LMFDB - Embedded newform 9248.2.a.r.1.2

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:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ 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)

$f(q)$	$=$	$q-1.34730 q^{3} +2.22668 q^{5} +1.46791 q^{7} -1.18479 q^{9} -1.87939 q^{11} +5.87939 q^{13} -3.00000 q^{15} +2.81521 q^{19} -1.97771 q^{21} +9.24897 q^{23} -0.0418891 q^{25} +5.63816 q^{27} -7.00774 q^{29} +5.26857 q^{31} +2.53209 q^{33} +3.26857 q^{35} -0.120615 q^{37} -7.92127 q^{39} +3.59627 q^{41} +6.29086 q^{43} -2.63816 q^{45} +2.33275 q^{47} -4.84524 q^{49} -3.92902 q^{53} -4.18479 q^{55} -3.79292 q^{57} +7.24897 q^{59} -3.22668 q^{61} -1.73917 q^{63} +13.0915 q^{65} +0.0145479 q^{67} -12.4611 q^{69} +1.67230 q^{71} -11.5253 q^{73} +0.0564370 q^{75} -2.75877 q^{77} +5.54664 q^{79} -4.04189 q^{81} +9.53714 q^{83} +9.44150 q^{87} +2.35504 q^{89} +8.63041 q^{91} -7.09833 q^{93} +6.26857 q^{95} +7.61081 q^{97} +2.22668 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 9 q^{7} + 12 q^{13} - 9 q^{15} + 12 q^{19} - 12 q^{21} + 15 q^{23} + 3 q^{25} + 3 q^{29} + 6 q^{31} + 3 q^{33} - 6 q^{37} - 15 q^{39} - 3 q^{41} + 3 q^{43} + 9 q^{45} - 12 q^{47} + 12 q^{49}+ \cdots + 27 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 + 9 * q^7 + 12 * q^13 - 9 * q^15 + 12 * q^19 - 12 * q^21 + 15 * q^23 + 3 * q^25 + 3 * q^29 + 6 * q^31 + 3 * q^33 - 6 * q^37 - 15 * q^39 - 3 * q^41 + 3 * q^43 + 9 * q^45 - 12 * q^47 + 12 * q^49 + 21 * q^53 - 9 * q^55 - 21 * q^57 + 9 * q^59 - 3 * q^61 + 9 * q^63 + 9 * q^65 + 18 * q^67 + 9 * q^71 + 12 * q^73 + 15 * q^75 + 3 * q^77 + 30 * q^79 - 9 * q^81 + 9 * q^83 - 12 * q^87 - 18 * q^89 + 33 * q^91 - 33 * q^93 + 9 * q^95 + 27 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.34730	−0.777862	−0.388931	−	0.921267i	$-0.627156\pi$
$3$	−1.34730	−0.777862	−0.388931	+	0.921267i	$0.627156\pi$
$4$	0	0
$5$	2.22668	0.995802	0.497901	−	0.867234i	$-0.334104\pi$
$5$	2.22668	0.995802	0.497901	+	0.867234i	$0.334104\pi$
$6$	0	0
$7$	1.46791	0.554818	0.277409	−	0.960752i	$-0.410524\pi$
$7$	1.46791	0.554818	0.277409	+	0.960752i	$0.410524\pi$
$8$	0	0
$9$	−1.18479	−0.394931
$10$	0	0
$11$	−1.87939	−0.566656	−0.283328	−	0.959023i	$-0.591439\pi$
$11$	−1.87939	−0.566656	−0.283328	+	0.959023i	$0.591439\pi$
$12$	0	0
$13$	5.87939	1.63065	0.815324	−	0.579005i	$-0.196559\pi$
$13$	5.87939	1.63065	0.815324	+	0.579005i	$0.196559\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	2.81521	0.645853	0.322926	−	0.946424i	$-0.395333\pi$
$19$	2.81521	0.645853	0.322926	+	0.946424i	$0.395333\pi$
$20$	0	0
$21$	−1.97771	−0.431572
$22$	0	0
$23$	9.24897	1.92854	0.964272	−	0.264915i	$-0.0853438\pi$
$23$	9.24897	1.92854	0.964272	+	0.264915i	$0.0853438\pi$
$24$	0	0
$25$	−0.0418891	−0.00837781
$26$	0	0
$27$	5.63816	1.08506
$28$	0	0
$29$	−7.00774	−1.30130	−0.650652	−	0.759376i	$-0.725504\pi$
$29$	−7.00774	−1.30130	−0.650652	+	0.759376i	$0.725504\pi$
$30$	0	0
$31$	5.26857	0.946263	0.473132	−	0.880992i	$-0.343124\pi$
$31$	5.26857	0.946263	0.473132	+	0.880992i	$0.343124\pi$
$32$	0	0
$33$	2.53209	0.440780
$34$	0	0
$35$	3.26857	0.552489
$36$	0	0
$37$	−0.120615	−0.0198289	−0.00991447	−	0.999951i	$-0.503156\pi$
$37$	−0.120615	−0.0198289	−0.00991447	+	0.999951i	$0.503156\pi$
$38$	0	0
$39$	−7.92127	−1.26842
$40$	0	0
$41$	3.59627	0.561642	0.280821	−	0.959760i	$-0.409393\pi$
$41$	3.59627	0.561642	0.280821	+	0.959760i	$0.409393\pi$
$42$	0	0
$43$	6.29086	0.959347	0.479674	−	0.877447i	$-0.340755\pi$
$43$	6.29086	0.959347	0.479674	+	0.877447i	$0.340755\pi$
$44$	0	0
$45$	−2.63816	−0.393273
$46$	0	0
$47$	2.33275	0.340266	0.170133	−	0.985421i	$-0.445580\pi$
$47$	2.33275	0.340266	0.170133	+	0.985421i	$0.445580\pi$
$48$	0	0
$49$	−4.84524	−0.692177
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.92902	−0.539692	−0.269846	−	0.962904i	$-0.586973\pi$
$53$	−3.92902	−0.539692	−0.269846	+	0.962904i	$0.586973\pi$
$54$	0	0
$55$	−4.18479	−0.564277
$56$	0	0
$57$	−3.79292	−0.502384
$58$	0	0
$59$	7.24897	0.943736	0.471868	−	0.881669i	$-0.343580\pi$
$59$	7.24897	0.943736	0.471868	+	0.881669i	$0.343580\pi$
$60$	0	0
$61$	−3.22668	−0.413134	−0.206567	−	0.978432i	$-0.566229\pi$
$61$	−3.22668	−0.413134	−0.206567	+	0.978432i	$0.566229\pi$
$62$	0	0
$63$	−1.73917	−0.219115
$64$	0	0
$65$	13.0915	1.62380
$66$	0	0
$67$	0.0145479	0.00177731	0.000888656	−	1.00000i	$-0.499717\pi$
$67$	0.0145479	0.00177731	0.000888656		1.00000i	$0.499717\pi$
$68$	0	0
$69$	−12.4611	−1.50014
$70$	0	0
$71$	1.67230	0.198466	0.0992330	−	0.995064i	$-0.468361\pi$
$71$	1.67230	0.198466	0.0992330	+	0.995064i	$0.468361\pi$
$72$	0	0
