LMFDB - Embedded newform 921.2.x.a.47.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [921,2,Mod(5,921)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(921, base_ring=CyclotomicField(306))

chi = DirichletCharacter(H, H._module([153, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("921.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 921 = 3 \cdot 307 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	921.x (of order $306$, degree $96$, minimal)

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:	no
Analytic conductor:	$7.35422202616$
Analytic rank:	$0$
Dimension:	$96$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{306}]$

Embedding invariants

Embedding label			47.1
Character	$\chi$	$=$	921.47
Dual form			921.2.x.a.98.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.72466 + 0.159813i) q^{3} +(0.306783 + 1.97633i) q^{4} +(-3.92744 + 2.90303i) q^{7} +(2.94892 - 0.551249i) q^{9} +O(q^{10})$	$q+(-1.72466 + 0.159813i) q^{3} +(0.306783 + 1.97633i) q^{4} +(-3.92744 + 2.90303i) q^{7} +(2.94892 - 0.551249i) q^{9} +(-0.844942 - 3.35948i) q^{12} +(-3.80594 - 4.93373i) q^{13} +(-3.81177 + 1.21261i) q^{16} +(-1.58766 + 7.24950i) q^{19} +(6.30956 - 5.63441i) q^{21} +(4.62429 - 1.90155i) q^{25} +(-4.99779 + 1.42199i) q^{27} +(-6.94223 - 6.87132i) q^{28} +(8.82487 - 6.11197i) q^{31} +(1.99413 + 5.65893i) q^{36} +(2.76697 - 11.4993i) q^{37} +(7.35244 + 7.90078i) q^{39} +(-5.54094 + 1.82555i) q^{43} +(6.38022 - 2.70052i) q^{48} +(4.94370 - 16.1109i) q^{49} +(8.58308 - 9.03539i) q^{52} +(1.57962 - 12.7567i) q^{57} +(-10.6733 + 8.40969i) q^{61} +(-9.98141 + 10.7258i) q^{63} +(-3.56591 - 7.16131i) q^{64} +(-2.30980 - 3.40929i) q^{67} +(-16.7829 + 2.25334i) q^{73} +(-7.67145 + 4.01856i) q^{75} +(-14.8145 - 0.913725i) q^{76} +(-0.547183 - 17.7601i) q^{79} +(8.39225 - 3.25117i) q^{81} +(13.0711 + 10.7412i) q^{84} +(29.2704 + 8.32814i) q^{91} +(-14.2431 + 11.9514i) q^{93} +(0.646302 + 2.95111i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 6 q^{4} + 18 q^{9}+O(q^{10}) $ 96 * q - 6 * q^4 + 18 * q^9	$ 96 q - 6 q^{4} + 18 q^{9} + 18 q^{12} + 12 q^{16} + 21 q^{31} + 18 q^{36} - 39 q^{43} + 36 q^{48} + 39 q^{61} + 48 q^{64} - 48 q^{67} + 51 q^{73} - 54 q^{81} + 45 q^{93}+O(q^{100}) $ 96 * q - 6 * q^4 + 18 * q^9 + 18 * q^12 + 12 * q^16 + 21 * q^31 + 18 * q^36 - 39 * q^43 + 36 * q^48 + 39 * q^61 + 48 * q^64 - 48 * q^67 + 51 * q^73 - 54 * q^81 + 45 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/921\mathbb{Z}\right)^\times$.

$n$	$308$	$619$
$\chi(n)$	$-1$	$e\left(\frac{287}{306}\right)$

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		−0.759405	−	0.650618i	$-0.774510\pi$
$2$		0			0		0.759405	+	0.650618i	$0.225490\pi$
$3$	−1.72466	+	0.159813i	−0.995734	+	0.0922684i
$3$	−1.72466	+	0.159813i	−0.995734	+	0.0922684i
$4$	0.306783	+	1.97633i	0.153392	+	0.988165i
$5$		0			0		0.981035	−	0.193831i	$-0.0620915\pi$
$5$		0			0		−0.981035	+	0.193831i	$0.937908\pi$
$6$		0			0
$7$	−3.92744	+	2.90303i	−1.48443	+	1.09724i	−0.513442	+	0.858124i	$0.671630\pi$
$7$	−3.92744	+	2.90303i	−1.48443	+	1.09724i	−0.970990	+	0.239119i	$0.923141\pi$
$8$		0			0
$9$	2.94892	−	0.551249i	0.982973	−	0.183750i
$10$		0			0
$11$		0			0		0.710727	−	0.703468i	$-0.248366\pi$
$11$		0			0		−0.710727	+	0.703468i	$0.751634\pi$
$12$	−0.844942	−	3.35948i	−0.243914	−	0.969797i
$13$	−3.80594	−	4.93373i	−1.05558	−	1.36837i	−0.926311	−	0.376761i	$-0.877038\pi$
$13$	−3.80594	−	4.93373i	−1.05558	−	1.36837i	−0.129268	−	0.991610i	$-0.541263\pi$
$14$		0			0
$15$		0			0
$16$	−3.81177	+	1.21261i	−0.952942	+	0.303153i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	−1.58766	+	7.24950i	−0.364235	+	1.66315i	0.330442	+	0.943826i	$0.392802\pi$
$19$	−1.58766	+	7.24950i	−0.364235	+	1.66315i	−0.694677	+	0.719322i	$0.744453\pi$
$20$		0			0
$21$	6.30956	−	5.63441i	1.37686	−	1.22953i
$22$		0			0
$23$		0			0		0.936132	−	0.351649i	$-0.114379\pi$
$23$		0			0		−0.936132	+	0.351649i	$0.885621\pi$
$24$		0			0
$25$	4.62429	−	1.90155i	0.924859	−	0.380311i
$26$		0			0
$27$	−4.99779	+	1.42199i	−0.961826	+	0.273663i
$28$	−6.94223	−	6.87132i	−1.31196	−	1.29856i
$29$		0			0		0.508865	−	0.860847i	$-0.330065\pi$
$29$		0			0		−0.508865	+	0.860847i	$0.669935\pi$
$30$		0			0
$31$	8.82487	−	6.11197i	1.58499	−	1.09774i	0.641118	−	0.767442i	$-0.278471\pi$
$31$	8.82487	−	6.11197i	1.58499	−	1.09774i	0.943876	−	0.330301i	$-0.107150\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.99413	+	5.65893i	0.332355	+	0.943154i
$37$	2.76697	−	11.4993i	0.454888	−	1.89047i	−0.000159202	−	1.00000i	$-0.500051\pi$
$37$	2.76697	−	11.4993i	0.454888	−	1.89047i	0.455047	−	0.890468i	$-0.349623\pi$
$38$		0			0
$39$	7.35244	+	7.90078i	1.17733	+	1.26514i
$40$		0			0
$41$		0			0		−0.958965	−	0.283523i	$-0.908497\pi$
$41$		0			0		0.958965	+	0.283523i	$0.0915033\pi$
$42$		0			0
$43$	−5.54094	+	1.82555i	−0.844985	+	0.278394i	−0.702748	−	0.711439i	$-0.748044\pi$
$43$	−5.54094	+	1.82555i	−0.844985	+	0.278394i	−0.142237	+	0.989833i	$0.545430\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.844768	−	0.535133i	$-0.820261\pi$
$47$		0			0		0.844768	+	0.535133i	$0.179739\pi$
$48$	6.38022	−	2.70052i	0.920906	−	0.389786i
$49$	4.94370	−	16.1109i	0.706243	−	2.30156i
$50$		0			0
$51$		0			0
$52$	8.58308	−	9.03539i	1.19026	−	1.25298i
$53$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$53$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.57962	−	12.7567i	0.209225	−	1.68966i
