LMFDB - Embedded newform 1336.2.a.c.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, 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("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.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:	$10.6680137100$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 13x^{7} + 8x^{6} + 56x^{5} - 15x^{4} - 81x^{3} + 2x^{2} + 13x + 1 $ x^9 - x^8 - 13x^7 + 8x^6 + 56x^5 - 15x^4 - 81x^3 + 2x^2 + 13*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.20474$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.20474 q^{3} -3.21786 q^{5} +0.280589 q^{7} +1.86086 q^{9} +O(q^{10})$	$q+2.20474 q^{3} -3.21786 q^{5} +0.280589 q^{7} +1.86086 q^{9} -1.36018 q^{11} -4.48782 q^{13} -7.09454 q^{15} +3.11192 q^{17} -2.88609 q^{19} +0.618625 q^{21} -4.94790 q^{23} +5.35465 q^{25} -2.51151 q^{27} -7.64940 q^{29} -3.54842 q^{31} -2.99884 q^{33} -0.902898 q^{35} +4.33486 q^{37} -9.89445 q^{39} +11.0085 q^{41} +1.44871 q^{43} -5.98798 q^{45} +2.09099 q^{47} -6.92127 q^{49} +6.86095 q^{51} -13.6086 q^{53} +4.37689 q^{55} -6.36305 q^{57} -11.1920 q^{59} +8.49589 q^{61} +0.522136 q^{63} +14.4412 q^{65} -1.87500 q^{67} -10.9088 q^{69} -1.74286 q^{71} -4.37801 q^{73} +11.8056 q^{75} -0.381653 q^{77} +10.6796 q^{79} -11.1198 q^{81} +0.318715 q^{83} -10.0137 q^{85} -16.8649 q^{87} +0.886287 q^{89} -1.25923 q^{91} -7.82333 q^{93} +9.28703 q^{95} -0.410493 q^{97} -2.53111 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} - 8 q^{5} + 2 q^{7}+O(q^{10}) $ 9 * q - q^3 - 8 * q^5 + 2 * q^7	$ 9 q - q^{3} - 8 q^{5} + 2 q^{7} - 10 q^{11} - 13 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 10 q^{27} - 25 q^{29} - q^{31} - 12 q^{33} - 17 q^{35} - 35 q^{37} - 4 q^{39} - 16 q^{41} + 9 q^{43} - 24 q^{45} - q^{47} - q^{49} - 10 q^{51} - 29 q^{53} + 9 q^{55} - 17 q^{57} - 14 q^{59} - 28 q^{61} + 4 q^{63} - 31 q^{65} + 19 q^{67} - 19 q^{69} - 9 q^{71} - 7 q^{75} - 33 q^{77} - 18 q^{79} - 27 q^{81} - 13 q^{83} - 36 q^{85} + 18 q^{87} - 21 q^{89} + 20 q^{91} - 35 q^{93} - 12 q^{95} + 2 q^{97} - 3 q^{99}+O(q^{100}) $ 9 * q - q^3 - 8 * q^5 + 2 * q^7 - 10 * q^11 - 13 * q^13 + 2 * q^15 - 8 * q^17 - q^19 - 19 * q^21 - 3 * q^23 + 3 * q^25 - 10 * q^27 - 25 * q^29 - q^31 - 12 * q^33 - 17 * q^35 - 35 * q^37 - 4 * q^39 - 16 * q^41 + 9 * q^43 - 24 * q^45 - q^47 - q^49 - 10 * q^51 - 29 * q^53 + 9 * q^55 - 17 * q^57 - 14 * q^59 - 28 * q^61 + 4 * q^63 - 31 * q^65 + 19 * q^67 - 19 * q^69 - 9 * q^71 - 7 * q^75 - 33 * q^77 - 18 * q^79 - 27 * q^81 - 13 * q^83 - 36 * q^85 + 18 * q^87 - 21 * q^89 + 20 * q^91 - 35 * q^93 - 12 * q^95 + 2 * q^97 - 3 * 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
$3$	2.20474	1.27290	0.636452	−	0.771316i	$-0.280401\pi$
$3$	2.20474	1.27290	0.636452	+	0.771316i	$0.280401\pi$
$4$	0	0
$5$	−3.21786	−1.43907	−0.719536	−	0.694455i	$-0.755646\pi$
$5$	−3.21786	−1.43907	−0.719536	+	0.694455i	$0.755646\pi$
$6$	0	0
$7$	0.280589	0.106053	0.0530264	−	0.998593i	$-0.483113\pi$
$7$	0.280589	0.106053	0.0530264	+	0.998593i	$0.483113\pi$
$8$	0	0
$9$	1.86086	0.620286
$10$	0	0
$11$	−1.36018	−0.410111	−0.205055	−	0.978750i	$-0.565737\pi$
$11$	−1.36018	−0.410111	−0.205055	+	0.978750i	$0.565737\pi$
$12$	0	0
$13$	−4.48782	−1.24470	−0.622348	−	0.782741i	$-0.713821\pi$
$13$	−4.48782	−1.24470	−0.622348	+	0.782741i	$0.713821\pi$
$14$	0	0
$15$	−7.09454	−1.83180
$16$	0	0
$17$	3.11192	0.754751	0.377375	−	0.926060i	$-0.376827\pi$
$17$	3.11192	0.754751	0.377375	+	0.926060i	$0.376827\pi$
$18$	0	0
$19$	−2.88609	−0.662113	−0.331057	−	0.943611i	$-0.607405\pi$
$19$	−2.88609	−0.662113	−0.331057	+	0.943611i	$0.607405\pi$
$20$	0	0
$21$	0.618625	0.134995
$22$	0	0
$23$	−4.94790	−1.03171	−0.515854	−	0.856676i	$-0.672525\pi$
$23$	−4.94790	−1.03171	−0.515854	+	0.856676i	$0.672525\pi$
$24$	0	0
$25$	5.35465	1.07093
$26$	0	0
$27$	−2.51151	−0.483340
$28$	0	0
$29$	−7.64940	−1.42046	−0.710229	−	0.703971i	$-0.751408\pi$
$29$	−7.64940	−1.42046	−0.710229	+	0.703971i	$0.751408\pi$
$30$	0	0
$31$	−3.54842	−0.637315	−0.318658	−	0.947870i	$-0.603232\pi$
$31$	−3.54842	−0.637315	−0.318658	+	0.947870i	$0.603232\pi$
$32$	0	0
$33$	−2.99884	−0.522032
$34$	0	0
$35$	−0.902898	−0.152618
$36$	0	0
$37$	4.33486	0.712647	0.356324	−	0.934363i	$-0.384030\pi$
$37$	4.33486	0.712647	0.356324	+	0.934363i	$0.384030\pi$
$38$	0	0
$39$	−9.89445	−1.58438
$40$	0	0
$41$	11.0085	1.71923	0.859616	−	0.510941i	$-0.170703\pi$
$41$	11.0085	1.71923	0.859616	+	0.510941i	$0.170703\pi$
$42$	0	0
$43$	1.44871	0.220927	0.110463	−	0.993880i	$-0.464767\pi$
$43$	1.44871	0.220927	0.110463	+	0.993880i	$0.464767\pi$
$44$	0	0
$45$	−5.98798	−0.892636
$46$	0	0
$47$	2.09099	0.305003	0.152501	−	0.988303i	$-0.451267\pi$
$47$	2.09099	0.305003	0.152501	+	0.988303i	$0.451267\pi$
$48$	0	0
$49$	−6.92127	−0.988753
