LMFDB - Embedded newform 1512.2.r.d.1009.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1512,2,Mod(505,1512)] 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("1512.505"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.r (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.508277025.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 $ x^8 - 3x^7 + 5x^6 - 15x^5 + 21x^4 + 3x^3 - 22x^2 + 3*x + 19
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1009.3
Root			$1.86526 + 0.199842i$ of defining polynomial
Character	$\chi$	$=$	1512.1009
Dual form			1512.2.r.d.505.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.468293 - 0.811107i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(2.48741 + 4.30833i) q^{11} +(-0.622156 + 1.07761i) q^{13} -5.22446 q^{17} +5.18622 q^{19} +(-1.00266 + 1.73666i) q^{23} +(2.06140 + 3.57046i) q^{25} +(3.43925 + 5.95695i) q^{29} +(2.86526 - 4.96277i) q^{31} -0.936586 q^{35} +9.73051 q^{37} +(-5.73705 + 9.93686i) q^{41} +(-4.80184 - 8.31704i) q^{43} +(-0.984753 - 1.70564i) q^{47} +(-0.500000 + 0.866025i) q^{49} +7.63418 q^{53} +4.65935 q^{55} +(2.43925 - 4.22490i) q^{59} +(1.52178 + 2.63580i) q^{61} +(0.582703 + 1.00927i) q^{65} +(0.573990 - 0.994179i) q^{67} +8.83749 q^{71} +6.10698 q^{73} +(2.48741 - 4.30833i) q^{77} +(6.05414 + 10.4861i) q^{79} +(-0.431332 - 0.747088i) q^{83} +(-2.44658 + 4.23760i) q^{85} +10.8480 q^{89} +1.24431 q^{91} +(2.42867 - 4.20658i) q^{95} +(-3.78521 - 6.55618i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{5} - 4 q^{7} + 6 q^{11} - 3 q^{13} + 16 q^{17} - 4 q^{19} + 5 q^{23} - 14 q^{25} - q^{29} + 11 q^{31} + 8 q^{35} + 54 q^{37} - 2 q^{41} - 11 q^{43} - 7 q^{47} - 4 q^{49} + 8 q^{53} + 12 q^{55}+ \cdots - q^{97}+O(q^{100}) $ 8 * q - 4 * q^5 - 4 * q^7 + 6 * q^11 - 3 * q^13 + 16 * q^17 - 4 * q^19 + 5 * q^23 - 14 * q^25 - q^29 + 11 * q^31 + 8 * q^35 + 54 * q^37 - 2 * q^41 - 11 * q^43 - 7 * q^47 - 4 * q^49 + 8 * q^53 + 12 * q^55 - 9 * q^59 - 7 * q^61 + 9 * q^65 - 12 * q^67 + 24 * q^71 + 26 * q^73 + 6 * q^77 - 22 * q^79 + 6 * q^83 - 11 * q^85 + 28 * q^89 + 6 * q^91 + 23 * q^95 - q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.468293	−	0.811107i	0.209427	−	0.362738i	−0.742107	−	0.670281i	$-0.766173\pi$
$5$	0.468293	−	0.811107i	0.209427	−	0.362738i	0.951534	+	0.307543i	$0.0995068\pi$
$6$		0			0
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.48741	+	4.30833i	0.749983	+	1.29901i	0.947830	+	0.318776i	$0.103272\pi$
$11$	2.48741	+	4.30833i	0.749983	+	1.29901i	−0.197847	+	0.980233i	$0.563395\pi$
$12$		0			0
$13$	−0.622156	+	1.07761i	−0.172555	+	0.298874i	−0.939312	−	0.343063i	$-0.888536\pi$
$13$	−0.622156	+	1.07761i	−0.172555	+	0.298874i	0.766757	+	0.641937i	$0.221869\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.22446			−1.26712			−0.633559	−	0.773694i	$-0.718407\pi$
$17$	−5.22446			−1.26712			−0.633559	+	0.773694i	$0.718407\pi$
$18$		0			0
$19$	5.18622			1.18980			0.594900	−	0.803800i	$-0.297192\pi$
$19$	5.18622			1.18980			0.594900	+	0.803800i	$0.297192\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.00266	+	1.73666i	−0.209069	+	0.362118i	−0.951422	−	0.307891i	$-0.900377\pi$
$23$	−1.00266	+	1.73666i	−0.209069	+	0.362118i	0.742352	+	0.670010i	$0.233710\pi$
$24$		0			0
$25$	2.06140	+	3.57046i	0.412281	+	0.714091i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.43925	+	5.95695i	0.638652	+	1.10618i	0.985729	+	0.168341i	$0.0538410\pi$
$29$	3.43925	+	5.95695i	0.638652	+	1.10618i	−0.347077	+	0.937837i	$0.612826\pi$
$30$		0			0
$31$	2.86526	−	4.96277i	0.514615	−	0.891340i	−0.485241	−	0.874381i	$-0.661268\pi$
