LMFDB - Embedded newform 2268.3.bg.c.2213.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,3,Mod(701,2268)] 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("2268.701"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2268.bg (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,8] 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:	$61.7985239569$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12745506816.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 $ x^8 - 8x^6 + 55x^4 - 72*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2213.3
Root			$1.00781 - 0.581861i$ of defining polynomial
Character	$\chi$	$=$	2268.2213
Dual form			2268.3.bg.c.701.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.01563 - 1.16372i) q^{5} +(-1.32288 + 2.29129i) q^{7} +(-7.70549 - 4.44876i) q^{11} +(-4.29150 - 7.43310i) q^{13} +20.1225i q^{17} +5.87451 q^{19} +(15.0540 - 8.69140i) q^{23} +(-9.79150 + 16.9594i) q^{25} +(-16.4820 - 9.51590i) q^{29} +(-2.35425 - 4.07768i) q^{31} +6.15784i q^{35} +39.7490 q^{37} +(6.04688 - 3.49117i) q^{41} +(17.8745 - 30.9596i) q^{43} +(63.5330 + 36.6808i) q^{47} +(-3.50000 - 6.06218i) q^{49} +46.6690i q^{53} -20.7085 q^{55} +(-20.8703 + 12.0495i) q^{59} +(36.6458 - 63.4723i) q^{61} +(-17.3001 - 9.98823i) q^{65} +(16.0405 + 27.7830i) q^{67} -114.843i q^{71} +41.2915 q^{73} +(20.3868 - 11.7703i) q^{77} +(13.3320 - 23.0917i) q^{79} +(117.828 + 68.0283i) q^{83} +(23.4170 + 40.5594i) q^{85} -145.753i q^{89} +22.7085 q^{91} +(11.8408 - 6.83629i) q^{95} +(32.9778 - 57.1192i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{13} - 80 q^{19} - 36 q^{25} - 40 q^{31} + 64 q^{37} + 16 q^{43} - 28 q^{49} - 208 q^{55} + 272 q^{61} - 168 q^{67} + 288 q^{73} - 232 q^{79} + 272 q^{85} + 224 q^{91} - 96 q^{97}+O(q^{100}) $ 8 * q + 8 * q^13 - 80 * q^19 - 36 * q^25 - 40 * q^31 + 64 * q^37 + 16 * q^43 - 28 * q^49 - 208 * q^55 + 272 * q^61 - 168 * q^67 + 288 * q^73 - 232 * q^79 + 272 * q^85 + 224 * q^91 - 96 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.01563	−	1.16372i	0.403125	−	0.232744i	−0.284706	−	0.958615i	$-0.591896\pi$
$5$	2.01563	−	1.16372i	0.403125	−	0.232744i	0.687832	+	0.725870i	$0.258563\pi$
$6$		0			0
$7$	−1.32288	+	2.29129i	−0.188982	+	0.327327i
$7$	−1.32288	+	2.29129i	−0.188982	+	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7.70549	−	4.44876i	−0.700499	−	0.404433i	0.107034	−	0.994255i	$-0.465864\pi$
$11$	−7.70549	−	4.44876i	−0.700499	−	0.404433i	−0.807533	+	0.589822i	$0.799198\pi$
$12$		0			0
$13$	−4.29150	−	7.43310i	−0.330116	−	0.571777i	0.652419	−	0.757859i	$-0.273754\pi$
$13$	−4.29150	−	7.43310i	−0.330116	−	0.571777i	−0.982534	+	0.186082i	$0.940421\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			20.1225i			1.18368i	0.806057	+	0.591838i	$0.201598\pi$
$17$			20.1225i			1.18368i	−0.806057	+	0.591838i	$0.798402\pi$
$18$		0			0
$19$	5.87451			0.309185			0.154592	−	0.987978i	$-0.450594\pi$
$19$	5.87451			0.309185			0.154592	+	0.987978i	$0.450594\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	15.0540	−	8.69140i	0.654520	−	0.377887i	−0.135666	−	0.990755i	$-0.543317\pi$
$23$	15.0540	−	8.69140i	0.654520	−	0.377887i	0.790186	+	0.612867i	$0.209984\pi$
$24$		0			0
$25$	−9.79150	+	16.9594i	−0.391660	+	0.678375i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−16.4820	−	9.51590i	−0.568346	−	0.328134i	0.188143	−	0.982142i	$-0.439753\pi$
$29$	−16.4820	−	9.51590i	−0.568346	−	0.328134i	−0.756488	+	0.654007i	$0.773087\pi$
$30$		0			0
$31$	−2.35425	−	4.07768i	−0.0759435	−	0.131538i	0.825553	−	0.564325i	$-0.190864\pi$
$31$	−2.35425	−	4.07768i	−0.0759435	−	0.131538i	−0.901496	+	0.432787i	$0.857530\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			6.15784i			0.175938i
$36$		0			0
$37$	39.7490			1.07430			0.537149	−	0.843487i	$-0.319501\pi$
$37$	39.7490			1.07430			0.537149	+	0.843487i	$0.319501\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.04688	−	3.49117i	0.147485	−	0.0851504i	−0.424442	−	0.905455i	$-0.639530\pi$
$41$	6.04688	−	3.49117i	0.147485	−	0.0851504i	0.571926	+	0.820305i	$0.306196\pi$
$42$		0			0
$43$	17.8745	−	30.9596i	0.415686	−	0.719990i	−0.579814	−	0.814749i	$-0.696875\pi$
$43$	17.8745	−	30.9596i	0.415686	−	0.719990i	0.995500	+	0.0947592i	$0.0302081\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	63.5330	+	36.6808i	1.35177	+	0.780443i	0.988497	−	0.151241i	$-0.0483270\pi$
$47$	63.5330	+	36.6808i	1.35177	+	0.780443i	0.363270	+	0.931684i	$0.381660\pi$
$48$		0			0
$49$	−3.50000	−	6.06218i	−0.0714286	−	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			46.6690i			0.880548i	0.897863	+	0.440274i	$0.145119\pi$
$53$			46.6690i			0.880548i	−0.897863	+	0.440274i	$0.854881\pi$
$54$		0			0
$55$	−20.7085			−0.376518
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−20.8703	+	12.0495i	−0.353734	+	0.204228i	−0.666328	−	0.745658i	$-0.732135\pi$
$59$	−20.8703	+	12.0495i	−0.353734	+	0.204228i	0.312595	+	0.949887i	$0.398802\pi$
$60$		0			0
$61$	36.6458	−	63.4723i	0.600750	−	1.04053i	−0.391958	−	0.919983i	$-0.628202\pi$
$61$	36.6458	−	63.4723i	0.600750	−	1.04053i	0.992708	−	0.120546i	$-0.0384646\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−17.3001	−	9.98823i	−0.266156	−	0.153665i
$66$		0			0
$67$	16.0405	+	27.7830i	0.239411	+	0.414672i	0.960545	−	0.278123i	$-0.0897124\pi$
$67$	16.0405	+	27.7830i	0.239411	+	0.414672i	−0.721135	+	0.692795i	$0.756379\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	114.843i		−	1.61751i	−0.588144	−	0.808756i	$-0.700141\pi$
$71$		−	114.843i		−	1.61751i	0.588144	−	0.808756i	$-0.299859\pi$
$72$		0			0
$73$	41.2915			0.565637			0.282819	−	0.959173i	$-0.408731\pi$
$73$	41.2915			0.565637			0.282819	+	0.959173i	$0.408731\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	20.3868	−	11.7703i	0.264764	−	0.152861i
$78$		0			0
$79$	13.3320	−	23.0917i	0.168760	−	0.292300i	−0.769224	−	0.638979i	$-0.779357\pi$
