LMFDB - Embedded newform 8464.2.a.ch.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8464,2,Mod(1,8464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8464, 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("8464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.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:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 $ x^15 - x^14 - 30x^13 + 28x^12 + 354x^11 - 302x^10 - 2111x^9 + 1596x^8 + 6777x^7 - 4292x^6 - 11506x^5 + 5297x^4 + 9480x^3 - 1825x^2 - 3136*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.593309$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.593309 q^{3} -2.48229 q^{5} -3.58382 q^{7} -2.64799 q^{9} +O(q^{10})$	$q+0.593309 q^{3} -2.48229 q^{5} -3.58382 q^{7} -2.64799 q^{9} +0.309711 q^{11} -1.08036 q^{13} -1.47276 q^{15} +5.71863 q^{17} +1.82587 q^{19} -2.12631 q^{21} +1.16177 q^{25} -3.35100 q^{27} -6.00622 q^{29} -6.94727 q^{31} +0.183754 q^{33} +8.89607 q^{35} -9.89141 q^{37} -0.640986 q^{39} -6.59824 q^{41} +2.87615 q^{43} +6.57307 q^{45} -12.3753 q^{47} +5.84373 q^{49} +3.39291 q^{51} +8.28037 q^{53} -0.768793 q^{55} +1.08331 q^{57} -3.48827 q^{59} -8.06229 q^{61} +9.48989 q^{63} +2.68176 q^{65} -1.48841 q^{67} -5.32836 q^{71} +8.52126 q^{73} +0.689285 q^{75} -1.10995 q^{77} +8.48841 q^{79} +5.95578 q^{81} -3.51157 q^{83} -14.1953 q^{85} -3.56354 q^{87} +1.08382 q^{89} +3.87180 q^{91} -4.12187 q^{93} -4.53235 q^{95} +1.05757 q^{97} -0.820111 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * 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
$3$	0.593309	0.342547	0.171273	−	0.985224i	$-0.445212\pi$
$3$	0.593309	0.342547	0.171273	+	0.985224i	$0.445212\pi$
$4$	0	0
$5$	−2.48229	−1.11011	−0.555057	−	0.831812i	$-0.687304\pi$
$5$	−2.48229	−1.11011	−0.555057	+	0.831812i	$0.687304\pi$
$6$	0	0
$7$	−3.58382	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$7$	−3.58382	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$8$	0	0
$9$	−2.64799	−0.882662
$10$	0	0
$11$	0.309711	0.0933815	0.0466907	−	0.998909i	$-0.485132\pi$
$11$	0.309711	0.0933815	0.0466907	+	0.998909i	$0.485132\pi$
$12$	0	0
$13$	−1.08036	−0.299637	−0.149819	−	0.988713i	$-0.547869\pi$
$13$	−1.08036	−0.299637	−0.149819	+	0.988713i	$0.547869\pi$
$14$	0	0
$15$	−1.47276	−0.380266
$16$	0	0
$17$	5.71863	1.38697	0.693485	−	0.720471i	$-0.256074\pi$
$17$	5.71863	1.38697	0.693485	+	0.720471i	$0.256074\pi$
$18$	0	0
$19$	1.82587	0.418884	0.209442	−	0.977821i	$-0.432835\pi$
$19$	1.82587	0.418884	0.209442	+	0.977821i	$0.432835\pi$
$20$	0	0
$21$	−2.12631	−0.463999
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	1.16177	0.232353
$26$	0	0
$27$	−3.35100	−0.644900
$28$	0	0
$29$	−6.00622	−1.11533	−0.557664	−	0.830067i	$-0.688302\pi$
$29$	−6.00622	−1.11533	−0.557664	+	0.830067i	$0.688302\pi$
$30$	0	0
$31$	−6.94727	−1.24777	−0.623883	−	0.781518i	$-0.714446\pi$
$31$	−6.94727	−1.24777	−0.623883	+	0.781518i	$0.714446\pi$
$32$	0	0
$33$	0.183754	0.0319875
$34$	0	0
$35$	8.89607	1.50371
$36$	0	0
$37$	−9.89141	−1.62614	−0.813069	−	0.582168i	$-0.802205\pi$
$37$	−9.89141	−1.62614	−0.813069	+	0.582168i	$0.802205\pi$
$38$	0	0
$39$	−0.640986	−0.102640
$40$	0	0
$41$	−6.59824	−1.03047	−0.515236	−	0.857048i	$-0.672296\pi$
$41$	−6.59824	−1.03047	−0.515236	+	0.857048i	$0.672296\pi$
$42$	0	0
$43$	2.87615	0.438608	0.219304	−	0.975657i	$-0.429621\pi$
$43$	2.87615	0.438608	0.219304	+	0.975657i	$0.429621\pi$
$44$	0	0
$45$	6.57307	0.979855
$46$	0	0
$47$	−12.3753	−1.80513	−0.902563	−	0.430557i	$-0.858317\pi$
$47$	−12.3753	−1.80513	−0.902563	+	0.430557i	$0.858317\pi$
$48$	0	0
$49$	5.84373	0.834819
$50$	0	0
$51$	3.39291	0.475102
$52$	0	0
$53$	8.28037	1.13740	0.568698	−	0.822546i	$-0.307447\pi$
$53$	8.28037	1.13740	0.568698	+	0.822546i	$0.307447\pi$
$54$	0	0
$55$	−0.768793	−0.103664
$56$	0	0
$57$	1.08331	0.143487
$58$	0	0
$59$	−3.48827	−0.454134	−0.227067	−	0.973879i	$-0.572914\pi$
$59$	−3.48827	−0.454134	−0.227067	+	0.973879i	$0.572914\pi$
$60$	0	0
$61$	−8.06229	−1.03227	−0.516135	−	0.856507i	$-0.672630\pi$
$61$	−8.06229	−1.03227	−0.516135	+	0.856507i	$0.672630\pi$
$62$	0	0
$63$	9.48989	1.19561
$64$	0	0
$65$	2.68176	0.332632
$66$	0	0
$67$	−1.48841	−0.181839	−0.0909193	−	0.995858i	$-0.528981\pi$
$67$	−1.48841	−0.181839	−0.0909193	+	0.995858i	$0.528981\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.32836	−0.632361	−0.316180	−	0.948699i	$-0.602400\pi$
$71$	−5.32836	−0.632361	−0.316180	+	0.948699i	$0.602400\pi$
$72$	0	0
$73$	8.52126	0.997338	0.498669	−	0.866792i	$-0.333822\pi$
