LMFDB - Embedded newform 927.2.ba.a.532.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [927,2,Mod(19,927)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(927, base_ring=CyclotomicField(102))

chi = DirichletCharacter(H, H._module([0, 80]))

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

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

chi := DirichletCharacter("927.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 927 = 3^{2} \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	927.ba (of order $51$, degree $32$, 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:	$7.40213226737$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{51}]$

Embedding invariants

Embedding label			532.1
Character	$\chi$	$=$	927.532
Dual form			927.2.ba.a.406.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.664710 - 1.88631i) q^{4} +(-1.75994 + 1.81500i) q^{7} +O(q^{10})$	$q+(-0.664710 - 1.88631i) q^{4} +(-1.75994 + 1.81500i) q^{7} +(1.60631 + 5.64559i) q^{13} +(-3.11632 + 2.50770i) q^{16} +(7.40586 - 2.35597i) q^{19} +(1.94893 + 4.60453i) q^{25} +(4.59351 + 2.11335i) q^{28} +(1.29845 + 0.503023i) q^{31} +(6.80861 - 4.21571i) q^{37} +(5.68433 - 3.05242i) q^{43} +(0.0187345 + 0.608070i) q^{49} +(9.58159 - 6.78267i) q^{52} +(-6.35891 + 8.42056i) q^{61} +(6.80174 + 4.21146i) q^{64} +(1.91344 - 0.481249i) q^{67} +(-0.399429 + 0.802161i) q^{73} +(-9.36684 - 12.4037i) q^{76} +(6.78375 + 13.6236i) q^{79} +(-13.0738 - 7.02046i) q^{91} +(2.60090 - 0.322061i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 2 q^{4} + 5 q^{7}+O(q^{10}) $ 32 * q - 2 * q^4 + 5 * q^7	$ 32 q - 2 q^{4} + 5 q^{7} - 10 q^{13} + 4 q^{16} - 7 q^{19} - 5 q^{25} - 10 q^{28} + 14 q^{31} - 22 q^{37} - 13 q^{43} + 18 q^{49} - 10 q^{52} - 28 q^{61} + 16 q^{64} - 16 q^{67} + 14 q^{73} - 28 q^{76} + 26 q^{79} + 212 q^{91} + 99 q^{97}+O(q^{100}) $ 32 * q - 2 * q^4 + 5 * q^7 - 10 * q^13 + 4 * q^16 - 7 * q^19 - 5 * q^25 - 10 * q^28 + 14 * q^31 - 22 * q^37 - 13 * q^43 + 18 * q^49 - 10 * q^52 - 28 * q^61 + 16 * q^64 - 16 * q^67 + 14 * q^73 - 28 * q^76 + 26 * q^79 + 212 * q^91 + 99 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/927\mathbb{Z}\right)^\times$.

$n$	$722$	$829$
$\chi(n)$	$1$	$e\left(\frac{35}{51}\right)$

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		0.577774	−	0.816197i	$-0.303922\pi$
$2$		0			0		−0.577774	+	0.816197i	$0.696078\pi$
$3$		0			0
$3$		0			0
$4$	−0.664710	−	1.88631i	−0.332355	−	0.943154i
$5$		0			0		−0.833602	−	0.552365i	$-0.813725\pi$
$5$		0			0		0.833602	+	0.552365i	$0.186275\pi$
$6$		0			0
$7$	−1.75994	+	1.81500i	−0.665197	+	0.686007i	−0.963270	−	0.268536i	$-0.913460\pi$
$7$	−1.75994	+	1.81500i	−0.665197	+	0.686007i	0.298073	+	0.954543i	$0.403656\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.417960	−	0.908465i	$-0.637255\pi$
$11$		0			0		0.417960	+	0.908465i	$0.362745\pi$
$12$		0			0
$13$	1.60631	+	5.64559i	0.445510	+	1.56580i	0.781479	+	0.623931i	$0.214465\pi$
$13$	1.60631	+	5.64559i	0.445510	+	1.56580i	−0.335970	+	0.941873i	$0.609064\pi$
$14$		0			0
$15$		0			0
$16$	−3.11632	+	2.50770i	−0.779081	+	0.626924i
$17$		0			0		0.122888	−	0.992421i	$-0.460784\pi$
$17$		0			0		−0.122888	+	0.992421i	$0.539216\pi$
$18$		0			0
$19$	7.40586	−	2.35597i	1.69902	−	0.540497i	0.712152	−	0.702025i	$-0.247721\pi$
$19$	7.40586	−	2.35597i	1.69902	−	0.540497i	0.986868	+	0.161528i	$0.0516421\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$23$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$24$		0			0
$25$	1.94893	+	4.60453i	0.389786	+	0.920906i
$26$		0			0
$27$		0			0
$28$	4.59351	+	2.11335i	0.868091	+	0.399385i
$29$		0			0		0.0615609	−	0.998103i	$-0.480392\pi$
$29$		0			0		−0.0615609	+	0.998103i	$0.519608\pi$
$30$		0			0
$31$	1.29845	+	0.503023i	0.233209	+	0.0903456i	0.475011	−	0.879980i	$-0.342444\pi$
$31$	1.29845	+	0.503023i	0.233209	+	0.0903456i	−0.241802	+	0.970326i	$0.577738\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.80861	−	4.21571i	1.11933	−	0.693059i	0.163371	−	0.986565i	$-0.447763\pi$
$37$	6.80861	−	4.21571i	1.11933	−	0.693059i	0.955957	+	0.293506i	$0.0948219\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.833602	−	0.552365i	$-0.186275\pi$
$41$		0			0		−0.833602	+	0.552365i	$0.813725\pi$
$42$		0			0
$43$	5.68433	−	3.05242i	0.866852	−	0.465490i	0.0215434	−	0.999768i	$-0.493142\pi$
$43$	5.68433	−	3.05242i	0.866852	−	0.465490i	0.845309	+	0.534278i	$0.179417\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	0.0187345	+	0.608070i	0.00267635	+	0.0868672i
$50$		0			0
$51$		0			0
