LMFDB - Embedded newform 2700.2.s.d.1549.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2700,2,Mod(1549,2700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2700.1549"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.1333317747165888577536.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 3x^{14} + 5x^{12} + 15x^{10} + 45x^{8} + 60x^{6} + 80x^{4} + 192x^{2} + 256 $ x^16 + 3x^14 + 5x^12 + 15x^10 + 45x^8 + 60x^6 + 80x^4 + 192*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{10} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1549.7
Root			$1.27069 + 0.620769i$ of defining polynomial
Character	$\chi$	$=$	2700.1549
Dual form			2700.2.s.d.2449.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.94604 - 1.70089i) q^{7} +(-2.20089 - 3.81206i) q^{11} +(-1.78679 - 1.03160i) q^{13} -1.40179i q^{17} +6.35717 q^{19} +(0.0933799 + 0.0539129i) q^{23} +(-4.54089 - 7.86505i) q^{29} +(-1.53160 + 2.65281i) q^{31} -1.95538i q^{37} +(-4.34788 + 7.53074i) q^{41} +(-6.17165 + 3.56320i) q^{43} +(-6.48096 + 3.74179i) q^{47} +(2.28608 - 3.95961i) q^{49} -13.0975i q^{53} +(6.58966 - 11.4136i) q^{59} +(0.862308 + 1.49356i) q^{61} +(-10.7177 - 6.18787i) q^{67} +7.50961 q^{71} -5.42037i q^{73} +(-12.9678 - 7.48698i) q^{77} +(4.71806 + 8.17193i) q^{79} +(-7.77167 + 4.48698i) q^{83} +4.01576 q^{89} -7.01858 q^{91} +(-1.88017 + 1.08551i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{11} + 16 q^{19} - 18 q^{29} - 4 q^{31} - 18 q^{41} + 18 q^{49} - 30 q^{59} + 2 q^{61} + 48 q^{71} - 14 q^{79} + 12 q^{89} - 44 q^{91}+O(q^{100}) $ 16 * q - 6 * q^11 + 16 * q^19 - 18 * q^29 - 4 * q^31 - 18 * q^41 + 18 * q^49 - 30 * q^59 + 2 * q^61 + 48 * q^71 - 14 * q^79 + 12 * q^89 - 44 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$-1$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.94604	−	1.70089i	1.11350	−	0.642878i	0.173764	−	0.984787i	$-0.444407\pi$
$7$	2.94604	−	1.70089i	1.11350	−	0.642878i	0.939733	+	0.341910i	$0.111074\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.20089	−	3.81206i	−0.663595	−	1.14938i	−0.979664	−	0.200644i	$-0.935697\pi$
$11$	−2.20089	−	3.81206i	−0.663595	−	1.14938i	0.316070	−	0.948736i	$-0.397637\pi$
$12$		0			0
$13$	−1.78679	−	1.03160i	−0.495565	−	0.286115i	0.231315	−	0.972879i	$-0.425697\pi$
$13$	−1.78679	−	1.03160i	−0.495565	−	0.286115i	−0.726880	+	0.686764i	$0.759031\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.40179i		−	0.339984i	−0.985445	−	0.169992i	$-0.945626\pi$
$17$		−	1.40179i		−	0.339984i	0.985445	−	0.169992i	$-0.0543741\pi$
$18$		0			0
$19$	6.35717			1.45843			0.729217	−	0.684283i	$-0.239885\pi$
$19$	6.35717			1.45843			0.729217	+	0.684283i	$0.239885\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.0933799	+	0.0539129i	0.0194711	+	0.0112416i	0.509704	−	0.860350i	$-0.329755\pi$
$23$	0.0933799	+	0.0539129i	0.0194711	+	0.0112416i	−0.490233	+	0.871591i	$0.663088\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.54089	−	7.86505i	−0.843222	−	1.46050i	−0.887156	−	0.461469i	$-0.847322\pi$
$29$	−4.54089	−	7.86505i	−0.843222	−	1.46050i	0.0439339	−	0.999034i	$-0.486011\pi$
$30$		0			0
$31$	−1.53160	+	2.65281i	−0.275084	+	0.476459i	−0.970156	−	0.242481i	$-0.922039\pi$
$31$	−1.53160	+	2.65281i	−0.275084	+	0.476459i	0.695073	+	0.718940i	$0.255372\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	1.95538i		−	0.321462i	−0.986998	−	0.160731i	$-0.948615\pi$
$37$		−	1.95538i		−	0.321462i	0.986998	−	0.160731i	$-0.0513852\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.34788	+	7.53074i	−0.679024	+	1.17610i	0.296251	+	0.955110i	$0.404264\pi$
$41$	−4.34788	+	7.53074i	−0.679024	+	1.17610i	−0.975275	+	0.220994i	$0.929070\pi$
$42$		0			0
$43$	−6.17165	+	3.56320i	−0.941167	+	0.543383i	−0.890326	−	0.455323i	$-0.849524\pi$
$43$	−6.17165	+	3.56320i	−0.941167	+	0.543383i	−0.0508414	+	0.998707i	$0.516190\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.48096	+	3.74179i	−0.945345	+	0.545795i	−0.891632	−	0.452761i	$-0.850439\pi$
$47$	−6.48096	+	3.74179i	−0.945345	+	0.545795i	−0.0537135	+	0.998556i	$0.517106\pi$
$48$		0			0
$49$	2.28608	−	3.95961i	0.326583	−	0.565659i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	13.0975i		−	1.79909i	−0.436833	−	0.899543i	$-0.643900\pi$
$53$		−	13.0975i		−	1.79909i	0.436833	−	0.899543i	$-0.356100\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.58966	−	11.4136i	0.857901	−	1.48593i	−0.0160267	−	0.999872i	$-0.505102\pi$
$59$	6.58966	−	11.4136i	0.857901	−	1.48593i	0.873928	−	0.486056i	$-0.161565\pi$
$60$		0			0
$61$	0.862308	+	1.49356i	0.110407	+	0.191231i	0.915935	−	0.401328i	$-0.131451\pi$
$61$	0.862308	+	1.49356i	0.110407	+	0.191231i	−0.805527	+	0.592559i	$0.798118\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−10.7177	−	6.18787i	−1.30938	−	0.755969i	−0.327385	−	0.944891i	$-0.606167\pi$
$67$	−10.7177	−	6.18787i	−1.30938	−	0.755969i	−0.981992	+	0.188922i	$0.939501\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.50961			0.891227			0.445614	−	0.895225i	$-0.352986\pi$
$71$	7.50961			0.891227			0.445614	+	0.895225i	$0.352986\pi$
$72$		0			0
$73$		−	5.42037i		−	0.634406i	−0.948358	−	0.317203i	$-0.897256\pi$
$73$		−	5.42037i		−	0.634406i	0.948358	−	0.317203i	$-0.102744\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.9678	−	7.48698i	−1.47782	−	0.853220i
$78$		0			0
$79$	4.71806	+	8.17193i	0.530824	+	0.919413i	0.999353	+	0.0359656i	$0.0114507\pi$
