LMFDB - Embedded newform 507.2.x.a.128.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(2,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(156))

chi = DirichletCharacter(H, H._module([78, 1]))

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

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

chi := DirichletCharacter("507.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.x (of order $156$, degree $48$, 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:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{156}]$

Embedding invariants

Embedding label			128.1
Character	$\chi$	$=$	507.128
Dual form			507.2.x.a.305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.346455 - 1.69705i) q^{3} +(1.92104 + 0.556435i) q^{4} +(4.15936 - 2.99643i) q^{7} +(-2.75994 - 1.17590i) q^{9} +O(q^{10})$	$q+(0.346455 - 1.69705i) q^{3} +(1.92104 + 0.556435i) q^{4} +(4.15936 - 2.99643i) q^{7} +(-2.75994 - 1.17590i) q^{9} +(1.60985 - 3.06731i) q^{12} +(-2.59808 + 2.50000i) q^{13} +(3.38076 + 2.13786i) q^{16} +(-2.13167 + 0.571179i) q^{19} +(-3.64406 - 8.09676i) q^{21} +(-4.11492 + 2.84032i) q^{25} +(-2.95175 + 4.27635i) q^{27} +(9.65760 - 3.44184i) q^{28} +(10.2992 - 1.88740i) q^{31} +(-4.64763 - 3.79467i) q^{36} +(-9.11192 + 7.75064i) q^{37} +(3.34250 + 5.27519i) q^{39} +(-0.928666 - 11.5036i) q^{43} +(4.79934 - 4.99664i) q^{48} +(6.10501 - 18.2867i) q^{49} +(-6.38209 + 3.35693i) q^{52} +(0.230791 + 3.81543i) q^{57} +(-3.35038 - 4.10347i) q^{61} +(-15.0031 + 3.37897i) q^{63} +(5.30498 + 5.98809i) q^{64} +(-6.71032 + 12.1828i) q^{67} +(4.13891 + 5.28293i) q^{73} +(3.39453 + 7.96726i) q^{75} +(-4.41284 - 0.0888796i) q^{76} +(17.1816 + 4.23488i) q^{79} +(6.23452 + 6.49082i) q^{81} +(-2.49504 - 17.5818i) q^{84} +(-3.31526 + 18.1834i) q^{91} +(0.365203 - 18.1322i) q^{93} +(-12.5008 + 11.5323i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) $ 48 * q + 10 * q^7 + 6 * q^9	$ 48 q + 10 q^{7} + 6 q^{9} - 8 q^{16} - 14 q^{19} - 18 q^{21} + 20 q^{28} + 14 q^{31} + 2 q^{37} + 24 q^{39} + 6 q^{43} - 18 q^{49} - 28 q^{52} - 12 q^{57} - 24 q^{63} - 32 q^{67} + 34 q^{73} + 30 q^{75} + 28 q^{76} + 18 q^{81} + 12 q^{84} - 2 q^{91} - 6 q^{93} + 38 q^{97}+O(q^{100}) $ 48 * q + 10 * q^7 + 6 * q^9 - 8 * q^16 - 14 * q^19 - 18 * q^21 + 20 * q^28 + 14 * q^31 + 2 * q^37 + 24 * q^39 + 6 * q^43 - 18 * q^49 - 28 * q^52 - 12 * q^57 - 24 * q^63 - 32 * q^67 + 34 * q^73 + 30 * q^75 + 28 * q^76 + 18 * q^81 + 12 * q^84 - 2 * q^91 - 6 * q^93 + 38 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{7}{156}\right)$

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		−0.990080	−	0.140502i	$-0.955128\pi$
$2$		0			0		0.990080	+	0.140502i	$0.0448718\pi$
$3$	0.346455	−	1.69705i	0.200026	−	0.979791i
$3$	0.346455	−	1.69705i	0.200026	−	0.979791i
$4$	1.92104	+	0.556435i	0.960518	+	0.278217i
$5$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$5$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$6$		0			0
$7$	4.15936	−	2.99643i	1.57209	−	1.13254i	0.641126	−	0.767436i	$-0.278468\pi$
$7$	4.15936	−	2.99643i	1.57209	−	1.13254i	0.930965	−	0.365109i	$-0.118968\pi$
$8$		0			0
$9$	−2.75994	−	1.17590i	−0.919979	−	0.391967i
$10$		0			0
$11$		0			0		0.373361	−	0.927686i	$-0.378205\pi$
$11$		0			0		−0.373361	+	0.927686i	$0.621795\pi$
$12$	1.60985	−	3.06731i	0.464723	−	0.885456i
$13$	−2.59808	+	2.50000i	−0.720577	+	0.693375i
$13$	−2.59808	+	2.50000i	−0.720577	+	0.693375i
$14$		0			0
$15$		0			0
$16$	3.38076	+	2.13786i	0.845190	+	0.534466i
$17$		0			0		0.160411	−	0.987050i	$-0.448718\pi$
$17$		0			0		−0.160411	+	0.987050i	$0.551282\pi$
$18$		0			0
$19$	−2.13167	+	0.571179i	−0.489039	+	0.131038i	−0.494908	−	0.868945i	$-0.664798\pi$
$19$	−2.13167	+	0.571179i	−0.489039	+	0.131038i	0.00586953	+	0.999983i	$0.498132\pi$
$20$		0			0
$21$	−3.64406	−	8.09676i	−0.795198	−	1.76686i
$22$		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
$25$	−4.11492	+	2.84032i	−0.822984	+	0.568065i
$26$		0			0
$27$	−2.95175	+	4.27635i	−0.568065	+	0.822984i
$28$	9.65760	−	3.44184i	1.82512	−	0.650447i
$29$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$29$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$30$		0			0
$31$	10.2992	−	1.88740i	1.84980	−	0.338988i	0.863550	−	0.504264i	$-0.168236\pi$
$31$	10.2992	−	1.88740i	1.84980	−	0.338988i	0.986247	+	0.165276i	$0.0528515\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−4.64763	−	3.79467i	−0.774605	−	0.632445i
$37$	−9.11192	+	7.75064i	−1.49799	+	1.27420i	−0.631098	+	0.775703i	$0.717395\pi$
$37$	−9.11192	+	7.75064i	−1.49799	+	1.27420i	−0.866893	+	0.498493i	$0.833887\pi$
$38$		0			0
$39$	3.34250	+	5.27519i	0.535229	+	0.844707i
$40$		0			0
$41$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$41$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$42$		0			0
$43$	−0.928666	−	11.5036i	−0.141620	−	1.75428i	−0.544254	−	0.838920i	$-0.683187\pi$
$43$	−0.928666	−	11.5036i	−0.141620	−	1.75428i	0.402634	−	0.915361i	$-0.368095\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$47$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$48$	4.79934	−	4.99664i	0.692724	−	0.721202i
$49$	6.10501	−	18.2867i	0.872144	−	2.61239i
$50$		0			0
$51$		0			0
$52$	−6.38209	+	3.35693i	−0.885036	+	0.465522i
$53$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$53$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.230791	+	3.81543i	0.0305690	+	0.505366i