$52$	9.58159	−	6.78267i	1.32873	−	0.940587i
$53$		0			0		−0.976848	−	0.213933i	$-0.931373\pi$
$53$		0			0		0.976848	+	0.213933i	$0.0686275\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.717912	−	0.696134i	$-0.245098\pi$
$59$		0			0		−0.717912	+	0.696134i	$0.754902\pi$
$60$		0			0
$61$	−6.35891	+	8.42056i	−0.814175	+	1.07814i	0.181328	+	0.983423i	$0.441960\pi$
$61$	−6.35891	+	8.42056i	−0.814175	+	1.07814i	−0.995504	+	0.0947200i	$0.969804\pi$
$62$		0			0
$63$		0			0
$64$	6.80174	+	4.21146i	0.850217	+	0.526432i
$65$		0			0
$66$		0			0
$67$	1.91344	−	0.481249i	0.233764	−	0.0587940i	−0.125254	−	0.992125i	$-0.539975\pi$
$67$	1.91344	−	0.481249i	0.233764	−	0.0587940i	0.359018	+	0.933331i	$0.383112\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.0615609	−	0.998103i	$-0.519608\pi$
$71$		0			0		0.0615609	+	0.998103i	$0.480392\pi$
$72$		0			0
$73$	−0.399429	+	0.802161i	−0.0467496	+	0.0938858i	−0.917306	−	0.398182i	$-0.869641\pi$
$73$	−0.399429	+	0.802161i	−0.0467496	+	0.0938858i	0.870557	+	0.492068i	$0.163759\pi$
$74$		0			0
$75$		0			0
$76$	−9.36684	−	12.4037i	−1.07445	−	1.42280i
$77$		0			0
$78$		0			0
$79$	6.78375	+	13.6236i	0.763232	+	1.53278i	0.844181	+	0.536058i	$0.180087\pi$
$79$	6.78375	+	13.6236i	0.763232	+	1.53278i	−0.0809495	+	0.996718i	$0.525795\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.243914	−	0.969797i	$-0.421569\pi$
$83$		0			0		−0.243914	+	0.969797i	$0.578431\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$89$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$90$		0			0
$91$	−13.0738	−	7.02046i	−1.37050	−	0.735945i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	2.60090	−	0.322061i	0.264082	−	0.0327004i	0.0102451	−	0.999948i	$-0.496739\pi$
$97$	2.60090	−	0.322061i	0.264082	−	0.0327004i	0.253837	+	0.967247i	$0.418307\pi$
$98$		0			0
$99$		0			0
$100$	7.39009	−	6.73696i	0.739009	−	0.673696i
$101$		0			0		0.577774	−	0.816197i	$-0.303922\pi$
$101$		0			0		−0.577774	+	0.816197i	$0.696078\pi$
$102$		0			0
$103$	9.62954	+	3.20498i	0.948827	+	0.315796i
$103$	9.62954	+	3.20498i	0.948827	+	0.315796i
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.943154	−	0.332355i	$-0.107843\pi$
$107$		0			0		−0.943154	+	0.332355i	$0.892157\pi$
$108$		0			0
$109$	−20.6874	+	2.56166i	−1.98150	+	0.245362i	−0.983616	+	0.180279i	$0.942300\pi$
$109$	−20.6874	+	2.56166i	−1.98150	+	0.245362i	−0.997880	+	0.0650837i	$0.979269\pi$
$110$		0			0
$111$		0			0
$112$	0.933080	−	10.0695i	0.0881678	−	0.951482i
$113$		0			0		−0.361242	−	0.932472i	$-0.617647\pi$
$113$		0			0		0.361242	+	0.932472i	$0.382353\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.15680	+	8.35345i	−0.650618	+	0.759405i
$122$		0			0
$123$		0			0
$124$	0.0857632	−	2.78365i	0.00770177	−	0.249979i
$125$		0			0
$126$		0			0
$127$	9.95260	+	19.9875i	0.883150	+	1.77360i	0.563636	+	0.826023i	$0.309402\pi$
$127$	9.95260	+	19.9875i	0.883150	+	1.77360i	0.319514	+	0.947581i	$0.396480\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.417960	−	0.908465i	$-0.362745\pi$
$131$		0			0		−0.417960	+	0.908465i	$0.637255\pi$
$132$		0			0
$133$	−8.75780	+	17.5880i	−0.759398	+	1.52508i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$137$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$138$		0			0
$139$	−16.3116	+	4.10252i	−1.38353	+	0.347972i	−0.862712	−	0.505696i	$-0.831236\pi$
$139$	−16.3116	+	4.10252i	−1.38353	+	0.347972i	−0.520818	+	0.853668i	$0.674373\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−12.4779	−	10.0409i	−1.02568	−	0.825358i
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	−2.76171	+	17.7913i	−0.224745	+	1.44783i	0.560247	+	0.828326i	$0.310706\pi$
$151$	−2.76171	+	17.7913i	−0.224745	+	1.44783i	−0.784992	+	0.619506i	$0.787333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.765133	−	24.8342i	−0.0610642	−	1.98198i	−0.160869	−	0.986976i	$-0.551430\pi$
$157$	−0.765133	−	24.8342i	−0.0610642	−	1.98198i	0.0998050	−	0.995007i	$-0.468178\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	18.0959	−	9.71727i	1.41738	−	0.761115i	0.427631	−	0.903954i	$-0.359348\pi$
$163$	18.0959	−	9.71727i	1.41738	−	0.761115i	0.989746	+	0.142838i	$0.0456229\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.526432	−	0.850217i	$-0.323529\pi$
$167$		0			0		−0.526432	+	0.850217i	$0.676471\pi$
$168$		0			0
$169$	−18.2396	+	11.2935i	−1.40305	+	0.868729i
$170$		0			0
$171$		0			0
$172$	−9.53624	−	8.69343i	−0.727131	−	0.662868i
$173$		0			0		−0.920906	−	0.389786i	$-0.872549\pi$
$173$		0			0		0.920906	+	0.389786i	$0.127451\pi$
