LMFDB - Embedded newform 882.4.g.ba.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,4,Mod(361,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("882.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.g (of order $3$, degree $2$, not 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:	$52.0396846251$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 98)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	882.361
Dual form			882.4.g.ba.667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-9.89949 - 17.1464i) q^{5} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-9.89949 - 17.1464i) q^{5} +8.00000 q^{8} +(-19.7990 + 34.2929i) q^{10} +(-7.00000 + 12.1244i) q^{11} +50.9117 q^{13} +(-8.00000 - 13.8564i) q^{16} +(0.707107 - 1.22474i) q^{17} +(0.707107 + 1.22474i) q^{19} +79.1960 q^{20} +28.0000 q^{22} +(70.0000 + 121.244i) q^{23} +(-133.500 + 231.229i) q^{25} +(-50.9117 - 88.1816i) q^{26} +286.000 q^{29} +(46.6690 - 80.8332i) q^{31} +(-16.0000 + 27.7128i) q^{32} -2.82843 q^{34} +(19.0000 + 32.9090i) q^{37} +(1.41421 - 2.44949i) q^{38} +(-79.1960 - 137.171i) q^{40} +125.865 q^{41} -34.0000 q^{43} +(-28.0000 - 48.4974i) q^{44} +(140.000 - 242.487i) q^{46} +(261.630 + 453.156i) q^{47} +534.000 q^{50} +(-101.823 + 176.363i) q^{52} +(-37.0000 + 64.0859i) q^{53} +277.186 q^{55} +(-286.000 - 495.367i) q^{58} +(217.082 - 375.997i) q^{59} +(-7.07107 - 12.2474i) q^{61} -186.676 q^{62} +64.0000 q^{64} +(-504.000 - 872.954i) q^{65} +(-342.000 + 592.361i) q^{67} +(2.82843 + 4.89898i) q^{68} -588.000 q^{71} +(135.057 - 233.926i) q^{73} +(38.0000 - 65.8179i) q^{74} -5.65685 q^{76} +(-610.000 - 1056.55i) q^{79} +(-158.392 + 274.343i) q^{80} +(-125.865 - 218.005i) q^{82} -422.850 q^{83} -28.0000 q^{85} +(34.0000 + 58.8897i) q^{86} +(-56.0000 + 96.9948i) q^{88} +(309.006 + 535.214i) q^{89} -560.000 q^{92} +(523.259 - 906.311i) q^{94} +(14.0000 - 24.2487i) q^{95} +1483.51 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8	$ 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 28 q^{11} - 32 q^{16} + 112 q^{22} + 280 q^{23} - 534 q^{25} + 1144 q^{29} - 64 q^{32} + 76 q^{37} - 136 q^{43} - 112 q^{44} + 560 q^{46} + 2136 q^{50} - 148 q^{53} - 1144 q^{58} + 256 q^{64} - 2016 q^{65} - 1368 q^{67} - 2352 q^{71} + 152 q^{74} - 2440 q^{79} - 112 q^{85} + 136 q^{86} - 224 q^{88} - 2240 q^{92} + 56 q^{95}+O(q^{100}) $ 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8 - 28 * q^11 - 32 * q^16 + 112 * q^22 + 280 * q^23 - 534 * q^25 + 1144 * q^29 - 64 * q^32 + 76 * q^37 - 136 * q^43 - 112 * q^44 + 560 * q^46 + 2136 * q^50 - 148 * q^53 - 1144 * q^58 + 256 * q^64 - 2016 * q^65 - 1368 * q^67 - 2352 * q^71 + 152 * q^74 - 2440 * q^79 - 112 * q^85 + 136 * q^86 - 224 * q^88 - 2240 * q^92 + 56 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	−1.00000	−	1.73205i	−0.353553	−	0.612372i
$2$	−1.00000	−	1.73205i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	−9.89949	−	17.1464i	−0.885438	−	1.53362i	−0.845211	−	0.534433i	$-0.820525\pi$
$5$	−9.89949	−	17.1464i	−0.885438	−	1.53362i	−0.0402266	−	0.999191i	$-0.512808\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	8.00000			0.353553
$9$		0			0
$10$	−19.7990	+	34.2929i	−0.626099	+	1.08444i
$11$	−7.00000	+	12.1244i	−0.191871	+	0.332330i	−0.945870	−	0.324545i	$-0.894789\pi$
$11$	−7.00000	+	12.1244i	−0.191871	+	0.332330i	0.753999	+	0.656875i	$0.228122\pi$
$12$		0			0
$13$	50.9117			1.08618			0.543091	−	0.839674i	$-0.317254\pi$
$13$	50.9117			1.08618			0.543091	+	0.839674i	$0.317254\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	0.707107	−	1.22474i	0.0100882	−	0.0174732i	−0.860937	−	0.508711i	$-0.830122\pi$
$17$	0.707107	−	1.22474i	0.0100882	−	0.0174732i	0.871025	+	0.491238i	$0.163455\pi$
$18$		0			0
$19$	0.707107	+	1.22474i	0.00853797	+	0.0147882i	0.870263	−	0.492588i	$-0.163949\pi$
$19$	0.707107	+	1.22474i	0.00853797	+	0.0147882i	−0.861725	+	0.507376i	$0.830616\pi$
$20$	79.1960			0.885438
$21$		0			0
$22$	28.0000			0.271346
$23$	70.0000	+	121.244i	0.634609	+	1.09918i	0.986598	+	0.163171i	$0.0521722\pi$
$23$	70.0000	+	121.244i	0.634609	+	1.09918i	−0.351989	+	0.936004i	$0.614494\pi$
$24$		0			0
$25$	−133.500	+	231.229i	−1.06800	+	1.84983i
$26$	−50.9117	−	88.1816i	−0.384023	−	0.665148i
$27$		0			0
$28$		0			0
$29$	286.000			1.83134			0.915670	−	0.401931i	$-0.131661\pi$
$29$	286.000			1.83134			0.915670	+	0.401931i	$0.131661\pi$
$30$		0			0
$31$	46.6690	−	80.8332i	0.270387	−	0.468325i	−0.698574	−	0.715538i	$-0.746182\pi$
$31$	46.6690	−	80.8332i	0.270387	−	0.468325i	0.968961	+	0.247213i	$0.0795149\pi$
$32$	−16.0000	+	27.7128i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−2.82843			−0.0142668
$35$		0			0
$36$		0			0
$37$	19.0000	+	32.9090i	0.0844211	+	0.146222i	0.905144	−	0.425104i	$-0.139763\pi$
$37$	19.0000	+	32.9090i	0.0844211	+	0.146222i	−0.820723	+	0.571326i	$0.806429\pi$
$38$	1.41421	−	2.44949i	0.00603726	−	0.0104568i
$39$		0			0
$40$	−79.1960	−	137.171i	−0.313050	−	0.542218i
$41$	125.865			0.479434			0.239717	−	0.970843i	$-0.422945\pi$
$41$	125.865			0.479434			0.239717	+	0.970843i	$0.422945\pi$
$42$		0			0
$43$	−34.0000			−0.120580			−0.0602901	−	0.998181i	$-0.519203\pi$
$43$	−34.0000			−0.120580			−0.0602901	+	0.998181i	$0.519203\pi$
$44$	−28.0000	−	48.4974i	−0.0959354	−	0.166165i
$45$		0			0
$46$	140.000	−	242.487i	0.448736	−	0.777234i
$47$	261.630	+	453.156i	0.811970	+	1.40637i	0.911483	+	0.411337i	$0.134938\pi$