$73$	8.52126	0.997338	0.498669	+	0.866792i	$0.333822\pi$
$74$	0	0
$75$	0.689285	0.0795918
$76$	0	0
$77$	−1.10995	−0.126490
$78$	0	0
$79$	8.48841	0.955020	0.477510	−	0.878626i	$-0.341539\pi$
$79$	8.48841	0.955020	0.477510	+	0.878626i	$0.341539\pi$
$80$	0	0
$81$	5.95578	0.661753
$82$	0	0
$83$	−3.51157	−0.385445	−0.192723	−	0.981253i	$-0.561732\pi$
$83$	−3.51157	−0.385445	−0.192723	+	0.981253i	$0.561732\pi$
$84$	0	0
$85$	−14.1953	−1.53970
$86$	0	0
$87$	−3.56354	−0.382052
$88$	0	0
$89$	1.08382	0.114885	0.0574424	−	0.998349i	$-0.481705\pi$
$89$	1.08382	0.114885	0.0574424	+	0.998349i	$0.481705\pi$
$90$	0	0
$91$	3.87180	0.405875
$92$	0	0
$93$	−4.12187	−0.427418
$94$	0	0
$95$	−4.53235	−0.465009
$96$	0	0
$97$	1.05757	0.107380	0.0536898	−	0.998558i	$-0.482902\pi$
$97$	1.05757	0.107380	0.0536898	+	0.998558i	$0.482902\pi$
$98$	0	0
$99$	−0.820111	−0.0824243
$100$	0	0
$101$	−6.92537	−0.689100	−0.344550	−	0.938768i	$-0.611969\pi$
$101$	−6.92537	−0.689100	−0.344550	+	0.938768i	$0.611969\pi$
$102$	0	0
$103$	15.6189	1.53898	0.769490	−	0.638659i	$-0.220511\pi$
$103$	15.6189	1.53898	0.769490	+	0.638659i	$0.220511\pi$
$104$	0	0
$105$	5.27811	0.515091
$106$	0	0
$107$	15.6594	1.51385	0.756924	−	0.653502i	$-0.226701\pi$
$107$	15.6594	1.51385	0.756924	+	0.653502i	$0.226701\pi$
$108$	0	0
$109$	−14.9341	−1.43042	−0.715212	−	0.698907i	$-0.753670\pi$
$109$	−14.9341	−1.43042	−0.715212	+	0.698907i	$0.753670\pi$
$110$	0	0
$111$	−5.86866	−0.557028
$112$	0	0
$113$	−16.4972	−1.55193	−0.775963	−	0.630778i	$-0.782736\pi$
$113$	−16.4972	−1.55193	−0.775963	+	0.630778i	$0.782736\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.86077	0.264478
$118$	0	0
$119$	−20.4945	−1.87873
$120$	0	0
$121$	−10.9041	−0.991280
$122$	0	0
$123$	−3.91479	−0.352985
$124$	0	0
$125$	9.52761	0.852176
$126$	0	0
$127$	15.6044	1.38467	0.692334	−	0.721578i	$-0.256583\pi$
$127$	15.6044	1.38467	0.692334	+	0.721578i	$0.256583\pi$
$128$	0	0
$129$	1.70644	0.150244
$130$	0	0
$131$	−8.58127	−0.749749	−0.374874	−	0.927076i	$-0.622314\pi$
$131$	−8.58127	−0.749749	−0.374874	+	0.927076i	$0.622314\pi$
$132$	0	0
$133$	−6.54359	−0.567401
$134$	0	0
$135$	8.31815	0.715912
$136$	0	0
$137$	−6.38302	−0.545338	−0.272669	−	0.962108i	$-0.587906\pi$
$137$	−6.38302	−0.545338	−0.272669	+	0.962108i	$0.587906\pi$
$138$	0	0
$139$	21.1827	1.79670	0.898348	−	0.439284i	$-0.144768\pi$
$139$	21.1827	1.79670	0.898348	+	0.439284i	$0.144768\pi$
$140$	0	0
$141$	−7.34238	−0.618340
$142$	0	0
$143$	−0.334599	−0.0279806
$144$	0	0
$145$	14.9092	1.23814
$146$	0	0
$147$	3.46714	0.285965
$148$	0	0
$149$	5.50002	0.450579	0.225289	−	0.974292i	$-0.427667\pi$
$149$	5.50002	0.450579	0.225289	+	0.974292i	$0.427667\pi$
$150$	0	0
$151$	0.944676	0.0768767	0.0384383	−	0.999261i	$-0.487762\pi$
$151$	0.944676	0.0768767	0.0384383	+	0.999261i	$0.487762\pi$
$152$	0	0
$153$	−15.1428	−1.22423
$154$	0	0
$155$	17.2451	1.38516
$156$	0	0
$157$	22.1040	1.76409	0.882047	−	0.471162i	$-0.156165\pi$
$157$	22.1040	1.76409	0.882047	+	0.471162i	$0.156165\pi$
$158$	0	0
$159$	4.91282	0.389612
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−19.9082	−1.55933	−0.779665	−	0.626197i	$-0.784611\pi$
$163$	−19.9082	−1.55933	−0.779665	+	0.626197i	$0.784611\pi$
$164$	0	0
$165$	−0.456132	−0.0355098
$166$	0	0
$167$	−6.38102	−0.493778	−0.246889	−	0.969044i	$-0.579408\pi$
$167$	−6.38102	−0.493778	−0.246889	+	0.969044i	$0.579408\pi$
$168$	0	0
$169$	−11.8328	−0.910217
$170$	0	0
$171$	−4.83488	−0.369733
$172$	0	0
$173$	0.982165	0.0746726	0.0373363	−	0.999303i	$-0.488113\pi$
$173$	0.982165	0.0746726	0.0373363	+	0.999303i	$0.488113\pi$
$174$	0	0
$175$	−4.16355	−0.314735
$176$	0	0
$177$	−2.06962	−0.155562
$178$	0	0
$179$	24.9804	1.86712	0.933561	−	0.358420i	$-0.116684\pi$
$179$	24.9804	1.86712	0.933561	+	0.358420i	$0.116684\pi$
$180$	0	0
$181$	−16.3447	−1.21489	−0.607446	−	0.794361i	$-0.707806\pi$
$181$	−16.3447	−1.21489	−0.607446	+	0.794361i	$0.707806\pi$
$182$	0	0
$183$	−4.78342	−0.353601
$184$	0	0
$185$	24.5533	1.80520
$186$	0	0
$187$	1.77112	0.129517
$188$	0	0
$189$	12.0094	0.873552
$190$	0	0
$191$	9.79988	0.709094	0.354547	−	0.935038i	$-0.384635\pi$
$191$	9.79988	0.709094	0.354547	+	0.935038i	$0.384635\pi$
$192$	0	0
$193$	19.9095	1.43312	0.716558	−	0.697528i	$-0.245717\pi$
$193$	19.9095	1.43312	0.716558	+	0.697528i	$0.245717\pi$
$194$	0	0
$195$	1.59111	0.113942
$196$	0	0
$197$	6.39454	0.455592	0.227796	−	0.973709i	$-0.426848\pi$
$197$	6.39454	0.455592	0.227796	+	0.973709i	$0.426848\pi$
$198$	0	0
$199$	15.7933	1.11956	0.559778	−	0.828643i	$-0.310886\pi$
