LMFDB - Embedded newform 2160.2.w.d.1457.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,2,Mod(593,2160)] 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("2160.593"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.w (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,4,0,0,0,0,0,-4] 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:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.12745506816.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{4} + 1 $ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1457.1
Root			$-0.323042 - 0.323042i$ of defining polynomial
Character	$\chi$	$=$	2160.1457
Dual form			2160.2.w.d.593.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.87083i) q^{5} +(2.79129 + 2.79129i) q^{7} +1.80341i q^{11} +(-2.79129 + 2.79129i) q^{13} +(1.87083 - 1.87083i) q^{17} +3.00000i q^{19} +(-0.578661 - 0.578661i) q^{23} +(-2.00000 + 4.58258i) q^{25} -6.83723 q^{29} +1.00000 q^{31} +(1.80341 - 8.64064i) q^{35} +(-5.00000 - 5.00000i) q^{37} -1.80341i q^{41} +(-7.79129 + 7.79129i) q^{43} +(-3.09557 + 3.09557i) q^{47} +8.58258i q^{49} +(7.41589 + 7.41589i) q^{53} +(3.37386 - 2.20871i) q^{55} -6.83723 q^{59} -1.00000 q^{61} +(8.64064 + 1.80341i) q^{65} +(0.582576 + 0.582576i) q^{67} +1.80341i q^{71} +(3.37386 - 3.37386i) q^{73} +(-5.03383 + 5.03383i) q^{77} -1.41742i q^{79} +(6.25857 + 6.25857i) q^{83} +(-5.79129 - 1.20871i) q^{85} -5.41022 q^{89} -15.5826 q^{91} +(5.61249 - 3.67423i) q^{95} +(5.58258 + 5.58258i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7} - 4 q^{13} - 16 q^{25} + 8 q^{31} - 40 q^{37} - 44 q^{43} - 28 q^{55} - 8 q^{61} - 32 q^{67} - 28 q^{73} - 28 q^{85} - 88 q^{91} + 8 q^{97}+O(q^{100}) $ 8 * q + 4 * q^7 - 4 * q^13 - 16 * q^25 + 8 * q^31 - 40 * q^37 - 44 * q^43 - 28 * q^55 - 8 * q^61 - 32 * q^67 - 28 * q^73 - 28 * q^85 - 88 * q^91 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.22474	−	1.87083i	−0.547723	−	0.836660i
$5$	−1.22474	−	1.87083i	−0.547723	−	0.836660i
$6$		0			0
$7$	2.79129	+	2.79129i	1.05501	+	1.05501i	0.998396	+	0.0566113i	$0.0180296\pi$
$7$	2.79129	+	2.79129i	1.05501	+	1.05501i	0.0566113	+	0.998396i	$0.481970\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.80341i			0.543747i	0.962333	+	0.271874i	$0.0876433\pi$
$11$			1.80341i			0.543747i	−0.962333	+	0.271874i	$0.912357\pi$
$12$		0			0
$13$	−2.79129	+	2.79129i	−0.774164	+	0.774164i	−0.978832	−	0.204668i	$-0.934389\pi$
$13$	−2.79129	+	2.79129i	−0.774164	+	0.774164i	0.204668	+	0.978832i	$0.434389\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.87083	−	1.87083i	0.453743	−	0.453743i	−0.442852	−	0.896595i	$-0.646033\pi$
$17$	1.87083	−	1.87083i	0.453743	−	0.453743i	0.896595	+	0.442852i	$0.146033\pi$
$18$		0			0
$19$			3.00000i			0.688247i	0.938924	+	0.344124i	$0.111824\pi$
$19$			3.00000i			0.688247i	−0.938924	+	0.344124i	$0.888176\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.578661	−	0.578661i	−0.120659	−	0.120659i	0.644199	−	0.764858i	$-0.277191\pi$
$23$	−0.578661	−	0.578661i	−0.120659	−	0.120659i	−0.764858	+	0.644199i	$0.777191\pi$
$24$		0			0
$25$	−2.00000	+	4.58258i	−0.400000	+	0.916515i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.83723			−1.26964			−0.634821	−	0.772659i	$-0.718926\pi$
$29$	−6.83723			−1.26964			−0.634821	+	0.772659i	$0.718926\pi$
$30$		0			0
$31$	1.00000			0.179605			0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000			0.179605			0.0898027	+	0.995960i	$0.471376\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.80341	−	8.64064i	0.304831	−	1.46053i
$36$		0			0
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.80341i		−	0.281645i	−0.990035	−	0.140822i	$-0.955025\pi$
$41$		−	1.80341i		−	0.281645i	0.990035	−	0.140822i	$-0.0449746\pi$
$42$		0			0
$43$	−7.79129	+	7.79129i	−1.18816	+	1.18816i	−0.210585	+	0.977576i	$0.567537\pi$
$43$	−7.79129	+	7.79129i	−1.18816	+	1.18816i	−0.977576	+	0.210585i	$0.932463\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.09557	+	3.09557i	−0.451536	+	0.451536i	−0.895864	−	0.444328i	$-0.853442\pi$
$47$	−3.09557	+	3.09557i	−0.451536	+	0.451536i	0.444328	+	0.895864i	$0.353442\pi$
$48$		0			0
$49$			8.58258i			1.22608i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.41589	+	7.41589i	1.01865	+	1.01865i	0.999823	+	0.0188284i	$0.00599361\pi$
$53$	7.41589	+	7.41589i	1.01865	+	1.01865i	0.0188284	+	0.999823i	$0.494006\pi$
$54$		0			0
$55$	3.37386	−	2.20871i	0.454932	−	0.297823i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.83723			−0.890132			−0.445066	−	0.895498i	$-0.646820\pi$
$59$	−6.83723			−0.890132			−0.445066	+	0.895498i	$0.646820\pi$
$60$		0			0
$61$	−1.00000			−0.128037			−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000			−0.128037			−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	8.64064	+	1.80341i	1.07174	+	0.223685i
$66$		0			0
$67$	0.582576	+	0.582576i	0.0711729	+	0.0711729i	0.741797	−	0.670624i	$-0.233974\pi$
$67$	0.582576	+	0.582576i	0.0711729	+	0.0711729i	−0.670624	+	0.741797i	$0.733974\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			1.80341i			0.214025i	0.994258	+	0.107012i	$0.0341285\pi$
$71$			1.80341i			0.214025i	−0.994258	+	0.107012i	$0.965872\pi$
$72$		0			0
$73$	3.37386	−	3.37386i	0.394881	−	0.394881i	−0.481542	−	0.876423i	$-0.659923\pi$
$73$	3.37386	−	3.37386i	0.394881	−	0.394881i	0.876423	+	0.481542i	$0.159923\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.03383	+	5.03383i	−0.573658	+	0.573658i
$78$		0			0
$79$		−	1.41742i		−	0.159473i	−0.996816	−	0.0797363i	$-0.974592\pi$
$79$		−	1.41742i		−	0.159473i	0.996816	−	0.0797363i	$-0.0254078\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.25857	+	6.25857i	0.686967	+	0.686967i	0.961560	−	0.274593i	$-0.0885432\pi$
