LMFDB - Embedded newform 7448.2.a.bm.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.98211824.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 15x^{3} + 8x^{2} - 9x + 1 $ x^6 - 3x^5 - 6x^4 + 15x^3 + 8x^2 - 9*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.511631$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.488369 q^{3} +3.51566 q^{5} -2.76150 q^{9} +O(q^{10})$	$q-0.488369 q^{3} +3.51566 q^{5} -2.76150 q^{9} -2.25811 q^{11} +2.21264 q^{13} -1.71694 q^{15} +0.716938 q^{17} +1.00000 q^{19} -7.53470 q^{23} +7.35984 q^{25} +2.81374 q^{27} -7.66391 q^{29} -0.473849 q^{31} +1.10279 q^{33} -0.268043 q^{37} -1.08059 q^{39} +11.9093 q^{41} -6.26213 q^{43} -9.70847 q^{45} -7.10998 q^{47} -0.350131 q^{51} -2.16643 q^{53} -7.93873 q^{55} -0.488369 q^{57} +0.0795286 q^{59} -15.1946 q^{61} +7.77889 q^{65} +7.97595 q^{67} +3.67972 q^{69} +0.764655 q^{71} +10.2795 q^{73} -3.59432 q^{75} -14.9272 q^{79} +6.91034 q^{81} +3.68887 q^{83} +2.52051 q^{85} +3.74282 q^{87} -2.22816 q^{89} +0.231413 q^{93} +3.51566 q^{95} -7.09236 q^{97} +6.23575 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - q^{5} + 3 q^{9}+O(q^{10}) $ 6 * q - 3 * q^3 - q^5 + 3 * q^9	$ 6 q - 3 q^{3} - q^{5} + 3 q^{9} + 3 q^{11} - 6 q^{13} - 8 q^{15} + 2 q^{17} + 6 q^{19} + 2 q^{23} + 5 q^{25} - 6 q^{27} - q^{29} - 14 q^{31} - 19 q^{33} - 7 q^{37} + 22 q^{39} + 21 q^{41} + q^{43} + 4 q^{45} - 23 q^{47} + 2 q^{51} - 15 q^{53} - 2 q^{55} - 3 q^{57} - 25 q^{59} - 21 q^{61} + 6 q^{65} + 2 q^{67} + 20 q^{69} + 13 q^{71} + 2 q^{73} - 24 q^{75} - 17 q^{79} - 2 q^{81} - 4 q^{83} + 40 q^{85} - 30 q^{87} + q^{89} + 6 q^{93} - q^{95} + 13 q^{97} + 41 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 - q^5 + 3 * q^9 + 3 * q^11 - 6 * q^13 - 8 * q^15 + 2 * q^17 + 6 * q^19 + 2 * q^23 + 5 * q^25 - 6 * q^27 - q^29 - 14 * q^31 - 19 * q^33 - 7 * q^37 + 22 * q^39 + 21 * q^41 + q^43 + 4 * q^45 - 23 * q^47 + 2 * q^51 - 15 * q^53 - 2 * q^55 - 3 * q^57 - 25 * q^59 - 21 * q^61 + 6 * q^65 + 2 * q^67 + 20 * q^69 + 13 * q^71 + 2 * q^73 - 24 * q^75 - 17 * q^79 - 2 * q^81 - 4 * q^83 + 40 * q^85 - 30 * q^87 + q^89 + 6 * q^93 - q^95 + 13 * q^97 + 41 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.488369	−0.281960	−0.140980	−	0.990012i	$-0.545025\pi$
$3$	−0.488369	−0.281960	−0.140980	+	0.990012i	$0.545025\pi$
$4$	0	0
$5$	3.51566	1.57225	0.786125	−	0.618068i	$-0.212084\pi$
$5$	3.51566	1.57225	0.786125	+	0.618068i	$0.212084\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.76150	−0.920499
$10$	0	0
$11$	−2.25811	−0.680845	−0.340423	−	0.940273i	$-0.610570\pi$
$11$	−2.25811	−0.680845	−0.340423	+	0.940273i	$0.610570\pi$
$12$	0	0
$13$	2.21264	0.613676	0.306838	−	0.951762i	$-0.400729\pi$
$13$	2.21264	0.613676	0.306838	+	0.951762i	$0.400729\pi$
$14$	0	0
$15$	−1.71694	−0.443312
$16$	0	0
$17$	0.716938	0.173883	0.0869415	−	0.996213i	$-0.472291\pi$
$17$	0.716938	0.173883	0.0869415	+	0.996213i	$0.472291\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.53470	−1.57109	−0.785547	−	0.618802i	$-0.787618\pi$
$23$	−7.53470	−1.57109	−0.785547	+	0.618802i	$0.787618\pi$
$24$	0	0
$25$	7.35984	1.47197
$26$	0	0
$27$	2.81374	0.541504
$28$	0	0
$29$	−7.66391	−1.42315	−0.711576	−	0.702609i	$-0.752018\pi$
$29$	−7.66391	−1.42315	−0.711576	+	0.702609i	$0.752018\pi$
$30$	0	0
$31$	−0.473849	−0.0851058	−0.0425529	−	0.999094i	$-0.513549\pi$
$31$	−0.473849	−0.0851058	−0.0425529	+	0.999094i	$0.513549\pi$
$32$	0	0
$33$	1.10279	0.191971
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.268043	−0.0440660	−0.0220330	−	0.999757i	$-0.507014\pi$
$37$	−0.268043	−0.0440660	−0.0220330	+	0.999757i	$0.507014\pi$
$38$	0	0
$39$	−1.08059	−0.173032
$40$	0	0
$41$	11.9093	1.85991	0.929956	−	0.367670i	$-0.119844\pi$
$41$	11.9093	1.85991	0.929956	+	0.367670i	$0.119844\pi$
$42$	0	0
$43$	−6.26213	−0.954966	−0.477483	−	0.878641i	$-0.658451\pi$
$43$	−6.26213	−0.954966	−0.477483	+	0.878641i	$0.658451\pi$
$44$	0	0
$45$	−9.70847	−1.44725
$46$	0	0
$47$	−7.10998	−1.03710	−0.518548	−	0.855048i	$-0.673527\pi$
$47$	−7.10998	−1.03710	−0.518548	+	0.855048i	$0.673527\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.350131	−0.0490281
$52$	0	0
$53$	−2.16643	−0.297583	−0.148791	−	0.988869i	$-0.547538\pi$
$53$	−2.16643	−0.297583	−0.148791	+	0.988869i	$0.547538\pi$
$54$	0	0
$55$	−7.93873	−1.07046
$56$	0	0
$57$	−0.488369	−0.0646861
$58$	0	0
$59$	0.0795286	0.0103537	0.00517687	−	0.999987i	$-0.498352\pi$
$59$	0.0795286	0.0103537	0.00517687	+	0.999987i	$0.498352\pi$
$60$	0	0
$61$	−15.1946	−1.94547	−0.972734	−	0.231925i	$-0.925498\pi$
$61$	−15.1946	−1.94547	−0.972734	+	0.231925i	$0.925498\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.77889	0.964852
