LMFDB - Embedded newform 3528.2.s.bf.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,2,Mod(361,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("3528.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3528.s (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:	$28.1712218331$
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:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	3528.361
Dual form			3528.2.s.bf.3313.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 - 2.44949i) q^{5} +O(q^{10})$	$q+(-1.41421 - 2.44949i) q^{5} +(-1.00000 + 1.73205i) q^{11} +5.65685 q^{13} +(-1.41421 + 2.44949i) q^{17} +(-2.82843 - 4.89898i) q^{19} +(-3.00000 - 5.19615i) q^{23} +(-1.50000 + 2.59808i) q^{25} -4.00000 q^{29} +(2.82843 - 4.89898i) q^{31} +(1.00000 + 1.73205i) q^{37} -2.82843 q^{41} -4.00000 q^{43} +(5.65685 + 9.79796i) q^{47} +(-6.00000 + 10.3923i) q^{53} +5.65685 q^{55} +(5.65685 - 9.79796i) q^{59} +(-2.82843 - 4.89898i) q^{61} +(-8.00000 - 13.8564i) q^{65} +(6.00000 - 10.3923i) q^{67} -6.00000 q^{71} +(-4.00000 - 6.92820i) q^{79} +11.3137 q^{83} +8.00000 q^{85} +(-4.24264 - 7.34847i) q^{89} +(-8.00000 + 13.8564i) q^{95} -11.3137 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 4 q^{11} - 12 q^{23} - 6 q^{25} - 16 q^{29} + 4 q^{37} - 16 q^{43} - 24 q^{53} - 32 q^{65} + 24 q^{67} - 24 q^{71} - 16 q^{79} + 32 q^{85} - 32 q^{95}+O(q^{100}) $ 4 * q - 4 * q^11 - 12 * q^23 - 6 * q^25 - 16 * q^29 + 4 * q^37 - 16 * q^43 - 24 * q^53 - 32 * q^65 + 24 * q^67 - 24 * q^71 - 16 * q^79 + 32 * q^85 - 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.41421	−	2.44949i	−0.632456	−	1.09545i	−0.987048	−	0.160424i	$-0.948714\pi$
$5$	−1.41421	−	2.44949i	−0.632456	−	1.09545i	0.354593	−	0.935021i	$-0.384620\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	5.65685			1.56893			0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685			1.56893			0.784465	+	0.620174i	$0.212938\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.41421	+	2.44949i	−0.342997	+	0.594089i	−0.984988	−	0.172624i	$-0.944775\pi$
$17$	−1.41421	+	2.44949i	−0.342997	+	0.594089i	0.641991	+	0.766712i	$0.278109\pi$
$18$		0			0
$19$	−2.82843	−	4.89898i	−0.648886	−	1.12390i	−0.983389	−	0.181509i	$-0.941902\pi$
$19$	−2.82843	−	4.89898i	−0.648886	−	1.12390i	0.334504	−	0.942394i	$-0.391431\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$	−1.50000	+	2.59808i	−0.300000	+	0.519615i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.00000			−0.742781			−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000			−0.742781			−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$	2.82843	−	4.89898i	0.508001	−	0.879883i	−0.491957	−	0.870620i	$-0.663718\pi$
$31$	2.82843	−	4.89898i	0.508001	−	0.879883i	0.999957	−	0.00926296i	$-0.00294853\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	0.936442	−	0.350823i	$-0.114098\pi$
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	−0.772043	+	0.635571i	$0.780765\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.82843			−0.441726			−0.220863	−	0.975305i	$-0.570887\pi$
$41$	−2.82843			−0.441726			−0.220863	+	0.975305i	$0.570887\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.65685	+	9.79796i	0.825137	+	1.42918i	0.901815	+	0.432123i	$0.142235\pi$
$47$	5.65685	+	9.79796i	0.825137	+	1.42918i	−0.0766776	+	0.997056i	$0.524431\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.00000	+	10.3923i	−0.824163	+	1.42749i	0.0783936	+	0.996922i	$0.475021\pi$
$53$	−6.00000	+	10.3923i	−0.824163	+	1.42749i	−0.902557	+	0.430570i	$0.858312\pi$
$54$		0			0
$55$	5.65685			0.762770
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.65685	−	9.79796i	0.736460	−	1.27559i	−0.217620	−	0.976034i	$-0.569829\pi$
$59$	5.65685	−	9.79796i	0.736460	−	1.27559i	0.954080	−	0.299552i	$-0.0968372\pi$
$60$		0			0
$61$	−2.82843	−	4.89898i	−0.362143	−	0.627250i	0.626170	−	0.779686i	$-0.284621\pi$
$61$	−2.82843	−	4.89898i	−0.362143	−	0.627250i	−0.988313	+	0.152436i	$0.951288\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.00000	−	13.8564i	−0.992278	−	1.71868i
$66$		0			0
$67$	6.00000	−	10.3923i	0.733017	−	1.26962i	−0.222571	−	0.974916i	$-0.571445\pi$
$67$	6.00000	−	10.3923i	0.733017	−	1.26962i	0.955588	−	0.294706i	$-0.0952216\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$73$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	0.548352	−	0.836247i	$-0.315255\pi$
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	−0.998388	+	0.0567635i	$0.981922\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.3137			1.24184			0.620920	−	0.783874i	$-0.286759\pi$
$83$	11.3137			1.24184			0.620920	+	0.783874i	$0.286759\pi$
$84$		0			0
$85$	8.00000			0.867722
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.24264	−	7.34847i	−0.449719	−	0.778936i	0.548648	−	0.836053i	$-0.315143\pi$
