LMFDB - Embedded newform 6300.2.dd.a.4049.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(1349,6300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6300, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))

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

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

chi := DirichletCharacter("6300.1349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.dd (of order $6$, 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:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4049.3
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	6300.4049
Dual form			6300.2.dd.a.1349.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.59808 + 0.500000i) q^{7} +O(q^{10})$	$q+(2.59808 + 0.500000i) q^{7} +(-3.67423 - 2.12132i) q^{11} +1.73205 q^{13} +(4.24264 + 2.44949i) q^{17} +(-4.50000 + 2.59808i) q^{19} -8.48528i q^{29} +(-1.50000 - 0.866025i) q^{31} +(-4.33013 + 2.50000i) q^{37} +12.2474 q^{41} +11.0000i q^{43} +(2.12132 - 1.22474i) q^{47} +(6.50000 + 2.59808i) q^{49} +(4.24264 - 7.34847i) q^{53} +(2.44949 - 4.24264i) q^{59} +(-3.00000 + 1.73205i) q^{61} +(6.06218 + 3.50000i) q^{67} +12.7279i q^{71} +(7.79423 - 13.5000i) q^{73} +(-8.48528 - 7.34847i) q^{77} +(5.50000 + 9.52628i) q^{79} -12.2474i q^{83} +(7.34847 + 12.7279i) q^{89} +(4.50000 + 0.866025i) q^{91} -3.46410 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 36 q^{19} - 12 q^{31} + 52 q^{49} - 24 q^{61} + 44 q^{79} + 36 q^{91}+O(q^{100}) $ 8 * q - 36 * q^19 - 12 * q^31 + 52 * q^49 - 24 * q^61 + 44 * q^79 + 36 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$.

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.59808	+	0.500000i	0.981981	+	0.188982i
$7$	2.59808	+	0.500000i	0.981981	+	0.188982i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.67423	−	2.12132i	−1.10782	−	0.639602i	−0.169559	−	0.985520i	$-0.554234\pi$
$11$	−3.67423	−	2.12132i	−1.10782	−	0.639602i	−0.938265	+	0.345918i	$0.887568\pi$
$12$		0			0
$13$	1.73205			0.480384			0.240192	−	0.970725i	$-0.422790\pi$
$13$	1.73205			0.480384			0.240192	+	0.970725i	$0.422790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.24264	+	2.44949i	1.02899	+	0.594089i	0.916696	−	0.399586i	$-0.130846\pi$
$17$	4.24264	+	2.44949i	1.02899	+	0.594089i	0.112296	+	0.993675i	$0.464180\pi$
$18$		0			0
$19$	−4.50000	+	2.59808i	−1.03237	+	0.596040i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−4.50000	+	2.59808i	−1.03237	+	0.596040i	−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	8.48528i		−	1.57568i	−0.615882	−	0.787839i	$-0.711200\pi$
$29$		−	8.48528i		−	1.57568i	0.615882	−	0.787839i	$-0.288800\pi$
$30$		0			0
$31$	−1.50000	−	0.866025i	−0.269408	−	0.155543i	0.359211	−	0.933257i	$-0.383046\pi$
$31$	−1.50000	−	0.866025i	−0.269408	−	0.155543i	−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.33013	+	2.50000i	−0.711868	+	0.410997i	−0.811752	−	0.584002i	$-0.801486\pi$
$37$	−4.33013	+	2.50000i	−0.711868	+	0.410997i	0.0998840	+	0.994999i	$0.468153\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	12.2474			1.91273			0.956365	−	0.292174i	$-0.0943788\pi$
$41$	12.2474			1.91273			0.956365	+	0.292174i	$0.0943788\pi$
$42$		0			0
$43$			11.0000i			1.67748i	0.544529	+	0.838742i	$0.316708\pi$
$43$			11.0000i			1.67748i	−0.544529	+	0.838742i	$0.683292\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.12132	−	1.22474i	0.309426	−	0.178647i	−0.337243	−	0.941417i	$-0.609495\pi$
$47$	2.12132	−	1.22474i	0.309426	−	0.178647i	0.646670	+	0.762770i	$0.276161\pi$
$48$		0			0
$49$	6.50000	+	2.59808i	0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.24264	−	7.34847i	0.582772	−	1.00939i	−0.412378	−	0.911013i	$-0.635302\pi$
$53$	4.24264	−	7.34847i	0.582772	−	1.00939i	0.995149	−	0.0983769i	$-0.0313651\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.44949	−	4.24264i	0.318896	−	0.552345i	−0.661362	−	0.750067i	$-0.730021\pi$
$59$	2.44949	−	4.24264i	0.318896	−	0.552345i	0.980258	+	0.197722i	$0.0633545\pi$
$60$		0			0
$61$	−3.00000	+	1.73205i	−0.384111	+	0.221766i	−0.679605	−	0.733578i	$-0.737849\pi$
$61$	−3.00000	+	1.73205i	−0.384111	+	0.221766i	0.295495	+	0.955344i	$0.404516\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.06218	+	3.50000i	0.740613	+	0.427593i	0.822292	−	0.569066i	$-0.192695\pi$
$67$	6.06218	+	3.50000i	0.740613	+	0.427593i	−0.0816792	+	0.996659i	$0.526028\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.7279i			1.51053i	0.655422	+	0.755263i	$0.272491\pi$
$71$			12.7279i			1.51053i	−0.655422	+	0.755263i	$0.727509\pi$
$72$		0			0
$73$	7.79423	−	13.5000i	0.912245	−	1.58006i	0.101361	−	0.994850i	$-0.467680\pi$
$73$	7.79423	−	13.5000i	0.912245	−	1.58006i	0.810885	−	0.585206i	$-0.198986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.48528	−	7.34847i	−0.966988	−	0.837436i
$78$		0			0
$79$	5.50000	+	9.52628i	0.618798	+	1.07179i	0.989705	+	0.143120i	$0.0457135\pi$
$79$	5.50000	+	9.52628i	0.618798	+	1.07179i	−0.370907	+	0.928670i	$0.620953\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	12.2474i		−	1.34433i	−0.740400	−	0.672166i	$-0.765364\pi$
$83$		−	12.2474i		−	1.34433i	0.740400	−	0.672166i	$-0.234636\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.34847	+	12.7279i	0.778936	+	1.34916i	0.932555	+	0.361027i	$0.117574\pi$