$66$	0	0
$67$	7.97595	0.974418	0.487209	−	0.873285i	$-0.338015\pi$
$67$	7.97595	0.974418	0.487209	+	0.873285i	$0.338015\pi$
$68$	0	0
$69$	3.67972	0.442986
$70$	0	0
$71$	0.764655	0.0907479	0.0453739	−	0.998970i	$-0.485552\pi$
$71$	0.764655	0.0907479	0.0453739	+	0.998970i	$0.485552\pi$
$72$	0	0
$73$	10.2795	1.20313	0.601564	−	0.798825i	$-0.294545\pi$
$73$	10.2795	1.20313	0.601564	+	0.798825i	$0.294545\pi$
$74$	0	0
$75$	−3.59432	−0.415036
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.9272	−1.67945	−0.839723	−	0.543015i	$-0.817283\pi$
$79$	−14.9272	−1.67945	−0.839723	+	0.543015i	$0.817283\pi$
$80$	0	0
$81$	6.91034	0.767816
$82$	0	0
$83$	3.68887	0.404906	0.202453	−	0.979292i	$-0.435109\pi$
$83$	3.68887	0.404906	0.202453	+	0.979292i	$0.435109\pi$
$84$	0	0
$85$	2.52051	0.273388
$86$	0	0
$87$	3.74282	0.401272
$88$	0	0
$89$	−2.22816	−0.236184	−0.118092	−	0.993003i	$-0.537678\pi$
$89$	−2.22816	−0.236184	−0.118092	+	0.993003i	$0.537678\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.231413	0.0239964
$94$	0	0
$95$	3.51566	0.360699
$96$	0	0
$97$	−7.09236	−0.720120	−0.360060	−	0.932929i	$-0.617244\pi$
$97$	−7.09236	−0.720120	−0.360060	+	0.932929i	$0.617244\pi$
$98$	0	0
$99$	6.23575	0.626717
$100$	0	0
$101$	−5.61873	−0.559085	−0.279542	−	0.960133i	$-0.590183\pi$
$101$	−5.61873	−0.559085	−0.279542	+	0.960133i	$0.590183\pi$
$102$	0	0
$103$	0.529025	0.0521263	0.0260632	−	0.999660i	$-0.491703\pi$
$103$	0.529025	0.0521263	0.0260632	+	0.999660i	$0.491703\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.1976	−1.56588	−0.782938	−	0.622099i	$-0.786280\pi$
$107$	−16.1976	−1.56588	−0.782938	+	0.622099i	$0.786280\pi$
$108$	0	0
$109$	9.24340	0.885357	0.442678	−	0.896680i	$-0.354028\pi$
$109$	9.24340	0.885357	0.442678	+	0.896680i	$0.354028\pi$
$110$	0	0
$111$	0.130904	0.0124249
$112$	0	0
$113$	−10.3591	−0.974498	−0.487249	−	0.873263i	$-0.662000\pi$
$113$	−10.3591	−0.974498	−0.487249	+	0.873263i	$0.662000\pi$
$114$	0	0
$115$	−26.4894	−2.47015
$116$	0	0
$117$	−6.11020	−0.564888
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.90095	−0.536450
$122$	0	0
$123$	−5.81611	−0.524421
$124$	0	0
$125$	8.29639	0.742052
$126$	0	0
$127$	−12.9388	−1.14813	−0.574066	−	0.818809i	$-0.694635\pi$
$127$	−12.9388	−1.14813	−0.574066	+	0.818809i	$0.694635\pi$
$128$	0	0
$129$	3.05823	0.269262
$130$	0	0
$131$	17.5212	1.53084	0.765419	−	0.643533i	$-0.222532\pi$
$131$	17.5212	1.53084	0.765419	+	0.643533i	$0.222532\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	9.89213	0.851379
$136$	0	0
$137$	7.30076	0.623746	0.311873	−	0.950124i	$-0.399044\pi$
$137$	7.30076	0.623746	0.311873	+	0.950124i	$0.399044\pi$
$138$	0	0
$139$	−8.40692	−0.713066	−0.356533	−	0.934283i	$-0.616041\pi$
$139$	−8.40692	−0.713066	−0.356533	+	0.934283i	$0.616041\pi$
$140$	0	0
$141$	3.47229	0.292420
$142$	0	0
$143$	−4.99638	−0.417819
$144$	0	0
$145$	−26.9437	−2.23755
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.99634	0.491239	0.245620	−	0.969366i	$-0.421009\pi$
$149$	5.99634	0.491239	0.245620	+	0.969366i	$0.421009\pi$
$150$	0	0
$151$	−22.0572	−1.79499	−0.897494	−	0.441027i	$-0.854614\pi$
$151$	−22.0572	−1.79499	−0.897494	+	0.441027i	$0.854614\pi$
$152$	0	0
$153$	−1.97982	−0.160059
$154$	0	0
$155$	−1.66589	−0.133808
$156$	0	0
$157$	−20.7625	−1.65703	−0.828513	−	0.559969i	$-0.810813\pi$
$157$	−20.7625	−1.65703	−0.828513	+	0.559969i	$0.810813\pi$
$158$	0	0
$159$	1.05802	0.0839064
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.0347	0.785982	0.392991	−	0.919542i	$-0.371440\pi$
$163$	10.0347	0.785982	0.392991	+	0.919542i	$0.371440\pi$
$164$	0	0
$165$	3.87703	0.301826
$166$	0	0
$167$	−11.4994	−0.889854	−0.444927	−	0.895567i	$-0.646770\pi$
$167$	−11.4994	−0.889854	−0.444927	+	0.895567i	$0.646770\pi$
$168$	0	0
$169$	−8.10422	−0.623401
$170$	0	0
$171$	−2.76150	−0.211177
$172$	0	0
$173$	−0.407066	−0.0309487	−0.0154743	−	0.999880i	$-0.504926\pi$
$173$	−0.407066	−0.0309487	−0.0154743	+	0.999880i	$0.504926\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.0388393	−0.00291934
$178$	0	0
$179$	0.0666375	0.00498072	0.00249036	−	0.999997i	$-0.499207\pi$
$179$	0.0666375	0.00498072	0.00249036	+	0.999997i	$0.499207\pi$
$180$	0	0
$181$	7.64418	0.568188	0.284094	−	0.958796i	$-0.408307\pi$
$181$	7.64418	0.568188	0.284094	+	0.958796i	$0.408307\pi$
$182$	0	0
$183$	7.42057	0.548544
$184$	0	0
$185$	−0.942348	−0.0692828
$186$	0	0
$187$	−1.61892	−0.118387
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.8535	−0.857687	−0.428844	−	0.903379i	$-0.641079\pi$
$191$	−11.8535	−0.857687	−0.428844	+	0.903379i	$0.641079\pi$
$192$	0	0
$193$	7.34976	0.529047	0.264523	−	0.964379i	$-0.414785\pi$
