LMFDB - Embedded newform 9196.2.a.q.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9196,2,Mod(1,9196)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9196.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9196 = 2^{2} \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9196.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-1,0,0,0,-2,0,9,0,0,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$73.4304296988$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 16x^{6} + 14x^{5} + 80x^{4} - 64x^{3} - 140x^{2} + 81x + 72 $ x^8 - x^7 - 16x^6 + 14x^5 + 80x^4 - 64x^3 - 140x^2 + 81x + 72
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.66154$ of defining polynomial
Character	$\chi$	$=$	9196.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.66154 q^{3} -1.44886 q^{5} +4.59395 q^{7} -0.239269 q^{9} -2.58959 q^{13} +2.40734 q^{15} +1.76666 q^{17} -1.00000 q^{19} -7.63305 q^{21} +3.19412 q^{23} -2.90081 q^{25} +5.38219 q^{27} +9.13043 q^{29} -8.95096 q^{31} -6.65597 q^{35} -11.2613 q^{37} +4.30272 q^{39} -3.21116 q^{41} +9.73920 q^{43} +0.346666 q^{45} +4.06579 q^{47} +14.1043 q^{49} -2.93539 q^{51} -7.05310 q^{53} +1.66154 q^{57} -3.60579 q^{59} -3.92847 q^{61} -1.09919 q^{63} +3.75195 q^{65} +4.93611 q^{67} -5.30717 q^{69} -10.4040 q^{71} -3.59116 q^{73} +4.81983 q^{75} +6.70441 q^{79} -8.22495 q^{81} +11.7228 q^{83} -2.55964 q^{85} -15.1706 q^{87} -15.0519 q^{89} -11.8964 q^{91} +14.8724 q^{93} +1.44886 q^{95} -1.56053 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{3} - 2 q^{7} + 9 q^{9} + 5 q^{13} - 9 q^{15} - 3 q^{17} - 8 q^{19} - 5 q^{21} + q^{23} - q^{27} - q^{29} + 3 q^{31} - 8 q^{35} - 23 q^{37} - 9 q^{39} - 10 q^{41} - 2 q^{43} + 9 q^{45} - 8 q^{47}+ \cdots + 6 q^{97}+O(q^{100}) $ 8 * q - q^3 - 2 * q^7 + 9 * q^9 + 5 * q^13 - 9 * q^15 - 3 * q^17 - 8 * q^19 - 5 * q^21 + q^23 - q^27 - q^29 + 3 * q^31 - 8 * q^35 - 23 * q^37 - 9 * q^39 - 10 * q^41 - 2 * q^43 + 9 * q^45 - 8 * q^47 + 22 * q^49 - 9 * q^51 + 6 * q^53 + q^57 - 11 * q^59 - 9 * q^61 - 32 * q^63 - 4 * q^65 + 9 * q^67 - 27 * q^69 - 3 * q^71 + 6 * q^73 + 30 * q^75 - 10 * q^79 - 24 * q^81 - 34 * q^83 - 25 * q^85 - 24 * q^87 - 11 * q^89 - 13 * q^91 + 45 * q^93 + 6 * 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
$3$	−1.66154	−0.959293	−0.479647	−	0.877462i	$-0.659235\pi$
$3$	−1.66154	−0.959293	−0.479647	+	0.877462i	$0.659235\pi$
$4$	0	0
$5$	−1.44886	−0.647949	−0.323974	−	0.946066i	$-0.605019\pi$
$5$	−1.44886	−0.647949	−0.323974	+	0.946066i	$0.605019\pi$
$6$	0	0
$7$	4.59395	1.73635	0.868174	−	0.496260i	$-0.165294\pi$
$7$	4.59395	1.73635	0.868174	+	0.496260i	$0.165294\pi$
$8$	0	0
$9$	−0.239269	−0.0797562
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.58959	−0.718224	−0.359112	−	0.933295i	$-0.616920\pi$
$13$	−2.58959	−0.718224	−0.359112	+	0.933295i	$0.616920\pi$
$14$	0	0
$15$	2.40734	0.621573
$16$	0	0
$17$	1.76666	0.428478	0.214239	−	0.976781i	$-0.431273\pi$
$17$	1.76666	0.428478	0.214239	+	0.976781i	$0.431273\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−7.63305	−1.66567
$22$	0	0
$23$	3.19412	0.666020	0.333010	−	0.942923i	$-0.391936\pi$
$23$	3.19412	0.666020	0.333010	+	0.942923i	$0.391936\pi$
$24$	0	0
$25$	−2.90081	−0.580163
$26$	0	0
$27$	5.38219	1.03580
$28$	0	0
$29$	9.13043	1.69548	0.847739	−	0.530414i	$-0.177963\pi$
$29$	9.13043	1.69548	0.847739	+	0.530414i	$0.177963\pi$
$30$	0	0
$31$	−8.95096	−1.60764	−0.803820	−	0.594873i	$-0.797202\pi$
$31$	−8.95096	−1.60764	−0.803820	+	0.594873i	$0.797202\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.65597	−1.12506
$36$	0	0
$37$	−11.2613	−1.85135	−0.925674	−	0.378323i	$-0.876501\pi$
$37$	−11.2613	−1.85135	−0.925674	+	0.378323i	$0.876501\pi$
$38$	0	0
$39$	4.30272	0.688987
$40$	0	0
$41$	−3.21116	−0.501499	−0.250750	−	0.968052i	$-0.580677\pi$
$41$	−3.21116	−0.501499	−0.250750	+	0.968052i	$0.580677\pi$
$42$	0	0
$43$	9.73920	1.48521	0.742607	−	0.669727i	$-0.233589\pi$
$43$	9.73920	1.48521	0.742607	+	0.669727i	$0.233589\pi$
$44$	0	0
$45$	0.346666	0.0516779
$46$	0	0
$47$	4.06579	0.593056	0.296528	−	0.955024i	$-0.404171\pi$
$47$	4.06579	0.593056	0.296528	+	0.955024i	$0.404171\pi$
$48$	0	0
$49$	14.1043	2.01491
$50$	0	0
$51$	−2.93539	−0.411036
$52$	0	0
$53$	−7.05310	−0.968817	−0.484409	−	0.874842i	$-0.660965\pi$
$53$	−7.05310	−0.968817	−0.484409	+	0.874842i	$0.660965\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.66154	0.220077
$58$	0	0
$59$	−3.60579	−0.469434	−0.234717	−	0.972064i	$-0.575416\pi$
$59$	−3.60579	−0.469434	−0.234717	+	0.972064i	$0.575416\pi$
$60$	0	0
$61$	−3.92847	−0.502988	−0.251494	−	0.967859i	$-0.580922\pi$
