LMFDB - Embedded newform 9248.2.a.bz.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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 40 x^{18} + 620 x^{16} - 4784 x^{14} + 19585 x^{12} - 41912 x^{10} + 43536 x^{8} - 20328 x^{6} + \cdots + 8 $ x^20 - 40x^18 + 620x^16 - 4784x^14 + 19585x^12 - 41912x^10 + 43536x^8 - 20328x^6 + 4208x^4 - 352*x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.12443$ 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)

$f(q)$	$=$	$q-2.15620 q^{3} +1.57755 q^{5} +2.46548 q^{7} +1.64918 q^{9} -5.90216 q^{11} -3.94016 q^{13} -3.40150 q^{15} +1.56671 q^{19} -5.31607 q^{21} -1.12968 q^{23} -2.51134 q^{25} +2.91263 q^{27} -8.23944 q^{29} +4.16791 q^{31} +12.7262 q^{33} +3.88942 q^{35} -8.28282 q^{37} +8.49576 q^{39} -8.29395 q^{41} +4.88567 q^{43} +2.60166 q^{45} +0.211521 q^{47} -0.921386 q^{49} -2.51889 q^{53} -9.31094 q^{55} -3.37813 q^{57} -6.97442 q^{59} -0.958971 q^{61} +4.06604 q^{63} -6.21579 q^{65} -2.23948 q^{67} +2.43582 q^{69} +10.3750 q^{71} +8.62894 q^{73} +5.41495 q^{75} -14.5517 q^{77} +11.2760 q^{79} -11.2277 q^{81} +13.3105 q^{83} +17.7658 q^{87} +9.61042 q^{89} -9.71440 q^{91} -8.98683 q^{93} +2.47156 q^{95} -10.2598 q^{97} -9.73374 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 28 q^{9} - 8 q^{13} + 16 q^{15} + 40 q^{19} - 32 q^{21} + 28 q^{25} + 32 q^{35} + 40 q^{43} + 32 q^{47} + 36 q^{49} - 40 q^{53} + 48 q^{55} + 8 q^{59} + 72 q^{67} - 48 q^{69} - 48 q^{77} + 36 q^{81}+ \cdots - 32 q^{93}+O(q^{100}) $ 20 * q + 28 * q^9 - 8 * q^13 + 16 * q^15 + 40 * q^19 - 32 * q^21 + 28 * q^25 + 32 * q^35 + 40 * q^43 + 32 * q^47 + 36 * q^49 - 40 * q^53 + 48 * q^55 + 8 * q^59 + 72 * q^67 - 48 * q^69 - 48 * q^77 + 36 * q^81 + 24 * q^83 + 96 * q^87 + 64 * 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
$3$	−2.15620	−1.24488	−0.622440	−	0.782667i	$-0.713859\pi$
$3$	−2.15620	−1.24488	−0.622440	+	0.782667i	$0.713859\pi$
$4$	0	0
$5$	1.57755	0.705501	0.352750	−	0.935718i	$-0.385247\pi$
$5$	1.57755	0.705501	0.352750	+	0.935718i	$0.385247\pi$
$6$	0	0
$7$	2.46548	0.931866	0.465933	−	0.884820i	$-0.345719\pi$
$7$	2.46548	0.931866	0.465933	+	0.884820i	$0.345719\pi$
$8$	0	0
$9$	1.64918	0.549728
$10$	0	0
$11$	−5.90216	−1.77957	−0.889784	−	0.456382i	$-0.849145\pi$
$11$	−5.90216	−1.77957	−0.889784	+	0.456382i	$0.849145\pi$
$12$	0	0
$13$	−3.94016	−1.09280	−0.546402	−	0.837523i	$-0.684003\pi$
$13$	−3.94016	−1.09280	−0.546402	+	0.837523i	$0.684003\pi$
$14$	0	0
$15$	−3.40150	−0.878264
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	1.56671	0.359427	0.179714	−	0.983719i	$-0.442483\pi$
$19$	1.56671	0.359427	0.179714	+	0.983719i	$0.442483\pi$
$20$	0	0
$21$	−5.31607	−1.16006
$22$	0	0
$23$	−1.12968	−0.235555	−0.117778	−	0.993040i	$-0.537577\pi$
$23$	−1.12968	−0.235555	−0.117778	+	0.993040i	$0.537577\pi$
$24$	0	0
$25$	−2.51134	−0.502269
$26$	0	0
$27$	2.91263	0.560535
$28$	0	0
$29$	−8.23944	−1.53002	−0.765012	−	0.644016i	$-0.777267\pi$
$29$	−8.23944	−1.53002	−0.765012	+	0.644016i	$0.777267\pi$
$30$	0	0
$31$	4.16791	0.748579	0.374289	−	0.927312i	$-0.377887\pi$
$31$	4.16791	0.748579	0.374289	+	0.927312i	$0.377887\pi$
$32$	0	0
$33$	12.7262	2.21535
$34$	0	0
$35$	3.88942	0.657432
$36$	0	0
$37$	−8.28282	−1.36169	−0.680844	−	0.732429i	$-0.738387\pi$
$37$	−8.28282	−1.36169	−0.680844	+	0.732429i	$0.738387\pi$
$38$	0	0
$39$	8.49576	1.36041
$40$	0	0
$41$	−8.29395	−1.29530	−0.647649	−	0.761939i	$-0.724247\pi$
$41$	−8.29395	−1.29530	−0.647649	+	0.761939i	$0.724247\pi$
$42$	0	0
$43$	4.88567	0.745058	0.372529	−	0.928021i	$-0.378491\pi$
$43$	4.88567	0.745058	0.372529	+	0.928021i	$0.378491\pi$
$44$	0	0
$45$	2.60166	0.387833
$46$	0	0
$47$	0.211521	0.0308535	0.0154267	−	0.999881i	$-0.495089\pi$
$47$	0.211521	0.0308535	0.0154267	+	0.999881i	$0.495089\pi$
$48$	0	0
$49$	−0.921386	−0.131627
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.51889	−0.345996	−0.172998	−	0.984922i	$-0.555345\pi$
$53$	−2.51889	−0.345996	−0.172998	+	0.984922i	$0.555345\pi$
$54$	0	0
$55$	−9.31094	−1.25549
$56$	0	0
$57$	−3.37813	−0.447444
$58$	0	0
$59$	−6.97442	−0.907992	−0.453996	−	0.891004i	$-0.650002\pi$
$59$	−6.97442	−0.907992	−0.453996	+	0.891004i	$0.650002\pi$
$60$	0	0
$61$	−0.958971	−0.122784	−0.0613918	−	0.998114i	$-0.519554\pi$
$61$	−0.958971	−0.122784	−0.0613918	+	0.998114i	$0.519554\pi$
$62$	0	0
$63$	4.06604	0.512272
$64$	0	0
$65$	−6.21579	−0.770973
$66$	0	0
$67$	−2.23948	−0.273597	−0.136798	−	0.990599i	$-0.543681\pi$
$67$	−2.23948	−0.273597	−0.136798	+	0.990599i	$0.543681\pi$
$68$	0	0
$69$	2.43582	0.293238
$70$	0	0
$71$	10.3750	1.23129	0.615643	−	0.788025i	$-0.288896\pi$
