LMFDB - Embedded newform 3483.2.a.r.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3483,2,Mod(1,3483)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3483, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3483.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3483 = 3^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3483.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:	$27.8118950240$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 100 x^{16} + 181 x^{15} - 1020 x^{14} - 619 x^{13} + 5458 x^{12} + \cdots + 9 $ x^19 - 4x^18 - 22x^17 + 100x^16 + 181x^15 - 1020x^14 - 619x^13 + 5458x^12 + 145x^11 - 16363x^10 + 4943x^9 + 27069x^8 - 13490x^7 - 22501x^6 + 13478x^5 + 7567x^4 - 4310x^3 - 855x^2 + 225x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 387)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$-1.77505$ of defining polynomial
Character	$\chi$	$=$	3483.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.77505 q^{2} +1.15081 q^{4} -3.69926 q^{5} -0.0897007 q^{7} -1.50735 q^{8} +O(q^{10})$	\(q+1.77505 q^{2} +1.15081 q^{4} -3.69926 q^{5} -0.0897007 q^{7} -1.50735 q^{8} -6.56638 q^{10} +4.81081 q^{11} +4.40415 q^{13} -0.159223 q^{14} -4.97726 q^{16} -4.51570 q^{17} +0.457142 q^{19} -4.25715 q^{20} +8.53944 q^{22} +6.27843 q^{23} +8.68453 q^{25} +7.81759 q^{26} -0.103229 q^{28} -4.82111 q^{29} -3.24460 q^{31} -5.82018 q^{32} -8.01560 q^{34} +0.331826 q^{35} -9.48421 q^{37} +0.811452 q^{38} +5.57609 q^{40} -7.08781 q^{41} +1.00000 q^{43} +5.53633 q^{44} +11.1445 q^{46} -3.97603 q^{47} -6.99195 q^{49} +15.4155 q^{50} +5.06835 q^{52} -8.18271 q^{53} -17.7964 q^{55} +0.135211 q^{56} -8.55773 q^{58} -9.78676 q^{59} +11.7377 q^{61} -5.75934 q^{62} -0.376622 q^{64} -16.2921 q^{65} -6.22052 q^{67} -5.19672 q^{68} +0.589009 q^{70} -13.8969 q^{71} +11.2453 q^{73} -16.8350 q^{74} +0.526085 q^{76} -0.431533 q^{77} -9.29367 q^{79} +18.4122 q^{80} -12.5812 q^{82} -4.03603 q^{83} +16.7047 q^{85} +1.77505 q^{86} -7.25159 q^{88} -8.46429 q^{89} -0.395055 q^{91} +7.22529 q^{92} -7.05766 q^{94} -1.69109 q^{95} -0.679148 q^{97} -12.4111 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 9 q^{5} + 7 q^{7} - 12 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 9 * q^5 + 7 * q^7 - 12 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 9 q^{5} + 7 q^{7} - 12 q^{8} - 7 q^{10} - 5 q^{11} - 5 q^{13} - 17 q^{14} + 24 q^{16} - 21 q^{17} - 4 q^{19} - 21 q^{20} - 20 q^{22} - 22 q^{23} + 10 q^{25} - 17 q^{26} - q^{28} - 30 q^{29} - 5 q^{31} - 48 q^{32} - 6 q^{34} - 53 q^{35} - q^{37} - 21 q^{38} + 16 q^{40} - 29 q^{41} + 19 q^{43} - 29 q^{44} - 32 q^{47} - 10 q^{49} + 11 q^{50} + q^{52} - 38 q^{53} + 2 q^{55} - 46 q^{56} + 30 q^{58} - 30 q^{59} - 10 q^{61} - 25 q^{62} + 14 q^{64} - 8 q^{65} + 3 q^{67} - 47 q^{68} + 56 q^{70} - 21 q^{71} + 8 q^{73} - 28 q^{74} - 36 q^{76} - 49 q^{77} + 4 q^{79} - 70 q^{80} - 4 q^{82} - 29 q^{83} - 4 q^{85} - 4 q^{86} - 47 q^{88} - 54 q^{89} + 4 q^{91} - 12 q^{92} - 23 q^{94} - 33 q^{95} - 4 q^{97} - 49 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 9 * q^5 + 7 * q^7 - 12 * q^8 - 7 * q^10 - 5 * q^11 - 5 * q^13 - 17 * q^14 + 24 * q^16 - 21 * q^17 - 4 * q^19 - 21 * q^20 - 20 * q^22 - 22 * q^23 + 10 * q^25 - 17 * q^26 - q^28 - 30 * q^29 - 5 * q^31 - 48 * q^32 - 6 * q^34 - 53 * q^35 - q^37 - 21 * q^38 + 16 * q^40 - 29 * q^41 + 19 * q^43 - 29 * q^44 - 32 * q^47 - 10 * q^49 + 11 * q^50 + q^52 - 38 * q^53 + 2 * q^55 - 46 * q^56 + 30 * q^58 - 30 * q^59 - 10 * q^61 - 25 * q^62 + 14 * q^64 - 8 * q^65 + 3 * q^67 - 47 * q^68 + 56 * q^70 - 21 * q^71 + 8 * q^73 - 28 * q^74 - 36 * q^76 - 49 * q^77 + 4 * q^79 - 70 * q^80 - 4 * q^82 - 29 * q^83 - 4 * q^85 - 4 * q^86 - 47 * q^88 - 54 * q^89 + 4 * q^91 - 12 * q^92 - 23 * q^94 - 33 * q^95 - 4 * q^97 - 49 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.77505	1.25515	0.627576	−	0.778555i	$-0.284047\pi$
$2$	1.77505	1.25515	0.627576	+	0.778555i	$0.284047\pi$
$3$	0	0
$3$	0	0
$4$	1.15081	0.575406
$5$	−3.69926	−1.65436	−0.827180	−	0.561937i	$-0.810056\pi$
$5$	−3.69926	−1.65436	−0.827180	+	0.561937i	$0.810056\pi$
$6$	0	0
$7$	−0.0897007	−0.0339037	−0.0169518	−	0.999856i	$-0.505396\pi$
$7$	−0.0897007	−0.0339037	−0.0169518	+	0.999856i	$0.505396\pi$
$8$	−1.50735	−0.532930
$9$	0	0
$10$	−6.56638	−2.07647
$11$	4.81081	1.45051	0.725256	−	0.688479i	$-0.241721\pi$
$11$	4.81081	1.45051	0.725256	+	0.688479i	$0.241721\pi$
$12$	0	0
$13$	4.40415	1.22149	0.610745	−	0.791827i	$-0.290870\pi$
$13$	4.40415	1.22149	0.610745	+	0.791827i	$0.290870\pi$
$14$	−0.159223	−0.0425543
$15$	0	0
$16$	−4.97726	−1.24431
$17$	−4.51570	−1.09522	−0.547609	−	0.836735i	$-0.684462\pi$
$17$	−4.51570	−1.09522	−0.547609	+	0.836735i	$0.684462\pi$
$18$	0	0
$19$	0.457142	0.104876	0.0524378	−	0.998624i	$-0.483301\pi$
$19$	0.457142	0.104876	0.0524378	+	0.998624i	$0.483301\pi$
$20$	−4.25715	−0.951929
$21$	0	0
$22$	8.53944	1.82061
$23$	6.27843	1.30914	0.654571	−	0.756000i	$-0.272849\pi$
$23$	6.27843	1.30914	0.654571	+	0.756000i	$0.272849\pi$
$24$	0	0
$25$	8.68453	1.73691
$26$	7.81759	1.53316
$27$	0	0
$28$	−0.103229	−0.0195084
$29$	−4.82111	−0.895258	−0.447629	−	0.894219i	$-0.647731\pi$
$29$	−4.82111	−0.895258	−0.447629	+	0.894219i	$0.647731\pi$
$30$	0	0
$31$	−3.24460	−0.582748	−0.291374	−	0.956609i	$-0.594112\pi$
$31$	−3.24460	−0.582748	−0.291374	+	0.956609i	$0.594112\pi$