$79$	13.3320	−	23.0917i	0.168760	−	0.292300i	0.937984	+	0.346678i	$0.112690\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	117.828	+	68.0283i	1.41962	+	0.819618i	0.996265	−	0.0863461i	$-0.0275191\pi$
$83$	117.828	+	68.0283i	1.41962	+	0.819618i	0.423355	+	0.905964i	$0.360852\pi$
$84$		0			0
$85$	23.4170	+	40.5594i	0.275494	+	0.477170i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	145.753i		−	1.63767i	−0.574028	−	0.818836i	$-0.694620\pi$
$89$		−	145.753i		−	1.63767i	0.574028	−	0.818836i	$-0.305380\pi$
$90$		0			0
$91$	22.7085			0.249544
$92$		0			0
$93$		0			0
$94$		0			0
$95$	11.8408	−	6.83629i	0.124640	−	0.0719610i
$96$		0			0
$97$	32.9778	−	57.1192i	0.339977	−	0.588857i	−0.644451	−	0.764646i	$-0.722914\pi$
$97$	32.9778	−	57.1192i	0.339977	−	0.588857i	0.984428	+	0.175788i	$0.0562474\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−80.2456	−	46.3298i	−0.794511	−	0.458711i	0.0470373	−	0.998893i	$-0.485022\pi$
$101$	−80.2456	−	46.3298i	−0.794511	−	0.458711i	−0.841548	+	0.540182i	$0.818355\pi$
$102$		0			0
$103$	96.5608	+	167.248i	0.937483	+	1.62377i	0.770145	+	0.637869i	$0.220184\pi$
$103$	96.5608	+	167.248i	0.937483	+	1.62377i	0.167339	+	0.985899i	$0.446483\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			47.2015i			0.441135i	0.975372	+	0.220568i	$0.0707910\pi$
$107$			47.2015i			0.441135i	−0.975372	+	0.220568i	$0.929209\pi$
$108$		0			0
$109$	208.162			1.90974			0.954872	−	0.297018i	$-0.0959921\pi$
$109$	208.162			1.90974			0.954872	+	0.297018i	$0.0959921\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−107.981	+	62.3428i	−0.955583	+	0.551706i	−0.894811	−	0.446446i	$-0.852690\pi$
$113$	−107.981	+	62.3428i	−0.955583	+	0.551706i	−0.0607721	+	0.998152i	$0.519356\pi$
$114$		0			0
$115$	20.2288	−	35.0372i	0.175902	−	0.304672i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−46.1064	−	26.6196i	−0.387449	−	0.223694i
$120$		0			0
$121$	−20.9170	−	36.2293i	−0.172868	−	0.299416i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			103.764i			0.830116i
$126$		0			0
$127$	51.5791			0.406134			0.203067	−	0.979165i	$-0.434909\pi$
$127$	51.5791			0.406134			0.203067	+	0.979165i	$0.434909\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−153.604	+	88.6832i	−1.17255	+	0.676971i	−0.954279	−	0.298918i	$-0.903374\pi$
$131$	−153.604	+	88.6832i	−1.17255	+	0.676971i	−0.218269	+	0.975889i	$0.570041\pi$
$132$		0			0
$133$	−7.77124	+	13.4602i	−0.0584304	+	0.101204i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−139.264	−	80.4040i	−1.01652	−	0.586891i	−0.103429	−	0.994637i	$-0.532982\pi$
$137$	−139.264	−	80.4040i	−1.01652	−	0.586891i	−0.913095	+	0.407746i	$0.866315\pi$
$138$		0			0
$139$	37.4170	+	64.8081i	0.269187	+	0.466246i	0.968652	−	0.248421i	$-0.0799117\pi$
$139$	37.4170	+	64.8081i	0.269187	+	0.466246i	−0.699465	+	0.714667i	$0.746578\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			76.3675i			0.534039i
$144$		0			0
$145$	−44.2954			−0.305486
$146$		0			0
$147$		0			0
$148$		0			0
$149$	227.490	−	131.342i	1.52678	−	0.881487i	0.527286	−	0.849688i	$-0.323209\pi$
$149$	227.490	−	131.342i	1.52678	−	0.881487i	0.999494	−	0.0317997i	$-0.0101239\pi$
$150$		0			0
$151$	−16.9595	+	29.3747i	−0.112314	+	0.194534i	−0.916703	−	0.399569i	$-0.869160\pi$
$151$	−16.9595	+	29.3747i	−0.112314	+	0.194534i	0.804389	+	0.594104i	$0.202493\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.49057	−	5.47938i	−0.0612295	−	0.0353508i
$156$		0			0
$157$	−81.8118	−	141.702i	−0.521094	−	0.902561i	−0.999699	−	0.0245310i	$-0.992191\pi$
$157$	−81.8118	−	141.702i	−0.521094	−	0.902561i	0.478605	−	0.878030i	$-0.341143\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			45.9906i			0.285656i
$162$		0			0
$163$	280.745			1.72236			0.861181	−	0.508298i	$-0.169725\pi$
$163$	280.745			1.72236			0.861181	+	0.508298i	$0.169725\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−65.8833	+	38.0377i	−0.394511	+	0.227771i	−0.684113	−	0.729376i	$-0.739810\pi$
$167$	−65.8833	+	38.0377i	−0.394511	+	0.227771i	0.289602	+	0.957147i	$0.406477\pi$
$168$		0			0
$169$	47.6660	−	82.5600i	0.282047	−	0.488520i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	133.321	+	76.9730i	0.770642	+	0.444930i	0.833104	−	0.553117i	$-0.186562\pi$
$173$	133.321	+	76.9730i	0.770642	+	0.444930i	−0.0624615	+	0.998047i	$0.519895\pi$
$174$		0			0
$175$	−25.9059	−	44.8703i	−0.148034	−	0.256402i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			265.689i			1.48430i	0.670236	+	0.742148i	$0.266193\pi$
$179$			265.689i			1.48430i	−0.670236	+	0.742148i	$0.733807\pi$
$180$		0			0
$181$	203.328			1.12336			0.561680	−	0.827355i	$-0.310155\pi$
$181$	203.328			1.12336			0.561680	+	0.827355i	$0.310155\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	80.1191	−	46.2568i	0.433076	−	0.250037i
$186$		0			0
$187$	89.5203	−	155.054i	0.478718	−	0.829164i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−184.783	−	106.684i	−0.967449	−	0.558557i	−0.0689915	−	0.997617i	$-0.521978\pi$
$191$	−184.783	−	106.684i	−0.967449	−	0.558557i	−0.898458	+	0.439060i	$0.855311\pi$
$192$		0			0
$193$	−8.25098	−	14.2911i	−0.0427512	−	0.0740473i	0.843858	−	0.536567i	$-0.180279\pi$
$193$	−8.25098	−	14.2911i	−0.0427512	−	0.0740473i	−0.886609	+	0.462519i	$0.846946\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	57.0437i		−	0.289562i	−0.989464	−	0.144781i	$-0.953752\pi$
$197$		−	57.0437i		−	0.289562i	0.989464	−	0.144781i	$-0.0462477\pi$
$198$		0			0
$199$	302.243			1.51881			0.759405	−	0.650618i	$-0.225490\pi$
$199$	302.243			1.51881			0.759405	+	0.650618i	$0.225490\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	43.6073	−	25.1767i	0.214814	−	0.124023i
$204$		0			0
$205$	8.12549	−	14.0738i	0.0396365	−	0.0686525i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−45.2659	−	26.1343i	−0.216583	−	0.125044i
$210$		0			0
$211$	71.4980	+	123.838i	0.338853	+	0.586911i	0.984217	−	0.176964i	$-0.0566277\pi$
