LMFDB - Embedded newform 225.3.g.e.82.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,3,Mod(82,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("225.82");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	225.g (of order $4$, degree $2$, 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:	$6.13080594811$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			82.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	225.82
Dual form			225.3.g.e.118.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.22474 + 1.22474i) q^{2} -1.00000i q^{4} +(4.89898 + 4.89898i) q^{7} +(6.12372 - 6.12372i) q^{8} +3.00000 q^{11} +(7.34847 - 7.34847i) q^{13} +12.0000i q^{14} +11.0000 q^{16} +(13.4722 + 13.4722i) q^{17} +5.00000i q^{19} +(3.67423 + 3.67423i) q^{22} +(17.1464 - 17.1464i) q^{23} +18.0000 q^{26} +(4.89898 - 4.89898i) q^{28} +30.0000i q^{29} -38.0000 q^{31} +(-11.0227 - 11.0227i) q^{32} +33.0000i q^{34} +(-19.5959 - 19.5959i) q^{37} +(-6.12372 + 6.12372i) q^{38} -57.0000 q^{41} +(-4.89898 + 4.89898i) q^{43} -3.00000i q^{44} +42.0000 q^{46} +(7.34847 + 7.34847i) q^{47} -1.00000i q^{49} +(-7.34847 - 7.34847i) q^{52} +(-31.8434 + 31.8434i) q^{53} +60.0000 q^{56} +(-36.7423 + 36.7423i) q^{58} -90.0000i q^{59} -28.0000 q^{61} +(-46.5403 - 46.5403i) q^{62} -71.0000i q^{64} +(47.7650 + 47.7650i) q^{67} +(13.4722 - 13.4722i) q^{68} -42.0000 q^{71} +(13.4722 - 13.4722i) q^{73} -48.0000i q^{74} +5.00000 q^{76} +(14.6969 + 14.6969i) q^{77} -80.0000i q^{79} +(-69.8105 - 69.8105i) q^{82} +(-111.452 + 111.452i) q^{83} -12.0000 q^{86} +(18.3712 - 18.3712i) q^{88} +15.0000i q^{89} +72.0000 q^{91} +(-17.1464 - 17.1464i) q^{92} +18.0000i q^{94} +(53.8888 + 53.8888i) q^{97} +(1.22474 - 1.22474i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{11} + 44 q^{16} + 72 q^{26} - 152 q^{31} - 228 q^{41} + 168 q^{46} + 240 q^{56} - 112 q^{61} - 168 q^{71} + 20 q^{76} - 48 q^{86} + 288 q^{91}+O(q^{100}) $ 4 * q + 12 * q^11 + 44 * q^16 + 72 * q^26 - 152 * q^31 - 228 * q^41 + 168 * q^46 + 240 * q^56 - 112 * q^61 - 168 * q^71 + 20 * q^76 - 48 * q^86 + 288 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	1.22474	+	1.22474i	0.612372	+	0.612372i	0.943564	−	0.331191i	$-0.107451\pi$
$2$	1.22474	+	1.22474i	0.612372	+	0.612372i	−0.331191	+	0.943564i	$0.607451\pi$
$3$		0			0
$3$		0			0
$4$		−	1.00000i		−	0.250000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	4.89898	+	4.89898i	0.699854	+	0.699854i	0.964379	−	0.264525i	$-0.0852151\pi$
$7$	4.89898	+	4.89898i	0.699854	+	0.699854i	−0.264525	+	0.964379i	$0.585215\pi$
$8$	6.12372	−	6.12372i	0.765466	−	0.765466i
$9$		0			0
$10$		0			0
$11$	3.00000			0.272727			0.136364	−	0.990659i	$-0.456458\pi$
$11$	3.00000			0.272727			0.136364	+	0.990659i	$0.456458\pi$
$12$		0			0
$13$	7.34847	−	7.34847i	0.565267	−	0.565267i	−0.365532	−	0.930799i	$-0.619113\pi$
$13$	7.34847	−	7.34847i	0.565267	−	0.565267i	0.930799	+	0.365532i	$0.119113\pi$
$14$			12.0000i			0.857143i
$15$		0			0
$16$	11.0000			0.687500
$17$	13.4722	+	13.4722i	0.792482	+	0.792482i	0.981897	−	0.189415i	$-0.0606592\pi$
$17$	13.4722	+	13.4722i	0.792482	+	0.792482i	−0.189415	+	0.981897i	$0.560659\pi$
$18$		0			0
$19$			5.00000i			0.263158i	0.991306	+	0.131579i	$0.0420047\pi$
$19$			5.00000i			0.263158i	−0.991306	+	0.131579i	$0.957995\pi$
$20$		0			0
$21$		0			0
$22$	3.67423	+	3.67423i	0.167011	+	0.167011i
$23$	17.1464	−	17.1464i	0.745497	−	0.745497i	−0.228133	−	0.973630i	$-0.573262\pi$
$23$	17.1464	−	17.1464i	0.745497	−	0.745497i	0.973630	+	0.228133i	$0.0732621\pi$
$24$		0			0
$25$		0			0
$26$	18.0000			0.692308
$27$		0			0
$28$	4.89898	−	4.89898i	0.174964	−	0.174964i
$29$			30.0000i			1.03448i	0.855840	+	0.517241i	$0.173041\pi$
$29$			30.0000i			1.03448i	−0.855840	+	0.517241i	$0.826959\pi$
$30$		0			0
$31$	−38.0000			−1.22581			−0.612903	−	0.790158i	$-0.709998\pi$
$31$	−38.0000			−1.22581			−0.612903	+	0.790158i	$0.709998\pi$
$32$	−11.0227	−	11.0227i	−0.344459	−	0.344459i
$33$		0			0
$34$			33.0000i			0.970588i
$35$		0			0
$36$		0			0
$37$	−19.5959	−	19.5959i	−0.529619	−	0.529619i	0.390839	−	0.920459i	$-0.372185\pi$
$37$	−19.5959	−	19.5959i	−0.529619	−	0.529619i	−0.920459	+	0.390839i	$0.872185\pi$
$38$	−6.12372	+	6.12372i	−0.161151	+	0.161151i
$39$		0			0
$40$		0			0
$41$	−57.0000			−1.39024			−0.695122	−	0.718892i	$-0.744650\pi$
$41$	−57.0000			−1.39024			−0.695122	+	0.718892i	$0.744650\pi$
$42$		0			0
$43$	−4.89898	+	4.89898i	−0.113930	+	0.113930i	−0.761773	−	0.647844i	$-0.775671\pi$
$43$	−4.89898	+	4.89898i	−0.113930	+	0.113930i	0.647844	+	0.761773i	$0.275671\pi$
$44$		−	3.00000i		−	0.0681818i
$45$		0			0
$46$	42.0000			0.913043
$47$	7.34847	+	7.34847i	0.156350	+	0.156350i	0.780947	−	0.624597i	$-0.214737\pi$
$47$	7.34847	+	7.34847i	0.156350	+	0.156350i	−0.624597	+	0.780947i	$0.714737\pi$
$48$		0			0
$49$		−	1.00000i		−	0.0204082i
$50$		0			0
$51$		0			0
$52$	−7.34847	−	7.34847i	−0.141317	−	0.141317i
$53$	−31.8434	+	31.8434i	−0.600818	+	0.600818i	−0.940530	−	0.339711i	$-0.889671\pi$
$53$	−31.8434	+	31.8434i	−0.600818	+	0.600818i	0.339711	+	0.940530i	$0.389671\pi$
$54$		0			0
$55$		0			0
$56$	60.0000			1.07143
$57$		0			0
$58$	−36.7423	+	36.7423i	−0.633489	+	0.633489i
$59$		−	90.0000i		−	1.52542i	−0.646738	−	0.762712i	$-0.723867\pi$
$59$		−	90.0000i		−	1.52542i	0.646738	−	0.762712i	$-0.276133\pi$
$60$		0			0
$61$	−28.0000			−0.459016			−0.229508	−	0.973307i	$-0.573712\pi$
$61$	−28.0000			−0.459016			−0.229508	+	0.973307i	$0.573712\pi$
$62$	−46.5403	−	46.5403i	−0.750650	−	0.750650i
$63$		0			0
$64$		−	71.0000i		−	1.10938i
$65$		0			0
$66$		0			0
$67$	47.7650	+	47.7650i	0.712911	+	0.712911i	0.967143	−	0.254232i	$-0.0818227\pi$
$67$	47.7650	+	47.7650i	0.712911	+	0.712911i	−0.254232	+	0.967143i	$0.581823\pi$
$68$	13.4722	−	13.4722i	0.198120	−	0.198120i
$69$		0			0
$70$		0			0
$71$	−42.0000			−0.591549			−0.295775	−	0.955258i	$-0.595578\pi$
$71$	−42.0000			−0.591549			−0.295775	+	0.955258i	$0.595578\pi$
$72$		0			0
$73$	13.4722	−	13.4722i	0.184551	−	0.184551i	−0.608785	−	0.793335i	$-0.708343\pi$
$73$	13.4722	−	13.4722i	0.184551	−	0.184551i	0.793335	+	0.608785i	$0.208343\pi$
$74$		−	48.0000i		−	0.648649i
$75$		0			0
$76$	5.00000			0.0657895