$193$	7.34976	0.529047	0.264523	+	0.964379i	$0.414785\pi$
$194$	0	0
$195$	−3.79897	−0.272050
$196$	0	0
$197$	20.2535	1.44300	0.721500	−	0.692414i	$-0.243453\pi$
$197$	20.2535	1.44300	0.721500	+	0.692414i	$0.243453\pi$
$198$	0	0
$199$	−16.8733	−1.19611	−0.598057	−	0.801453i	$-0.704060\pi$
$199$	−16.8733	−1.19611	−0.598057	+	0.801453i	$0.704060\pi$
$200$	0	0
$201$	−3.89521	−0.274747
$202$	0	0
$203$	0	0
$204$	0	0
$205$	41.8688	2.92425
$206$	0	0
$207$	20.8070	1.44619
$208$	0	0
$209$	−2.25811	−0.156197
$210$	0	0
$211$	11.9286	0.821197	0.410598	−	0.911816i	$-0.365320\pi$
$211$	11.9286	0.821197	0.410598	+	0.911816i	$0.365320\pi$
$212$	0	0
$213$	−0.373434	−0.0255873
$214$	0	0
$215$	−22.0155	−1.50145
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5.02020	−0.339234
$220$	0	0
$221$	1.58633	0.106708
$222$	0	0
$223$	23.2754	1.55863	0.779317	−	0.626630i	$-0.215566\pi$
$223$	23.2754	1.55863	0.779317	+	0.626630i	$0.215566\pi$
$224$	0	0
$225$	−20.3242	−1.35494
$226$	0	0
$227$	1.56606	0.103943	0.0519714	−	0.998649i	$-0.483450\pi$
$227$	1.56606	0.103943	0.0519714	+	0.998649i	$0.483450\pi$
$228$	0	0
$229$	−13.8669	−0.916350	−0.458175	−	0.888862i	$-0.651497\pi$
$229$	−13.8669	−0.916350	−0.458175	+	0.888862i	$0.651497\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.3117	−0.937590	−0.468795	−	0.883307i	$-0.655312\pi$
$233$	−14.3117	−0.937590	−0.468795	+	0.883307i	$0.655312\pi$
$234$	0	0
$235$	−24.9962	−1.63057
$236$	0	0
$237$	7.29000	0.473537
$238$	0	0
$239$	27.4532	1.77580	0.887899	−	0.460037i	$-0.152164\pi$
$239$	27.4532	1.77580	0.887899	+	0.460037i	$0.152164\pi$
$240$	0	0
$241$	8.58000	0.552686	0.276343	−	0.961059i	$-0.410877\pi$
$241$	8.58000	0.552686	0.276343	+	0.961059i	$0.410877\pi$
$242$	0	0
$243$	−11.8160	−0.757997
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.21264	0.140787
$248$	0	0
$249$	−1.80153	−0.114167
$250$	0	0
$251$	3.23766	0.204359	0.102180	−	0.994766i	$-0.467418\pi$
$251$	3.23766	0.204359	0.102180	+	0.994766i	$0.467418\pi$
$252$	0	0
$253$	17.0142	1.06967
$254$	0	0
$255$	−1.23094	−0.0770844
$256$	0	0
$257$	−1.41758	−0.0884263	−0.0442132	−	0.999022i	$-0.514078\pi$
$257$	−1.41758	−0.0884263	−0.0442132	+	0.999022i	$0.514078\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	21.1639	1.31001
$262$	0	0
$263$	−2.57536	−0.158803	−0.0794017	−	0.996843i	$-0.525301\pi$
$263$	−2.57536	−0.158803	−0.0794017	+	0.996843i	$0.525301\pi$
$264$	0	0
$265$	−7.61644	−0.467874
$266$	0	0
$267$	1.08816	0.0665946
$268$	0	0
$269$	10.9509	0.667685	0.333843	−	0.942629i	$-0.391655\pi$
$269$	10.9509	0.667685	0.333843	+	0.942629i	$0.391655\pi$
$270$	0	0
$271$	−23.1518	−1.40637	−0.703186	−	0.711006i	$-0.748240\pi$
$271$	−23.1518	−1.40637	−0.703186	+	0.711006i	$0.748240\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−16.6193	−1.00218
$276$	0	0
$277$	−14.0972	−0.847017	−0.423508	−	0.905892i	$-0.639202\pi$
$277$	−14.0972	−0.847017	−0.423508	+	0.905892i	$0.639202\pi$
$278$	0	0
$279$	1.30853	0.0783398
$280$	0	0
$281$	0.341082	0.0203473	0.0101736	−	0.999948i	$-0.496762\pi$
$281$	0.341082	0.0203473	0.0101736	+	0.999948i	$0.496762\pi$
$282$	0	0
$283$	−9.97496	−0.592950	−0.296475	−	0.955041i	$-0.595811\pi$
$283$	−9.97496	−0.592950	−0.296475	+	0.955041i	$0.595811\pi$
$284$	0	0
$285$	−1.71694	−0.101703
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.4860	−0.969765
$290$	0	0
$291$	3.46369	0.203045
$292$	0	0
$293$	21.6737	1.26619	0.633097	−	0.774073i	$-0.281784\pi$
$293$	21.6737	1.26619	0.633097	+	0.774073i	$0.281784\pi$
$294$	0	0
$295$	0.279595	0.0162787
$296$	0	0
$297$	−6.35372	−0.368680
$298$	0	0
$299$	−16.6716	−0.964143
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.74402	0.157640
$304$	0	0
$305$	−53.4189	−3.05876
$306$	0	0
$307$	−20.6231	−1.17702	−0.588510	−	0.808490i	$-0.700285\pi$
$307$	−20.6231	−1.17702	−0.588510	+	0.808490i	$0.700285\pi$
$308$	0	0
$309$	−0.258359	−0.0146975
$310$	0	0
$311$	−7.92634	−0.449461	−0.224731	−	0.974421i	$-0.572150\pi$
$311$	−7.92634	−0.449461	−0.224731	+	0.974421i	$0.572150\pi$
$312$	0	0
$313$	10.7662	0.608542	0.304271	−	0.952586i	$-0.401587\pi$
$313$	10.7662	0.608542	0.304271	+	0.952586i	$0.401587\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.02183	−0.450551	−0.225275	−	0.974295i	$-0.572328\pi$
$317$	−8.02183	−0.450551	−0.225275	+	0.974295i	$0.572328\pi$
$318$	0	0
$319$	17.3059	0.968947
$320$	0	0
$321$	7.91039	0.441515
$322$	0	0
$323$	0.716938	0.0398915
$324$	0	0
$325$	16.2847	0.903312
$326$	0	0
$327$	−4.51419	−0.249635
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12.5703	0.690928	0.345464	−	0.938432i	$-0.387722\pi$
$331$	12.5703	0.690928	0.345464	+	0.938432i	$0.387722\pi$
$332$	0	0
$333$	0.740200	0.0405627
