LMFDB - Embedded newform 9248.2.a.bv.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9248,2,Mod(1,9248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9248, 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("9248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9248 = 2^{5} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9248.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:	$73.8456517893$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.12.10455582754471936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 18x^{10} + 122x^{8} - 384x^{6} + 553x^{4} - 294x^{2} + 4 $ x^12 - 18x^10 + 122x^8 - 384x^6 + 553x^4 - 294*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.44664$ of defining polynomial
Character	$\chi$	$=$	9248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.610563 q^{3} -0.408946 q^{5} +3.61232 q^{7} -2.62721 q^{9} +4.04633 q^{11} -3.62721 q^{13} +0.249687 q^{15} -6.83553 q^{19} -2.20555 q^{21} +8.26922 q^{23} -4.83276 q^{25} +3.43577 q^{27} -8.36703 q^{29} +0.176556 q^{31} -2.47054 q^{33} -1.47725 q^{35} +2.41948 q^{37} +2.21464 q^{39} -1.41421 q^{41} -11.0807 q^{43} +1.07439 q^{45} +10.2172 q^{47} +6.04888 q^{49} +1.42166 q^{53} -1.65473 q^{55} +4.17352 q^{57} +5.72237 q^{59} +3.23737 q^{61} -9.49034 q^{63} +1.48333 q^{65} +12.5579 q^{67} -5.04888 q^{69} +9.61584 q^{71} +6.83456 q^{73} +2.95071 q^{75} +14.6167 q^{77} +1.39768 q^{79} +5.78389 q^{81} +7.44931 q^{83} +5.10860 q^{87} +13.0489 q^{89} -13.1027 q^{91} -0.107798 q^{93} +2.79536 q^{95} -7.36176 q^{97} -10.6306 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 20 q^{9} + 8 q^{13} + 32 q^{21} + 52 q^{25} + 8 q^{33} + 28 q^{49} + 24 q^{53} - 16 q^{69} + 4 q^{81} + 112 q^{89} - 32 q^{93}+O(q^{100}) $ 12 * q + 20 * q^9 + 8 * q^13 + 32 * q^21 + 52 * q^25 + 8 * q^33 + 28 * q^49 + 24 * q^53 - 16 * q^69 + 4 * q^81 + 112 * q^89 - 32 * q^93

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
$3$	−0.610563	−0.352509	−0.176254	−	0.984345i	$-0.556398\pi$
$3$	−0.610563	−0.352509	−0.176254	+	0.984345i	$0.556398\pi$
$4$	0	0
$5$	−0.408946	−0.182886	−0.0914431	−	0.995810i	$-0.529148\pi$
$5$	−0.408946	−0.182886	−0.0914431	+	0.995810i	$0.529148\pi$
$6$	0	0
$7$	3.61232	1.36533	0.682665	−	0.730732i	$-0.260821\pi$
$7$	3.61232	1.36533	0.682665	+	0.730732i	$0.260821\pi$
$8$	0	0
$9$	−2.62721	−0.875738
$10$	0	0
$11$	4.04633	1.22001	0.610007	−	0.792396i	$-0.291167\pi$
$11$	4.04633	1.22001	0.610007	+	0.792396i	$0.291167\pi$
$12$	0	0
$13$	−3.62721	−1.00601	−0.503004	−	0.864284i	$-0.667772\pi$
$13$	−3.62721	−1.00601	−0.503004	+	0.864284i	$0.667772\pi$
$14$	0	0
$15$	0.249687	0.0644690
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−6.83553	−1.56818	−0.784089	−	0.620649i	$-0.786869\pi$
$19$	−6.83553	−1.56818	−0.784089	+	0.620649i	$0.786869\pi$
$20$	0	0
$21$	−2.20555	−0.481290
$22$	0	0
$23$	8.26922	1.72425	0.862125	−	0.506695i	$-0.169133\pi$
$23$	8.26922	1.72425	0.862125	+	0.506695i	$0.169133\pi$
$24$	0	0
$25$	−4.83276	−0.966553
$26$	0	0
$27$	3.43577	0.661214
$28$	0	0
$29$	−8.36703	−1.55372	−0.776859	−	0.629675i	$-0.783188\pi$
$29$	−8.36703	−1.55372	−0.776859	+	0.629675i	$0.783188\pi$
$30$	0	0
$31$	0.176556	0.0317103	0.0158552	−	0.999874i	$-0.494953\pi$
$31$	0.176556	0.0317103	0.0158552	+	0.999874i	$0.494953\pi$
$32$	0	0
$33$	−2.47054	−0.430066
$34$	0	0
$35$	−1.47725	−0.249700
$36$	0	0
$37$	2.41948	0.397760	0.198880	−	0.980024i	$-0.436270\pi$
$37$	2.41948	0.397760	0.198880	+	0.980024i	$0.436270\pi$
$38$	0	0
$39$	2.21464	0.354626
$40$	0	0
$41$	−1.41421	−0.220863	−0.110432	−	0.993884i	$-0.535223\pi$
$41$	−1.41421	−0.220863	−0.110432	+	0.993884i	$0.535223\pi$
$42$	0	0
$43$	−11.0807	−1.68978	−0.844892	−	0.534937i	$-0.820336\pi$
$43$	−11.0807	−1.68978	−0.844892	+	0.534937i	$0.820336\pi$
$44$	0	0
$45$	1.07439	0.160160
$46$	0	0
$47$	10.2172	1.49033	0.745165	−	0.666880i	$-0.232371\pi$
$47$	10.2172	1.49033	0.745165	+	0.666880i	$0.232371\pi$
$48$	0	0
$49$	6.04888	0.864125
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.42166	0.195280	0.0976402	−	0.995222i	$-0.468871\pi$
$53$	1.42166	0.195280	0.0976402	+	0.995222i	$0.468871\pi$
$54$	0	0
$55$	−1.65473	−0.223124
$56$	0	0
$57$	4.17352	0.552796
$58$	0	0
$59$	5.72237	0.744990	0.372495	−	0.928034i	$-0.378502\pi$
$59$	5.72237	0.744990	0.372495	+	0.928034i	$0.378502\pi$
$60$	0	0
$61$	3.23737	0.414503	0.207252	−	0.978288i	$-0.433548\pi$
$61$	3.23737	0.414503	0.207252	+	0.978288i	$0.433548\pi$
$62$	0	0
$63$	−9.49034	−1.19567
$64$	0	0
$65$	1.48333	0.183985
$66$	0	0
$67$	12.5579	1.53419	0.767096	−	0.641532i	$-0.221701\pi$
$67$	12.5579	1.53419	0.767096	+	0.641532i	$0.221701\pi$
$68$	0	0