$79$	4.71806	+	8.17193i	0.530824	+	0.919413i	−0.468529	+	0.883448i	$0.655216\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.77167	+	4.48698i	−0.853052	+	0.492510i	−0.861679	−	0.507453i	$-0.830587\pi$
$83$	−7.77167	+	4.48698i	−0.853052	+	0.492510i	0.00862744	+	0.999963i	$0.497254\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.01576			0.425670			0.212835	−	0.977088i	$-0.431730\pi$
$89$	4.01576			0.425670			0.212835	+	0.977088i	$0.431730\pi$
$90$		0			0
$91$	−7.01858			−0.735747
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.88017	+	1.08551i	−0.190902	+	0.110217i	−0.592405	−	0.805641i	$-0.701821\pi$
$97$	−1.88017	+	1.08551i	−0.190902	+	0.110217i	0.401503	+	0.915858i	$0.368488\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.48698	+	2.57552i	0.147960	+	0.256274i	0.930473	−	0.366360i	$-0.119396\pi$
$101$	1.48698	+	2.57552i	0.147960	+	0.256274i	−0.782513	+	0.622634i	$0.786063\pi$
$102$		0			0
$103$	−5.74640	−	3.31768i	−0.566209	−	0.326901i	0.189425	−	0.981895i	$-0.439338\pi$
$103$	−5.74640	−	3.31768i	−0.566209	−	0.326901i	−0.755634	+	0.654994i	$0.772671\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			13.7878i			1.33292i	0.745541	+	0.666459i	$0.232191\pi$
$107$			13.7878i			1.33292i	−0.745541	+	0.666459i	$0.767809\pi$
$108$		0			0
$109$	18.1347			1.73699			0.868495	−	0.495699i	$-0.165088\pi$
$109$	18.1347			1.73699			0.868495	+	0.495699i	$0.165088\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.0708250	−	0.0408909i	−0.00666266	−	0.00384669i	0.496665	−	0.867942i	$-0.334558\pi$
$113$	−0.0708250	−	0.0408909i	−0.00666266	−	0.00384669i	−0.503328	+	0.864096i	$0.667891\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.38429	−	4.12972i	−0.218568	−	0.378571i
$120$		0			0
$121$	−4.18787	+	7.25361i	−0.380716	+	0.659419i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	21.5811i		−	1.91501i	−0.288410	−	0.957507i	$-0.593127\pi$
$127$		−	21.5811i		−	1.91501i	0.288410	−	0.957507i	$-0.406873\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.39391	+	9.34253i	−0.471268	+	0.816260i	−0.999460	−	0.0328649i	$-0.989537\pi$
$131$	−5.39391	+	9.34253i	−0.471268	+	0.816260i	0.528192	+	0.849125i	$0.322870\pi$
$132$		0			0
$133$	18.7284	−	10.8129i	1.62396	−	0.937594i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.6997	−	6.75481i	0.999570	−	0.577102i	0.0914491	−	0.995810i	$-0.470850\pi$
$137$	11.6997	−	6.75481i	0.999570	−	0.577102i	0.908121	+	0.418708i	$0.137517\pi$
$138$		0			0
$139$	7.30499	−	12.6526i	0.619601	−	1.07318i	−0.369958	−	0.929049i	$-0.620628\pi$
$139$	7.30499	−	12.6526i	0.619601	−	1.07318i	0.989559	−	0.144132i	$-0.0460389\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			9.08178i			0.759457i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.0669350	+	0.115935i	−0.00548353	+	0.00949775i	−0.868754	−	0.495244i	$-0.835079\pi$
$149$	−0.0669350	+	0.115935i	−0.00548353	+	0.00949775i	0.863271	+	0.504741i	$0.168412\pi$
$150$		0			0
$151$	−3.07249	−	5.32171i	−0.250036	−	0.433075i	0.713500	−	0.700656i	$-0.247109\pi$
$151$	−3.07249	−	5.32171i	−0.250036	−	0.433075i	−0.963535	+	0.267581i	$0.913776\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	12.7953	+	7.38736i	1.02117	+	0.589575i	0.914444	−	0.404713i	$-0.132628\pi$
$157$	12.7953	+	7.38736i	1.02117	+	0.589575i	0.106730	+	0.994288i	$0.465962\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.366801			0.0289079
$162$		0			0
$163$		−	15.9451i		−	1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$163$		−	15.9451i		−	1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−17.7529	−	10.2497i	−1.37376	−	0.793143i	−0.382364	−	0.924012i	$-0.624890\pi$
$167$	−17.7529	−	10.2497i	−1.37376	−	0.793143i	−0.991400	+	0.130869i	$0.958223\pi$
$168$		0			0
$169$	−4.37160	−	7.57183i	−0.336277	−	0.582448i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−19.5647	+	11.2957i	−1.48748	+	0.858796i	−0.999898	−	0.0142821i	$-0.995454\pi$
$173$	−19.5647	+	11.2957i	−1.48748	+	0.858796i	−0.487580	+	0.873078i	$0.662120\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	20.5879			1.53881			0.769407	−	0.638759i	$-0.220552\pi$
$179$	20.5879			1.53881			0.769407	+	0.638759i	$0.220552\pi$
$180$		0			0
$181$	−8.32830			−0.619038			−0.309519	−	0.950893i	$-0.600168\pi$
$181$	−8.32830			−0.619038			−0.309519	+	0.950893i	$0.600168\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−5.34370	+	3.08519i	−0.390770	+	0.225611i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.39217	−	7.60747i	−0.317807	−	0.550457i	0.662224	−	0.749306i	$-0.269613\pi$
$191$	−4.39217	−	7.60747i	−0.317807	−	0.550457i	−0.980030	+	0.198849i	$0.936280\pi$
$192$		0			0
$193$	−6.26202	−	3.61538i	−0.450750	−	0.260241i	0.257397	−	0.966306i	$-0.417135\pi$
$193$	−6.26202	−	3.61538i	−0.450750	−	0.260241i	−0.708147	+	0.706065i	$0.750469\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	4.17932i		−	0.297764i	−0.988855	−	0.148882i	$-0.952432\pi$
$197$		−	4.17932i		−	0.297764i	0.988855	−	0.148882i	$-0.0475675\pi$
$198$		0			0
$199$	−15.6518			−1.10953			−0.554763	−	0.832009i	$-0.687191\pi$
$199$	−15.6518			−1.10953			−0.554763	+	0.832009i	$0.687191\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−26.7553	−	15.4472i	−1.87785	−	1.08418i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−13.9914	−	24.2339i	−0.967809	−	1.67629i
$210$		0			0