$174$		0			0
$175$	−11.7872	−	4.56640i	−0.891032	−	0.345187i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$179$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$180$		0			0
$181$	−8.47454	−	20.0219i	−0.629907	−	1.48822i	−0.857513	−	0.514462i	$-0.827992\pi$
$181$	−8.47454	−	20.0219i	−0.629907	−	1.48822i	0.227606	−	0.973753i	$-0.426910\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.988165	−	0.153392i	$-0.0490196\pi$
$191$		0			0		−0.988165	+	0.153392i	$0.950980\pi$
$192$		0			0
$193$	−5.72230	−	20.1118i	−0.411900	−	1.44768i	−0.839326	−	0.543629i	$-0.817050\pi$
$193$	−5.72230	−	20.1118i	−0.411900	−	1.44768i	0.427425	−	0.904051i	$-0.359421\pi$
$194$		0			0
$195$		0			0
$196$	1.13456	−	0.439529i	0.0810397	−	0.0313949i
$197$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$197$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$198$		0			0
$199$	19.0477	−	19.6436i	1.35025	−	1.39250i	0.500093	−	0.865972i	$-0.333299\pi$
$199$	19.0477	−	19.6436i	1.35025	−	1.39250i	0.850161	−	0.526523i	$-0.176505\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−19.1632	−	13.5653i	−1.32873	−	0.940587i
$209$		0			0
$210$		0			0
$211$	11.1485	−	16.8248i	0.767496	−	1.15827i	−0.215832	−	0.976430i	$-0.569247\pi$
$211$	11.1485	−	16.8248i	0.767496	−	1.15827i	0.983329	−	0.181837i	$-0.0582045\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.19819	+	1.47140i	−0.217107	+	0.0998853i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−12.8335	−	1.58913i	−0.859393	−	0.106416i	−0.318924	−	0.947780i	$-0.603322\pi$
$223$	−12.8335	−	1.58913i	−0.859393	−	0.106416i	−0.540469	+	0.841364i	$0.681753\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.999526	−	0.0307951i	$-0.990196\pi$
$227$		0			0		0.999526	+	0.0307951i	$0.00980392\pi$
$228$		0			0
$229$	1.97936	+	21.3607i	0.130800	+	1.41156i	0.767798	+	0.640692i	$0.221352\pi$
$229$	1.97936	+	21.3607i	0.130800	+	1.41156i	−0.636998	+	0.770865i	$0.719824\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$233$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.920906	−	0.389786i	$-0.872549\pi$
$239$		0			0		0.920906	+	0.389786i	$0.127451\pi$
$240$		0			0
$241$	−5.19887	−	23.7388i	−0.334888	−	1.52915i	−0.775441	−	0.631420i	$-0.782472\pi$
$241$	−5.19887	−	23.7388i	−0.334888	−	1.52915i	0.440552	−	0.897727i	$-0.354783\pi$
$242$		0			0
$243$		0			0
$244$	20.1106	+	6.39765i	1.28745	+	0.409567i
$245$		0			0
$246$		0			0
$247$	25.1969	+	38.0260i	1.60324	+	2.41954i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.626924	−	0.779081i	$-0.284314\pi$
$251$		0			0		−0.626924	+	0.779081i	$0.715686\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	3.42293	−	15.6296i	0.213933	−	0.976848i
$257$		0			0		−0.577774	−	0.816197i	$-0.696078\pi$
$257$		0			0		0.577774	+	0.816197i	$0.303922\pi$
$258$		0			0
$259$	−4.33125	+	19.7771i	−0.269131	+	1.22889i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−2.17967	−	3.28945i	−0.133144	−	0.200935i
$269$		0			0		0.759405	−	0.650618i	$-0.225490\pi$
$269$		0			0		−0.759405	+	0.650618i	$0.774510\pi$
$270$		0			0
$271$	−30.4240	−	9.67856i	−1.84812	−	0.587931i	−0.998129	−	0.0611393i	$-0.980527\pi$
$271$	−30.4240	−	9.67856i	−1.84812	−	0.587931i	−0.849995	−	0.526791i	$-0.823395\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−31.6275	+	1.95072i	−1.90031	+	0.117207i	−0.969007	−	0.247034i	$-0.920544\pi$
$277$	−31.6275	+	1.95072i	−1.90031	+	0.117207i	−0.931305	+	0.364241i	$0.881328\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.417960	−	0.908465i	$-0.362745\pi$
$281$		0			0		−0.417960	+	0.908465i	$0.637255\pi$
$282$		0			0
$283$	4.29662	−	12.1929i	0.255408	−	0.724794i	−0.743048	−	0.669238i	$-0.766621\pi$
$283$	4.29662	−	12.1929i	0.255408	−	0.724794i	0.998456	−	0.0555561i	$-0.0176932\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.4865	−	4.14653i	−0.969797	−	0.243914i
$290$		0			0
$291$		0			0
$292$	1.77863	+	0.220242i	0.104086	+	0.0128887i
$293$		0			0		−0.626924	−	0.779081i	$-0.715686\pi$
$293$		0			0		0.626924	+	0.779081i	$0.284314\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−4.46395	+	15.6892i	−0.257298	+	0.904308i
$302$		0			0
$303$		0			0
$304$	−17.1710	+	25.9136i	−0.984823	+	1.48625i
$305$		0			0
$306$		0			0
$307$	24.4990	+	17.3425i	1.39823	+	0.989787i	0.997281	+	0.0736860i	$0.0234763\pi$
$307$	24.4990	+	17.3425i	1.39823	+	0.989787i	0.400949	+	0.916101i	$0.368681\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.577774	−	0.816197i	$-0.303922\pi$
$311$		0			0		−0.577774	+	0.816197i	$0.696078\pi$
$312$		0			0