$61$	−3.92847	−0.502988	−0.251494	+	0.967859i	$0.580922\pi$
$62$	0	0
$63$	−1.09919	−0.138484
$64$	0	0
$65$	3.75195	0.465372
$66$	0	0
$67$	4.93611	0.603042	0.301521	−	0.953460i	$-0.402506\pi$
$67$	4.93611	0.603042	0.301521	+	0.953460i	$0.402506\pi$
$68$	0	0
$69$	−5.30717	−0.638908
$70$	0	0
$71$	−10.4040	−1.23472	−0.617361	−	0.786680i	$-0.711798\pi$
$71$	−10.4040	−1.23472	−0.617361	+	0.786680i	$0.711798\pi$
$72$	0	0
$73$	−3.59116	−0.420314	−0.210157	−	0.977668i	$-0.567397\pi$
$73$	−3.59116	−0.420314	−0.210157	+	0.977668i	$0.567397\pi$
$74$	0	0
$75$	4.81983	0.556546
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.70441	0.754305	0.377152	−	0.926151i	$-0.376903\pi$
$79$	6.70441	0.754305	0.377152	+	0.926151i	$0.376903\pi$
$80$	0	0
$81$	−8.22495	−0.913883
$82$	0	0
$83$	11.7228	1.28675	0.643375	−	0.765551i	$-0.277534\pi$
$83$	11.7228	1.28675	0.643375	+	0.765551i	$0.277534\pi$
$84$	0	0
$85$	−2.55964	−0.277632
$86$	0	0
$87$	−15.1706	−1.62646
$88$	0	0
$89$	−15.0519	−1.59550	−0.797751	−	0.602987i	$-0.793977\pi$
$89$	−15.0519	−1.59550	−0.797751	+	0.602987i	$0.793977\pi$
$90$	0	0
$91$	−11.8964	−1.24709
$92$	0	0
$93$	14.8724	1.54220
$94$	0	0
$95$	1.44886	0.148650
$96$	0	0
$97$	−1.56053	−0.158448	−0.0792241	−	0.996857i	$-0.525244\pi$
$97$	−1.56053	−0.158448	−0.0792241	+	0.996857i	$0.525244\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.77889	0.973036	0.486518	−	0.873670i	$-0.338267\pi$
$101$	9.77889	0.973036	0.486518	+	0.873670i	$0.338267\pi$
$102$	0	0
$103$	0.0628117	0.00618902	0.00309451	−	0.999995i	$-0.499015\pi$
$103$	0.0628117	0.00618902	0.00309451	+	0.999995i	$0.499015\pi$
$104$	0	0
$105$	11.0592	1.07927
$106$	0	0
$107$	−5.89837	−0.570217	−0.285108	−	0.958495i	$-0.592030\pi$
$107$	−5.89837	−0.570217	−0.285108	+	0.958495i	$0.592030\pi$
$108$	0	0
$109$	8.45790	0.810120	0.405060	−	0.914290i	$-0.367251\pi$
$109$	8.45790	0.810120	0.405060	+	0.914290i	$0.367251\pi$
$110$	0	0
$111$	18.7112	1.77599
$112$	0	0
$113$	−19.7345	−1.85647	−0.928233	−	0.372000i	$-0.878672\pi$
$113$	−19.7345	−1.85647	−0.928233	+	0.372000i	$0.878672\pi$
$114$	0	0
$115$	−4.62782	−0.431547
$116$	0	0
$117$	0.619608	0.0572828
$118$	0	0
$119$	8.11594	0.743987
$120$	0	0
$121$	0	0
$122$	0	0
$123$	5.33549	0.481085
$124$	0	0
$125$	11.4471	1.02386
$126$	0	0
$127$	−5.16531	−0.458347	−0.229174	−	0.973386i	$-0.573602\pi$
$127$	−5.16531	−0.458347	−0.229174	+	0.973386i	$0.573602\pi$
$128$	0	0
$129$	−16.1821	−1.42476
$130$	0	0
$131$	6.22880	0.544213	0.272106	−	0.962267i	$-0.412280\pi$
$131$	6.22880	0.544213	0.272106	+	0.962267i	$0.412280\pi$
$132$	0	0
$133$	−4.59395	−0.398346
$134$	0	0
$135$	−7.79802	−0.671147
$136$	0	0
$137$	−17.8957	−1.52893	−0.764466	−	0.644665i	$-0.776997\pi$
$137$	−17.8957	−1.52893	−0.764466	+	0.644665i	$0.776997\pi$
$138$	0	0
$139$	16.0945	1.36512	0.682560	−	0.730830i	$-0.260867\pi$
$139$	16.0945	1.36512	0.682560	+	0.730830i	$0.260867\pi$
$140$	0	0
$141$	−6.75549	−0.568915
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−13.2287	−1.09858
$146$	0	0
$147$	−23.4350	−1.93289
$148$	0	0
$149$	−14.0357	−1.14985	−0.574923	−	0.818207i	$-0.694968\pi$
$149$	−14.0357	−1.14985	−0.574923	+	0.818207i	$0.694968\pi$
$150$	0	0
$151$	19.8156	1.61257	0.806287	−	0.591525i	$-0.201474\pi$
$151$	19.8156	1.61257	0.806287	+	0.591525i	$0.201474\pi$
$152$	0	0
$153$	−0.422706	−0.0341738
$154$	0	0
$155$	12.9687	1.04167
$156$	0	0
$157$	−8.61314	−0.687403	−0.343702	−	0.939079i	$-0.611681\pi$
$157$	−8.61314	−0.687403	−0.343702	+	0.939079i	$0.611681\pi$
$158$	0	0
$159$	11.7190	0.929380
$160$	0	0
$161$	14.6736	1.15644
$162$	0	0
$163$	7.53736	0.590371	0.295186	−	0.955440i	$-0.404618\pi$
$163$	7.53736	0.590371	0.295186	+	0.955440i	$0.404618\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.12864	−0.396866	−0.198433	−	0.980114i	$-0.563585\pi$
$167$	−5.12864	−0.396866	−0.198433	+	0.980114i	$0.563585\pi$
$168$	0	0
$169$	−6.29401	−0.484155
$170$	0	0
$171$	0.239269	0.0182973
$172$	0	0
$173$	10.3490	0.786822	0.393411	−	0.919363i	$-0.371295\pi$
$173$	10.3490	0.786822	0.393411	+	0.919363i	$0.371295\pi$
$174$	0	0
$175$	−13.3262	−1.00736
$176$	0	0
$177$	5.99118	0.450325
$178$	0	0
$179$	15.0594	1.12560	0.562798	−	0.826595i	$-0.309725\pi$
$179$	15.0594	1.12560	0.562798	+	0.826595i	$0.309725\pi$
$180$	0	0
$181$	13.5630	1.00813	0.504063	−	0.863667i	$-0.331838\pi$
$181$	13.5630	1.00813	0.504063	+	0.863667i	$0.331838\pi$
$182$	0	0
$183$	6.52732	0.482513
$184$	0	0
$185$	16.3160	1.19958
$186$	0	0
$187$	0	0
$188$	0	0
$189$	24.7255	1.79851
$190$	0	0
$191$	−24.7232	−1.78891	−0.894453	−	0.447161i	$-0.852435\pi$
$191$	−24.7232	−1.78891	−0.894453	+	0.447161i	$0.852435\pi$