$73$	−11.5253	−1.34893	−0.674466	−	0.738306i	$-0.735626\pi$
$73$	−11.5253	−1.34893	−0.674466	+	0.738306i	$0.735626\pi$
$74$	0	0
$75$	0.0564370	0.00651678
$76$	0	0
$77$	−2.75877	−0.314391
$78$	0	0
$79$	5.54664	0.624045	0.312023	−	0.950075i	$-0.398993\pi$
$79$	5.54664	0.624045	0.312023	+	0.950075i	$0.398993\pi$
$80$	0	0
$81$	−4.04189	−0.449099
$82$	0	0
$83$	9.53714	1.04684	0.523419	−	0.852076i	$-0.324656\pi$
$83$	9.53714	1.04684	0.523419	+	0.852076i	$0.324656\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.44150	1.01224
$88$	0	0
$89$	2.35504	0.249633	0.124817	−	0.992180i	$-0.460166\pi$
$89$	2.35504	0.249633	0.124817	+	0.992180i	$0.460166\pi$
$90$	0	0
$91$	8.63041	0.904713
$92$	0	0
$93$	−7.09833	−0.736062
$94$	0	0
$95$	6.26857	0.643142
$96$	0	0
$97$	7.61081	0.772761	0.386381	−	0.922339i	$-0.373725\pi$
$97$	7.61081	0.772761	0.386381	+	0.922339i	$0.373725\pi$
$98$	0	0
$99$	2.22668	0.223790
$100$	0	0
$101$	−13.7861	−1.37177	−0.685885	−	0.727710i	$-0.740585\pi$
$101$	−13.7861	−1.37177	−0.685885	+	0.727710i	$0.740585\pi$
$102$	0	0
$103$	−16.0496	−1.58142	−0.790709	−	0.612193i	$-0.790288\pi$
$103$	−16.0496	−1.58142	−0.790709	+	0.612193i	$0.790288\pi$
$104$	0	0
$105$	−4.40373	−0.429760
$106$	0	0
$107$	−19.3628	−1.87187	−0.935935	−	0.352172i	$-0.885443\pi$
$107$	−19.3628	−1.87187	−0.935935	+	0.352172i	$0.885443\pi$
$108$	0	0
$109$	13.7520	1.31720	0.658600	−	0.752494i	$-0.271149\pi$
$109$	13.7520	1.31720	0.658600	+	0.752494i	$0.271149\pi$
$110$	0	0
$111$	0.162504	0.0154242
$112$	0	0
$113$	8.15570	0.767223	0.383612	−	0.923494i	$-0.374680\pi$
$113$	8.15570	0.767223	0.383612	+	0.923494i	$0.374680\pi$
$114$	0	0
$115$	20.5945	1.92045
$116$	0	0
$117$	−6.96585	−0.643993
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.46791	−0.678901
$122$	0	0
$123$	−4.84524	−0.436880
$124$	0	0
$125$	−11.2267	−1.00414
$126$	0	0
$127$	−20.1061	−1.78412	−0.892062	−	0.451912i	$-0.850742\pi$
$127$	−20.1061	−1.78412	−0.892062	+	0.451912i	$0.850742\pi$
$128$	0	0
$129$	−8.47565	−0.746240
$130$	0	0
$131$	5.80066	0.506806	0.253403	−	0.967361i	$-0.418450\pi$
$131$	5.80066	0.506806	0.253403	+	0.967361i	$0.418450\pi$
$132$	0	0
$133$	4.13247	0.358331
$134$	0	0
$135$	12.5544	1.08051
$136$	0	0
$137$	19.5749	1.67240	0.836199	−	0.548426i	$-0.184773\pi$
$137$	19.5749	1.67240	0.836199	+	0.548426i	$0.184773\pi$
$138$	0	0
$139$	20.4611	1.73549	0.867744	−	0.497011i	$-0.165569\pi$
$139$	20.4611	1.73549	0.867744	+	0.497011i	$0.165569\pi$
$140$	0	0
$141$	−3.14290	−0.264680
$142$	0	0
$143$	−11.0496	−0.924016
$144$	0	0
$145$	−15.6040	−1.29584
$146$	0	0
$147$	6.52797	0.538418
$148$	0	0
$149$	12.8033	1.04889	0.524446	−	0.851444i	$-0.324273\pi$
$149$	12.8033	1.04889	0.524446	+	0.851444i	$0.324273\pi$
$150$	0	0
$151$	11.6604	0.948914	0.474457	−	0.880279i	$-0.342645\pi$
$151$	11.6604	0.948914	0.474457	+	0.880279i	$0.342645\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11.7314	0.942291
$156$	0	0
$157$	−6.63816	−0.529783	−0.264891	−	0.964278i	$-0.585336\pi$
$157$	−6.63816	−0.529783	−0.264891	+	0.964278i	$0.585336\pi$
$158$	0	0
$159$	5.29355	0.419806
$160$	0	0
$161$	13.5767	1.06999
$162$	0	0
$163$	0.0564370	0.00442049	0.00221024	−	0.999998i	$-0.499296\pi$
$163$	0.0564370	0.00442049	0.00221024	+	0.999998i	$0.499296\pi$
$164$	0	0
$165$	5.63816	0.438930
$166$	0	0
$167$	−5.88444	−0.455351	−0.227676	−	0.973737i	$-0.573113\pi$
$167$	−5.88444	−0.455351	−0.227676	+	0.973737i	$0.573113\pi$
$168$	0	0
$169$	21.5672	1.65901
$170$	0	0
$171$	−3.33544	−0.255067
$172$	0	0
$173$	17.0523	1.29646	0.648232	−	0.761443i	$-0.275509\pi$
$173$	17.0523	1.29646	0.648232	+	0.761443i	$0.275509\pi$
$174$	0	0
$175$	−0.0614894	−0.00464816
$176$	0	0
$177$	−9.76651	−0.734096
$178$	0	0
$179$	−22.9959	−1.71879	−0.859396	−	0.511310i	$-0.829160\pi$
$179$	−22.9959	−1.71879	−0.859396	+	0.511310i	$0.829160\pi$
$180$	0	0
$181$	5.91622	0.439749	0.219875	−	0.975528i	$-0.429435\pi$
$181$	5.91622	0.439749	0.219875	+	0.975528i	$0.429435\pi$
$182$	0	0
$183$	4.34730	0.321361
$184$	0	0
$185$	−0.268571	−0.0197457
$186$	0	0
$187$	0	0
$188$	0	0
$189$	8.27631	0.602013
$190$	0	0
$191$	−14.0642	−1.01765	−0.508824	−	0.860871i	$-0.669920\pi$
$191$	−14.0642	−1.01765	−0.508824	+	0.860871i	$0.669920\pi$
$192$	0	0
$193$	19.8794	1.43095	0.715475	−	0.698639i	$-0.246210\pi$
$193$	19.8794	1.43095	0.715475	+	0.698639i	$0.246210\pi$
$194$	0	0
$195$	−17.6382	−1.26309
$196$	0	0
$197$	−9.60401	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$197$	−9.60401	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$198$	0	0
$199$	22.6459	1.60533	0.802663	−	0.596433i	$-0.203416\pi$
$199$	22.6459	1.60533	0.802663	+	0.596433i	$0.203416\pi$