$50$	0	0
$51$	6.86095	0.960726
$52$	0	0
$53$	−13.6086	−1.86929	−0.934643	−	0.355587i	$-0.884281\pi$
$53$	−13.6086	−1.86929	−0.934643	+	0.355587i	$0.884281\pi$
$54$	0	0
$55$	4.37689	0.590179
$56$	0	0
$57$	−6.36305	−0.842807
$58$	0	0
$59$	−11.1920	−1.45707	−0.728534	−	0.685010i	$-0.759798\pi$
$59$	−11.1920	−1.45707	−0.728534	+	0.685010i	$0.759798\pi$
$60$	0	0
$61$	8.49589	1.08779	0.543894	−	0.839154i	$-0.316950\pi$
$61$	8.49589	1.08779	0.543894	+	0.839154i	$0.316950\pi$
$62$	0	0
$63$	0.522136	0.0657830
$64$	0	0
$65$	14.4412	1.79121
$66$	0	0
$67$	−1.87500	−0.229068	−0.114534	−	0.993419i	$-0.536537\pi$
$67$	−1.87500	−0.229068	−0.114534	+	0.993419i	$0.536537\pi$
$68$	0	0
$69$	−10.9088	−1.31327
$70$	0	0
$71$	−1.74286	−0.206840	−0.103420	−	0.994638i	$-0.532979\pi$
$71$	−1.74286	−0.206840	−0.103420	+	0.994638i	$0.532979\pi$
$72$	0	0
$73$	−4.37801	−0.512408	−0.256204	−	0.966623i	$-0.582472\pi$
$73$	−4.37801	−0.512408	−0.256204	+	0.966623i	$0.582472\pi$
$74$	0	0
$75$	11.8056	1.36319
$76$	0	0
$77$	−0.381653	−0.0434934
$78$	0	0
$79$	10.6796	1.20155	0.600775	−	0.799418i	$-0.294859\pi$
$79$	10.6796	1.20155	0.600775	+	0.799418i	$0.294859\pi$
$80$	0	0
$81$	−11.1198	−1.23553
$82$	0	0
$83$	0.318715	0.0349835	0.0174918	−	0.999847i	$-0.494432\pi$
$83$	0.318715	0.0349835	0.0174918	+	0.999847i	$0.494432\pi$
$84$	0	0
$85$	−10.0137	−1.08614
$86$	0	0
$87$	−16.8649	−1.80811
$88$	0	0
$89$	0.886287	0.0939462	0.0469731	−	0.998896i	$-0.485042\pi$
$89$	0.886287	0.0939462	0.0469731	+	0.998896i	$0.485042\pi$
$90$	0	0
$91$	−1.25923	−0.132003
$92$	0	0
$93$	−7.82333	−0.811241
$94$	0	0
$95$	9.28703	0.952829
$96$	0	0
$97$	−0.410493	−0.0416792	−0.0208396	−	0.999783i	$-0.506634\pi$
$97$	−0.410493	−0.0416792	−0.0208396	+	0.999783i	$0.506634\pi$
$98$	0	0
$99$	−2.53111	−0.254386
$100$	0	0
$101$	−16.2198	−1.61393	−0.806966	−	0.590597i	$-0.798892\pi$
$101$	−16.2198	−1.61393	−0.806966	+	0.590597i	$0.798892\pi$
$102$	0	0
$103$	−0.874849	−0.0862014	−0.0431007	−	0.999071i	$-0.513724\pi$
$103$	−0.874849	−0.0862014	−0.0431007	+	0.999071i	$0.513724\pi$
$104$	0	0
$105$	−1.99065	−0.194268
$106$	0	0
$107$	3.00737	0.290734	0.145367	−	0.989378i	$-0.453564\pi$
$107$	3.00737	0.290734	0.145367	+	0.989378i	$0.453564\pi$
$108$	0	0
$109$	18.6363	1.78504	0.892519	−	0.451011i	$-0.148936\pi$
$109$	18.6363	1.78504	0.892519	+	0.451011i	$0.148936\pi$
$110$	0	0
$111$	9.55723	0.907132
$112$	0	0
$113$	6.23506	0.586545	0.293272	−	0.956029i	$-0.405256\pi$
$113$	6.23506	0.586545	0.293272	+	0.956029i	$0.405256\pi$
$114$	0	0
$115$	15.9217	1.48470
$116$	0	0
$117$	−8.35119	−0.772067
$118$	0	0
$119$	0.873170	0.0800434
$120$	0	0
$121$	−9.14990	−0.831809
$122$	0	0
$123$	24.2707	2.18842
$124$	0	0
$125$	−1.14121	−0.102073
$126$	0	0
$127$	14.9645	1.32789	0.663944	−	0.747782i	$-0.268881\pi$
$127$	14.9645	1.32789	0.663944	+	0.747782i	$0.268881\pi$
$128$	0	0
$129$	3.19403	0.281219
$130$	0	0
$131$	−10.2474	−0.895320	−0.447660	−	0.894204i	$-0.647743\pi$
$131$	−10.2474	−0.895320	−0.447660	+	0.894204i	$0.647743\pi$
$132$	0	0
$133$	−0.809805	−0.0702190
$134$	0	0
$135$	8.08169	0.695561
$136$	0	0
$137$	−1.60764	−0.137350	−0.0686748	−	0.997639i	$-0.521877\pi$
$137$	−1.60764	−0.137350	−0.0686748	+	0.997639i	$0.521877\pi$
$138$	0	0
$139$	20.4478	1.73436	0.867179	−	0.497997i	$-0.165931\pi$
$139$	20.4478	1.73436	0.867179	+	0.497997i	$0.165931\pi$
$140$	0	0
$141$	4.61008	0.388239
$142$	0	0
$143$	6.10425	0.510463
$144$	0	0
$145$	24.6147	2.04414
$146$	0	0
$147$	−15.2596	−1.25859
$148$	0	0
$149$	−1.75007	−0.143371	−0.0716857	−	0.997427i	$-0.522838\pi$
$149$	−1.75007	−0.143371	−0.0716857	+	0.997427i	$0.522838\pi$
$150$	0	0
$151$	1.32648	0.107948	0.0539738	−	0.998542i	$-0.482811\pi$
$151$	1.32648	0.107948	0.0539738	+	0.998542i	$0.482811\pi$
$152$	0	0
$153$	5.79083	0.468161
$154$	0	0
$155$	11.4183	0.917143
$156$	0	0
$157$	−2.37847	−0.189823	−0.0949114	−	0.995486i	$-0.530257\pi$
$157$	−2.37847	−0.189823	−0.0949114	+	0.995486i	$0.530257\pi$
$158$	0	0
$159$	−30.0034	−2.37942
$160$	0	0
$161$	−1.38833	−0.109416
$162$	0	0
$163$	5.57772	0.436881	0.218440	−	0.975850i	$-0.429903\pi$
$163$	5.57772	0.436881	0.218440	+	0.975850i	$0.429903\pi$
$164$	0	0
$165$	9.64987	0.751242
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	7.14050	0.549269
$170$	0	0
$171$	−5.37059	−0.410700
$172$	0	0
$173$	−12.9767	−0.986599	−0.493300	−	0.869859i	$-0.664209\pi$
$173$	−12.9767	−0.986599	−0.493300	+	0.869859i	$0.664209\pi$
$174$	0	0
$175$	1.50246	0.113575
$176$	0	0
$177$	−24.6753	−1.85471
$178$	0	0
$179$	22.1209	1.65339	0.826696	−	0.562649i	$-0.190218\pi$
$179$	22.1209	1.65339	0.826696	+	0.562649i	$0.190218\pi$
$180$	0	0
$181$	1.94925	0.144887	0.0724434	−	0.997373i	$-0.476920\pi$
$181$	1.94925	0.144887	0.0724434	+	0.997373i	$0.476920\pi$
$182$	0	0
$183$	18.7312	1.38465