$32$	−5.82018	−1.02887
$33$	0	0
$34$	−8.01560	−1.37466
$35$	0.331826	0.0560889
$36$	0	0
$37$	−9.48421	−1.55919	−0.779597	−	0.626282i	$-0.784576\pi$
$37$	−9.48421	−1.55919	−0.779597	+	0.626282i	$0.784576\pi$
$38$	0.811452	0.131635
$39$	0	0
$40$	5.57609	0.881658
$41$	−7.08781	−1.10693	−0.553465	−	0.832873i	$-0.686695\pi$
$41$	−7.08781	−1.10693	−0.553465	+	0.832873i	$0.686695\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	5.53633	0.834634
$45$	0	0
$46$	11.1445	1.64317
$47$	−3.97603	−0.579963	−0.289982	−	0.957032i	$-0.593649\pi$
$47$	−3.97603	−0.579963	−0.289982	+	0.957032i	$0.593649\pi$
$48$	0	0
$49$	−6.99195	−0.998851
$50$	15.4155	2.18008
$51$	0	0
$52$	5.06835	0.702853
$53$	−8.18271	−1.12398	−0.561991	−	0.827143i	$-0.689964\pi$
$53$	−8.18271	−1.12398	−0.561991	+	0.827143i	$0.689964\pi$
$54$	0	0
$55$	−17.7964	−2.39967
$56$	0.135211	0.0180683
$57$	0	0
$58$	−8.55773	−1.12368
$59$	−9.78676	−1.27413	−0.637064	−	0.770811i	$-0.719851\pi$
$59$	−9.78676	−1.27413	−0.637064	+	0.770811i	$0.719851\pi$
$60$	0	0
$61$	11.7377	1.50286	0.751431	−	0.659812i	$-0.229364\pi$
$61$	11.7377	1.50286	0.751431	+	0.659812i	$0.229364\pi$
$62$	−5.75934	−0.731437
$63$	0	0
$64$	−0.376622	−0.0470778
$65$	−16.2921	−2.02079
$66$	0	0
$67$	−6.22052	−0.759958	−0.379979	−	0.924995i	$-0.624069\pi$
$67$	−6.22052	−0.759958	−0.379979	+	0.924995i	$0.624069\pi$
$68$	−5.19672	−0.630195
$69$	0	0
$70$	0.589009	0.0704001
$71$	−13.8969	−1.64925	−0.824627	−	0.565676i	$-0.808615\pi$
$71$	−13.8969	−1.64925	−0.824627	+	0.565676i	$0.808615\pi$
$72$	0	0
$73$	11.2453	1.31616	0.658078	−	0.752949i	$-0.271369\pi$
$73$	11.2453	1.31616	0.658078	+	0.752949i	$0.271369\pi$
$74$	−16.8350	−1.95703
$75$	0	0
$76$	0.526085	0.0603461
$77$	−0.431533	−0.0491777
$78$	0	0
$79$	−9.29367	−1.04562	−0.522810	−	0.852449i	$-0.675116\pi$
$79$	−9.29367	−1.04562	−0.522810	+	0.852449i	$0.675116\pi$
$80$	18.4122	2.05854
$81$	0	0
$82$	−12.5812	−1.38936
$83$	−4.03603	−0.443012	−0.221506	−	0.975159i	$-0.571097\pi$
$83$	−4.03603	−0.443012	−0.221506	+	0.975159i	$0.571097\pi$
$84$	0	0
$85$	16.7047	1.81188
$86$	1.77505	0.191409
$87$	0	0
$88$	−7.25159	−0.773022
$89$	−8.46429	−0.897213	−0.448607	−	0.893729i	$-0.648080\pi$
$89$	−8.46429	−0.897213	−0.448607	+	0.893729i	$0.648080\pi$
$90$	0	0
$91$	−0.395055	−0.0414130
$92$	7.22529	0.753288
$93$	0	0
$94$	−7.05766	−0.727942
$95$	−1.69109	−0.173502
$96$	0	0
$97$	−0.679148	−0.0689570	−0.0344785	−	0.999405i	$-0.510977\pi$
$97$	−0.679148	−0.0689570	−0.0344785	+	0.999405i	$0.510977\pi$
$98$	−12.4111	−1.25371
$99$	0	0
$100$	9.99427	0.999427
$101$	−3.63139	−0.361336	−0.180668	−	0.983544i	$-0.557826\pi$
$101$	−3.63139	−0.361336	−0.180668	+	0.983544i	$0.557826\pi$
$102$	0	0
$103$	−5.45946	−0.537937	−0.268968	−	0.963149i	$-0.586683\pi$
$103$	−5.45946	−0.537937	−0.268968	+	0.963149i	$0.586683\pi$
$104$	−6.63861	−0.650969
$105$	0	0
$106$	−14.5247	−1.41077
$107$	2.04706	0.197896	0.0989482	−	0.995093i	$-0.468452\pi$
$107$	2.04706	0.197896	0.0989482	+	0.995093i	$0.468452\pi$
$108$	0	0
$109$	−1.75892	−0.168474	−0.0842371	−	0.996446i	$-0.526845\pi$
$109$	−1.75892	−0.168474	−0.0842371	+	0.996446i	$0.526845\pi$
$110$	−31.5896	−3.01195
$111$	0	0
$112$	0.446463	0.0421868
$113$	13.4213	1.26257	0.631285	−	0.775551i	$-0.282528\pi$
$113$	13.4213	1.26257	0.631285	+	0.775551i	$0.282528\pi$
$114$	0	0
$115$	−23.2255	−2.16579
$116$	−5.54819	−0.515137
$117$	0	0
$118$	−17.3720	−1.59922
$119$	0.405061	0.0371319
$120$	0	0
$121$	12.1439	1.10399
$122$	20.8351	1.88632
$123$	0	0
$124$	−3.73393	−0.335317
$125$	−13.6301	−1.21911
$126$	0	0
$127$	5.69493	0.505344	0.252672	−	0.967552i	$-0.418691\pi$
$127$	5.69493	0.505344	0.252672	+	0.967552i	$0.418691\pi$
$128$	10.9718	0.969783
$129$	0	0
$130$	−28.9193	−2.53639
$131$	0.420816	0.0367668	0.0183834	−	0.999831i	$-0.494148\pi$
$131$	0.420816	0.0367668	0.0183834	+	0.999831i	$0.494148\pi$
$132$	0	0
$133$	−0.0410060	−0.00355567
$134$	−11.0418	−0.953862
$135$	0	0
$136$	6.80675	0.583674
$137$	13.0653	1.11624	0.558120	−	0.829760i	$-0.311523\pi$
$137$	13.0653	1.11624	0.558120	+	0.829760i	$0.311523\pi$
$138$	0	0
$139$	7.42236	0.629557	0.314778	−	0.949165i	$-0.398070\pi$
$139$	7.42236	0.629557	0.314778	+	0.949165i	$0.398070\pi$
$140$	0.381870	0.0322739
$141$	0	0
$142$	−24.6677	−2.07006
$143$	21.1875	1.77179
$144$	0	0
$145$	17.8346	1.48108
$146$	19.9609	1.65198
$147$	0	0
$148$	−10.9145	−0.897170
$149$	2.57480	0.210936	0.105468	−	0.994423i	$-0.466366\pi$
$149$	2.57480	0.210936	0.105468	+	0.994423i	$0.466366\pi$
$150$	0	0
$151$	18.1236	1.47488	0.737440	−	0.675413i	$-0.236035\pi$
$151$	18.1236	1.47488	0.737440	+	0.675413i	$0.236035\pi$
$152$	−0.689075	−0.0558914
$153$	0	0
$154$	−0.765993	−0.0617255
$155$	12.0026	0.964074
$156$	0	0
$157$	6.70671	0.535254	0.267627	−	0.963523i	$-0.413761\pi$
$157$	6.70671	0.535254	0.267627	+	0.963523i	$0.413761\pi$
$158$	−16.4968	−1.31241
$159$	0	0
$160$	21.5304	1.70213
$161$	−0.563179	−0.0443847
$162$	0	0
$163$	1.31744	0.103190	0.0515949	−	0.998668i	$-0.483570\pi$
$163$	1.31744	0.103190	0.0515949	+	0.998668i	$0.483570\pi$
$164$	−8.15674	−0.636934
$165$	0	0
$166$	−7.16417	−0.556048
$167$	−23.7986	−1.84159	−0.920795	−	0.390047i	$-0.872459\pi$
$167$	−23.7986	−1.84159	−0.920795	+	0.390047i	$0.872459\pi$
$168$	0	0
$169$	6.39651	0.492039
$170$	29.6518	2.27419
$171$	0	0
$172$	1.15081	0.0877486