$199$	15.7933	1.11956	0.559778	+	0.828643i	$0.310886\pi$
$200$	0	0
$201$	−0.883088	−0.0622882
$202$	0	0
$203$	21.5252	1.51077
$204$	0	0
$205$	16.3788	1.14394
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.565493	0.0391160
$210$	0	0
$211$	7.48312	0.515160	0.257580	−	0.966257i	$-0.417075\pi$
$211$	7.48312	0.515160	0.257580	+	0.966257i	$0.417075\pi$
$212$	0	0
$213$	−3.16136	−0.216613
$214$	0	0
$215$	−7.13943	−0.486905
$216$	0	0
$217$	24.8977	1.69017
$218$	0	0
$219$	5.05574	0.341635
$220$	0	0
$221$	−6.17816	−0.415588
$222$	0	0
$223$	19.4545	1.30277	0.651385	−	0.758747i	$-0.274188\pi$
$223$	19.4545	1.30277	0.651385	+	0.758747i	$0.274188\pi$
$224$	0	0
$225$	−3.07634	−0.205089
$226$	0	0
$227$	−19.0195	−1.26237	−0.631183	−	0.775634i	$-0.717430\pi$
$227$	−19.0195	−1.26237	−0.631183	+	0.775634i	$0.717430\pi$
$228$	0	0
$229$	−4.93124	−0.325865	−0.162933	−	0.986637i	$-0.552095\pi$
$229$	−4.93124	−0.325865	−0.162933	+	0.986637i	$0.552095\pi$
$230$	0	0
$231$	−0.658542	−0.0433289
$232$	0	0
$233$	−12.6337	−0.827661	−0.413831	−	0.910354i	$-0.635809\pi$
$233$	−12.6337	−0.827661	−0.413831	+	0.910354i	$0.635809\pi$
$234$	0	0
$235$	30.7191	2.00390
$236$	0	0
$237$	5.03624	0.327139
$238$	0	0
$239$	5.81067	0.375861	0.187930	−	0.982182i	$-0.439822\pi$
$239$	5.81067	0.375861	0.187930	+	0.982182i	$0.439822\pi$
$240$	0	0
$241$	−12.1708	−0.783993	−0.391996	−	0.919967i	$-0.628215\pi$
$241$	−12.1708	−0.783993	−0.391996	+	0.919967i	$0.628215\pi$
$242$	0	0
$243$	13.5866	0.871581
$244$	0	0
$245$	−14.5058	−0.926744
$246$	0	0
$247$	−1.97260	−0.125513
$248$	0	0
$249$	−2.08345	−0.132033
$250$	0	0
$251$	23.5690	1.48766	0.743831	−	0.668368i	$-0.233007\pi$
$251$	23.5690	1.48766	0.743831	+	0.668368i	$0.233007\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−8.42219	−0.527418
$256$	0	0
$257$	20.5339	1.28087	0.640434	−	0.768013i	$-0.278754\pi$
$257$	20.5339	1.28087	0.640434	+	0.768013i	$0.278754\pi$
$258$	0	0
$259$	35.4490	2.20269
$260$	0	0
$261$	15.9044	0.984457
$262$	0	0
$263$	−3.49432	−0.215469	−0.107734	−	0.994180i	$-0.534360\pi$
$263$	−3.49432	−0.215469	−0.107734	+	0.994180i	$0.534360\pi$
$264$	0	0
$265$	−20.5543	−1.26264
$266$	0	0
$267$	0.643040	0.0393534
$268$	0	0
$269$	0.136318	0.00831147	0.00415573	−	0.999991i	$-0.498677\pi$
$269$	0.136318	0.00831147	0.00415573	+	0.999991i	$0.498677\pi$
$270$	0	0
$271$	15.9988	0.971858	0.485929	−	0.873998i	$-0.338481\pi$
$271$	15.9988	0.971858	0.485929	+	0.873998i	$0.338481\pi$
$272$	0	0
$273$	2.29717	0.139031
$274$	0	0
$275$	0.359812	0.0216975
$276$	0	0
$277$	25.2361	1.51629	0.758147	−	0.652084i	$-0.226105\pi$
$277$	25.2361	1.51629	0.758147	+	0.652084i	$0.226105\pi$
$278$	0	0
$279$	18.3963	1.10136
$280$	0	0
$281$	−13.8972	−0.829037	−0.414518	−	0.910041i	$-0.636050\pi$
$281$	−13.8972	−0.829037	−0.414518	+	0.910041i	$0.636050\pi$
$282$	0	0
$283$	6.32674	0.376086	0.188043	−	0.982161i	$-0.439786\pi$
$283$	6.32674	0.376086	0.188043	+	0.982161i	$0.439786\pi$
$284$	0	0
$285$	−2.68908	−0.159287
$286$	0	0
$287$	23.6469	1.39583
$288$	0	0
$289$	15.7027	0.923687
$290$	0	0
$291$	0.627464	0.0367826
$292$	0	0
$293$	1.42772	0.0834080	0.0417040	−	0.999130i	$-0.486721\pi$
$293$	1.42772	0.0834080	0.0417040	+	0.999130i	$0.486721\pi$
$294$	0	0
$295$	8.65889	0.504140
$296$	0	0
$297$	−1.03784	−0.0602217
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−10.3076	−0.594119
$302$	0	0
$303$	−4.10888	−0.236049
$304$	0	0
$305$	20.0129	1.14594
$306$	0	0
$307$	−20.2169	−1.15384	−0.576918	−	0.816802i	$-0.695745\pi$
$307$	−20.2169	−1.15384	−0.576918	+	0.816802i	$0.695745\pi$
$308$	0	0
$309$	9.26685	0.527172
$310$	0	0
$311$	9.14913	0.518799	0.259400	−	0.965770i	$-0.416475\pi$
$311$	9.14913	0.518799	0.259400	+	0.965770i	$0.416475\pi$
$312$	0	0
$313$	−19.0420	−1.07632	−0.538159	−	0.842844i	$-0.680880\pi$
$313$	−19.0420	−1.07632	−0.538159	+	0.842844i	$0.680880\pi$
$314$	0	0
$315$	−23.5567	−1.32727
$316$	0	0
$317$	15.0855	0.847284	0.423642	−	0.905830i	$-0.360751\pi$
$317$	15.0855	0.847284	0.423642	+	0.905830i	$0.360751\pi$
$318$	0	0
$319$	−1.86020	−0.104151
$320$	0	0
$321$	9.29084	0.518564
$322$	0	0
$323$	10.4415	0.580980
$324$	0	0
$325$	−1.25512	−0.0696217
$326$	0	0
$327$	−8.86051	−0.489987
$328$	0	0
$329$	44.3509	2.44514
$330$	0	0
$331$	9.86945	0.542474	0.271237	−	0.962513i	$-0.412567\pi$
$331$	9.86945	0.542474	0.271237	+	0.962513i	$0.412567\pi$
$332$	0	0
$333$	26.1923	1.43533
$334$	0	0
$335$	3.69467	0.201862
$336$	0	0
$337$	9.39577	0.511820	0.255910	−	0.966701i	$-0.417625\pi$
$337$	9.39577	0.511820	0.255910	+	0.966701i	$0.417625\pi$