$200$	0	0
$201$	−0.0196004	−0.00138250
$202$	0	0
$203$	−10.2867	−0.721988
$204$	0	0
$205$	8.00774	0.559285
$206$	0	0
$207$	−10.9581	−0.761641
$208$	0	0
$209$	−5.29086	−0.365976
$210$	0	0
$211$	−0.645897	−0.0444653	−0.0222327	−	0.999753i	$-0.507077\pi$
$211$	−0.645897	−0.0444653	−0.0222327	+	0.999753i	$0.507077\pi$
$212$	0	0
$213$	−2.25309	−0.154379
$214$	0	0
$215$	14.0077	0.955320
$216$	0	0
$217$	7.73379	0.525004
$218$	0	0
$219$	15.5280	1.04928
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2.32770	0.155874	0.0779370	−	0.996958i	$-0.475167\pi$
$223$	2.32770	0.155874	0.0779370	+	0.996958i	$0.475167\pi$
$224$	0	0
$225$	0.0496299	0.00330866
$226$	0	0
$227$	5.66044	0.375697	0.187848	−	0.982198i	$-0.439849\pi$
$227$	5.66044	0.375697	0.187848	+	0.982198i	$0.439849\pi$
$228$	0	0
$229$	10.3969	0.687048	0.343524	−	0.939144i	$-0.388379\pi$
$229$	10.3969	0.687048	0.343524	+	0.939144i	$0.388379\pi$
$230$	0	0
$231$	3.71688	0.244553
$232$	0	0
$233$	−22.0155	−1.44228	−0.721141	−	0.692788i	$-0.756382\pi$
$233$	−22.0155	−1.44228	−0.721141	+	0.692788i	$0.756382\pi$
$234$	0	0
$235$	5.19429	0.338838
$236$	0	0
$237$	−7.47296	−0.485421
$238$	0	0
$239$	5.33544	0.345121	0.172560	−	0.984999i	$-0.444796\pi$
$239$	5.33544	0.345121	0.172560	+	0.984999i	$0.444796\pi$
$240$	0	0
$241$	−8.69459	−0.560068	−0.280034	−	0.959990i	$-0.590346\pi$
$241$	−8.69459	−0.560068	−0.280034	+	0.959990i	$0.590346\pi$
$242$	0	0
$243$	−11.4688	−0.735727
$244$	0	0
$245$	−10.7888	−0.689271
$246$	0	0
$247$	16.5517	1.05316
$248$	0	0
$249$	−12.8494	−0.814295
$250$	0	0
$251$	25.1293	1.58615	0.793073	−	0.609126i	$-0.208480\pi$
$251$	25.1293	1.58615	0.793073	+	0.609126i	$0.208480\pi$
$252$	0	0
$253$	−17.3824	−1.09282
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.5476	−1.53124	−0.765618	−	0.643296i	$-0.777567\pi$
$257$	−24.5476	−1.53124	−0.765618	+	0.643296i	$0.777567\pi$
$258$	0	0
$259$	−0.177052	−0.0110015
$260$	0	0
$261$	8.30272	0.513925
$262$	0	0
$263$	−16.0128	−0.987391	−0.493696	−	0.869635i	$-0.664354\pi$
$263$	−16.0128	−0.987391	−0.493696	+	0.869635i	$0.664354\pi$
$264$	0	0
$265$	−8.74867	−0.537426
$266$	0	0
$267$	−3.17293	−0.194180
$268$	0	0
$269$	3.81016	0.232309	0.116155	−	0.993231i	$-0.462943\pi$
$269$	3.81016	0.232309	0.116155	+	0.993231i	$0.462943\pi$
$270$	0	0
$271$	−28.0770	−1.70555	−0.852777	−	0.522275i	$-0.825084\pi$
$271$	−28.0770	−1.70555	−0.852777	+	0.522275i	$0.825084\pi$
$272$	0	0
$273$	−11.6277	−0.703742
$274$	0	0
$275$	0.0787257	0.00474734
$276$	0	0
$277$	23.6905	1.42342	0.711711	−	0.702472i	$-0.247920\pi$
$277$	23.6905	1.42342	0.711711	+	0.702472i	$0.247920\pi$
$278$	0	0
$279$	−6.24216	−0.373709
$280$	0	0
$281$	−0.117926	−0.00703490	−0.00351745	−	0.999994i	$-0.501120\pi$
$281$	−0.117926	−0.00703490	−0.00351745	+	0.999994i	$0.501120\pi$
$282$	0	0
$283$	19.8675	1.18100	0.590501	−	0.807037i	$-0.298930\pi$
$283$	19.8675	1.18100	0.590501	+	0.807037i	$0.298930\pi$
$284$	0	0
$285$	−8.44562	−0.500275
$286$	0	0
$287$	5.27900	0.311610
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−10.2540	−0.601101
$292$	0	0
$293$	−8.43882	−0.493001	−0.246500	−	0.969143i	$-0.579281\pi$
$293$	−8.43882	−0.493001	−0.246500	+	0.969143i	$0.579281\pi$
$294$	0	0
$295$	16.1411	0.939774
$296$	0	0
$297$	−10.5963	−0.614858
$298$	0	0
$299$	54.3783	3.14478
$300$	0	0
$301$	9.23442	0.532263
$302$	0	0
$303$	18.5740	1.06705
$304$	0	0
$305$	−7.18479	−0.411400
$306$	0	0
$307$	30.2226	1.72489	0.862446	−	0.506148i	$-0.168931\pi$
$307$	30.2226	1.72489	0.862446	+	0.506148i	$0.168931\pi$
$308$	0	0
$309$	21.6236	1.23012
$310$	0	0
$311$	−7.78611	−0.441510	−0.220755	−	0.975329i	$-0.570852\pi$
$311$	−7.78611	−0.441510	−0.220755	+	0.975329i	$0.570852\pi$
$312$	0	0
$313$	1.64084	0.0927460	0.0463730	−	0.998924i	$-0.485234\pi$
$313$	1.64084	0.0927460	0.0463730	+	0.998924i	$0.485234\pi$
$314$	0	0
$315$	−3.87258	−0.218195
$316$	0	0
$317$	13.4142	0.753414	0.376707	−	0.926332i	$-0.377056\pi$
$317$	13.4142	0.753414	0.376707	+	0.926332i	$0.377056\pi$
$318$	0	0
$319$	13.1702	0.737392
$320$	0	0
$321$	26.0874	1.45606
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−0.246282	−0.0136613
$326$	0	0
$327$	−18.5280	−1.02460
$328$	0	0
$329$	3.42427	0.188786
$330$	0	0
$331$	−29.7229	−1.63372	−0.816858	−	0.576838i	$-0.804286\pi$
$331$	−29.7229	−1.63372	−0.816858	+	0.576838i	$0.804286\pi$
$332$	0	0
$333$	0.142903	0.00783106
$334$	0	0
$335$	0.0323936	0.00176985
$336$	0	0
$337$	−15.0669	−0.820744	−0.410372	−	0.911918i	$-0.634601\pi$
$337$	−15.0669	−0.820744	−0.410372	+	0.911918i	$0.634601\pi$
$338$	0	0
$339$	−10.9881	−0.596794
$340$	0	0