$71$	10.3750	1.23129	0.615643	+	0.788025i	$0.288896\pi$
$72$	0	0
$73$	8.62894	1.00994	0.504971	−	0.863136i	$-0.331503\pi$
$73$	8.62894	1.00994	0.504971	+	0.863136i	$0.331503\pi$
$74$	0	0
$75$	5.41495	0.625265
$76$	0	0
$77$	−14.5517	−1.65832
$78$	0	0
$79$	11.2760	1.26865	0.634323	−	0.773068i	$-0.281279\pi$
$79$	11.2760	1.26865	0.634323	+	0.773068i	$0.281279\pi$
$80$	0	0
$81$	−11.2277	−1.24753
$82$	0	0
$83$	13.3105	1.46102	0.730509	−	0.682903i	$-0.239283\pi$
$83$	13.3105	1.46102	0.730509	+	0.682903i	$0.239283\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	17.7658	1.90470
$88$	0	0
$89$	9.61042	1.01870	0.509351	−	0.860559i	$-0.329885\pi$
$89$	9.61042	1.01870	0.509351	+	0.860559i	$0.329885\pi$
$90$	0	0
$91$	−9.71440	−1.01835
$92$	0	0
$93$	−8.98683	−0.931891
$94$	0	0
$95$	2.47156	0.253576
$96$	0	0
$97$	−10.2598	−1.04173	−0.520865	−	0.853639i	$-0.674390\pi$
$97$	−10.2598	−1.04173	−0.520865	+	0.853639i	$0.674390\pi$
$98$	0	0
$99$	−9.73374	−0.978278
$100$	0	0
$101$	11.1934	1.11378	0.556891	−	0.830586i	$-0.311994\pi$
$101$	11.1934	1.11378	0.556891	+	0.830586i	$0.311994\pi$
$102$	0	0
$103$	8.09875	0.797994	0.398997	−	0.916952i	$-0.369358\pi$
$103$	8.09875	0.797994	0.398997	+	0.916952i	$0.369358\pi$
$104$	0	0
$105$	−8.38635	−0.818424
$106$	0	0
$107$	−11.1857	−1.08136	−0.540679	−	0.841229i	$-0.681833\pi$
$107$	−11.1857	−1.08136	−0.540679	+	0.841229i	$0.681833\pi$
$108$	0	0
$109$	17.5301	1.67908	0.839539	−	0.543300i	$-0.182825\pi$
$109$	17.5301	1.67908	0.839539	+	0.543300i	$0.182825\pi$
$110$	0	0
$111$	17.8594	1.69514
$112$	0	0
$113$	2.98824	0.281110	0.140555	−	0.990073i	$-0.455111\pi$
$113$	2.98824	0.281110	0.140555	+	0.990073i	$0.455111\pi$
$114$	0	0
$115$	−1.78213	−0.166184
$116$	0	0
$117$	−6.49804	−0.600744
$118$	0	0
$119$	0	0
$120$	0	0
$121$	23.8355	2.16686
$122$	0	0
$123$	17.8834	1.61249
$124$	0	0
$125$	−11.8495	−1.05985
$126$	0	0
$127$	4.75600	0.422027	0.211013	−	0.977483i	$-0.432324\pi$
$127$	4.75600	0.422027	0.211013	+	0.977483i	$0.432324\pi$
$128$	0	0
$129$	−10.5345	−0.927508
$130$	0	0
$131$	−5.59636	−0.488956	−0.244478	−	0.969655i	$-0.578617\pi$
$131$	−5.59636	−0.488956	−0.244478	+	0.969655i	$0.578617\pi$
$132$	0	0
$133$	3.86269	0.334938
$134$	0	0
$135$	4.59481	0.395458
$136$	0	0
$137$	7.94054	0.678406	0.339203	−	0.940713i	$-0.389843\pi$
$137$	7.94054	0.678406	0.339203	+	0.940713i	$0.389843\pi$
$138$	0	0
$139$	−20.4306	−1.73290	−0.866452	−	0.499260i	$-0.833605\pi$
$139$	−20.4306	−1.73290	−0.866452	+	0.499260i	$0.833605\pi$
$140$	0	0
$141$	−0.456080	−0.0384089
$142$	0	0
$143$	23.2554	1.94472
$144$	0	0
$145$	−12.9981	−1.07943
$146$	0	0
$147$	1.98669	0.163859
$148$	0	0
$149$	19.7486	1.61787	0.808933	−	0.587901i	$-0.200046\pi$
$149$	19.7486	1.61787	0.808933	+	0.587901i	$0.200046\pi$
$150$	0	0
$151$	−1.85708	−0.151127	−0.0755634	−	0.997141i	$-0.524076\pi$
$151$	−1.85708	−0.151127	−0.0755634	+	0.997141i	$0.524076\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.57508	0.528123
$156$	0	0
$157$	16.3091	1.30161	0.650804	−	0.759246i	$-0.274432\pi$
$157$	16.3091	1.30161	0.650804	+	0.759246i	$0.274432\pi$
$158$	0	0
$159$	5.43122	0.430724
$160$	0	0
$161$	−2.78521	−0.219506
$162$	0	0
$163$	14.0979	1.10423	0.552115	−	0.833768i	$-0.313821\pi$
$163$	14.0979	1.10423	0.552115	+	0.833768i	$0.313821\pi$
$164$	0	0
$165$	20.0762	1.56293
$166$	0	0
$167$	−23.3158	−1.80423	−0.902116	−	0.431493i	$-0.857987\pi$
$167$	−23.3158	−1.80423	−0.902116	+	0.431493i	$0.857987\pi$
$168$	0	0
$169$	2.52485	0.194219
$170$	0	0
$171$	2.58379	0.197587
$172$	0	0
$173$	−18.6895	−1.42094	−0.710469	−	0.703728i	$-0.751517\pi$
$173$	−18.6895	−1.42094	−0.710469	+	0.703728i	$0.751517\pi$
$174$	0	0
$175$	−6.19168	−0.468047
$176$	0	0
$177$	15.0382	1.13034
$178$	0	0
$179$	15.1882	1.13522	0.567610	−	0.823298i	$-0.307868\pi$
$179$	15.1882	1.13522	0.567610	+	0.823298i	$0.307868\pi$
$180$	0	0
$181$	22.5025	1.67260	0.836300	−	0.548272i	$-0.184714\pi$
$181$	22.5025	1.67260	0.836300	+	0.548272i	$0.184714\pi$
$182$	0	0
$183$	2.06773	0.152851
$184$	0	0
$185$	−13.0665	−0.960671
$186$	0	0
$187$	0	0
$188$	0	0
$189$	7.18104	0.522343
$190$	0	0
$191$	−11.0257	−0.797793	−0.398896	−	0.916996i	$-0.630607\pi$
$191$	−11.0257	−0.797793	−0.398896	+	0.916996i	$0.630607\pi$
$192$	0	0
$193$	−5.15917	−0.371365	−0.185683	−	0.982610i	$-0.559450\pi$
$193$	−5.15917	−0.371365	−0.185683	+	0.982610i	$0.559450\pi$
$194$	0	0
$195$	13.4025	0.959770
$196$	0	0
$197$	−2.94652	−0.209931	−0.104966	−	0.994476i	$-0.533473\pi$
$197$	−2.94652	−0.209931	−0.104966	+	0.994476i	$0.533473\pi$
$198$	0	0
$199$	14.9188	1.05757	0.528784	−	0.848757i	$-0.322648\pi$