$211$	71.4980	+	123.838i	0.338853	+	0.586911i	−0.645364	+	0.763875i	$0.723294\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	83.2038i		−	0.386995i
$216$		0			0
$217$	12.4575			0.0574079
$218$		0			0
$219$		0			0
$220$		0			0
$221$	149.573	−	86.3558i	0.676799	−	0.390750i
$222$		0			0
$223$	83.8745	−	145.275i	0.376119	−	0.651457i	−0.614375	−	0.789014i	$-0.710592\pi$
$223$	83.8745	−	145.275i	0.376119	−	0.651457i	0.990494	+	0.137557i	$0.0439251\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−53.1203	−	30.6690i	−0.234010	−	0.135106i	0.378411	−	0.925638i	$-0.376471\pi$
$227$	−53.1203	−	30.6690i	−0.234010	−	0.135106i	−0.612421	+	0.790532i	$0.709804\pi$
$228$		0			0
$229$	22.7935	+	39.4795i	0.0995348	+	0.172399i	0.911492	−	0.411317i	$-0.134931\pi$
$229$	22.7935	+	39.4795i	0.0995348	+	0.172399i	−0.811957	+	0.583717i	$0.801598\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	299.390i		−	1.28494i	−0.766313	−	0.642468i	$-0.777911\pi$
$233$		−	299.390i		−	1.28494i	0.766313	−	0.642468i	$-0.222089\pi$
$234$		0			0
$235$	170.745			0.726575
$236$		0			0
$237$		0			0
$238$		0			0
$239$	127.215	−	73.4476i	0.532280	−	0.307312i	−0.209664	−	0.977773i	$-0.567237\pi$
$239$	127.215	−	73.4476i	0.532280	−	0.307312i	0.741944	+	0.670461i	$0.233904\pi$
$240$		0			0
$241$	176.808	−	306.240i	0.733642	−	1.27071i	−0.221674	−	0.975121i	$-0.571152\pi$
$241$	176.808	−	306.240i	0.733642	−	1.27071i	0.955316	−	0.295585i	$-0.0955146\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−14.1094	−	8.14605i	−0.0575893	−	0.0332492i
$246$		0			0
$247$	−25.2105	−	43.6658i	−0.102067	−	0.176785i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	289.908i		−	1.15501i	−0.816386	−	0.577506i	$-0.804026\pi$
$251$		−	289.908i		−	1.15501i	0.816386	−	0.577506i	$-0.195974\pi$
$252$		0			0
$253$	−154.664			−0.611320
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−420.001	+	242.488i	−1.63425	+	0.943532i	−0.651482	+	0.758664i	$0.725852\pi$
$257$	−420.001	+	242.488i	−1.63425	+	0.943532i	−0.982763	+	0.184867i	$0.940814\pi$
$258$		0			0
$259$	−52.5830	+	91.0764i	−0.203023	+	0.351646i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	181.719	+	104.915i	0.690945	+	0.398917i	0.803966	−	0.594675i	$-0.202719\pi$
$263$	181.719	+	104.915i	0.690945	+	0.398917i	−0.113021	+	0.993593i	$0.536053\pi$
$264$		0			0
$265$	54.3098	+	94.0673i	0.204943	+	0.354971i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			157.004i			0.583656i	0.956471	+	0.291828i	$0.0942636\pi$
$269$			157.004i			0.583656i	−0.956471	+	0.291828i	$0.905736\pi$
$270$		0			0
$271$	159.203			0.587463			0.293732	−	0.955888i	$-0.405103\pi$
$271$	159.203			0.587463			0.293732	+	0.955888i	$0.405103\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	150.897	−	87.1202i	0.548715	−	0.316801i
$276$		0			0
$277$	−13.7530	+	23.8208i	−0.0496496	+	0.0859957i	−0.889782	−	0.456385i	$-0.849144\pi$
$277$	−13.7530	+	23.8208i	−0.0496496	+	0.0859957i	0.840133	+	0.542381i	$0.182477\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	435.643	+	251.518i	1.55033	+	0.895083i	0.998114	+	0.0613811i	$0.0195505\pi$
$281$	435.643	+	251.518i	1.55033	+	0.895083i	0.552215	+	0.833702i	$0.313783\pi$
$282$		0			0
$283$	101.225	+	175.327i	0.357685	+	0.619528i	0.987574	−	0.157157i	$-0.0502329\pi$
$283$	101.225	+	175.327i	0.357685	+	0.619528i	−0.629889	+	0.776685i	$0.716900\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			18.4735i			0.0643676i
$288$		0			0
$289$	−115.915			−0.401090
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.86998	+	5.69844i	−0.0336860	+	0.0194486i	−0.516748	−	0.856137i	$-0.672858\pi$
$293$	−9.86998	+	5.69844i	−0.0336860	+	0.0194486i	0.483062	+	0.875586i	$0.339524\pi$
$294$		0			0
$295$	−28.0445	+	48.5744i	−0.0950660	+	0.164659i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−129.208	−	74.5984i	−0.432134	−	0.249493i
$300$		0			0
$301$	47.2915	+	81.9113i	0.157115	+	0.272131i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	170.582i		−	0.559285i
$306$		0			0
$307$	−9.96342			−0.0324541			−0.0162271	−	0.999868i	$-0.505165\pi$
$307$	−9.96342			−0.0324541			−0.0162271	+	0.999868i	$0.505165\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−132.317	+	76.3934i	−0.425457	+	0.245638i	−0.697410	−	0.716673i	$-0.745664\pi$
$311$	−132.317	+	76.3934i	−0.425457	+	0.245638i	0.271952	+	0.962311i	$0.412331\pi$
$312$		0			0
$313$	254.535	−	440.867i	0.813210	−	1.40852i	−0.0973968	−	0.995246i	$-0.531052\pi$
$313$	254.535	−	440.867i	0.813210	−	1.40852i	0.910606	−	0.413275i	$-0.135615\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	228.413	+	131.874i	0.720544	+	0.416006i	0.814953	−	0.579527i	$-0.196763\pi$
$317$	228.413	+	131.874i	0.720544	+	0.416006i	−0.0944088	+	0.995534i	$0.530096\pi$
$318$		0			0
$319$	84.6680	+	146.649i	0.265417	+	0.459715i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			118.210i			0.365975i
$324$		0			0
$325$	168.081			0.517172
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−168.093	+	97.0483i	−0.510920	+	0.294980i
$330$		0			0
$331$	48.2510	−	83.5732i	0.145773	−	0.252487i	−0.783888	−	0.620902i	$-0.786766\pi$
$331$	48.2510	−	83.5732i	0.145773	−	0.252487i	0.929661	+	0.368416i	$0.120100\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	64.6634	+	37.3334i	0.193025	+	0.111443i
$336$		0			0
$337$	106.708	+	184.825i	0.316642	+	0.548441i	0.979785	−	0.200052i	$-0.0641111\pi$
$337$	106.708	+	184.825i	0.316642	+	0.548441i	−0.663143	+	0.748493i	$0.730778\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			41.8940i			0.122856i
$342$		0			0
$343$	18.5203			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−111.298	+	64.2580i	−0.320744	+	0.185182i	−0.651724	−	0.758456i	$-0.725954\pi$
$347$	−111.298	+	64.2580i	−0.320744	+	0.185182i	0.330980	+	0.943638i	$0.392621\pi$
$348$		0			0
$349$	−186.520	+	323.063i	−0.534442	+	0.925681i	0.464748	+	0.885443i	$0.346145\pi$