$341$	−9.90167	−0.536206
$342$	0	0
$343$	−17.3878	−0.938851
$344$	0	0
$345$	−27.7469	−1.49384
$346$	0	0
$347$	10.6955	0.574166	0.287083	−	0.957906i	$-0.407314\pi$
$347$	10.6955	0.574166	0.287083	+	0.957906i	$0.407314\pi$
$348$	0	0
$349$	20.7733	1.11197	0.555985	−	0.831192i	$-0.312341\pi$
$349$	20.7733	1.11197	0.555985	+	0.831192i	$0.312341\pi$
$350$	0	0
$351$	33.1489	1.76936
$352$	0	0
$353$	−22.7520	−1.21096	−0.605482	−	0.795859i	$-0.707020\pi$
$353$	−22.7520	−1.21096	−0.605482	+	0.795859i	$0.707020\pi$
$354$	0	0
$355$	3.72369	0.197633
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.4020	−0.601773	−0.300887	−	0.953660i	$-0.597283\pi$
$359$	−11.4020	−0.601773	−0.300887	+	0.953660i	$0.597283\pi$
$360$	0	0
$361$	−11.0746	−0.582874
$362$	0	0
$363$	10.0615	0.528091
$364$	0	0
$365$	−25.6631	−1.34327
$366$	0	0
$367$	16.3200	0.851895	0.425947	−	0.904748i	$-0.359941\pi$
$367$	16.3200	0.851895	0.425947	+	0.904748i	$0.359941\pi$
$368$	0	0
$369$	−4.26083	−0.221810
$370$	0	0
$371$	−5.76744	−0.299431
$372$	0	0
$373$	−33.9094	−1.75576	−0.877881	−	0.478878i	$-0.841044\pi$
$373$	−33.9094	−1.75576	−0.877881	+	0.478878i	$0.841044\pi$
$374$	0	0
$375$	15.1257	0.781086
$376$	0	0
$377$	−41.2012	−2.12197
$378$	0	0
$379$	28.3354	1.45549	0.727747	−	0.685846i	$-0.240568\pi$
$379$	28.3354	1.45549	0.727747	+	0.685846i	$0.240568\pi$
$380$	0	0
$381$	27.0888	1.38780
$382$	0	0
$383$	−19.9290	−1.01833	−0.509163	−	0.860670i	$-0.670045\pi$
$383$	−19.9290	−1.01833	−0.509163	+	0.860670i	$0.670045\pi$
$384$	0	0
$385$	−6.14290	−0.313071
$386$	0	0
$387$	−7.45336	−0.378876
$388$	0	0
$389$	1.06418	0.0539560	0.0269780	−	0.999636i	$-0.491412\pi$
$389$	1.06418	0.0539560	0.0269780	+	0.999636i	$0.491412\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−7.81521	−0.394225
$394$	0	0
$395$	12.3506	0.621426
$396$	0	0
$397$	−13.1584	−0.660400	−0.330200	−	0.943911i	$-0.607116\pi$
$397$	−13.1584	−0.660400	−0.330200	+	0.943911i	$0.607116\pi$
$398$	0	0
$399$	−5.56767	−0.278732
$400$	0	0
$401$	29.4492	1.47063	0.735313	−	0.677728i	$-0.237035\pi$
$401$	29.4492	1.47063	0.735313	+	0.677728i	$0.237035\pi$
$402$	0	0
$403$	30.9760	1.54302
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	0.226682	0.0112362
$408$	0	0
$409$	−20.3286	−1.00519	−0.502593	−	0.864523i	$-0.667621\pi$
$409$	−20.3286	−1.00519	−0.502593	+	0.864523i	$0.667621\pi$
$410$	0	0
$411$	−26.3732	−1.30089
$412$	0	0
$413$	10.6408	0.523602
$414$	0	0
$415$	21.2362	1.04244
$416$	0	0
$417$	−27.5672	−1.34997
$418$	0	0
$419$	−2.37639	−0.116094	−0.0580472	−	0.998314i	$-0.518487\pi$
$419$	−2.37639	−0.116094	−0.0580472	+	0.998314i	$0.518487\pi$
$420$	0	0
$421$	4.52435	0.220503	0.110252	−	0.993904i	$-0.464834\pi$
$421$	4.52435	0.220503	0.110252	+	0.993904i	$0.464834\pi$
$422$	0	0
$423$	−2.76382	−0.134382
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.73648	−0.229214
$428$	0	0
$429$	14.8871	0.718757
$430$	0	0
$431$	8.87670	0.427575	0.213788	−	0.976880i	$-0.431420\pi$
$431$	8.87670	0.427575	0.213788	+	0.976880i	$0.431420\pi$
$432$	0	0
$433$	33.9445	1.63127	0.815634	−	0.578568i	$-0.196388\pi$
$433$	33.9445	1.63127	0.815634	+	0.578568i	$0.196388\pi$
$434$	0	0
$435$	21.0232	1.00799
$436$	0	0
$437$	26.0378	1.24556
$438$	0	0
$439$	16.9240	0.807737	0.403868	−	0.914817i	$-0.367665\pi$
$439$	16.9240	0.807737	0.403868	+	0.914817i	$0.367665\pi$
$440$	0	0
$441$	5.74060	0.273362
$442$	0	0
$443$	21.7365	1.03273	0.516366	−	0.856368i	$-0.327285\pi$
$443$	21.7365	1.03273	0.516366	+	0.856368i	$0.327285\pi$
$444$	0	0
$445$	5.24392	0.248586
$446$	0	0
$447$	−17.2499	−0.815892
$448$	0	0
$449$	−20.8057	−0.981882	−0.490941	−	0.871193i	$-0.663347\pi$
$449$	−20.8057	−0.981882	−0.490941	+	0.871193i	$0.663347\pi$
$450$	0	0
$451$	−6.75877	−0.318258
$452$	0	0
$453$	−15.7101	−0.738124
$454$	0	0
$455$	19.2172	0.900916
$456$	0	0
$457$	12.7374	0.595831	0.297916	−	0.954592i	$-0.403709\pi$
$457$	12.7374	0.595831	0.297916	+	0.954592i	$0.403709\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	30.5722	1.42389	0.711945	−	0.702235i	$-0.247814\pi$
$461$	30.5722	1.42389	0.711945	+	0.702235i	$0.247814\pi$
$462$	0	0
$463$	−41.6732	−1.93672	−0.968360	−	0.249558i	$-0.919715\pi$
$463$	−41.6732	−1.93672	−0.968360	+	0.249558i	$0.919715\pi$
$464$	0	0
$465$	−15.8057	−0.732972
$466$	0	0
$467$	−22.3533	−1.03439	−0.517193	−	0.855869i	$-0.673023\pi$
$467$	−22.3533	−1.03439	−0.517193	+	0.855869i	$0.673023\pi$
$468$	0	0
$469$	0.0213551	0.000986085 0
$470$	0	0
$471$	8.94356	0.412098
$472$	0	0
$473$	−11.8229	−0.543620
$474$	0	0
$475$	−0.117926	−0.00541083
$476$	0	0
$477$	4.65507	0.213141
$478$	0	0
$479$	−5.99050	−0.273713	−0.136857	−	0.990591i	$-0.543700\pi$