$334$	0	0
$335$	28.0407	1.53203
$336$	0	0
$337$	25.7552	1.40297	0.701487	−	0.712683i	$-0.252520\pi$
$337$	25.7552	1.40297	0.701487	+	0.712683i	$0.252520\pi$
$338$	0	0
$339$	5.05904	0.274770
$340$	0	0
$341$	1.07000	0.0579439
$342$	0	0
$343$	0	0
$344$	0	0
$345$	12.9366	0.696484
$346$	0	0
$347$	6.18123	0.331826	0.165913	−	0.986140i	$-0.446943\pi$
$347$	6.18123	0.331826	0.165913	+	0.986140i	$0.446943\pi$
$348$	0	0
$349$	−8.99215	−0.481339	−0.240669	−	0.970607i	$-0.577367\pi$
$349$	−8.99215	−0.481339	−0.240669	+	0.970607i	$0.577367\pi$
$350$	0	0
$351$	6.22579	0.332308
$352$	0	0
$353$	17.0593	0.907977	0.453989	−	0.891007i	$-0.350001\pi$
$353$	17.0593	0.907977	0.453989	+	0.891007i	$0.350001\pi$
$354$	0	0
$355$	2.68827	0.142678
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.6501	0.878757	0.439379	−	0.898302i	$-0.355199\pi$
$359$	16.6501	0.878757	0.439379	+	0.898302i	$0.355199\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.88184	0.151257
$364$	0	0
$365$	36.1393	1.89162
$366$	0	0
$367$	−33.1117	−1.72842	−0.864209	−	0.503134i	$-0.832180\pi$
$367$	−33.1117	−1.72842	−0.864209	+	0.503134i	$0.832180\pi$
$368$	0	0
$369$	−32.8873	−1.71205
$370$	0	0
$371$	0	0
$372$	0	0
$373$	33.2511	1.72167	0.860837	−	0.508880i	$-0.169940\pi$
$373$	33.2511	1.72167	0.860837	+	0.508880i	$0.169940\pi$
$374$	0	0
$375$	−4.05170	−0.209229
$376$	0	0
$377$	−16.9575	−0.873355
$378$	0	0
$379$	−28.1510	−1.44602	−0.723011	−	0.690837i	$-0.757242\pi$
$379$	−28.1510	−1.44602	−0.723011	+	0.690837i	$0.757242\pi$
$380$	0	0
$381$	6.31891	0.323727
$382$	0	0
$383$	−22.7648	−1.16323	−0.581614	−	0.813465i	$-0.697579\pi$
$383$	−22.7648	−1.16323	−0.581614	+	0.813465i	$0.697579\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	17.2929	0.879045
$388$	0	0
$389$	−3.90959	−0.198224	−0.0991121	−	0.995076i	$-0.531600\pi$
$389$	−3.90959	−0.198224	−0.0991121	+	0.995076i	$0.531600\pi$
$390$	0	0
$391$	−5.40192	−0.273187
$392$	0	0
$393$	−8.55683	−0.431635
$394$	0	0
$395$	−52.4790	−2.64051
$396$	0	0
$397$	6.80489	0.341527	0.170764	−	0.985312i	$-0.445377\pi$
$397$	6.80489	0.341527	0.170764	+	0.985312i	$0.445377\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.05449	−0.452160	−0.226080	−	0.974109i	$-0.572591\pi$
$401$	−9.05449	−0.452160	−0.226080	+	0.974109i	$0.572591\pi$
$402$	0	0
$403$	−1.04846	−0.0522274
$404$	0	0
$405$	24.2944	1.20720
$406$	0	0
$407$	0.605270	0.0300021
$408$	0	0
$409$	−0.697334	−0.0344810	−0.0172405	−	0.999851i	$-0.505488\pi$
$409$	−0.697334	−0.0344810	−0.0172405	+	0.999851i	$0.505488\pi$
$410$	0	0
$411$	−3.56547	−0.175872
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.9688	0.636613
$416$	0	0
$417$	4.10568	0.201056
$418$	0	0
$419$	9.91019	0.484145	0.242072	−	0.970258i	$-0.422173\pi$
$419$	9.91019	0.484145	0.242072	+	0.970258i	$0.422173\pi$
$420$	0	0
$421$	−23.5825	−1.14934	−0.574670	−	0.818385i	$-0.694870\pi$
$421$	−23.5825	−1.14934	−0.574670	+	0.818385i	$0.694870\pi$
$422$	0	0
$423$	19.6342	0.954646
$424$	0	0
$425$	5.27655	0.255950
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2.44008	0.117808
$430$	0	0
$431$	1.43559	0.0691498	0.0345749	−	0.999402i	$-0.488992\pi$
$431$	1.43559	0.0691498	0.0345749	+	0.999402i	$0.488992\pi$
$432$	0	0
$433$	37.4216	1.79837	0.899185	−	0.437569i	$-0.144161\pi$
$433$	37.4216	1.79837	0.899185	+	0.437569i	$0.144161\pi$
$434$	0	0
$435$	13.1585	0.630900
$436$	0	0
$437$	−7.53470	−0.360434
$438$	0	0
$439$	29.2865	1.39777	0.698885	−	0.715234i	$-0.253680\pi$
$439$	29.2865	1.39777	0.698885	+	0.715234i	$0.253680\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.27913	0.108285	0.0541424	−	0.998533i	$-0.482757\pi$
$443$	2.27913	0.108285	0.0541424	+	0.998533i	$0.482757\pi$
$444$	0	0
$445$	−7.83344	−0.371341
$446$	0	0
$447$	−2.92843	−0.138510
$448$	0	0
$449$	5.22719	0.246686	0.123343	−	0.992364i	$-0.460638\pi$
$449$	5.22719	0.246686	0.123343	+	0.992364i	$0.460638\pi$
$450$	0	0
$451$	−26.8924	−1.26631
$452$	0	0
$453$	10.7720	0.506115
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−32.2735	−1.50969	−0.754845	−	0.655903i	$-0.772288\pi$
$457$	−32.2735	−1.50969	−0.754845	+	0.655903i	$0.772288\pi$
$458$	0	0
$459$	2.01728	0.0941584
$460$	0	0
$461$	26.1333	1.21715	0.608576	−	0.793496i	$-0.291741\pi$
$461$	26.1333	1.21715	0.608576	+	0.793496i	$0.291741\pi$
$462$	0	0
$463$	6.42014	0.298369	0.149185	−	0.988809i	$-0.452335\pi$
$463$	6.42014	0.298369	0.149185	+	0.988809i	$0.452335\pi$
$464$	0	0
$465$	0.813570	0.0377284
$466$	0	0
$467$	21.8516	1.01117	0.505586	−	0.862776i	$-0.331276\pi$
$467$	21.8516	1.01117	0.505586	+	0.862776i	$0.331276\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	10.1398	0.467215
$472$	0	0
$473$	14.1406	0.650184