$173$	4.46029	0.339110	0.169555	−	0.985521i	$-0.445767\pi$
$173$	4.46029	0.339110	0.169555	+	0.985521i	$0.445767\pi$
$174$	0	0
$175$	−0.779009	−0.0588875
$176$	−23.9446	−1.80489
$177$	0	0
$178$	−15.0246	−1.12614
$179$	14.0146	1.04750	0.523749	−	0.851873i	$-0.324533\pi$
$179$	14.0146	1.04750	0.523749	+	0.851873i	$0.324533\pi$
$180$	0	0
$181$	−22.6669	−1.68481	−0.842407	−	0.538842i	$-0.818862\pi$
$181$	−22.6669	−1.68481	−0.842407	+	0.538842i	$0.818862\pi$
$182$	−0.701244	−0.0519796
$183$	0	0
$184$	−9.46381	−0.697681
$185$	35.0846	2.57947
$186$	0	0
$187$	−21.7241	−1.58863
$188$	−4.57566	−0.333714
$189$	0	0
$190$	−3.00177	−0.217771
$191$	−3.85778	−0.279139	−0.139570	−	0.990212i	$-0.544572\pi$
$191$	−3.85778	−0.279139	−0.139570	+	0.990212i	$0.544572\pi$
$192$	0	0
$193$	−14.0501	−1.01135	−0.505673	−	0.862725i	$-0.668756\pi$
$193$	−14.0501	−1.01135	−0.505673	+	0.862725i	$0.668756\pi$
$194$	−1.20552	−0.0865516
$195$	0	0
$196$	−8.04642	−0.574745
$197$	−1.78785	−0.127379	−0.0636897	−	0.997970i	$-0.520287\pi$
$197$	−1.78785	−0.127379	−0.0636897	+	0.997970i	$0.520287\pi$
$198$	0	0
$199$	−17.5450	−1.24373	−0.621865	−	0.783125i	$-0.713625\pi$
$199$	−17.5450	−1.24373	−0.621865	+	0.783125i	$0.713625\pi$
$200$	−13.0907	−0.925650
$201$	0	0
$202$	−6.44590	−0.453532
$203$	0.432457	0.0303525
$204$	0	0
$205$	26.2197	1.83126
$206$	−9.69083	−0.675192
$207$	0	0
$208$	−21.9206	−1.51992
$209$	2.19922	0.152124
$210$	0	0
$211$	−11.0540	−0.760987	−0.380493	−	0.924784i	$-0.624246\pi$
$211$	−11.0540	−0.760987	−0.380493	+	0.924784i	$0.624246\pi$
$212$	−9.41676	−0.646746
$213$	0	0
$214$	3.63363	0.248390
$215$	−3.69926	−0.252288
$216$	0	0
$217$	0.291043	0.0197573
$218$	−3.12218	−0.211461
$219$	0	0
$220$	−20.4803	−1.38078
$221$	−19.8878	−1.33780
$222$	0	0
$223$	−2.76254	−0.184994	−0.0924968	−	0.995713i	$-0.529485\pi$
$223$	−2.76254	−0.184994	−0.0924968	+	0.995713i	$0.529485\pi$
$224$	0.522075	0.0348826
$225$	0	0
$226$	23.8235	1.58472
$227$	−9.91423	−0.658030	−0.329015	−	0.944325i	$-0.606717\pi$
$227$	−9.91423	−0.658030	−0.329015	+	0.944325i	$0.606717\pi$
$228$	0	0
$229$	5.11558	0.338047	0.169024	−	0.985612i	$-0.445939\pi$
$229$	5.11558	0.338047	0.169024	+	0.985612i	$0.445939\pi$
$230$	−41.2266	−2.71840
$231$	0	0
$232$	7.26712	0.477110
$233$	−14.9600	−0.980063	−0.490032	−	0.871705i	$-0.663015\pi$
$233$	−14.9600	−0.980063	−0.490032	+	0.871705i	$0.663015\pi$
$234$	0	0
$235$	14.7084	0.959468
$236$	−11.2627	−0.733141
$237$	0	0
$238$	0.719005	0.0466062
$239$	11.9196	0.771015	0.385507	−	0.922705i	$-0.374026\pi$
$239$	11.9196	0.771015	0.385507	+	0.922705i	$0.374026\pi$
$240$	0	0
$241$	17.6997	1.14014	0.570068	−	0.821598i	$-0.306917\pi$
$241$	17.6997	1.14014	0.570068	+	0.821598i	$0.306917\pi$
$242$	21.5560	1.38567
$243$	0	0
$244$	13.5079	0.864756
$245$	25.8651	1.65246
$246$	0	0
$247$	2.01332	0.128105
$248$	4.89076	0.310564
$249$	0	0
$250$	−24.1941	−1.53017
$251$	−20.2883	−1.28058	−0.640292	−	0.768131i	$-0.721187\pi$
$251$	−20.2883	−1.28058	−0.640292	+	0.768131i	$0.721187\pi$
$252$	0	0
$253$	30.2043	1.89893
$254$	10.1088	0.634283
$255$	0	0
$256$	20.2288	1.26430
$257$	−26.9062	−1.67836	−0.839182	−	0.543851i	$-0.816966\pi$
$257$	−26.9062	−1.67836	−0.839182	+	0.543851i	$0.816966\pi$
$258$	0	0
$259$	0.850740	0.0528624
$260$	−18.7491	−1.16277
$261$	0	0
$262$	0.746970	0.0461480
$263$	−19.7521	−1.21797	−0.608983	−	0.793184i	$-0.708422\pi$
$263$	−19.7521	−1.21797	−0.608983	+	0.793184i	$0.708422\pi$
$264$	0	0
$265$	30.2700	1.85947
$266$	−0.0727878	−0.00446291
$267$	0	0
$268$	−7.15865	−0.437284
$269$	−12.3187	−0.751085	−0.375542	−	0.926805i	$-0.622544\pi$
$269$	−12.3187	−0.751085	−0.375542	+	0.926805i	$0.622544\pi$
$270$	0	0
$271$	10.3245	0.627168	0.313584	−	0.949560i	$-0.398470\pi$
$271$	10.3245	0.627168	0.313584	+	0.949560i	$0.398470\pi$
$272$	22.4758	1.36279
$273$	0	0
$274$	23.1915	1.40105
$275$	41.7796	2.51941
$276$	0	0
$277$	21.2126	1.27454	0.637269	−	0.770641i	$-0.280064\pi$
$277$	21.2126	1.27454	0.637269	+	0.770641i	$0.280064\pi$
$278$	13.1751	0.790189
$279$	0	0
$280$	−0.500180	−0.0298914
$281$	−5.50677	−0.328506	−0.164253	−	0.986418i	$-0.552521\pi$
$281$	−5.50677	−0.328506	−0.164253	+	0.986418i	$0.552521\pi$
$282$	0	0
$283$	−9.51678	−0.565714	−0.282857	−	0.959162i	$-0.591282\pi$
$283$	−9.51678	−0.565714	−0.282857	+	0.959162i	$0.591282\pi$
$284$	−15.9927	−0.948991
$285$	0	0
$286$	37.6089	2.22386
$287$	0.635781	0.0375290
$288$	0	0
$289$	3.39152	0.199501
$290$	31.6573	1.85898
$291$	0	0
$292$	12.9412	0.757325
$293$	−6.60529	−0.385885	−0.192943	−	0.981210i	$-0.561803\pi$
$293$	−6.60529	−0.385885	−0.192943	+	0.981210i	$0.561803\pi$
$294$	0	0
$295$	36.2038	2.10787
$296$	14.2961	0.830941
$297$	0	0
$298$	4.57040	0.264756
$299$	27.6511	1.59911
$300$	0	0
$301$	−0.0897007	−0.00517026
$302$	32.1704	1.85120
$303$	0	0
$304$	−2.27531	−0.130498
$305$	−43.4209	−2.48627
$306$	0	0
$307$	9.30227	0.530908	0.265454	−	0.964123i	$-0.414478\pi$
$307$	9.30227	0.530908	0.265454	+	0.964123i	$0.414478\pi$
$308$	−0.496613	−0.0282972
$309$	0	0
$310$	21.3053	1.21006
$311$	−22.7720	−1.29128	−0.645641	−	0.763641i	$-0.723410\pi$
$311$	−22.7720	−1.29128	−0.645641	+	0.763641i	$0.723410\pi$
$312$	0	0
$313$	9.63688	0.544709	0.272354	−	0.962197i	$-0.412198\pi$
$313$	9.63688	0.544709	0.272354	+	0.962197i	$0.412198\pi$
$314$	11.9048	0.671825
$315$	0	0
$316$	−10.6953	−0.601656
$317$	33.0980	1.85897	0.929485	−	0.368859i	$-0.120251\pi$