$192$	0	0
$193$	2.03890	0.146763	0.0733817	−	0.997304i	$-0.476621\pi$
$193$	2.03890	0.146763	0.0733817	+	0.997304i	$0.476621\pi$
$194$	0	0
$195$	−6.23403	−0.446428
$196$	0	0
$197$	22.0345	1.56989	0.784946	−	0.619564i	$-0.212691\pi$
$197$	22.0345	1.56989	0.784946	+	0.619564i	$0.212691\pi$
$198$	0	0
$199$	2.89588	0.205284	0.102642	−	0.994718i	$-0.467270\pi$
$199$	2.89588	0.205284	0.102642	+	0.994718i	$0.467270\pi$
$200$	0	0
$201$	−8.20157	−0.578494
$202$	0	0
$203$	41.9447	2.94394
$204$	0	0
$205$	4.65252	0.324946
$206$	0	0
$207$	−0.764252	−0.0531192
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−24.9872	−1.72019	−0.860093	−	0.510136i	$-0.829595\pi$
$211$	−24.9872	−1.72019	−0.860093	+	0.510136i	$0.829595\pi$
$212$	0	0
$213$	17.2866	1.18446
$214$	0	0
$215$	−14.1107	−0.962342
$216$	0	0
$217$	−41.1202	−2.79142
$218$	0	0
$219$	5.96687	0.403204
$220$	0	0
$221$	−4.57493	−0.307743
$222$	0	0
$223$	−3.54435	−0.237347	−0.118673	−	0.992933i	$-0.537864\pi$
$223$	−3.54435	−0.237347	−0.118673	+	0.992933i	$0.537864\pi$
$224$	0	0
$225$	0.694073	0.0462716
$226$	0	0
$227$	−13.2298	−0.878093	−0.439047	−	0.898464i	$-0.644684\pi$
$227$	−13.2298	−0.878093	−0.439047	+	0.898464i	$0.644684\pi$
$228$	0	0
$229$	18.9951	1.25523	0.627615	−	0.778524i	$-0.284031\pi$
$229$	18.9951	1.25523	0.627615	+	0.778524i	$0.284031\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.0699	−1.70790	−0.853948	−	0.520358i	$-0.825798\pi$
$233$	−26.0699	−1.70790	−0.853948	+	0.520358i	$0.825798\pi$
$234$	0	0
$235$	−5.89074	−0.384270
$236$	0	0
$237$	−11.1397	−0.723600
$238$	0	0
$239$	9.56674	0.618821	0.309410	−	0.950929i	$-0.399868\pi$
$239$	9.56674	0.618821	0.309410	+	0.950929i	$0.399868\pi$
$240$	0	0
$241$	9.46361	0.609605	0.304802	−	0.952416i	$-0.401410\pi$
$241$	9.46361	0.609605	0.304802	+	0.952416i	$0.401410\pi$
$242$	0	0
$243$	−2.48045	−0.159121
$244$	0	0
$245$	−20.4352	−1.30556
$246$	0	0
$247$	2.58959	0.164772
$248$	0	0
$249$	−19.4780	−1.23437
$250$	0	0
$251$	21.9956	1.38835	0.694174	−	0.719807i	$-0.255770\pi$
$251$	21.9956	1.38835	0.694174	+	0.719807i	$0.255770\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4.25295	0.266330
$256$	0	0
$257$	−14.8922	−0.928948	−0.464474	−	0.885587i	$-0.653757\pi$
$257$	−14.8922	−0.928948	−0.464474	+	0.885587i	$0.653757\pi$
$258$	0	0
$259$	−51.7338	−3.21458
$260$	0	0
$261$	−2.18462	−0.135225
$262$	0	0
$263$	−5.52825	−0.340886	−0.170443	−	0.985368i	$-0.554520\pi$
$263$	−5.52825	−0.340886	−0.170443	+	0.985368i	$0.554520\pi$
$264$	0	0
$265$	10.2189	0.627744
$266$	0	0
$267$	25.0095	1.53055
$268$	0	0
$269$	−2.34695	−0.143096	−0.0715481	−	0.997437i	$-0.522794\pi$
$269$	−2.34695	−0.143096	−0.0715481	+	0.997437i	$0.522794\pi$
$270$	0	0
$271$	−7.88969	−0.479265	−0.239632	−	0.970864i	$-0.577027\pi$
$271$	−7.88969	−0.479265	−0.239632	+	0.970864i	$0.577027\pi$
$272$	0	0
$273$	19.7665	1.19632
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.5692	1.05563	0.527816	−	0.849359i	$-0.323011\pi$
$277$	17.5692	1.05563	0.527816	+	0.849359i	$0.323011\pi$
$278$	0	0
$279$	2.14168	0.128219
$280$	0	0
$281$	−17.9900	−1.07320	−0.536598	−	0.843838i	$-0.680291\pi$
$281$	−17.9900	−1.07320	−0.536598	+	0.843838i	$0.680291\pi$
$282$	0	0
$283$	−18.6489	−1.10856	−0.554282	−	0.832329i	$-0.687007\pi$
$283$	−18.6489	−1.10856	−0.554282	+	0.832329i	$0.687007\pi$
$284$	0	0
$285$	−2.40734	−0.142599
$286$	0	0
$287$	−14.7519	−0.870778
$288$	0	0
$289$	−13.8789	−0.816407
$290$	0	0
$291$	2.59290	0.151998
$292$	0	0
$293$	12.3540	0.721727	0.360863	−	0.932619i	$-0.382482\pi$
$293$	12.3540	0.721727	0.360863	+	0.932619i	$0.382482\pi$
$294$	0	0
$295$	5.22427	0.304169
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.27146	−0.478351
$300$	0	0
$301$	44.7414	2.57885
$302$	0	0
$303$	−16.2481	−0.933427
$304$	0	0
$305$	5.69178	0.325911
$306$	0	0
$307$	−8.11292	−0.463029	−0.231514	−	0.972831i	$-0.574368\pi$
$307$	−8.11292	−0.463029	−0.231514	+	0.972831i	$0.574368\pi$
$308$	0	0
$309$	−0.104364	−0.00593709
$310$	0	0
$311$	17.8515	1.01227	0.506134	−	0.862455i	$-0.331074\pi$
$311$	17.8515	1.01227	0.506134	+	0.862455i	$0.331074\pi$
$312$	0	0
$313$	−7.17643	−0.405636	−0.202818	−	0.979216i	$-0.565010\pi$
$313$	−7.17643	−0.405636	−0.202818	+	0.979216i	$0.565010\pi$
$314$	0	0
$315$	1.59256	0.0897308
$316$	0	0
$317$	−17.0526	−0.957771	−0.478885	−	0.877877i	$-0.658959\pi$
$317$	−17.0526	−0.957771	−0.478885	+	0.877877i	$0.658959\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	9.80040	0.547005
$322$	0	0
$323$	−1.76666	−0.0982996
$324$	0	0
$325$	7.51192	0.416686
$326$	0	0
$327$	−14.0532	−0.777143
$328$	0	0
$329$	18.6780	1.02975
$330$	0	0
