LMFDB - Embedded newform 4600.2.e.w.4049.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 18x^{8} + 117x^{6} + 333x^{4} + 396x^{2} + 144 $ x^10 + 18x^8 + 117x^6 + 333x^4 + 396x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.3
Root			$-1.83957i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.w.4049.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83957i q^{3} -3.97272i q^{7} -0.384010 q^{9} +O(q^{10})$	$q-1.83957i q^{3} -3.97272i q^{7} -0.384010 q^{9} +2.10339 q^{11} -5.35673i q^{13} +1.29567i q^{17} -2.10339 q^{19} -7.30809 q^{21} +1.00000i q^{23} -4.81229i q^{27} -6.03130 q^{29} -8.32489 q^{31} -3.86933i q^{33} +5.10131i q^{37} -9.85407 q^{39} -8.33994 q^{41} -7.78253i q^{43} +11.3964i q^{47} -8.78253 q^{49} +2.38347 q^{51} +0.573664i q^{53} +3.86933i q^{57} +9.17951 q^{59} +13.5149 q^{61} +1.52557i q^{63} -15.8314i q^{67} +1.83957 q^{69} +14.9449 q^{71} -8.36459i q^{73} -8.35619i q^{77} +9.41149 q^{79} -10.0046 q^{81} -1.51206i q^{83} +11.0950i q^{87} -10.5903 q^{89} -21.2808 q^{91} +15.3142i q^{93} -0.337451i q^{97} -0.807724 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{9}+O(q^{10}) $ 10 * q - 6 * q^9	$ 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) $ 10 * q - 6 * q^9 - 24 * q^29 - 36 * q^31 - 18 * q^39 - 12 * q^41 - 30 * q^49 - 12 * q^51 + 2 * q^59 + 20 * q^61 + 16 * q^71 - 54 * q^81 - 28 * q^89 - 92 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.83957i	− 1.06208i	−0.847348	−	0.531038i	$-0.821802\pi$
$3$	− 1.83957i	− 1.06208i	0.847348	−	0.531038i	$-0.178198\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.97272i	− 1.50155i	−0.660559	−	0.750774i	$-0.729681\pi$
$7$	− 3.97272i	− 1.50155i	0.660559	−	0.750774i	$-0.270319\pi$
$8$	0	0
$9$	−0.384010	−0.128003
$10$	0	0
$11$	2.10339	0.634196	0.317098	−	0.948393i	$-0.397292\pi$
$11$	2.10339	0.634196	0.317098	+	0.948393i	$0.397292\pi$
$12$	0	0
$13$	− 5.35673i	− 1.48569i	−0.669463	−	0.742845i	$-0.733476\pi$
$13$	− 5.35673i	− 1.48569i	0.669463	−	0.742845i	$-0.266524\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.29567i	0.314246i	0.987579	+	0.157123i	$0.0502218\pi$
$17$	1.29567i	0.314246i	−0.987579	+	0.157123i	$0.949778\pi$
$18$	0	0
$19$	−2.10339	−0.482551	−0.241276	−	0.970457i	$-0.577566\pi$
$19$	−2.10339	−0.482551	−0.241276	+	0.970457i	$0.577566\pi$
$20$	0	0
$21$	−7.30809	−1.59476
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.81229i	− 0.926126i
$28$	0	0
$29$	−6.03130	−1.11998	−0.559992	−	0.828498i	$-0.689196\pi$
$29$	−6.03130	−1.11998	−0.559992	+	0.828498i	$0.689196\pi$
$30$	0	0
$31$	−8.32489	−1.49519	−0.747597	−	0.664153i	$-0.768793\pi$
$31$	−8.32489	−1.49519	−0.747597	+	0.664153i	$0.768793\pi$
$32$	0	0
$33$	− 3.86933i	− 0.673564i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.10131i	0.838650i	0.907836	+	0.419325i	$0.137733\pi$
$37$	5.10131i	0.838650i	−0.907836	+	0.419325i	$0.862267\pi$
$38$	0	0
$39$	−9.85407	−1.57791
$40$	0	0
$41$	−8.33994	−1.30248	−0.651240	−	0.758872i	$-0.725751\pi$
$41$	−8.33994	−1.30248	−0.651240	+	0.758872i	$0.725751\pi$
$42$	0	0
$43$	− 7.78253i	− 1.18682i	−0.804899	−	0.593412i	$-0.797780\pi$
$43$	− 7.78253i	− 1.18682i	0.804899	−	0.593412i	$-0.202220\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.3964i	1.66234i	0.556018	+	0.831171i	$0.312329\pi$
$47$	11.3964i	1.66234i	−0.556018	+	0.831171i	$0.687671\pi$
$48$	0	0
$49$	−8.78253	−1.25465
$50$	0	0
$51$	2.38347	0.333752
$52$	0	0
$53$	0.573664i	0.0787988i	0.999224	+	0.0393994i	$0.0125445\pi$
$53$	0.573664i	0.0787988i	−0.999224	+	0.0393994i	$0.987456\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.86933i	0.512505i
$58$	0	0
$59$	9.17951	1.19507	0.597535	−	0.801843i	$-0.296147\pi$
$59$	9.17951	1.19507	0.597535	+	0.801843i	$0.296147\pi$
$60$	0	0
$61$	13.5149	1.73040	0.865201	−	0.501425i	$-0.167191\pi$
$61$	13.5149	1.73040	0.865201	+	0.501425i	$0.167191\pi$
$62$	0	0
$63$	1.52557i	0.192203i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 15.8314i	− 1.93411i	−0.254569	−	0.967055i	$-0.581934\pi$
$67$	− 15.8314i	− 1.93411i	0.254569	−	0.967055i	$-0.418066\pi$
$68$	0	0
$69$	1.83957	0.221458
$70$	0	0
$71$	14.9449	1.77363	0.886817	−	0.462121i	$-0.152911\pi$
$71$	14.9449	1.77363	0.886817	+	0.462121i	$0.152911\pi$
$72$	0	0
$73$	− 8.36459i	− 0.979002i	−0.872003	−	0.489501i	$-0.837179\pi$
$73$	− 8.36459i	− 0.979002i	0.872003	−	0.489501i	$-0.162821\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 8.35619i	− 0.952276i
$78$	0	0
$79$	9.41149	1.05887	0.529437	−	0.848349i	$-0.322403\pi$
$79$	9.41149	1.05887	0.529437	+	0.848349i	$0.322403\pi$
$80$	0	0
$81$	−10.0046	−1.11162
$82$	0	0
