LMFDB - Embedded newform 5808.2.a.ci.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5808,2,Mod(1,5808)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5808, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5808.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5808 = 2^{4} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5808.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$46.3771134940$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	5808.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.618034 q^{5} -1.00000 q^{7} +1.00000 q^{9} -0.236068 q^{13} -0.618034 q^{15} +1.14590 q^{17} +5.85410 q^{19} -1.00000 q^{21} -0.236068 q^{23} -4.61803 q^{25} +1.00000 q^{27} +6.00000 q^{29} +6.09017 q^{31} +0.618034 q^{35} -6.23607 q^{37} -0.236068 q^{39} -0.236068 q^{41} -6.70820 q^{43} -0.618034 q^{45} +10.0902 q^{47} -6.00000 q^{49} +1.14590 q^{51} -0.381966 q^{53} +5.85410 q^{57} -7.38197 q^{59} +11.5623 q^{61} -1.00000 q^{63} +0.145898 q^{65} -1.85410 q^{67} -0.236068 q^{69} -10.3262 q^{71} +5.70820 q^{73} -4.61803 q^{75} +11.0000 q^{79} +1.00000 q^{81} +1.47214 q^{83} -0.708204 q^{85} +6.00000 q^{87} -8.23607 q^{89} +0.236068 q^{91} +6.09017 q^{93} -3.61803 q^{95} +7.85410 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{13} + q^{15} + 9 q^{17} + 5 q^{19} - 2 q^{21} + 4 q^{23} - 7 q^{25} + 2 q^{27} + 12 q^{29} + q^{31} - q^{35} - 8 q^{37} + 4 q^{39} + 4 q^{41} + q^{45}+ \cdots + 9 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 + 4 * q^13 + q^15 + 9 * q^17 + 5 * q^19 - 2 * q^21 + 4 * q^23 - 7 * q^25 + 2 * q^27 + 12 * q^29 + q^31 - q^35 - 8 * q^37 + 4 * q^39 + 4 * q^41 + q^45 + 9 * q^47 - 12 * q^49 + 9 * q^51 - 3 * q^53 + 5 * q^57 - 17 * q^59 + 3 * q^61 - 2 * q^63 + 7 * q^65 + 3 * q^67 + 4 * q^69 - 5 * q^71 - 2 * q^73 - 7 * q^75 + 22 * q^79 + 2 * q^81 - 6 * q^83 + 12 * q^85 + 12 * q^87 - 12 * q^89 - 4 * q^91 + q^93 - 5 * q^95 + 9 * 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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.618034	−0.276393	−0.138197	−	0.990405i	$-0.544131\pi$
$5$	−0.618034	−0.276393	−0.138197	+	0.990405i	$0.544131\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.236068	−0.0654735	−0.0327367	−	0.999464i	$-0.510422\pi$
$13$	−0.236068	−0.0654735	−0.0327367	+	0.999464i	$0.510422\pi$
$14$	0	0
$15$	−0.618034	−0.159576
$16$	0	0
$17$	1.14590	0.277921	0.138961	−	0.990298i	$-0.455624\pi$
$17$	1.14590	0.277921	0.138961	+	0.990298i	$0.455624\pi$
$18$	0	0
$19$	5.85410	1.34302	0.671512	−	0.740994i	$-0.265645\pi$
$19$	5.85410	1.34302	0.671512	+	0.740994i	$0.265645\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−0.236068	−0.0492236	−0.0246118	−	0.999697i	$-0.507835\pi$
$23$	−0.236068	−0.0492236	−0.0246118	+	0.999697i	$0.507835\pi$
$24$	0	0
$25$	−4.61803	−0.923607
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	6.09017	1.09383	0.546913	−	0.837189i	$-0.315803\pi$
$31$	6.09017	1.09383	0.546913	+	0.837189i	$0.315803\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.618034	0.104467
$36$	0	0
$37$	−6.23607	−1.02520	−0.512602	−	0.858627i	$-0.671318\pi$
$37$	−6.23607	−1.02520	−0.512602	+	0.858627i	$0.671318\pi$
$38$	0	0
$39$	−0.236068	−0.0378011
$40$	0	0
$41$	−0.236068	−0.0368676	−0.0184338	−	0.999830i	$-0.505868\pi$
$41$	−0.236068	−0.0368676	−0.0184338	+	0.999830i	$0.505868\pi$
$42$	0	0
$43$	−6.70820	−1.02299	−0.511496	−	0.859286i	$-0.670908\pi$
$43$	−6.70820	−1.02299	−0.511496	+	0.859286i	$0.670908\pi$
$44$	0	0
$45$	−0.618034	−0.0921311
$46$	0	0
$47$	10.0902	1.47180	0.735901	−	0.677089i	$-0.236759\pi$
$47$	10.0902	1.47180	0.735901	+	0.677089i	$0.236759\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	1.14590	0.160458
$52$	0	0
$53$	−0.381966	−0.0524671	−0.0262335	−	0.999656i	$-0.508351\pi$
$53$	−0.381966	−0.0524671	−0.0262335	+	0.999656i	$0.508351\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.85410	0.775395
$58$	0	0
$59$	−7.38197	−0.961050	−0.480525	−	0.876981i	$-0.659554\pi$
$59$	−7.38197	−0.961050	−0.480525	+	0.876981i	$0.659554\pi$
$60$	0	0
$61$	11.5623	1.48040	0.740201	−	0.672386i	$-0.234730\pi$
$61$	11.5623	1.48040	0.740201	+	0.672386i	$0.234730\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0.145898	0.0180964
$66$	0	0
$67$	−1.85410	−0.226515	−0.113257	−	0.993566i	$-0.536128\pi$
$67$	−1.85410	−0.226515	−0.113257	+	0.993566i	$0.536128\pi$
$68$	0	0
$69$	−0.236068	−0.0284192
$70$	0	0
$71$	−10.3262	−1.22550	−0.612749	−	0.790277i	$-0.709937\pi$
$71$	−10.3262	−1.22550	−0.612749	+	0.790277i	$0.709937\pi$
$72$	0	0
$73$	5.70820	0.668095	0.334047	−	0.942556i	$-0.391585\pi$
$73$	5.70820	0.668095	0.334047	+	0.942556i	$0.391585\pi$
$74$	0	0
$75$	−4.61803	−0.533245
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.47214	0.161588	0.0807940	−	0.996731i	$-0.474254\pi$