$89$	−4.24264	−	7.34847i	−0.449719	−	0.778936i	−0.998367	+	0.0571170i	$0.981809\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.00000	+	13.8564i	−0.820783	+	1.42164i
$96$		0			0
$97$	−11.3137			−1.14873			−0.574367	−	0.818598i	$-0.694752\pi$
$97$	−11.3137			−1.14873			−0.574367	+	0.818598i	$0.694752\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.24264	+	7.34847i	−0.422159	+	0.731200i	−0.996150	−	0.0876610i	$-0.972061\pi$
$101$	−4.24264	+	7.34847i	−0.422159	+	0.731200i	0.573992	+	0.818861i	$0.305394\pi$
$102$		0			0
$103$	8.48528	+	14.6969i	0.836080	+	1.44813i	0.893148	+	0.449762i	$0.148491\pi$
$103$	8.48528	+	14.6969i	0.836080	+	1.44813i	−0.0570688	+	0.998370i	$0.518175\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.00000	−	15.5885i	−0.870063	−	1.50699i	−0.861931	−	0.507026i	$-0.830745\pi$
$107$	−9.00000	−	15.5885i	−0.870063	−	1.50699i	−0.00813215	−	0.999967i	$-0.502589\pi$
$108$		0			0
$109$	−7.00000	+	12.1244i	−0.670478	+	1.16130i	0.307290	+	0.951616i	$0.400578\pi$
$109$	−7.00000	+	12.1244i	−0.670478	+	1.16130i	−0.977769	+	0.209687i	$0.932756\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.00000			−0.752577			−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000			−0.752577			−0.376288	+	0.926503i	$0.622800\pi$
$114$		0			0
$115$	−8.48528	+	14.6969i	−0.791257	+	1.37050i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.65685			−0.505964
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.65685	−	9.79796i	−0.494242	−	0.856052i	0.505736	−	0.862688i	$-0.331221\pi$
$131$	−5.65685	−	9.79796i	−0.494242	−	0.856052i	−0.999978	+	0.00663646i	$0.997888\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.00000	+	6.92820i	−0.341743	+	0.591916i	−0.984757	−	0.173939i	$-0.944351\pi$
$137$	−4.00000	+	6.92820i	−0.341743	+	0.591916i	0.643013	+	0.765855i	$0.277684\pi$
$138$		0			0
$139$	−11.3137			−0.959616			−0.479808	−	0.877373i	$-0.659294\pi$
$139$	−11.3137			−0.959616			−0.479808	+	0.877373i	$0.659294\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.65685	+	9.79796i	−0.473050	+	0.819346i
$144$		0			0
$145$	5.65685	+	9.79796i	0.469776	+	0.813676i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.0000	−	17.3205i	−0.819232	−	1.41895i	−0.906249	−	0.422744i	$-0.861067\pi$
$149$	−10.0000	−	17.3205i	−0.819232	−	1.41895i	0.0870170	−	0.996207i	$-0.472267\pi$
$150$		0			0
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	−0.656101	−	0.754673i	$-0.727796\pi$
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	0.981617	+	0.190864i	$0.0611289\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−16.0000			−1.28515
$156$		0			0
$157$	2.82843	−	4.89898i	0.225733	−	0.390981i	−0.730806	−	0.682585i	$-0.760856\pi$
$157$	2.82843	−	4.89898i	0.225733	−	0.390981i	0.956539	+	0.291604i	$0.0941890\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.00000	+	3.46410i	0.156652	+	0.271329i	0.933659	−	0.358162i	$-0.116597\pi$
$163$	2.00000	+	3.46410i	0.156652	+	0.271329i	−0.777007	+	0.629492i	$0.783263\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−22.6274			−1.75096			−0.875481	−	0.483252i	$-0.839455\pi$
$167$	−22.6274			−1.75096			−0.875481	+	0.483252i	$0.839455\pi$
$168$		0			0
$169$	19.0000			1.46154
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.41421	+	2.44949i	0.107521	+	0.186231i	0.914765	−	0.403986i	$-0.132375\pi$
$173$	1.41421	+	2.44949i	0.107521	+	0.186231i	−0.807245	+	0.590217i	$0.799042\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−11.0000	+	19.0526i	−0.822179	+	1.42406i	0.0818780	+	0.996642i	$0.473908\pi$
$179$	−11.0000	+	19.0526i	−0.822179	+	1.42406i	−0.904057	+	0.427413i	$0.859425\pi$
$180$		0			0
$181$	−16.9706			−1.26141			−0.630706	−	0.776022i	$-0.717235\pi$
$181$	−16.9706			−1.26141			−0.630706	+	0.776022i	$0.717235\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.82843	−	4.89898i	0.207950	−	0.360180i
$186$		0			0
$187$	−2.82843	−	4.89898i	−0.206835	−	0.358249i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−9.00000	−	15.5885i	−0.651217	−	1.12794i	−0.982828	−	0.184525i	$-0.940925\pi$
$191$	−9.00000	−	15.5885i	−0.651217	−	1.12794i	0.331611	−	0.943416i	$-0.392408\pi$
$192$		0			0
$193$	−11.0000	+	19.0526i	−0.791797	+	1.37143i	0.133056	+	0.991109i	$0.457521\pi$
$193$	−11.0000	+	19.0526i	−0.791797	+	1.37143i	−0.924853	+	0.380325i	$0.875812\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.00000			0.284988			0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000			0.284988			0.142494	+	0.989796i	$0.454488\pi$
$198$		0			0
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.00000	+	6.92820i	0.279372	+	0.483887i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.3137			0.782586
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.65685	+	9.79796i	0.385794	+	0.668215i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.00000	+	13.8564i	−0.538138	+	0.932083i
$222$		0			0
$223$	−11.3137			−0.757622			−0.378811	−	0.925474i	$-0.623667\pi$