$83$	− 1.51206i	− 0.165970i	−0.996551	−	0.0829849i	$-0.973555\pi$
$83$	− 1.51206i	− 0.165970i	0.996551	−	0.0829849i	$-0.0264453\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.0950i	1.18951i
$88$	0	0
$89$	−10.5903	−1.12256	−0.561282	−	0.827624i	$-0.689692\pi$
$89$	−10.5903	−1.12256	−0.561282	+	0.827624i	$0.689692\pi$
$90$	0	0
$91$	−21.2808	−2.23084
$92$	0	0
$93$	15.3142i	1.58801i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 0.337451i	− 0.0342630i	−0.999853	−	0.0171315i	$-0.994547\pi$
$97$	− 0.337451i	− 0.0342630i	0.999853	−	0.0171315i	$-0.00545339\pi$
$98$	0	0
$99$	−0.807724	−0.0811793
$100$	0	0
$101$	−6.19119	−0.616047	−0.308023	−	0.951379i	$-0.599667\pi$
$101$	−6.19119	−0.616047	−0.308023	+	0.951379i	$0.599667\pi$
$102$	0	0
$103$	− 12.6267i	− 1.24414i	−0.782961	−	0.622071i	$-0.786292\pi$
$103$	− 12.6267i	− 1.24414i	0.782961	−	0.622071i	$-0.213708\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.7940i	1.23684i	0.785848	+	0.618419i	$0.212227\pi$
$107$	12.7940i	1.23684i	−0.785848	+	0.618419i	$0.787773\pi$
$108$	0	0
$109$	−14.3165	−1.37127	−0.685635	−	0.727945i	$-0.740476\pi$
$109$	−14.3165	−1.37127	−0.685635	+	0.727945i	$0.740476\pi$
$110$	0	0
$111$	9.38421	0.890710
$112$	0	0
$113$	− 4.59463i	− 0.432226i	−0.976368	−	0.216113i	$-0.930662\pi$
$113$	− 4.59463i	− 0.432226i	0.976368	−	0.216113i	$-0.0693380\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.05704i	0.190173i
$118$	0	0
$119$	5.14733	0.471855
$120$	0	0
$121$	−6.57574	−0.597795
$122$	0	0
$123$	15.3419i	1.38333i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.6623i	1.12360i	0.827273	+	0.561801i	$0.189891\pi$
$127$	12.6623i	1.12360i	−0.827273	+	0.561801i	$0.810109\pi$
$128$	0	0
$129$	−14.3165	−1.26050
$130$	0	0
$131$	4.68370	0.409217	0.204609	−	0.978844i	$-0.434408\pi$
$131$	4.68370	0.409217	0.204609	+	0.978844i	$0.434408\pi$
$132$	0	0
$133$	8.35619i	0.724574i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 0.468120i	− 0.0399942i	−0.999800	−	0.0199971i	$-0.993634\pi$
$137$	− 0.468120i	− 0.0399942i	0.999800	−	0.0199971i	$-0.00636570\pi$
$138$	0	0
$139$	−0.739205	−0.0626985	−0.0313493	−	0.999508i	$-0.509980\pi$
$139$	−0.739205	−0.0626985	−0.0313493	+	0.999508i	$0.509980\pi$
$140$	0	0
$141$	20.9645	1.76553
$142$	0	0
$143$	− 11.2673i	− 0.942220i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	16.1561i	1.33253i
$148$	0	0
$149$	−14.5903	−1.19528	−0.597640	−	0.801765i	$-0.703895\pi$
$149$	−14.5903	−1.19528	−0.597640	+	0.801765i	$0.703895\pi$
$150$	0	0
$151$	21.6885	1.76498	0.882491	−	0.470329i	$-0.155865\pi$
$151$	21.6885	1.76498	0.882491	+	0.470329i	$0.155865\pi$
$152$	0	0
$153$	− 0.497550i	− 0.0402245i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.71253i	0.615527i	0.951463	+	0.307763i	$0.0995805\pi$
$157$	7.71253i	0.615527i	−0.951463	+	0.307763i	$0.900420\pi$
$158$	0	0
$159$	1.05529	0.0836902
$160$	0	0
$161$	3.97272	0.313094
$162$	0	0
$163$	4.69177i	0.367488i	0.982974	+	0.183744i	$0.0588217\pi$
$163$	4.69177i	0.367488i	−0.982974	+	0.183744i	$0.941178\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.6738i	0.903343i	0.892184	+	0.451671i	$0.149172\pi$
$167$	11.6738i	0.903343i	−0.892184	+	0.451671i	$0.850828\pi$
$168$	0	0
$169$	−15.6946	−1.20728
$170$	0	0
$171$	0.807724	0.0617682
$172$	0	0
$173$	8.31723i	0.632347i	0.948701	+	0.316174i	$0.102398\pi$
$173$	8.31723i	0.632347i	−0.948701	+	0.316174i	$0.897602\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 16.8863i	− 1.26925i
$178$	0	0
$179$	4.07517	0.304592	0.152296	−	0.988335i	$-0.451333\pi$
$179$	4.07517	0.304592	0.152296	+	0.988335i	$0.451333\pi$
$180$	0	0
$181$	2.52349	0.187569	0.0937846	−	0.995593i	$-0.470103\pi$
$181$	2.52349	0.187569	0.0937846	+	0.995593i	$0.470103\pi$
$182$	0	0
$183$	− 24.8615i	− 1.83782i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.72530i	0.199293i
$188$	0	0
$189$	−19.1179	−1.39062
$190$	0	0
$191$	−23.9380	−1.73210	−0.866048	−	0.499961i	$-0.833348\pi$
$191$	−23.9380	−1.73210	−0.866048	+	0.499961i	$0.833348\pi$
$192$	0	0
$193$	− 18.7733i	− 1.35133i	−0.737207	−	0.675667i	$-0.763856\pi$
$193$	− 18.7733i	− 1.35133i	0.737207	−	0.675667i	$-0.236144\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.9052i	1.13320i	0.823993	+	0.566600i	$0.191741\pi$
$197$	15.9052i	1.13320i	−0.823993	+	0.566600i	$0.808259\pi$
$198$	0	0
$199$	−9.87544	−0.700052	−0.350026	−	0.936740i	$-0.613827\pi$
$199$	−9.87544	−0.700052	−0.350026	+	0.936740i	$0.613827\pi$
$200$	0	0
$201$	−29.1229	−2.05417
$202$	0	0
$203$	23.9607i	1.68171i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 0.384010i	− 0.0266906i
$208$	0	0
$209$	−4.42426	−0.306032
$210$	0	0