$83$	1.47214	0.161588	0.0807940	+	0.996731i	$0.474254\pi$
$84$	0	0
$85$	−0.708204	−0.0768155
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−8.23607	−0.873021	−0.436511	−	0.899699i	$-0.643786\pi$
$89$	−8.23607	−0.873021	−0.436511	+	0.899699i	$0.643786\pi$
$90$	0	0
$91$	0.236068	0.0247466
$92$	0	0
$93$	6.09017	0.631521
$94$	0	0
$95$	−3.61803	−0.371202
$96$	0	0
$97$	7.85410	0.797463	0.398732	−	0.917068i	$-0.369451\pi$
$97$	7.85410	0.797463	0.398732	+	0.917068i	$0.369451\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.2361	1.01853	0.509263	−	0.860611i	$-0.329918\pi$
$101$	10.2361	1.01853	0.509263	+	0.860611i	$0.329918\pi$
$102$	0	0
$103$	10.9443	1.07837	0.539186	−	0.842187i	$-0.318732\pi$
$103$	10.9443	1.07837	0.539186	+	0.842187i	$0.318732\pi$
$104$	0	0
$105$	0.618034	0.0603139
$106$	0	0
$107$	−11.4721	−1.10905	−0.554527	−	0.832166i	$-0.687101\pi$
$107$	−11.4721	−1.10905	−0.554527	+	0.832166i	$0.687101\pi$
$108$	0	0
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	0	0
$111$	−6.23607	−0.591901
$112$	0	0
$113$	13.4721	1.26735	0.633676	−	0.773599i	$-0.281545\pi$
$113$	13.4721	1.26735	0.633676	+	0.773599i	$0.281545\pi$
$114$	0	0
$115$	0.145898	0.0136051
$116$	0	0
$117$	−0.236068	−0.0218245
$118$	0	0
$119$	−1.14590	−0.105044
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−0.236068	−0.0212855
$124$	0	0
$125$	5.94427	0.531672
$126$	0	0
$127$	7.70820	0.683992	0.341996	−	0.939701i	$-0.388897\pi$
$127$	7.70820	0.683992	0.341996	+	0.939701i	$0.388897\pi$
$128$	0	0
$129$	−6.70820	−0.590624
$130$	0	0
$131$	−11.7984	−1.03083	−0.515414	−	0.856941i	$-0.672362\pi$
$131$	−11.7984	−1.03083	−0.515414	+	0.856941i	$0.672362\pi$
$132$	0	0
$133$	−5.85410	−0.507615
$134$	0	0
$135$	−0.618034	−0.0531919
$136$	0	0
$137$	9.76393	0.834189	0.417095	−	0.908863i	$-0.363048\pi$
$137$	9.76393	0.834189	0.417095	+	0.908863i	$0.363048\pi$
$138$	0	0
$139$	14.5623	1.23516	0.617579	−	0.786509i	$-0.288113\pi$
$139$	14.5623	1.23516	0.617579	+	0.786509i	$0.288113\pi$
$140$	0	0
$141$	10.0902	0.849746
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.70820	−0.307950
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	4.23607	0.347032	0.173516	−	0.984831i	$-0.444487\pi$
$149$	4.23607	0.347032	0.173516	+	0.984831i	$0.444487\pi$
$150$	0	0
$151$	1.05573	0.0859139	0.0429570	−	0.999077i	$-0.486322\pi$
$151$	1.05573	0.0859139	0.0429570	+	0.999077i	$0.486322\pi$
$152$	0	0
$153$	1.14590	0.0926404
$154$	0	0
$155$	−3.76393	−0.302326
$156$	0	0
$157$	15.7082	1.25365	0.626826	−	0.779160i	$-0.284354\pi$
$157$	15.7082	1.25365	0.626826	+	0.779160i	$0.284354\pi$
$158$	0	0
$159$	−0.381966	−0.0302919
$160$	0	0
$161$	0.236068	0.0186048
$162$	0	0
$163$	5.14590	0.403058	0.201529	−	0.979483i	$-0.435409\pi$
$163$	5.14590	0.403058	0.201529	+	0.979483i	$0.435409\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0344	0.931253	0.465627	−	0.884981i	$-0.345829\pi$
$167$	12.0344	0.931253	0.465627	+	0.884981i	$0.345829\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	0	0
$171$	5.85410	0.447674
$172$	0	0
$173$	18.0344	1.37113	0.685567	−	0.728010i	$-0.259555\pi$
$173$	18.0344	1.37113	0.685567	+	0.728010i	$0.259555\pi$
$174$	0	0
$175$	4.61803	0.349091
$176$	0	0
$177$	−7.38197	−0.554863
$178$	0	0
$179$	8.52786	0.637402	0.318701	−	0.947855i	$-0.396753\pi$
$179$	8.52786	0.637402	0.318701	+	0.947855i	$0.396753\pi$
$180$	0	0
$181$	2.52786	0.187895	0.0939473	−	0.995577i	$-0.470051\pi$
$181$	2.52786	0.187895	0.0939473	+	0.995577i	$0.470051\pi$
$182$	0	0
$183$	11.5623	0.854710
$184$	0	0
$185$	3.85410	0.283359
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	0.819660	0.0593085	0.0296543	−	0.999560i	$-0.490559\pi$
$191$	0.819660	0.0593085	0.0296543	+	0.999560i	$0.490559\pi$
$192$	0	0
$193$	3.14590	0.226447	0.113223	−	0.993570i	$-0.463882\pi$
$193$	3.14590	0.226447	0.113223	+	0.993570i	$0.463882\pi$
$194$	0	0
$195$	0.145898	0.0104480
$196$	0	0
$197$	−13.0344	−0.928666	−0.464333	−	0.885661i	$-0.653706\pi$
$197$	−13.0344	−0.928666	−0.464333	+	0.885661i	$0.653706\pi$
$198$	0	0
$199$	−6.70820	−0.475532	−0.237766	−	0.971322i	$-0.576415\pi$
$199$	−6.70820	−0.475532	−0.237766	+	0.971322i	$0.576415\pi$
$200$	0	0
$201$	−1.85410	−0.130778
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	0.145898	0.0101900
$206$	0	0
$207$	−0.236068	−0.0164079
$208$	0	0
$209$	0	0
$210$	0	0
$211$	3.61803	0.249076	0.124538	−	0.992215i	$-0.460255\pi$
$211$	3.61803	0.249076	0.124538	+	0.992215i	$0.460255\pi$
$212$	0	0
$213$	−10.3262	−0.707542
$214$	0	0
$215$	4.14590	0.282748
$216$	0	0
$217$	−6.09017	−0.413428
$218$	0	0
$219$	5.70820	0.385725
$220$	0	0
$221$	−0.270510	−0.0181965
$222$	0	0
$223$	−7.18034	−0.480831	−0.240416	−	0.970670i	$-0.577284\pi$
$223$	−7.18034	−0.480831	−0.240416	+	0.970670i	$0.577284\pi$
