LMFDB - Embedded newform 9675.2.a.cv.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9675,2,Mod(1,9675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9675, 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("9675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9675 = 3^{2} \cdot 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9675.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:	$77.2552639556$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 13x^{8} + 58x^{6} - 103x^{4} + 65x^{2} - 13 $ x^10 - 13x^8 + 58x^6 - 103x^4 + 65x^2 - 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.643165$ of defining polynomial
Character	$\chi$	$=$	9675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.643165 q^{2} -1.58634 q^{4} -1.33434 q^{7} +2.30661 q^{8} +O(q^{10})$	$q-0.643165 q^{2} -1.58634 q^{4} -1.33434 q^{7} +2.30661 q^{8} -2.30661 q^{11} -1.25200 q^{13} +0.858202 q^{14} +1.68915 q^{16} -3.88304 q^{17} +3.02349 q^{19} +1.48353 q^{22} +5.45053 q^{23} +0.805240 q^{26} +2.11672 q^{28} +5.11641 q^{29} -8.24302 q^{31} -5.69962 q^{32} +2.49744 q^{34} +5.59592 q^{37} -1.94460 q^{38} -6.75897 q^{41} +1.00000 q^{43} +3.65906 q^{44} -3.50559 q^{46} +3.52804 q^{47} -5.21953 q^{49} +1.98609 q^{52} -4.31733 q^{53} -3.07780 q^{56} -3.29069 q^{58} +6.05267 q^{59} +0.0482823 q^{61} +5.30162 q^{62} +0.287490 q^{64} +10.7760 q^{67} +6.15982 q^{68} +9.57072 q^{71} -8.93217 q^{73} -3.59910 q^{74} -4.79629 q^{76} +3.07780 q^{77} +10.3473 q^{79} +4.34713 q^{82} -13.3799 q^{83} -0.643165 q^{86} -5.32043 q^{88} -8.70358 q^{89} +1.67059 q^{91} -8.64640 q^{92} -2.26911 q^{94} +16.0546 q^{97} +3.35702 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6 q^{4}+O(q^{10}) $ 10 * q + 6 * q^4	$ 10 q + 6 q^{4} - 4 q^{13} - 10 q^{16} - 10 q^{19} - 2 q^{22} - 16 q^{31} - 8 q^{34} + 10 q^{37} + 10 q^{43} - 28 q^{46} - 26 q^{49} + 36 q^{52} - 16 q^{58} - 32 q^{61} - 28 q^{64} - 14 q^{67} + 4 q^{73} - 24 q^{76} - 18 q^{79} + 20 q^{82} - 56 q^{88} - 44 q^{91} - 12 q^{94} + 4 q^{97}+O(q^{100}) $ 10 * q + 6 * q^4 - 4 * q^13 - 10 * q^16 - 10 * q^19 - 2 * q^22 - 16 * q^31 - 8 * q^34 + 10 * q^37 + 10 * q^43 - 28 * q^46 - 26 * q^49 + 36 * q^52 - 16 * q^58 - 32 * q^61 - 28 * q^64 - 14 * q^67 + 4 * q^73 - 24 * q^76 - 18 * q^79 + 20 * q^82 - 56 * q^88 - 44 * q^91 - 12 * q^94 + 4 * q^97

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 $

$2$	−0.643165	−0.454786	−0.227393	−	0.973803i	$-0.573020\pi$
$2$	−0.643165	−0.454786	−0.227393	+	0.973803i	$0.573020\pi$
$3$	0	0
$3$	0	0
$4$	−1.58634	−0.793170
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.33434	−0.504334	−0.252167	−	0.967684i	$-0.581143\pi$
$7$	−1.33434	−0.504334	−0.252167	+	0.967684i	$0.581143\pi$
$8$	2.30661	0.815509
$9$	0	0
$10$	0	0
$11$	−2.30661	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$11$	−2.30661	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$12$	0	0
$13$	−1.25200	−0.347241	−0.173621	−	0.984813i	$-0.555547\pi$
$13$	−1.25200	−0.347241	−0.173621	+	0.984813i	$0.555547\pi$
$14$	0.858202	0.229364
$15$	0	0
$16$	1.68915	0.422288
$17$	−3.88304	−0.941776	−0.470888	−	0.882193i	$-0.656066\pi$
$17$	−3.88304	−0.941776	−0.470888	+	0.882193i	$0.656066\pi$
$18$	0	0
$19$	3.02349	0.693637	0.346818	−	0.937932i	$-0.387262\pi$
$19$	3.02349	0.693637	0.346818	+	0.937932i	$0.387262\pi$
$20$	0	0
$21$	0	0
$22$	1.48353	0.316289
$23$	5.45053	1.13651	0.568257	−	0.822851i	$-0.307618\pi$
$23$	5.45053	1.13651	0.568257	+	0.822851i	$0.307618\pi$
$24$	0	0
$25$	0	0
$26$	0.805240	0.157921
$27$	0	0
$28$	2.11672	0.400022
$29$	5.11641	0.950094	0.475047	−	0.879961i	$-0.342431\pi$
$29$	5.11641	0.950094	0.475047	+	0.879961i	$0.342431\pi$
$30$	0	0
$31$	−8.24302	−1.48049	−0.740245	−	0.672337i	$-0.765291\pi$
$31$	−8.24302	−1.48049	−0.740245	+	0.672337i	$0.765291\pi$
$32$	−5.69962	−1.00756
$33$	0	0
$34$	2.49744	0.428307
$35$	0	0
$36$	0	0
$37$	5.59592	0.919964	0.459982	−	0.887928i	$-0.347856\pi$
$37$	5.59592	0.919964	0.459982	+	0.887928i	$0.347856\pi$
$38$	−1.94460	−0.315456
$39$	0	0
$40$	0	0
$41$	−6.75897	−1.05557	−0.527787	−	0.849377i	$-0.676978\pi$
$41$	−6.75897	−1.05557	−0.527787	+	0.849377i	$0.676978\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	3.65906	0.551624
$45$	0	0
$46$	−3.50559	−0.516871
$47$	3.52804	0.514617	0.257309	−	0.966329i	$-0.417164\pi$
$47$	3.52804	0.514617	0.257309	+	0.966329i	$0.417164\pi$
$48$	0	0
$49$	−5.21953	−0.745647
$50$	0	0
$51$	0	0
$52$	1.98609	0.275421
$53$	−4.31733	−0.593032	−0.296516	−	0.955028i	$-0.595825\pi$
$53$	−4.31733	−0.593032	−0.296516	+	0.955028i	$0.595825\pi$
$54$	0	0
$55$	0	0
$56$	−3.07780	−0.411289
$57$	0	0
$58$	−3.29069	−0.432089
$59$	6.05267	0.787991	0.393995	−	0.919112i	$-0.371093\pi$
$59$	6.05267	0.787991	0.393995	+	0.919112i	$0.371093\pi$
$60$	0	0