$69$	−5.04888	−0.607813
$70$	0	0
$71$	9.61584	1.14119	0.570595	−	0.821231i	$-0.306713\pi$
$71$	9.61584	1.14119	0.570595	+	0.821231i	$0.306713\pi$
$72$	0	0
$73$	6.83456	0.799925	0.399962	−	0.916532i	$-0.369023\pi$
$73$	6.83456	0.799925	0.399962	+	0.916532i	$0.369023\pi$
$74$	0	0
$75$	2.95071	0.340718
$76$	0	0
$77$	14.6167	1.66572
$78$	0	0
$79$	1.39768	0.157251	0.0786257	−	0.996904i	$-0.474947\pi$
$79$	1.39768	0.157251	0.0786257	+	0.996904i	$0.474947\pi$
$80$	0	0
$81$	5.78389	0.642654
$82$	0	0
$83$	7.44931	0.817668	0.408834	−	0.912609i	$-0.365936\pi$
$83$	7.44931	0.817668	0.408834	+	0.912609i	$0.365936\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.10860	0.547699
$88$	0	0
$89$	13.0489	1.38318	0.691589	−	0.722291i	$-0.256911\pi$
$89$	13.0489	1.38318	0.691589	+	0.722291i	$0.256911\pi$
$90$	0	0
$91$	−13.1027	−1.37353
$92$	0	0
$93$	−0.107798	−0.0111782
$94$	0	0
$95$	2.79536	0.286798
$96$	0	0
$97$	−7.36176	−0.747473	−0.373737	−	0.927535i	$-0.621924\pi$
$97$	−7.36176	−0.747473	−0.373737	+	0.927535i	$0.621924\pi$
$98$	0	0
$99$	−10.6306	−1.06841
$100$	0	0
$101$	−4.37279	−0.435109	−0.217554	−	0.976048i	$-0.569808\pi$
$101$	−4.37279	−0.435109	−0.217554	+	0.976048i	$0.569808\pi$
$102$	0	0
$103$	−7.08522	−0.698127	−0.349064	−	0.937099i	$-0.613500\pi$
$103$	−7.08522	−0.698127	−0.349064	+	0.937099i	$0.613500\pi$
$104$	0	0
$105$	0.901951	0.0880214
$106$	0	0
$107$	−1.83169	−0.177076	−0.0885380	−	0.996073i	$-0.528219\pi$
$107$	−1.83169	−0.177076	−0.0885380	+	0.996073i	$0.528219\pi$
$108$	0	0
$109$	15.5072	1.48532	0.742661	−	0.669668i	$-0.233563\pi$
$109$	15.5072	1.48532	0.742661	+	0.669668i	$0.233563\pi$
$110$	0	0
$111$	−1.47725	−0.140214
$112$	0	0
$113$	−15.8470	−1.49076	−0.745382	−	0.666637i	$-0.767733\pi$
$113$	−15.8470	−1.49076	−0.745382	+	0.666637i	$0.767733\pi$
$114$	0	0
$115$	−3.38166	−0.315342
$116$	0	0
$117$	9.52946	0.880999
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.37279	0.488435
$122$	0	0
$123$	0.863466	0.0778561
$124$	0	0
$125$	4.02107	0.359655
$126$	0	0
$127$	−15.1483	−1.34419	−0.672097	−	0.740463i	$-0.734606\pi$
$127$	−15.1483	−1.34419	−0.672097	+	0.740463i	$0.734606\pi$
$128$	0	0
$129$	6.76544	0.595664
$130$	0	0
$131$	−0.963674	−0.0841966	−0.0420983	−	0.999113i	$-0.513404\pi$
$131$	−0.963674	−0.0841966	−0.0420983	+	0.999113i	$0.513404\pi$
$132$	0	0
$133$	−24.6921	−2.14108
$134$	0	0
$135$	−1.40504	−0.120927
$136$	0	0
$137$	2.47054	0.211072	0.105536	−	0.994415i	$-0.466344\pi$
$137$	2.47054	0.211072	0.105536	+	0.994415i	$0.466344\pi$
$138$	0	0
$139$	4.39944	0.373156	0.186578	−	0.982440i	$-0.440260\pi$
$139$	4.39944	0.373156	0.186578	+	0.982440i	$0.440260\pi$
$140$	0	0
$141$	−6.23824	−0.525354
$142$	0	0
$143$	−14.6769	−1.22734
$144$	0	0
$145$	3.42166	0.284154
$146$	0	0
$147$	−3.69322	−0.304612
$148$	0	0
$149$	−8.84333	−0.724473	−0.362237	−	0.932086i	$-0.617987\pi$
$149$	−8.84333	−0.724473	−0.362237	+	0.932086i	$0.617987\pi$
$150$	0	0
$151$	4.60922	0.375093	0.187547	−	0.982256i	$-0.439946\pi$
$151$	4.60922	0.375093	0.187547	+	0.982256i	$0.439946\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.0722017	−0.00579938
$156$	0	0
$157$	17.2544	1.37705	0.688527	−	0.725211i	$-0.258258\pi$
$157$	17.2544	1.37705	0.688527	+	0.725211i	$0.258258\pi$
$158$	0	0
$159$	−0.868015	−0.0688380
$160$	0	0
$161$	29.8711	2.35417
$162$	0	0
$163$	1.95719	0.153299	0.0766495	−	0.997058i	$-0.475578\pi$
$163$	1.95719	0.153299	0.0766495	+	0.997058i	$0.475578\pi$
$164$	0	0
$165$	1.01032	0.0786531
$166$	0	0
$167$	12.9261	1.00025	0.500126	−	0.865953i	$-0.333287\pi$
$167$	12.9261	1.00025	0.500126	+	0.865953i	$0.333287\pi$
$168$	0	0
$169$	0.156674	0.0120519
$170$	0	0
$171$	17.9584	1.37331
$172$	0	0
$173$	8.36703	0.636133	0.318067	−	0.948068i	$-0.396966\pi$
$173$	8.36703	0.636133	0.318067	+	0.948068i	$0.396966\pi$
$174$	0	0
$175$	−17.4575	−1.31966
$176$	0	0
$177$	−3.49387	−0.262615
$178$	0	0
$179$	11.6944	0.874083	0.437042	−	0.899441i	$-0.356026\pi$
$179$	11.6944	0.874083	0.437042	+	0.899441i	$0.356026\pi$
$180$	0	0
$181$	−7.83983	−0.582730	−0.291365	−	0.956612i	$-0.594109\pi$
$181$	−7.83983	−0.582730	−0.291365	+	0.956612i	$0.594109\pi$
$182$	0	0
$183$	−1.97662	−0.146116
$184$	0	0
$185$	−0.989437	−0.0727449
$186$	0	0
$187$	0	0
$188$	0	0
$189$	12.4111	0.902775
$190$	0	0
$191$	10.0397	0.726448	0.363224	−	0.931702i	$-0.381676\pi$
$191$	10.0397	0.726448	0.363224	+	0.931702i	$0.381676\pi$
$192$	0	0
$193$	24.6230	1.77240	0.886202	−	0.463300i	$-0.153335\pi$
$193$	24.6230	1.77240	0.886202	+	0.463300i	$0.153335\pi$
$194$	0	0
$195$	−0.905669	−0.0648563
$196$	0	0
$197$	−0.699638	−0.0498471	−0.0249236	−	0.999689i	$-0.507934\pi$
$197$	−0.699638	−0.0498471	−0.0249236	+	0.999689i	$0.507934\pi$