$224$	0	0
$225$	−4.61803	−0.307869
$226$	0	0
$227$	13.1803	0.874810	0.437405	−	0.899265i	$-0.355898\pi$
$227$	13.1803	0.874810	0.437405	+	0.899265i	$0.355898\pi$
$228$	0	0
$229$	0.472136	0.0311996	0.0155998	−	0.999878i	$-0.495034\pi$
$229$	0.472136	0.0311996	0.0155998	+	0.999878i	$0.495034\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.14590	0.271607	0.135803	−	0.990736i	$-0.456638\pi$
$233$	4.14590	0.271607	0.135803	+	0.990736i	$0.456638\pi$
$234$	0	0
$235$	−6.23607	−0.406796
$236$	0	0
$237$	11.0000	0.714527
$238$	0	0
$239$	0.381966	0.0247073	0.0123537	−	0.999924i	$-0.496068\pi$
$239$	0.381966	0.0247073	0.0123537	+	0.999924i	$0.496068\pi$
$240$	0	0
$241$	−8.29180	−0.534122	−0.267061	−	0.963680i	$-0.586052\pi$
$241$	−8.29180	−0.534122	−0.267061	+	0.963680i	$0.586052\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.70820	0.236908
$246$	0	0
$247$	−1.38197	−0.0879324
$248$	0	0
$249$	1.47214	0.0932928
$250$	0	0
$251$	21.9787	1.38728	0.693642	−	0.720320i	$-0.256005\pi$
$251$	21.9787	1.38728	0.693642	+	0.720320i	$0.256005\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−0.708204	−0.0443495
$256$	0	0
$257$	29.7426	1.85530	0.927648	−	0.373457i	$-0.121828\pi$
$257$	29.7426	1.85530	0.927648	+	0.373457i	$0.121828\pi$
$258$	0	0
$259$	6.23607	0.387490
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−15.2705	−0.941620	−0.470810	−	0.882235i	$-0.656038\pi$
$263$	−15.2705	−0.941620	−0.470810	+	0.882235i	$0.656038\pi$
$264$	0	0
$265$	0.236068	0.0145015
$266$	0	0
$267$	−8.23607	−0.504039
$268$	0	0
$269$	−25.4164	−1.54967	−0.774833	−	0.632166i	$-0.782166\pi$
$269$	−25.4164	−1.54967	−0.774833	+	0.632166i	$0.782166\pi$
$270$	0	0
$271$	18.6180	1.13097	0.565483	−	0.824760i	$-0.308690\pi$
$271$	18.6180	1.13097	0.565483	+	0.824760i	$0.308690\pi$
$272$	0	0
$273$	0.236068	0.0142875
$274$	0	0
$275$	0	0
$276$	0	0
$277$	29.2148	1.75535	0.877673	−	0.479260i	$-0.159095\pi$
$277$	29.2148	1.75535	0.877673	+	0.479260i	$0.159095\pi$
$278$	0	0
$279$	6.09017	0.364609
$280$	0	0
$281$	−24.7639	−1.47729	−0.738646	−	0.674093i	$-0.764535\pi$
$281$	−24.7639	−1.47729	−0.738646	+	0.674093i	$0.764535\pi$
$282$	0	0
$283$	−5.70820	−0.339318	−0.169659	−	0.985503i	$-0.554267\pi$
$283$	−5.70820	−0.339318	−0.169659	+	0.985503i	$0.554267\pi$
$284$	0	0
$285$	−3.61803	−0.214314
$286$	0	0
$287$	0.236068	0.0139347
$288$	0	0
$289$	−15.6869	−0.922760
$290$	0	0
$291$	7.85410	0.460416
$292$	0	0
$293$	21.6525	1.26495	0.632476	−	0.774580i	$-0.282039\pi$
$293$	21.6525	1.26495	0.632476	+	0.774580i	$0.282039\pi$
$294$	0	0
$295$	4.56231	0.265628
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.0557281	0.00322284
$300$	0	0
$301$	6.70820	0.386654
$302$	0	0
$303$	10.2361	0.588047
$304$	0	0
$305$	−7.14590	−0.409173
$306$	0	0
$307$	27.9787	1.59683	0.798415	−	0.602108i	$-0.205672\pi$
$307$	27.9787	1.59683	0.798415	+	0.602108i	$0.205672\pi$
$308$	0	0
$309$	10.9443	0.622598
$310$	0	0
$311$	−11.6525	−0.660751	−0.330376	−	0.943850i	$-0.607175\pi$
$311$	−11.6525	−0.660751	−0.330376	+	0.943850i	$0.607175\pi$
$312$	0	0
$313$	−2.52786	−0.142883	−0.0714417	−	0.997445i	$-0.522760\pi$
$313$	−2.52786	−0.142883	−0.0714417	+	0.997445i	$0.522760\pi$
$314$	0	0
$315$	0.618034	0.0348223
$316$	0	0
$317$	−6.81966	−0.383030	−0.191515	−	0.981490i	$-0.561340\pi$
$317$	−6.81966	−0.383030	−0.191515	+	0.981490i	$0.561340\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−11.4721	−0.640312
$322$	0	0
$323$	6.70820	0.373254
$324$	0	0
$325$	1.09017	0.0604717
$326$	0	0
$327$	12.0000	0.663602
$328$	0	0
$329$	−10.0902	−0.556289
$330$	0	0
$331$	−16.7082	−0.918366	−0.459183	−	0.888342i	$-0.651858\pi$
$331$	−16.7082	−0.918366	−0.459183	+	0.888342i	$0.651858\pi$
$332$	0	0
$333$	−6.23607	−0.341734
$334$	0	0
$335$	1.14590	0.0626071
$336$	0	0
$337$	18.1803	0.990346	0.495173	−	0.868794i	$-0.335105\pi$
$337$	18.1803	0.990346	0.495173	+	0.868794i	$0.335105\pi$
$338$	0	0
$339$	13.4721	0.731706
$340$	0	0
$341$	0	0
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	0.145898	0.00785489
$346$	0	0
$347$	−1.52786	−0.0820200	−0.0410100	−	0.999159i	$-0.513058\pi$
$347$	−1.52786	−0.0820200	−0.0410100	+	0.999159i	$0.513058\pi$
$348$	0	0
$349$	−12.7082	−0.680255	−0.340127	−	0.940379i	$-0.610470\pi$
$349$	−12.7082	−0.680255	−0.340127	+	0.940379i	$0.610470\pi$
$350$	0	0
$351$	−0.236068	−0.0126004
$352$	0	0
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	0	0
$355$	6.38197	0.338720
$356$	0	0
$357$	−1.14590	−0.0606474
$358$	0	0
$359$	−9.70820	−0.512379	−0.256190	−	0.966627i	$-0.582467\pi$
$359$	−9.70820	−0.512379	−0.256190	+	0.966627i	$0.582467\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.52786	−0.184657
$366$	0	0
$367$	−22.1459	−1.15601	−0.578003	−	0.816034i	$-0.696168\pi$
$367$	−22.1459	−1.15601	−0.578003	+	0.816034i	$0.696168\pi$