$61$	0.0482823	0.00618191	0.00309095	−	0.999995i	$-0.499016\pi$
$61$	0.0482823	0.00618191	0.00309095	+	0.999995i	$0.499016\pi$
$62$	5.30162	0.673307
$63$	0	0
$64$	0.287490	0.0359362
$65$	0	0
$66$	0	0
$67$	10.7760	1.31649	0.658246	−	0.752803i	$-0.271299\pi$
$67$	10.7760	1.31649	0.658246	+	0.752803i	$0.271299\pi$
$68$	6.15982	0.746988
$69$	0	0
$70$	0	0
$71$	9.57072	1.13584	0.567918	−	0.823085i	$-0.307749\pi$
$71$	9.57072	1.13584	0.567918	+	0.823085i	$0.307749\pi$
$72$	0	0
$73$	−8.93217	−1.04543	−0.522716	−	0.852507i	$-0.675081\pi$
$73$	−8.93217	−1.04543	−0.522716	+	0.852507i	$0.675081\pi$
$74$	−3.59910	−0.418387
$75$	0	0
$76$	−4.79629	−0.550172
$77$	3.07780	0.350748
$78$	0	0
$79$	10.3473	1.16416	0.582079	−	0.813132i	$-0.302239\pi$
$79$	10.3473	1.16416	0.582079	+	0.813132i	$0.302239\pi$
$80$	0	0
$81$	0	0
$82$	4.34713	0.480060
$83$	−13.3799	−1.46864	−0.734319	−	0.678805i	$-0.762498\pi$
$83$	−13.3799	−1.46864	−0.734319	+	0.678805i	$0.762498\pi$
$84$	0	0
$85$	0	0
$86$	−0.643165	−0.0693542
$87$	0	0
$88$	−5.32043	−0.567160
$89$	−8.70358	−0.922577	−0.461289	−	0.887250i	$-0.652613\pi$
$89$	−8.70358	−0.922577	−0.461289	+	0.887250i	$0.652613\pi$
$90$	0	0
$91$	1.67059	0.175126
$92$	−8.64640	−0.901449
$93$	0	0
$94$	−2.26911	−0.234041
$95$	0	0
$96$	0	0
$97$	16.0546	1.63010	0.815051	−	0.579389i	$-0.196709\pi$
$97$	16.0546	1.63010	0.815051	+	0.579389i	$0.196709\pi$
$98$	3.35702	0.339110
$99$	0	0
$100$	0	0
$101$	15.4555	1.53788	0.768942	−	0.639318i	$-0.220783\pi$
$101$	15.4555	1.53788	0.768942	+	0.639318i	$0.220783\pi$
$102$	0	0
$103$	5.77773	0.569296	0.284648	−	0.958632i	$-0.408123\pi$
$103$	5.77773	0.569296	0.284648	+	0.958632i	$0.408123\pi$
$104$	−2.88786	−0.283178
$105$	0	0
$106$	2.77676	0.269703
$107$	−9.85293	−0.952519	−0.476259	−	0.879305i	$-0.658008\pi$
$107$	−9.85293	−0.952519	−0.476259	+	0.879305i	$0.658008\pi$
$108$	0	0
$109$	15.9246	1.52530	0.762651	−	0.646810i	$-0.223897\pi$
$109$	15.9246	1.52530	0.762651	+	0.646810i	$0.223897\pi$
$110$	0	0
$111$	0	0
$112$	−2.25390	−0.212974
$113$	5.87866	0.553018	0.276509	−	0.961011i	$-0.410822\pi$
$113$	5.87866	0.553018	0.276509	+	0.961011i	$0.410822\pi$
$114$	0	0
$115$	0	0
$116$	−8.11636	−0.753585
$117$	0	0
$118$	−3.89287	−0.358367
$119$	5.18131	0.474970
$120$	0	0
$121$	−5.67957	−0.516324
$122$	−0.0310534	−0.00281145
$123$	0	0
$124$	13.0762	1.17428
$125$	0	0
$126$	0	0
$127$	−12.6041	−1.11843	−0.559215	−	0.829023i	$-0.688897\pi$
$127$	−12.6041	−1.11843	−0.559215	+	0.829023i	$0.688897\pi$
$128$	11.2143	0.991216
$129$	0	0
$130$	0	0
$131$	−10.2030	−0.891440	−0.445720	−	0.895172i	$-0.647052\pi$
$131$	−10.2030	−0.891440	−0.445720	+	0.895172i	$0.647052\pi$
$132$	0	0
$133$	−4.03437	−0.349825
$134$	−6.93071	−0.598722
$135$	0	0
$136$	−8.95665	−0.768027
$137$	18.8663	1.61185	0.805927	−	0.592015i	$-0.201667\pi$
$137$	18.8663	1.61185	0.805927	+	0.592015i	$0.201667\pi$
$138$	0	0
$139$	−12.6952	−1.07679	−0.538397	−	0.842692i	$-0.680970\pi$
$139$	−12.6952	−1.07679	−0.538397	+	0.842692i	$0.680970\pi$
$140$	0	0
$141$	0	0
$142$	−6.15555	−0.516562
$143$	2.88786	0.241495
$144$	0	0
$145$	0	0
$146$	5.74486	0.475448
$147$	0	0
$148$	−8.87703	−0.729688
$149$	19.1204	1.56640	0.783201	−	0.621768i	$-0.213585\pi$
$149$	19.1204	1.56640	0.783201	+	0.621768i	$0.213585\pi$
$150$	0	0
$151$	−0.321731	−0.0261821	−0.0130910	−	0.999914i	$-0.504167\pi$
$151$	−0.321731	−0.0261821	−0.0130910	+	0.999914i	$0.504167\pi$
$152$	6.97401	0.565667
$153$	0	0
$154$	−1.97953	−0.159515
$155$	0	0
$156$	0	0
$157$	9.59895	0.766080	0.383040	−	0.923732i	$-0.374877\pi$
$157$	9.59895	0.766080	0.383040	+	0.923732i	$0.374877\pi$
$158$	−6.65500	−0.529443
$159$	0	0
$160$	0	0
$161$	−7.27288	−0.573183
$162$	0	0
$163$	−9.53674	−0.746976	−0.373488	−	0.927635i	$-0.621838\pi$
$163$	−9.53674	−0.746976	−0.373488	+	0.927635i	$0.621838\pi$
$164$	10.7220	0.837249
$165$	0	0
$166$	8.60549	0.667916
$167$	9.36567	0.724738	0.362369	−	0.932035i	$-0.381968\pi$
$167$	9.36567	0.724738	0.362369	+	0.932035i	$0.381968\pi$
$168$	0	0
$169$	−11.4325	−0.879423
$170$	0	0
$171$	0	0
$172$	−1.58634	−0.120957
$173$	11.0791	0.842327	0.421164	−	0.906985i	$-0.361622\pi$
$173$	11.0791	0.842327	0.421164	+	0.906985i	$0.361622\pi$
$174$	0	0
$175$	0	0
$176$	−3.89621	−0.293688
$177$	0	0
$178$	5.59783	0.419575
$179$	−17.6771	−1.32125	−0.660625	−	0.750716i	$-0.729709\pi$
$179$	−17.6771	−1.32125	−0.660625	+	0.750716i	$0.729709\pi$
$180$	0	0
$181$	−11.6919	−0.869050	−0.434525	−	0.900660i	$-0.643084\pi$
$181$	−11.6919	−0.869050	−0.434525	+	0.900660i	$0.643084\pi$
$182$	−1.07447	−0.0796447
$183$	0	0
$184$	12.5722	0.926838
$185$	0	0
$186$	0	0
$187$	8.95665	0.654975
$188$	−5.59667	−0.408179
$189$	0	0
$190$	0	0
$191$	−20.5290	−1.48542	−0.742712	−	0.669611i	$-0.766461\pi$
$191$	−20.5290	−1.48542	−0.742712	+	0.669611i	$0.766461\pi$