$47$	261.630	+	453.156i	0.811970	+	1.40637i	−0.0995134	+	0.995036i	$0.531729\pi$
$48$		0			0
$49$		0			0
$50$	534.000			1.51038
$51$		0			0
$52$	−101.823	+	176.363i	−0.271545	+	0.470330i
$53$	−37.0000	+	64.0859i	−0.0958932	+	0.166092i	−0.909981	−	0.414650i	$-0.863904\pi$
$53$	−37.0000	+	64.0859i	−0.0958932	+	0.166092i	0.814088	+	0.580742i	$0.197237\pi$
$54$		0			0
$55$	277.186			0.679559
$56$		0			0
$57$		0			0
$58$	−286.000	−	495.367i	−0.647477	−	1.12146i
$59$	217.082	−	375.997i	0.479011	−	0.829671i	−0.520699	−	0.853740i	$-0.674329\pi$
$59$	217.082	−	375.997i	0.479011	−	0.829671i	0.999710	+	0.0240689i	$0.00766211\pi$
$60$		0			0
$61$	−7.07107	−	12.2474i	−0.0148419	−	0.0257070i	0.858509	−	0.512798i	$-0.171391\pi$
$61$	−7.07107	−	12.2474i	−0.0148419	−	0.0257070i	−0.873351	+	0.487091i	$0.838058\pi$
$62$	−186.676			−0.382385
$63$		0			0
$64$	64.0000			0.125000
$65$	−504.000	−	872.954i	−0.961746	−	1.66579i
$66$		0			0
$67$	−342.000	+	592.361i	−0.623611	+	1.08013i	0.365196	+	0.930930i	$0.381002\pi$
$67$	−342.000	+	592.361i	−0.623611	+	1.08013i	−0.988808	+	0.149196i	$0.952332\pi$
$68$	2.82843	+	4.89898i	0.00504408	+	0.00873660i
$69$		0			0
$70$		0			0
$71$	−588.000			−0.982856			−0.491428	−	0.870918i	$-0.663525\pi$
$71$	−588.000			−0.982856			−0.491428	+	0.870918i	$0.663525\pi$
$72$		0			0
$73$	135.057	−	233.926i	0.216538	−	0.375055i	−0.737209	−	0.675664i	$-0.763857\pi$
$73$	135.057	−	233.926i	0.216538	−	0.375055i	0.953747	+	0.300610i	$0.0971902\pi$
$74$	38.0000	−	65.8179i	0.0596947	−	0.103394i
$75$		0			0
$76$	−5.65685			−0.00853797
$77$		0			0
$78$		0			0
$79$	−610.000	−	1056.55i	−0.868739	−	1.50470i	−0.863286	−	0.504715i	$-0.831598\pi$
$79$	−610.000	−	1056.55i	−0.868739	−	1.50470i	−0.00545246	−	0.999985i	$-0.501736\pi$
$80$	−158.392	+	274.343i	−0.221359	+	0.383406i
$81$		0			0
$82$	−125.865	−	218.005i	−0.169506	−	0.293592i
$83$	−422.850			−0.559202			−0.279601	−	0.960116i	$-0.590202\pi$
$83$	−422.850			−0.559202			−0.279601	+	0.960116i	$0.590202\pi$
$84$		0			0
$85$	−28.0000			−0.0357297
$86$	34.0000	+	58.8897i	0.0426316	+	0.0738400i
$87$		0			0
$88$	−56.0000	+	96.9948i	−0.0678366	+	0.117496i
$89$	309.006	+	535.214i	0.368028	+	0.637444i	0.989257	−	0.146185i	$-0.0466996\pi$
$89$	309.006	+	535.214i	0.368028	+	0.637444i	−0.621229	+	0.783629i	$0.713366\pi$
$90$		0			0
$91$		0			0
$92$	−560.000			−0.634609
$93$		0			0
$94$	523.259	−	906.311i	0.574149	−	0.994456i
$95$	14.0000	−	24.2487i	0.0151197	−	0.0261881i
$96$		0			0
$97$	1483.51			1.55286			0.776431	−	0.630202i	$-0.217028\pi$
$97$	1483.51			1.55286			0.776431	+	0.630202i	$0.217028\pi$
$98$		0			0
$99$		0			0
$100$	−534.000	−	924.915i	−0.534000	−	0.924915i
$101$	564.271	−	977.346i	0.555912	−	0.962867i	−0.441920	−	0.897054i	$-0.645703\pi$
$101$	564.271	−	977.346i	0.555912	−	0.962867i	0.997832	−	0.0658130i	$-0.0209641\pi$
$102$		0			0
$103$	434.164	+	751.993i	0.415334	+	0.719380i	0.995463	−	0.0951446i	$-0.0303314\pi$
$103$	434.164	+	751.993i	0.415334	+	0.719380i	−0.580129	+	0.814524i	$0.696998\pi$
$104$	407.294			0.384023
$105$		0			0
$106$	148.000			0.135613
$107$	−842.000	−	1458.39i	−0.760740	−	1.31764i	−0.942469	−	0.334292i	$-0.891503\pi$
$107$	−842.000	−	1458.39i	−0.760740	−	1.31764i	0.181729	−	0.983349i	$-0.441831\pi$
$108$		0			0
$109$	409.000	−	708.409i	0.359405	−	0.622507i	−0.628457	−	0.777844i	$-0.716313\pi$
$109$	409.000	−	708.409i	0.359405	−	0.622507i	0.987861	+	0.155337i	$0.0496465\pi$
$110$	−277.186	−	480.100i	−0.240260	−	0.416143i
$111$		0			0
$112$		0			0
$113$	540.000			0.449548			0.224774	−	0.974411i	$-0.427836\pi$
$113$	540.000			0.449548			0.224774	+	0.974411i	$0.427836\pi$
$114$		0			0
$115$	1385.93	−	2400.50i	1.12381	−	1.94650i
$116$	−572.000	+	990.733i	−0.457835	+	0.792994i
$117$		0			0
$118$	−868.327			−0.677424
$119$		0			0
$120$		0			0
$121$	567.500	+	982.939i	0.426371	+	0.738496i
$122$	−14.1421	+	24.4949i	−0.0104948	+	0.0181776i
$123$		0			0
$124$	186.676	+	323.333i	0.135194	+	0.234162i
$125$	2811.46			2.01171
$126$		0			0
$127$	1720.00			1.20177			0.600887	−	0.799334i	$-0.294814\pi$
$127$	1720.00			1.20177			0.600887	+	0.799334i	$0.294814\pi$
$128$	−64.0000	−	110.851i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−1008.00	+	1745.91i	−0.680057	+	1.17789i
$131$	−867.620	−	1502.76i	−0.578659	−	1.00227i	−0.995634	−	0.0933484i	$-0.970243\pi$
$131$	−867.620	−	1502.76i	−0.578659	−	1.00227i	0.416975	−	0.908918i	$-0.363090\pi$
$132$		0			0
$133$		0			0
$134$	1368.00			0.881919
$135$		0			0
$136$	5.65685	−	9.79796i	0.00356670	−	0.00617771i
$137$	414.000	−	717.069i	0.258178	−	0.447178i	−0.707576	−	0.706638i	$-0.750211\pi$
$137$	414.000	−	717.069i	0.258178	−	0.447178i	0.965754	+	0.259460i	$0.0835445\pi$
$138$		0			0
$139$	−425.678			−0.259752			−0.129876	−	0.991530i	$-0.541458\pi$
$139$	−425.678			−0.259752			−0.129876	+	0.991530i	$0.541458\pi$
$140$		0			0
$141$		0			0
$142$	588.000	+	1018.45i	0.347492	+	0.601874i
$143$	−356.382	+	617.271i	−0.208407	+	0.360971i
$144$		0			0
$145$	−2831.26	−	4903.88i	−1.62154	−	2.80859i
$146$	−540.230			−0.306231
$147$		0			0
$148$	−152.000			−0.0844211
$149$	1025.00	+	1775.35i	0.563566	+	0.976124i	0.997182	+	0.0750264i	$0.0239041\pi$
$149$	1025.00	+	1775.35i	0.563566	+	0.976124i	−0.433616	+	0.901098i	$0.642763\pi$
$150$		0			0
$151$	236.000	−	408.764i	0.127188	−	0.220296i	−0.795398	−	0.606087i	$-0.792738\pi$