$368$	0	0
$369$	−0.236068	−0.0122892
$370$	0	0
$371$	0.381966	0.0198307
$372$	0	0
$373$	−0.888544	−0.0460071	−0.0230035	−	0.999735i	$-0.507323\pi$
$373$	−0.888544	−0.0460071	−0.0230035	+	0.999735i	$0.507323\pi$
$374$	0	0
$375$	5.94427	0.306961
$376$	0	0
$377$	−1.41641	−0.0729487
$378$	0	0
$379$	24.8885	1.27844	0.639219	−	0.769024i	$-0.279258\pi$
$379$	24.8885	1.27844	0.639219	+	0.769024i	$0.279258\pi$
$380$	0	0
$381$	7.70820	0.394903
$382$	0	0
$383$	12.7082	0.649359	0.324679	−	0.945824i	$-0.394744\pi$
$383$	12.7082	0.649359	0.324679	+	0.945824i	$0.394744\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.70820	−0.340997
$388$	0	0
$389$	−36.7426	−1.86293	−0.931463	−	0.363836i	$-0.881467\pi$
$389$	−36.7426	−1.86293	−0.931463	+	0.363836i	$0.881467\pi$
$390$	0	0
$391$	−0.270510	−0.0136803
$392$	0	0
$393$	−11.7984	−0.595149
$394$	0	0
$395$	−6.79837	−0.342063
$396$	0	0
$397$	−18.7082	−0.938938	−0.469469	−	0.882949i	$-0.655555\pi$
$397$	−18.7082	−0.938938	−0.469469	+	0.882949i	$0.655555\pi$
$398$	0	0
$399$	−5.85410	−0.293072
$400$	0	0
$401$	−31.6869	−1.58237	−0.791185	−	0.611577i	$-0.790535\pi$
$401$	−31.6869	−1.58237	−0.791185	+	0.611577i	$0.790535\pi$
$402$	0	0
$403$	−1.43769	−0.0716166
$404$	0	0
$405$	−0.618034	−0.0307104
$406$	0	0
$407$	0	0
$408$	0	0
$409$	6.47214	0.320027	0.160013	−	0.987115i	$-0.448846\pi$
$409$	6.47214	0.320027	0.160013	+	0.987115i	$0.448846\pi$
$410$	0	0
$411$	9.76393	0.481619
$412$	0	0
$413$	7.38197	0.363243
$414$	0	0
$415$	−0.909830	−0.0446618
$416$	0	0
$417$	14.5623	0.713119
$418$	0	0
$419$	−31.4508	−1.53647	−0.768237	−	0.640165i	$-0.778866\pi$
$419$	−31.4508	−1.53647	−0.768237	+	0.640165i	$0.778866\pi$
$420$	0	0
$421$	10.5066	0.512059	0.256030	−	0.966669i	$-0.417586\pi$
$421$	10.5066	0.512059	0.256030	+	0.966669i	$0.417586\pi$
$422$	0	0
$423$	10.0902	0.490601
$424$	0	0
$425$	−5.29180	−0.256690
$426$	0	0
$427$	−11.5623	−0.559539
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.90983	−0.284666	−0.142333	−	0.989819i	$-0.545460\pi$
$431$	−5.90983	−0.284666	−0.142333	+	0.989819i	$0.545460\pi$
$432$	0	0
$433$	35.3050	1.69665	0.848324	−	0.529478i	$-0.177612\pi$
$433$	35.3050	1.69665	0.848324	+	0.529478i	$0.177612\pi$
$434$	0	0
$435$	−3.70820	−0.177795
$436$	0	0
$437$	−1.38197	−0.0661084
$438$	0	0
$439$	23.2918	1.11166	0.555828	−	0.831297i	$-0.312401\pi$
$439$	23.2918	1.11166	0.555828	+	0.831297i	$0.312401\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	31.5967	1.50121	0.750603	−	0.660753i	$-0.229763\pi$
$443$	31.5967	1.50121	0.750603	+	0.660753i	$0.229763\pi$
$444$	0	0
$445$	5.09017	0.241297
$446$	0	0
$447$	4.23607	0.200359
$448$	0	0
$449$	−9.05573	−0.427366	−0.213683	−	0.976903i	$-0.568546\pi$
$449$	−9.05573	−0.427366	−0.213683	+	0.976903i	$0.568546\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	1.05573	0.0496024
$454$	0	0
$455$	−0.145898	−0.00683981
$456$	0	0
$457$	−23.9787	−1.12168	−0.560838	−	0.827925i	$-0.689521\pi$
$457$	−23.9787	−1.12168	−0.560838	+	0.827925i	$0.689521\pi$
$458$	0	0
$459$	1.14590	0.0534859
$460$	0	0
$461$	−9.27051	−0.431771	−0.215885	−	0.976419i	$-0.569264\pi$
$461$	−9.27051	−0.431771	−0.215885	+	0.976419i	$0.569264\pi$
$462$	0	0
$463$	−1.72949	−0.0803762	−0.0401881	−	0.999192i	$-0.512796\pi$
$463$	−1.72949	−0.0803762	−0.0401881	+	0.999192i	$0.512796\pi$
$464$	0	0
$465$	−3.76393	−0.174548
$466$	0	0
$467$	−20.8885	−0.966607	−0.483303	−	0.875453i	$-0.660563\pi$
$467$	−20.8885	−0.966607	−0.483303	+	0.875453i	$0.660563\pi$
$468$	0	0
$469$	1.85410	0.0856145
$470$	0	0
$471$	15.7082	0.723796
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−27.0344	−1.24043
$476$	0	0
$477$	−0.381966	−0.0174890
$478$	0	0
$479$	−28.3820	−1.29681	−0.648403	−	0.761298i	$-0.724563\pi$
$479$	−28.3820	−1.29681	−0.648403	+	0.761298i	$0.724563\pi$
$480$	0	0
$481$	1.47214	0.0671236
$482$	0	0
$483$	0.236068	0.0107415
$484$	0	0
$485$	−4.85410	−0.220413
$486$	0	0
$487$	12.7082	0.575864	0.287932	−	0.957651i	$-0.407032\pi$
$487$	12.7082	0.575864	0.287932	+	0.957651i	$0.407032\pi$
$488$	0	0
$489$	5.14590	0.232706
$490$	0	0
$491$	−17.9098	−0.808259	−0.404130	−	0.914702i	$-0.632426\pi$
$491$	−17.9098	−0.808259	−0.404130	+	0.914702i	$0.632426\pi$
$492$	0	0
$493$	6.87539	0.309652
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.3262	0.463195
$498$	0	0
$499$	18.1459	0.812322	0.406161	−	0.913802i	$-0.366867\pi$
$499$	18.1459	0.812322	0.406161	+	0.913802i	$0.366867\pi$
$500$	0	0
$501$	12.0344	0.537659
$502$	0	0
$503$	8.65248	0.385795	0.192897	−	0.981219i	$-0.438212\pi$
$503$	8.65248	0.385795	0.192897	+	0.981219i	$0.438212\pi$
$504$	0	0
$505$	−6.32624	−0.281514
$506$	0	0
$507$	−12.9443	−0.574875
$508$	0	0
$509$	38.7426	1.71724	0.858619	−	0.512615i	$-0.171323\pi$