$89$	7.34847	+	12.7279i	0.778936	+	1.34916i	−0.153619	+	0.988130i	$0.549093\pi$
$90$		0			0
$91$	4.50000	+	0.866025i	0.471728	+	0.0907841i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.46410			−0.351726			−0.175863	−	0.984415i	$-0.556272\pi$
$97$	−3.46410			−0.351726			−0.175863	+	0.984415i	$0.556272\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.67423	−	6.36396i	0.365600	−	0.633238i	−0.623272	−	0.782005i	$-0.714197\pi$
$101$	3.67423	−	6.36396i	0.365600	−	0.633238i	0.988872	+	0.148767i	$0.0475305\pi$
$102$		0			0
$103$	−4.33013	−	7.50000i	−0.426660	−	0.738997i	0.569914	−	0.821705i	$-0.306977\pi$
$103$	−4.33013	−	7.50000i	−0.426660	−	0.738997i	−0.996574	+	0.0827075i	$0.973643\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.24264	+	7.34847i	0.410152	+	0.710403i	0.994906	−	0.100807i	$-0.0321425\pi$
$107$	4.24264	+	7.34847i	0.410152	+	0.710403i	−0.584754	+	0.811210i	$0.698809\pi$
$108$		0			0
$109$	−2.50000	+	4.33013i	−0.239457	+	0.414751i	−0.960558	−	0.278078i	$-0.910303\pi$
$109$	−2.50000	+	4.33013i	−0.239457	+	0.414751i	0.721102	+	0.692829i	$0.243636\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.24264			−0.399114			−0.199557	−	0.979886i	$-0.563950\pi$
$113$	−4.24264			−0.399114			−0.199557	+	0.979886i	$0.563950\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.79796	+	8.48528i	0.898177	+	0.777844i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$			5.00000i			0.443678i	0.975083	+	0.221839i	$0.0712060\pi$
$127$			5.00000i			0.443678i	−0.975083	+	0.221839i	$0.928794\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.67423	−	6.36396i	−0.321019	−	0.556022i	0.659679	−	0.751547i	$-0.270692\pi$
$131$	−3.67423	−	6.36396i	−0.321019	−	0.556022i	−0.980699	+	0.195525i	$0.937359\pi$
$132$		0			0
$133$	−12.9904	+	4.50000i	−1.12641	+	0.390199i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$			1.73205i			0.146911i	0.997299	+	0.0734553i	$0.0234026\pi$
$139$			1.73205i			0.146911i	−0.997299	+	0.0734553i	$0.976597\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.36396	−	3.67423i	−0.532181	−	0.307255i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	14.6969	−	8.48528i	1.20402	−	0.695141i	0.242574	−	0.970133i	$-0.422008\pi$
$149$	14.6969	−	8.48528i	1.20402	−	0.695141i	0.961447	+	0.274992i	$0.0886751\pi$
$150$		0			0
$151$	1.00000	−	1.73205i	0.0813788	−	0.140952i	−0.822464	−	0.568818i	$-0.807401\pi$
$151$	1.00000	−	1.73205i	0.0813788	−	0.140952i	0.903842	+	0.427865i	$0.140734\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.19615	−	9.00000i	0.414698	−	0.718278i	−0.580699	−	0.814119i	$-0.697221\pi$
$157$	5.19615	−	9.00000i	0.414698	−	0.718278i	0.995397	+	0.0958404i	$0.0305539\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	19.0526	−	11.0000i	1.49231	−	0.861586i	0.492350	−	0.870397i	$-0.336138\pi$
$163$	19.0526	−	11.0000i	1.49231	−	0.861586i	0.999961	+	0.00881059i	$0.00280453\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			17.1464i			1.32683i	0.748251	+	0.663415i	$0.230894\pi$
$167$			17.1464i			1.32683i	−0.748251	+	0.663415i	$0.769106\pi$
$168$		0			0
$169$	−10.0000			−0.769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.0227	+	6.36396i	0.823876	+	0.475665i	0.851751	−	0.523947i	$-0.175541\pi$
$179$	11.0227	+	6.36396i	0.823876	+	0.475665i	−0.0278755	+	0.999611i	$0.508874\pi$
$180$		0			0
$181$		−	5.19615i		−	0.386227i	−0.981176	−	0.193113i	$-0.938141\pi$
$181$		−	5.19615i		−	0.386227i	0.981176	−	0.193113i	$-0.0618586\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−10.3923	−	18.0000i	−0.759961	−	1.31629i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.67423	−	2.12132i	0.265858	−	0.153493i	−0.361146	−	0.932509i	$-0.617614\pi$
$191$	3.67423	−	2.12132i	0.265858	−	0.153493i	0.627004	+	0.779016i	$0.284281\pi$
$192$		0			0
$193$	16.4545	+	9.50000i	1.18442	+	0.683825i	0.957033	−	0.289980i	$-0.0936485\pi$
$193$	16.4545	+	9.50000i	1.18442	+	0.683825i	0.227387	+	0.973805i	$0.426982\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.48528			−0.604551			−0.302276	−	0.953221i	$-0.597746\pi$
$197$	−8.48528			−0.604551			−0.302276	+	0.953221i	$0.597746\pi$
$198$		0			0
$199$	12.0000	+	6.92820i	0.850657	+	0.491127i	0.860873	−	0.508821i	$-0.169918\pi$
$199$	12.0000	+	6.92820i	0.850657	+	0.491127i	−0.0102152	+	0.999948i	$0.503252\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.24264	−	22.0454i	0.297775	−	1.54728i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	22.0454			1.52491
$210$		0			0
$211$	−2.00000			−0.137686			−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000			−0.137686			−0.0688428	+	0.997628i	$0.521931\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.46410	−	3.00000i	−0.235159	−	0.203653i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.34847	+	4.24264i	0.494312	+	0.285391i
$222$		0			0
$223$	20.7846			1.39184			0.695920	−	0.718119i	$-0.254997\pi$
$223$	20.7846			1.39184			0.695920	+	0.718119i	$0.254997\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.0919	−	11.0227i	−1.26717	−	0.731603i	−0.292721	−	0.956198i	$-0.594561\pi$
$227$	−19.0919	−	11.0227i	−1.26717	−	0.731603i	−0.974452	+	0.224595i	$0.927894\pi$
$228$		0			0
$229$	−1.50000	+	0.866025i	−0.0991228	+	0.0572286i	−0.548742	−	0.835992i	$-0.684893\pi$