$184$	0	0
$185$	−13.9490	−1.02555
$186$	0	0
$187$	−4.23278	−0.309531
$188$	0	0
$189$	−0.704702	−0.0512595
$190$	0	0
$191$	−16.6460	−1.20447	−0.602233	−	0.798321i	$-0.705722\pi$
$191$	−16.6460	−1.20447	−0.602233	+	0.798321i	$0.705722\pi$
$192$	0	0
$193$	−5.10268	−0.367299	−0.183650	−	0.982992i	$-0.558791\pi$
$193$	−5.10268	−0.367299	−0.183650	+	0.982992i	$0.558791\pi$
$194$	0	0
$195$	31.8390	2.28004
$196$	0	0
$197$	−10.2557	−0.730691	−0.365346	−	0.930872i	$-0.619049\pi$
$197$	−10.2557	−0.730691	−0.365346	+	0.930872i	$0.619049\pi$
$198$	0	0
$199$	15.7661	1.11763	0.558814	−	0.829293i	$-0.311256\pi$
$199$	15.7661	1.11763	0.558814	+	0.829293i	$0.311256\pi$
$200$	0	0
$201$	−4.13388	−0.291581
$202$	0	0
$203$	−2.14634	−0.150643
$204$	0	0
$205$	−35.4237	−2.47410
$206$	0	0
$207$	−9.20733	−0.639954
$208$	0	0
$209$	3.92561	0.271540
$210$	0	0
$211$	0.671854	0.0462523	0.0231262	−	0.999733i	$-0.492638\pi$
$211$	0.671854	0.0462523	0.0231262	+	0.999733i	$0.492638\pi$
$212$	0	0
$213$	−3.84255	−0.263287
$214$	0	0
$215$	−4.66176	−0.317930
$216$	0	0
$217$	−0.995649	−0.0675890
$218$	0	0
$219$	−9.65236	−0.652246
$220$	0	0
$221$	−13.9657	−0.939435
$222$	0	0
$223$	18.2343	1.22106	0.610530	−	0.791993i	$-0.290956\pi$
$223$	18.2343	1.22106	0.610530	+	0.791993i	$0.290956\pi$
$224$	0	0
$225$	9.96423	0.664282
$226$	0	0
$227$	−1.88243	−0.124941	−0.0624706	−	0.998047i	$-0.519898\pi$
$227$	−1.88243	−0.124941	−0.0624706	+	0.998047i	$0.519898\pi$
$228$	0	0
$229$	−16.8224	−1.11166	−0.555829	−	0.831296i	$-0.687599\pi$
$229$	−16.8224	−1.11166	−0.555829	+	0.831296i	$0.687599\pi$
$230$	0	0
$231$	−0.841444	−0.0553629
$232$	0	0
$233$	−22.6477	−1.48370	−0.741849	−	0.670567i	$-0.766051\pi$
$233$	−22.6477	−1.48370	−0.741849	+	0.670567i	$0.766051\pi$
$234$	0	0
$235$	−6.72853	−0.438921
$236$	0	0
$237$	23.5457	1.52946
$238$	0	0
$239$	−19.2622	−1.24597	−0.622985	−	0.782234i	$-0.714080\pi$
$239$	−19.2622	−1.24597	−0.622985	+	0.782234i	$0.714080\pi$
$240$	0	0
$241$	1.26699	0.0816138	0.0408069	−	0.999167i	$-0.487007\pi$
$241$	1.26699	0.0816138	0.0408069	+	0.999167i	$0.487007\pi$
$242$	0	0
$243$	−16.9816	−1.08937
$244$	0	0
$245$	22.2717	1.42289
$246$	0	0
$247$	12.9522	0.824130
$248$	0	0
$249$	0.702682	0.0445307
$250$	0	0
$251$	−23.2385	−1.46680	−0.733400	−	0.679798i	$-0.762068\pi$
$251$	−23.2385	−1.46680	−0.733400	+	0.679798i	$0.762068\pi$
$252$	0	0
$253$	6.73005	0.423115
$254$	0	0
$255$	−22.0776	−1.38255
$256$	0	0
$257$	2.30407	0.143724	0.0718619	−	0.997415i	$-0.477106\pi$
$257$	2.30407	0.143724	0.0718619	+	0.997415i	$0.477106\pi$
$258$	0	0
$259$	1.21632	0.0755782
$260$	0	0
$261$	−14.2344	−0.881089
$262$	0	0
$263$	−6.11509	−0.377073	−0.188536	−	0.982066i	$-0.560374\pi$
$263$	−6.11509	−0.377073	−0.188536	+	0.982066i	$0.560374\pi$
$264$	0	0
$265$	43.7907	2.69004
$266$	0	0
$267$	1.95403	0.119585
$268$	0	0
$269$	6.79787	0.414473	0.207237	−	0.978291i	$-0.433553\pi$
$269$	6.79787	0.414473	0.207237	+	0.978291i	$0.433553\pi$
$270$	0	0
$271$	−22.1828	−1.34751	−0.673755	−	0.738955i	$-0.735320\pi$
$271$	−22.1828	−1.34751	−0.673755	+	0.738955i	$0.735320\pi$
$272$	0	0
$273$	−2.77628	−0.168028
$274$	0	0
$275$	−7.28330	−0.439200
$276$	0	0
$277$	5.27067	0.316684	0.158342	−	0.987384i	$-0.449385\pi$
$277$	5.27067	0.316684	0.158342	+	0.987384i	$0.449385\pi$
$278$	0	0
$279$	−6.60310	−0.395318
$280$	0	0
$281$	24.2548	1.44692	0.723461	−	0.690365i	$-0.242550\pi$
$281$	24.2548	1.44692	0.723461	+	0.690365i	$0.242550\pi$
$282$	0	0
$283$	5.24894	0.312017	0.156008	−	0.987756i	$-0.450137\pi$
$283$	5.24894	0.312017	0.156008	+	0.987756i	$0.450137\pi$
$284$	0	0
$285$	20.4754	1.21286
$286$	0	0
$287$	3.08885	0.182329
$288$	0	0
$289$	−7.31597	−0.430351
$290$	0	0
$291$	−0.905028	−0.0530537
$292$	0	0
$293$	−5.02485	−0.293555	−0.146778	−	0.989170i	$-0.546890\pi$
$293$	−5.02485	−0.293555	−0.146778	+	0.989170i	$0.546890\pi$
$294$	0	0
$295$	36.0142	2.09683
$296$	0	0
$297$	3.41611	0.198223
$298$	0	0
$299$	22.2053	1.28416
$300$	0	0
$301$	0.406494	0.0234299
$302$	0	0
$303$	−35.7604	−2.05438
$304$	0	0
$305$	−27.3386	−1.56540
$306$	0	0
$307$	18.0803	1.03190	0.515950	−	0.856619i	$-0.327439\pi$
$307$	18.0803	1.03190	0.515950	+	0.856619i	$0.327439\pi$
$308$	0	0
$309$	−1.92881	−0.109726
$310$	0	0
$311$	2.07065	0.117416	0.0587079	−	0.998275i	$-0.481302\pi$
$311$	2.07065	0.117416	0.0587079	+	0.998275i	$0.481302\pi$
$312$	0	0
$313$	16.0361	0.906413	0.453207	−	0.891406i	$-0.350280\pi$
$313$	16.0361	0.906413	0.453207	+	0.891406i	$0.350280\pi$
$314$	0	0
$315$	−1.68016	−0.0946665
$316$	0	0
$317$	23.3332	1.31052	0.655262	−	0.755401i	$-0.272558\pi$
$317$	23.3332	1.31052	0.655262	+	0.755401i	$0.272558\pi$
$318$	0	0
$319$	10.4046	0.582545
$320$	0	0
$321$	6.63046	0.370076
$322$	0	0
$323$	−8.98126	−0.499731
$324$	0	0
$325$	−24.0307	−1.33298
