LMFDB - Embedded newform 2700.2.bf.f.557.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2700,2,Mod(557,2700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2700.557"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 10, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.bf (of order $12$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,0,0,12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$8$ over $\Q(\zeta_{12})$
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			557.5
Character	$\chi$	$=$	2700.557
Dual form			2700.2.bf.f.1493.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0937177 - 0.349759i) q^{7} +(-3.66769 + 2.11754i) q^{11} +(0.219136 + 0.817828i) q^{13} +(-0.443477 + 0.443477i) q^{17} -2.85410i q^{19} +(-0.980206 + 0.262645i) q^{23} +(0.511841 + 0.886534i) q^{29} +(4.24803 - 7.35780i) q^{31} +(4.97633 + 4.97633i) q^{37} +(-8.74911 - 5.05130i) q^{41} +(-10.2869 - 2.75637i) q^{43} +(-7.58648 - 2.03279i) q^{47} +(5.94863 + 3.43444i) q^{49} +(-7.52500 - 7.52500i) q^{53} +(6.10629 - 10.5764i) q^{59} +(-5.68247 - 9.84233i) q^{61} +(2.84062 - 0.761142i) q^{67} -2.71873i q^{71} +(-2.35741 + 2.35741i) q^{73} +(0.396902 + 1.48126i) q^{77} +(6.28431 - 3.62825i) q^{79} +(-3.86932 + 14.4405i) q^{83} -3.50111 q^{89} +0.306580 q^{91} +(-1.57063 + 5.86166i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 12 q^{11} + 8 q^{31} - 60 q^{41} + 52 q^{61} + 120 q^{91}+O(q^{100}) $ 32 * q + 12 * q^11 + 8 * q^31 - 60 * q^41 + 52 * q^61 + 120 * q^91

Character values

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

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.0937177	−	0.349759i	0.0354219	−	0.132196i	−0.945951	−	0.324311i	$-0.894868\pi$
$7$	0.0937177	−	0.349759i	0.0354219	−	0.132196i	0.981373	+	0.192114i	$0.0615344\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.66769	+	2.11754i	−1.10585	+	0.638462i	−0.937751	−	0.347308i	$-0.887096\pi$
$11$	−3.66769	+	2.11754i	−1.10585	+	0.638462i	−0.168098	+	0.985770i	$0.553763\pi$
$12$		0			0
$13$	0.219136	+	0.817828i	0.0607775	+	0.226825i	0.989633	−	0.143616i	$-0.0458732\pi$
$13$	0.219136	+	0.817828i	0.0607775	+	0.226825i	−0.928856	+	0.370441i	$0.879207\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.443477	+	0.443477i	−0.107559	+	0.107559i	−0.758838	−	0.651279i	$-0.774233\pi$
$17$	−0.443477	+	0.443477i	−0.107559	+	0.107559i	0.651279	+	0.758838i	$0.274233\pi$
$18$		0			0
$19$		−	2.85410i		−	0.654776i	−0.944890	−	0.327388i	$-0.893832\pi$
$19$		−	2.85410i		−	0.654776i	0.944890	−	0.327388i	$-0.106168\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.980206	+	0.262645i	−0.204387	+	0.0547653i	−0.359560	−	0.933122i	$-0.617073\pi$
$23$	−0.980206	+	0.262645i	−0.204387	+	0.0547653i	0.155173	+	0.987887i	$0.450407\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.511841	+	0.886534i	0.0950465	+	0.164625i	0.909628	−	0.415424i	$-0.136367\pi$
$29$	0.511841	+	0.886534i	0.0950465	+	0.164625i	−0.814582	+	0.580049i	$0.803033\pi$
$30$		0			0
$31$	4.24803	−	7.35780i	0.762968	−	1.32150i	−0.178346	−	0.983968i	$-0.557075\pi$
$31$	4.24803	−	7.35780i	0.762968	−	1.32150i	0.941314	−	0.337532i	$-0.109592\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.97633	+	4.97633i	0.818104	+	0.818104i	0.985833	−	0.167729i	$-0.0536434\pi$
$37$	4.97633	+	4.97633i	0.818104	+	0.818104i	−0.167729	+	0.985833i	$0.553643\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.74911	−	5.05130i	−1.36638	−	0.788880i	−0.375917	−	0.926653i	$-0.622672\pi$
$41$	−8.74911	−	5.05130i	−1.36638	−	0.788880i	−0.990464	+	0.137773i	$0.956006\pi$
$42$		0			0
$43$	−10.2869	−	2.75637i	−1.56874	−	0.420342i	−0.633322	−	0.773889i	$-0.718309\pi$
$43$	−10.2869	−	2.75637i	−1.56874	−	0.420342i	−0.935417	+	0.353547i	$0.884976\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.58648	−	2.03279i	−1.10660	−	0.296513i	−0.341152	−	0.940008i	$-0.610817\pi$
$47$	−7.58648	−	2.03279i	−1.10660	−	0.296513i	−0.765450	+	0.643495i	$0.777484\pi$
$48$		0			0
$49$	5.94863	+	3.43444i	0.849804	+	0.490635i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.52500	−	7.52500i	−1.03364	−	1.03364i	−0.999414	−	0.0342238i	$-0.989104\pi$
$53$	−7.52500	−	7.52500i	−1.03364	−	1.03364i	−0.0342238	−	0.999414i	$-0.510896\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.10629	−	10.5764i	0.794971	−	1.37693i	−0.127887	−	0.991789i	$-0.540820\pi$
$59$	6.10629	−	10.5764i	0.794971	−	1.37693i	0.922858	−	0.385141i	$-0.125847\pi$
$60$		0			0
$61$	−5.68247	−	9.84233i	−0.727566	−	1.26018i	−0.957909	−	0.287072i	$-0.907318\pi$
$61$	−5.68247	−	9.84233i	−0.727566	−	1.26018i	0.230343	−	0.973109i	$-0.426015\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.84062	−	0.761142i	0.347037	−	0.0929883i	−0.0810902	−	0.996707i	$-0.525840\pi$
$67$	2.84062	−	0.761142i	0.347037	−	0.0929883i	0.428127	+	0.903718i	$0.359174\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.71873i		−	0.322654i	−0.986901	−	0.161327i	$-0.948423\pi$
$71$		−	2.71873i		−	0.322654i	0.986901	−	0.161327i	$-0.0515773\pi$
$72$		0			0
$73$	−2.35741	+	2.35741i	−0.275914	+	0.275914i	−0.831475	−	0.555561i	$-0.812503\pi$
$73$	−2.35741	+	2.35741i	−0.275914	+	0.275914i	0.555561	+	0.831475i	$0.312503\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.396902	+	1.48126i	0.0452312	+	0.168805i
$78$		0			0
$79$	6.28431	−	3.62825i	0.707040	−	0.408210i	−0.102924	−	0.994689i	$-0.532820\pi$