$83$	6.25857	+	6.25857i	0.686967	+	0.686967i	−0.274593	+	0.961560i	$0.588543\pi$
$84$		0			0
$85$	−5.79129	−	1.20871i	−0.628153	−	0.131103i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.41022			−0.573482			−0.286741	−	0.958008i	$-0.592572\pi$
$89$	−5.41022			−0.573482			−0.286741	+	0.958008i	$0.592572\pi$
$90$		0			0
$91$	−15.5826			−1.63350
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.61249	−	3.67423i	0.575829	−	0.376969i
$96$		0			0
$97$	5.58258	+	5.58258i	0.566825	+	0.566825i	0.931238	−	0.364413i	$-0.118730\pi$
$97$	5.58258	+	5.58258i	0.566825	+	0.566825i	−0.364413	+	0.931238i	$0.618730\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			3.60681i			0.358891i	0.983768	+	0.179446i	$0.0574304\pi$
$101$			3.60681i			0.358891i	−0.983768	+	0.179446i	$0.942570\pi$
$102$		0			0
$103$	−5.58258	+	5.58258i	−0.550068	+	0.550068i	−0.926460	−	0.376393i	$-0.877164\pi$
$103$	−5.58258	+	5.58258i	−0.550068	+	0.550068i	0.376393	+	0.926460i	$0.377164\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.74166	−	3.74166i	0.361720	−	0.361720i	−0.502726	−	0.864446i	$-0.667670\pi$
$107$	3.74166	−	3.74166i	0.361720	−	0.361720i	0.864446	+	0.502726i	$0.167670\pi$
$108$		0			0
$109$			13.7477i			1.31679i	0.752671	+	0.658397i	$0.228765\pi$
$109$			13.7477i			1.31679i	−0.752671	+	0.658397i	$0.771235\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.15732	−	1.15732i	−0.108872	−	0.108872i	0.650572	−	0.759444i	$-0.274529\pi$
$113$	−1.15732	−	1.15732i	−0.108872	−	0.108872i	−0.759444	+	0.650572i	$0.774529\pi$
$114$		0			0
$115$	−0.373864	+	1.79129i	−0.0348630	+	0.167038i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	10.4440			0.957404
$120$		0			0
$121$	7.74773			0.704339
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0227	−	1.87083i	0.985901	−	0.167332i
$126$		0			0
$127$	5.00000	+	5.00000i	0.443678	+	0.443678i	0.893246	−	0.449568i	$-0.148422\pi$
$127$	5.00000	+	5.00000i	0.443678	+	0.443678i	−0.449568	+	0.893246i	$0.648422\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			22.3151i			1.94968i	0.222907	+	0.974840i	$0.428445\pi$
$131$			22.3151i			1.94968i	−0.222907	+	0.974840i	$0.571555\pi$
$132$		0			0
$133$	−8.37386	+	8.37386i	−0.726106	+	0.726106i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.1183	−	14.1183i	1.20621	−	1.20621i	0.233959	−	0.972246i	$-0.424832\pi$
$137$	14.1183	−	14.1183i	1.20621	−	1.20621i	0.972246	−	0.233959i	$-0.0751683\pi$
$138$		0			0
$139$			15.1652i			1.28629i	0.765744	+	0.643146i	$0.222371\pi$
$139$			15.1652i			1.28629i	−0.765744	+	0.643146i	$0.777629\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.03383	−	5.03383i	−0.420950	−	0.420950i
$144$		0			0
$145$	8.37386	+	12.7913i	0.695412	+	1.06226i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−19.0847			−1.56348			−0.781739	−	0.623606i	$-0.785667\pi$
$149$	−19.0847			−1.56348			−0.781739	+	0.623606i	$0.785667\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.22474	−	1.87083i	−0.0983739	−	0.150269i
$156$		0			0
$157$	7.79129	+	7.79129i	0.621812	+	0.621812i	0.945995	−	0.324182i	$-0.105089\pi$
$157$	7.79129	+	7.79129i	0.621812	+	0.621812i	−0.324182	+	0.945995i	$0.605089\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	3.23042i		−	0.254593i
$162$		0			0
$163$	5.00000	−	5.00000i	0.391630	−	0.391630i	−0.483638	−	0.875268i	$-0.660685\pi$
$163$	5.00000	−	5.00000i	0.391630	−	0.391630i	0.875268	+	0.483638i	$0.160685\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−15.5453	+	15.5453i	−1.20293	+	1.20293i	−0.229660	+	0.973271i	$0.573761\pi$
$167$	−15.5453	+	15.5453i	−1.20293	+	1.20293i	−0.973271	+	0.229660i	$0.926239\pi$
$168$		0			0
$169$		−	2.58258i		−	0.198660i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.6688	−	11.6688i	−0.887161	−	0.887161i	0.107088	−	0.994250i	$-0.465847\pi$
$173$	−11.6688	−	11.6688i	−0.887161	−	0.887161i	−0.994250	+	0.107088i	$0.965847\pi$
$174$		0			0
$175$	−18.3739	+	7.20871i	−1.38893	+	0.544927i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−17.7477			−1.31918			−0.659589	−	0.751626i	$-0.729270\pi$
$181$	−17.7477			−1.31918			−0.659589	+	0.751626i	$0.729270\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.23042	+	15.4779i	−0.237505	+	1.13796i
$186$		0			0
$187$	3.37386	+	3.37386i	0.246721	+	0.246721i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	18.7083i		−	1.35368i	−0.736128	−	0.676842i	$-0.763348\pi$
$191$		−	18.7083i		−	1.35368i	0.736128	−	0.676842i	$-0.236652\pi$
$192$		0			0
$193$	−2.79129	+	2.79129i	−0.200921	+	0.200921i	−0.800395	−	0.599473i	$-0.795377\pi$
$193$	−2.79129	+	2.79129i	−0.200921	+	0.200921i	0.599473	+	0.800395i	$0.295377\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.87083	−	1.87083i	0.133291	−	0.133291i	−0.637314	−	0.770605i	$-0.719954\pi$
$197$	1.87083	−	1.87083i	0.133291	−	0.133291i	0.770605	+	0.637314i	$0.219954\pi$
$198$		0			0
$199$			10.7477i			0.761886i	0.924598	+	0.380943i	$0.124401\pi$
$199$			10.7477i			0.761886i	−0.924598	+	0.380943i	$0.875599\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−19.0847	−	19.0847i	−1.33948	−	1.33948i
$204$		0			0
$205$	−3.37386	+	2.20871i	−0.235641	+	0.154263i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.41022			−0.374233
$210$		0			0
$211$	17.7477			1.22180			0.610902	−	0.791706i	$-0.290807\pi$
$211$	17.7477			1.22180			0.610902	+	0.791706i	$0.290807\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	24.1185	+	5.03383i	1.64487	+	0.343304i
$216$		0			0
$217$	2.79129	+	2.79129i	0.189485	+	0.189485i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			10.4440i			0.702542i
$222$		0			0