$192$	0	0
$193$	−14.9974	−1.07954	−0.539768	−	0.841814i	$-0.681488\pi$
$193$	−14.9974	−1.07954	−0.539768	+	0.841814i	$0.681488\pi$
$194$	−10.3258	−0.741348
$195$	0	0
$196$	8.27995	0.591425
$197$	−7.33128	−0.522332	−0.261166	−	0.965294i	$-0.584107\pi$
$197$	−7.33128	−0.522332	−0.261166	+	0.965294i	$0.584107\pi$
$198$	0	0
$199$	−10.3706	−0.735154	−0.367577	−	0.929993i	$-0.619813\pi$
$199$	−10.3706	−0.735154	−0.367577	+	0.929993i	$0.619813\pi$
$200$	0	0
$201$	0	0
$202$	−9.94046	−0.699408
$203$	−6.82704	−0.479164
$204$	0	0
$205$	0	0
$206$	−3.71603	−0.258908
$207$	0	0
$208$	−2.11481	−0.146636
$209$	−6.97401	−0.482402
$210$	0	0
$211$	−8.90388	−0.612968	−0.306484	−	0.951876i	$-0.599153\pi$
$211$	−8.90388	−0.612968	−0.306484	+	0.951876i	$0.599153\pi$
$212$	6.84876	0.470375
$213$	0	0
$214$	6.33706	0.433192
$215$	0	0
$216$	0	0
$217$	10.9990	0.746662
$218$	−10.2422	−0.693687
$219$	0	0
$220$	0	0
$221$	4.86156	0.327024
$222$	0	0
$223$	13.3669	0.895115	0.447557	−	0.894255i	$-0.352294\pi$
$223$	13.3669	0.895115	0.447557	+	0.894255i	$0.352294\pi$
$224$	7.60524	0.508146
$225$	0	0
$226$	−3.78095	−0.251505
$227$	−10.8891	−0.722737	−0.361369	−	0.932423i	$-0.617690\pi$
$227$	−10.8891	−0.722737	−0.361369	+	0.932423i	$0.617690\pi$
$228$	0	0
$229$	−20.1384	−1.33079	−0.665393	−	0.746494i	$-0.731736\pi$
$229$	−20.1384	−1.33079	−0.665393	+	0.746494i	$0.731736\pi$
$230$	0	0
$231$	0	0
$232$	11.8015	0.774809
$233$	22.3394	1.46350	0.731751	−	0.681572i	$-0.238704\pi$
$233$	22.3394	1.46350	0.731751	+	0.681572i	$0.238704\pi$
$234$	0	0
$235$	0	0
$236$	−9.60159	−0.625010
$237$	0	0
$238$	−3.33243	−0.216010
$239$	15.0681	0.974671	0.487336	−	0.873215i	$-0.337969\pi$
$239$	15.0681	0.974671	0.487336	+	0.873215i	$0.337969\pi$
$240$	0	0
$241$	−29.3511	−1.89067	−0.945335	−	0.326102i	$-0.894265\pi$
$241$	−29.3511	−1.89067	−0.945335	+	0.326102i	$0.894265\pi$
$242$	3.65290	0.234817
$243$	0	0
$244$	−0.0765920	−0.00490330
$245$	0	0
$246$	0	0
$247$	−3.78540	−0.240859
$248$	−19.0134	−1.20735
$249$	0	0
$250$	0	0
$251$	6.31265	0.398451	0.199225	−	0.979954i	$-0.436157\pi$
$251$	6.31265	0.398451	0.199225	+	0.979954i	$0.436157\pi$
$252$	0	0
$253$	−12.5722	−0.790410
$254$	8.10649	0.508646
$255$	0	0
$256$	−7.78764	−0.486727
$257$	10.6569	0.664762	0.332381	−	0.943145i	$-0.392148\pi$
$257$	10.6569	0.664762	0.332381	+	0.943145i	$0.392148\pi$
$258$	0	0
$259$	−7.46688	−0.463969
$260$	0	0
$261$	0	0
$262$	6.56221	0.405414
$263$	14.0338	0.865361	0.432680	−	0.901547i	$-0.357568\pi$
$263$	14.0338	0.865361	0.432680	+	0.901547i	$0.357568\pi$
$264$	0	0
$265$	0	0
$266$	2.59477	0.159095
$267$	0	0
$268$	−17.0943	−1.04420
$269$	5.62958	0.343242	0.171621	−	0.985163i	$-0.445100\pi$
$269$	5.62958	0.343242	0.171621	+	0.985163i	$0.445100\pi$
$270$	0	0
$271$	−15.5540	−0.944839	−0.472419	−	0.881374i	$-0.656619\pi$
$271$	−15.5540	−0.944839	−0.472419	+	0.881374i	$0.656619\pi$
$272$	−6.55904	−0.397700
$273$	0	0
$274$	−12.1341	−0.733049
$275$	0	0
$276$	0	0
$277$	3.94356	0.236946	0.118473	−	0.992957i	$-0.462200\pi$
$277$	3.94356	0.236946	0.118473	+	0.992957i	$0.462200\pi$
$278$	8.16511	0.489711
$279$	0	0
$280$	0	0
$281$	−22.1731	−1.32274	−0.661368	−	0.750062i	$-0.730024\pi$
$281$	−22.1731	−1.32274	−0.661368	+	0.750062i	$0.730024\pi$
$282$	0	0
$283$	−6.17588	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$283$	−6.17588	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$284$	−15.1824	−0.900910
$285$	0	0
$286$	−1.85737	−0.109829
$287$	9.01878	0.532362
$288$	0	0
$289$	−1.92198	−0.113058
$290$	0	0
$291$	0	0
$292$	14.1695	0.829205
$293$	23.4684	1.37104	0.685518	−	0.728056i	$-0.259576\pi$
$293$	23.4684	1.37104	0.685518	+	0.728056i	$0.259576\pi$
$294$	0	0
$295$	0	0
$296$	12.9076	0.750239
$297$	0	0
$298$	−12.2976	−0.712378
$299$	−6.82405	−0.394645
$300$	0	0
$301$	−1.33434	−0.0769102
$302$	0.206926	0.0119072
$303$	0	0
$304$	5.10713	0.292914
$305$	0	0
$306$	0	0
$307$	−2.19283	−0.125152	−0.0625758	−	0.998040i	$-0.519932\pi$
$307$	−2.19283	−0.125152	−0.0625758	+	0.998040i	$0.519932\pi$
$308$	−4.88244	−0.278203
$309$	0	0
$310$	0	0
$311$	8.67149	0.491715	0.245858	−	0.969306i	$-0.420930\pi$
$311$	8.67149	0.491715	0.245858	+	0.969306i	$0.420930\pi$
$312$	0	0
$313$	−21.8927	−1.23745	−0.618723	−	0.785609i	$-0.712350\pi$
$313$	−21.8927	−1.23745	−0.618723	+	0.785609i	$0.712350\pi$
$314$	−6.17371	−0.348402
$315$	0	0
$316$	−16.4143	−0.923375
$317$	16.4418	0.923462	0.461731	−	0.887020i	$-0.347228\pi$
$317$	16.4418	0.923462	0.461731	+	0.887020i	$0.347228\pi$
$318$	0	0
$319$	−11.8015	−0.660760
$320$	0	0
$321$	0	0
$322$	4.67766	0.260676
$323$	−11.7404	−0.653251
$324$	0	0
$325$	0	0
$326$	6.13370	0.339714
$327$	0	0
$328$	−15.5903	−0.860830
$329$	−4.70761	−0.259539
$330$	0	0
$331$	−0.831494	−0.0457030	−0.0228515	−	0.999739i	$-0.507274\pi$
