$718$	0	0
$719$	−35.2705	−1.31537	−0.657684	−	0.753294i	$-0.728464\pi$
$719$	−35.2705	−1.31537	−0.657684	+	0.753294i	$0.728464\pi$
$720$	0	0
$721$	−10.2361	−0.381211
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.23607	−0.343019
$726$	0	0
$727$	−28.2016	−1.04594	−0.522970	−	0.852351i	$-0.675176\pi$
$727$	−28.2016	−1.04594	−0.522970	+	0.852351i	$0.675176\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−29.4164	−1.08652	−0.543260	−	0.839565i	$-0.682810\pi$
$733$	−29.4164	−1.08652	−0.543260	+	0.839565i	$0.682810\pi$
$734$	0	0
$735$	10.7082	0.394978
$736$	0	0
$737$	−41.3050	−1.52149
$738$	0	0
$739$	13.8885	0.510898	0.255449	−	0.966822i	$-0.417777\pi$
$739$	13.8885	0.510898	0.255449	+	0.966822i	$0.417777\pi$
$740$	0	0
$741$	−21.2705	−0.781392
$742$	0	0
$743$	−33.6312	−1.23381	−0.616904	−	0.787038i	$-0.711613\pi$
$743$	−33.6312	−1.23381	−0.616904	+	0.787038i	$0.711613\pi$
$744$	0	0
$745$	−7.85410	−0.287752
$746$	0	0
$747$	−2.65248	−0.0970490
$748$	0	0
$749$	−11.2361	−0.410557
$750$	0	0
$751$	47.0132	1.71553	0.857767	−	0.514038i	$-0.171851\pi$
$751$	47.0132	1.71553	0.857767	+	0.514038i	$0.171851\pi$
$752$	0	0
$753$	9.94427	0.362389
$754$	0	0
$755$	2.56231	0.0932519
$756$	0	0
$757$	−17.8885	−0.650170	−0.325085	−	0.945685i	$-0.605393\pi$
$757$	−17.8885	−0.650170	−0.325085	+	0.945685i	$0.605393\pi$
$758$	0	0
$759$	4.61803	0.167624
$760$	0	0
$761$	46.8673	1.69894	0.849468	−	0.527640i	$-0.176923\pi$
$761$	46.8673	1.69894	0.849468	+	0.527640i	$0.176923\pi$
$762$	0	0
$763$	−7.14590	−0.258699
$764$	0	0
$765$	−2.32624	−0.0841053
$766$	0	0
$767$	12.1115	0.437319
$768$	0	0
$769$	−6.58359	−0.237410	−0.118705	−	0.992930i	$-0.537874\pi$
$769$	−6.58359	−0.237410	−0.118705	+	0.992930i	$0.537874\pi$
$770$	0	0
$771$	12.6525	0.455668
$772$	0	0
$773$	28.9443	1.04105	0.520527	−	0.853845i	$-0.325736\pi$
$773$	28.9443	1.04105	0.520527	+	0.853845i	$0.325736\pi$
$774$	0	0
$775$	−9.09017	−0.326529
$776$	0	0
$777$	−6.47214	−0.232187
$778$	0	0
$779$	−6.16718	−0.220962
$780$	0	0
$781$	11.6738	0.417720
$782$	0	0
$783$	−50.5410	−1.80619
$784$	0	0
$785$	−3.70820	−0.132351
$786$	0	0
$787$	2.87539	0.102497	0.0512483	−	0.998686i	$-0.483680\pi$
$787$	2.87539	0.102497	0.0512483	+	0.998686i	$0.483680\pi$
$788$	0	0
$789$	33.5623	1.19485
$790$	0	0
$791$	0.652476	0.0231994
$792$	0	0
$793$	66.1246	2.34815
$794$	0	0
$795$	−0.763932	−0.0270939
$796$	0	0
$797$	13.7082	0.485569	0.242785	−	0.970080i	$-0.421939\pi$
$797$	13.7082	0.485569	0.242785	+	0.970080i	$0.421939\pi$
$798$	0	0
$799$	22.5836	0.798950
$800$	0	0
$801$	4.00000	0.141333
$802$	0	0
$803$	9.23607	0.325934
$804$	0	0
$805$	−0.618034	−0.0217828
$806$	0	0
$807$	22.9443	0.807677
$808$	0	0
$809$	−46.7426	−1.64338	−0.821692	−	0.569932i	$-0.806970\pi$
$809$	−46.7426	−1.64338	−0.821692	+	0.569932i	$0.806970\pi$