$198$	0	0
$199$	22.7185	1.61047	0.805236	−	0.592954i	$-0.202039\pi$
$199$	22.7185	1.61047	0.805236	+	0.592954i	$0.202039\pi$
$200$	0	0
$201$	−7.66739	−0.540816
$202$	0	0
$203$	−30.2244	−2.12134
$204$	0	0
$205$	0.578337	0.0403928
$206$	0	0
$207$	−21.7250	−1.50999
$208$	0	0
$209$	−27.6588	−1.91320
$210$	0	0
$211$	−9.05633	−0.623464	−0.311732	−	0.950170i	$-0.600909\pi$
$211$	−9.05633	−0.623464	−0.311732	+	0.950170i	$0.600909\pi$
$212$	0	0
$213$	−5.87108	−0.402280
$214$	0	0
$215$	4.53139	0.309038
$216$	0	0
$217$	0.637776	0.0432950
$218$	0	0
$219$	−4.17293	−0.281980
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−18.7797	−1.25758	−0.628789	−	0.777576i	$-0.716449\pi$
$223$	−18.7797	−1.25758	−0.628789	+	0.777576i	$0.716449\pi$
$224$	0	0
$225$	12.6967	0.846447
$226$	0	0
$227$	6.48858	0.430662	0.215331	−	0.976541i	$-0.430917\pi$
$227$	6.48858	0.430662	0.215331	+	0.976541i	$0.430917\pi$
$228$	0	0
$229$	−13.3622	−0.883001	−0.441500	−	0.897261i	$-0.645554\pi$
$229$	−13.3622	−0.883001	−0.441500	+	0.897261i	$0.645554\pi$
$230$	0	0
$231$	−8.92438	−0.587181
$232$	0	0
$233$	−4.76984	−0.312483	−0.156241	−	0.987719i	$-0.549938\pi$
$233$	−4.76984	−0.312483	−0.156241	+	0.987719i	$0.549938\pi$
$234$	0	0
$235$	−4.17828	−0.272561
$236$	0	0
$237$	−0.853372	−0.0554325
$238$	0	0
$239$	−3.13198	−0.202591	−0.101295	−	0.994856i	$-0.532299\pi$
$239$	−3.13198	−0.202591	−0.101295	+	0.994856i	$0.532299\pi$
$240$	0	0
$241$	28.2693	1.82099	0.910494	−	0.413522i	$-0.135702\pi$
$241$	28.2693	1.82099	0.910494	+	0.413522i	$0.135702\pi$
$242$	0	0
$243$	−13.8387	−0.887755
$244$	0	0
$245$	−2.47366	−0.158037
$246$	0	0
$247$	24.7939	1.57760
$248$	0	0
$249$	−4.54827	−0.288235
$250$	0	0
$251$	−13.9629	−0.881333	−0.440667	−	0.897671i	$-0.645258\pi$
$251$	−13.9629	−0.881333	−0.440667	+	0.897671i	$0.645258\pi$
$252$	0	0
$253$	33.4600	2.10361
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.31386	0.206713	0.103357	−	0.994644i	$-0.467042\pi$
$257$	3.31386	0.206713	0.103357	+	0.994644i	$0.467042\pi$
$258$	0	0
$259$	8.73995	0.543074
$260$	0	0
$261$	21.9820	1.36065
$262$	0	0
$263$	2.88229	0.177730	0.0888648	−	0.996044i	$-0.471676\pi$
$263$	2.88229	0.177730	0.0888648	+	0.996044i	$0.471676\pi$
$264$	0	0
$265$	−0.581383	−0.0357141
$266$	0	0
$267$	−7.96716	−0.487582
$268$	0	0
$269$	15.2707	0.931071	0.465536	−	0.885029i	$-0.345862\pi$
$269$	15.2707	0.931071	0.465536	+	0.885029i	$0.345862\pi$
$270$	0	0
$271$	3.13198	0.190254	0.0951270	−	0.995465i	$-0.469674\pi$
$271$	3.13198	0.190254	0.0951270	+	0.995465i	$0.469674\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	−19.5550	−1.17921
$276$	0	0
$277$	17.8084	1.07001	0.535003	−	0.844850i	$-0.320311\pi$
$277$	17.8084	1.07001	0.535003	+	0.844850i	$0.320311\pi$
$278$	0	0
$279$	−0.463849	−0.0277699
$280$	0	0
$281$	17.8328	1.06381	0.531907	−	0.846803i	$-0.321476\pi$
$281$	17.8328	1.06381	0.531907	+	0.846803i	$0.321476\pi$
$282$	0	0
$283$	−12.8452	−0.763569	−0.381784	−	0.924251i	$-0.624690\pi$
$283$	−12.8452	−0.763569	−0.381784	+	0.924251i	$0.624690\pi$
$284$	0	0
$285$	−1.70674	−0.101099
$286$	0	0
$287$	−5.10860	−0.301551
$288$	0	0
$289$	0	0
$290$	0	0
$291$	4.49482	0.263491
$292$	0	0
$293$	8.09775	0.473076	0.236538	−	0.971622i	$-0.423987\pi$
$293$	8.09775	0.473076	0.236538	+	0.971622i	$0.423987\pi$
$294$	0	0
$295$	−2.34014	−0.136248
$296$	0	0
$297$	13.9022	0.806690
$298$	0	0
$299$	−29.9942	−1.73461
$300$	0	0
$301$	−40.0269	−2.30711
$302$	0	0
$303$	2.66986	0.153380
$304$	0	0
$305$	−1.32391	−0.0758069
$306$	0	0
$307$	−28.2478	−1.61219	−0.806093	−	0.591789i	$-0.798422\pi$
$307$	−28.2478	−1.61219	−0.806093	+	0.591789i	$0.798422\pi$
$308$	0	0
$309$	4.32597	0.246096
$310$	0	0
$311$	20.1508	1.14264	0.571322	−	0.820726i	$-0.306431\pi$
$311$	20.1508	1.14264	0.571322	+	0.820726i	$0.306431\pi$
$312$	0	0
$313$	−26.6335	−1.50542	−0.752709	−	0.658354i	$-0.771253\pi$
$313$	−26.6335	−1.50542	−0.752709	+	0.658354i	$0.771253\pi$
$314$	0	0
$315$	3.88104	0.218672
$316$	0	0
$317$	−4.95722	−0.278425	−0.139213	−	0.990263i	$-0.544457\pi$
$317$	−4.95722	−0.278425	−0.139213	+	0.990263i	$0.544457\pi$
$318$	0	0
$319$	−33.8558	−1.89556
$320$	0	0
$321$	1.11836	0.0624208
$322$	0	0
$323$	0	0
$324$	0	0
$325$	17.5295	0.972360
$326$	0	0
$327$	−9.46813	−0.523589
$328$	0	0
$329$	36.9078	2.03479
$330$	0	0
$331$	10.5813	0.581600	0.290800	−	0.956784i	$-0.406079\pi$
$331$	10.5813	0.581600	0.290800	+	0.956784i	$0.406079\pi$
$332$	0	0
$333$	−6.35649	−0.348334
$334$	0	0
$335$	−5.13551	−0.280583
$336$	0	0
$337$	1.03946	0.0566232	0.0283116	−	0.999599i	$-0.490987\pi$
$337$	1.03946	0.0566232	0.0283116	+	0.999599i	$0.490987\pi$
$338$	0	0