$331$	−0.831494	−0.0457030	−0.0228515	+	0.999739i	$0.507274\pi$
$332$	21.2251	1.16488
$333$	0	0
$334$	−6.02367	−0.329601
$335$	0	0
$336$	0	0
$337$	18.8853	1.02875	0.514375	−	0.857566i	$-0.328024\pi$
$337$	18.8853	1.02875	0.514375	+	0.857566i	$0.328024\pi$
$338$	7.35298	0.399950
$339$	0	0
$340$	0	0
$341$	19.0134	1.02963
$342$	0	0
$343$	16.3050	0.880389
$344$	2.30661	0.124364
$345$	0	0
$346$	−7.12568	−0.383079
$347$	−3.17303	−0.170337	−0.0851687	−	0.996367i	$-0.527143\pi$
$347$	−3.17303	−0.170337	−0.0851687	+	0.996367i	$0.527143\pi$
$348$	0	0
$349$	−9.10505	−0.487382	−0.243691	−	0.969853i	$-0.578358\pi$
$349$	−9.10505	−0.487382	−0.243691	+	0.969853i	$0.578358\pi$
$350$	0	0
$351$	0	0
$352$	13.1468	0.700725
$353$	21.1860	1.12762	0.563809	−	0.825905i	$-0.309335\pi$
$353$	21.1860	1.12762	0.563809	+	0.825905i	$0.309335\pi$
$354$	0	0
$355$	0	0
$356$	13.8068	0.731760
$357$	0	0
$358$	11.3693	0.600886
$359$	33.8868	1.78848	0.894238	−	0.447591i	$-0.147718\pi$
$359$	33.8868	1.78848	0.894238	+	0.447591i	$0.147718\pi$
$360$	0	0
$361$	−9.85849	−0.518868
$362$	7.51980	0.395232
$363$	0	0
$364$	−2.65013	−0.138904
$365$	0	0
$366$	0	0
$367$	−19.2326	−1.00393	−0.501966	−	0.864887i	$-0.667390\pi$
$367$	−19.2326	−1.00393	−0.501966	+	0.864887i	$0.667390\pi$
$368$	9.20677	0.479936
$369$	0	0
$370$	0	0
$371$	5.76080	0.299086
$372$	0	0
$373$	23.8910	1.23703	0.618516	−	0.785772i	$-0.287734\pi$
$373$	23.8910	1.23703	0.618516	+	0.785772i	$0.287734\pi$
$374$	−5.76060	−0.297874
$375$	0	0
$376$	8.13780	0.419675
$377$	−6.40573	−0.329912
$378$	0	0
$379$	−28.6048	−1.46933	−0.734666	−	0.678429i	$-0.762661\pi$
$379$	−28.6048	−1.46933	−0.734666	+	0.678429i	$0.762661\pi$
$380$	0	0
$381$	0	0
$382$	13.2035	0.675550
$383$	−11.2121	−0.572910	−0.286455	−	0.958094i	$-0.592477\pi$
$383$	−11.2121	−0.572910	−0.286455	+	0.958094i	$0.592477\pi$
$384$	0	0
$385$	0	0
$386$	9.64580	0.490958
$387$	0	0
$388$	−25.4681	−1.29295
$389$	10.2641	0.520410	0.260205	−	0.965553i	$-0.416210\pi$
$389$	10.2641	0.520410	0.260205	+	0.965553i	$0.416210\pi$
$390$	0	0
$391$	−21.1647	−1.07034
$392$	−12.0394	−0.608082
$393$	0	0
$394$	4.71522	0.237549
$395$	0	0
$396$	0	0
$397$	−9.52302	−0.477947	−0.238973	−	0.971026i	$-0.576811\pi$
$397$	−9.52302	−0.477947	−0.238973	+	0.971026i	$0.576811\pi$
$398$	6.67002	0.334338
$399$	0	0
$400$	0	0
$401$	−14.5099	−0.724592	−0.362296	−	0.932063i	$-0.618007\pi$
$401$	−14.5099	−0.724592	−0.362296	+	0.932063i	$0.618007\pi$
$402$	0	0
$403$	10.3202	0.514088
$404$	−24.5177	−1.21980
$405$	0	0
$406$	4.39091	0.217917
$407$	−12.9076	−0.639806
$408$	0	0
$409$	−22.1375	−1.09463	−0.547313	−	0.836928i	$-0.684349\pi$
$409$	−22.1375	−1.09463	−0.547313	+	0.836928i	$0.684349\pi$
$410$	0	0
$411$	0	0
$412$	−9.16544	−0.451549
$413$	−8.07634	−0.397411
$414$	0	0
$415$	0	0
$416$	7.13590	0.349866
$417$	0	0
$418$	4.48544	0.219390
$419$	18.0673	0.882646	0.441323	−	0.897348i	$-0.354509\pi$
$419$	18.0673	0.882646	0.441323	+	0.897348i	$0.354509\pi$
$420$	0	0
$421$	3.48989	0.170087	0.0850435	−	0.996377i	$-0.472897\pi$
$421$	3.48989	0.170087	0.0850435	+	0.996377i	$0.472897\pi$
$422$	5.72666	0.278769
$423$	0	0
$424$	−9.95839	−0.483622
$425$	0	0
$426$	0	0
$427$	−0.0644250	−0.00311775
$428$	15.6301	0.755509
$429$	0	0
$430$	0	0
$431$	−11.6312	−0.560257	−0.280129	−	0.959963i	$-0.590377\pi$
$431$	−11.6312	−0.560257	−0.280129	+	0.959963i	$0.590377\pi$
$432$	0	0
$433$	0.996806	0.0479035	0.0239517	−	0.999713i	$-0.492375\pi$
$433$	0.996806	0.0479035	0.0239517	+	0.999713i	$0.492375\pi$
$434$	−7.07418	−0.339571
$435$	0	0
$436$	−25.2619	−1.20982
$437$	16.4797	0.788329
$438$	0	0
$439$	−7.98670	−0.381185	−0.190592	−	0.981669i	$-0.561041\pi$
$439$	−7.98670	−0.381185	−0.190592	+	0.981669i	$0.561041\pi$
$440$	0	0
$441$	0	0
$442$	−3.12678	−0.148726
$443$	3.28658	0.156150	0.0780750	−	0.996947i	$-0.475123\pi$
$443$	3.28658	0.156150	0.0780750	+	0.996947i	$0.475123\pi$
$444$	0	0
$445$	0	0
$446$	−8.59713	−0.407086
$447$	0	0
$448$	−0.383610	−0.0181239
$449$	−16.0435	−0.757139	−0.378570	−	0.925573i	$-0.623584\pi$
$449$	−16.0435	−0.757139	−0.378570	+	0.925573i	$0.623584\pi$
$450$	0	0
$451$	15.5903	0.734118
$452$	−9.32555	−0.438637
$453$	0	0
$454$	7.00350	0.328691
$455$	0	0
$456$	0	0
$457$	−5.20719	−0.243582	−0.121791	−	0.992556i	$-0.538864\pi$
$457$	−5.20719	−0.243582	−0.121791	+	0.992556i	$0.538864\pi$
$458$	12.9523	0.605223
$459$	0	0
$460$	0	0
$461$	−38.8040	−1.80728	−0.903641	−	0.428290i	$-0.859116\pi$
$461$	−38.8040	−1.80728	−0.903641	+	0.428290i	$0.859116\pi$
$462$	0	0
$463$	−36.9251	−1.71605	−0.858027	−	0.513605i	$-0.828310\pi$
$463$	−36.9251	−1.71605	−0.858027	+	0.513605i	$0.828310\pi$
$464$	8.64239	0.401213
$465$	0	0
$466$	−14.3679	−0.665580
$467$	−36.5217	−1.69002	−0.845011	−	0.534749i	$-0.820406\pi$
$467$	−36.5217	−1.69002	−0.845011	+	0.534749i	$0.820406\pi$