$77$	14.6969	+	14.6969i	0.190869	+	0.190869i
$78$		0			0
$79$		−	80.0000i		−	1.01266i	−0.862340	−	0.506329i	$-0.831002\pi$
$79$		−	80.0000i		−	1.01266i	0.862340	−	0.506329i	$-0.168998\pi$
$80$		0			0
$81$		0			0
$82$	−69.8105	−	69.8105i	−0.851347	−	0.851347i
$83$	−111.452	+	111.452i	−1.34279	+	1.34279i	−0.449525	+	0.893268i	$0.648407\pi$
$83$	−111.452	+	111.452i	−1.34279	+	1.34279i	−0.893268	+	0.449525i	$0.851593\pi$
$84$		0			0
$85$		0			0
$86$	−12.0000			−0.139535
$87$		0			0
$88$	18.3712	−	18.3712i	0.208763	−	0.208763i
$89$			15.0000i			0.168539i	0.996443	+	0.0842697i	$0.0268557\pi$
$89$			15.0000i			0.168539i	−0.996443	+	0.0842697i	$0.973144\pi$
$90$		0			0
$91$	72.0000			0.791209
$92$	−17.1464	−	17.1464i	−0.186374	−	0.186374i
$93$		0			0
$94$			18.0000i			0.191489i
$95$		0			0
$96$		0			0
$97$	53.8888	+	53.8888i	0.555554	+	0.555554i	0.928039	−	0.372484i	$-0.121494\pi$
$97$	53.8888	+	53.8888i	0.555554	+	0.555554i	−0.372484	+	0.928039i	$0.621494\pi$
$98$	1.22474	−	1.22474i	0.0124974	−	0.0124974i
$99$		0			0
$100$		0			0
$101$	48.0000			0.475248			0.237624	−	0.971357i	$-0.423631\pi$
$101$	48.0000			0.475248			0.237624	+	0.971357i	$0.423631\pi$
$102$		0			0
$103$	−90.6311	+	90.6311i	−0.879914	+	0.879914i	−0.993525	−	0.113611i	$-0.963758\pi$
$103$	−90.6311	+	90.6311i	−0.879914	+	0.879914i	0.113611	+	0.993525i	$0.463758\pi$
$104$		−	90.0000i		−	0.865385i
$105$		0			0
$106$	−78.0000			−0.735849
$107$	25.7196	+	25.7196i	0.240370	+	0.240370i	0.817003	−	0.576633i	$-0.195634\pi$
$107$	25.7196	+	25.7196i	0.240370	+	0.240370i	−0.576633	+	0.817003i	$0.695634\pi$
$108$		0			0
$109$			40.0000i			0.366972i	0.983022	+	0.183486i	$0.0587383\pi$
$109$			40.0000i			0.366972i	−0.983022	+	0.183486i	$0.941262\pi$
$110$		0			0
$111$		0			0
$112$	53.8888	+	53.8888i	0.481150	+	0.481150i
$113$	84.5074	−	84.5074i	0.747853	−	0.747853i	−0.226223	−	0.974076i	$-0.572638\pi$
$113$	84.5074	−	84.5074i	0.747853	−	0.747853i	0.974076	+	0.226223i	$0.0726377\pi$
$114$		0			0
$115$		0			0
$116$	30.0000			0.258621
$117$		0			0
$118$	110.227	−	110.227i	0.934127	−	0.934127i
$119$			132.000i			1.10924i
$120$		0			0
$121$	−112.000			−0.925620
$122$	−34.2929	−	34.2929i	−0.281089	−	0.281089i
$123$		0			0
$124$			38.0000i			0.306452i
$125$		0			0
$126$		0			0
$127$	−154.318	−	154.318i	−1.21510	−	1.21510i	−0.969327	−	0.245774i	$-0.920958\pi$
$127$	−154.318	−	154.318i	−1.21510	−	1.21510i	−0.245774	−	0.969327i	$-0.579042\pi$
$128$	42.8661	−	42.8661i	0.334891	−	0.334891i
$129$		0			0
$130$		0			0
$131$	−162.000			−1.23664			−0.618321	−	0.785926i	$-0.712187\pi$
$131$	−162.000			−1.23664			−0.618321	+	0.785926i	$0.712187\pi$
$132$		0			0
$133$	−24.4949	+	24.4949i	−0.184172	+	0.184172i
$134$			117.000i			0.873134i
$135$		0			0
$136$	165.000			1.21324
$137$	50.2145	+	50.2145i	0.366529	+	0.366529i	0.866210	−	0.499680i	$-0.166549\pi$
$137$	50.2145	+	50.2145i	0.366529	+	0.366529i	−0.499680	+	0.866210i	$0.666549\pi$
$138$		0			0
$139$			185.000i			1.33094i	0.746427	+	0.665468i	$0.231768\pi$
$139$			185.000i			1.33094i	−0.746427	+	0.665468i	$0.768232\pi$
$140$		0			0
$141$		0			0
$142$	−51.4393	−	51.4393i	−0.362248	−	0.362248i
$143$	22.0454	−	22.0454i	0.154164	−	0.154164i
$144$		0			0
$145$		0			0
$146$	33.0000			0.226027
$147$		0			0
$148$	−19.5959	+	19.5959i	−0.132405	+	0.132405i
$149$			150.000i			1.00671i	0.864079	+	0.503356i	$0.167901\pi$
$149$			150.000i			1.00671i	−0.864079	+	0.503356i	$0.832099\pi$
$150$		0			0
$151$	52.0000			0.344371			0.172185	−	0.985065i	$-0.444917\pi$
$151$	52.0000			0.344371			0.172185	+	0.985065i	$0.444917\pi$
$152$	30.6186	+	30.6186i	0.201438	+	0.201438i
$153$		0			0
$154$			36.0000i			0.233766i
$155$		0			0
$156$		0			0
$157$	188.611	+	188.611i	1.20134	+	1.20134i	0.973759	+	0.227584i	$0.0730825\pi$
$157$	188.611	+	188.611i	1.20134	+	1.20134i	0.227584	+	0.973759i	$0.426918\pi$
$158$	97.9796	−	97.9796i	0.620124	−	0.620124i
$159$		0			0
$160$		0			0
$161$	168.000			1.04348
$162$		0			0
$163$	99.2043	−	99.2043i	0.608616	−	0.608616i	−0.333969	−	0.942584i	$-0.608388\pi$
$163$	99.2043	−	99.2043i	0.608616	−	0.608616i	0.942584	+	0.333969i	$0.108388\pi$
$164$			57.0000i			0.347561i
$165$		0			0
$166$	−273.000			−1.64458
$167$	−17.1464	−	17.1464i	−0.102673	−	0.102673i	0.653904	−	0.756577i	$-0.273130\pi$
$167$	−17.1464	−	17.1464i	−0.102673	−	0.102673i	−0.756577	+	0.653904i	$0.773130\pi$
$168$		0			0
$169$			61.0000i			0.360947i
$170$		0			0
$171$		0			0
$172$	4.89898	+	4.89898i	0.0284824	+	0.0284824i
$173$	78.3837	−	78.3837i	0.453085	−	0.453085i	−0.443292	−	0.896377i	$-0.646190\pi$
$173$	78.3837	−	78.3837i	0.453085	−	0.453085i	0.896377	+	0.443292i	$0.146190\pi$
$174$		0			0
$175$		0			0
$176$	33.0000			0.187500
$177$		0			0
$178$	−18.3712	+	18.3712i	−0.103209	+	0.103209i
$179$		−	195.000i		−	1.08939i	−0.838636	−	0.544693i	$-0.816646\pi$
$179$		−	195.000i		−	1.08939i	0.838636	−	0.544693i	$-0.183354\pi$
$180$		0			0
$181$	262.000			1.44751			0.723757	−	0.690055i	$-0.242414\pi$
$181$	262.000			1.44751			0.723757	+	0.690055i	$0.242414\pi$
$182$	88.1816	+	88.1816i	0.484514	+	0.484514i
$183$		0			0
$184$		−	210.000i		−	1.14130i
$185$		0			0
$186$		0			0
$187$	40.4166	+	40.4166i	0.216131	+	0.216131i
$188$	7.34847	−	7.34847i	0.0390876	−	0.0390876i
$189$		0			0
$190$		0			0
$191$	18.0000			0.0942408			0.0471204	−	0.998889i	$-0.484996\pi$
$191$	18.0000			0.0942408			0.0471204	+	0.998889i	$0.484996\pi$
$192$		0			0
$193$	86.9569	−	86.9569i	0.450554	−	0.450554i	−0.444984	−	0.895538i	$-0.646791\pi$
$193$	86.9569	−	86.9569i	0.450554	−	0.450554i	0.895538	+	0.444984i	$0.146791\pi$
$194$			132.000i			0.680412i
$195$		0			0
$196$	−1.00000			−0.00510204
$197$	129.823	+	129.823i	0.659000	+	0.659000i	0.955143	−	0.296144i	$-0.0957007\pi$
$197$	129.823	+	129.823i	0.659000	+	0.659000i	−0.296144	+	0.955143i	$0.595701\pi$
$198$		0			0
$199$		−	200.000i		−	1.00503i	−0.864570	−	0.502513i	$-0.832409\pi$
$199$		−	200.000i		−	1.00503i	0.864570	−	0.502513i	$-0.167591\pi$
$200$		0			0
$201$		0			0
$202$	58.7878	+	58.7878i	0.291028	+	0.291028i
$203$	−146.969	+	146.969i	−0.723987	+	0.723987i
$204$		0			0
$205$		0			0
$206$	−222.000			−1.07767
$207$		0			0
$208$	80.8332	−	80.8332i	0.388621	−	0.388621i