$211$	−7.06980	−0.486705	−0.243353	−	0.969938i	$-0.578247\pi$
$211$	−7.06980	−0.486705	−0.243353	+	0.969938i	$0.578247\pi$
$212$	0	0
$213$	− 27.4922i	− 1.88373i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	33.0725i	2.24511i
$218$	0	0
$219$	−15.3872	−1.03977
$220$	0	0
$221$	6.94055	0.466872
$222$	0	0
$223$	− 8.58817i	− 0.575106i	−0.957765	−	0.287553i	$-0.907158\pi$
$223$	− 8.58817i	− 0.575106i	0.957765	−	0.287553i	$-0.0928418\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.2068i	0.942937i	0.881883	+	0.471469i	$0.156276\pi$
$227$	14.2068i	0.942937i	−0.881883	+	0.471469i	$0.843724\pi$
$228$	0	0
$229$	7.49721	0.495429	0.247715	−	0.968833i	$-0.420320\pi$
$229$	7.49721	0.495429	0.247715	+	0.968833i	$0.420320\pi$
$230$	0	0
$231$	−15.3718	−1.01139
$232$	0	0
$233$	16.9301i	1.10912i	0.832142	+	0.554562i	$0.187114\pi$
$233$	16.9301i	1.10912i	−0.832142	+	0.554562i	$0.812886\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 17.3131i	− 1.12460i
$238$	0	0
$239$	17.7850	1.15042	0.575208	−	0.818007i	$-0.304921\pi$
$239$	17.7850	1.15042	0.575208	+	0.818007i	$0.304921\pi$
$240$	0	0
$241$	−9.55263	−0.615339	−0.307669	−	0.951493i	$-0.599549\pi$
$241$	−9.55263	−0.615339	−0.307669	+	0.951493i	$0.599549\pi$
$242$	0	0
$243$	3.96721i	0.254497i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	11.2673i	0.716922i
$248$	0	0
$249$	−2.78153	−0.176272
$250$	0	0
$251$	−13.2075	−0.833651	−0.416826	−	0.908986i	$-0.636857\pi$
$251$	−13.2075	−0.833651	−0.416826	+	0.908986i	$0.636857\pi$
$252$	0	0
$253$	2.10339i	0.132239i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.41506i	− 0.213025i	−0.994311	−	0.106513i	$-0.966032\pi$
$257$	− 3.41506i	− 0.213025i	0.994311	−	0.106513i	$-0.0339685\pi$
$258$	0	0
$259$	20.2661	1.25927
$260$	0	0
$261$	2.31608	0.143362
$262$	0	0
$263$	− 18.6589i	− 1.15056i	−0.817957	−	0.575279i	$-0.804894\pi$
$263$	− 18.6589i	− 1.15056i	0.817957	−	0.575279i	$-0.195106\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	19.4815i	1.19225i
$268$	0	0
$269$	24.2224	1.47687	0.738434	−	0.674326i	$-0.235566\pi$
$269$	24.2224	1.47687	0.738434	+	0.674326i	$0.235566\pi$
$270$	0	0
$271$	−10.1201	−0.614749	−0.307375	−	0.951589i	$-0.599450\pi$
$271$	−10.1201	−0.614749	−0.307375	+	0.951589i	$0.599450\pi$
$272$	0	0
$273$	39.1475i	2.36932i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 23.5298i	− 1.41377i	−0.707329	−	0.706884i	$-0.750100\pi$
$277$	− 23.5298i	− 1.41377i	0.707329	−	0.706884i	$-0.249900\pi$
$278$	0	0
$279$	3.19684	0.191390
$280$	0	0
$281$	1.77314	0.105776	0.0528882	−	0.998600i	$-0.483157\pi$
$281$	1.77314	0.105776	0.0528882	+	0.998600i	$0.483157\pi$
$282$	0	0
$283$	− 5.96641i	− 0.354666i	−0.984151	−	0.177333i	$-0.943253\pi$
$283$	− 5.96641i	− 0.354666i	0.984151	−	0.177333i	$-0.0567470\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	33.1323i	1.95574i
$288$	0	0
$289$	15.3212	0.901250
$290$	0	0
$291$	−0.620765	−0.0363899
$292$	0	0
$293$	− 12.2404i	− 0.715090i	−0.933896	−	0.357545i	$-0.883614\pi$
$293$	− 12.2404i	− 0.715090i	0.933896	−	0.357545i	$-0.116386\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 10.1221i	− 0.587346i
$298$	0	0
$299$	5.35673	0.309788
$300$	0	0
$301$	−30.9178	−1.78207
$302$	0	0
$303$	11.3891i	0.654288i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 8.82713i	− 0.503791i	−0.967754	−	0.251896i	$-0.918946\pi$
$307$	− 8.82713i	− 0.503791i	0.967754	−	0.251896i	$-0.0810539\pi$
$308$	0	0
$309$	−23.2276	−1.32137
$310$	0	0
$311$	−13.9202	−0.789345	−0.394672	−	0.918822i	$-0.629142\pi$
$311$	−13.9202	−0.789345	−0.394672	+	0.918822i	$0.629142\pi$
$312$	0	0
$313$	− 23.5030i	− 1.32847i	−0.747523	−	0.664235i	$-0.768757\pi$
$313$	− 23.5030i	− 1.32847i	0.747523	−	0.664235i	$-0.231243\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.3279i	0.748570i	0.927314	+	0.374285i	$0.122112\pi$
$317$	13.3279i	0.748570i	−0.927314	+	0.374285i	$0.877888\pi$
$318$	0	0
$319$	−12.6862	−0.710290
$320$	0	0
$321$	23.5354	1.31362
$322$	0	0
$323$	− 2.72530i	− 0.151640i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	26.3362i	1.45639i
$328$	0	0
$329$	45.2749	2.49609
$330$	0	0
$331$	−0.270137	−0.0148481	−0.00742404	−	0.999972i	$-0.502363\pi$
$331$	−0.270137	−0.0148481	−0.00742404	+	0.999972i	$0.502363\pi$
$332$	0	0
$333$	− 1.95896i	− 0.107350i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 20.0205i	− 1.09058i	−0.838246	−	0.545292i	$-0.816419\pi$
$337$	− 20.0205i	− 1.09058i	0.838246	−	0.545292i	$-0.183581\pi$
$338$	0	0
$339$	−8.45213	−0.459057
$340$	0	0
$341$	−17.5105	−0.948247
$342$	0	0
$343$	7.08149i	0.382364i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.1156i	− 0.811448i	−0.913996	−	0.405724i	$-0.867019\pi$