$349$	−186.520	+	323.063i	−0.534442	+	0.925681i	−0.999190	+	0.0402378i	$0.987188\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	348.658	+	201.298i	0.987701	+	0.570249i	0.904586	−	0.426291i	$-0.140180\pi$
$353$	348.658	+	201.298i	0.987701	+	0.570249i	0.0831146	+	0.996540i	$0.473513\pi$
$354$		0			0
$355$	−133.646	−	231.481i	−0.376467	−	0.652060i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			379.296i			1.05653i	0.849078	+	0.528267i	$0.177158\pi$
$359$			379.296i			1.05653i	−0.849078	+	0.528267i	$0.822842\pi$
$360$		0			0
$361$	−326.490			−0.904405
$362$		0			0
$363$		0			0
$364$		0			0
$365$	83.2282	−	48.0518i	0.228022	−	0.131649i
$366$		0			0
$367$	82.8784	−	143.550i	0.225827	−	0.391144i	−0.730740	−	0.682655i	$-0.760825\pi$
$367$	82.8784	−	143.550i	0.225827	−	0.391144i	0.956567	+	0.291512i	$0.0941583\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−106.932	−	61.7373i	−0.288227	−	0.166408i
$372$		0			0
$373$	42.1660	+	73.0337i	0.113046	+	0.195801i	0.916997	−	0.398895i	$-0.130606\pi$
$373$	42.1660	+	73.0337i	0.113046	+	0.195801i	−0.803951	+	0.594695i	$0.797273\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			163.350i			0.433289i
$378$		0			0
$379$	537.652			1.41861			0.709304	−	0.704903i	$-0.249009\pi$
$379$	537.652			1.41861			0.709304	+	0.704903i	$0.249009\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	364.567	−	210.483i	0.951873	−	0.549564i	0.0582107	−	0.998304i	$-0.481460\pi$
$383$	364.567	−	210.483i	0.951873	−	0.549564i	0.893662	+	0.448740i	$0.148127\pi$
$384$		0			0
$385$	27.3948	−	47.4491i	0.0711552	−	0.123244i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−276.118	−	159.417i	−0.709816	−	0.409812i	0.101177	−	0.994868i	$-0.467739\pi$
$389$	−276.118	−	159.417i	−0.709816	−	0.409812i	−0.810993	+	0.585056i	$0.801072\pi$
$390$		0			0
$391$	174.893	+	302.923i	0.447296	+	0.774740i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	62.0591i		−	0.157112i
$396$		0			0
$397$	135.712			0.341845			0.170922	−	0.985284i	$-0.445325\pi$
$397$	135.712			0.341845			0.170922	+	0.985284i	$0.445325\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−369.417	+	213.283i	−0.921239	+	0.531877i	−0.884030	−	0.467430i	$-0.845180\pi$
$401$	−369.417	+	213.283i	−0.921239	+	0.531877i	−0.0372086	+	0.999308i	$0.511847\pi$
$402$		0			0
$403$	−20.2065	+	34.9987i	−0.0501403	+	0.0868455i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−306.285	−	176.834i	−0.752544	−	0.434482i
$408$		0			0
$409$	−270.557	−	468.618i	−0.661508	−	1.14577i	−0.980219	−	0.197914i	$-0.936583\pi$
$409$	−270.557	−	468.618i	−0.661508	−	1.14577i	0.318711	−	0.947852i	$-0.396750\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	63.7598i		−	0.154382i
$414$		0			0
$415$	316.664			0.763046
$416$		0			0
$417$		0			0
$418$		0			0
$419$	292.392	−	168.813i	0.697833	−	0.402894i	−0.108707	−	0.994074i	$-0.534671\pi$
$419$	292.392	−	168.813i	0.697833	−	0.402894i	0.806540	+	0.591180i	$0.201338\pi$
$420$		0			0
$421$	−183.162	+	317.246i	−0.435064	+	0.753553i	−0.997301	−	0.0734229i	$-0.976608\pi$
$421$	−183.162	+	317.246i	−0.435064	+	0.753553i	0.562237	+	0.826976i	$0.309941\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−341.265	−	197.030i	−0.802977	−	0.463599i
$426$		0			0
$427$	96.9555	+	167.932i	0.227062	+	0.393283i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	301.760i		−	0.700139i	−0.936724	−	0.350069i	$-0.886158\pi$
$431$		−	301.760i		−	0.700139i	0.936724	−	0.350069i	$-0.113842\pi$
$432$		0			0
$433$	−40.2431			−0.0929402			−0.0464701	−	0.998920i	$-0.514797\pi$
$433$	−40.2431			−0.0929402			−0.0464701	+	0.998920i	$0.514797\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	88.4346	−	51.0577i	0.202367	−	0.116837i
$438$		0			0
$439$	−395.409	+	684.869i	−0.900704	+	1.56007i	−0.0741221	+	0.997249i	$0.523615\pi$
$439$	−395.409	+	684.869i	−0.900704	+	1.56007i	−0.826582	+	0.562816i	$0.809718\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	166.813	+	96.3098i	0.376554	+	0.217404i	0.676318	−	0.736610i	$-0.263575\pi$
$443$	166.813	+	96.3098i	0.376554	+	0.217404i	−0.299764	+	0.954013i	$0.596908\pi$
$444$		0			0
$445$	−169.616	−	293.783i	−0.381159	−	0.660186i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			196.690i			0.438063i	0.975718	+	0.219032i	$0.0702897\pi$
$449$			196.690i			0.438063i	−0.975718	+	0.219032i	$0.929710\pi$
$450$		0			0
$451$	−62.1255			−0.137751
$452$		0			0
$453$		0			0
$454$		0			0
$455$	45.7718	−	26.4264i	0.100597	−	0.0580799i
$456$		0			0
$457$	324.664	−	562.335i	0.710425	−	1.23049i	−0.254273	−	0.967132i	$-0.581836\pi$
$457$	324.664	−	562.335i	0.710425	−	1.23049i	0.964698	−	0.263359i	$-0.0848304\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−732.876	−	423.126i	−1.58975	−	0.917844i	−0.993347	−	0.115161i	$-0.963261\pi$
$461$	−732.876	−	423.126i	−1.58975	−	0.917844i	−0.596406	−	0.802683i	$-0.703405\pi$
$462$		0			0
$463$	−1.95948	−	3.39392i	−0.00423214	−	0.00733028i	0.863902	−	0.503661i	$-0.168014\pi$
$463$	−1.95948	−	3.39392i	−0.00423214	−	0.00733028i	−0.868134	+	0.496330i	$0.834680\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			835.199i			1.78844i	0.447632	+	0.894218i	$0.352267\pi$
$467$			835.199i			1.78844i	−0.447632	+	0.894218i	$0.647733\pi$
$468$		0			0
$469$	−84.8784			−0.180977
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−275.464	+	159.039i	−0.582375	+	0.336235i
$474$		0			0
$475$	−57.5203	+	99.6280i	−0.121095	+	0.209743i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−353.530	−	204.111i	−0.738059	−	0.426118i	0.0833043	−	0.996524i	$-0.473453\pi$
$479$	−353.530	−	204.111i	−0.738059	−	0.426118i	−0.821363	+	0.570406i	$0.806786\pi$
$480$		0			0
$481$	−170.583	−	295.458i	−0.354642	−	0.614259i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	153.508i		−	0.316511i
$486$		0			0
$487$	−373.255			−0.766437			−0.383219	−	0.923658i	$-0.625184\pi$