$209$			15.0000i			0.0717703i
$210$		0			0
$211$	−103.000			−0.488152			−0.244076	−	0.969756i	$-0.578485\pi$
$211$	−103.000			−0.488152			−0.244076	+	0.969756i	$0.578485\pi$
$212$	31.8434	+	31.8434i	0.150205	+	0.150205i
$213$		0			0
$214$			63.0000i			0.294393i
$215$		0			0
$216$		0			0
$217$	−186.161	−	186.161i	−0.857886	−	0.857886i
$218$	−48.9898	+	48.9898i	−0.224724	+	0.224724i
$219$		0			0
$220$		0			0
$221$	198.000			0.895928
$222$		0			0
$223$	−200.858	+	200.858i	−0.900709	+	0.900709i	−0.995497	−	0.0947882i	$-0.969783\pi$
$223$	−200.858	+	200.858i	−0.900709	+	0.900709i	0.0947882	+	0.995497i	$0.469783\pi$
$224$		−	108.000i		−	0.482143i
$225$		0			0
$226$	207.000			0.915929
$227$	−274.343	−	274.343i	−1.20856	−	1.20856i	−0.971494	−	0.237065i	$-0.923815\pi$
$227$	−274.343	−	274.343i	−1.20856	−	1.20856i	−0.237065	−	0.971494i	$-0.576185\pi$
$228$		0			0
$229$			20.0000i			0.0873362i	0.999046	+	0.0436681i	$0.0139044\pi$
$229$			20.0000i			0.0873362i	−0.999046	+	0.0436681i	$0.986096\pi$
$230$		0			0
$231$		0			0
$232$	183.712	+	183.712i	0.791861	+	0.791861i
$233$	102.879	−	102.879i	0.441539	−	0.441539i	−0.450990	−	0.892529i	$-0.648929\pi$
$233$	102.879	−	102.879i	0.441539	−	0.441539i	0.892529	+	0.450990i	$0.148929\pi$
$234$		0			0
$235$		0			0
$236$	−90.0000			−0.381356
$237$		0			0
$238$	−161.666	+	161.666i	−0.679270	+	0.679270i
$239$		−	210.000i		−	0.878661i	−0.898325	−	0.439331i	$-0.855216\pi$
$239$		−	210.000i		−	0.878661i	0.898325	−	0.439331i	$-0.144784\pi$
$240$		0			0
$241$	−43.0000			−0.178423			−0.0892116	−	0.996013i	$-0.528435\pi$
$241$	−43.0000			−0.178423			−0.0892116	+	0.996013i	$0.528435\pi$
$242$	−137.171	−	137.171i	−0.566824	−	0.566824i
$243$		0			0
$244$			28.0000i			0.114754i
$245$		0			0
$246$		0			0
$247$	36.7423	+	36.7423i	0.148754	+	0.148754i
$248$	−232.702	+	232.702i	−0.938313	+	0.938313i
$249$		0			0
$250$		0			0
$251$	123.000			0.490040			0.245020	−	0.969518i	$-0.421205\pi$
$251$	123.000			0.490040			0.245020	+	0.969518i	$0.421205\pi$
$252$		0			0
$253$	51.4393	−	51.4393i	0.203317	−	0.203317i
$254$		−	378.000i		−	1.48819i
$255$		0			0
$256$	−179.000			−0.699219
$257$	−4.89898	−	4.89898i	−0.0190622	−	0.0190622i	0.697511	−	0.716574i	$-0.254291\pi$
$257$	−4.89898	−	4.89898i	−0.0190622	−	0.0190622i	−0.716574	+	0.697511i	$0.754291\pi$
$258$		0			0
$259$		−	192.000i		−	0.741313i
$260$		0			0
$261$		0			0
$262$	−198.409	−	198.409i	−0.757285	−	0.757285i
$263$	−7.34847	+	7.34847i	−0.0279409	+	0.0279409i	−0.720939	−	0.692998i	$-0.756289\pi$
$263$	−7.34847	+	7.34847i	−0.0279409	+	0.0279409i	0.692998	+	0.720939i	$0.256289\pi$
$264$		0			0
$265$		0			0
$266$	−60.0000			−0.225564
$267$		0			0
$268$	47.7650	−	47.7650i	0.178228	−	0.178228i
$269$			120.000i			0.446097i	0.974807	+	0.223048i	$0.0716008\pi$
$269$			120.000i			0.446097i	−0.974807	+	0.223048i	$0.928399\pi$
$270$		0			0
$271$	−58.0000			−0.214022			−0.107011	−	0.994258i	$-0.534128\pi$
$271$	−58.0000			−0.214022			−0.107011	+	0.994258i	$0.534128\pi$
$272$	148.194	+	148.194i	0.544831	+	0.544831i
$273$		0			0
$274$			123.000i			0.448905i
$275$		0			0
$276$		0			0
$277$	−276.792	−	276.792i	−0.999250	−	0.999250i	0.000749391	−	1.00000i	$-0.499761\pi$
$277$	−276.792	−	276.792i	−0.999250	−	0.999250i	−1.00000		0.000749391i	$0.999761\pi$
$278$	−226.578	+	226.578i	−0.815028	+	0.815028i
$279$		0			0
$280$		0			0
$281$	−462.000			−1.64413			−0.822064	−	0.569395i	$-0.807178\pi$
$281$	−462.000			−1.64413			−0.822064	+	0.569395i	$0.807178\pi$
$282$		0			0
$283$	−72.2599	+	72.2599i	−0.255336	+	0.255336i	−0.823154	−	0.567818i	$-0.807788\pi$
$283$	−72.2599	+	72.2599i	−0.255336	+	0.255336i	0.567818	+	0.823154i	$0.307788\pi$
$284$			42.0000i			0.147887i
$285$		0			0
$286$	54.0000			0.188811
$287$	−279.242	−	279.242i	−0.972968	−	0.972968i
$288$		0			0
$289$			74.0000i			0.256055i
$290$		0			0
$291$		0			0
$292$	−13.4722	−	13.4722i	−0.0461376	−	0.0461376i
$293$	4.89898	−	4.89898i	0.0167201	−	0.0167201i	−0.698697	−	0.715417i	$-0.746237\pi$
$293$	4.89898	−	4.89898i	0.0167201	−	0.0167201i	0.715417	+	0.698697i	$0.246237\pi$
$294$		0			0
$295$		0			0
$296$	−240.000			−0.810811
$297$		0			0
$298$	−183.712	+	183.712i	−0.616482	+	0.616482i
$299$		−	252.000i		−	0.842809i
$300$		0			0
$301$	−48.0000			−0.159468
$302$	63.6867	+	63.6867i	0.210883	+	0.210883i
$303$		0			0
$304$			55.0000i			0.180921i
$305$		0			0
$306$		0			0
$307$	−86.9569	−	86.9569i	−0.283247	−	0.283247i	0.551155	−	0.834403i	$-0.314187\pi$
$307$	−86.9569	−	86.9569i	−0.283247	−	0.283247i	−0.834403	+	0.551155i	$0.814187\pi$
$308$	14.6969	−	14.6969i	0.0477173	−	0.0477173i
$309$		0			0
$310$		0			0
$311$	528.000			1.69775			0.848875	−	0.528594i	$-0.177281\pi$
$311$	528.000			1.69775			0.848875	+	0.528594i	$0.177281\pi$
$312$		0			0
$313$	191.060	−	191.060i	0.610416	−	0.610416i	−0.332638	−	0.943054i	$-0.607939\pi$
$313$	191.060	−	191.060i	0.610416	−	0.610416i	0.943054	+	0.332638i	$0.107939\pi$
$314$			462.000i			1.47134i
$315$		0			0
$316$	−80.0000			−0.253165
$317$	−200.858	−	200.858i	−0.633622	−	0.633622i	0.315353	−	0.948975i	$-0.397877\pi$
$317$	−200.858	−	200.858i	−0.633622	−	0.633622i	−0.948975	+	0.315353i	$0.897877\pi$
$318$		0			0
$319$			90.0000i			0.282132i
$320$		0			0
$321$		0			0
$322$	205.757	+	205.757i	0.638997	+	0.638997i
$323$	−67.3610	+	67.3610i	−0.208548	+	0.208548i
$324$		0			0
$325$		0			0
$326$	243.000			0.745399
$327$		0			0
$328$	−349.052	+	349.052i	−1.06418	+	1.06418i
$329$			72.0000i			0.218845i
$330$		0			0
$331$	−313.000			−0.945619			−0.472810	−	0.881165i	$-0.656760\pi$
$331$	−313.000			−0.945619			−0.472810	+	0.881165i	$0.656760\pi$
$332$	111.452	+	111.452i	0.335698	+	0.335698i
$333$		0			0
$334$		−	42.0000i		−	0.125749i
$335$		0			0
$336$		0			0
$337$	194.734	+	194.734i	0.577847	+	0.577847i	0.934310	−	0.356463i	$-0.116017\pi$
$337$	194.734	+	194.734i	0.577847	+	0.577847i	−0.356463	+	0.934310i	$0.616017\pi$
$338$	−74.7094	+	74.7094i	−0.221034	+	0.221034i
$339$		0			0
$340$		0			0
$341$	−114.000			−0.334311
$342$		0			0
$343$	244.949	−	244.949i	0.714137	−	0.714137i
$344$			60.0000i			0.174419i
$345$		0			0
$346$	192.000			0.554913
$347$	99.2043	+	99.2043i	0.285891	+	0.285891i	0.835453	−	0.549562i	$-0.185205\pi$