$58$		0			0
$59$		0			0		0.998683	−	0.0513107i	$-0.0163399\pi$
$59$		0			0		−0.998683	+	0.0513107i	$0.983660\pi$
$60$		0			0
$61$	−10.6733	+	8.40969i	−1.36657	+	1.07675i	−0.378032	+	0.925792i	$0.623399\pi$
$61$	−10.6733	+	8.40969i	−1.36657	+	1.07675i	−0.988540	+	0.150958i	$0.951764\pi$
$62$		0			0
$63$	−9.98141	+	10.7258i	−1.25754	+	1.35132i
$64$	−3.56591	−	7.16131i	−0.445738	−	0.895163i
$65$		0			0
$66$		0			0
$67$	−2.30980	−	3.40929i	−0.282186	−	0.416511i	0.660173	−	0.751114i	$-0.270483\pi$
$67$	−2.30980	−	3.40929i	−0.282186	−	0.416511i	−0.942359	+	0.334602i	$0.891398\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.978993	−	0.203893i	$-0.0653595\pi$
$71$		0			0		−0.978993	+	0.203893i	$0.934641\pi$
$72$		0			0
$73$	−16.7829	+	2.25334i	−1.96428	+	0.263734i	−0.999999	−	0.00113145i	$-0.999640\pi$
$73$	−16.7829	+	2.25334i	−1.96428	+	0.263734i	−0.964285	+	0.264865i	$0.914673\pi$
$74$		0			0
$75$	−7.67145	+	4.01856i	−0.885823	+	0.464024i
$76$	−14.8145	−	0.913725i	−1.69934	−	0.104811i
$77$		0			0
$78$		0			0
$79$	−0.547183	−	17.7601i	−0.0615629	−	1.99817i	−0.0604680	−	0.998170i	$-0.519259\pi$
$79$	−0.547183	−	17.7601i	−0.0615629	−	1.99817i	−0.00109494	−	0.999999i	$-0.500349\pi$
$80$		0			0
$81$	8.39225	−	3.25117i	0.932472	−	0.361242i
$82$		0			0
$83$		0			0		−0.233944	−	0.972250i	$-0.575163\pi$
$83$		0			0		0.233944	+	0.972250i	$0.424837\pi$
$84$	13.0711	+	10.7412i	1.42618	+	1.17197i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.899692	−	0.436525i	$-0.856209\pi$
$89$		0			0		0.899692	+	0.436525i	$0.143791\pi$
$90$		0			0
$91$	29.2704	+	8.32814i	3.06837	+	0.873027i
$92$		0			0
$93$	−14.2431	+	11.9514i	−1.47695	+	1.23930i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.646302	+	2.95111i	0.0656220	+	0.299639i	0.998102	−	0.0615823i	$-0.0196147\pi$
$97$	0.646302	+	2.95111i	0.0656220	+	0.299639i	−0.932480	+	0.361222i	$0.882360\pi$
$98$		0			0
$99$		0			0
$100$	5.17676	+	8.55577i	0.517676	+	0.855577i
$101$		0			0		0.998103	−	0.0615609i	$-0.0196078\pi$
$101$		0			0		−0.998103	+	0.0615609i	$0.980392\pi$
$102$		0			0
$103$	−2.26841	−	2.33938i	−0.223514	−	0.230506i	0.596843	−	0.802358i	$-0.296421\pi$
$103$	−2.26841	−	2.33938i	−0.223514	−	0.230506i	−0.820357	+	0.571852i	$0.806225\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.986539	−	0.163529i	$-0.947712\pi$
$107$		0			0		0.986539	+	0.163529i	$0.0522876\pi$
$108$	−4.34357	−	9.44105i	−0.417960	−	0.908465i
$109$	3.00262	+	11.4405i	0.287599	+	1.09580i	0.940894	+	0.338701i	$0.109988\pi$
$109$	3.00262	+	11.4405i	0.287599	+	1.09580i	−0.653295	+	0.757104i	$0.726614\pi$
$110$		0			0
$111$	−2.93436	+	20.2745i	−0.278517	+	1.92437i
$112$	11.4502	−	15.8281i	1.08195	−	1.49562i
$113$		0			0		−0.881012	−	0.473094i	$-0.843137\pi$
$113$		0			0		0.881012	+	0.473094i	$0.156863\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−13.9431	−	12.4512i	−1.28904	−	1.15111i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.112931	−	10.9994i	0.0102665	−	0.999947i
$122$		0			0
$123$		0			0
$124$	14.7866	+	15.5658i	1.32788	+	1.39785i
$125$		0			0
$126$		0			0
$127$	9.58422	−	5.66544i	0.850462	−	0.502726i	−0.0172166	−	0.999852i	$-0.505480\pi$
$127$	9.58422	−	5.66544i	0.850462	−	0.502726i	0.867679	+	0.497126i	$0.165611\pi$
$128$		0			0
$129$	9.26450	−	4.03398i	0.815694	−	0.355172i
$130$		0			0
$131$		0			0		−0.885823	−	0.464024i	$-0.846405\pi$
$131$		0			0		0.885823	+	0.464024i	$0.153595\pi$
$132$		0			0
$133$	−14.8101	−	33.0810i	−1.28420	−	2.86849i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.999947	−	0.0102665i	$-0.00326797\pi$
$137$		0			0		−0.999947	+	0.0102665i	$0.996732\pi$
$138$		0			0
$139$	−13.0953	+	2.30906i	−1.11073	+	0.195852i	−0.698769	−	0.715348i	$-0.746268\pi$
$139$	−13.0953	+	2.30906i	−1.11073	+	0.195852i	−0.411964	+	0.911200i	$0.635157\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−10.5721	+	5.67712i	−0.881012	+	0.473094i
$145$		0			0
$146$		0			0
$147$	−5.95147	+	28.5760i	−0.490869	+	2.35691i
$148$	23.5752	+	1.94067i	1.93787	+	0.159522i
$149$		0			0		−0.779081	−	0.626924i	$-0.784314\pi$
$149$		0			0		0.779081	+	0.626924i	$0.215686\pi$
$150$		0			0
$151$	−18.9783	−	14.9534i	−1.54443	−	1.21689i	−0.885460	−	0.464716i	$-0.846157\pi$
$151$	−18.9783	−	14.9534i	−1.54443	−	1.21689i	−0.658968	−	0.752171i	$-0.729007\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−13.3589	+	16.9547i	−1.06957	+	1.35746i
$157$	−2.91822	+	12.6997i	−0.232899	+	1.01354i	0.716033	+	0.698066i	$0.245956\pi$
$157$	−2.91822	+	12.6997i	−0.232899	+	1.01354i	−0.948933	+	0.315479i	$0.897835\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	11.9364	+	22.2284i	0.934931	+	1.74106i	0.602957	+	0.797773i	$0.293989\pi$
$163$	11.9364	+	22.2284i	0.934931	+	1.74106i	0.331973	+	0.943289i	$0.392286\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.552365	−	0.833602i	$-0.313725\pi$
$167$		0			0		−0.552365	+	0.833602i	$0.686275\pi$
$168$		0			0
$169$	−6.55634	+	24.9808i	−0.504334	+	1.92160i
$170$		0			0
$171$	−0.685619	+	22.2534i	−0.0524306	+	1.70176i
$172$	−5.30776	−	10.3907i	−0.404713	−	0.792282i
$173$		0			0		−0.993630	−	0.112693i	$-0.964052\pi$
$173$		0			0		0.993630	+	0.112693i	$0.0359477\pi$
$174$		0			0
$175$	−12.6414	+	20.8927i	−0.955597	+	1.57934i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.969797	−	0.243914i	$-0.921569\pi$