$347$	− 15.1156i	− 0.811448i	0.913996	−	0.405724i	$-0.132981\pi$
$348$	0	0
$349$	14.5673	0.779771	0.389886	−	0.920863i	$-0.372515\pi$
$349$	14.5673	0.779771	0.389886	+	0.920863i	$0.372515\pi$
$350$	0	0
$351$	−25.7782	−1.37594
$352$	0	0
$353$	− 18.5376i	− 0.986656i	−0.869843	−	0.493328i	$-0.835780\pi$
$353$	− 18.5376i	− 0.986656i	0.869843	−	0.493328i	$-0.164220\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 9.46886i	− 0.501145i
$358$	0	0
$359$	24.1834	1.27635	0.638175	−	0.769891i	$-0.279689\pi$
$359$	24.1834	1.27635	0.638175	+	0.769891i	$0.279689\pi$
$360$	0	0
$361$	−14.5757	−0.767144
$362$	0	0
$363$	12.0965i	0.634903i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.79604i	0.302551i	0.988492	+	0.151275i	$0.0483380\pi$
$367$	5.79604i	0.302551i	−0.988492	+	0.151275i	$0.951662\pi$
$368$	0	0
$369$	3.20262	0.166722
$370$	0	0
$371$	2.27901	0.118320
$372$	0	0
$373$	25.5753i	1.32424i	0.749399	+	0.662119i	$0.230343\pi$
$373$	25.5753i	1.32424i	−0.749399	+	0.662119i	$0.769657\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.3081i	1.66395i
$378$	0	0
$379$	9.82223	0.504534	0.252267	−	0.967658i	$-0.418824\pi$
$379$	9.82223	0.504534	0.252267	+	0.967658i	$0.418824\pi$
$380$	0	0
$381$	23.2932	1.19335
$382$	0	0
$383$	33.9797i	1.73628i	0.496319	+	0.868141i	$0.334685\pi$
$383$	33.9797i	1.73628i	−0.496319	+	0.868141i	$0.665315\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.98857i	0.151918i
$388$	0	0
$389$	24.6744	1.25104	0.625521	−	0.780207i	$-0.284886\pi$
$389$	24.6744	1.25104	0.625521	+	0.780207i	$0.284886\pi$
$390$	0	0
$391$	−1.29567	−0.0655247
$392$	0	0
$393$	− 8.61599i	− 0.434619i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 3.17340i	− 0.159268i	−0.996824	−	0.0796342i	$-0.974625\pi$
$397$	− 3.17340i	− 0.159268i	0.996824	−	0.0796342i	$-0.0253752\pi$
$398$	0	0
$399$	15.3718	0.769552
$400$	0	0
$401$	4.07720	0.203606	0.101803	−	0.994805i	$-0.467539\pi$
$401$	4.07720	0.203606	0.101803	+	0.994805i	$0.467539\pi$
$402$	0	0
$403$	44.5942i	2.22140i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.7301i	0.531869i
$408$	0	0
$409$	−18.5426	−0.916871	−0.458435	−	0.888728i	$-0.651590\pi$
$409$	−18.5426	−0.916871	−0.458435	+	0.888728i	$0.651590\pi$
$410$	0	0
$411$	−0.861138	−0.0424768
$412$	0	0
$413$	− 36.4676i	− 1.79445i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.35982i	0.0665906i
$418$	0	0
$419$	3.22587	0.157594	0.0787970	−	0.996891i	$-0.474892\pi$
$419$	3.22587	0.157594	0.0787970	+	0.996891i	$0.474892\pi$
$420$	0	0
$421$	18.0595	0.880167	0.440084	−	0.897957i	$-0.354949\pi$
$421$	18.0595	0.880167	0.440084	+	0.897957i	$0.354949\pi$
$422$	0	0
$423$	− 4.37635i	− 0.212785i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 53.6909i	− 2.59828i
$428$	0	0
$429$	−20.7270	−1.00071
$430$	0	0
$431$	−9.62458	−0.463600	−0.231800	−	0.972763i	$-0.574461\pi$
$431$	−9.62458	−0.463600	−0.231800	+	0.972763i	$0.574461\pi$
$432$	0	0
$433$	1.83990i	0.0884200i	0.999022	+	0.0442100i	$0.0140771\pi$
$433$	1.83990i	0.0884200i	−0.999022	+	0.0442100i	$0.985923\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.10339i	− 0.100619i
$438$	0	0
$439$	−6.73907	−0.321638	−0.160819	−	0.986984i	$-0.551414\pi$
$439$	−6.73907	−0.321638	−0.160819	+	0.986984i	$0.551414\pi$
$440$	0	0
$441$	3.37258	0.160599
$442$	0	0
$443$	− 11.4099i	− 0.542103i	−0.962565	−	0.271051i	$-0.912629\pi$
$443$	− 11.4099i	− 0.542103i	0.962565	−	0.271051i	$-0.0873714\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	26.8398i	1.26948i
$448$	0	0
$449$	−27.8081	−1.31235	−0.656173	−	0.754610i	$-0.727826\pi$
$449$	−27.8081	−1.31235	−0.656173	+	0.754610i	$0.727826\pi$
$450$	0	0
$451$	−17.5422	−0.826028
$452$	0	0
$453$	− 39.8974i	− 1.87454i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.6835i	1.24820i	0.781343	+	0.624102i	$0.214535\pi$
$457$	26.6835i	1.24820i	−0.781343	+	0.624102i	$0.785465\pi$
$458$	0	0
$459$	6.23513	0.291031
$460$	0	0
$461$	−23.0398	−1.07307	−0.536534	−	0.843879i	$-0.680267\pi$
$461$	−23.0398	−1.07307	−0.536534	+	0.843879i	$0.680267\pi$
$462$	0	0
$463$	23.3998i	1.08748i	0.839253	+	0.543740i	$0.182992\pi$
$463$	23.3998i	1.08748i	−0.839253	+	0.543740i	$0.817008\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.29380i	0.198693i	0.995053	+	0.0993467i	$0.0316753\pi$
$467$	4.29380i	0.198693i	−0.995053	+	0.0993467i	$0.968325\pi$
$468$	0	0
$469$	−62.8936	−2.90416
$470$	0	0
$471$	14.1877	0.653735
$472$	0	0
$473$	− 16.3697i	− 0.752680i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.220293i	− 0.0100865i
$478$	0	0
$479$	26.6694	1.21856	0.609278	−	0.792957i	$-0.291459\pi$