$339$	9.67561	0.525507
$340$	0	0
$341$	0.714402	0.0386870
$342$	0	0
$343$	−3.43577	−0.185514
$344$	0	0
$345$	2.06472	0.111161
$346$	0	0
$347$	−4.04633	−0.217218	−0.108609	−	0.994085i	$-0.534640\pi$
$347$	−4.04633	−0.217218	−0.108609	+	0.994085i	$0.534640\pi$
$348$	0	0
$349$	6.57834	0.352130	0.176065	−	0.984379i	$-0.443663\pi$
$349$	6.57834	0.352130	0.176065	+	0.984379i	$0.443663\pi$
$350$	0	0
$351$	−12.4623	−0.665186
$352$	0	0
$353$	9.73501	0.518142	0.259071	−	0.965858i	$-0.416584\pi$
$353$	9.73501	0.518142	0.259071	+	0.965858i	$0.416584\pi$
$354$	0	0
$355$	−3.93236	−0.208708
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.2244	1.59518	0.797591	−	0.603198i	$-0.206107\pi$
$359$	30.2244	1.59518	0.797591	+	0.603198i	$0.206107\pi$
$360$	0	0
$361$	27.7244	1.45918
$362$	0	0
$363$	−3.28042	−0.172178
$364$	0	0
$365$	−2.79497	−0.146295
$366$	0	0
$367$	21.3719	1.11560	0.557802	−	0.829974i	$-0.311645\pi$
$367$	21.3719	1.11560	0.557802	+	0.829974i	$0.311645\pi$
$368$	0	0
$369$	3.71544	0.193418
$370$	0	0
$371$	5.13551	0.266622
$372$	0	0
$373$	−30.5472	−1.58167	−0.790836	−	0.612028i	$-0.790354\pi$
$373$	−30.5472	−1.58167	−0.790836	+	0.612028i	$0.790354\pi$
$374$	0	0
$375$	−2.45512	−0.126782
$376$	0	0
$377$	30.3490	1.56305
$378$	0	0
$379$	21.9314	1.12654	0.563270	−	0.826273i	$-0.309543\pi$
$379$	21.9314	1.12654	0.563270	+	0.826273i	$0.309543\pi$
$380$	0	0
$381$	9.24899	0.473840
$382$	0	0
$383$	−25.0436	−1.27967	−0.639834	−	0.768513i	$-0.720997\pi$
$383$	−25.0436	−1.27967	−0.639834	+	0.768513i	$0.720997\pi$
$384$	0	0
$385$	−5.97742	−0.304638
$386$	0	0
$387$	29.1112	1.47981
$388$	0	0
$389$	18.5189	0.938945	0.469473	−	0.882947i	$-0.344444\pi$
$389$	18.5189	0.938945	0.469473	+	0.882947i	$0.344444\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0.588384	0.0296800
$394$	0	0
$395$	−0.571576	−0.0287591
$396$	0	0
$397$	−23.7560	−1.19228	−0.596139	−	0.802881i	$-0.703299\pi$
$397$	−23.7560	−1.19228	−0.596139	+	0.802881i	$0.703299\pi$
$398$	0	0
$399$	15.0761	0.754749
$400$	0	0
$401$	10.8556	0.542104	0.271052	−	0.962565i	$-0.412628\pi$
$401$	10.8556	0.542104	0.271052	+	0.962565i	$0.412628\pi$
$402$	0	0
$403$	−0.640405	−0.0319008
$404$	0	0
$405$	−2.36530	−0.117533
$406$	0	0
$407$	9.79002	0.485273
$408$	0	0
$409$	1.32391	0.0654632	0.0327316	−	0.999464i	$-0.489579\pi$
$409$	1.32391	0.0654632	0.0327316	+	0.999464i	$0.489579\pi$
$410$	0	0
$411$	−1.50842	−0.0744048
$412$	0	0
$413$	20.6711	1.01716
$414$	0	0
$415$	−3.04637	−0.149540
$416$	0	0
$417$	−2.68614	−0.131541
$418$	0	0
$419$	16.7959	0.820533	0.410266	−	0.911966i	$-0.365436\pi$
$419$	16.7959	0.820533	0.410266	+	0.911966i	$0.365436\pi$
$420$	0	0
$421$	−28.9794	−1.41237	−0.706185	−	0.708028i	$-0.749585\pi$
$421$	−28.9794	−1.41237	−0.706185	+	0.708028i	$0.749585\pi$
$422$	0	0
$423$	−26.8427	−1.30514
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11.6944	0.565933
$428$	0	0
$429$	8.96117	0.432649
$430$	0	0
$431$	−17.9361	−0.863952	−0.431976	−	0.901885i	$-0.642184\pi$
$431$	−17.9361	−0.863952	−0.431976	+	0.901885i	$0.642184\pi$
$432$	0	0
$433$	5.69670	0.273766	0.136883	−	0.990587i	$-0.456292\pi$
$433$	5.69670	0.273766	0.136883	+	0.990587i	$0.456292\pi$
$434$	0	0
$435$	−2.08914	−0.100167
$436$	0	0
$437$	−56.5245	−2.70393
$438$	0	0
$439$	4.48034	0.213835	0.106917	−	0.994268i	$-0.465902\pi$
$439$	4.48034	0.213835	0.106917	+	0.994268i	$0.465902\pi$
$440$	0	0
$441$	−15.8917	−0.756747
$442$	0	0
$443$	7.52151	0.357358	0.178679	−	0.983907i	$-0.442818\pi$
$443$	7.52151	0.357358	0.178679	+	0.983907i	$0.442818\pi$
$444$	0	0
$445$	−5.33629	−0.252964
$446$	0	0
$447$	5.39941	0.255383
$448$	0	0
$449$	19.3951	0.915311	0.457656	−	0.889130i	$-0.348689\pi$
$449$	19.3951	0.915311	0.457656	+	0.889130i	$0.348689\pi$
$450$	0	0
$451$	−5.72237	−0.269456
$452$	0	0
$453$	−2.81422	−0.132224
$454$	0	0
$455$	5.35828	0.251200
$456$	0	0
$457$	7.45998	0.348963	0.174481	−	0.984660i	$-0.444175\pi$
$457$	7.45998	0.348963	0.174481	+	0.984660i	$0.444175\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18.2439	−0.849701	−0.424851	−	0.905263i	$-0.639673\pi$
$461$	−18.2439	−0.849701	−0.424851	+	0.905263i	$0.639673\pi$
$462$	0	0
$463$	25.2933	1.17548	0.587739	−	0.809050i	$-0.300018\pi$
$463$	25.2933	1.17548	0.587739	+	0.809050i	$0.300018\pi$
$464$	0	0
$465$	0.0440837	0.00204433
$466$	0	0
$467$	−5.28608	−0.244611	−0.122305	−	0.992493i	$-0.539029\pi$
$467$	−5.28608	−0.244611	−0.122305	+	0.992493i	$0.539029\pi$
$468$	0	0
$469$	45.3632	2.09468
$470$	0	0
$471$	−10.5349	−0.485423
$472$	0	0
$473$	−44.8360	−2.06156
$474$	0	0
$475$	33.0345	1.51573
$476$	0	0
$477$	−3.73501	−0.171014
$478$	0	0
$479$	22.8206	1.04270	0.521350	−	0.853343i	$-0.325429\pi$