$151$	236.000	−	408.764i	0.127188	−	0.220296i	0.922586	+	0.385791i	$0.126071\pi$
$152$	5.65685	+	9.79796i	0.00301863	+	0.00522842i
$153$		0			0
$154$		0			0
$155$	−1848.00			−0.957645
$156$		0			0
$157$	1105.92	−	1915.50i	0.562176	−	0.973717i	−0.435130	−	0.900367i	$-0.643298\pi$
$157$	1105.92	−	1915.50i	0.562176	−	0.973717i	0.997306	−	0.0733498i	$-0.0233690\pi$
$158$	−1220.00	+	2113.10i	−0.614291	+	1.06398i
$159$		0			0
$160$	633.568			0.313050
$161$		0			0
$162$		0			0
$163$	−1643.00	−	2845.76i	−0.789507	−	1.36747i	−0.926269	−	0.376863i	$-0.877003\pi$
$163$	−1643.00	−	2845.76i	−0.789507	−	1.36747i	0.136762	−	0.990604i	$-0.456331\pi$
$164$	−251.730	+	436.009i	−0.119859	+	0.207601i
$165$		0			0
$166$	422.850	+	732.397i	0.197708	+	0.342440i
$167$	1490.58			0.690686			0.345343	−	0.938476i	$-0.387763\pi$
$167$	1490.58			0.690686			0.345343	+	0.938476i	$0.387763\pi$
$168$		0			0
$169$	395.000			0.179791
$170$	28.0000	+	48.4974i	0.0126324	+	0.0218799i
$171$		0			0
$172$	68.0000	−	117.779i	0.0301451	−	0.0522128i
$173$	−1035.20	−	1793.03i	−0.454943	−	0.787984i	0.543742	−	0.839252i	$-0.317007\pi$
$173$	−1035.20	−	1793.03i	−0.454943	−	0.787984i	−0.998685	+	0.0512682i	$0.983674\pi$
$174$		0			0
$175$		0			0
$176$	224.000			0.0959354
$177$		0			0
$178$	618.011	−	1070.43i	0.260235	−	0.450741i
$179$	270.000	−	467.654i	0.112742	−	0.195274i	−0.804133	−	0.594449i	$-0.797370\pi$
$179$	270.000	−	467.654i	0.112742	−	0.195274i	0.916875	+	0.399175i	$0.130703\pi$
$180$		0			0
$181$	3784.44			1.55412			0.777058	−	0.629429i	$-0.216711\pi$
$181$	3784.44			1.55412			0.777058	+	0.629429i	$0.216711\pi$
$182$		0			0
$183$		0			0
$184$	560.000	+	969.948i	0.224368	+	0.388617i
$185$	376.181	−	651.564i	0.149499	−	0.258940i
$186$		0			0
$187$	9.89949	+	17.1464i	0.00387124	+	0.00670519i
$188$	−2093.04			−0.811970
$189$		0			0
$190$	−56.0000			−0.0213825
$191$	514.000	+	890.274i	0.194721	+	0.337267i	0.946809	−	0.321796i	$-0.104286\pi$
$191$	514.000	+	890.274i	0.194721	+	0.337267i	−0.752088	+	0.659063i	$0.770953\pi$
$192$		0			0
$193$	−2296.00	+	3976.79i	−0.856320	+	1.48319i	0.0190956	+	0.999818i	$0.493921\pi$
$193$	−2296.00	+	3976.79i	−0.856320	+	1.48319i	−0.875415	+	0.483372i	$0.839412\pi$
$194$	−1483.51	−	2569.51i	−0.549020	−	0.950930i
$195$		0			0
$196$		0			0
$197$	−794.000			−0.287158			−0.143579	−	0.989639i	$-0.545861\pi$
$197$	−794.000			−0.287158			−0.143579	+	0.989639i	$0.545861\pi$
$198$		0			0
$199$	−1243.09	+	2153.10i	−0.442817	+	0.766981i	−0.997897	−	0.0648153i	$-0.979354\pi$
$199$	−1243.09	+	2153.10i	−0.442817	+	0.766981i	0.555080	+	0.831797i	$0.312688\pi$
$200$	−1068.00	+	1849.83i	−0.377595	+	0.654014i
$201$		0			0
$202$	−2257.08			−0.786178
$203$		0			0
$204$		0			0
$205$	−1246.00	−	2158.14i	−0.424509	−	0.735272i
$206$	868.327	−	1503.99i	0.293686	−	0.508678i
$207$		0			0
$208$	−407.294	−	705.453i	−0.135773	−	0.235165i
$209$	−19.7990			−0.00655275
$210$		0			0
$211$	−2748.00			−0.896588			−0.448294	−	0.893886i	$-0.647968\pi$
$211$	−2748.00			−0.896588			−0.448294	+	0.893886i	$0.647968\pi$
$212$	−148.000	−	256.344i	−0.0479466	−	0.0830460i
$213$		0			0
$214$	−1684.00	+	2916.77i	−0.537925	+	0.931713i
$215$	336.583	+	582.979i	0.106766	+	0.184925i
$216$		0			0
$217$		0			0
$218$	−1636.00			−0.508275
$219$		0			0
$220$	−554.372	+	960.200i	−0.169890	+	0.294258i
$221$	36.0000	−	62.3538i	0.0109576	−	0.0189791i
$222$		0			0
$223$	−3428.05			−1.02941			−0.514707	−	0.857366i	$-0.672099\pi$
$223$	−3428.05			−1.02941			−0.514707	+	0.857366i	$0.672099\pi$
$224$		0			0
$225$		0			0
$226$	−540.000	−	935.307i	−0.158939	−	0.275291i
$227$	2645.29	−	4581.77i	0.773453	−	1.33966i	−0.162207	−	0.986757i	$-0.551861\pi$
$227$	2645.29	−	4581.77i	0.773453	−	1.33966i	0.935660	−	0.352903i	$-0.114805\pi$
$228$		0			0
$229$	1374.62	+	2380.90i	0.396669	+	0.687051i	0.993313	−	0.115456i	$-0.0368328\pi$
$229$	1374.62	+	2380.90i	0.396669	+	0.687051i	−0.596644	+	0.802506i	$0.703499\pi$
$230$	−5543.72			−1.58931
$231$		0			0
$232$	2288.00			0.647477
$233$	36.0000	+	62.3538i	0.0101221	+	0.0175319i	0.871042	−	0.491208i	$-0.163445\pi$
$233$	36.0000	+	62.3538i	0.0101221	+	0.0175319i	−0.860920	+	0.508740i	$0.830111\pi$
$234$		0			0
$235$	5180.00	−	8972.02i	1.43790	−	2.49051i
$236$	868.327	+	1503.99i	0.239505	+	0.414836i
$237$		0			0
$238$		0			0
$239$	−4308.00			−1.16595			−0.582974	−	0.812491i	$-0.698111\pi$
$239$	−4308.00			−1.16595			−0.582974	+	0.812491i	$0.698111\pi$
$240$		0			0
$241$	770.039	−	1333.75i	0.205820	−	0.356490i	−0.744574	−	0.667540i	$-0.767347\pi$
$241$	770.039	−	1333.75i	0.205820	−	0.356490i	0.950394	+	0.311050i	$0.100681\pi$
$242$	1135.00	−	1965.88i	0.301490	−	0.522196i
$243$		0			0
$244$	56.5685			0.0148419
$245$		0			0
$246$		0			0
$247$	36.0000	+	62.3538i	0.00927379	+	0.0160627i
$248$	373.352	−	646.665i	0.0955964	−	0.165578i
$249$		0			0
$250$	−2811.46	−	4869.59i	−0.711249	−	1.23192i
$251$	−931.967			−0.234363			−0.117182	−	0.993110i	$-0.537386\pi$
$251$	−931.967			−0.234363			−0.117182	+	0.993110i	$0.537386\pi$
$252$		0			0
$253$	−1960.00			−0.487052
$254$	−1720.00	−	2979.13i	−0.424891	−	0.735933i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	468.812	+	812.006i	0.113789	+	0.197088i	0.917295	−	0.398209i	$-0.130368\pi$
$257$	468.812	+	812.006i	0.113789	+	0.197088i	−0.803506	+	0.595296i	$0.797035\pi$
$258$		0			0
$259$		0			0
$260$	4032.00			0.961746
$261$		0			0
$262$	−1735.24	+	3005.52i	−0.409174	+	0.708709i