$313$	−4.84440	−	13.7474i	−0.273822	−	0.777050i	−0.996323	−	0.0856799i	$-0.972694\pi$
$313$	−4.84440	−	13.7474i	−0.273822	−	0.777050i	0.722501	−	0.691370i	$-0.242993\pi$
$314$		0			0
$315$		0			0
$316$	21.1891	−	21.8520i	1.19198	−	1.22927i
$317$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$317$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−22.8647	+	18.3991i	−1.26830	+	1.02060i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−7.76754	−	1.45201i	−0.426943	−	0.0798095i	−0.0341073	−	0.999418i	$-0.510859\pi$
$331$	−7.76754	−	1.45201i	−0.426943	−	0.0798095i	−0.392836	+	0.919609i	$0.628506\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	32.4836	+	14.9448i	1.76949	+	0.814097i	0.979629	+	0.200816i	$0.0643594\pi$
$337$	32.4836	+	14.9448i	1.76949	+	0.814097i	0.789865	+	0.613280i	$0.210150\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−14.2150	−	12.9587i	−0.767538	−	0.699703i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.303153	−	0.952942i	$-0.401961\pi$
$347$		0			0		−0.303153	+	0.952942i	$0.598039\pi$
$348$		0			0
$349$	22.2469	+	25.9667i	1.19085	+	1.38996i	0.901522	+	0.432734i	$0.142451\pi$
$349$	22.2469	+	25.9667i	1.19085	+	1.38996i	0.289327	+	0.957230i	$0.406568\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.717912	−	0.696134i	$-0.245098\pi$
$353$		0			0		−0.717912	+	0.696134i	$0.754902\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.976848	−	0.213933i	$-0.931373\pi$
$359$		0			0		0.976848	+	0.213933i	$0.0686275\pi$
$360$		0			0
$361$	33.7884	−	23.9183i	1.77834	−	1.25886i
$362$		0			0
$363$		0			0
$364$	−4.55251	+	29.3277i	−0.238616	+	1.53719i
$365$		0			0
$366$		0			0
$367$	−11.2992	−	9.09242i	−0.589813	−	0.474621i	0.285367	−	0.958418i	$-0.407885\pi$
$367$	−11.2992	−	9.09242i	−0.589813	−	0.474621i	−0.875180	+	0.483798i	$0.839257\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	30.5700	+	18.9281i	1.58285	+	0.980061i	0.983783	+	0.179364i	$0.0574041\pi$
$373$	30.5700	+	18.9281i	1.58285	+	0.980061i	0.599070	+	0.800697i	$0.295537\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	7.47422	−	17.6586i	0.383925	−	0.907059i	−0.609522	−	0.792769i	$-0.708639\pi$
$379$	7.47422	−	17.6586i	0.383925	−	0.907059i	0.993447	−	0.114290i	$-0.0364593\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.0615609	−	0.998103i	$-0.480392\pi$
$383$		0			0		−0.0615609	+	0.998103i	$0.519608\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−2.33635	−	4.69203i	−0.118610	−	0.238202i
$389$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$389$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.70630	+	10.9922i	0.0856367	+	0.551681i	0.991704	+	0.128546i	$0.0410310\pi$
$397$	1.70630	+	10.9922i	0.0856367	+	0.551681i	−0.906067	+	0.423135i	$0.860930\pi$
$398$		0			0
$399$		0			0
$400$	−17.6202	−	9.46187i	−0.881012	−	0.473094i
$401$		0			0		−0.417960	−	0.908465i	$-0.637255\pi$
$401$		0			0		0.417960	+	0.908465i	$0.362745\pi$
$402$		0			0
$403$	−0.754146	+	8.13853i	−0.0375667	+	0.405409i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−12.1976	+	11.1196i	−0.603135	+	0.549830i	−0.916618	−	0.399764i	$-0.869092\pi$
$409$	−12.1976	+	11.1196i	−0.603135	+	0.549830i	0.313483	+	0.949594i	$0.398504\pi$
$410$		0			0
$411$		0			0
$412$	−0.355273	−	20.2947i	−0.0175030	−	0.999847i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.717912	−	0.696134i	$-0.754902\pi$
$419$		0			0		0.717912	+	0.696134i	$0.245098\pi$
$420$		0			0
$421$	1.95638	−	21.1127i	0.0953480	−	1.02897i	−0.804847	−	0.593482i	$-0.797753\pi$
$421$	1.95638	−	21.1127i	0.0953480	−	1.02897i	0.900195	−	0.435487i	$-0.143424\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−4.09201	−	26.3612i	−0.198026	−	1.27571i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.303153	−	0.952942i	$-0.598039\pi$
$431$		0			0		0.303153	+	0.952942i	$0.401961\pi$
$432$		0			0
$433$	−0.174681	+	5.66969i	−0.00839464	+	0.272468i	0.986001	+	0.166742i	$0.0533246\pi$
$433$	−0.174681	+	5.66969i	−0.00839464	+	0.272468i	−0.994395	+	0.105726i	$0.966283\pi$
$434$		0			0
$435$		0			0
$436$	18.5832	+	37.3201i	0.889974	+	1.78731i
$437$		0			0
$438$		0			0
$439$	4.82466	+	6.38888i	0.230268	+	0.304925i	0.898452	−	0.439072i	$-0.144693\pi$
$439$	4.82466	+	6.38888i	0.230268	+	0.304925i	−0.668184	+	0.743996i	$0.732928\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$443$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−19.6145	+	4.93324i	−0.926697	+	0.233074i
$449$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$449$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.39365	+	1.92616i	0.111970	+	0.0901020i	0.681288	−	0.732015i	$-0.261420\pi$