$211$	12.0770	−	20.9179i	0.831412	−	1.44005i	−0.0655057	−	0.997852i	$-0.520866\pi$
$211$	12.0770	−	20.9179i	0.831412	−	1.44005i	0.896918	−	0.442196i	$-0.145801\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			10.4204i			0.707381i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.44609	+	2.50470i	−0.0972743	+	0.168484i
$222$		0			0
$223$	1.25263	−	0.723206i	0.0838823	−	0.0484295i	−0.457472	−	0.889224i	$-0.651245\pi$
$223$	1.25263	−	0.723206i	0.0838823	−	0.0484295i	0.541354	+	0.840794i	$0.317912\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	17.7755	−	10.2627i	1.17980	−	0.681158i	0.223832	−	0.974628i	$-0.428143\pi$
$227$	17.7755	−	10.2627i	1.17980	−	0.681158i	0.955969	+	0.293469i	$0.0948098\pi$
$228$		0			0
$229$	10.4743	−	18.1420i	0.692160	−	1.19886i	−0.278969	−	0.960300i	$-0.589993\pi$
$229$	10.4743	−	18.1420i	0.692160	−	1.19886i	0.971129	−	0.238556i	$-0.0766741\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			9.90112i			0.648644i	0.945947	+	0.324322i	$0.105136\pi$
$233$			9.90112i			0.648644i	−0.945947	+	0.324322i	$0.894864\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.05391	+	5.28953i	−0.197541	+	0.342151i	−0.947731	−	0.319072i	$-0.896629\pi$
$239$	−3.05391	+	5.28953i	−0.197541	+	0.342151i	0.750189	+	0.661223i	$0.229962\pi$
$240$		0			0
$241$	0.148392	+	0.257022i	0.00955875	+	0.0165562i	0.870765	−	0.491699i	$-0.163624\pi$
$241$	0.148392	+	0.257022i	0.00955875	+	0.0165562i	−0.861206	+	0.508255i	$0.830291\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−11.3589	−	6.55806i	−0.722749	−	0.417279i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.51643			0.285075			0.142537	−	0.989789i	$-0.454474\pi$
$251$	4.51643			0.285075			0.142537	+	0.989789i	$0.454474\pi$
$252$		0			0
$253$		−	0.474626i		−	0.0298395i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.7764	−	6.79911i	−0.734591	−	0.424117i	0.0855081	−	0.996337i	$-0.472749\pi$
$257$	−11.7764	−	6.79911i	−0.734591	−	0.424117i	−0.820100	+	0.572221i	$0.806082\pi$
$258$		0			0
$259$	−3.32589	−	5.76061i	−0.206661	−	0.357947i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.16921	−	3.56179i	0.380409	−	0.219630i	−0.297587	−	0.954695i	$-0.596182\pi$
$263$	6.16921	−	3.56179i	0.380409	−	0.219630i	0.677996	+	0.735065i	$0.262848\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.80358			0.536764			0.268382	−	0.963313i	$-0.413511\pi$
$269$	8.80358			0.536764			0.268382	+	0.963313i	$0.413511\pi$
$270$		0			0
$271$	−9.95885			−0.604957			−0.302478	−	0.953156i	$-0.597814\pi$
$271$	−9.95885			−0.604957			−0.302478	+	0.953156i	$0.597814\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	7.73303	−	4.46467i	0.464633	−	0.268256i	−0.249358	−	0.968411i	$-0.580219\pi$
$277$	7.73303	−	4.46467i	0.464633	−	0.268256i	0.713990	+	0.700156i	$0.246886\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.0402	+	27.7825i	0.956879	+	1.65736i	0.730008	+	0.683439i	$0.239516\pi$
$281$	16.0402	+	27.7825i	0.956879	+	1.65736i	0.226872	+	0.973925i	$0.427150\pi$
$282$		0			0
$283$	5.51437	+	3.18373i	0.327796	+	0.189253i	0.654862	−	0.755749i	$-0.272727\pi$
$283$	5.51437	+	3.18373i	0.327796	+	0.189253i	−0.327066	+	0.945001i	$0.606060\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			29.5811i			1.74612i
$288$		0			0
$289$	15.0350			0.884411
$290$		0			0
$291$		0			0
$292$		0			0
$293$	24.1631	+	13.9506i	1.41162	+	0.815000i	0.995541	−	0.0943273i	$-0.0300700\pi$
$293$	24.1631	+	13.9506i	1.41162	+	0.815000i	0.416081	+	0.909328i	$0.363403\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.111233	−	0.192662i	−0.00643278	−	0.0111419i
$300$		0			0
$301$	−12.1213	+	20.9946i	−0.698658	+	1.21011i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			12.2054i			0.696597i	0.937384	+	0.348299i	$0.113240\pi$
$307$			12.2054i			0.696597i	−0.937384	+	0.348299i	$0.886760\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.45123	+	4.24565i	−0.138996	+	0.240749i	−0.927117	−	0.374772i	$-0.877721\pi$
$311$	−2.45123	+	4.24565i	−0.138996	+	0.240749i	0.788121	+	0.615521i	$0.211054\pi$
$312$		0			0
$313$	−15.2322	+	8.79429i	−0.860972	+	0.497083i	−0.864338	−	0.502912i	$-0.832262\pi$
$313$	−15.2322	+	8.79429i	−0.860972	+	0.497083i	0.00336551	+	0.999994i	$0.498929\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	16.9500	−	9.78608i	0.952007	−	0.549641i	0.0583028	−	0.998299i	$-0.481431\pi$
$317$	16.9500	−	9.78608i	0.952007	−	0.549641i	0.893704	+	0.448658i	$0.148098\pi$
$318$		0			0
$319$	−19.9880	+	34.6203i	−1.11912	+	1.93836i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	8.91140i		−	0.495844i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.7288	+	22.0469i	−0.701759	+	1.21548i
$330$		0			0
$331$	−7.29537	−	12.6360i	−0.400990	−	0.694535i	0.592856	−	0.805309i	$-0.298000\pi$
$331$	−7.29537	−	12.6360i	−0.400990	−	0.694535i	−0.993846	+	0.110774i	$0.964667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−8.58108	−	4.95429i	−0.467441	−	0.269877i	0.247727	−	0.968830i	$-0.420317\pi$
$337$	−8.58108	−	4.95429i	−0.467441	−	0.269877i	−0.715168	+	0.698953i	$0.753650\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	13.4836			0.730176
$342$		0			0
$343$			8.25897i			0.445943i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.93739	−	5.16000i	−0.479784	−	0.277004i	0.240542	−	0.970639i	$-0.422675\pi$
$347$	−8.93739	−	5.16000i	−0.479784	−	0.277004i	−0.720327	+	0.693635i	$0.756008\pi$
$348$		0			0