$263$	3570.00	−	6183.42i	0.837018	−	1.44976i	−0.0553595	−	0.998466i	$-0.517630\pi$
$263$	3570.00	−	6183.42i	0.837018	−	1.44976i	0.892377	−	0.451291i	$-0.149036\pi$
$264$		0			0
$265$	1465.13			0.339630
$266$		0			0
$267$		0			0
$268$	−1368.00	−	2369.45i	−0.311806	−	0.540063i
$269$	2305.17	−	3992.67i	0.522485	−	0.904971i	−0.477172	−	0.878810i	$-0.658338\pi$
$269$	2305.17	−	3992.67i	0.522485	−	0.904971i	0.999658	−	0.0261615i	$-0.00832843\pi$
$270$		0			0
$271$	1182.28	+	2047.77i	0.265013	+	0.459016i	0.967567	−	0.252614i	$-0.0812904\pi$
$271$	1182.28	+	2047.77i	0.265013	+	0.459016i	−0.702554	+	0.711630i	$0.747957\pi$
$272$	−22.6274			−0.00504408
$273$		0			0
$274$	−1656.00			−0.365119
$275$	−1869.00	−	3237.20i	−0.409836	−	0.709857i
$276$		0			0
$277$	−2003.00	+	3469.30i	−0.434472	+	0.752527i	−0.997252	−	0.0740794i	$-0.976398\pi$
$277$	−2003.00	+	3469.30i	−0.434472	+	0.752527i	0.562781	+	0.826606i	$0.309731\pi$
$278$	425.678	+	737.296i	0.0918363	+	0.159065i
$279$		0			0
$280$		0			0
$281$	5984.00			1.27038			0.635188	−	0.772358i	$-0.280923\pi$
$281$	5984.00			1.27038			0.635188	+	0.772358i	$0.280923\pi$
$282$		0			0
$283$	2464.27	−	4268.24i	0.517617	−	0.896538i	−0.482174	−	0.876075i	$-0.660153\pi$
$283$	2464.27	−	4268.24i	0.517617	−	0.896538i	0.999791	−	0.0204627i	$-0.00651393\pi$
$284$	1176.00	−	2036.89i	0.245714	−	0.425589i
$285$		0			0
$286$	1425.53			0.294731
$287$		0			0
$288$		0			0
$289$	2455.50	+	4253.05i	0.499796	+	0.865673i
$290$	−5662.51	+	9807.76i	−1.14660	+	1.98597i
$291$		0			0
$292$	540.230	+	935.705i	0.108269	+	0.187527i
$293$	−1971.41			−0.393076			−0.196538	−	0.980496i	$-0.562970\pi$
$293$	−1971.41			−0.393076			−0.196538	+	0.980496i	$0.562970\pi$
$294$		0			0
$295$	−8596.00			−1.69654
$296$	152.000	+	263.272i	0.0298474	+	0.0516972i
$297$		0			0
$298$	2050.00	−	3550.70i	0.398501	−	0.690224i
$299$	3563.82	+	6172.71i	0.689301	+	1.19390i
$300$		0			0
$301$		0			0
$302$	−944.000			−0.179871
$303$		0			0
$304$	11.3137	−	19.5959i	0.00213449	−	0.00369705i
$305$	−140.000	+	242.487i	−0.0262832	+	0.0455238i
$306$		0			0
$307$	−4767.31			−0.886270			−0.443135	−	0.896455i	$-0.646134\pi$
$307$	−4767.31			−0.886270			−0.443135	+	0.896455i	$0.646134\pi$
$308$		0			0
$309$		0			0
$310$	1848.00	+	3200.83i	0.338579	+	0.586435i
$311$	−3388.46	+	5868.98i	−0.617819	+	1.07009i	0.372064	+	0.928207i	$0.378650\pi$
$311$	−3388.46	+	5868.98i	−0.617819	+	1.07009i	−0.989883	+	0.141887i	$0.954683\pi$
$312$		0			0
$313$	3095.01	+	5360.71i	0.558914	+	0.968068i	0.997587	+	0.0694210i	$0.0221152\pi$
$313$	3095.01	+	5360.71i	0.558914	+	0.968068i	−0.438673	+	0.898647i	$0.644551\pi$
$314$	−4423.66			−0.795037
$315$		0			0
$316$	4880.00			0.868739
$317$	4913.00	+	8509.57i	0.870478	+	1.50771i	0.861503	+	0.507753i	$0.169524\pi$
$317$	4913.00	+	8509.57i	0.870478	+	1.50771i	0.00897524	+	0.999960i	$0.497143\pi$
$318$		0			0
$319$	−2002.00	+	3467.57i	−0.351381	+	0.608609i
$320$	−633.568	−	1097.37i	−0.110680	−	0.191703i
$321$		0			0
$322$		0			0
$323$	2.00000			0.000344529
$324$		0			0
$325$	−6796.71	+	11772.2i	−1.16004	+	2.00925i
$326$	−3286.00	+	5691.52i	−0.558266	+	0.966945i
$327$		0			0
$328$	1006.92			0.169506
$329$		0			0
$330$		0			0
$331$	2869.00	+	4969.25i	0.476418	+	0.825181i	0.999635	−	0.0270189i	$-0.00860142\pi$
$331$	2869.00	+	4969.25i	0.476418	+	0.825181i	−0.523216	+	0.852200i	$0.675268\pi$
$332$	845.700	−	1464.79i	0.139801	−	0.242142i
$333$		0			0
$334$	−1490.58	−	2581.76i	−0.244195	−	0.422957i
$335$	13542.5			2.20868
$336$		0			0
$337$	−2254.00			−0.364342			−0.182171	−	0.983267i	$-0.558312\pi$
$337$	−2254.00			−0.364342			−0.182171	+	0.983267i	$0.558312\pi$
$338$	−395.000	−	684.160i	−0.0635656	−	0.110099i
$339$		0			0
$340$	56.0000	−	96.9948i	0.00893243	−	0.0154714i
$341$	653.367	+	1131.66i	0.103759	+	0.179716i
$342$		0			0
$343$		0			0
$344$	−272.000			−0.0426316
$345$		0			0
$346$	−2070.41	+	3586.05i	−0.321693	+	0.557189i
$347$	−993.000	+	1719.93i	−0.153623	+	0.266082i	−0.932557	−	0.361024i	$-0.882427\pi$
$347$	−993.000	+	1719.93i	−0.153623	+	0.266082i	0.778934	+	0.627106i	$0.215761\pi$
$348$		0			0
$349$	6771.25			1.03856			0.519279	−	0.854605i	$-0.326200\pi$
$349$	6771.25			1.03856			0.519279	+	0.854605i	$0.326200\pi$
$350$		0			0
$351$		0			0
$352$	−224.000	−	387.979i	−0.0339183	−	0.0587482i
$353$	3496.64	−	6056.36i	0.527217	−	0.913166i	−0.472280	−	0.881449i	$-0.656569\pi$
$353$	3496.64	−	6056.36i	0.527217	−	0.913166i	0.999497	−	0.0317177i	$-0.0100978\pi$
$354$		0			0
$355$	5820.90	+	10082.1i	0.870258	+	1.50733i
$356$	−2472.05			−0.368028
$357$		0			0
$358$	−1080.00			−0.159441
$359$	2972.00	+	5147.66i	0.436925	+	0.756777i	0.997451	−	0.0713606i	$-0.0227341\pi$
$359$	2972.00	+	5147.66i	0.436925	+	0.756777i	−0.560525	+	0.828137i	$0.689401\pi$
$360$		0			0
$361$	3428.50	−	5938.34i	0.499854	−	0.865773i
$362$	−3784.44	−	6554.83i	−0.549463	−	0.951697i
$363$		0			0
$364$		0			0
$365$	−5348.00			−0.766924
$366$		0			0
$367$	421.436	−	729.948i	0.0599421	−	0.103823i	−0.834497	−	0.551012i	$-0.814242\pi$
$367$	421.436	−	729.948i	0.0599421	−	0.103823i	0.894439	+	0.447190i	$0.147575\pi$
$368$	1120.00	−	1939.90i	0.158652	−	0.274794i
$369$		0			0
$370$	−1504.72			−0.211424
$371$		0			0
$372$		0			0
$373$	2863.00	+	4958.86i	0.397428	+	0.688365i	0.993408	−	0.114634i	$-0.0365696\pi$
$373$	2863.00	+	4958.86i	0.397428	+	0.688365i	−0.595980	+	0.802999i	$0.703236\pi$