$179$		0			0		0.969797	+	0.243914i	$0.0784314\pi$
$180$		0			0
$181$	1.76124	+	15.5290i	0.130912	+	1.15426i	0.874294	+	0.485397i	$0.161325\pi$
$181$	1.76124	+	15.5290i	0.130912	+	1.15426i	−0.743382	+	0.668867i	$0.766780\pi$
$182$		0			0
$183$	17.0638	−	16.2096i	1.26139	−	1.19825i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	15.5004	−	20.0936i	1.12749	−	1.46159i
$190$		0			0
$191$		0			0		0.370796	−	0.928714i	$-0.379085\pi$
$191$		0			0		−0.370796	+	0.928714i	$0.620915\pi$
$192$	7.29446	+	11.7810i	0.526432	+	0.850217i
$193$	−11.4306	−	12.5388i	−0.822792	−	0.902559i	0.173823	−	0.984777i	$-0.444388\pi$
$193$	−11.4306	−	12.5388i	−0.822792	−	0.902559i	−0.996614	+	0.0822176i	$0.973800\pi$
$194$		0			0
$195$		0			0
$196$	33.3572	+	4.82782i	2.38266	+	0.344845i
$197$		0			0		−0.263774	−	0.964585i	$-0.584967\pi$
$197$		0			0		0.263774	+	0.964585i	$0.415033\pi$
$198$		0			0
$199$	−15.8606	+	7.29705i	−1.12433	+	0.517274i	−0.890491	−	0.455001i	$-0.849639\pi$
$199$	−15.8606	+	7.29705i	−1.12433	+	0.517274i	−0.233840	+	0.972275i	$0.575129\pi$
$200$		0			0
$201$	4.52847	+	5.51074i	0.319414	+	0.388698i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	20.4901	+	14.1911i	1.42073	+	0.983976i
$209$		0			0
$210$		0			0
$211$	−3.69657	+	11.6199i	−0.254482	+	0.799950i	0.737697	+	0.675132i	$0.235913\pi$
$211$	−3.69657	+	11.6199i	−0.254482	+	0.799950i	−0.992180	+	0.124818i	$0.960165\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−16.9159	+	49.6233i	−1.14833	+	3.36865i
$218$		0			0
$219$	28.5846	−	6.56838i	1.93157	−	0.443850i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	4.17370	−	5.07902i	0.279492	−	0.340117i	−0.613909	−	0.789377i	$-0.710404\pi$
$223$	4.17370	−	5.07902i	0.279492	−	0.340117i	0.893401	+	0.449260i	$0.148312\pi$
$224$		0			0
$225$	12.5884	−	8.15667i	0.839229	−	0.543778i
$226$		0			0
$227$		0			0		−0.482114	−	0.876109i	$-0.660131\pi$
$227$		0			0		0.482114	+	0.876109i	$0.339869\pi$
$228$	25.6960	−	0.791685i	1.70176	−	0.0524306i
$229$	−30.1774	+	1.23998i	−1.99418	+	0.0819402i	−0.999729	−	0.0232641i	$-0.992594\pi$
$229$	−30.1774	+	1.23998i	−1.99418	+	0.0819402i	−0.994451	+	0.105204i	$0.966450\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.586123	−	0.810222i	$-0.699346\pi$
$233$		0			0		0.586123	+	0.810222i	$0.300654\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	3.78201	+	30.5428i	0.245668	+	1.98397i
$238$		0			0
$239$		0			0		−0.626924	−	0.779081i	$-0.715686\pi$
$239$		0			0		0.626924	+	0.779081i	$0.284314\pi$
$240$		0			0
$241$	−5.06621	−	30.5635i	−0.326343	−	1.96877i	−0.228316	−	0.973587i	$-0.573322\pi$
$241$	−5.06621	−	30.5635i	−0.326343	−	1.96877i	−0.0980274	−	0.995184i	$-0.531253\pi$
$242$		0			0
$243$	−13.9542	+	6.94837i	−0.895163	+	0.445738i
$244$	−19.8947	−	18.5140i	−1.27363	−	1.18524i
$245$		0			0
$246$		0			0
$247$	41.8096	−	19.7581i	2.66028	−	1.25718i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.996629	−	0.0820408i	$-0.0261438\pi$
$251$		0			0		−0.996629	+	0.0820408i	$0.973856\pi$
$252$	−24.2599	−	16.4361i	−1.52823	−	1.03537i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	13.0592	−	9.24438i	0.816197	−	0.577774i
$257$		0			0		0.312920	−	0.949779i	$-0.398693\pi$
$257$		0			0		−0.312920	+	0.949779i	$0.601307\pi$
$258$		0			0
$259$	22.5156	+	53.1953i	1.39905	+	3.30539i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.594410	−	0.804162i	$-0.297386\pi$
$263$		0			0		−0.594410	+	0.804162i	$0.702614\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	6.02928	−	5.61084i	0.368297	−	0.342736i
$269$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$269$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$270$		0			0
$271$	−14.1143	+	4.97368i	−0.857382	+	0.302130i	−0.724730	−	0.689033i	$-0.758035\pi$
$271$	−14.1143	+	4.97368i	−0.857382	+	0.302130i	−0.132652	+	0.991163i	$0.542349\pi$
$272$		0			0
$273$	−51.8125	−	9.68543i	−3.13583	−	0.586189i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	3.76426	−	8.40816i	0.226172	−	0.505197i	−0.763972	−	0.645250i	$-0.776753\pi$
$277$	3.76426	−	8.40816i	0.226172	−	0.505197i	0.990144	+	0.140053i	$0.0447271\pi$
$278$		0			0
$279$	22.6546	−	22.8884i	1.35630	−	1.37029i
$280$		0			0
$281$		0			0		−0.998683	−	0.0513107i	$-0.983660\pi$
$281$		0			0		0.998683	+	0.0513107i	$0.0163399\pi$
$282$		0			0
$283$	−21.9241	−	10.9169i	−1.30325	−	0.648944i	−0.346012	−	0.938230i	$-0.612464\pi$
$283$	−21.9241	−	10.9169i	−1.30325	−	0.648944i	−0.957243	+	0.289286i	$0.906582\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.50000	+	14.7224i	−0.500000	+	0.866025i
$290$		0			0
$291$	−1.58628	−	4.98637i	−0.0929894	−	0.292306i
$292$	−9.60205	−	32.4772i	−0.561918	−	1.90058i
$293$		0			0		0.0718047	−	0.997419i	$-0.477124\pi$
$293$		0			0		−0.0718047	+	0.997419i	$0.522876\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−10.2955	−	13.9285i	−0.594410	−	0.804162i
$301$	16.4621	−	23.2553i	0.948858	−	1.34041i
$302$		0			0
$303$		0			0
$304$	−2.73901	−	29.5586i	−0.157093	−	1.69530i
$305$		0			0
$306$		0			0
$307$	−2.47703	−	17.3454i	−0.141371	−	0.989957i
$307$	−2.47703	−	17.3454i	−0.141371	−	0.989957i
$308$		0			0
$309$	4.28611	+	3.67212i	0.243829	+	0.208899i
$310$		0			0
$311$		0			0		−0.153392	−	0.988165i	$-0.549020\pi$
$311$		0			0		0.153392	+	0.988165i	$0.450980\pi$
$312$		0			0
$313$	15.3654	+	10.8770i	0.868506	+	0.614803i	0.923464	−	0.383686i	$-0.125345\pi$