$810$	0	0
$811$	−21.8197	−0.766192	−0.383096	−	0.923709i	$-0.625142\pi$
$811$	−21.8197	−0.766192	−0.383096	+	0.923709i	$0.625142\pi$
$812$	0	0
$813$	−49.0689	−1.72092
$814$	0	0
$815$	−1.38197	−0.0484082
$816$	0	0
$817$	0	0
$818$	0	0
$819$	1.67376	0.0584860
$820$	0	0
$821$	−33.0557	−1.15365	−0.576826	−	0.816867i	$-0.695709\pi$
$821$	−33.0557	−1.15365	−0.576826	+	0.816867i	$0.695709\pi$
$822$	0	0
$823$	−25.4164	−0.885960	−0.442980	−	0.896531i	$-0.646079\pi$
$823$	−25.4164	−0.885960	−0.442980	+	0.896531i	$0.646079\pi$
$824$	0	0
$825$	−4.61803	−0.160779
$826$	0	0
$827$	−21.7082	−0.754868	−0.377434	−	0.926036i	$-0.623194\pi$
$827$	−21.7082	−0.754868	−0.377434	+	0.926036i	$0.623194\pi$
$828$	0	0
$829$	18.9443	0.657962	0.328981	−	0.944337i	$-0.393295\pi$
$829$	18.9443	0.657962	0.328981	+	0.944337i	$0.393295\pi$
$830$	0	0
$831$	−47.5967	−1.65111
$832$	0	0
$833$	−40.3050	−1.39648
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	−49.7426	−1.71936
$838$	0	0
$839$	−33.0132	−1.13974	−0.569870	−	0.821735i	$-0.693007\pi$
$839$	−33.0132	−1.13974	−0.569870	+	0.821735i	$0.693007\pi$
$840$	0	0
$841$	56.3050	1.94155
$842$	0	0
$843$	−36.8328	−1.26859
$844$	0	0
$845$	37.2705	1.28214
$846$	0	0
$847$	−1.76393	−0.0606094
$848$	0	0
$849$	−43.5967	−1.49624
$850$	0	0
$851$	−6.47214	−0.221862
$852$	0	0
$853$	−10.7984	−0.369729	−0.184865	−	0.982764i	$-0.559185\pi$
$853$	−10.7984	−0.369729	−0.184865	+	0.982764i	$0.559185\pi$
$854$	0	0
$855$	0.708204	0.0242201
$856$	0	0
$857$	−6.58359	−0.224891	−0.112446	−	0.993658i	$-0.535868\pi$
$857$	−6.58359	−0.224891	−0.112446	+	0.993658i	$0.535868\pi$
$858$	0	0
$859$	24.0689	0.821220	0.410610	−	0.911811i	$-0.365316\pi$
$859$	24.0689	0.821220	0.410610	+	0.911811i	$0.365316\pi$
$860$	0	0
$861$	−3.32624	−0.113358
$862$	0	0
$863$	−32.7639	−1.11530	−0.557649	−	0.830077i	$-0.688296\pi$
$863$	−32.7639	−1.11530	−0.557649	+	0.830077i	$0.688296\pi$
$864$	0	0
$865$	−1.43769	−0.0488831
$866$	0	0
$867$	−32.5066	−1.10398
$868$	0	0
$869$	−4.36068	−0.147926
$870$	0	0
$871$	102.610	3.47680
$872$	0	0
$873$	−4.72949	−0.160069
$874$	0	0
$875$	0.618034	0.0208934
$876$	0	0
$877$	18.7426	0.632894	0.316447	−	0.948610i	$-0.397510\pi$
$877$	18.7426	0.632894	0.316447	+	0.948610i	$0.397510\pi$
$878$	0	0
$879$	32.1803	1.08542
$880$	0	0
$881$	8.58359	0.289189	0.144594	−	0.989491i	$-0.453812\pi$
$881$	8.58359	0.289189	0.144594	+	0.989491i	$0.453812\pi$
$882$	0	0
$883$	−15.5623	−0.523713	−0.261857	−	0.965107i	$-0.584335\pi$
$883$	−15.5623	−0.523713	−0.261857	+	0.965107i	$0.584335\pi$
$884$	0	0
$885$	2.76393	0.0929086
$886$	0	0
$887$	5.16718	0.173497	0.0867485	−	0.996230i	$-0.472352\pi$
$887$	5.16718	0.173497	0.0867485	+	0.996230i	$0.472352\pi$
$888$	0	0
$889$	−10.0000	−0.335389
$890$	0	0
$891$	−22.0000	−0.737028