$338$	0	0
$339$	−9.78793	−0.531607
$340$	0	0
$341$	−2.15165	−0.116518
$342$	0	0
$343$	4.14384	0.223747
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.4905	0.563161	0.281581	−	0.959538i	$-0.409141\pi$
$347$	10.4905	0.563161	0.281581	+	0.959538i	$0.409141\pi$
$348$	0	0
$349$	−30.7716	−1.64717	−0.823583	−	0.567196i	$-0.808028\pi$
$349$	−30.7716	−1.64717	−0.823583	+	0.567196i	$0.808028\pi$
$350$	0	0
$351$	3.62028	0.193236
$352$	0	0
$353$	−15.0753	−0.802378	−0.401189	−	0.915995i	$-0.631403\pi$
$353$	−15.0753	−0.802378	−0.401189	+	0.915995i	$0.631403\pi$
$354$	0	0
$355$	13.2265	0.701992
$356$	0	0
$357$	−12.1596	−0.643552
$358$	0	0
$359$	1.36983	0.0722970	0.0361485	−	0.999346i	$-0.488491\pi$
$359$	1.36983	0.0722970	0.0361485	+	0.999346i	$0.488491\pi$
$360$	0	0
$361$	−15.6662	−0.824536
$362$	0	0
$363$	−6.46948	−0.339560
$364$	0	0
$365$	−21.1523	−1.10716
$366$	0	0
$367$	31.6797	1.65367	0.826834	−	0.562447i	$-0.190140\pi$
$367$	31.6797	1.65367	0.826834	+	0.562447i	$0.190140\pi$
$368$	0	0
$369$	17.4720	0.909558
$370$	0	0
$371$	−29.6753	−1.54067
$372$	0	0
$373$	−21.1443	−1.09481	−0.547404	−	0.836868i	$-0.684384\pi$
$373$	−21.1443	−1.09481	−0.547404	+	0.836868i	$0.684384\pi$
$374$	0	0
$375$	5.65281	0.291910
$376$	0	0
$377$	6.48887	0.334194
$378$	0	0
$379$	−23.0257	−1.18275	−0.591375	−	0.806396i	$-0.701415\pi$
$379$	−23.0257	−1.18275	−0.591375	+	0.806396i	$0.701415\pi$
$380$	0	0
$381$	9.25823	0.474313
$382$	0	0
$383$	−11.9058	−0.608358	−0.304179	−	0.952615i	$-0.598382\pi$
$383$	−11.9058	−0.608358	−0.304179	+	0.952615i	$0.598382\pi$
$384$	0	0
$385$	2.75521	0.140419
$386$	0	0
$387$	−7.61599	−0.387143
$388$	0	0
$389$	−23.1143	−1.17194	−0.585970	−	0.810333i	$-0.699286\pi$
$389$	−23.1143	−1.17194	−0.585970	+	0.810333i	$0.699286\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−5.09134	−0.256824
$394$	0	0
$395$	−21.0707	−1.06018
$396$	0	0
$397$	−6.16095	−0.309209	−0.154605	−	0.987976i	$-0.549410\pi$
$397$	−6.16095	−0.309209	−0.154605	+	0.987976i	$0.549410\pi$
$398$	0	0
$399$	−3.88237	−0.194361
$400$	0	0
$401$	−10.3034	−0.514527	−0.257264	−	0.966341i	$-0.582821\pi$
$401$	−10.3034	−0.514527	−0.257264	+	0.966341i	$0.582821\pi$
$402$	0	0
$403$	7.50554	0.373877
$404$	0	0
$405$	−14.7840	−0.734622
$406$	0	0
$407$	−3.06348	−0.151851
$408$	0	0
$409$	0.00814518	0.000402753 0	0.000201377	−	1.00000i	$-0.499936\pi$
$409$	0.00814518	0.000402753 0	0.000201377		1.00000i	$0.499936\pi$
$410$	0	0
$411$	−3.78710	−0.186804
$412$	0	0
$413$	12.5013	0.615149
$414$	0	0
$415$	8.71675	0.427888
$416$	0	0
$417$	12.5679	0.615453
$418$	0	0
$419$	26.8859	1.31346	0.656730	−	0.754126i	$-0.271939\pi$
$419$	26.8859	1.31346	0.656730	+	0.754126i	$0.271939\pi$
$420$	0	0
$421$	3.30579	0.161115	0.0805573	−	0.996750i	$-0.474330\pi$
$421$	3.30579	0.161115	0.0805573	+	0.996750i	$0.474330\pi$
$422$	0	0
$423$	32.7697	1.59332
$424$	0	0
$425$	6.64370	0.322267
$426$	0	0
$427$	28.8938	1.39827
$428$	0	0
$429$	−0.198520	−0.00958466
$430$	0	0
$431$	12.2487	0.589997	0.294999	−	0.955498i	$-0.404681\pi$
$431$	12.2487	0.589997	0.294999	+	0.955498i	$0.404681\pi$
$432$	0	0
$433$	2.62340	0.126072	0.0630362	−	0.998011i	$-0.479922\pi$
$433$	2.62340	0.126072	0.0630362	+	0.998011i	$0.479922\pi$
$434$	0	0
$435$	8.84575	0.424121
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−6.11062	−0.291644	−0.145822	−	0.989311i	$-0.546583\pi$
$439$	−6.11062	−0.291644	−0.145822	+	0.989311i	$0.546583\pi$
$440$	0	0
$441$	−15.4741	−0.736863
$442$	0	0
$443$	24.2860	1.15386	0.576932	−	0.816792i	$-0.304250\pi$
$443$	24.2860	1.15386	0.576932	+	0.816792i	$0.304250\pi$
$444$	0	0
$445$	−2.69036	−0.127535
$446$	0	0
$447$	3.26321	0.154344
$448$	0	0
$449$	35.8960	1.69404	0.847018	−	0.531564i	$-0.178395\pi$
$449$	35.8960	1.69404	0.847018	+	0.531564i	$0.178395\pi$
$450$	0	0
$451$	−2.04355	−0.0962270
$452$	0	0
$453$	0.560485	0.0263339
$454$	0	0
$455$	−9.61094	−0.450568
$456$	0	0
$457$	−27.2842	−1.27630	−0.638151	−	0.769911i	$-0.720301\pi$
$457$	−27.2842	−1.27630	−0.638151	+	0.769911i	$0.720301\pi$
$458$	0	0
$459$	−19.1631	−0.894457
$460$	0	0
$461$	−3.54919	−0.165302	−0.0826510	−	0.996579i	$-0.526339\pi$
$461$	−3.54919	−0.165302	−0.0826510	+	0.996579i	$0.526339\pi$
$462$	0	0
$463$	−24.4936	−1.13831	−0.569157	−	0.822229i	$-0.692731\pi$
$463$	−24.4936	−1.13831	−0.569157	+	0.822229i	$0.692731\pi$
$464$	0	0
$465$	10.2317	0.474483
$466$	0	0
$467$	31.8164	1.47229	0.736145	−	0.676824i	$-0.236644\pi$
$467$	31.8164	1.47229	0.736145	+	0.676824i	$0.236644\pi$
$468$	0	0
$469$	5.33420	0.246310