$349$	8.56767	+	14.8396i	0.458617	+	0.794348i	0.998888	−	0.0471429i	$-0.0150116\pi$
$349$	8.56767	+	14.8396i	0.458617	+	0.794348i	−0.540271	+	0.841491i	$0.681678\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.4845	−	6.63055i	0.611256	−	0.352909i	−0.162201	−	0.986758i	$-0.551859\pi$
$353$	11.4845	−	6.63055i	0.611256	−	0.352909i	0.773457	+	0.633849i	$0.218526\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	14.2817			0.753758			0.376879	−	0.926263i	$-0.376997\pi$
$359$	14.2817			0.753758			0.376879	+	0.926263i	$0.376997\pi$
$360$		0			0
$361$	21.4136			1.12703
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−8.29749	+	4.79056i	−0.433125	+	0.250065i	−0.700677	−	0.713478i	$-0.747119\pi$
$367$	−8.29749	+	4.79056i	−0.433125	+	0.250065i	0.267552	+	0.963543i	$0.413785\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−22.2775	−	38.5858i	−1.15659	−	2.00328i
$372$		0			0
$373$	20.8763	+	12.0529i	1.08093	+	0.624076i	0.931148	−	0.364642i	$-0.118809\pi$
$373$	20.8763	+	12.0529i	1.08093	+	0.624076i	0.149784	+	0.988719i	$0.452142\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			18.7376i			0.965033i
$378$		0			0
$379$	−2.73973			−0.140730			−0.0703651	−	0.997521i	$-0.522416\pi$
$379$	−2.73973			−0.140730			−0.0703651	+	0.997521i	$0.522416\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	18.4924	+	10.6766i	0.944917	+	0.545548i	0.891498	−	0.453024i	$-0.149655\pi$
$383$	18.4924	+	10.6766i	0.944917	+	0.545548i	0.0534187	+	0.998572i	$0.482988\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.33485	+	12.7043i	0.371892	+	0.644136i	0.989857	−	0.142070i	$-0.0453759\pi$
$389$	7.33485	+	12.7043i	0.371892	+	0.644136i	−0.617965	+	0.786206i	$0.712043\pi$
$390$		0			0
$391$	0.0755745	−	0.130899i	0.00382197	−	0.00661984i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	6.43065i		−	0.322745i	−0.986894	−	0.161373i	$-0.948408\pi$
$397$		−	6.43065i		−	0.322745i	0.986894	−	0.161373i	$-0.0515921\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.03748	−	15.6534i	0.451310	−	0.781693i	−0.547157	−	0.837030i	$-0.684290\pi$
$401$	9.03748	−	15.6534i	0.451310	−	0.781693i	0.998468	+	0.0553373i	$0.0176234\pi$
$402$		0			0
$403$	5.47329	−	3.16000i	0.272644	−	0.157411i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7.45402	+	4.30358i	−0.369482	+	0.213320i
$408$		0			0
$409$	−9.39532	+	16.2732i	−0.464569	+	0.804657i	−0.999182	−	0.0404403i	$-0.987124\pi$
$409$	−9.39532	+	16.2732i	−0.464569	+	0.804657i	0.534613	+	0.845097i	$0.320457\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	44.8333i		−	2.20610i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−19.8297	+	34.3461i	−0.968745	+	1.67792i	−0.269547	+	0.962987i	$0.586874\pi$
$419$	−19.8297	+	34.3461i	−0.968745	+	1.67792i	−0.699198	+	0.714928i	$0.746460\pi$
$420$		0			0
$421$	18.7682	+	32.5076i	0.914708	+	1.58432i	0.807328	+	0.590103i	$0.200913\pi$
$421$	18.7682	+	32.5076i	0.914708	+	1.58432i	0.107380	+	0.994218i	$0.465754\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	5.08078	+	2.93339i	0.245876	+	0.141957i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.7357			−0.806132			−0.403066	−	0.915171i	$-0.632055\pi$
$431$	−16.7357			−0.806132			−0.403066	+	0.915171i	$0.632055\pi$
$432$		0			0
$433$			21.0008i			1.00924i	0.863343	+	0.504618i	$0.168367\pi$
$433$			21.0008i			1.00924i	−0.863343	+	0.504618i	$0.831633\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.593631	+	0.342733i	0.0283972	+	0.0163952i
$438$		0			0
$439$	−3.01576	−	5.22345i	−0.143934	−	0.249302i	0.785041	−	0.619444i	$-0.212642\pi$
$439$	−3.01576	−	5.22345i	−0.143934	−	0.249302i	−0.928975	+	0.370143i	$0.879309\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.47803	+	3.16274i	−0.260269	+	0.150266i	−0.624457	−	0.781059i	$-0.714680\pi$
$443$	−5.47803	+	3.16274i	−0.260269	+	0.150266i	0.364188	+	0.931325i	$0.381346\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.6594			−0.880593			−0.440296	−	0.897853i	$-0.645127\pi$
$449$	−18.6594			−0.880593			−0.440296	+	0.897853i	$0.645127\pi$
$450$		0			0
$451$	38.2769			1.80239
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	23.8889	−	13.7923i	1.11748	−	0.645176i	0.176722	−	0.984261i	$-0.443451\pi$
$457$	23.8889	−	13.7923i	1.11748	−	0.645176i	0.940756	+	0.339085i	$0.110117\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.76442	−	11.7163i	−0.315051	−	0.545684i	0.664398	−	0.747379i	$-0.268688\pi$
$461$	−6.76442	−	11.7163i	−0.315051	−	0.545684i	−0.979448	+	0.201696i	$0.935355\pi$
$462$		0			0
$463$	−23.8020	−	13.7421i	−1.10617	−	0.638650i	−0.168338	−	0.985729i	$-0.553840\pi$
$463$	−23.8020	−	13.7421i	−1.10617	−	0.638650i	−0.937836	+	0.347079i	$0.887173\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			32.7515i			1.51556i	0.652511	+	0.757779i	$0.273716\pi$
$467$			32.7515i			1.51556i	−0.652511	+	0.757779i	$0.726284\pi$
$468$		0			0
$469$	−42.0997			−1.94398
$470$		0			0
$471$		0			0
$472$		0			0
$473$	27.1663	+	15.6845i	1.24911	+	0.721172i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.92783	−	5.07116i	−0.133776	−	0.231707i	0.791353	−	0.611359i	$-0.209377\pi$
$479$	−2.92783	−	5.07116i	−0.133776	−	0.231707i	−0.925129	+	0.379652i	$0.876044\pi$
$480$		0			0
$481$	−2.01717	+	3.49384i	−0.0919750	+	0.159305i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			16.4864i			0.747070i	0.927616	+	0.373535i	$0.121854\pi$