$58$		0			0
$59$		0			0		0.219715	−	0.975564i	$-0.429487\pi$
$59$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$60$		0			0
$61$	−3.35038	−	4.10347i	−0.428972	−	0.525395i	0.514213	−	0.857662i	$-0.328084\pi$
$61$	−3.35038	−	4.10347i	−0.428972	−	0.525395i	−0.943185	+	0.332267i	$0.892187\pi$
$62$		0			0
$63$	−15.0031	+	3.37897i	−1.89021	+	0.425711i
$64$	5.30498	+	5.98809i	0.663123	+	0.748511i
$65$		0			0
$66$		0			0
$67$	−6.71032	+	12.1828i	−0.819796	+	1.48836i	0.0520846	+	0.998643i	$0.483413\pi$
$67$	−6.71032	+	12.1828i	−0.819796	+	1.48836i	−0.871880	+	0.489719i	$0.837099\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.894635	−	0.446798i	$-0.852564\pi$
$71$		0			0		0.894635	+	0.446798i	$0.147436\pi$
$72$		0			0
$73$	4.13891	+	5.28293i	0.484423	+	0.618320i	0.966144	−	0.258005i	$-0.0830650\pi$
$73$	4.13891	+	5.28293i	0.484423	+	0.618320i	−0.481721	+	0.876325i	$0.659988\pi$
$74$		0			0
$75$	3.39453	+	7.96726i	0.391967	+	0.919979i
$76$	−4.41284	−	0.0888796i	−0.506187	−	0.0101952i
$77$		0			0
$78$		0			0
$79$	17.1816	+	4.23488i	1.93308	+	0.476462i	0.989302	+	0.145885i	$0.0466029\pi$
$79$	17.1816	+	4.23488i	1.93308	+	0.476462i	0.943779	+	0.330577i	$0.107243\pi$
$80$		0			0
$81$	6.23452	+	6.49082i	0.692724	+	0.721202i
$82$		0			0
$83$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$83$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$84$	−2.49504	−	17.5818i	−0.272232	−	1.91834i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$89$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$90$		0			0
$91$	−3.31526	+	18.1834i	−0.347533	+	1.90613i
$92$		0			0
$93$	0.365203	−	18.1322i	0.0378698	−	1.88022i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−12.5008	+	11.5323i	−1.26926	+	1.17093i	−0.292578	+	0.956242i	$0.594513\pi$
$97$	−12.5008	+	11.5323i	−1.26926	+	1.17093i	−0.976684	+	0.214684i	$0.931128\pi$
$98$		0			0
$99$		0			0
$100$	−9.48536	+	3.16668i	−0.948536	+	0.316668i
$101$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$101$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$102$		0			0
$103$	5.14682	−	5.80956i	0.507131	−	0.572433i	−0.438062	−	0.898945i	$-0.644335\pi$
$103$	5.14682	−	5.80956i	0.507131	−	0.572433i	0.945193	+	0.326512i	$0.105873\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$107$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$108$	−8.04993	+	6.57257i	−0.774605	+	0.632445i
$109$	−2.80535	+	15.3083i	−0.268704	+	1.46627i	0.521407	+	0.853308i	$0.325407\pi$
$109$	−2.80535	+	15.3083i	−0.268704	+	1.46627i	−0.790112	+	0.612963i	$0.789977\pi$
$110$		0			0
$111$	9.99633	+	18.1486i	0.948810	+	1.72259i
$112$	20.4678	−	1.23807i	1.93402	−	0.116987i
$113$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$113$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	10.1103	−	3.84477i	0.934696	−	0.355449i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.93323	−	7.61997i	−0.721202	−	0.692724i
$122$		0			0
$123$		0			0
$124$	20.8354	+	2.10508i	1.87108	+	0.189042i
$125$		0			0
$126$		0			0
$127$	−9.20038	+	2.66492i	−0.816402	+	0.236474i	−0.660044	−	0.751227i	$-0.729462\pi$
$127$	−9.20038	+	2.66492i	−0.816402	+	0.236474i	−0.156358	+	0.987701i	$0.549975\pi$
$128$		0			0
$129$	−19.8439	−	2.40948i	−1.74716	−	0.212143i
$130$		0			0
$131$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$131$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$132$		0			0
$133$	−7.15489	+	8.76315i	−0.620407	+	0.759861i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.647915	−	0.761712i	$-0.275641\pi$
$137$		0			0		−0.647915	+	0.761712i	$0.724359\pi$
$138$		0			0
$139$	−0.586046	+	14.5426i	−0.0497078	+	1.23349i	0.761280	+	0.648423i	$0.224571\pi$
$139$	−0.586046	+	14.5426i	−0.0497078	+	1.23349i	−0.810988	+	0.585063i	$0.801070\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−6.81678	−	9.87581i	−0.568065	−	0.822984i
$145$		0			0
$146$		0			0
$147$	−28.9183	−	16.6960i	−2.38514	−	1.37706i
$148$	−21.8171	+	9.81906i	−1.79335	+	0.807122i
$149$		0			0		0.999797	−	0.0201371i	$-0.00641026\pi$
$149$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$150$		0			0
$151$	−0.225804	+	0.373525i	−0.0183757	+	0.0303970i	−0.864834	−	0.502058i	$-0.832577\pi$
$151$	−0.225804	+	0.373525i	−0.0183757	+	0.0303970i	0.846458	+	0.532455i	$0.178730\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	3.48577	+	11.9937i	0.279085	+	0.960267i
$157$	14.2283	+	7.46761i	1.13555	+	0.595980i	0.924381	−	0.381470i	$-0.124582\pi$
$157$	14.2283	+	7.46761i	1.13555	+	0.595980i	0.211165	+	0.977451i	$0.432274\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−22.2812	−	7.94071i	−1.74520	−	0.621964i	−0.746156	−	0.665771i	$-0.768103\pi$
$163$	−22.2812	−	7.94071i	−1.74520	−	0.621964i	−0.999040	+	0.0438070i	$0.986051\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.140502	−	0.990080i	$-0.455128\pi$
$167$		0			0		−0.140502	+	0.990080i	$0.544872\pi$
$168$		0			0
$169$	0.500000	−	12.9904i	0.0384615	−	0.999260i
$170$		0			0
$171$	6.55493	+	0.930211i	0.501268	+	0.0711350i
$172$	4.61700	−	22.6156i	0.352043	−	1.72442i
$173$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$173$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$174$		0			0
$175$	−8.60460	+	24.1440i	−0.650447	+	1.82512i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.160411	−	0.987050i	$-0.551282\pi$