$223$	0.582576	−	0.582576i	0.0390122	−	0.0390122i	−0.687332	−	0.726344i	$-0.741218\pi$
$223$	0.582576	−	0.582576i	0.0390122	−	0.0390122i	0.726344	+	0.687332i	$0.241218\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.8036	−	11.8036i	0.783435	−	0.783435i	−0.196974	−	0.980409i	$-0.563111\pi$
$227$	11.8036	−	11.8036i	0.783435	−	0.783435i	0.980409	+	0.196974i	$0.0631115\pi$
$228$		0			0
$229$		−	24.1652i		−	1.59688i	−0.602076	−	0.798439i	$-0.705659\pi$
$229$		−	24.1652i		−	1.59688i	0.602076	−	0.798439i	$-0.294341\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.42157	−	9.42157i	−0.617227	−	0.617227i	0.327592	−	0.944819i	$-0.393763\pi$
$233$	−9.42157	−	9.42157i	−0.617227	−	0.617227i	−0.944819	+	0.327592i	$0.893763\pi$
$234$		0			0
$235$	9.58258	+	2.00000i	0.625098	+	0.130466i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.2474			−0.792222			−0.396111	−	0.918203i	$-0.629640\pi$
$239$	−12.2474			−0.792222			−0.396111	+	0.918203i	$0.629640\pi$
$240$		0			0
$241$	15.7477			1.01440			0.507200	−	0.861828i	$-0.330680\pi$
$241$	15.7477			1.01440			0.507200	+	0.861828i	$0.330680\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	16.0565	−	10.5115i	1.02581	−	0.671553i
$246$		0			0
$247$	−8.37386	−	8.37386i	−0.532816	−	0.532816i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			7.21362i			0.455320i	0.973741	+	0.227660i	$0.0731075\pi$
$251$			7.21362i			0.455320i	−0.973741	+	0.227660i	$0.926893\pi$
$252$		0			0
$253$	1.04356	−	1.04356i	0.0656081	−	0.0656081i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.96640	+	4.96640i	−0.309796	+	0.309796i	−0.844830	−	0.535035i	$-0.820299\pi$
$257$	−4.96640	+	4.96640i	−0.309796	+	0.309796i	0.535035	+	0.844830i	$0.320299\pi$
$258$		0			0
$259$		−	27.9129i		−	1.73442i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−17.9274	−	17.9274i	−1.10545	−	1.10545i	−0.993741	−	0.111707i	$-0.964368\pi$
$263$	−17.9274	−	17.9274i	−1.10545	−	1.10545i	−0.111707	−	0.993741i	$-0.535632\pi$
$264$		0			0
$265$	4.79129	−	22.9564i	0.294326	−	1.41020i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	25.9219			1.58049			0.790243	−	0.612793i	$-0.209954\pi$
$269$	25.9219			1.58049			0.790243	+	0.612793i	$0.209954\pi$
$270$		0			0
$271$	1.00000			0.0607457			0.0303728	−	0.999539i	$-0.490331\pi$
$271$	1.00000			0.0607457			0.0303728	+	0.999539i	$0.490331\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.26424	−	3.60681i	−0.498353	−	0.217499i
$276$		0			0
$277$	−17.7913	−	17.7913i	−1.06897	−	1.06897i	−0.997438	−	0.0715369i	$-0.977210\pi$
$277$	−17.7913	−	17.7913i	−1.06897	−	1.06897i	−0.0715369	−	0.997438i	$-0.522790\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			24.1185i			1.43879i	0.694602	+	0.719395i	$0.255581\pi$
$281$			24.1185i			1.43879i	−0.694602	+	0.719395i	$0.744419\pi$
$282$		0			0
$283$	17.7913	−	17.7913i	1.05758	−	1.05758i	0.0593447	−	0.998238i	$-0.481099\pi$
$283$	17.7913	−	17.7913i	1.05758	−	1.05758i	0.998238	−	0.0593447i	$-0.0189011\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.03383	−	5.03383i	0.297137	−	0.297137i
$288$		0			0
$289$			10.0000i			0.588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.83156	−	4.83156i	−0.282263	−	0.282263i	0.551748	−	0.834011i	$-0.313961\pi$
$293$	−4.83156	−	4.83156i	−0.282263	−	0.282263i	−0.834011	+	0.551748i	$0.813961\pi$
$294$		0			0
$295$	8.37386	+	12.7913i	0.487545	+	0.744738i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.23042			0.186820
$300$		0			0
$301$	−43.4955			−2.50704
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.22474	+	1.87083i	0.0701287	+	0.107123i
$306$		0			0
$307$	−11.7477	−	11.7477i	−0.670478	−	0.670478i	0.287348	−	0.957826i	$-0.407226\pi$
$307$	−11.7477	−	11.7477i	−0.670478	−	0.670478i	−0.957826	+	0.287348i	$0.907226\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			27.7253i			1.57216i	0.618126	+	0.786079i	$0.287892\pi$
$311$			27.7253i			1.57216i	−0.618126	+	0.786079i	$0.712108\pi$
$312$		0			0
$313$	1.16515	−	1.16515i	0.0658583	−	0.0658583i	−0.673411	−	0.739269i	$-0.735171\pi$
$313$	1.16515	−	1.16515i	0.0658583	−	0.0658583i	0.739269	+	0.673411i	$0.235171\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.29784	−	3.29784i	0.185225	−	0.185225i	−0.608403	−	0.793628i	$-0.708190\pi$
$317$	3.29784	−	3.29784i	0.185225	−	0.185225i	0.793628	+	0.608403i	$0.208190\pi$
$318$		0			0
$319$		−	12.3303i		−	0.690364i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.61249	+	5.61249i	0.312287	+	0.312287i
$324$		0			0
$325$	−7.20871	−	18.3739i	−0.399867	−	1.01920i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−17.2813			−0.952747
$330$		0			0
$331$	14.7477			0.810608			0.405304	−	0.914182i	$-0.367166\pi$
$331$	14.7477			0.810608			0.405304	+	0.914182i	$0.367166\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.376393	−	1.80341i	0.0205645	−	0.0985306i
$336$		0			0
$337$	22.7913	+	22.7913i	1.24152	+	1.24152i	0.959370	+	0.282150i	$0.0910477\pi$
$337$	22.7913	+	22.7913i	1.24152	+	1.24152i	0.282150	+	0.959370i	$0.408952\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			1.80341i			0.0976599i
$342$		0			0
$343$	−4.41742	+	4.41742i	−0.238518	+	0.238518i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.9891	−	15.9891i	0.858340	−	0.858340i	−0.132802	−	0.991143i	$-0.542398\pi$
$347$	15.9891	−	15.9891i	0.858340	−	0.858340i	0.991143	+	0.132802i	$0.0423976\pi$
$348$		0			0
$349$		−	28.5826i		−	1.52999i	−0.644036	−	0.764995i	$-0.722741\pi$
$349$		−	28.5826i		−	1.52999i	0.644036	−	0.764995i	$-0.277259\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.58434	−	2.58434i	−0.137550	−	0.137550i	0.634979	−	0.772529i	$-0.281009\pi$