$229$	−1.50000	+	0.866025i	−0.0991228	+	0.0572286i	0.449619	+	0.893220i	$0.351560\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.12132	−	3.67423i	−0.138972	−	0.240707i	0.788136	−	0.615502i	$-0.211047\pi$
$233$	−2.12132	−	3.67423i	−0.138972	−	0.240707i	−0.927108	+	0.374795i	$0.877713\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			4.24264i			0.274434i	0.990541	+	0.137217i	$0.0438157\pi$
$239$			4.24264i			0.274434i	−0.990541	+	0.137217i	$0.956184\pi$
$240$		0			0
$241$	12.0000	+	6.92820i	0.772988	+	0.446285i	0.833939	−	0.551856i	$-0.186080\pi$
$241$	12.0000	+	6.92820i	0.772988	+	0.446285i	−0.0609515	+	0.998141i	$0.519414\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−7.79423	+	4.50000i	−0.495935	+	0.286328i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.79796			0.618442			0.309221	−	0.950990i	$-0.399932\pi$
$251$	9.79796			0.618442			0.309221	+	0.950990i	$0.399932\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.36396	+	3.67423i	−0.396973	+	0.229192i	−0.685177	−	0.728377i	$-0.740275\pi$
$257$	−6.36396	+	3.67423i	−0.396973	+	0.229192i	0.288204	+	0.957569i	$0.406942\pi$
$258$		0			0
$259$	−12.5000	+	4.33013i	−0.776712	+	0.269061i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.7279	−	22.0454i	0.784837	−	1.35938i	−0.144259	−	0.989540i	$-0.546080\pi$
$263$	12.7279	−	22.0454i	0.784837	−	1.35938i	0.929096	−	0.369838i	$-0.120587\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.12372	−	10.6066i	0.373370	−	0.646696i	−0.616712	−	0.787189i	$-0.711536\pi$
$269$	6.12372	−	10.6066i	0.373370	−	0.646696i	0.990082	+	0.140493i	$0.0448688\pi$
$270$		0			0
$271$	−24.0000	+	13.8564i	−1.45790	+	0.841717i	−0.998908	−	0.0467255i	$-0.985121\pi$
$271$	−24.0000	+	13.8564i	−1.45790	+	0.841717i	−0.458988	+	0.888442i	$0.651788\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.52628	−	5.50000i	−0.572379	−	0.330463i	0.185720	−	0.982603i	$-0.440538\pi$
$277$	−9.52628	−	5.50000i	−0.572379	−	0.330463i	−0.758099	+	0.652140i	$0.773872\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	16.9706i		−	1.01238i	−0.862422	−	0.506189i	$-0.831054\pi$
$281$		−	16.9706i		−	1.01238i	0.862422	−	0.506189i	$-0.168946\pi$
$282$		0			0
$283$	−4.33013	+	7.50000i	−0.257399	+	0.445829i	−0.965544	−	0.260238i	$-0.916199\pi$
$283$	−4.33013	+	7.50000i	−0.257399	+	0.445829i	0.708145	+	0.706067i	$0.249532\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	31.8198	+	6.12372i	1.87826	+	0.361472i
$288$		0			0
$289$	3.50000	+	6.06218i	0.205882	+	0.356599i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			24.4949i			1.43101i	0.698609	+	0.715504i	$0.253803\pi$
$293$			24.4949i			1.43101i	−0.698609	+	0.715504i	$0.746197\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−5.50000	+	28.5788i	−0.317015	+	1.64726i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	19.0526			1.08739			0.543693	−	0.839284i	$-0.317025\pi$
$307$	19.0526			1.08739			0.543693	+	0.839284i	$0.317025\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.57321	+	14.8492i	−0.486142	+	0.842023i	−0.999873	−	0.0159282i	$-0.994930\pi$
$311$	−8.57321	+	14.8492i	−0.486142	+	0.842023i	0.513731	+	0.857951i	$0.328263\pi$
$312$		0			0
$313$	7.79423	+	13.5000i	0.440556	+	0.763065i	0.997731	−	0.0673300i	$-0.0214480\pi$
$313$	7.79423	+	13.5000i	0.440556	+	0.763065i	−0.557175	+	0.830395i	$0.688115\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.24264	−	7.34847i	−0.238290	−	0.412731i	0.721933	−	0.691963i	$-0.243254\pi$
$317$	−4.24264	−	7.34847i	−0.238290	−	0.412731i	−0.960224	+	0.279231i	$0.909920\pi$
$318$		0			0
$319$	−18.0000	+	31.1769i	−1.00781	+	1.74557i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−25.4558			−1.41640
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.12372	−	2.12132i	0.337612	−	0.116952i
$330$		0			0
$331$	2.50000	+	4.33013i	0.137412	+	0.238005i	0.926516	−	0.376254i	$-0.122788\pi$
$331$	2.50000	+	4.33013i	0.137412	+	0.238005i	−0.789104	+	0.614260i	$0.789455\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$			7.00000i			0.381314i	0.981657	+	0.190657i	$0.0610619\pi$
$337$			7.00000i			0.381314i	−0.981657	+	0.190657i	$0.938938\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.67423	+	6.36396i	0.198971	+	0.344628i
$342$		0			0
$343$	15.5885	+	10.0000i	0.841698	+	0.539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.24264	−	7.34847i	0.227757	−	0.394486i	−0.729386	−	0.684102i	$-0.760194\pi$
$347$	4.24264	−	7.34847i	0.227757	−	0.394486i	0.957143	+	0.289616i	$0.0935275\pi$
$348$		0			0
$349$			24.2487i			1.29800i	0.760787	+	0.649002i	$0.224813\pi$
$349$			24.2487i			1.29800i	−0.760787	+	0.649002i	$0.775187\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−23.3345	−	13.4722i	−1.24197	−	0.717053i	−0.272476	−	0.962163i	$-0.587843\pi$
$353$	−23.3345	−	13.4722i	−1.24197	−	0.717053i	−0.969495	+	0.245110i	$0.921176\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	14.6969	−	8.48528i	0.775675	−	0.447836i	−0.0592205	−	0.998245i	$-0.518862\pi$
$359$	14.6969	−	8.48528i	0.775675	−	0.447836i	0.834895	+	0.550409i	$0.185528\pi$
$360$		0			0
$361$	4.00000	−	6.92820i	0.210526	−	0.364642i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−12.9904	+	22.5000i	−0.678092	+	1.17449i	0.297462	+	0.954734i	$0.403860\pi$
$367$	−12.9904	+	22.5000i	−0.678092	+	1.17449i	−0.975555	+	0.219757i	$0.929474\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	14.6969	−	16.9706i	0.763027	−	0.881068i