$487$	−373.255			−0.766437			−0.383219	+	0.923658i	$0.625184\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	284.299	−	164.140i	0.579021	−	0.334298i	−0.181723	−	0.983350i	$-0.558167\pi$
$491$	284.299	−	164.140i	0.579021	−	0.334298i	0.760744	+	0.649052i	$0.224834\pi$
$492$		0			0
$493$	191.484	−	331.659i	0.388405	−	0.672737i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	263.139	+	151.924i	0.529455	+	0.305681i
$498$		0			0
$499$	−143.203	−	248.034i	−0.286979	−	0.497062i	0.686108	−	0.727500i	$-0.259318\pi$
$499$	−143.203	−	248.034i	−0.286979	−	0.497062i	−0.973087	+	0.230437i	$0.925984\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	798.493i		−	1.58746i	−0.608270	−	0.793730i	$-0.708136\pi$
$503$		−	798.493i		−	1.58746i	0.608270	−	0.793730i	$-0.291864\pi$
$504$		0			0
$505$	−215.660			−0.427050
$506$		0			0
$507$		0			0
$508$		0			0
$509$	732.921	−	423.152i	1.43992	−	0.831340i	0.442080	−	0.896976i	$-0.354241\pi$
$509$	732.921	−	423.152i	1.43992	−	0.831340i	0.997844	+	0.0656358i	$0.0209076\pi$
$510$		0			0
$511$	−54.6235	+	94.6107i	−0.106895	+	0.185148i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	389.261	+	224.740i	0.755846	+	0.436388i
$516$		0			0
$517$	−326.369	−	565.287i	−0.631274	−	1.09340i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	256.938i		−	0.493162i	−0.969122	−	0.246581i	$-0.920693\pi$
$521$		−	256.938i		−	0.493162i	0.969122	−	0.246581i	$-0.0793072\pi$
$522$		0			0
$523$	43.5138			0.0832003			0.0416002	−	0.999134i	$-0.486754\pi$
$523$	43.5138			0.0832003			0.0416002	+	0.999134i	$0.486754\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	82.0531	−	47.3734i	0.155698	−	0.0898925i
$528$		0			0
$529$	−113.419	+	196.447i	−0.214403	+	0.371356i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−51.9004	−	29.9647i	−0.0973741	−	0.0562189i
$534$		0			0
$535$	54.9294	+	95.1405i	0.102672	+	0.177833i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			62.2827i			0.115552i
$540$		0			0
$541$	−609.660			−1.12691			−0.563457	−	0.826146i	$-0.690529\pi$
$541$	−609.660			−1.12691			−0.563457	+	0.826146i	$0.690529\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	419.577	−	242.243i	0.769866	−	0.444482i
$546$		0			0
$547$	−501.915	+	869.342i	−0.917578	+	1.58929i	−0.114495	+	0.993424i	$0.536525\pi$
$547$	−501.915	+	869.342i	−0.917578	+	1.58929i	−0.803083	+	0.595867i	$0.796808\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−96.8238	−	55.9012i	−0.175724	−	0.101454i
$552$		0			0
$553$	35.2732	+	61.0950i	0.0637852	+	0.110479i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	10.4948i		−	0.0188416i	−0.999956	−	0.00942081i	$-0.997001\pi$
$557$		−	10.4948i		−	0.0188416i	0.999956	−	0.00942081i	$-0.00299878\pi$
$558$		0			0
$559$	−306.834			−0.548898
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−833.992	+	481.505i	−1.48134	+	0.855250i	−0.999776	−	0.0211638i	$-0.993263\pi$
$563$	−833.992	+	481.505i	−1.48134	+	0.855250i	−0.481560	+	0.876413i	$0.659930\pi$
$564$		0			0
$565$	−145.099	+	251.319i	−0.256813	+	0.444813i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	381.972	+	220.531i	0.671303	+	0.387577i	0.796570	−	0.604546i	$-0.206645\pi$
$569$	381.972	+	220.531i	0.671303	+	0.387577i	−0.125267	+	0.992123i	$0.539979\pi$
$570$		0			0
$571$	481.575	+	834.113i	0.843389	+	1.46079i	0.887013	+	0.461745i	$0.152776\pi$
$571$	481.575	+	834.113i	0.843389	+	1.46079i	−0.0436239	+	0.999048i	$0.513890\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			340.408i			0.592013i
$576$		0			0
$577$	−408.065			−0.707219			−0.353609	−	0.935393i	$-0.615046\pi$
$577$	−408.065			−0.707219			−0.353609	+	0.935393i	$0.615046\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−311.745	+	179.986i	−0.536566	+	0.309786i
$582$		0			0
$583$	207.620	−	359.608i	0.356123	−	0.616823i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	557.190	+	321.694i	0.949216	+	0.548030i	0.892837	−	0.450379i	$-0.148711\pi$
$587$	557.190	+	321.694i	0.949216	+	0.548030i	0.0563789	+	0.998409i	$0.482045\pi$
$588$		0			0
$589$	−13.8301	−	23.9544i	−0.0234806	−	0.0406695i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			95.5621i			0.161150i	0.996749	+	0.0805751i	$0.0256757\pi$
$593$			95.5621i			0.161150i	−0.996749	+	0.0805751i	$0.974324\pi$
$594$		0			0
$595$	−123.911			−0.208254
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−627.833	+	362.480i	−1.04814	+	0.605141i	−0.922127	−	0.386888i	$-0.873550\pi$
$599$	−627.833	+	362.480i	−1.04814	+	0.605141i	−0.126009	+	0.992029i	$0.540217\pi$
$600$		0			0
$601$	−499.620	+	865.367i	−0.831314	+	1.43988i	0.0656829	+	0.997841i	$0.479077\pi$
$601$	−499.620	+	865.367i	−0.831314	+	1.43988i	−0.896997	+	0.442037i	$0.854256\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−84.3217	−	48.6831i	−0.139375	−	0.0804680i
$606$		0			0
$607$	−231.793	−	401.478i	−0.381867	−	0.661414i	0.609462	−	0.792815i	$-0.291386\pi$
$607$	−231.793	−	401.478i	−0.381867	−	0.661414i	−0.991329	+	0.131402i	$0.958052\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	629.663i		−	1.03055i
$612$		0			0
$613$	−945.393			−1.54224			−0.771120	−	0.636690i	$-0.780303\pi$
$613$	−945.393			−1.54224			−0.771120	+	0.636690i	$0.780303\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−300.008	+	173.210i	−0.486237	+	0.280729i	−0.723012	−	0.690836i	$-0.757243\pi$
$617$	−300.008	+	173.210i	−0.486237	+	0.280729i	0.236775	+	0.971564i	$0.423910\pi$
$618$		0			0
$619$	569.292	−	986.042i	0.919695	−	1.59296i	0.119818	−	0.992796i	$-0.461769\pi$
$619$	569.292	−	986.042i	0.919695	−	1.59296i	0.799877	−	0.600163i	$-0.204898\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	333.961	+	192.813i	0.536054	+	0.309491i
$624$		0			0
$625$	−124.035	−	214.834i	−0.198455	−	0.343735i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			799.850i			1.27162i
$630$		0			0
$631$	−1006.22			−1.59464			−0.797321	−	0.603555i	$-0.793750\pi$
$631$	−1006.22			−1.59464			−0.797321	+	0.603555i	$0.793750\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	103.964	−	60.0237i	0.163723	−	0.0945255i