$31$	2.86526	−	4.96277i	0.514615	−	0.891340i	0.999856	−	0.0169594i	$-0.00539860\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.936586			−0.158312
$36$		0			0
$37$	9.73051			1.59969			0.799843	−	0.600209i	$-0.204916\pi$
$37$	9.73051			1.59969			0.799843	+	0.600209i	$0.204916\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.73705	+	9.93686i	−0.895976	+	1.55188i	−0.0633848	+	0.997989i	$0.520190\pi$
$41$	−5.73705	+	9.93686i	−0.895976	+	1.55188i	−0.832592	+	0.553887i	$0.813144\pi$
$42$		0			0
$43$	−4.80184	−	8.31704i	−0.732274	−	1.26834i	−0.955909	−	0.293663i	$-0.905126\pi$
$43$	−4.80184	−	8.31704i	−0.732274	−	1.26834i	0.223635	−	0.974673i	$-0.428208\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.984753	−	1.70564i	−0.143641	−	0.248793i	0.785224	−	0.619212i	$-0.212548\pi$
$47$	−0.984753	−	1.70564i	−0.143641	−	0.248793i	−0.928865	+	0.370418i	$0.879214\pi$
$48$		0			0
$49$	−0.500000	+	0.866025i	−0.0714286	+	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.63418			1.04864			0.524318	−	0.851523i	$-0.324320\pi$
$53$	7.63418			1.04864			0.524318	+	0.851523i	$0.324320\pi$
$54$		0			0
$55$	4.65935			0.628267
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.43925	−	4.22490i	0.317563	−	0.550035i	−0.662416	−	0.749136i	$-0.730469\pi$
$59$	2.43925	−	4.22490i	0.317563	−	0.550035i	0.979979	+	0.199101i	$0.0638022\pi$
$60$		0			0
$61$	1.52178	+	2.63580i	0.194844	+	0.337480i	0.946849	−	0.321677i	$-0.104247\pi$
$61$	1.52178	+	2.63580i	0.194844	+	0.337480i	−0.752005	+	0.659157i	$0.770913\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.582703	+	1.00927i	0.0722754	+	0.125185i
$66$		0			0
$67$	0.573990	−	0.994179i	0.0701240	−	0.121458i	−0.828831	−	0.559498i	$-0.810994\pi$
$67$	0.573990	−	0.994179i	0.0701240	−	0.121458i	0.898955	+	0.438040i	$0.144327\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	8.83749			1.04882			0.524409	−	0.851467i	$-0.324286\pi$
$71$	8.83749			1.04882			0.524409	+	0.851467i	$0.324286\pi$
$72$		0			0
$73$	6.10698			0.714768			0.357384	−	0.933958i	$-0.383669\pi$
$73$	6.10698			0.714768			0.357384	+	0.933958i	$0.383669\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.48741	−	4.30833i	0.283467	−	0.490979i
$78$		0			0
$79$	6.05414	+	10.4861i	0.681144	+	1.17978i	0.974632	+	0.223813i	$0.0718505\pi$
$79$	6.05414	+	10.4861i	0.681144	+	1.17978i	−0.293488	+	0.955963i	$0.594816\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.431332	−	0.747088i	−0.0473448	−	0.0820036i	0.841382	−	0.540441i	$-0.181743\pi$
$83$	−0.431332	−	0.747088i	−0.0473448	−	0.0820036i	−0.888727	+	0.458438i	$0.848409\pi$
$84$		0			0
$85$	−2.44658	+	4.23760i	−0.265369	+	0.459632i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.8480			1.14989			0.574943	−	0.818194i	$-0.305024\pi$
$89$	10.8480			1.14989			0.574943	+	0.818194i	$0.305024\pi$
$90$		0			0
$91$	1.24431			0.130439
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.42867	−	4.20658i	0.249176	−	0.431586i
$96$		0			0
$97$	−3.78521	−	6.55618i	−0.384330	−	0.665680i	0.607346	−	0.794438i	$-0.292234\pi$
$97$	−3.78521	−	6.55618i	−0.384330	−	0.665680i	−0.991676	+	0.128758i	$0.958901\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.52970	+	4.38156i	0.251714	+	0.435982i	0.963998	−	0.265910i	$-0.0856724\pi$
$101$	2.52970	+	4.38156i	0.251714	+	0.435982i	−0.712284	+	0.701892i	$0.752339\pi$
$102$		0			0
$103$	−0.119496	+	0.206973i	−0.0117743	+	0.0203936i	−0.871853	−	0.489769i	$-0.837081\pi$
$103$	−0.119496	+	0.206973i	−0.0117743	+	0.0203936i	0.860078	+	0.510162i	$0.170415\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.25496			−0.894710			−0.447355	−	0.894356i	$-0.647634\pi$
$107$	−9.25496			−0.894710			−0.447355	+	0.894356i	$0.647634\pi$
$108$		0			0