$479$	−5.99050	−0.273713	−0.136857	+	0.990591i	$0.543700\pi$
$480$	0	0
$481$	−0.709141	−0.0323340
$482$	0	0
$483$	−18.2918	−0.832305
$484$	0	0
$485$	16.9469	0.769517
$486$	0	0
$487$	29.9786	1.35846	0.679231	−	0.733925i	$-0.262313\pi$
$487$	29.9786	1.35846	0.679231	+	0.733925i	$0.262313\pi$
$488$	0	0
$489$	−0.0760373	−0.00343853
$490$	0	0
$491$	5.66550	0.255680	0.127840	−	0.991795i	$-0.459196\pi$
$491$	5.66550	0.255680	0.127840	+	0.991795i	$0.459196\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.95811	0.222851
$496$	0	0
$497$	2.45479	0.110113
$498$	0	0
$499$	29.6236	1.32613	0.663067	−	0.748560i	$-0.269254\pi$
$499$	29.6236	1.32613	0.663067	+	0.748560i	$0.269254\pi$
$500$	0	0
$501$	7.92808	0.354201
$502$	0	0
$503$	28.7374	1.28134	0.640669	−	0.767817i	$-0.278657\pi$
$503$	28.7374	1.28134	0.640669	+	0.767817i	$0.278657\pi$
$504$	0	0
$505$	−30.6973	−1.36601
$506$	0	0
$507$	−29.0574	−1.29048
$508$	0	0
$509$	17.5699	0.778770	0.389385	−	0.921075i	$-0.372688\pi$
$509$	17.5699	0.778770	0.389385	+	0.921075i	$0.372688\pi$
$510$	0	0
$511$	−16.9181	−0.748412
$512$	0	0
$513$	15.8726	0.700791
$514$	0	0
$515$	−35.7374	−1.57478
$516$	0	0
$517$	−4.38413	−0.192814
$518$	0	0
$519$	−22.9745	−1.00847
$520$	0	0
$521$	−6.42190	−0.281349	−0.140674	−	0.990056i	$-0.544927\pi$
$521$	−6.42190	−0.281349	−0.140674	+	0.990056i	$0.544927\pi$
$522$	0	0
$523$	7.02498	0.307181	0.153590	−	0.988135i	$-0.450916\pi$
$523$	7.02498	0.307181	0.153590	+	0.988135i	$0.450916\pi$
$524$	0	0
$525$	0.0828445	0.00361563
$526$	0	0
$527$	0	0
$528$	0	0
$529$	62.5435	2.71928
$530$	0	0
$531$	−8.58853	−0.372710
$532$	0	0
$533$	21.1438	0.915841
$534$	0	0
$535$	−43.1147	−1.86401
$536$	0	0
$537$	30.9823	1.33698
$538$	0	0
$539$	9.10607	0.392226
$540$	0	0
$541$	−0.0692302	−0.00297644	−0.00148822	−	0.999999i	$-0.500474\pi$
$541$	−0.0692302	−0.00297644	−0.00148822	+	0.999999i	$0.500474\pi$
$542$	0	0
$543$	−7.97090	−0.342064
$544$	0	0
$545$	30.6212	1.31167
$546$	0	0
$547$	19.3756	0.828440	0.414220	−	0.910177i	$-0.364054\pi$
$547$	19.3756	0.828440	0.414220	+	0.910177i	$0.364054\pi$
$548$	0	0
$549$	3.82295	0.163159
$550$	0	0
$551$	−19.7282	−0.840451
$552$	0	0
$553$	8.14197	0.346232
$554$	0	0
$555$	0.361844	0.0153594
$556$	0	0
$557$	14.2695	0.604618	0.302309	−	0.953210i	$-0.402243\pi$
$557$	14.2695	0.604618	0.302309	+	0.953210i	$0.402243\pi$
$558$	0	0
$559$	36.9864	1.56436
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.60576	0.0676748	0.0338374	−	0.999427i	$-0.489227\pi$
$563$	1.60576	0.0676748	0.0338374	+	0.999427i	$0.489227\pi$
$564$	0	0
$565$	18.1601	0.764003
$566$	0	0
$567$	−5.93313	−0.249168
$568$	0	0
$569$	−7.29860	−0.305973	−0.152987	−	0.988228i	$-0.548889\pi$
$569$	−7.29860	−0.305973	−0.152987	+	0.988228i	$0.548889\pi$
$570$	0	0
$571$	−3.54933	−0.148535	−0.0742673	−	0.997238i	$-0.523662\pi$
$571$	−3.54933	−0.148535	−0.0742673	+	0.997238i	$0.523662\pi$
$572$	0	0
$573$	18.9486	0.791590
$574$	0	0
$575$	−0.387431	−0.0161570
$576$	0	0
$577$	40.6382	1.69179	0.845894	−	0.533351i	$-0.179067\pi$
$577$	40.6382	1.69179	0.845894	+	0.533351i	$0.179067\pi$
$578$	0	0
$579$	−26.7834	−1.11308
$580$	0	0
$581$	13.9997	0.580804
$582$	0	0
$583$	7.38413	0.305820
$584$	0	0
$585$	−15.5107	−0.641290
$586$	0	0
$587$	−24.4415	−1.00881	−0.504404	−	0.863468i	$-0.668288\pi$
$587$	−24.4415	−1.00881	−0.504404	+	0.863468i	$0.668288\pi$
$588$	0	0
$589$	14.8321	0.611147
$590$	0	0
$591$	12.9394	0.532258
$592$	0	0
$593$	−42.9881	−1.76531	−0.882656	−	0.470020i	$-0.844247\pi$
$593$	−42.9881	−1.76531	−0.882656	+	0.470020i	$0.844247\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−30.5107	−1.24872
$598$	0	0
$599$	36.2864	1.48262	0.741311	−	0.671161i	$-0.234204\pi$
$599$	36.2864	1.48262	0.741311	+	0.671161i	$0.234204\pi$
$600$	0	0
$601$	−22.7989	−0.929987	−0.464993	−	0.885314i	$-0.653943\pi$
$601$	−22.7989	−0.929987	−0.464993	+	0.885314i	$0.653943\pi$
$602$	0	0
$603$	−0.0172363	−0.000701915 0
$604$	0	0
$605$	−16.6287	−0.676051
$606$	0	0
$607$	−21.2003	−0.860493	−0.430246	−	0.902712i	$-0.641573\pi$
$607$	−21.2003	−0.860493	−0.430246	+	0.902712i	$0.641573\pi$
$608$	0	0
$609$	13.8593	0.561607
$610$	0	0
$611$	13.7151	0.554855
$612$	0	0
$613$	19.1274	0.772549	0.386275	−	0.922384i	$-0.373762\pi$
$613$	19.1274	0.772549	0.386275	+	0.922384i	$0.373762\pi$
$614$	0	0
$615$	−10.7888	−0.435046
$616$	0	0
$617$	37.4293	1.50685	0.753424	−	0.657535i	$-0.228401\pi$
$617$	37.4293	1.50685	0.753424	+	0.657535i	$0.228401\pi$
$618$	0	0
$619$	−26.0702	−1.04785	−0.523924	−	0.851765i	$-0.675532\pi$
$619$	−26.0702	−1.04785	−0.523924	+	0.851765i	$0.675532\pi$