$474$	0	0
$475$	7.35984	0.337693
$476$	0	0
$477$	5.98260	0.273924
$478$	0	0
$479$	−28.9775	−1.32402	−0.662008	−	0.749497i	$-0.730296\pi$
$479$	−28.9775	−1.32402	−0.662008	+	0.749497i	$0.730296\pi$
$480$	0	0
$481$	−0.593083	−0.0270423
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−24.9343	−1.13221
$486$	0	0
$487$	−23.8101	−1.07894	−0.539469	−	0.842006i	$-0.681375\pi$
$487$	−23.8101	−1.07894	−0.539469	+	0.842006i	$0.681375\pi$
$488$	0	0
$489$	−4.90066	−0.221616
$490$	0	0
$491$	23.5199	1.06144	0.530718	−	0.847548i	$-0.321922\pi$
$491$	23.5199	1.06144	0.530718	+	0.847548i	$0.321922\pi$
$492$	0	0
$493$	−5.49455	−0.247462
$494$	0	0
$495$	21.9228	0.985355
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12.4365	0.556734	0.278367	−	0.960475i	$-0.410207\pi$
$499$	12.4365	0.556734	0.278367	+	0.960475i	$0.410207\pi$
$500$	0	0
$501$	5.61597	0.250903
$502$	0	0
$503$	−21.3309	−0.951099	−0.475549	−	0.879689i	$-0.657751\pi$
$503$	−21.3309	−0.951099	−0.475549	+	0.879689i	$0.657751\pi$
$504$	0	0
$505$	−19.7535	−0.879021
$506$	0	0
$507$	3.95785	0.175774
$508$	0	0
$509$	−42.7523	−1.89496	−0.947481	−	0.319814i	$-0.896380\pi$
$509$	−42.7523	−1.89496	−0.947481	+	0.319814i	$0.896380\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.81374	0.124230
$514$	0	0
$515$	1.85987	0.0819556
$516$	0	0
$517$	16.0551	0.706102
$518$	0	0
$519$	0.198799	0.00872629
$520$	0	0
$521$	−24.9707	−1.09398	−0.546992	−	0.837138i	$-0.684227\pi$
$521$	−24.9707	−1.09398	−0.546992	+	0.837138i	$0.684227\pi$
$522$	0	0
$523$	−15.9078	−0.695599	−0.347800	−	0.937569i	$-0.613071\pi$
$523$	−15.9078	−0.695599	−0.347800	+	0.937569i	$0.613071\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.339721	−0.0147985
$528$	0	0
$529$	33.7717	1.46834
$530$	0	0
$531$	−0.219618	−0.00953060
$532$	0	0
$533$	26.3509	1.14138
$534$	0	0
$535$	−56.9450	−2.46195
$536$	0	0
$537$	−0.0325437	−0.00140437
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8.68783	−0.373519	−0.186759	−	0.982406i	$-0.559798\pi$
$541$	−8.68783	−0.373519	−0.186759	+	0.982406i	$0.559798\pi$
$542$	0	0
$543$	−3.73318	−0.160206
$544$	0	0
$545$	32.4966	1.39200
$546$	0	0
$547$	−26.9468	−1.15216	−0.576081	−	0.817393i	$-0.695419\pi$
$547$	−26.9468	−1.15216	−0.576081	+	0.817393i	$0.695419\pi$
$548$	0	0
$549$	41.9598	1.79080
$550$	0	0
$551$	−7.66391	−0.326494
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.460213	0.0195350
$556$	0	0
$557$	−42.0535	−1.78187	−0.890933	−	0.454135i	$-0.849948\pi$
$557$	−42.0535	−1.78187	−0.890933	+	0.454135i	$0.849948\pi$
$558$	0	0
$559$	−13.8559	−0.586040
$560$	0	0
$561$	0.790632	0.0333805
$562$	0	0
$563$	−25.9858	−1.09517	−0.547586	−	0.836749i	$-0.684453\pi$
$563$	−25.9858	−1.09517	−0.547586	+	0.836749i	$0.684453\pi$
$564$	0	0
$565$	−36.4189	−1.53215
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.8816	−0.456182	−0.228091	−	0.973640i	$-0.573248\pi$
$569$	−10.8816	−0.456182	−0.228091	+	0.973640i	$0.573248\pi$
$570$	0	0
$571$	11.6649	0.488163	0.244081	−	0.969755i	$-0.421514\pi$
$571$	11.6649	0.488163	0.244081	+	0.969755i	$0.421514\pi$
$572$	0	0
$573$	5.78887	0.241834
$574$	0	0
$575$	−55.4542	−2.31260
$576$	0	0
$577$	35.1757	1.46439	0.732193	−	0.681097i	$-0.238497\pi$
$577$	35.1757	1.46439	0.732193	+	0.681097i	$0.238497\pi$
$578$	0	0
$579$	−3.58939	−0.149170
$580$	0	0
$581$	0	0
$582$	0	0
$583$	4.89204	0.202608
$584$	0	0
$585$	−21.4814	−0.888145
$586$	0	0
$587$	−17.9581	−0.741212	−0.370606	−	0.928790i	$-0.620850\pi$
$587$	−17.9581	−0.741212	−0.370606	+	0.928790i	$0.620850\pi$
$588$	0	0
$589$	−0.473849	−0.0195246
$590$	0	0
$591$	−9.89117	−0.406868
$592$	0	0
$593$	−7.85554	−0.322588	−0.161294	−	0.986906i	$-0.551567\pi$
$593$	−7.85554	−0.322588	−0.161294	+	0.986906i	$0.551567\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.24039	0.337257
$598$	0	0
$599$	−14.7639	−0.603236	−0.301618	−	0.953429i	$-0.597527\pi$
$599$	−14.7639	−0.603236	−0.301618	+	0.953429i	$0.597527\pi$
$600$	0	0
$601$	−6.67351	−0.272218	−0.136109	−	0.990694i	$-0.543460\pi$
$601$	−6.67351	−0.272218	−0.136109	+	0.990694i	$0.543460\pi$
$602$	0	0
$603$	−22.0256	−0.896950
$604$	0	0
$605$	−20.7457	−0.843433
$606$	0	0
$607$	37.8693	1.53707	0.768534	−	0.639809i	$-0.220986\pi$
$607$	37.8693	1.53707	0.768534	+	0.639809i	$0.220986\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.7318	−0.636442
$612$	0	0
$613$	17.6730	0.713805	0.356902	−	0.934142i	$-0.383833\pi$
$613$	17.6730	0.713805	0.356902	+	0.934142i	$0.383833\pi$
$614$	0	0
$615$	−20.4475	−0.824521
$616$	0	0
$617$	−15.9720	−0.643009	−0.321505	−	0.946908i	$-0.604189\pi$
$617$	−15.9720	−0.643009	−0.321505	+	0.946908i	$0.604189\pi$
$618$	0	0
$619$	34.8682	1.40147	0.700735	−	0.713422i	$-0.252856\pi$