$372$		0			0
$373$	6.06218	−	3.50000i	0.313888	−	0.181223i	−0.334777	−	0.942297i	$-0.608661\pi$
$373$	6.06218	−	3.50000i	0.313888	−	0.181223i	0.648665	+	0.761074i	$0.275328\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	14.6969i		−	0.756931i
$378$		0			0
$379$	5.00000			0.256833			0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000			0.256833			0.128416	+	0.991720i	$0.459011\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.48528	−	4.89898i	0.433578	−	0.250326i	−0.267292	−	0.963616i	$-0.586129\pi$
$383$	8.48528	−	4.89898i	0.433578	−	0.250326i	0.700870	+	0.713289i	$0.252795\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.67423	+	2.12132i	0.186291	+	0.107555i	0.590245	−	0.807224i	$-0.299031\pi$
$389$	3.67423	+	2.12132i	0.186291	+	0.107555i	−0.403954	+	0.914779i	$0.632364\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	4.33013	+	7.50000i	0.217323	+	0.376414i	0.953989	−	0.299843i	$-0.0969342\pi$
$397$	4.33013	+	7.50000i	0.217323	+	0.376414i	−0.736666	+	0.676257i	$0.763601\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.34847	−	4.24264i	0.366965	−	0.211867i	−0.305167	−	0.952299i	$-0.598712\pi$
$401$	7.34847	−	4.24264i	0.366965	−	0.211867i	0.672132	+	0.740432i	$0.265379\pi$
$402$		0			0
$403$	−2.59808	−	1.50000i	−0.129419	−	0.0747203i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	21.2132			1.05150
$408$		0			0
$409$	7.50000	+	4.33013i	0.370851	+	0.214111i	0.673830	−	0.738886i	$-0.264648\pi$
$409$	7.50000	+	4.33013i	0.370851	+	0.214111i	−0.302979	+	0.952997i	$0.597981\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.48528	−	9.79796i	0.417533	−	0.482126i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	26.9444			1.31632			0.658160	−	0.752878i	$-0.271335\pi$
$419$	26.9444			1.31632			0.658160	+	0.752878i	$0.271335\pi$
$420$		0			0
$421$	35.0000			1.70580			0.852898	−	0.522078i	$-0.174843\pi$
$421$	35.0000			1.70580			0.852898	+	0.522078i	$0.174843\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−8.66025	+	3.00000i	−0.419099	+	0.145180i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.3712	−	10.6066i	−0.884908	−	0.510902i	−0.0126347	−	0.999920i	$-0.504022\pi$
$431$	−18.3712	−	10.6066i	−0.884908	−	0.510902i	−0.872274	+	0.489018i	$0.837355\pi$
$432$		0			0
$433$	−12.1244			−0.582659			−0.291330	−	0.956623i	$-0.594098\pi$
$433$	−12.1244			−0.582659			−0.291330	+	0.956623i	$0.594098\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	36.0000	−	20.7846i	1.71819	−	0.991995i	0.795956	−	0.605355i	$-0.206969\pi$
$439$	36.0000	−	20.7846i	1.71819	−	0.991995i	0.922231	−	0.386640i	$-0.126365\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$443$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			12.7279i			0.600668i	0.953834	+	0.300334i	$0.0970981\pi$
$449$			12.7279i			0.600668i	−0.953834	+	0.300334i	$0.902902\pi$
$450$		0			0
$451$	−45.0000	−	25.9808i	−2.11897	−	1.22339i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.2583	−	6.50000i	0.526642	−	0.304057i	−0.213006	−	0.977051i	$-0.568325\pi$
$457$	11.2583	−	6.50000i	0.526642	−	0.304057i	0.739648	+	0.672994i	$0.234992\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.89898			0.228168			0.114084	−	0.993471i	$-0.463607\pi$
$461$	4.89898			0.228168			0.114084	+	0.993471i	$0.463607\pi$
$462$		0			0
$463$		−	29.0000i		−	1.34774i	−0.738848	−	0.673872i	$-0.764630\pi$
$463$		−	29.0000i		−	1.34774i	0.738848	−	0.673872i	$-0.235370\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.36396	−	3.67423i	0.294489	−	0.170023i	−0.345476	−	0.938428i	$-0.612282\pi$
$467$	6.36396	−	3.67423i	0.294489	−	0.170023i	0.639965	+	0.768404i	$0.278949\pi$
$468$		0			0
$469$	14.0000	+	12.1244i	0.646460	+	0.559851i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	23.3345	−	40.4166i	1.07292	−	1.85836i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.44949	−	4.24264i	0.111920	−	0.193851i	−0.804624	−	0.593784i	$-0.797633\pi$
$479$	2.44949	−	4.24264i	0.111920	−	0.193851i	0.916544	+	0.399933i	$0.130967\pi$
$480$		0			0
$481$	−7.50000	+	4.33013i	−0.341971	+	0.197437i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−6.06218	−	3.50000i	−0.274703	−	0.158600i	0.356320	−	0.934364i	$-0.384031\pi$
$487$	−6.06218	−	3.50000i	−0.274703	−	0.158600i	−0.631023	+	0.775764i	$0.717365\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			33.9411i			1.53174i	0.642995	+	0.765871i	$0.277692\pi$
$491$			33.9411i			1.53174i	−0.642995	+	0.765871i	$0.722308\pi$
$492$		0			0
$493$	20.7846	−	36.0000i	0.936092	−	1.62136i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.36396	+	33.0681i	−0.285463	+	1.48331i
$498$		0			0
$499$	−17.5000	−	30.3109i	−0.783408	−	1.35690i	−0.929946	−	0.367697i	$-0.880146\pi$
$499$	−17.5000	−	30.3109i	−0.783408	−	1.35690i	0.146538	−	0.989205i	$-0.453187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			2.44949i			0.109217i	0.998508	+	0.0546087i	$0.0173911\pi$
$503$			2.44949i			0.109217i	−0.998508	+	0.0546087i	$0.982609\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.57321	−	14.8492i	−0.380001	−	0.658181i	0.611061	−	0.791584i	$-0.290743\pi$
$509$	−8.57321	−	14.8492i	−0.380001	−	0.658181i	−0.991062	+	0.133402i	$0.957410\pi$
$510$		0			0
$511$	27.0000	−	31.1769i	1.19441	−	1.37919i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−10.3923			−0.457053