$223$	−11.3137			−0.757622			−0.378811	+	0.925474i	$0.623667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	8.48528	+	14.6969i	0.560723	+	0.971201i	0.997434	+	0.0715988i	$0.0228101\pi$
$229$	8.48528	+	14.6969i	0.560723	+	0.971201i	−0.436710	+	0.899602i	$0.643857\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.00000	−	6.92820i	−0.262049	−	0.453882i	0.704737	−	0.709468i	$-0.251065\pi$
$233$	−4.00000	−	6.92820i	−0.262049	−	0.453882i	−0.966786	+	0.255586i	$0.917731\pi$
$234$		0			0
$235$	16.0000	−	27.7128i	1.04372	−	1.80778i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.00000			−0.129369			−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000			−0.129369			−0.0646846	+	0.997906i	$0.520604\pi$
$240$		0			0
$241$	−5.65685	+	9.79796i	−0.364390	+	0.631142i	−0.988678	−	0.150052i	$-0.952056\pi$
$241$	−5.65685	+	9.79796i	−0.364390	+	0.631142i	0.624288	+	0.781194i	$0.285389\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−16.0000	−	27.7128i	−1.01806	−	1.76332i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	22.6274			1.42823			0.714115	−	0.700028i	$-0.246829\pi$
$251$	22.6274			1.42823			0.714115	+	0.700028i	$0.246829\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.7279	+	22.0454i	0.793946	+	1.37515i	0.923506	+	0.383583i	$0.125310\pi$
$257$	12.7279	+	22.0454i	0.793946	+	1.37515i	−0.129560	+	0.991572i	$0.541357\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.00000	−	8.66025i	0.308313	−	0.534014i	−0.669680	−	0.742650i	$-0.733569\pi$
$263$	5.00000	−	8.66025i	0.308313	−	0.534014i	0.977993	+	0.208635i	$0.0669022\pi$
$264$		0			0
$265$	33.9411			2.08499
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.5563	−	26.9444i	0.948487	−	1.64283i	0.199874	−	0.979822i	$-0.435947\pi$
$269$	15.5563	−	26.9444i	0.948487	−	1.64283i	0.748614	−	0.663007i	$-0.230720\pi$
$270$		0			0
$271$	−8.48528	−	14.6969i	−0.515444	−	0.892775i	−0.999839	−	0.0179261i	$-0.994294\pi$
$271$	−8.48528	−	14.6969i	−0.515444	−	0.892775i	0.484395	−	0.874849i	$-0.339040\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.00000	−	5.19615i	−0.180907	−	0.313340i
$276$		0			0
$277$	−7.00000	+	12.1244i	−0.420589	+	0.728482i	−0.995997	−	0.0893846i	$-0.971510\pi$
$277$	−7.00000	+	12.1244i	−0.420589	+	0.728482i	0.575408	+	0.817867i	$0.304843\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−24.0000			−1.43172			−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000			−1.43172			−0.715860	+	0.698244i	$0.753965\pi$
$282$		0			0
$283$	14.1421	−	24.4949i	0.840663	−	1.45607i	−0.0486726	−	0.998815i	$-0.515499\pi$
$283$	14.1421	−	24.4949i	0.840663	−	1.45607i	0.889335	−	0.457256i	$-0.151168\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4.50000	+	7.79423i	0.264706	+	0.458484i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	19.7990			1.15667			0.578335	−	0.815800i	$-0.303703\pi$
$293$	19.7990			1.15667			0.578335	+	0.815800i	$0.303703\pi$
$294$		0			0
$295$	−32.0000			−1.86311
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.9706	−	29.3939i	−0.981433	−	1.69989i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.00000	+	13.8564i	−0.458079	+	0.793416i
$306$		0			0
$307$	16.9706			0.968561			0.484281	−	0.874913i	$-0.339081\pi$
$307$	16.9706			0.968561			0.484281	+	0.874913i	$0.339081\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.65685	+	9.79796i	−0.320771	+	0.555591i	−0.980647	−	0.195783i	$-0.937275\pi$
$311$	−5.65685	+	9.79796i	−0.320771	+	0.555591i	0.659877	+	0.751374i	$0.270609\pi$
$312$		0			0
$313$	−16.9706	−	29.3939i	−0.959233	−	1.66144i	−0.724370	−	0.689412i	$-0.757869\pi$
$313$	−16.9706	−	29.3939i	−0.959233	−	1.66144i	−0.234863	−	0.972028i	$-0.575464\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.00000	−	10.3923i	−0.336994	−	0.583690i	0.646872	−	0.762598i	$-0.276077\pi$
$317$	−6.00000	−	10.3923i	−0.336994	−	0.583690i	−0.983866	+	0.178908i	$0.942743\pi$
$318$		0			0
$319$	4.00000	−	6.92820i	0.223957	−	0.387905i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.0000			0.890264
$324$		0			0
$325$	−8.48528	+	14.6969i	−0.470679	+	0.815239i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	0.998298	+	0.0583130i	$0.0185721\pi$
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	−0.448649	+	0.893708i	$0.648095\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−33.9411			−1.85440
$336$		0			0
$337$	2.00000			0.108947			0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000			0.108947			0.0544735	+	0.998515i	$0.482652\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5.65685	+	9.79796i	0.306336	+	0.530589i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.00000	−	1.73205i	0.0536828	−	0.0929814i	−0.837935	−	0.545770i	$-0.816237\pi$
$347$	1.00000	−	1.73205i	0.0536828	−	0.0929814i	0.891618	+	0.452788i	$0.149571\pi$
$348$		0			0
$349$	16.9706			0.908413			0.454207	−	0.890896i	$-0.349923\pi$