$331$	3.96498	0.217935	0.108968	−	0.994045i	$-0.465246\pi$
$331$	3.96498	0.217935	0.108968	+	0.994045i	$0.465246\pi$
$332$	0	0
$333$	2.69448	0.147656
$334$	0	0
$335$	−7.15172	−0.390740
$336$	0	0
$337$	−5.53239	−0.301368	−0.150684	−	0.988582i	$-0.548148\pi$
$337$	−5.53239	−0.301368	−0.150684	+	0.988582i	$0.548148\pi$
$338$	0	0
$339$	32.7898	1.78089
$340$	0	0
$341$	0	0
$342$	0	0
$343$	32.6370	1.76223
$344$	0	0
$345$	7.68933	0.413980
$346$	0	0
$347$	10.2158	0.548414	0.274207	−	0.961671i	$-0.411585\pi$
$347$	10.2158	0.548414	0.274207	+	0.961671i	$0.411585\pi$
$348$	0	0
$349$	−20.4024	−1.09212	−0.546058	−	0.837747i	$-0.683872\pi$
$349$	−20.4024	−1.09212	−0.546058	+	0.837747i	$0.683872\pi$
$350$	0	0
$351$	−13.9377	−0.743938
$352$	0	0
$353$	9.96981	0.530640	0.265320	−	0.964160i	$-0.414522\pi$
$353$	9.96981	0.530640	0.265320	+	0.964160i	$0.414522\pi$
$354$	0	0
$355$	15.0738	0.800036
$356$	0	0
$357$	−13.4850	−0.713702
$358$	0	0
$359$	−27.0934	−1.42993	−0.714966	−	0.699159i	$-0.753558\pi$
$359$	−27.0934	−1.42993	−0.714966	+	0.699159i	$0.753558\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.20308	0.272342
$366$	0	0
$367$	−33.7848	−1.76355	−0.881776	−	0.471669i	$-0.843652\pi$
$367$	−33.7848	−1.76355	−0.881776	+	0.471669i	$0.843652\pi$
$368$	0	0
$369$	0.768330	0.0399977
$370$	0	0
$371$	−32.4015	−1.68220
$372$	0	0
$373$	0.216005	0.0111843	0.00559217	−	0.999984i	$-0.498220\pi$
$373$	0.216005	0.0111843	0.00559217	+	0.999984i	$0.498220\pi$
$374$	0	0
$375$	−19.0200	−0.982186
$376$	0	0
$377$	−23.6441	−1.21773
$378$	0	0
$379$	17.2235	0.884713	0.442357	−	0.896839i	$-0.354143\pi$
$379$	17.2235	0.884713	0.442357	+	0.896839i	$0.354143\pi$
$380$	0	0
$381$	8.58240	0.439689
$382$	0	0
$383$	12.0944	0.617997	0.308999	−	0.951063i	$-0.400006\pi$
$383$	12.0944	0.617997	0.308999	+	0.951063i	$0.400006\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.33028	−0.118455
$388$	0	0
$389$	−30.8585	−1.56459	−0.782294	−	0.622910i	$-0.785950\pi$
$389$	−30.8585	−1.56459	−0.782294	+	0.622910i	$0.785950\pi$
$390$	0	0
$391$	5.64292	0.285375
$392$	0	0
$393$	−10.3494	−0.522060
$394$	0	0
$395$	−9.71373	−0.488751
$396$	0	0
$397$	−26.0141	−1.30561	−0.652806	−	0.757525i	$-0.726408\pi$
$397$	−26.0141	−1.30561	−0.652806	+	0.757525i	$0.726408\pi$
$398$	0	0
$399$	7.63305	0.382130
$400$	0	0
$401$	22.0666	1.10195	0.550977	−	0.834520i	$-0.314255\pi$
$401$	22.0666	1.10195	0.550977	+	0.834520i	$0.314255\pi$
$402$	0	0
$403$	23.1793	1.15464
$404$	0	0
$405$	11.9168	0.592149
$406$	0	0
$407$	0	0
$408$	0	0
$409$	36.2610	1.79299	0.896495	−	0.443054i	$-0.146105\pi$
$409$	36.2610	1.79299	0.896495	+	0.443054i	$0.146105\pi$
$410$	0	0
$411$	29.7345	1.46669
$412$	0	0
$413$	−16.5648	−0.815100
$414$	0	0
$415$	−16.9847	−0.833748
$416$	0	0
$417$	−26.7418	−1.30955
$418$	0	0
$419$	21.1442	1.03296	0.516481	−	0.856299i	$-0.327242\pi$
$419$	21.1442	1.03296	0.516481	+	0.856299i	$0.327242\pi$
$420$	0	0
$421$	−34.1410	−1.66393	−0.831966	−	0.554827i	$-0.812784\pi$
$421$	−34.1410	−1.66393	−0.831966	+	0.554827i	$0.812784\pi$
$422$	0	0
$423$	−0.972815	−0.0472999
$424$	0	0
$425$	−5.12475	−0.248587
$426$	0	0
$427$	−18.0472	−0.873363
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.5709	−0.942697	−0.471348	−	0.881947i	$-0.656233\pi$
$431$	−19.5709	−0.942697	−0.471348	+	0.881947i	$0.656233\pi$
$432$	0	0
$433$	5.46838	0.262794	0.131397	−	0.991330i	$-0.458054\pi$
$433$	5.46838	0.262794	0.131397	+	0.991330i	$0.458054\pi$
$434$	0	0
$435$	21.9801	1.05386
$436$	0	0
$437$	−3.19412	−0.152795
$438$	0	0
$439$	−33.0797	−1.57881	−0.789404	−	0.613875i	$-0.789610\pi$
$439$	−33.0797	−1.57881	−0.789404	+	0.613875i	$0.789610\pi$
$440$	0	0
$441$	−3.37472	−0.160701
$442$	0	0
$443$	−23.7656	−1.12914	−0.564570	−	0.825385i	$-0.690958\pi$
$443$	−23.7656	−1.12914	−0.564570	+	0.825385i	$0.690958\pi$
$444$	0	0
$445$	21.8081	1.03380
$446$	0	0
$447$	23.3209	1.10304
$448$	0	0
$449$	12.3197	0.581403	0.290701	−	0.956814i	$-0.406111\pi$
$449$	12.3197	0.581403	0.290701	+	0.956814i	$0.406111\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−32.9246	−1.54693
$454$	0	0
$455$	17.2362	0.808048
$456$	0	0
$457$	28.0122	1.31036	0.655178	−	0.755474i	$-0.272594\pi$
$457$	28.0122	1.31036	0.655178	+	0.755474i	$0.272594\pi$
$458$	0	0
$459$	9.50850	0.443819
$460$	0	0
$461$	3.06738	0.142862	0.0714311	−	0.997446i	$-0.477243\pi$
$461$	3.06738	0.142862	0.0714311	+	0.997446i	$0.477243\pi$
$462$	0	0
$463$	−5.14378	−0.239052	−0.119526	−	0.992831i	$-0.538137\pi$
$463$	−5.14378	−0.239052	−0.119526	+	0.992831i	$0.538137\pi$
$464$	0	0
$465$	−21.5480	−0.999265
$466$	0	0
$467$	−37.7978	−1.74908	−0.874538	−	0.484957i	$-0.838835\pi$