$470$	0	0
$471$	13.1145	0.604285
$472$	0	0
$473$	0.890775	0.0409579
$474$	0	0
$475$	2.12124	0.0973290
$476$	0	0
$477$	−21.9263	−1.00394
$478$	0	0
$479$	−15.5961	−0.712603	−0.356301	−	0.934371i	$-0.615962\pi$
$479$	−15.5961	−0.712603	−0.356301	+	0.934371i	$0.615962\pi$
$480$	0	0
$481$	10.6863	0.487251
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.62519	−0.119204
$486$	0	0
$487$	−33.5151	−1.51871	−0.759357	−	0.650675i	$-0.774486\pi$
$487$	−33.5151	−1.51871	−0.759357	+	0.650675i	$0.774486\pi$
$488$	0	0
$489$	−11.8117	−0.534144
$490$	0	0
$491$	−16.5885	−0.748626	−0.374313	−	0.927302i	$-0.622121\pi$
$491$	−16.5885	−0.748626	−0.374313	+	0.927302i	$0.622121\pi$
$492$	0	0
$493$	−34.3473	−1.54693
$494$	0	0
$495$	2.03575	0.0915003
$496$	0	0
$497$	19.0959	0.856567
$498$	0	0
$499$	16.5815	0.742288	0.371144	−	0.928575i	$-0.378966\pi$
$499$	16.5815	0.742288	0.371144	+	0.928575i	$0.378966\pi$
$500$	0	0
$501$	−3.78591	−0.169142
$502$	0	0
$503$	−27.1612	−1.21106	−0.605530	−	0.795823i	$-0.707039\pi$
$503$	−27.1612	−1.21106	−0.605530	+	0.795823i	$0.707039\pi$
$504$	0	0
$505$	17.1908	0.764980
$506$	0	0
$507$	−7.02052	−0.311792
$508$	0	0
$509$	9.51627	0.421801	0.210901	−	0.977508i	$-0.432360\pi$
$509$	9.51627	0.421801	0.210901	+	0.977508i	$0.432360\pi$
$510$	0	0
$511$	−30.5386	−1.35095
$512$	0	0
$513$	−6.11849	−0.270138
$514$	0	0
$515$	−38.7707	−1.70844
$516$	0	0
$517$	−3.83278	−0.168565
$518$	0	0
$519$	0.582727	0.0255789
$520$	0	0
$521$	9.97611	0.437061	0.218531	−	0.975830i	$-0.429874\pi$
$521$	9.97611	0.437061	0.218531	+	0.975830i	$0.429874\pi$
$522$	0	0
$523$	−5.37220	−0.234910	−0.117455	−	0.993078i	$-0.537474\pi$
$523$	−5.37220	−0.234910	−0.117455	+	0.993078i	$0.537474\pi$
$524$	0	0
$525$	−2.47027	−0.107811
$526$	0	0
$527$	−39.7288	−1.73061
$528$	0	0
$529$	0	0
$530$	0	0
$531$	9.23688	0.400846
$532$	0	0
$533$	7.12846	0.308768
$534$	0	0
$535$	−38.8711	−1.68054
$536$	0	0
$537$	14.8211	0.639576
$538$	0	0
$539$	1.80987	0.0779566
$540$	0	0
$541$	32.8030	1.41031	0.705155	−	0.709053i	$-0.250877\pi$
$541$	32.8030	1.41031	0.705155	+	0.709053i	$0.250877\pi$
$542$	0	0
$543$	−9.69745	−0.416157
$544$	0	0
$545$	37.0707	1.58793
$546$	0	0
$547$	−41.5457	−1.77636	−0.888182	−	0.459491i	$-0.848032\pi$
$547$	−41.5457	−1.77636	−0.888182	+	0.459491i	$0.848032\pi$
$548$	0	0
$549$	21.3488	0.911145
$550$	0	0
$551$	−10.9666	−0.467193
$552$	0	0
$553$	−30.4209	−1.29363
$554$	0	0
$555$	14.5677	0.618365
$556$	0	0
$557$	32.8796	1.39316	0.696578	−	0.717481i	$-0.254705\pi$
$557$	32.8796	1.39316	0.696578	+	0.717481i	$0.254705\pi$
$558$	0	0
$559$	−3.10727	−0.131423
$560$	0	0
$561$	1.05082	0.0443658
$562$	0	0
$563$	8.55565	0.360578	0.180289	−	0.983614i	$-0.442297\pi$
$563$	8.55565	0.360578	0.180289	+	0.983614i	$0.442297\pi$
$564$	0	0
$565$	40.9509	1.72282
$566$	0	0
$567$	−21.3444	−0.896381
$568$	0	0
$569$	24.1498	1.01241	0.506207	−	0.862412i	$-0.331047\pi$
$569$	24.1498	1.01241	0.506207	+	0.862412i	$0.331047\pi$
$570$	0	0
$571$	20.6468	0.864043	0.432021	−	0.901863i	$-0.357801\pi$
$571$	20.6468	0.864043	0.432021	+	0.901863i	$0.357801\pi$
$572$	0	0
$573$	5.81435	0.242898
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−6.28441	−0.261623	−0.130812	−	0.991407i	$-0.541758\pi$
$577$	−6.28441	−0.261623	−0.130812	+	0.991407i	$0.541758\pi$
$578$	0	0
$579$	11.8125	0.490909
$580$	0	0
$581$	12.5848	0.522107
$582$	0	0
$583$	2.56453	0.106212
$584$	0	0
$585$	−7.10127	−0.293601
$586$	0	0
$587$	−32.2070	−1.32932	−0.664662	−	0.747144i	$-0.731424\pi$
$587$	−32.2070	−1.32932	−0.664662	+	0.747144i	$0.731424\pi$
$588$	0	0
$589$	−12.6848	−0.522669
$590$	0	0
$591$	3.79393	0.156062
$592$	0	0
$593$	−6.20462	−0.254793	−0.127397	−	0.991852i	$-0.540662\pi$
$593$	−6.20462	−0.254793	−0.127397	+	0.991852i	$0.540662\pi$
$594$	0	0
$595$	50.8733	2.08560
$596$	0	0
$597$	9.37029	0.383500
$598$	0	0
$599$	−2.67037	−0.109108	−0.0545542	−	0.998511i	$-0.517374\pi$
$599$	−2.67037	−0.109108	−0.0545542	+	0.998511i	$0.517374\pi$
$600$	0	0
$601$	0.828318	0.0337878	0.0168939	−	0.999857i	$-0.494622\pi$
$601$	0.828318	0.0337878	0.0168939	+	0.999857i	$0.494622\pi$
$602$	0	0
$603$	3.94129	0.160502
$604$	0	0
$605$	27.0671	1.10043
$606$	0	0
$607$	−17.2415	−0.699810	−0.349905	−	0.936785i	$-0.613786\pi$
$607$	−17.2415	−0.699810	−0.349905	+	0.936785i	$0.613786\pi$
$608$	0	0
$609$	12.7711	0.517510
$610$	0	0
$611$	13.3698	0.540883
$612$	0	0
$613$	−20.9368	−0.845630	−0.422815	−	0.906216i	$-0.638958\pi$
$613$	−20.9368	−0.845630	−0.422815	+	0.906216i	$0.638958\pi$
$614$	0	0
$615$	9.71765	0.391854
$616$	0	0