$487$			16.4864i			0.747070i	−0.927616	+	0.373535i	$0.878146\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.85302	+	11.8698i	−0.309272	+	0.535676i	−0.978203	−	0.207649i	$-0.933419\pi$
$491$	−6.85302	+	11.8698i	−0.309272	+	0.535676i	0.668931	+	0.743324i	$0.266752\pi$
$492$		0			0
$493$	−11.0251	+	6.36537i	−0.496548	+	0.286682i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	22.1236	−	12.7731i	0.992379	−	0.572950i
$498$		0			0
$499$	2.23482	−	3.87082i	0.100044	−	0.173282i	−0.811658	−	0.584132i	$-0.801435\pi$
$499$	2.23482	−	3.87082i	0.100044	−	0.173282i	0.911703	+	0.410851i	$0.134768\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			18.1010i			0.807084i	0.914961	+	0.403542i	$0.132221\pi$
$503$			18.1010i			0.807084i	−0.914961	+	0.403542i	$0.867779\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.6044	−	18.3674i	0.470033	−	0.814120i	−0.529380	−	0.848385i	$-0.677575\pi$
$509$	10.6044	−	18.3674i	0.470033	−	0.814120i	0.999413	+	0.0342644i	$0.0109088\pi$
$510$		0			0
$511$	−9.21947	−	15.9686i	−0.407846	−	0.706409i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	28.5278	+	16.4705i	1.25465	+	0.724374i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	28.4507			1.24645			0.623224	−	0.782043i	$-0.285822\pi$
$521$	28.4507			1.24645			0.623224	+	0.782043i	$0.285822\pi$
$522$		0			0
$523$			26.5111i			1.15925i	0.814884	+	0.579625i	$0.196801\pi$
$523$			26.5111i			1.15925i	−0.814884	+	0.579625i	$0.803199\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.71868	+	2.14698i	0.161988	+	0.0935240i
$528$		0			0
$529$	−11.4942	−	19.9085i	−0.499747	−	0.865588i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	15.5374	−	8.97055i	0.673001	−	0.388558i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−20.1257			−0.866876
$540$		0			0
$541$	−14.3132			−0.615372			−0.307686	−	0.951488i	$-0.599555\pi$
$541$	−14.3132			−0.615372			−0.307686	+	0.951488i	$0.599555\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−11.8222	+	6.82556i	−0.505482	+	0.291840i	−0.730974	−	0.682405i	$-0.760934\pi$
$547$	−11.8222	+	6.82556i	−0.505482	+	0.291840i	0.225493	+	0.974245i	$0.427601\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−28.8672	−	49.9994i	−1.22978	−	2.13005i
$552$		0			0
$553$	27.7992	+	16.0499i	1.18214	+	0.682509i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	23.4665i		−	0.994306i	−0.867663	−	0.497153i	$-0.834379\pi$
$557$		−	23.4665i		−	0.994306i	0.867663	−	0.497153i	$-0.165621\pi$
$558$		0			0
$559$	14.7032			0.621880
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.504418	−	0.291226i	−0.0212587	−	0.0122737i	0.489333	−	0.872097i	$-0.337240\pi$
$563$	−0.504418	−	0.291226i	−0.0212587	−	0.0122737i	−0.510592	+	0.859823i	$0.670574\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.77994	+	4.81500i	0.116541	+	0.201855i	0.918395	−	0.395665i	$-0.129486\pi$
$569$	2.77994	+	4.81500i	0.116541	+	0.201855i	−0.801854	+	0.597521i	$0.796153\pi$
$570$		0			0
$571$	11.3706	−	19.6945i	0.475845	−	0.824188i	−0.523772	−	0.851858i	$-0.675476\pi$
$571$	11.3706	−	19.6945i	0.475845	−	0.824188i	0.999617	+	0.0276708i	$0.00880903\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			30.5522i			1.27191i	0.771728	+	0.635953i	$0.219393\pi$
$577$			30.5522i			1.27191i	−0.771728	+	0.635953i	$0.780607\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−15.2638	+	26.4376i	−0.633247	+	1.09682i
$582$		0			0
$583$	−49.9286	+	28.8263i	−2.06783	+	1.19386i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	17.6353	−	10.1817i	0.727886	−	0.420245i	−0.0897626	−	0.995963i	$-0.528611\pi$
$587$	17.6353	−	10.1817i	0.727886	−	0.420245i	0.817648	+	0.575718i	$0.195278\pi$
$588$		0			0
$589$	−9.73664	+	16.8644i	−0.401191	+	0.694884i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	15.5199i		−	0.637326i	−0.947868	−	0.318663i	$-0.896766\pi$
$593$		−	15.5199i		−	0.637326i	0.947868	−	0.318663i	$-0.103234\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−14.6288	+	25.3379i	−0.597717	+	1.03528i	0.395440	+	0.918492i	$0.370592\pi$
$599$	−14.6288	+	25.3379i	−0.597717	+	1.03528i	−0.993157	+	0.116785i	$0.962741\pi$
$600$		0			0
$601$	9.15254	+	15.8527i	0.373340	+	0.646644i	0.990077	−	0.140526i	$-0.0448792\pi$
$601$	9.15254	+	15.8527i	0.373340	+	0.646644i	−0.616737	+	0.787169i	$0.711546\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	14.5845	+	8.42037i	0.591967	+	0.341772i	0.765875	−	0.642990i	$-0.222306\pi$
$607$	14.5845	+	8.42037i	0.591967	+	0.341772i	−0.173908	+	0.984762i	$0.555640\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.4401			0.624640
$612$		0			0
$613$			13.5954i			0.549113i	0.961571	+	0.274556i	$0.0885310\pi$
$613$			13.5954i			0.549113i	−0.961571	+	0.274556i	$0.911469\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	25.3861	+	14.6567i	1.02201	+	0.590056i	0.914685	−	0.404168i	$-0.132439\pi$
$617$	25.3861	+	14.6567i	1.02201	+	0.590056i	0.107322	+	0.994224i	$0.465772\pi$
$618$		0			0
$619$	6.08211	+	10.5345i	0.244461	+	0.423418i	0.961980	−	0.273121i	$-0.0880558\pi$
$619$	6.08211	+	10.5345i	0.244461	+	0.423418i	−0.717519	+	0.696539i	$0.754722\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	11.8306	−	6.83038i	0.473982	−	0.273653i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.74103			−0.109292
$630$		0			0
$631$	5.45471			0.217148			0.108574	−	0.994088i	$-0.465371\pi$