$457$	2.39365	+	1.92616i	0.111970	+	0.0901020i	−0.569318	+	0.822117i	$0.692793\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.999526	−	0.0307951i	$-0.00980392\pi$
$461$		0			0		−0.999526	+	0.0307951i	$0.990196\pi$
$462$		0			0
$463$	34.2391	−	24.2374i	1.59123	−	1.12641i	0.666676	−	0.745348i	$-0.267717\pi$
$463$	34.2391	−	24.2374i	1.59123	−	1.12641i	0.924551	−	0.381058i	$-0.124440\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.988165	−	0.153392i	$-0.950980\pi$
$467$		0			0		0.988165	+	0.153392i	$0.0490196\pi$
$468$		0			0
$469$	−2.49408	+	4.31987i	−0.115166	+	0.199473i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	25.2816	+	29.5089i	1.16000	+	1.35396i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.243914	−	0.969797i	$-0.578431\pi$
$479$		0			0		0.243914	+	0.969797i	$0.421569\pi$
$480$		0			0
$481$	34.7369	+	31.6669i	1.58387	+	1.44388i
$482$		0			0
$483$		0			0
$484$	20.5144	+	7.94732i	0.932472	+	0.361242i
$485$		0			0
$486$		0			0
$487$	32.3286	+	14.8735i	1.46495	+	0.673983i	0.979714	−	0.200401i	$-0.0642245\pi$
$487$	32.3286	+	14.8735i	1.46495	+	0.673983i	0.485234	+	0.874384i	$0.338734\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$491$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−5.30782	+	1.68854i	−0.238328	+	0.0758177i
$497$		0			0
$498$		0			0
$499$	−0.0825766	+	0.0664492i	−0.00369664	+	0.00297467i	−0.628770	−	0.777591i	$-0.716441\pi$
$499$	−0.0825766	+	0.0664492i	−0.00369664	+	0.00297467i	0.625074	+	0.780566i	$0.285069\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.473094	−	0.881012i	$-0.343137\pi$
$503$		0			0		−0.473094	+	0.881012i	$0.656863\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	31.0870	−	32.0596i	1.37926	−	1.42241i
$509$		0			0		−0.122888	−	0.992421i	$-0.539216\pi$
$509$		0			0		0.122888	+	0.992421i	$0.460784\pi$
$510$		0			0
$511$	−0.752952	−	2.13672i	−0.0333086	−	0.0945230i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.122888	−	0.992421i	$-0.539216\pi$
$521$		0			0		0.122888	+	0.992421i	$0.460784\pi$
$522$		0			0
$523$	10.0561	−	35.3435i	0.439722	−	1.54546i	−0.352847	−	0.935681i	$-0.614786\pi$
$523$	10.0561	−	35.3435i	0.439722	−	1.54546i	0.792569	−	0.609782i	$-0.208743\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	22.6084	−	4.22624i	0.982973	−	0.183750i
$530$		0			0
$531$		0			0
$532$	38.9979	+	4.82898i	1.69077	+	0.209363i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−15.3882	+	43.6686i	−0.661591	+	1.87746i	−0.239537	+	0.970887i	$0.576996\pi$
$541$	−15.3882	+	43.6686i	−0.661591	+	1.87746i	−0.422053	+	0.906571i	$0.638691\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	23.7235	−	1.46322i	1.01434	−	0.0625626i	0.454151	−	0.890925i	$-0.349943\pi$
$547$	23.7235	−	1.46322i	1.01434	−	0.0625626i	0.560193	+	0.828362i	$0.310727\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−36.6659	−	11.6643i	−1.55919	−	0.496015i
$554$		0			0
$555$		0			0
$556$	18.5811	+	28.0417i	0.788014	+	1.18923i
$557$		0			0		−0.798017	−	0.602635i	$-0.794118\pi$
$557$		0			0		0.798017	+	0.602635i	$0.205882\pi$
$558$		0			0
$559$	26.3635	+	27.1882i	1.11506	+	1.14994i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.988165	−	0.153392i	$-0.950980\pi$
$563$		0			0		0.988165	+	0.153392i	$0.0490196\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.999526	−	0.0307951i	$-0.00980392\pi$
$569$		0			0		−0.999526	+	0.0307951i	$0.990196\pi$
$570$		0			0
$571$	−21.6825	−	37.5551i	−0.907383	−	1.57163i	−0.817686	−	0.575664i	$-0.804744\pi$
$571$	−21.6825	−	37.5551i	−0.907383	−	1.57163i	−0.0896964	−	0.995969i	$-0.528590\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−23.9336	−	36.1195i	−0.996371	−	1.50367i	−0.858232	−	0.513262i	$-0.828437\pi$
$577$	−23.9336	−	36.1195i	−0.996371	−	1.50367i	−0.138139	−	0.990413i	$-0.544112\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$587$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$588$		0			0
$589$	10.8013	+	0.666199i	0.445058	+	0.0274503i
$590$		0			0
$591$		0			0
$592$	−10.6461	+	30.2114i	−0.437552	+	1.24168i
$593$		0			0		0.920906	−	0.389786i	$-0.127451\pi$
$593$		0			0		−0.920906	+	0.389786i	$0.872549\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.303153	−	0.952942i	$-0.598039\pi$
$599$		0			0		0.303153	+	0.952942i	$0.401961\pi$
$600$		0			0
$601$	−47.5791	−	5.89157i	−1.94079	−	0.240322i	−0.944651	−	0.328076i	$-0.893600\pi$
$601$	−47.5791	−	5.89157i	−1.94079	−	0.240322i	−0.996142	+	0.0877534i	$0.972031\pi$