$479$	22.8206	1.04270	0.521350	+	0.853343i	$0.325429\pi$
$480$	0	0
$481$	−8.77597	−0.400150
$482$	0	0
$483$	−18.2382	−0.829865
$484$	0	0
$485$	3.01056	0.136703
$486$	0	0
$487$	−8.90962	−0.403733	−0.201867	−	0.979413i	$-0.564701\pi$
$487$	−8.90962	−0.403733	−0.201867	+	0.979413i	$0.564701\pi$
$488$	0	0
$489$	−1.19499	−0.0540392
$490$	0	0
$491$	−27.2699	−1.23067	−0.615337	−	0.788264i	$-0.710980\pi$
$491$	−27.2699	−1.23067	−0.615337	+	0.788264i	$0.710980\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.34733	0.195398
$496$	0	0
$497$	34.7355	1.55810
$498$	0	0
$499$	23.6674	1.05950	0.529750	−	0.848154i	$-0.322286\pi$
$499$	23.6674	1.05950	0.529750	+	0.848154i	$0.322286\pi$
$500$	0	0
$501$	−7.89220	−0.352598
$502$	0	0
$503$	16.9426	0.755433	0.377717	−	0.925921i	$-0.376709\pi$
$503$	16.9426	0.755433	0.377717	+	0.925921i	$0.376709\pi$
$504$	0	0
$505$	1.78823	0.0795754
$506$	0	0
$507$	−0.0956594	−0.00424838
$508$	0	0
$509$	−30.0766	−1.33312	−0.666562	−	0.745450i	$-0.732235\pi$
$509$	−30.0766	−1.33312	−0.666562	+	0.745450i	$0.732235\pi$
$510$	0	0
$511$	24.6886	1.09216
$512$	0	0
$513$	−23.4853	−1.03690
$514$	0	0
$515$	2.89747	0.127678
$516$	0	0
$517$	41.3421	1.81822
$518$	0	0
$519$	−5.10860	−0.224242
$520$	0	0
$521$	−26.0223	−1.14006	−0.570029	−	0.821625i	$-0.693068\pi$
$521$	−26.0223	−1.14006	−0.570029	+	0.821625i	$0.693068\pi$
$522$	0	0
$523$	17.4168	0.761584	0.380792	−	0.924661i	$-0.375651\pi$
$523$	17.4168	0.761584	0.380792	+	0.924661i	$0.375651\pi$
$524$	0	0
$525$	10.6589	0.465193
$526$	0	0
$527$	0	0
$528$	0	0
$529$	45.3799	1.97304
$530$	0	0
$531$	−15.0339	−0.652416
$532$	0	0
$533$	5.12965	0.222190
$534$	0	0
$535$	0.749062	0.0323848
$536$	0	0
$537$	−7.14019	−0.308122
$538$	0	0
$539$	24.4757	1.05425
$540$	0	0
$541$	−18.5722	−0.798479	−0.399240	−	0.916847i	$-0.630726\pi$
$541$	−18.5722	−0.798479	−0.399240	+	0.916847i	$0.630726\pi$
$542$	0	0
$543$	4.78671	0.205417
$544$	0	0
$545$	−6.34162	−0.271645
$546$	0	0
$547$	22.9907	0.983012	0.491506	−	0.870874i	$-0.336447\pi$
$547$	22.9907	0.983012	0.491506	+	0.870874i	$0.336447\pi$
$548$	0	0
$549$	−8.50527	−0.362996
$550$	0	0
$551$	57.1931	2.43651
$552$	0	0
$553$	5.04888	0.214700
$554$	0	0
$555$	0.604114	0.0256432
$556$	0	0
$557$	−11.6272	−0.492661	−0.246330	−	0.969186i	$-0.579225\pi$
$557$	−11.6272	−0.492661	−0.246330	+	0.969186i	$0.579225\pi$
$558$	0	0
$559$	40.1919	1.69994
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.0703	−0.845862	−0.422931	−	0.906162i	$-0.638999\pi$
$563$	−20.0703	−0.845862	−0.422931	+	0.906162i	$0.638999\pi$
$564$	0	0
$565$	6.48059	0.272640
$566$	0	0
$567$	20.8933	0.877435
$568$	0	0
$569$	31.2544	1.31025	0.655127	−	0.755519i	$-0.272615\pi$
$569$	31.2544	1.31025	0.655127	+	0.755519i	$0.272615\pi$
$570$	0	0
$571$	35.1596	1.47138	0.735691	−	0.677318i	$-0.236858\pi$
$571$	35.1596	1.47138	0.735691	+	0.677318i	$0.236858\pi$
$572$	0	0
$573$	−6.12987	−0.256079
$574$	0	0
$575$	−39.9632	−1.66658
$576$	0	0
$577$	21.3139	0.887308	0.443654	−	0.896198i	$-0.353682\pi$
$577$	21.3139	0.887308	0.443654	+	0.896198i	$0.353682\pi$
$578$	0	0
$579$	−15.0339	−0.624787
$580$	0	0
$581$	26.9093	1.11639
$582$	0	0
$583$	5.75252	0.238245
$584$	0	0
$585$	−3.89704	−0.161123
$586$	0	0
$587$	15.1483	0.625237	0.312619	−	0.949879i	$-0.398794\pi$
$587$	15.1483	0.625237	0.312619	+	0.949879i	$0.398794\pi$
$588$	0	0
$589$	−1.20685	−0.0497274
$590$	0	0
$591$	0.427173	0.0175715
$592$	0	0
$593$	47.5466	1.95251	0.976253	−	0.216632i	$-0.0695070\pi$
$593$	47.5466	1.95251	0.976253	+	0.216632i	$0.0695070\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13.8711	−0.567705
$598$	0	0
$599$	−5.85766	−0.239337	−0.119669	−	0.992814i	$-0.538183\pi$
$599$	−5.85766	−0.239337	−0.119669	+	0.992814i	$0.538183\pi$
$600$	0	0
$601$	13.6841	0.558184	0.279092	−	0.960264i	$-0.409967\pi$
$601$	13.6841	0.558184	0.279092	+	0.960264i	$0.409967\pi$
$602$	0	0
$603$	−32.9923	−1.34355
$604$	0	0
$605$	−2.19718	−0.0893281
$606$	0	0
$607$	11.5795	0.469997	0.234998	−	0.971996i	$-0.424492\pi$
$607$	11.5795	0.469997	0.234998	+	0.971996i	$0.424492\pi$
$608$	0	0
$609$	18.4539	0.747790
$610$	0	0
$611$	−37.0599	−1.49928
$612$	0	0
$613$	−36.1744	−1.46107	−0.730535	−	0.682876i	$-0.760729\pi$
$613$	−36.1744	−1.46107	−0.730535	+	0.682876i	$0.760729\pi$
$614$	0	0
$615$	−0.353111	−0.0142388
$616$	0	0
$617$	−8.41616	−0.338822	−0.169411	−	0.985546i	$-0.554187\pi$
$617$	−8.41616	−0.338822	−0.169411	+	0.985546i	$0.554187\pi$
$618$	0	0
$619$	39.9186	1.60446	0.802231	−	0.597014i	$-0.203646\pi$
$619$	39.9186	1.60446	0.802231	+	0.597014i	$0.203646\pi$
$620$	0	0
$621$	28.4111	1.14010
$622$	0	0
$623$	47.1368	1.88849