$326$	0	0
$327$	41.0882	2.27218
$328$	0	0
$329$	0.586710	0.0323464
$330$	0	0
$331$	16.7936	0.923062	0.461531	−	0.887124i	$-0.347300\pi$
$331$	16.7936	0.923062	0.461531	+	0.887124i	$0.347300\pi$
$332$	0	0
$333$	8.06656	0.442045
$334$	0	0
$335$	6.03349	0.329645
$336$	0	0
$337$	−8.25845	−0.449867	−0.224933	−	0.974374i	$-0.572216\pi$
$337$	−8.25845	−0.449867	−0.224933	+	0.974374i	$0.572216\pi$
$338$	0	0
$339$	13.7467	0.746616
$340$	0	0
$341$	4.82650	0.261370
$342$	0	0
$343$	−3.90616	−0.210913
$344$	0	0
$345$	35.1031	1.88988
$346$	0	0
$347$	8.39297	0.450558	0.225279	−	0.974294i	$-0.427671\pi$
$347$	8.39297	0.450558	0.225279	+	0.974294i	$0.427671\pi$
$348$	0	0
$349$	8.67895	0.464573	0.232287	−	0.972647i	$-0.425379\pi$
$349$	8.67895	0.464573	0.232287	+	0.972647i	$0.425379\pi$
$350$	0	0
$351$	11.2712	0.601612
$352$	0	0
$353$	18.0591	0.961190	0.480595	−	0.876943i	$-0.340421\pi$
$353$	18.0591	0.961190	0.480595	+	0.876943i	$0.340421\pi$
$354$	0	0
$355$	5.60829	0.297657
$356$	0	0
$357$	1.92511	0.101888
$358$	0	0
$359$	−21.7799	−1.14950	−0.574751	−	0.818329i	$-0.694901\pi$
$359$	−21.7799	−1.14950	−0.574751	+	0.818329i	$0.694901\pi$
$360$	0	0
$361$	−10.6705	−0.561606
$362$	0	0
$363$	−20.1731	−1.05881
$364$	0	0
$365$	14.0879	0.737392
$366$	0	0
$367$	−32.8229	−1.71334	−0.856671	−	0.515863i	$-0.827472\pi$
$367$	−32.8229	−1.71334	−0.856671	+	0.515863i	$0.827472\pi$
$368$	0	0
$369$	20.4852	1.06642
$370$	0	0
$371$	−3.81843	−0.198243
$372$	0	0
$373$	−32.5895	−1.68742	−0.843711	−	0.536798i	$-0.819634\pi$
$373$	−32.5895	−1.68742	−0.843711	+	0.536798i	$0.819634\pi$
$374$	0	0
$375$	−2.51606	−0.129929
$376$	0	0
$377$	34.3291	1.76804
$378$	0	0
$379$	9.61233	0.493752	0.246876	−	0.969047i	$-0.420596\pi$
$379$	9.61233	0.493752	0.246876	+	0.969047i	$0.420596\pi$
$380$	0	0
$381$	32.9929	1.69028
$382$	0	0
$383$	−13.0082	−0.664689	−0.332344	−	0.943158i	$-0.607840\pi$
$383$	−13.0082	−0.664689	−0.332344	+	0.943158i	$0.607840\pi$
$384$	0	0
$385$	1.22811	0.0625901
$386$	0	0
$387$	2.69585	0.137038
$388$	0	0
$389$	−2.33467	−0.118372	−0.0591862	−	0.998247i	$-0.518851\pi$
$389$	−2.33467	−0.118372	−0.0591862	+	0.998247i	$0.518851\pi$
$390$	0	0
$391$	−15.3974	−0.778682
$392$	0	0
$393$	−22.5928	−1.13966
$394$	0	0
$395$	−34.3656	−1.72912
$396$	0	0
$397$	−14.2452	−0.714946	−0.357473	−	0.933923i	$-0.616362\pi$
$397$	−14.2452	−0.714946	−0.357473	+	0.933923i	$0.616362\pi$
$398$	0	0
$399$	−1.78540	−0.0893820
$400$	0	0
$401$	24.9598	1.24643	0.623215	−	0.782050i	$-0.285826\pi$
$401$	24.9598	1.24643	0.623215	+	0.782050i	$0.285826\pi$
$402$	0	0
$403$	15.9247	0.793264
$404$	0	0
$405$	35.7819	1.77802
$406$	0	0
$407$	−5.89621	−0.292264
$408$	0	0
$409$	−2.98733	−0.147714	−0.0738570	−	0.997269i	$-0.523531\pi$
$409$	−2.98733	−0.147714	−0.0738570	+	0.997269i	$0.523531\pi$
$410$	0	0
$411$	−3.54441	−0.174833
$412$	0	0
$413$	−3.14034	−0.154526
$414$	0	0
$415$	−1.02558	−0.0503438
$416$	0	0
$417$	45.0819	2.20767
$418$	0	0
$419$	7.70015	0.376177	0.188088	−	0.982152i	$-0.439771\pi$
$419$	7.70015	0.376177	0.188088	+	0.982152i	$0.439771\pi$
$420$	0	0
$421$	−6.31712	−0.307878	−0.153939	−	0.988080i	$-0.549196\pi$
$421$	−6.31712	−0.307878	−0.153939	+	0.988080i	$0.549196\pi$
$422$	0	0
$423$	3.89104	0.189189
$424$	0	0
$425$	16.6632	0.808285
$426$	0	0
$427$	2.38386	0.115363
$428$	0	0
$429$	13.4583	0.649771
$430$	0	0
$431$	14.1044	0.679385	0.339692	−	0.940537i	$-0.389677\pi$
$431$	14.1044	0.679385	0.339692	+	0.940537i	$0.389677\pi$
$432$	0	0
$433$	−37.5703	−1.80552	−0.902758	−	0.430149i	$-0.858461\pi$
$433$	−37.5703	−1.80552	−0.902758	+	0.430149i	$0.858461\pi$
$434$	0	0
$435$	54.2689	2.60200
$436$	0	0
$437$	14.2801	0.683108
$438$	0	0
$439$	27.7302	1.32349	0.661745	−	0.749729i	$-0.269816\pi$
$439$	27.7302	1.32349	0.661745	+	0.749729i	$0.269816\pi$
$440$	0	0
$441$	−12.8795	−0.613309
$442$	0	0
$443$	26.7039	1.26874	0.634370	−	0.773029i	$-0.281260\pi$
$443$	26.7039	1.26874	0.634370	+	0.773029i	$0.281260\pi$
$444$	0	0
$445$	−2.85195	−0.135195
$446$	0	0
$447$	−3.85845	−0.182498
$448$	0	0
$449$	−14.2415	−0.672097	−0.336048	−	0.941845i	$-0.609091\pi$
$449$	−14.2415	−0.672097	−0.336048	+	0.941845i	$0.609091\pi$
$450$	0	0
$451$	−14.9735	−0.705076
$452$	0	0
$453$	2.92454	0.137407
$454$	0	0
$455$	4.05204	0.189963
$456$	0	0
$457$	−23.7482	−1.11089	−0.555447	−	0.831552i	$-0.687453\pi$
$457$	−23.7482	−1.11089	−0.555447	+	0.831552i	$0.687453\pi$
$458$	0	0
$459$	−7.81561	−0.364801
$460$	0	0
$461$	33.5863	1.56427	0.782136	−	0.623108i	$-0.214130\pi$
$461$	33.5863	1.56427	0.782136	+	0.623108i	$0.214130\pi$
$462$	0	0
$463$	−18.2496	−0.848133	−0.424066	−	0.905631i	$-0.639398\pi$
$463$	−18.2496	−0.848133	−0.424066	+	0.905631i	$0.639398\pi$
$464$	0	0
$465$	25.1744	1.16744
$466$	0	0
$467$	−20.5039	−0.948806	−0.474403	−	0.880308i	$-0.657336\pi$