$467$	−37.7978	−1.74908	−0.874538	+	0.484957i	$0.838835\pi$
$468$	0	0
$469$	22.6762	1.04709
$470$	0	0
$471$	14.3111	0.659421
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.90081	0.133098
$476$	0	0
$477$	1.68758	0.0772692
$478$	0	0
$479$	−34.7304	−1.58687	−0.793436	−	0.608654i	$-0.791710\pi$
$479$	−34.7304	−1.58687	−0.793436	+	0.608654i	$0.791710\pi$
$480$	0	0
$481$	29.1622	1.32968
$482$	0	0
$483$	−24.3809	−1.10937
$484$	0	0
$485$	2.26099	0.102666
$486$	0	0
$487$	−8.23272	−0.373060	−0.186530	−	0.982449i	$-0.559724\pi$
$487$	−8.23272	−0.373060	−0.186530	+	0.982449i	$0.559724\pi$
$488$	0	0
$489$	−12.5237	−0.566339
$490$	0	0
$491$	−29.1977	−1.31767	−0.658836	−	0.752287i	$-0.728951\pi$
$491$	−29.1977	−1.31767	−0.658836	+	0.752287i	$0.728951\pi$
$492$	0	0
$493$	16.1304	0.726475
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−47.7952	−2.14391
$498$	0	0
$499$	−1.92969	−0.0863848	−0.0431924	−	0.999067i	$-0.513753\pi$
$499$	−1.92969	−0.0863848	−0.0431924	+	0.999067i	$0.513753\pi$
$500$	0	0
$501$	8.52147	0.380711
$502$	0	0
$503$	14.1733	0.631957	0.315979	−	0.948766i	$-0.397667\pi$
$503$	14.1733	0.631957	0.315979	+	0.948766i	$0.397667\pi$
$504$	0	0
$505$	−14.1682	−0.630477
$506$	0	0
$507$	10.4578	0.464447
$508$	0	0
$509$	−9.19561	−0.407588	−0.203794	−	0.979014i	$-0.565327\pi$
$509$	−9.19561	−0.407588	−0.203794	+	0.979014i	$0.565327\pi$
$510$	0	0
$511$	−16.4976	−0.729811
$512$	0	0
$513$	−5.38219	−0.237629
$514$	0	0
$515$	−0.0910052	−0.00401017
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−17.1954	−0.754794
$520$	0	0
$521$	−0.941850	−0.0412632	−0.0206316	−	0.999787i	$-0.506568\pi$
$521$	−0.941850	−0.0412632	−0.0206316	+	0.999787i	$0.506568\pi$
$522$	0	0
$523$	13.1301	0.574140	0.287070	−	0.957910i	$-0.407319\pi$
$523$	13.1301	0.574140	0.287070	+	0.957910i	$0.407319\pi$
$524$	0	0
$525$	22.1420	0.966358
$526$	0	0
$527$	−15.8133	−0.688838
$528$	0	0
$529$	−12.7976	−0.556418
$530$	0	0
$531$	0.862751	0.0374402
$532$	0	0
$533$	8.31560	0.360189
$534$	0	0
$535$	8.54589	0.369471
$536$	0	0
$537$	−25.0219	−1.07978
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−31.4618	−1.35265	−0.676324	−	0.736604i	$-0.736428\pi$
$541$	−31.4618	−1.35265	−0.676324	+	0.736604i	$0.736428\pi$
$542$	0	0
$543$	−22.5355	−0.967089
$544$	0	0
$545$	−12.2543	−0.524916
$546$	0	0
$547$	−12.6219	−0.539674	−0.269837	−	0.962906i	$-0.586970\pi$
$547$	−12.6219	−0.539674	−0.269837	+	0.962906i	$0.586970\pi$
$548$	0	0
$549$	0.939958	0.0401164
$550$	0	0
$551$	−9.13043	−0.388969
$552$	0	0
$553$	30.7997	1.30974
$554$	0	0
$555$	−27.1098	−1.15075
$556$	0	0
$557$	−4.11613	−0.174406	−0.0872030	−	0.996191i	$-0.527793\pi$
$557$	−4.11613	−0.174406	−0.0872030	+	0.996191i	$0.527793\pi$
$558$	0	0
$559$	−25.2206	−1.06672
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.4128	−1.19746	−0.598729	−	0.800952i	$-0.704327\pi$
$563$	−28.4128	−1.19746	−0.598729	+	0.800952i	$0.704327\pi$
$564$	0	0
$565$	28.5925	1.20289
$566$	0	0
$567$	−37.7850	−1.58682
$568$	0	0
$569$	−22.9359	−0.961523	−0.480761	−	0.876851i	$-0.659640\pi$
$569$	−22.9359	−0.961523	−0.480761	+	0.876851i	$0.659640\pi$
$570$	0	0
$571$	−5.63667	−0.235888	−0.117944	−	0.993020i	$-0.537630\pi$
$571$	−5.63667	−0.235888	−0.117944	+	0.993020i	$0.537630\pi$
$572$	0	0
$573$	41.0787	1.71609
$574$	0	0
$575$	−9.26554	−0.386400
$576$	0	0
$577$	36.0204	1.49955	0.749774	−	0.661694i	$-0.230162\pi$
$577$	36.0204	1.49955	0.749774	+	0.661694i	$0.230162\pi$
$578$	0	0
$579$	−3.38773	−0.140789
$580$	0	0
$581$	53.8541	2.23425
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−0.897723	−0.0371163
$586$	0	0
$587$	−25.4878	−1.05199	−0.525997	−	0.850487i	$-0.676308\pi$
$587$	−25.4878	−1.05199	−0.525997	+	0.850487i	$0.676308\pi$
$588$	0	0
$589$	8.95096	0.368818
$590$	0	0
$591$	−36.6113	−1.50599
$592$	0	0
$593$	−12.0275	−0.493908	−0.246954	−	0.969027i	$-0.579430\pi$
$593$	−12.0275	−0.493908	−0.246954	+	0.969027i	$0.579430\pi$
$594$	0	0
$595$	−11.7588	−0.482065
$596$	0	0
$597$	−4.81164	−0.196927
$598$	0	0
$599$	−21.3143	−0.870879	−0.435439	−	0.900218i	$-0.643407\pi$
$599$	−21.3143	−0.870879	−0.435439	+	0.900218i	$0.643407\pi$
$600$	0	0
$601$	10.2895	0.419719	0.209860	−	0.977731i	$-0.432699\pi$
$601$	10.2895	0.419719	0.209860	+	0.977731i	$0.432699\pi$
$602$	0	0
$603$	−1.18106	−0.0480963
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−15.1270	−0.613984	−0.306992	−	0.951712i	$-0.599323\pi$
$607$	−15.1270	−0.613984	−0.306992	+	0.951712i	$0.599323\pi$
$608$	0	0
$609$	−69.6930	−2.82410
$610$	0	0
$611$	−10.5287	−0.425947
$612$	0	0
$613$	−34.8184	−1.40630	−0.703151	−	0.711040i	$-0.748224\pi$