$631$	5.45471			0.217148			0.108574	+	0.994088i	$0.465371\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−8.16948	+	4.71665i	−0.323687	+	0.186881i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.839996	−	1.45492i	−0.0331779	−	0.0574657i	0.848960	−	0.528458i	$-0.177229\pi$
$641$	−0.839996	−	1.45492i	−0.0331779	−	0.0574657i	−0.882138	+	0.470992i	$0.843896\pi$
$642$		0			0
$643$	4.92202	+	2.84173i	0.194106	+	0.112067i	0.593903	−	0.804537i	$-0.297586\pi$
$643$	4.92202	+	2.84173i	0.194106	+	0.112067i	−0.399797	+	0.916603i	$0.630920\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	34.3064i		−	1.34872i	−0.738401	−	0.674361i	$-0.764419\pi$
$647$		−	34.3064i		−	1.34872i	0.738401	−	0.674361i	$-0.235581\pi$
$648$		0			0
$649$	−58.0126			−2.27719
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.9377	+	6.31486i	0.428024	+	0.247120i	0.698504	−	0.715606i	$-0.253849\pi$
$653$	10.9377	+	6.31486i	0.428024	+	0.247120i	−0.270480	+	0.962725i	$0.587183\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.16697	+	5.48535i	0.123368	+	0.213679i	0.921094	−	0.389341i	$-0.127297\pi$
$659$	3.16697	+	5.48535i	0.123368	+	0.213679i	−0.797726	+	0.603020i	$0.793964\pi$
$660$		0			0
$661$	18.8448	−	32.6401i	0.732978	−	1.26955i	−0.222628	−	0.974904i	$-0.571463\pi$
$661$	18.8448	−	32.6401i	0.732978	−	1.26955i	0.955605	−	0.294651i	$-0.0952033\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	0.979251i		−	0.0379167i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3.79570	−	6.57434i	0.146531	−	0.253800i
$672$		0			0
$673$	26.5477	−	15.3273i	1.02334	−	0.590824i	0.108268	−	0.994122i	$-0.465469\pi$
$673$	26.5477	−	15.3273i	1.02334	−	0.590824i	0.915069	+	0.403298i	$0.132136\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	25.0035	−	14.4358i	0.960964	−	0.554813i	0.0644945	−	0.997918i	$-0.479456\pi$
$677$	25.0035	−	14.4358i	0.960964	−	0.554813i	0.896470	+	0.443105i	$0.146123\pi$
$678$		0			0
$679$	−3.69269	+	6.39593i	−0.141712	+	0.245453i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			47.0352i			1.79975i	0.436147	+	0.899875i	$0.356343\pi$
$683$			47.0352i			1.79975i	−0.436147	+	0.899875i	$0.643657\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−13.5114	+	23.4025i	−0.514745	+	0.891564i
$690$		0			0
$691$	−0.425186	−	0.736443i	−0.0161748	−	0.0280156i	0.857825	−	0.513942i	$-0.171815\pi$
$691$	−0.425186	−	0.736443i	−0.0161748	−	0.0280156i	−0.874000	+	0.485927i	$0.838482\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	10.5565	+	6.09480i	0.399856	+	0.230857i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	35.6821			1.34770			0.673848	−	0.738870i	$-0.264640\pi$
$701$	35.6821			1.34770			0.673848	+	0.738870i	$0.264640\pi$
$702$		0			0
$703$		−	12.4307i		−	0.468831i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8.76138	+	5.05838i	0.329506	+	0.190240i
$708$		0			0
$709$	7.69543	+	13.3289i	0.289008	+	0.500576i	0.973573	−	0.228375i	$-0.0733413\pi$
$709$	7.69543	+	13.3289i	0.289008	+	0.500576i	−0.684565	+	0.728951i	$0.740008\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.286042	+	0.165146i	−0.0107123	+	0.00618477i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	25.5846			0.954144			0.477072	−	0.878864i	$-0.341698\pi$
$719$	25.5846			0.954144			0.477072	+	0.878864i	$0.341698\pi$
$720$		0			0
$721$	−22.5721			−0.840630
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−5.31509	+	3.06867i	−0.197126	+	0.113811i	−0.595314	−	0.803493i	$-0.702972\pi$
$727$	−5.31509	+	3.06867i	−0.197126	+	0.113811i	0.398188	+	0.917304i	$0.369639\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.99486	+	8.65135i	0.184741	+	0.319982i
$732$		0			0
$733$	−4.44306	−	2.56520i	−0.164108	−	0.0947478i	0.415697	−	0.909503i	$-0.363538\pi$
$733$	−4.44306	−	2.56520i	−0.164108	−	0.0947478i	−0.579805	+	0.814756i	$0.696871\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			54.4754i			2.00663i
$738$		0			0
$739$	−18.8784			−0.694454			−0.347227	−	0.937781i	$-0.612877\pi$
$739$	−18.8784			−0.694454			−0.347227	+	0.937781i	$0.612877\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	19.0152	+	10.9784i	0.697600	+	0.402759i	0.806453	−	0.591298i	$-0.201384\pi$
$743$	19.0152	+	10.9784i	0.697600	+	0.402759i	−0.108853	+	0.994058i	$0.534718\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	23.4516	+	40.6194i	0.856904	+	1.48420i
$750$		0			0
$751$	1.58037	−	2.73728i	0.0576686	−	0.0998849i	−0.835750	−	0.549110i	$-0.814967\pi$
$751$	1.58037	−	2.73728i	0.0576686	−	0.0998849i	0.893418	+	0.449225i	$0.148300\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	39.1753i		−	1.42385i	−0.702255	−	0.711926i	$-0.747823\pi$
$757$		−	39.1753i		−	1.42385i	0.702255	−	0.711926i	$-0.252177\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.03916	−	8.72807i	0.182669	−	0.316392i	−0.760119	−	0.649783i	$-0.774860\pi$
$761$	5.03916	−	8.72807i	0.182669	−	0.316392i	0.942789	+	0.333391i	$0.108193\pi$
$762$		0			0
$763$	53.4255	−	30.8452i	1.93413	−	1.11667i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−23.5486	+	13.5958i	−0.850292	+	0.490916i
$768$		0			0
$769$	10.5790	−	18.3233i	0.381487	−	0.660755i	−0.609788	−	0.792565i	$-0.708745\pi$
$769$	10.5790	−	18.3233i	0.381487	−	0.660755i	0.991275	+	0.131810i	$0.0420787\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	43.2543i		−	1.55575i	−0.628420	−	0.777874i	$-0.716298\pi$
$773$		−	43.2543i		−	1.55575i	0.628420	−	0.777874i	$-0.283702\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.6402	+	47.8742i	−0.990312	+	1.71527i