$619$	34.8682	1.40147	0.700735	+	0.713422i	$0.252856\pi$
$620$	0	0
$621$	−21.2007	−0.850753
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−7.63195	−0.305278
$626$	0	0
$627$	1.10279	0.0440412
$628$	0	0
$629$	−0.192170	−0.00766233
$630$	0	0
$631$	−14.4788	−0.576393	−0.288197	−	0.957571i	$-0.593056\pi$
$631$	−14.4788	−0.576393	−0.288197	+	0.957571i	$0.593056\pi$
$632$	0	0
$633$	−5.82555	−0.231545
$634$	0	0
$635$	−45.4883	−1.80515
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.11159	−0.0835333
$640$	0	0
$641$	28.5138	1.12623	0.563113	−	0.826380i	$-0.309604\pi$
$641$	28.5138	1.12623	0.563113	+	0.826380i	$0.309604\pi$
$642$	0	0
$643$	−23.7793	−0.937763	−0.468882	−	0.883261i	$-0.655343\pi$
$643$	−23.7793	−0.937763	−0.468882	+	0.883261i	$0.655343\pi$
$644$	0	0
$645$	10.7517	0.423348
$646$	0	0
$647$	35.5860	1.39903	0.699514	−	0.714619i	$-0.253400\pi$
$647$	35.5860	1.39903	0.699514	+	0.714619i	$0.253400\pi$
$648$	0	0
$649$	−0.179584	−0.00704929
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.24174	−0.361657	−0.180829	−	0.983515i	$-0.557878\pi$
$653$	−9.24174	−0.361657	−0.180829	+	0.983515i	$0.557878\pi$
$654$	0	0
$655$	61.5986	2.40686
$656$	0	0
$657$	−28.3869	−1.10748
$658$	0	0
$659$	−23.4715	−0.914320	−0.457160	−	0.889384i	$-0.651133\pi$
$659$	−23.4715	−0.914320	−0.457160	+	0.889384i	$0.651133\pi$
$660$	0	0
$661$	−10.1636	−0.395317	−0.197658	−	0.980271i	$-0.563334\pi$
$661$	−10.1636	−0.395317	−0.197658	+	0.980271i	$0.563334\pi$
$662$	0	0
$663$	−0.774714	−0.0300874
$664$	0	0
$665$	0	0
$666$	0	0
$667$	57.7453	2.23591
$668$	0	0
$669$	−11.3670	−0.439472
$670$	0	0
$671$	34.3110	1.32456
$672$	0	0
$673$	−11.0320	−0.425251	−0.212626	−	0.977134i	$-0.568201\pi$
$673$	−11.0320	−0.425251	−0.212626	+	0.977134i	$0.568201\pi$
$674$	0	0
$675$	20.7087	0.797076
$676$	0	0
$677$	−34.5867	−1.32928	−0.664638	−	0.747165i	$-0.731414\pi$
$677$	−34.5867	−1.32928	−0.664638	+	0.747165i	$0.731414\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−0.764814	−0.0293077
$682$	0	0
$683$	−26.0946	−0.998480	−0.499240	−	0.866464i	$-0.666387\pi$
$683$	−26.0946	−0.998480	−0.499240	+	0.866464i	$0.666387\pi$
$684$	0	0
$685$	25.6670	0.980685
$686$	0	0
$687$	6.77216	0.258374
$688$	0	0
$689$	−4.79354	−0.182619
$690$	0	0
$691$	8.51398	0.323887	0.161944	−	0.986800i	$-0.448224\pi$
$691$	8.51398	0.323887	0.161944	+	0.986800i	$0.448224\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−29.5559	−1.12112
$696$	0	0
$697$	8.53820	0.323407
$698$	0	0
$699$	6.98939	0.264363
$700$	0	0
$701$	−7.20119	−0.271985	−0.135993	−	0.990710i	$-0.543422\pi$
$701$	−7.20119	−0.271985	−0.135993	+	0.990710i	$0.543422\pi$
$702$	0	0
$703$	−0.268043	−0.0101094
$704$	0	0
$705$	12.2074	0.459757
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−20.9748	−0.787726	−0.393863	−	0.919169i	$-0.628861\pi$
$709$	−20.9748	−0.787726	−0.393863	+	0.919169i	$0.628861\pi$
$710$	0	0
$711$	41.2215	1.54593
$712$	0	0
$713$	3.57031	0.133709
$714$	0	0
$715$	−17.5656	−0.656915
$716$	0	0
$717$	−13.4073	−0.500704
$718$	0	0
$719$	29.7635	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$719$	29.7635	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4.19021	−0.155835
$724$	0	0
$725$	−56.4052	−2.09484
$726$	0	0
$727$	21.9856	0.815401	0.407700	−	0.913116i	$-0.366331\pi$
$727$	21.9856	0.815401	0.407700	+	0.913116i	$0.366331\pi$
$728$	0	0
$729$	−14.9605	−0.554091
$730$	0	0
$731$	−4.48956	−0.166052
$732$	0	0
$733$	41.0702	1.51696	0.758482	−	0.651694i	$-0.225942\pi$
$733$	41.0702	1.51696	0.758482	+	0.651694i	$0.225942\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18.0106	−0.663428
$738$	0	0
$739$	21.2697	0.782417	0.391209	−	0.920302i	$-0.372057\pi$
$739$	21.2697	0.782417	0.391209	+	0.920302i	$0.372057\pi$
$740$	0	0
$741$	−1.08059	−0.0396963
$742$	0	0
$743$	−6.24375	−0.229061	−0.114531	−	0.993420i	$-0.536536\pi$
$743$	−6.24375	−0.229061	−0.114531	+	0.993420i	$0.536536\pi$
$744$	0	0
$745$	21.0811	0.772351
$746$	0	0
$747$	−10.1868	−0.372715
$748$	0	0
$749$	0	0
$750$	0	0
$751$	18.3670	0.670222	0.335111	−	0.942179i	$-0.391226\pi$
$751$	18.3670	0.670222	0.335111	+	0.942179i	$0.391226\pi$
$752$	0	0
$753$	−1.58117	−0.0576211
$754$	0	0
$755$	−77.5455	−2.82217
$756$	0	0
$757$	−3.63834	−0.132238	−0.0661188	−	0.997812i	$-0.521062\pi$
$757$	−3.63834	−0.132238	−0.0661188	+	0.997812i	$0.521062\pi$
$758$	0	0
$759$	−8.30919	−0.301605
$760$	0	0
$761$	−5.98449	−0.216938	−0.108469	−	0.994100i	$-0.534595\pi$
$761$	−5.98449	−0.216938	−0.108469	+	0.994100i	$0.534595\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6.96037	−0.251653
$766$	0	0
$767$	0.175968	0.00635385
$768$	0	0
$769$	−45.7519	−1.64986	−0.824928	−	0.565238i	$-0.808784\pi$
