LMFDB - Embedded newform 4655.2.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4655,2,Mod(1,4655)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4655, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4655.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4655 = 5 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4655.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:	$37.1703621409$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.51658$ of defining polynomial
Character	$\chi$	$=$	4655.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.816594 q^{2} -1.53844 q^{3} -1.33317 q^{4} +1.00000 q^{5} -1.25628 q^{6} -2.72185 q^{8} -0.633188 q^{9} +0.816594 q^{10} -3.03316 q^{11} +2.05101 q^{12} +4.57160 q^{13} -1.53844 q^{15} +0.443701 q^{16} +1.07689 q^{17} -0.517058 q^{18} -1.00000 q^{19} -1.33317 q^{20} -2.47686 q^{22} +4.11005 q^{23} +4.18742 q^{24} +1.00000 q^{25} +3.73315 q^{26} +5.58946 q^{27} -1.07689 q^{29} -1.25628 q^{30} -5.58946 q^{31} +5.80602 q^{32} +4.66635 q^{33} +0.879381 q^{34} +0.844150 q^{36} +0.0947438 q^{37} -0.816594 q^{38} -7.03316 q^{39} -2.72185 q^{40} -10.6663 q^{41} +5.03316 q^{43} +4.04373 q^{44} -0.633188 q^{45} +3.35624 q^{46} +12.2995 q^{47} -0.682609 q^{48} +0.816594 q^{50} -1.65673 q^{51} -6.09474 q^{52} -4.09474 q^{53} +4.56432 q^{54} -3.03316 q^{55} +1.53844 q^{57} -0.879381 q^{58} +1.39997 q^{59} +2.05101 q^{60} +5.69951 q^{61} -4.56432 q^{62} +3.85376 q^{64} +4.57160 q^{65} +3.81051 q^{66} +5.28168 q^{67} -1.43568 q^{68} -6.32308 q^{69} -5.67692 q^{71} +1.72344 q^{72} -9.07689 q^{73} +0.0773672 q^{74} -1.53844 q^{75} +1.33317 q^{76} -5.74324 q^{78} -5.39997 q^{79} +0.443701 q^{80} -6.69951 q^{81} -8.71008 q^{82} -1.95627 q^{83} +1.07689 q^{85} +4.11005 q^{86} +1.65673 q^{87} +8.25581 q^{88} +2.18949 q^{89} -0.517058 q^{90} -5.47941 q^{92} +8.59907 q^{93} +10.0437 q^{94} -1.00000 q^{95} -8.93225 q^{96} +2.16106 q^{97} +1.92056 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{3} + 8 q^{4} + 4 q^{5} - 12 q^{8} + 8 q^{9} - 2 q^{10} + 4 q^{11} - 6 q^{12} - 2 q^{13} - 2 q^{15} + 4 q^{16} - 4 q^{17} - 34 q^{18} - 4 q^{19} + 8 q^{20} + 4 q^{22} - 8 q^{23} + 24 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^3 + 8 * q^4 + 4 * q^5 - 12 * q^8 + 8 * q^9 - 2 * q^10 + 4 * q^11 - 6 * q^12 - 2 * q^13 - 2 * q^15 + 4 * q^16 - 4 * q^17 - 34 * q^18 - 4 * q^19 + 8 * q^20 + 4 * q^22 - 8 * q^23 + 24 * q^24 + 4 * q^25 - 4 * q^26 + 4 * q^27 + 4 * q^29 - 4 * q^31 - 6 * q^32 - 8 * q^33 + 4 * q^34 + 40 * q^36 - 6 * q^37 + 2 * q^38 - 12 * q^39 - 12 * q^40 - 16 * q^41 + 4 * q^43 + 24 * q^44 + 8 * q^45 + 12 * q^47 - 38 * q^48 - 2 * q^50 - 36 * q^51 - 18 * q^52 - 10 * q^53 + 20 * q^54 + 4 * q^55 + 2 * q^57 - 4 * q^58 - 6 * q^60 - 20 * q^61 - 20 * q^62 - 4 * q^64 - 2 * q^65 + 28 * q^66 - 18 * q^67 - 4 * q^68 - 28 * q^69 - 20 * q^71 - 52 * q^72 - 28 * q^73 + 32 * q^74 - 2 * q^75 - 8 * q^76 + 12 * q^78 - 16 * q^79 + 4 * q^80 + 16 * q^81 - 16 * q^82 - 4 * q^85 - 8 * q^86 + 36 * q^87 - 12 * q^88 - 4 * q^89 - 34 * q^90 - 28 * q^92 - 40 * q^93 + 48 * q^94 - 4 * q^95 + 52 * q^96 - 30 * q^97 - 4 * 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 $

$2$	0.816594	0.577419	0.288710	−	0.957417i	$-0.406774\pi$
$2$	0.816594	0.577419	0.288710	+	0.957417i	$0.406774\pi$
$3$	−1.53844	−0.888221	−0.444111	−	0.895972i	$-0.646480\pi$
$3$	−1.53844	−0.888221	−0.444111	+	0.895972i	$0.646480\pi$
$4$	−1.33317	−0.666587
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.25628	−0.512876
$7$	0	0
$7$	0	0
$8$	−2.72185	−0.962319
$9$	−0.633188	−0.211063
$10$	0.816594	0.258230
$11$	−3.03316	−0.914532	−0.457266	−	0.889330i	$-0.651171\pi$
$11$	−3.03316	−0.914532	−0.457266	+	0.889330i	$0.651171\pi$
$12$	2.05101	0.592077
$13$	4.57160	1.26793	0.633967	−	0.773360i	$-0.281425\pi$
$13$	4.57160	1.26793	0.633967	+	0.773360i	$0.281425\pi$
$14$	0	0
$15$	−1.53844	−0.397225
$16$	0.443701	0.110925
$17$	1.07689	0.261184	0.130592	−	0.991436i	$-0.458312\pi$
$17$	1.07689	0.261184	0.130592	+	0.991436i	$0.458312\pi$
$18$	−0.517058	−0.121872
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.33317	−0.298107
$21$	0	0
$22$	−2.47686	−0.528068
$23$	4.11005	0.857004	0.428502	−	0.903541i	$-0.359041\pi$
$23$	4.11005	0.857004	0.428502	+	0.903541i	$0.359041\pi$
$24$	4.18742	0.854753
$25$	1.00000	0.200000
$26$	3.73315	0.732130
$27$	5.58946	1.07569
$28$	0	0
$29$	−1.07689	−0.199973	−0.0999866	−	0.994989i	$-0.531880\pi$
$29$	−1.07689	−0.199973	−0.0999866	+	0.994989i	$0.531880\pi$
$30$	−1.25628	−0.229365
$31$	−5.58946	−1.00390	−0.501948	−	0.864898i	$-0.667383\pi$
$31$	−5.58946	−1.00390	−0.501948	+	0.864898i	$0.667383\pi$
$32$	5.80602	1.02637
$33$	4.66635	0.812307
$34$	0.879381	0.150813
$35$	0	0
$36$	0.844150	0.140692
$37$	0.0947438	0.0155758	0.00778789	−	0.999970i	$-0.497521\pi$
$37$	0.0947438	0.0155758	0.00778789	+	0.999970i	$0.497521\pi$
$38$	−0.816594	−0.132469
$39$	−7.03316	−1.12621
$40$	−2.72185	−0.430362
$41$	−10.6663	−1.66580	−0.832902	−	0.553421i	$-0.813322\pi$
$41$	−10.6663	−1.66580	−0.832902	+	0.553421i	$0.813322\pi$
$42$	0	0
$43$	5.03316	0.767550	0.383775	−	0.923427i	$-0.374624\pi$
$43$	5.03316	0.767550	0.383775	+	0.923427i	$0.374624\pi$
$44$	4.04373	0.609615
$45$	−0.633188	−0.0943901
$46$	3.35624	0.494851
$47$	12.2995	1.79407	0.897036	−	0.441958i	$-0.145716\pi$
$47$	12.2995	1.79407	0.897036	+	0.441958i	$0.145716\pi$
$48$	−0.682609	−0.0985261
$49$	0	0
$50$	0.816594	0.115484
$51$	−1.65673	−0.231989
$52$	−6.09474	−0.845189
$53$	−4.09474	−0.562456	−0.281228	−	0.959641i	$-0.590742\pi$
$53$	−4.09474	−0.562456	−0.281228	+	0.959641i	$0.590742\pi$
$54$	4.56432	0.621125
$55$	−3.03316	−0.408991
$56$	0	0