$79$	6.28431	−	3.62825i	0.707040	−	0.408210i	0.809964	+	0.586479i	$0.199486\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.86932	+	14.4405i	−0.424713	+	1.58505i	0.339834	+	0.940485i	$0.389629\pi$
$83$	−3.86932	+	14.4405i	−0.424713	+	1.58505i	−0.764548	+	0.644567i	$0.777038\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.50111			−0.371116			−0.185558	−	0.982633i	$-0.559409\pi$
$89$	−3.50111			−0.371116			−0.185558	+	0.982633i	$0.559409\pi$
$90$		0			0
$91$	0.306580			0.0321383
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.57063	+	5.86166i	−0.159473	+	0.595161i	0.839208	+	0.543811i	$0.183019\pi$
$97$	−1.57063	+	5.86166i	−0.159473	+	0.595161i	−0.998681	+	0.0513503i	$0.983647\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.97102	+	4.60207i	−0.793146	+	0.457923i	−0.841069	−	0.540928i	$-0.818073\pi$
$101$	−7.97102	+	4.60207i	−0.793146	+	0.457923i	0.0479231	+	0.998851i	$0.484740\pi$
$102$		0			0
$103$	−3.94411	−	14.7196i	−0.388625	−	1.45037i	−0.832373	−	0.554216i	$-0.813018\pi$
$103$	−3.94411	−	14.7196i	−0.388625	−	1.45037i	0.443748	−	0.896152i	$-0.353649\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.0927119	−	0.0927119i	0.00896280	−	0.00896280i	−0.702611	−	0.711574i	$-0.747983\pi$
$107$	0.0927119	−	0.0927119i	0.00896280	−	0.00896280i	0.711574	+	0.702611i	$0.247983\pi$
$108$		0			0
$109$			4.38761i			0.420257i	0.977674	+	0.210129i	$0.0673883\pi$
$109$			4.38761i			0.420257i	−0.977674	+	0.210129i	$0.932612\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.93899	+	2.12725i	−0.746837	+	0.200114i	−0.612115	−	0.790769i	$-0.709681\pi$
$113$	−7.93899	+	2.12725i	−0.746837	+	0.200114i	−0.134722	+	0.990883i	$0.543014\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.113548	+	0.196672i	0.0104090	+	0.0180289i
$120$		0			0
$121$	3.46795	−	6.00667i	0.315269	−	0.546061i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−9.86606	−	9.86606i	−0.875471	−	0.875471i	0.117591	−	0.993062i	$-0.462483\pi$
$127$	−9.86606	−	9.86606i	−0.875471	−	0.875471i	−0.993062	+	0.117591i	$0.962483\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−13.4118	−	7.74329i	−1.17179	−	0.676534i	−0.217690	−	0.976018i	$-0.569852\pi$
$131$	−13.4118	−	7.74329i	−1.17179	−	0.676534i	−0.954101	+	0.299483i	$0.903186\pi$
$132$		0			0
$133$	−0.998248	−	0.267480i	−0.0865591	−	0.0231934i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.5453	+	3.89739i	1.24269	+	0.332977i	0.819507	−	0.573070i	$-0.194248\pi$
$137$	14.5453	+	3.89739i	1.24269	+	0.332977i	0.423179	+	0.906046i	$0.360914\pi$
$138$		0			0
$139$	−2.19342	−	1.26637i	−0.186043	−	0.107412i	0.404086	−	0.914721i	$-0.367590\pi$
$139$	−2.19342	−	1.26637i	−0.186043	−	0.107412i	−0.590129	+	0.807309i	$0.700923\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.53551	−	2.53551i	−0.212030	−	0.212030i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.25871	−	10.8404i	0.512733	−	0.888080i	−0.487158	−	0.873314i	$-0.661966\pi$
$149$	6.25871	−	10.8404i	0.512733	−	0.888080i	0.999891	−	0.0147659i	$-0.00470032\pi$
$150$		0			0
$151$	−3.73432	−	6.46804i	−0.303895	−	0.526361i	0.673120	−	0.739533i	$-0.264954\pi$
$151$	−3.73432	−	6.46804i	−0.303895	−	0.526361i	−0.977015	+	0.213172i	$0.931621\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−13.7214	+	3.67663i	−1.09509	+	0.293427i	−0.760762	−	0.649031i	$-0.775174\pi$
$157$	−13.7214	+	3.67663i	−1.09509	+	0.293427i	−0.334323	+	0.942458i	$0.608508\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.367450i			0.0289591i
$162$		0			0
$163$	−0.845478	+	0.845478i	−0.0662229	+	0.0662229i	−0.739443	−	0.673220i	$-0.764911\pi$
$163$	−0.845478	+	0.845478i	−0.0662229	+	0.0662229i	0.673220	+	0.739443i	$0.264911\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.86380	+	21.8840i	0.453754	+	1.69343i	0.691725	+	0.722161i	$0.256851\pi$
$167$	5.86380	+	21.8840i	0.453754	+	1.69343i	−0.237971	+	0.971272i	$0.576482\pi$
$168$		0			0
$169$	10.6375	−	6.14157i	0.818270	−	0.472428i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.94946	+	18.4716i	−0.376300	+	1.40437i	0.475136	+	0.879912i	$0.342399\pi$
$173$	−4.94946	+	18.4716i	−0.376300	+	1.40437i	−0.851436	+	0.524459i	$0.824268\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.34494			−0.623730			−0.311865	−	0.950126i	$-0.600954\pi$
$179$	−8.34494			−0.623730			−0.311865	+	0.950126i	$0.600954\pi$
$180$		0			0
$181$	−10.9499			−0.813901			−0.406951	−	0.913450i	$-0.633408\pi$
$181$	−10.9499			−0.813901			−0.406951	+	0.913450i	$0.633408\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.687454	−	2.56561i	0.0502716	−	0.187616i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.7169	−	7.91945i	0.992519	−	0.573031i	0.0864928	−	0.996252i	$-0.472434\pi$
$191$	13.7169	−	7.91945i	0.992519	−	0.573031i	0.906026	+	0.423221i	$0.139101\pi$
$192$		0			0
$193$	−4.15986	−	15.5248i	−0.299433	−	1.11750i	−0.937632	−	0.347629i	$-0.886987\pi$
$193$	−4.15986	−	15.5248i	−0.299433	−	1.11750i	0.638199	−	0.769871i	$-0.279680\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.5320	−	17.5320i	1.24910	−	1.24910i	0.292986	−	0.956117i	$-0.405351\pi$
$197$	17.5320	−	17.5320i	1.24910	−	1.24910i	0.956117	−	0.292986i	$-0.0946488\pi$
$198$		0			0
$199$		−	4.00216i		−	0.283705i	−0.989888	−	0.141853i	$-0.954694\pi$
$199$		−	4.00216i		−	0.283705i	0.989888	−	0.141853i	$-0.0453059\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.358042	−	0.0959371i	0.0251296	−	0.00673346i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.04368	+	10.4680i	0.418050	+	0.724083i
$210$		0			0
$211$	−2.30619	+	3.99444i	−0.158765	+	0.274989i	−0.934424	−	0.356164i	$-0.884085\pi$