$620$	0	0
$621$	52.1471	2.09259
$622$	0	0
$623$	3.45699	0.138501
$624$	0	0
$625$	−24.7888	−0.991552
$626$	0	0
$627$	7.12836	0.284679
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1.24216	−0.0494497	−0.0247249	−	0.999694i	$-0.507871\pi$
$631$	−1.24216	−0.0494497	−0.0247249	+	0.999694i	$0.507871\pi$
$632$	0	0
$633$	0.870214	0.0345879
$634$	0	0
$635$	−44.7698	−1.77664
$636$	0	0
$637$	−28.4870	−1.12870
$638$	0	0
$639$	−1.98133	−0.0783803
$640$	0	0
$641$	0.154763	0.00611277	0.00305639	−	0.999995i	$-0.499027\pi$
$641$	0.154763	0.00611277	0.00305639	+	0.999995i	$0.499027\pi$
$642$	0	0
$643$	30.5594	1.20515	0.602573	−	0.798064i	$-0.294142\pi$
$643$	30.5594	1.20515	0.602573	+	0.798064i	$0.294142\pi$
$644$	0	0
$645$	−18.8726	−0.743107
$646$	0	0
$647$	32.4593	1.27611	0.638054	−	0.769991i	$-0.279740\pi$
$647$	32.4593	1.27611	0.638054	+	0.769991i	$0.279740\pi$
$648$	0	0
$649$	−13.6236	−0.534773
$650$	0	0
$651$	−10.4197	−0.408381
$652$	0	0
$653$	−7.80840	−0.305566	−0.152783	−	0.988260i	$-0.548824\pi$
$653$	−7.80840	−0.305566	−0.152783	+	0.988260i	$0.548824\pi$
$654$	0	0
$655$	12.9162	0.504679
$656$	0	0
$657$	13.6551	0.532735
$658$	0	0
$659$	42.0069	1.63636	0.818179	−	0.574964i	$-0.194984\pi$
$659$	42.0069	1.63636	0.818179	+	0.574964i	$0.194984\pi$
$660$	0	0
$661$	28.5084	1.10885	0.554424	−	0.832235i	$-0.312939\pi$
$661$	28.5084	1.10885	0.554424	+	0.832235i	$0.312939\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.20170	0.356827
$666$	0	0
$667$	−64.8144	−2.50962
$668$	0	0
$669$	−3.13610	−0.121248
$670$	0	0
$671$	6.06418	0.234105
$672$	0	0
$673$	31.5885	1.21765	0.608824	−	0.793305i	$-0.291642\pi$
$673$	31.5885	1.21765	0.608824	+	0.793305i	$0.291642\pi$
$674$	0	0
$675$	−0.236177	−0.00909046
$676$	0	0
$677$	−44.3405	−1.70414	−0.852072	−	0.523425i	$-0.824654\pi$
$677$	−44.3405	−1.70414	−0.852072	+	0.523425i	$0.824654\pi$
$678$	0	0
$679$	11.1720	0.428742
$680$	0	0
$681$	−7.62630	−0.292240
$682$	0	0
$683$	15.6313	0.598117	0.299058	−	0.954235i	$-0.403327\pi$
$683$	15.6313	0.598117	0.299058	+	0.954235i	$0.403327\pi$
$684$	0	0
$685$	43.5871	1.66538
$686$	0	0
$687$	−14.0077	−0.534429
$688$	0	0
$689$	−23.1002	−0.880047
$690$	0	0
$691$	6.88207	0.261806	0.130903	−	0.991395i	$-0.458212\pi$
$691$	6.88207	0.261806	0.130903	+	0.991395i	$0.458212\pi$
$692$	0	0
$693$	3.26857	0.124163
$694$	0	0
$695$	45.5604	1.72820
$696$	0	0
$697$	0	0
$698$	0	0
$699$	29.6614	1.12190
$700$	0	0
$701$	38.4097	1.45072	0.725358	−	0.688372i	$-0.241674\pi$
$701$	38.4097	1.45072	0.725358	+	0.688372i	$0.241674\pi$
$702$	0	0
$703$	−0.339556	−0.0128066
$704$	0	0
$705$	−6.99825	−0.263569
$706$	0	0
$707$	−20.2368	−0.761083
$708$	0	0
$709$	1.95306	0.0733487	0.0366743	−	0.999327i	$-0.488324\pi$
$709$	1.95306	0.0733487	0.0366743	+	0.999327i	$0.488324\pi$
$710$	0	0
$711$	−6.57161	−0.246455
$712$	0	0
$713$	48.7289	1.82491
$714$	0	0
$715$	−24.6040	−0.920138
$716$	0	0
$717$	−7.18841	−0.268456
$718$	0	0
$719$	−21.3841	−0.797494	−0.398747	−	0.917061i	$-0.630555\pi$
$719$	−21.3841	−0.797494	−0.398747	+	0.917061i	$0.630555\pi$
$720$	0	0
$721$	−23.5594	−0.877399
$722$	0	0
$723$	11.7142	0.435656
$724$	0	0
$725$	0.293548	0.0109021
$726$	0	0
$727$	−25.8780	−0.959760	−0.479880	−	0.877334i	$-0.659320\pi$
$727$	−25.8780	−0.959760	−0.479880	+	0.877334i	$0.659320\pi$
$728$	0	0
$729$	27.5776	1.02139
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−2.98309	−0.110183	−0.0550914	−	0.998481i	$-0.517545\pi$
$733$	−2.98309	−0.110183	−0.0550914	+	0.998481i	$0.517545\pi$
$734$	0	0
$735$	14.5357	0.536158
$736$	0	0
$737$	−0.0273411	−0.00100712
$738$	0	0
$739$	9.75196	0.358732	0.179366	−	0.983782i	$-0.442595\pi$
$739$	9.75196	0.358732	0.179366	+	0.983782i	$0.442595\pi$
$740$	0	0
$741$	−22.3000	−0.819212
$742$	0	0
$743$	15.0746	0.553034	0.276517	−	0.961009i	$-0.410820\pi$
$743$	15.0746	0.553034	0.276517	+	0.961009i	$0.410820\pi$
$744$	0	0
$745$	28.5090	1.04449
$746$	0	0
$747$	−11.2995	−0.413428
$748$	0	0
$749$	−28.4228	−1.03855
$750$	0	0
$751$	42.9231	1.56629	0.783144	−	0.621841i	$-0.213615\pi$
$751$	42.9231	1.56629	0.783144	+	0.621841i	$0.213615\pi$
$752$	0	0
$753$	−33.8566	−1.23380
$754$	0	0
$755$	25.9641	0.944930
$756$	0	0
$757$	32.0648	1.16541	0.582707	−	0.812682i	$-0.301993\pi$
$757$	32.0648	1.16541	0.582707	+	0.812682i	$0.301993\pi$
$758$	0	0
$759$	23.4192	0.850064
$760$	0	0
$761$	−21.0556	−0.763265	−0.381633	−	0.924314i	$-0.624638\pi$
$761$	−21.0556	−0.763265	−0.381633	+	0.924314i	$0.624638\pi$
$762$	0	0
$763$	20.1867	0.730806
$764$	0	0
$765$	0	0
$766$	0	0
$767$	42.6195	1.53890
$768$	0	0
$769$	−48.4475	−1.74706	−0.873531	−	0.486769i	$-0.838175\pi$