$57$	1.53844	0.203772
$58$	−0.879381	−0.115468
$59$	1.39997	0.182261	0.0911304	−	0.995839i	$-0.470952\pi$
$59$	1.39997	0.182261	0.0911304	+	0.995839i	$0.470952\pi$
$60$	2.05101	0.264785
$61$	5.69951	0.729747	0.364874	−	0.931057i	$-0.381112\pi$
$61$	5.69951	0.729747	0.364874	+	0.931057i	$0.381112\pi$
$62$	−4.56432	−0.579669
$63$	0	0
$64$	3.85376	0.481720
$65$	4.57160	0.567038
$66$	3.81051	0.469042
$67$	5.28168	0.645260	0.322630	−	0.946525i	$-0.395433\pi$
$67$	5.28168	0.645260	0.322630	+	0.946525i	$0.395433\pi$
$68$	−1.43568	−0.174102
$69$	−6.32308	−0.761210
$70$	0	0
$71$	−5.67692	−0.673726	−0.336863	−	0.941554i	$-0.609366\pi$
$71$	−5.67692	−0.673726	−0.336863	+	0.941554i	$0.609366\pi$
$72$	1.72344	0.203110
$73$	−9.07689	−1.06237	−0.531185	−	0.847256i	$-0.678253\pi$
$73$	−9.07689	−1.06237	−0.531185	+	0.847256i	$0.678253\pi$
$74$	0.0773672	0.00899375
$75$	−1.53844	−0.177644
$76$	1.33317	0.152926
$77$	0	0
$78$	−5.74324	−0.650294
$79$	−5.39997	−0.607544	−0.303772	−	0.952745i	$-0.598246\pi$
$79$	−5.39997	−0.607544	−0.303772	+	0.952745i	$0.598246\pi$
$80$	0.443701	0.0496073
$81$	−6.69951	−0.744390
$82$	−8.71008	−0.961867
$83$	−1.95627	−0.214729	−0.107364	−	0.994220i	$-0.534241\pi$
$83$	−1.95627	−0.214729	−0.107364	+	0.994220i	$0.534241\pi$
$84$	0	0
$85$	1.07689	0.116805
$86$	4.11005	0.443198
$87$	1.65673	0.177621
$88$	8.25581	0.880072
$89$	2.18949	0.232085	0.116043	−	0.993244i	$-0.462979\pi$
$89$	2.18949	0.232085	0.116043	+	0.993244i	$0.462979\pi$
$90$	−0.517058	−0.0545027
$91$	0	0
$92$	−5.47941	−0.571268
$93$	8.59907	0.891682
$94$	10.0437	1.03593
$95$	−1.00000	−0.102598
$96$	−8.93225	−0.911644
$97$	2.16106	0.219423	0.109711	−	0.993963i	$-0.465007\pi$
$97$	2.16106	0.219423	0.109711	+	0.993963i	$0.465007\pi$
$98$	0	0
$99$	1.92056	0.193024
$100$	−1.33317	−0.133317
$101$	−12.5869	−1.25244	−0.626222	−	0.779645i	$-0.715400\pi$
$101$	−12.5869	−1.25244	−0.626222	+	0.779645i	$0.715400\pi$
$102$	−1.35288	−0.133955
$103$	6.20479	0.611376	0.305688	−	0.952132i	$-0.401113\pi$
$103$	6.20479	0.611376	0.305688	+	0.952132i	$0.401113\pi$
$104$	−12.4432	−1.22016
$105$	0	0
$106$	−3.34374	−0.324773
$107$	−12.5481	−1.21307	−0.606533	−	0.795058i	$-0.707440\pi$
$107$	−12.5481	−1.21307	−0.606533	+	0.795058i	$0.707440\pi$
$108$	−7.45172	−0.717042
$109$	−15.8096	−1.51428	−0.757140	−	0.653252i	$-0.773404\pi$
$109$	−15.8096	−1.51428	−0.757140	+	0.653252i	$0.773404\pi$
$110$	−2.47686	−0.236159
$111$	−0.145758	−0.0138347
$112$	0	0
$113$	−5.49472	−0.516899	−0.258450	−	0.966025i	$-0.583212\pi$
$113$	−5.49472	−0.516899	−0.258450	+	0.966025i	$0.583212\pi$
$114$	1.25628	0.117662
$115$	4.11005	0.383264
$116$	1.43568	0.133300
$117$	−2.89469	−0.267614
$118$	1.14321	0.105241
$119$	0	0
$120$	4.18742	0.382257
$121$	−1.79994	−0.163631
$122$	4.65418	0.421370
$123$	16.4096	1.47960
$124$	7.45172	0.669184
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.61533	−0.764487	−0.382244	−	0.924062i	$-0.624848\pi$
$127$	−8.61533	−0.764487	−0.382244	+	0.924062i	$0.624848\pi$
$128$	−8.46509	−0.748215
$129$	−7.74324	−0.681754
$130$	3.73315	0.327419
$131$	2.15378	0.188176	0.0940882	−	0.995564i	$-0.470006\pi$
$131$	2.15378	0.188176	0.0940882	+	0.995564i	$0.470006\pi$
$132$	−6.22105	−0.541473
$133$	0	0
$134$	4.31299	0.372586
$135$	5.58946	0.481064
$136$	−2.93113	−0.251342
$137$	2.18949	0.187061	0.0935303	−	0.995616i	$-0.470185\pi$
$137$	2.18949	0.187061	0.0935303	+	0.995616i	$0.470185\pi$
$138$	−5.16339	−0.439537
$139$	−22.5196	−1.91009	−0.955045	−	0.296460i	$-0.904194\pi$
$139$	−22.5196	−1.91009	−0.955045	+	0.296460i	$0.904194\pi$
$140$	0	0
$141$	−18.9222	−1.59353
$142$	−4.63574	−0.389022
$143$	−13.8664	−1.15957
$144$	−0.280946	−0.0234122
$145$	−1.07689	−0.0894308
$146$	−7.41213	−0.613433
$147$	0	0
$148$	−0.126310	−0.0103826
$149$	9.78697	0.801780	0.400890	−	0.916126i	$-0.368701\pi$
$149$	9.78697	0.801780	0.400890	+	0.916126i	$0.368701\pi$
$150$	−1.25628	−0.102575
$151$	5.87683	0.478250	0.239125	−	0.970989i	$-0.423139\pi$
$151$	5.87683	0.478250	0.239125	+	0.970989i	$0.423139\pi$
$152$	2.72185	0.220771
$153$	−0.681874	−0.0551262
$154$	0	0
$155$	−5.58946	−0.448956
$156$	9.37643	0.750715
$157$	−10.1895	−0.813210	−0.406605	−	0.913604i	$-0.633287\pi$
$157$	−10.1895	−0.813210	−0.406605	+	0.913604i	$0.633287\pi$
$158$	−4.40959	−0.350808
$159$	6.29954	0.499586
$160$	5.80602	0.459007
$161$	0	0
$162$	−5.47078	−0.429825
$163$	−10.3659	−0.811916	−0.405958	−	0.913892i	$-0.633062\pi$
$163$	−10.3659	−0.811916	−0.405958	+	0.913892i	$0.633062\pi$
$164$	14.2201	1.11040
$165$	4.66635	0.363275
$166$	−1.59748	−0.123988
$167$	15.6048	1.20753	0.603766	−	0.797161i	$-0.293666\pi$
$167$	15.6048	1.20753	0.603766	+	0.797161i	$0.293666\pi$
$168$	0	0
$169$	7.89956	0.607659
$170$	0.879381	0.0674455
$171$	0.633188	0.0484211
$172$	−6.71008	−0.511639
$173$	−0.571604	−0.0434583	−0.0217291	−	0.999764i	$-0.506917\pi$
$173$	−0.571604	−0.0434583	−0.0217291	+	0.999764i	$0.506917\pi$
$174$	1.35288	0.102562
$175$	0	0
$176$	−1.34582	−0.101445
$177$	−2.15378	−0.161888
$178$	1.78792	0.134010
$179$	24.8864	1.86010	0.930050	−	0.367433i	$-0.119763\pi$
$179$	24.8864	1.86010	0.930050	+	0.367433i	$0.119763\pi$
$180$	0.844150	0.0629192
$181$	−6.95372	−0.516866	−0.258433	−	0.966029i	$-0.583206\pi$
$181$	−6.95372	−0.516866	−0.258433	+	0.966029i	$0.583206\pi$
$182$	0	0
$183$	−8.76838	−0.648177
$184$	−11.1869	−0.824712
$185$	0.0947438	0.00696570
$186$	7.02195	0.514875
$187$	−3.26638	−0.238861
$188$	−16.3974	−1.19590
$189$	0	0
$190$	−0.816594	−0.0592420
$191$	−3.91254	−0.283102	−0.141551	−	0.989931i	$-0.545209\pi$
$191$	−3.91254	−0.283102	−0.141551	+	0.989931i	$0.545209\pi$
$192$	−5.92880	−0.427874