$317$	33.0980	1.85897	0.929485	+	0.368859i	$0.120251\pi$
$318$	0	0
$319$	−23.1934	−1.29858
$320$	1.39322	0.0778836
$321$	0	0
$322$	−0.999673	−0.0557096
$323$	−2.06432	−0.114862
$324$	0	0
$325$	38.2480	2.12162
$326$	2.33852	0.129519
$327$	0	0
$328$	10.6838	0.589916
$329$	0.356653	0.0196629
$330$	0	0
$331$	12.1690	0.668870	0.334435	−	0.942419i	$-0.391455\pi$
$331$	12.1690	0.668870	0.334435	+	0.942419i	$0.391455\pi$
$332$	−4.64472	−0.254912
$333$	0	0
$334$	−42.2437	−2.31147
$335$	23.0113	1.25724
$336$	0	0
$337$	−3.68855	−0.200928	−0.100464	−	0.994941i	$-0.532033\pi$
$337$	−3.68855	−0.200928	−0.100464	+	0.994941i	$0.532033\pi$
$338$	11.3541	0.617584
$339$	0	0
$340$	19.2240	1.04257
$341$	−15.6092	−0.845283
$342$	0	0
$343$	1.25509	0.0677684
$344$	−1.50735	−0.0812711
$345$	0	0
$346$	7.91726	0.425634
$347$	21.9004	1.17567	0.587837	−	0.808979i	$-0.299979\pi$
$347$	21.9004	1.17567	0.587837	+	0.808979i	$0.299979\pi$
$348$	0	0
$349$	−7.90695	−0.423250	−0.211625	−	0.977351i	$-0.567875\pi$
$349$	−7.90695	−0.423250	−0.211625	+	0.977351i	$0.567875\pi$
$350$	−1.38278	−0.0739128
$351$	0	0
$352$	−27.9998	−1.49239
$353$	9.43640	0.502249	0.251124	−	0.967955i	$-0.419200\pi$
$353$	9.43640	0.502249	0.251124	+	0.967955i	$0.419200\pi$
$354$	0	0
$355$	51.4081	2.72846
$356$	−9.74081	−0.516262
$357$	0	0
$358$	24.8766	1.31477
$359$	20.4163	1.07753	0.538765	−	0.842456i	$-0.318891\pi$
$359$	20.4163	1.07753	0.538765	+	0.842456i	$0.318891\pi$
$360$	0	0
$361$	−18.7910	−0.989001
$362$	−40.2349	−2.11470
$363$	0	0
$364$	−0.454634	−0.0238293
$365$	−41.5991	−2.17740
$366$	0	0
$367$	4.02929	0.210327	0.105164	−	0.994455i	$-0.466463\pi$
$367$	4.02929	0.210327	0.105164	+	0.994455i	$0.466463\pi$
$368$	−31.2493	−1.62898
$369$	0	0
$370$	62.2769	3.23762
$371$	0.733995	0.0381071
$372$	0	0
$373$	9.28855	0.480943	0.240471	−	0.970656i	$-0.422698\pi$
$373$	9.28855	0.480943	0.240471	+	0.970656i	$0.422698\pi$
$374$	−38.5615	−1.99397
$375$	0	0
$376$	5.99328	0.309080
$377$	−21.2329	−1.09355
$378$	0	0
$379$	29.2337	1.50164	0.750818	−	0.660509i	$-0.229660\pi$
$379$	29.2337	1.50164	0.750818	+	0.660509i	$0.229660\pi$
$380$	−1.94613	−0.0998341
$381$	0	0
$382$	−6.84776	−0.350362
$383$	−13.8573	−0.708076	−0.354038	−	0.935231i	$-0.615192\pi$
$383$	−13.8573	−0.708076	−0.354038	+	0.935231i	$0.615192\pi$
$384$	0	0
$385$	1.59635	0.0813576
$386$	−24.9396	−1.26939
$387$	0	0
$388$	−0.781572	−0.0396783
$389$	−10.9434	−0.554850	−0.277425	−	0.960747i	$-0.589481\pi$
$389$	−10.9434	−0.554850	−0.277425	+	0.960747i	$0.589481\pi$
$390$	0	0
$391$	−28.3515	−1.43380
$392$	10.5393	0.532317
$393$	0	0
$394$	−3.17354	−0.159880
$395$	34.3797	1.72983
$396$	0	0
$397$	−32.9079	−1.65160	−0.825799	−	0.563964i	$-0.809276\pi$
$397$	−32.9079	−1.65160	−0.825799	+	0.563964i	$0.809276\pi$
$398$	−31.1432	−1.56107
$399$	0	0
$400$	−43.2251	−2.16126
$401$	12.8058	0.639493	0.319746	−	0.947503i	$-0.396402\pi$
$401$	12.8058	0.639493	0.319746	+	0.947503i	$0.396402\pi$
$402$	0	0
$403$	−14.2897	−0.711821
$404$	−4.17904	−0.207915
$405$	0	0
$406$	0.767634	0.0380970
$407$	−45.6267	−2.26163
$408$	0	0
$409$	−20.8369	−1.03032	−0.515160	−	0.857094i	$-0.672268\pi$
$409$	−20.8369	−1.03032	−0.515160	+	0.857094i	$0.672268\pi$
$410$	46.5413	2.29851
$411$	0	0
$412$	−6.28281	−0.309532
$413$	0.877879	0.0431976
$414$	0	0
$415$	14.9303	0.732902
$416$	−25.6329	−1.25676
$417$	0	0
$418$	3.90374	0.190938
$419$	−18.2465	−0.891400	−0.445700	−	0.895182i	$-0.647045\pi$
$419$	−18.2465	−0.891400	−0.445700	+	0.895182i	$0.647045\pi$
$420$	0	0
$421$	20.6352	1.00570	0.502849	−	0.864374i	$-0.332285\pi$
$421$	20.6352	1.00570	0.502849	+	0.864374i	$0.332285\pi$
$422$	−19.6214	−0.955154
$423$	0	0
$424$	12.3342	0.599003
$425$	−39.2167	−1.90229
$426$	0	0
$427$	−1.05288	−0.0509525
$428$	2.35578	0.113871
$429$	0	0
$430$	−6.56638	−0.316659
$431$	21.7596	1.04812	0.524062	−	0.851680i	$-0.324416\pi$
$431$	21.7596	1.04812	0.524062	+	0.851680i	$0.324416\pi$
$432$	0	0
$433$	28.7635	1.38229	0.691144	−	0.722717i	$-0.257107\pi$
$433$	28.7635	1.38229	0.691144	+	0.722717i	$0.257107\pi$
$434$	0.516617	0.0247984
$435$	0	0
$436$	−2.02419	−0.0969411
$437$	2.87013	0.137297
$438$	0	0
$439$	−26.3586	−1.25803	−0.629015	−	0.777393i	$-0.716542\pi$
$439$	−26.3586	−1.25803	−0.629015	+	0.777393i	$0.716542\pi$
$440$	26.8255	1.27886
$441$	0	0
$442$	−35.3019	−1.67914
$443$	15.0840	0.716664	0.358332	−	0.933594i	$-0.383346\pi$
$443$	15.0840	0.716664	0.358332	+	0.933594i	$0.383346\pi$
$444$	0	0
$445$	31.3116	1.48431
$446$	−4.90366	−0.232195
$447$	0	0
$448$	0.0337833	0.00159611
$449$	29.9757	1.41464	0.707320	−	0.706893i	$-0.249904\pi$
$449$	29.9757	1.41464	0.707320	+	0.706893i	$0.249904\pi$
$450$	0	0
$451$	−34.0981	−1.60562
$452$	15.4454	0.726490
$453$	0	0
$454$	−17.5983	−0.825928
$455$	1.46141	0.0685120
$456$	0	0
$457$	−21.3108	−0.996878	−0.498439	−	0.866925i	$-0.666093\pi$
$457$	−21.3108	−0.996878	−0.498439	+	0.866925i	$0.666093\pi$
$458$	9.08043	0.424301
$459$	0	0
$460$	−26.7282	−1.24621
$461$	−8.76963	−0.408442	−0.204221	−	0.978925i	$-0.565466\pi$
$461$	−8.76963	−0.408442	−0.204221	+	0.978925i	$0.565466\pi$
$462$	0	0
$463$	31.5067	1.46424	0.732121	−	0.681174i	$-0.238530\pi$
$463$	31.5067	1.46424	0.732121	+	0.681174i	$0.238530\pi$
$464$	23.9959	1.11398
$465$	0	0
$466$	−26.5548	−1.23013
$467$	−19.0342	−0.880796	−0.440398	−	0.897803i	$-0.645163\pi$