$468$	0	0
$469$	−14.3788	−0.663952
$470$	0	0
$471$	0	0
$472$	13.9611	0.642613
$473$	−2.30661	−0.106058
$474$	0	0
$475$	0	0
$476$	−8.21931	−0.376732
$477$	0	0
$478$	−9.69124	−0.443267
$479$	−5.12380	−0.234113	−0.117056	−	0.993125i	$-0.537346\pi$
$479$	−5.12380	−0.234113	−0.117056	+	0.993125i	$0.537346\pi$
$480$	0	0
$481$	−7.00608	−0.319450
$482$	18.8776	0.859850
$483$	0	0
$484$	9.00972	0.409533
$485$	0	0
$486$	0	0
$487$	10.9905	0.498029	0.249014	−	0.968500i	$-0.419893\pi$
$487$	10.9905	0.498029	0.249014	+	0.968500i	$0.419893\pi$
$488$	0.111368	0.00504140
$489$	0	0
$490$	0	0
$491$	27.5018	1.24114	0.620568	−	0.784152i	$-0.286902\pi$
$491$	27.5018	1.24114	0.620568	+	0.784152i	$0.286902\pi$
$492$	0	0
$493$	−19.8672	−0.894776
$494$	2.43464	0.109540
$495$	0	0
$496$	−13.9237	−0.625193
$497$	−12.7706	−0.572841
$498$	0	0
$499$	13.8799	0.621348	0.310674	−	0.950517i	$-0.399445\pi$
$499$	13.8799	0.621348	0.310674	+	0.950517i	$0.399445\pi$
$500$	0	0
$501$	0	0
$502$	−4.06007	−0.181210
$503$	−32.2039	−1.43590	−0.717951	−	0.696094i	$-0.754920\pi$
$503$	−32.2039	−1.43590	−0.717951	+	0.696094i	$0.754920\pi$
$504$	0	0
$505$	0	0
$506$	8.08602	0.359467
$507$	0	0
$508$	19.9943	0.887104
$509$	−30.1266	−1.33534	−0.667668	−	0.744459i	$-0.732707\pi$
$509$	−30.1266	−1.33534	−0.667668	+	0.744459i	$0.732707\pi$
$510$	0	0
$511$	11.9186	0.527247
$512$	−17.4199	−0.769859
$513$	0	0
$514$	−6.85417	−0.302324
$515$	0	0
$516$	0	0
$517$	−8.13780	−0.357900
$518$	4.80243	0.211007
$519$	0	0
$520$	0	0
$521$	−15.1821	−0.665141	−0.332570	−	0.943078i	$-0.607916\pi$
$521$	−15.1821	−0.665141	−0.332570	+	0.943078i	$0.607916\pi$
$522$	0	0
$523$	37.1021	1.62236	0.811180	−	0.584797i	$-0.198826\pi$
$523$	37.1021	1.62236	0.811180	+	0.584797i	$0.198826\pi$
$524$	16.1854	0.707063
$525$	0	0
$526$	−9.02604	−0.393554
$527$	32.0080	1.39429
$528$	0	0
$529$	6.70832	0.291666
$530$	0	0
$531$	0	0
$532$	6.39989	0.277470
$533$	8.46221	0.366539
$534$	0	0
$535$	0	0
$536$	24.8559	1.07361
$537$	0	0
$538$	−3.62075	−0.156101
$539$	12.0394	0.518574
$540$	0	0
$541$	−13.0878	−0.562687	−0.281344	−	0.959607i	$-0.590780\pi$
$541$	−13.0878	−0.562687	−0.281344	+	0.959607i	$0.590780\pi$
$542$	10.0038	0.429699
$543$	0	0
$544$	22.1319	0.948895
$545$	0	0
$546$	0	0
$547$	29.7783	1.27323	0.636614	−	0.771183i	$-0.280334\pi$
$547$	29.7783	1.27323	0.636614	+	0.771183i	$0.280334\pi$
$548$	−29.9283	−1.27847
$549$	0	0
$550$	0	0
$551$	15.4694	0.659020
$552$	0	0
$553$	−13.8068	−0.587125
$554$	−2.53636	−0.107760
$555$	0	0
$556$	20.1389	0.854080
$557$	−31.3627	−1.32888	−0.664440	−	0.747341i	$-0.731330\pi$
$557$	−31.3627	−1.32888	−0.664440	+	0.747341i	$0.731330\pi$
$558$	0	0
$559$	−1.25200	−0.0529538
$560$	0	0
$561$	0	0
$562$	14.2609	0.601562
$563$	20.2672	0.854159	0.427079	−	0.904214i	$-0.359543\pi$
$563$	20.2672	0.854159	0.427079	+	0.904214i	$0.359543\pi$
$564$	0	0
$565$	0	0
$566$	3.97211	0.166960
$567$	0	0
$568$	22.0759	0.926284
$569$	26.9433	1.12952	0.564761	−	0.825254i	$-0.308968\pi$
$569$	26.9433	1.12952	0.564761	+	0.825254i	$0.308968\pi$
$570$	0	0
$571$	−21.0542	−0.881089	−0.440545	−	0.897731i	$-0.645215\pi$
$571$	−21.0542	−0.881089	−0.440545	+	0.897731i	$0.645215\pi$
$572$	−4.58113	−0.191547
$573$	0	0
$574$	−5.80056	−0.242111
$575$	0	0
$576$	0	0
$577$	−39.3849	−1.63961	−0.819807	−	0.572639i	$-0.805920\pi$
$577$	−39.3849	−1.63961	−0.819807	+	0.572639i	$0.805920\pi$
$578$	1.23615	0.0514170
$579$	0	0
$580$	0	0
$581$	17.8534	0.740684
$582$	0	0
$583$	9.95839	0.412435
$584$	−20.6030	−0.852559
$585$	0	0
$586$	−15.0940	−0.623528
$587$	−43.9502	−1.81402	−0.907010	−	0.421109i	$-0.861641\pi$
$587$	−43.9502	−1.81402	−0.907010	+	0.421109i	$0.861641\pi$
$588$	0	0
$589$	−24.9227	−1.02692
$590$	0	0
$591$	0	0
$592$	9.45236	0.388490
$593$	−37.1270	−1.52462	−0.762312	−	0.647210i	$-0.775936\pi$
$593$	−37.1270	−1.52462	−0.762312	+	0.647210i	$0.775936\pi$
$594$	0	0
$595$	0	0
$596$	−30.3314	−1.24242
$597$	0	0
$598$	4.38899	0.179479
$599$	14.3002	0.584292	0.292146	−	0.956374i	$-0.405631\pi$
$599$	14.3002	0.584292	0.292146	+	0.956374i	$0.405631\pi$
$600$	0	0
$601$	−32.3855	−1.32103	−0.660515	−	0.750812i	$-0.729662\pi$
$601$	−32.3855	−1.32103	−0.660515	+	0.750812i	$0.729662\pi$
$602$	0.858202	0.0349777
$603$	0	0
$604$	0.510374	0.0207668
$605$	0	0
$606$	0	0
$607$	−15.6944	−0.637017	−0.318509	−	0.947920i	$-0.603182\pi$
$607$	−15.6944	−0.637017	−0.318509	+	0.947920i	$0.603182\pi$
$608$	−17.2327	−0.698880
$609$	0	0
$610$	0	0
$611$	−4.41709	−0.178696
$612$	0	0
$613$	4.07911	0.164754	0.0823769	−	0.996601i	$-0.473749\pi$
$613$	4.07911	0.164754	0.0823769	+	0.996601i	$0.473749\pi$
$614$	1.41035	0.0569172
$615$	0	0
$616$	7.09928	0.286038
$617$	−24.4536	−0.984465	−0.492232	−	0.870464i	$-0.663819\pi$
$617$	−24.4536	−0.984465	−0.492232	+	0.870464i	$0.663819\pi$