$193$	−9.20734	−0.662759	−0.331379	−	0.943498i	$-0.607514\pi$
$193$	−9.20734	−0.662759	−0.331379	+	0.943498i	$0.607514\pi$
$194$	1.76471	0.126699
$195$	−7.03316	−0.503655
$196$	0	0
$197$	−3.84622	−0.274032	−0.137016	−	0.990569i	$-0.543751\pi$
$197$	−3.84622	−0.274032	−0.137016	+	0.990569i	$0.543751\pi$
$198$	1.56832	0.111456
$199$	18.2864	1.29629	0.648145	−	0.761517i	$-0.275545\pi$
$199$	18.2864	1.29629	0.648145	+	0.761517i	$0.275545\pi$
$200$	−2.72185	−0.192464
$201$	−8.12557	−0.573134
$202$	−10.2784	−0.723185
$203$	0	0
$204$	2.20871	0.154641
$205$	−10.6663	−0.744970
$206$	5.06680	0.353020
$207$	−2.60243	−0.180882
$208$	2.02842	0.140646
$209$	3.03316	0.209808
$210$	0	0
$211$	−19.2970	−1.32846	−0.664230	−	0.747529i	$-0.731240\pi$
$211$	−19.2970	−1.32846	−0.664230	+	0.747529i	$0.731240\pi$
$212$	5.45901	0.374926
$213$	8.73362	0.598418
$214$	−10.2467	−0.700448
$215$	5.03316	0.343259
$216$	−15.2137	−1.03516
$217$	0	0
$218$	−12.9100	−0.874375
$219$	13.9643	0.943619
$220$	4.04373	0.272628
$221$	4.92311	0.331164
$222$	−0.119025	−0.00798844
$223$	−0.615334	−0.0412058	−0.0206029	−	0.999788i	$-0.506559\pi$
$223$	−0.615334	−0.0412058	−0.0206029	+	0.999788i	$0.506559\pi$
$224$	0	0
$225$	−0.633188	−0.0422126
$226$	−4.48695	−0.298468
$227$	−17.4712	−1.15960	−0.579801	−	0.814758i	$-0.696870\pi$
$227$	−17.4712	−1.15960	−0.579801	+	0.814758i	$0.696870\pi$
$228$	−2.05101	−0.135832
$229$	−9.69951	−0.640962	−0.320481	−	0.947255i	$-0.603844\pi$
$229$	−9.69951	−0.640962	−0.320481	+	0.947255i	$0.603844\pi$
$230$	3.35624	0.221304
$231$	0	0
$232$	2.93113	0.192438
$233$	−8.15378	−0.534172	−0.267086	−	0.963673i	$-0.586061\pi$
$233$	−8.15378	−0.534172	−0.267086	+	0.963673i	$0.586061\pi$
$234$	−2.36378	−0.154525
$235$	12.2995	0.802333
$236$	−1.86641	−0.121493
$237$	8.30756	0.539634
$238$	0	0
$239$	24.5991	1.59118	0.795591	−	0.605834i	$-0.207161\pi$
$239$	24.5991	1.59118	0.795591	+	0.605834i	$0.207161\pi$
$240$	−0.682609	−0.0440622
$241$	−25.7738	−1.66024	−0.830120	−	0.557585i	$-0.811728\pi$
$241$	−25.7738	−1.66024	−0.830120	+	0.557585i	$0.811728\pi$
$242$	−1.46982	−0.0944838
$243$	−6.46156	−0.414509
$244$	−7.59844	−0.486440
$245$	0	0
$246$	13.4000	0.854351
$247$	−4.57160	−0.290884
$248$	15.2137	0.966069
$249$	3.00961	0.190727
$250$	0.816594	0.0516459
$251$	−12.5991	−0.795246	−0.397623	−	0.917549i	$-0.630165\pi$
$251$	−12.5991	−0.795246	−0.397623	+	0.917549i	$0.630165\pi$
$252$	0	0
$253$	−12.4664	−0.783758
$254$	−7.03523	−0.441430
$255$	−1.65673	−0.103749
$256$	−14.6201	−0.913754
$257$	−0.182203	−0.0113655	−0.00568275	−	0.999984i	$-0.501809\pi$
$257$	−0.182203	−0.0113655	−0.00568275	+	0.999984i	$0.501809\pi$
$258$	−6.32308	−0.393658
$259$	0	0
$260$	−6.09474	−0.377980
$261$	0.681874	0.0422069
$262$	1.75876	0.108657
$263$	7.37643	0.454850	0.227425	−	0.973796i	$-0.426969\pi$
$263$	7.37643	0.454850	0.227425	+	0.973796i	$0.426969\pi$
$264$	−12.7011	−0.781699
$265$	−4.09474	−0.251538
$266$	0	0
$267$	−3.36841	−0.206143
$268$	−7.04140	−0.430122
$269$	−4.70206	−0.286689	−0.143345	−	0.989673i	$-0.545786\pi$
$269$	−4.70206	−0.286689	−0.143345	+	0.989673i	$0.545786\pi$
$270$	4.56432	0.277776
$271$	−9.92056	−0.602631	−0.301316	−	0.953524i	$-0.597426\pi$
$271$	−9.92056	−0.602631	−0.301316	+	0.953524i	$0.597426\pi$
$272$	0.477817	0.0289719
$273$	0	0
$274$	1.78792	0.108012
$275$	−3.03316	−0.182906
$276$	8.42977	0.507412
$277$	−22.6297	−1.35969	−0.679843	−	0.733358i	$-0.737952\pi$
$277$	−22.6297	−1.35969	−0.679843	+	0.733358i	$0.737952\pi$
$278$	−18.3894	−1.10292
$279$	3.53918	0.211885
$280$	0	0
$281$	−3.95386	−0.235868	−0.117934	−	0.993021i	$-0.537627\pi$
$281$	−3.95386	−0.235868	−0.117934	+	0.993021i	$0.537627\pi$
$282$	−15.4517	−0.920137
$283$	13.0638	0.776561	0.388280	−	0.921541i	$-0.373069\pi$
$283$	13.0638	0.776561	0.388280	+	0.921541i	$0.373069\pi$
$284$	7.56832	0.449097
$285$	1.53844	0.0911296
$286$	−11.3232	−0.669556
$287$	0	0
$288$	−3.67631	−0.216628
$289$	−15.8403	−0.931783
$290$	−0.879381	−0.0516391
$291$	−3.32468	−0.194896
$292$	12.1011	0.708162
$293$	9.69463	0.566366	0.283183	−	0.959066i	$-0.408610\pi$
$293$	9.69463	0.566366	0.283183	+	0.959066i	$0.408610\pi$
$294$	0	0
$295$	1.39997	0.0815095
$296$	−0.257878	−0.0149889
$297$	−16.9537	−0.983755
$298$	7.99198	0.462963
$299$	18.7895	1.08663
$300$	2.05101	0.118415
$301$	0	0
$302$	4.79899	0.276151
$303$	19.3643	1.11245
$304$	−0.443701	−0.0254480
$305$	5.69951	0.326353
$306$	−0.556814	−0.0318309
$307$	28.5481	1.62932	0.814662	−	0.579936i	$-0.196923\pi$
$307$	28.5481	1.62932	0.814662	+	0.579936i	$0.196923\pi$
$308$	0	0
$309$	−9.54573	−0.543038
$310$	−4.56432	−0.259236
$311$	−25.4070	−1.44070	−0.720350	−	0.693610i	$-0.756019\pi$
$311$	−25.4070	−1.44070	−0.720350	+	0.693610i	$0.756019\pi$
$312$	19.1432	1.08377
$313$	−16.7999	−0.949589	−0.474794	−	0.880097i	$-0.657478\pi$
$313$	−16.7999	−0.949589	−0.474794	+	0.880097i	$0.657478\pi$
$314$	−8.32068	−0.469563
$315$	0	0
$316$	7.19910	0.404981
$317$	−31.7505	−1.78329	−0.891643	−	0.452738i	$-0.850447\pi$
$317$	−31.7505	−1.78329	−0.891643	+	0.452738i	$0.850447\pi$
$318$	5.14416	0.288470
$319$	3.26638	0.182882
$320$	3.85376	0.215432
$321$	19.3045	1.07747
$322$	0	0
$323$	−1.07689	−0.0599197
$324$	8.93161	0.496201
$325$	4.57160	0.253587
$326$	−8.46470	−0.468816
$327$	24.3221	1.34502
$328$	29.0322	1.60304
$329$	0	0
$330$	3.81051	0.209762
$331$	−9.96429	−0.547687	−0.273843	−	0.961774i	$-0.588295\pi$
$331$	−9.96429	−0.547687	−0.273843	+	0.961774i	$0.588295\pi$
$332$	2.60805	0.143135
$333$	−0.0599906	−0.00328747
$334$	12.7428	0.697253
$335$	5.28168	0.288569
$336$	0	0
$337$	23.1918	1.26334	0.631669	−	0.775238i	$-0.282370\pi$
$337$	23.1918	1.26334	0.631669	+	0.775238i	$0.282370\pi$