$467$	−19.0342	−0.880796	−0.440398	+	0.897803i	$0.645163\pi$
$468$	0	0
$469$	0.557985	0.0257654
$470$	26.1081	1.20428
$471$	0	0
$472$	14.7521	0.679021
$473$	4.81081	0.221201
$474$	0	0
$475$	3.97007	0.182159
$476$	0.466149	0.0213659
$477$	0	0
$478$	21.1579	0.967741
$479$	−36.2190	−1.65489	−0.827443	−	0.561550i	$-0.810205\pi$
$479$	−36.2190	−1.65489	−0.827443	+	0.561550i	$0.810205\pi$
$480$	0	0
$481$	−41.7698	−1.90454
$482$	31.4178	1.43104
$483$	0	0
$484$	13.9753	0.635241
$485$	2.51235	0.114080
$486$	0	0
$487$	17.1775	0.778386	0.389193	−	0.921156i	$-0.372754\pi$
$487$	17.1775	0.778386	0.389193	+	0.921156i	$0.372754\pi$
$488$	−17.6929	−0.800920
$489$	0	0
$490$	45.9119	2.07409
$491$	−42.5417	−1.91988	−0.959940	−	0.280207i	$-0.909597\pi$
$491$	−42.5417	−1.91988	−0.959940	+	0.280207i	$0.909597\pi$
$492$	0	0
$493$	21.7707	0.980502
$494$	3.57375	0.160791
$495$	0	0
$496$	16.1492	0.725121
$497$	1.24656	0.0559158
$498$	0	0
$499$	8.29507	0.371338	0.185669	−	0.982612i	$-0.440555\pi$
$499$	8.29507	0.371338	0.185669	+	0.982612i	$0.440555\pi$
$500$	−15.6856	−0.701483
$501$	0	0
$502$	−36.0128	−1.60733
$503$	25.8405	1.15217	0.576085	−	0.817390i	$-0.304580\pi$
$503$	25.8405	1.15217	0.576085	+	0.817390i	$0.304580\pi$
$504$	0	0
$505$	13.4334	0.597780
$506$	53.6142	2.38344
$507$	0	0
$508$	6.55380	0.290778
$509$	24.6849	1.09414	0.547069	−	0.837088i	$-0.315744\pi$
$509$	24.6849	1.09414	0.547069	+	0.837088i	$0.315744\pi$
$510$	0	0
$511$	−1.00871	−0.0446226
$512$	13.9636	0.617109
$513$	0	0
$514$	−47.7599	−2.10660
$515$	20.1960	0.889941
$516$	0	0
$517$	−19.1279	−0.841244
$518$	1.51011	0.0663503
$519$	0	0
$520$	24.5579	1.07694
$521$	39.1146	1.71364	0.856822	−	0.515613i	$-0.172436\pi$
$521$	39.1146	1.71364	0.856822	+	0.515613i	$0.172436\pi$
$522$	0	0
$523$	2.94374	0.128721	0.0643603	−	0.997927i	$-0.479499\pi$
$523$	2.94374	0.128721	0.0643603	+	0.997927i	$0.479499\pi$
$524$	0.484280	0.0211559
$525$	0	0
$526$	−35.0610	−1.52873
$527$	14.6516	0.638235
$528$	0	0
$529$	16.4186	0.713854
$530$	53.7308	2.33392
$531$	0	0
$532$	−0.0471902	−0.00204595
$533$	−31.2158	−1.35210
$534$	0	0
$535$	−7.57260	−0.327392
$536$	9.37653	0.405004
$537$	0	0
$538$	−21.8663	−0.942725
$539$	−33.6369	−1.44885
$540$	0	0
$541$	−9.17458	−0.394446	−0.197223	−	0.980359i	$-0.563192\pi$
$541$	−9.17458	−0.394446	−0.197223	+	0.980359i	$0.563192\pi$
$542$	18.3265	0.787191
$543$	0	0
$544$	26.2822	1.12684
$545$	6.50671	0.278717
$546$	0	0
$547$	−17.8333	−0.762498	−0.381249	−	0.924472i	$-0.624506\pi$
$547$	−17.8333	−0.762498	−0.381249	+	0.924472i	$0.624506\pi$
$548$	15.0357	0.642292
$549$	0	0
$550$	74.1610	3.16224
$551$	−2.20393	−0.0938908
$552$	0	0
$553$	0.833649	0.0354504
$554$	37.6534	1.59974
$555$	0	0
$556$	8.54175	0.362251
$557$	−46.8931	−1.98693	−0.993463	−	0.114154i	$-0.963584\pi$
$557$	−46.8931	−1.98693	−0.993463	+	0.114154i	$0.963584\pi$
$558$	0	0
$559$	4.40415	0.186276
$560$	−1.65158	−0.0697922
$561$	0	0
$562$	−9.77480	−0.412325
$563$	−28.5293	−1.20237	−0.601183	−	0.799111i	$-0.705304\pi$
$563$	−28.5293	−1.20237	−0.601183	+	0.799111i	$0.705304\pi$
$564$	0	0
$565$	−49.6489	−2.08874
$566$	−16.8928	−0.710056
$567$	0	0
$568$	20.9475	0.878937
$569$	14.4016	0.603747	0.301874	−	0.953348i	$-0.402388\pi$
$569$	14.4016	0.603747	0.301874	+	0.953348i	$0.402388\pi$
$570$	0	0
$571$	9.84307	0.411920	0.205960	−	0.978560i	$-0.433968\pi$
$571$	9.84307	0.411920	0.205960	+	0.978560i	$0.433968\pi$
$572$	24.3828	1.01950
$573$	0	0
$574$	1.12855	0.0471046
$575$	54.5252	2.27386
$576$	0	0
$577$	37.4871	1.56061	0.780304	−	0.625400i	$-0.215064\pi$
$577$	37.4871	1.56061	0.780304	+	0.625400i	$0.215064\pi$
$578$	6.02012	0.250404
$579$	0	0
$580$	20.5242	0.852222
$581$	0.362035	0.0150197
$582$	0	0
$583$	−39.3654	−1.63035
$584$	−16.9506	−0.701419
$585$	0	0
$586$	−11.7247	−0.484345
$587$	11.6667	0.481536	0.240768	−	0.970583i	$-0.422601\pi$
$587$	11.6667	0.481536	0.240768	+	0.970583i	$0.422601\pi$
$588$	0	0
$589$	−1.48325	−0.0611160
$590$	64.2636	2.64569
$591$	0	0
$592$	47.2053	1.94013
$593$	15.7994	0.648805	0.324402	−	0.945919i	$-0.394837\pi$
$593$	15.7994	0.648805	0.324402	+	0.945919i	$0.394837\pi$
$594$	0	0
$595$	−1.49843	−0.0614295
$596$	2.96311	0.121374
$597$	0	0
$598$	49.0822	2.00712
$599$	−18.5381	−0.757445	−0.378722	−	0.925510i	$-0.623637\pi$
$599$	−18.5381	−0.757445	−0.378722	+	0.925510i	$0.623637\pi$
$600$	0	0
$601$	40.6099	1.65651	0.828256	−	0.560350i	$-0.189333\pi$
$601$	40.6099	1.65651	0.828256	+	0.560350i	$0.189333\pi$
$602$	−0.159223	−0.00648946
$603$	0	0
$604$	20.8569	0.848654
$605$	−44.9233	−1.82639
$606$	0	0
$607$	−8.77395	−0.356124	−0.178062	−	0.984019i	$-0.556983\pi$
$607$	−8.77395	−0.356124	−0.178062	+	0.984019i	$0.556983\pi$
$608$	−2.66065	−0.107904
$609$	0	0
$610$	−77.0744	−3.12065
$611$	−17.5110	−0.708420
$612$	0	0
$613$	−13.3507	−0.539230	−0.269615	−	0.962968i	$-0.586897\pi$
$613$	−13.3507	−0.539230	−0.269615	+	0.962968i	$0.586897\pi$
$614$	16.5120	0.666371
$615$	0	0
$616$	0.650472	0.0262083
$617$	28.4966	1.14723	0.573615	−	0.819125i	$-0.305541\pi$
$617$	28.4966	1.14723	0.573615	+	0.819125i	$0.305541\pi$
$618$	0	0
$619$	6.31760	0.253926	0.126963	−	0.991907i	$-0.459477\pi$
$619$	6.31760	0.253926	0.126963	+	0.991907i	$0.459477\pi$
$620$	13.8128	0.554734
$621$	0	0
$622$	−40.4215	−1.62075
$623$	0.759253	0.0304188
$624$	0	0
$625$	6.99847	0.279939
$626$	17.1060	0.683692
$627$	0	0