$892$	0	0
$893$	−6.87539	−0.230076
$894$	0	0
$895$	−2.18034	−0.0728807
$896$	0	0
$897$	−11.4721	−0.383043
$898$	0	0
$899$	83.9574	2.80014
$900$	0	0
$901$	2.87539	0.0957931
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.1459	−0.403743
$906$	0	0
$907$	−7.12461	−0.236569	−0.118284	−	0.992980i	$-0.537739\pi$
$907$	−7.12461	−0.236569	−0.118284	+	0.992980i	$0.537739\pi$
$908$	0	0
$909$	0.111456	0.00369677
$910$	0	0
$911$	−36.0689	−1.19502	−0.597508	−	0.801863i	$-0.703842\pi$
$911$	−36.0689	−1.19502	−0.597508	+	0.801863i	$0.703842\pi$
$912$	0	0
$913$	19.8197	0.655935
$914$	0	0
$915$	15.0902	0.498866
$916$	0	0
$917$	−1.81966	−0.0600905
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−46.0344	−1.51689
$922$	0	0
$923$	−29.0000	−0.954547
$924$	0	0
$925$	6.47214	0.212803
$926$	0	0
$927$	6.32624	0.207781
$928$	0	0
$929$	−3.52786	−0.115745	−0.0578727	−	0.998324i	$-0.518432\pi$
$929$	−3.52786	−0.115745	−0.0578727	+	0.998324i	$0.518432\pi$
$930$	0	0
$931$	12.2705	0.402150
$932$	0	0
$933$	6.47214	0.211888
$934$	0	0
$935$	17.3820	0.568451
$936$	0	0
$937$	−36.7984	−1.20215	−0.601075	−	0.799192i	$-0.705261\pi$
$937$	−36.7984	−1.20215	−0.601075	+	0.799192i	$0.705261\pi$
$938$	0	0
$939$	−20.7082	−0.675787
$940$	0	0
$941$	−22.4934	−0.733265	−0.366632	−	0.930366i	$-0.619489\pi$
$941$	−22.4934	−0.733265	−0.366632	+	0.930366i	$0.619489\pi$
$942$	0	0
$943$	−3.32624	−0.108317
$944$	0	0
$945$	3.38197	0.110015
$946$	0	0
$947$	54.6869	1.77709	0.888543	−	0.458793i	$-0.151718\pi$
$947$	54.6869	1.77709	0.888543	+	0.458793i	$0.151718\pi$
$948$	0	0
$949$	−22.9443	−0.744803
$950$	0	0
$951$	17.9443	0.581883
$952$	0	0
$953$	3.79837	0.123041	0.0615207	−	0.998106i	$-0.480405\pi$
$953$	3.79837	0.123041	0.0615207	+	0.998106i	$0.480405\pi$
$954$	0	0
$955$	13.7082	0.443587
$956$	0	0
$957$	42.6525	1.37876
$958$	0	0
$959$	−6.38197	−0.206084
$960$	0	0
$961$	51.6312	1.66552
$962$	0	0
$963$	6.94427	0.223776
$964$	0	0
$965$	0.763932	0.0245918
$966$	0	0
$967$	16.5410	0.531923	0.265962	−	0.963984i	$-0.414311\pi$
$967$	16.5410	0.531923	0.265962	+	0.963984i	$0.414311\pi$
$968$	0	0
$969$	18.2705	0.586933
$970$	0	0
$971$	−34.2705	−1.09979	−0.549896	−	0.835233i	$-0.685333\pi$
$971$	−34.2705	−1.09979	−0.549896	+	0.835233i	$0.685333\pi$
$972$	0	0
$973$	−7.88854	−0.252895
$974$	0	0
$975$	11.4721	0.367402
$976$	0	0
$977$	−23.5623	−0.753825	−0.376912	−	0.926249i	$-0.623014\pi$
$977$	−23.5623	−0.753825	−0.376912	+	0.926249i	$0.623014\pi$
$978$	0	0
$979$	−29.8885	−0.955242
$980$	0	0
$981$	4.41641	0.141005
$982$	0	0
$983$	14.2705	0.455159	0.227579	−	0.973760i	$-0.426919\pi$
$983$	14.2705	0.455159	0.227579	+	0.973760i	$0.426919\pi$
$984$	0	0
$985$	−22.5623	−0.718895
$986$	0	0
$987$	−3.70820	−0.118033
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−27.5066	−0.873775	−0.436888	−	0.899516i	$-0.643919\pi$