$338$	6.45074	0.350874
$339$	8.45331	0.459121
$340$	−1.43568	−0.0778607
$341$	16.9537	0.918095
$342$	0.517058	0.0279593
$343$	0	0
$344$	−13.6995	−0.738628
$345$	−6.32308	−0.340423
$346$	−0.466769	−0.0250936
$347$	21.1352	1.13460	0.567298	−	0.823512i	$-0.307989\pi$
$347$	21.1352	1.13460	0.567298	+	0.823512i	$0.307989\pi$
$348$	−2.20871	−0.118400
$349$	5.04628	0.270121	0.135061	−	0.990837i	$-0.456877\pi$
$349$	5.04628	0.270121	0.135061	+	0.990837i	$0.456877\pi$
$350$	0	0
$351$	25.5528	1.36391
$352$	−17.6106	−0.938648
$353$	−12.5634	−0.668680	−0.334340	−	0.942452i	$-0.608513\pi$
$353$	−12.5634	−0.668680	−0.334340	+	0.942452i	$0.608513\pi$
$354$	−1.75876	−0.0934772
$355$	−5.67692	−0.301300
$356$	−2.91897	−0.154705
$357$	0	0
$358$	20.3221	1.07406
$359$	−17.6534	−0.931709	−0.465855	−	0.884861i	$-0.654253\pi$
$359$	−17.6534	−0.931709	−0.465855	+	0.884861i	$0.654253\pi$
$360$	1.72344	0.0908335
$361$	1.00000	0.0526316
$362$	−5.67837	−0.298448
$363$	2.76911	0.145341
$364$	0	0
$365$	−9.07689	−0.475106
$366$	−7.16021	−0.374270
$367$	−4.43409	−0.231457	−0.115729	−	0.993281i	$-0.536920\pi$
$367$	−4.43409	−0.231457	−0.115729	+	0.993281i	$0.536920\pi$
$368$	1.82363	0.0950634
$369$	6.75381	0.351589
$370$	0.0773672	0.00402213
$371$	0	0
$372$	−11.4641	−0.594384
$373$	−20.8735	−1.08079	−0.540396	−	0.841411i	$-0.681725\pi$
$373$	−20.8735	−1.08079	−0.540396	+	0.841411i	$0.681725\pi$
$374$	−2.66730	−0.137923
$375$	−1.53844	−0.0794449
$376$	−33.4775	−1.72647
$377$	−4.92311	−0.253553
$378$	0	0
$379$	−6.79994	−0.349290	−0.174645	−	0.984632i	$-0.555878\pi$
$379$	−6.79994	−0.349290	−0.174645	+	0.984632i	$0.555878\pi$
$380$	1.33317	0.0683904
$381$	13.2542	0.679034
$382$	−3.19496	−0.163468
$383$	10.9078	0.557363	0.278681	−	0.960384i	$-0.410103\pi$
$383$	10.9078	0.557363	0.278681	+	0.960384i	$0.410103\pi$
$384$	13.0231	0.664581
$385$	0	0
$386$	−7.51866	−0.382690
$387$	−3.18694	−0.162001
$388$	−2.88107	−0.146264
$389$	20.9326	1.06132	0.530662	−	0.847584i	$-0.321943\pi$
$389$	20.9326	1.06132	0.530662	+	0.847584i	$0.321943\pi$
$390$	−5.74324	−0.290820
$391$	4.42607	0.223836
$392$	0	0
$393$	−3.31347	−0.167142
$394$	−3.14080	−0.158231
$395$	−5.39997	−0.271702
$396$	−2.56044	−0.128667
$397$	−13.1221	−0.658578	−0.329289	−	0.944229i	$-0.606809\pi$
$397$	−13.1221	−0.658578	−0.329289	+	0.944229i	$0.606809\pi$
$398$	14.9326	0.748503
$399$	0	0
$400$	0.443701	0.0221850
$401$	−11.5528	−0.576919	−0.288459	−	0.957492i	$-0.593143\pi$
$401$	−11.5528	−0.576919	−0.288459	+	0.957492i	$0.593143\pi$
$402$	−6.63530	−0.330939
$403$	−25.5528	−1.27288
$404$	16.7805	0.834863
$405$	−6.69951	−0.332901
$406$	0	0
$407$	−0.287373	−0.0142445
$408$	4.50938	0.223248
$409$	18.9433	0.936686	0.468343	−	0.883547i	$-0.344851\pi$
$409$	18.9433	0.936686	0.468343	+	0.883547i	$0.344851\pi$
$410$	−8.71008	−0.430160
$411$	−3.36841	−0.166151
$412$	−8.27207	−0.407536
$413$	0	0
$414$	−2.12513	−0.104445
$415$	−1.95627	−0.0960295
$416$	26.5428	1.30137
$417$	34.6452	1.69658
$418$	2.47686	0.121147
$419$	−27.2664	−1.33205	−0.666025	−	0.745930i	$-0.732006\pi$
$419$	−27.2664	−1.33205	−0.666025	+	0.745930i	$0.732006\pi$
$420$	0	0
$421$	6.66635	0.324898	0.162449	−	0.986717i	$-0.448061\pi$
$421$	6.66635	0.324898	0.162449	+	0.986717i	$0.448061\pi$
$422$	−15.7578	−0.767078
$423$	−7.78792	−0.378662
$424$	11.1453	0.541263
$425$	1.07689	0.0522368
$426$	7.13183	0.345538
$427$	0	0
$428$	16.7287	0.808614
$429$	21.3327	1.02995
$430$	4.11005	0.198204
$431$	−32.4548	−1.56329	−0.781646	−	0.623723i	$-0.785619\pi$
$431$	−32.4548	−1.56329	−0.781646	+	0.623723i	$0.785619\pi$
$432$	2.48005	0.119321
$433$	−33.8168	−1.62513	−0.812567	−	0.582868i	$-0.801930\pi$
$433$	−33.8168	−1.62513	−0.812567	+	0.582868i	$0.801930\pi$
$434$	0	0
$435$	1.65673	0.0794343
$436$	21.0769	1.00940
$437$	−4.11005	−0.196610
$438$	11.4032	0.544864
$439$	27.8453	1.32898	0.664491	−	0.747296i	$-0.268648\pi$
$439$	27.8453	1.32898	0.664491	+	0.747296i	$0.268648\pi$
$440$	8.25581	0.393580
$441$	0	0
$442$	4.02018	0.191221
$443$	22.7754	1.08209	0.541047	−	0.840992i	$-0.318028\pi$
$443$	22.7754	1.08209	0.541047	+	0.840992i	$0.318028\pi$
$444$	0.194321	0.00922206
$445$	2.18949	0.103792
$446$	−0.502478	−0.0237930
$447$	−15.0567	−0.712158
$448$	0	0
$449$	24.8507	1.17278	0.586389	−	0.810029i	$-0.300549\pi$
$449$	24.8507	1.17278	0.586389	+	0.810029i	$0.300549\pi$
$450$	−0.517058	−0.0243743
$451$	32.3527	1.52343
$452$	7.32541	0.344558
$453$	−9.04118	−0.424792
$454$	−14.2669	−0.669577
$455$	0	0
$456$	−4.18742	−0.196094
$457$	−7.33270	−0.343009	−0.171505	−	0.985183i	$-0.554863\pi$
$457$	−7.33270	−0.343009	−0.171505	+	0.985183i	$0.554863\pi$
$458$	−7.92056	−0.370104
$459$	6.01923	0.280953
$460$	−5.47941	−0.255479
$461$	−10.3076	−0.480071	−0.240035	−	0.970764i	$-0.577159\pi$
$461$	−10.3076	−0.480071	−0.240035	+	0.970764i	$0.577159\pi$
$462$	0	0
$463$	7.80249	0.362613	0.181306	−	0.983427i	$-0.441967\pi$
$463$	7.80249	0.362613	0.181306	+	0.983427i	$0.441967\pi$
$464$	−0.477817	−0.0221821
$465$	8.59907	0.398772
$466$	−6.65833	−0.308441
$467$	7.37643	0.341340	0.170670	−	0.985328i	$-0.445407\pi$
$467$	7.37643	0.341340	0.170670	+	0.985328i	$0.445407\pi$
$468$	3.85912	0.178388
$469$	0	0
$470$	10.0437	0.463283
$471$	15.6760	0.722310
$472$	−3.81051	−0.175393
$473$	−15.2664	−0.701949
$474$	6.78390	0.311595
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	2.59274	0.118714
$478$	20.0875	0.918779
$479$	16.1458	0.737719	0.368859	−	0.929485i	$-0.379748\pi$
$479$	16.1458	0.737719	0.368859	+	0.929485i	$0.379748\pi$
$480$	−8.93225	−0.407699
$481$	0.433131	0.0197491
$482$	−21.0468	−0.958654
$483$	0	0
$484$	2.39964	0.109074