$602$		0			0
$603$		0			0
$604$	35.3955	−	6.61657i	1.44022	−	0.269224i
$605$		0			0
$606$		0			0
$607$	−11.4929	+	5.28755i	−0.466481	+	0.214615i	−0.637191	−	0.770706i	$-0.719904\pi$
$607$	−11.4929	+	5.28755i	−0.466481	+	0.214615i	0.170710	+	0.985321i	$0.445394\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	16.7494	−	25.2773i	0.676500	−	1.02094i	−0.320774	−	0.947156i	$-0.603943\pi$
$613$	16.7494	−	25.2773i	0.676500	−	1.02094i	0.997274	−	0.0737856i	$-0.0235081\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	−24.4533			−0.982863			−0.491431	−	0.870916i	$-0.663526\pi$
$619$	−24.4533			−0.982863			−0.491431	+	0.870916i	$0.663526\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−17.4033	+	17.9478i	−0.696134	+	0.717912i
$626$		0			0
$627$		0			0
$628$	−46.3363	+	17.9508i	−1.84902	+	0.716314i
$629$		0			0
$630$		0			0
$631$	−12.1829	−	42.8184i	−0.484993	−	1.70457i	−0.688272	−	0.725453i	$-0.741630\pi$
$631$	−12.1829	−	42.8184i	−0.484993	−	1.70457i	0.203279	−	0.979121i	$-0.434840\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−3.40282	+	1.08252i	−0.134825	+	0.0428908i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$641$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$642$		0			0
$643$	18.9285	+	44.7204i	0.746467	+	1.76360i	0.638746	+	0.769418i	$0.279453\pi$
$643$	18.9285	+	44.7204i	0.746467	+	1.76360i	0.107721	+	0.994181i	$0.465645\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.0615609	−	0.998103i	$-0.480392\pi$
$647$		0			0		−0.0615609	+	0.998103i	$0.519608\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−30.3583	−	27.6752i	−1.18892	−	1.08384i
$653$		0			0		0.976848	−	0.213933i	$-0.0686275\pi$
$653$		0			0		−0.976848	+	0.213933i	$0.931373\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.833602	−	0.552365i	$-0.186275\pi$
$659$		0			0		−0.833602	+	0.552365i	$0.813725\pi$
$660$		0			0
$661$	−1.39470	+	0.748937i	−0.0542474	+	0.0291303i	−0.499993	−	0.866029i	$-0.666664\pi$
$661$	−1.39470	+	0.748937i	−0.0542474	+	0.0291303i	0.445746	+	0.895160i	$0.352939\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.51600	+	9.76621i	−0.0584374	+	0.376460i	0.940854	+	0.338813i	$0.110025\pi$
$673$	−1.51600	+	9.76621i	−0.0584374	+	0.376460i	−0.999291	+	0.0376467i	$0.988014\pi$
$674$		0			0
$675$		0			0
$676$	33.4270	+	26.8986i	1.28565	+	1.03456i
$677$		0			0		0.717912	−	0.696134i	$-0.245098\pi$
$677$		0			0		−0.717912	+	0.696134i	$0.754902\pi$
$678$		0			0
$679$	−3.99290	+	5.28746i	−0.153233	+	0.202914i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.303153	−	0.952942i	$-0.401961\pi$
$683$		0			0		−0.303153	+	0.952942i	$0.598039\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−10.0597	+	23.7669i	−0.383521	+	0.906104i
$689$		0			0
$690$		0			0
$691$	0.692633	−	1.39100i	0.0263490	−	0.0529160i	−0.881598	−	0.472002i	$-0.843532\pi$
$691$	0.692633	−	1.39100i	0.0263490	−	0.0529160i	0.907947	+	0.419086i	$0.137649\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.778551	+	25.2697i	−0.0294265	+	0.955105i
$701$		0			0		0.243914	−	0.969797i	$-0.421569\pi$
$701$		0			0		−0.243914	+	0.969797i	$0.578431\pi$
$702$		0			0
$703$	40.4915	−	47.2619i	1.52717	−	1.78252i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	11.2012	+	6.01493i	0.420670	+	0.225895i	0.669748	−	0.742588i	$-0.266402\pi$
$709$	11.2012	+	6.01493i	0.420670	+	0.225895i	−0.249078	+	0.968483i	$0.580127\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.577774	−	0.816197i	$-0.303922\pi$
$719$		0			0		−0.577774	+	0.816197i	$0.696078\pi$
$720$		0			0
$721$	−22.7645	+	11.8371i	−0.847795	+	0.440835i
$722$		0			0
$723$		0			0
$724$	−32.1344	+	29.2943i	−1.19426	+	1.08872i
$725$		0			0
$726$		0			0
$727$	−50.9914	+	6.31410i	−1.89117	+	0.234177i	−0.982890	−	0.184195i	$-0.941032\pi$
$727$	−50.9914	+	6.31410i	−1.89117	+	0.234177i	−0.908276	+	0.418372i	$0.862601\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−41.1593	−	22.1021i	−1.52025	−	0.816359i	−0.520947	−	0.853589i	$-0.674421\pi$
$733$	−41.1593	−	22.1021i	−1.52025	−	0.816359i	−0.999307	+	0.0372299i	$0.988147\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	35.1908	−	41.0749i	1.29451	−	1.51096i	0.570165	−	0.821530i	$-0.306879\pi$
$739$	35.1908	−	41.0749i	1.29451	−	1.51096i	0.724349	−	0.689433i	$-0.242140\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$743$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	21.3651	−	42.9069i	0.779624	−	1.56570i	−0.0443147	−	0.999018i	$-0.514110\pi$