$211$	−2.30619	+	3.99444i	−0.158765	+	0.274989i	0.775659	+	0.631153i	$0.217418\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.17534	−	2.17534i	−0.147672	−	0.147672i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.459870	−	0.265506i	−0.0309342	−	0.0178599i
$222$		0			0
$223$	−10.9565	−	2.93579i	−0.733703	−	0.196595i	−0.127425	−	0.991848i	$-0.540671\pi$
$223$	−10.9565	−	2.93579i	−0.733703	−	0.196595i	−0.606278	+	0.795253i	$0.707338\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.8460	−	5.04978i	−1.25085	−	0.335165i	−0.428188	−	0.903690i	$-0.640848\pi$
$227$	−18.8460	−	5.04978i	−1.25085	−	0.335165i	−0.822667	+	0.568524i	$0.807515\pi$
$228$		0			0
$229$	9.00807	+	5.20081i	0.595270	+	0.343679i	0.767179	−	0.641434i	$-0.221660\pi$
$229$	9.00807	+	5.20081i	0.595270	+	0.343679i	−0.171908	+	0.985113i	$0.554993\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−18.6324	−	18.6324i	−1.22065	−	1.22065i	−0.967402	−	0.253245i	$-0.918502\pi$
$233$	−18.6324	−	18.6324i	−1.22065	−	1.22065i	−0.253245	−	0.967402i	$-0.581498\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	11.2711	−	19.5222i	0.729069	−	1.26278i	−0.228208	−	0.973612i	$-0.573287\pi$
$239$	11.2711	−	19.5222i	0.729069	−	1.26278i	0.957277	−	0.289172i	$-0.0933800\pi$
$240$		0			0
$241$	−8.47496	−	14.6791i	−0.545920	−	0.945561i	−0.998548	−	0.0538621i	$-0.982847\pi$
$241$	−8.47496	−	14.6791i	−0.545920	−	0.945561i	0.452628	−	0.891699i	$-0.350486\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.33416	−	0.625438i	0.148519	−	0.0397956i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	20.0265i		−	1.26406i	−0.774943	−	0.632031i	$-0.782222\pi$
$251$		−	20.0265i		−	1.26406i	0.774943	−	0.632031i	$-0.217778\pi$
$252$		0			0
$253$	3.03893	−	3.03893i	0.191056	−	0.191056i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.69234	+	10.0479i	0.167943	+	0.626773i	0.997646	+	0.0685683i	$0.0218431\pi$
$257$	2.69234	+	10.0479i	0.167943	+	0.626773i	−0.829703	+	0.558205i	$0.811490\pi$
$258$		0			0
$259$	2.20689	−	1.27415i	0.137129	−	0.0791717i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.36216	+	16.2798i	−0.268982	+	1.00386i	0.690785	+	0.723060i	$0.257265\pi$
$263$	−4.36216	+	16.2798i	−0.268982	+	1.00386i	−0.959767	+	0.280796i	$0.909402\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.2671			−0.625995			−0.312998	−	0.949754i	$-0.601333\pi$
$269$	−10.2671			−0.625995			−0.312998	+	0.949754i	$0.601333\pi$
$270$		0			0
$271$	−7.86673			−0.477870			−0.238935	−	0.971036i	$-0.576798\pi$
$271$	−7.86673			−0.477870			−0.238935	+	0.971036i	$0.576798\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.34209	+	12.4729i	−0.200807	+	0.749421i	0.789880	+	0.613261i	$0.210143\pi$
$277$	−3.34209	+	12.4729i	−0.200807	+	0.749421i	−0.990687	+	0.136160i	$0.956524\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.778091	−	0.449231i	0.0464170	−	0.0267989i	−0.476612	−	0.879114i	$-0.658135\pi$
$281$	0.778091	−	0.449231i	0.0464170	−	0.0267989i	0.523029	+	0.852315i	$0.324802\pi$
$282$		0			0
$283$	1.30977	+	4.88814i	0.0778580	+	0.290570i	0.993866	−	0.110590i	$-0.0352739\pi$
$283$	1.30977	+	4.88814i	0.0778580	+	0.290570i	−0.916008	+	0.401160i	$0.868607\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.58668	+	2.58668i	−0.152687	+	0.152687i
$288$		0			0
$289$			16.6067i			0.976862i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	25.4523	−	6.81992i	1.48694	−	0.398424i	0.578237	−	0.815869i	$-0.303741\pi$
$293$	25.4523	−	6.81992i	1.48694	−	0.398424i	0.908702	+	0.417445i	$0.137074\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.429597	−	0.744085i	−0.0248443	−	0.0430315i
$300$		0			0
$301$	−1.92813	+	3.33962i	−0.111136	+	0.192492i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	0.731912	+	0.731912i	0.0417724	+	0.0417724i	0.727684	−	0.685912i	$-0.240597\pi$
$307$	0.731912	+	0.731912i	0.0417724	+	0.0417724i	−0.685912	+	0.727684i	$0.740597\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	14.1899	+	8.19252i	0.804633	+	0.464555i	0.845089	−	0.534626i	$-0.179548\pi$
$311$	14.1899	+	8.19252i	0.804633	+	0.464555i	−0.0404554	+	0.999181i	$0.512881\pi$
$312$		0			0
$313$	9.78893	+	2.62294i	0.553303	+	0.148257i	0.524628	−	0.851332i	$-0.324204\pi$
$313$	9.78893	+	2.62294i	0.553303	+	0.148257i	0.0286753	+	0.999589i	$0.490871\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.83587	+	0.759869i	0.159278	+	0.0426785i	0.337577	−	0.941298i	$-0.390393\pi$
$317$	2.83587	+	0.759869i	0.159278	+	0.0426785i	−0.178299	+	0.983976i	$0.557059\pi$
$318$		0			0
$319$	−3.75454	−	2.16769i	−0.210214	−	0.121367i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.26573	+	1.26573i	0.0704270	+	0.0704270i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.42197	+	2.46293i	−0.0783960	+	0.135786i
$330$		0			0
$331$	16.0229	+	27.7524i	0.880696	+	1.52541i	0.850568	+	0.525865i	$0.176258\pi$
$331$	16.0229	+	27.7524i	0.880696	+	1.52541i	0.0301284	+	0.999546i	$0.490408\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	4.47628	−	1.19941i	0.243838	−	0.0653363i	−0.134830	−	0.990869i	$-0.543049\pi$
$337$	4.47628	−	1.19941i	0.243838	−	0.0653363i	0.378668	+	0.925532i	$0.376382\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			35.9815i			1.94851i
$342$		0			0
$343$	3.55101	−	3.55101i	0.191736	−	0.191736i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.70113	−	13.8128i	−0.198687	−	0.741511i	−0.991281	−	0.131761i	$-0.957937\pi$