$485$	2.16106	0.0981288
$486$	−5.27647	−0.239345
$487$	−8.40566	−0.380897	−0.190448	−	0.981697i	$-0.560994\pi$
$487$	−8.40566	−0.380897	−0.190448	+	0.981697i	$0.560994\pi$
$488$	−15.5132	−0.702250
$489$	15.9473	0.721162
$490$	0	0
$491$	−36.0855	−1.62852	−0.814259	−	0.580502i	$-0.802856\pi$
$491$	−36.0855	−1.62852	−0.814259	+	0.580502i	$0.802856\pi$
$492$	−21.8768	−0.986284
$493$	−1.15969	−0.0522298
$494$	−3.73315	−0.167962
$495$	1.92056	0.0863228
$496$	−2.48005	−0.111357
$497$	0	0
$498$	2.45763	0.110129
$499$	−3.30035	−0.147744	−0.0738720	−	0.997268i	$-0.523536\pi$
$499$	−3.30035	−0.147744	−0.0738720	+	0.997268i	$0.523536\pi$
$500$	−1.33317	−0.0596214
$501$	−24.0071	−1.07256
$502$	−10.2883	−0.459191
$503$	−3.53530	−0.157631	−0.0788157	−	0.996889i	$-0.525114\pi$
$503$	−3.53530	−0.157631	−0.0788157	+	0.996889i	$0.525114\pi$
$504$	0	0
$505$	−12.5869	−0.560110
$506$	−10.1800	−0.452557
$507$	−12.1530	−0.539736
$508$	11.4857	0.509597
$509$	−22.4096	−0.993287	−0.496644	−	0.867955i	$-0.665434\pi$
$509$	−22.4096	−0.993287	−0.496644	+	0.867955i	$0.665434\pi$
$510$	−1.35288	−0.0599065
$511$	0	0
$512$	4.99151	0.220596
$513$	−5.58946	−0.246781
$514$	−0.148786	−0.00656266
$515$	6.20479	0.273416
$516$	10.3231	0.454448
$517$	−37.3065	−1.64074
$518$	0	0
$519$	0.879381	0.0386006
$520$	−12.4432	−0.545671
$521$	−34.4402	−1.50885	−0.754426	−	0.656385i	$-0.772085\pi$
$521$	−34.4402	−1.50885	−0.754426	+	0.656385i	$0.772085\pi$
$522$	0.556814	0.0243711
$523$	30.4451	1.33127	0.665635	−	0.746277i	$-0.268161\pi$
$523$	30.4451	1.33127	0.665635	+	0.746277i	$0.268161\pi$
$524$	−2.87136	−0.125436
$525$	0	0
$526$	6.02355	0.262639
$527$	−6.01923	−0.262202
$528$	2.07046	0.0901053
$529$	−6.10750	−0.265543
$530$	−3.34374	−0.145243
$531$	−0.886445	−0.0384685
$532$	0	0
$533$	−48.7623	−2.11213
$534$	−2.75062	−0.119031
$535$	−12.5481	−0.542500
$536$	−14.3759	−0.620946
$537$	−38.2864	−1.65218
$538$	−3.83967	−0.165540
$539$	0	0
$540$	−7.45172	−0.320671
$541$	−12.2794	−0.527931	−0.263965	−	0.964532i	$-0.585030\pi$
$541$	−12.2794	−0.527931	−0.263965	+	0.964532i	$0.585030\pi$
$542$	−8.10107	−0.347971
$543$	10.6979	0.459091
$544$	6.25244	0.268071
$545$	−15.8096	−0.677207
$546$	0	0
$547$	0.250928	0.0107289	0.00536446	−	0.999986i	$-0.498292\pi$
$547$	0.250928	0.0107289	0.00536446	+	0.999986i	$0.498292\pi$
$548$	−2.91897	−0.124692
$549$	−3.60886	−0.154022
$550$	−2.47686	−0.105614
$551$	1.07689	0.0458770
$552$	17.2105	0.732527
$553$	0	0
$554$	−18.4793	−0.785109
$555$	−0.145758	−0.00618708
$556$	30.0226	1.27324
$557$	1.67596	0.0710128	0.0355064	−	0.999369i	$-0.488696\pi$
$557$	1.67596	0.0710128	0.0355064	+	0.999369i	$0.488696\pi$
$558$	2.89007	0.122347
$559$	23.0096	0.973203
$560$	0	0
$561$	5.02514	0.212162
$562$	−3.22870	−0.136195
$563$	−20.6605	−0.870737	−0.435368	−	0.900252i	$-0.643382\pi$
$563$	−20.6605	−0.870737	−0.435368	+	0.900252i	$0.643382\pi$
$564$	25.2265	1.06223
$565$	−5.49472	−0.231164
$566$	10.6678	0.448401
$567$	0	0
$568$	15.4517	0.648340
$569$	40.9673	1.71744	0.858720	−	0.512445i	$-0.171260\pi$
$569$	40.9673	1.71744	0.858720	+	0.512445i	$0.171260\pi$
$570$	1.25628	0.0526200
$571$	24.2070	1.01303	0.506515	−	0.862231i	$-0.330933\pi$
$571$	24.2070	1.01303	0.506515	+	0.862231i	$0.330933\pi$
$572$	18.4863	0.772952
$573$	6.01923	0.251457
$574$	0	0
$575$	4.11005	0.171401
$576$	−2.44016	−0.101673
$577$	−40.2864	−1.67715	−0.838573	−	0.544790i	$-0.816609\pi$
$577$	−40.2864	−1.67715	−0.838573	+	0.544790i	$0.816609\pi$
$578$	−12.9351	−0.538029
$579$	14.1650	0.588677
$580$	1.43568	0.0596134
$581$	0	0
$582$	−2.71491	−0.112537
$583$	12.4200	0.514384
$584$	24.7059	1.02234
$585$	−2.89469	−0.119681
$586$	7.91658	0.327031
$587$	34.7754	1.43534	0.717668	−	0.696385i	$-0.245210\pi$
$587$	34.7754	1.43534	0.717668	+	0.696385i	$0.245210\pi$
$588$	0	0
$589$	5.58946	0.230310
$590$	1.14321	0.0470651
$591$	5.91720	0.243401
$592$	0.0420379	0.00172775
$593$	35.0864	1.44082	0.720412	−	0.693546i	$-0.243953\pi$
$593$	35.0864	1.44082	0.720412	+	0.693546i	$0.243953\pi$
$594$	−13.8443	−0.568039
$595$	0	0
$596$	−13.0477	−0.534456
$597$	−28.1326	−1.15139
$598$	15.3434	0.627439
$599$	30.8201	1.25928	0.629638	−	0.776889i	$-0.283203\pi$
$599$	30.8201	1.25928	0.629638	+	0.776889i	$0.283203\pi$
$600$	4.18742	0.170951
$601$	7.10654	0.289882	0.144941	−	0.989440i	$-0.453701\pi$
$601$	7.10654	0.289882	0.144941	+	0.989440i	$0.453701\pi$
$602$	0	0
$603$	−3.34430	−0.136190
$604$	−7.83484	−0.318795
$605$	−1.79994	−0.0731781
$606$	15.8127	0.642349
$607$	−2.01531	−0.0817987	−0.0408994	−	0.999163i	$-0.513022\pi$
$607$	−2.01531	−0.0817987	−0.0408994	+	0.999163i	$0.513022\pi$
$608$	−5.80602	−0.235465
$609$	0	0
$610$	4.65418	0.188442
$611$	56.2286	2.27477
$612$	0.909056	0.0367464
$613$	−11.6096	−0.468909	−0.234455	−	0.972127i	$-0.575330\pi$
$613$	−11.6096	−0.468909	−0.234455	+	0.972127i	$0.575330\pi$
$614$	23.3122	0.940803
$615$	16.4096	0.661698
$616$	0	0
$617$	12.4307	0.500442	0.250221	−	0.968189i	$-0.419497\pi$
$617$	12.4307	0.500442	0.250221	+	0.968189i	$0.419497\pi$
$618$	−7.79499	−0.313560
$619$	−3.85424	−0.154915	−0.0774575	−	0.996996i	$-0.524680\pi$
$619$	−3.85424	−0.154915	−0.0774575	+	0.996996i	$0.524680\pi$
$620$	7.45172	0.299268
$621$	22.9729	0.921873
$622$	−20.7472	−0.831888
$623$	0	0
$624$	−3.12062	−0.124925
$625$	1.00000	0.0400000
$626$	−13.7187	−0.548311
$627$	−4.66635	−0.186356
$628$	13.5844	0.542075
$629$	0.102029	0.00406814
$630$	0	0
$631$	−16.8794	−0.671958	−0.335979	−	0.941870i	$-0.609067\pi$
$631$	−16.8794	−0.671958	−0.335979	+	0.941870i	$0.609067\pi$
$632$	14.6979	0.584652
$633$	29.6873	1.17997
$634$	−25.9273	−1.02970
$635$	−8.61533	−0.341889
$636$	−8.39838	−0.333017