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.44949	+	4.24264i	−0.107314	+	0.185873i	−0.914681	−	0.404176i	$-0.867558\pi$
$521$	−2.44949	+	4.24264i	−0.107314	+	0.185873i	0.807367	+	0.590049i	$0.200892\pi$
$522$		0			0
$523$	4.33013	+	7.50000i	0.189343	+	0.327952i	0.945031	−	0.326979i	$-0.106031\pi$
$523$	4.33013	+	7.50000i	0.189343	+	0.327952i	−0.755688	+	0.654932i	$0.772697\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.24264	−	7.34847i	−0.184812	−	0.320104i
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	21.2132			0.918846
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−18.3712	−	23.3345i	−0.791302	−	1.00509i
$540$		0			0
$541$	9.50000	+	16.4545i	0.408437	+	0.707433i	0.994715	−	0.102677i	$-0.0327407\pi$
$541$	9.50000	+	16.4545i	0.408437	+	0.707433i	−0.586278	+	0.810110i	$0.699407\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			2.00000i			0.0855138i	0.999086	+	0.0427569i	$0.0136141\pi$
$547$			2.00000i			0.0855138i	−0.999086	+	0.0427569i	$0.986386\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	22.0454	+	38.1838i	0.939166	+	1.62668i
$552$		0			0
$553$	9.52628	+	27.5000i	0.405099	+	1.16942i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.6066	+	18.3712i	−0.449416	+	0.778412i	−0.998348	−	0.0574555i	$-0.981701\pi$
$557$	−10.6066	+	18.3712i	−0.449416	+	0.778412i	0.548932	+	0.835867i	$0.315035\pi$
$558$		0			0
$559$			19.0526i			0.805837i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.36396	−	3.67423i	−0.268209	−	0.154851i	0.359864	−	0.933005i	$-0.382823\pi$
$563$	−6.36396	−	3.67423i	−0.268209	−	0.154851i	−0.628073	+	0.778154i	$0.716156\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.3712	−	10.6066i	0.770160	−	0.444652i	−0.0627719	−	0.998028i	$-0.519994\pi$
$569$	18.3712	−	10.6066i	0.770160	−	0.444652i	0.832932	+	0.553376i	$0.186661\pi$
$570$		0			0
$571$	3.50000	−	6.06218i	0.146470	−	0.253694i	−0.783450	−	0.621455i	$-0.786542\pi$
$571$	3.50000	−	6.06218i	0.146470	−	0.253694i	0.929921	+	0.367760i	$0.119875\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	19.9186	−	34.5000i	0.829222	−	1.43625i	−0.0694283	−	0.997587i	$-0.522117\pi$
$577$	19.9186	−	34.5000i	0.829222	−	1.43625i	0.898650	−	0.438667i	$-0.144549\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	6.12372	−	31.8198i	0.254055	−	1.32011i
$582$		0			0
$583$	−31.1769	+	18.0000i	−1.29122	+	0.745484i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			14.6969i			0.606608i	0.952894	+	0.303304i	$0.0980897\pi$
$587$			14.6969i			0.606608i	−0.952894	+	0.303304i	$0.901910\pi$
$588$		0			0
$589$	9.00000			0.370839
$590$		0			0
$591$		0			0
$592$		0			0
$593$	2.12132	−	1.22474i	0.0871122	−	0.0502942i	−0.455811	−	0.890077i	$-0.650651\pi$
$593$	2.12132	−	1.22474i	0.0871122	−	0.0502942i	0.542923	+	0.839782i	$0.317317\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	14.6969	+	8.48528i	0.600501	+	0.346699i	0.769238	−	0.638962i	$-0.220636\pi$
$599$	14.6969	+	8.48528i	0.600501	+	0.346699i	−0.168738	+	0.985661i	$0.553969\pi$
$600$		0			0
$601$			43.3013i			1.76630i	0.469095	+	0.883148i	$0.344580\pi$
$601$			43.3013i			1.76630i	−0.469095	+	0.883148i	$0.655420\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	12.9904	+	22.5000i	0.527263	+	0.913247i	0.999495	+	0.0317724i	$0.0101152\pi$
$607$	12.9904	+	22.5000i	0.527263	+	0.913247i	−0.472232	+	0.881474i	$0.656551\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.67423	−	2.12132i	0.148644	−	0.0858194i
$612$		0			0
$613$	6.92820	+	4.00000i	0.279827	+	0.161558i	0.633345	−	0.773869i	$-0.281681\pi$
$613$	6.92820	+	4.00000i	0.279827	+	0.161558i	−0.353518	+	0.935428i	$0.615015\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−21.2132			−0.854011			−0.427006	−	0.904249i	$-0.640432\pi$
$617$	−21.2132			−0.854011			−0.427006	+	0.904249i	$0.640432\pi$
$618$		0			0
$619$	−7.50000	−	4.33013i	−0.301450	−	0.174042i	0.341644	−	0.939829i	$-0.389016\pi$
$619$	−7.50000	−	4.33013i	−0.301450	−	0.174042i	−0.643094	+	0.765787i	$0.722350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.7279	+	36.7423i	0.509933	+	1.47205i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−24.4949			−0.976676
$630$		0			0
$631$	−22.0000			−0.875806			−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000			−0.875806			−0.437903	+	0.899022i	$0.644279\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	11.2583	+	4.50000i	0.446071	+	0.178296i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.34847	−	4.24264i	−0.290247	−	0.167574i	0.347806	−	0.937566i	$-0.386927\pi$
$641$	−7.34847	−	4.24264i	−0.290247	−	0.167574i	−0.638053	+	0.769992i	$0.720260\pi$
$642$		0			0
$643$	−32.9090			−1.29780			−0.648901	−	0.760872i	$-0.724771\pi$
$643$	−32.9090			−1.29780			−0.648901	+	0.760872i	$0.724771\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	19.0919	+	11.0227i	0.750579	+	0.433347i	0.825903	−	0.563812i	$-0.190666\pi$
$647$	19.0919	+	11.0227i	0.750579	+	0.433347i	−0.0753238	+	0.997159i	$0.523999\pi$
$648$		0			0
$649$	−18.0000	+	10.3923i	−0.706562	+	0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10.6066	−	18.3712i	−0.415068	−	0.718920i	0.580367	−	0.814355i	$-0.302909\pi$
$653$	−10.6066	−	18.3712i	−0.415068	−	0.718920i	−0.995436	+	0.0954353i	$0.969576\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			16.9706i			0.661079i	0.943792	+	0.330540i	$0.107231\pi$