$751$	21.3651	−	42.9069i	0.779624	−	1.56570i	0.823939	−	0.566679i	$-0.191772\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−49.1302	+	12.3567i	−1.78567	+	0.449113i	−0.988061	−	0.154065i	$-0.950763\pi$
$757$	−49.1302	+	12.3567i	−1.78567	+	0.449113i	−0.797606	+	0.603178i	$0.793901\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.759405	−	0.650618i	$-0.225490\pi$
$761$		0			0		−0.759405	+	0.650618i	$0.774510\pi$
$762$		0			0
$763$	31.7593	−	42.0561i	1.14976	−	1.52253i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2.78515	−	17.9422i	0.100435	−	0.647013i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	2.78515	−	17.9422i	0.100435	−	0.647013i	0.983928	−	0.178568i	$-0.0571464\pi$
$770$		0			0
$771$		0			0
$772$	−34.1334	+	24.1625i	−1.22849	+	0.869629i
$773$		0			0		−0.577774	−	0.816197i	$-0.696078\pi$
$773$		0			0		0.577774	+	0.816197i	$0.303922\pi$
$774$		0			0
$775$	0.214408	+	6.95912i	0.00770177	+	0.249979i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.58324	−	1.84796i	−0.0565442	−	0.0659987i
$785$		0			0
$786$		0			0
$787$	21.2708	−	13.1703i	0.758223	−	0.469472i	−0.0920887	−	0.995751i	$-0.529354\pi$
$787$	21.2708	−	13.1703i	0.758223	−	0.469472i	0.850312	+	0.526279i	$0.176413\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−57.7534	−	22.3738i	−2.05088	−	0.794516i
$794$		0			0
$795$		0			0
$796$	−49.7150	−	22.8725i	−1.76210	−	0.810695i
$797$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$797$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.988165	−	0.153392i	$-0.0490196\pi$
$809$		0			0		−0.988165	+	0.153392i	$0.950980\pi$
$810$		0			0
$811$	15.4711	+	54.3754i	0.543265	+	1.90938i	0.388579	+	0.921415i	$0.372966\pi$
$811$	15.4711	+	54.3754i	0.543265	+	1.90938i	0.154686	+	0.987964i	$0.450563\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	34.9059	−	35.9979i	1.22120	−	1.25941i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$821$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$822$		0			0
$823$	25.8386			0.900675			0.450338	−	0.892858i	$-0.351304\pi$
$823$	25.8386			0.900675			0.450338	+	0.892858i	$0.351304\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$827$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$828$		0			0
$829$	−21.3973	+	32.2917i	−0.743158	+	1.12154i	0.245211	+	0.969470i	$0.421143\pi$
$829$	−21.3973	+	32.2917i	−0.743158	+	1.12154i	−0.988370	+	0.152069i	$0.951406\pi$
$830$		0			0
$831$		0			0
$832$	−12.8505	+	45.1647i	−0.445510	+	1.56580i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.988165	−	0.153392i	$-0.0490196\pi$
$839$		0			0		−0.988165	+	0.153392i	$0.950980\pi$
$840$		0			0
$841$	−28.7802	−	3.56376i	−0.992421	−	0.122888i
$842$		0			0
$843$		0			0
$844$	−39.1473	−	9.84595i	−1.34751	−	0.338911i
$845$		0			0
$846$		0			0
$847$	−2.56597	−	27.6912i	−0.0881678	−	0.951482i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	56.6014	+	3.49106i	1.93800	+	0.119532i	0.983756	−	0.179510i	$-0.0574512\pi$
$853$	56.6014	+	3.49106i	1.93800	+	0.119532i	0.954240	+	0.299041i	$0.0966669\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.920906	−	0.389786i	$-0.872549\pi$
$857$		0			0		0.920906	+	0.389786i	$0.127451\pi$
$858$		0			0
$859$	9.70633	+	44.3204i	0.331176	+	1.51219i	0.784094	+	0.620642i	$0.213128\pi$
$859$	9.70633	+	44.3204i	0.331176	+	1.51219i	−0.452918	+	0.891552i	$0.649617\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.526432	−	0.850217i	$-0.323529\pi$
$863$		0			0		−0.526432	+	0.850217i	$0.676471\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	4.90139	+	5.05472i	0.166364	+	0.171569i
$869$		0			0
$870$		0			0
$871$	5.79051	+	10.0294i	0.196204	+	0.339835i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	11.9697	−	54.6553i	0.404188	−	1.84558i	−0.118366	−	0.992970i	$-0.537765\pi$
$877$	11.9697	−	54.6553i	0.404188	−	1.84558i	0.522553	−	0.852607i	$-0.324980\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$881$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$882$		0			0
$883$	−29.3587	−	30.2772i	−0.987999	−	1.01891i	−0.999820	−	0.0189779i	$-0.993959\pi$
$883$	−29.3587	−	30.2772i	−0.987999	−	1.01891i	0.0118206	−	0.999930i	$-0.496237\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.759405	−	0.650618i	$-0.225490\pi$
$887$		0			0		−0.759405	+	0.650618i	$0.774510\pi$
$888$		0			0
$889$	−53.7934	−	17.1129i	−1.80417	−	0.573949i
$890$		0			0
$891$		0			0
$892$	5.53295	+	25.2642i	0.185257	+	0.845908i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−31.3901	−	7.89492i	−1.04229	−	0.262147i	−0.315442	−	0.948945i	$-0.602153\pi$