$479$	26.6694	1.21856	0.609278	+	0.792957i	$0.291459\pi$
$480$	0	0
$481$	27.3264	1.24597
$482$	0	0
$483$	− 7.30809i	− 0.332530i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8.67114i	− 0.392927i	−0.980511	−	0.196463i	$-0.937054\pi$
$487$	− 8.67114i	− 0.392927i	0.980511	−	0.196463i	$-0.0629457\pi$
$488$	0	0
$489$	8.63083	0.390300
$490$	0	0
$491$	−25.0900	−1.13229	−0.566147	−	0.824304i	$-0.691566\pi$
$491$	−25.0900	−1.13229	−0.566147	+	0.824304i	$0.691566\pi$
$492$	0	0
$493$	− 7.81456i	− 0.351950i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 59.3720i	− 2.66320i
$498$	0	0
$499$	34.9323	1.56379	0.781893	−	0.623412i	$-0.214254\pi$
$499$	34.9323	1.56379	0.781893	+	0.623412i	$0.214254\pi$
$500$	0	0
$501$	21.4747	0.959418
$502$	0	0
$503$	− 25.7782i	− 1.14939i	−0.818367	−	0.574695i	$-0.805121\pi$
$503$	− 25.7782i	− 1.14939i	0.818367	−	0.574695i	$-0.194879\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	28.8713i	1.28222i
$508$	0	0
$509$	−4.21667	−0.186900	−0.0934502	−	0.995624i	$-0.529790\pi$
$509$	−4.21667	−0.186900	−0.0934502	+	0.995624i	$0.529790\pi$
$510$	0	0
$511$	−33.2302	−1.47002
$512$	0	0
$513$	10.1221i	0.446903i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	23.9712i	1.05425i
$518$	0	0
$519$	15.3001	0.671600
$520$	0	0
$521$	29.3112	1.28415	0.642073	−	0.766644i	$-0.278075\pi$
$521$	29.3112	1.28415	0.642073	+	0.766644i	$0.278075\pi$
$522$	0	0
$523$	31.1549i	1.36231i	0.732140	+	0.681154i	$0.238522\pi$
$523$	31.1549i	1.36231i	−0.732140	+	0.681154i	$0.761478\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 10.7863i	− 0.469858i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−3.52502	−0.152973
$532$	0	0
$533$	44.6748i	1.93508i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 7.49655i	− 0.323500i
$538$	0	0
$539$	−18.4731	−0.795692
$540$	0	0
$541$	15.7393	0.676683	0.338342	−	0.941023i	$-0.390134\pi$
$541$	15.7393	0.676683	0.338342	+	0.941023i	$0.390134\pi$
$542$	0	0
$543$	− 4.64212i	− 0.199213i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 39.2935i	− 1.68007i	−0.542534	−	0.840034i	$-0.682535\pi$
$547$	− 39.2935i	− 1.68007i	0.542534	−	0.840034i	$-0.317465\pi$
$548$	0	0
$549$	−5.18985	−0.221497
$550$	0	0
$551$	12.6862	0.540450
$552$	0	0
$553$	− 37.3892i	− 1.58995i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.1005i	1.61437i	0.590297	+	0.807186i	$0.299010\pi$
$557$	38.1005i	1.61437i	−0.590297	+	0.807186i	$0.700990\pi$
$558$	0	0
$559$	−41.6889	−1.76325
$560$	0	0
$561$	5.01337	0.211665
$562$	0	0
$563$	− 32.6868i	− 1.37758i	−0.724959	−	0.688792i	$-0.758141\pi$
$563$	− 32.6868i	− 1.37758i	0.724959	−	0.688792i	$-0.241859\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	39.7454i	1.66915i
$568$	0	0
$569$	−21.0256	−0.881438	−0.440719	−	0.897645i	$-0.645276\pi$
$569$	−21.0256	−0.881438	−0.440719	+	0.897645i	$0.645276\pi$
$570$	0	0
$571$	−7.43876	−0.311303	−0.155651	−	0.987812i	$-0.549748\pi$
$571$	−7.43876	−0.311303	−0.155651	+	0.987812i	$0.549748\pi$
$572$	0	0
$573$	44.0357i	1.83962i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 30.4148i	− 1.26618i	−0.774077	−	0.633091i	$-0.781786\pi$
$577$	− 30.4148i	− 1.26618i	0.774077	−	0.633091i	$-0.218214\pi$
$578$	0	0
$579$	−34.5348	−1.43522
$580$	0	0
$581$	−6.00698	−0.249212
$582$	0	0
$583$	1.20664i	0.0499739i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 33.7544i	− 1.39319i	−0.717464	−	0.696596i	$-0.754697\pi$
$587$	− 33.7544i	− 1.39319i	0.717464	−	0.696596i	$-0.245303\pi$
$588$	0	0
$589$	17.5105	0.721508
$590$	0	0
$591$	29.2587	1.20354
$592$	0	0
$593$	− 27.3980i	− 1.12510i	−0.826763	−	0.562550i	$-0.809820\pi$
$593$	− 27.3980i	− 1.12510i	0.826763	−	0.562550i	$-0.190180\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	18.1666i	0.743507i
$598$	0	0
$599$	−37.2927	−1.52374	−0.761869	−	0.647731i	$-0.775718\pi$
$599$	−37.2927	−1.52374	−0.761869	+	0.647731i	$0.775718\pi$
$600$	0	0
$601$	−2.26039	−0.0922032	−0.0461016	−	0.998937i	$-0.514680\pi$
$601$	−2.26039	−0.0922032	−0.0461016	+	0.998937i	$0.514680\pi$
$602$	0	0
$603$	6.07941i	0.247573i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 23.4782i	− 0.952951i	−0.879188	−	0.476476i	$-0.841914\pi$
$607$	− 23.4782i	− 0.952951i	0.879188	−	0.476476i	$-0.158086\pi$
$608$	0	0
$609$	44.0773	1.78610
$610$	0	0
$611$	61.0477	2.46972
$612$	0	0
$613$	25.0418i	1.01143i	0.862701	+	0.505714i	$0.168771\pi$
$613$	25.0418i	1.01143i	−0.862701	+	0.505714i	$0.831229\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 18.2705i	− 0.735541i	−0.929916	−	0.367771i	$-0.880121\pi$
$617$	− 18.2705i	− 0.735541i	0.929916	−	0.367771i	$-0.119879\pi$
$618$	0	0
$619$	−7.17743	−0.288485	−0.144243	−	0.989542i	$-0.546075\pi$