$509$	38.7426	1.71724	0.858619	+	0.512615i	$0.171323\pi$
$510$	0	0
$511$	−5.70820	−0.252516
$512$	0	0
$513$	5.85410	0.258465
$514$	0	0
$515$	−6.76393	−0.298054
$516$	0	0
$517$	0	0
$518$	0	0
$519$	18.0344	0.791624
$520$	0	0
$521$	8.94427	0.391856	0.195928	−	0.980618i	$-0.437228\pi$
$521$	8.94427	0.391856	0.195928	+	0.980618i	$0.437228\pi$
$522$	0	0
$523$	18.2705	0.798914	0.399457	−	0.916752i	$-0.369199\pi$
$523$	18.2705	0.798914	0.399457	+	0.916752i	$0.369199\pi$
$524$	0	0
$525$	4.61803	0.201548
$526$	0	0
$527$	6.97871	0.303998
$528$	0	0
$529$	−22.9443	−0.997577
$530$	0	0
$531$	−7.38197	−0.320350
$532$	0	0
$533$	0.0557281	0.00241385
$534$	0	0
$535$	7.09017	0.306535
$536$	0	0
$537$	8.52786	0.368004
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.49342	0.322167	0.161084	−	0.986941i	$-0.448501\pi$
$541$	7.49342	0.322167	0.161084	+	0.986941i	$0.448501\pi$
$542$	0	0
$543$	2.52786	0.108481
$544$	0	0
$545$	−7.41641	−0.317684
$546$	0	0
$547$	−30.7426	−1.31446	−0.657230	−	0.753690i	$-0.728272\pi$
$547$	−30.7426	−1.31446	−0.657230	+	0.753690i	$0.728272\pi$
$548$	0	0
$549$	11.5623	0.493467
$550$	0	0
$551$	35.1246	1.49636
$552$	0	0
$553$	−11.0000	−0.467768
$554$	0	0
$555$	3.85410	0.163598
$556$	0	0
$557$	−37.6312	−1.59448	−0.797242	−	0.603659i	$-0.793709\pi$
$557$	−37.6312	−1.59448	−0.797242	+	0.603659i	$0.793709\pi$
$558$	0	0
$559$	1.58359	0.0669788
$560$	0	0
$561$	0	0
$562$	0	0
$563$	40.5967	1.71095	0.855474	−	0.517845i	$-0.173266\pi$
$563$	40.5967	1.71095	0.855474	+	0.517845i	$0.173266\pi$
$564$	0	0
$565$	−8.32624	−0.350287
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−34.1803	−1.43291	−0.716457	−	0.697631i	$-0.754237\pi$
$569$	−34.1803	−1.43291	−0.716457	+	0.697631i	$0.754237\pi$
$570$	0	0
$571$	−9.09017	−0.380412	−0.190206	−	0.981744i	$-0.560916\pi$
$571$	−9.09017	−0.380412	−0.190206	+	0.981744i	$0.560916\pi$
$572$	0	0
$573$	0.819660	0.0342418
$574$	0	0
$575$	1.09017	0.0454632
$576$	0	0
$577$	31.7082	1.32003	0.660015	−	0.751253i	$-0.270550\pi$
$577$	31.7082	1.32003	0.660015	+	0.751253i	$0.270550\pi$
$578$	0	0
$579$	3.14590	0.130739
$580$	0	0
$581$	−1.47214	−0.0610745
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0.145898	0.00603214
$586$	0	0
$587$	2.12461	0.0876921	0.0438461	−	0.999038i	$-0.486039\pi$
$587$	2.12461	0.0876921	0.0438461	+	0.999038i	$0.486039\pi$
$588$	0	0
$589$	35.6525	1.46903
$590$	0	0
$591$	−13.0344	−0.536165
$592$	0	0
$593$	14.0344	0.576325	0.288163	−	0.957581i	$-0.406956\pi$
$593$	14.0344	0.576325	0.288163	+	0.957581i	$0.406956\pi$
$594$	0	0
$595$	0.708204	0.0290335
$596$	0	0
$597$	−6.70820	−0.274549
$598$	0	0
$599$	−12.6525	−0.516966	−0.258483	−	0.966016i	$-0.583223\pi$
$599$	−12.6525	−0.516966	−0.258483	+	0.966016i	$0.583223\pi$
$600$	0	0
$601$	6.88854	0.280990	0.140495	−	0.990081i	$-0.455131\pi$
$601$	6.88854	0.280990	0.140495	+	0.990081i	$0.455131\pi$
$602$	0	0
$603$	−1.85410	−0.0755049
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−16.5623	−0.672243	−0.336122	−	0.941819i	$-0.609115\pi$
$607$	−16.5623	−0.672243	−0.336122	+	0.941819i	$0.609115\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	−2.38197	−0.0963640
$612$	0	0
$613$	14.2918	0.577240	0.288620	−	0.957444i	$-0.406804\pi$
$613$	14.2918	0.577240	0.288620	+	0.957444i	$0.406804\pi$
$614$	0	0
$615$	0.145898	0.00588318
$616$	0	0
$617$	11.1803	0.450104	0.225052	−	0.974347i	$-0.427745\pi$
$617$	11.1803	0.450104	0.225052	+	0.974347i	$0.427745\pi$
$618$	0	0
$619$	−24.1246	−0.969650	−0.484825	−	0.874611i	$-0.661117\pi$
$619$	−24.1246	−0.969650	−0.484825	+	0.874611i	$0.661117\pi$
$620$	0	0
$621$	−0.236068	−0.00947308
$622$	0	0
$623$	8.23607	0.329971
$624$	0	0
$625$	19.4164	0.776656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.14590	−0.284926
$630$	0	0
$631$	−19.2148	−0.764928	−0.382464	−	0.923970i	$-0.624924\pi$
$631$	−19.2148	−0.764928	−0.382464	+	0.923970i	$0.624924\pi$
$632$	0	0
$633$	3.61803	0.143804
$634$	0	0
$635$	−4.76393	−0.189051
$636$	0	0
$637$	1.41641	0.0561201
$638$	0	0
$639$	−10.3262	−0.408500
$640$	0	0
$641$	−25.0902	−0.991002	−0.495501	−	0.868607i	$-0.665016\pi$
$641$	−25.0902	−0.991002	−0.495501	+	0.868607i	$0.665016\pi$
$642$	0	0
$643$	−20.8541	−0.822406	−0.411203	−	0.911544i	$-0.634891\pi$
$643$	−20.8541	−0.822406	−0.411203	+	0.911544i	$0.634891\pi$
$644$	0	0
$645$	4.14590	0.163245
$646$	0	0
$647$	−45.0344	−1.77049	−0.885243	−	0.465128i	$-0.846008\pi$
$647$	−45.0344	−1.77049	−0.885243	+	0.465128i	$0.846008\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−6.09017	−0.238693
$652$	0	0
$653$	−5.61803	−0.219851	−0.109925	−	0.993940i	$-0.535061\pi$
$653$	−5.61803	−0.219851	−0.109925	+	0.993940i	$0.535061\pi$
$654$	0	0
$655$	7.29180	0.284914
$656$	0	0
$657$	5.70820	0.222698
$658$	0	0
$659$	−41.1246	−1.60199	−0.800994	−	0.598673i	$-0.795695\pi$