$347$	99.2043	+	99.2043i	0.285891	+	0.285891i	−0.549562	+	0.835453i	$0.685205\pi$
$348$		0			0
$349$		−	100.000i		−	0.286533i	−0.989684	−	0.143266i	$-0.954239\pi$
$349$		−	100.000i		−	0.286533i	0.989684	−	0.143266i	$-0.0457606\pi$
$350$		0			0
$351$		0			0
$352$	−33.0681	−	33.0681i	−0.0939435	−	0.0939435i
$353$	396.817	−	396.817i	1.12413	−	1.12413i	0.133014	−	0.991114i	$-0.457534\pi$
$353$	396.817	−	396.817i	1.12413	−	1.12413i	0.991114	−	0.133014i	$-0.0424656\pi$
$354$		0			0
$355$		0			0
$356$	15.0000			0.0421348
$357$		0			0
$358$	238.825	−	238.825i	0.667110	−	0.667110i
$359$		−	540.000i		−	1.50418i	−0.659061	−	0.752089i	$-0.729046\pi$
$359$		−	540.000i		−	1.50418i	0.659061	−	0.752089i	$-0.270954\pi$
$360$		0			0
$361$	336.000			0.930748
$362$	320.883	+	320.883i	0.886418	+	0.886418i
$363$		0			0
$364$		−	72.0000i		−	0.197802i
$365$		0			0
$366$		0			0
$367$	−44.0908	−	44.0908i	−0.120138	−	0.120138i	0.644481	−	0.764620i	$-0.277073\pi$
$367$	−44.0908	−	44.0908i	−0.120138	−	0.120138i	−0.764620	+	0.644481i	$0.777073\pi$
$368$	188.611	−	188.611i	0.512529	−	0.512529i
$369$		0			0
$370$		0			0
$371$	−312.000			−0.840970
$372$		0			0
$373$	350.277	−	350.277i	0.939081	−	0.939081i	−0.0591675	−	0.998248i	$-0.518845\pi$
$373$	350.277	−	350.277i	0.939081	−	0.939081i	0.998248	+	0.0591675i	$0.0188446\pi$
$374$			99.0000i			0.264706i
$375$		0			0
$376$	90.0000			0.239362
$377$	220.454	+	220.454i	0.584759	+	0.584759i
$378$		0			0
$379$		−	505.000i		−	1.33245i	−0.745749	−	0.666227i	$-0.767908\pi$
$379$		−	505.000i		−	1.33245i	0.745749	−	0.666227i	$-0.232092\pi$
$380$		0			0
$381$		0			0
$382$	22.0454	+	22.0454i	0.0577105	+	0.0577105i
$383$	−142.070	+	142.070i	−0.370941	+	0.370941i	−0.867820	−	0.496879i	$-0.834479\pi$
$383$	−142.070	+	142.070i	−0.370941	+	0.370941i	0.496879	+	0.867820i	$0.334479\pi$
$384$		0			0
$385$		0			0
$386$	213.000			0.551813
$387$		0			0
$388$	53.8888	−	53.8888i	0.138889	−	0.138889i
$389$			690.000i			1.77378i	0.461982	+	0.886889i	$0.347139\pi$
$389$			690.000i			1.77378i	−0.461982	+	0.886889i	$0.652861\pi$
$390$		0			0
$391$	462.000			1.18159
$392$	−6.12372	−	6.12372i	−0.0156217	−	0.0156217i
$393$		0			0
$394$			318.000i			0.807107i
$395$		0			0
$396$		0			0
$397$	421.312	+	421.312i	1.06124	+	1.06124i	0.997998	+	0.0632416i	$0.0201439\pi$
$397$	421.312	+	421.312i	1.06124	+	1.06124i	0.0632416	+	0.997998i	$0.479856\pi$
$398$	244.949	−	244.949i	0.615450	−	0.615450i
$399$		0			0
$400$		0			0
$401$	573.000			1.42893			0.714464	−	0.699672i	$-0.246671\pi$
$401$	573.000			1.42893			0.714464	+	0.699672i	$0.246671\pi$
$402$		0			0
$403$	−279.242	+	279.242i	−0.692908	+	0.692908i
$404$		−	48.0000i		−	0.118812i
$405$		0			0
$406$	−360.000			−0.886700
$407$	−58.7878	−	58.7878i	−0.144442	−	0.144442i
$408$		0			0
$409$			365.000i			0.892421i	0.894928	+	0.446210i	$0.147227\pi$
$409$			365.000i			0.892421i	−0.894928	+	0.446210i	$0.852773\pi$
$410$		0			0
$411$		0			0
$412$	90.6311	+	90.6311i	0.219978	+	0.219978i
$413$	440.908	−	440.908i	1.06757	−	1.06757i
$414$		0			0
$415$		0			0
$416$	−162.000			−0.389423
$417$		0			0
$418$	−18.3712	+	18.3712i	−0.0439502	+	0.0439502i
$419$			645.000i			1.53938i	0.638418	+	0.769690i	$0.279589\pi$
$419$			645.000i			1.53938i	−0.638418	+	0.769690i	$0.720411\pi$
$420$		0			0
$421$	−8.00000			−0.0190024			−0.00950119	−	0.999955i	$-0.503024\pi$
$421$	−8.00000			−0.0190024			−0.00950119	+	0.999955i	$0.503024\pi$
$422$	−126.149	−	126.149i	−0.298931	−	0.298931i
$423$		0			0
$424$			390.000i			0.919811i
$425$		0			0
$426$		0			0
$427$	−137.171	−	137.171i	−0.321245	−	0.321245i
$428$	25.7196	−	25.7196i	0.0600926	−	0.0600926i
$429$		0			0
$430$		0			0
$431$	−12.0000			−0.0278422			−0.0139211	−	0.999903i	$-0.504431\pi$
$431$	−12.0000			−0.0278422			−0.0139211	+	0.999903i	$0.504431\pi$
$432$		0			0
$433$	−439.683	+	439.683i	−1.01544	+	1.01544i	−0.0155561	+	0.999879i	$0.504952\pi$
$433$	−439.683	+	439.683i	−1.01544	+	1.01544i	−0.999879	+	0.0155561i	$0.995048\pi$
$434$		−	456.000i		−	1.05069i
$435$		0			0
$436$	40.0000			0.0917431
$437$	85.7321	+	85.7321i	0.196183	+	0.196183i
$438$		0			0
$439$			10.0000i			0.0227790i	0.999935	+	0.0113895i	$0.00362548\pi$
$439$			10.0000i			0.0227790i	−0.999935	+	0.0113895i	$0.996375\pi$
$440$		0			0
$441$		0			0
$442$	242.499	+	242.499i	0.548641	+	0.548641i
$443$	−86.9569	+	86.9569i	−0.196291	+	0.196291i	−0.798408	−	0.602117i	$-0.794324\pi$
$443$	−86.9569	+	86.9569i	−0.196291	+	0.196291i	0.602117	+	0.798408i	$0.294324\pi$
$444$		0			0
$445$		0			0
$446$	−492.000			−1.10314
$447$		0			0
$448$	347.828	−	347.828i	0.776401	−	0.776401i
$449$			75.0000i			0.167038i	0.996506	+	0.0835189i	$0.0266159\pi$
$449$			75.0000i			0.167038i	−0.996506	+	0.0835189i	$0.973384\pi$
$450$		0			0
$451$	−171.000			−0.379157
$452$	−84.5074	−	84.5074i	−0.186963	−	0.186963i
$453$		0			0
$454$		−	672.000i		−	1.48018i
$455$		0			0
$456$		0			0
$457$	−209.431	−	209.431i	−0.458274	−	0.458274i	0.439814	−	0.898089i	$-0.355044\pi$
$457$	−209.431	−	209.431i	−0.458274	−	0.458274i	−0.898089	+	0.439814i	$0.855044\pi$
$458$	−24.4949	+	24.4949i	−0.0534823	+	0.0534823i
$459$		0			0
$460$		0			0
$461$	228.000			0.494577			0.247289	−	0.968942i	$-0.420460\pi$
$461$	228.000			0.494577			0.247289	+	0.968942i	$0.420460\pi$
$462$		0			0
$463$	436.009	−	436.009i	0.941704	−	0.941704i	−0.0566875	−	0.998392i	$-0.518054\pi$
$463$	436.009	−	436.009i	0.941704	−	0.941704i	0.998392	+	0.0566875i	$0.0180539\pi$
$464$			330.000i			0.711207i
$465$		0			0
$466$	252.000			0.540773
$467$	533.989	+	533.989i	1.14344	+	1.14344i	0.987816	+	0.155629i	$0.0497406\pi$
$467$	533.989	+	533.989i	1.14344	+	1.14344i	0.155629	+	0.987816i	$0.450259\pi$
$468$		0			0
$469$			468.000i			0.997868i
$470$		0			0
$471$		0			0
$472$	−551.135	−	551.135i	−1.16766	−	1.16766i
$473$	−14.6969	+	14.6969i	−0.0310718	+	0.0310718i
$474$		0			0
$475$		0			0
$476$	132.000			0.277311
$477$		0			0
$478$	257.196	−	257.196i	0.538068	−	0.538068i
$479$		−	270.000i		−	0.563674i	−0.959462	−	0.281837i	$-0.909056\pi$
$479$		−	270.000i		−	0.563674i	0.959462	−	0.281837i	$-0.0909438\pi$
$480$		0			0
$481$	−288.000			−0.598753
$482$	−52.6640	−	52.6640i	−0.109261	−	0.109261i