$109$	−10.5453			−1.01005			−0.505026	−	0.863104i	$-0.668517\pi$
$109$	−10.5453			−1.01005			−0.505026	+	0.863104i	$0.668517\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.06406	+	8.77122i	−0.476387	+	0.825127i	−0.999634	−	0.0270545i	$-0.991387\pi$
$113$	−5.06406	+	8.77122i	−0.476387	+	0.825127i	0.523247	+	0.852181i	$0.324721\pi$
$114$		0			0
$115$	0.939078	+	1.62653i	0.0875695	+	0.151675i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.61223	+	4.52452i	0.239463	+	0.414762i
$120$		0			0
$121$	−6.87445	+	11.9069i	−0.624950	+	1.08245i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.54429			0.764225
$126$		0			0
$127$	6.52720			0.579196			0.289598	−	0.957148i	$-0.406478\pi$
$127$	6.52720			0.579196			0.289598	+	0.957148i	$0.406478\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.76748	+	3.06136i	−0.154425	+	0.267472i	−0.932850	−	0.360266i	$-0.882686\pi$
$131$	−1.76748	+	3.06136i	−0.154425	+	0.267472i	0.778424	+	0.627738i	$0.216019\pi$
$132$		0			0
$133$	−2.59311	−	4.49140i	−0.224851	−	0.389454i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.48603	+	4.30593i	0.212396	+	0.367881i	0.952464	−	0.304651i	$-0.0985400\pi$
$137$	2.48603	+	4.30593i	0.212396	+	0.367881i	−0.740068	+	0.672532i	$0.765207\pi$
$138$		0			0
$139$	−4.87566	+	8.44490i	−0.413548	+	0.716287i	−0.995275	−	0.0970976i	$-0.969044\pi$
$139$	−4.87566	+	8.44490i	−0.413548	+	0.716287i	0.581726	+	0.813385i	$0.302377\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.19024			−0.517654
$144$		0			0
$145$	6.44230			0.535004
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.39108	+	11.0697i	−0.523578	+	0.906863i	0.476046	+	0.879421i	$0.342070\pi$
$149$	−6.39108	+	11.0697i	−0.523578	+	0.906863i	−0.999623	+	0.0274426i	$0.991264\pi$
$150$		0			0
$151$	−11.4781	−	19.8807i	−0.934078	−	1.61787i	−0.776271	−	0.630400i	$-0.782891\pi$
$151$	−11.4781	−	19.8807i	−0.934078	−	1.61787i	−0.157807	−	0.987470i	$-0.550442\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.68356	−	4.64806i	−0.215549	−	0.373341i
$156$		0			0
$157$	−8.25489	+	14.2979i	−0.658812	+	1.14110i	0.322112	+	0.946702i	$0.395607\pi$
$157$	−8.25489	+	14.2979i	−0.658812	+	1.14110i	−0.980924	+	0.194394i	$0.937726\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.00532			0.158041
$162$		0			0
$163$	17.2245			1.34912			0.674562	−	0.738218i	$-0.264333\pi$
$163$	17.2245			1.34912			0.674562	+	0.738218i	$0.264333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.56406	−	11.3693i	0.507943	−	0.879782i	−0.492015	−	0.870587i	$-0.663740\pi$
$167$	6.56406	−	11.3693i	0.507943	−	0.879782i	0.999958	−	0.00919564i	$-0.00292711\pi$
$168$		0			0
$169$	5.72584	+	9.91745i	0.440449	+	0.762881i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.49412	−	16.4443i	−0.721824	−	1.25024i	−0.960268	−	0.279080i	$-0.909971\pi$
$173$	−9.49412	−	16.4443i	−0.721824	−	1.25024i	0.238444	−	0.971156i	$-0.423363\pi$
$174$		0			0
$175$	2.06140	−	3.57046i	0.155827	−	0.269901i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−7.61835			−0.569422			−0.284711	−	0.958613i	$-0.591898\pi$
$179$	−7.61835			−0.569422			−0.284711	+	0.958613i	$0.591898\pi$
$180$		0			0
$181$	−9.27737			−0.689581			−0.344791	−	0.938680i	$-0.612050\pi$
$181$	−9.27737			−0.689581			−0.344791	+	0.938680i	$0.612050\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.55673	−	7.89249i	0.335018	−	0.580267i
$186$		0			0
$187$	−12.9954	−	22.5087i	−0.950317	−	1.64600i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.80119	+	8.31591i	0.347402	+	0.601718i	0.985787	−	0.167999i	$-0.0537306\pi$
$191$	4.80119	+	8.31591i	0.347402	+	0.601718i	−0.638385	+	0.769717i	$0.720397\pi$
$192$		0			0