$637$	0	0
$638$	2.66730	0.105600
$639$	3.59456	0.142199
$640$	−8.46509	−0.334612
$641$	43.4855	1.71757	0.858787	−	0.512332i	$-0.171218\pi$
$641$	43.4855	1.71757	0.858787	+	0.512332i	$0.171218\pi$
$642$	15.7639	0.622153
$643$	−11.0689	−0.436514	−0.218257	−	0.975891i	$-0.570037\pi$
$643$	−11.0689	−0.436514	−0.218257	+	0.975891i	$0.570037\pi$
$644$	0	0
$645$	−7.74324	−0.304890
$646$	−0.879381	−0.0345988
$647$	−5.61766	−0.220853	−0.110427	−	0.993884i	$-0.535222\pi$
$647$	−5.61766	−0.220853	−0.110427	+	0.993884i	$0.535222\pi$
$648$	18.2351	0.716341
$649$	−4.24634	−0.166683
$650$	3.73315	0.146426
$651$	0	0
$652$	13.8195	0.541213
$653$	44.5211	1.74224	0.871122	−	0.491066i	$-0.163393\pi$
$653$	44.5211	1.74224	0.871122	+	0.491066i	$0.163393\pi$
$654$	19.8613	0.776639
$655$	2.15378	0.0841551
$656$	−4.73267	−0.184780
$657$	5.74738	0.224227
$658$	0	0
$659$	−6.89154	−0.268456	−0.134228	−	0.990950i	$-0.542856\pi$
$659$	−6.89154	−0.268456	−0.134228	+	0.990950i	$0.542856\pi$
$660$	−6.22105	−0.242154
$661$	44.3989	1.72692	0.863458	−	0.504421i	$-0.168294\pi$
$661$	44.3989	1.72692	0.863458	+	0.504421i	$0.168294\pi$
$662$	−8.13678	−0.316245
$663$	−7.57393	−0.294147
$664$	5.32468	0.206637
$665$	0	0
$666$	−0.0489880	−0.00189825
$667$	−4.42607	−0.171378
$668$	−20.8039	−0.804926
$669$	0.946657	0.0365999
$670$	4.31299	0.166625
$671$	−17.2875	−0.667377
$672$	0	0
$673$	38.6214	1.48875	0.744374	−	0.667763i	$-0.232748\pi$
$673$	38.6214	1.48875	0.744374	+	0.667763i	$0.232748\pi$
$674$	18.9383	0.729476
$675$	5.58946	0.215138
$676$	−10.5315	−0.405057
$677$	−43.3917	−1.66768	−0.833840	−	0.552006i	$-0.813862\pi$
$677$	−43.3917	−1.66768	−0.833840	+	0.552006i	$0.813862\pi$
$678$	6.90293	0.265105
$679$	0	0
$680$	−2.93113	−0.112404
$681$	26.8784	1.02998
$682$	13.8443	0.530126
$683$	−34.7123	−1.32823	−0.664114	−	0.747631i	$-0.731191\pi$
$683$	−34.7123	−1.32823	−0.664114	+	0.747631i	$0.731191\pi$
$684$	−0.844150	−0.0322769
$685$	2.18949	0.0836560
$686$	0	0
$687$	14.9222	0.569316
$688$	2.23322	0.0851406
$689$	−18.7195	−0.713158
$690$	−5.16339	−0.196567
$691$	−2.94570	−0.112060	−0.0560299	−	0.998429i	$-0.517844\pi$
$691$	−2.94570	−0.112060	−0.0560299	+	0.998429i	$0.517844\pi$
$692$	0.762048	0.0289687
$693$	0	0
$694$	17.2589	0.655138
$695$	−22.5196	−0.854218
$696$	−4.50938	−0.170928
$697$	−11.4865	−0.435081
$698$	4.12076	0.155973
$699$	12.5441	0.474463
$700$	0	0
$701$	18.5920	0.702210	0.351105	−	0.936336i	$-0.385806\pi$
$701$	18.5920	0.702210	0.351105	+	0.936336i	$0.385806\pi$
$702$	20.8663	0.787546
$703$	−0.0947438	−0.00357333
$704$	−11.6891	−0.440549
$705$	−18.9222	−0.712650
$706$	−10.2592	−0.386109
$707$	0	0
$708$	2.87136	0.107912
$709$	39.8443	1.49638	0.748192	−	0.663482i	$-0.230922\pi$
$709$	39.8443	1.49638	0.748192	+	0.663482i	$0.230922\pi$
$710$	−4.63574	−0.173976
$711$	3.41920	0.128230
$712$	−5.95946	−0.223340
$713$	−22.9729	−0.860344
$714$	0	0
$715$	−13.8664	−0.518574
$716$	−33.1780	−1.23992
$717$	−37.8443	−1.41332
$718$	−14.4156	−0.537987
$719$	−21.2321	−0.791824	−0.395912	−	0.918288i	$-0.629572\pi$
$719$	−21.2321	−0.791824	−0.395912	+	0.918288i	$0.629572\pi$
$720$	−0.280946	−0.0104702
$721$	0	0
$722$	0.816594	0.0303905
$723$	39.6516	1.47466
$724$	9.27052	0.344536
$725$	−1.07689	−0.0399947
$726$	2.26124	0.0839225
$727$	49.1658	1.82346	0.911729	−	0.410792i	$-0.134748\pi$
$727$	49.1658	1.82346	0.911729	+	0.410792i	$0.134748\pi$
$728$	0	0
$729$	30.0393	1.11257
$730$	−7.41213	−0.274335
$731$	5.42015	0.200472
$732$	11.6898	0.432066
$733$	−1.60498	−0.0592815	−0.0296407	−	0.999561i	$-0.509436\pi$
$733$	−1.60498	−0.0592815	−0.0296407	+	0.999561i	$0.509436\pi$
$734$	−3.62085	−0.133648
$735$	0	0
$736$	23.8630	0.879603
$737$	−16.0202	−0.590111
$738$	5.51512	0.203014
$739$	8.13264	0.299164	0.149582	−	0.988749i	$-0.452207\pi$
$739$	8.13264	0.299164	0.149582	+	0.988749i	$0.452207\pi$
$740$	−0.126310	−0.00464324
$741$	7.03316	0.258370
$742$	0	0
$743$	−42.9261	−1.57481	−0.787403	−	0.616439i	$-0.788575\pi$
$743$	−42.9261	−1.57481	−0.787403	+	0.616439i	$0.788575\pi$
$744$	−23.4054	−0.858083
$745$	9.78697	0.358567
$746$	−17.0452	−0.624070
$747$	1.23869	0.0453212
$748$	4.35465	0.159222
$749$	0	0
$750$	−1.25628	−0.0458730
$751$	4.05685	0.148037	0.0740183	−	0.997257i	$-0.476418\pi$
$751$	4.05685	0.148037	0.0740183	+	0.997257i	$0.476418\pi$
$752$	5.45731	0.199008
$753$	19.3830	0.706355
$754$	−4.02018	−0.146406
$755$	5.87683	0.213880
$756$	0	0
$757$	22.6151	0.821960	0.410980	−	0.911644i	$-0.365187\pi$
$757$	22.6151	0.821960	0.410980	+	0.911644i	$0.365187\pi$
$758$	−5.55279	−0.201687
$759$	19.1789	0.696151
$760$	2.72185	0.0987319
$761$	41.3609	1.49933	0.749666	−	0.661817i	$-0.230214\pi$
$761$	41.3609	1.49933	0.749666	+	0.661817i	$0.230214\pi$
$762$	10.8233	0.392087
$763$	0	0
$764$	5.21610	0.188712
$765$	−0.681874	−0.0246532
$766$	8.90725	0.321832
$767$	6.40011	0.231095
$768$	22.4922	0.811616
$769$	−17.9196	−0.646197	−0.323099	−	0.946365i	$-0.604725\pi$
$769$	−17.9196	−0.646197	−0.323099	+	0.946365i	$0.604725\pi$
$770$	0	0
$771$	0.280309	0.0100951
$772$	12.2750	0.441787
$773$	21.9043	0.787843	0.393921	−	0.919144i	$-0.371118\pi$
$773$	21.9043	0.787843	0.393921	+	0.919144i	$0.371118\pi$
$774$	−2.60243	−0.0935426
$775$	−5.58946	−0.200779
$776$	−5.88209	−0.211155
$777$	0	0
$778$	17.0934	0.612829
$779$	10.6663	0.382162
$780$	9.37643	0.335730
$781$	17.2190	0.616144
$782$	3.61430	0.129247
$783$	−6.01923	−0.215110
$784$	0	0
$785$	−10.1895	−0.363678
$786$	−2.70576	−0.0965112
$787$	31.6354	1.12768	0.563840	−	0.825884i	$-0.309324\pi$
$787$	31.6354	1.12768	0.563840	+	0.825884i	$0.309324\pi$
$788$	5.12768	0.182666
$789$	−11.3482	−0.404007