$617$	14.4417	0.581403	0.290701	−	0.956814i	$-0.406111\pi$
$617$	14.4417	0.581403	0.290701	+	0.956814i	$0.406111\pi$
$618$	0	0
$619$	1.08672	0.0436788	0.0218394	−	0.999761i	$-0.493048\pi$
$619$	1.08672	0.0436788	0.0218394	+	0.999761i	$0.493048\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.88421	−0.155618
$624$	0	0
$625$	−29.4591	−1.17837
$626$	0	0
$627$	0.335512	0.0133991
$628$	0	0
$629$	−56.5653	−2.25540
$630$	0	0
$631$	−10.1753	−0.405071	−0.202536	−	0.979275i	$-0.564918\pi$
$631$	−10.1753	−0.405071	−0.202536	+	0.979275i	$0.564918\pi$
$632$	0	0
$633$	4.43980	0.176466
$634$	0	0
$635$	−38.7347	−1.53714
$636$	0	0
$637$	−6.31332	−0.250143
$638$	0	0
$639$	14.1094	0.558160
$640$	0	0
$641$	29.0504	1.14742	0.573710	−	0.819058i	$-0.305504\pi$
$641$	29.0504	1.14742	0.573710	+	0.819058i	$0.305504\pi$
$642$	0	0
$643$	38.2701	1.50922	0.754612	−	0.656171i	$-0.227825\pi$
$643$	38.2701	1.50922	0.754612	+	0.656171i	$0.227825\pi$
$644$	0	0
$645$	−4.23588	−0.166788
$646$	0	0
$647$	−5.00577	−0.196797	−0.0983985	−	0.995147i	$-0.531372\pi$
$647$	−5.00577	−0.196797	−0.0983985	+	0.995147i	$0.531372\pi$
$648$	0	0
$649$	−1.08036	−0.0424077
$650$	0	0
$651$	14.7720	0.578962
$652$	0	0
$653$	−1.09064	−0.0426802	−0.0213401	−	0.999772i	$-0.506793\pi$
$653$	−1.09064	−0.0426802	−0.0213401	+	0.999772i	$0.506793\pi$
$654$	0	0
$655$	21.3012	0.832307
$656$	0	0
$657$	−22.5642	−0.880312
$658$	0	0
$659$	−24.7118	−0.962634	−0.481317	−	0.876547i	$-0.659841\pi$
$659$	−24.7118	−0.962634	−0.481317	+	0.876547i	$0.659841\pi$
$660$	0	0
$661$	33.5858	1.30633	0.653167	−	0.757214i	$-0.273440\pi$
$661$	33.5858	1.30633	0.653167	+	0.757214i	$0.273440\pi$
$662$	0	0
$663$	−3.66556	−0.142358
$664$	0	0
$665$	16.2431	0.629880
$666$	0	0
$667$	0	0
$668$	0	0
$669$	11.5425	0.446260
$670$	0	0
$671$	−2.49698	−0.0963949
$672$	0	0
$673$	−9.20521	−0.354835	−0.177417	−	0.984136i	$-0.556774\pi$
$673$	−9.20521	−0.354835	−0.177417	+	0.984136i	$0.556774\pi$
$674$	0	0
$675$	−3.89307	−0.149844
$676$	0	0
$677$	12.0216	0.462028	0.231014	−	0.972950i	$-0.425796\pi$
$677$	12.0216	0.462028	0.231014	+	0.972950i	$0.425796\pi$
$678$	0	0
$679$	−3.79013	−0.145452
$680$	0	0
$681$	−11.2844	−0.432419
$682$	0	0
$683$	−33.1023	−1.26663	−0.633313	−	0.773896i	$-0.718305\pi$
$683$	−33.1023	−1.26663	−0.633313	+	0.773896i	$0.718305\pi$
$684$	0	0
$685$	15.8445	0.605388
$686$	0	0
$687$	−2.92575	−0.111624
$688$	0	0
$689$	−8.94577	−0.340807
$690$	0	0
$691$	28.0797	1.06820	0.534102	−	0.845420i	$-0.320650\pi$
$691$	28.0797	1.06820	0.534102	+	0.845420i	$0.320650\pi$
$692$	0	0
$693$	2.93913	0.111648
$694$	0	0
$695$	−52.5817	−1.99454
$696$	0	0
$697$	−37.7329	−1.42923
$698$	0	0
$699$	−7.49568	−0.283513
$700$	0	0
$701$	3.84360	0.145171	0.0725854	−	0.997362i	$-0.476875\pi$
$701$	3.84360	0.145171	0.0725854	+	0.997362i	$0.476875\pi$
$702$	0	0
$703$	−18.0604	−0.681163
$704$	0	0
$705$	18.2259	0.686428
$706$	0	0
$707$	24.8193	0.933424
$708$	0	0
$709$	4.63312	0.174000	0.0870002	−	0.996208i	$-0.472272\pi$
$709$	4.63312	0.174000	0.0870002	+	0.996208i	$0.472272\pi$
$710$	0	0
$711$	−22.4772	−0.842960
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0.830572	0.0310616
$716$	0	0
$717$	3.44752	0.128750
$718$	0	0
$719$	−30.9053	−1.15257	−0.576286	−	0.817248i	$-0.695499\pi$
$719$	−30.9053	−1.15257	−0.576286	+	0.817248i	$0.695499\pi$
$720$	0	0
$721$	−55.9754	−2.08463
$722$	0	0
$723$	−7.22107	−0.268554
$724$	0	0
$725$	−6.97782	−0.259150
$726$	0	0
$727$	−20.4685	−0.759134	−0.379567	−	0.925164i	$-0.623927\pi$
$727$	−20.4685	−0.759134	−0.379567	+	0.925164i	$0.623927\pi$
$728$	0	0
$729$	−9.80629	−0.363196
$730$	0	0
$731$	16.4476	0.608337
$732$	0	0
$733$	26.8697	0.992456	0.496228	−	0.868192i	$-0.334718\pi$
$733$	26.8697	0.992456	0.496228	+	0.868192i	$0.334718\pi$
$734$	0	0
$735$	−8.60644	−0.317453
$736$	0	0
$737$	−0.460978	−0.0169804
$738$	0	0
$739$	32.8142	1.20709	0.603544	−	0.797329i	$-0.293755\pi$
$739$	32.8142	1.20709	0.603544	+	0.797329i	$0.293755\pi$
$740$	0	0
$741$	−1.17036	−0.0429942
$742$	0	0
$743$	18.5291	0.679767	0.339883	−	0.940468i	$-0.389612\pi$
$743$	18.5291	0.679767	0.339883	+	0.940468i	$0.389612\pi$
$744$	0	0
$745$	−13.6526	−0.500194
$746$	0	0
$747$	9.29860	0.340218
$748$	0	0
$749$	−56.1203	−2.05059
$750$	0	0
$751$	−20.4057	−0.744615	−0.372307	−	0.928109i	$-0.621433\pi$
$751$	−20.4057	−0.744615	−0.372307	+	0.928109i	$0.621433\pi$
$752$	0	0
$753$	13.9837	0.509594
$754$	0	0
$755$	−2.34496	−0.0853419
$756$	0	0
$757$	−5.53188	−0.201060	−0.100530	−	0.994934i	$-0.532054\pi$