$179$		0			0		0.160411	+	0.987050i	$0.448718\pi$
$180$		0			0
$181$	11.7729	−	22.4315i	0.875076	−	1.66732i	0.139135	−	0.990273i	$-0.455568\pi$
$181$	11.7729	−	22.4315i	0.875076	−	1.66732i	0.735941	−	0.677045i	$-0.236740\pi$
$182$		0			0
$183$	−8.12454	+	4.26408i	−0.600583	+	0.315210i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.536391	+	26.6316i	0.0390167	+	1.93716i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	12.0000	−	6.92820i	0.866025	−	0.500000i
$193$	2.79145	−	3.87483i	0.200933	−	0.278916i	−0.698836	−	0.715281i	$-0.746298\pi$
$193$	2.79145	−	3.87483i	0.200933	−	0.278916i	0.899770	+	0.436365i	$0.143734\pi$
$194$		0			0
$195$		0			0
$196$	21.9033	−	31.7324i	1.56452	−	2.26660i
$197$		0			0		0.941967	−	0.335705i	$-0.108974\pi$
$197$		0			0		−0.941967	+	0.335705i	$0.891026\pi$
$198$		0			0
$199$	−24.3818	−	0.982554i	−1.72838	−	0.0696514i	−0.843979	−	0.536376i	$-0.819793\pi$
$199$	−24.3818	−	0.982554i	−1.72838	−	0.0696514i	−0.884403	+	0.466725i	$0.845434\pi$
$200$		0			0
$201$	18.3499	+	15.6085i	1.29430	+	1.10094i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−14.1281	+	2.89757i	−0.979610	+	0.200910i
$209$		0			0
$210$		0			0
$211$	−7.08273	−	24.4524i	−0.487595	−	1.68337i	−0.706569	−	0.707644i	$-0.749758\pi$
$211$	−7.08273	−	24.4524i	−0.487595	−	1.68337i	0.218974	−	0.975731i	$-0.429729\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	37.1828	−	38.7113i	2.52413	−	2.62790i
$218$		0			0
$219$	10.3993	−	5.19363i	0.702721	−	0.350953i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	8.44221	−	9.15120i	0.565332	−	0.612810i	−0.385428	−	0.922738i	$-0.625946\pi$
$223$	8.44221	−	9.15120i	0.565332	−	0.612810i	0.950760	+	0.309928i	$0.100305\pi$
$224$		0			0
$225$	14.6969	−	3.00039i	0.979791	−	0.200026i
$226$		0			0
$227$		0			0		0.875918	−	0.482459i	$-0.160256\pi$
$227$		0			0		−0.875918	+	0.482459i	$0.839744\pi$
$228$	−1.67968	+	7.45800i	−0.111240	+	0.493918i
$229$	26.7420	+	4.90066i	1.76716	+	0.323844i	0.962784	−	0.270273i	$-0.0871140\pi$
$229$	26.7420	+	4.90066i	1.76716	+	0.323844i	0.804378	+	0.594118i	$0.202499\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$233$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	13.1394	−	27.6908i	0.853498	−	1.79871i
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	11.1700	+	12.1080i	0.719520	+	0.779946i	0.982780	−	0.184778i	$-0.0591567\pi$
$241$	11.1700	+	12.1080i	0.719520	+	0.779946i	−0.263260	+	0.964725i	$0.584798\pi$
$242$		0			0
$243$	13.1752	−	8.33150i	0.845190	−	0.534466i
$244$	−4.15289	−	9.74718i	−0.265861	−	0.623999i
$245$		0			0
$246$		0			0
$247$	4.11029	−	6.81314i	0.261532	−	0.433510i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.903450	−	0.428693i	$-0.141026\pi$
$251$		0			0		−0.903450	+	0.428693i	$0.858974\pi$
$252$	−30.7016	−	1.85711i	−1.93402	−	0.116987i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	6.85908	+	14.4552i	0.428693	+	0.903450i
$257$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$257$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$258$		0			0
$259$	−14.6755	+	59.5410i	−0.911893	+	3.69970i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.919979	−	0.391967i	$-0.128205\pi$
$263$		0			0		−0.919979	+	0.391967i	$0.871795\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−19.6697	+	19.6697i	−1.20152	+	1.20152i
$269$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$269$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$270$		0			0
$271$	−1.45620	−	0.802082i	−0.0884580	−	0.0487230i	0.437615	−	0.899162i	$-0.355823\pi$
$271$	−1.45620	−	0.802082i	−0.0884580	−	0.0487230i	−0.526073	+	0.850439i	$0.676336\pi$
$272$		0			0
$273$	29.7094	+	11.9259i	1.79810	+	0.721786i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.30419	+	1.06484i	−0.0783612	+	0.0639800i	−0.670816	−	0.741623i	$-0.734056\pi$
$277$	−1.30419	+	1.06484i	−0.0783612	+	0.0639800i	0.592455	+	0.805603i	$0.298159\pi$
$278$		0			0
$279$	−30.6446	−	6.90175i	−1.83465	−	0.413197i
$280$		0			0
$281$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$281$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$282$		0			0
$283$	2.62324	−	32.4947i	0.155936	−	1.93161i	−0.168500	−	0.985702i	$-0.553892\pi$
$283$	2.62324	−	32.4947i	0.155936	−	1.93161i	0.324435	−	0.945908i	$-0.394826\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.1251	−	5.38336i	−0.948536	−	0.316668i
$290$		0			0
$291$	15.2399	+	25.2098i	0.893377	+	1.47783i
$292$	5.01139	+	12.4517i	0.293269	+	0.728682i
$293$		0			0		−0.994935	−	0.100522i	$-0.967949\pi$
$293$		0			0		0.994935	+	0.100522i	$0.0320513\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	2.08776	+	17.1942i	0.120537	+	0.992709i
$301$	−38.3324	−	45.0649i	−2.20944	−	2.59750i
$302$		0			0
$303$		0			0
$304$	−8.42777	−	2.62620i	−0.483366	−	0.150623i
$305$		0			0
$306$		0			0
$307$	−2.87397	−	15.6827i	−0.164026	−	0.895061i	−0.957927	−	0.287014i	$-0.907338\pi$
$307$	−2.87397	−	15.6827i	−0.164026	−	0.895061i	0.793900	−	0.608048i	$-0.208047\pi$
$308$		0			0
$309$	−8.07596	−	10.7471i	−0.459425	−	0.611384i
$310$		0			0
$311$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$311$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$312$		0			0
$313$	9.09592	+	13.1777i	0.514132	+	0.744849i	0.990961	−	0.134147i	$-0.0428295\pi$