$353$	−2.58434	−	2.58434i	−0.137550	−	0.137550i	−0.772529	+	0.634979i	$0.781009\pi$
$354$		0			0
$355$	3.37386	−	2.20871i	0.179066	−	0.117226i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.1015			0.797025			0.398513	−	0.917163i	$-0.369526\pi$
$359$	15.1015			0.797025			0.398513	+	0.917163i	$0.369526\pi$
$360$		0			0
$361$	10.0000			0.526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.4440	−	2.17980i	−0.546666	−	0.114096i
$366$		0			0
$367$	−13.9564	−	13.9564i	−0.728520	−	0.728520i	0.241805	−	0.970325i	$-0.422261\pi$
$367$	−13.9564	−	13.9564i	−0.728520	−	0.728520i	−0.970325	+	0.241805i	$0.922261\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			41.3998i			2.14937i
$372$		0			0
$373$	5.58258	−	5.58258i	0.289055	−	0.289055i	−0.547652	−	0.836706i	$-0.684478\pi$
$373$	5.58258	−	5.58258i	0.289055	−	0.289055i	0.836706	+	0.547652i	$0.184478\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	19.0847	−	19.0847i	0.982911	−	0.982911i
$378$		0			0
$379$			19.7477i			1.01437i	0.861836	+	0.507186i	$0.169314\pi$
$379$			19.7477i			1.01437i	−0.861836	+	0.507186i	$0.830686\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.98888	−	5.98888i	−0.306017	−	0.306017i	0.537345	−	0.843362i	$-0.319427\pi$
$383$	−5.98888	−	5.98888i	−0.306017	−	0.306017i	−0.843362	+	0.537345i	$0.819427\pi$
$384$		0			0
$385$	15.5826	+	3.25227i	0.794162	+	0.165751i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	19.0847			0.967632			0.483816	−	0.875170i	$-0.339250\pi$
$389$	19.0847			0.967632			0.483816	+	0.875170i	$0.339250\pi$
$390$		0			0
$391$	−2.16515			−0.109496
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.65176	+	1.73598i	−0.133424	+	0.0873468i
$396$		0			0
$397$	−15.1216	−	15.1216i	−0.758931	−	0.758931i	0.217197	−	0.976128i	$-0.430309\pi$
$397$	−15.1216	−	15.1216i	−0.758931	−	0.758931i	−0.976128	+	0.217197i	$0.930309\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			18.7083i			0.934247i	0.884192	+	0.467124i	$0.154710\pi$
$401$			18.7083i			0.934247i	−0.884192	+	0.467124i	$0.845290\pi$
$402$		0			0
$403$	−2.79129	+	2.79129i	−0.139044	+	0.139044i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.01703	−	9.01703i	0.446958	−	0.446958i
$408$		0			0
$409$			5.83485i			0.288515i	0.989540	+	0.144257i	$0.0460793\pi$
$409$			5.83485i			0.288515i	−0.989540	+	0.144257i	$0.953921\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−19.0847	−	19.0847i	−0.939096	−	0.939096i
$414$		0			0
$415$	4.04356	−	19.3739i	0.198491	−	0.951026i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−32.7591			−1.60039			−0.800194	−	0.599741i	$-0.795270\pi$
$419$	−32.7591			−1.60039			−0.800194	+	0.599741i	$0.795270\pi$
$420$		0			0
$421$	12.2523			0.597139			0.298570	−	0.954388i	$-0.403490\pi$
$421$	12.2523			0.597139			0.298570	+	0.954388i	$0.403490\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.83156	+	12.3149i	0.234365	+	0.597359i
$426$		0			0
$427$	−2.79129	−	2.79129i	−0.135080	−	0.135080i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			27.7253i			1.33548i	0.744394	+	0.667741i	$0.232739\pi$
$431$			27.7253i			1.33548i	−0.744394	+	0.667741i	$0.767261\pi$
$432$		0			0
$433$	18.3739	−	18.3739i	0.882992	−	0.882992i	−0.110846	−	0.993838i	$-0.535356\pi$
$433$	18.3739	−	18.3739i	0.882992	−	0.882992i	0.993838	+	0.110846i	$0.0353561\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.73598	−	1.73598i	0.0830433	−	0.0830433i
$438$		0			0
$439$		−	1.41742i		−	0.0676500i	−0.999428	−	0.0338250i	$-0.989231\pi$
$439$		−	1.41742i		−	0.0676500i	0.999428	−	0.0338250i	$-0.0107689\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.6688	+	11.6688i	0.554401	+	0.554401i	0.927708	−	0.373307i	$-0.121776\pi$
$443$	11.6688	+	11.6688i	0.554401	+	0.554401i	−0.373307	+	0.927708i	$0.621776\pi$
$444$		0			0
$445$	6.62614	+	10.1216i	0.314109	+	0.479809i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−20.5117			−0.968007			−0.484003	−	0.875066i	$-0.660818\pi$
$449$	−20.5117			−0.968007			−0.484003	+	0.875066i	$0.660818\pi$
$450$		0			0
$451$	3.25227			0.153144
$452$		0			0
$453$		0			0
$454$		0			0
$455$	19.0847	+	29.1523i	0.894704	+	1.36668i
$456$		0			0
$457$	5.58258	+	5.58258i	0.261142	+	0.261142i	0.825518	−	0.564376i	$-0.190883\pi$
$457$	5.58258	+	5.58258i	0.261142	+	0.261142i	−0.564376	+	0.825518i	$0.690883\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			29.5287i			1.37529i	0.726047	+	0.687645i	$0.241355\pi$
$461$			29.5287i			1.37529i	−0.726047	+	0.687645i	$0.758645\pi$
$462$		0			0
$463$	26.1652	−	26.1652i	1.21600	−	1.21600i	0.246976	−	0.969022i	$-0.420563\pi$
$463$	26.1652	−	26.1652i	1.21600	−	1.21600i	0.969022	−	0.246976i	$-0.0794369\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.0511	−	24.0511i	1.11295	−	1.11295i	0.120202	−	0.992749i	$-0.461646\pi$
$467$	24.0511	−	24.0511i	1.11295	−	1.11295i	0.992749	−	0.120202i	$-0.0383542\pi$
$468$		0			0
$469$			3.25227i			0.150176i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14.0509	−	14.0509i	−0.646059	−	0.646059i
$474$		0			0
$475$	−13.7477	−	6.00000i	−0.630789	−	0.275299i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.0847			−0.872001			−0.436001	−	0.899946i	$-0.643605\pi$
$479$	−19.0847			−0.872001			−0.436001	+	0.899946i	$0.643605\pi$
$480$		0			0
$481$	27.9129			1.27272
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.60681	−	17.2813i	0.163777	−	0.784702i
$486$		0			0
$487$	13.3739	+	13.3739i	0.606028	+	0.606028i	0.941906	−	0.335878i	$-0.109033\pi$
$487$	13.3739	+	13.3739i	0.606028	+	0.606028i	−0.335878	+	0.941906i	$0.609033\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	29.5287i		−	1.33261i	−0.745678	−	0.666306i	$-0.767874\pi$