$313$	15.3654	+	10.8770i	0.868506	+	0.614803i	−0.0549575	+	0.998489i	$0.517502\pi$
$314$		0			0
$315$		0			0
$316$	34.9320	−	6.52992i	1.96508	−	0.367337i
$317$		0			0		−0.871113	−	0.491083i	$-0.836601\pi$
$317$		0			0		0.871113	+	0.491083i	$0.163399\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	9.00000	+	15.5885i	0.500000	+	0.866025i
$325$	−26.9816	−	15.5778i	−1.49667	−	0.864101i
$326$		0			0
$327$	−7.00686	−	19.2512i	−0.387480	−	1.06459i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−3.19647	+	6.41937i	−0.175694	+	0.352840i	−0.965453	−	0.260576i	$-0.916088\pi$
$331$	−3.19647	+	6.41937i	−0.175694	+	0.352840i	0.789760	+	0.613416i	$0.210205\pi$
$332$		0			0
$333$	1.82063	−	35.4357i	0.0997698	−	1.94186i
$334$		0			0
$335$		0			0
$336$	−17.2182	+	29.1281i	−0.939332	+	1.58907i
$337$	−31.4068	−	14.0606i	−1.71084	−	0.765928i	−0.997905	−	0.0646965i	$-0.979392\pi$
$337$	−31.4068	−	14.0606i	−1.71084	−	0.765928i	−0.712934	−	0.701231i	$-0.752634\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	15.9922	+	45.3827i	0.863499	+	2.45043i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.351649	−	0.936132i	$-0.385621\pi$
$347$		0			0		−0.351649	+	0.936132i	$0.614379\pi$
$348$		0			0
$349$	32.2368	−	0.662021i	1.72560	−	0.0354372i	0.852425	−	0.522849i	$-0.175131\pi$
$349$	32.2368	−	0.662021i	1.72560	−	0.0354372i	0.873172	+	0.487412i	$0.162059\pi$
$350$		0			0
$351$	26.0371	+	19.2457i	1.38975	+	1.02726i
$352$		0			0
$353$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$353$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.688727	−	0.725021i	$-0.258170\pi$
$359$		0			0		−0.688727	+	0.725021i	$0.741830\pi$
$360$		0			0
$361$	−32.7737	−	15.0783i	−1.72493	−	0.793594i
$362$		0			0
$363$	1.56309	+	18.9883i	0.0820408	+	0.996629i
$364$	−7.47950	+	60.4029i	−0.392032	+	3.16597i
$365$		0			0
$366$		0			0
$367$	−14.6682	−	31.0391i	−0.765675	−	1.62023i	−0.784284	−	0.620402i	$-0.786969\pi$
$367$	−14.6682	−	31.0391i	−0.765675	−	1.62023i	0.0186089	−	0.999827i	$-0.494076\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−27.9895	−	24.4827i	−1.45119	−	1.26937i
$373$	24.0618	−	3.98849i	1.24588	−	0.206516i	0.496133	−	0.868246i	$-0.334753\pi$
$373$	24.0618	−	3.98849i	1.24588	−	0.206516i	0.749742	+	0.661730i	$0.230178\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−3.59243	+	9.27312i	−0.184531	+	0.476328i	−0.993810	−	0.111096i	$-0.964564\pi$
$379$	−3.59243	+	9.27312i	−0.184531	+	0.476328i	0.809279	+	0.587424i	$0.199858\pi$
$380$		0			0
$381$	−15.6241	+	11.3027i	−0.800448	+	0.579052i
$382$		0			0
$383$		0			0		−0.998103	−	0.0615609i	$-0.980392\pi$
$383$		0			0		0.998103	+	0.0615609i	$0.0196078\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−15.3335	+	8.43784i	−0.779443	+	0.428919i
$388$	−5.63409	+	2.18266i	−0.286027	+	0.110808i
$389$		0			0		−0.543778	−	0.839229i	$-0.683007\pi$
$389$		0			0		0.543778	+	0.839229i	$0.316993\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	11.5984	+	3.95372i	0.582105	+	0.198431i	0.598074	−	0.801441i	$-0.295933\pi$
$397$	11.5984	+	3.95372i	0.582105	+	0.198431i	−0.0159690	+	0.999872i	$0.505083\pi$
$398$		0			0
$399$	30.8292	+	54.6867i	1.54339	+	2.73776i
$400$	−15.3209	+	12.8558i	−0.766044	+	0.642788i
$401$		0			0		−0.293353	−	0.956004i	$-0.594771\pi$
$401$		0			0		0.293353	+	0.956004i	$0.405229\pi$
$402$		0			0
$403$	−63.7418	−	20.2777i	−3.17520	−	1.01011i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.02345	+	11.0447i	−0.0506062	+	0.546128i	0.932258	+	0.361795i	$0.117836\pi$
$409$	−1.02345	+	11.0447i	−0.0506062	+	0.546128i	−0.982864	+	0.184333i	$0.940988\pi$
$410$		0			0
$411$		0			0
$412$	3.92748	−	5.20082i	0.193493	−	0.256226i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	22.2160	−	6.07517i	1.08792	−	0.297502i
$418$		0			0
$419$		0			0		0.586123	−	0.810222i	$-0.300654\pi$
$419$		0			0		−0.586123	+	0.810222i	$0.699346\pi$
$420$		0			0
$421$	25.0872	−	22.8700i	1.22268	−	1.11462i	0.232823	−	0.972519i	$-0.425204\pi$
$421$	25.0872	−	22.8700i	1.22268	−	1.11462i	0.989853	−	0.142097i	$-0.0453845\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	17.5050	−	64.0134i	0.847127	−	3.09783i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.688727	−	0.725021i	$-0.741830\pi$
$431$		0			0		0.688727	+	0.725021i	$0.258170\pi$
$432$	17.3261	−	11.4807i	0.833602	−	0.552365i
$433$	−40.0316	+	4.54020i	−1.92379	+	0.218188i	−0.990150	−	0.140014i	$-0.955285\pi$
$433$	−40.0316	+	4.54020i	−1.92379	+	0.218188i	−0.933644	+	0.358202i	$0.883390\pi$
$434$		0			0
$435$		0			0
$436$	−21.6891	+	9.44394i	−1.03872	+	0.452283i
$437$		0			0
$438$		0			0
$439$	1.57341	+	0.952008i	0.0750948	+	0.0454369i	0.554724	−	0.832034i	$-0.312824\pi$
$439$	1.57341	+	0.952008i	0.0750948	+	0.0454369i	−0.479630	+	0.877471i	$0.659229\pi$
$440$		0			0
$441$	5.69744	−	50.2351i	0.271307	−	2.39215i
$442$		0			0
$443$		0			0		−0.999526	−	0.0307951i	$-0.990196\pi$
$443$		0			0		0.999526	+	0.0307951i	$0.00980392\pi$
$444$	−40.9694	+	0.420633i	−1.94432	+	0.0199624i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	34.7944	+	17.7736i	1.64388	+	0.839726i
$449$		0			0		−0.876109	−	0.482114i	$-0.839869\pi$
$449$		0			0		0.876109	+	0.482114i	$0.160131\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	35.1208	+	22.7565i	1.65012	+	1.06919i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	40.4101	+	9.28571i	1.89030	+	0.434367i	0.999788	−	0.0205877i	$-0.00655374\pi$