$199$	14.9188	1.05757	0.528784	+	0.848757i	$0.322648\pi$
$200$	0	0
$201$	4.82877	0.340595
$202$	0	0
$203$	−20.3142	−1.42578
$204$	0	0
$205$	−13.0841	−0.913833
$206$	0	0
$207$	−1.86305	−0.129491
$208$	0	0
$209$	−9.24696	−0.639626
$210$	0	0
$211$	−17.5904	−1.21097	−0.605485	−	0.795856i	$-0.707021\pi$
$211$	−17.5904	−1.21097	−0.605485	+	0.795856i	$0.707021\pi$
$212$	0	0
$213$	−22.3705	−1.53280
$214$	0	0
$215$	7.70738	0.525639
$216$	0	0
$217$	10.2759	0.697575
$218$	0	0
$219$	−18.6057	−1.25726
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1.90859	−0.127809	−0.0639043	−	0.997956i	$-0.520355\pi$
$223$	−1.90859	−0.127809	−0.0639043	+	0.997956i	$0.520355\pi$
$224$	0	0
$225$	−4.14167	−0.276111
$226$	0	0
$227$	−19.8120	−1.31497	−0.657484	−	0.753468i	$-0.728379\pi$
$227$	−19.8120	−1.31497	−0.657484	+	0.753468i	$0.728379\pi$
$228$	0	0
$229$	−5.52285	−0.364960	−0.182480	−	0.983210i	$-0.558412\pi$
$229$	−5.52285	−0.364960	−0.182480	+	0.983210i	$0.558412\pi$
$230$	0	0
$231$	31.3763	2.06441
$232$	0	0
$233$	−25.1830	−1.64979	−0.824897	−	0.565283i	$-0.808767\pi$
$233$	−25.1830	−1.64979	−0.824897	+	0.565283i	$0.808767\pi$
$234$	0	0
$235$	0.333684	0.0217671
$236$	0	0
$237$	−24.3132	−1.57931
$238$	0	0
$239$	30.3838	1.96536	0.982681	−	0.185305i	$-0.0593272\pi$
$239$	30.3838	1.96536	0.982681	+	0.185305i	$0.0593272\pi$
$240$	0	0
$241$	8.80077	0.566907	0.283454	−	0.958986i	$-0.408520\pi$
$241$	8.80077	0.566907	0.283454	+	0.958986i	$0.408520\pi$
$242$	0	0
$243$	15.4713	0.992487
$244$	0	0
$245$	−1.45353	−0.0928626
$246$	0	0
$247$	−6.17308	−0.392784
$248$	0	0
$249$	−28.7001	−1.81879
$250$	0	0
$251$	13.3007	0.839530	0.419765	−	0.907633i	$-0.362112\pi$
$251$	13.3007	0.839530	0.419765	+	0.907633i	$0.362112\pi$
$252$	0	0
$253$	6.66757	0.419186
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.07113	−0.253950	−0.126975	−	0.991906i	$-0.540527\pi$
$257$	−4.07113	−0.253950	−0.126975	+	0.991906i	$0.540527\pi$
$258$	0	0
$259$	−20.4212	−1.26891
$260$	0	0
$261$	−13.5883	−0.841097
$262$	0	0
$263$	8.00344	0.493513	0.246757	−	0.969078i	$-0.420635\pi$
$263$	8.00344	0.493513	0.246757	+	0.969078i	$0.420635\pi$
$264$	0	0
$265$	−3.97367	−0.244100
$266$	0	0
$267$	−20.7220	−1.26816
$268$	0	0
$269$	−24.5174	−1.49485	−0.747426	−	0.664345i	$-0.768711\pi$
$269$	−24.5174	−1.49485	−0.747426	+	0.664345i	$0.768711\pi$
$270$	0	0
$271$	−2.94219	−0.178725	−0.0893626	−	0.995999i	$-0.528483\pi$
$271$	−2.94219	−0.178725	−0.0893626	+	0.995999i	$0.528483\pi$
$272$	0	0
$273$	20.9462	1.26772
$274$	0	0
$275$	14.8224	0.893822
$276$	0	0
$277$	−8.24285	−0.495265	−0.247632	−	0.968854i	$-0.579653\pi$
$277$	−8.24285	−0.495265	−0.247632	+	0.968854i	$0.579653\pi$
$278$	0	0
$279$	6.87365	0.411515
$280$	0	0
$281$	2.97000	0.177176	0.0885878	−	0.996068i	$-0.471765\pi$
$281$	2.97000	0.177176	0.0885878	+	0.996068i	$0.471765\pi$
$282$	0	0
$283$	7.29636	0.433724	0.216862	−	0.976202i	$-0.430418\pi$
$283$	7.29636	0.433724	0.216862	+	0.976202i	$0.430418\pi$
$284$	0	0
$285$	−5.32916	−0.315672
$286$	0	0
$287$	−20.4486	−1.20704
$288$	0	0
$289$	0	0
$290$	0	0
$291$	22.1222	1.29683
$292$	0	0
$293$	−1.35734	−0.0792964	−0.0396482	−	0.999214i	$-0.512624\pi$
$293$	−1.35734	−0.0792964	−0.0396482	+	0.999214i	$0.512624\pi$
$294$	0	0
$295$	−11.0025	−0.640589
$296$	0	0
$297$	−17.1908	−0.997511
$298$	0	0
$299$	4.45113	0.257415
$300$	0	0
$301$	12.0455	0.694294
$302$	0	0
$303$	−24.1351	−1.38653
$304$	0	0
$305$	−1.51282	−0.0866239
$306$	0	0
$307$	−25.9552	−1.48134	−0.740672	−	0.671867i	$-0.765493\pi$
$307$	−25.9552	−1.48134	−0.740672	+	0.671867i	$0.765493\pi$
$308$	0	0
$309$	−17.4625	−0.993407
$310$	0	0
$311$	5.79539	0.328626	0.164313	−	0.986408i	$-0.447459\pi$
$311$	5.79539	0.328626	0.164313	+	0.986408i	$0.447459\pi$
$312$	0	0
$313$	−3.63081	−0.205226	−0.102613	−	0.994721i	$-0.532720\pi$
$313$	−3.63081	−0.205226	−0.102613	+	0.994721i	$0.532720\pi$
$314$	0	0
$315$	6.41436	0.361408
$316$	0	0
$317$	−1.56313	−0.0877942	−0.0438971	−	0.999036i	$-0.513977\pi$
$317$	−1.56313	−0.0877942	−0.0438971	+	0.999036i	$0.513977\pi$
$318$	0	0
$319$	48.6305	2.72278
$320$	0	0
$321$	24.1185	1.34616
$322$	0	0
$323$	0	0
$324$	0	0
$325$	9.89510	0.548881
$326$	0	0
$327$	−37.7983	−2.09025
$328$	0	0
$329$	0.521501	0.0287513
$330$	0	0
$331$	14.1104	0.775576	0.387788	−	0.921749i	$-0.373239\pi$
$331$	14.1104	0.775576	0.387788	+	0.921749i	$0.373239\pi$
$332$	0	0
$333$	−13.6599	−0.748557
$334$	0	0
$335$	−3.53289	−0.193023
$336$	0	0
$337$	30.6088	1.66737	0.833685	−	0.552241i	$-0.186227\pi$
$337$	30.6088	1.66737	0.833685	+	0.552241i	$0.186227\pi$
$338$	0	0
$339$	−6.44324	−0.349949
$340$	0	0
$341$	−24.5997	−1.33215