$659$	−41.1246	−1.60199	−0.800994	+	0.598673i	$0.795695\pi$
$660$	0	0
$661$	36.5623	1.42211	0.711054	−	0.703137i	$-0.248218\pi$
$661$	36.5623	1.42211	0.711054	+	0.703137i	$0.248218\pi$
$662$	0	0
$663$	−0.270510	−0.0105057
$664$	0	0
$665$	3.61803	0.140301
$666$	0	0
$667$	−1.41641	−0.0548435
$668$	0	0
$669$	−7.18034	−0.277608
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−35.8328	−1.38125	−0.690627	−	0.723211i	$-0.742665\pi$
$673$	−35.8328	−1.38125	−0.690627	+	0.723211i	$0.742665\pi$
$674$	0	0
$675$	−4.61803	−0.177748
$676$	0	0
$677$	−13.5279	−0.519918	−0.259959	−	0.965620i	$-0.583709\pi$
$677$	−13.5279	−0.519918	−0.259959	+	0.965620i	$0.583709\pi$
$678$	0	0
$679$	−7.85410	−0.301413
$680$	0	0
$681$	13.1803	0.505072
$682$	0	0
$683$	−9.06888	−0.347011	−0.173506	−	0.984833i	$-0.555509\pi$
$683$	−9.06888	−0.347011	−0.173506	+	0.984833i	$0.555509\pi$
$684$	0	0
$685$	−6.03444	−0.230564
$686$	0	0
$687$	0.472136	0.0180131
$688$	0	0
$689$	0.0901699	0.00343520
$690$	0	0
$691$	−1.34752	−0.0512622	−0.0256311	−	0.999671i	$-0.508160\pi$
$691$	−1.34752	−0.0512622	−0.0256311	+	0.999671i	$0.508160\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.00000	−0.341389
$696$	0	0
$697$	−0.270510	−0.0102463
$698$	0	0
$699$	4.14590	0.156812
$700$	0	0
$701$	34.7984	1.31432	0.657158	−	0.753753i	$-0.271758\pi$
$701$	34.7984	1.31432	0.657158	+	0.753753i	$0.271758\pi$
$702$	0	0
$703$	−36.5066	−1.37687
$704$	0	0
$705$	−6.23607	−0.234864
$706$	0	0
$707$	−10.2361	−0.384967
$708$	0	0
$709$	11.2148	0.421180	0.210590	−	0.977574i	$-0.432462\pi$
$709$	11.2148	0.421180	0.210590	+	0.977574i	$0.432462\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	0	0
$713$	−1.43769	−0.0538421
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.381966	0.0142648
$718$	0	0
$719$	38.4853	1.43526	0.717630	−	0.696425i	$-0.245227\pi$
$719$	38.4853	1.43526	0.717630	+	0.696425i	$0.245227\pi$
$720$	0	0
$721$	−10.9443	−0.407586
$722$	0	0
$723$	−8.29180	−0.308375
$724$	0	0
$725$	−27.7082	−1.02906
$726$	0	0
$727$	9.14590	0.339203	0.169601	−	0.985513i	$-0.445752\pi$
$727$	9.14590	0.339203	0.169601	+	0.985513i	$0.445752\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.68692	−0.284311
$732$	0	0
$733$	0.403252	0.0148945	0.00744723	−	0.999972i	$-0.497629\pi$
$733$	0.403252	0.0148945	0.00744723	+	0.999972i	$0.497629\pi$
$734$	0	0
$735$	3.70820	0.136779
$736$	0	0
$737$	0	0
$738$	0	0
$739$	3.00000	0.110357	0.0551784	−	0.998477i	$-0.482427\pi$
$739$	3.00000	0.110357	0.0551784	+	0.998477i	$0.482427\pi$
$740$	0	0
$741$	−1.38197	−0.0507678
$742$	0	0
$743$	−42.8885	−1.57343	−0.786714	−	0.617318i	$-0.788219\pi$
$743$	−42.8885	−1.57343	−0.786714	+	0.617318i	$0.788219\pi$
$744$	0	0
$745$	−2.61803	−0.0959173
$746$	0	0
$747$	1.47214	0.0538626
$748$	0	0
$749$	11.4721	0.419183
$750$	0	0
$751$	−16.1459	−0.589172	−0.294586	−	0.955625i	$-0.595182\pi$
$751$	−16.1459	−0.589172	−0.294586	+	0.955625i	$0.595182\pi$
$752$	0	0
$753$	21.9787	0.800949
$754$	0	0
$755$	−0.652476	−0.0237460
$756$	0	0
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−29.2918	−1.06183	−0.530913	−	0.847426i	$-0.678151\pi$
$761$	−29.2918	−1.06183	−0.530913	+	0.847426i	$0.678151\pi$
$762$	0	0
$763$	−12.0000	−0.434429
$764$	0	0
$765$	−0.708204	−0.0256052
$766$	0	0
$767$	1.74265	0.0629233
$768$	0	0
$769$	34.5066	1.24434	0.622170	−	0.782883i	$-0.286251\pi$
$769$	34.5066	1.24434	0.622170	+	0.782883i	$0.286251\pi$
$770$	0	0
$771$	29.7426	1.07116
$772$	0	0
$773$	27.1803	0.977609	0.488804	−	0.872393i	$-0.337433\pi$
$773$	27.1803	0.977609	0.488804	+	0.872393i	$0.337433\pi$
$774$	0	0
$775$	−28.1246	−1.01027
$776$	0	0
$777$	6.23607	0.223718
$778$	0	0
$779$	−1.38197	−0.0495141
$780$	0	0
$781$	0	0
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	−9.70820	−0.346501
$786$	0	0
$787$	−9.70820	−0.346060	−0.173030	−	0.984917i	$-0.555356\pi$
$787$	−9.70820	−0.346060	−0.173030	+	0.984917i	$0.555356\pi$
$788$	0	0
$789$	−15.2705	−0.543645
$790$	0	0
$791$	−13.4721	−0.479014
$792$	0	0
$793$	−2.72949	−0.0969270
$794$	0	0
$795$	0.236068	0.00837247
$796$	0	0
$797$	18.5410	0.656757	0.328378	−	0.944546i	$-0.393498\pi$
$797$	18.5410	0.656757	0.328378	+	0.944546i	$0.393498\pi$
$798$	0	0
$799$	11.5623	0.409045
$800$	0	0
$801$	−8.23607	−0.291007
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−0.145898	−0.00514223
$806$	0	0
$807$	−25.4164	−0.894700
$808$	0	0
$809$	−27.0902	−0.952440	−0.476220	−	0.879326i	$-0.657993\pi$
$809$	−27.0902	−0.952440	−0.476220	+	0.879326i	$0.657993\pi$
$810$	0	0
$811$	36.5623	1.28388	0.641938	−	0.766756i	$-0.278131\pi$
$811$	36.5623	1.28388	0.641938	+	0.766756i	$0.278131\pi$
$812$	0	0
$813$	18.6180	0.652963
$814$	0	0
$815$	−3.18034	−0.111402
$816$	0	0
$817$	−39.2705	−1.37390
$818$	0	0
$819$	0.236068	0.00824888
$820$	0	0