$347$	−3.70113	−	13.8128i	−0.198687	−	0.741511i	0.792594	−	0.609749i	$-0.208730\pi$
$348$		0			0
$349$	−26.8777	+	15.5178i	−1.43873	+	0.830651i	−0.997762	−	0.0668713i	$-0.978698\pi$
$349$	−26.8777	+	15.5178i	−1.43873	+	0.830651i	−0.440969	+	0.897523i	$0.645365\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.70376	+	6.35852i	−0.0906819	+	0.338430i	−0.996329	−	0.0856018i	$-0.972719\pi$
$353$	−1.70376	+	6.35852i	−0.0906819	+	0.338430i	0.905647	+	0.424031i	$0.139385\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.5812			−1.13901			−0.569506	−	0.821987i	$-0.692865\pi$
$359$	−21.5812			−1.13901			−0.569506	+	0.821987i	$0.692865\pi$
$360$		0			0
$361$	10.8541			0.571269
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	7.64906	−	28.5467i	0.399278	−	1.49012i	−0.415093	−	0.909779i	$-0.636251\pi$
$367$	7.64906	−	28.5467i	0.399278	−	1.49012i	0.814371	−	0.580345i	$-0.197082\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.33716	+	1.92671i	−0.173257	+	0.100030i
$372$		0			0
$373$	7.94485	+	29.6506i	0.411369	+	1.53525i	0.791999	+	0.610522i	$0.209040\pi$
$373$	7.94485	+	29.6506i	0.411369	+	1.53525i	−0.380631	+	0.924727i	$0.624293\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.612870	+	0.612870i	−0.0315644	+	0.0315644i
$378$		0			0
$379$		−	27.4359i		−	1.40929i	−0.709560	−	0.704645i	$-0.751106\pi$
$379$		−	27.4359i		−	1.40929i	0.709560	−	0.704645i	$-0.248894\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	17.4030	−	4.66313i	0.889253	−	0.238275i	0.214858	−	0.976645i	$-0.431071\pi$
$383$	17.4030	−	4.66313i	0.889253	−	0.238275i	0.674395	+	0.738371i	$0.264404\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.4239	+	28.4471i	0.832725	+	1.44232i	0.895869	+	0.444318i	$0.146554\pi$
$389$	16.4239	+	28.4471i	0.832725	+	1.44232i	−0.0631434	+	0.998004i	$0.520113\pi$
$390$		0			0
$391$	0.318221	−	0.551175i	0.0160931	−	0.0278741i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	22.4277	+	22.4277i	1.12561	+	1.12561i	0.990882	+	0.134729i	$0.0430164\pi$
$397$	22.4277	+	22.4277i	1.12561	+	1.12561i	0.134729	+	0.990882i	$0.456984\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−17.5401	−	10.1268i	−0.875912	−	0.505708i	−0.00660386	−	0.999978i	$-0.502102\pi$
$401$	−17.5401	−	10.1268i	−0.875912	−	0.505708i	−0.869308	+	0.494270i	$0.835435\pi$
$402$		0			0
$403$	6.94831	+	1.86179i	0.346120	+	0.0927426i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−28.7892	−	7.71405i	−1.42703	−	0.382371i
$408$		0			0
$409$	−3.50934	−	2.02612i	−0.173526	−	0.100185i	0.410722	−	0.911761i	$-0.365277\pi$
$409$	−3.50934	−	2.02612i	−0.173526	−	0.100185i	−0.584247	+	0.811576i	$0.698610\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.12692	−	3.12692i	−0.153866	−	0.153866i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.21902	+	3.84346i	−0.108406	+	0.187765i	−0.915125	−	0.403171i	$-0.867908\pi$
$419$	−2.21902	+	3.84346i	−0.108406	+	0.187765i	0.806718	+	0.590936i	$0.201241\pi$
$420$		0			0
$421$	7.06970	+	12.2451i	0.344556	+	0.596788i	0.985273	−	0.170989i	$-0.0546961\pi$
$421$	7.06970	+	12.2451i	0.344556	+	0.596788i	−0.640717	+	0.767777i	$0.721363\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3.97499	+	1.06510i	−0.192363	+	0.0515436i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			9.06378i			0.436587i	0.975883	+	0.218294i	$0.0700490\pi$
$431$			9.06378i			0.436587i	−0.975883	+	0.218294i	$0.929951\pi$
$432$		0			0
$433$	6.33694	−	6.33694i	0.304534	−	0.304534i	−0.538251	−	0.842785i	$-0.680915\pi$
$433$	6.33694	−	6.33694i	0.304534	−	0.304534i	0.842785	+	0.538251i	$0.180915\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.749617	+	2.79761i	0.0358590	+	0.133828i
$438$		0			0
$439$	1.33899	−	0.773069i	0.0639067	−	0.0368966i	−0.467706	−	0.883884i	$-0.654919\pi$
$439$	1.33899	−	0.773069i	0.0639067	−	0.0368966i	0.531613	+	0.846987i	$0.321586\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.95825	−	22.2365i	0.283085	−	1.05649i	−0.667142	−	0.744930i	$-0.732483\pi$
$443$	5.95825	−	22.2365i	0.283085	−	1.05649i	0.950227	−	0.311557i	$-0.100851\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.58683			0.358045			0.179022	−	0.983845i	$-0.442707\pi$
$449$	7.58683			0.358045			0.179022	+	0.983845i	$0.442707\pi$
$450$		0			0
$451$	42.7853			2.01468
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.682064	+	2.54550i	−0.0319056	+	0.119073i	−0.980042	−	0.198790i	$-0.936299\pi$
$457$	−0.682064	+	2.54550i	−0.0319056	+	0.119073i	0.948136	+	0.317864i	$0.102965\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.70782	−	4.45011i	0.358989	−	0.207262i	−0.309648	−	0.950851i	$-0.600211\pi$
$461$	7.70782	−	4.45011i	0.358989	−	0.207262i	0.668637	+	0.743589i	$0.266878\pi$
$462$		0			0
$463$	8.33790	+	31.1175i	0.387495	+	1.44615i	0.834196	+	0.551468i	$0.185932\pi$
$463$	8.33790	+	31.1175i	0.387495	+	1.44615i	−0.446701	+	0.894683i	$0.647401\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10.9540	−	10.9540i	0.506891	−	0.506891i	−0.406680	−	0.913571i	$-0.633313\pi$
$467$	10.9540	−	10.9540i	0.506891	−	0.506891i	0.913571	+	0.406680i	$0.133313\pi$
$468$		0			0
$469$		−	1.06486i		−	0.0491709i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	43.5659	−	11.6734i	2.00316	−	0.536745i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.16179	+	10.6725i	0.281539	+	0.487641i	0.971764	−	0.235955i	$-0.0758216\pi$
$479$	6.16179	+	10.6725i	0.281539	+	0.487641i	−0.690225	+	0.723595i	$0.742488\pi$
$480$		0			0