$374$	19.7990	−	34.2929i	0.00273738	−	0.00474129i
$375$		0			0
$376$	2093.04	+	3625.24i	0.287075	+	0.497228i
$377$	14560.7			1.98917
$378$		0			0
$379$	10330.0			1.40004			0.700022	−	0.714122i	$-0.253174\pi$
$379$	10330.0			1.40004			0.700022	+	0.714122i	$0.253174\pi$
$380$	56.0000	+	96.9948i	0.00755984	+	0.0130940i
$381$		0			0
$382$	1028.00	−	1780.55i	0.137689	−	0.238484i
$383$	502.046	+	869.569i	0.0669800	+	0.116013i	0.897571	−	0.440871i	$-0.145330\pi$
$383$	502.046	+	869.569i	0.0669800	+	0.116013i	−0.830591	+	0.556884i	$0.811997\pi$
$384$		0			0
$385$		0			0
$386$	9184.00			1.21102
$387$		0			0
$388$	−2967.02	+	5139.03i	−0.388216	+	0.672409i
$389$	2605.00	−	4511.99i	0.339534	−	0.588090i	−0.644811	−	0.764342i	$-0.723064\pi$
$389$	2605.00	−	4511.99i	0.339534	−	0.588090i	0.984345	+	0.176252i	$0.0563973\pi$
$390$		0			0
$391$	197.990			0.0256081
$392$		0			0
$393$		0			0
$394$	794.000	+	1375.25i	0.101526	+	0.175848i
$395$	−12077.4	+	20918.6i	−1.53843	+	2.66464i
$396$		0			0
$397$	36.7696	+	63.6867i	0.00464839	+	0.00805125i	0.868340	−	0.495969i	$-0.165187\pi$
$397$	36.7696	+	63.6867i	0.00464839	+	0.00805125i	−0.863692	+	0.504020i	$0.831854\pi$
$398$	4972.37			0.626238
$399$		0			0
$400$	4272.00			0.534000
$401$	−249.000	−	431.281i	−0.0310086	−	0.0537085i	0.850105	−	0.526614i	$-0.176539\pi$
$401$	−249.000	−	431.281i	−0.0310086	−	0.0537085i	−0.881113	+	0.472905i	$0.843205\pi$
$402$		0			0
$403$	2376.00	−	4115.35i	0.293690	−	0.508686i
$404$	2257.08	+	3909.39i	0.277956	+	0.481434i
$405$		0			0
$406$		0			0
$407$	−532.000			−0.0647918
$408$		0			0
$409$	−1677.96	+	2906.32i	−0.202861	+	0.351365i	−0.949449	−	0.313921i	$-0.898357\pi$
$409$	−1677.96	+	2906.32i	−0.202861	+	0.351365i	0.746588	+	0.665286i	$0.231691\pi$
$410$	−2492.00	+	4316.27i	−0.300173	+	0.519916i
$411$		0			0
$412$	−3473.31			−0.415334
$413$		0			0
$414$		0			0
$415$	4186.00	+	7250.36i	0.495139	+	0.857606i
$416$	−814.587	+	1410.91i	−0.0960058	+	0.166287i
$417$		0			0
$418$	19.7990	+	34.2929i	0.00231675	+	0.00401272i
$419$	−14545.2			−1.69589			−0.847946	−	0.530082i	$-0.822161\pi$
$419$	−14545.2			−1.69589			−0.847946	+	0.530082i	$0.822161\pi$
$420$		0			0
$421$	10854.0			1.25651			0.628256	−	0.778007i	$-0.283769\pi$
$421$	10854.0			1.25651			0.628256	+	0.778007i	$0.283769\pi$
$422$	2748.00	+	4759.68i	0.316992	+	0.549046i
$423$		0			0
$424$	−296.000	+	512.687i	−0.0339034	+	0.0587224i
$425$	188.798	+	327.007i	0.0215483	+	0.0373227i
$426$		0			0
$427$		0			0
$428$	6736.00			0.760740
$429$		0			0
$430$	673.166	−	1165.96i	0.0754952	−	0.130762i
$431$	−2682.00	+	4645.36i	−0.299739	+	0.519163i	−0.976076	−	0.217429i	$-0.930233\pi$
$431$	−2682.00	+	4645.36i	−0.299739	+	0.519163i	0.676337	+	0.736592i	$0.263566\pi$
$432$		0			0
$433$	−6487.00			−0.719966			−0.359983	−	0.932959i	$-0.617217\pi$
$433$	−6487.00			−0.719966			−0.359983	+	0.932959i	$0.617217\pi$
$434$		0			0
$435$		0			0
$436$	1636.00	+	2833.64i	0.179702	+	0.311253i
$437$	−98.9949	+	171.464i	−0.0108365	+	0.0187694i
$438$		0			0
$439$	−6966.42	−	12066.2i	−0.757378	−	1.31182i	−0.944183	−	0.329420i	$-0.893147\pi$
$439$	−6966.42	−	12066.2i	−0.757378	−	1.31182i	0.186806	−	0.982397i	$-0.440187\pi$
$440$	2217.49			0.240260
$441$		0			0
$442$	−144.000			−0.0154963
$443$	2998.00	+	5192.69i	0.321533	+	0.556912i	0.980805	−	0.194993i	$-0.0624684\pi$
$443$	2998.00	+	5192.69i	0.321533	+	0.556912i	−0.659271	+	0.751905i	$0.729135\pi$
$444$		0			0
$445$	6118.00	−	10596.7i	0.651733	−	1.12883i
$446$	3428.05	+	5937.56i	0.363953	+	0.630385i
$447$		0			0
$448$		0			0
$449$	−2622.00			−0.275590			−0.137795	−	0.990461i	$-0.544001\pi$
$449$	−2622.00			−0.275590			−0.137795	+	0.990461i	$0.544001\pi$
$450$		0			0
$451$	−881.055	+	1526.03i	−0.0919895	+	0.159330i
$452$	−1080.00	+	1870.61i	−0.112387	+	0.194660i
$453$		0			0
$454$	−10581.1			−1.09383
$455$		0			0
$456$		0			0
$457$	−5604.00	−	9706.41i	−0.573619	−	0.993538i	−0.996190	−	0.0872080i	$-0.972206\pi$
$457$	−5604.00	−	9706.41i	−0.573619	−	0.993538i	0.422571	−	0.906330i	$-0.361128\pi$
$458$	2749.23	−	4761.81i	0.280487	−	0.485818i
$459$		0			0
$460$	5543.72	+	9602.00i	0.561907	+	0.973251i
$461$	−9786.36			−0.988712			−0.494356	−	0.869260i	$-0.664596\pi$
$461$	−9786.36			−0.988712			−0.494356	+	0.869260i	$0.664596\pi$
$462$		0			0
$463$	3952.00			0.396685			0.198342	−	0.980133i	$-0.436444\pi$
$463$	3952.00			0.396685			0.198342	+	0.980133i	$0.436444\pi$
$464$	−2288.00	−	3962.93i	−0.228918	−	0.396497i
$465$		0			0
$466$	72.0000	−	124.708i	0.00715737	−	0.0123969i
$467$	8753.27	+	15161.1i	0.867352	+	1.50230i	0.864693	+	0.502301i	$0.167513\pi$
$467$	8753.27	+	15161.1i	0.867352	+	1.50230i	0.00265876	+	0.999996i	$0.499154\pi$
$468$		0			0
$469$		0			0
$470$	−20720.0			−2.03349
$471$		0			0
$472$	1736.65	−	3007.97i	0.169356	−	0.293333i
$473$	238.000	−	412.228i	0.0231358	−	0.0400724i
$474$		0			0
$475$	−377.595			−0.0364742
$476$		0			0
$477$		0			0
$478$	4308.00	+	7461.67i	0.412225	+	0.713994i
$479$	1144.10	−	1981.64i	0.109134	−	0.189026i	−0.806286	−	0.591526i	$-0.798526\pi$
$479$	1144.10	−	1981.64i	0.109134	−	0.189026i	0.915420	+	0.402501i	$0.131859\pi$
$480$		0			0
$481$	967.322	+	1675.45i	0.0916967	+	0.158823i
$482$	−3080.16			−0.291073
$483$		0			0
$484$	−4540.00			−0.426371
$485$	−14686.0	−	25436.9i	−1.37496	−	2.38151i
$486$		0			0
$487$	−486.000	+	841.777i	−0.0452213	+	0.0783256i	−0.887750	−	0.460326i	$-0.847733\pi$