$457$	40.4101	+	9.28571i	1.89030	+	0.434367i	0.890515	+	0.454955i	$0.150345\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.759405	−	0.650618i	$-0.225490\pi$
$461$		0			0		−0.759405	+	0.650618i	$0.774510\pi$
$462$		0			0
$463$	−21.4308	+	27.1992i	−0.995974	+	1.26405i	−0.0316027	+	0.999501i	$0.510061\pi$
$463$	−21.4308	+	27.1992i	−0.995974	+	1.26405i	−0.964371	+	0.264553i	$0.914775\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.978993	−	0.203893i	$-0.934641\pi$
$467$		0			0		0.978993	+	0.203893i	$0.0653595\pi$
$468$	20.3301	−	31.3760i	0.939758	−	1.45036i
$469$	18.9689	+	6.68437i	0.875901	+	0.308656i
$470$		0			0
$471$	3.00337	−	22.3690i	0.138388	−	1.03071i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	6.44349	+	36.5428i	0.295647	+	1.67670i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.454905	−	0.890540i	$-0.650327\pi$
$479$		0			0		0.454905	+	0.890540i	$0.349673\pi$
$480$		0			0
$481$	−67.2652	+	30.1140i	−3.06703	+	1.37308i
$482$		0			0
$483$		0			0
$484$	21.7731	−	3.15125i	0.989688	−	0.143239i
$485$		0			0
$486$		0			0
$487$	−14.4978	−	24.5260i	−0.656959	−	1.11138i	−0.985949	−	0.167045i	$-0.946577\pi$
$487$	−14.4978	−	24.5260i	−0.656959	−	1.11138i	0.328990	−	0.944333i	$-0.393292\pi$
$488$		0			0
$489$	−24.1386	−	36.4289i	−1.09159	−	1.64737i
$490$		0			0
$491$		0			0		0.193831	−	0.981035i	$-0.437908\pi$
$491$		0			0		−0.193831	+	0.981035i	$0.562092\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−26.2269	+	33.9986i	−1.17762	+	1.52658i
$497$		0			0
$498$		0			0
$499$	−15.2596	−	24.6451i	−0.683114	−	1.10327i	−0.988527	−	0.151047i	$-0.951736\pi$
$499$	−15.2596	−	24.6451i	−0.683114	−	1.10327i	0.305413	−	0.952220i	$-0.401206\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.989688	−	0.143239i	$-0.954248\pi$
$503$		0			0		0.989688	+	0.143239i	$0.0457516\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	7.31520	−	44.1313i	0.324880	−	1.95994i
$508$	14.1371	+	17.2035i	0.627230	+	0.763283i
$509$		0			0		−0.798017	−	0.602635i	$-0.794118\pi$
$509$		0			0		0.798017	+	0.602635i	$0.205882\pi$
$510$		0			0
$511$	59.3721	−	57.5711i	2.62647	−	2.54679i
$512$		0			0
$513$	−2.37393	−	38.4891i	−0.104811	−	1.69934i
$514$		0			0
$515$		0			0
$516$	10.8147	+	17.0722i	0.476089	+	0.751560i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$521$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$522$		0			0
$523$	1.64976	−	5.79830i	0.0721390	−	0.253542i	−0.917365	−	0.398047i	$-0.869688\pi$
$523$	1.64976	−	5.79830i	0.0721390	−	0.253542i	0.989504	+	0.144504i	$0.0461588\pi$
$524$		0			0
$525$	18.4631	−	38.0531i	0.805797	−	1.66078i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	17.3118	−	15.1427i	0.752685	−	0.658380i
$530$		0			0
$531$		0			0
$532$	60.8355	−	39.4183i	2.63755	−	1.70900i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−6.11268	−	45.5271i	−0.262804	−	1.95736i	−0.287845	−	0.957677i	$-0.592939\pi$
$541$	−6.11268	−	45.5271i	−0.262804	−	1.95736i	0.0250408	−	0.999686i	$-0.492028\pi$
$542$		0			0
$543$	−5.51929	−	26.5009i	−0.236855	−	1.13726i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−38.7236	+	26.2352i	−1.65570	+	1.12174i	−0.822567	+	0.568668i	$0.807459\pi$
$547$	−38.7236	+	26.2352i	−1.65570	+	1.12174i	−0.833134	+	0.553071i	$0.813456\pi$
$548$		0			0
$549$	−26.8388	+	30.6831i	−1.14545	+	1.30952i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	53.7072	+	68.1633i	2.28386	+	2.89860i
$554$		0			0
$555$		0			0
$556$	−8.58091	−	25.1724i	−0.363912	−	1.06755i
$557$		0			0		−0.992421	−	0.122888i	$-0.960784\pi$
$557$		0			0		0.992421	+	0.122888i	$0.0392157\pi$
$558$		0			0
$559$	30.0953	+	20.3896i	1.27289	+	0.862386i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.816197	−	0.577774i	$-0.196078\pi$
$563$		0			0		−0.816197	+	0.577774i	$0.803922\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−23.5218	+	37.1318i	−0.987822	+	1.55939i
$568$		0			0
$569$		0			0		−0.370796	−	0.928714i	$-0.620915\pi$
$569$		0			0		0.370796	+	0.928714i	$0.379085\pi$
$570$		0			0
$571$	−6.14595	−	18.6543i	−0.257200	−	0.780657i	−0.994466	−	0.105055i	$-0.966498\pi$
$571$	−6.14595	−	18.6543i	−0.257200	−	0.780657i	0.737266	−	0.675602i	$-0.236116\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−14.4632	−	19.1524i	−0.602635	−	0.798017i
$577$	−2.63690	−	0.634496i	−0.109776	−	0.0264144i	0.178684	−	0.983907i	$-0.442816\pi$
$577$	−2.63690	−	0.634496i	−0.109776	−	0.0264144i	−0.288459	+	0.957492i	$0.593143\pi$
$578$		0			0
$579$	21.7178	+	19.7984i	0.902559	+	0.822792i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.273663	−	0.961826i	$-0.588235\pi$
$587$		0			0		0.273663	+	0.961826i	$0.411765\pi$
$588$	−58.3015	−	2.99543i	−2.40431	−	0.123530i
$589$	30.2978	+	73.6796i	1.24840	+	3.03592i
$590$		0			0
$591$		0			0
$592$	3.39707	+	47.1878i	0.139619	+	1.93941i
$593$		0			0		−0.666073	−	0.745886i	$-0.732026\pi$
$593$		0			0		0.666073	+	0.745886i	$0.267974\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	26.1881	−	15.1197i	1.07181	−	0.618808i
$598$		0			0
$599$		0			0		−0.283523	−	0.958965i	$-0.591503\pi$
$599$		0			0		0.283523	+	0.958965i	$0.408497\pi$
$600$		0			0
$601$	32.4589	−	25.0392i	1.32402	−	1.02137i	0.326942	−	0.945044i	$-0.393982\pi$
$601$	32.4589	−	25.0392i	1.32402	−	1.02137i	0.997083	−	0.0763255i	$-0.0243188\pi$
$602$		0			0
$603$	−8.69077	−	8.78046i	−0.353915	−	0.357568i
$604$	23.7306	−	42.0947i	0.965583	−	1.71281i
$605$		0			0