$769$	−48.4475	−1.74706	−0.873531	+	0.486769i	$0.838175\pi$
$770$	0	0
$771$	33.0729	1.19109
$772$	0	0
$773$	−31.1702	−1.12112	−0.560558	−	0.828115i	$-0.689413\pi$
$773$	−31.1702	−1.12112	−0.560558	+	0.828115i	$0.689413\pi$
$774$	0	0
$775$	−0.220696	−0.00792762
$776$	0	0
$777$	0.238541	0.00855762
$778$	0	0
$779$	10.1242	0.362738
$780$	0	0
$781$	−3.14290	−0.112462
$782$	0	0
$783$	−39.5107	−1.41200
$784$	0	0
$785$	−14.7811	−0.527559
$786$	0	0
$787$	−34.5536	−1.23170	−0.615851	−	0.787863i	$-0.711188\pi$
$787$	−34.5536	−1.23170	−0.615851	+	0.787863i	$0.711188\pi$
$788$	0	0
$789$	21.5740	0.768054
$790$	0	0
$791$	11.9718	0.425670
$792$	0	0
$793$	−18.9709	−0.673677
$794$	0	0
$795$	11.7870	0.418043
$796$	0	0
$797$	23.5936	0.835727	0.417864	−	0.908510i	$-0.362779\pi$
$797$	23.5936	0.835727	0.417864	+	0.908510i	$0.362779\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−2.79023	−0.0985879
$802$	0	0
$803$	21.6604	0.764380
$804$	0	0
$805$	30.2309	1.06550
$806$	0	0
$807$	−5.13341	−0.180705
$808$	0	0
$809$	35.1162	1.23462	0.617309	−	0.786720i	$-0.288223\pi$
$809$	35.1162	1.23462	0.617309	+	0.786720i	$0.288223\pi$
$810$	0	0
$811$	−10.0601	−0.353256	−0.176628	−	0.984278i	$-0.556519\pi$
$811$	−10.0601	−0.353256	−0.176628	+	0.984278i	$0.556519\pi$
$812$	0	0
$813$	37.8280	1.32669
$814$	0	0
$815$	0.125667	0.00440193
$816$	0	0
$817$	17.7101	0.619597
$818$	0	0
$819$	−10.2253	−0.357299
$820$	0	0
$821$	−3.33687	−0.116457	−0.0582287	−	0.998303i	$-0.518545\pi$
$821$	−3.33687	−0.116457	−0.0582287	+	0.998303i	$0.518545\pi$
$822$	0	0
$823$	−15.3405	−0.534736	−0.267368	−	0.963595i	$-0.586154\pi$
$823$	−15.3405	−0.534736	−0.267368	+	0.963595i	$0.586154\pi$
$824$	0	0
$825$	−0.106067	−0.00369277
$826$	0	0
$827$	−2.77156	−0.0963767	−0.0481884	−	0.998838i	$-0.515345\pi$
$827$	−2.77156	−0.0963767	−0.0481884	+	0.998838i	$0.515345\pi$
$828$	0	0
$829$	18.3259	0.636486	0.318243	−	0.948009i	$-0.396907\pi$
$829$	18.3259	0.636486	0.318243	+	0.948009i	$0.396907\pi$
$830$	0	0
$831$	−31.9181	−1.10723
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−13.1028	−0.453440
$836$	0	0
$837$	29.7050	1.02676
$838$	0	0
$839$	35.3814	1.22150	0.610752	−	0.791822i	$-0.290867\pi$
$839$	35.3814	1.22150	0.610752	+	0.791822i	$0.290867\pi$
$840$	0	0
$841$	20.1084	0.693394
$842$	0	0
$843$	0.158882	0.00547218
$844$	0	0
$845$	48.0232	1.65205
$846$	0	0
$847$	−10.9622	−0.376667
$848$	0	0
$849$	−26.7674	−0.918656
$850$	0	0
$851$	−1.11556	−0.0382410
$852$	0	0
$853$	−23.4679	−0.803526	−0.401763	−	0.915744i	$-0.631602\pi$
$853$	−23.4679	−0.803526	−0.401763	+	0.915744i	$0.631602\pi$
$854$	0	0
$855$	−7.42696	−0.253997
$856$	0	0
$857$	20.3013	0.693479	0.346739	−	0.937962i	$-0.387289\pi$
$857$	20.3013	0.693479	0.346739	+	0.937962i	$0.387289\pi$
$858$	0	0
$859$	−11.6827	−0.398610	−0.199305	−	0.979938i	$-0.563868\pi$
$859$	−11.6827	−0.398610	−0.199305	+	0.979938i	$0.563868\pi$
$860$	0	0
$861$	−7.11238	−0.242389
$862$	0	0
$863$	25.1985	0.857768	0.428884	−	0.903360i	$-0.358907\pi$
$863$	25.1985	0.857768	0.428884	+	0.903360i	$0.358907\pi$
$864$	0	0
$865$	37.9701	1.29102
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−10.4243	−0.353619
$870$	0	0
$871$	0.0855328	0.00289817
$872$	0	0
$873$	−9.01724	−0.305187
$874$	0	0
$875$	−16.4798	−0.557118
$876$	0	0
$877$	−26.8648	−0.907161	−0.453581	−	0.891215i	$-0.649854\pi$
$877$	−26.8648	−0.907161	−0.453581	+	0.891215i	$0.649854\pi$
$878$	0	0
$879$	11.3696	0.383487
$880$	0	0
$881$	−22.8639	−0.770304	−0.385152	−	0.922853i	$-0.625851\pi$
$881$	−22.8639	−0.770304	−0.385152	+	0.922853i	$0.625851\pi$
$882$	0	0
$883$	34.7547	1.16959	0.584794	−	0.811182i	$-0.301176\pi$
$883$	34.7547	1.16959	0.584794	+	0.811182i	$0.301176\pi$
$884$	0	0
$885$	−21.7469	−0.731014
$886$	0	0
$887$	28.3577	0.952159	0.476080	−	0.879402i	$-0.342057\pi$
$887$	28.3577	0.952159	0.476080	+	0.879402i	$0.342057\pi$
$888$	0	0
$889$	−29.5139	−0.989865
$890$	0	0
$891$	7.59627	0.254485
$892$	0	0
$893$	6.56717	0.219762
$894$	0	0
$895$	−51.2045	−1.71158
$896$	0	0
$897$	−73.2636	−2.44620
$898$	0	0
$899$	−36.9208	−1.23138
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−12.4415	−0.414027
$904$	0	0
$905$	13.1735	0.437903
$906$	0	0
$907$	32.0060	1.06274	0.531371	−	0.847139i	$-0.321677\pi$
$907$	32.0060	1.06274	0.531371	+	0.847139i	$0.321677\pi$
$908$	0	0
$909$	16.3337	0.541754
$910$	0	0
$911$	24.7297	0.819331	0.409665	−	0.912236i	$-0.365645\pi$
$911$	24.7297	0.819331	0.409665	+	0.912236i	$0.365645\pi$
$912$	0	0
$913$	−17.9240	−0.593197
$914$	0	0
$915$	9.68004	0.320012
$916$	0	0
$917$	8.51485	0.281185
$918$	0	0
$919$	8.54252	0.281792	0.140896	−	0.990024i	$-0.455002\pi$