$342$	0	0
$343$	−19.5301	−1.05452
$344$	0	0
$345$	3.84262	0.206880
$346$	0	0
$347$	31.0919	1.66910	0.834549	−	0.550933i	$-0.185728\pi$
$347$	31.0919	1.66910	0.834549	+	0.550933i	$0.185728\pi$
$348$	0	0
$349$	27.8189	1.48911	0.744557	−	0.667559i	$-0.232661\pi$
$349$	27.8189	1.48911	0.744557	+	0.667559i	$0.232661\pi$
$350$	0	0
$351$	−11.4762	−0.612555
$352$	0	0
$353$	4.21923	0.224567	0.112283	−	0.993676i	$-0.464184\pi$
$353$	4.21923	0.224567	0.112283	+	0.993676i	$0.464184\pi$
$354$	0	0
$355$	16.3671	0.868673
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.62787	−0.191472	−0.0957358	−	0.995407i	$-0.530520\pi$
$359$	−3.62787	−0.191472	−0.0957358	+	0.995407i	$0.530520\pi$
$360$	0	0
$361$	−16.5454	−0.870812
$362$	0	0
$363$	−51.3940	−2.69749
$364$	0	0
$365$	13.6126	0.712514
$366$	0	0
$367$	7.79468	0.406879	0.203440	−	0.979087i	$-0.434788\pi$
$367$	7.79468	0.406879	0.203440	+	0.979087i	$0.434788\pi$
$368$	0	0
$369$	−13.6782	−0.712061
$370$	0	0
$371$	−6.21028	−0.322422
$372$	0	0
$373$	7.31810	0.378917	0.189458	−	0.981889i	$-0.439327\pi$
$373$	7.31810	0.378917	0.189458	+	0.981889i	$0.439327\pi$
$374$	0	0
$375$	25.5499	1.31939
$376$	0	0
$377$	32.4647	1.67202
$378$	0	0
$379$	17.2876	0.888004	0.444002	−	0.896026i	$-0.353558\pi$
$379$	17.2876	0.888004	0.444002	+	0.896026i	$0.353558\pi$
$380$	0	0
$381$	−10.2549	−0.525373
$382$	0	0
$383$	22.9908	1.17478	0.587389	−	0.809305i	$-0.300156\pi$
$383$	22.9908	1.17478	0.587389	+	0.809305i	$0.300156\pi$
$384$	0	0
$385$	−22.9560	−1.16994
$386$	0	0
$387$	8.05737	0.409579
$388$	0	0
$389$	−2.74492	−0.139173	−0.0695866	−	0.997576i	$-0.522168\pi$
$389$	−2.74492	−0.139173	−0.0695866	+	0.997576i	$0.522168\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	12.0668	0.608692
$394$	0	0
$395$	17.7884	0.895030
$396$	0	0
$397$	32.2334	1.61775	0.808874	−	0.587983i	$-0.200078\pi$
$397$	32.2334	1.61775	0.808874	+	0.587983i	$0.200078\pi$
$398$	0	0
$399$	−8.32873	−0.416958
$400$	0	0
$401$	33.7526	1.68552	0.842762	−	0.538286i	$-0.180928\pi$
$401$	33.7526	1.68552	0.842762	+	0.538286i	$0.180928\pi$
$402$	0	0
$403$	−16.4222	−0.818050
$404$	0	0
$405$	−17.7123	−0.880131
$406$	0	0
$407$	48.8865	2.42322
$408$	0	0
$409$	−18.0760	−0.893798	−0.446899	−	0.894584i	$-0.647472\pi$
$409$	−18.0760	−0.893798	−0.446899	+	0.894584i	$0.647472\pi$
$410$	0	0
$411$	−17.1214	−0.844534
$412$	0	0
$413$	−17.1953	−0.846126
$414$	0	0
$415$	20.9980	1.03075
$416$	0	0
$417$	44.0525	2.15726
$418$	0	0
$419$	12.8385	0.627201	0.313601	−	0.949555i	$-0.398465\pi$
$419$	12.8385	0.627201	0.313601	+	0.949555i	$0.398465\pi$
$420$	0	0
$421$	−13.4991	−0.657906	−0.328953	−	0.944346i	$-0.606696\pi$
$421$	−13.4991	−0.657906	−0.328953	+	0.944346i	$0.606696\pi$
$422$	0	0
$423$	0.348837	0.0169610
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.36433	−0.114418
$428$	0	0
$429$	−50.1433	−2.42094
$430$	0	0
$431$	6.69704	0.322585	0.161292	−	0.986907i	$-0.448434\pi$
$431$	6.69704	0.322585	0.161292	+	0.986907i	$0.448434\pi$
$432$	0	0
$433$	1.24299	0.0597342	0.0298671	−	0.999554i	$-0.490492\pi$
$433$	1.24299	0.0597342	0.0298671	+	0.999554i	$0.490492\pi$
$434$	0	0
$435$	28.0265	1.34377
$436$	0	0
$437$	−1.76988	−0.0846650
$438$	0	0
$439$	−40.4205	−1.92917	−0.964583	−	0.263781i	$-0.915030\pi$
$439$	−40.4205	−1.92917	−0.964583	+	0.263781i	$0.915030\pi$
$440$	0	0
$441$	−1.51953	−0.0723588
$442$	0	0
$443$	11.2390	0.533979	0.266990	−	0.963699i	$-0.413971\pi$
$443$	11.2390	0.533979	0.266990	+	0.963699i	$0.413971\pi$
$444$	0	0
$445$	15.1609	0.718696
$446$	0	0
$447$	−42.5818	−2.01405
$448$	0	0
$449$	−29.5756	−1.39576	−0.697878	−	0.716216i	$-0.745872\pi$
$449$	−29.5756	−1.39576	−0.697878	+	0.716216i	$0.745872\pi$
$450$	0	0
$451$	48.9522	2.30507
$452$	0	0
$453$	4.00422	0.188135
$454$	0	0
$455$	−15.3249	−0.718444
$456$	0	0
$457$	−18.9754	−0.887630	−0.443815	−	0.896118i	$-0.646375\pi$
$457$	−18.9754	−0.887630	−0.443815	+	0.896118i	$0.646375\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−39.2233	−1.82681	−0.913405	−	0.407051i	$-0.866557\pi$
$461$	−39.2233	−1.82681	−0.913405	+	0.407051i	$0.866557\pi$
$462$	0	0
$463$	39.7484	1.84726	0.923632	−	0.383280i	$-0.125205\pi$
$463$	39.7484	1.84726	0.923632	+	0.383280i	$0.125205\pi$
$464$	0	0
$465$	−14.1772	−0.657450
$466$	0	0
$467$	25.9493	1.20079	0.600395	−	0.799703i	$-0.295010\pi$
$467$	25.9493	1.20079	0.600395	+	0.799703i	$0.295010\pi$
$468$	0	0
$469$	−5.52141	−0.254955
$470$	0	0
$471$	−35.1656	−1.62035
$472$	0	0
$473$	−28.8360	−1.32588
$474$	0	0
$475$	−3.93454	−0.180529
$476$	0	0
$477$	−4.15411	−0.190204
$478$	0	0
$479$	17.3613	0.793258	0.396629	−	0.917979i	$-0.370180\pi$