$483$		0			0
$484$			112.000i			0.231405i
$485$		0			0
$486$		0			0
$487$	−80.8332	−	80.8332i	−0.165982	−	0.165982i	0.619229	−	0.785211i	$-0.287445\pi$
$487$	−80.8332	−	80.8332i	−0.165982	−	0.165982i	−0.785211	+	0.619229i	$0.787445\pi$
$488$	−171.464	+	171.464i	−0.351361	+	0.351361i
$489$		0			0
$490$		0			0
$491$	−582.000			−1.18534			−0.592668	−	0.805447i	$-0.701925\pi$
$491$	−582.000			−1.18534			−0.592668	+	0.805447i	$0.701925\pi$
$492$		0			0
$493$	−404.166	+	404.166i	−0.819809	+	0.819809i
$494$			90.0000i			0.182186i
$495$		0			0
$496$	−418.000			−0.842742
$497$	−205.757	−	205.757i	−0.413998	−	0.413998i
$498$		0			0
$499$			250.000i			0.501002i	0.968116	+	0.250501i	$0.0805953\pi$
$499$			250.000i			0.501002i	−0.968116	+	0.250501i	$0.919405\pi$
$500$		0			0
$501$		0			0
$502$	150.644	+	150.644i	0.300087	+	0.300087i
$503$	−460.504	+	460.504i	−0.915515	+	0.915515i	−0.996699	−	0.0811841i	$-0.974130\pi$
$503$	−460.504	+	460.504i	−0.915515	+	0.915515i	0.0811841	+	0.996699i	$0.474130\pi$
$504$		0			0
$505$		0			0
$506$	126.000			0.249012
$507$		0			0
$508$	−154.318	+	154.318i	−0.303775	+	0.303775i
$509$		−	390.000i		−	0.766208i	−0.923705	−	0.383104i	$-0.874855\pi$
$509$		−	390.000i		−	0.766208i	0.923705	−	0.383104i	$-0.125145\pi$
$510$		0			0
$511$	132.000			0.258317
$512$	−390.694	−	390.694i	−0.763073	−	0.763073i
$513$		0			0
$514$		−	12.0000i		−	0.0233463i
$515$		0			0
$516$		0			0
$517$	22.0454	+	22.0454i	0.0426410	+	0.0426410i
$518$	235.151	−	235.151i	0.453959	−	0.453959i
$519$		0			0
$520$		0			0
$521$	183.000			0.351248			0.175624	−	0.984457i	$-0.443806\pi$
$521$	183.000			0.351248			0.175624	+	0.984457i	$0.443806\pi$
$522$		0			0
$523$	−476.426	+	476.426i	−0.910948	+	0.910948i	−0.996347	−	0.0853989i	$-0.972784\pi$
$523$	−476.426	+	476.426i	−0.910948	+	0.910948i	0.0853989	+	0.996347i	$0.472784\pi$
$524$			162.000i			0.309160i
$525$		0			0
$526$	−18.0000			−0.0342205
$527$	−511.943	−	511.943i	−0.971430	−	0.971430i
$528$		0			0
$529$		−	59.0000i		−	0.111531i
$530$		0			0
$531$		0			0
$532$	24.4949	+	24.4949i	0.0460430	+	0.0460430i
$533$	−418.863	+	418.863i	−0.785859	+	0.785859i
$534$		0			0
$535$		0			0
$536$	585.000			1.09142
$537$		0			0
$538$	−146.969	+	146.969i	−0.273177	+	0.273177i
$539$		−	3.00000i		−	0.00556586i
$540$		0			0
$541$	−568.000			−1.04991			−0.524954	−	0.851131i	$-0.675917\pi$
$541$	−568.000			−1.04991			−0.524954	+	0.851131i	$0.675917\pi$
$542$	−71.0352	−	71.0352i	−0.131061	−	0.131061i
$543$		0			0
$544$		−	297.000i		−	0.545956i
$545$		0			0
$546$		0			0
$547$	−160.442	−	160.442i	−0.293312	−	0.293312i	0.545075	−	0.838387i	$-0.316501\pi$
$547$	−160.442	−	160.442i	−0.293312	−	0.293312i	−0.838387	+	0.545075i	$0.816501\pi$
$548$	50.2145	−	50.2145i	0.0916324	−	0.0916324i
$549$		0			0
$550$		0			0
$551$	−150.000			−0.272232
$552$		0			0
$553$	391.918	−	391.918i	0.708713	−	0.708713i
$554$		−	678.000i		−	1.22383i
$555$		0			0
$556$	185.000			0.332734
$557$	−66.1362	−	66.1362i	−0.118736	−	0.118736i	0.645242	−	0.763978i	$-0.276757\pi$
$557$	−66.1362	−	66.1362i	−0.118736	−	0.118736i	−0.763978	+	0.645242i	$0.776757\pi$
$558$		0			0
$559$			72.0000i			0.128801i
$560$		0			0
$561$		0			0
$562$	−565.832	−	565.832i	−1.00682	−	1.00682i
$563$	−191.060	+	191.060i	−0.339361	+	0.339361i	−0.856127	−	0.516766i	$-0.827136\pi$
$563$	−191.060	+	191.060i	−0.339361	+	0.339361i	0.516766	+	0.856127i	$0.327136\pi$
$564$		0			0
$565$		0			0
$566$	−177.000			−0.312721
$567$		0			0
$568$	−257.196	+	257.196i	−0.452811	+	0.452811i
$569$			45.0000i			0.0790861i	0.999218	+	0.0395431i	$0.0125902\pi$
$569$			45.0000i			0.0790861i	−0.999218	+	0.0395431i	$0.987410\pi$
$570$		0			0
$571$	542.000			0.949212			0.474606	−	0.880198i	$-0.342591\pi$
$571$	542.000			0.949212			0.474606	+	0.880198i	$0.342591\pi$
$572$	−22.0454	−	22.0454i	−0.0385409	−	0.0385409i
$573$		0			0
$574$		−	684.000i		−	1.19164i
$575$		0			0
$576$		0			0
$577$	549.910	+	549.910i	0.953051	+	0.953051i	0.998946	−	0.0458952i	$-0.0146140\pi$
$577$	549.910	+	549.910i	0.953051	+	0.953051i	−0.0458952	+	0.998946i	$0.514614\pi$
$578$	−90.6311	+	90.6311i	−0.156801	+	0.156801i
$579$		0			0
$580$		0			0
$581$	−1092.00			−1.87952
$582$		0			0
$583$	−95.5301	+	95.5301i	−0.163860	+	0.163860i
$584$		−	165.000i		−	0.282534i
$585$		0			0
$586$	12.0000			0.0204778
$587$	417.638	+	417.638i	0.711479	+	0.711479i	0.966845	−	0.255366i	$-0.0821959\pi$
$587$	417.638	+	417.638i	0.711479	+	0.711479i	−0.255366	+	0.966845i	$0.582196\pi$
$588$		0			0
$589$		−	190.000i		−	0.322581i
$590$		0			0
$591$		0			0
$592$	−215.555	−	215.555i	−0.364113	−	0.364113i
$593$	−331.906	+	331.906i	−0.559706	+	0.559706i	−0.929224	−	0.369517i	$-0.879523\pi$
$593$	−331.906	+	331.906i	−0.559706	+	0.559706i	0.369517	+	0.929224i	$0.379523\pi$
$594$		0			0
$595$		0			0
$596$	150.000			0.251678
$597$		0			0
$598$	308.636	−	308.636i	0.516113	−	0.516113i
$599$		−	900.000i		−	1.50250i	−0.660015	−	0.751252i	$-0.729450\pi$
$599$		−	900.000i		−	1.50250i	0.660015	−	0.751252i	$-0.270550\pi$
$600$		0			0
$601$	577.000			0.960067			0.480033	−	0.877250i	$-0.340625\pi$
$601$	577.000			0.960067			0.480033	+	0.877250i	$0.340625\pi$
$602$	−58.7878	−	58.7878i	−0.0976541	−	0.0976541i
$603$		0			0
$604$		−	52.0000i		−	0.0860927i
$605$		0			0
$606$		0			0
$607$	127.373	+	127.373i	0.209841	+	0.209841i	0.804200	−	0.594359i	$-0.202594\pi$
$607$	127.373	+	127.373i	0.209841	+	0.209841i	−0.594359	+	0.804200i	$0.702594\pi$
$608$	55.1135	−	55.1135i	0.0906472	−	0.0906472i
$609$		0			0
$610$		0			0
$611$	108.000			0.176759
$612$		0			0
$613$	−237.601	+	237.601i	−0.387603	+	0.387603i	−0.873832	−	0.486229i	$-0.838372\pi$
$613$	−237.601	+	237.601i	−0.387603	+	0.387603i	0.486229	+	0.873832i	$0.338372\pi$
$614$		−	213.000i		−	0.346906i
$615$		0			0
$616$	180.000			0.292208
$617$	656.463	+	656.463i	1.06396	+	1.06396i	0.997810	+	0.0661502i	$0.0210717\pi$
$617$	656.463	+	656.463i	1.06396	+	1.06396i	0.0661502	+	0.997810i	$0.478928\pi$
$618$		0			0
$619$		−	470.000i		−	0.759289i	−0.925132	−	0.379645i	$-0.876046\pi$
$619$		−	470.000i		−	0.759289i	0.925132	−	0.379645i	$-0.123954\pi$
$620$		0			0
$621$		0			0
$622$	646.665	+	646.665i	1.03965	+	1.03965i