$790$	−4.40959	−0.156886
$791$	0	0
$792$	−5.22748	−0.185750
$793$	26.0559	0.925272
$794$	−10.7154	−0.380275
$795$	6.29954	0.223422
$796$	−24.3790	−0.864090
$797$	15.4486	0.547217	0.273608	−	0.961841i	$-0.411783\pi$
$797$	15.4486	0.547217	0.273608	+	0.961841i	$0.411783\pi$
$798$	0	0
$799$	13.2452	0.468583
$800$	5.80602	0.205274
$801$	−1.38636	−0.0489845
$802$	−9.43394	−0.333124
$803$	27.5317	0.971571
$804$	10.8328	0.382044
$805$	0	0
$806$	−20.8663	−0.734983
$807$	7.23385	0.254644
$808$	34.2597	1.20525
$809$	−32.9326	−1.15785	−0.578924	−	0.815382i	$-0.696527\pi$
$809$	−32.9326	−1.15785	−0.578924	+	0.815382i	$0.696527\pi$
$810$	−5.47078	−0.192224
$811$	25.5895	0.898567	0.449284	−	0.893389i	$-0.351679\pi$
$811$	25.5895	0.898567	0.449284	+	0.893389i	$0.351679\pi$
$812$	0	0
$813$	15.2622	0.535270
$814$	−0.234667	−0.00822508
$815$	−10.3659	−0.363100
$816$	−0.735094	−0.0257334
$817$	−5.03316	−0.176088
$818$	15.4690	0.540860
$819$	0	0
$820$	14.2201	0.496587
$821$	17.8674	0.623575	0.311788	−	0.950152i	$-0.399072\pi$
$821$	17.8674	0.623575	0.311788	+	0.950152i	$0.399072\pi$
$822$	−2.75062	−0.0959389
$823$	−18.4533	−0.643242	−0.321621	−	0.946868i	$-0.604228\pi$
$823$	−18.4533	−0.643242	−0.321621	+	0.946868i	$0.604228\pi$
$824$	−16.8885	−0.588339
$825$	4.66635	0.162461
$826$	0	0
$827$	47.5460	1.65334	0.826668	−	0.562690i	$-0.190233\pi$
$827$	47.5460	1.65334	0.826668	+	0.562690i	$0.190233\pi$
$828$	3.46950	0.120573
$829$	−19.0308	−0.660965	−0.330483	−	0.943812i	$-0.607212\pi$
$829$	−19.0308	−0.660965	−0.330483	+	0.943812i	$0.607212\pi$
$830$	−1.59748	−0.0554493
$831$	34.8145	1.20770
$832$	17.6179	0.610790
$833$	0	0
$834$	28.2911	0.979640
$835$	15.6048	0.540025
$836$	−4.04373	−0.139855
$837$	−31.2420	−1.07988
$838$	−22.2656	−0.769151
$839$	−3.04022	−0.104960	−0.0524801	−	0.998622i	$-0.516713\pi$
$839$	−3.04022	−0.104960	−0.0524801	+	0.998622i	$0.516713\pi$
$840$	0	0
$841$	−27.8403	−0.960011
$842$	5.44370	0.187602
$843$	6.08280	0.209503
$844$	25.7262	0.885534
$845$	7.89956	0.271753
$846$	−6.35957	−0.218647
$847$	0	0
$848$	−1.81684	−0.0623906
$849$	−20.0979	−0.689758
$850$	0.879381	0.0301625
$851$	0.389401	0.0133485
$852$	−11.6434	−0.398898
$853$	18.2201	0.623844	0.311922	−	0.950108i	$-0.399027\pi$
$853$	18.2201	0.623844	0.311922	+	0.950108i	$0.399027\pi$
$854$	0	0
$855$	0.633188	0.0216546
$856$	34.1539	1.16736
$857$	−19.3611	−0.661363	−0.330682	−	0.943742i	$-0.607279\pi$
$857$	−19.3611	−0.661363	−0.330682	+	0.943742i	$0.607279\pi$
$858$	17.4202	0.594714
$859$	−3.44129	−0.117415	−0.0587077	−	0.998275i	$-0.518698\pi$
$859$	−3.44129	−0.117415	−0.0587077	+	0.998275i	$0.518698\pi$
$860$	−6.71008	−0.228812
$861$	0	0
$862$	−26.5024	−0.902674
$863$	−26.5471	−0.903674	−0.451837	−	0.892101i	$-0.649231\pi$
$863$	−26.5471	−0.903674	−0.451837	+	0.892101i	$0.649231\pi$
$864$	32.4525	1.10406
$865$	−0.571604	−0.0194351
$866$	−27.6146	−0.938383
$867$	24.3694	0.827630
$868$	0	0
$869$	16.3790	0.555619
$870$	1.35288	0.0458669
$871$	24.1458	0.818148
$872$	43.0312	1.45722
$873$	−1.36836	−0.0463120
$874$	−3.35624	−0.113527
$875$	0	0
$876$	−18.6168	−0.629004
$877$	20.5495	0.693908	0.346954	−	0.937882i	$-0.387216\pi$
$877$	20.5495	0.693908	0.346954	+	0.937882i	$0.387216\pi$
$878$	22.7383	0.767380
$879$	−14.9146	−0.503059
$880$	−1.34582	−0.0453674
$881$	31.1911	1.05085	0.525427	−	0.850839i	$-0.323906\pi$
$881$	31.1911	1.05085	0.525427	+	0.850839i	$0.323906\pi$
$882$	0	0
$883$	14.1861	0.477401	0.238701	−	0.971093i	$-0.423279\pi$
$883$	14.1861	0.477401	0.238701	+	0.971093i	$0.423279\pi$
$884$	−6.56336	−0.220750
$885$	−2.15378	−0.0723985
$886$	18.5983	0.624822
$887$	−46.6046	−1.56483	−0.782415	−	0.622757i	$-0.786012\pi$
$887$	−46.6046	−1.56483	−0.782415	+	0.622757i	$0.786012\pi$
$888$	0.396732	0.0133134
$889$	0	0
$890$	1.78792	0.0599313
$891$	20.3207	0.680768
$892$	0.820347	0.0274672
$893$	−12.2995	−0.411588
$894$	−12.2952	−0.411214
$895$	24.8864	0.831862
$896$	0	0
$897$	−28.9066	−0.965164
$898$	20.2930	0.677185
$899$	6.01923	0.200752
$900$	0.844150	0.0281383
$901$	−4.40959	−0.146905
$902$	26.4191	0.879658
$903$	0	0
$904$	14.9558	0.497422
$905$	−6.95372	−0.231150
$906$	−7.38297	−0.245283
$907$	−0.754028	−0.0250371	−0.0125185	−	0.999922i	$-0.503985\pi$
$907$	−0.754028	−0.0250371	−0.0125185	+	0.999922i	$0.503985\pi$
$908$	23.2921	0.772976
$909$	7.96988	0.264344
$910$	0	0
$911$	−53.1413	−1.76065	−0.880325	−	0.474371i	$-0.842675\pi$
$911$	−53.1413	−1.76065	−0.880325	+	0.474371i	$0.842675\pi$
$912$	0.682609	0.0226034
$913$	5.93368	0.196376
$914$	−5.98784	−0.198060
$915$	−8.76838	−0.289874
$916$	12.9311	0.427257
$917$	0	0
$918$	4.91527	0.162228
$919$	37.4202	1.23438	0.617189	−	0.786815i	$-0.288272\pi$
$919$	37.4202	1.23438	0.617189	+	0.786815i	$0.288272\pi$
$920$	−11.1869	−0.368822
$921$	−43.9196	−1.44720
$922$	−8.41709	−0.277202
$923$	−25.9526	−0.854241
$924$	0	0
$925$	0.0947438	0.00311516
$926$	6.37147	0.209379
$927$	−3.92880	−0.129039
$928$	−6.25244	−0.205247
$929$	19.1981	0.629871	0.314935	−	0.949113i	$-0.398017\pi$
$929$	19.1981	0.629871	0.314935	+	0.949113i	$0.398017\pi$
$930$	7.02195	0.230259
$931$	0	0
$932$	10.8704	0.356072
$933$	39.0873	1.27966
$934$	6.02355	0.197096
$935$	−3.26638	−0.106822
$936$	7.87890	0.257530
$937$	20.0095	0.653681	0.326840	−	0.945080i	$-0.394016\pi$
$937$	20.0095	0.653681	0.326840	+	0.945080i	$0.394016\pi$
$938$	0	0
$939$	25.8458	0.843445
$940$	−16.3974	−0.534825
$941$	−15.3327	−0.499832	−0.249916	−	0.968268i	$-0.580403\pi$
$941$	−15.3327	−0.499832	−0.249916	+	0.968268i	$0.580403\pi$
$942$	12.8009	0.417076
$943$	−43.8392	−1.42760
$944$	0.621168	0.0202173
$945$	0	0
$946$	−12.4664	−0.405319