$628$	7.71816	0.307988
$629$	42.8278	1.70766
$630$	0	0
$631$	−8.72945	−0.347514	−0.173757	−	0.984789i	$-0.555591\pi$
$631$	−8.72945	−0.347514	−0.173757	+	0.984789i	$0.555591\pi$
$632$	14.0089	0.557242
$633$	0	0
$634$	58.7508	2.33329
$635$	−21.0670	−0.836020
$636$	0	0
$637$	−30.7936	−1.22009
$638$	−41.1696	−1.62992
$639$	0	0
$640$	−40.5877	−1.60437
$641$	−25.9003	−1.02300	−0.511499	−	0.859284i	$-0.670910\pi$
$641$	−25.9003	−1.02300	−0.511499	+	0.859284i	$0.670910\pi$
$642$	0	0
$643$	−25.0923	−0.989542	−0.494771	−	0.869023i	$-0.664748\pi$
$643$	−25.0923	−0.989542	−0.494771	+	0.869023i	$0.664748\pi$
$644$	−0.648113	−0.0255392
$645$	0	0
$646$	−3.66427	−0.144169
$647$	−39.8334	−1.56601	−0.783006	−	0.622014i	$-0.786315\pi$
$647$	−39.8334	−1.56601	−0.783006	+	0.622014i	$0.786315\pi$
$648$	0	0
$649$	−47.0822	−1.84814
$650$	67.8922	2.66295
$651$	0	0
$652$	1.51612	0.0593760
$653$	28.3610	1.10985	0.554926	−	0.831900i	$-0.312747\pi$
$653$	28.3610	1.10985	0.554926	+	0.831900i	$0.312747\pi$
$654$	0	0
$655$	−1.55671	−0.0608256
$656$	35.2778	1.37737
$657$	0	0
$658$	0.633077	0.0246799
$659$	14.7536	0.574717	0.287358	−	0.957823i	$-0.407223\pi$
$659$	14.7536	0.574717	0.287358	+	0.957823i	$0.407223\pi$
$660$	0	0
$661$	−34.4159	−1.33862	−0.669312	−	0.742982i	$-0.733411\pi$
$661$	−34.4159	−1.33862	−0.669312	+	0.742982i	$0.733411\pi$
$662$	21.6006	0.839533
$663$	0	0
$664$	6.08373	0.236094
$665$	0.151692	0.00588236
$666$	0	0
$667$	−30.2690	−1.17202
$668$	−27.3877	−1.05966
$669$	0	0
$670$	40.8463	1.57803
$671$	56.4679	2.17992
$672$	0	0
$673$	0.485034	0.0186967	0.00934835	−	0.999956i	$-0.497024\pi$
$673$	0.485034	0.0186967	0.00934835	+	0.999956i	$0.497024\pi$
$674$	−6.54737	−0.252195
$675$	0	0
$676$	7.36118	0.283122
$677$	−15.2160	−0.584797	−0.292398	−	0.956297i	$-0.594453\pi$
$677$	−15.2160	−0.584797	−0.292398	+	0.956297i	$0.594453\pi$
$678$	0	0
$679$	0.0609201	0.00233790
$680$	−25.1800	−0.965607
$681$	0	0
$682$	−27.7071	−1.06096
$683$	−15.8107	−0.604978	−0.302489	−	0.953153i	$-0.597818\pi$
$683$	−15.8107	−0.604978	−0.302489	+	0.953153i	$0.597818\pi$
$684$	0	0
$685$	−48.3318	−1.84666
$686$	2.22785	0.0850596
$687$	0	0
$688$	−4.97726	−0.189756
$689$	−36.0379	−1.37293
$690$	0	0
$691$	21.1215	0.803500	0.401750	−	0.915749i	$-0.368402\pi$
$691$	21.1215	0.803500	0.401750	+	0.915749i	$0.368402\pi$
$692$	5.13296	0.195126
$693$	0	0
$694$	38.8744	1.47565
$695$	−27.4573	−1.04151
$696$	0	0
$697$	32.0064	1.21233
$698$	−14.0353	−0.531242
$699$	0	0
$700$	−0.896493	−0.0338842
$701$	40.9055	1.54498	0.772491	−	0.635026i	$-0.219011\pi$
$701$	40.9055	1.54498	0.772491	+	0.635026i	$0.219011\pi$
$702$	0	0
$703$	−4.33563	−0.163522
$704$	−1.81186	−0.0682869
$705$	0	0
$706$	16.7501	0.630399
$707$	0.325738	0.0122506
$708$	0	0
$709$	32.0272	1.20281	0.601404	−	0.798945i	$-0.294608\pi$
$709$	32.0272	1.20281	0.601404	+	0.798945i	$0.294608\pi$
$710$	91.2522	3.42463
$711$	0	0
$712$	12.7587	0.478152
$713$	−20.3710	−0.762900
$714$	0	0
$715$	−78.3781	−2.93118
$716$	16.1281	0.602736
$717$	0	0
$718$	36.2400	1.35246
$719$	9.63404	0.359289	0.179644	−	0.983732i	$-0.442505\pi$
$719$	9.63404	0.359289	0.179644	+	0.983732i	$0.442505\pi$
$720$	0	0
$721$	0.489718	0.0182380
$722$	−33.3551	−1.24135
$723$	0	0
$724$	−26.0853	−0.969452
$725$	−41.8691	−1.55498
$726$	0	0
$727$	−11.9907	−0.444711	−0.222356	−	0.974966i	$-0.571375\pi$
$727$	−11.9907	−0.444711	−0.222356	+	0.974966i	$0.571375\pi$
$728$	0.595488	0.0220702
$729$	0	0
$730$	−73.8406	−2.73296
$731$	−4.51570	−0.167019
$732$	0	0
$733$	0.0564342	0.00208445	0.00104222	−	0.999999i	$-0.499668\pi$
$733$	0.0564342	0.00208445	0.00104222	+	0.999999i	$0.499668\pi$
$734$	7.15221	0.263993
$735$	0	0
$736$	−36.5416	−1.34694
$737$	−29.9257	−1.10233
$738$	0	0
$739$	−6.30302	−0.231860	−0.115930	−	0.993257i	$-0.536985\pi$
$739$	−6.30302	−0.231860	−0.115930	+	0.993257i	$0.536985\pi$
$740$	40.3757	1.48424
$741$	0	0
$742$	1.30288	0.0478302
$743$	−52.6136	−1.93021	−0.965103	−	0.261872i	$-0.915660\pi$
$743$	−52.6136	−1.93021	−0.965103	+	0.261872i	$0.915660\pi$
$744$	0	0
$745$	−9.52485	−0.348964
$746$	16.4877	0.603656
$747$	0	0
$748$	−25.0004	−0.914105
$749$	−0.183622	−0.00670942
$750$	0	0
$751$	−2.21252	−0.0807358	−0.0403679	−	0.999185i	$-0.512853\pi$
$751$	−2.21252	−0.0807358	−0.0403679	+	0.999185i	$0.512853\pi$
$752$	19.7897	0.721657
$753$	0	0
$754$	−37.6895	−1.37257
$755$	−67.0440	−2.43998
$756$	0	0
$757$	−43.0907	−1.56616	−0.783079	−	0.621923i	$-0.786352\pi$
$757$	−43.0907	−1.56616	−0.783079	+	0.621923i	$0.786352\pi$
$758$	51.8914	1.88478
$759$	0	0
$760$	2.54907	0.0924645
$761$	41.6973	1.51153	0.755763	−	0.654846i	$-0.227266\pi$
$761$	41.6973	1.51153	0.755763	+	0.654846i	$0.227266\pi$
$762$	0	0
$763$	0.157777	0.00571189
$764$	−4.43958	−0.160618
$765$	0	0
$766$	−24.5975	−0.888743
$767$	−43.1023	−1.55634
$768$	0	0
$769$	36.7504	1.32525	0.662627	−	0.748950i	$-0.269442\pi$
$769$	36.7504	1.32525	0.662627	+	0.748950i	$0.269442\pi$
$770$	2.83361	0.102116
$771$	0	0
$772$	−16.1690	−0.581934
$773$	−10.3042	−0.370617	−0.185309	−	0.982680i	$-0.559329\pi$
$773$	−10.3042	−0.370617	−0.185309	+	0.982680i	$0.559329\pi$
$774$	0	0
$775$	−28.1779	−1.01218
$776$	1.02372	0.0367493
$777$	0	0
$778$	−19.4250	−0.696421
$779$	−3.24014	−0.116090
$780$	0	0
$781$	−66.8551	−2.39226
$782$	−50.3254	−1.79963
$783$	0	0
$784$	34.8007	1.24288
$785$	−24.8099	−0.885502