$991$	−27.5066	−0.873775	−0.436888	+	0.899516i	$0.643919\pi$
$992$	0	0
$993$	31.1246	0.987710
$994$	0	0
$995$	−2.00000	−0.0634043
$996$	0	0
$997$	57.1935	1.81134	0.905668	−	0.423987i	$-0.139370\pi$
$997$	57.1935	1.81134	0.905668	+	0.423987i	$0.139370\pi$
$998$	0	0
$999$	35.4164	1.12053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.a.l.1.1		2
4.3	odd	2		230.2.a.c.1.2	✓	2
5.4	even	2		9200.2.a.bu.1.2		2
8.3	odd	2		7360.2.a.bh.1.1		2
8.5	even	2		7360.2.a.bn.1.2		2
12.11	even	2		2070.2.a.u.1.1		2
20.3	even	4		1150.2.b.i.599.2		4
20.7	even	4		1150.2.b.i.599.3		4
20.19	odd	2		1150.2.a.j.1.1		2
92.91	even	2		5290.2.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.c.1.2	✓	2	4.3	odd	2
1150.2.a.j.1.1		2	20.19	odd	2
1150.2.b.i.599.2		4	20.3	even	4
1150.2.b.i.599.3		4	20.7	even	4
1840.2.a.l.1.1		2	1.1	even	1	trivial
2070.2.a.u.1.1		2	12.11	even	2
5290.2.a.o.1.2		2	92.91	even	2
7360.2.a.bh.1.1		2	8.3	odd	2
7360.2.a.bn.1.2		2	8.5	even	2
9200.2.a.bu.1.2		2	5.4	even	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} +1.00000 q^{5} +0.618034 q^{7} -0.381966 q^{9} +2.85410 q^{11} -7.09017 q^{13} -1.61803 q^{15} +6.09017 q^{17} -1.85410 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +5.47214 q^{27} -9.23607 q^{29} -9.09017 q^{31} -4.61803 q^{33} +0.618034 q^{35} +6.47214 q^{37} +11.4721 q^{39} +3.32624 q^{41} -0.381966 q^{45} +3.70820 q^{47} -6.61803 q^{49} -9.85410 q^{51} +0.472136 q^{53} +2.85410 q^{55} +3.00000 q^{57} -1.70820 q^{59} -9.32624 q^{61} -0.236068 q^{63} -7.09017 q^{65} -14.4721 q^{67} +1.61803 q^{69} +4.09017 q^{71} +3.23607 q^{73} -1.61803 q^{75} +1.76393 q^{77} -1.52786 q^{79} -7.70820 q^{81} +6.94427 q^{83} +6.09017 q^{85} +14.9443 q^{87} -10.4721 q^{89} -4.38197 q^{91} +14.7082 q^{93} -1.85410 q^{95} +12.3820 q^{97} -1.09017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{5} - q^{7} - 3 q^{9} - q^{11} - 3 q^{13} - q^{15} + q^{17} + 3 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 14 q^{29} - 7 q^{31} - 7 q^{33} - q^{35} + 4 q^{37} + 14 q^{39}+ \cdots + 9 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^5 - q^7 - 3 * q^9 - q^11 - 3 * q^13 - q^15 + q^17 + 3 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 - 14 * q^29 - 7 * q^31 - 7 * q^33 - q^35 + 4 * q^37 + 14 * q^39 - 9 * q^41 - 3 * q^45 - 6 * q^47 - 11 * q^49 - 13 * q^51 - 8 * q^53 - q^55 + 6 * q^57 + 10 * q^59 - 3 * q^61 + 4 * q^63 - 3 * q^65 - 20 * q^67 + q^69 - 3 * q^71 + 2 * q^73 - q^75 + 8 * q^77 - 12 * q^79 - 2 * q^81 - 4 * q^83 + q^85 + 12 * q^87 - 12 * q^89 - 11 * q^91 + 16 * q^93 + 3 * q^95 + 27 * q^97 + 9 * q^99

Introduction

L-functions

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Database

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