$780$		0			0
$781$	−16.5279	−	28.6271i	−0.591414	−	1.02436i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1.02189	−	0.589986i	−0.0364263	−	0.0210307i	0.481676	−	0.876349i	$-0.340028\pi$
$787$	−1.02189	−	0.589986i	−0.0364263	−	0.0210307i	−0.518103	+	0.855318i	$0.673361\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.278204			−0.00989180
$792$		0			0
$793$		−	3.55823i		−	0.126357i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.989706	+	0.571407i	0.0350572	+	0.0202403i	0.517426	−	0.855728i	$-0.326890\pi$
$797$	0.989706	+	0.571407i	0.0350572	+	0.0202403i	−0.482369	+	0.875968i	$0.660224\pi$
$798$		0			0
$799$	5.24519	+	9.08494i	0.185562	+	0.321402i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−20.6628	+	11.9297i	−0.729173	+	0.420988i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−11.3723			−0.399828			−0.199914	−	0.979813i	$-0.564066\pi$
$809$	−11.3723			−0.399828			−0.199914	+	0.979813i	$0.564066\pi$
$810$		0			0
$811$	2.08590			0.0732459			0.0366230	−	0.999329i	$-0.488340\pi$
$811$	2.08590			0.0732459			0.0366230	+	0.999329i	$0.488340\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−39.2342	+	22.6519i	−1.37263	+	0.792489i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.89564	+	13.6757i	0.275560	+	0.477284i	0.970276	−	0.242000i	$-0.0778034\pi$
$821$	7.89564	+	13.6757i	0.275560	+	0.477284i	−0.694716	+	0.719284i	$0.744470\pi$
$822$		0			0
$823$	−6.34003	−	3.66042i	−0.221000	−	0.127594i	0.385413	−	0.922744i	$-0.374059\pi$
$823$	−6.34003	−	3.66042i	−0.221000	−	0.127594i	−0.606413	+	0.795150i	$0.707392\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	14.8401i		−	0.516040i	−0.966140	−	0.258020i	$-0.916930\pi$
$827$		−	14.8401i		−	0.516040i	0.966140	−	0.258020i	$-0.0830701\pi$
$828$		0			0
$829$	23.3346			0.810444			0.405222	−	0.914218i	$-0.367194\pi$
$829$	23.3346			0.810444			0.405222	+	0.914218i	$0.367194\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.55054	−	3.20461i	−0.192315	−	0.111033i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	7.33826	+	12.7102i	0.253345	+	0.438806i	0.964445	−	0.264285i	$-0.0851359\pi$
$839$	7.33826	+	12.7102i	0.253345	+	0.438806i	−0.711100	+	0.703091i	$0.751803\pi$
$840$		0			0
$841$	−26.7394	+	46.3140i	−0.922048	+	1.59703i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$			28.4925i			0.979014i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.105420	−	0.182593i	0.00361375	−	0.00625920i
$852$		0			0
$853$	21.2369	−	12.2611i	0.727136	−	0.419812i	−0.0902375	−	0.995920i	$-0.528763\pi$
$853$	21.2369	−	12.2611i	0.727136	−	0.419812i	0.817374	+	0.576108i	$0.195429\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.83929	−	2.79396i	0.165307	−	0.0954400i	−0.415064	−	0.909792i	$-0.636241\pi$
$857$	4.83929	−	2.79396i	0.165307	−	0.0954400i	0.580371	+	0.814352i	$0.302908\pi$
$858$		0			0
$859$	−20.0309	+	34.6946i	−0.683447	+	1.18376i	0.290476	+	0.956882i	$0.406186\pi$
$859$	−20.0309	+	34.6946i	−0.683447	+	1.18376i	−0.973922	+	0.226882i	$0.927147\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			27.2157i			0.926432i	0.886246	+	0.463216i	$0.153305\pi$
$863$			27.2157i			0.926432i	−0.886246	+	0.463216i	$0.846695\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	20.7679	−	35.9711i	0.704503	−	1.22024i
$870$		0			0
$871$	12.7668	+	22.1128i	0.432588	+	0.749264i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−20.6806	−	11.9399i	−0.698334	−	0.403183i	0.108393	−	0.994108i	$-0.465430\pi$
$877$	−20.6806	−	11.9399i	−0.698334	−	0.403183i	−0.806727	+	0.590925i	$0.798763\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−28.3585			−0.955421			−0.477710	−	0.878517i	$-0.658533\pi$
$881$	−28.3585			−0.955421			−0.477710	+	0.878517i	$0.658533\pi$
$882$		0			0
$883$			28.3449i			0.953881i	0.878936	+	0.476941i	$0.158254\pi$
$883$			28.3449i			0.953881i	−0.878936	+	0.476941i	$0.841746\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.446080	−	0.257544i	−0.0149779	−	0.00864749i	0.492492	−	0.870317i	$-0.336086\pi$
$887$	−0.446080	−	0.257544i	−0.0149779	−	0.00864749i	−0.507470	+	0.861669i	$0.669419\pi$
$888$		0			0
$889$	−36.7072	−	63.5787i	−1.23112	−	2.13236i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−41.2005	+	23.7871i	−1.37872	+	0.796007i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	27.8193			0.927827
$900$		0			0
$901$	−18.3600			−0.611660
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.87426	+	1.08211i	−0.0622339	+	0.0359308i	−0.530794	−	0.847501i	$-0.678106\pi$
$907$	−1.87426	+	1.08211i	−0.0622339	+	0.0359308i	0.468560	+	0.883432i	$0.344773\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	19.2332	+	33.3129i	0.637226	+	1.10371i	0.986039	+	0.166515i	$0.0532513\pi$
$911$	19.2332	+	33.3129i	0.637226	+	1.10371i	−0.348813	+	0.937192i	$0.613415\pi$
$912$		0			0
$913$	34.2093	+	19.7507i	1.13216	+	0.653654i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			36.6979i			1.21187i
$918$		0			0
$919$	−18.6992			−0.616830			−0.308415	−	0.951252i	$-0.599799\pi$
$919$	−18.6992			−0.616830			−0.308415	+	0.951252i	$0.599799\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−13.4181	−	7.74693i	−0.441661	−	0.254993i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.10335	+	1.91106i	0.0361999	+	0.0627000i	0.883558	−	0.468322i	$-0.155141\pi$