$786$	0	0
$787$	−6.74070	−0.240280	−0.120140	−	0.992757i	$-0.538334\pi$
$787$	−6.74070	−0.240280	−0.120140	+	0.992757i	$0.538334\pi$
$788$	−2.05748	−0.0732948
$789$	0	0
$790$	61.0258	2.17120
$791$	−1.20390	−0.0428057
$792$	0	0
$793$	51.6947	1.83573
$794$	−58.4132	−2.07301
$795$	0	0
$796$	−20.1910	−0.715649
$797$	37.9463	1.34413	0.672063	−	0.740494i	$-0.265408\pi$
$797$	37.9463	1.34413	0.672063	+	0.740494i	$0.265408\pi$
$798$	0	0
$799$	17.9545	0.635186
$800$	−50.5456	−1.78706
$801$	0	0
$802$	22.7310	0.802660
$803$	54.0987	1.90910
$804$	0	0
$805$	2.08335	0.0734283
$806$	−25.3650	−0.893443
$807$	0	0
$808$	5.47378	0.192567
$809$	13.5355	0.475884	0.237942	−	0.971279i	$-0.423527\pi$
$809$	13.5355	0.475884	0.237942	+	0.971279i	$0.423527\pi$
$810$	0	0
$811$	−21.9654	−0.771309	−0.385654	−	0.922643i	$-0.626024\pi$
$811$	−21.9654	−0.771309	−0.385654	+	0.922643i	$0.626024\pi$
$812$	0.497677	0.0174650
$813$	0	0
$814$	−80.9898	−2.83869
$815$	−4.87355	−0.170713
$816$	0	0
$817$	0.457142	0.0159934
$818$	−36.9867	−1.29321
$819$	0	0
$820$	30.1739	1.05372
$821$	−8.97895	−0.313367	−0.156684	−	0.987649i	$-0.550080\pi$
$821$	−8.97895	−0.313367	−0.156684	+	0.987649i	$0.550080\pi$
$822$	0	0
$823$	35.6733	1.24349	0.621746	−	0.783219i	$-0.286424\pi$
$823$	35.6733	1.24349	0.621746	+	0.783219i	$0.286424\pi$
$824$	8.22934	0.286683
$825$	0	0
$826$	1.55828	0.0542196
$827$	14.9924	0.521337	0.260668	−	0.965428i	$-0.416057\pi$
$827$	14.9924	0.521337	0.260668	+	0.965428i	$0.416057\pi$
$828$	0	0
$829$	3.42729	0.119035	0.0595174	−	0.998227i	$-0.481044\pi$
$829$	3.42729	0.119035	0.0595174	+	0.998227i	$0.481044\pi$
$830$	26.5021	0.919903
$831$	0	0
$832$	−1.65870	−0.0575050
$833$	31.5735	1.09396
$834$	0	0
$835$	88.0372	3.04665
$836$	2.53089	0.0875328
$837$	0	0
$838$	−32.3885	−1.11884
$839$	56.0444	1.93487	0.967433	−	0.253127i	$-0.0814590\pi$
$839$	56.0444	1.93487	0.967433	+	0.253127i	$0.0814590\pi$
$840$	0	0
$841$	−5.75688	−0.198513
$842$	36.6286	1.26230
$843$	0	0
$844$	−12.7210	−0.437876
$845$	−23.6624	−0.814010
$846$	0	0
$847$	−1.08931	−0.0374292
$848$	40.7274	1.39859
$849$	0	0
$850$	−69.6118	−2.38766
$851$	−59.5459	−2.04121
$852$	0	0
$853$	−20.8382	−0.713488	−0.356744	−	0.934202i	$-0.616113\pi$
$853$	−20.8382	−0.713488	−0.356744	+	0.934202i	$0.616113\pi$
$854$	−1.86892	−0.0639532
$855$	0	0
$856$	−3.08564	−0.105465
$857$	−22.2946	−0.761570	−0.380785	−	0.924664i	$-0.624346\pi$
$857$	−22.2946	−0.761570	−0.380785	+	0.924664i	$0.624346\pi$
$858$	0	0
$859$	30.3621	1.03594	0.517970	−	0.855399i	$-0.326688\pi$
$859$	30.3621	1.03594	0.517970	+	0.855399i	$0.326688\pi$
$860$	−4.25715	−0.145168
$861$	0	0
$862$	38.6245	1.31555
$863$	11.2552	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$863$	11.2552	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$864$	0	0
$865$	−16.4998	−0.561010
$866$	51.0568	1.73498
$867$	0	0
$868$	0.334936	0.0113685
$869$	−44.7101	−1.51669
$870$	0	0
$871$	−27.3961	−0.928281
$872$	2.65132	0.0897849
$873$	0	0
$874$	5.09464	0.172329
$875$	1.22263	0.0413323
$876$	0	0
$877$	3.62407	0.122376	0.0611881	−	0.998126i	$-0.480511\pi$
$877$	3.62407	0.122376	0.0611881	+	0.998126i	$0.480511\pi$
$878$	−46.7880	−1.57902
$879$	0	0
$880$	88.5774	2.98594
$881$	44.6822	1.50538	0.752692	−	0.658373i	$-0.228755\pi$
$881$	44.6822	1.50538	0.752692	+	0.658373i	$0.228755\pi$
$882$	0	0
$883$	54.4600	1.83273	0.916363	−	0.400348i	$-0.131111\pi$
$883$	54.4600	1.83273	0.916363	+	0.400348i	$0.131111\pi$
$884$	−22.8871	−0.769777
$885$	0	0
$886$	26.7749	0.899522
$887$	−7.06446	−0.237201	−0.118601	−	0.992942i	$-0.537841\pi$
$887$	−7.06446	−0.237201	−0.118601	+	0.992942i	$0.537841\pi$
$888$	0	0
$889$	−0.510839	−0.0171330
$890$	55.5798	1.86304
$891$	0	0
$892$	−3.17917	−0.106446
$893$	−1.81761	−0.0608240
$894$	0	0
$895$	−51.8435	−1.73294
$896$	−0.984182	−0.0328792
$897$	0	0
$898$	53.2085	1.77559
$899$	15.6426	0.521710
$900$	0	0
$901$	36.9506	1.23100
$902$	−60.5259	−2.01529
$903$	0	0
$904$	−20.2306	−0.672861
$905$	83.8506	2.78729
$906$	0	0
$907$	−28.0393	−0.931031	−0.465516	−	0.885040i	$-0.654131\pi$
$907$	−28.0393	−0.931031	−0.465516	+	0.885040i	$0.654131\pi$
$908$	−11.4094	−0.378635
$909$	0	0
$910$	2.59408	0.0859930
$911$	43.0260	1.42552	0.712758	−	0.701411i	$-0.247446\pi$
$911$	43.0260	1.42552	0.712758	+	0.701411i	$0.247446\pi$
$912$	0	0
$913$	−19.4166	−0.642595
$914$	−37.8278	−1.25123
$915$	0	0
$916$	5.88708	0.194514
$917$	−0.0377475	−0.00124653
$918$	0	0
$919$	9.18522	0.302993	0.151496	−	0.988458i	$-0.451591\pi$
$919$	9.18522	0.302993	0.151496	+	0.988458i	$0.451591\pi$
$920$	35.0091	1.15422
$921$	0	0
$922$	−15.5666	−0.512657
$923$	−61.2038	−2.01455
$924$	0	0
$925$	−82.3659	−2.70817
$926$	55.9261	1.83785
$927$	0	0
$928$	28.0598	0.921107
$929$	−0.530102	−0.0173921	−0.00869604	−	0.999962i	$-0.502768\pi$
$929$	−0.530102	−0.0173921	−0.00869604	+	0.999962i	$0.502768\pi$
$930$	0	0
$931$	−3.19632	−0.104755
$932$	−17.2162	−0.563934
$933$	0	0
$934$	−33.7866	−1.10553
$935$	80.3633	2.62816
$936$	0	0
$937$	−36.8282	−1.20313	−0.601563	−	0.798825i	$-0.705455\pi$
$937$	−36.8282	−1.20313	−0.601563	+	0.798825i	$0.705455\pi$
$938$	0.990453	0.0323394
$939$	0	0
$940$	16.9266	0.552084
$941$	27.2115	0.887069	0.443535	−	0.896257i	$-0.353724\pi$
$941$	27.2115	0.887069	0.443535	+	0.896257i	$0.353724\pi$
$942$	0	0
$943$	−44.5003	−1.44913
$944$	48.7112	1.58542
$945$	0	0
$946$	8.53944	0.277641