$481$	−2.97929	+	5.16028i	−0.135844	+	0.235289i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−14.7058	−	14.7058i	−0.666383	−	0.666383i	0.290494	−	0.956877i	$-0.406180\pi$
$487$	−14.7058	−	14.7058i	−0.666383	−	0.666383i	−0.956877	+	0.290494i	$0.906180\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	26.3427	+	15.2089i	1.18883	+	0.686370i	0.958040	−	0.286634i	$-0.0925365\pi$
$491$	26.3427	+	15.2089i	1.18883	+	0.686370i	0.230787	+	0.973004i	$0.425870\pi$
$492$		0			0
$493$	−0.620147	−	0.166168i	−0.0279300	−	0.00748382i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.950900	−	0.254793i	−0.0426537	−	0.0114290i
$498$		0			0
$499$	−9.76122	−	5.63564i	−0.436972	−	0.252286i	0.265340	−	0.964155i	$-0.414516\pi$
$499$	−9.76122	−	5.63564i	−0.436972	−	0.252286i	−0.702312	+	0.711869i	$0.747849\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−10.7406	−	10.7406i	−0.478899	−	0.478899i	0.425881	−	0.904779i	$-0.359964\pi$
$503$	−10.7406	−	10.7406i	−0.478899	−	0.478899i	−0.904779	+	0.425881i	$0.859964\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.89519	+	15.4069i	−0.394272	+	0.682900i	−0.993008	−	0.118047i	$-0.962337\pi$
$509$	−8.89519	+	15.4069i	−0.394272	+	0.682900i	0.598736	+	0.800947i	$0.295670\pi$
$510$		0			0
$511$	0.603595	+	1.04546i	0.0267015	+	0.0462483i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	32.1294	−	8.60903i	1.41305	−	0.378625i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			7.18091i			0.314601i	0.987551	+	0.157301i	$0.0502792\pi$
$521$			7.18091i			0.314601i	−0.987551	+	0.157301i	$0.949721\pi$
$522$		0			0
$523$	−19.7187	+	19.7187i	−0.862240	+	0.862240i	−0.991598	−	0.129358i	$-0.958708\pi$
$523$	−19.7187	+	19.7187i	−0.862240	+	0.862240i	0.129358	+	0.991598i	$0.458708\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.37911	+	5.14691i	0.0600750	+	0.224203i
$528$		0			0
$529$	−19.0268	+	10.9851i	−0.827251	+	0.477613i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.21385	−	8.26219i	0.0958923	−	0.357875i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−29.0903			−1.25301
$540$		0			0
$541$	−36.2098			−1.55678			−0.778390	−	0.627781i	$-0.783963\pi$
$541$	−36.2098			−1.55678			−0.778390	+	0.627781i	$0.783963\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−3.09306	+	11.5435i	−0.132250	+	0.493562i	−0.999994	−	0.00344834i	$-0.998902\pi$
$547$	−3.09306	+	11.5435i	−0.132250	+	0.493562i	0.867744	+	0.497011i	$0.165569\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.53026	−	1.46085i	0.107793	−	0.0622341i
$552$		0			0
$553$	−0.680062	−	2.53803i	−0.0289192	−	0.107928i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7.42031	+	7.42031i	−0.314409	+	0.314409i	−0.846615	−	0.532206i	$-0.821363\pi$
$557$	−7.42031	+	7.42031i	−0.314409	+	0.314409i	0.532206	+	0.846615i	$0.321363\pi$
$558$		0			0
$559$		−	9.01694i		−	0.381376i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	22.8806	−	6.13083i	0.964300	−	0.258383i	0.257881	−	0.966177i	$-0.416976\pi$
$563$	22.8806	−	6.13083i	0.964300	−	0.258383i	0.706420	+	0.707793i	$0.250309\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	19.9559	+	34.5646i	0.836593	+	1.44902i	0.892727	+	0.450599i	$0.148789\pi$
$569$	19.9559	+	34.5646i	0.836593	+	1.44902i	−0.0561334	+	0.998423i	$0.517877\pi$
$570$		0			0
$571$	−1.46164	+	2.53164i	−0.0611677	+	0.105946i	−0.894988	−	0.446091i	$-0.852816\pi$
$571$	−1.46164	+	2.53164i	−0.0611677	+	0.105946i	0.833820	+	0.552037i	$0.186149\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	14.0335	+	14.0335i	0.584223	+	0.584223i	0.936061	−	0.351838i	$-0.114443\pi$
$577$	14.0335	+	14.0335i	0.584223	+	0.584223i	−0.351838	+	0.936061i	$0.614443\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.68808	+	2.70666i	0.194494	+	0.112291i
$582$		0			0
$583$	43.5338	+	11.6649i	1.80299	+	0.483109i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−44.0450	−	11.8018i	−1.81793	−	0.487114i	−0.821403	−	0.570348i	$-0.806808\pi$
$587$	−44.0450	−	11.8018i	−1.81793	−	0.487114i	−0.996530	+	0.0832347i	$0.973475\pi$
$588$		0			0
$589$	−20.9999	−	12.1243i	−0.865286	−	0.499573i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11.4862	−	11.4862i	−0.471680	−	0.471680i	0.430778	−	0.902458i	$-0.358239\pi$
$593$	−11.4862	−	11.4862i	−0.471680	−	0.471680i	−0.902458	+	0.430778i	$0.858239\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10.8461	−	18.7860i	0.443160	−	0.767575i	−0.554762	−	0.832009i	$-0.687191\pi$
$599$	10.8461	−	18.7860i	0.443160	−	0.767575i	0.997922	+	0.0644337i	$0.0205241\pi$
$600$		0			0
$601$	17.6688	+	30.6032i	0.720724	+	1.24833i	0.960710	+	0.277554i	$0.0895238\pi$
$601$	17.6688	+	30.6032i	0.720724	+	1.24833i	−0.239986	+	0.970776i	$0.577143\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−0.910852	+	0.244062i	−0.0369703	+	0.00990617i	−0.277257	−	0.960796i	$-0.589425\pi$
$607$	−0.910852	+	0.244062i	−0.0369703	+	0.00990617i	0.240286	+	0.970702i	$0.422759\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	6.64989i		−	0.269026i
$612$		0			0
$613$	−0.288873	+	0.288873i	−0.0116675	+	0.0116675i	−0.712916	−	0.701249i	$-0.752626\pi$
$613$	−0.288873	+	0.288873i	−0.0116675	+	0.0116675i	0.701249	+	0.712916i	$0.252626\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.78241	−	10.3841i	−0.112016	−	0.418048i	0.887031	−	0.461710i	$-0.152764\pi$
$617$	−2.78241	−	10.3841i	−0.112016	−	0.418048i	−0.999046	+	0.0436626i	$0.986097\pi$
$618$		0			0
$619$	19.6462	−	11.3428i	0.789649	−	0.455904i	−0.0501897	−	0.998740i	$-0.515983\pi$