$613$	−34.8184	−1.40630	−0.703151	+	0.711040i	$0.748224\pi$
$614$	0	0
$615$	−7.73037	−0.311718
$616$	0	0
$617$	3.70286	0.149072	0.0745358	−	0.997218i	$-0.476252\pi$
$617$	3.70286	0.149072	0.0745358	+	0.997218i	$0.476252\pi$
$618$	0	0
$619$	−20.4316	−0.821216	−0.410608	−	0.911812i	$-0.634684\pi$
$619$	−20.4316	−0.821216	−0.410608	+	0.911812i	$0.634684\pi$
$620$	0	0
$621$	17.1914	0.689865
$622$	0	0
$623$	−69.1478	−2.77035
$624$	0	0
$625$	−2.08121	−0.0832486
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−19.8949	−0.793262
$630$	0	0
$631$	−10.5535	−0.420129	−0.210064	−	0.977688i	$-0.567367\pi$
$631$	−10.5535	−0.420129	−0.210064	+	0.977688i	$0.567367\pi$
$632$	0	0
$633$	41.5173	1.65016
$634$	0	0
$635$	7.48380	0.296985
$636$	0	0
$637$	−36.5245	−1.44715
$638$	0	0
$639$	2.48934	0.0984767
$640$	0	0
$641$	−27.4572	−1.08450	−0.542248	−	0.840219i	$-0.682427\pi$
$641$	−27.4572	−1.08450	−0.542248	+	0.840219i	$0.682427\pi$
$642$	0	0
$643$	−25.0333	−0.987215	−0.493608	−	0.869685i	$-0.664322\pi$
$643$	−25.0333	−0.987215	−0.493608	+	0.869685i	$0.664322\pi$
$644$	0	0
$645$	23.4456	0.923169
$646$	0	0
$647$	19.0899	0.750503	0.375252	−	0.926923i	$-0.377556\pi$
$647$	19.0899	0.750503	0.375252	+	0.926923i	$0.377556\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	68.3231	2.67779
$652$	0	0
$653$	−29.7809	−1.16542	−0.582708	−	0.812682i	$-0.698007\pi$
$653$	−29.7809	−1.16542	−0.582708	+	0.812682i	$0.698007\pi$
$654$	0	0
$655$	−9.02464	−0.352622
$656$	0	0
$657$	0.859252	0.0335226
$658$	0	0
$659$	−21.0160	−0.818666	−0.409333	−	0.912385i	$-0.634238\pi$
$659$	−21.0160	−0.818666	−0.409333	+	0.912385i	$0.634238\pi$
$660$	0	0
$661$	−14.8988	−0.579497	−0.289748	−	0.957103i	$-0.593572\pi$
$661$	−14.8988	−0.579497	−0.289748	+	0.957103i	$0.593572\pi$
$662$	0	0
$663$	7.60145	0.295216
$664$	0	0
$665$	6.65597	0.258107
$666$	0	0
$667$	29.1637	1.12922
$668$	0	0
$669$	5.88909	0.227685
$670$	0	0
$671$	0	0
$672$	0	0
$673$	23.4358	0.903384	0.451692	−	0.892174i	$-0.350821\pi$
$673$	23.4358	0.903384	0.451692	+	0.892174i	$0.350821\pi$
$674$	0	0
$675$	−15.6127	−0.600934
$676$	0	0
$677$	11.9827	0.460532	0.230266	−	0.973128i	$-0.426040\pi$
$677$	11.9827	0.460532	0.230266	+	0.973128i	$0.426040\pi$
$678$	0	0
$679$	−7.16901	−0.275121
$680$	0	0
$681$	21.9819	0.842349
$682$	0	0
$683$	1.59889	0.0611797	0.0305899	−	0.999532i	$-0.490261\pi$
$683$	1.59889	0.0611797	0.0305899	+	0.999532i	$0.490261\pi$
$684$	0	0
$685$	25.9283	0.990669
$686$	0	0
$687$	−31.5612	−1.20413
$688$	0	0
$689$	18.2646	0.695827
$690$	0	0
$691$	−9.25201	−0.351963	−0.175981	−	0.984393i	$-0.556310\pi$
$691$	−9.25201	−0.351963	−0.175981	+	0.984393i	$0.556310\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−23.3187	−0.884527
$696$	0	0
$697$	−5.67304	−0.214882
$698$	0	0
$699$	43.3163	1.63837
$700$	0	0
$701$	−13.3459	−0.504066	−0.252033	−	0.967719i	$-0.581099\pi$
$701$	−13.3459	−0.504066	−0.252033	+	0.967719i	$0.581099\pi$
$702$	0	0
$703$	11.2613	0.424728
$704$	0	0
$705$	9.78773	0.368627
$706$	0	0
$707$	44.9237	1.68953
$708$	0	0
$709$	26.7172	1.00339	0.501694	−	0.865045i	$-0.332710\pi$
$709$	26.7172	1.00339	0.501694	+	0.865045i	$0.332710\pi$
$710$	0	0
$711$	−1.60415	−0.0601605
$712$	0	0
$713$	−28.5904	−1.07072
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.8956	−0.593631
$718$	0	0
$719$	−25.3343	−0.944809	−0.472404	−	0.881382i	$-0.656614\pi$
$719$	−25.3343	−0.944809	−0.472404	+	0.881382i	$0.656614\pi$
$720$	0	0
$721$	0.288554	0.0107463
$722$	0	0
$723$	−15.7242	−0.584790
$724$	0	0
$725$	−26.4857	−0.983653
$726$	0	0
$727$	11.6846	0.433358	0.216679	−	0.976243i	$-0.430477\pi$
$727$	11.6846	0.433358	0.216679	+	0.976243i	$0.430477\pi$
$728$	0	0
$729$	28.7962	1.06653
$730$	0	0
$731$	17.2059	0.636382
$732$	0	0
$733$	−20.5372	−0.758559	−0.379279	−	0.925282i	$-0.623828\pi$
$733$	−20.5372	−0.758559	−0.379279	+	0.925282i	$0.623828\pi$
$734$	0	0
$735$	33.9540	1.25241
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−11.7218	−0.431193	−0.215597	−	0.976483i	$-0.569170\pi$
$739$	−11.7218	−0.431193	−0.215597	+	0.976483i	$0.569170\pi$
$740$	0	0
$741$	−4.30272	−0.158064
$742$	0	0
$743$	26.8400	0.984666	0.492333	−	0.870407i	$-0.336144\pi$
$743$	26.8400	0.984666	0.492333	+	0.870407i	$0.336144\pi$
$744$	0	0
$745$	20.3357	0.745041
$746$	0	0
$747$	−2.80491	−0.102626
$748$	0	0
$749$	−27.0968	−0.990095
$750$	0	0
$751$	26.5399	0.968456	0.484228	−	0.874942i	$-0.339101\pi$
$751$	26.5399	0.968456	0.484228	+	0.874942i	$0.339101\pi$
$752$	0	0
$753$	−36.5466	−1.33183
$754$	0	0
$755$	−28.7100	−1.04486
$756$	0	0
$757$	−29.8915	−1.08642	−0.543212	−	0.839596i	$-0.682792\pi$