$947$	3.43102	0.111493	0.0557466	−	0.998445i	$-0.482246\pi$
$947$	3.43102	0.111493	0.0557466	+	0.998445i	$0.482246\pi$
$948$	0	0
$949$	49.5257	1.60767
$950$	7.04708	0.228638
$951$	0	0
$952$	−0.610570	−0.0197887
$953$	−27.0518	−0.876293	−0.438146	−	0.898904i	$-0.644365\pi$
$953$	−27.0518	−0.876293	−0.438146	+	0.898904i	$0.644365\pi$
$954$	0	0
$955$	14.2709	0.461797
$956$	13.7172	0.443647
$957$	0	0
$958$	−64.2905	−2.07713
$959$	−1.17196	−0.0378447
$960$	0	0
$961$	−20.4726	−0.660405
$962$	−74.1437	−2.39049
$963$	0	0
$964$	20.3690	0.656041
$965$	51.9749	1.67313
$966$	0	0
$967$	−21.4428	−0.689554	−0.344777	−	0.938685i	$-0.612045\pi$
$967$	−21.4428	−0.689554	−0.344777	+	0.938685i	$0.612045\pi$
$968$	−18.3051	−0.588348
$969$	0	0
$970$	4.45955	0.143187
$971$	27.1957	0.872751	0.436376	−	0.899765i	$-0.356262\pi$
$971$	27.1957	0.872751	0.436376	+	0.899765i	$0.356262\pi$
$972$	0	0
$973$	−0.665791	−0.0213443
$974$	30.4909	0.976992
$975$	0	0
$976$	−58.4217	−1.87003
$977$	−19.9057	−0.636839	−0.318420	−	0.947950i	$-0.603152\pi$
$977$	−19.9057	−0.636839	−0.318420	+	0.947950i	$0.603152\pi$
$978$	0	0
$979$	−40.7201	−1.30142
$980$	29.7658	0.950834
$981$	0	0
$982$	−75.5137	−2.40974
$983$	−8.10258	−0.258432	−0.129216	−	0.991616i	$-0.541246\pi$
$983$	−8.10258	−0.258432	−0.129216	+	0.991616i	$0.541246\pi$
$984$	0	0
$985$	6.61374	0.210731
$986$	38.6441	1.23068
$987$	0	0
$988$	2.31696	0.0737122
$989$	6.27843	0.199642
$990$	0	0
$991$	−33.1910	−1.05435	−0.527174	−	0.849757i	$-0.676748\pi$
$991$	−33.1910	−1.05435	−0.527174	+	0.849757i	$0.676748\pi$
$992$	18.8842	0.599573
$993$	0	0
$994$	2.21271	0.0701828
$995$	64.9034	2.05758
$996$	0	0
$997$	2.34684	0.0743251	0.0371626	−	0.999309i	$-0.488168\pi$
$997$	2.34684	0.0743251	0.0371626	+	0.999309i	$0.488168\pi$
$998$	14.7242	0.466086
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3483.2.a.r.1.16		19
3.2	odd	2		3483.2.a.s.1.4		19
9.2	odd	6		387.2.f.c.130.16	✓	38
9.4	even	3		1161.2.f.c.775.4		38
9.5	odd	6		387.2.f.c.259.16	yes	38
9.7	even	3		1161.2.f.c.388.4		38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
387.2.f.c.130.16	✓	38	9.2	odd	6
387.2.f.c.259.16	yes	38	9.5	odd	6
1161.2.f.c.388.4		38	9.7	even	3
1161.2.f.c.775.4		38	9.4	even	3
3483.2.a.r.1.16		19	1.1	even	1	trivial
3483.2.a.s.1.4		19	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3483 = 3^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3483.a (trivial)

\(f(q)\)	\(=\)	\(q+1.77505 q^{2} +1.15081 q^{4} -3.69926 q^{5} -0.0897007 q^{7} -1.50735 q^{8} +O(q^{10})\)	\(q+1.77505 q^{2} +1.15081 q^{4} -3.69926 q^{5} -0.0897007 q^{7} -1.50735 q^{8} -6.56638 q^{10} +4.81081 q^{11} +4.40415 q^{13} -0.159223 q^{14} -4.97726 q^{16} -4.51570 q^{17} +0.457142 q^{19} -4.25715 q^{20} +8.53944 q^{22} +6.27843 q^{23} +8.68453 q^{25} +7.81759 q^{26} -0.103229 q^{28} -4.82111 q^{29} -3.24460 q^{31} -5.82018 q^{32} -8.01560 q^{34} +0.331826 q^{35} -9.48421 q^{37} +0.811452 q^{38} +5.57609 q^{40} -7.08781 q^{41} +1.00000 q^{43} +5.53633 q^{44} +11.1445 q^{46} -3.97603 q^{47} -6.99195 q^{49} +15.4155 q^{50} +5.06835 q^{52} -8.18271 q^{53} -17.7964 q^{55} +0.135211 q^{56} -8.55773 q^{58} -9.78676 q^{59} +11.7377 q^{61} -5.75934 q^{62} -0.376622 q^{64} -16.2921 q^{65} -6.22052 q^{67} -5.19672 q^{68} +0.589009 q^{70} -13.8969 q^{71} +11.2453 q^{73} -16.8350 q^{74} +0.526085 q^{76} -0.431533 q^{77} -9.29367 q^{79} +18.4122 q^{80} -12.5812 q^{82} -4.03603 q^{83} +16.7047 q^{85} +1.77505 q^{86} -7.25159 q^{88} -8.46429 q^{89} -0.395055 q^{91} +7.22529 q^{92} -7.05766 q^{94} -1.69109 q^{95} -0.679148 q^{97} -12.4111 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 9 q^{5} + 7 q^{7} - 12 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 9 * q^5 + 7 * q^7 - 12 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 9 q^{5} + 7 q^{7} - 12 q^{8} - 7 q^{10} - 5 q^{11} - 5 q^{13} - 17 q^{14} + 24 q^{16} - 21 q^{17} - 4 q^{19} - 21 q^{20} - 20 q^{22} - 22 q^{23} + 10 q^{25} - 17 q^{26} - q^{28} - 30 q^{29} - 5 q^{31} - 48 q^{32} - 6 q^{34} - 53 q^{35} - q^{37} - 21 q^{38} + 16 q^{40} - 29 q^{41} + 19 q^{43} - 29 q^{44} - 32 q^{47} - 10 q^{49} + 11 q^{50} + q^{52} - 38 q^{53} + 2 q^{55} - 46 q^{56} + 30 q^{58} - 30 q^{59} - 10 q^{61} - 25 q^{62} + 14 q^{64} - 8 q^{65} + 3 q^{67} - 47 q^{68} + 56 q^{70} - 21 q^{71} + 8 q^{73} - 28 q^{74} - 36 q^{76} - 49 q^{77} + 4 q^{79} - 70 q^{80} - 4 q^{82} - 29 q^{83} - 4 q^{85} - 4 q^{86} - 47 q^{88} - 54 q^{89} + 4 q^{91} - 12 q^{92} - 23 q^{94} - 33 q^{95} - 4 q^{97} - 49 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 9 * q^5 + 7 * q^7 - 12 * q^8 - 7 * q^10 - 5 * q^11 - 5 * q^13 - 17 * q^14 + 24 * q^16 - 21 * q^17 - 4 * q^19 - 21 * q^20 - 20 * q^22 - 22 * q^23 + 10 * q^25 - 17 * q^26 - q^28 - 30 * q^29 - 5 * q^31 - 48 * q^32 - 6 * q^34 - 53 * q^35 - q^37 - 21 * q^38 + 16 * q^40 - 29 * q^41 + 19 * q^43 - 29 * q^44 - 32 * q^47 - 10 * q^49 + 11 * q^50 + q^52 - 38 * q^53 + 2 * q^55 - 46 * q^56 + 30 * q^58 - 30 * q^59 - 10 * q^61 - 25 * q^62 + 14 * q^64 - 8 * q^65 + 3 * q^67 - 47 * q^68 + 56 * q^70 - 21 * q^71 + 8 * q^73 - 28 * q^74 - 36 * q^76 - 49 * q^77 + 4 * q^79 - 70 * q^80 - 4 * q^82 - 29 * q^83 - 4 * q^85 - 4 * q^86 - 47 * q^88 - 54 * q^89 + 4 * q^91 - 12 * q^92 - 23 * q^94 - 33 * q^95 - 4 * q^97 - 49 * q^98

Introduction

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