$313$	9.09592	+	13.1777i	0.514132	+	0.744849i	−0.476830	+	0.878996i	$0.658214\pi$
$314$		0			0
$315$		0			0
$316$	30.6500	+	17.6958i	1.72420	+	0.995467i
$317$		0			0		0.911900	−	0.410413i	$-0.134615\pi$
$317$		0			0		−0.911900	+	0.410413i	$0.865385\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	8.36502	+	15.9382i	0.464723	+	0.885456i
$325$	3.59006	−	17.6667i	0.199141	−	0.979971i
$326$		0			0
$327$	25.0070	+	10.0645i	1.38289	+	0.556566i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	7.68797	+	10.6717i	0.422569	+	0.586570i	0.967801	−	0.251715i	$-0.0809945\pi$
$331$	7.68797	+	10.6717i	0.422569	+	0.586570i	−0.545233	+	0.838285i	$0.683559\pi$
$332$		0			0
$333$	34.2623	−	10.6766i	1.87756	−	0.585072i
$334$		0			0
$335$		0			0
$336$	4.99009	−	35.1637i	0.272232	−	1.91834i
$337$		−	10.7699i		−	0.586674i	−0.956009	−	0.293337i	$-0.905234\pi$
$337$		−	10.7699i		−	0.586674i	0.956009	−	0.293337i	$-0.0947658\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18.7264	−	60.0951i	−1.01113	−	3.24483i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$347$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$348$		0			0
$349$	−1.70149	+	4.22766i	−0.0910784	+	0.226301i	−0.966299	−	0.257423i	$-0.917127\pi$
$349$	−1.70149	+	4.22766i	−0.0910784	+	0.226301i	0.875220	+	0.483724i	$0.160716\pi$
$350$		0			0
$351$	−3.02200	−	18.4897i	−0.161302	−	0.986905i
$352$		0			0
$353$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$353$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.410413	−	0.911900i	$-0.634615\pi$
$359$		0			0		0.410413	+	0.911900i	$0.365385\pi$
$360$		0			0
$361$	−12.2367	+	7.06487i	−0.644037	+	0.371835i
$362$		0			0
$363$	−15.6799	+	10.8231i	−0.822984	+	0.568065i
$364$	−16.4866	+	33.0862i	−0.864132	+	1.73419i
$365$		0			0
$366$		0			0
$367$	−6.61341	+	4.96966i	−0.345217	+	0.259414i	−0.759181	−	0.650879i	$-0.774400\pi$
$367$	−6.61341	+	4.96966i	−0.345217	+	0.259414i	0.413964	+	0.910293i	$0.364144\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	10.7909	−	34.6294i	0.559485	−	1.79545i
$373$	−30.7204	−	23.0849i	−1.59064	−	1.19529i	−0.856249	−	0.516563i	$-0.827211\pi$
$373$	−30.7204	−	23.0849i	−1.59064	−	1.19529i	−0.734391	−	0.678726i	$-0.762532\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	30.5364	−	20.1821i	1.56855	−	1.03669i	0.596291	−	0.802768i	$-0.296640\pi$
$379$	30.5364	−	20.1821i	1.56855	−	1.03669i	0.972256	−	0.233918i	$-0.0751546\pi$
$380$		0			0
$381$	1.33498	+	16.5368i	0.0683933	+	0.847203i
$382$		0			0
$383$		0			0		0.100522	−	0.994935i	$-0.467949\pi$
$383$		0			0		−0.100522	+	0.994935i	$0.532051\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−10.9640	+	32.8412i	−0.557332	+	1.66941i
$388$	−30.4314	+	15.1981i	−1.54492	+	0.771564i
$389$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$389$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7.60418	−	33.7635i	0.381643	−	1.69454i	−0.292765	−	0.956184i	$-0.594576\pi$
$397$	7.60418	−	33.7635i	0.381643	−	1.69454i	0.674408	−	0.738359i	$-0.264399\pi$
$398$		0			0
$399$	12.3926	+	15.1782i	0.620407	+	0.759861i
$400$	−19.9838	+	0.805319i	−0.999189	+	0.0402659i
$401$		0			0		0.975564	−	0.219715i	$-0.0705128\pi$
$401$		0			0		−0.975564	+	0.219715i	$0.929487\pi$
$402$		0			0
$403$	−22.0397	+	30.6517i	−1.09787	+	1.52687i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	36.1661	+	18.0621i	1.78830	+	0.893111i	0.908877	+	0.417064i	$0.136941\pi$
$409$	36.1661	+	18.0621i	1.78830	+	0.893111i	0.879420	+	0.476047i	$0.157931\pi$
$410$		0			0
$411$		0			0
$412$	13.1199	−	8.29650i	0.646370	−	0.408739i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	24.4764	+	6.03290i	1.19862	+	0.295432i
$418$		0			0
$419$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$419$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$420$		0			0
$421$	−40.3050	−	2.43800i	−1.96435	−	0.118821i	−0.971405	−	0.237428i	$-0.923695\pi$
$421$	−40.3050	−	2.43800i	−1.96435	−	0.118821i	−0.992941	+	0.118607i	$0.962157\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−26.2312	−	7.02863i	−1.26942	−	0.340139i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.0201371	−	0.999797i	$-0.493590\pi$
$431$		0			0		−0.0201371	+	0.999797i	$0.506410\pi$
$432$	−19.1214	+	8.14687i	−0.919979	+	0.391967i
$433$	−9.69185	−	15.3264i	−0.465760	−	0.736541i	0.527858	−	0.849332i	$-0.322995\pi$
$433$	−9.69185	−	15.3264i	−0.465760	−	0.736541i	−0.993619	+	0.112791i	$0.964021\pi$
$434$		0			0
$435$		0			0
$436$	−13.9073	+	27.8468i	−0.666037	+	1.33362i
$437$		0			0
$438$		0			0
$439$	−15.9974	−	7.59088i	−0.763516	−	0.362293i	0.00678165	−	0.999977i	$-0.497841\pi$
$439$	−15.9974	−	7.59088i	−0.763516	−	0.362293i	−0.770298	+	0.637684i	$0.779893\pi$
$440$		0			0
$441$	−38.3528	+	43.2914i	−1.82632	+	2.06149i
$442$		0			0
$443$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$443$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$444$	9.10479	+	40.4264i	0.432094	+	1.91855i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	40.0082	+	9.01060i	1.89021	+	0.425711i
$449$		0			0		−0.482459	−	0.875918i	$-0.660256\pi$
$449$		0			0		0.482459	+	0.875918i	$0.339744\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	0.555659	+	0.512609i	0.0261071	+	0.0240845i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	19.1009	+	38.2462i	0.893502	+	1.78908i	0.459771	+	0.888037i	$0.347931\pi$