$757$	−29.8915	−1.08642	−0.543212	+	0.839596i	$0.682792\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−51.1360	−1.85368	−0.926840	−	0.375457i	$-0.877486\pi$
$761$	−51.1360	−1.85368	−0.926840	+	0.375457i	$0.877486\pi$
$762$	0	0
$763$	38.8552	1.40665
$764$	0	0
$765$	0.612441	0.0221428
$766$	0	0
$767$	9.33752	0.337158
$768$	0	0
$769$	45.7305	1.64909	0.824543	−	0.565800i	$-0.191433\pi$
$769$	45.7305	1.64909	0.824543	+	0.565800i	$0.191433\pi$
$770$	0	0
$771$	24.7440	0.891134
$772$	0	0
$773$	17.9908	0.647085	0.323543	−	0.946214i	$-0.395126\pi$
$773$	17.9908	0.647085	0.323543	+	0.946214i	$0.395126\pi$
$774$	0	0
$775$	25.9651	0.932692
$776$	0	0
$777$	85.9581	3.08373
$778$	0	0
$779$	3.21116	0.115052
$780$	0	0
$781$	0	0
$782$	0	0
$783$	49.1417	1.75618
$784$	0	0
$785$	12.4792	0.445402
$786$	0	0
$787$	44.9337	1.60171	0.800857	−	0.598856i	$-0.204378\pi$
$787$	44.9337	1.60171	0.800857	+	0.598856i	$0.204378\pi$
$788$	0	0
$789$	9.18543	0.327010
$790$	0	0
$791$	−90.6592	−3.22347
$792$	0	0
$793$	10.1731	0.361258
$794$	0	0
$795$	−16.9792	−0.602190
$796$	0	0
$797$	−48.4413	−1.71588	−0.857940	−	0.513750i	$-0.828256\pi$
$797$	−48.4413	−1.71588	−0.857940	+	0.513750i	$0.828256\pi$
$798$	0	0
$799$	7.18286	0.254111
$800$	0	0
$801$	3.60145	0.127251
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−21.2600	−0.749315
$806$	0	0
$807$	3.89957	0.137271
$808$	0	0
$809$	−38.2139	−1.34353	−0.671765	−	0.740765i	$-0.734463\pi$
$809$	−38.2139	−1.34353	−0.671765	+	0.740765i	$0.734463\pi$
$810$	0	0
$811$	−10.4246	−0.366058	−0.183029	−	0.983107i	$-0.558590\pi$
$811$	−10.4246	−0.366058	−0.183029	+	0.983107i	$0.558590\pi$
$812$	0	0
$813$	13.1091	0.459756
$814$	0	0
$815$	−10.9206	−0.382530
$816$	0	0
$817$	−9.73920	−0.340731
$818$	0	0
$819$	2.84644	0.0994628
$820$	0	0
$821$	−5.42933	−0.189485	−0.0947424	−	0.995502i	$-0.530203\pi$
$821$	−5.42933	−0.189485	−0.0947424	+	0.995502i	$0.530203\pi$
$822$	0	0
$823$	−6.09596	−0.212492	−0.106246	−	0.994340i	$-0.533883\pi$
$823$	−6.09596	−0.212492	−0.106246	+	0.994340i	$0.533883\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.47854	−0.0861874	−0.0430937	−	0.999071i	$-0.513721\pi$
$827$	−2.47854	−0.0861874	−0.0430937	+	0.999071i	$0.513721\pi$
$828$	0	0
$829$	−1.82226	−0.0632897	−0.0316448	−	0.999499i	$-0.510075\pi$
$829$	−1.82226	−0.0632897	−0.0316448	+	0.999499i	$0.510075\pi$
$830$	0	0
$831$	−29.1920	−1.01266
$832$	0	0
$833$	24.9176	0.863343
$834$	0	0
$835$	7.43067	0.257149
$836$	0	0
$837$	−48.1757	−1.66520
$838$	0	0
$839$	−44.5711	−1.53877	−0.769383	−	0.638788i	$-0.779436\pi$
$839$	−44.5711	−1.53877	−0.769383	+	0.638788i	$0.779436\pi$
$840$	0	0
$841$	54.3647	1.87464
$842$	0	0
$843$	29.8913	1.02951
$844$	0	0
$845$	9.11913	0.313708
$846$	0	0
$847$	0	0
$848$	0	0
$849$	30.9861	1.06344
$850$	0	0
$851$	−35.9699	−1.23303
$852$	0	0
$853$	1.56811	0.0536911	0.0268455	−	0.999640i	$-0.491454\pi$
$853$	1.56811	0.0536911	0.0268455	+	0.999640i	$0.491454\pi$
$854$	0	0
$855$	−0.346666	−0.0118557
$856$	0	0
$857$	5.77727	0.197348	0.0986740	−	0.995120i	$-0.468540\pi$
$857$	5.77727	0.197348	0.0986740	+	0.995120i	$0.468540\pi$
$858$	0	0
$859$	14.9131	0.508827	0.254414	−	0.967095i	$-0.418118\pi$
$859$	14.9131	0.508827	0.254414	+	0.967095i	$0.418118\pi$
$860$	0	0
$861$	24.5110	0.835331
$862$	0	0
$863$	−26.4882	−0.901668	−0.450834	−	0.892608i	$-0.648873\pi$
$863$	−26.4882	−0.901668	−0.450834	+	0.892608i	$0.648873\pi$
$864$	0	0
$865$	−14.9943	−0.509820
$866$	0	0
$867$	23.0604	0.783173
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−12.7825	−0.433119
$872$	0	0
$873$	0.373387	0.0126372
$874$	0	0
$875$	52.5876	1.77778
$876$	0	0
$877$	20.5045	0.692388	0.346194	−	0.938163i	$-0.387474\pi$
$877$	20.5045	0.692388	0.346194	+	0.938163i	$0.387474\pi$
$878$	0	0
$879$	−20.5267	−0.692348
$880$	0	0
$881$	−24.2668	−0.817570	−0.408785	−	0.912631i	$-0.634047\pi$
$881$	−24.2668	−0.817570	−0.408785	+	0.912631i	$0.634047\pi$
$882$	0	0
$883$	11.5929	0.390133	0.195067	−	0.980790i	$-0.437508\pi$
$883$	11.5929	0.390133	0.195067	+	0.980790i	$0.437508\pi$
$884$	0	0
$885$	−8.68036	−0.291787
$886$	0	0
$887$	5.95973	0.200108	0.100054	−	0.994982i	$-0.468098\pi$
$887$	5.95973	0.200108	0.100054	+	0.994982i	$0.468098\pi$
$888$	0	0
$889$	−23.7292	−0.795850
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.06579	−0.136056
$894$	0	0
$895$	−21.8190	−0.729328
$896$	0	0
$897$	13.7434	0.458879
$898$	0	0
$899$	−81.7260	−2.72572
$900$	0	0
$901$	−12.4604	−0.415117
$902$	0	0
$903$	−74.3398	−2.47387
$904$	0	0
$905$	−19.6508	−0.653214
$906$	0	0
$907$	−49.2797	−1.63630	−0.818152	−	0.575002i	$-0.805001\pi$
$907$	−49.2797	−1.63630	−0.818152	+	0.575002i	$0.805001\pi$