$349$	16.9706			0.908413			0.454207	+	0.890896i	$0.349923\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.41421	+	2.44949i	−0.0752710	+	0.130373i	−0.901204	−	0.433395i	$-0.857316\pi$
$353$	−1.41421	+	2.44949i	−0.0752710	+	0.130373i	0.825933	+	0.563768i	$0.190649\pi$
$354$		0			0
$355$	8.48528	+	14.6969i	0.450352	+	0.780033i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3.00000	+	5.19615i	0.158334	+	0.274242i	0.934268	−	0.356572i	$-0.116054\pi$
$359$	3.00000	+	5.19615i	0.158334	+	0.274242i	−0.775934	+	0.630814i	$0.782721\pi$
$360$		0			0
$361$	−6.50000	+	11.2583i	−0.342105	+	0.592544i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−22.6274			−1.16537
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.65685	−	9.79796i	−0.289052	−	0.500652i	0.684532	−	0.728983i	$-0.260007\pi$
$383$	−5.65685	−	9.79796i	−0.289052	−	0.500652i	−0.973584	+	0.228331i	$0.926673\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	14.0000	−	24.2487i	0.709828	−	1.22946i	−0.255092	−	0.966917i	$-0.582106\pi$
$389$	14.0000	−	24.2487i	0.709828	−	1.22946i	0.964921	−	0.262542i	$-0.0845608\pi$
$390$		0			0
$391$	16.9706			0.858238
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.3137	+	19.5959i	−0.569254	+	0.985978i
$396$		0			0
$397$	−8.48528	−	14.6969i	−0.425864	−	0.737618i	0.570637	−	0.821203i	$-0.306696\pi$
$397$	−8.48528	−	14.6969i	−0.425864	−	0.737618i	−0.996501	+	0.0835845i	$0.973363\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	0.992932	+	0.118686i	$0.0378683\pi$
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	−0.393680	+	0.919247i	$0.628798\pi$
$402$		0			0
$403$	16.0000	−	27.7128i	0.797017	−	1.38047i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.00000			−0.198273
$408$		0			0
$409$	16.9706	−	29.3939i	0.839140	−	1.45343i	−0.0514740	−	0.998674i	$-0.516392\pi$
$409$	16.9706	−	29.3939i	0.839140	−	1.45343i	0.890614	−	0.454759i	$-0.150275\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−16.0000	−	27.7128i	−0.785409	−	1.36037i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	11.3137			0.552711			0.276355	−	0.961056i	$-0.410873\pi$
$419$	11.3137			0.552711			0.276355	+	0.961056i	$0.410873\pi$
$420$		0			0
$421$	30.0000			1.46211			0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000			1.46211			0.731055	+	0.682318i	$0.239028\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.24264	−	7.34847i	−0.205798	−	0.356453i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.00000	+	15.5885i	−0.433515	+	0.750870i	−0.997173	−	0.0751385i	$-0.976060\pi$
$431$	−9.00000	+	15.5885i	−0.433515	+	0.750870i	0.563658	+	0.826008i	$0.309393\pi$
$432$		0			0
$433$	−11.3137			−0.543702			−0.271851	−	0.962339i	$-0.587636\pi$
$433$	−11.3137			−0.543702			−0.271851	+	0.962339i	$0.587636\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−16.9706	+	29.3939i	−0.811812	+	1.40610i
$438$		0			0
$439$	−16.9706	−	29.3939i	−0.809961	−	1.40289i	−0.912890	−	0.408205i	$-0.866155\pi$
$439$	−16.9706	−	29.3939i	−0.809961	−	1.40289i	0.102930	−	0.994689i	$-0.467178\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.00000	+	1.73205i	0.0475114	+	0.0822922i	0.888803	−	0.458289i	$-0.151538\pi$
$443$	1.00000	+	1.73205i	0.0475114	+	0.0822922i	−0.841292	+	0.540581i	$0.818204\pi$
$444$		0			0
$445$	−12.0000	+	20.7846i	−0.568855	+	0.985285i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	32.0000			1.51017			0.755087	−	0.655625i	$-0.227595\pi$
$449$	32.0000			1.51017			0.755087	+	0.655625i	$0.227595\pi$
$450$		0			0
$451$	2.82843	−	4.89898i	0.133185	−	0.230684i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.00000	−	5.19615i	−0.140334	−	0.243066i	0.787288	−	0.616585i	$-0.211484\pi$
$457$	−3.00000	−	5.19615i	−0.140334	−	0.243066i	−0.927622	+	0.373519i	$0.878151\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.1421			−0.658665			−0.329332	−	0.944214i	$-0.606824\pi$
$461$	−14.1421			−0.658665			−0.329332	+	0.944214i	$0.606824\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.00000	−	6.92820i	0.183920	−	0.318559i
$474$		0			0
$475$	16.9706			0.778663
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.65685	−	9.79796i	0.258468	−	0.447680i	−0.707364	−	0.706850i	$-0.750116\pi$
$479$	5.65685	−	9.79796i	0.258468	−	0.447680i	0.965832	+	0.259170i	$0.0834489\pi$
$480$		0			0
$481$	5.65685	+	9.79796i	0.257930	+	0.446748i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.0000	+	27.7128i	0.726523	+	1.25837i
$486$		0			0
$487$	−12.0000	+	20.7846i	−0.543772	+	0.941841i	0.454911	+	0.890537i	$0.349671\pi$
$487$	−12.0000	+	20.7846i	−0.543772	+	0.941841i	−0.998683	+	0.0513038i	$0.983662\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.00000			−0.0902587			−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000			−0.0902587			−0.0451294	+	0.998981i	$0.514370\pi$
$492$		0			0
$493$	5.65685	−	9.79796i	0.254772	−	0.441278i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	10.0000	+	17.3205i	0.447661	+	0.775372i	0.998233	−	0.0594153i	$-0.0189236\pi$