$606$		0			0
$607$	−21.7883	−	29.4768i	−0.884359	−	1.19643i	−0.979597	−	0.200972i	$-0.935590\pi$
$607$	−21.7883	−	29.4768i	−0.884359	−	1.19643i	0.0952379	−	0.995455i	$-0.469639\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			25.7182i			1.03875i	0.854547	+	0.519374i	$0.173835\pi$
$613$			25.7182i			1.03875i	−0.854547	+	0.519374i	$0.826165\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$617$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$618$		0			0
$619$	−14.3389	+	2.83307i	−0.576331	+	0.113871i	−0.473446	−	0.880823i	$-0.656990\pi$
$619$	−14.3389	+	2.83307i	−0.576331	+	0.113871i	−0.102884	+	0.994693i	$0.532807\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−37.6064	−	21.2003i	−1.50546	−	0.848690i
$625$	17.7682	−	17.5867i	0.710727	−	0.703468i
$626$		0			0
$627$		0			0
$628$	−25.9940	−	1.87132i	−1.03727	−	0.0746739i
$629$		0			0
$630$		0			0
$631$	16.9114	+	29.2914i	0.673232	+	1.16607i	0.976982	+	0.213320i	$0.0684278\pi$
$631$	16.9114	+	29.2914i	0.673232	+	1.16607i	−0.303750	+	0.952752i	$0.598239\pi$
$632$		0			0
$633$	4.51831	−	20.6312i	0.179587	−	0.820018i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−98.3025	+	36.9264i	−3.89489	+	1.46308i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$641$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$642$		0			0
$643$	−9.20208	+	15.5672i	−0.362895	+	0.613909i	−0.985708	−	0.168461i	$-0.946120\pi$
$643$	−9.20208	+	15.5672i	−0.362895	+	0.613909i	0.622814	+	0.782370i	$0.285989\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$647$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	21.2437	−	88.2868i	0.832607	−	3.46023i
$652$	−40.2688	+	30.4096i	−1.57705	+	1.19093i
$653$		0			0		−0.681247	−	0.732053i	$-0.738562\pi$
$653$		0			0		0.681247	+	0.732053i	$0.261438\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−48.2491	+	15.8965i	−1.88238	+	0.620180i
$658$		0			0
$659$		0			0		0.928714	−	0.370796i	$-0.120915\pi$
$659$		0			0		−0.928714	+	0.370796i	$0.879085\pi$
$660$		0			0
$661$	19.7604	+	12.5176i	0.768590	+	0.486877i	0.860851	−	0.508858i	$-0.169932\pi$
$661$	19.7604	+	12.5176i	0.768590	+	0.486877i	−0.0922603	+	0.995735i	$0.529409\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−6.38653	+	9.42661i	−0.246918	+	0.364454i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−36.5764	+	1.87923i	−1.40992	+	0.0724392i	−0.741303	−	0.671171i	$-0.765792\pi$
$673$	−36.5764	+	1.87923i	−1.40992	+	0.0724392i	−0.668613	+	0.743610i	$0.733112\pi$
$674$		0			0
$675$	−20.4073	+	16.0793i	−0.785476	+	0.618892i
$676$	−51.3818	−	5.29380i	−1.97622	−	0.203608i
$677$		0			0		0.681247	−	0.732053i	$-0.261438\pi$
$677$		0			0		−0.681247	+	0.732053i	$0.738562\pi$
$678$		0			0
$679$	−11.1055	−	9.71405i	−0.426189	−	0.372791i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.899692	−	0.436525i	$-0.143791\pi$
$683$		0			0		−0.899692	+	0.436525i	$0.856209\pi$
$684$	−44.1904	+	5.47195i	−1.68966	+	0.209225i
$685$		0			0
$686$		0			0
$687$	51.8477	−	6.96131i	1.97811	−	0.265590i
$688$	18.9071	−	13.6776i	0.720826	−	0.521453i
$689$		0			0
$690$		0			0
$691$	1.06323	−	6.84942i	0.0404470	−	0.260564i	−0.959316	−	0.282335i	$-0.908891\pi$
$691$	1.06323	−	6.84942i	0.0404470	−	0.260564i	0.999763	+	0.0217706i	$0.00693034\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−45.1691	−	18.5740i	−1.70723	−	0.702030i
$701$		0			0		0.427265	−	0.904126i	$-0.359477\pi$
$701$		0			0		−0.427265	+	0.904126i	$0.640523\pi$
$702$		0			0
$703$	78.9709	+	38.3161i	2.97844	+	1.44512i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.195603	+	9.52479i	−0.00734602	+	0.357711i	0.979988	+	0.199056i	$0.0637876\pi$
$709$	−0.195603	+	9.52479i	−0.00734602	+	0.357711i	−0.987334	+	0.158655i	$0.949284\pi$
$710$		0			0
$711$	−11.4038	−	52.0715i	−0.427677	−	1.95283i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$719$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$720$		0			0
$721$	15.7004	+	2.60249i	0.584712	+	0.0969218i
$722$		0			0
$723$	13.6220	+	51.9021i	0.506606	+	1.93026i
$724$	−30.1502	+	8.24484i	−1.12052	+	0.306417i
$725$		0			0
$726$		0			0
$727$	45.8211	+	24.6054i	1.69941	+	0.912564i	0.974702	+	0.223509i	$0.0717511\pi$
$727$	45.8211	+	24.6054i	1.69941	+	0.912564i	0.724709	+	0.689056i	$0.241974\pi$
$728$		0			0
$729$	22.9559	−	14.2137i	0.850217	−	0.526432i
$730$		0			0
$731$		0			0
$732$	37.2704	+	28.7509i	1.37755	+	1.06266i
$733$	49.5396	+	21.5707i	1.82979	+	0.796731i	0.940995	+	0.338420i	$0.109893\pi$
$733$	49.5396	+	21.5707i	1.82979	+	0.796731i	0.888792	+	0.458311i	$0.151545\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−6.53560	+	4.33064i	−0.240416	+	0.159305i	−0.666800	−	0.745237i	$-0.732336\pi$
$739$	−6.53560	+	4.33064i	−0.240416	+	0.159305i	0.426384	+	0.904542i	$0.359787\pi$
$740$		0			0
$741$	−68.9499	+	40.7577i	−2.53294	+	1.49727i
$742$		0			0
$743$		0			0		0.916855	−	0.399220i	$-0.130719\pi$
$743$		0			0		−0.916855	+	0.399220i	$0.869281\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	6.35389	−	0.0652354i	0.231857	−	0.00238047i	0.105731	−	0.994395i	$-0.466282\pi$
$751$	6.35389	−	0.0652354i	0.231857	−	0.00238047i	0.126126	+	0.992014i	$0.459746\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	44.4668	+	24.4696i	1.61724	+	0.889951i
$757$	32.8368	+	4.40881i	1.19347	+	0.160241i	0.702983	−	0.711207i	$-0.251851\pi$