$479$	17.3613	0.793258	0.396629	+	0.917979i	$0.370180\pi$
$480$	0	0
$481$	32.6356	1.48806
$482$	0	0
$483$	6.00547	0.273258
$484$	0	0
$485$	−16.1854	−0.734941
$486$	0	0
$487$	1.08027	0.0489517	0.0244759	−	0.999700i	$-0.492208\pi$
$487$	1.08027	0.0489517	0.0244759	+	0.999700i	$0.492208\pi$
$488$	0	0
$489$	−30.3978	−1.37463
$490$	0	0
$491$	−19.1493	−0.864194	−0.432097	−	0.901827i	$-0.642226\pi$
$491$	−19.1493	−0.864194	−0.432097	+	0.901827i	$0.642226\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−15.3554	−0.690176
$496$	0	0
$497$	25.5794	1.14739
$498$	0	0
$499$	−13.8789	−0.621306	−0.310653	−	0.950523i	$-0.600548\pi$
$499$	−13.8789	−0.621306	−0.310653	+	0.950523i	$0.600548\pi$
$500$	0	0
$501$	50.2735	2.24605
$502$	0	0
$503$	−10.1670	−0.453323	−0.226661	−	0.973974i	$-0.572781\pi$
$503$	−10.1670	−0.453323	−0.226661	+	0.973974i	$0.572781\pi$
$504$	0	0
$505$	17.6581	0.785774
$506$	0	0
$507$	−5.44408	−0.241780
$508$	0	0
$509$	2.58258	0.114471	0.0572354	−	0.998361i	$-0.481771\pi$
$509$	2.58258	0.114471	0.0572354	+	0.998361i	$0.481771\pi$
$510$	0	0
$511$	21.2745	0.941130
$512$	0	0
$513$	4.56324	0.201472
$514$	0	0
$515$	12.7762	0.562985
$516$	0	0
$517$	−1.24843	−0.0549059
$518$	0	0
$519$	40.2983	1.76890
$520$	0	0
$521$	−4.72330	−0.206932	−0.103466	−	0.994633i	$-0.532993\pi$
$521$	−4.72330	−0.206932	−0.103466	+	0.994633i	$0.532993\pi$
$522$	0	0
$523$	8.50844	0.372048	0.186024	−	0.982545i	$-0.440440\pi$
$523$	8.50844	0.372048	0.186024	+	0.982545i	$0.440440\pi$
$524$	0	0
$525$	13.3505	0.582663
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−21.7238	−0.944514
$530$	0	0
$531$	−11.5021	−0.499148
$532$	0	0
$533$	32.6795	1.41551
$534$	0	0
$535$	−17.6459	−0.762899
$536$	0	0
$537$	−32.7488	−1.41321
$538$	0	0
$539$	5.43817	0.234238
$540$	0	0
$541$	−0.320440	−0.0137768	−0.00688839	−	0.999976i	$-0.502193\pi$
$541$	−0.320440	−0.0137768	−0.00688839	+	0.999976i	$0.502193\pi$
$542$	0	0
$543$	−48.5199	−2.08219
$544$	0	0
$545$	27.6545	1.18459
$546$	0	0
$547$	20.0987	0.859358	0.429679	−	0.902982i	$-0.358627\pi$
$547$	20.0987	0.859358	0.429679	+	0.902982i	$0.358627\pi$
$548$	0	0
$549$	−1.58152	−0.0674976
$550$	0	0
$551$	−12.9088	−0.549933
$552$	0	0
$553$	27.8007	1.18221
$554$	0	0
$555$	28.1740	1.19592
$556$	0	0
$557$	29.5029	1.25008	0.625039	−	0.780594i	$-0.285083\pi$
$557$	29.5029	1.25008	0.625039	+	0.780594i	$0.285083\pi$
$558$	0	0
$559$	−19.2503	−0.814202
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.3687	0.479132	0.239566	−	0.970880i	$-0.422995\pi$
$563$	11.3687	0.479132	0.239566	+	0.970880i	$0.422995\pi$
$564$	0	0
$565$	4.71409	0.198323
$566$	0	0
$567$	−27.6818	−1.16253
$568$	0	0
$569$	6.38076	0.267495	0.133748	−	0.991015i	$-0.457299\pi$
$569$	6.38076	0.267495	0.133748	+	0.991015i	$0.457299\pi$
$570$	0	0
$571$	27.1399	1.13577	0.567884	−	0.823108i	$-0.307762\pi$
$571$	27.1399	1.13577	0.567884	+	0.823108i	$0.307762\pi$
$572$	0	0
$573$	23.7736	0.993157
$574$	0	0
$575$	2.83702	0.118312
$576$	0	0
$577$	3.41114	0.142008	0.0710038	−	0.997476i	$-0.477380\pi$
$577$	3.41114	0.142008	0.0710038	+	0.997476i	$0.477380\pi$
$578$	0	0
$579$	11.1242	0.462305
$580$	0	0
$581$	32.8169	1.36147
$582$	0	0
$583$	14.8669	0.615724
$584$	0	0
$585$	−10.2510	−0.423825
$586$	0	0
$587$	−20.1025	−0.829721	−0.414860	−	0.909885i	$-0.636170\pi$
$587$	−20.1025	−0.829721	−0.414860	+	0.909885i	$0.636170\pi$
$588$	0	0
$589$	6.52990	0.269060
$590$	0	0
$591$	6.35329	0.261339
$592$	0	0
$593$	−1.39793	−0.0574060	−0.0287030	−	0.999588i	$-0.509138\pi$
$593$	−1.39793	−0.0574060	−0.0287030	+	0.999588i	$0.509138\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−32.1679	−1.31655
$598$	0	0
$599$	−14.6858	−0.600047	−0.300024	−	0.953932i	$-0.596995\pi$
$599$	−14.6858	−0.600047	−0.300024	+	0.953932i	$0.596995\pi$
$600$	0	0
$601$	21.3490	0.870845	0.435422	−	0.900226i	$-0.356599\pi$
$601$	21.3490	0.870845	0.435422	+	0.900226i	$0.356599\pi$
$602$	0	0
$603$	−3.69332	−0.150404
$604$	0	0
$605$	37.6016	1.52872
$606$	0	0
$607$	34.4365	1.39773	0.698867	−	0.715251i	$-0.253688\pi$
$607$	34.4365	1.39773	0.698867	+	0.715251i	$0.253688\pi$
$608$	0	0
$609$	43.8014	1.77492
$610$	0	0
$611$	−0.833426	−0.0337168
$612$	0	0
$613$	−15.6670	−0.632783	−0.316391	−	0.948629i	$-0.602471\pi$
$613$	−15.6670	−0.632783	−0.316391	+	0.948629i	$0.602471\pi$
$614$	0	0
$615$	28.2119	1.13761
$616$	0	0
$617$	−9.86658	−0.397213	−0.198607	−	0.980079i	$-0.563642\pi$
$617$	−9.86658	−0.397213	−0.198607	+	0.980079i	$0.563642\pi$
$618$	0	0
$619$	−17.7457	−0.713259	−0.356629	−	0.934246i	$-0.616074\pi$
$619$	−17.7457	−0.713259	−0.356629	+	0.934246i	$0.616074\pi$
$620$	0	0