$467$	−20.5039	−0.948806	−0.474403	+	0.880308i	$0.657336\pi$
$468$	0	0
$469$	−0.526105	−0.0242933
$470$	0	0
$471$	−5.24390	−0.241626
$472$	0	0
$473$	−1.97052	−0.0906045
$474$	0	0
$475$	−15.4540	−0.709077
$476$	0	0
$477$	−25.3237	−1.15949
$478$	0	0
$479$	4.31949	0.197362	0.0986812	−	0.995119i	$-0.468538\pi$
$479$	4.31949	0.197362	0.0986812	+	0.995119i	$0.468538\pi$
$480$	0	0
$481$	−19.4541	−0.887029
$482$	0	0
$483$	−3.06089	−0.139275
$484$	0	0
$485$	1.32091	0.0599794
$486$	0	0
$487$	33.2355	1.50604	0.753021	−	0.657996i	$-0.228596\pi$
$487$	33.2355	1.50604	0.753021	+	0.657996i	$0.228596\pi$
$488$	0	0
$489$	12.2974	0.556108
$490$	0	0
$491$	−26.7607	−1.20769	−0.603846	−	0.797101i	$-0.706366\pi$
$491$	−26.7607	−1.20769	−0.603846	+	0.797101i	$0.706366\pi$
$492$	0	0
$493$	−23.8043	−1.07209
$494$	0	0
$495$	8.14476	0.366080
$496$	0	0
$497$	−0.489028	−0.0219359
$498$	0	0
$499$	2.38547	0.106788	0.0533942	−	0.998574i	$-0.482996\pi$
$499$	2.38547	0.106788	0.0533942	+	0.998574i	$0.482996\pi$
$500$	0	0
$501$	2.20474	0.0985003
$502$	0	0
$503$	35.6697	1.59044	0.795218	−	0.606324i	$-0.207357\pi$
$503$	35.6697	1.59044	0.795218	+	0.606324i	$0.207357\pi$
$504$	0	0
$505$	52.1932	2.32257
$506$	0	0
$507$	15.7429	0.699167
$508$	0	0
$509$	17.8185	0.789793	0.394896	−	0.918726i	$-0.370781\pi$
$509$	17.8185	0.789793	0.394896	+	0.918726i	$0.370781\pi$
$510$	0	0
$511$	−1.22842	−0.0543423
$512$	0	0
$513$	7.24843	0.320026
$514$	0	0
$515$	2.81514	0.124050
$516$	0	0
$517$	−2.84413	−0.125085
$518$	0	0
$519$	−28.6102	−1.25585
$520$	0	0
$521$	−27.0885	−1.18677	−0.593385	−	0.804919i	$-0.702209\pi$
$521$	−27.0885	−1.18677	−0.593385	+	0.804919i	$0.702209\pi$
$522$	0	0
$523$	−1.66136	−0.0726463	−0.0363232	−	0.999340i	$-0.511565\pi$
$523$	−1.66136	−0.0726463	−0.0363232	+	0.999340i	$0.511565\pi$
$524$	0	0
$525$	3.31252	0.144570
$526$	0	0
$527$	−11.0424	−0.481014
$528$	0	0
$529$	1.48170	0.0644217
$530$	0	0
$531$	−20.8266	−0.903799
$532$	0	0
$533$	−49.4039	−2.13992
$534$	0	0
$535$	−9.67731	−0.418387
$536$	0	0
$537$	48.7707	2.10461
$538$	0	0
$539$	9.41420	0.405498
$540$	0	0
$541$	−38.8694	−1.67113	−0.835563	−	0.549395i	$-0.814858\pi$
$541$	−38.8694	−1.67113	−0.835563	+	0.549395i	$0.814858\pi$
$542$	0	0
$543$	4.29759	0.184427
$544$	0	0
$545$	−59.9692	−2.56880
$546$	0	0
$547$	−30.5457	−1.30604	−0.653020	−	0.757341i	$-0.726498\pi$
$547$	−30.5457	−1.30604	−0.653020	+	0.757341i	$0.726498\pi$
$548$	0	0
$549$	15.8096	0.674739
$550$	0	0
$551$	22.0768	0.940504
$552$	0	0
$553$	2.99659	0.127428
$554$	0	0
$555$	−30.7539	−1.30543
$556$	0	0
$557$	−33.7632	−1.43059	−0.715296	−	0.698822i	$-0.753708\pi$
$557$	−33.7632	−1.43059	−0.715296	+	0.698822i	$0.753708\pi$
$558$	0	0
$559$	−6.50156	−0.274987
$560$	0	0
$561$	−9.33216	−0.394004
$562$	0	0
$563$	−47.1945	−1.98901	−0.994507	−	0.104675i	$-0.966620\pi$
$563$	−47.1945	−1.98901	−0.994507	+	0.104675i	$0.966620\pi$
$564$	0	0
$565$	−20.0636	−0.844081
$566$	0	0
$567$	−3.12009	−0.131032
$568$	0	0
$569$	−12.7810	−0.535807	−0.267903	−	0.963446i	$-0.586331\pi$
$569$	−12.7810	−0.535807	−0.267903	+	0.963446i	$0.586331\pi$
$570$	0	0
$571$	−32.8881	−1.37632	−0.688162	−	0.725557i	$-0.741582\pi$
$571$	−32.8881	−1.37632	−0.688162	+	0.725557i	$0.741582\pi$
$572$	0	0
$573$	−36.7001	−1.53317
$574$	0	0
$575$	−26.4942	−1.10489
$576$	0	0
$577$	−29.9396	−1.24640	−0.623201	−	0.782062i	$-0.714168\pi$
$577$	−29.9396	−1.24640	−0.623201	+	0.782062i	$0.714168\pi$
$578$	0	0
$579$	−11.2501	−0.467537
$580$	0	0
$581$	0.0894280	0.00371010
$582$	0	0
$583$	18.5102	0.766615
$584$	0	0
$585$	26.8730	1.11106
$586$	0	0
$587$	−21.4120	−0.883769	−0.441884	−	0.897072i	$-0.645690\pi$
$587$	−21.4120	−0.883769	−0.441884	+	0.897072i	$0.645690\pi$
$588$	0	0
$589$	10.2410	0.421975
$590$	0	0
$591$	−22.6112	−0.930100
$592$	0	0
$593$	19.6033	0.805010	0.402505	−	0.915418i	$-0.368140\pi$
$593$	19.6033	0.805010	0.402505	+	0.915418i	$0.368140\pi$
$594$	0	0
$595$	−2.80974	−0.115188
$596$	0	0
$597$	34.7601	1.42263
$598$	0	0
$599$	11.3558	0.463986	0.231993	−	0.972717i	$-0.425475\pi$
$599$	11.3558	0.463986	0.231993	+	0.972717i	$0.425475\pi$
$600$	0	0
$601$	33.8270	1.37983	0.689915	−	0.723890i	$-0.257648\pi$
$601$	33.8270	1.37983	0.689915	+	0.723890i	$0.257648\pi$
$602$	0	0
$603$	−3.48911	−0.142087
$604$	0	0
$605$	29.4431	1.19703
$606$	0	0
$607$	42.8543	1.73940	0.869702	−	0.493577i	$-0.164311\pi$
$607$	42.8543	1.73940	0.869702	+	0.493577i	$0.164311\pi$
$608$	0	0
$609$	−4.73211	−0.191755
$610$	0	0
$611$	−9.38399	−0.379636
$612$	0	0
$613$	5.73144	0.231491	0.115745	−	0.993279i	$-0.463074\pi$
$613$	5.73144	0.231491	0.115745	+	0.993279i	$0.463074\pi$
$614$	0	0
$615$	−78.0999	−3.14929
$616$	0	0
$617$	−21.8209	−0.878478	−0.439239	−	0.898370i	$-0.644752\pi$
$617$	−21.8209	−0.878478	−0.439239	+	0.898370i	$0.644752\pi$