$457$	19.1009	+	38.2462i	0.893502	+	1.78908i	0.433731	+	0.901043i	$0.357197\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.373361	−	0.927686i	$-0.621795\pi$
$461$		0			0		0.373361	+	0.927686i	$0.378205\pi$
$462$		0			0
$463$	−25.8044	−	20.2165i	−1.19923	−	0.939539i	−0.200044	−	0.979787i	$-0.564108\pi$
$463$	−25.8044	−	20.2165i	−1.19923	−	0.939539i	−0.999190	+	0.0402476i	$0.987185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$467$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$468$	21.5616	−	1.76023i	0.996684	−	0.0813665i
$469$	8.59419	+	70.7795i	0.396843	+	3.26829i
$470$		0			0
$471$	17.6024	−	21.5590i	0.811074	−	0.993386i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	7.14932	−	8.40499i	0.328033	−	0.385647i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.335705	−	0.941967i	$-0.608974\pi$
$479$		0			0		0.335705	+	0.941967i	$0.391026\pi$
$480$		0			0
$481$	4.29688	−	42.9166i	0.195921	−	1.95683i
$482$		0			0
$483$		0			0
$484$	−11.0000	−	19.0526i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	−19.7943	+	0.398680i	−0.896967	+	0.0180659i	−0.466481	−	0.884531i	$-0.654478\pi$
$487$	−19.7943	+	0.398680i	−0.896967	+	0.0180659i	−0.430486	+	0.902597i	$0.641658\pi$
$488$		0			0
$489$	−21.1952	+	35.0611i	−0.958479	+	1.58552i
$490$		0			0
$491$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$491$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	38.8543	+	15.6375i	1.74461	+	0.702144i
$497$		0			0
$498$		0			0
$499$	−8.30887	+	18.4616i	−0.371956	+	0.826453i	0.626921	+	0.779083i	$0.284315\pi$
$499$	−8.30887	+	18.4616i	−0.371956	+	0.826453i	−0.998877	+	0.0473700i	$0.984916\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.278217	−	0.960518i	$-0.410256\pi$
$503$		0			0		−0.278217	+	0.960518i	$0.589744\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−21.8721	−	5.34910i	−0.971372	−	0.237562i
$508$	−19.1571			−0.849960
$509$		0			0		−0.990080	−	0.140502i	$-0.955128\pi$
$509$		0			0		0.990080	+	0.140502i	$0.0448718\pi$
$510$		0			0
$511$	33.0452	+	9.57164i	1.46183	+	0.423425i
$512$		0			0
$513$	3.84960	−	10.8017i	0.169964	−	0.476909i
$514$		0			0
$515$		0			0
$516$	−36.7801	−	15.6705i	−1.61915	−	0.689857i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$521$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$522$		0			0
$523$	38.5992	+	24.4087i	1.68783	+	1.06732i	0.873284	+	0.487212i	$0.161986\pi$
$523$	38.5992	+	24.4087i	1.68783	+	1.06732i	0.814542	+	0.580105i	$0.196988\pi$
$524$		0			0
$525$	37.9924	+	22.9672i	1.65812	+	1.00237i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−18.6209	+	12.8531i	−0.807319	+	0.557252i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7.29680	+	23.4163i	−0.313714	+	1.00674i	0.654382	+	0.756164i	$0.272929\pi$
$541$	−7.29680	+	23.4163i	−0.313714	+	1.00674i	−0.968096	+	0.250579i	$0.919379\pi$
$542$		0			0
$543$	−33.9885	−	27.7507i	−1.45859	−	1.19090i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−4.14888	+	34.1691i	−0.177393	+	1.46096i	0.583514	+	0.812103i	$0.301677\pi$
$547$	−4.14888	+	34.1691i	−0.177393	+	1.46096i	−0.760907	+	0.648861i	$0.775246\pi$
$548$		0			0
$549$	4.42157	+	15.2650i	0.188708	+	0.651496i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	84.1540	−	33.8691i	3.57859	−	1.44026i
$554$		0			0
$555$		0			0
$556$	−9.21782	+	27.6107i	−0.390923	+	1.17096i
$557$		0			0		0.894635	−	0.446798i	$-0.147436\pi$
$557$		0			0		−0.894635	+	0.446798i	$0.852564\pi$
$558$		0			0
$559$	31.1717	+	27.5655i	1.31842	+	1.16590i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$563$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	45.3809	+	8.31636i	1.90582	+	0.349254i
$568$		0			0
$569$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$569$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$570$		0			0
$571$	−27.7442	−	31.3168i	−1.16106	−	1.31057i	−0.942293	−	0.334790i	$-0.891335\pi$
$571$	−27.7442	−	31.3168i	−1.16106	−	1.31057i	−0.218768	−	0.975777i	$-0.570204\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−7.60003	−	22.7649i	−0.316668	−	0.948536i
$577$	−33.7602	−	33.7602i	−1.40546	−	1.40546i	−0.781292	−	0.624165i	$-0.785439\pi$
$577$	−33.7602	−	33.7602i	−1.40546	−	1.40546i	−0.624165	−	0.781292i	$-0.714561\pi$
$578$		0			0
$579$	−5.60866	−	6.07968i	−0.233088	−	0.252663i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$587$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$588$	−46.2629	−	48.1648i	−1.90785	−	1.98628i
$589$	−20.8765	+	9.90603i	−0.860202	+	0.408171i
$590$		0			0
$591$		0			0
$592$	−47.3750	+	6.72300i	−1.94710	+	0.276314i
$593$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$593$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−10.1146	+	41.0367i	−0.413965	+	1.67952i
$598$		0			0
$599$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$599$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$600$		0			0
$601$	35.9262	−	15.3067i	1.46546	−	0.624374i	0.495713	−	0.868487i	$-0.334907\pi$
$601$	35.9262	−	15.3067i	1.46546	−	0.624374i	0.969747	+	0.244113i	$0.0784967\pi$
$602$		0			0
$603$	32.8458	−	25.7330i	1.33758	−	1.04793i
$604$	−0.641620	+	0.591910i	−0.0261071	+	0.0240845i
$605$		0			0
$606$		0			0
$607$	46.6670	−	15.5797i	1.89416	−	0.632362i	0.929525	−	0.368758i	$-0.120217\pi$