$769$	−45.7519	−1.64986	−0.824928	+	0.565238i	$0.808784\pi$
$770$	0	0
$771$	0.692303	0.0249327
$772$	0	0
$773$	33.5347	1.20616	0.603080	−	0.797681i	$-0.293940\pi$
$773$	33.5347	1.20616	0.603080	+	0.797681i	$0.293940\pi$
$774$	0	0
$775$	−3.48745	−0.125273
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.9093	0.426693
$780$	0	0
$781$	−1.72667	−0.0617853
$782$	0	0
$783$	−21.5642	−0.770643
$784$	0	0
$785$	−72.9938	−2.60526
$786$	0	0
$787$	−3.51073	−0.125144	−0.0625719	−	0.998040i	$-0.519930\pi$
$787$	−3.51073	−0.125144	−0.0625719	+	0.998040i	$0.519930\pi$
$788$	0	0
$789$	1.25772	0.0447762
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−33.6202	−1.19389
$794$	0	0
$795$	3.71963	0.131922
$796$	0	0
$797$	−15.1264	−0.535806	−0.267903	−	0.963446i	$-0.586331\pi$
$797$	−15.1264	−0.535806	−0.267903	+	0.963446i	$0.586331\pi$
$798$	0	0
$799$	−5.09741	−0.180334
$800$	0	0
$801$	6.15305	0.217407
$802$	0	0
$803$	−23.2123	−0.819144
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.34806	−0.188261
$808$	0	0
$809$	−2.38201	−0.0837470	−0.0418735	−	0.999123i	$-0.513333\pi$
$809$	−2.38201	−0.0837470	−0.0418735	+	0.999123i	$0.513333\pi$
$810$	0	0
$811$	−25.9215	−0.910228	−0.455114	−	0.890433i	$-0.650401\pi$
$811$	−25.9215	−0.910228	−0.455114	+	0.890433i	$0.650401\pi$
$812$	0	0
$813$	11.3066	0.396541
$814$	0	0
$815$	35.2787	1.23576
$816$	0	0
$817$	−6.26213	−0.219084
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−53.8970	−1.88102	−0.940509	−	0.339769i	$-0.889651\pi$
$821$	−53.8970	−1.88102	−0.940509	+	0.339769i	$0.889651\pi$
$822$	0	0
$823$	12.9577	0.451677	0.225838	−	0.974165i	$-0.427488\pi$
$823$	12.9577	0.451677	0.225838	+	0.974165i	$0.427488\pi$
$824$	0	0
$825$	8.11636	0.282575
$826$	0	0
$827$	29.8634	1.03845	0.519226	−	0.854637i	$-0.326220\pi$
$827$	29.8634	1.03845	0.519226	+	0.854637i	$0.326220\pi$
$828$	0	0
$829$	−39.7956	−1.38216	−0.691079	−	0.722780i	$-0.742864\pi$
$829$	−39.7956	−1.38216	−0.691079	+	0.722780i	$0.742864\pi$
$830$	0	0
$831$	6.88462	0.238825
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−40.4281	−1.39907
$836$	0	0
$837$	−1.33329	−0.0460851
$838$	0	0
$839$	−41.5209	−1.43346	−0.716731	−	0.697350i	$-0.754362\pi$
$839$	−41.5209	−1.43346	−0.716731	+	0.697350i	$0.754362\pi$
$840$	0	0
$841$	29.7356	1.02536
$842$	0	0
$843$	−0.166574	−0.00573711
$844$	0	0
$845$	−28.4916	−0.980142
$846$	0	0
$847$	0	0
$848$	0	0
$849$	4.87147	0.167188
$850$	0	0
$851$	2.01962	0.0692318
$852$	0	0
$853$	15.0014	0.513637	0.256818	−	0.966460i	$-0.417326\pi$
$853$	15.0014	0.513637	0.256818	+	0.966460i	$0.417326\pi$
$854$	0	0
$855$	−9.70847	−0.332023
$856$	0	0
$857$	21.3427	0.729053	0.364527	−	0.931193i	$-0.381231\pi$
$857$	21.3427	0.729053	0.364527	+	0.931193i	$0.381231\pi$
$858$	0	0
$859$	23.7738	0.811151	0.405576	−	0.914062i	$-0.367071\pi$
$859$	23.7738	0.811151	0.405576	+	0.914062i	$0.367071\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.5643	−0.870217	−0.435109	−	0.900378i	$-0.643290\pi$
$863$	−25.5643	−0.870217	−0.435109	+	0.900378i	$0.643290\pi$
$864$	0	0
$865$	−1.43111	−0.0486590
$866$	0	0
$867$	8.05125	0.273435
$868$	0	0
$869$	33.7073	1.14344
$870$	0	0
$871$	17.6479	0.597977
$872$	0	0
$873$	19.5855	0.662869
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−10.1296	−0.342053	−0.171026	−	0.985266i	$-0.554708\pi$
$877$	−10.1296	−0.342053	−0.171026	+	0.985266i	$0.554708\pi$
$878$	0	0
$879$	−10.5848	−0.357016
$880$	0	0
$881$	33.6185	1.13264	0.566319	−	0.824186i	$-0.308367\pi$
$881$	33.6185	1.13264	0.566319	+	0.824186i	$0.308367\pi$
$882$	0	0
$883$	32.8755	1.10635	0.553174	−	0.833066i	$-0.313416\pi$
$883$	32.8755	1.10635	0.553174	+	0.833066i	$0.313416\pi$
$884$	0	0
$885$	−0.136546	−0.00458993
$886$	0	0
$887$	44.6026	1.49761	0.748804	−	0.662792i	$-0.230629\pi$
$887$	44.6026	1.49761	0.748804	+	0.662792i	$0.230629\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−15.6043	−0.522764
$892$	0	0
$893$	−7.10998	−0.237926
$894$	0	0
$895$	0.234275	0.00783094
$896$	0	0
$897$	8.14189	0.271850
$898$	0	0
$899$	3.63154	0.121119
$900$	0	0
$901$	−1.55320	−0.0517446
$902$	0	0
$903$	0	0
$904$	0	0
$905$	26.8743	0.893333
$906$	0	0
$907$	−51.6897	−1.71633	−0.858164	−	0.513376i	$-0.828395\pi$
$907$	−51.6897	−1.71633	−0.858164	+	0.513376i	$0.828395\pi$
$908$	0	0
$909$	15.5161	0.514637
$910$	0	0
$911$	28.0623	0.929745	0.464872	−	0.885378i	$-0.346100\pi$
$911$	28.0623	0.929745	0.464872	+	0.885378i	$0.346100\pi$
$912$	0	0
$913$	−8.32986	−0.275678
$914$	0	0
$915$	26.0882	0.862448
$916$	0	0
$917$	0	0
$918$	0	0
$919$	21.7561	0.717669	0.358834	−	0.933401i	$-0.383174\pi$
$919$	21.7561	0.717669	0.358834	+	0.933401i	$0.383174\pi$
$920$	0	0