$947$	−29.8217	−0.969076	−0.484538	−	0.874770i	$-0.661012\pi$
$947$	−29.8217	−0.969076	−0.484538	+	0.874770i	$0.661012\pi$
$948$	−11.0754	−0.359713
$949$	−41.4959	−1.34702
$950$	−0.816594	−0.0264938
$951$	48.8464	1.58395
$952$	0	0
$953$	31.4938	1.02018	0.510091	−	0.860120i	$-0.329612\pi$
$953$	31.4938	1.02018	0.510091	+	0.860120i	$0.329612\pi$
$954$	2.11722	0.0685475
$955$	−3.91254	−0.126607
$956$	−32.7948	−1.06066
$957$	−5.02514	−0.162440
$958$	13.1845	0.425973
$959$	0	0
$960$	−5.92880	−0.191351
$961$	0.242050	0.00780806
$962$	0.353692	0.0114035
$963$	7.94528	0.256033
$964$	34.3610	1.10669
$965$	−9.20734	−0.296395
$966$	0	0
$967$	1.16143	0.0373490	0.0186745	−	0.999826i	$-0.494055\pi$
$967$	1.16143	0.0373490	0.0186745	+	0.999826i	$0.494055\pi$
$968$	4.89917	0.157465
$969$	1.65673	0.0532220
$970$	1.76471	0.0566615
$971$	−18.4557	−0.592272	−0.296136	−	0.955146i	$-0.595698\pi$
$971$	−18.4557	−0.592272	−0.296136	+	0.955146i	$0.595698\pi$
$972$	8.61438	0.276306
$973$	0	0
$974$	−6.86401	−0.219937
$975$	−7.03316	−0.225241
$976$	2.52888	0.0809474
$977$	5.62735	0.180035	0.0900175	−	0.995940i	$-0.471308\pi$
$977$	5.62735	0.180035	0.0900175	+	0.995940i	$0.471308\pi$
$978$	13.0225	0.416413
$979$	−6.64107	−0.212249
$980$	0	0
$981$	10.0104	0.319608
$982$	−29.4672	−0.940338
$983$	25.1228	0.801293	0.400647	−	0.916233i	$-0.368786\pi$
$983$	25.1228	0.801293	0.400647	+	0.916233i	$0.368786\pi$
$984$	−44.6644	−1.42385
$985$	−3.84622	−0.122551
$986$	−0.946996	−0.0301585
$987$	0	0
$988$	6.09474	0.193900
$989$	20.6865	0.657793
$990$	1.56832	0.0498445
$991$	−41.8749	−1.33020	−0.665100	−	0.746754i	$-0.731611\pi$
$991$	−41.8749	−1.33020	−0.665100	+	0.746754i	$0.731611\pi$
$992$	−32.4525	−1.03037
$993$	15.3295	0.486467
$994$	0	0
$995$	18.2864	0.579718
$996$	−4.01234	−0.127136
$997$	37.0346	1.17290	0.586449	−	0.809986i	$-0.300525\pi$
$997$	37.0346	1.17290	0.586449	+	0.809986i	$0.300525\pi$
$998$	−2.69505	−0.0853102
$999$	0.529566	0.0167547

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4655.2.a.y.1.3		4
7.6	odd	2		95.2.a.b.1.3	✓	4
21.20	even	2		855.2.a.m.1.2		4
28.27	even	2		1520.2.a.t.1.2		4
35.13	even	4		475.2.b.e.324.4		8
35.27	even	4		475.2.b.e.324.5		8
35.34	odd	2		475.2.a.i.1.2		4
56.13	odd	2		6080.2.a.cc.1.2		4
56.27	even	2		6080.2.a.ch.1.3		4
105.104	even	2		4275.2.a.bo.1.3		4
133.132	even	2		1805.2.a.p.1.2		4
140.139	even	2		7600.2.a.cf.1.3		4
665.664	even	2		9025.2.a.bf.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.3	✓	4	7.6	odd	2
475.2.a.i.1.2		4	35.34	odd	2
475.2.b.e.324.4		8	35.13	even	4
475.2.b.e.324.5		8	35.27	even	4
855.2.a.m.1.2		4	21.20	even	2
1520.2.a.t.1.2		4	28.27	even	2
1805.2.a.p.1.2		4	133.132	even	2
4275.2.a.bo.1.3		4	105.104	even	2
4655.2.a.y.1.3		4	1.1	even	1	trivial
6080.2.a.cc.1.2		4	56.13	odd	2
6080.2.a.ch.1.3		4	56.27	even	2
7600.2.a.cf.1.3		4	140.139	even	2
9025.2.a.bf.1.3		4	665.664	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:	\( N \)	\(=\)	\( 4655 = 5 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4655.a (trivial)

\(f(q)\)	\(=\)	\(q+0.816594 q^{2} -1.53844 q^{3} -1.33317 q^{4} +1.00000 q^{5} -1.25628 q^{6} -2.72185 q^{8} -0.633188 q^{9} +0.816594 q^{10} -3.03316 q^{11} +2.05101 q^{12} +4.57160 q^{13} -1.53844 q^{15} +0.443701 q^{16} +1.07689 q^{17} -0.517058 q^{18} -1.00000 q^{19} -1.33317 q^{20} -2.47686 q^{22} +4.11005 q^{23} +4.18742 q^{24} +1.00000 q^{25} +3.73315 q^{26} +5.58946 q^{27} -1.07689 q^{29} -1.25628 q^{30} -5.58946 q^{31} +5.80602 q^{32} +4.66635 q^{33} +0.879381 q^{34} +0.844150 q^{36} +0.0947438 q^{37} -0.816594 q^{38} -7.03316 q^{39} -2.72185 q^{40} -10.6663 q^{41} +5.03316 q^{43} +4.04373 q^{44} -0.633188 q^{45} +3.35624 q^{46} +12.2995 q^{47} -0.682609 q^{48} +0.816594 q^{50} -1.65673 q^{51} -6.09474 q^{52} -4.09474 q^{53} +4.56432 q^{54} -3.03316 q^{55} +1.53844 q^{57} -0.879381 q^{58} +1.39997 q^{59} +2.05101 q^{60} +5.69951 q^{61} -4.56432 q^{62} +3.85376 q^{64} +4.57160 q^{65} +3.81051 q^{66} +5.28168 q^{67} -1.43568 q^{68} -6.32308 q^{69} -5.67692 q^{71} +1.72344 q^{72} -9.07689 q^{73} +0.0773672 q^{74} -1.53844 q^{75} +1.33317 q^{76} -5.74324 q^{78} -5.39997 q^{79} +0.443701 q^{80} -6.69951 q^{81} -8.71008 q^{82} -1.95627 q^{83} +1.07689 q^{85} +4.11005 q^{86} +1.65673 q^{87} +8.25581 q^{88} +2.18949 q^{89} -0.517058 q^{90} -5.47941 q^{92} +8.59907 q^{93} +10.0437 q^{94} -1.00000 q^{95} -8.93225 q^{96} +2.16106 q^{97} +1.92056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{3} + 8 q^{4} + 4 q^{5} - 12 q^{8} + 8 q^{9} - 2 q^{10} + 4 q^{11} - 6 q^{12} - 2 q^{13} - 2 q^{15} + 4 q^{16} - 4 q^{17} - 34 q^{18} - 4 q^{19} + 8 q^{20} + 4 q^{22} - 8 q^{23} + 24 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^3 + 8 * q^4 + 4 * q^5 - 12 * q^8 + 8 * q^9 - 2 * q^10 + 4 * q^11 - 6 * q^12 - 2 * q^13 - 2 * q^15 + 4 * q^16 - 4 * q^17 - 34 * q^18 - 4 * q^19 + 8 * q^20 + 4 * q^22 - 8 * q^23 + 24 * q^24 + 4 * q^25 - 4 * q^26 + 4 * q^27 + 4 * q^29 - 4 * q^31 - 6 * q^32 - 8 * q^33 + 4 * q^34 + 40 * q^36 - 6 * q^37 + 2 * q^38 - 12 * q^39 - 12 * q^40 - 16 * q^41 + 4 * q^43 + 24 * q^44 + 8 * q^45 + 12 * q^47 - 38 * q^48 - 2 * q^50 - 36 * q^51 - 18 * q^52 - 10 * q^53 + 20 * q^54 + 4 * q^55 + 2 * q^57 - 4 * q^58 - 6 * q^60 - 20 * q^61 - 20 * q^62 - 4 * q^64 - 2 * q^65 + 28 * q^66 - 18 * q^67 - 4 * q^68 - 28 * q^69 - 20 * q^71 - 52 * q^72 - 28 * q^73 + 32 * q^74 - 2 * q^75 - 8 * q^76 + 12 * q^78 - 16 * q^79 + 4 * q^80 + 16 * q^81 - 16 * q^82 - 4 * q^85 - 8 * q^86 + 36 * q^87 - 12 * q^88 - 4 * q^89 - 34 * q^90 - 28 * q^92 - 40 * q^93 + 48 * q^94 - 4 * q^95 + 52 * q^96 - 30 * q^97 - 4 * q^99

Introduction

L-functions

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