$908$	0	0
$909$	−2.33978	−0.0776056
$910$	0	0
$911$	−21.2880	−0.705304	−0.352652	−	0.935755i	$-0.614720\pi$
$911$	−21.2880	−0.705304	−0.352652	+	0.935755i	$0.614720\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−9.45716	−0.312644
$916$	0	0
$917$	28.6148	0.944943
$918$	0	0
$919$	−7.49120	−0.247112	−0.123556	−	0.992338i	$-0.539430\pi$
$919$	−7.49120	−0.247112	−0.123556	+	0.992338i	$0.539430\pi$
$920$	0	0
$921$	13.4800	0.444180
$922$	0	0
$923$	26.9420	0.886806
$924$	0	0
$925$	32.6669	1.07408
$926$	0	0
$927$	−0.0150289	−0.000493613 0
$928$	0	0
$929$	−7.42600	−0.243639	−0.121820	−	0.992552i	$-0.538873\pi$
$929$	−7.42600	−0.243639	−0.121820	+	0.992552i	$0.538873\pi$
$930$	0	0
$931$	−14.1043	−0.462251
$932$	0	0
$933$	−29.6611	−0.971062
$934$	0	0
$935$	0	0
$936$	0	0
$937$	34.0952	1.11384	0.556921	−	0.830566i	$-0.311983\pi$
$937$	34.0952	1.11384	0.556921	+	0.830566i	$0.311983\pi$
$938$	0	0
$939$	11.9240	0.389124
$940$	0	0
$941$	43.5658	1.42020	0.710102	−	0.704099i	$-0.248649\pi$
$941$	43.5658	1.42020	0.710102	+	0.704099i	$0.248649\pi$
$942$	0	0
$943$	−10.2568	−0.334009
$944$	0	0
$945$	−35.8237	−1.16535
$946$	0	0
$947$	−30.0314	−0.975890	−0.487945	−	0.872874i	$-0.662253\pi$
$947$	−30.0314	−0.975890	−0.487945	+	0.872874i	$0.662253\pi$
$948$	0	0
$949$	9.29964	0.301879
$950$	0	0
$951$	28.3337	0.918783
$952$	0	0
$953$	−27.1162	−0.878380	−0.439190	−	0.898394i	$-0.644735\pi$
$953$	−27.1162	−0.878380	−0.439190	+	0.898394i	$0.644735\pi$
$954$	0	0
$955$	35.8204	1.15912
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−82.2118	−2.65476
$960$	0	0
$961$	49.1196	1.58450
$962$	0	0
$963$	1.41129	0.0454783
$964$	0	0
$965$	−2.95408	−0.0950952
$966$	0	0
$967$	13.3979	0.430846	0.215423	−	0.976521i	$-0.430887\pi$
$967$	13.3979	0.430846	0.215423	+	0.976521i	$0.430887\pi$
$968$	0	0
$969$	2.93539	0.0942982
$970$	0	0
$971$	−2.02489	−0.0649818	−0.0324909	−	0.999472i	$-0.510344\pi$
$971$	−2.02489	−0.0649818	−0.0324909	+	0.999472i	$0.510344\pi$
$972$	0	0
$973$	73.9374	2.37032
$974$	0	0
$975$	−12.4814	−0.399725
$976$	0	0
$977$	14.0785	0.450413	0.225206	−	0.974311i	$-0.427694\pi$
$977$	14.0785	0.450413	0.225206	+	0.974311i	$0.427694\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−2.02371	−0.0646121
$982$	0	0
$983$	60.3551	1.92503	0.962514	−	0.271232i	$-0.0874311\pi$
$983$	60.3551	1.92503	0.962514	+	0.271232i	$0.0874311\pi$
$984$	0	0
$985$	−31.9248	−1.01721
$986$	0	0
$987$	−31.0343	−0.987834
$988$	0	0
$989$	31.1082	0.989182
$990$	0	0
$991$	−22.2644	−0.707252	−0.353626	−	0.935387i	$-0.615052\pi$
$991$	−22.2644	−0.707252	−0.353626	+	0.935387i	$0.615052\pi$
$992$	0	0
$993$	−6.58800	−0.209064
$994$	0	0
$995$	−4.19572	−0.133013
$996$	0	0
$997$	−15.3521	−0.486207	−0.243103	−	0.970000i	$-0.578165\pi$
$997$	−15.3521	−0.486207	−0.243103	+	0.970000i	$0.578165\pi$
$998$	0	0
$999$	−60.6105	−1.91763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.q.1.3	✓	8
11.10	odd	2		9196.2.a.r.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9196.2.a.q.1.3	✓	8	1.1	even	1	trivial
9196.2.a.r.1.3	yes	8	11.10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\(q-1.66154 q^{3} -1.44886 q^{5} +4.59395 q^{7} -0.239269 q^{9} -2.58959 q^{13} +2.40734 q^{15} +1.76666 q^{17} -1.00000 q^{19} -7.63305 q^{21} +3.19412 q^{23} -2.90081 q^{25} +5.38219 q^{27} +9.13043 q^{29} -8.95096 q^{31} -6.65597 q^{35} -11.2613 q^{37} +4.30272 q^{39} -3.21116 q^{41} +9.73920 q^{43} +0.346666 q^{45} +4.06579 q^{47} +14.1043 q^{49} -2.93539 q^{51} -7.05310 q^{53} +1.66154 q^{57} -3.60579 q^{59} -3.92847 q^{61} -1.09919 q^{63} +3.75195 q^{65} +4.93611 q^{67} -5.30717 q^{69} -10.4040 q^{71} -3.59116 q^{73} +4.81983 q^{75} +6.70441 q^{79} -8.22495 q^{81} +11.7228 q^{83} -2.55964 q^{85} -15.1706 q^{87} -15.0519 q^{89} -11.8964 q^{91} +14.8724 q^{93} +1.44886 q^{95} -1.56053 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} - 2 q^{7} + 9 q^{9} + 5 q^{13} - 9 q^{15} - 3 q^{17} - 8 q^{19} - 5 q^{21} + q^{23} - q^{27} - q^{29} + 3 q^{31} - 8 q^{35} - 23 q^{37} - 9 q^{39} - 10 q^{41} - 2 q^{43} + 9 q^{45} - 8 q^{47}+ \cdots + 6 q^{97}+O(q^{100}) \) 8 * q - q^3 - 2 * q^7 + 9 * q^9 + 5 * q^13 - 9 * q^15 - 3 * q^17 - 8 * q^19 - 5 * q^21 + q^23 - q^27 - q^29 + 3 * q^31 - 8 * q^35 - 23 * q^37 - 9 * q^39 - 10 * q^41 - 2 * q^43 + 9 * q^45 - 8 * q^47 + 22 * q^49 - 9 * q^51 + 6 * q^53 + q^57 - 11 * q^59 - 9 * q^61 - 32 * q^63 - 4 * q^65 + 9 * q^67 - 27 * q^69 - 3 * q^71 + 6 * q^73 + 30 * q^75 - 10 * q^79 - 24 * q^81 - 34 * q^83 - 25 * q^85 - 24 * q^87 - 11 * q^89 - 13 * q^91 + 45 * q^93 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more