$659$			16.9706i			0.661079i	−0.943792	+	0.330540i	$0.892769\pi$
$660$		0			0
$661$	4.50000	+	2.59808i	0.175030	+	0.101053i	0.584955	−	0.811065i	$-0.301112\pi$
$661$	4.50000	+	2.59808i	0.175030	+	0.101053i	−0.409926	+	0.912119i	$0.634445\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.6969			0.567369
$672$		0			0
$673$		−	19.0000i		−	0.732396i	−0.930537	−	0.366198i	$-0.880659\pi$
$673$		−	19.0000i		−	0.732396i	0.930537	−	0.366198i	$-0.119341\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	25.4558	−	14.6969i	0.978348	−	0.564849i	0.0765767	−	0.997064i	$-0.475601\pi$
$677$	25.4558	−	14.6969i	0.978348	−	0.564849i	0.901771	+	0.432214i	$0.142268\pi$
$678$		0			0
$679$	−9.00000	−	1.73205i	−0.345388	−	0.0664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	4.24264	−	7.34847i	0.162340	−	0.281181i	−0.773367	−	0.633958i	$-0.781429\pi$
$683$	4.24264	−	7.34847i	0.162340	−	0.281181i	0.935708	+	0.352777i	$0.114762\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.34847	−	12.7279i	0.279954	−	0.484895i
$690$		0			0
$691$	16.5000	−	9.52628i	0.627690	−	0.362397i	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	16.5000	−	9.52628i	0.627690	−	0.362397i	0.779857	+	0.625958i	$0.215292\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	51.9615	+	30.0000i	1.96818	+	1.13633i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	33.9411i		−	1.28194i	−0.767567	−	0.640969i	$-0.778533\pi$
$701$		−	33.9411i		−	1.28194i	0.767567	−	0.640969i	$-0.221467\pi$
$702$		0			0
$703$	12.9904	−	22.5000i	0.489942	−	0.848604i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.7279	−	14.6969i	0.478683	−	0.552735i
$708$		0			0
$709$	8.00000	+	13.8564i	0.300446	+	0.520388i	0.976237	−	0.216705i	$-0.0695310\pi$
$709$	8.00000	+	13.8564i	0.300446	+	0.520388i	−0.675791	+	0.737093i	$0.736198\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13.4722	+	23.3345i	0.502428	+	0.870231i	0.999996	+	0.00280593i	$0.000893157\pi$
$719$	13.4722	+	23.3345i	0.502428	+	0.870231i	−0.497568	+	0.867425i	$0.665774\pi$
$720$		0			0
$721$	−7.50000	−	21.6506i	−0.279315	−	0.806312i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−39.8372			−1.47748			−0.738739	−	0.673991i	$-0.764579\pi$
$727$	−39.8372			−1.47748			−0.738739	+	0.673991i	$0.764579\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−26.9444	+	46.6690i	−0.996574	+	1.72612i
$732$		0			0
$733$	−4.33013	−	7.50000i	−0.159937	−	0.277019i	0.774909	−	0.632073i	$-0.217796\pi$
$733$	−4.33013	−	7.50000i	−0.159937	−	0.277019i	−0.934846	+	0.355054i	$0.884462\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−14.8492	−	25.7196i	−0.546979	−	0.947395i
$738$		0			0
$739$	−9.50000	+	16.4545i	−0.349463	+	0.605288i	−0.986154	−	0.165831i	$-0.946969\pi$
$739$	−9.50000	+	16.4545i	−0.349463	+	0.605288i	0.636691	+	0.771119i	$0.280303\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.7279			0.466942			0.233471	−	0.972364i	$-0.424992\pi$
$743$	12.7279			0.466942			0.233471	+	0.972364i	$0.424992\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7.34847	+	21.2132i	0.268507	+	0.775114i
$750$		0			0
$751$	2.50000	+	4.33013i	0.0912263	+	0.158009i	0.908027	−	0.418911i	$-0.137588\pi$
$751$	2.50000	+	4.33013i	0.0912263	+	0.158009i	−0.816801	+	0.576919i	$0.804255\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	14.0000i		−	0.508839i	−0.967094	−	0.254419i	$-0.918116\pi$
$757$		−	14.0000i		−	0.508839i	0.967094	−	0.254419i	$-0.0818843\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17.1464	−	29.6985i	−0.621558	−	1.07657i	−0.989196	−	0.146600i	$-0.953167\pi$
$761$	−17.1464	−	29.6985i	−0.621558	−	1.07657i	0.367638	−	0.929969i	$-0.380166\pi$
$762$		0			0
$763$	−8.66025	+	10.0000i	−0.313522	+	0.362024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.24264	−	7.34847i	0.153193	−	0.265338i
$768$		0			0
$769$			15.5885i			0.562134i	0.959688	+	0.281067i	$0.0906883\pi$
$769$			15.5885i			0.562134i	−0.959688	+	0.281067i	$0.909312\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−36.0624	−	20.8207i	−1.29708	−	0.748867i	−0.317178	−	0.948366i	$-0.602735\pi$
$773$	−36.0624	−	20.8207i	−1.29708	−	0.748867i	−0.979898	+	0.199499i	$0.936069\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−55.1135	+	31.8198i	−1.97465	+	1.14006i
$780$		0			0
$781$	27.0000	−	46.7654i	0.966136	−	1.67340i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−8.66025	+	15.0000i	−0.308705	+	0.534692i	−0.978079	−	0.208233i	$-0.933229\pi$
$787$	−8.66025	+	15.0000i	−0.308705	+	0.534692i	0.669375	+	0.742925i	$0.266562\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−11.0227	−	2.12132i	−0.391922	−	0.0754255i
$792$		0			0
$793$	−5.19615	+	3.00000i	−0.184521	+	0.106533i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	12.0000			0.424529
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−57.2756	+	33.0681i	−2.02121	+	1.16695i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.3712	+	10.6066i	0.645896	+	0.372908i	0.786882	−	0.617103i	$-0.211694\pi$
$809$	18.3712	+	10.6066i	0.645896	+	0.372908i	−0.140986	+	0.990012i	$0.545027\pi$
$810$		0			0
$811$			31.1769i			1.09477i	0.836881	+	0.547385i	$0.184377\pi$