$619$	19.6462	−	11.3428i	0.789649	−	0.455904i	0.839839	+	0.542835i	$0.182649\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.328115	+	1.22454i	−0.0131457	+	0.0490603i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.41378			−0.175989
$630$		0			0
$631$	19.9499			0.794194			0.397097	−	0.917777i	$-0.370018\pi$
$631$	19.9499			0.794194			0.397097	+	0.917777i	$0.370018\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.50522	+	5.61757i	−0.0596391	+	0.222576i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	25.3896	−	14.6587i	1.00283	−	0.578983i	0.0937447	−	0.995596i	$-0.470116\pi$
$641$	25.3896	−	14.6587i	1.00283	−	0.578983i	0.909084	+	0.416613i	$0.136783\pi$
$642$		0			0
$643$	−3.89715	−	14.5444i	−0.153689	−	0.573574i	−0.999214	−	0.0396389i	$-0.987379\pi$
$643$	−3.89715	−	14.5444i	−0.153689	−	0.573574i	0.845525	−	0.533935i	$-0.179287\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	15.8603	−	15.8603i	0.623531	−	0.623531i	−0.322901	−	0.946433i	$-0.604658\pi$
$647$	15.8603	−	15.8603i	0.623531	−	0.623531i	0.946433	+	0.322901i	$0.104658\pi$
$648$		0			0
$649$			51.7212i			2.03024i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	21.9133	−	5.87165i	0.857533	−	0.229775i	0.196844	−	0.980435i	$-0.436931\pi$
$653$	21.9133	−	5.87165i	0.857533	−	0.229775i	0.660689	+	0.750659i	$0.270264\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.73561	−	16.8626i	−0.379246	−	0.656873i	0.611707	−	0.791084i	$-0.290483\pi$
$659$	−9.73561	−	16.8626i	−0.379246	−	0.656873i	−0.990953	+	0.134212i	$0.957150\pi$
$660$		0			0
$661$	−17.0942	+	29.6081i	−0.664889	+	1.15162i	0.314426	+	0.949282i	$0.398188\pi$
$661$	−17.0942	+	29.6081i	−0.664889	+	1.15162i	−0.979315	+	0.202340i	$0.935146\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.734553	−	0.734553i	−0.0284420	−	0.0284420i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	41.6831	+	24.0657i	1.60916	+	0.929047i
$672$		0			0
$673$	28.5892	+	7.66046i	1.10203	+	0.295289i	0.763592	−	0.645699i	$-0.223434\pi$
$673$	28.5892	+	7.66046i	1.10203	+	0.295289i	0.338441	+	0.940988i	$0.390100\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	28.0185	+	7.50754i	1.07684	+	0.288538i	0.753300	−	0.657677i	$-0.228461\pi$
$677$	28.0185	+	7.50754i	1.07684	+	0.288538i	0.323539	+	0.946215i	$0.395127\pi$
$678$		0			0
$679$	1.90297	+	1.09868i	0.0730294	+	0.0421635i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	19.6240	+	19.6240i	0.750892	+	0.750892i	0.974646	−	0.223754i	$-0.0718311\pi$
$683$	19.6240	+	19.6240i	0.750892	+	0.750892i	−0.223754	+	0.974646i	$0.571831\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.50515	−	7.80316i	0.171633	−	0.297277i
$690$		0			0
$691$	1.96293	+	3.39990i	0.0746734	+	0.129338i	0.900944	−	0.433935i	$-0.142875\pi$
$691$	1.96293	+	3.39990i	0.0746734	+	0.129338i	−0.826271	+	0.563273i	$0.809542\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	6.12016	−	1.63989i	0.231818	−	0.0621153i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			22.9088i			0.865253i	0.901573	+	0.432627i	$0.142413\pi$
$701$			22.9088i			0.865253i	−0.901573	+	0.432627i	$0.857587\pi$
$702$		0			0
$703$	14.2030	−	14.2030i	0.535675	−	0.535675i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.862590	+	3.21923i	0.0324410	+	0.121072i
$708$		0			0
$709$	−22.4427	+	12.9573i	−0.842854	+	0.486622i	−0.858233	−	0.513260i	$-0.828438\pi$
$709$	−22.4427	+	12.9573i	−0.842854	+	0.486622i	0.0153793	+	0.999882i	$0.495104\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.23145	+	8.32788i	−0.0835684	+	0.311882i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−41.9600			−1.56484			−0.782422	−	0.622748i	$-0.786016\pi$
$719$	−41.9600			−1.56484			−0.782422	+	0.622748i	$0.786016\pi$
$720$		0			0
$721$	−5.51795			−0.205499
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	9.45700	−	35.2940i	0.350741	−	1.30898i	−0.535020	−	0.844839i	$-0.679696\pi$
$727$	9.45700	−	35.2940i	0.350741	−	1.30898i	0.885761	−	0.464142i	$-0.153637\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.78439	−	3.33962i	0.213943	−	0.123520i
$732$		0			0
$733$	5.51909	+	20.5975i	0.203852	+	0.760787i	0.989796	+	0.142490i	$0.0455107\pi$
$733$	5.51909	+	20.5975i	0.203852	+	0.760787i	−0.785944	+	0.618298i	$0.787823\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.80676	+	8.80676i	−0.324401	+	0.324401i
$738$		0			0
$739$			11.8977i			0.437663i	0.975763	+	0.218832i	$0.0702245\pi$
$739$			11.8977i			0.437663i	−0.975763	+	0.218832i	$0.929775\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−20.5075	+	5.49496i	−0.752346	+	0.201591i	−0.614559	−	0.788871i	$-0.710666\pi$
$743$	−20.5075	+	5.49496i	−0.752346	+	0.201591i	−0.137788	+	0.990462i	$0.543999\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.0237381	−	0.0411156i	−0.000867371	−	0.00150233i
$750$		0			0
$751$	−25.4044	+	44.0017i	−0.927019	+	1.60564i	−0.138738	+	0.990329i	$0.544304\pi$
$751$	−25.4044	+	44.0017i	−0.927019	+	1.60564i	−0.788281	+	0.615315i	$0.789029\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−30.3660	−	30.3660i	−1.10367	−	1.10367i	−0.993964	−	0.109705i	$-0.965009\pi$
$757$	−30.3660	−	30.3660i	−1.10367	−	1.10367i	−0.109705	−	0.993964i	$-0.534991\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.14551	+	1.23871i	0.0777747	+	0.0449033i	0.538383	−	0.842700i	$-0.319035\pi$
$761$	2.14551	+	1.23871i	0.0777747	+	0.0449033i	−0.460608	+	0.887604i	$0.652369\pi$
$762$		0			0
$763$	1.53461	+	0.411197i	0.0555565	+	0.0148863i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	9.98778	+	2.67622i	0.360638	+	0.0966326i