$636$		0			0
$637$	−30.0405	+	52.0317i	−0.0471594	+	0.0816824i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	103.741	+	59.8952i	0.161843	+	0.0934402i	0.578734	−	0.815516i	$-0.303547\pi$
$641$	103.741	+	59.8952i	0.161843	+	0.0934402i	−0.416891	+	0.908957i	$0.636880\pi$
$642$		0			0
$643$	467.808	+	810.267i	0.727539	+	1.26014i	0.957920	+	0.287035i	$0.0926695\pi$
$643$	467.808	+	810.267i	0.727539	+	1.26014i	−0.230381	+	0.973101i	$0.573997\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	571.143i		−	0.882755i	−0.897321	−	0.441378i	$-0.854490\pi$
$647$		−	571.143i		−	0.882755i	0.897321	−	0.441378i	$-0.145510\pi$
$648$		0			0
$649$	214.421			0.330387
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−532.601	+	307.497i	−0.815621	+	0.470899i	−0.848904	−	0.528547i	$-0.822737\pi$
$653$	−532.601	+	307.497i	−0.815621	+	0.470899i	0.0332829	+	0.999446i	$0.489404\pi$
$654$		0			0
$655$	−206.405	+	357.504i	−0.315122	+	0.545808i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−830.110	−	479.264i	−1.25965	−	0.727259i	−0.286644	−	0.958037i	$-0.592540\pi$
$659$	−830.110	−	479.264i	−1.25965	−	0.727259i	−0.973007	+	0.230778i	$0.925873\pi$
$660$		0			0
$661$	−397.893	−	689.171i	−0.601956	−	1.04262i	−0.992525	−	0.122044i	$-0.961055\pi$
$661$	−397.893	−	689.171i	−0.601956	−	1.04262i	0.390569	−	0.920574i	$-0.372278\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			36.1743i			0.0543974i
$666$		0			0
$667$	−330.826			−0.495991
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−564.747	+	326.057i	−0.841649	+	0.485926i
$672$		0			0
$673$	−63.7569	+	110.430i	−0.0947353	+	0.164086i	−0.909498	−	0.415708i	$-0.863534\pi$
$673$	−63.7569	+	110.430i	−0.0947353	+	0.164086i	0.814763	+	0.579794i	$0.196867\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−26.4113	−	15.2486i	−0.0390122	−	0.0225237i	0.480367	−	0.877068i	$-0.340503\pi$
$677$	−26.4113	−	15.2486i	−0.0390122	−	0.0225237i	−0.519379	+	0.854544i	$0.673837\pi$
$678$		0			0
$679$	87.2510	+	151.123i	0.128499	+	0.222567i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1094.05i		−	1.60183i	−0.598776	−	0.800917i	$-0.704346\pi$
$683$		−	1094.05i		−	1.60183i	0.598776	−	0.800917i	$-0.295654\pi$
$684$		0			0
$685$	−374.272			−0.546382
$686$		0			0
$687$		0			0
$688$		0			0
$689$	346.896	−	200.280i	0.503477	−	0.290683i
$690$		0			0
$691$	−55.8745	+	96.7775i	−0.0808604	+	0.140054i	−0.903620	−	0.428336i	$-0.859100\pi$
$691$	−55.8745	+	96.7775i	−0.0808604	+	0.140054i	0.822759	+	0.568390i	$0.192433\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	150.837	+	87.0860i	0.217032	+	0.125304i
$696$		0			0
$697$	70.2510	+	121.678i	0.100791	+	0.174574i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	71.8328i		−	0.102472i	−0.998687	−	0.0512360i	$-0.983684\pi$
$701$		−	71.8328i		−	0.102472i	0.998687	−	0.0512360i	$-0.0163161\pi$
$702$		0			0
$703$	233.506			0.332156
$704$		0			0
$705$		0			0
$706$		0			0
$707$	212.310	−	122.577i	0.300297	−	0.173376i
$708$		0			0
$709$	−90.9882	+	157.596i	−0.128333	+	0.222280i	−0.923031	−	0.384726i	$-0.874296\pi$
$709$	−90.9882	+	157.596i	−0.128333	+	0.222280i	0.794698	+	0.607005i	$0.207629\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−70.8815	−	40.9235i	−0.0994131	−	0.0573962i
$714$		0			0
$715$	88.8706	+	153.928i	0.124295	+	0.215284i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			733.616i			1.02033i	0.860077	+	0.510164i	$0.170415\pi$
$719$			733.616i			1.02033i	−0.860077	+	0.510164i	$0.829585\pi$
$720$		0			0
$721$	−510.952			−0.708671
$722$		0			0
$723$		0			0
$724$		0			0
$725$	322.767	−	186.350i	0.445197	−	0.257034i
$726$		0			0
$727$	65.2248	−	112.973i	0.0897178	−	0.155396i	−0.817674	−	0.575681i	$-0.804737\pi$
$727$	65.2248	−	112.973i	0.0897178	−	0.155396i	0.907392	+	0.420286i	$0.138070\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	622.984	+	359.680i	0.852235	+	0.492038i
$732$		0			0
$733$	−419.907	−	727.301i	−0.572861	−	0.992224i	−0.996270	−	0.0862860i	$-0.972500\pi$
$733$	−419.907	−	727.301i	−0.572861	−	0.992224i	0.423409	−	0.905938i	$-0.360833\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	285.442i		−	0.387302i
$738$		0			0
$739$	−260.907			−0.353054			−0.176527	−	0.984296i	$-0.556486\pi$
$739$	−260.907			−0.353054			−0.176527	+	0.984296i	$0.556486\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−124.448	+	71.8503i	−0.167495	+	0.0967030i	−0.581404	−	0.813615i	$-0.697496\pi$
$743$	−124.448	+	71.8503i	−0.167495	+	0.0967030i	0.413909	+	0.910318i	$0.364163\pi$
$744$		0			0
$745$	305.690	−	529.471i	0.410322	−	0.710699i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−108.152	−	62.4417i	−0.144395	−	0.0833667i
$750$		0			0
$751$	44.0732	+	76.3370i	0.0586860	+	0.101647i	0.893876	−	0.448315i	$-0.147976\pi$
$751$	44.0732	+	76.3370i	0.0586860	+	0.101647i	−0.835190	+	0.549962i	$0.814642\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			78.9445i			0.104562i
$756$		0			0
$757$	−519.660			−0.686473			−0.343237	−	0.939249i	$-0.611523\pi$
$757$	−519.660			−0.686473			−0.343237	+	0.939249i	$0.611523\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	221.637	−	127.962i	0.291245	−	0.168150i	−0.347258	−	0.937769i	$-0.612887\pi$
$761$	221.637	−	127.962i	0.291245	−	0.168150i	0.638503	+	0.769619i	$0.279554\pi$
$762$		0			0
$763$	−275.373	+	476.959i	−0.360908	+	0.625110i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	179.130	+	103.421i	0.233546	+	0.134838i
$768$		0			0
$769$	−601.316	−	1041.51i	−0.781946	−	1.35437i	−0.930807	−	0.365512i	$-0.880894\pi$
$769$	−601.316	−	1041.51i	−0.781946	−	1.35437i	0.148861	−	0.988858i	$-0.452439\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			100.613i			0.130158i	0.997880	+	0.0650792i	$0.0207300\pi$
$773$			100.613i			0.130158i	−0.997880	+	0.0650792i	$0.979270\pi$
$774$		0			0
$775$	92.2065			0.118976
$776$		0			0
$777$		0			0