$618$	0	0
$619$	−40.5226	−1.62874	−0.814370	−	0.580346i	$-0.802917\pi$
$619$	−40.5226	−1.62874	−0.814370	+	0.580346i	$0.802917\pi$
$620$	0	0
$621$	12.4267	0.498666
$622$	0	0
$623$	0.248682	0.00996325
$624$	0	0
$625$	−23.1010	−0.924040
$626$	0	0
$627$	8.65492	0.345644
$628$	0	0
$629$	13.4897	0.537871
$630$	0	0
$631$	15.7067	0.625274	0.312637	−	0.949873i	$-0.398788\pi$
$631$	15.7067	0.625274	0.312637	+	0.949873i	$0.398788\pi$
$632$	0	0
$633$	1.48126	0.0588748
$634$	0	0
$635$	−48.1539	−1.91093
$636$	0	0
$637$	31.0614	1.23070
$638$	0	0
$639$	−3.24322	−0.128300
$640$	0	0
$641$	−10.2313	−0.404114	−0.202057	−	0.979374i	$-0.564763\pi$
$641$	−10.2313	−0.404114	−0.202057	+	0.979374i	$0.564763\pi$
$642$	0	0
$643$	3.55170	0.140066	0.0700328	−	0.997545i	$-0.477690\pi$
$643$	3.55170	0.140066	0.0700328	+	0.997545i	$0.477690\pi$
$644$	0	0
$645$	−10.2780	−0.404694
$646$	0	0
$647$	−2.72914	−0.107293	−0.0536467	−	0.998560i	$-0.517084\pi$
$647$	−2.72914	−0.107293	−0.0536467	+	0.998560i	$0.517084\pi$
$648$	0	0
$649$	15.2231	0.597559
$650$	0	0
$651$	−2.19514	−0.0860344
$652$	0	0
$653$	−46.2617	−1.81036	−0.905179	−	0.425030i	$-0.860264\pi$
$653$	−46.2617	−1.81036	−0.905179	+	0.425030i	$0.860264\pi$
$654$	0	0
$655$	32.9748	1.28843
$656$	0	0
$657$	−8.14686	−0.317839
$658$	0	0
$659$	−15.1161	−0.588838	−0.294419	−	0.955676i	$-0.595126\pi$
$659$	−15.1161	−0.588838	−0.294419	+	0.955676i	$0.595126\pi$
$660$	0	0
$661$	10.8605	0.422426	0.211213	−	0.977440i	$-0.432259\pi$
$661$	10.8605	0.422426	0.211213	+	0.977440i	$0.432259\pi$
$662$	0	0
$663$	−30.7907	−1.19581
$664$	0	0
$665$	2.60584	0.101050
$666$	0	0
$667$	37.8484	1.46550
$668$	0	0
$669$	40.2019	1.55429
$670$	0	0
$671$	−11.5560	−0.446113
$672$	0	0
$673$	−31.3590	−1.20880	−0.604401	−	0.796680i	$-0.706587\pi$
$673$	−31.3590	−1.20880	−0.604401	+	0.796680i	$0.706587\pi$
$674$	0	0
$675$	−13.4482	−0.517623
$676$	0	0
$677$	−42.1030	−1.61815	−0.809075	−	0.587705i	$-0.800031\pi$
$677$	−42.1030	−1.61815	−0.809075	+	0.587705i	$0.800031\pi$
$678$	0	0
$679$	−0.115180	−0.00442020
$680$	0	0
$681$	−4.15026	−0.159038
$682$	0	0
$683$	22.4556	0.859240	0.429620	−	0.903010i	$-0.358648\pi$
$683$	22.4556	0.859240	0.429620	+	0.903010i	$0.358648\pi$
$684$	0	0
$685$	5.17315	0.197656
$686$	0	0
$687$	−37.0890	−1.41504
$688$	0	0
$689$	61.0729	2.32669
$690$	0	0
$691$	2.63351	0.100184	0.0500918	−	0.998745i	$-0.484049\pi$
$691$	2.63351	0.100184	0.0500918	+	0.998745i	$0.484049\pi$
$692$	0	0
$693$	−0.710202	−0.0269783
$694$	0	0
$695$	−65.7981	−2.49587
$696$	0	0
$697$	34.2574	1.29759
$698$	0	0
$699$	−49.9321	−1.88861
$700$	0	0
$701$	−10.8687	−0.410503	−0.205252	−	0.978709i	$-0.565801\pi$
$701$	−10.8687	−0.410503	−0.205252	+	0.978709i	$0.565801\pi$
$702$	0	0
$703$	−12.5108	−0.471853
$704$	0	0
$705$	−14.8346	−0.558704
$706$	0	0
$707$	−4.55111	−0.171162
$708$	0	0
$709$	−32.9437	−1.23723	−0.618614	−	0.785695i	$-0.712305\pi$
$709$	−32.9437	−1.23723	−0.618614	+	0.785695i	$0.712305\pi$
$710$	0	0
$711$	19.8732	0.745305
$712$	0	0
$713$	17.5572	0.657523
$714$	0	0
$715$	−19.6427	−0.734594
$716$	0	0
$717$	−42.4681	−1.58600
$718$	0	0
$719$	17.7649	0.662518	0.331259	−	0.943540i	$-0.392527\pi$
$719$	17.7649	0.662518	0.331259	+	0.943540i	$0.392527\pi$
$720$	0	0
$721$	−0.245473	−0.00914190
$722$	0	0
$723$	2.79337	0.103887
$724$	0	0
$725$	−40.9598	−1.52121
$726$	0	0
$727$	41.0256	1.52155	0.760777	−	0.649014i	$-0.224818\pi$
$727$	41.0256	1.52155	0.760777	+	0.649014i	$0.224818\pi$
$728$	0	0
$729$	−4.08069	−0.151137
$730$	0	0
$731$	4.50828	0.166745
$732$	0	0
$733$	1.39154	0.0513978	0.0256989	−	0.999670i	$-0.491819\pi$
$733$	1.39154	0.0513978	0.0256989	+	0.999670i	$0.491819\pi$
$734$	0	0
$735$	49.1032	1.81120
$736$	0	0
$737$	2.55034	0.0939431
$738$	0	0
$739$	−10.2422	−0.376766	−0.188383	−	0.982096i	$-0.560325\pi$
$739$	−10.2422	−0.376766	−0.188383	+	0.982096i	$0.560325\pi$
$740$	0	0
$741$	28.5562	1.04904
$742$	0	0
$743$	−42.2689	−1.55070	−0.775348	−	0.631534i	$-0.782426\pi$
$743$	−42.2689	−1.55070	−0.775348	+	0.631534i	$0.782426\pi$
$744$	0	0
$745$	5.63149	0.206322
$746$	0	0
$747$	0.593083	0.0216998
$748$	0	0
$749$	0.843836	0.0308331
$750$	0	0
$751$	−29.0769	−1.06103	−0.530516	−	0.847675i	$-0.678002\pi$
$751$	−29.0769	−1.06103	−0.530516	+	0.847675i	$0.678002\pi$
$752$	0	0
$753$	−51.2347	−1.86710
$754$	0	0
$755$	−4.26844	−0.155344
$756$	0	0
$757$	22.6682	0.823889	0.411945	−	0.911209i	$-0.364850\pi$
$757$	22.6682	0.823889	0.411945	+	0.911209i	$0.364850\pi$
$758$	0	0
$759$	14.8380	0.538585
$760$	0	0
$761$	17.7722	0.644243	0.322122	−	0.946698i	$-0.395604\pi$
$761$	17.7722	0.644243	0.322122	+	0.946698i	$0.395604\pi$
$762$	0	0
$763$	5.22915	0.189308
$764$	0	0
$765$	−18.6341	−0.673718
$766$	0	0
$767$	50.2274	1.81361
$768$	0	0
$769$	21.2344	0.765730	0.382865	−	0.923804i	$-0.374937\pi$