$619$	−7.17743	−0.288485	−0.144243	+	0.989542i	$0.546075\pi$
$620$	0	0
$621$	4.81229	0.193111
$622$	0	0
$623$	42.0721i	1.68558i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.13872i	0.325029i
$628$	0	0
$629$	−6.60960	−0.263542
$630$	0	0
$631$	−28.7621	−1.14500	−0.572500	−	0.819904i	$-0.694026\pi$
$631$	−28.7621	−1.14500	−0.572500	+	0.819904i	$0.694026\pi$
$632$	0	0
$633$	13.0054i	0.516917i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	47.0457i	1.86402i
$638$	0	0
$639$	−5.73900	−0.227031
$640$	0	0
$641$	−40.7383	−1.60907	−0.804533	−	0.593907i	$-0.797585\pi$
$641$	−40.7383	−1.60907	−0.804533	+	0.593907i	$0.797585\pi$
$642$	0	0
$643$	8.24043i	0.324971i	0.986711	+	0.162485i	$0.0519510\pi$
$643$	8.24043i	0.324971i	−0.986711	+	0.162485i	$0.948049\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 33.1134i	− 1.30182i	−0.759154	−	0.650911i	$-0.774387\pi$
$647$	− 33.1134i	− 1.30182i	0.759154	−	0.650911i	$-0.225613\pi$
$648$	0	0
$649$	19.3081	0.757909
$650$	0	0
$651$	60.8391	2.38447
$652$	0	0
$653$	− 14.5452i	− 0.569199i	−0.958647	−	0.284599i	$-0.908139\pi$
$653$	− 14.5452i	− 0.569199i	0.958647	−	0.284599i	$-0.0918605\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.21209i	0.125316i
$658$	0	0
$659$	24.4498	0.952427	0.476214	−	0.879330i	$-0.342009\pi$
$659$	24.4498	0.952427	0.476214	+	0.879330i	$0.342009\pi$
$660$	0	0
$661$	1.43400	0.0557763	0.0278882	−	0.999611i	$-0.491122\pi$
$661$	1.43400	0.0557763	0.0278882	+	0.999611i	$0.491122\pi$
$662$	0	0
$663$	− 12.7676i	− 0.495853i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 6.03130i	− 0.233533i
$668$	0	0
$669$	−15.7985	−0.610806
$670$	0	0
$671$	28.4271	1.09742
$672$	0	0
$673$	3.39873i	0.131011i	0.997852	+	0.0655056i	$0.0208660\pi$
$673$	3.39873i	0.131011i	−0.997852	+	0.0655056i	$0.979134\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 15.7751i	− 0.606287i	−0.952945	−	0.303144i	$-0.901964\pi$
$677$	− 15.7751i	− 0.606287i	0.952945	−	0.303144i	$-0.0980362\pi$
$678$	0	0
$679$	−1.34060	−0.0514475
$680$	0	0
$681$	26.1343	1.00147
$682$	0	0
$683$	4.99979i	0.191312i	0.995414	+	0.0956559i	$0.0304948\pi$
$683$	4.99979i	0.191312i	−0.995414	+	0.0956559i	$0.969505\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 13.7916i	− 0.526183i
$688$	0	0
$689$	3.07296	0.117071
$690$	0	0
$691$	40.6864	1.54779	0.773893	−	0.633317i	$-0.218307\pi$
$691$	40.6864	1.54779	0.773893	+	0.633317i	$0.218307\pi$
$692$	0	0
$693$	3.20886i	0.121895i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 10.8058i	− 0.409298i
$698$	0	0
$699$	31.1440	1.17797
$700$	0	0
$701$	−1.30372	−0.0492407	−0.0246204	−	0.999697i	$-0.507838\pi$
$701$	−1.30372	−0.0492407	−0.0246204	+	0.999697i	$0.507838\pi$
$702$	0	0
$703$	− 10.7301i	− 0.404692i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24.5959i	0.925024i
$708$	0	0
$709$	−52.5228	−1.97253	−0.986267	−	0.165157i	$-0.947187\pi$
$709$	−52.5228	−1.97253	−0.986267	+	0.165157i	$0.947187\pi$
$710$	0	0
$711$	−3.61411	−0.135540
$712$	0	0
$713$	− 8.32489i	− 0.311770i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 32.7167i	− 1.22183i
$718$	0	0
$719$	24.2329	0.903735	0.451868	−	0.892085i	$-0.350758\pi$
$719$	24.2329	0.903735	0.451868	+	0.892085i	$0.350758\pi$
$720$	0	0
$721$	−50.1622	−1.86814
$722$	0	0
$723$	17.5727i	0.653536i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	37.0943i	1.37575i	0.725829	+	0.687875i	$0.241456\pi$
$727$	37.0943i	1.37575i	−0.725829	+	0.687875i	$0.758544\pi$
$728$	0	0
$729$	−22.7158	−0.841324
$730$	0	0
$731$	10.0836	0.372954
$732$	0	0
$733$	− 46.3065i	− 1.71037i	−0.518325	−	0.855184i	$-0.673444\pi$
$733$	− 46.3065i	− 1.71037i	0.518325	−	0.855184i	$-0.326556\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 33.2996i	− 1.22660i
$738$	0	0
$739$	32.3516	1.19007	0.595037	−	0.803698i	$-0.297137\pi$
$739$	32.3516	1.19007	0.595037	+	0.803698i	$0.297137\pi$
$740$	0	0
$741$	20.7270	0.761425
$742$	0	0
$743$	1.73343i	0.0635935i	0.999494	+	0.0317967i	$0.0101229\pi$
$743$	1.73343i	0.0635935i	−0.999494	+	0.0317967i	$0.989877\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.580645i	0.0212447i
$748$	0	0
$749$	50.8268	1.85717
$750$	0	0
$751$	42.9983	1.56903	0.784515	−	0.620110i	$-0.212912\pi$
$751$	42.9983	1.56903	0.784515	+	0.620110i	$0.212912\pi$
$752$	0	0
$753$	24.2961i	0.885400i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	48.3219i	1.75629i	0.478397	+	0.878144i	$0.341218\pi$
$757$	48.3219i	1.75629i	−0.478397	+	0.878144i	$0.658782\pi$
$758$	0	0
$759$	3.86933	0.140448
$760$	0	0
$761$	19.2528	0.697915	0.348958	−	0.937139i	$-0.386536\pi$