$757$	−5.53188	−0.201060	−0.100530	+	0.994934i	$0.532054\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	15.5746	0.564579	0.282289	−	0.959329i	$-0.408906\pi$
$761$	15.5746	0.564579	0.282289	+	0.959329i	$0.408906\pi$
$762$	0	0
$763$	53.5209	1.93759
$764$	0	0
$765$	37.5889	1.35903
$766$	0	0
$767$	3.76858	0.136075
$768$	0	0
$769$	9.76741	0.352222	0.176111	−	0.984370i	$-0.443648\pi$
$769$	9.76741	0.352222	0.176111	+	0.984370i	$0.443648\pi$
$770$	0	0
$771$	12.1829	0.438758
$772$	0	0
$773$	16.0970	0.578968	0.289484	−	0.957183i	$-0.406516\pi$
$773$	16.0970	0.578968	0.289484	+	0.957183i	$0.406516\pi$
$774$	0	0
$775$	−8.07110	−0.289922
$776$	0	0
$777$	21.0322	0.754525
$778$	0	0
$779$	−12.0475	−0.431648
$780$	0	0
$781$	−1.65025	−0.0590508
$782$	0	0
$783$	20.1268	0.719274
$784$	0	0
$785$	−54.8686	−1.95835
$786$	0	0
$787$	11.3012	0.402845	0.201422	−	0.979504i	$-0.435444\pi$
$787$	11.3012	0.402845	0.201422	+	0.979504i	$0.435444\pi$
$788$	0	0
$789$	−2.07321	−0.0738082
$790$	0	0
$791$	59.1229	2.10217
$792$	0	0
$793$	8.71016	0.309307
$794$	0	0
$795$	−12.1950	−0.432513
$796$	0	0
$797$	49.4017	1.74990	0.874949	−	0.484216i	$-0.160895\pi$
$797$	49.4017	1.74990	0.874949	+	0.484216i	$0.160895\pi$
$798$	0	0
$799$	−70.7699	−2.50366
$800$	0	0
$801$	−2.86994	−0.101404
$802$	0	0
$803$	2.63913	0.0931329
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.0808788	0.00284707
$808$	0	0
$809$	−55.1319	−1.93834	−0.969168	−	0.246401i	$-0.920752\pi$
$809$	−55.1319	−1.93834	−0.969168	+	0.246401i	$0.920752\pi$
$810$	0	0
$811$	−50.0399	−1.75714	−0.878570	−	0.477614i	$-0.841502\pi$
$811$	−50.0399	−1.75714	−0.878570	+	0.477614i	$0.841502\pi$
$812$	0	0
$813$	9.49222	0.332907
$814$	0	0
$815$	49.4179	1.73103
$816$	0	0
$817$	5.25147	0.183726
$818$	0	0
$819$	−10.2525	−0.358251
$820$	0	0
$821$	−19.1147	−0.667109	−0.333554	−	0.942731i	$-0.608248\pi$
$821$	−19.1147	−0.667109	−0.333554	+	0.942731i	$0.608248\pi$
$822$	0	0
$823$	0.799459	0.0278674	0.0139337	−	0.999903i	$-0.495565\pi$
$823$	0.799459	0.0278674	0.0139337	+	0.999903i	$0.495565\pi$
$824$	0	0
$825$	0.213479	0.00743240
$826$	0	0
$827$	26.1707	0.910046	0.455023	−	0.890480i	$-0.349631\pi$
$827$	26.1707	0.910046	0.455023	+	0.890480i	$0.349631\pi$
$828$	0	0
$829$	−23.0541	−0.800703	−0.400352	−	0.916362i	$-0.631112\pi$
$829$	−23.0541	−0.800703	−0.400352	+	0.916362i	$0.631112\pi$
$830$	0	0
$831$	14.9728	0.519401
$832$	0	0
$833$	33.4181	1.15787
$834$	0	0
$835$	15.8395	0.548150
$836$	0	0
$837$	23.2803	0.804684
$838$	0	0
$839$	−53.7809	−1.85672	−0.928361	−	0.371679i	$-0.878782\pi$
$839$	−53.7809	−1.85672	−0.928361	+	0.371679i	$0.878782\pi$
$840$	0	0
$841$	7.07471	0.243955
$842$	0	0
$843$	−8.24532	−0.283984
$844$	0	0
$845$	29.3725	1.01045
$846$	0	0
$847$	39.0782	1.34274
$848$	0	0
$849$	3.75371	0.128827
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−28.8558	−0.988003	−0.494002	−	0.869461i	$-0.664466\pi$
$853$	−28.8558	−0.988003	−0.494002	+	0.869461i	$0.664466\pi$
$854$	0	0
$855$	12.0016	0.410445
$856$	0	0
$857$	−4.88284	−0.166794	−0.0833972	−	0.996516i	$-0.526577\pi$
$857$	−4.88284	−0.166794	−0.0833972	+	0.996516i	$0.526577\pi$
$858$	0	0
$859$	23.7917	0.811764	0.405882	−	0.913926i	$-0.366964\pi$
$859$	23.7917	0.811764	0.405882	+	0.913926i	$0.366964\pi$
$860$	0	0
$861$	14.0299	0.478138
$862$	0	0
$863$	30.3772	1.03405	0.517026	−	0.855970i	$-0.327039\pi$
$863$	30.3772	1.03405	0.517026	+	0.855970i	$0.327039\pi$
$864$	0	0
$865$	−2.43802	−0.0828951
$866$	0	0
$867$	9.31654	0.316406
$868$	0	0
$869$	2.62896	0.0891812
$870$	0	0
$871$	1.60802	0.0544856
$872$	0	0
$873$	−2.80042	−0.0947799
$874$	0	0
$875$	−34.1452	−1.15432
$876$	0	0
$877$	−12.8762	−0.434798	−0.217399	−	0.976083i	$-0.569757\pi$
$877$	−12.8762	−0.434798	−0.217399	+	0.976083i	$0.569757\pi$
$878$	0	0
$879$	0.847076	0.0285711
$880$	0	0
$881$	−23.7139	−0.798943	−0.399472	−	0.916746i	$-0.630806\pi$
$881$	−23.7139	−0.798943	−0.399472	+	0.916746i	$0.630806\pi$
$882$	0	0
$883$	−17.8223	−0.599769	−0.299885	−	0.953976i	$-0.596948\pi$
$883$	−17.8223	−0.599769	−0.299885	+	0.953976i	$0.596948\pi$
$884$	0	0
$885$	5.13739	0.172692
$886$	0	0
$887$	−33.1050	−1.11156	−0.555778	−	0.831331i	$-0.687580\pi$
$887$	−33.1050	−1.11156	−0.555778	+	0.831331i	$0.687580\pi$
$888$	0	0
$889$	−55.9233	−1.87561
$890$	0	0
$891$	1.84457	0.0617955
$892$	0	0
$893$	−22.5958	−0.756138
$894$	0	0
$895$	−62.0086	−2.07272
$896$	0	0
$897$	0	0
$898$	0	0
$899$	41.7268	1.39167
$900$	0	0
$901$	47.3524	1.57754
$902$	0	0
$903$	−6.11557	−0.203514
$904$	0	0
$905$	40.5723	1.34867
$906$	0	0
$907$	−54.0149	−1.79354	−0.896768	−	0.442502i	$-0.854091\pi$