$778$		0			0
$779$	35.5224	−	20.5089i	0.0456000	−	0.0263272i
$780$		0			0
$781$	−510.911	+	884.924i	−0.654176	+	1.13307i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−329.804	−	190.412i	−0.420132	−	0.242563i
$786$		0			0
$787$	−368.907	−	638.966i	−0.468751	−	0.811901i	0.530611	−	0.847616i	$-0.321963\pi$
$787$	−368.907	−	638.966i	−0.468751	−	0.811901i	−0.999362	+	0.0357147i	$0.988629\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	329.887i		−	0.417051i
$792$		0			0
$793$	−629.061			−0.793268
$794$		0			0
$795$		0			0
$796$		0			0
$797$	647.045	−	373.571i	0.811850	−	0.468722i	−0.0357477	−	0.999361i	$-0.511381\pi$
$797$	647.045	−	373.571i	0.811850	−	0.468722i	0.847598	+	0.530639i	$0.178048\pi$
$798$		0			0
$799$	−738.110	+	1278.44i	−0.923792	+	1.60005i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−318.171	−	183.696i	−0.396228	−	0.228762i
$804$		0			0
$805$	53.5203	+	92.6998i	0.0664848	+	0.115155i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1156.21i			1.42919i	0.699538	+	0.714595i	$0.253389\pi$
$809$			1156.21i			1.42919i	−0.699538	+	0.714595i	$0.746611\pi$
$810$		0			0
$811$	−194.081			−0.239311			−0.119655	−	0.992815i	$-0.538179\pi$
$811$	−194.081			−0.239311			−0.119655	+	0.992815i	$0.538179\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	565.877	−	326.709i	0.694328	−	0.400870i
$816$		0			0
$817$	105.004	−	181.872i	0.128524	−	0.222610i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−93.8794	−	54.2013i	−0.114348	−	0.0660187i	0.441735	−	0.897145i	$-0.354363\pi$
$821$	−93.8794	−	54.2013i	−0.114348	−	0.0660187i	−0.556083	+	0.831127i	$0.687696\pi$
$822$		0			0
$823$	546.409	+	946.408i	0.663924	+	1.14995i	0.979576	+	0.201074i	$0.0644433\pi$
$823$	546.409	+	946.408i	0.663924	+	1.14995i	−0.315652	+	0.948875i	$0.602223\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			18.9283i			0.0228879i	0.999935	+	0.0114440i	$0.00364281\pi$
$827$			18.9283i			0.0228879i	−0.999935	+	0.0114440i	$0.996357\pi$
$828$		0			0
$829$	18.1778			0.0219274			0.0109637	−	0.999940i	$-0.496510\pi$
$829$	18.1778			0.0219274			0.0109637	+	0.999940i	$0.496510\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	121.986	−	70.4288i	0.146442	−	0.0845483i
$834$		0			0
$835$	−88.5307	+	153.340i	−0.106025	+	0.183640i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	257.613	+	148.733i	0.307047	+	0.177274i	0.645604	−	0.763672i	$-0.276606\pi$
$839$	257.613	+	148.733i	0.307047	+	0.177274i	−0.338557	+	0.940946i	$0.609939\pi$
$840$		0			0
$841$	−239.395	−	414.645i	−0.284656	−	0.493038i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	221.880i		−	0.262580i
$846$		0			0
$847$	110.682			0.130676
$848$		0			0
$849$		0			0
$850$		0			0
$851$	598.380	−	345.475i	0.703149	−	0.405963i
$852$		0			0
$853$	−140.063	+	242.596i	−0.164200	+	0.284403i	−0.936371	−	0.351012i	$-0.885838\pi$
$853$	−140.063	+	242.596i	−0.164200	+	0.284403i	0.772171	+	0.635415i	$0.219171\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−370.867	−	214.120i	−0.432750	−	0.249849i	0.267767	−	0.963484i	$-0.413714\pi$
$857$	−370.867	−	214.120i	−0.432750	−	0.249849i	−0.700518	+	0.713635i	$0.747048\pi$
$858$		0			0
$859$	267.549	+	463.408i	0.311466	+	0.539474i	0.978680	−	0.205392i	$-0.0658468\pi$
$859$	267.549	+	463.408i	0.311466	+	0.539474i	−0.667214	+	0.744866i	$0.732513\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1628.73i			1.88729i	0.330964	+	0.943643i	$0.392626\pi$
$863$			1628.73i			1.88729i	−0.330964	+	0.943643i	$0.607374\pi$
$864$		0			0
$865$	358.301			0.414220
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−205.459	+	118.622i	−0.236432	+	0.136504i
$870$		0			0
$871$	137.676	−	238.462i	0.158066	−	0.273779i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−237.754	−	137.267i	−0.271719	−	0.156877i
$876$		0			0
$877$	−310.705	−	538.156i	−0.354281	−	0.613633i	0.632714	−	0.774386i	$-0.281941\pi$
$877$	−310.705	−	538.156i	−0.354281	−	0.613633i	−0.986995	+	0.160753i	$0.948608\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1011.60i			1.14824i	0.818773	+	0.574118i	$0.194655\pi$
$881$			1011.60i			1.14824i	−0.818773	+	0.574118i	$0.805345\pi$
$882$		0			0
$883$	400.235			0.453268			0.226634	−	0.973980i	$-0.427228\pi$
$883$	400.235			0.453268			0.226634	+	0.973980i	$0.427228\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	396.030	−	228.648i	0.446482	−	0.257777i	−0.259861	−	0.965646i	$-0.583677\pi$
$887$	396.030	−	228.648i	0.446482	−	0.257777i	0.706343	+	0.707869i	$0.250344\pi$
$888$		0			0
$889$	−68.2327	+	118.182i	−0.0767522	+	0.132939i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	373.225	+	215.482i	0.417945	+	0.241301i
$894$		0			0
$895$	309.188	+	535.530i	0.345462	+	0.598357i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			89.6112i			0.0996787i
$900$		0			0
$901$	−939.098			−1.04228
$902$		0			0
$903$		0			0
$904$		0			0
$905$	409.833	−	236.617i	0.452854	−	0.261456i
$906$		0			0
$907$	−543.793	+	941.878i	−0.599552	+	1.03845i	0.393335	+	0.919395i	$0.371321\pi$
$907$	−543.793	+	941.878i	−0.599552	+	1.03845i	−0.992887	+	0.119059i	$0.962012\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	641.310	+	370.261i	0.703963	+	0.406433i	0.808822	−	0.588054i	$-0.200106\pi$
$911$	641.310	+	370.261i	0.703963	+	0.406433i	−0.104859	+	0.994487i	$0.533439\pi$
$912$		0			0
$913$	−605.284	−	1048.38i	−0.662961	−	1.14828i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	469.267i		−	0.511742i
$918$		0			0
$919$	1079.91			1.17509			0.587547	−	0.809190i	$-0.300094\pi$
$919$	1079.91			1.17509			0.587547	+	0.809190i	$0.300094\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−853.642	+	492.851i	−0.924856	+	0.533966i
$924$		0			0
$925$	−389.203	+	674.119i	−0.420760	+	0.728777i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	126.389	+	72.9707i	0.136048	+	0.0785475i	0.566479	−	0.824076i	$-0.308305\pi$