$487$	−486.000	+	841.777i	−0.0452213	+	0.0783256i	0.842529	+	0.538651i	$0.181066\pi$
$488$	−56.5685	−	97.9796i	−0.00524741	−	0.00908879i
$489$		0			0
$490$		0			0
$491$	7404.00			0.680525			0.340263	−	0.940330i	$-0.389484\pi$
$491$	7404.00			0.680525			0.340263	+	0.940330i	$0.389484\pi$
$492$		0			0
$493$	202.233	−	350.277i	0.0184748	−	0.0319994i
$494$	72.0000	−	124.708i	0.00655756	−	0.0113580i
$495$		0			0
$496$	−1493.41			−0.135194
$497$		0			0
$498$		0			0
$499$	6122.00	+	10603.6i	0.549215	+	0.951269i	0.998329	+	0.0577938i	$0.0184066\pi$
$499$	6122.00	+	10603.6i	0.549215	+	0.951269i	−0.449113	+	0.893475i	$0.648260\pi$
$500$	−5622.91	+	9739.17i	−0.502929	+	0.871098i
$501$		0			0
$502$	931.967	+	1614.21i	0.0828600	+	0.143518i
$503$	2415.48			0.214117			0.107058	−	0.994253i	$-0.465857\pi$
$503$	2415.48			0.214117			0.107058	+	0.994253i	$0.465857\pi$
$504$		0			0
$505$	−22344.0			−1.96890
$506$	1960.00	+	3394.82i	0.172199	+	0.298257i
$507$		0			0
$508$	−3440.00	+	5958.25i	−0.300444	+	0.520383i
$509$	−2853.88	−	4943.07i	−0.248519	−	0.430447i	0.714596	−	0.699537i	$-0.246610\pi$
$509$	−2853.88	−	4943.07i	−0.248519	−	0.430447i	−0.963115	+	0.269090i	$0.913277\pi$
$510$		0			0
$511$		0			0
$512$	512.000			0.0441942
$513$		0			0
$514$	937.624	−	1624.01i	0.0804607	−	0.139362i
$515$	8596.00	−	14888.7i	0.735505	−	1.27393i
$516$		0			0
$517$	−7325.63			−0.623173
$518$		0			0
$519$		0			0
$520$	−4032.00	−	6983.63i	−0.340029	−	0.588947i
$521$	0.707107	−	1.22474i	5.94605e−5	−	0.000102989i	−0.865996	−	0.500051i	$-0.833314\pi$
$521$	0.707107	−	1.22474i	5.94605e−5	−	0.000102989i	0.866055	+	0.499949i	$0.166648\pi$
$522$		0			0
$523$	6128.49	+	10614.9i	0.512391	+	0.887487i	0.999897	+	0.0143672i	$0.00457337\pi$
$523$	6128.49	+	10614.9i	0.512391	+	0.887487i	−0.487506	+	0.873120i	$0.662093\pi$
$524$	6940.96			0.578659
$525$		0			0
$526$	−14280.0			−1.18372
$527$	−66.0000	−	114.315i	−0.00545542	−	0.00944906i
$528$		0			0
$529$	−3716.50	+	6437.17i	−0.305457	+	0.529068i
$530$	−1465.13	−	2537.67i	−0.120077	−	0.207980i
$531$		0			0
$532$		0			0
$533$	6408.00			0.520753
$534$		0			0
$535$	−16670.7	+	28874.6i	−1.34718	+	2.33338i
$536$	−2736.00	+	4738.89i	−0.220480	+	0.381882i
$537$		0			0
$538$	−9220.67			−0.738906
$539$		0			0
$540$		0			0
$541$	−1025.00	−	1775.35i	−0.0814569	−	0.141088i	0.822419	−	0.568882i	$-0.192624\pi$
$541$	−1025.00	−	1775.35i	−0.0814569	−	0.141088i	−0.903876	+	0.427795i	$0.859291\pi$
$542$	2364.57	−	4095.55i	0.187393	−	0.324573i
$543$		0			0
$544$	22.6274	+	39.1918i	0.00178335	+	0.00308885i
$545$	−16195.6			−1.27292
$546$		0			0
$547$	14554.0			1.13763			0.568815	−	0.822465i	$-0.307402\pi$
$547$	14554.0			1.13763			0.568815	+	0.822465i	$0.307402\pi$
$548$	1656.00	+	2868.28i	0.129089	+	0.223589i
$549$		0			0
$550$	−3738.00	+	6474.41i	−0.289798	+	0.501945i
$551$	202.233	+	350.277i	0.0156359	+	0.0270822i
$552$		0			0
$553$		0			0
$554$	8012.00			0.614435
$555$		0			0
$556$	851.357	−	1474.59i	0.0649381	−	0.112476i
$557$	3477.00	−	6022.34i	0.264498	−	0.458123i	−0.702934	−	0.711255i	$-0.748127\pi$
$557$	3477.00	−	6022.34i	0.264498	−	0.458123i	0.967432	+	0.253131i	$0.0814605\pi$
$558$		0			0
$559$	−1731.00			−0.130972
$560$		0			0
$561$		0			0
$562$	−5984.00	−	10364.6i	−0.449146	−	0.777943i
$563$	−818.123	+	1417.03i	−0.0612429	+	0.106076i	−0.895021	−	0.446024i	$-0.852840\pi$
$563$	−818.123	+	1417.03i	−0.0612429	+	0.106076i	0.833778	+	0.552099i	$0.186173\pi$
$564$		0			0
$565$	−5345.73	−	9259.07i	−0.398047	−	0.689437i
$566$	−9857.07			−0.732020
$567$		0			0
$568$	−4704.00			−0.347492
$569$	−3571.00	−	6185.15i	−0.263100	−	0.455703i	0.703964	−	0.710236i	$-0.251412\pi$
$569$	−3571.00	−	6185.15i	−0.263100	−	0.455703i	−0.967064	+	0.254533i	$0.918078\pi$
$570$		0			0
$571$	10303.0	−	17845.3i	0.755109	−	1.30789i	−0.190211	−	0.981743i	$-0.560917\pi$
$571$	10303.0	−	17845.3i	0.755109	−	1.30789i	0.945320	−	0.326144i	$-0.105749\pi$
$572$	−1425.53	−	2469.09i	−0.104203	−	0.180485i
$573$		0			0
$574$		0			0
$575$	−37380.0			−2.71105
$576$		0			0
$577$	4401.74	−	7624.04i	0.317585	−	0.550074i	−0.662398	−	0.749152i	$-0.730461\pi$
$577$	4401.74	−	7624.04i	0.317585	−	0.550074i	0.979984	+	0.199078i	$0.0637946\pi$
$578$	4911.00	−	8506.10i	0.353409	−	0.612123i
$579$		0			0
$580$	22650.0			1.62154
$581$		0			0
$582$		0			0
$583$	−518.000	−	897.202i	−0.0367982	−	0.0637364i
$584$	1080.46	−	1871.41i	0.0765577	−	0.132602i
$585$		0			0
$586$	1971.41	+	3414.59i	0.138973	+	0.240709i
$587$	−6503.97			−0.457321			−0.228661	−	0.973506i	$-0.573435\pi$
$587$	−6503.97			−0.457321			−0.228661	+	0.973506i	$0.573435\pi$
$588$		0			0
$589$	132.000			0.00923424
$590$	8596.00	+	14888.7i	0.599816	+	1.03891i
$591$		0			0
$592$	304.000	−	526.543i	0.0211053	−	0.0365554i
$593$	−11570.4	−	20040.5i	−0.801246	−	1.38780i	−0.918796	−	0.394732i	$-0.870837\pi$
$593$	−11570.4	−	20040.5i	−0.801246	−	1.38780i	0.117550	−	0.993067i	$-0.462496\pi$
$594$		0			0
$595$		0			0
$596$	−8200.00			−0.563566
$597$		0			0
$598$	7127.64	−	12345.4i	0.487409	−	0.844218i
$599$	−5648.00	+	9782.62i	−0.385260	+	0.667291i	−0.991805	−	0.127759i	$-0.959222\pi$
$599$	−5648.00	+	9782.62i	−0.385260	+	0.667291i	0.606545	+	0.795049i	$0.292555\pi$
$600$		0			0
$601$	8727.11			0.592323			0.296162	−	0.955138i	$-0.404293\pi$
$601$	8727.11			0.592323			0.296162	+	0.955138i	$0.404293\pi$
$602$		0			0
$603$		0			0
$604$	944.000	+	1635.06i	0.0635941	+	0.110148i