$607$	46.6670	−	15.5797i	1.89416	−	0.632362i	0.964631	−	0.263604i	$-0.0849112\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−4.44573	−	19.7396i	−0.179561	−	0.797275i	−0.980181	−	0.198103i	$-0.936522\pi$
$613$	−4.44573	−	19.7396i	−0.179561	−	0.797275i	0.800620	−	0.599172i	$-0.204504\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.975564	−	0.219715i	$-0.929487\pi$
$617$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$618$		0			0
$619$	3.68408	−	0.222846i	0.148076	−	0.00895692i	0.0138256	−	0.999904i	$-0.495599\pi$
$619$	3.68408	−	0.222846i	0.148076	−	0.00895692i	0.134250	+	0.990948i	$0.457137\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.0225545	+	24.9800i	0.000902901	+	1.00000i
$625$	8.86512	−	23.3754i	0.354605	−	0.935016i
$626$		0			0
$627$		0			0
$628$	23.1779	+	22.2627i	0.924900	+	0.888379i
$629$		0			0
$630$		0			0
$631$	−45.9806	−	4.64559i	−1.83046	−	0.184938i	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−45.9806	−	4.64559i	−1.83046	−	0.184938i	−0.954652	+	0.297725i	$0.903772\pi$
$632$		0			0
$633$	−43.9508	+	3.54807i	−1.74689	+	0.141023i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	29.8555	+	62.7728i	1.18292	+	2.48715i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.600742	−	0.799443i	$-0.294872\pi$
$641$		0			0		−0.600742	+	0.799443i	$0.705128\pi$
$642$		0			0
$643$	−20.2154	+	30.5867i	−0.797217	+	1.20622i	0.177795	+	0.984068i	$0.443104\pi$
$643$	−20.2154	+	30.5867i	−0.797217	+	1.20622i	−0.975012	+	0.222154i	$0.928691\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.600742	−	0.799443i	$-0.705128\pi$
$647$		0			0		0.600742	+	0.799443i	$0.294872\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−52.8128	−	76.5126i	−2.06990	−	2.99877i
$652$	−38.3844	−	27.6524i	−1.50325	−	1.08295i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−5.21094	−	19.4475i	−0.203298	−	0.758719i
$658$		0			0
$659$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$659$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$660$		0			0
$661$	−2.91095	−	28.8116i	−0.113223	−	1.12064i	−0.878764	−	0.477256i	$-0.841632\pi$
$661$	−2.91095	−	28.8116i	−0.113223	−	1.12064i	0.765542	−	0.643386i	$-0.222471\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−12.6052	−	17.4973i	−0.487344	−	0.676485i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	6.71227	+	1.37032i	0.258739	+	0.0528220i	0.327644	−	0.944801i	$-0.393745\pi$
$673$	6.71227	+	1.37032i	0.258739	+	0.0528220i	−0.0689049	+	0.997623i	$0.521950\pi$
$674$		0			0
$675$		−	25.9808i		−	1.00000i
$676$	8.18882	−	24.6768i	0.314955	−	0.949107i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	−17.4396	+	85.4246i	−0.669269	+	3.27830i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.811378	−	0.584522i	$-0.198718\pi$
$683$		0			0		−0.811378	+	0.584522i	$0.801282\pi$
$684$	12.0747	+	5.43436i	0.461686	+	0.207788i
$685$		0			0
$686$		0			0
$687$	17.5815	−	43.6846i	0.670778	−	1.66667i
$688$	21.4535	−	40.8763i	0.817908	−	1.55839i
$689$		0			0
$690$		0			0
$691$	17.9705	−	1.81563i	0.683630	−	0.0690697i	0.247411	−	0.968911i	$-0.420420\pi$
$691$	17.9705	−	1.81563i	0.683630	−	0.0690697i	0.436218	+	0.899841i	$0.356318\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−29.9643	+	41.5936i	−1.13254	+	1.57209i
$701$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$701$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$702$		0			0
$703$	14.9966	−	21.7263i	0.565608	−	0.819425i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	12.2340	+	8.08570i	0.459457	+	0.303665i	0.759789	−	0.650170i	$-0.225302\pi$
$709$	12.2340	+	8.08570i	0.459457	+	0.303665i	−0.300331	+	0.953835i	$0.597097\pi$
$710$		0			0
$711$	−42.4403	−	31.8919i	−1.59164	−	1.19604i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.0804666	−	0.996757i	$-0.525641\pi$
$719$		0			0		0.0804666	+	0.996757i	$0.474359\pi$
$720$		0			0
$721$	3.99954	−	39.5862i	0.148951	−	1.47427i
$722$		0			0
$723$	24.4178	−	14.7611i	0.908107	−	0.548970i
$724$	35.0979	−	36.5408i	1.30440	−	1.35803i
$725$		0			0
$726$		0			0
$727$	35.9318	+	13.6271i	1.33264	+	0.505403i	0.915089	−	0.403252i	$-0.132120\pi$
$727$	35.9318	+	13.6271i	1.33264	+	0.505403i	0.417548	+	0.908655i	$0.362889\pi$
$728$		0			0
$729$	−9.57433	−	25.2454i	−0.354605	−	0.935016i
$730$		0			0
$731$		0			0
$732$	−17.9802	+	3.67069i	−0.664568	+	0.135672i
$733$	0.945119	+	15.6247i	0.0349088	+	0.577110i	0.973012	+	0.230755i	$0.0741195\pi$
$733$	0.945119	+	15.6247i	0.0349088	+	0.577110i	−0.938103	+	0.346356i	$0.887419\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	51.4562	−	11.5889i	1.89285	−	0.426305i	0.893124	−	0.449810i	$-0.148508\pi$
$739$	51.4562	−	11.5889i	1.89285	−	0.426305i	0.999724	+	0.0235054i	$0.00748270\pi$
$740$		0			0
$741$	−10.1382	−	9.33581i	−0.372436	−	0.342959i
$742$		0			0
$743$		0			0		0.482459	−	0.875918i	$-0.339744\pi$
$743$		0			0		−0.482459	+	0.875918i	$0.660256\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−3.46070	−	8.12255i	−0.126283	−	0.296396i	0.844823	−	0.535046i	$-0.179706\pi$
$751$	−3.46070	−	8.12255i	−0.126283	−	0.296396i	−0.971106	+	0.238649i	$0.923295\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−13.7883	+	51.4587i	−0.501477	+	1.87154i
$757$	−4.92735	−	5.12991i	−0.179088	−	0.186450i	0.625608	−	0.780138i	$-0.284851\pi$