$491$		−	29.5287i		−	1.33261i	0.745678	−	0.666306i	$-0.232126\pi$
$492$		0			0
$493$	−12.7913	+	12.7913i	−0.576091	+	0.576091i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.03383	+	5.03383i	−0.225798	+	0.225798i
$498$		0			0
$499$		−	9.33030i		−	0.417682i	−0.977950	−	0.208841i	$-0.933031\pi$
$499$		−	9.33030i		−	0.417682i	0.977950	−	0.208841i	$-0.0669691\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.8261	−	12.8261i	−0.571888	−	0.571888i	0.360768	−	0.932656i	$-0.382515\pi$
$503$	−12.8261	−	12.8261i	−0.571888	−	0.571888i	−0.932656	+	0.360768i	$0.882515\pi$
$504$		0			0
$505$	6.74773	−	4.41742i	0.300270	−	0.196573i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	27.3489			1.21222			0.606110	−	0.795381i	$-0.292729\pi$
$509$	27.3489			1.21222			0.606110	+	0.795381i	$0.292729\pi$
$510$		0			0
$511$	18.8348			0.833205
$512$		0			0
$513$		0			0
$514$		0			0
$515$	17.2813	+	3.60681i	0.761504	+	0.158935i
$516$		0			0
$517$	−5.58258	−	5.58258i	−0.245521	−	0.245521i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			9.01703i			0.395043i	0.980299	+	0.197522i	$0.0632893\pi$
$521$			9.01703i			0.395043i	−0.980299	+	0.197522i	$0.936711\pi$
$522$		0			0
$523$	−1.62614	+	1.62614i	−0.0711060	+	0.0711060i	−0.741765	−	0.670659i	$-0.766011\pi$
$523$	−1.62614	+	1.62614i	−0.0711060	+	0.0711060i	0.670659	+	0.741765i	$0.266011\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.87083	−	1.87083i	0.0814946	−	0.0814946i
$528$		0			0
$529$		−	22.3303i		−	0.970883i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	5.03383	+	5.03383i	0.218039	+	0.218039i
$534$		0			0
$535$	−11.5826	−	2.41742i	−0.500758	−	0.104514i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−15.4779			−0.666679
$540$		0			0
$541$	35.4955			1.52607			0.763034	−	0.646358i	$-0.223709\pi$
$541$	35.4955			1.52607			0.763034	+	0.646358i	$0.223709\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	25.7196	−	16.8375i	1.10171	−	0.721237i
$546$		0			0
$547$	1.04356	+	1.04356i	0.0446194	+	0.0446194i	0.729064	−	0.684445i	$-0.239955\pi$
$547$	1.04356	+	1.04356i	0.0446194	+	0.0446194i	−0.684445	+	0.729064i	$0.739955\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	20.5117i		−	0.873827i
$552$		0			0
$553$	3.95644	−	3.95644i	0.168245	−	0.168245i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.3598	−	11.3598i	0.481331	−	0.481331i	−0.424226	−	0.905557i	$-0.639454\pi$
$557$	11.3598	−	11.3598i	0.481331	−	0.481331i	0.905557	+	0.424226i	$0.139454\pi$
$558$		0			0
$559$		−	43.4955i		−	1.83966i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.15732	+	1.15732i	0.0487753	+	0.0487753i	0.731074	−	0.682298i	$-0.239020\pi$
$563$	1.15732	+	1.15732i	0.0487753	+	0.0487753i	−0.682298	+	0.731074i	$0.739020\pi$
$564$		0			0
$565$	−0.747727	+	3.58258i	−0.0314571	+	0.150720i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−17.6577			−0.740248			−0.370124	−	0.928982i	$-0.620685\pi$
$569$	−17.6577			−0.740248			−0.370124	+	0.928982i	$0.620685\pi$
$570$		0			0
$571$	−32.4955			−1.35989			−0.679946	−	0.733262i	$-0.737997\pi$
$571$	−32.4955			−1.35989			−0.679946	+	0.733262i	$0.737997\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.80908	−	1.49444i	0.158850	−	0.0623223i
$576$		0			0
$577$	3.37386	+	3.37386i	0.140456	+	0.140456i	0.773839	−	0.633383i	$-0.218334\pi$
$577$	3.37386	+	3.37386i	0.140456	+	0.140456i	−0.633383	+	0.773839i	$0.718334\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			34.9389i			1.44951i
$582$		0			0
$583$	−13.3739	+	13.3739i	−0.553889	+	0.553889i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−19.5285	+	19.5285i	−0.806027	+	0.806027i	−0.984030	−	0.178003i	$-0.943036\pi$
$587$	−19.5285	+	19.5285i	−0.806027	+	0.806027i	0.178003	+	0.984030i	$0.443036\pi$
$588$		0			0
$589$			3.00000i			0.123613i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	22.5174	+	22.5174i	0.924677	+	0.924677i	0.997355	−	0.0726780i	$-0.0231545\pi$
$593$	22.5174	+	22.5174i	0.924677	+	0.924677i	−0.0726780	+	0.997355i	$0.523155\pi$
$594$		0			0
$595$	−12.7913	−	19.5390i	−0.524392	−	0.801022i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−17.6577			−0.721473			−0.360736	−	0.932668i	$-0.617475\pi$
$599$	−17.6577			−0.721473			−0.360736	+	0.932668i	$0.617475\pi$
$600$		0			0
$601$	12.2523			0.499781			0.249890	−	0.968274i	$-0.419605\pi$
$601$	12.2523			0.499781			0.249890	+	0.968274i	$0.419605\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.48899	−	14.4947i	−0.385782	−	0.589292i
$606$		0			0
$607$	8.95644	+	8.95644i	0.363531	+	0.363531i	0.865111	−	0.501580i	$-0.167248\pi$
$607$	8.95644	+	8.95644i	0.363531	+	0.363531i	−0.501580	+	0.865111i	$0.667248\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	17.2813i		−	0.699126i
$612$		0			0
$613$	10.0000	−	10.0000i	0.403896	−	0.403896i	−0.475707	−	0.879604i	$-0.657808\pi$
$613$	10.0000	−	10.0000i	0.403896	−	0.403896i	0.879604	+	0.475707i	$0.157808\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−10.3766	+	10.3766i	−0.417747	+	0.417747i	−0.884427	−	0.466680i	$-0.845450\pi$
$617$	−10.3766	+	10.3766i	−0.417747	+	0.417747i	0.466680	+	0.884427i	$0.345450\pi$
$618$		0			0
$619$		−	10.4174i		−	0.418712i	−0.977840	−	0.209356i	$-0.932863\pi$
$619$		−	10.4174i		−	0.418712i	0.977840	−	0.209356i	$-0.0671367\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−15.1015	−	15.1015i	−0.605028	−	0.605028i
$624$		0			0
$625$	−17.0000	−	18.3303i	−0.680000	−	0.733212i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−18.7083			−0.745948
$630$		0			0
$631$	−12.2523			−0.487755			−0.243878	−	0.969806i	$-0.578420\pi$