$621$	−3.29034	−0.132037
$622$	0	0
$623$	23.6944	0.949294
$624$	0	0
$625$	−6.13642	−0.245457
$626$	0	0
$627$	19.9383	0.796258
$628$	0	0
$629$	0	0
$630$	0	0
$631$	41.4514	1.65015	0.825076	−	0.565021i	$-0.191132\pi$
$631$	41.4514	1.65015	0.825076	+	0.565021i	$0.191132\pi$
$632$	0	0
$633$	37.9283	1.50751
$634$	0	0
$635$	7.50282	0.297740
$636$	0	0
$637$	3.63041	0.143842
$638$	0	0
$639$	17.1103	0.676872
$640$	0	0
$641$	9.06623	0.358095	0.179047	−	0.983840i	$-0.442698\pi$
$641$	9.06623	0.358095	0.179047	+	0.983840i	$0.442698\pi$
$642$	0	0
$643$	−41.1618	−1.62326	−0.811632	−	0.584169i	$-0.801420\pi$
$643$	−41.1618	−1.62326	−0.811632	+	0.584169i	$0.801420\pi$
$644$	0	0
$645$	−16.6186	−0.654357
$646$	0	0
$647$	8.21161	0.322832	0.161416	−	0.986886i	$-0.448394\pi$
$647$	8.21161	0.322832	0.161416	+	0.986886i	$0.448394\pi$
$648$	0	0
$649$	41.1641	1.61583
$650$	0	0
$651$	−22.1569	−0.868397
$652$	0	0
$653$	−5.12970	−0.200741	−0.100370	−	0.994950i	$-0.532003\pi$
$653$	−5.12970	−0.200741	−0.100370	+	0.994950i	$0.532003\pi$
$654$	0	0
$655$	−8.82852	−0.344959
$656$	0	0
$657$	14.2307	0.555193
$658$	0	0
$659$	−14.7505	−0.574596	−0.287298	−	0.957841i	$-0.592757\pi$
$659$	−14.7505	−0.574596	−0.287298	+	0.957841i	$0.592757\pi$
$660$	0	0
$661$	22.1806	0.862725	0.431362	−	0.902179i	$-0.358033\pi$
$661$	22.1806	0.862725	0.431362	+	0.902179i	$0.358033\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.09358	0.236299
$666$	0	0
$667$	9.30795	0.360405
$668$	0	0
$669$	4.11529	0.159106
$670$	0	0
$671$	5.66000	0.218502
$672$	0	0
$673$	37.2607	1.43630	0.718148	−	0.695891i	$-0.244990\pi$
$673$	37.2607	1.43630	0.718148	+	0.695891i	$0.244990\pi$
$674$	0	0
$675$	−7.31461	−0.281539
$676$	0	0
$677$	10.4097	0.400075	0.200038	−	0.979788i	$-0.435894\pi$
$677$	10.4097	0.400075	0.200038	+	0.979788i	$0.435894\pi$
$678$	0	0
$679$	−25.2955	−0.970752
$680$	0	0
$681$	42.7186	1.63698
$682$	0	0
$683$	−0.741019	−0.0283543	−0.0141772	−	0.999899i	$-0.504513\pi$
$683$	−0.741019	−0.0283543	−0.0141772	+	0.999899i	$0.504513\pi$
$684$	0	0
$685$	12.5266	0.478616
$686$	0	0
$687$	11.9083	0.454331
$688$	0	0
$689$	9.92483	0.378106
$690$	0	0
$691$	−36.2813	−1.38021	−0.690104	−	0.723710i	$-0.742435\pi$
$691$	−36.2813	−1.38021	−0.690104	+	0.723710i	$0.742435\pi$
$692$	0	0
$693$	−23.9984	−0.911624
$694$	0	0
$695$	−32.2303	−1.22256
$696$	0	0
$697$	0	0
$698$	0	0
$699$	54.2995	2.05380
$700$	0	0
$701$	−31.4714	−1.18866	−0.594329	−	0.804222i	$-0.702582\pi$
$701$	−31.4714	−1.18866	−0.594329	+	0.804222i	$0.702582\pi$
$702$	0	0
$703$	−12.9768	−0.489428
$704$	0	0
$705$	−0.719488	−0.0270975
$706$	0	0
$707$	27.5971	1.03789
$708$	0	0
$709$	24.6786	0.926825	0.463412	−	0.886143i	$-0.346625\pi$
$709$	24.6786	0.926825	0.463412	+	0.886143i	$0.346625\pi$
$710$	0	0
$711$	18.5961	0.697410
$712$	0	0
$713$	−4.70842	−0.176332
$714$	0	0
$715$	36.6866	1.37200
$716$	0	0
$717$	−65.5134	−2.44664
$718$	0	0
$719$	9.43430	0.351840	0.175920	−	0.984404i	$-0.443710\pi$
$719$	9.43430	0.351840	0.175920	+	0.984404i	$0.443710\pi$
$720$	0	0
$721$	19.9674	0.743623
$722$	0	0
$723$	−18.9762	−0.705732
$724$	0	0
$725$	20.6921	0.768484
$726$	0	0
$727$	18.1106	0.671685	0.335843	−	0.941918i	$-0.390979\pi$
$727$	18.1106	0.671685	0.335843	+	0.941918i	$0.390979\pi$
$728$	0	0
$729$	0.323978	0.0119992
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−16.5979	−0.613056	−0.306528	−	0.951862i	$-0.599167\pi$
$733$	−16.5979	−0.613056	−0.306528	+	0.951862i	$0.599167\pi$
$734$	0	0
$735$	3.13410	0.115603
$736$	0	0
$737$	13.2178	0.486884
$738$	0	0
$739$	12.4648	0.458527	0.229263	−	0.973364i	$-0.426368\pi$
$739$	12.4648	0.458527	0.229263	+	0.973364i	$0.426368\pi$
$740$	0	0
$741$	13.3104	0.488969
$742$	0	0
$743$	−17.7169	−0.649970	−0.324985	−	0.945719i	$-0.605359\pi$
$743$	−17.7169	−0.649970	−0.324985	+	0.945719i	$0.605359\pi$
$744$	0	0
$745$	31.1543	1.14140
$746$	0	0
$747$	21.9515	0.803162
$748$	0	0
$749$	−27.5781	−1.00768
$750$	0	0
$751$	−46.5548	−1.69881	−0.849404	−	0.527743i	$-0.823038\pi$
$751$	−46.5548	−1.69881	−0.849404	+	0.527743i	$0.823038\pi$
$752$	0	0
$753$	−28.6788	−1.04511
$754$	0	0
$755$	−2.92963	−0.106620
$756$	0	0
$757$	20.6174	0.749354	0.374677	−	0.927155i	$-0.377754\pi$
$757$	20.6174	0.749354	0.374677	+	0.927155i	$0.377754\pi$
$758$	0	0
$759$	−14.3766	−0.521837
$760$	0	0
$761$	39.7886	1.44234	0.721168	−	0.692760i	$-0.243606\pi$
$761$	39.7886	1.44234	0.721168	+	0.692760i	$0.243606\pi$
$762$	0	0
$763$	43.2202	1.56467
$764$	0	0
$765$	0	0
$766$	0	0
$767$	27.4803	0.992257
$768$	0	0
$769$	35.0869	1.26527	0.632633	−	0.774451i	$-0.281974\pi$
$769$	35.0869	1.26527	0.632633	+	0.774451i	$0.281974\pi$