$757$	−4.92735	−	5.12991i	−0.179088	−	0.186450i	−0.804696	+	0.593688i	$0.797671\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.990080	−	0.140502i	$-0.0448718\pi$
$761$		0			0		−0.990080	+	0.140502i	$0.955128\pi$
$762$		0			0
$763$	34.2018	+	72.0789i	1.23819	+	2.60943i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	26.9075	−	6.63211i	0.970942	−	0.239316i
$769$	1.10509	−	54.8671i	0.0398504	−	1.97856i	−0.124713	−	0.992193i	$-0.539801\pi$
$769$	1.10509	−	54.8671i	0.0398504	−	1.97856i	0.164563	−	0.986367i	$-0.447378\pi$
$770$		0			0
$771$		0			0
$772$	7.51857	−	5.89043i	0.270599	−	0.212001i
$773$		0			0		0.735006	−	0.678061i	$-0.237179\pi$
$773$		0			0		−0.735006	+	0.678061i	$0.762821\pi$
$774$		0			0
$775$	−37.0197	+	37.0197i	−1.32979	+	1.32979i
$776$		0			0
$777$	95.9594	+	45.5333i	3.44252	+	1.63350i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	59.7341	−	48.7714i	2.13336	−	1.74183i
$785$		0			0
$786$		0			0
$787$	18.7249	+	33.9957i	0.667472	+	1.21181i	0.966278	+	0.257500i	$0.0828988\pi$
$787$	18.7249	+	33.9957i	0.667472	+	1.21181i	−0.298806	+	0.954314i	$0.596588\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	18.9632	+	2.28518i	0.673403	+	0.0811491i
$794$		0			0
$795$		0			0
$796$	−46.2916	−	15.4544i	−1.64076	−	0.547767i
$797$		0			0		−0.721202	−	0.692724i	$-0.756410\pi$
$797$		0			0		0.721202	+	0.692724i	$0.243590\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	26.5657	+	40.1950i	0.936901	+	1.41757i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$809$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$810$		0			0
$811$	45.8937	+	14.3011i	1.61155	+	0.502178i	0.965343	−	0.260986i	$-0.0840475\pi$
$811$	45.8937	+	14.3011i	1.61155	+	0.502178i	0.646205	+	0.763164i	$0.276355\pi$
$812$		0			0
$813$	−1.86568	+	2.19336i	−0.0654323	+	0.0769245i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	8.55022	+	23.9914i	0.299134	+	0.839354i
$818$		0			0
$819$	30.5317	−	46.2865i	1.06686	−	1.61738i
$820$		0			0
$821$		0			0		−0.811378	−	0.584522i	$-0.801282\pi$
$821$		0			0		0.811378	+	0.584522i	$0.198718\pi$
$822$		0			0
$823$	−47.2362	−	27.2719i	−1.64655	−	0.950637i	−0.978428	−	0.206587i	$-0.933764\pi$
$823$	−47.2362	−	27.2719i	−1.64655	−	0.950637i	−0.668124	−	0.744050i	$-0.732902\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$827$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$828$		0			0
$829$	−19.4894	+	30.8200i	−0.676894	+	1.07042i	0.315690	+	0.948862i	$0.397764\pi$
$829$	−19.4894	+	30.8200i	−0.676894	+	1.07042i	−0.992584	+	0.121560i	$0.961210\pi$
$830$		0			0
$831$	1.35524	+	2.58219i	0.0470127	+	0.0895753i
$832$	−28.7530	−	2.29505i	−0.996830	−	0.0795666i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−22.3296	+	49.6143i	−0.771823	+	1.71492i
$838$		0			0
$839$		0			0		−0.941967	−	0.335705i	$-0.891026\pi$
$839$		0			0		0.941967	+	0.335705i	$0.108974\pi$
$840$		0			0
$841$	8.06831	−	27.8550i	0.278217	−	0.960518i
$842$		0			0
$843$		0			0
$844$		−	50.9151i		−	1.75257i
$845$		0			0
$846$		0			0
$847$	−55.8299	−	7.92283i	−1.91834	−	0.272232i
$848$		0			0
$849$	−54.2362	−	15.7097i	−1.86138	−	0.539156i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−17.7842	−	8.00403i	−0.608920	−	0.274053i	0.0824022	−	0.996599i	$-0.473741\pi$
$853$	−17.7842	−	8.00403i	−0.608920	−	0.274053i	−0.691322	+	0.722546i	$0.742972\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$857$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$858$		0			0
$859$	−45.5157	+	23.8885i	−1.55298	+	0.815065i	−0.999838	−	0.0180234i	$-0.994263\pi$
$859$	−45.5157	+	23.8885i	−1.55298	+	0.815065i	−0.553140	+	0.833089i	$0.686570\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$863$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−14.7224	+	25.5000i	−0.500000	+	0.866025i
$868$	92.9698	−	53.6761i	3.15560	−	1.82189i
$869$		0			0
$870$		0			0
$871$	−13.0230	−	48.4275i	−0.441267	−	1.64090i
$872$		0			0
$873$	48.0622	−	17.1287i	1.62666	−	0.579719i
$874$		0			0
$875$		0			0
$876$	22.8674	−	4.19060i	0.772618	−	0.141587i
$877$	−23.0576	−	19.6129i	−0.778600	−	0.662280i	0.167603	−	0.985855i	$-0.446397\pi$
$877$	−23.0576	−	19.6129i	−0.778600	−	0.662280i	−0.946203	+	0.323574i	$0.895115\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$881$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$882$		0			0
$883$	1.53887	−	0.186853i	0.0517872	−	0.00628810i	−0.0946021	−	0.995515i	$-0.530158\pi$
$883$	1.53887	−	0.186853i	0.0517872	−	0.00628810i	0.146389	+	0.989227i	$0.453235\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$887$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$888$		0			0
$889$	−30.2824	+	38.6527i	−1.01564	+	1.29637i
$890$		0			0
$891$		0			0
$892$	21.3098	−	12.8823i	0.713506	−	0.431330i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	29.9027	+	2.41400i	0.996757	+	0.0804666i
$901$		0			0
$902$		0			0
$903$	−89.7577	+	49.4389i	−2.98695	+	1.64522i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	14.3275	−	0.577378i	0.475736	−	0.0191715i	0.198760	−	0.980048i	$-0.436309\pi$
$907$	14.3275	−	0.577378i	0.475736	−	0.0191715i	0.276976	+	0.960877i	$0.410668\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$911$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$912$	−7.37662	+	13.3925i	−0.244264	+	0.443469i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	48.6455	+	24.2945i	1.60729	+	0.802714i