$761$	19.2528	0.697915	0.348958	+	0.937139i	$0.386536\pi$
$762$	0	0
$763$	56.8754i	2.05903i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 49.1722i	− 1.77550i
$768$	0	0
$769$	−9.05816	−0.326645	−0.163323	−	0.986573i	$-0.552221\pi$
$769$	−9.05816	−0.326645	−0.163323	+	0.986573i	$0.552221\pi$
$770$	0	0
$771$	−6.28223	−0.226249
$772$	0	0
$773$	− 29.1276i	− 1.04765i	−0.851827	−	0.523824i	$-0.824505\pi$
$773$	− 29.1276i	− 1.04765i	0.851827	−	0.523824i	$-0.175495\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 37.2809i	− 1.33744i
$778$	0	0
$779$	17.5422	0.628513
$780$	0	0
$781$	31.4350	1.12483
$782$	0	0
$783$	29.0244i	1.03725i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 23.8771i	− 0.851128i	−0.904928	−	0.425564i	$-0.860076\pi$
$787$	− 23.8771i	− 0.851128i	0.904928	−	0.425564i	$-0.139924\pi$
$788$	0	0
$789$	−34.3243	−1.22198
$790$	0	0
$791$	−18.2532	−0.649008
$792$	0	0
$793$	− 72.3956i	− 2.57084i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.8896i	− 0.704526i	−0.935901	−	0.352263i	$-0.885412\pi$
$797$	− 19.8896i	− 0.704526i	0.935901	−	0.352263i	$-0.114588\pi$
$798$	0	0
$799$	−14.7660	−0.522383
$800$	0	0
$801$	4.06677	0.143692
$802$	0	0
$803$	− 17.5940i	− 0.620879i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 44.5588i	− 1.56854i
$808$	0	0
$809$	−26.3598	−0.926760	−0.463380	−	0.886160i	$-0.653363\pi$
$809$	−26.3598	−0.926760	−0.463380	+	0.886160i	$0.653363\pi$
$810$	0	0
$811$	−14.2482	−0.500323	−0.250161	−	0.968204i	$-0.580484\pi$
$811$	−14.2482	−0.500323	−0.250161	+	0.968204i	$0.580484\pi$
$812$	0	0
$813$	18.6165i	0.652910i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16.3697i	0.572703i
$818$	0	0
$819$	8.17205	0.285555
$820$	0	0
$821$	5.56520	0.194227	0.0971134	−	0.995273i	$-0.469039\pi$
$821$	5.56520	0.194227	0.0971134	+	0.995273i	$0.469039\pi$
$822$	0	0
$823$	34.1312i	1.18974i	0.803823	+	0.594869i	$0.202796\pi$
$823$	34.1312i	1.18974i	−0.803823	+	0.594869i	$0.797204\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23.5085i	− 0.817472i	−0.912653	−	0.408736i	$-0.865970\pi$
$827$	− 23.5085i	− 0.817472i	0.912653	−	0.408736i	$-0.134030\pi$
$828$	0	0
$829$	31.7442	1.10252	0.551260	−	0.834333i	$-0.314147\pi$
$829$	31.7442	1.10252	0.551260	+	0.834333i	$0.314147\pi$
$830$	0	0
$831$	−43.2846	−1.50153
$832$	0	0
$833$	− 11.3792i	− 0.394267i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	40.0618i	1.38474i
$838$	0	0
$839$	−24.5409	−0.847247	−0.423623	−	0.905838i	$-0.639242\pi$
$839$	−24.5409	−0.847247	−0.423623	+	0.905838i	$0.639242\pi$
$840$	0	0
$841$	7.37661	0.254366
$842$	0	0
$843$	− 3.26180i	− 0.112343i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.1236i	0.897618i
$848$	0	0
$849$	−10.9756	−0.376682
$850$	0	0
$851$	−5.10131	−0.174871
$852$	0	0
$853$	23.7499i	0.813183i	0.913610	+	0.406591i	$0.133283\pi$
$853$	23.7499i	0.813183i	−0.913610	+	0.406591i	$0.866717\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 43.0643i	− 1.47105i	−0.677499	−	0.735524i	$-0.736936\pi$
$857$	− 43.0643i	− 1.47105i	0.677499	−	0.735524i	$-0.263064\pi$
$858$	0	0
$859$	25.0591	0.855006	0.427503	−	0.904014i	$-0.359393\pi$
$859$	25.0591	0.855006	0.427503	+	0.904014i	$0.359393\pi$
$860$	0	0
$861$	60.9490	2.07714
$862$	0	0
$863$	− 46.0967i	− 1.56915i	−0.620033	−	0.784575i	$-0.712881\pi$
$863$	− 46.0967i	− 1.56915i	0.620033	−	0.784575i	$-0.287119\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 28.1845i	− 0.957195i
$868$	0	0
$869$	19.7960	0.671535
$870$	0	0
$871$	−84.8044	−2.87349
$872$	0	0
$873$	0.129585i	0.00438578i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 35.1644i	− 1.18742i	−0.804680	−	0.593709i	$-0.797663\pi$
$877$	− 35.1644i	− 1.18742i	0.804680	−	0.593709i	$-0.202337\pi$
$878$	0	0
$879$	−22.5170	−0.759480
$880$	0	0
$881$	−38.8350	−1.30838	−0.654192	−	0.756329i	$-0.726991\pi$
$881$	−38.8350	−1.30838	−0.654192	+	0.756329i	$0.726991\pi$
$882$	0	0
$883$	− 10.5954i	− 0.356562i	−0.983980	−	0.178281i	$-0.942946\pi$
$883$	− 10.5954i	− 0.356562i	0.983980	−	0.178281i	$-0.0570536\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 26.8733i	− 0.902316i	−0.892444	−	0.451158i	$-0.851011\pi$
$887$	− 26.8733i	− 0.902316i	0.892444	−	0.451158i	$-0.148989\pi$
$888$	0	0
$889$	50.3040	1.68714
$890$	0	0
$891$	−21.0435	−0.704984
$892$	0	0
$893$	− 23.9712i	− 0.802165i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 9.85407i	− 0.329018i
$898$	0	0
$899$	50.2099	1.67459
$900$	0	0
$901$	−0.743278	−0.0247622
$902$	0	0
$903$	56.8754i	1.89270i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	58.3210i	1.93652i	0.249949	+	0.968259i	$0.419586\pi$
$907$	58.3210i	1.93652i	−0.249949	+	0.968259i	$0.580414\pi$