$605$	11235.9	−	19461.2i	0.755050	−	1.30779i
$606$		0			0
$607$	−9868.38	−	17092.5i	−0.659877	−	1.14294i	−0.980647	−	0.195783i	$-0.937275\pi$
$607$	−9868.38	−	17092.5i	−0.659877	−	1.14294i	0.320770	−	0.947157i	$-0.396058\pi$
$608$	−45.2548			−0.00301863
$609$		0			0
$610$	560.000			0.0371701
$611$	13320.0	+	23070.9i	0.881947	+	1.52758i
$612$		0			0
$613$	−8481.00	+	14689.5i	−0.558800	+	0.967870i	0.438797	+	0.898586i	$0.355405\pi$
$613$	−8481.00	+	14689.5i	−0.558800	+	0.967870i	−0.997597	+	0.0692837i	$0.977929\pi$
$614$	4767.31	+	8257.23i	0.313344	+	0.542727i
$615$		0			0
$616$		0			0
$617$	19034.0			1.24194			0.620972	−	0.783832i	$-0.286738\pi$
$617$	19034.0			1.24194			0.620972	+	0.783832i	$0.286738\pi$
$618$		0			0
$619$	9338.76	−	16175.2i	0.606392	−	1.05030i	−0.385438	−	0.922734i	$-0.625950\pi$
$619$	9338.76	−	16175.2i	0.606392	−	1.05030i	0.991830	−	0.127568i	$-0.0407169\pi$
$620$	3696.00	−	6401.66i	0.239411	−	0.414672i
$621$		0			0
$622$	13553.8			0.873728
$623$		0			0
$624$		0			0
$625$	−11144.5	−	19302.8i	−0.713248	−	1.23538i
$626$	6190.01	−	10721.4i	0.395212	−	0.684527i
$627$		0			0
$628$	4423.66	+	7662.00i	0.281088	+	0.486859i
$629$	53.7401			0.00340661
$630$		0			0
$631$	−14716.0			−0.928423			−0.464211	−	0.885724i	$-0.653662\pi$
$631$	−14716.0			−0.928423			−0.464211	+	0.885724i	$0.653662\pi$
$632$	−4880.00	−	8452.41i	−0.307146	−	0.531992i
$633$		0			0
$634$	9826.00	−	17019.1i	0.615521	−	1.06611i
$635$	−17027.1	−	29491.9i	−1.06410	−	1.84307i
$636$		0			0
$637$		0			0
$638$	8008.00			0.496928
$639$		0			0
$640$	−1267.14	+	2194.74i	−0.0782624	+	0.135554i
$641$	2365.00	−	4096.30i	0.145728	−	0.252409i	−0.783916	−	0.620867i	$-0.786781\pi$
$641$	2365.00	−	4096.30i	0.145728	−	0.252409i	0.929644	+	0.368458i	$0.120114\pi$
$642$		0			0
$643$	19056.5			1.16877			0.584383	−	0.811478i	$-0.301337\pi$
$643$	19056.5			1.16877			0.584383	+	0.811478i	$0.301337\pi$
$644$		0			0
$645$		0			0
$646$	−2.00000	−	3.46410i	−0.000121810	−	0.000210980i
$647$	4671.15	−	8090.66i	0.283836	−	0.491618i	−0.688490	−	0.725245i	$-0.741726\pi$
$647$	4671.15	−	8090.66i	0.283836	−	0.491618i	0.972326	+	0.233627i	$0.0750596\pi$
$648$		0			0
$649$	3039.14	+	5263.95i	0.183816	+	0.318379i
$650$	27186.8			1.64055
$651$		0			0
$652$	13144.0			0.789507
$653$	1887.00	+	3268.38i	0.113084	+	0.195868i	0.917012	−	0.398859i	$-0.130594\pi$
$653$	1887.00	+	3268.38i	0.113084	+	0.195868i	−0.803928	+	0.594727i	$0.797260\pi$
$654$		0			0
$655$	−17178.0	+	29753.2i	−1.02473	+	1.77489i
$656$	−1006.92	−	1744.04i	−0.0599293	−	0.103801i
$657$		0			0
$658$		0			0
$659$	21150.0			1.25021			0.625104	−	0.780541i	$-0.285057\pi$
$659$	21150.0			1.25021			0.625104	+	0.780541i	$0.285057\pi$
$660$		0			0
$661$	5188.75	−	8987.18i	0.305324	−	0.528836i	−0.672010	−	0.740542i	$-0.734569\pi$
$661$	5188.75	−	8987.18i	0.305324	−	0.528836i	0.977333	+	0.211706i	$0.0679020\pi$
$662$	5738.00	−	9938.51i	0.336879	−	0.583491i
$663$		0			0
$664$	−3382.80			−0.197708
$665$		0			0
$666$		0			0
$667$	20020.0	+	34675.7i	1.16219	+	2.01296i
$668$	−2981.16	+	5163.52i	−0.172672	+	0.299076i
$669$		0			0
$670$	−13542.5	−	23456.3i	−0.780885	−	1.35253i
$671$	197.990			0.0113909
$672$		0			0
$673$	−1164.00			−0.0666700			−0.0333350	−	0.999444i	$-0.510613\pi$
$673$	−1164.00			−0.0666700			−0.0333350	+	0.999444i	$0.510613\pi$
$674$	2254.00	+	3904.04i	0.128814	+	0.223113i
$675$		0			0
$676$	−790.000	+	1368.32i	−0.0449477	+	0.0778516i
$677$	13576.5	+	23515.1i	0.770732	+	1.33495i	0.937162	+	0.348893i	$0.113442\pi$
$677$	13576.5	+	23515.1i	0.770732	+	1.33495i	−0.166431	+	0.986053i	$0.553224\pi$
$678$		0			0
$679$		0			0
$680$	−224.000			−0.0126324
$681$		0			0
$682$	1306.73	−	2263.33i	0.0733686	−	0.127078i
$683$	−8298.00	+	14372.6i	−0.464882	+	0.805199i	−0.999196	−	0.0400871i	$-0.987236\pi$
$683$	−8298.00	+	14372.6i	−0.464882	+	0.805199i	0.534315	+	0.845286i	$0.320570\pi$
$684$		0			0
$685$	−16393.6			−0.914403
$686$		0			0
$687$		0			0
$688$	272.000	+	471.118i	0.0150725	+	0.0261064i
$689$	−1883.73	+	3262.72i	−0.104157	+	0.180406i
$690$		0			0
$691$	−5649.08	−	9784.49i	−0.311000	−	0.538668i	0.667579	−	0.744539i	$-0.267331\pi$
$691$	−5649.08	−	9784.49i	−0.311000	−	0.538668i	−0.978579	+	0.205871i	$0.933997\pi$
$692$	8281.63			0.454943
$693$		0			0
$694$	3972.00			0.217255
$695$	4214.00	+	7298.86i	0.229994	+	0.398362i
$696$		0			0
$697$	89.0000	−	154.153i	0.00483661	−	0.00837725i
$698$	−6771.25	−	11728.2i	−0.367186	−	0.635985i
$699$		0			0
$700$		0			0
$701$	2754.00			0.148384			0.0741920	−	0.997244i	$-0.476362\pi$
$701$	2754.00			0.148384			0.0741920	+	0.997244i	$0.476362\pi$
$702$		0			0
$703$	−26.8701	+	46.5403i	−0.00144157	+	0.00249687i
$704$	−448.000	+	775.959i	−0.0239839	+	0.0415413i
$705$		0			0
$706$	−13986.6			−0.745597
$707$		0			0
$708$		0			0
$709$	−14717.0	−	25490.6i	−0.779561	−	1.35024i	−0.932195	−	0.361956i	$-0.882109\pi$
$709$	−14717.0	−	25490.6i	−0.779561	−	1.35024i	0.152634	−	0.988283i	$-0.451224\pi$
$710$	11641.8	−	20164.2i	0.615365	−	1.06584i
$711$		0			0
$712$	2472.05	+	4281.71i	0.130118	+	0.225370i
$713$	13067.3			0.686361
$714$		0			0
$715$	14112.0			0.738124
$716$	1080.00	+	1870.61i	0.0563708	+	0.0976371i
$717$		0			0
$718$	5944.00	−	10295.3i	0.308953	−	0.535122i
$719$	−8834.59	−	15302.0i	−0.458240	−	0.793695i	0.540628	−	0.841262i	$-0.318187\pi$
$719$	−8834.59	−	15302.0i	−0.458240	−	0.793695i	−0.998868	+	0.0475666i	$0.984853\pi$