$757$	32.8368	+	4.40881i	1.19347	+	0.160241i	0.490489	+	0.871447i	$0.336818\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.203893	−	0.978993i	$-0.434641\pi$
$761$		0			0		−0.203893	+	0.978993i	$0.565359\pi$
$762$		0			0
$763$	−45.0049	−	36.2153i	−1.62929	−	1.31108i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−21.0452	+	18.0305i	−0.759405	+	0.650618i
$769$	35.7845	−	36.9040i	1.29042	−	1.33079i	0.375684	−	0.926748i	$-0.377408\pi$
$769$	35.7845	−	36.9040i	1.29042	−	1.33079i	0.914739	−	0.404045i	$-0.132396\pi$
$770$		0			0
$771$		0			0
$772$	21.2740	−	26.4373i	0.765669	−	0.951499i
$773$		0			0		0.0820408	−	0.996629i	$-0.473856\pi$
$773$		0			0		−0.0820408	+	0.996629i	$0.526144\pi$
$774$		0			0
$775$	29.1866	−	45.0445i	1.04841	−	1.61805i
$776$		0			0
$777$	−47.3332	−	88.1456i	−1.69807	−	3.16221i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.692057	+	67.4060i	0.0247163	+	2.40736i
$785$		0			0
$786$		0			0
$787$	−13.6079	−	1.54334i	−0.485068	−	0.0550142i	−0.133249	−	0.991083i	$-0.542541\pi$
$787$	−13.6079	−	1.54334i	−0.485068	−	0.0550142i	−0.351818	+	0.936068i	$0.614437\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	82.1130	+	20.6522i	2.91592	+	0.733383i
$794$		0			0
$795$		0			0
$796$	−19.2872	−	29.1073i	−0.683616	−	1.03168i
$797$		0			0		0.725021	−	0.688727i	$-0.241830\pi$
$797$		0			0		−0.725021	+	0.688727i	$0.758170\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−9.50179	+	10.6404i	−0.335102	+	0.375256i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.810222	−	0.586123i	$-0.800654\pi$
$809$		0			0		0.810222	+	0.586123i	$0.199346\pi$
$810$		0			0
$811$	10.3328	+	37.7857i	0.362835	+	1.32684i	0.881631	+	0.471939i	$0.156446\pi$
$811$	10.3328	+	37.7857i	0.362835	+	1.32684i	−0.518796	+	0.854898i	$0.673620\pi$
$812$		0			0
$813$	23.5475	−	10.8336i	0.825847	−	0.379950i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−4.43718	−	43.0674i	−0.155237	−	1.50674i
$818$		0			0
$819$	90.9069	+	8.42376i	3.17654	+	0.294350i
$820$		0			0
$821$		0			0		0.855577	−	0.517676i	$-0.173203\pi$
$821$		0			0		−0.855577	+	0.517676i	$0.826797\pi$
$822$		0			0
$823$	−16.0295	−	25.3044i	−0.558754	−	0.882056i	0.441127	−	0.897445i	$-0.354579\pi$
$823$	−16.0295	−	25.3044i	−0.558754	−	0.882056i	−0.999882	+	0.0153881i	$0.995102\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.956004	−	0.293353i	$-0.0947712\pi$
$827$		0			0		−0.956004	+	0.293353i	$0.905229\pi$
$828$		0			0
$829$	44.9338	−	25.3310i	1.56061	−	0.879783i	0.562001	−	0.827136i	$-0.310032\pi$
$829$	44.9338	−	25.3310i	1.56061	−	0.879783i	0.998613	−	0.0526472i	$-0.0167659\pi$
$830$		0			0
$831$	−5.14834	+	15.1028i	−0.178594	+	0.523911i
$832$	−21.7603	+	44.8487i	−0.754403	+	1.55485i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−35.4137	+	43.0953i	−1.22408	+	1.48959i
$838$		0			0
$839$		0			0		0.839229	−	0.543778i	$-0.183007\pi$
$839$		0			0		−0.839229	+	0.543778i	$0.816993\pi$
$840$		0			0
$841$	−13.9813	−	25.4072i	−0.482114	−	0.876109i
$842$		0			0
$843$		0			0
$844$	−24.0989	−	3.74084i	−0.829518	−	0.128765i
$845$		0			0
$846$		0			0
$847$	31.4882	+	43.5274i	1.08195	+	1.49562i
$848$		0			0
$849$	39.5564	+	15.3242i	1.35757	+	0.525926i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−35.8946	−	44.6064i	−1.22901	−	1.52729i	−0.768809	−	0.639479i	$-0.779150\pi$
$853$	−35.8946	−	44.6064i	−1.22901	−	1.52729i	−0.460200	−	0.887815i	$-0.652222\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$857$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$858$		0			0
$859$	−3.43144	+	33.3057i	−0.117079	+	1.13638i	0.757952	+	0.652310i	$0.226200\pi$
$859$	−3.43144	+	33.3057i	−0.117079	+	1.13638i	−0.875032	+	0.484066i	$0.839160\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.322654	−	0.946517i	$-0.604575\pi$
$863$		0			0		0.322654	+	0.946517i	$0.395425\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	12.3068	−	26.7496i	0.417960	−	0.908465i
$868$	−103.262	−	18.2078i	−3.50493	−	0.618013i
$869$		0			0
$870$		0			0
$871$	−8.02957	+	24.3715i	−0.272072	+	0.825796i
$872$		0			0
$873$	3.53269	+	8.34630i	0.119563	+	0.282479i
$874$		0			0
$875$		0			0
$876$	21.7506	+	54.4776i	0.734884	+	1.84063i
$877$	−33.5975	+	45.4532i	−1.13451	+	1.53485i	−0.326963	+	0.945037i	$0.606025\pi$
$877$	−33.5975	+	45.4532i	−1.13451	+	1.53485i	−0.807543	+	0.589808i	$0.799203\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.936132	−	0.351649i	$-0.885621\pi$
$881$		0			0		0.936132	+	0.351649i	$0.114379\pi$
$882$		0			0
$883$	35.0512	+	46.4153i	1.17957	+	1.56200i	0.753708	+	0.657210i	$0.228263\pi$
$883$	35.0512	+	46.4153i	1.17957	+	1.56200i	0.425859	+	0.904789i	$0.359972\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.982973	−	0.183750i	$-0.941176\pi$
$887$		0			0		0.982973	+	0.183750i	$0.0588235\pi$
$888$		0			0
$889$	−21.1945	+	50.0740i	−0.710840	+	1.67943i
$890$		0			0
$891$		0			0
$892$	11.3183	+	6.69046i	0.378963	+	0.224013i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	19.9822	+	22.3766i	0.666073	+	0.745886i
$901$		0			0
$902$		0			0
$903$	−24.6750	+	42.7384i	−0.821133	+	1.42224i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−4.12494	+	57.2985i	−0.136967	+	1.90256i	0.231361	+	0.972868i	$0.425682\pi$
$907$	−4.12494	+	57.2985i	−0.136967	+	1.90256i	−0.368327	+	0.929696i	$0.620069\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.491083	−	0.871113i	$-0.336601\pi$
$911$		0			0		−0.491083	+	0.871113i	$0.663399\pi$