$618$	0	0
$619$	25.5505	1.02696	0.513480	−	0.858101i	$-0.328356\pi$
$619$	25.5505	1.02696	0.513480	+	0.858101i	$0.328356\pi$
$620$	0	0
$621$	0	0
$622$	−5.57720	−0.223625
$623$	11.6135	0.465287
$624$	0	0
$625$	0	0
$626$	14.0806	0.562774
$627$	0	0
$628$	−15.2272	−0.607631
$629$	−21.7292	−0.866400
$630$	0	0
$631$	−48.8985	−1.94662	−0.973309	−	0.229499i	$-0.926291\pi$
$631$	−48.8985	−1.94662	−0.973309	+	0.229499i	$0.926291\pi$
$632$	23.8671	0.949381
$633$	0	0
$634$	−10.5748	−0.419978
$635$	0	0
$636$	0	0
$637$	6.53484	0.258920
$638$	7.59034	0.300504
$639$	0	0
$640$	0	0
$641$	−5.53179	−0.218492	−0.109246	−	0.994015i	$-0.534844\pi$
$641$	−5.53179	−0.218492	−0.109246	+	0.994015i	$0.534844\pi$
$642$	0	0
$643$	−10.1529	−0.400389	−0.200195	−	0.979756i	$-0.564157\pi$
$643$	−10.1529	−0.400389	−0.200195	+	0.979756i	$0.564157\pi$
$644$	11.5373	0.454631
$645$	0	0
$646$	7.55098	0.297089
$647$	−21.6339	−0.850516	−0.425258	−	0.905072i	$-0.639817\pi$
$647$	−21.6339	−0.850516	−0.425258	+	0.905072i	$0.639817\pi$
$648$	0	0
$649$	−13.9611	−0.548023
$650$	0	0
$651$	0	0
$652$	15.1285	0.592478
$653$	−38.5870	−1.51003	−0.755013	−	0.655710i	$-0.772370\pi$
$653$	−38.5870	−1.51003	−0.755013	+	0.655710i	$0.772370\pi$
$654$	0	0
$655$	0	0
$656$	−11.4169	−0.445756
$657$	0	0
$658$	3.02777	0.118035
$659$	−2.07264	−0.0807384	−0.0403692	−	0.999185i	$-0.512853\pi$
$659$	−2.07264	−0.0807384	−0.0403692	+	0.999185i	$0.512853\pi$
$660$	0	0
$661$	1.02218	0.0397582	0.0198791	−	0.999802i	$-0.493672\pi$
$661$	1.02218	0.0397582	0.0198791	+	0.999802i	$0.493672\pi$
$662$	0.534787	0.0207851
$663$	0	0
$664$	−30.8622	−1.19769
$665$	0	0
$666$	0	0
$667$	27.8872	1.07980
$668$	−14.8571	−0.574840
$669$	0	0
$670$	0	0
$671$	−0.111368	−0.00429932
$672$	0	0
$673$	−3.89719	−0.150226	−0.0751128	−	0.997175i	$-0.523932\pi$
$673$	−3.89719	−0.150226	−0.0751128	+	0.997175i	$0.523932\pi$
$674$	−12.1464	−0.467861
$675$	0	0
$676$	18.1358	0.697532
$677$	32.8769	1.26356	0.631781	−	0.775147i	$-0.282324\pi$
$677$	32.8769	1.26356	0.631781	+	0.775147i	$0.282324\pi$
$678$	0	0
$679$	−21.4224	−0.822116
$680$	0	0
$681$	0	0
$682$	−12.2288	−0.468263
$683$	2.95808	0.113188	0.0565939	−	0.998397i	$-0.481976\pi$
$683$	2.95808	0.113188	0.0565939	+	0.998397i	$0.481976\pi$
$684$	0	0
$685$	0	0
$686$	−10.4868	−0.400389
$687$	0	0
$688$	1.68915	0.0643983
$689$	5.40529	0.205925
$690$	0	0
$691$	−12.9757	−0.493618	−0.246809	−	0.969064i	$-0.579382\pi$
$691$	−12.9757	−0.493618	−0.246809	+	0.969064i	$0.579382\pi$
$692$	−17.5752	−0.668108
$693$	0	0
$694$	2.04078	0.0774671
$695$	0	0
$696$	0	0
$697$	26.2454	0.994115
$698$	5.85604	0.221655
$699$	0	0
$700$	0	0
$701$	−36.2621	−1.36960	−0.684801	−	0.728731i	$-0.740111\pi$
$701$	−36.2621	−1.36960	−0.684801	+	0.728731i	$0.740111\pi$
$702$	0	0
$703$	16.9192	0.638121
$704$	−0.663126	−0.0249925
$705$	0	0
$706$	−13.6261	−0.512825
$707$	−20.6230	−0.775607
$708$	0	0
$709$	−24.1008	−0.905126	−0.452563	−	0.891732i	$-0.649490\pi$
$709$	−24.1008	−0.905126	−0.452563	+	0.891732i	$0.649490\pi$
$710$	0	0
$711$	0	0
$712$	−20.0757	−0.752370
$713$	−44.9289	−1.68260
$714$	0	0
$715$	0	0
$716$	28.0419	1.04797
$717$	0	0
$718$	−21.7948	−0.813374
$719$	29.8002	1.11136	0.555681	−	0.831396i	$-0.312458\pi$
$719$	29.8002	1.11136	0.555681	+	0.831396i	$0.312458\pi$
$720$	0	0
$721$	−7.70947	−0.287115
$722$	6.34063	0.235974
$723$	0	0
$724$	18.5473	0.689304
$725$	0	0
$726$	0	0
$727$	−24.8763	−0.922611	−0.461306	−	0.887241i	$-0.652619\pi$
$727$	−24.8763	−0.922611	−0.461306	+	0.887241i	$0.652619\pi$
$728$	3.85340	0.142816
$729$	0	0
$730$	0	0
$731$	−3.88304	−0.143620
$732$	0	0
$733$	−23.6472	−0.873430	−0.436715	−	0.899600i	$-0.643858\pi$
$733$	−23.6472	−0.873430	−0.436715	+	0.899600i	$0.643858\pi$
$734$	12.3697	0.456575
$735$	0	0
$736$	−31.0659	−1.14511
$737$	−24.8559	−0.915578
$738$	0	0
$739$	35.3165	1.29914	0.649569	−	0.760302i	$-0.274949\pi$
$739$	35.3165	1.29914	0.649569	+	0.760302i	$0.274949\pi$
$740$	0	0
$741$	0	0
$742$	−3.70514	−0.136020
$743$	−27.9426	−1.02511	−0.512556	−	0.858654i	$-0.671301\pi$
$743$	−27.9426	−1.02511	−0.512556	+	0.858654i	$0.671301\pi$
$744$	0	0
$745$	0	0
$746$	−15.3659	−0.562585
$747$	0	0
$748$	−14.2083	−0.519507
$749$	13.1472	0.480387
$750$	0	0
$751$	11.7598	0.429121	0.214561	−	0.976711i	$-0.431168\pi$
$751$	11.7598	0.429121	0.214561	+	0.976711i	$0.431168\pi$
$752$	5.95939	0.217317
$753$	0	0
$754$	4.11994	0.150039
$755$	0	0
$756$	0	0
$757$	−22.4192	−0.814839	−0.407420	−	0.913241i	$-0.633571\pi$
$757$	−22.4192	−0.814839	−0.407420	+	0.913241i	$0.633571\pi$
$758$	18.3976	0.668232
$759$	0	0
$760$	0	0
$761$	−31.9155	−1.15694	−0.578469	−	0.815704i	$-0.696350\pi$
$761$	−31.9155	−1.15694	−0.578469	+	0.815704i	$0.696350\pi$
$762$	0	0
$763$	−21.2489	−0.769262
$764$	32.5659	1.17819
$765$	0	0