$768$		0			0
$769$	−23.4800	−	13.5562i	−0.846712	−	0.488849i	0.0128282	−	0.999918i	$-0.495917\pi$
$769$	−23.4800	−	13.5562i	−0.846712	−	0.488849i	−0.859540	+	0.511068i	$0.829250\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	32.7753	+	32.7753i	1.17885	+	1.17885i	0.980038	+	0.198809i	$0.0637072\pi$
$773$	32.7753	+	32.7753i	1.17885	+	1.17885i	0.198809	+	0.980038i	$0.436293\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−14.4169	+	24.9708i	−0.516540	+	0.894673i
$780$		0			0
$781$	5.75702	+	9.97145i	0.206002	+	0.356806i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−27.9327	+	7.48454i	−0.995693	+	0.266795i	−0.719640	−	0.694347i	$-0.755693\pi$
$787$	−27.9327	+	7.48454i	−0.995693	+	0.266795i	−0.276053	+	0.961142i	$0.589027\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			2.97609i			0.105818i
$792$		0			0
$793$	6.80410	−	6.80410i	0.241621	−	0.241621i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.58414	−	5.91209i	−0.0561131	−	0.209417i	0.932177	−	0.362002i	$-0.117907\pi$
$797$	−1.58414	−	5.91209i	−0.0561131	−	0.209417i	−0.988290	+	0.152585i	$0.951240\pi$
$798$		0			0
$799$	4.26592	−	2.46293i	0.150917	−	0.0871323i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.65433	−	13.6382i	0.128959	−	0.481280i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	48.6392			1.71006			0.855032	−	0.518574i	$-0.173537\pi$
$809$	48.6392			1.71006			0.855032	+	0.518574i	$0.173537\pi$
$810$		0			0
$811$	−14.8894			−0.522838			−0.261419	−	0.965225i	$-0.584190\pi$
$811$	−14.8894			−0.522838			−0.261419	+	0.965225i	$0.584190\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−7.86696	+	29.3599i	−0.275230	+	1.02717i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13.5079	−	7.79880i	0.471429	−	0.272180i	−0.245409	−	0.969420i	$-0.578922\pi$
$821$	13.5079	−	7.79880i	0.471429	−	0.272180i	0.716838	+	0.697240i	$0.245589\pi$
$822$		0			0
$823$	3.93044	+	14.6686i	0.137006	+	0.511315i	0.999982	+	0.00606868i	$0.00193173\pi$
$823$	3.93044	+	14.6686i	0.137006	+	0.511315i	−0.862975	+	0.505246i	$0.831402\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	19.9628	−	19.9628i	0.694175	−	0.694175i	−0.268973	−	0.963148i	$-0.586684\pi$
$827$	19.9628	−	19.9628i	0.694175	−	0.694175i	0.963148	+	0.268973i	$0.0866842\pi$
$828$		0			0
$829$		−	30.9272i		−	1.07415i	−0.843535	−	0.537074i	$-0.819530\pi$
$829$		−	30.9272i		−	1.07415i	0.843535	−	0.537074i	$-0.180470\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.16117	+	1.11498i	−0.144176	+	0.0386319i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−4.62581	−	8.01213i	−0.159701	−	0.276609i	0.775060	−	0.631887i	$-0.217720\pi$
$839$	−4.62581	−	8.01213i	−0.159701	−	0.276609i	−0.934761	+	0.355278i	$0.884386\pi$
$840$		0			0
$841$	13.9760	−	24.2072i	0.481932	−	0.834731i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1.77588	−	1.77588i	−0.0610199	−	0.0610199i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.18484	−	3.57082i	−0.212014	−	0.122406i
$852$		0			0
$853$	−13.9837	−	3.74693i	−0.478794	−	0.128293i	0.0113469	−	0.999936i	$-0.496388\pi$
$853$	−13.9837	−	3.74693i	−0.478794	−	0.128293i	−0.490141	+	0.871643i	$0.663055\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	34.3258	+	9.19758i	1.17255	+	0.314183i	0.791967	−	0.610564i	$-0.209057\pi$
$857$	34.3258	+	9.19758i	1.17255	+	0.314183i	0.380581	+	0.924748i	$0.375724\pi$
$858$		0			0
$859$	22.0478	+	12.7293i	0.752261	+	0.434318i	0.826510	−	0.562922i	$-0.190323\pi$
$859$	22.0478	+	12.7293i	0.752261	+	0.434318i	−0.0742492	+	0.997240i	$0.523656\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12.7951	−	12.7951i	−0.435551	−	0.435551i	0.454960	−	0.890512i	$-0.349653\pi$
$863$	−12.7951	−	12.7951i	−0.435551	−	0.435551i	−0.890512	+	0.454960i	$0.849653\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−15.3659	+	26.6146i	−0.521253	+	0.902837i
$870$		0			0
$871$	1.24497	+	2.15634i	0.0421841	+	0.0730649i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	5.31783	−	1.42491i	0.179570	−	0.0481157i	−0.167913	−	0.985802i	$-0.553703\pi$
$877$	5.31783	−	1.42491i	0.179570	−	0.0481157i	0.347484	+	0.937686i	$0.387036\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	28.8632i		−	0.972425i	−0.873840	−	0.486213i	$-0.838378\pi$
$881$		−	28.8632i		−	0.972425i	0.873840	−	0.486213i	$-0.161622\pi$
$882$		0			0
$883$	−4.77501	+	4.77501i	−0.160692	+	0.160692i	−0.782873	−	0.622181i	$-0.786247\pi$
$883$	−4.77501	+	4.77501i	−0.160692	+	0.160692i	0.622181	+	0.782873i	$0.286247\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.7501	+	47.5840i	0.428107	+	1.59772i	0.757044	+	0.653364i	$0.226643\pi$
$887$	12.7501	+	47.5840i	0.428107	+	1.59772i	−0.328937	+	0.944352i	$0.606690\pi$
$888$		0			0
$889$	−4.37537	+	2.52612i	−0.146745	+	0.0847233i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5.80179	+	21.6526i	−0.194150	+	0.724576i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.69726			0.290070
$900$		0			0
$901$	6.67432			0.222354
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.10724	+	7.86434i	−0.0699698	+	0.261131i	−0.992046	−	0.125878i	$-0.959825\pi$
$907$	−2.10724	+	7.86434i	−0.0699698	+	0.261131i	0.922076	+	0.387009i	$0.126492\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.07444	−	5.23913i	0.300650	−	0.173580i	−0.342085	−	0.939669i	$-0.611133\pi$
$911$	9.07444	−	5.23913i	0.300650	−	0.173580i	0.642735	+	0.766089i	$0.277800\pi$
$912$		0			0