$907$	−54.0149	−1.79354	−0.896768	+	0.442502i	$0.854091\pi$
$908$	0	0
$909$	18.3383	0.608242
$910$	0	0
$911$	41.2595	1.36699	0.683493	−	0.729957i	$-0.260460\pi$
$911$	41.2595	1.36699	0.683493	+	0.729957i	$0.260460\pi$
$912$	0	0
$913$	−1.08757	−0.0359935
$914$	0	0
$915$	11.8738	0.392537
$916$	0	0
$917$	30.7537	1.01558
$918$	0	0
$919$	−41.8698	−1.38116	−0.690579	−	0.723257i	$-0.742644\pi$
$919$	−41.8698	−1.38116	−0.690579	+	0.723257i	$0.742644\pi$
$920$	0	0
$921$	−11.9948	−0.395243
$922$	0	0
$923$	5.75654	0.189479
$924$	0	0
$925$	−11.4915	−0.377838
$926$	0	0
$927$	−41.3587	−1.35840
$928$	0	0
$929$	−3.30456	−0.108419	−0.0542095	−	0.998530i	$-0.517264\pi$
$929$	−3.30456	−0.108419	−0.0542095	+	0.998530i	$0.517264\pi$
$930$	0	0
$931$	10.6699	0.349692
$932$	0	0
$933$	5.42826	0.177713
$934$	0	0
$935$	−4.39644	−0.143779
$936$	0	0
$937$	−11.5552	−0.377493	−0.188746	−	0.982026i	$-0.560442\pi$
$937$	−11.5552	−0.377493	−0.188746	+	0.982026i	$0.560442\pi$
$938$	0	0
$939$	−11.2978	−0.368689
$940$	0	0
$941$	11.0702	0.360878	0.180439	−	0.983586i	$-0.442248\pi$
$941$	11.0702	0.360878	0.180439	+	0.983586i	$0.442248\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−29.8107	−0.969743
$946$	0	0
$947$	48.9486	1.59062	0.795308	−	0.606206i	$-0.207309\pi$
$947$	48.9486	1.59062	0.795308	+	0.606206i	$0.207309\pi$
$948$	0	0
$949$	−9.20601	−0.298840
$950$	0	0
$951$	8.95034	0.290235
$952$	0	0
$953$	−6.11432	−0.198062	−0.0990311	−	0.995084i	$-0.531574\pi$
$953$	−6.11432	−0.198062	−0.0990311	+	0.995084i	$0.531574\pi$
$954$	0	0
$955$	−24.3261	−0.787175
$956$	0	0
$957$	−1.10367	−0.0356766
$958$	0	0
$959$	22.8756	0.738691
$960$	0	0
$961$	17.2645	0.556920
$962$	0	0
$963$	−41.4658	−1.33622
$964$	0	0
$965$	−49.4211	−1.59092
$966$	0	0
$967$	8.52591	0.274175	0.137087	−	0.990559i	$-0.456226\pi$
$967$	8.52591	0.274175	0.137087	+	0.990559i	$0.456226\pi$
$968$	0	0
$969$	6.19502	0.199013
$970$	0	0
$971$	26.0062	0.834579	0.417290	−	0.908774i	$-0.362980\pi$
$971$	26.0062	0.834579	0.417290	+	0.908774i	$0.362980\pi$
$972$	0	0
$973$	−75.9150	−2.43372
$974$	0	0
$975$	−0.744675	−0.0238487
$976$	0	0
$977$	8.31555	0.266038	0.133019	−	0.991113i	$-0.457533\pi$
$977$	8.31555	0.266038	0.133019	+	0.991113i	$0.457533\pi$
$978$	0	0
$979$	0.335671	0.0107281
$980$	0	0
$981$	39.5452	1.26258
$982$	0	0
$983$	−15.2291	−0.485733	−0.242867	−	0.970060i	$-0.578088\pi$
$983$	−15.2291	−0.485733	−0.242867	+	0.970060i	$0.578088\pi$
$984$	0	0
$985$	−15.8731	−0.505759
$986$	0	0
$987$	26.3138	0.837576
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−24.7796	−0.787150	−0.393575	−	0.919293i	$-0.628762\pi$
$991$	−24.7796	−0.787150	−0.393575	+	0.919293i	$0.628762\pi$
$992$	0	0
$993$	5.85563	0.185823
$994$	0	0
$995$	−39.2035	−1.24283
$996$	0	0
$997$	−45.5183	−1.44158	−0.720790	−	0.693153i	$-0.756221\pi$
$997$	−45.5183	−1.44158	−0.720790	+	0.693153i	$0.756221\pi$
$998$	0	0
$999$	33.1461	1.04870

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.ch.1.9		15
4.3	odd	2		4232.2.a.ba.1.7		15
23.7	odd	22		368.2.m.e.49.2		30
23.10	odd	22		368.2.m.e.353.2		30
23.22	odd	2		8464.2.a.cg.1.9		15
92.7	even	22		184.2.i.b.49.2	✓	30
92.79	even	22		184.2.i.b.169.2	yes	30
92.91	even	2		4232.2.a.bb.1.7		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.49.2	✓	30	92.7	even	22
184.2.i.b.169.2	yes	30	92.79	even	22
368.2.m.e.49.2		30	23.7	odd	22
368.2.m.e.353.2		30	23.10	odd	22
4232.2.a.ba.1.7		15	4.3	odd	2
4232.2.a.bb.1.7		15	92.91	even	2
8464.2.a.cg.1.9		15	23.22	odd	2
8464.2.a.ch.1.9		15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+0.593309 q^{3} -2.48229 q^{5} -3.58382 q^{7} -2.64799 q^{9} +O(q^{10})\)	\(q+0.593309 q^{3} -2.48229 q^{5} -3.58382 q^{7} -2.64799 q^{9} +0.309711 q^{11} -1.08036 q^{13} -1.47276 q^{15} +5.71863 q^{17} +1.82587 q^{19} -2.12631 q^{21} +1.16177 q^{25} -3.35100 q^{27} -6.00622 q^{29} -6.94727 q^{31} +0.183754 q^{33} +8.89607 q^{35} -9.89141 q^{37} -0.640986 q^{39} -6.59824 q^{41} +2.87615 q^{43} +6.57307 q^{45} -12.3753 q^{47} +5.84373 q^{49} +3.39291 q^{51} +8.28037 q^{53} -0.768793 q^{55} +1.08331 q^{57} -3.48827 q^{59} -8.06229 q^{61} +9.48989 q^{63} +2.68176 q^{65} -1.48841 q^{67} -5.32836 q^{71} +8.52126 q^{73} +0.689285 q^{75} -1.10995 q^{77} +8.48841 q^{79} +5.95578 q^{81} -3.51157 q^{83} -14.1953 q^{85} -3.56354 q^{87} +1.08382 q^{89} +3.87180 q^{91} -4.12187 q^{93} -4.53235 q^{95} +1.05757 q^{97} -0.820111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * q^99

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