$499$	10.0000	+	17.3205i	0.447661	+	0.775372i	−0.550572	+	0.834788i	$0.685590\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	11.3137			0.504453			0.252227	−	0.967668i	$-0.418837\pi$
$503$	11.3137			0.504453			0.252227	+	0.967668i	$0.418837\pi$
$504$		0			0
$505$	24.0000			1.06799
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.41421	−	2.44949i	−0.0626839	−	0.108572i	0.832980	−	0.553303i	$-0.186633\pi$
$509$	−1.41421	−	2.44949i	−0.0626839	−	0.108572i	−0.895664	+	0.444731i	$0.853299\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	24.0000	−	41.5692i	1.05757	−	1.83176i
$516$		0			0
$517$	−22.6274			−0.995153
$518$		0			0
$519$		0			0
$520$		0			0
$521$	15.5563	−	26.9444i	0.681536	−	1.18046i	−0.292976	−	0.956120i	$-0.594646\pi$
$521$	15.5563	−	26.9444i	0.681536	−	1.18046i	0.974512	−	0.224335i	$-0.0720211\pi$
$522$		0			0
$523$	11.3137	+	19.5959i	0.494714	+	0.856870i	0.999981	−	0.00609311i	$-0.00193951\pi$
$523$	11.3137	+	19.5959i	0.494714	+	0.856870i	−0.505268	+	0.862963i	$0.668606\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.00000	+	13.8564i	0.348485	+	0.603595i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−16.0000			−0.693037
$534$		0			0
$535$	−25.4558	+	44.0908i	−1.10055	+	1.90621i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	5.00000	+	8.66025i	0.214967	+	0.372333i	0.953262	−	0.302144i	$-0.0977023\pi$
$541$	5.00000	+	8.66025i	0.214967	+	0.372333i	−0.738296	+	0.674477i	$0.764369\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	39.5980			1.69619
$546$		0			0
$547$	−4.00000			−0.171028			−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000			−0.171028			−0.0855138	+	0.996337i	$0.527253\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	11.3137	+	19.5959i	0.481980	+	0.834814i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6.00000	−	10.3923i	0.254228	−	0.440336i	−0.710457	−	0.703740i	$-0.751512\pi$
$557$	6.00000	−	10.3923i	0.254228	−	0.440336i	0.964686	+	0.263404i	$0.0848453\pi$
$558$		0			0
$559$	−22.6274			−0.957038
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.9706	−	29.3939i	0.715224	−	1.23880i	−0.247649	−	0.968850i	$-0.579658\pi$
$563$	16.9706	−	29.3939i	0.715224	−	1.23880i	0.962873	−	0.269954i	$-0.0870086\pi$
$564$		0			0
$565$	11.3137	+	19.5959i	0.475971	+	0.824406i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	22.0000	−	38.1051i	0.920671	−	1.59465i	0.122292	−	0.992494i	$-0.460975\pi$
$571$	22.0000	−	38.1051i	0.920671	−	1.59465i	0.798379	−	0.602155i	$-0.205691\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	18.0000			0.750652
$576$		0			0
$577$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$577$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−12.0000	−	20.7846i	−0.496989	−	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7.07107	−	12.2474i	−0.290374	−	0.502942i	0.683524	−	0.729928i	$-0.260446\pi$
$593$	−7.07107	−	12.2474i	−0.290374	−	0.502942i	−0.973898	+	0.226985i	$0.927113\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	11.0000	−	19.0526i	0.449448	−	0.778466i	−0.548902	−	0.835887i	$-0.684954\pi$
$599$	11.0000	−	19.0526i	0.449448	−	0.778466i	0.998350	+	0.0574201i	$0.0182874\pi$
$600$		0			0
$601$	−22.6274			−0.922992			−0.461496	−	0.887142i	$-0.652687\pi$
$601$	−22.6274			−0.922992			−0.461496	+	0.887142i	$0.652687\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.89949	−	17.1464i	0.402472	−	0.697101i
$606$		0			0
$607$	5.65685	+	9.79796i	0.229605	+	0.397687i	0.957691	−	0.287799i	$-0.0929234\pi$
$607$	5.65685	+	9.79796i	0.229605	+	0.397687i	−0.728086	+	0.685485i	$0.759590\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	32.0000	+	55.4256i	1.29458	+	2.24228i
$612$		0			0
$613$	3.00000	−	5.19615i	0.121169	−	0.209871i	−0.799060	−	0.601251i	$-0.794669\pi$
$613$	3.00000	−	5.19615i	0.121169	−	0.209871i	0.920229	+	0.391381i	$0.128002\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	48.0000			1.93241			0.966204	−	0.257780i	$-0.0829910\pi$
$617$	48.0000			1.93241			0.966204	+	0.257780i	$0.0829910\pi$
$618$		0			0
$619$	−5.65685	+	9.79796i	−0.227368	+	0.393813i	−0.957027	−	0.289998i	$-0.906345\pi$
$619$	−5.65685	+	9.79796i	−0.227368	+	0.393813i	0.729659	+	0.683811i	$0.239679\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15.5000	+	26.8468i	0.620000	+	1.07387i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.65685			−0.225554
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	22.6274	+	39.1918i	0.897942	+	1.55528i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	20.0000	−	34.6410i	0.789953	−	1.36824i	−0.136043	−	0.990703i	$-0.543438\pi$
$641$	20.0000	−	34.6410i	0.789953	−	1.36824i	0.925995	−	0.377535i	$-0.123228\pi$
$642$		0			0
$643$	−16.9706			−0.669254			−0.334627	−	0.942351i	$-0.608610\pi$