$913$	−16.3869	−	61.1567i	−0.542327	−	2.02399i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.96521	+	3.96521i	−0.130943	+	0.130943i
$918$		0			0
$919$		−	7.87147i		−	0.259656i	−0.991537	−	0.129828i	$-0.958558\pi$
$919$		−	7.87147i		−	0.259656i	0.991537	−	0.129828i	$-0.0414425\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.22345	−	0.595772i	0.0731858	−	0.0196101i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4.96149	+	8.59355i	0.162781	+	0.281945i	0.935865	−	0.352358i	$-0.114620\pi$
$929$	4.96149	+	8.59355i	0.162781	+	0.281945i	−0.773084	+	0.634304i	$0.781287\pi$
$930$		0			0
$931$	9.80225	−	16.9780i	0.321256	−	0.556431i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−19.0704	−	19.0704i	−0.623002	−	0.623002i	0.323296	−	0.946298i	$-0.395209\pi$
$937$	−19.0704	−	19.0704i	−0.623002	−	0.623002i	−0.946298	+	0.323296i	$0.895209\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	7.88359	+	4.55159i	0.256998	+	0.148378i	0.622964	−	0.782250i	$-0.285928\pi$
$941$	7.88359	+	4.55159i	0.256998	+	0.148378i	−0.365967	+	0.930628i	$0.619262\pi$
$942$		0			0
$943$	9.90262	+	2.65340i	0.322474	+	0.0864066i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.48040	+	2.54027i	0.308072	+	0.0825476i	0.409543	−	0.912291i	$-0.365688\pi$
$947$	9.48040	+	2.54027i	0.308072	+	0.0825476i	−0.101471	+	0.994838i	$0.532355\pi$
$948$		0			0
$949$	−2.44455	−	1.41136i	−0.0793535	−	0.0458148i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−34.0291	−	34.0291i	−1.10231	−	1.10231i	−0.994131	−	0.108179i	$-0.965498\pi$
$953$	−34.0291	−	34.0291i	−1.10231	−	1.10231i	−0.108179	−	0.994131i	$-0.534502\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.72630	−	4.72208i	0.0880367	−	0.152484i
$960$		0			0
$961$	−20.5915	−	35.6655i	−0.664242	−	1.15050i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−42.2686	+	11.3258i	−1.35927	+	0.364214i	−0.863547	−	0.504268i	$-0.831762\pi$
$967$	−42.2686	+	11.3258i	−1.35927	+	0.364214i	−0.495719	+	0.868483i	$0.665096\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	35.4120i		−	1.13643i	−0.822882	−	0.568213i	$-0.807635\pi$
$971$		−	35.4120i		−	1.13643i	0.822882	−	0.568213i	$-0.192365\pi$
$972$		0			0
$973$	−0.648486	+	0.648486i	−0.0207895	+	0.0207895i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.01386	−	18.7120i	−0.160408	−	0.598649i	−0.998581	−	0.0532459i	$-0.983043\pi$
$977$	−5.01386	−	18.7120i	−0.160408	−	0.598649i	0.838174	−	0.545403i	$-0.183623\pi$
$978$		0			0
$979$	12.8410	−	7.41373i	0.410399	−	0.236944i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.84879	+	10.6318i	−0.0908624	+	0.339103i	−0.996360	−	0.0852489i	$-0.972831\pi$
$983$	−2.84879	+	10.6318i	−0.0908624	+	0.339103i	0.905497	+	0.424352i	$0.139498\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	10.8072			0.343650
$990$		0			0
$991$	−44.8001			−1.42312			−0.711561	−	0.702624i	$-0.752011\pi$
$991$	−44.8001			−1.42312			−0.711561	+	0.702624i	$0.752011\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	11.8187	−	44.1081i	0.374303	−	1.39692i	−0.480057	−	0.877237i	$-0.659384\pi$
$997$	11.8187	−	44.1081i	0.374303	−	1.39692i	0.854360	−	0.519681i	$-0.173949\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.bf.f.557.5		32
3.2	odd	2		900.2.be.f.257.3	✓	32
5.2	odd	4	inner	2700.2.bf.f.2393.5		32
5.3	odd	4	inner	2700.2.bf.f.2393.4		32
5.4	even	2	inner	2700.2.bf.f.557.4		32
9.2	odd	6	inner	2700.2.bf.f.2357.4		32
9.7	even	3		900.2.be.f.857.2	yes	32
15.2	even	4		900.2.be.f.293.7	yes	32
15.8	even	4		900.2.be.f.293.2	yes	32
15.14	odd	2		900.2.be.f.257.6	yes	32
45.2	even	12	inner	2700.2.bf.f.1493.4		32
45.7	odd	12		900.2.be.f.893.6	yes	32
45.29	odd	6	inner	2700.2.bf.f.2357.5		32
45.34	even	6		900.2.be.f.857.7	yes	32
45.38	even	12	inner	2700.2.bf.f.1493.5		32
45.43	odd	12		900.2.be.f.893.3	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.be.f.257.3	✓	32	3.2	odd	2
900.2.be.f.257.6	yes	32	15.14	odd	2
900.2.be.f.293.2	yes	32	15.8	even	4
900.2.be.f.293.7	yes	32	15.2	even	4
900.2.be.f.857.2	yes	32	9.7	even	3
900.2.be.f.857.7	yes	32	45.34	even	6
900.2.be.f.893.3	yes	32	45.43	odd	12
900.2.be.f.893.6	yes	32	45.7	odd	12
2700.2.bf.f.557.4		32	5.4	even	2	inner
2700.2.bf.f.557.5		32	1.1	even	1	trivial
2700.2.bf.f.1493.4		32	45.2	even	12	inner
2700.2.bf.f.1493.5		32	45.38	even	12	inner
2700.2.bf.f.2357.4		32	9.2	odd	6	inner
2700.2.bf.f.2357.5		32	45.29	odd	6	inner
2700.2.bf.f.2393.4		32	5.3	odd	4	inner
2700.2.bf.f.2393.5		32	5.2	odd	4	inner

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.bf (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.0937177 - 0.349759i) q^{7} +(-3.66769 + 2.11754i) q^{11} +(0.219136 + 0.817828i) q^{13} +(-0.443477 + 0.443477i) q^{17} -2.85410i q^{19} +(-0.980206 + 0.262645i) q^{23} +(0.511841 + 0.886534i) q^{29} +(4.24803 - 7.35780i) q^{31} +(4.97633 + 4.97633i) q^{37} +(-8.74911 - 5.05130i) q^{41} +(-10.2869 - 2.75637i) q^{43} +(-7.58648 - 2.03279i) q^{47} +(5.94863 + 3.43444i) q^{49} +(-7.52500 - 7.52500i) q^{53} +(6.10629 - 10.5764i) q^{59} +(-5.68247 - 9.84233i) q^{61} +(2.84062 - 0.761142i) q^{67} -2.71873i q^{71} +(-2.35741 + 2.35741i) q^{73} +(0.396902 + 1.48126i) q^{77} +(6.28431 - 3.62825i) q^{79} +(-3.86932 + 14.4405i) q^{83} -3.50111 q^{89} +0.306580 q^{91} +(-1.57063 + 5.86166i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 12 q^{11} + 8 q^{31} - 60 q^{41} + 52 q^{61} + 120 q^{91}+O(q^{100}) \) 32 * q + 12 * q^11 + 8 * q^31 - 60 * q^41 + 52 * q^61 + 120 * q^91

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