$624$	0	0
$625$	22.5194	0.900777
$626$	0	0
$627$	16.8874	0.674419
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−24.8661	−0.989905	−0.494952	−	0.868920i	$-0.664814\pi$
$631$	−24.8661	−0.989905	−0.494952	+	0.868920i	$0.664814\pi$
$632$	0	0
$633$	5.52946	0.219776
$634$	0	0
$635$	6.19484	0.245835
$636$	0	0
$637$	−21.9406	−0.869317
$638$	0	0
$639$	−25.2629	−0.999384
$640$	0	0
$641$	16.8032	0.663686	0.331843	−	0.943335i	$-0.392330\pi$
$641$	16.8032	0.663686	0.331843	+	0.943335i	$0.392330\pi$
$642$	0	0
$643$	26.4628	1.04359	0.521795	−	0.853071i	$-0.325263\pi$
$643$	26.4628	1.04359	0.521795	+	0.853071i	$0.325263\pi$
$644$	0	0
$645$	−2.76670	−0.108939
$646$	0	0
$647$	36.7380	1.44432	0.722161	−	0.691725i	$-0.243149\pi$
$647$	36.7380	1.44432	0.722161	+	0.691725i	$0.243149\pi$
$648$	0	0
$649$	23.1546	0.908898
$650$	0	0
$651$	−0.389402	−0.0152619
$652$	0	0
$653$	28.5408	1.11689	0.558443	−	0.829543i	$-0.311399\pi$
$653$	28.5408	1.11689	0.558443	+	0.829543i	$0.311399\pi$
$654$	0	0
$655$	0.394091	0.0153984
$656$	0	0
$657$	−17.9558	−0.700524
$658$	0	0
$659$	6.76333	0.263462	0.131731	−	0.991286i	$-0.457947\pi$
$659$	6.76333	0.263462	0.131731	+	0.991286i	$0.457947\pi$
$660$	0	0
$661$	−16.1461	−0.628011	−0.314005	−	0.949421i	$-0.601671\pi$
$661$	−16.1461	−0.628011	−0.314005	+	0.949421i	$0.601671\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.0978	0.391574
$666$	0	0
$667$	−69.1888	−2.67900
$668$	0	0
$669$	11.4662	0.443307
$670$	0	0
$671$	13.0995	0.505700
$672$	0	0
$673$	−13.9747	−0.538687	−0.269343	−	0.963044i	$-0.586807\pi$
$673$	−13.9747	−0.538687	−0.269343	+	0.963044i	$0.586807\pi$
$674$	0	0
$675$	−16.6042	−0.639098
$676$	0	0
$677$	−19.0695	−0.732900	−0.366450	−	0.930438i	$-0.619427\pi$
$677$	−19.0695	−0.732900	−0.366450	+	0.930438i	$0.619427\pi$
$678$	0	0
$679$	−26.5931	−1.02055
$680$	0	0
$681$	−3.96169	−0.151812
$682$	0	0
$683$	17.1490	0.656188	0.328094	−	0.944645i	$-0.393594\pi$
$683$	17.1490	0.656188	0.328094	+	0.944645i	$0.393594\pi$
$684$	0	0
$685$	−1.01032	−0.0386022
$686$	0	0
$687$	8.15848	0.311265
$688$	0	0
$689$	−5.15667	−0.196454
$690$	0	0
$691$	6.48858	0.246837	0.123419	−	0.992355i	$-0.460614\pi$
$691$	6.48858	0.246837	0.123419	+	0.992355i	$0.460614\pi$
$692$	0	0
$693$	−38.4011	−1.45874
$694$	0	0
$695$	−1.79913	−0.0682451
$696$	0	0
$697$	0	0
$698$	0	0
$699$	2.91229	0.110153
$700$	0	0
$701$	−21.9406	−0.828684	−0.414342	−	0.910121i	$-0.635988\pi$
$701$	−21.9406	−0.828684	−0.414342	+	0.910121i	$0.635988\pi$
$702$	0	0
$703$	−16.5384	−0.623759
$704$	0	0
$705$	2.55110	0.0960801
$706$	0	0
$707$	−15.7959	−0.594067
$708$	0	0
$709$	−28.6092	−1.07444	−0.537220	−	0.843442i	$-0.680525\pi$
$709$	−28.6092	−1.07444	−0.537220	+	0.843442i	$0.680525\pi$
$710$	0	0
$711$	−3.67201	−0.137711
$712$	0	0
$713$	1.45998	0.0546765
$714$	0	0
$715$	6.00206	0.224464
$716$	0	0
$717$	1.91227	0.0714150
$718$	0	0
$719$	−50.7257	−1.89175	−0.945874	−	0.324533i	$-0.894793\pi$
$719$	−50.7257	−1.89175	−0.945874	+	0.324533i	$0.894793\pi$
$720$	0	0
$721$	−25.5941	−0.953174
$722$	0	0
$723$	−17.2602	−0.641914
$724$	0	0
$725$	40.4359	1.50175
$726$	0	0
$727$	6.40836	0.237673	0.118836	−	0.992914i	$-0.462084\pi$
$727$	6.40836	0.237673	0.118836	+	0.992914i	$0.462084\pi$
$728$	0	0
$729$	−8.90225	−0.329713
$730$	0	0
$731$	0	0
$732$	0	0
$733$	41.4983	1.53277	0.766387	−	0.642379i	$-0.222052\pi$
$733$	41.4983	1.53277	0.766387	+	0.642379i	$0.222052\pi$
$734$	0	0
$735$	1.51033	0.0557093
$736$	0	0
$737$	50.8134	1.87174
$738$	0	0
$739$	11.6944	0.430187	0.215093	−	0.976593i	$-0.430994\pi$
$739$	11.6944	0.430187	0.215093	+	0.976593i	$0.430994\pi$
$740$	0	0
$741$	−15.1382	−0.556117
$742$	0	0
$743$	−33.5044	−1.22916	−0.614579	−	0.788855i	$-0.710674\pi$
$743$	−33.5044	−1.22916	−0.614579	+	0.788855i	$0.710674\pi$
$744$	0	0
$745$	3.61644	0.132496
$746$	0	0
$747$	−19.5709	−0.716062
$748$	0	0
$749$	−6.61665	−0.241767
$750$	0	0
$751$	−7.27570	−0.265494	−0.132747	−	0.991150i	$-0.542380\pi$
$751$	−7.27570	−0.265494	−0.132747	+	0.991150i	$0.542380\pi$
$752$	0	0
$753$	8.52526	0.310678
$754$	0	0
$755$	−1.88492	−0.0685994
$756$	0	0
$757$	51.3210	1.86529	0.932647	−	0.360791i	$-0.117493\pi$
$757$	51.3210	1.86529	0.932647	+	0.360791i	$0.117493\pi$
$758$	0	0
$759$	−20.4294	−0.741541
$760$	0	0
$761$	2.68614	0.0973723	0.0486862	−	0.998814i	$-0.484497\pi$
$761$	2.68614	0.0973723	0.0486862	+	0.998814i	$0.484497\pi$
$762$	0	0
$763$	56.0171	2.02795
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−20.7563	−0.749466
$768$	0	0
$769$	14.0594	0.506996	0.253498	−	0.967336i	$-0.418419\pi$
$769$	14.0594	0.506996	0.253498	+	0.967336i	$0.418419\pi$
$770$	0	0
$771$	−2.02332	−0.0728682
$772$	0	0