$908$	0	0
$909$	2.37748	0.0788561
$910$	0	0
$911$	40.3287	1.33615	0.668074	−	0.744095i	$-0.267119\pi$
$911$	40.3287	1.33615	0.668074	+	0.744095i	$0.267119\pi$
$912$	0	0
$913$	− 3.18045i	− 0.105257i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 18.6071i	− 0.614459i
$918$	0	0
$919$	37.6333	1.24141	0.620704	−	0.784045i	$-0.286847\pi$
$919$	37.6333	1.24141	0.620704	+	0.784045i	$0.286847\pi$
$920$	0	0
$921$	−16.2381	−0.535064
$922$	0	0
$923$	− 80.0559i	− 2.63507i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.84877i	0.159254i
$928$	0	0
$929$	−30.7190	−1.00786	−0.503929	−	0.863745i	$-0.668113\pi$
$929$	−30.7190	−1.00786	−0.503929	+	0.863745i	$0.668113\pi$
$930$	0	0
$931$	18.4731	0.605431
$932$	0	0
$933$	25.6072i	0.838343i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 40.5843i	− 1.32583i	−0.748694	−	0.662915i	$-0.769319\pi$
$937$	− 40.5843i	− 1.32583i	0.748694	−	0.662915i	$-0.230681\pi$
$938$	0	0
$939$	−43.2355	−1.41094
$940$	0	0
$941$	36.5104	1.19020	0.595102	−	0.803650i	$-0.297112\pi$
$941$	36.5104	1.19020	0.595102	+	0.803650i	$0.297112\pi$
$942$	0	0
$943$	− 8.33994i	− 0.271586i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.9681i	1.13631i	0.822921	+	0.568155i	$0.192343\pi$
$947$	34.9681i	1.13631i	−0.822921	+	0.568155i	$0.807657\pi$
$948$	0	0
$949$	−44.8069	−1.45449
$950$	0	0
$951$	24.5176	0.795038
$952$	0	0
$953$	− 35.7164i	− 1.15697i	−0.815694	−	0.578484i	$-0.803645\pi$
$953$	− 35.7164i	− 1.15697i	0.815694	−	0.578484i	$-0.196355\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	23.3371i	0.754382i
$958$	0	0
$959$	−1.85971	−0.0600532
$960$	0	0
$961$	38.3038	1.23561
$962$	0	0
$963$	− 4.91301i	− 0.158320i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 43.0422i	− 1.38414i	−0.721829	−	0.692072i	$-0.756698\pi$
$967$	− 43.0422i	− 1.38414i	0.721829	−	0.692072i	$-0.243302\pi$
$968$	0	0
$969$	−5.01337	−0.161053
$970$	0	0
$971$	−9.76196	−0.313276	−0.156638	−	0.987656i	$-0.550066\pi$
$971$	−9.76196	−0.313276	−0.156638	+	0.987656i	$0.550066\pi$
$972$	0	0
$973$	2.93666i	0.0941449i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.0469i	0.769329i	0.923056	+	0.384665i	$0.125683\pi$
$977$	24.0469i	0.769329i	−0.923056	+	0.384665i	$0.874317\pi$
$978$	0	0
$979$	−22.2754	−0.711926
$980$	0	0
$981$	5.49768	0.175527
$982$	0	0
$983$	19.2418i	0.613717i	0.951755	+	0.306859i	$0.0992778\pi$
$983$	19.2418i	0.613717i	−0.951755	+	0.306859i	$0.900722\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 83.2862i	− 2.65103i
$988$	0	0
$989$	7.78253	0.247470
$990$	0	0
$991$	−38.1944	−1.21328	−0.606642	−	0.794975i	$-0.707484\pi$
$991$	−38.1944	−1.21328	−0.606642	+	0.794975i	$0.707484\pi$
$992$	0	0
$993$	0.496936i	0.0157698i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 41.5981i	− 1.31742i	−0.752395	−	0.658712i	$-0.771101\pi$
$997$	− 41.5981i	− 1.31742i	0.752395	−	0.658712i	$-0.228899\pi$
$998$	0	0
$999$	24.5490	0.776696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.w.4049.3		10
5.2	odd	4		4600.2.a.bd.1.2	✓	5
5.3	odd	4		4600.2.a.bf.1.4	yes	5
5.4	even	2	inner	4600.2.e.w.4049.8		10
20.3	even	4		9200.2.a.ct.1.2		5
20.7	even	4		9200.2.a.cv.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bd.1.2	✓	5	5.2	odd	4
4600.2.a.bf.1.4	yes	5	5.3	odd	4
4600.2.e.w.4049.3		10	1.1	even	1	trivial
4600.2.e.w.4049.8		10	5.4	even	2	inner
9200.2.a.ct.1.2		5	20.3	even	4
9200.2.a.cv.1.4		5	20.7	even	4

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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.83957i q^{3} -3.97272i q^{7} -0.384010 q^{9} +O(q^{10})\)	\(q-1.83957i q^{3} -3.97272i q^{7} -0.384010 q^{9} +2.10339 q^{11} -5.35673i q^{13} +1.29567i q^{17} -2.10339 q^{19} -7.30809 q^{21} +1.00000i q^{23} -4.81229i q^{27} -6.03130 q^{29} -8.32489 q^{31} -3.86933i q^{33} +5.10131i q^{37} -9.85407 q^{39} -8.33994 q^{41} -7.78253i q^{43} +11.3964i q^{47} -8.78253 q^{49} +2.38347 q^{51} +0.573664i q^{53} +3.86933i q^{57} +9.17951 q^{59} +13.5149 q^{61} +1.52557i q^{63} -15.8314i q^{67} +1.83957 q^{69} +14.9449 q^{71} -8.36459i q^{73} -8.35619i q^{77} +9.41149 q^{79} -10.0046 q^{81} -1.51206i q^{83} +11.0950i q^{87} -10.5903 q^{89} -21.2808 q^{91} +15.3142i q^{93} -0.337451i q^{97} -0.807724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{9}+O(q^{10}) \) 10 * q - 6 * q^9	\( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) \) 10 * q - 6 * q^9 - 24 * q^29 - 36 * q^31 - 18 * q^39 - 12 * q^41 - 30 * q^49 - 12 * q^51 + 2 * q^59 + 20 * q^61 + 16 * q^71 - 54 * q^81 - 28 * q^89 - 92 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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