$907$	−31.3901	−	7.89492i	−1.04229	−	0.262147i	−0.726848	+	0.686798i	$0.759016\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.626924	−	0.779081i	$-0.715686\pi$
$911$		0			0		0.626924	+	0.779081i	$0.284314\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	38.9772	−	17.9324i	1.28784	−	0.592502i
$917$		0			0
$918$		0			0
$919$	16.2644	−	57.1634i	0.536513	−	1.88565i	0.0780147	−	0.996952i	$-0.475142\pi$
$919$	16.2644	−	57.1634i	0.536513	−	1.88565i	0.458498	−	0.888695i	$-0.348388\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	32.6809	+	23.1343i	1.07454	+	0.760651i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.577774	−	0.816197i	$-0.303922\pi$
$929$		0			0		−0.577774	+	0.816197i	$0.696078\pi$
$930$		0			0
$931$	1.57134	+	4.45915i	0.0514987	+	0.146143i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−57.0451	+	22.0994i	−1.86358	+	0.721956i	−0.917987	+	0.396610i	$0.870186\pi$
$937$	−57.0451	+	22.0994i	−1.86358	+	0.721956i	−0.945595	+	0.325346i	$0.894519\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$941$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.243914	−	0.969797i	$-0.421569\pi$
$947$		0			0		−0.243914	+	0.969797i	$0.578431\pi$
$948$		0			0
$949$	−5.17027	−	0.966491i	−0.167834	−	0.0313736i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.943154	−	0.332355i	$-0.892157\pi$
$953$		0			0		0.943154	+	0.332355i	$0.107843\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−21.4763	−	19.5783i	−0.692785	−	0.631557i
$962$		0			0
$963$		0			0
$964$	−41.3229	+	25.5860i	−1.33092	+	0.824071i
$965$		0			0
$966$		0			0
$967$	−5.75049	−	6.71200i	−0.184923	−	0.215843i	0.659236	−	0.751936i	$-0.270880\pi$
$967$	−5.75049	−	6.71200i	−0.184923	−	0.215843i	−0.844159	+	0.536093i	$0.819900\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.717912	−	0.696134i	$-0.245098\pi$
$971$		0			0		−0.717912	+	0.696134i	$0.754902\pi$
$972$		0			0
$973$	21.2614	−	36.8258i	0.681608	−	1.18058i
$974$		0			0
$975$		0			0
$976$	−1.29978	−	42.1874i	−0.0416049	−	1.35039i
$977$		0			0		−0.976848	−	0.213933i	$-0.931373\pi$
$977$		0			0		0.976848	+	0.213933i	$0.0686275\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	54.9801	−	72.8054i	1.74915	−	2.31625i
$989$		0			0
$990$		0			0
$991$	−8.41170	−	5.20830i	−0.267206	−	0.165447i	0.386289	−	0.922378i	$-0.373757\pi$
$991$	−8.41170	−	5.20830i	−0.267206	−	0.165447i	−0.653495	+	0.756931i	$0.726698\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	24.2185	−	57.2185i	0.767008	−	1.81213i	0.218835	−	0.975762i	$-0.429774\pi$
$997$	24.2185	−	57.2185i	0.767008	−	1.81213i	0.548172	−	0.836365i	$-0.315324\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.ba.a.532.1	yes	32
3.2	odd	2	CM	927.2.ba.a.532.1	yes	32
103.97	even	51	inner	927.2.ba.a.406.1	✓	32
309.200	odd	102	inner	927.2.ba.a.406.1	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.ba.a.406.1	✓	32	103.97	even	51	inner
927.2.ba.a.406.1	✓	32	309.200	odd	102	inner
927.2.ba.a.532.1	yes	32	1.1	even	1	trivial
927.2.ba.a.532.1	yes	32	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 927 = 3^{2} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	927.ba (of order \(51\), degree \(32\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.664710 - 1.88631i) q^{4} +(-1.75994 + 1.81500i) q^{7} +O(q^{10})\)	\(q+(-0.664710 - 1.88631i) q^{4} +(-1.75994 + 1.81500i) q^{7} +(1.60631 + 5.64559i) q^{13} +(-3.11632 + 2.50770i) q^{16} +(7.40586 - 2.35597i) q^{19} +(1.94893 + 4.60453i) q^{25} +(4.59351 + 2.11335i) q^{28} +(1.29845 + 0.503023i) q^{31} +(6.80861 - 4.21571i) q^{37} +(5.68433 - 3.05242i) q^{43} +(0.0187345 + 0.608070i) q^{49} +(9.58159 - 6.78267i) q^{52} +(-6.35891 + 8.42056i) q^{61} +(6.80174 + 4.21146i) q^{64} +(1.91344 - 0.481249i) q^{67} +(-0.399429 + 0.802161i) q^{73} +(-9.36684 - 12.4037i) q^{76} +(6.78375 + 13.6236i) q^{79} +(-13.0738 - 7.02046i) q^{91} +(2.60090 - 0.322061i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{4} + 5 q^{7}+O(q^{10}) \) 32 * q - 2 * q^4 + 5 * q^7	\( 32 q - 2 q^{4} + 5 q^{7} - 10 q^{13} + 4 q^{16} - 7 q^{19} - 5 q^{25} - 10 q^{28} + 14 q^{31} - 22 q^{37} - 13 q^{43} + 18 q^{49} - 10 q^{52} - 28 q^{61} + 16 q^{64} - 16 q^{67} + 14 q^{73} - 28 q^{76} + 26 q^{79} + 212 q^{91} + 99 q^{97}+O(q^{100}) \) 32 * q - 2 * q^4 + 5 * q^7 - 10 * q^13 + 4 * q^16 - 7 * q^19 - 5 * q^25 - 10 * q^28 + 14 * q^31 - 22 * q^37 - 13 * q^43 + 18 * q^49 - 10 * q^52 - 28 * q^61 + 16 * q^64 - 16 * q^67 + 14 * q^73 - 28 * q^76 + 26 * q^79 + 212 * q^91 + 99 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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