$821$	−40.5967	−1.41684	−0.708418	−	0.705793i	$-0.750591\pi$
$821$	−40.5967	−1.41684	−0.708418	+	0.705793i	$0.750591\pi$
$822$	0	0
$823$	27.8328	0.970191	0.485095	−	0.874461i	$-0.338785\pi$
$823$	27.8328	0.970191	0.485095	+	0.874461i	$0.338785\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.6525	−0.370423	−0.185211	−	0.982699i	$-0.559297\pi$
$827$	−10.6525	−0.370423	−0.185211	+	0.982699i	$0.559297\pi$
$828$	0	0
$829$	−31.3951	−1.09040	−0.545199	−	0.838307i	$-0.683546\pi$
$829$	−31.3951	−1.09040	−0.545199	+	0.838307i	$0.683546\pi$
$830$	0	0
$831$	29.2148	1.01345
$832$	0	0
$833$	−6.87539	−0.238218
$834$	0	0
$835$	−7.43769	−0.257392
$836$	0	0
$837$	6.09017	0.210507
$838$	0	0
$839$	17.8328	0.615657	0.307829	−	0.951442i	$-0.400398\pi$
$839$	17.8328	0.615657	0.307829	+	0.951442i	$0.400398\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−24.7639	−0.852915
$844$	0	0
$845$	8.00000	0.275208
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−5.70820	−0.195905
$850$	0	0
$851$	1.47214	0.0504642
$852$	0	0
$853$	55.8328	1.91168	0.955840	−	0.293889i	$-0.0949496\pi$
$853$	55.8328	1.91168	0.955840	+	0.293889i	$0.0949496\pi$
$854$	0	0
$855$	−3.61803	−0.123734
$856$	0	0
$857$	−27.7639	−0.948398	−0.474199	−	0.880418i	$-0.657262\pi$
$857$	−27.7639	−0.948398	−0.474199	+	0.880418i	$0.657262\pi$
$858$	0	0
$859$	34.4164	1.17427	0.587136	−	0.809488i	$-0.300255\pi$
$859$	34.4164	1.17427	0.587136	+	0.809488i	$0.300255\pi$
$860$	0	0
$861$	0.236068	0.00804518
$862$	0	0
$863$	−0.111456	−0.00379401	−0.00189701	−	0.999998i	$-0.500604\pi$
$863$	−0.111456	−0.00379401	−0.00189701	+	0.999998i	$0.500604\pi$
$864$	0	0
$865$	−11.1459	−0.378972
$866$	0	0
$867$	−15.6869	−0.532756
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0.437694	0.0148307
$872$	0	0
$873$	7.85410	0.265821
$874$	0	0
$875$	−5.94427	−0.200953
$876$	0	0
$877$	57.9443	1.95664	0.978320	−	0.207101i	$-0.0664030\pi$
$877$	57.9443	1.95664	0.978320	+	0.207101i	$0.0664030\pi$
$878$	0	0
$879$	21.6525	0.730320
$880$	0	0
$881$	6.20163	0.208938	0.104469	−	0.994528i	$-0.466686\pi$
$881$	6.20163	0.208938	0.104469	+	0.994528i	$0.466686\pi$
$882$	0	0
$883$	−1.05573	−0.0355281	−0.0177640	−	0.999842i	$-0.505655\pi$
$883$	−1.05573	−0.0355281	−0.0177640	+	0.999842i	$0.505655\pi$
$884$	0	0
$885$	4.56231	0.153360
$886$	0	0
$887$	−54.4853	−1.82944	−0.914719	−	0.404092i	$-0.867588\pi$
$887$	−54.4853	−1.82944	−0.914719	+	0.404092i	$0.867588\pi$
$888$	0	0
$889$	−7.70820	−0.258525
$890$	0	0
$891$	0	0
$892$	0	0
$893$	59.0689	1.97666
$894$	0	0
$895$	−5.27051	−0.176174
$896$	0	0
$897$	0.0557281	0.00186071
$898$	0	0
$899$	36.5410	1.21871
$900$	0	0
$901$	−0.437694	−0.0145817
$902$	0	0
$903$	6.70820	0.223235
$904$	0	0
$905$	−1.56231	−0.0519328
$906$	0	0
$907$	42.3951	1.40771	0.703853	−	0.710345i	$-0.251461\pi$
$907$	42.3951	1.40771	0.703853	+	0.710345i	$0.251461\pi$
$908$	0	0
$909$	10.2361	0.339509
$910$	0	0
$911$	−38.5967	−1.27877	−0.639384	−	0.768888i	$-0.720810\pi$
$911$	−38.5967	−1.27877	−0.639384	+	0.768888i	$0.720810\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−7.14590	−0.236236
$916$	0	0
$917$	11.7984	0.389617
$918$	0	0
$919$	25.6525	0.846197	0.423099	−	0.906084i	$-0.360942\pi$
$919$	25.6525	0.846197	0.423099	+	0.906084i	$0.360942\pi$
$920$	0	0
$921$	27.9787	0.921930
$922$	0	0
$923$	2.43769	0.0802377
$924$	0	0
$925$	28.7984	0.946885
$926$	0	0
$927$	10.9443	0.359457
$928$	0	0
$929$	−12.7082	−0.416943	−0.208471	−	0.978028i	$-0.566849\pi$
$929$	−12.7082	−0.416943	−0.208471	+	0.978028i	$0.566849\pi$
$930$	0	0
$931$	−35.1246	−1.15116
$932$	0	0
$933$	−11.6525	−0.381485
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−41.6525	−1.36073	−0.680364	−	0.732875i	$-0.738178\pi$
$937$	−41.6525	−1.36073	−0.680364	+	0.732875i	$0.738178\pi$
$938$	0	0
$939$	−2.52786	−0.0824937
$940$	0	0
$941$	14.4508	0.471084	0.235542	−	0.971864i	$-0.424313\pi$
$941$	14.4508	0.471084	0.235542	+	0.971864i	$0.424313\pi$
$942$	0	0
$943$	0.0557281	0.00181476
$944$	0	0
$945$	0.618034	0.0201046
$946$	0	0
$947$	32.3951	1.05270	0.526350	−	0.850268i	$-0.323560\pi$
$947$	32.3951	1.05270	0.526350	+	0.850268i	$0.323560\pi$
$948$	0	0
$949$	−1.34752	−0.0437425
$950$	0	0
$951$	−6.81966	−0.221143
$952$	0	0
$953$	−11.3475	−0.367582	−0.183791	−	0.982965i	$-0.558837\pi$
$953$	−11.3475	−0.367582	−0.183791	+	0.982965i	$0.558837\pi$
$954$	0	0
$955$	−0.506578	−0.0163925
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.76393	−0.315294
$960$	0	0
$961$	6.09017	0.196457
$962$	0	0
$963$	−11.4721	−0.369684
$964$	0	0
$965$	−1.94427	−0.0625883
$966$	0	0
$967$	43.9230	1.41247	0.706234	−	0.707978i	$-0.250393\pi$
$967$	43.9230	1.41247	0.706234	+	0.707978i	$0.250393\pi$
$968$	0	0
$969$	6.70820	0.215499
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	0	0