$912$	9.44773	+	50.5409i	0.312846	+	1.67358i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−11.7085	−	59.2602i	−0.386861	−	1.95801i
$917$		0			0
$918$		0			0
$919$	−30.3755	+	35.4544i	−1.00199	+	1.16953i	−0.0164935	+	0.999864i	$0.505250\pi$
$919$	−30.3755	+	35.4544i	−1.00199	+	1.16953i	−0.985501	+	0.169669i	$0.945730\pi$
$920$		0			0
$921$	7.04407	+	29.5192i	0.232110	+	0.972690i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−9.07118	−	58.4375i	−0.298259	−	1.92141i
$926$		0			0
$927$	−7.97895	−	5.64818i	−0.262063	−	0.185511i
$928$		0			0
$929$		0			0		0.526432	−	0.850217i	$-0.323529\pi$
$929$		0			0		−0.526432	+	0.850217i	$0.676471\pi$
$930$		0			0
$931$	108.947	+	61.4181i	3.57060	+	2.01290i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−56.5372	+	17.9858i	−1.84699	+	0.587570i	−0.848716	+	0.528849i	$0.822624\pi$
$937$	−56.5372	+	17.9858i	−1.84699	+	0.587570i	−0.998274	+	0.0587215i	$0.981298\pi$
$938$		0			0
$939$	−28.2385	−	16.3035i	−0.921528	−	0.532045i
$940$		0			0
$941$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$941$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.0513107	−	0.998683i	$-0.483660\pi$
$947$		0			0		−0.0513107	+	0.998683i	$0.516340\pi$
$948$	−59.2023	+	16.8445i	−1.92280	+	0.547085i
$949$	74.9920	+	74.2260i	2.43434	+	2.40948i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.920906	−	0.389786i	$-0.872549\pi$
$953$		0			0		0.920906	+	0.389786i	$0.127451\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	29.6210	−	78.8547i	0.955518	−	2.54370i
$962$		0			0
$963$		0			0
$964$	58.8494	−	19.3889i	1.89541	−	0.624474i
$965$		0			0
$966$		0			0
$967$	22.2878	+	8.11208i	0.716726	+	0.260867i	0.674535	−	0.738243i	$-0.264344\pi$
$967$	22.2878	+	8.11208i	0.716726	+	0.260867i	0.0421907	+	0.999110i	$0.486566\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.949779	−	0.312920i	$-0.898693\pi$
$971$		0			0		0.949779	+	0.312920i	$0.101307\pi$
$972$	−18.0132	−	25.4465i	−0.577774	−	0.816197i
$973$	44.7279	−	47.0849i	1.43391	−	1.50947i
$974$		0			0
$975$	49.0236	+	22.5544i	1.57001	+	0.722320i
$976$	30.4864	−	44.9983i	0.975845	−	1.44036i
$977$		0			0		−0.0820408	−	0.996629i	$-0.526144\pi$
$977$		0			0		0.0820408	+	0.996629i	$0.473856\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	15.1611	+	32.0820i	0.484056	+	1.02430i
$982$		0			0
$983$		0			0		−0.994734	−	0.102486i	$-0.967320\pi$
$983$		0			0		0.994734	+	0.102486i	$0.0326797\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	51.8750	+	76.5682i	1.65036	+	2.43596i
$989$		0			0
$990$		0			0
$991$	−48.0169	+	5.94578i	−1.52531	+	0.188874i	−0.841287	−	0.540589i	$-0.818201\pi$
$991$	−48.0169	+	5.94578i	−1.52531	+	0.188874i	−0.684020	+	0.729463i	$0.739770\pi$
$992$		0			0
$993$	4.48692	−	11.5821i	0.142388	−	0.367546i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−61.6415	−	3.80191i	−1.95220	−	0.120408i	−0.963248	−	0.268614i	$-0.913434\pi$
$997$	−61.6415	−	3.80191i	−1.95220	−	0.120408i	−0.988956	+	0.148206i	$0.952650\pi$
$998$		0			0
$999$	2.52313	+	61.4056i	0.0798284	+	1.94279i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	921.2.x.a.47.1	✓	96
3.2	odd	2	CM	921.2.x.a.47.1	✓	96
307.98	odd	306	inner	921.2.x.a.98.1	yes	96
921.98	even	306	inner	921.2.x.a.98.1	yes	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
921.2.x.a.47.1	✓	96	1.1	even	1	trivial
921.2.x.a.47.1	✓	96	3.2	odd	2	CM
921.2.x.a.98.1	yes	96	307.98	odd	306	inner
921.2.x.a.98.1	yes	96	921.98	even	306	inner

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:	\( N \)	\(=\)	\( 921 = 3 \cdot 307 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	921.x (of order \(306\), degree \(96\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.72466 + 0.159813i) q^{3} +(0.306783 + 1.97633i) q^{4} +(-3.92744 + 2.90303i) q^{7} +(2.94892 - 0.551249i) q^{9} +O(q^{10})\)	\(q+(-1.72466 + 0.159813i) q^{3} +(0.306783 + 1.97633i) q^{4} +(-3.92744 + 2.90303i) q^{7} +(2.94892 - 0.551249i) q^{9} +(-0.844942 - 3.35948i) q^{12} +(-3.80594 - 4.93373i) q^{13} +(-3.81177 + 1.21261i) q^{16} +(-1.58766 + 7.24950i) q^{19} +(6.30956 - 5.63441i) q^{21} +(4.62429 - 1.90155i) q^{25} +(-4.99779 + 1.42199i) q^{27} +(-6.94223 - 6.87132i) q^{28} +(8.82487 - 6.11197i) q^{31} +(1.99413 + 5.65893i) q^{36} +(2.76697 - 11.4993i) q^{37} +(7.35244 + 7.90078i) q^{39} +(-5.54094 + 1.82555i) q^{43} +(6.38022 - 2.70052i) q^{48} +(4.94370 - 16.1109i) q^{49} +(8.58308 - 9.03539i) q^{52} +(1.57962 - 12.7567i) q^{57} +(-10.6733 + 8.40969i) q^{61} +(-9.98141 + 10.7258i) q^{63} +(-3.56591 - 7.16131i) q^{64} +(-2.30980 - 3.40929i) q^{67} +(-16.7829 + 2.25334i) q^{73} +(-7.67145 + 4.01856i) q^{75} +(-14.8145 - 0.913725i) q^{76} +(-0.547183 - 17.7601i) q^{79} +(8.39225 - 3.25117i) q^{81} +(13.0711 + 10.7412i) q^{84} +(29.2704 + 8.32814i) q^{91} +(-14.2431 + 11.9514i) q^{93} +(0.646302 + 2.95111i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6 q^{4} + 18 q^{9}+O(q^{10}) \) 96 * q - 6 * q^4 + 18 * q^9	\( 96 q - 6 q^{4} + 18 q^{9} + 18 q^{12} + 12 q^{16} + 21 q^{31} + 18 q^{36} - 39 q^{43} + 36 q^{48} + 39 q^{61} + 48 q^{64} - 48 q^{67} + 51 q^{73} - 54 q^{81} + 45 q^{93}+O(q^{100}) \) 96 * q - 6 * q^4 + 18 * q^9 + 18 * q^12 + 12 * q^16 + 21 * q^31 + 18 * q^36 - 39 * q^43 + 36 * q^48 + 39 * q^61 + 48 * q^64 - 48 * q^67 + 51 * q^73 - 54 * q^81 + 45 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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