$919$	8.54252	0.281792	0.140896	+	0.990024i	$0.455002\pi$
$920$	0	0
$921$	−40.7187	−1.34173
$922$	0	0
$923$	9.83212	0.323628
$924$	0	0
$925$	0.00505244	0.000166123 0
$926$	0	0
$927$	19.0155	0.624550
$928$	0	0
$929$	−39.3182	−1.28999	−0.644994	−	0.764188i	$-0.723140\pi$
$929$	−39.3182	−1.28999	−0.644994	+	0.764188i	$0.723140\pi$
$930$	0	0
$931$	−13.6403	−0.447044
$932$	0	0
$933$	10.4902	0.343434
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.39868	0.241704	0.120852	−	0.992671i	$-0.461437\pi$
$937$	7.39868	0.241704	0.120852	+	0.992671i	$0.461437\pi$
$938$	0	0
$939$	−2.21070	−0.0721436
$940$	0	0
$941$	−0.517541	−0.0168714	−0.00843568	−	0.999964i	$-0.502685\pi$
$941$	−0.517541	−0.0168714	−0.00843568	+	0.999964i	$0.502685\pi$
$942$	0	0
$943$	33.2618	1.08315
$944$	0	0
$945$	18.4287	0.599486
$946$	0	0
$947$	13.9135	0.452129	0.226065	−	0.974112i	$-0.427414\pi$
$947$	13.9135	0.452129	0.226065	+	0.974112i	$0.427414\pi$
$948$	0	0
$949$	−67.7616	−2.19963
$950$	0	0
$951$	−18.0729	−0.586052
$952$	0	0
$953$	−32.3628	−1.04833	−0.524167	−	0.851616i	$-0.675623\pi$
$953$	−32.3628	−1.04833	−0.524167	+	0.851616i	$0.675623\pi$
$954$	0	0
$955$	−31.3164	−1.01338
$956$	0	0
$957$	−17.7442	−0.573589
$958$	0	0
$959$	28.7342	0.927877
$960$	0	0
$961$	−3.24216	−0.104586
$962$	0	0
$963$	22.9409	0.739259
$964$	0	0
$965$	44.2651	1.42494
$966$	0	0
$967$	−20.6058	−0.662637	−0.331318	−	0.943519i	$-0.607493\pi$
$967$	−20.6058	−0.662637	−0.331318	+	0.943519i	$0.607493\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.9753	0.641039	0.320520	−	0.947242i	$-0.396142\pi$
$971$	19.9753	0.641039	0.320520	+	0.947242i	$0.396142\pi$
$972$	0	0
$973$	30.0351	0.962881
$974$	0	0
$975$	0.331815	0.0106266
$976$	0	0
$977$	19.7929	0.633232	0.316616	−	0.948554i	$-0.397453\pi$
$977$	19.7929	0.633232	0.316616	+	0.948554i	$0.397453\pi$
$978$	0	0
$979$	−4.42602	−0.141456
$980$	0	0
$981$	−16.2932	−0.520203
$982$	0	0
$983$	−29.4492	−0.939285	−0.469643	−	0.882857i	$-0.655617\pi$
$983$	−29.4492	−0.939285	−0.469643	+	0.882857i	$0.655617\pi$
$984$	0	0
$985$	−21.3851	−0.681385
$986$	0	0
$987$	−4.61350	−0.146849
$988$	0	0
$989$	58.1840	1.85014
$990$	0	0
$991$	16.0615	0.510210	0.255105	−	0.966913i	$-0.417890\pi$
$991$	16.0615	0.510210	0.255105	+	0.966913i	$0.417890\pi$
$992$	0	0
$993$	40.0455	1.27081
$994$	0	0
$995$	50.4252	1.59859
$996$	0	0
$997$	10.7965	0.341930	0.170965	−	0.985277i	$-0.445312\pi$
$997$	10.7965	0.341930	0.170965	+	0.985277i	$0.445312\pi$
$998$	0	0
$999$	−0.680045	−0.0215157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9248.2.a.r.1.2	✓	3
4.3	odd	2		9248.2.a.w.1.2	yes	3
17.16	even	2		9248.2.a.v.1.2	yes	3
68.67	odd	2		9248.2.a.s.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9248.2.a.r.1.2	✓	3	1.1	even	1	trivial
9248.2.a.s.1.2	yes	3	68.67	odd	2
9248.2.a.v.1.2	yes	3	17.16	even	2
9248.2.a.w.1.2	yes	3	4.3	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:	\( N \)	\(=\)	\( 9248 = 2^{5} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9248.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34730 q^{3} +2.22668 q^{5} +1.46791 q^{7} -1.18479 q^{9} -1.87939 q^{11} +5.87939 q^{13} -3.00000 q^{15} +2.81521 q^{19} -1.97771 q^{21} +9.24897 q^{23} -0.0418891 q^{25} +5.63816 q^{27} -7.00774 q^{29} +5.26857 q^{31} +2.53209 q^{33} +3.26857 q^{35} -0.120615 q^{37} -7.92127 q^{39} +3.59627 q^{41} +6.29086 q^{43} -2.63816 q^{45} +2.33275 q^{47} -4.84524 q^{49} -3.92902 q^{53} -4.18479 q^{55} -3.79292 q^{57} +7.24897 q^{59} -3.22668 q^{61} -1.73917 q^{63} +13.0915 q^{65} +0.0145479 q^{67} -12.4611 q^{69} +1.67230 q^{71} -11.5253 q^{73} +0.0564370 q^{75} -2.75877 q^{77} +5.54664 q^{79} -4.04189 q^{81} +9.53714 q^{83} +9.44150 q^{87} +2.35504 q^{89} +8.63041 q^{91} -7.09833 q^{93} +6.26857 q^{95} +7.61081 q^{97} +2.22668 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 9 q^{7} + 12 q^{13} - 9 q^{15} + 12 q^{19} - 12 q^{21} + 15 q^{23} + 3 q^{25} + 3 q^{29} + 6 q^{31} + 3 q^{33} - 6 q^{37} - 15 q^{39} - 3 q^{41} + 3 q^{43} + 9 q^{45} - 12 q^{47} + 12 q^{49}+ \cdots + 27 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 + 9 * q^7 + 12 * q^13 - 9 * q^15 + 12 * q^19 - 12 * q^21 + 15 * q^23 + 3 * q^25 + 3 * q^29 + 6 * q^31 + 3 * q^33 - 6 * q^37 - 15 * q^39 - 3 * q^41 + 3 * q^43 + 9 * q^45 - 12 * q^47 + 12 * q^49 + 21 * q^53 - 9 * q^55 - 21 * q^57 + 9 * q^59 - 3 * q^61 + 9 * q^63 + 9 * q^65 + 18 * q^67 + 9 * q^71 + 12 * q^73 + 15 * q^75 + 3 * q^77 + 30 * q^79 - 9 * q^81 + 9 * q^83 - 12 * q^87 - 18 * q^89 + 33 * q^91 - 33 * q^93 + 9 * q^95 + 27 * q^97

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