$770$	0	0
$771$	8.77817	0.316138
$772$	0	0
$773$	−15.8480	−0.570014	−0.285007	−	0.958525i	$-0.591996\pi$
$773$	−15.8480	−0.570014	−0.285007	+	0.958525i	$0.591996\pi$
$774$	0	0
$775$	−10.4671	−0.375988
$776$	0	0
$777$	44.0320	1.57964
$778$	0	0
$779$	−12.9942	−0.465565
$780$	0	0
$781$	−61.2349	−2.19116
$782$	0	0
$783$	−23.9984	−0.857633
$784$	0	0
$785$	25.7284	0.918286
$786$	0	0
$787$	19.7284	0.703240	0.351620	−	0.936143i	$-0.385631\pi$
$787$	19.7284	0.703240	0.351620	+	0.936143i	$0.385631\pi$
$788$	0	0
$789$	−17.2570	−0.614365
$790$	0	0
$791$	7.36747	0.261957
$792$	0	0
$793$	3.77850	0.134178
$794$	0	0
$795$	8.56801	0.303876
$796$	0	0
$797$	5.27058	0.186693	0.0933467	−	0.995634i	$-0.470243\pi$
$797$	5.27058	0.186693	0.0933467	+	0.995634i	$0.470243\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	15.8494	0.560009
$802$	0	0
$803$	−50.9294	−1.79726
$804$	0	0
$805$	−4.39381	−0.154861
$806$	0	0
$807$	52.8643	1.86091
$808$	0	0
$809$	10.4484	0.367346	0.183673	−	0.982987i	$-0.441201\pi$
$809$	10.4484	0.367346	0.183673	+	0.982987i	$0.441201\pi$
$810$	0	0
$811$	11.7259	0.411753	0.205877	−	0.978578i	$-0.433995\pi$
$811$	11.7259	0.411753	0.205877	+	0.978578i	$0.433995\pi$
$812$	0	0
$813$	6.34393	0.222491
$814$	0	0
$815$	22.2401	0.779035
$816$	0	0
$817$	7.65442	0.267794
$818$	0	0
$819$	−16.0208	−0.559813
$820$	0	0
$821$	37.0630	1.29351	0.646754	−	0.762699i	$-0.276126\pi$
$821$	37.0630	1.29351	0.646754	+	0.762699i	$0.276126\pi$
$822$	0	0
$823$	9.45683	0.329645	0.164822	−	0.986323i	$-0.447295\pi$
$823$	9.45683	0.329645	0.164822	+	0.986323i	$0.447295\pi$
$824$	0	0
$825$	−31.9599	−1.11270
$826$	0	0
$827$	−7.97078	−0.277171	−0.138586	−	0.990350i	$-0.544256\pi$
$827$	−7.97078	−0.277171	−0.138586	+	0.990350i	$0.544256\pi$
$828$	0	0
$829$	12.3271	0.428136	0.214068	−	0.976819i	$-0.431329\pi$
$829$	12.3271	0.428136	0.214068	+	0.976819i	$0.431329\pi$
$830$	0	0
$831$	17.7732	0.616546
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−36.7818	−1.27289
$836$	0	0
$837$	12.1396	0.419605
$838$	0	0
$839$	38.2843	1.32172	0.660860	−	0.750509i	$-0.270192\pi$
$839$	38.2843	1.32172	0.660860	+	0.750509i	$0.270192\pi$
$840$	0	0
$841$	38.8883	1.34098
$842$	0	0
$843$	−6.40391	−0.220562
$844$	0	0
$845$	3.98307	0.137022
$846$	0	0
$847$	58.7661	2.01923
$848$	0	0
$849$	−15.7324	−0.539934
$850$	0	0
$851$	9.35696	0.320752
$852$	0	0
$853$	−7.04137	−0.241092	−0.120546	−	0.992708i	$-0.538465\pi$
$853$	−7.04137	−0.241092	−0.120546	+	0.992708i	$0.538465\pi$
$854$	0	0
$855$	4.07605	0.139398
$856$	0	0
$857$	54.4516	1.86003	0.930016	−	0.367520i	$-0.119793\pi$
$857$	54.4516	1.86003	0.930016	+	0.367520i	$0.119793\pi$
$858$	0	0
$859$	5.59907	0.191038	0.0955189	−	0.995428i	$-0.469549\pi$
$859$	5.59907	0.191038	0.0955189	+	0.995428i	$0.469549\pi$
$860$	0	0
$861$	44.0912	1.50262
$862$	0	0
$863$	−14.3588	−0.488779	−0.244389	−	0.969677i	$-0.578587\pi$
$863$	−14.3588	−0.488779	−0.244389	+	0.969677i	$0.578587\pi$
$864$	0	0
$865$	−29.4836	−1.00247
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−66.5526	−2.25764
$870$	0	0
$871$	8.82392	0.298987
$872$	0	0
$873$	−16.9204	−0.572668
$874$	0	0
$875$	−29.2148	−0.987639
$876$	0	0
$877$	32.1960	1.08718	0.543591	−	0.839350i	$-0.317064\pi$
$877$	32.1960	1.08718	0.543591	+	0.839350i	$0.317064\pi$
$878$	0	0
$879$	2.92668	0.0987145
$880$	0	0
$881$	36.2745	1.22212	0.611060	−	0.791584i	$-0.290743\pi$
$881$	36.2745	1.22212	0.611060	+	0.791584i	$0.290743\pi$
$882$	0	0
$883$	15.5374	0.522876	0.261438	−	0.965220i	$-0.415803\pi$
$883$	15.5374	0.522876	0.261438	+	0.965220i	$0.415803\pi$
$884$	0	0
$885$	23.7235	0.797457
$886$	0	0
$887$	−40.9925	−1.37639	−0.688197	−	0.725524i	$-0.741597\pi$
$887$	−40.9925	−1.37639	−0.688197	+	0.725524i	$0.741597\pi$
$888$	0	0
$889$	11.7258	0.393272
$890$	0	0
$891$	66.2679	2.22006
$892$	0	0
$893$	0.331391	0.0110896
$894$	0	0
$895$	23.9601	0.800898
$896$	0	0
$897$	−9.59751	−0.320451
$898$	0	0
$899$	−34.3412	−1.14534
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−25.9726	−0.864313
$904$	0	0
$905$	35.4988	1.18002
$906$	0	0
$907$	26.5196	0.880568	0.440284	−	0.897859i	$-0.354878\pi$
$907$	26.5196	0.880568	0.440284	+	0.897859i	$0.354878\pi$
$908$	0	0
$909$	18.4599	0.612277
$910$	0	0
$911$	36.7889	1.21887	0.609435	−	0.792836i	$-0.291397\pi$
$911$	36.7889	1.21887	0.609435	+	0.792836i	$0.291397\pi$
$912$	0	0
$913$	−78.5608	−2.59998
$914$	0	0
$915$	3.26194	0.107836
$916$	0	0
$917$	−13.7977	−0.455641
$918$	0	0
$919$	−39.2016	−1.29314	−0.646570	−	0.762854i	$-0.723797\pi$
$919$	−39.2016	−1.29314	−0.646570	+	0.762854i	$0.723797\pi$
$920$	0	0
$921$	55.9646	1.84410