$769$	21.2344	0.765730	0.382865	+	0.923804i	$0.374937\pi$
$770$	0	0
$771$	5.07986	0.182947
$772$	0	0
$773$	17.2055	0.618839	0.309419	−	0.950926i	$-0.399865\pi$
$773$	17.2055	0.618839	0.309419	+	0.950926i	$0.399865\pi$
$774$	0	0
$775$	−19.0005	−0.682520
$776$	0	0
$777$	2.68165	0.0962038
$778$	0	0
$779$	−31.7713	−1.13833
$780$	0	0
$781$	2.37061	0.0848271
$782$	0	0
$783$	19.2115	0.686564
$784$	0	0
$785$	7.65360	0.273169
$786$	0	0
$787$	−24.1207	−0.859811	−0.429905	−	0.902874i	$-0.641453\pi$
$787$	−24.1207	−0.859811	−0.429905	+	0.902874i	$0.641453\pi$
$788$	0	0
$789$	−13.4822	−0.479978
$790$	0	0
$791$	1.74949	0.0622047
$792$	0	0
$793$	−38.1280	−1.35396
$794$	0	0
$795$	96.5468	3.42416
$796$	0	0
$797$	28.2741	1.00152	0.500759	−	0.865586i	$-0.333054\pi$
$797$	28.2741	1.00152	0.500759	+	0.865586i	$0.333054\pi$
$798$	0	0
$799$	6.50699	0.230201
$800$	0	0
$801$	1.64925	0.0582735
$802$	0	0
$803$	5.95490	0.210144
$804$	0	0
$805$	4.46745	0.157457
$806$	0	0
$807$	14.9875	0.527585
$808$	0	0
$809$	−31.0619	−1.09208	−0.546039	−	0.837760i	$-0.683865\pi$
$809$	−31.0619	−1.09208	−0.546039	+	0.837760i	$0.683865\pi$
$810$	0	0
$811$	−28.2286	−0.991239	−0.495619	−	0.868540i	$-0.665059\pi$
$811$	−28.2286	−0.991239	−0.495619	+	0.868540i	$0.665059\pi$
$812$	0	0
$813$	−48.9072	−1.71525
$814$	0	0
$815$	−17.9483	−0.628703
$816$	0	0
$817$	−4.18111	−0.146279
$818$	0	0
$819$	−2.34325	−0.0818799
$820$	0	0
$821$	−21.4382	−0.748199	−0.374100	−	0.927389i	$-0.622048\pi$
$821$	−21.4382	−0.748199	−0.374100	+	0.927389i	$0.622048\pi$
$822$	0	0
$823$	29.6744	1.03438	0.517192	−	0.855870i	$-0.326977\pi$
$823$	29.6744	1.03438	0.517192	+	0.855870i	$0.326977\pi$
$824$	0	0
$825$	−16.0578	−0.559059
$826$	0	0
$827$	−24.1934	−0.841286	−0.420643	−	0.907226i	$-0.638196\pi$
$827$	−24.1934	−0.841286	−0.420643	+	0.907226i	$0.638196\pi$
$828$	0	0
$829$	0.207584	0.00720969	0.00360484	−	0.999994i	$-0.498853\pi$
$829$	0.207584	0.00720969	0.00360484	+	0.999994i	$0.498853\pi$
$830$	0	0
$831$	11.6204	0.403108
$832$	0	0
$833$	−21.5384	−0.746262
$834$	0	0
$835$	−3.21786	−0.111359
$836$	0	0
$837$	8.91189	0.308040
$838$	0	0
$839$	46.2349	1.59621	0.798103	−	0.602521i	$-0.205837\pi$
$839$	46.2349	1.59621	0.798103	+	0.602521i	$0.205837\pi$
$840$	0	0
$841$	29.5133	1.01770
$842$	0	0
$843$	53.4755	1.84179
$844$	0	0
$845$	−22.9771	−0.790438
$846$	0	0
$847$	−2.56736	−0.0882157
$848$	0	0
$849$	11.5725	0.397168
$850$	0	0
$851$	−21.4485	−0.735244
$852$	0	0
$853$	22.1137	0.757159	0.378580	−	0.925569i	$-0.376413\pi$
$853$	22.1137	0.757159	0.378580	+	0.925569i	$0.376413\pi$
$854$	0	0
$855$	17.2818	0.591026
$856$	0	0
$857$	−9.30846	−0.317971	−0.158986	−	0.987281i	$-0.550822\pi$
$857$	−9.30846	−0.317971	−0.158986	+	0.987281i	$0.550822\pi$
$858$	0	0
$859$	−43.3453	−1.47892	−0.739462	−	0.673198i	$-0.764920\pi$
$859$	−43.3453	−1.47892	−0.739462	+	0.673198i	$0.764920\pi$
$860$	0	0
$861$	6.81011	0.232088
$862$	0	0
$863$	34.5330	1.17552	0.587758	−	0.809037i	$-0.300011\pi$
$863$	34.5330	1.17552	0.587758	+	0.809037i	$0.300011\pi$
$864$	0	0
$865$	41.7572	1.41979
$866$	0	0
$867$	−16.1298	−0.547796
$868$	0	0
$869$	−14.5262	−0.492769
$870$	0	0
$871$	8.41465	0.285120
$872$	0	0
$873$	−0.763869	−0.0258530
$874$	0	0
$875$	−0.320210	−0.0108251
$876$	0	0
$877$	2.22059	0.0749840	0.0374920	−	0.999297i	$-0.488063\pi$
$877$	2.22059	0.0749840	0.0374920	+	0.999297i	$0.488063\pi$
$878$	0	0
$879$	−11.0785	−0.373668
$880$	0	0
$881$	−10.9398	−0.368570	−0.184285	−	0.982873i	$-0.558997\pi$
$881$	−10.9398	−0.368570	−0.184285	+	0.982873i	$0.558997\pi$
$882$	0	0
$883$	−2.95035	−0.0992870	−0.0496435	−	0.998767i	$-0.515809\pi$
$883$	−2.95035	−0.0992870	−0.0496435	+	0.998767i	$0.515809\pi$
$884$	0	0
$885$	79.4017	2.66906
$886$	0	0
$887$	56.5510	1.89880	0.949399	−	0.314074i	$-0.101694\pi$
$887$	56.5510	1.89880	0.949399	+	0.314074i	$0.101694\pi$
$888$	0	0
$889$	4.19889	0.140826
$890$	0	0
$891$	15.1249	0.506705
$892$	0	0
$893$	−6.03478	−0.201946
$894$	0	0
$895$	−71.1820	−2.37935
$896$	0	0
$897$	48.9567	1.63462
$898$	0	0
$899$	27.1433	0.905279
$900$	0	0
$901$	−42.3489	−1.41085
$902$	0	0
$903$	0.896211	0.0298240
$904$	0	0
$905$	−6.27243	−0.208503
$906$	0	0
$907$	34.2518	1.13731	0.568656	−	0.822575i	$-0.307464\pi$
$907$	34.2518	1.13731	0.568656	+	0.822575i	$0.307464\pi$
$908$	0	0
$909$	−30.1828	−1.00110
$910$	0	0
$911$	−35.6389	−1.18077	−0.590386	−	0.807121i	$-0.701024\pi$
$911$	−35.6389	−1.18077	−0.590386	+	0.807121i	$0.701024\pi$
$912$	0	0
$913$	−0.433511	−0.0143471
$914$	0	0
$915$	−60.2744	−1.99261
$916$	0	0
$917$	−2.87531	−0.0949512
$918$	0	0
$919$	6.73412	0.222138	0.111069	−	0.993813i	$-0.464573\pi$
$919$	6.73412	0.222138	0.111069	+	0.993813i	$0.464573\pi$
$920$	0	0
$921$	39.8623	1.31351
$922$	0	0
$923$	7.82164	0.257452
$924$	0	0
$925$	23.2117	0.763195