$623$	−73.4847	+	73.4847i	−0.117953	+	0.117953i
$624$		0			0
$625$		0			0
$626$	468.000			0.747604
$627$		0			0
$628$	188.611	−	188.611i	0.300336	−	0.300336i
$629$		−	528.000i		−	0.839428i
$630$		0			0
$631$	−838.000			−1.32805			−0.664025	−	0.747710i	$-0.731153\pi$
$631$	−838.000			−1.32805			−0.664025	+	0.747710i	$0.731153\pi$
$632$	−489.898	−	489.898i	−0.775155	−	0.775155i
$633$		0			0
$634$		−	492.000i		−	0.776025i
$635$		0			0
$636$		0			0
$637$	−7.34847	−	7.34847i	−0.0115361	−	0.0115361i
$638$	−110.227	+	110.227i	−0.172770	+	0.172770i
$639$		0			0
$640$		0			0
$641$	618.000			0.964119			0.482059	−	0.876139i	$-0.339889\pi$
$641$	618.000			0.964119			0.482059	+	0.876139i	$0.339889\pi$
$642$		0			0
$643$	−249.848	+	249.848i	−0.388566	+	0.388566i	−0.874176	−	0.485610i	$-0.838598\pi$
$643$	−249.848	+	249.848i	−0.388566	+	0.388566i	0.485610	+	0.874176i	$0.338598\pi$
$644$		−	168.000i		−	0.260870i
$645$		0			0
$646$	−165.000			−0.255418
$647$	−421.312	−	421.312i	−0.651178	−	0.651178i	0.302099	−	0.953277i	$-0.402313\pi$
$647$	−421.312	−	421.312i	−0.651178	−	0.651178i	−0.953277	+	0.302099i	$0.902313\pi$
$648$		0			0
$649$		−	270.000i		−	0.416025i
$650$		0			0
$651$		0			0
$652$	−99.2043	−	99.2043i	−0.152154	−	0.152154i
$653$	519.292	−	519.292i	0.795240	−	0.795240i	−0.187101	−	0.982341i	$-0.559909\pi$
$653$	519.292	−	519.292i	0.795240	−	0.795240i	0.982341	+	0.187101i	$0.0599090\pi$
$654$		0			0
$655$		0			0
$656$	−627.000			−0.955793
$657$		0			0
$658$	−88.1816	+	88.1816i	−0.134015	+	0.134015i
$659$			885.000i			1.34294i	0.741030	+	0.671472i	$0.234338\pi$
$659$			885.000i			1.34294i	−0.741030	+	0.671472i	$0.765662\pi$
$660$		0			0
$661$	922.000			1.39486			0.697428	−	0.716655i	$-0.254328\pi$
$661$	922.000			1.39486			0.697428	+	0.716655i	$0.254328\pi$
$662$	−383.345	−	383.345i	−0.579071	−	0.579071i
$663$		0			0
$664$			1365.00i			2.05572i
$665$		0			0
$666$		0			0
$667$	514.393	+	514.393i	0.771204	+	0.771204i
$668$	−17.1464	+	17.1464i	−0.0256683	+	0.0256683i
$669$		0			0
$670$		0			0
$671$	−84.0000			−0.125186
$672$		0			0
$673$	−78.3837	+	78.3837i	−0.116469	+	0.116469i	−0.762939	−	0.646470i	$-0.776244\pi$
$673$	−78.3837	+	78.3837i	−0.116469	+	0.116469i	0.646470	+	0.762939i	$0.276244\pi$
$674$			477.000i			0.707715i
$675$		0			0
$676$	61.0000			0.0902367
$677$	705.453	+	705.453i	1.04203	+	1.04203i	0.999077	+	0.0429510i	$0.0136759\pi$
$677$	705.453	+	705.453i	1.04203	+	1.04203i	0.0429510	+	0.999077i	$0.486324\pi$
$678$		0			0
$679$			528.000i			0.777614i
$680$		0			0
$681$		0			0
$682$	−139.621	−	139.621i	−0.204723	−	0.204723i
$683$	133.497	−	133.497i	0.195457	−	0.195457i	−0.602592	−	0.798049i	$-0.705865\pi$
$683$	133.497	−	133.497i	0.195457	−	0.195457i	0.798049	+	0.602592i	$0.205865\pi$
$684$		0			0
$685$		0			0
$686$	600.000			0.874636
$687$		0			0
$688$	−53.8888	+	53.8888i	−0.0783267	+	0.0783267i
$689$			468.000i			0.679245i
$690$		0			0
$691$	−893.000			−1.29233			−0.646165	−	0.763198i	$-0.723628\pi$
$691$	−893.000			−1.29233			−0.646165	+	0.763198i	$0.723628\pi$
$692$	−78.3837	−	78.3837i	−0.113271	−	0.113271i
$693$		0			0
$694$			243.000i			0.350144i
$695$		0			0
$696$		0			0
$697$	−767.915	−	767.915i	−1.10174	−	1.10174i
$698$	122.474	−	122.474i	0.175465	−	0.175465i
$699$		0			0
$700$		0			0
$701$	−402.000			−0.573466			−0.286733	−	0.958010i	$-0.592569\pi$
$701$	−402.000			−0.573466			−0.286733	+	0.958010i	$0.592569\pi$
$702$		0			0
$703$	97.9796	−	97.9796i	0.139374	−	0.139374i
$704$		−	213.000i		−	0.302557i
$705$		0			0
$706$	972.000			1.37677
$707$	235.151	+	235.151i	0.332604	+	0.332604i
$708$		0			0
$709$		−	1060.00i		−	1.49506i	−0.664226	−	0.747532i	$-0.731239\pi$
$709$		−	1060.00i		−	1.49506i	0.664226	−	0.747532i	$-0.268761\pi$
$710$		0			0
$711$		0			0
$712$	91.8559	+	91.8559i	0.129011	+	0.129011i
$713$	−651.564	+	651.564i	−0.913835	+	0.913835i
$714$		0			0
$715$		0			0
$716$	−195.000			−0.272346
$717$		0			0
$718$	661.362	−	661.362i	0.921117	−	0.921117i
$719$			1320.00i			1.83588i	0.396716	+	0.917942i	$0.370150\pi$
$719$			1320.00i			1.83588i	−0.396716	+	0.917942i	$0.629850\pi$
$720$		0			0
$721$	−888.000			−1.23162
$722$	411.514	+	411.514i	0.569964	+	0.569964i
$723$		0			0
$724$		−	262.000i		−	0.361878i
$725$		0			0
$726$		0			0
$727$	−31.8434	−	31.8434i	−0.0438011	−	0.0438011i	0.684867	−	0.728668i	$-0.259860\pi$
$727$	−31.8434	−	31.8434i	−0.0438011	−	0.0438011i	−0.728668	+	0.684867i	$0.759860\pi$
$728$	440.908	−	440.908i	0.605643	−	0.605643i
$729$		0			0
$730$		0			0
$731$	−132.000			−0.180575
$732$		0			0
$733$	815.680	−	815.680i	1.11280	−	1.11280i	0.120026	−	0.992771i	$-0.461702\pi$
$733$	815.680	−	815.680i	1.11280	−	1.11280i	0.992771	−	0.120026i	$-0.0382978\pi$
$734$		−	108.000i		−	0.147139i
$735$		0			0
$736$	−378.000			−0.513587
$737$	143.295	+	143.295i	0.194430	+	0.194430i
$738$		0			0
$739$			710.000i			0.960758i	0.877061	+	0.480379i	$0.159501\pi$
$739$			710.000i			0.960758i	−0.877061	+	0.480379i	$0.840499\pi$
$740$		0			0
$741$		0			0
$742$	−382.120	−	382.120i	−0.514987	−	0.514987i
$743$	800.983	−	800.983i	1.07804	−	1.07804i	0.0813540	−	0.996685i	$-0.474076\pi$
$743$	800.983	−	800.983i	1.07804	−	1.07804i	0.996685	−	0.0813540i	$-0.0259244\pi$
$744$		0			0
$745$		0			0
$746$	858.000			1.15013
$747$		0			0
$748$	40.4166	−	40.4166i	0.0540329	−	0.0540329i
$749$			252.000i			0.336449i
$750$		0			0
$751$	502.000			0.668442			0.334221	−	0.942495i	$-0.391527\pi$
$751$	502.000			0.668442			0.334221	+	0.942495i	$0.391527\pi$
$752$	80.8332	+	80.8332i	0.107491	+	0.107491i
$753$		0			0
$754$			540.000i			0.716180i
$755$		0			0
$756$		0			0
$757$	4.89898	+	4.89898i	0.00647157	+	0.00647157i	0.710335	−	0.703864i	$-0.248543\pi$
$757$	4.89898	+	4.89898i	0.00647157	+	0.00647157i	−0.703864	+	0.710335i	$0.748543\pi$
$758$	618.496	−	618.496i	0.815958	−	0.815958i
$759$		0			0
$760$		0			0
$761$	−747.000			−0.981603			−0.490802	−	0.871271i	$-0.663296\pi$
$761$	−747.000			−0.981603			−0.490802	+	0.871271i	$0.663296\pi$
$762$		0			0
$763$	−195.959	+	195.959i	−0.256827	+	0.256827i
$764$		−	18.0000i		−	0.0235602i
$765$		0			0
$766$	−348.000			−0.454308
$767$	−661.362	−	661.362i	−0.862271	−	0.862271i
$768$		0			0