$921$	10.0717	0.331873
$922$	0	0
$923$	1.69191	0.0556898
$924$	0	0
$925$	−1.97275	−0.0648638
$926$	0	0
$927$	−1.46090	−0.0479822
$928$	0	0
$929$	52.4361	1.72037	0.860186	−	0.509980i	$-0.170347\pi$
$929$	52.4361	1.72037	0.860186	+	0.509980i	$0.170347\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.87098	0.126730
$934$	0	0
$935$	−5.69158	−0.186135
$936$	0	0
$937$	−6.48443	−0.211837	−0.105919	−	0.994375i	$-0.533778\pi$
$937$	−6.48443	−0.211837	−0.105919	+	0.994375i	$0.533778\pi$
$938$	0	0
$939$	−5.25788	−0.171585
$940$	0	0
$941$	5.12436	0.167049	0.0835247	−	0.996506i	$-0.473382\pi$
$941$	5.12436	0.167049	0.0835247	+	0.996506i	$0.473382\pi$
$942$	0	0
$943$	−89.7327	−2.92210
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.3046	−0.562325	−0.281163	−	0.959660i	$-0.590720\pi$
$947$	−17.3046	−0.562325	−0.281163	+	0.959660i	$0.590720\pi$
$948$	0	0
$949$	22.7449	0.738331
$950$	0	0
$951$	3.91761	0.127037
$952$	0	0
$953$	−52.4552	−1.69919	−0.849595	−	0.527436i	$-0.823153\pi$
$953$	−52.4552	−1.69919	−0.849595	+	0.527436i	$0.823153\pi$
$954$	0	0
$955$	−41.6727	−1.34850
$956$	0	0
$957$	−8.45169	−0.273204
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.7755	−0.992757
$962$	0	0
$963$	44.7295	1.44139
$964$	0	0
$965$	25.8392	0.831794
$966$	0	0
$967$	39.0111	1.25451	0.627256	−	0.778813i	$-0.284178\pi$
$967$	39.0111	1.25451	0.627256	+	0.778813i	$0.284178\pi$
$968$	0	0
$969$	−0.350131	−0.0112478
$970$	0	0
$971$	25.7693	0.826975	0.413488	−	0.910510i	$-0.364311\pi$
$971$	25.7693	0.826975	0.413488	+	0.910510i	$0.364311\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−7.95294	−0.254698
$976$	0	0
$977$	−3.35129	−0.107217	−0.0536087	−	0.998562i	$-0.517072\pi$
$977$	−3.35129	−0.107217	−0.0536087	+	0.998562i	$0.517072\pi$
$978$	0	0
$979$	5.03142	0.160805
$980$	0	0
$981$	−25.5256	−0.814970
$982$	0	0
$983$	−25.3969	−0.810037	−0.405018	−	0.914309i	$-0.632735\pi$
$983$	−25.3969	−0.810037	−0.405018	+	0.914309i	$0.632735\pi$
$984$	0	0
$985$	71.2043	2.26876
$986$	0	0
$987$	0	0
$988$	0	0
$989$	47.1833	1.50034
$990$	0	0
$991$	−35.6136	−1.13130	−0.565651	−	0.824645i	$-0.691375\pi$
$991$	−35.6136	−1.13130	−0.565651	+	0.824645i	$0.691375\pi$
$992$	0	0
$993$	−6.13896	−0.194814
$994$	0	0
$995$	−59.3206	−1.88059
$996$	0	0
$997$	−29.0273	−0.919303	−0.459651	−	0.888099i	$-0.652026\pi$
$997$	−29.0273	−0.919303	−0.459651	+	0.888099i	$0.652026\pi$
$998$	0	0
$999$	−0.754203	−0.0238619

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bm.1.4	✓	6
7.6	odd	2		7448.2.a.bn.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bm.1.4	✓	6	1.1	even	1	trivial
7448.2.a.bn.1.3	yes	6	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-0.488369 q^{3} +3.51566 q^{5} -2.76150 q^{9} +O(q^{10})\)	\(q-0.488369 q^{3} +3.51566 q^{5} -2.76150 q^{9} -2.25811 q^{11} +2.21264 q^{13} -1.71694 q^{15} +0.716938 q^{17} +1.00000 q^{19} -7.53470 q^{23} +7.35984 q^{25} +2.81374 q^{27} -7.66391 q^{29} -0.473849 q^{31} +1.10279 q^{33} -0.268043 q^{37} -1.08059 q^{39} +11.9093 q^{41} -6.26213 q^{43} -9.70847 q^{45} -7.10998 q^{47} -0.350131 q^{51} -2.16643 q^{53} -7.93873 q^{55} -0.488369 q^{57} +0.0795286 q^{59} -15.1946 q^{61} +7.77889 q^{65} +7.97595 q^{67} +3.67972 q^{69} +0.764655 q^{71} +10.2795 q^{73} -3.59432 q^{75} -14.9272 q^{79} +6.91034 q^{81} +3.68887 q^{83} +2.52051 q^{85} +3.74282 q^{87} -2.22816 q^{89} +0.231413 q^{93} +3.51566 q^{95} -7.09236 q^{97} +6.23575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - q^{5} + 3 q^{9}+O(q^{10}) \) 6 * q - 3 * q^3 - q^5 + 3 * q^9	\( 6 q - 3 q^{3} - q^{5} + 3 q^{9} + 3 q^{11} - 6 q^{13} - 8 q^{15} + 2 q^{17} + 6 q^{19} + 2 q^{23} + 5 q^{25} - 6 q^{27} - q^{29} - 14 q^{31} - 19 q^{33} - 7 q^{37} + 22 q^{39} + 21 q^{41} + q^{43} + 4 q^{45} - 23 q^{47} + 2 q^{51} - 15 q^{53} - 2 q^{55} - 3 q^{57} - 25 q^{59} - 21 q^{61} + 6 q^{65} + 2 q^{67} + 20 q^{69} + 13 q^{71} + 2 q^{73} - 24 q^{75} - 17 q^{79} - 2 q^{81} - 4 q^{83} + 40 q^{85} - 30 q^{87} + q^{89} + 6 q^{93} - q^{95} + 13 q^{97} + 41 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 - q^5 + 3 * q^9 + 3 * q^11 - 6 * q^13 - 8 * q^15 + 2 * q^17 + 6 * q^19 + 2 * q^23 + 5 * q^25 - 6 * q^27 - q^29 - 14 * q^31 - 19 * q^33 - 7 * q^37 + 22 * q^39 + 21 * q^41 + q^43 + 4 * q^45 - 23 * q^47 + 2 * q^51 - 15 * q^53 - 2 * q^55 - 3 * q^57 - 25 * q^59 - 21 * q^61 + 6 * q^65 + 2 * q^67 + 20 * q^69 + 13 * q^71 + 2 * q^73 - 24 * q^75 - 17 * q^79 - 2 * q^81 - 4 * q^83 + 40 * q^85 - 30 * q^87 + q^89 + 6 * q^93 - q^95 + 13 * q^97 + 41 * q^99

Introduction

L-functions

Modular forms

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Database

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