$926$	0	0
$927$	−1.62797	−0.0534695
$928$	0	0
$929$	−49.5260	−1.62490	−0.812448	−	0.583034i	$-0.801865\pi$
$929$	−49.5260	−1.62490	−0.812448	+	0.583034i	$0.801865\pi$
$930$	0	0
$931$	19.9754	0.654667
$932$	0	0
$933$	4.56523	0.149459
$934$	0	0
$935$	13.6205	0.445438
$936$	0	0
$937$	−47.9096	−1.56514	−0.782568	−	0.622565i	$-0.786091\pi$
$937$	−47.9096	−1.56514	−0.782568	+	0.622565i	$0.786091\pi$
$938$	0	0
$939$	35.3553	1.15378
$940$	0	0
$941$	−38.2710	−1.24760	−0.623799	−	0.781585i	$-0.714412\pi$
$941$	−38.2710	−1.24760	−0.623799	+	0.781585i	$0.714412\pi$
$942$	0	0
$943$	−54.4687	−1.77375
$944$	0	0
$945$	2.26764	0.0737662
$946$	0	0
$947$	31.6373	1.02807	0.514037	−	0.857768i	$-0.328150\pi$
$947$	31.6373	1.02807	0.514037	+	0.857768i	$0.328150\pi$
$948$	0	0
$949$	19.6477	0.637792
$950$	0	0
$951$	51.4436	1.66817
$952$	0	0
$953$	6.47995	0.209906	0.104953	−	0.994477i	$-0.466531\pi$
$953$	6.47995	0.209906	0.104953	+	0.994477i	$0.466531\pi$
$954$	0	0
$955$	53.5647	1.73331
$956$	0	0
$957$	22.9394	0.741524
$958$	0	0
$959$	−0.451085	−0.0145663
$960$	0	0
$961$	−18.4087	−0.593829
$962$	0	0
$963$	5.59629	0.180338
$964$	0	0
$965$	16.4197	0.528570
$966$	0	0
$967$	45.5625	1.46519	0.732596	−	0.680664i	$-0.238309\pi$
$967$	45.5625	1.46519	0.732596	+	0.680664i	$0.238309\pi$
$968$	0	0
$969$	−19.8013	−0.636109
$970$	0	0
$971$	54.6875	1.75501	0.877503	−	0.479571i	$-0.159208\pi$
$971$	54.6875	1.75501	0.877503	+	0.479571i	$0.159208\pi$
$972$	0	0
$973$	5.73743	0.183933
$974$	0	0
$975$	−52.9813	−1.69676
$976$	0	0
$977$	−7.95887	−0.254627	−0.127313	−	0.991863i	$-0.540635\pi$
$977$	−7.95887	−0.254627	−0.127313	+	0.991863i	$0.540635\pi$
$978$	0	0
$979$	−1.20551	−0.0385283
$980$	0	0
$981$	34.6796	1.10723
$982$	0	0
$983$	16.5922	0.529209	0.264605	−	0.964357i	$-0.414759\pi$
$983$	16.5922	0.529209	0.264605	+	0.964357i	$0.414759\pi$
$984$	0	0
$985$	33.0016	1.05152
$986$	0	0
$987$	1.29354	0.0411738
$988$	0	0
$989$	−7.16809	−0.227932
$990$	0	0
$991$	20.3367	0.646017	0.323008	−	0.946396i	$-0.395306\pi$
$991$	20.3367	0.646017	0.323008	+	0.946396i	$0.395306\pi$
$992$	0	0
$993$	37.0255	1.17497
$994$	0	0
$995$	−50.7331	−1.60835
$996$	0	0
$997$	2.82191	0.0893707	0.0446853	−	0.999001i	$-0.485771\pi$
$997$	2.82191	0.0893707	0.0446853	+	0.999001i	$0.485771\pi$
$998$	0	0
$999$	−10.8870	−0.344451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.c.1.9	✓	9
4.3	odd	2		2672.2.a.m.1.1		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.c.1.9	✓	9	1.1	even	1	trivial
2672.2.a.m.1.1		9	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:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q+2.20474 q^{3} -3.21786 q^{5} +0.280589 q^{7} +1.86086 q^{9} +O(q^{10})\)	\(q+2.20474 q^{3} -3.21786 q^{5} +0.280589 q^{7} +1.86086 q^{9} -1.36018 q^{11} -4.48782 q^{13} -7.09454 q^{15} +3.11192 q^{17} -2.88609 q^{19} +0.618625 q^{21} -4.94790 q^{23} +5.35465 q^{25} -2.51151 q^{27} -7.64940 q^{29} -3.54842 q^{31} -2.99884 q^{33} -0.902898 q^{35} +4.33486 q^{37} -9.89445 q^{39} +11.0085 q^{41} +1.44871 q^{43} -5.98798 q^{45} +2.09099 q^{47} -6.92127 q^{49} +6.86095 q^{51} -13.6086 q^{53} +4.37689 q^{55} -6.36305 q^{57} -11.1920 q^{59} +8.49589 q^{61} +0.522136 q^{63} +14.4412 q^{65} -1.87500 q^{67} -10.9088 q^{69} -1.74286 q^{71} -4.37801 q^{73} +11.8056 q^{75} -0.381653 q^{77} +10.6796 q^{79} -11.1198 q^{81} +0.318715 q^{83} -10.0137 q^{85} -16.8649 q^{87} +0.886287 q^{89} -1.25923 q^{91} -7.82333 q^{93} +9.28703 q^{95} -0.410493 q^{97} -2.53111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} - 8 q^{5} + 2 q^{7}+O(q^{10}) \) 9 * q - q^3 - 8 * q^5 + 2 * q^7	\( 9 q - q^{3} - 8 q^{5} + 2 q^{7} - 10 q^{11} - 13 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 10 q^{27} - 25 q^{29} - q^{31} - 12 q^{33} - 17 q^{35} - 35 q^{37} - 4 q^{39} - 16 q^{41} + 9 q^{43} - 24 q^{45} - q^{47} - q^{49} - 10 q^{51} - 29 q^{53} + 9 q^{55} - 17 q^{57} - 14 q^{59} - 28 q^{61} + 4 q^{63} - 31 q^{65} + 19 q^{67} - 19 q^{69} - 9 q^{71} - 7 q^{75} - 33 q^{77} - 18 q^{79} - 27 q^{81} - 13 q^{83} - 36 q^{85} + 18 q^{87} - 21 q^{89} + 20 q^{91} - 35 q^{93} - 12 q^{95} + 2 q^{97} - 3 q^{99}+O(q^{100}) \) 9 * q - q^3 - 8 * q^5 + 2 * q^7 - 10 * q^11 - 13 * q^13 + 2 * q^15 - 8 * q^17 - q^19 - 19 * q^21 - 3 * q^23 + 3 * q^25 - 10 * q^27 - 25 * q^29 - q^31 - 12 * q^33 - 17 * q^35 - 35 * q^37 - 4 * q^39 - 16 * q^41 + 9 * q^43 - 24 * q^45 - q^47 - q^49 - 10 * q^51 - 29 * q^53 + 9 * q^55 - 17 * q^57 - 14 * q^59 - 28 * q^61 + 4 * q^63 - 31 * q^65 + 19 * q^67 - 19 * q^69 - 9 * q^71 - 7 * q^75 - 33 * q^77 - 18 * q^79 - 27 * q^81 - 13 * q^83 - 36 * q^85 + 18 * q^87 - 21 * q^89 + 20 * q^91 - 35 * q^93 - 12 * q^95 + 2 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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