$769$			1255.00i			1.63199i	0.578059	+	0.815995i	$0.303810\pi$
$769$			1255.00i			1.63199i	−0.578059	+	0.815995i	$0.696190\pi$
$770$		0			0
$771$		0			0
$772$	−86.9569	−	86.9569i	−0.112638	−	0.112638i
$773$	17.1464	−	17.1464i	0.0221817	−	0.0221817i	−0.695929	−	0.718111i	$-0.745007\pi$
$773$	17.1464	−	17.1464i	0.0221817	−	0.0221817i	0.718111	+	0.695929i	$0.245007\pi$
$774$		0			0
$775$		0			0
$776$	660.000			0.850515
$777$		0			0
$778$	−845.074	+	845.074i	−1.08621	+	1.08621i
$779$		−	285.000i		−	0.365854i
$780$		0			0
$781$	−126.000			−0.161332
$782$	565.832	+	565.832i	0.723570	+	0.723570i
$783$		0			0
$784$		−	11.0000i		−	0.0140306i
$785$		0			0
$786$		0			0
$787$	592.777	+	592.777i	0.753210	+	0.753210i	0.975077	−	0.221867i	$-0.0712150\pi$
$787$	592.777	+	592.777i	0.753210	+	0.753210i	−0.221867	+	0.975077i	$0.571215\pi$
$788$	129.823	−	129.823i	0.164750	−	0.164750i
$789$		0			0
$790$		0			0
$791$	828.000			1.04678
$792$		0			0
$793$	−205.757	+	205.757i	−0.259467	+	0.259467i
$794$			1032.00i			1.29975i
$795$		0			0
$796$	−200.000			−0.251256
$797$	−605.024	−	605.024i	−0.759127	−	0.759127i	0.217037	−	0.976163i	$-0.430361\pi$
$797$	−605.024	−	605.024i	−0.759127	−	0.759127i	−0.976163	+	0.217037i	$0.930361\pi$
$798$		0			0
$799$			198.000i			0.247810i
$800$		0			0
$801$		0			0
$802$	701.779	+	701.779i	0.875036	+	0.875036i
$803$	40.4166	−	40.4166i	0.0503320	−	0.0503320i
$804$		0			0
$805$		0			0
$806$	−684.000			−0.848635
$807$		0			0
$808$	293.939	−	293.939i	0.363786	−	0.363786i
$809$		−	90.0000i		−	0.111248i	−0.998452	−	0.0556242i	$-0.982285\pi$
$809$		−	90.0000i		−	0.111248i	0.998452	−	0.0556242i	$-0.0177149\pi$
$810$		0			0
$811$	−278.000			−0.342787			−0.171393	−	0.985203i	$-0.554827\pi$
$811$	−278.000			−0.342787			−0.171393	+	0.985203i	$0.554827\pi$
$812$	146.969	+	146.969i	0.180997	+	0.180997i
$813$		0			0
$814$		−	144.000i		−	0.176904i
$815$		0			0
$816$		0			0
$817$	−24.4949	−	24.4949i	−0.0299815	−	0.0299815i
$818$	−447.032	+	447.032i	−0.546494	+	0.546494i
$819$		0			0
$820$		0			0
$821$	−42.0000			−0.0511571			−0.0255786	−	0.999673i	$-0.508143\pi$
$821$	−42.0000			−0.0511571			−0.0255786	+	0.999673i	$0.508143\pi$
$822$		0			0
$823$	778.938	−	778.938i	0.946461	−	0.946461i	−0.0521765	−	0.998638i	$-0.516616\pi$
$823$	778.938	−	778.938i	0.946461	−	0.946461i	0.998638	+	0.0521765i	$0.0166158\pi$
$824$			1110.00i			1.34709i
$825$		0			0
$826$	1080.00			1.30751
$827$	674.834	+	674.834i	0.816003	+	0.816003i	0.985526	−	0.169523i	$-0.0542228\pi$
$827$	674.834	+	674.834i	0.816003	+	0.816003i	−0.169523	+	0.985526i	$0.554223\pi$
$828$		0			0
$829$			170.000i			0.205066i	0.994730	+	0.102533i	$0.0326948\pi$
$829$			170.000i			0.205066i	−0.994730	+	0.102533i	$0.967305\pi$
$830$		0			0
$831$		0			0
$832$	−521.741	−	521.741i	−0.627093	−	0.627093i
$833$	13.4722	−	13.4722i	0.0161731	−	0.0161731i
$834$		0			0
$835$		0			0
$836$	15.0000			0.0179426
$837$		0			0
$838$	−789.960	+	789.960i	−0.942674	+	0.942674i
$839$		−	360.000i		−	0.429082i	−0.976715	−	0.214541i	$-0.931174\pi$
$839$		−	360.000i		−	0.429082i	0.976715	−	0.214541i	$-0.0688256\pi$
$840$		0			0
$841$	−59.0000			−0.0701546
$842$	−9.79796	−	9.79796i	−0.0116365	−	0.0116365i
$843$		0			0
$844$			103.000i			0.122038i
$845$		0			0
$846$		0			0
$847$	−548.686	−	548.686i	−0.647799	−	0.647799i
$848$	−350.277	+	350.277i	−0.413063	+	0.413063i
$849$		0			0
$850$		0			0
$851$	−672.000			−0.789659
$852$		0			0
$853$	521.741	−	521.741i	0.611655	−	0.611655i	−0.331722	−	0.943377i	$-0.607630\pi$
$853$	521.741	−	521.741i	0.611655	−	0.611655i	0.943377	+	0.331722i	$0.107630\pi$
$854$		−	336.000i		−	0.393443i
$855$		0			0
$856$	315.000			0.367991
$857$	−280.467	−	280.467i	−0.327266	−	0.327266i	0.524280	−	0.851546i	$-0.324334\pi$
$857$	−280.467	−	280.467i	−0.327266	−	0.327266i	−0.851546	+	0.524280i	$0.824334\pi$
$858$		0			0
$859$		−	335.000i		−	0.389988i	−0.980804	−	0.194994i	$-0.937531\pi$
$859$		−	335.000i		−	0.389988i	0.980804	−	0.194994i	$-0.0624688\pi$
$860$		0			0
$861$		0			0
$862$	−14.6969	−	14.6969i	−0.0170498	−	0.0170498i
$863$	849.973	−	849.973i	0.984905	−	0.984905i	−0.0149828	−	0.999888i	$-0.504769\pi$
$863$	849.973	−	849.973i	0.984905	−	0.984905i	0.999888	+	0.0149828i	$0.00476936\pi$
$864$		0			0
$865$		0			0
$866$	−1077.00			−1.24365
$867$		0			0
$868$	−186.161	+	186.161i	−0.214471	+	0.214471i
$869$		−	240.000i		−	0.276180i
$870$		0			0
$871$	702.000			0.805970
$872$	244.949	+	244.949i	0.280905	+	0.280905i
$873$		0			0
$874$			210.000i			0.240275i
$875$		0			0
$876$		0			0
$877$	151.868	+	151.868i	0.173168	+	0.173168i	0.788370	−	0.615202i	$-0.210925\pi$
$877$	151.868	+	151.868i	0.173168	+	0.173168i	−0.615202	+	0.788370i	$0.710925\pi$
$878$	−12.2474	+	12.2474i	−0.0139493	+	0.0139493i
$879$		0			0
$880$		0			0
$881$	−162.000			−0.183882			−0.0919410	−	0.995764i	$-0.529307\pi$
$881$	−162.000			−0.183882			−0.0919410	+	0.995764i	$0.529307\pi$
$882$		0			0
$883$	295.164	−	295.164i	0.334274	−	0.334274i	−0.519933	−	0.854207i	$-0.674043\pi$
$883$	295.164	−	295.164i	0.334274	−	0.334274i	0.854207	+	0.519933i	$0.174043\pi$
$884$		−	198.000i		−	0.223982i
$885$		0			0
$886$	−213.000			−0.240406
$887$	509.494	+	509.494i	0.574401	+	0.574401i	0.933355	−	0.358954i	$-0.116867\pi$
$887$	509.494	+	509.494i	0.574401	+	0.574401i	−0.358954	+	0.933355i	$0.616867\pi$
$888$		0			0
$889$		−	1512.00i		−	1.70079i
$890$		0			0
$891$		0			0
$892$	200.858	+	200.858i	0.225177	+	0.225177i
$893$	−36.7423	+	36.7423i	−0.0411448	+	0.0411448i
$894$		0			0
$895$		0			0
$896$	420.000			0.468750
$897$		0			0
$898$	−91.8559	+	91.8559i	−0.102289	+	0.102289i
$899$		−	1140.00i		−	1.26808i
$900$		0			0
$901$	−858.000			−0.952275
$902$	−209.431	−	209.431i	−0.232186	−	0.232186i
$903$		0			0
$904$		−	1035.00i		−	1.14491i
$905$		0			0
$906$		0			0
$907$	494.797	+	494.797i	0.545531	+	0.545531i	0.925145	−	0.379614i	$-0.123943\pi$
$907$	494.797	+	494.797i	0.545531	+	0.545531i	−0.379614	+	0.925145i	$0.623943\pi$
$908$	−274.343	+	274.343i	−0.302140	+	0.302140i
$909$		0			0
$910$		0			0
$911$	828.000			0.908891			0.454446	−	0.890774i	$-0.349837\pi$
$911$	828.000			0.908891			0.454446	+	0.890774i	$0.349837\pi$
$912$		0			0
$913$	−334.355	+	334.355i	−0.366216	+	0.366216i