$766$	7.21121	0.260552
$767$	−7.57793	−0.273623
$768$	0	0
$769$	−37.5867	−1.35541	−0.677706	−	0.735333i	$-0.737026\pi$
$769$	−37.5867	−1.35541	−0.677706	+	0.735333i	$0.737026\pi$
$770$	0	0
$771$	0	0
$772$	23.7910	0.856256
$773$	−11.3390	−0.407836	−0.203918	−	0.978988i	$-0.565368\pi$
$773$	−11.3390	−0.407836	−0.203918	+	0.978988i	$0.565368\pi$
$774$	0	0
$775$	0	0
$776$	37.0318	1.32936
$777$	0	0
$778$	−6.60150	−0.236675
$779$	−20.4357	−0.732185
$780$	0	0
$781$	−22.0759	−0.789937
$782$	13.6124	0.486777
$783$	0	0
$784$	−8.81657	−0.314878
$785$	0	0
$786$	0	0
$787$	12.7368	0.454017	0.227008	−	0.973893i	$-0.427106\pi$
$787$	12.7368	0.454017	0.227008	+	0.973893i	$0.427106\pi$
$788$	11.6299	0.414298
$789$	0	0
$790$	0	0
$791$	−7.84415	−0.278906
$792$	0	0
$793$	−0.0604492	−0.00214662
$794$	6.12487	0.217363
$795$	0	0
$796$	16.4513	0.583102
$797$	7.51825	0.266310	0.133155	−	0.991095i	$-0.457489\pi$
$797$	7.51825	0.266310	0.133155	+	0.991095i	$0.457489\pi$
$798$	0	0
$799$	−13.6995	−0.484654
$800$	0	0
$801$	0	0
$802$	9.33229	0.329534
$803$	20.6030	0.727065
$804$	0	0
$805$	0	0
$806$	−6.63761	−0.233800
$807$	0	0
$808$	35.6499	1.25416
$809$	−10.3706	−0.364611	−0.182306	−	0.983242i	$-0.558356\pi$
$809$	−10.3706	−0.364611	−0.182306	+	0.983242i	$0.558356\pi$
$810$	0	0
$811$	−15.8669	−0.557163	−0.278582	−	0.960413i	$-0.589864\pi$
$811$	−15.8669	−0.557163	−0.278582	+	0.960413i	$0.589864\pi$
$812$	10.8300	0.380059
$813$	0	0
$814$	8.30171	0.290975
$815$	0	0
$816$	0	0
$817$	3.02349	0.105779
$818$	14.2380	0.497821
$819$	0	0
$820$	0	0
$821$	40.7878	1.42350	0.711752	−	0.702431i	$-0.247902\pi$
$821$	40.7878	1.42350	0.711752	+	0.702431i	$0.247902\pi$
$822$	0	0
$823$	8.33164	0.290423	0.145211	−	0.989401i	$-0.453614\pi$
$823$	8.33164	0.290423	0.145211	+	0.989401i	$0.453614\pi$
$824$	13.3269	0.464266
$825$	0	0
$826$	5.19441	0.180737
$827$	23.5733	0.819725	0.409862	−	0.912147i	$-0.365577\pi$
$827$	23.5733	0.819725	0.409862	+	0.912147i	$0.365577\pi$
$828$	0	0
$829$	−17.2563	−0.599337	−0.299668	−	0.954043i	$-0.596876\pi$
$829$	−17.2563	−0.599337	−0.299668	+	0.954043i	$0.596876\pi$
$830$	0	0
$831$	0	0
$832$	−0.359936	−0.0124786
$833$	20.2677	0.702233
$834$	0	0
$835$	0	0
$836$	11.0631	0.382627
$837$	0	0
$838$	−11.6203	−0.401415
$839$	−51.0992	−1.76414	−0.882070	−	0.471118i	$-0.843851\pi$
$839$	−51.0992	−1.76414	−0.882070	+	0.471118i	$0.843851\pi$
$840$	0	0
$841$	−2.82234	−0.0973221
$842$	−2.24458	−0.0773532
$843$	0	0
$844$	14.1246	0.486188
$845$	0	0
$846$	0	0
$847$	7.57848	0.260400
$848$	−7.29263	−0.250430
$849$	0	0
$850$	0	0
$851$	30.5008	1.04555
$852$	0	0
$853$	45.5197	1.55857	0.779283	−	0.626672i	$-0.215584\pi$
$853$	45.5197	1.55857	0.779283	+	0.626672i	$0.215584\pi$
$854$	0.0414359	0.00141791
$855$	0	0
$856$	−22.7268	−0.776787
$857$	13.4209	0.458448	0.229224	−	0.973374i	$-0.426381\pi$
$857$	13.4209	0.458448	0.229224	+	0.973374i	$0.426381\pi$
$858$	0	0
$859$	−4.29696	−0.146610	−0.0733051	−	0.997310i	$-0.523355\pi$
$859$	−4.29696	−0.146610	−0.0733051	+	0.997310i	$0.523355\pi$
$860$	0	0
$861$	0	0
$862$	7.48080	0.254797
$863$	41.5929	1.41584	0.707920	−	0.706293i	$-0.249634\pi$
$863$	41.5929	1.41584	0.707920	+	0.706293i	$0.249634\pi$
$864$	0	0
$865$	0	0
$866$	−0.641111	−0.0217858
$867$	0	0
$868$	−17.4482	−0.592229
$869$	−23.8671	−0.809635
$870$	0	0
$871$	−13.4915	−0.457141
$872$	36.7319	1.24390
$873$	0	0
$874$	−10.5991	−0.358521
$875$	0	0
$876$	0	0
$877$	32.9754	1.11350	0.556751	−	0.830680i	$-0.312048\pi$
$877$	32.9754	1.11350	0.556751	+	0.830680i	$0.312048\pi$
$878$	5.13677	0.173357
$879$	0	0
$880$	0	0
$881$	−31.2769	−1.05375	−0.526873	−	0.849944i	$-0.676636\pi$
$881$	−31.2769	−1.05375	−0.526873	+	0.849944i	$0.676636\pi$
$882$	0	0
$883$	33.2793	1.11994	0.559969	−	0.828513i	$-0.310813\pi$
$883$	33.2793	1.11994	0.559969	+	0.828513i	$0.310813\pi$
$884$	−7.71208	−0.259385
$885$	0	0
$886$	−2.11381	−0.0710149
$887$	25.8681	0.868567	0.434283	−	0.900776i	$-0.357002\pi$
$887$	25.8681	0.868567	0.434283	+	0.900776i	$0.357002\pi$
$888$	0	0
$889$	16.8181	0.564062
$890$	0	0
$891$	0	0
$892$	−21.2045	−0.709978
$893$	10.6670	0.356958
$894$	0	0
$895$	0	0
$896$	−14.9638	−0.499904
$897$	0	0
$898$	10.3186	0.344336
$899$	−42.1747	−1.40660
$900$	0	0
$901$	16.7644	0.558503
$902$	−10.0271	−0.333867
$903$	0	0
$904$	13.5598	0.450991
$905$	0	0
$906$	0	0
$907$	−44.2479	−1.46923	−0.734613	−	0.678486i	$-0.762636\pi$
$907$	−44.2479	−1.46923	−0.734613	+	0.678486i	$0.762636\pi$
$908$	17.2739	0.573253
$909$	0	0
$910$	0	0
$911$	−25.8278	−0.855715	−0.427857	−	0.903846i	$-0.640731\pi$
$911$	−25.8278	−0.855715	−0.427857	+	0.903846i	$0.640731\pi$
$912$	0	0
$913$	30.8622	1.02139
$914$	3.34908	0.110778
$915$	0	0
$916$	31.9464	1.05554
$917$	13.6143	0.449583
$918$	0	0
$919$	−42.5667	−1.40415	−0.702073	−	0.712105i	$-0.747742\pi$