$922$	0	0
$923$	−40.8791	−1.34555
$924$	0	0
$925$	20.8010	0.683933
$926$	0	0
$927$	13.3563	0.438679
$928$	0	0
$929$	−20.0588	−0.658108	−0.329054	−	0.944311i	$-0.606730\pi$
$929$	−20.0588	−0.658108	−0.329054	+	0.944311i	$0.606730\pi$
$930$	0	0
$931$	−1.44354	−0.0473102
$932$	0	0
$933$	−12.4960	−0.409101
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−20.5652	−0.671835	−0.335917	−	0.941891i	$-0.609046\pi$
$937$	−20.5652	−0.671835	−0.335917	+	0.941891i	$0.609046\pi$
$938$	0	0
$939$	7.82874	0.255481
$940$	0	0
$941$	2.14891	0.0700526	0.0350263	−	0.999386i	$-0.488849\pi$
$941$	2.14891	0.0700526	0.0350263	+	0.999386i	$0.488849\pi$
$942$	0	0
$943$	9.36953	0.305114
$944$	0	0
$945$	11.3284	0.368514
$946$	0	0
$947$	49.1259	1.59638	0.798188	−	0.602408i	$-0.205792\pi$
$947$	49.1259	1.59638	0.798188	+	0.602408i	$0.205792\pi$
$948$	0	0
$949$	−33.9994	−1.10367
$950$	0	0
$951$	3.37042	0.109293
$952$	0	0
$953$	−0.461467	−0.0149484	−0.00747419	−	0.999972i	$-0.502379\pi$
$953$	−0.461467	−0.0149484	−0.00747419	+	0.999972i	$0.502379\pi$
$954$	0	0
$955$	−17.3936	−0.562843
$956$	0	0
$957$	−104.857	−3.38954
$958$	0	0
$959$	19.5773	0.632183
$960$	0	0
$961$	−13.6285	−0.439630
$962$	0	0
$963$	−18.4472	−0.594452
$964$	0	0
$965$	−8.13883	−0.261998
$966$	0	0
$967$	35.2346	1.13307	0.566534	−	0.824038i	$-0.308284\pi$
$967$	35.2346	1.13307	0.566534	+	0.824038i	$0.308284\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.11831	−0.164254	−0.0821271	−	0.996622i	$-0.526171\pi$
$971$	−5.11831	−0.164254	−0.0821271	+	0.996622i	$0.526171\pi$
$972$	0	0
$973$	−50.3714	−1.61483
$974$	0	0
$975$	−21.3358	−0.683292
$976$	0	0
$977$	−34.6117	−1.10732	−0.553662	−	0.832741i	$-0.686770\pi$
$977$	−34.6117	−1.10732	−0.553662	+	0.832741i	$0.686770\pi$
$978$	0	0
$979$	−56.7223	−1.81285
$980$	0	0
$981$	28.9103	0.923035
$982$	0	0
$983$	−17.7230	−0.565277	−0.282638	−	0.959227i	$-0.591210\pi$
$983$	−17.7230	−0.565277	−0.282638	+	0.959227i	$0.591210\pi$
$984$	0	0
$985$	−4.64828	−0.148107
$986$	0	0
$987$	−1.12446	−0.0357919
$988$	0	0
$989$	−5.51926	−0.175502
$990$	0	0
$991$	33.7595	1.07241	0.536203	−	0.844089i	$-0.319858\pi$
$991$	33.7595	1.07241	0.536203	+	0.844089i	$0.319858\pi$
$992$	0	0
$993$	−30.4247	−0.965499
$994$	0	0
$995$	23.5351	0.746114
$996$	0	0
$997$	8.61257	0.272763	0.136381	−	0.990656i	$-0.456453\pi$
$997$	8.61257	0.272763	0.136381	+	0.990656i	$0.456453\pi$
$998$	0	0
$999$	−24.1248	−0.763274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9248.2.a.bz.1.5		20
4.3	odd	2		9248.2.a.by.1.16		20
17.10	odd	16		544.2.bb.f.321.1	yes	20
17.12	odd	16		544.2.bb.f.161.1	yes	20
17.16	even	2	inner	9248.2.a.bz.1.16		20
68.27	even	16		544.2.bb.e.321.5	yes	20
68.63	even	16		544.2.bb.e.161.5	✓	20
68.67	odd	2		9248.2.a.by.1.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.bb.e.161.5	✓	20	68.63	even	16
544.2.bb.e.321.5	yes	20	68.27	even	16
544.2.bb.f.161.1	yes	20	17.12	odd	16
544.2.bb.f.321.1	yes	20	17.10	odd	16
9248.2.a.by.1.5		20	68.67	odd	2
9248.2.a.by.1.16		20	4.3	odd	2
9248.2.a.bz.1.5		20	1.1	even	1	trivial
9248.2.a.bz.1.16		20	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)

\(f(q)\)	\(=\)	\(q-2.15620 q^{3} +1.57755 q^{5} +2.46548 q^{7} +1.64918 q^{9} -5.90216 q^{11} -3.94016 q^{13} -3.40150 q^{15} +1.56671 q^{19} -5.31607 q^{21} -1.12968 q^{23} -2.51134 q^{25} +2.91263 q^{27} -8.23944 q^{29} +4.16791 q^{31} +12.7262 q^{33} +3.88942 q^{35} -8.28282 q^{37} +8.49576 q^{39} -8.29395 q^{41} +4.88567 q^{43} +2.60166 q^{45} +0.211521 q^{47} -0.921386 q^{49} -2.51889 q^{53} -9.31094 q^{55} -3.37813 q^{57} -6.97442 q^{59} -0.958971 q^{61} +4.06604 q^{63} -6.21579 q^{65} -2.23948 q^{67} +2.43582 q^{69} +10.3750 q^{71} +8.62894 q^{73} +5.41495 q^{75} -14.5517 q^{77} +11.2760 q^{79} -11.2277 q^{81} +13.3105 q^{83} +17.7658 q^{87} +9.61042 q^{89} -9.71440 q^{91} -8.98683 q^{93} +2.47156 q^{95} -10.2598 q^{97} -9.73374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 28 q^{9} - 8 q^{13} + 16 q^{15} + 40 q^{19} - 32 q^{21} + 28 q^{25} + 32 q^{35} + 40 q^{43} + 32 q^{47} + 36 q^{49} - 40 q^{53} + 48 q^{55} + 8 q^{59} + 72 q^{67} - 48 q^{69} - 48 q^{77} + 36 q^{81}+ \cdots - 32 q^{93}+O(q^{100}) \) 20 * q + 28 * q^9 - 8 * q^13 + 16 * q^15 + 40 * q^19 - 32 * q^21 + 28 * q^25 + 32 * q^35 + 40 * q^43 + 32 * q^47 + 36 * q^49 - 40 * q^53 + 48 * q^55 + 8 * q^59 + 72 * q^67 - 48 * q^69 - 48 * q^77 + 36 * q^81 + 24 * q^83 + 96 * q^87 + 64 * q^89 - 32 * q^93

Introduction

L-functions

Modular forms

Varieties

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Database

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