$643$	−16.9706			−0.669254			−0.334627	+	0.942351i	$0.608610\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$	11.3137	+	19.5959i	0.444102	+	0.769207i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.0000	+	17.3205i	0.391330	+	0.677804i	0.992625	−	0.121223i	$-0.0386817\pi$
$653$	10.0000	+	17.3205i	0.391330	+	0.677804i	−0.601295	+	0.799027i	$0.705348\pi$
$654$		0			0
$655$	−16.0000	+	27.7128i	−0.625172	+	1.08283i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.00000			−0.233727			−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000			−0.233727			−0.116863	+	0.993148i	$0.537284\pi$
$660$		0			0
$661$	−2.82843	+	4.89898i	−0.110013	+	0.190548i	−0.915775	−	0.401691i	$-0.868423\pi$
$661$	−2.82843	+	4.89898i	−0.110013	+	0.190548i	0.805762	+	0.592239i	$0.201756\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	12.0000	+	20.7846i	0.464642	+	0.804783i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	11.3137			0.436761
$672$		0			0
$673$	−6.00000			−0.231283			−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000			−0.231283			−0.115642	+	0.993291i	$0.536892\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.24264	−	7.34847i	−0.163058	−	0.282425i	0.772906	−	0.634521i	$-0.218802\pi$
$677$	−4.24264	−	7.34847i	−0.163058	−	0.282425i	−0.935964	+	0.352096i	$0.885469\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.00000	+	12.1244i	−0.267848	+	0.463926i	−0.968306	−	0.249768i	$-0.919646\pi$
$683$	−7.00000	+	12.1244i	−0.267848	+	0.463926i	0.700458	+	0.713693i	$0.252979\pi$
$684$		0			0
$685$	22.6274			0.864549
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−33.9411	+	58.7878i	−1.29305	+	2.23964i
$690$		0			0
$691$	11.3137	+	19.5959i	0.430394	+	0.745464i	0.996907	−	0.0785887i	$-0.0250414\pi$
$691$	11.3137	+	19.5959i	0.430394	+	0.745464i	−0.566513	+	0.824053i	$0.691708\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	16.0000	+	27.7128i	0.606915	+	1.05121i
$696$		0			0
$697$	4.00000	−	6.92820i	0.151511	−	0.262424i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.00000			0.151078			0.0755390	−	0.997143i	$-0.475932\pi$
$701$	4.00000			0.151078			0.0755390	+	0.997143i	$0.475932\pi$
$702$		0			0
$703$	5.65685	−	9.79796i	0.213352	−	0.369537i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−19.0000	−	32.9090i	−0.713560	−	1.23592i	−0.963512	−	0.267664i	$-0.913748\pi$
$709$	−19.0000	−	32.9090i	−0.713560	−	1.23592i	0.249952	−	0.968258i	$-0.419585\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−33.9411			−1.27111
$714$		0			0
$715$	32.0000			1.19673
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.65685	+	9.79796i	0.210965	+	0.365402i	0.952017	−	0.306046i	$-0.0990060\pi$
$719$	5.65685	+	9.79796i	0.210965	+	0.365402i	−0.741052	+	0.671448i	$0.765673\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.00000	−	10.3923i	0.222834	−	0.385961i
$726$		0			0
$727$	−39.5980			−1.46861			−0.734304	−	0.678821i	$-0.762491\pi$
$727$	−39.5980			−1.46861			−0.734304	+	0.678821i	$0.762491\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.65685	−	9.79796i	0.209226	−	0.362391i
$732$		0			0
$733$	−8.48528	−	14.6969i	−0.313411	−	0.542844i	0.665687	−	0.746231i	$-0.268138\pi$
$733$	−8.48528	−	14.6969i	−0.313411	−	0.542844i	−0.979098	+	0.203387i	$0.934805\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12.0000	+	20.7846i	0.442026	+	0.765611i
$738$		0			0
$739$	18.0000	−	31.1769i	0.662141	−	1.14686i	−0.317911	−	0.948120i	$-0.602981\pi$
$739$	18.0000	−	31.1769i	0.662141	−	1.14686i	0.980052	−	0.198741i	$-0.0636852\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	26.0000			0.953847			0.476924	−	0.878945i	$-0.341752\pi$
$743$	26.0000			0.953847			0.476924	+	0.878945i	$0.341752\pi$
$744$		0			0
$745$	−28.2843	+	48.9898i	−1.03626	+	1.79485i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−24.0000	−	41.5692i	−0.875772	−	1.51688i	−0.855938	−	0.517079i	$-0.827019\pi$
$751$	−24.0000	−	41.5692i	−0.875772	−	1.51688i	−0.0198348	−	0.999803i	$-0.506314\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−22.6274			−0.823496
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15.5563	−	26.9444i	−0.563917	−	0.976733i	−0.997150	−	0.0754510i	$-0.975960\pi$
$761$	−15.5563	−	26.9444i	−0.563917	−	0.976733i	0.433232	−	0.901282i	$-0.357373\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	32.0000	−	55.4256i	1.15545	−	2.00130i
$768$		0			0
$769$	−33.9411			−1.22395			−0.611974	−	0.790878i	$-0.709624\pi$
$769$	−33.9411			−1.22395			−0.611974	+	0.790878i	$0.709624\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15.5563	+	26.9444i	−0.559523	+	0.969122i	0.438013	+	0.898969i	$0.355682\pi$
$773$	−15.5563	+	26.9444i	−0.559523	+	0.969122i	−0.997536	+	0.0701537i	$0.977651\pi$
$774$		0			0
$775$	8.48528	+	14.6969i	0.304800	+	0.527930i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.00000	+	13.8564i	0.286630	+	0.496457i
$780$		0			0