$929$	1.10335	+	1.91106i	0.0361999	+	0.0627000i	−0.847358	+	0.531022i	$0.821808\pi$
$930$		0			0
$931$	14.5330	−	25.1719i	0.476300	−	0.824976i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			21.3429i			0.697242i	0.937264	+	0.348621i	$0.113350\pi$
$937$			21.3429i			0.697242i	−0.937264	+	0.348621i	$0.886650\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.589661	+	1.02132i	−0.0192224	+	0.0332942i	−0.875477	−	0.483261i	$-0.839452\pi$
$941$	−0.589661	+	1.02132i	−0.0192224	+	0.0332942i	0.856254	+	0.516555i	$0.172786\pi$
$942$		0			0
$943$	−0.812008	+	0.468813i	−0.0264426	+	0.0152667i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20.5152	+	11.8445i	−0.666655	+	0.384894i	−0.794808	−	0.606861i	$-0.792428\pi$
$947$	−20.5152	+	11.8445i	−0.666655	+	0.384894i	0.128153	+	0.991754i	$0.459095\pi$
$948$		0			0
$949$	−5.59166	+	9.68504i	−0.181513	+	0.314390i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			12.8125i			0.415038i	0.978231	+	0.207519i	$0.0665389\pi$
$953$			12.8125i			0.415038i	−0.978231	+	0.207519i	$0.933461\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	22.9784	−	39.7998i	0.742012	−	1.28520i
$960$		0			0
$961$	10.8084	+	18.7207i	0.348658	+	0.603893i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−9.46264	−	5.46326i	−0.304298	−	0.175686i	0.340074	−	0.940399i	$-0.389548\pi$
$967$	−9.46264	−	5.46326i	−0.304298	−	0.175686i	−0.644372	+	0.764712i	$0.722881\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−15.7933			−0.506831			−0.253415	−	0.967358i	$-0.581554\pi$
$971$	−15.7933			−0.506831			−0.253415	+	0.967358i	$0.581554\pi$
$972$		0			0
$973$		−	49.7001i		−	1.59331i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−32.4742	−	18.7490i	−1.03894	−	0.599833i	−0.119409	−	0.992845i	$-0.538100\pi$
$977$	−32.4742	−	18.7490i	−1.03894	−	0.599833i	−0.919533	+	0.393012i	$0.871433\pi$
$978$		0			0
$979$	−8.83826	−	15.3083i	−0.282472	−	0.489256i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20.3795	−	11.7661i	0.650004	−	0.375280i	−0.138454	−	0.990369i	$-0.544213\pi$
$983$	20.3795	−	11.7661i	0.650004	−	0.375280i	0.788458	+	0.615089i	$0.210880\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.768410			−0.0244340
$990$		0			0
$991$	46.7019			1.48353			0.741767	−	0.670658i	$-0.233988\pi$
$991$	46.7019			1.48353			0.741767	+	0.670658i	$0.233988\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	37.1161	−	21.4290i	1.17548	−	0.678663i	0.220514	−	0.975384i	$-0.429226\pi$
$997$	37.1161	−	21.4290i	1.17548	−	0.678663i	0.954964	+	0.296721i	$0.0958931\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.s.d.1549.7		16
3.2	odd	2		900.2.s.d.49.5		16
5.2	odd	4		2700.2.i.d.901.4		8
5.3	odd	4		2700.2.i.e.901.1		8
5.4	even	2	inner	2700.2.s.d.1549.2		16
9.2	odd	6		900.2.s.d.349.4		16
9.4	even	3		8100.2.d.s.649.7		8
9.5	odd	6		8100.2.d.q.649.7		8
9.7	even	3	inner	2700.2.s.d.2449.2		16
15.2	even	4		900.2.i.e.301.4	yes	8
15.8	even	4		900.2.i.d.301.1	✓	8
15.14	odd	2		900.2.s.d.49.4		16
45.2	even	12		900.2.i.e.601.4	yes	8
45.4	even	6		8100.2.d.s.649.2		8
45.7	odd	12		2700.2.i.d.1801.4		8
45.13	odd	12		8100.2.a.y.1.4		4
45.14	odd	6		8100.2.d.q.649.2		8
45.22	odd	12		8100.2.a.ba.1.1		4
45.23	even	12		8100.2.a.x.1.4		4
45.29	odd	6		900.2.s.d.349.5		16
45.32	even	12		8100.2.a.z.1.1		4
45.34	even	6	inner	2700.2.s.d.2449.7		16
45.38	even	12		900.2.i.d.601.1	yes	8
45.43	odd	12		2700.2.i.e.1801.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.i.d.301.1	✓	8	15.8	even	4
900.2.i.d.601.1	yes	8	45.38	even	12
900.2.i.e.301.4	yes	8	15.2	even	4
900.2.i.e.601.4	yes	8	45.2	even	12
900.2.s.d.49.4		16	15.14	odd	2
900.2.s.d.49.5		16	3.2	odd	2
900.2.s.d.349.4		16	9.2	odd	6
900.2.s.d.349.5		16	45.29	odd	6
2700.2.i.d.901.4		8	5.2	odd	4
2700.2.i.d.1801.4		8	45.7	odd	12
2700.2.i.e.901.1		8	5.3	odd	4
2700.2.i.e.1801.1		8	45.43	odd	12
2700.2.s.d.1549.2		16	5.4	even	2	inner
2700.2.s.d.1549.7		16	1.1	even	1	trivial
2700.2.s.d.2449.2		16	9.7	even	3	inner
2700.2.s.d.2449.7		16	45.34	even	6	inner
8100.2.a.x.1.4		4	45.23	even	12
8100.2.a.y.1.4		4	45.13	odd	12
8100.2.a.z.1.1		4	45.32	even	12
8100.2.a.ba.1.1		4	45.22	odd	12
8100.2.d.q.649.2		8	45.14	odd	6
8100.2.d.q.649.7		8	9.5	odd	6
8100.2.d.s.649.2		8	45.4	even	6
8100.2.d.s.649.7		8	9.4	even	3

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.94604 - 1.70089i) q^{7} +(-2.20089 - 3.81206i) q^{11} +(-1.78679 - 1.03160i) q^{13} -1.40179i q^{17} +6.35717 q^{19} +(0.0933799 + 0.0539129i) q^{23} +(-4.54089 - 7.86505i) q^{29} +(-1.53160 + 2.65281i) q^{31} -1.95538i q^{37} +(-4.34788 + 7.53074i) q^{41} +(-6.17165 + 3.56320i) q^{43} +(-6.48096 + 3.74179i) q^{47} +(2.28608 - 3.95961i) q^{49} -13.0975i q^{53} +(6.58966 - 11.4136i) q^{59} +(0.862308 + 1.49356i) q^{61} +(-10.7177 - 6.18787i) q^{67} +7.50961 q^{71} -5.42037i q^{73} +(-12.9678 - 7.48698i) q^{77} +(4.71806 + 8.17193i) q^{79} +(-7.77167 + 4.48698i) q^{83} +4.01576 q^{89} -7.01858 q^{91} +(-1.88017 + 1.08551i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{11} + 16 q^{19} - 18 q^{29} - 4 q^{31} - 18 q^{41} + 18 q^{49} - 30 q^{59} + 2 q^{61} + 48 q^{71} - 14 q^{79} + 12 q^{89} - 44 q^{91}+O(q^{100}) \) 16 * q - 6 * q^11 + 16 * q^19 - 18 * q^29 - 4 * q^31 - 18 * q^41 + 18 * q^49 - 30 * q^59 + 2 * q^61 + 48 * q^71 - 14 * q^79 + 12 * q^89 - 44 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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