$773$	−31.6555	−1.13857	−0.569284	−	0.822141i	$-0.692780\pi$
$773$	−31.6555	−1.13857	−0.569284	+	0.822141i	$0.692780\pi$
$774$	0	0
$775$	−0.853251	−0.0306497
$776$	0	0
$777$	−5.33629	−0.191438
$778$	0	0
$779$	9.66690	0.346353
$780$	0	0
$781$	38.9089	1.39227
$782$	0	0
$783$	−28.7472	−1.02734
$784$	0	0
$785$	−7.05613	−0.251844
$786$	0	0
$787$	40.0207	1.42658	0.713291	−	0.700868i	$-0.247204\pi$
$787$	40.0207	1.42658	0.713291	+	0.700868i	$0.247204\pi$
$788$	0	0
$789$	−1.75982	−0.0626512
$790$	0	0
$791$	−57.2446	−2.03538
$792$	0	0
$793$	−11.7426	−0.416993
$794$	0	0
$795$	0.354971	0.0125895
$796$	0	0
$797$	32.6761	1.15745	0.578723	−	0.815524i	$-0.303551\pi$
$797$	32.6761	1.15745	0.578723	+	0.815524i	$0.303551\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−34.2822	−1.21130
$802$	0	0
$803$	27.6549	0.975920
$804$	0	0
$805$	−12.2157	−0.430545
$806$	0	0
$807$	−9.32373	−0.328211
$808$	0	0
$809$	−4.97647	−0.174964	−0.0874818	−	0.996166i	$-0.527882\pi$
$809$	−4.97647	−0.174964	−0.0874818	+	0.996166i	$0.527882\pi$
$810$	0	0
$811$	22.5121	0.790507	0.395254	−	0.918572i	$-0.370657\pi$
$811$	22.5121	0.790507	0.395254	+	0.918572i	$0.370657\pi$
$812$	0	0
$813$	−1.91227	−0.0670662
$814$	0	0
$815$	−0.800385	−0.0280363
$816$	0	0
$817$	75.7422	2.64988
$818$	0	0
$819$	34.4235	1.20285
$820$	0	0
$821$	11.7768	0.411014	0.205507	−	0.978656i	$-0.434116\pi$
$821$	11.7768	0.411014	0.205507	+	0.978656i	$0.434116\pi$
$822$	0	0
$823$	−48.2176	−1.68076	−0.840380	−	0.541997i	$-0.817668\pi$
$823$	−48.2176	−1.68076	−0.840380	+	0.541997i	$0.817668\pi$
$824$	0	0
$825$	11.9395	0.415681
$826$	0	0
$827$	−53.6253	−1.86474	−0.932368	−	0.361511i	$-0.882261\pi$
$827$	−53.6253	−1.86474	−0.932368	+	0.361511i	$0.882261\pi$
$828$	0	0
$829$	4.91995	0.170877	0.0854385	−	0.996343i	$-0.472771\pi$
$829$	4.91995	0.170877	0.0854385	+	0.996343i	$0.472771\pi$
$830$	0	0
$831$	−10.8732	−0.377186
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5.28608	−0.182932
$836$	0	0
$837$	0.606604	0.0209673
$838$	0	0
$839$	−12.9261	−0.446259	−0.223129	−	0.974789i	$-0.571627\pi$
$839$	−12.9261	−0.446259	−0.223129	+	0.974789i	$0.571627\pi$
$840$	0	0
$841$	41.0071	1.41404
$842$	0	0
$843$	−10.8880	−0.375003
$844$	0	0
$845$	−0.0640713	−0.00220412
$846$	0	0
$847$	19.4082	0.666875
$848$	0	0
$849$	7.84281	0.269165
$850$	0	0
$851$	20.0072	0.685838
$852$	0	0
$853$	30.0241	1.02801	0.514003	−	0.857788i	$-0.328162\pi$
$853$	30.0241	1.02801	0.514003	+	0.857788i	$0.328162\pi$
$854$	0	0
$855$	−7.34401	−0.251160
$856$	0	0
$857$	−35.4637	−1.21142	−0.605709	−	0.795687i	$-0.707110\pi$
$857$	−35.4637	−1.21142	−0.605709	+	0.795687i	$0.707110\pi$
$858$	0	0
$859$	39.9722	1.36383	0.681917	−	0.731429i	$-0.261146\pi$
$859$	39.9722	1.36383	0.681917	+	0.731429i	$0.261146\pi$
$860$	0	0
$861$	3.11912	0.106299
$862$	0	0
$863$	28.7472	0.978565	0.489282	−	0.872125i	$-0.337259\pi$
$863$	28.7472	0.978565	0.489282	+	0.872125i	$0.337259\pi$
$864$	0	0
$865$	−3.42166	−0.116340
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.65548	0.191849
$870$	0	0
$871$	−45.5502	−1.54341
$872$	0	0
$873$	19.3409	0.654591
$874$	0	0
$875$	14.5254	0.491048
$876$	0	0
$877$	−5.76090	−0.194532	−0.0972659	−	0.995258i	$-0.531010\pi$
$877$	−5.76090	−0.194532	−0.0972659	+	0.995258i	$0.531010\pi$
$878$	0	0
$879$	−4.94419	−0.166763
$880$	0	0
$881$	12.4230	0.418542	0.209271	−	0.977858i	$-0.432891\pi$
$881$	12.4230	0.418542	0.209271	+	0.977858i	$0.432891\pi$
$882$	0	0
$883$	23.4520	0.789221	0.394611	−	0.918848i	$-0.370880\pi$
$883$	23.4520	0.789221	0.394611	+	0.918848i	$0.370880\pi$
$884$	0	0
$885$	1.42880	0.0480287
$886$	0	0
$887$	3.73782	0.125504	0.0627519	−	0.998029i	$-0.480012\pi$
$887$	3.73782	0.125504	0.0627519	+	0.998029i	$0.480012\pi$
$888$	0	0
$889$	−54.7206	−1.83527
$890$	0	0
$891$	23.4035	0.784047
$892$	0	0
$893$	−69.8399	−2.33710
$894$	0	0
$895$	−4.78239	−0.159858
$896$	0	0
$897$	18.3133	0.611465
$898$	0	0
$899$	−1.47725	−0.0492689
$900$	0	0
$901$	0	0
$902$	0	0
$903$	24.4389	0.813277
$904$	0	0
$905$	3.20607	0.106573
$906$	0	0
$907$	8.22461	0.273094	0.136547	−	0.990634i	$-0.456400\pi$
$907$	8.22461	0.273094	0.136547	+	0.990634i	$0.456400\pi$
$908$	0	0
$909$	11.4882	0.381041
$910$	0	0
$911$	−25.7415	−0.852853	−0.426427	−	0.904522i	$-0.640228\pi$
$911$	−25.7415	−0.852853	−0.426427	+	0.904522i	$0.640228\pi$
$912$	0	0
$913$	30.1424	0.997566
$914$	0	0
$915$	0.808331	0.0267226
$916$	0	0
$917$	−3.48110	−0.114956
$918$	0	0
$919$	3.95324	0.130405	0.0652027	−	0.997872i	$-0.479231\pi$
$919$	3.95324	0.130405	0.0652027	+	0.997872i	$0.479231\pi$
$920$	0	0
$921$	17.2470	0.568310
$922$	0	0
$923$	−34.8787	−1.14805
$924$	0	0
$925$	−11.6928	−0.384456