$720$		0			0
$721$		0			0
$722$	−13714.0			−0.706901
$723$		0			0
$724$	−7568.87	+	13109.7i	−0.388529	+	0.672952i
$725$	−38181.0	+	66131.4i	−1.95587	+	3.38767i
$726$		0			0
$727$	28445.5			1.45115			0.725574	−	0.688144i	$-0.241574\pi$
$727$	28445.5			1.45115			0.725574	+	0.688144i	$0.241574\pi$
$728$		0			0
$729$		0			0
$730$	5348.00	+	9263.01i	0.271148	+	0.469643i
$731$	−24.0416	+	41.6413i	−0.00121643	+	0.00210692i
$732$		0			0
$733$	−11170.9	−	19348.5i	−0.562900	−	0.974971i	−0.997242	−	0.0742230i	$-0.976352\pi$
$733$	−11170.9	−	19348.5i	−0.562900	−	0.974971i	0.434342	−	0.900748i	$-0.356981\pi$
$734$	−1685.74			−0.0847710
$735$		0			0
$736$	−4480.00			−0.224368
$737$	−4788.00	−	8293.06i	−0.239306	−	0.414490i
$738$		0			0
$739$	−10335.0	+	17900.7i	−0.514451	+	0.891055i	0.485409	+	0.874287i	$0.338671\pi$
$739$	−10335.0	+	17900.7i	−0.514451	+	0.891055i	−0.999859	+	0.0167675i	$0.994662\pi$
$740$	1504.72	+	2606.26i	0.0747496	+	0.129470i
$741$		0			0
$742$		0			0
$743$	25400.0			1.25415			0.627076	−	0.778958i	$-0.284251\pi$
$743$	25400.0			1.25415			0.627076	+	0.778958i	$0.284251\pi$
$744$		0			0
$745$	20294.0	−	35150.2i	0.998004	−	1.72859i
$746$	5726.00	−	9917.72i	0.281024	−	0.486748i
$747$		0			0
$748$	−79.1960			−0.00387124
$749$		0			0
$750$		0			0
$751$	−14590.0	−	25270.6i	−0.708917	−	1.22788i	−0.965259	−	0.261294i	$-0.915851\pi$
$751$	−14590.0	−	25270.6i	−0.708917	−	1.22788i	0.256342	−	0.966586i	$-0.417483\pi$
$752$	4186.07	−	7250.49i	0.202992	−	0.351593i
$753$		0			0
$754$	−14560.7	−	25219.9i	−0.703277	−	1.21811i
$755$	−9345.12			−0.450469
$756$		0			0
$757$	−26206.0			−1.25822			−0.629110	−	0.777316i	$-0.716581\pi$
$757$	−26206.0			−1.25822			−0.629110	+	0.777316i	$0.716581\pi$
$758$	−10330.0	−	17892.1i	−0.494990	−	0.857348i
$759$		0			0
$760$	112.000	−	193.990i	0.00534561	−	0.00925888i
$761$	3431.59	+	5943.69i	0.163463	+	0.283125i	0.936108	−	0.351712i	$-0.114400\pi$
$761$	3431.59	+	5943.69i	0.163463	+	0.283125i	−0.772646	+	0.634838i	$0.781067\pi$
$762$		0			0
$763$		0			0
$764$	−4112.00			−0.194721
$765$		0			0
$766$	1004.09	−	1739.14i	0.0473620	−	0.0820334i
$767$	11052.0	−	19142.6i	0.520293	−	0.901174i
$768$		0			0
$769$	−9058.04			−0.424761			−0.212380	−	0.977187i	$-0.568122\pi$
$769$	−9058.04			−0.424761			−0.212380	+	0.977187i	$0.568122\pi$
$770$		0			0
$771$		0			0
$772$	−9184.00	−	15907.2i	−0.428160	−	0.741595i
$773$	−66.4680	+	115.126i	−0.00309274	+	0.00535679i	−0.867568	−	0.497319i	$-0.834318\pi$
$773$	−66.4680	+	115.126i	−0.00309274	+	0.00535679i	0.864475	+	0.502676i	$0.167651\pi$
$774$		0			0
$775$	12460.6	+	21582.5i	0.577547	+	1.00034i
$776$	11868.1			0.549020
$777$		0			0
$778$	−10420.0			−0.480174
$779$	89.0000	+	154.153i	0.00409340	+	0.00708997i
$780$		0			0
$781$	4116.00	−	7129.12i	0.188581	−	0.326633i
$782$	−197.990	−	342.929i	−0.00905384	−	0.0156817i
$783$		0			0
$784$		0			0
$785$	−43792.0			−1.99109
$786$		0			0

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.g (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-9.89949 - 17.1464i) q^{5} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-9.89949 - 17.1464i) q^{5} +8.00000 q^{8} +(-19.7990 + 34.2929i) q^{10} +(-7.00000 + 12.1244i) q^{11} +50.9117 q^{13} +(-8.00000 - 13.8564i) q^{16} +(0.707107 - 1.22474i) q^{17} +(0.707107 + 1.22474i) q^{19} +79.1960 q^{20} +28.0000 q^{22} +(70.0000 + 121.244i) q^{23} +(-133.500 + 231.229i) q^{25} +(-50.9117 - 88.1816i) q^{26} +286.000 q^{29} +(46.6690 - 80.8332i) q^{31} +(-16.0000 + 27.7128i) q^{32} -2.82843 q^{34} +(19.0000 + 32.9090i) q^{37} +(1.41421 - 2.44949i) q^{38} +(-79.1960 - 137.171i) q^{40} +125.865 q^{41} -34.0000 q^{43} +(-28.0000 - 48.4974i) q^{44} +(140.000 - 242.487i) q^{46} +(261.630 + 453.156i) q^{47} +534.000 q^{50} +(-101.823 + 176.363i) q^{52} +(-37.0000 + 64.0859i) q^{53} +277.186 q^{55} +(-286.000 - 495.367i) q^{58} +(217.082 - 375.997i) q^{59} +(-7.07107 - 12.2474i) q^{61} -186.676 q^{62} +64.0000 q^{64} +(-504.000 - 872.954i) q^{65} +(-342.000 + 592.361i) q^{67} +(2.82843 + 4.89898i) q^{68} -588.000 q^{71} +(135.057 - 233.926i) q^{73} +(38.0000 - 65.8179i) q^{74} -5.65685 q^{76} +(-610.000 - 1056.55i) q^{79} +(-158.392 + 274.343i) q^{80} +(-125.865 - 218.005i) q^{82} -422.850 q^{83} -28.0000 q^{85} +(34.0000 + 58.8897i) q^{86} +(-56.0000 + 96.9948i) q^{88} +(309.006 + 535.214i) q^{89} -560.000 q^{92} +(523.259 - 906.311i) q^{94} +(14.0000 - 24.2487i) q^{95} +1483.51 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8	\( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 28 q^{11} - 32 q^{16} + 112 q^{22} + 280 q^{23} - 534 q^{25} + 1144 q^{29} - 64 q^{32} + 76 q^{37} - 136 q^{43} - 112 q^{44} + 560 q^{46} + 2136 q^{50} - 148 q^{53} - 1144 q^{58} + 256 q^{64} - 2016 q^{65} - 1368 q^{67} - 2352 q^{71} + 152 q^{74} - 2440 q^{79} - 112 q^{85} + 136 q^{86} - 224 q^{88} - 2240 q^{92} + 56 q^{95}+O(q^{100}) \) 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8 - 28 * q^11 - 32 * q^16 + 112 * q^22 + 280 * q^23 - 534 * q^25 + 1144 * q^29 - 64 * q^32 + 76 * q^37 - 136 * q^43 - 112 * q^44 + 560 * q^46 + 2136 * q^50 - 148 * q^53 - 1144 * q^58 + 256 * q^64 - 2016 * q^65 - 1368 * q^67 - 2352 * q^71 + 152 * q^74 - 2440 * q^79 - 112 * q^85 + 136 * q^86 - 224 * q^88 - 2240 * q^92 + 56 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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