$919$	−42.5667	−1.40415	−0.702073	+	0.712105i	$0.747742\pi$
$920$	0	0
$921$	0	0
$922$	24.9574	0.821927
$923$	−11.9825	−0.394409
$924$	0	0
$925$	0	0
$926$	23.7489	0.780437
$927$	0	0
$928$	−29.1616	−0.957275
$929$	36.7900	1.20704	0.603520	−	0.797348i	$-0.293764\pi$
$929$	36.7900	1.20704	0.603520	+	0.797348i	$0.293764\pi$
$930$	0	0
$931$	−15.7812	−0.517208
$932$	−35.4378	−1.16080
$933$	0	0
$934$	23.4895	0.768599
$935$	0	0
$936$	0	0
$937$	−23.0303	−0.752368	−0.376184	−	0.926545i	$-0.622764\pi$
$937$	−23.0303	−0.752368	−0.376184	+	0.926545i	$0.622764\pi$
$938$	9.24794	0.301956
$939$	0	0
$940$	0	0
$941$	9.13277	0.297720	0.148860	−	0.988858i	$-0.452440\pi$
$941$	9.13277	0.297720	0.148860	+	0.988858i	$0.452440\pi$
$942$	0	0
$943$	−36.8400	−1.19968
$944$	10.2239	0.332759
$945$	0	0
$946$	1.48353	0.0482337
$947$	37.2639	1.21091	0.605457	−	0.795878i	$-0.292990\pi$
$947$	37.2639	1.21091	0.605457	+	0.795878i	$0.292990\pi$
$948$	0	0
$949$	11.1831	0.363017
$950$	0	0
$951$	0	0
$952$	11.9512	0.387342
$953$	−39.7614	−1.28800	−0.643999	−	0.765026i	$-0.722726\pi$
$953$	−39.7614	−1.28800	−0.643999	+	0.765026i	$0.722726\pi$
$954$	0	0
$955$	0	0
$956$	−23.9030	−0.773080
$957$	0	0
$958$	3.29545	0.106471
$959$	−25.1741	−0.812913
$960$	0	0
$961$	36.9474	1.19185
$962$	4.50606	0.145281
$963$	0	0
$964$	46.5608	1.49962
$965$	0	0
$966$	0	0
$967$	2.01815	0.0648993	0.0324497	−	0.999473i	$-0.489669\pi$
$967$	2.01815	0.0648993	0.0324497	+	0.999473i	$0.489669\pi$
$968$	−13.1005	−0.421067
$969$	0	0
$970$	0	0
$971$	−5.09972	−0.163658	−0.0818289	−	0.996646i	$-0.526076\pi$
$971$	−5.09972	−0.163658	−0.0818289	+	0.996646i	$0.526076\pi$
$972$	0	0
$973$	16.9397	0.543063
$974$	−7.06872	−0.226496
$975$	0	0
$976$	0.0815560	0.00261054
$977$	−5.90918	−0.189051	−0.0945257	−	0.995522i	$-0.530133\pi$
$977$	−5.90918	−0.189051	−0.0945257	+	0.995522i	$0.530133\pi$
$978$	0	0
$979$	20.0757	0.641623
$980$	0	0
$981$	0	0
$982$	−17.6882	−0.564452
$983$	9.96798	0.317929	0.158965	−	0.987284i	$-0.449184\pi$
$983$	9.96798	0.317929	0.158965	+	0.987284i	$0.449184\pi$
$984$	0	0
$985$	0	0
$986$	12.7779	0.406931
$987$	0	0
$988$	6.00493	0.191042
$989$	5.45053	0.173317
$990$	0	0
$991$	−1.26844	−0.0402933	−0.0201467	−	0.999797i	$-0.506413\pi$
$991$	−1.26844	−0.0402933	−0.0201467	+	0.999797i	$0.506413\pi$
$992$	46.9821	1.49168
$993$	0	0
$994$	8.21361	0.260520
$995$	0	0
$996$	0	0
$997$	20.0616	0.635358	0.317679	−	0.948198i	$-0.397096\pi$
$997$	20.0616	0.635358	0.317679	+	0.948198i	$0.397096\pi$
$998$	−8.92703	−0.282580
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9675.2.a.cv.1.5	✓	10
3.2	odd	2	inner	9675.2.a.cv.1.6	yes	10
5.4	even	2		9675.2.a.cw.1.6	yes	10
15.14	odd	2		9675.2.a.cw.1.5	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9675.2.a.cv.1.5	✓	10	1.1	even	1	trivial
9675.2.a.cv.1.6	yes	10	3.2	odd	2	inner
9675.2.a.cw.1.5	yes	10	15.14	odd	2
9675.2.a.cw.1.6	yes	10	5.4	even	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:	\( N \)	\(=\)	\( 9675 = 3^{2} \cdot 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9675.a (trivial)

\(f(q)\)	\(=\)	\(q-0.643165 q^{2} -1.58634 q^{4} -1.33434 q^{7} +2.30661 q^{8} +O(q^{10})\)	\(q-0.643165 q^{2} -1.58634 q^{4} -1.33434 q^{7} +2.30661 q^{8} -2.30661 q^{11} -1.25200 q^{13} +0.858202 q^{14} +1.68915 q^{16} -3.88304 q^{17} +3.02349 q^{19} +1.48353 q^{22} +5.45053 q^{23} +0.805240 q^{26} +2.11672 q^{28} +5.11641 q^{29} -8.24302 q^{31} -5.69962 q^{32} +2.49744 q^{34} +5.59592 q^{37} -1.94460 q^{38} -6.75897 q^{41} +1.00000 q^{43} +3.65906 q^{44} -3.50559 q^{46} +3.52804 q^{47} -5.21953 q^{49} +1.98609 q^{52} -4.31733 q^{53} -3.07780 q^{56} -3.29069 q^{58} +6.05267 q^{59} +0.0482823 q^{61} +5.30162 q^{62} +0.287490 q^{64} +10.7760 q^{67} +6.15982 q^{68} +9.57072 q^{71} -8.93217 q^{73} -3.59910 q^{74} -4.79629 q^{76} +3.07780 q^{77} +10.3473 q^{79} +4.34713 q^{82} -13.3799 q^{83} -0.643165 q^{86} -5.32043 q^{88} -8.70358 q^{89} +1.67059 q^{91} -8.64640 q^{92} -2.26911 q^{94} +16.0546 q^{97} +3.35702 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{4}+O(q^{10}) \) 10 * q + 6 * q^4	\( 10 q + 6 q^{4} - 4 q^{13} - 10 q^{16} - 10 q^{19} - 2 q^{22} - 16 q^{31} - 8 q^{34} + 10 q^{37} + 10 q^{43} - 28 q^{46} - 26 q^{49} + 36 q^{52} - 16 q^{58} - 32 q^{61} - 28 q^{64} - 14 q^{67} + 4 q^{73} - 24 q^{76} - 18 q^{79} + 20 q^{82} - 56 q^{88} - 44 q^{91} - 12 q^{94} + 4 q^{97}+O(q^{100}) \) 10 * q + 6 * q^4 - 4 * q^13 - 10 * q^16 - 10 * q^19 - 2 * q^22 - 16 * q^31 - 8 * q^34 + 10 * q^37 + 10 * q^43 - 28 * q^46 - 26 * q^49 + 36 * q^52 - 16 * q^58 - 32 * q^61 - 28 * q^64 - 14 * q^67 + 4 * q^73 - 24 * q^76 - 18 * q^79 + 20 * q^82 - 56 * q^88 - 44 * q^91 - 12 * q^94 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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