$631$	−12.2523			−0.487755			−0.243878	+	0.969806i	$0.578420\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.23042	−	15.4779i	0.128195	−	0.614220i
$636$		0			0
$637$	−23.9564	−	23.9564i	−0.949189	−	0.949189i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	37.4166i		−	1.47787i	−0.673779	−	0.738933i	$-0.735330\pi$
$641$		−	37.4166i		−	1.47787i	0.673779	−	0.738933i	$-0.264670\pi$
$642$		0			0
$643$	2.79129	−	2.79129i	0.110078	−	0.110078i	−0.649923	−	0.760000i	$-0.725199\pi$
$643$	2.79129	−	2.79129i	0.110078	−	0.110078i	0.760000	+	0.649923i	$0.225199\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.53939	−	3.53939i	0.139148	−	0.139148i	−0.634102	−	0.773250i	$-0.718630\pi$
$647$	3.53939	−	3.53939i	0.139148	−	0.139148i	0.773250	+	0.634102i	$0.218630\pi$
$648$		0			0
$649$		−	12.3303i		−	0.484007i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19.9330	−	19.9330i	−0.780040	−	0.780040i	0.199797	−	0.979837i	$-0.435972\pi$
$653$	−19.9330	−	19.9330i	−0.780040	−	0.780040i	−0.979837	+	0.199797i	$0.935972\pi$
$654$		0			0
$655$	41.7477	−	27.3303i	1.63122	−	1.06788i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.98320			−0.155164			−0.0775818	−	0.996986i	$-0.524720\pi$
$659$	−3.98320			−0.155164			−0.0775818	+	0.996986i	$0.524720\pi$
$660$		0			0
$661$	−31.4955			−1.22503			−0.612516	−	0.790459i	$-0.709842\pi$
$661$	−31.4955			−1.22503			−0.612516	+	0.790459i	$0.709842\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	25.9219	+	5.41022i	1.00521	+	0.209799i
$666$		0			0
$667$	3.95644	+	3.95644i	0.153194	+	0.153194i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1.80341i		−	0.0696197i
$672$		0			0
$673$	9.53901	−	9.53901i	0.367702	−	0.367702i	−0.498937	−	0.866639i	$-0.666276\pi$
$673$	9.53901	−	9.53901i	0.367702	−	0.367702i	0.866639	+	0.498937i	$0.166276\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−17.4161	+	17.4161i	−0.669356	+	0.669356i	−0.957567	−	0.288211i	$-0.906940\pi$
$677$	−17.4161	+	17.4161i	−0.669356	+	0.669356i	0.288211	+	0.957567i	$0.406940\pi$
$678$		0			0
$679$			31.1652i			1.19601i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.41589	−	7.41589i	−0.283761	−	0.283761i	0.550846	−	0.834607i	$-0.314305\pi$
$683$	−7.41589	−	7.41589i	−0.283761	−	0.283761i	−0.834607	+	0.550846i	$0.814305\pi$
$684$		0			0
$685$	−43.7042	−	9.12159i	−1.66985	−	0.348518i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−41.3998			−1.57721
$690$		0			0
$691$	−15.7477			−0.599072			−0.299536	−	0.954085i	$-0.596832\pi$
$691$	−15.7477			−0.599072			−0.299536	+	0.954085i	$0.596832\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	28.3714	−	18.5734i	1.07619	−	0.704531i
$696$		0			0
$697$	−3.37386	−	3.37386i	−0.127794	−	0.127794i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			40.3492i			1.52397i	0.647597	+	0.761983i	$0.275774\pi$
$701$			40.3492i			1.52397i	−0.647597	+	0.761983i	$0.724226\pi$
$702$		0			0
$703$	15.0000	−	15.0000i	0.565736	−	0.565736i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.0677	+	10.0677i	−0.378633	+	0.378633i
$708$		0			0
$709$			30.6606i			1.15148i	0.817632	+	0.575742i	$0.195287\pi$
$709$			30.6606i			1.15148i	−0.817632	+	0.575742i	$0.804713\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.578661	−	0.578661i	−0.0216710	−	0.0216710i
$714$		0			0
$715$	−3.25227	+	15.5826i	−0.121628	+	0.582755i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.1015			−0.563190			−0.281595	−	0.959533i	$-0.590863\pi$
$719$	−15.1015			−0.563190			−0.281595	+	0.959533i	$0.590863\pi$
$720$		0			0
$721$	−31.1652			−1.16065
$722$		0			0
$723$		0			0
$724$		0			0
$725$	13.6745	−	31.3321i	0.507857	−	1.16365i
$726$		0			0
$727$	−18.8348	−	18.8348i	−0.698546	−	0.698546i	0.265551	−	0.964097i	$-0.414446\pi$
$727$	−18.8348	−	18.8348i	−0.698546	−	0.698546i	−0.964097	+	0.265551i	$0.914446\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			29.1523i			1.07824i
$732$		0			0
$733$	35.5826	−	35.5826i	1.31427	−	1.31427i	0.396039	−	0.918234i	$-0.370385\pi$
$733$	35.5826	−	35.5826i	1.31427	−	1.31427i	0.918234	−	0.396039i	$-0.129615\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.05062	+	1.05062i	−0.0387001	+	0.0387001i
$738$		0			0
$739$		−	30.4955i		−	1.12179i	−0.827886	−	0.560897i	$-0.810456\pi$
$739$		−	30.4955i		−	1.12179i	0.827886	−	0.560897i	$-0.189544\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	27.0792	+	27.0792i	0.993441	+	0.993441i	0.999979	−	0.00653793i	$-0.00208110\pi$
$743$	27.0792	+	27.0792i	0.993441	+	0.993441i	−0.00653793	+	0.999979i	$0.502081\pi$
$744$		0			0
$745$	23.3739	+	35.7042i	0.856352	+	1.30810i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	20.8881			0.763234
$750$		0			0
$751$	17.7477			0.647624			0.323812	−	0.946121i	$-0.395036\pi$
$751$	17.7477			0.647624			0.323812	+	0.946121i	$0.395036\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.44949	+	3.74166i	0.0891461	+	0.136173i
$756$		0			0
$757$	−5.00000	−	5.00000i	−0.181728	−	0.181728i	0.610380	−	0.792108i	$-0.291017\pi$
$757$	−5.00000	−	5.00000i	−0.181728	−	0.181728i	−0.792108	+	0.610380i	$0.791017\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	27.7253i		−	1.00504i	−0.864565	−	0.502521i	$-0.832406\pi$
$761$		−	27.7253i		−	1.00504i	0.864565	−	0.502521i	$-0.167594\pi$
$762$		0			0
$763$	−38.3739	+	38.3739i	−1.38923	+	1.38923i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	19.0847	−	19.0847i	0.689108	−	0.689108i
$768$		0			0
$769$			9.33030i			0.336459i	0.985748	+	0.168230i	$0.0538050\pi$
$769$			9.33030i			0.336459i	−0.985748	+	0.168230i	$0.946195\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−29.3265	−	29.3265i	−1.05480	−	1.05480i	−0.998409	−	0.0563904i	$-0.982041\pi$