$811$			31.1769i			1.09477i	−0.836881	+	0.547385i	$0.815623\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−28.5788	−	49.5000i	−0.999847	−	1.73179i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.3712	−	10.6066i	0.641158	−	0.370173i	−0.143902	−	0.989592i	$-0.545965\pi$
$821$	18.3712	−	10.6066i	0.641158	−	0.370173i	0.785061	+	0.619419i	$0.212632\pi$
$822$		0			0
$823$	22.5167	+	13.0000i	0.784881	+	0.453152i	0.838157	−	0.545428i	$-0.183633\pi$
$823$	22.5167	+	13.0000i	0.784881	+	0.453152i	−0.0532760	+	0.998580i	$0.516966\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−29.6985			−1.03272			−0.516359	−	0.856372i	$-0.672713\pi$
$827$	−29.6985			−1.03272			−0.516359	+	0.856372i	$0.672713\pi$
$828$		0			0
$829$	−31.5000	−	18.1865i	−1.09404	−	0.631644i	−0.159391	−	0.987216i	$-0.550953\pi$
$829$	−31.5000	−	18.1865i	−1.09404	−	0.631644i	−0.934649	+	0.355571i	$0.884286\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	21.2132	+	26.9444i	0.734994	+	0.933568i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−53.8888			−1.86045			−0.930224	−	0.366993i	$-0.880387\pi$
$839$	−53.8888			−1.86045			−0.930224	+	0.366993i	$0.880387\pi$
$840$		0			0
$841$	−43.0000			−1.48276
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	6.06218	+	17.5000i	0.208299	+	0.601307i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−25.9808			−0.889564			−0.444782	−	0.895639i	$-0.646719\pi$
$853$	−25.9808			−0.889564			−0.444782	+	0.895639i	$0.646719\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−33.9411	−	19.5959i	−1.15941	−	0.669384i	−0.208245	−	0.978077i	$-0.566775\pi$
$857$	−33.9411	−	19.5959i	−1.15941	−	0.669384i	−0.951162	+	0.308693i	$0.900108\pi$
$858$		0			0
$859$	−39.0000	+	22.5167i	−1.33066	+	0.768259i	−0.985401	−	0.170248i	$-0.945543\pi$
$859$	−39.0000	+	22.5167i	−1.33066	+	0.768259i	−0.345262	+	0.938506i	$0.612210\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	19.0919	+	33.0681i	0.649895	+	1.12565i	0.983148	+	0.182814i	$0.0585206\pi$
$863$	19.0919	+	33.0681i	0.649895	+	1.12565i	−0.333252	+	0.942838i	$0.608146\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	46.6690i		−	1.58314i
$870$		0			0
$871$	10.5000	+	6.06218i	0.355779	+	0.205409i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−24.2487	+	14.0000i	−0.818821	+	0.472746i	−0.850010	−	0.526767i	$-0.823404\pi$
$877$	−24.2487	+	14.0000i	−0.818821	+	0.472746i	0.0311889	+	0.999514i	$0.490071\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	29.3939			0.990305			0.495152	−	0.868806i	$-0.335112\pi$
$881$	29.3939			0.990305			0.495152	+	0.868806i	$0.335112\pi$
$882$		0			0
$883$		−	1.00000i		−	0.0336527i	−0.999858	−	0.0168263i	$-0.994644\pi$
$883$		−	1.00000i		−	0.0336527i	0.999858	−	0.0168263i	$-0.00535624\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−48.7904	+	28.1691i	−1.63822	+	0.945827i	−0.656774	+	0.754087i	$0.728080\pi$
$887$	−48.7904	+	28.1691i	−1.63822	+	0.945827i	−0.981446	+	0.191740i	$0.938587\pi$
$888$		0			0
$889$	−2.50000	+	12.9904i	−0.0838473	+	0.435683i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.36396	+	11.0227i	−0.212962	+	0.368861i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−7.34847	+	12.7279i	−0.245085	+	0.424500i
$900$		0			0
$901$	36.0000	−	20.7846i	1.19933	−	0.692436i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−6.06218	−	3.50000i	−0.201291	−	0.116216i	0.395966	−	0.918265i	$-0.370410\pi$
$907$	−6.06218	−	3.50000i	−0.201291	−	0.116216i	−0.597258	+	0.802049i	$0.703743\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	8.48528i		−	0.281130i	−0.990071	−	0.140565i	$-0.955108\pi$
$911$		−	8.48528i		−	0.281130i	0.990071	−	0.140565i	$-0.0448919\pi$
$912$		0			0
$913$	−25.9808	+	45.0000i	−0.859838	+	1.48928i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.36396	−	18.3712i	−0.210157	−	0.606670i
$918$		0			0
$919$	−20.5000	−	35.5070i	−0.676233	−	1.17127i	−0.976107	−	0.217291i	$-0.930278\pi$
$919$	−20.5000	−	35.5070i	−0.676233	−	1.17127i	0.299874	−	0.953979i	$-0.403055\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			22.0454i			0.725633i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13.4722	−	23.3345i	−0.442008	−	0.765581i	0.555830	−	0.831296i	$-0.312401\pi$
$929$	−13.4722	−	23.3345i	−0.442008	−	0.765581i	−0.997838	+	0.0657150i	$0.979067\pi$
$930$		0			0
$931$	−36.0000	+	5.19615i	−1.17985	+	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−19.0526			−0.622420			−0.311210	−	0.950341i	$-0.600734\pi$
$937$	−19.0526			−0.622420			−0.311210	+	0.950341i	$0.600734\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−9.79796	+	16.9706i	−0.319404	+	0.553225i	−0.980364	−	0.197197i	$-0.936816\pi$
$941$	−9.79796	+	16.9706i	−0.319404	+	0.553225i	0.660960	+	0.750422i	$0.270149\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	27.5772	+	47.7650i	0.896137	+	1.55216i	0.832391	+	0.554189i	$0.186972\pi$
$947$	27.5772	+	47.7650i	0.896137	+	1.55216i	0.0637469	+	0.997966i	$0.479695\pi$
$948$		0			0
$949$	13.5000	−	23.3827i	0.438229	−	0.759034i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	33.9411			1.09946			0.549730	−	0.835342i	$-0.314730\pi$
$953$	33.9411			1.09946			0.549730	+	0.835342i	$0.314730\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−14.0000	−	24.2487i	−0.451613	−	0.782216i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	43.0000i		−	1.38279i	−0.722478	−	0.691393i	$-0.756997\pi$