$926$	0	0
$927$	18.6144	0.611376
$928$	0	0
$929$	−22.8490	−0.749651	−0.374825	−	0.927095i	$-0.622297\pi$
$929$	−22.8490	−0.749651	−0.374825	+	0.927095i	$0.622297\pi$
$930$	0	0
$931$	−41.3473	−1.35510
$932$	0	0
$933$	−12.3033	−0.402792
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.4394	−0.602388	−0.301194	−	0.953563i	$-0.597385\pi$
$937$	−18.4394	−0.602388	−0.301194	+	0.953563i	$0.597385\pi$
$938$	0	0
$939$	16.2615	0.530673
$940$	0	0
$941$	−6.01162	−0.195973	−0.0979866	−	0.995188i	$-0.531240\pi$
$941$	−6.01162	−0.195973	−0.0979866	+	0.995188i	$0.531240\pi$
$942$	0	0
$943$	−11.6944	−0.380823
$944$	0	0
$945$	−5.07547	−0.165105
$946$	0	0
$947$	−2.11898	−0.0688577	−0.0344288	−	0.999407i	$-0.510961\pi$
$947$	−2.11898	−0.0688577	−0.0344288	+	0.999407i	$0.510961\pi$
$948$	0	0
$949$	−24.7904	−0.804730
$950$	0	0
$951$	3.02669	0.0981472
$952$	0	0
$953$	13.1255	0.425177	0.212588	−	0.977142i	$-0.431811\pi$
$953$	13.1255	0.425177	0.212588	+	0.977142i	$0.431811\pi$
$954$	0	0
$955$	−4.10570	−0.132857
$956$	0	0
$957$	20.6711	0.668201
$958$	0	0
$959$	8.92438	0.288183
$960$	0	0
$961$	−30.9688	−0.998994
$962$	0	0
$963$	4.81224	0.155072
$964$	0	0
$965$	−10.0695	−0.324148
$966$	0	0
$967$	−28.3200	−0.910709	−0.455355	−	0.890310i	$-0.650488\pi$
$967$	−28.3200	−0.910709	−0.455355	+	0.890310i	$0.650488\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	57.6296	1.84942	0.924711	−	0.380670i	$-0.124307\pi$
$971$	57.6296	1.84942	0.924711	+	0.380670i	$0.124307\pi$
$972$	0	0
$973$	15.8922	0.509481
$974$	0	0
$975$	−10.7028	−0.342765
$976$	0	0
$977$	6.43223	0.205785	0.102893	−	0.994692i	$-0.467190\pi$
$977$	6.43223	0.205785	0.102893	+	0.994692i	$0.467190\pi$
$978$	0	0
$979$	52.8001	1.68750
$980$	0	0
$981$	−40.7408	−1.30075
$982$	0	0
$983$	−19.0317	−0.607018	−0.303509	−	0.952829i	$-0.598158\pi$
$983$	−19.0317	−0.607018	−0.303509	+	0.952829i	$0.598158\pi$
$984$	0	0
$985$	0.286114	0.00911635
$986$	0	0
$987$	−22.5345	−0.717282
$988$	0	0
$989$	−91.6284	−2.91361
$990$	0	0
$991$	21.9760	0.698090	0.349045	−	0.937106i	$-0.386506\pi$
$991$	21.9760	0.698090	0.349045	+	0.937106i	$0.386506\pi$
$992$	0	0
$993$	−6.46054	−0.205019
$994$	0	0
$995$	−9.29064	−0.294533
$996$	0	0
$997$	2.50354	0.0792879	0.0396440	−	0.999214i	$-0.487378\pi$
$997$	2.50354	0.0792879	0.0396440	+	0.999214i	$0.487378\pi$
$998$	0	0
$999$	8.31277	0.263005

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9248.2.a.bv.1.5		12
4.3	odd	2	inner	9248.2.a.bv.1.7		12
17.2	even	8		544.2.o.i.225.3	✓	12
17.9	even	8		544.2.o.i.353.3	yes	12
17.16	even	2	inner	9248.2.a.bv.1.8		12
68.19	odd	8		544.2.o.i.225.4	yes	12
68.43	odd	8		544.2.o.i.353.4	yes	12
68.67	odd	2	inner	9248.2.a.bv.1.6		12
136.19	odd	8		1088.2.o.w.769.3		12
136.43	odd	8		1088.2.o.w.897.3		12
136.53	even	8		1088.2.o.w.769.4		12
136.77	even	8		1088.2.o.w.897.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.o.i.225.3	✓	12	17.2	even	8
544.2.o.i.225.4	yes	12	68.19	odd	8
544.2.o.i.353.3	yes	12	17.9	even	8
544.2.o.i.353.4	yes	12	68.43	odd	8
1088.2.o.w.769.3		12	136.19	odd	8
1088.2.o.w.769.4		12	136.53	even	8
1088.2.o.w.897.3		12	136.43	odd	8
1088.2.o.w.897.4		12	136.77	even	8
9248.2.a.bv.1.5		12	1.1	even	1	trivial
9248.2.a.bv.1.6		12	68.67	odd	2	inner
9248.2.a.bv.1.7		12	4.3	odd	2	inner
9248.2.a.bv.1.8		12	17.16	even	2	inner

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:	\( N \)	\(=\)	\( 9248 = 2^{5} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9248.a (trivial)

\(f(q)\)	\(=\)	\(q-0.610563 q^{3} -0.408946 q^{5} +3.61232 q^{7} -2.62721 q^{9} +4.04633 q^{11} -3.62721 q^{13} +0.249687 q^{15} -6.83553 q^{19} -2.20555 q^{21} +8.26922 q^{23} -4.83276 q^{25} +3.43577 q^{27} -8.36703 q^{29} +0.176556 q^{31} -2.47054 q^{33} -1.47725 q^{35} +2.41948 q^{37} +2.21464 q^{39} -1.41421 q^{41} -11.0807 q^{43} +1.07439 q^{45} +10.2172 q^{47} +6.04888 q^{49} +1.42166 q^{53} -1.65473 q^{55} +4.17352 q^{57} +5.72237 q^{59} +3.23737 q^{61} -9.49034 q^{63} +1.48333 q^{65} +12.5579 q^{67} -5.04888 q^{69} +9.61584 q^{71} +6.83456 q^{73} +2.95071 q^{75} +14.6167 q^{77} +1.39768 q^{79} +5.78389 q^{81} +7.44931 q^{83} +5.10860 q^{87} +13.0489 q^{89} -13.1027 q^{91} -0.107798 q^{93} +2.79536 q^{95} -7.36176 q^{97} -10.6306 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 20 q^{9} + 8 q^{13} + 32 q^{21} + 52 q^{25} + 8 q^{33} + 28 q^{49} + 24 q^{53} - 16 q^{69} + 4 q^{81} + 112 q^{89} - 32 q^{93}+O(q^{100}) \) 12 * q + 20 * q^9 + 8 * q^13 + 32 * q^21 + 52 * q^25 + 8 * q^33 + 28 * q^49 + 24 * q^53 - 16 * q^69 + 4 * q^81 + 112 * q^89 - 32 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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