$973$	−14.5623	−0.466846
$974$	0	0
$975$	1.09017	0.0349134
$976$	0	0
$977$	−0.596748	−0.0190917	−0.00954583	−	0.999954i	$-0.503039\pi$
$977$	−0.596748	−0.0190917	−0.00954583	+	0.999954i	$0.503039\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	12.0000	0.383131
$982$	0	0
$983$	8.11146	0.258715	0.129358	−	0.991598i	$-0.458708\pi$
$983$	8.11146	0.258715	0.129358	+	0.991598i	$0.458708\pi$
$984$	0	0
$985$	8.05573	0.256677
$986$	0	0
$987$	−10.0902	−0.321174
$988$	0	0
$989$	1.58359	0.0503553
$990$	0	0
$991$	−3.74265	−0.118889	−0.0594445	−	0.998232i	$-0.518933\pi$
$991$	−3.74265	−0.118889	−0.0594445	+	0.998232i	$0.518933\pi$
$992$	0	0
$993$	−16.7082	−0.530219
$994$	0	0
$995$	4.14590	0.131434
$996$	0	0
$997$	21.2016	0.671462	0.335731	−	0.941958i	$-0.391017\pi$
$997$	21.2016	0.671462	0.335731	+	0.941958i	$0.391017\pi$
$998$	0	0
$999$	−6.23607	−0.197300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5808.2.a.ci.1.1		2
4.3	odd	2		363.2.a.i.1.2		2
11.7	odd	10		528.2.y.b.49.1		4
11.8	odd	10		528.2.y.b.97.1		4
11.10	odd	2		5808.2.a.cj.1.1		2
12.11	even	2		1089.2.a.l.1.1		2
20.19	odd	2		9075.2.a.u.1.1		2
44.3	odd	10		363.2.e.f.130.1		4
44.7	even	10		33.2.e.b.16.1	✓	4
44.15	odd	10		363.2.e.f.148.1		4
44.19	even	10		33.2.e.b.31.1	yes	4
44.27	odd	10		363.2.e.b.124.1		4
44.31	odd	10		363.2.e.b.202.1		4
44.35	even	10		363.2.e.k.202.1		4
44.39	even	10		363.2.e.k.124.1		4
44.43	even	2		363.2.a.d.1.1		2
132.95	odd	10		99.2.f.a.82.1		4
132.107	odd	10		99.2.f.a.64.1		4
132.131	odd	2		1089.2.a.t.1.2		2
220.7	odd	20		825.2.bx.d.49.2		8
220.19	even	10		825.2.n.c.526.1		4
220.63	odd	20		825.2.bx.d.724.2		8
220.107	odd	20		825.2.bx.d.724.1		8
220.139	even	10		825.2.n.c.676.1		4
220.183	odd	20		825.2.bx.d.49.1		8
220.219	even	2		9075.2.a.cb.1.2		2
396.7	even	30		891.2.n.c.676.1		8
396.95	odd	30		891.2.n.b.379.1		8
396.139	even	30		891.2.n.c.379.1		8
396.151	even	30		891.2.n.c.757.1		8
396.227	odd	30		891.2.n.b.676.1		8
396.239	odd	30		891.2.n.b.460.1		8
396.283	even	30		891.2.n.c.460.1		8
396.371	odd	30		891.2.n.b.757.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.e.b.16.1	✓	4	44.7	even	10
33.2.e.b.31.1	yes	4	44.19	even	10
99.2.f.a.64.1		4	132.107	odd	10
99.2.f.a.82.1		4	132.95	odd	10
363.2.a.d.1.1		2	44.43	even	2
363.2.a.i.1.2		2	4.3	odd	2
363.2.e.b.124.1		4	44.27	odd	10
363.2.e.b.202.1		4	44.31	odd	10
363.2.e.f.130.1		4	44.3	odd	10
363.2.e.f.148.1		4	44.15	odd	10
363.2.e.k.124.1		4	44.39	even	10
363.2.e.k.202.1		4	44.35	even	10
528.2.y.b.49.1		4	11.7	odd	10
528.2.y.b.97.1		4	11.8	odd	10
825.2.n.c.526.1		4	220.19	even	10
825.2.n.c.676.1		4	220.139	even	10
825.2.bx.d.49.1		8	220.183	odd	20
825.2.bx.d.49.2		8	220.7	odd	20
825.2.bx.d.724.1		8	220.107	odd	20
825.2.bx.d.724.2		8	220.63	odd	20
891.2.n.b.379.1		8	396.95	odd	30
891.2.n.b.460.1		8	396.239	odd	30
891.2.n.b.676.1		8	396.227	odd	30
891.2.n.b.757.1		8	396.371	odd	30
891.2.n.c.379.1		8	396.139	even	30
891.2.n.c.460.1		8	396.283	even	30
891.2.n.c.676.1		8	396.7	even	30
891.2.n.c.757.1		8	396.151	even	30
1089.2.a.l.1.1		2	12.11	even	2
1089.2.a.t.1.2		2	132.131	odd	2
5808.2.a.ci.1.1		2	1.1	even	1	trivial
5808.2.a.cj.1.1		2	11.10	odd	2
9075.2.a.u.1.1		2	20.19	odd	2
9075.2.a.cb.1.2		2	220.219	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5808.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.618034 q^{5} -1.00000 q^{7} +1.00000 q^{9} -0.236068 q^{13} -0.618034 q^{15} +1.14590 q^{17} +5.85410 q^{19} -1.00000 q^{21} -0.236068 q^{23} -4.61803 q^{25} +1.00000 q^{27} +6.00000 q^{29} +6.09017 q^{31} +0.618034 q^{35} -6.23607 q^{37} -0.236068 q^{39} -0.236068 q^{41} -6.70820 q^{43} -0.618034 q^{45} +10.0902 q^{47} -6.00000 q^{49} +1.14590 q^{51} -0.381966 q^{53} +5.85410 q^{57} -7.38197 q^{59} +11.5623 q^{61} -1.00000 q^{63} +0.145898 q^{65} -1.85410 q^{67} -0.236068 q^{69} -10.3262 q^{71} +5.70820 q^{73} -4.61803 q^{75} +11.0000 q^{79} +1.00000 q^{81} +1.47214 q^{83} -0.708204 q^{85} +6.00000 q^{87} -8.23607 q^{89} +0.236068 q^{91} +6.09017 q^{93} -3.61803 q^{95} +7.85410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{13} + q^{15} + 9 q^{17} + 5 q^{19} - 2 q^{21} + 4 q^{23} - 7 q^{25} + 2 q^{27} + 12 q^{29} + q^{31} - q^{35} - 8 q^{37} + 4 q^{39} + 4 q^{41} + q^{45}+ \cdots + 9 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 + 4 * q^13 + q^15 + 9 * q^17 + 5 * q^19 - 2 * q^21 + 4 * q^23 - 7 * q^25 + 2 * q^27 + 12 * q^29 + q^31 - q^35 - 8 * q^37 + 4 * q^39 + 4 * q^41 + q^45 + 9 * q^47 - 12 * q^49 + 9 * q^51 - 3 * q^53 + 5 * q^57 - 17 * q^59 + 3 * q^61 - 2 * q^63 + 7 * q^65 + 3 * q^67 + 4 * q^69 - 5 * q^71 - 2 * q^73 - 7 * q^75 + 22 * q^79 + 2 * q^81 - 6 * q^83 + 12 * q^85 + 12 * q^87 - 12 * q^89 - 4 * q^91 + q^93 - 5 * q^95 + 9 * q^97

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