$773$	−29.3265	−	29.3265i	−1.05480	−	1.05480i	−0.0563904	−	0.998409i	$-0.517959\pi$
$774$		0			0
$775$	−2.00000	+	4.58258i	−0.0718421	+	0.164611i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.41022			0.193841
$780$		0			0
$781$	−3.25227			−0.116375
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5.03383	−	24.1185i	0.179665	−	0.860826i
$786$		0			0
$787$	5.46099	+	5.46099i	0.194663	+	0.194663i	0.797708	−	0.603044i	$-0.206046\pi$
$787$	5.46099	+	5.46099i	0.194663	+	0.194663i	−0.603044	+	0.797708i	$0.706046\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	6.46084i		−	0.229721i
$792$		0			0
$793$	2.79129	−	2.79129i	0.0991215	−	0.0991215i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.96640	+	4.96640i	−0.175919	+	0.175919i	−0.789574	−	0.613655i	$-0.789699\pi$
$797$	−4.96640	+	4.96640i	−0.175919	+	0.175919i	0.613655	+	0.789574i	$0.289699\pi$
$798$		0			0
$799$			11.5826i			0.409762i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6.08445	+	6.08445i	0.214715	+	0.214715i
$804$		0			0
$805$	−6.04356	+	3.95644i	−0.213008	+	0.139446i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	35.6132			1.25209			0.626046	−	0.779786i	$-0.284672\pi$
$809$	35.6132			1.25209			0.626046	+	0.779786i	$0.284672\pi$
$810$		0			0
$811$	31.4955			1.10595			0.552977	−	0.833196i	$-0.313492\pi$
$811$	31.4955			1.10595			0.552977	+	0.833196i	$0.313492\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−15.4779	−	3.23042i	−0.542166	−	0.113157i
$816$		0			0
$817$	−23.3739	−	23.3739i	−0.817748	−	0.817748i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	22.3151i		−	0.778802i	−0.921068	−	0.389401i	$-0.872682\pi$
$821$		−	22.3151i		−	0.778802i	0.921068	−	0.389401i	$-0.127318\pi$
$822$		0			0
$823$	1.04356	−	1.04356i	0.0363762	−	0.0363762i	−0.688685	−	0.725061i	$-0.741812\pi$
$823$	1.04356	−	1.04356i	0.0363762	−	0.0363762i	0.725061	+	0.688685i	$0.241812\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−16.9723	+	16.9723i	−0.590185	+	0.590185i	−0.937681	−	0.347496i	$-0.887032\pi$
$827$	−16.9723	+	16.9723i	−0.590185	+	0.590185i	0.347496	+	0.937681i	$0.387032\pi$
$828$		0			0
$829$		−	7.25227i		−	0.251882i	−0.992038	−	0.125941i	$-0.959805\pi$
$829$		−	7.25227i		−	0.251882i	0.992038	−	0.125941i	$-0.0401950\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	16.0565	+	16.0565i	0.556326	+	0.556326i
$834$		0			0
$835$	48.1216	+	10.0436i	1.66532	+	0.347572i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−54.6978			−1.88838			−0.944190	−	0.329402i	$-0.893153\pi$
$839$	−54.6978			−1.88838			−0.944190	+	0.329402i	$0.893153\pi$
$840$		0			0
$841$	17.7477			0.611991
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.83156	+	3.16300i	−0.166211	+	0.108810i
$846$		0			0
$847$	21.6261	+	21.6261i	0.743083	+	0.743083i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			5.78661i			0.198362i
$852$		0			0
$853$	−32.3303	+	32.3303i	−1.10697	+	1.10697i	−0.113422	+	0.993547i	$0.536181\pi$
$853$	−32.3303	+	32.3303i	−1.10697	+	1.10697i	−0.993547	+	0.113422i	$0.963819\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	8.70806	−	8.70806i	0.297462	−	0.297462i	−0.542557	−	0.840019i	$-0.682544\pi$
$857$	8.70806	−	8.70806i	0.297462	−	0.297462i	0.840019	+	0.542557i	$0.182544\pi$
$858$		0			0
$859$			27.6606i			0.943768i	0.881661	+	0.471884i	$0.156426\pi$
$859$			27.6606i			0.943768i	−0.881661	+	0.471884i	$0.843574\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	33.6075	+	33.6075i	1.14401	+	1.14401i	0.987709	+	0.156303i	$0.0499577\pi$
$863$	33.6075	+	33.6075i	1.14401	+	1.14401i	0.156303	+	0.987709i	$0.450042\pi$
$864$		0			0
$865$	−7.53901	+	36.1216i	−0.256334	+	1.22817i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.55619			0.0867129
$870$		0			0
$871$	−3.25227			−0.110199
$872$		0			0
$873$		0			0
$874$		0			0
$875$	35.9896	+	25.5455i	1.21667	+	0.863596i
$876$		0			0
$877$	7.79129	+	7.79129i	0.263093	+	0.263093i	0.826309	−	0.563216i	$-0.190436\pi$
$877$	7.79129	+	7.79129i	0.263093	+	0.263093i	−0.563216	+	0.826309i	$0.690436\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			13.2981i			0.448023i	0.974586	+	0.224012i	$0.0719154\pi$
$881$			13.2981i			0.448023i	−0.974586	+	0.224012i	$0.928085\pi$
$882$		0			0
$883$	11.6261	−	11.6261i	0.391251	−	0.391251i	−0.483882	−	0.875133i	$-0.660774\pi$
$883$	11.6261	−	11.6261i	0.391251	−	0.391251i	0.875133	+	0.483882i	$0.160774\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.85403	+	5.85403i	−0.196559	+	0.196559i	−0.798523	−	0.601964i	$-0.794385\pi$
$887$	−5.85403	+	5.85403i	−0.196559	+	0.196559i	0.601964	+	0.798523i	$0.294385\pi$
$888$		0			0
$889$			27.9129i			0.936168i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.28672	−	9.28672i	−0.310768	−	0.310768i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−6.83723			−0.228034
$900$		0			0
$901$	27.7477			0.924411
$902$		0			0
$903$		0			0
$904$		0			0
$905$	21.7364	+	33.2030i	0.722544	+	1.10370i
$906$		0			0
$907$	27.9129	+	27.9129i	0.926832	+	0.926832i	0.997500	−	0.0706680i	$-0.0225131\pi$
$907$	27.9129	+	27.9129i	0.926832	+	0.926832i	−0.0706680	+	0.997500i	$0.522513\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			16.9049i			0.560084i	0.959988	+	0.280042i	$0.0903483\pi$
$911$			16.9049i			0.560084i	−0.959988	+	0.280042i	$0.909652\pi$
$912$		0			0
$913$	−11.2867	+	11.2867i	−0.373537	+	0.373537i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−62.2879	+	62.2879i	−2.05693	+	2.05693i
$918$		0			0
$919$			24.0000i			0.791687i	0.918318	+	0.395843i	$0.129548\pi$
$919$			24.0000i			0.791687i	−0.918318	+	0.395843i	$0.870452\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.03383	−	5.03383i	−0.165690	−	0.165690i