$781$	6.00000	−	10.3923i	0.214697	−	0.371866i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−16.0000			−0.571064
$786$		0			0
$787$	16.9706	−	29.3939i	0.604935	−	1.04778i	−0.387126	−	0.922027i	$-0.626532\pi$
$787$	16.9706	−	29.3939i	0.604935	−	1.04778i	0.992062	−	0.125752i	$-0.0401343\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−16.0000	−	27.7128i	−0.568177	−	0.984111i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−25.4558			−0.901692			−0.450846	−	0.892602i	$-0.648878\pi$
$797$	−25.4558			−0.901692			−0.450846	+	0.892602i	$0.648878\pi$
$798$		0			0
$799$	−32.0000			−1.13208
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$809$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$810$		0			0
$811$	−22.6274			−0.794556			−0.397278	−	0.917698i	$-0.630045\pi$
$811$	−22.6274			−0.794556			−0.397278	+	0.917698i	$0.630045\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.65685	−	9.79796i	0.198151	−	0.343208i
$816$		0			0
$817$	11.3137	+	19.5959i	0.395817	+	0.685574i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.0000	+	31.1769i	0.628204	+	1.08808i	0.987912	+	0.155017i	$0.0495431\pi$
$821$	18.0000	+	31.1769i	0.628204	+	1.08808i	−0.359708	+	0.933065i	$0.617124\pi$
$822$		0			0
$823$	−16.0000	+	27.7128i	−0.557725	+	0.966008i	0.439961	+	0.898017i	$0.354992\pi$
$823$	−16.0000	+	27.7128i	−0.557725	+	0.966008i	−0.997686	+	0.0679910i	$0.978341\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	50.0000			1.73867			0.869335	−	0.494223i	$-0.164547\pi$
$827$	50.0000			1.73867			0.869335	+	0.494223i	$0.164547\pi$
$828$		0			0
$829$	2.82843	−	4.89898i	0.0982353	−	0.170149i	−0.812719	−	0.582656i	$-0.802014\pi$
$829$	2.82843	−	4.89898i	0.0982353	−	0.170149i	0.910954	+	0.412507i	$0.135347\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	32.0000	+	55.4256i	1.10741	+	1.91808i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	45.2548			1.56237			0.781185	−	0.624299i	$-0.214615\pi$
$839$	45.2548			1.56237			0.781185	+	0.624299i	$0.214615\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−26.8701	−	46.5403i	−0.924358	−	1.60104i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.00000	−	10.3923i	0.205677	−	0.356244i
$852$		0			0
$853$	16.9706			0.581061			0.290531	−	0.956866i	$-0.406168\pi$
$853$	16.9706			0.581061			0.290531	+	0.956866i	$0.406168\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.07107	−	12.2474i	0.241543	−	0.418365i	−0.719611	−	0.694377i	$-0.755680\pi$
$857$	7.07107	−	12.2474i	0.241543	−	0.418365i	0.961154	+	0.276013i	$0.0890132\pi$
$858$		0			0
$859$	−25.4558	−	44.0908i	−0.868542	−	1.50436i	−0.863486	−	0.504372i	$-0.831724\pi$
$859$	−25.4558	−	44.0908i	−0.868542	−	1.50436i	−0.00505571	−	0.999987i	$-0.501609\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.00000	+	15.5885i	0.306364	+	0.530637i	0.977564	−	0.210639i	$-0.0675543\pi$
$863$	9.00000	+	15.5885i	0.306364	+	0.530637i	−0.671200	+	0.741276i	$0.734221\pi$
$864$		0			0
$865$	4.00000	−	6.92820i	0.136004	−	0.235566i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	16.0000			0.542763
$870$		0			0
$871$	33.9411	−	58.7878i	1.15005	−	1.99195i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	23.0000	+	39.8372i	0.776655	+	1.34521i	0.933860	+	0.357640i	$0.116418\pi$
$877$	23.0000	+	39.8372i	0.776655	+	1.34521i	−0.157205	+	0.987566i	$0.550248\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−8.48528			−0.285876			−0.142938	−	0.989732i	$-0.545655\pi$
$881$	−8.48528			−0.285876			−0.142938	+	0.989732i	$0.545655\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−22.6274	−	39.1918i	−0.759754	−	1.31593i	−0.942976	−	0.332861i	$-0.891986\pi$
$887$	−22.6274	−	39.1918i	−0.759754	−	1.31593i	0.183221	−	0.983072i	$-0.441347\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	32.0000	−	55.4256i	1.07084	−	1.85475i
$894$		0			0
$895$	62.2254			2.07997
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.3137	+	19.5959i	−0.377333	+	0.653560i
$900$		0			0
$901$	−16.9706	−	29.3939i	−0.565371	−	0.979252i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	24.0000	+	41.5692i	0.797787	+	1.38181i
$906$		0			0
$907$	−10.0000	+	17.3205i	−0.332045	+	0.575118i	−0.982913	−	0.184073i	$-0.941072\pi$
$907$	−10.0000	+	17.3205i	−0.332045	+	0.575118i	0.650868	+	0.759191i	$0.274405\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−46.0000			−1.52405			−0.762024	−	0.647549i	$-0.775794\pi$
$911$	−46.0000			−1.52405			−0.762024	+	0.647549i	$0.775794\pi$
$912$		0			0
$913$	−11.3137	+	19.5959i	−0.374429	+	0.648530i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−8.00000	−	13.8564i	−0.263896	−	0.457081i	0.703378	−	0.710816i	$-0.251674\pi$
$919$	−8.00000	−	13.8564i	−0.263896	−	0.457081i	−0.967274	+	0.253735i	$0.918341\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−33.9411			−1.11719
$924$		0			0
$925$	−6.00000			−0.197279