$967$		−	43.0000i		−	1.38279i	0.722478	−	0.691393i	$-0.243003\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.89898	+	8.48528i	0.157216	+	0.272306i	0.933864	−	0.357629i	$-0.116415\pi$
$971$	4.89898	+	8.48528i	0.157216	+	0.272306i	−0.776648	+	0.629935i	$0.783082\pi$
$972$		0			0
$973$	−0.866025	+	4.50000i	−0.0277635	+	0.144263i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.0919	−	33.0681i	0.610803	−	1.05794i	−0.380302	−	0.924862i	$-0.624180\pi$
$977$	19.0919	−	33.0681i	0.610803	−	1.05794i	0.991105	−	0.133080i	$-0.0424868\pi$
$978$		0			0
$979$		−	62.3538i		−	1.99284i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−46.6690	−	26.9444i	−1.48851	−	0.859392i	−0.488597	−	0.872509i	$-0.662491\pi$
$983$	−46.6690	−	26.9444i	−1.48851	−	0.859392i	−0.999914	+	0.0131168i	$0.995825\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−6.50000	+	11.2583i	−0.206479	+	0.357633i	−0.950603	−	0.310409i	$-0.899534\pi$
$991$	−6.50000	+	11.2583i	−0.206479	+	0.357633i	0.744124	+	0.668042i	$0.232867\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	25.1147	−	43.5000i	0.795392	−	1.37766i	−0.127198	−	0.991877i	$-0.540599\pi$
$997$	25.1147	−	43.5000i	0.795392	−	1.37766i	0.922590	−	0.385782i	$-0.126068\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.dd.a.4049.3		8
3.2	odd	2	inner	6300.2.dd.a.4049.4		8
5.2	odd	4		6300.2.ch.a.4301.1		4
5.3	odd	4		252.2.t.a.17.1	✓	4
5.4	even	2	inner	6300.2.dd.a.4049.1		8
7.5	odd	6	inner	6300.2.dd.a.1349.2		8
15.2	even	4		6300.2.ch.a.4301.2		4
15.8	even	4		252.2.t.a.17.2	yes	4
15.14	odd	2	inner	6300.2.dd.a.4049.2		8
20.3	even	4		1008.2.bt.a.17.1		4
21.5	even	6	inner	6300.2.dd.a.1349.1		8
35.3	even	12		1764.2.f.a.881.2		4
35.12	even	12		6300.2.ch.a.1601.2		4
35.13	even	4		1764.2.t.a.521.2		4
35.18	odd	12		1764.2.f.a.881.4		4
35.19	odd	6	inner	6300.2.dd.a.1349.4		8
35.23	odd	12		1764.2.t.a.1097.1		4
35.33	even	12		252.2.t.a.89.2	yes	4
45.13	odd	12		2268.2.w.h.269.1		4
45.23	even	12		2268.2.w.h.269.2		4
45.38	even	12		2268.2.bm.g.1025.1		4
45.43	odd	12		2268.2.bm.g.1025.2		4
60.23	odd	4		1008.2.bt.a.17.2		4
105.23	even	12		1764.2.t.a.1097.2		4
105.38	odd	12		1764.2.f.a.881.3		4
105.47	odd	12		6300.2.ch.a.1601.1		4
105.53	even	12		1764.2.f.a.881.1		4
105.68	odd	12		252.2.t.a.89.1	yes	4
105.83	odd	4		1764.2.t.a.521.1		4
105.89	even	6	inner	6300.2.dd.a.1349.3		8
140.3	odd	12		7056.2.k.a.881.1		4
140.103	odd	12		1008.2.bt.a.593.2		4
140.123	even	12		7056.2.k.a.881.3		4
315.68	odd	12		2268.2.bm.g.593.2		4
315.103	even	12		2268.2.bm.g.593.1		4
315.173	odd	12		2268.2.w.h.1349.1		4
315.313	even	12		2268.2.w.h.1349.2		4
420.143	even	12		7056.2.k.a.881.4		4
420.263	odd	12		7056.2.k.a.881.2		4
420.383	even	12		1008.2.bt.a.593.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.t.a.17.1	✓	4	5.3	odd	4
252.2.t.a.17.2	yes	4	15.8	even	4
252.2.t.a.89.1	yes	4	105.68	odd	12
252.2.t.a.89.2	yes	4	35.33	even	12
1008.2.bt.a.17.1		4	20.3	even	4
1008.2.bt.a.17.2		4	60.23	odd	4
1008.2.bt.a.593.1		4	420.383	even	12
1008.2.bt.a.593.2		4	140.103	odd	12
1764.2.f.a.881.1		4	105.53	even	12
1764.2.f.a.881.2		4	35.3	even	12
1764.2.f.a.881.3		4	105.38	odd	12
1764.2.f.a.881.4		4	35.18	odd	12
1764.2.t.a.521.1		4	105.83	odd	4
1764.2.t.a.521.2		4	35.13	even	4
1764.2.t.a.1097.1		4	35.23	odd	12
1764.2.t.a.1097.2		4	105.23	even	12
2268.2.w.h.269.1		4	45.13	odd	12
2268.2.w.h.269.2		4	45.23	even	12
2268.2.w.h.1349.1		4	315.173	odd	12
2268.2.w.h.1349.2		4	315.313	even	12
2268.2.bm.g.593.1		4	315.103	even	12
2268.2.bm.g.593.2		4	315.68	odd	12
2268.2.bm.g.1025.1		4	45.38	even	12
2268.2.bm.g.1025.2		4	45.43	odd	12
6300.2.ch.a.1601.1		4	105.47	odd	12
6300.2.ch.a.1601.2		4	35.12	even	12
6300.2.ch.a.4301.1		4	5.2	odd	4
6300.2.ch.a.4301.2		4	15.2	even	4
6300.2.dd.a.1349.1		8	21.5	even	6	inner
6300.2.dd.a.1349.2		8	7.5	odd	6	inner
6300.2.dd.a.1349.3		8	105.89	even	6	inner
6300.2.dd.a.1349.4		8	35.19	odd	6	inner
6300.2.dd.a.4049.1		8	5.4	even	2	inner
6300.2.dd.a.4049.2		8	15.14	odd	2	inner
6300.2.dd.a.4049.3		8	1.1	even	1	trivial
6300.2.dd.a.4049.4		8	3.2	odd	2	inner
7056.2.k.a.881.1		4	140.3	odd	12
7056.2.k.a.881.2		4	420.263	odd	12
7056.2.k.a.881.3		4	140.123	even	12
7056.2.k.a.881.4		4	420.143	even	12

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:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.dd (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.59808 + 0.500000i) q^{7} +O(q^{10})\)	\(q+(2.59808 + 0.500000i) q^{7} +(-3.67423 - 2.12132i) q^{11} +1.73205 q^{13} +(4.24264 + 2.44949i) q^{17} +(-4.50000 + 2.59808i) q^{19} -8.48528i q^{29} +(-1.50000 - 0.866025i) q^{31} +(-4.33013 + 2.50000i) q^{37} +12.2474 q^{41} +11.0000i q^{43} +(2.12132 - 1.22474i) q^{47} +(6.50000 + 2.59808i) q^{49} +(4.24264 - 7.34847i) q^{53} +(2.44949 - 4.24264i) q^{59} +(-3.00000 + 1.73205i) q^{61} +(6.06218 + 3.50000i) q^{67} +12.7279i q^{71} +(7.79423 - 13.5000i) q^{73} +(-8.48528 - 7.34847i) q^{77} +(5.50000 + 9.52628i) q^{79} -12.2474i q^{83} +(7.34847 + 12.7279i) q^{89} +(4.50000 + 0.866025i) q^{91} -3.46410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 36 q^{19} - 12 q^{31} + 52 q^{49} - 24 q^{61} + 44 q^{79} + 36 q^{91}+O(q^{100}) \) 8 * q - 36 * q^19 - 12 * q^31 + 52 * q^49 - 24 * q^61 + 44 * q^79 + 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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