$924$		0			0
$925$	32.9129	−	12.9129i	1.08217	−	0.424573i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−32.7591			−1.07479			−0.537396	−	0.843330i	$-0.680592\pi$
$929$	−32.7591			−1.07479			−0.537396	+	0.843330i	$0.680592\pi$
$930$		0			0
$931$	−25.7477			−0.843848
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.17980	−	10.4440i	0.0712870	−	0.341557i
$936$		0			0
$937$	26.7477	+	26.7477i	0.873810	+	0.873810i	0.992885	−	0.119075i	$-0.0379929\pi$
$937$	26.7477	+	26.7477i	0.873810	+	0.873810i	−0.119075	+	0.992885i	$0.537993\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			29.5287i			0.962609i	0.876554	+	0.481304i	$0.159837\pi$
$941$			29.5287i			0.962609i	−0.876554	+	0.481304i	$0.840163\pi$
$942$		0			0
$943$	−1.04356	+	1.04356i	−0.0339830	+	0.0339830i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17.2139	−	17.2139i	0.559375	−	0.559375i	−0.369754	−	0.929129i	$-0.620558\pi$
$947$	17.2139	−	17.2139i	0.559375	−	0.559375i	0.929129	+	0.369754i	$0.120558\pi$
$948$		0			0
$949$			18.8348i			0.611405i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	16.5003	+	16.5003i	0.534499	+	0.534499i	0.921908	−	0.387409i	$-0.126630\pi$
$953$	16.5003	+	16.5003i	0.534499	+	0.534499i	−0.387409	+	0.921908i	$0.626630\pi$
$954$		0			0
$955$	−35.0000	+	22.9129i	−1.13257	+	0.741443i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	78.8164			2.54511
$960$		0			0
$961$	−30.0000			−0.967742
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.64064	+	1.80341i	0.278152	+	0.0580537i
$966$		0			0
$967$	17.3303	+	17.3303i	0.557305	+	0.557305i	0.928539	−	0.371234i	$-0.121065\pi$
$967$	17.3303	+	17.3303i	0.557305	+	0.557305i	−0.371234	+	0.928539i	$0.621065\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	18.7083i		−	0.600377i	−0.953880	−	0.300189i	$-0.902950\pi$
$971$		−	18.7083i		−	0.600377i	0.953880	−	0.300189i	$-0.0970497\pi$
$972$		0			0
$973$	−42.3303	+	42.3303i	−1.35705	+	1.35705i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	35.8547	−	35.8547i	1.14709	−	1.14709i	0.159972	−	0.987121i	$-0.448859\pi$
$977$	35.8547	−	35.8547i	1.14709	−	1.14709i	0.987121	−	0.159972i	$-0.0511406\pi$
$978$		0			0
$979$		−	9.75682i		−	0.311829i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−19.6633	−	19.6633i	−0.627163	−	0.627163i	0.320190	−	0.947353i	$-0.396253\pi$
$983$	−19.6633	−	19.6633i	−0.627163	−	0.627163i	−0.947353	+	0.320190i	$0.896253\pi$
$984$		0			0
$985$	−5.79129	−	1.20871i	−0.184526	−	0.0385128i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9.01703			0.286725
$990$		0			0
$991$	−45.7477			−1.45322			−0.726612	−	0.687048i	$-0.758906\pi$
$991$	−45.7477			−1.45322			−0.726612	+	0.687048i	$0.758906\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	20.1072	−	13.1632i	0.637440	−	0.417302i
$996$		0			0
$997$	12.2087	+	12.2087i	0.386654	+	0.386654i	0.873492	−	0.486838i	$-0.161850\pi$
$997$	12.2087	+	12.2087i	0.386654	+	0.386654i	−0.486838	+	0.873492i	$0.661850\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.w.d.1457.1		8
3.2	odd	2	inner	2160.2.w.d.1457.4		8
4.3	odd	2		135.2.f.a.107.1	yes	8
5.3	odd	4	inner	2160.2.w.d.593.3		8
12.11	even	2		135.2.f.a.107.4	yes	8
15.8	even	4	inner	2160.2.w.d.593.2		8
20.3	even	4		135.2.f.a.53.4	yes	8
20.7	even	4		675.2.f.i.593.1		8
20.19	odd	2		675.2.f.i.107.4		8
36.7	odd	6		405.2.m.c.377.4		16
36.11	even	6		405.2.m.c.377.1		16
36.23	even	6		405.2.m.c.107.4		16
36.31	odd	6		405.2.m.c.107.1		16
60.23	odd	4		135.2.f.a.53.1	✓	8
60.47	odd	4		675.2.f.i.593.4		8
60.59	even	2		675.2.f.i.107.1		8
180.23	odd	12		405.2.m.c.188.4		16
180.43	even	12		405.2.m.c.53.4		16
180.83	odd	12		405.2.m.c.53.1		16
180.103	even	12		405.2.m.c.188.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.f.a.53.1	✓	8	60.23	odd	4
135.2.f.a.53.4	yes	8	20.3	even	4
135.2.f.a.107.1	yes	8	4.3	odd	2
135.2.f.a.107.4	yes	8	12.11	even	2
405.2.m.c.53.1		16	180.83	odd	12
405.2.m.c.53.4		16	180.43	even	12
405.2.m.c.107.1		16	36.31	odd	6
405.2.m.c.107.4		16	36.23	even	6
405.2.m.c.188.1		16	180.103	even	12
405.2.m.c.188.4		16	180.23	odd	12
405.2.m.c.377.1		16	36.11	even	6
405.2.m.c.377.4		16	36.7	odd	6
675.2.f.i.107.1		8	60.59	even	2
675.2.f.i.107.4		8	20.19	odd	2
675.2.f.i.593.1		8	20.7	even	4
675.2.f.i.593.4		8	60.47	odd	4
2160.2.w.d.593.2		8	15.8	even	4	inner
2160.2.w.d.593.3		8	5.3	odd	4	inner
2160.2.w.d.1457.1		8	1.1	even	1	trivial
2160.2.w.d.1457.4		8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.87083i) q^{5} +(2.79129 + 2.79129i) q^{7} +1.80341i q^{11} +(-2.79129 + 2.79129i) q^{13} +(1.87083 - 1.87083i) q^{17} +3.00000i q^{19} +(-0.578661 - 0.578661i) q^{23} +(-2.00000 + 4.58258i) q^{25} -6.83723 q^{29} +1.00000 q^{31} +(1.80341 - 8.64064i) q^{35} +(-5.00000 - 5.00000i) q^{37} -1.80341i q^{41} +(-7.79129 + 7.79129i) q^{43} +(-3.09557 + 3.09557i) q^{47} +8.58258i q^{49} +(7.41589 + 7.41589i) q^{53} +(3.37386 - 2.20871i) q^{55} -6.83723 q^{59} -1.00000 q^{61} +(8.64064 + 1.80341i) q^{65} +(0.582576 + 0.582576i) q^{67} +1.80341i q^{71} +(3.37386 - 3.37386i) q^{73} +(-5.03383 + 5.03383i) q^{77} -1.41742i q^{79} +(6.25857 + 6.25857i) q^{83} +(-5.79129 - 1.20871i) q^{85} -5.41022 q^{89} -15.5826 q^{91} +(5.61249 - 3.67423i) q^{95} +(5.58258 + 5.58258i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7} - 4 q^{13} - 16 q^{25} + 8 q^{31} - 40 q^{37} - 44 q^{43} - 28 q^{55} - 8 q^{61} - 32 q^{67} - 28 q^{73} - 28 q^{85} - 88 q^{91} + 8 q^{97}+O(q^{100}) \) 8 * q + 4 * q^7 - 4 * q^13 - 16 * q^25 + 8 * q^31 - 40 * q^37 - 44 * q^43 - 28 * q^55 - 8 * q^61 - 32 * q^67 - 28 * q^73 - 28 * q^85 - 88 * q^91 + 8 * q^97

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