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.3848	+	31.8434i	0.603185	+	1.04475i	0.992336	+	0.123573i	$0.0394352\pi$
$929$	18.3848	+	31.8434i	0.603185	+	1.04475i	−0.389151	+	0.921174i	$0.627231\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.00000	+	13.8564i	−0.261628	+	0.453153i
$936$		0			0
$937$	45.2548			1.47841			0.739205	−	0.673480i	$-0.235201\pi$
$937$	45.2548			1.47841			0.739205	+	0.673480i	$0.235201\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.89949	−	17.1464i	0.322714	−	0.558958i	−0.658333	−	0.752727i	$-0.728738\pi$
$941$	9.89949	−	17.1464i	0.322714	−	0.558958i	0.981047	+	0.193769i	$0.0620714\pi$
$942$		0			0
$943$	8.48528	+	14.6969i	0.276319	+	0.478598i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21.0000	−	36.3731i	−0.682408	−	1.18197i	−0.974244	−	0.225497i	$-0.927599\pi$
$947$	−21.0000	−	36.3731i	−0.682408	−	1.18197i	0.291835	−	0.956469i	$-0.405734\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	24.0000			0.777436			0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000			0.777436			0.388718	+	0.921357i	$0.372918\pi$
$954$		0			0
$955$	−25.4558	+	44.0908i	−0.823732	+	1.42675i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.0161290	−	0.0279363i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	62.2254			2.00311
$966$		0			0
$967$	24.0000			0.771788			0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000			0.771788			0.385894	+	0.922543i	$0.373893\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	11.3137	+	19.5959i	0.363074	+	0.628863i	0.988465	−	0.151449i	$-0.0483938\pi$
$971$	11.3137	+	19.5959i	0.363074	+	0.628863i	−0.625391	+	0.780312i	$0.715060\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$	16.9706			0.542382
$980$		0			0
$981$		0			0
$982$		0			0
$983$	11.3137	−	19.5959i	0.360851	−	0.625013i	−0.627250	−	0.778818i	$-0.715820\pi$
$983$	11.3137	−	19.5959i	0.360851	−	0.625013i	0.988101	+	0.153805i	$0.0491529\pi$
$984$		0			0
$985$	−5.65685	−	9.79796i	−0.180242	−	0.312189i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.0000	+	20.7846i	0.381578	+	0.660912i
$990$		0			0
$991$	−28.0000	+	48.4974i	−0.889449	+	1.54057i	−0.0489218	+	0.998803i	$0.515578\pi$
$991$	−28.0000	+	48.4974i	−0.889449	+	1.54057i	−0.840528	+	0.541769i	$0.817755\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	19.7990	−	34.2929i	0.627040	−	1.08607i	−0.361102	−	0.932526i	$-0.617599\pi$
$997$	19.7990	−	34.2929i	0.627040	−	1.08607i	0.988143	−	0.153539i	$-0.0490672\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.bf.361.1		4
3.2	odd	2		3528.2.s.bi.361.2		4
7.2	even	3	inner	3528.2.s.bf.3313.1		4
7.3	odd	6		3528.2.a.bi.1.1	yes	2
7.4	even	3		3528.2.a.bi.1.2	yes	2
7.5	odd	6	inner	3528.2.s.bf.3313.2		4
7.6	odd	2	inner	3528.2.s.bf.361.2		4
21.2	odd	6		3528.2.s.bi.3313.2		4
21.5	even	6		3528.2.s.bi.3313.1		4
21.11	odd	6		3528.2.a.bf.1.1	✓	2
21.17	even	6		3528.2.a.bf.1.2	yes	2
21.20	even	2		3528.2.s.bi.361.1		4
28.3	even	6		7056.2.a.ck.1.1		2
28.11	odd	6		7056.2.a.ck.1.2		2
84.11	even	6		7056.2.a.cq.1.1		2
84.59	odd	6		7056.2.a.cq.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3528.2.a.bf.1.1	✓	2	21.11	odd	6
3528.2.a.bf.1.2	yes	2	21.17	even	6
3528.2.a.bi.1.1	yes	2	7.3	odd	6
3528.2.a.bi.1.2	yes	2	7.4	even	3
3528.2.s.bf.361.1		4	1.1	even	1	trivial
3528.2.s.bf.361.2		4	7.6	odd	2	inner
3528.2.s.bf.3313.1		4	7.2	even	3	inner
3528.2.s.bf.3313.2		4	7.5	odd	6	inner
3528.2.s.bi.361.1		4	21.20	even	2
3528.2.s.bi.361.2		4	3.2	odd	2
3528.2.s.bi.3313.1		4	21.5	even	6
3528.2.s.bi.3313.2		4	21.2	odd	6
7056.2.a.ck.1.1		2	28.3	even	6
7056.2.a.ck.1.2		2	28.11	odd	6
7056.2.a.cq.1.1		2	84.11	even	6
7056.2.a.cq.1.2		2	84.59	odd	6

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:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 - 2.44949i) q^{5} +O(q^{10})\)	\(q+(-1.41421 - 2.44949i) q^{5} +(-1.00000 + 1.73205i) q^{11} +5.65685 q^{13} +(-1.41421 + 2.44949i) q^{17} +(-2.82843 - 4.89898i) q^{19} +(-3.00000 - 5.19615i) q^{23} +(-1.50000 + 2.59808i) q^{25} -4.00000 q^{29} +(2.82843 - 4.89898i) q^{31} +(1.00000 + 1.73205i) q^{37} -2.82843 q^{41} -4.00000 q^{43} +(5.65685 + 9.79796i) q^{47} +(-6.00000 + 10.3923i) q^{53} +5.65685 q^{55} +(5.65685 - 9.79796i) q^{59} +(-2.82843 - 4.89898i) q^{61} +(-8.00000 - 13.8564i) q^{65} +(6.00000 - 10.3923i) q^{67} -6.00000 q^{71} +(-4.00000 - 6.92820i) q^{79} +11.3137 q^{83} +8.00000 q^{85} +(-4.24264 - 7.34847i) q^{89} +(-8.00000 + 13.8564i) q^{95} -11.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 4 q^{11} - 12 q^{23} - 6 q^{25} - 16 q^{29} + 4 q^{37} - 16 q^{43} - 24 q^{53} - 32 q^{65} + 24 q^{67} - 24 q^{71} - 16 q^{79} + 32 q^{85} - 32 q^{95}+O(q^{100}) \) 4 * q - 4 * q^11 - 12 * q^23 - 6 * q^25 - 16 * q^29 + 4 * q^37 - 16 * q^43 - 24 * q^53 - 32 * q^65 + 24 * q^67 - 24 * q^71 - 16 * q^79 + 32 * q^85 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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