$929$	126.389	+	72.9707i	0.136048	+	0.0785475i	−0.430431	+	0.902623i	$0.641638\pi$
$930$		0			0
$931$	−20.5608	−	35.6123i	−0.0220846	−	0.0382517i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	416.707i		−	0.445676i
$936$		0			0
$937$	−406.972			−0.434336			−0.217168	−	0.976134i	$-0.569682\pi$
$937$	−406.972			−0.434336			−0.217168	+	0.976134i	$0.569682\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	359.026	−	207.284i	0.381537	−	0.220281i	−0.296950	−	0.954893i	$-0.595969\pi$
$941$	359.026	−	207.284i	0.381537	−	0.220281i	0.678487	+	0.734613i	$0.262636\pi$
$942$		0			0
$943$	60.6863	−	105.112i	0.0643545	−	0.111465i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−582.910	−	336.543i	−0.615533	−	0.355378i	0.159595	−	0.987183i	$-0.448981\pi$
$947$	−582.910	−	336.543i	−0.615533	−	0.355378i	−0.775128	+	0.631805i	$0.782315\pi$
$948$		0			0
$949$	−177.203	−	306.924i	−0.186726	−	0.323418i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	68.0024i		−	0.0713562i	−0.999363	−	0.0356781i	$-0.988641\pi$
$953$		−	68.0024i		−	0.0713562i	0.999363	−	0.0356781i	$-0.0113591\pi$
$954$		0			0
$955$	−496.604			−0.520004
$956$		0			0
$957$		0			0
$958$		0			0
$959$	368.458	−	212.729i	0.384210	−	0.221824i
$960$		0			0
$961$	469.415	−	813.051i	0.488465	−	0.846046i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−33.2618	−	19.2037i	−0.0344682	−	0.0199002i
$966$		0			0
$967$	604.624	+	1047.24i	0.625257	+	1.08298i	0.988491	+	0.151279i	$0.0483392\pi$
$967$	604.624	+	1047.24i	0.625257	+	1.08298i	−0.363234	+	0.931698i	$0.618327\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1441.24i		−	1.48429i	−0.670240	−	0.742144i	$-0.733809\pi$
$971$		−	1441.24i		−	1.48429i	0.670240	−	0.742144i	$-0.266191\pi$
$972$		0			0
$973$	−197.992			−0.203486
$974$		0			0
$975$		0			0
$976$		0			0
$977$	41.7998	−	24.1332i	0.0427839	−	0.0247013i	−0.478456	−	0.878112i	$-0.658803\pi$
$977$	41.7998	−	24.1332i	0.0427839	−	0.0247013i	0.521239	+	0.853411i	$0.325470\pi$
$978$		0			0
$979$	−648.420	+	1123.10i	−0.662328	+	1.14719i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.0513	+	9.26720i	0.0163289	+	0.00942747i	0.508142	−	0.861273i	$-0.330332\pi$
$983$	16.0513	+	9.26720i	0.0163289	+	0.00942747i	−0.491813	+	0.870701i	$0.663666\pi$
$984$		0			0
$985$	−66.3830	−	114.979i	−0.0673939	−	0.116730i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	621.418i		−	0.628330i
$990$		0			0
$991$	77.2549			0.0779565			0.0389783	−	0.999240i	$-0.487590\pi$
$991$	77.2549			0.0779565			0.0389783	+	0.999240i	$0.487590\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	609.209	−	351.727i	0.612270	−	0.353494i
$996$		0			0
$997$	758.674	−	1314.06i	0.760957	−	1.31802i	−0.181400	−	0.983409i	$-0.558063\pi$
$997$	758.674	−	1314.06i	0.760957	−	1.31802i	0.942358	−	0.334607i	$-0.108604\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.3.bg.c.2213.3		8
3.2	odd	2	inner	2268.3.bg.c.2213.2		8
9.2	odd	6		252.3.c.a.197.2	✓	4
9.4	even	3	inner	2268.3.bg.c.701.2		8
9.5	odd	6	inner	2268.3.bg.c.701.3		8
9.7	even	3		252.3.c.a.197.3	yes	4
36.7	odd	6		1008.3.d.c.449.3		4
36.11	even	6		1008.3.d.c.449.2		4
63.2	odd	6		1764.3.bk.d.557.2		8
63.11	odd	6		1764.3.bk.d.1745.3		8
63.16	even	3		1764.3.bk.d.557.3		8
63.20	even	6		1764.3.c.f.197.3		4
63.25	even	3		1764.3.bk.d.1745.2		8
63.34	odd	6		1764.3.c.f.197.2		4
63.38	even	6		1764.3.bk.e.1745.2		8
63.47	even	6		1764.3.bk.e.557.3		8
63.52	odd	6		1764.3.bk.e.1745.3		8
63.61	odd	6		1764.3.bk.e.557.2		8
72.11	even	6		4032.3.d.e.449.3		4
72.29	odd	6		4032.3.d.h.449.3		4
72.43	odd	6		4032.3.d.e.449.2		4
72.61	even	6		4032.3.d.h.449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.c.a.197.2	✓	4	9.2	odd	6
252.3.c.a.197.3	yes	4	9.7	even	3
1008.3.d.c.449.2		4	36.11	even	6
1008.3.d.c.449.3		4	36.7	odd	6
1764.3.c.f.197.2		4	63.34	odd	6
1764.3.c.f.197.3		4	63.20	even	6
1764.3.bk.d.557.2		8	63.2	odd	6
1764.3.bk.d.557.3		8	63.16	even	3
1764.3.bk.d.1745.2		8	63.25	even	3
1764.3.bk.d.1745.3		8	63.11	odd	6
1764.3.bk.e.557.2		8	63.61	odd	6
1764.3.bk.e.557.3		8	63.47	even	6
1764.3.bk.e.1745.2		8	63.38	even	6
1764.3.bk.e.1745.3		8	63.52	odd	6
2268.3.bg.c.701.2		8	9.4	even	3	inner
2268.3.bg.c.701.3		8	9.5	odd	6	inner
2268.3.bg.c.2213.2		8	3.2	odd	2	inner
2268.3.bg.c.2213.3		8	1.1	even	1	trivial
4032.3.d.e.449.2		4	72.43	odd	6
4032.3.d.e.449.3		4	72.11	even	6
4032.3.d.h.449.2		4	72.61	even	6
4032.3.d.h.449.3		4	72.29	odd	6

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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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2268.bg (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.01563 - 1.16372i) q^{5} +(-1.32288 + 2.29129i) q^{7} +(-7.70549 - 4.44876i) q^{11} +(-4.29150 - 7.43310i) q^{13} +20.1225i q^{17} +5.87451 q^{19} +(15.0540 - 8.69140i) q^{23} +(-9.79150 + 16.9594i) q^{25} +(-16.4820 - 9.51590i) q^{29} +(-2.35425 - 4.07768i) q^{31} +6.15784i q^{35} +39.7490 q^{37} +(6.04688 - 3.49117i) q^{41} +(17.8745 - 30.9596i) q^{43} +(63.5330 + 36.6808i) q^{47} +(-3.50000 - 6.06218i) q^{49} +46.6690i q^{53} -20.7085 q^{55} +(-20.8703 + 12.0495i) q^{59} +(36.6458 - 63.4723i) q^{61} +(-17.3001 - 9.98823i) q^{65} +(16.0405 + 27.7830i) q^{67} -114.843i q^{71} +41.2915 q^{73} +(20.3868 - 11.7703i) q^{77} +(13.3320 - 23.0917i) q^{79} +(117.828 + 68.0283i) q^{83} +(23.4170 + 40.5594i) q^{85} -145.753i q^{89} +22.7085 q^{91} +(11.8408 - 6.83629i) q^{95} +(32.9778 - 57.1192i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{13} - 80 q^{19} - 36 q^{25} - 40 q^{31} + 64 q^{37} + 16 q^{43} - 28 q^{49} - 208 q^{55} + 272 q^{61} - 168 q^{67} + 288 q^{73} - 232 q^{79} + 272 q^{85} + 224 q^{91} - 96 q^{97}+O(q^{100}) \) 8 * q + 8 * q^13 - 80 * q^19 - 36 * q^25 - 40 * q^31 + 64 * q^37 + 16 * q^43 - 28 * q^49 - 208 * q^55 + 272 * q^61 - 168 * q^67 + 288 * q^73 - 232 * q^79 + 272 * q^85 + 224 * q^91 - 96 * q^97

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