$917$		0			0
$918$		0			0
$919$	−21.2013	+	13.4069i	−0.699366	+	0.442253i	−0.836260	−	0.548333i	$-0.815263\pi$
$919$	−21.2013	+	13.4069i	−0.699366	+	0.442253i	0.136894	+	0.990586i	$0.456288\pi$
$920$		0			0
$921$	−27.6101	−	0.556098i	−0.909782	−	0.0183240i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	15.4805	−	57.7741i	0.508996	−	1.89960i
$926$		0			0
$927$	−21.0364	+	9.98188i	−0.690925	+	0.327848i
$928$		0			0
$929$		0			0		−0.140502	−	0.990080i	$-0.544872\pi$
$929$		0			0		0.140502	+	0.990080i	$0.455128\pi$
$930$		0			0
$931$	−2.56886	+	42.4683i	−0.0841910	+	1.39184i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−18.5737	+	4.57802i	−0.606778	+	0.149557i	−0.530721	−	0.847546i	$-0.678079\pi$
$937$	−18.5737	+	4.57802i	−0.606778	+	0.149557i	−0.0760561	+	0.997104i	$0.524233\pi$
$938$		0			0
$939$	25.5145	−	10.8707i	0.832635	−	0.354753i
$940$		0			0
$941$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$941$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.875918	−	0.482459i	$-0.839744\pi$
$947$		0			0		0.875918	+	0.482459i	$0.160256\pi$
$948$	40.6495	−	45.8838i	1.32023	−	1.49024i
$949$	−23.9605	−	3.37817i	−0.777791	−	0.109660i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.774605	−	0.632445i	$-0.217949\pi$
$953$		0			0		−0.774605	+	0.632445i	$0.782051\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	73.5264	−	27.8849i	2.37182	−	0.899513i
$962$		0			0
$963$		0			0
$964$	14.7206	+	29.4753i	0.474117	+	0.949336i
$965$		0			0
$966$		0			0
$967$	6.90659	+	11.4249i	0.222101	+	0.367400i	0.946883	−	0.321578i	$-0.104213\pi$
$967$	6.90659	+	11.4249i	0.222101	+	0.367400i	−0.724782	+	0.688978i	$0.758059\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$971$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$972$	29.9460	−	8.67396i	0.960518	−	0.278217i
$973$	41.1383	+	62.2439i	1.31883	+	1.99545i
$974$		0			0
$975$	−28.7374	−	12.2132i	−0.920333	−	0.391136i
$976$	−2.55417	−	21.0355i	−0.0817570	−	0.673330i
$977$		0			0		−0.647915	−	0.761712i	$-0.724359\pi$
$977$		0			0		0.647915	+	0.761712i	$0.275641\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	25.7436	−	38.9512i	0.821931	−	1.24362i
$982$		0			0
$983$		0			0		−0.180255	−	0.983620i	$-0.557692\pi$
$983$		0			0		0.180255	+	0.983620i	$0.442308\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	11.6871	−	10.8012i	0.371816	−	0.343631i
$989$		0			0
$990$		0			0
$991$	19.7007	+	34.1226i	0.625813	+	1.08394i	0.988383	+	0.151983i	$0.0485659\pi$
$991$	19.7007	+	34.1226i	0.625813	+	1.08394i	−0.362570	+	0.931956i	$0.618101\pi$
$992$		0			0
$993$	20.7739	−	9.34958i	0.659240	−	0.296700i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−14.7909	−	2.40376i	−0.468433	−	0.0761278i	−0.0783868	−	0.996923i	$-0.524977\pi$
$997$	−14.7909	−	2.40376i	−0.468433	−	0.0761278i	−0.390046	+	0.920795i	$0.627541\pi$
$998$		0			0
$999$	−6.24830	−	61.8437i	−0.197688	−	1.95665i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.a.128.1	✓	48
3.2	odd	2	CM	507.2.x.a.128.1	✓	48
169.136	odd	156	inner	507.2.x.a.305.1	yes	48
507.305	even	156	inner	507.2.x.a.305.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.a.128.1	✓	48	1.1	even	1	trivial
507.2.x.a.128.1	✓	48	3.2	odd	2	CM
507.2.x.a.305.1	yes	48	169.136	odd	156	inner
507.2.x.a.305.1	yes	48	507.305	even	156	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.x (of order \(156\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\(q+(0.346455 - 1.69705i) q^{3} +(1.92104 + 0.556435i) q^{4} +(4.15936 - 2.99643i) q^{7} +(-2.75994 - 1.17590i) q^{9} +O(q^{10})\)	\(q+(0.346455 - 1.69705i) q^{3} +(1.92104 + 0.556435i) q^{4} +(4.15936 - 2.99643i) q^{7} +(-2.75994 - 1.17590i) q^{9} +(1.60985 - 3.06731i) q^{12} +(-2.59808 + 2.50000i) q^{13} +(3.38076 + 2.13786i) q^{16} +(-2.13167 + 0.571179i) q^{19} +(-3.64406 - 8.09676i) q^{21} +(-4.11492 + 2.84032i) q^{25} +(-2.95175 + 4.27635i) q^{27} +(9.65760 - 3.44184i) q^{28} +(10.2992 - 1.88740i) q^{31} +(-4.64763 - 3.79467i) q^{36} +(-9.11192 + 7.75064i) q^{37} +(3.34250 + 5.27519i) q^{39} +(-0.928666 - 11.5036i) q^{43} +(4.79934 - 4.99664i) q^{48} +(6.10501 - 18.2867i) q^{49} +(-6.38209 + 3.35693i) q^{52} +(0.230791 + 3.81543i) q^{57} +(-3.35038 - 4.10347i) q^{61} +(-15.0031 + 3.37897i) q^{63} +(5.30498 + 5.98809i) q^{64} +(-6.71032 + 12.1828i) q^{67} +(4.13891 + 5.28293i) q^{73} +(3.39453 + 7.96726i) q^{75} +(-4.41284 - 0.0888796i) q^{76} +(17.1816 + 4.23488i) q^{79} +(6.23452 + 6.49082i) q^{81} +(-2.49504 - 17.5818i) q^{84} +(-3.31526 + 18.1834i) q^{91} +(0.365203 - 18.1322i) q^{93} +(-12.5008 + 11.5323i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \) 48 * q + 10 * q^7 + 6 * q^9	\( 48 q + 10 q^{7} + 6 q^{9} - 8 q^{16} - 14 q^{19} - 18 q^{21} + 20 q^{28} + 14 q^{31} + 2 q^{37} + 24 q^{39} + 6 q^{43} - 18 q^{49} - 28 q^{52} - 12 q^{57} - 24 q^{63} - 32 q^{67} + 34 q^{73} + 30 q^{75} + 28 q^{76} + 18 q^{81} + 12 q^{84} - 2 q^{91} - 6 q^{93} + 38 q^{97}+O(q^{100}) \) 48 * q + 10 * q^7 + 6 * q^9 - 8 * q^16 - 14 * q^19 - 18 * q^21 + 20 * q^28 + 14 * q^31 + 2 * q^37 + 24 * q^39 + 6 * q^43 - 18 * q^49 - 28 * q^52 - 12 * q^57 - 24 * q^63 - 32 * q^67 + 34 * q^73 + 30 * q^75 + 28 * q^76 + 18 * q^81 + 12 * q^84 - 2 * q^91 - 6 * q^93 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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