$914$		−	513.000i		−	0.561269i
$915$		0			0
$916$	20.0000			0.0218341
$917$	−793.635	−	793.635i	−0.865469	−	0.865469i
$918$		0			0
$919$		−	1270.00i		−	1.38194i	−0.722885	−	0.690968i	$-0.757184\pi$
$919$		−	1270.00i		−	1.38194i	0.722885	−	0.690968i	$-0.242816\pi$
$920$		0			0
$921$		0			0
$922$	279.242	+	279.242i	0.302865	+	0.302865i
$923$	−308.636	+	308.636i	−0.334383	+	0.334383i
$924$		0			0
$925$		0			0
$926$	1068.00			1.15335
$927$		0			0
$928$	330.681	−	330.681i	0.356337	−	0.356337i
$929$		−	570.000i		−	0.613563i	−0.951780	−	0.306781i	$-0.900748\pi$
$929$		−	570.000i		−	0.613563i	0.951780	−	0.306781i	$-0.0992521\pi$
$930$		0			0
$931$	5.00000			0.00537057
$932$	−102.879	−	102.879i	−0.110385	−	0.110385i
$933$		0			0
$934$			1308.00i			1.40043i
$935$		0			0
$936$		0			0
$937$	317.209	+	317.209i	0.338537	+	0.338537i	0.855816	−	0.517280i	$-0.173055\pi$
$937$	317.209	+	317.209i	0.338537	+	0.338537i	−0.517280	+	0.855816i	$0.673055\pi$
$938$	−573.181	+	573.181i	−0.611067	+	0.611067i
$939$		0			0
$940$		0			0
$941$	1518.00			1.61318			0.806589	−	0.591113i	$-0.201311\pi$
$941$	1518.00			1.61318			0.806589	+	0.591113i	$0.201311\pi$
$942$		0			0
$943$	−977.346	+	977.346i	−1.03642	+	1.03642i
$944$		−	990.000i		−	1.04873i
$945$		0			0
$946$	−36.0000			−0.0380550
$947$	436.009	+	436.009i	0.460411	+	0.460411i	0.898790	−	0.438379i	$-0.144447\pi$
$947$	436.009	+	436.009i	0.460411	+	0.460411i	−0.438379	+	0.898790i	$0.644447\pi$
$948$		0			0
$949$		−	198.000i		−	0.208641i
$950$		0			0
$951$		0			0
$952$	808.332	+	808.332i	0.849088	+	0.849088i
$953$	−429.885	+	429.885i	−0.451087	+	0.451087i	−0.895715	−	0.444629i	$-0.853336\pi$
$953$	−429.885	+	429.885i	−0.451087	+	0.451087i	0.444629	+	0.895715i	$0.353336\pi$
$954$		0			0
$955$		0			0
$956$	−210.000			−0.219665
$957$		0			0
$958$	330.681	−	330.681i	0.345179	−	0.345179i
$959$			492.000i			0.513034i
$960$		0			0
$961$	483.000			0.502601
$962$	−352.727	−	352.727i	−0.366660	−	0.366660i
$963$		0			0
$964$			43.0000i			0.0446058i
$965$		0			0
$966$		0			0
$967$	−533.989	−	533.989i	−0.552212	−	0.552212i	0.374867	−	0.927079i	$-0.377688\pi$
$967$	−533.989	−	533.989i	−0.552212	−	0.552212i	−0.927079	+	0.374867i	$0.877688\pi$
$968$	−685.857	+	685.857i	−0.708530	+	0.708530i
$969$		0			0
$970$		0			0
$971$	−1467.00			−1.51081			−0.755407	−	0.655256i	$-0.772561\pi$
$971$	−1467.00			−1.51081			−0.755407	+	0.655256i	$0.772561\pi$
$972$		0			0
$973$	−906.311	+	906.311i	−0.931461	+	0.931461i
$974$		−	198.000i		−	0.203285i
$975$		0			0
$976$	−308.000			−0.315574
$977$	−1039.81	−	1039.81i	−1.06429	−	1.06429i	−0.997786	−	0.0665006i	$-0.978817\pi$
$977$	−1039.81	−	1039.81i	−1.06429	−	1.06429i	−0.0665006	−	0.997786i	$-0.521183\pi$
$978$		0			0
$979$			45.0000i			0.0459653i
$980$		0			0
$981$		0			0
$982$	−712.802	−	712.802i	−0.725867	−	0.725867i
$983$	715.251	−	715.251i	0.727621	−	0.727621i	−0.242525	−	0.970145i	$-0.577976\pi$
$983$	715.251	−	715.251i	0.727621	−	0.727621i	0.970145	+	0.242525i	$0.0779755\pi$
$984$		0			0
$985$		0			0
$986$	−990.000			−1.00406
$987$		0			0
$988$	36.7423	−	36.7423i	0.0371886	−	0.0371886i
$989$			168.000i			0.169869i
$990$		0			0
$991$	1832.00			1.84864			0.924319	−	0.381621i	$-0.124634\pi$
$991$	1832.00			1.84864			0.924319	+	0.381621i	$0.124634\pi$
$992$	418.863	+	418.863i	0.422241	+	0.422241i
$993$		0			0
$994$		−	504.000i		−	0.507042i
$995$		0			0
$996$		0			0
$997$	−1293.33	−	1293.33i	−1.29722	−	1.29722i	−0.930220	−	0.367002i	$-0.880384\pi$
$997$	−1293.33	−	1293.33i	−1.29722	−	1.29722i	−0.367002	−	0.930220i	$-0.619616\pi$
$998$	−306.186	+	306.186i	−0.306800	+	0.306800i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.3.g.e.82.2		4
3.2	odd	2		25.3.c.a.7.1	✓	4
5.2	odd	4	inner	225.3.g.e.118.1		4
5.3	odd	4	inner	225.3.g.e.118.2		4
5.4	even	2	inner	225.3.g.e.82.1		4
12.11	even	2		400.3.p.j.257.1		4
15.2	even	4		25.3.c.a.18.2	yes	4
15.8	even	4		25.3.c.a.18.1	yes	4
15.14	odd	2		25.3.c.a.7.2	yes	4
60.23	odd	4		400.3.p.j.193.1		4
60.47	odd	4		400.3.p.j.193.2		4
60.59	even	2		400.3.p.j.257.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.3.c.a.7.1	✓	4	3.2	odd	2
25.3.c.a.7.2	yes	4	15.14	odd	2
25.3.c.a.18.1	yes	4	15.8	even	4
25.3.c.a.18.2	yes	4	15.2	even	4
225.3.g.e.82.1		4	5.4	even	2	inner
225.3.g.e.82.2		4	1.1	even	1	trivial
225.3.g.e.118.1		4	5.2	odd	4	inner
225.3.g.e.118.2		4	5.3	odd	4	inner
400.3.p.j.193.1		4	60.23	odd	4
400.3.p.j.193.2		4	60.47	odd	4
400.3.p.j.257.1		4	12.11	even	2
400.3.p.j.257.2		4	60.59	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{2} -1.00000i q^{4} +(4.89898 + 4.89898i) q^{7} +(6.12372 - 6.12372i) q^{8} +3.00000 q^{11} +(7.34847 - 7.34847i) q^{13} +12.0000i q^{14} +11.0000 q^{16} +(13.4722 + 13.4722i) q^{17} +5.00000i q^{19} +(3.67423 + 3.67423i) q^{22} +(17.1464 - 17.1464i) q^{23} +18.0000 q^{26} +(4.89898 - 4.89898i) q^{28} +30.0000i q^{29} -38.0000 q^{31} +(-11.0227 - 11.0227i) q^{32} +33.0000i q^{34} +(-19.5959 - 19.5959i) q^{37} +(-6.12372 + 6.12372i) q^{38} -57.0000 q^{41} +(-4.89898 + 4.89898i) q^{43} -3.00000i q^{44} +42.0000 q^{46} +(7.34847 + 7.34847i) q^{47} -1.00000i q^{49} +(-7.34847 - 7.34847i) q^{52} +(-31.8434 + 31.8434i) q^{53} +60.0000 q^{56} +(-36.7423 + 36.7423i) q^{58} -90.0000i q^{59} -28.0000 q^{61} +(-46.5403 - 46.5403i) q^{62} -71.0000i q^{64} +(47.7650 + 47.7650i) q^{67} +(13.4722 - 13.4722i) q^{68} -42.0000 q^{71} +(13.4722 - 13.4722i) q^{73} -48.0000i q^{74} +5.00000 q^{76} +(14.6969 + 14.6969i) q^{77} -80.0000i q^{79} +(-69.8105 - 69.8105i) q^{82} +(-111.452 + 111.452i) q^{83} -12.0000 q^{86} +(18.3712 - 18.3712i) q^{88} +15.0000i q^{89} +72.0000 q^{91} +(-17.1464 - 17.1464i) q^{92} +18.0000i q^{94} +(53.8888 + 53.8888i) q^{97} +(1.22474 - 1.22474i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{11} + 44 q^{16} + 72 q^{26} - 152 q^{31} - 228 q^{41} + 168 q^{46} + 240 q^{56} - 112 q^{61} - 168 q^{71} + 20 q^{76} - 48 q^{86} + 288 q^{91}+O(q^{100}) \) 4 * q + 12 * q^11 + 44 * q^16 + 72 * q^26 - 152 * q^31 - 228 * q^41 + 168 * q^46 + 240 * q^56 - 112 * q^61 - 168 * q^71 + 20 * q^76 - 48 * q^86 + 288 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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