$193$	7.54155	−	13.0624i	0.542853	−	0.940249i	−0.455886	−	0.890038i	$-0.650677\pi$
$193$	7.54155	−	13.0624i	0.542853	−	0.940249i	0.998739	−	0.0502103i	$-0.0159892\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	11.0712			0.788788			0.394394	−	0.918942i	$-0.370955\pi$
$197$	11.0712			0.788788			0.394394	+	0.918942i	$0.370955\pi$
$198$		0			0
$199$	18.3368			1.29986			0.649929	−	0.759995i	$-0.274799\pi$
$199$	18.3368			1.29986			0.649929	+	0.759995i	$0.274799\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.43925	−	5.95695i	0.241388	−	0.418096i
$204$		0			0
$205$	5.37324	+	9.30672i	0.375283	+	0.650010i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	12.9003	+	22.3439i	0.892331	+	1.54556i
$210$		0			0
$211$	−1.30911	+	2.26744i	−0.0901227	+	0.156097i	−0.907563	−	0.419917i	$-0.862059\pi$
$211$	−1.30911	+	2.26744i	−0.0901227	+	0.156097i	0.817440	+	0.576014i	$0.195393\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.99468			−0.613432
$216$		0			0
$217$	−5.73051			−0.389013
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.25043	−	5.62991i	0.218648	−	0.378709i
$222$		0			0
$223$	12.5442	+	21.7272i	0.840023	+	1.45496i	0.889875	+	0.456205i	$0.150792\pi$
$223$	12.5442	+	21.7272i	0.840023	+	1.45496i	−0.0498520	+	0.998757i	$0.515875\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.7781	−	22.1323i	−0.848113	−	1.46898i	−0.882890	−	0.469581i	$-0.844405\pi$
$227$	−12.7781	−	22.1323i	−0.848113	−	1.46898i	0.0347761	−	0.999395i	$-0.488928\pi$
$228$		0			0
$229$	−2.73657	+	4.73987i	−0.180837	+	0.313220i	−0.942166	−	0.335147i	$-0.891214\pi$
$229$	−2.73657	+	4.73987i	−0.180837	+	0.313220i	0.761329	+	0.648366i	$0.224547\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.6774			−0.699500			−0.349750	−	0.936843i	$-0.613733\pi$
$233$	−10.6774			−0.699500			−0.349750	+	0.936843i	$0.613733\pi$
$234$		0			0
$235$	−1.84461			−0.120329
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.50000	−	4.33013i	0.161712	−	0.280093i	−0.773771	−	0.633465i	$-0.781632\pi$
$239$	2.50000	−	4.33013i	0.161712	−	0.280093i	0.935483	+	0.353373i	$0.114965\pi$
$240$		0			0
$241$	−10.1684	−	17.6121i	−0.655003	−	1.13450i	−0.981893	−	0.189436i	$-0.939334\pi$
$241$	−10.1684	−	17.6121i	−0.655003	−	1.13450i	0.326890	−	0.945062i	$-0.393999\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.468293	+	0.811107i	0.0299181	+	0.0518197i
$246$		0			0
$247$	−3.22664	+	5.58871i	−0.205306	+	0.355601i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−19.8151			−1.25072			−0.625358	−	0.780338i	$-0.715047\pi$
$251$	−19.8151			−1.25072			−0.625358	+	0.780338i	$0.715047\pi$
$252$		0			0
$253$	−9.97613			−0.627194
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.969506	+	1.67923i	−0.0604761	+	0.104748i	−0.894678	−	0.446711i	$-0.852595\pi$
$257$	−0.969506	+	1.67923i	−0.0604761	+	0.104748i	0.834202	+	0.551459i	$0.185929\pi$
$258$		0			0
$259$	−4.86526	−	8.42687i	−0.302312	−	0.523620i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.67768	−	11.5661i	−0.411763	−	0.713195i	0.583320	−	0.812243i	$-0.301754\pi$
$263$	−6.67768	−	11.5661i	−0.411763	−	0.713195i	−0.995083	+	0.0990481i	$0.968420\pi$
$264$		0			0
$265$	3.57503	−	6.19214i	0.219613	−	0.380380i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.15040			−0.253055			−0.126527	−	0.991963i	$-0.540383\pi$
$269$	−4.15040			−0.253055			−0.126527	+	0.991963i	$0.540383\pi$
$270$		0			0
$271$	−16.6050			−1.00868			−0.504341	−	0.863505i	$-0.668264\pi$
$271$	−16.6050			−1.00868			−0.504341	+	0.863505i	$0.668264\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.2551	+	17.7624i	−0.618407	+	1.07111i
$276$		0			0
$277$	−8.55414	−	14.8162i	−0.513968	−	0.890219i	−0.999869	−	0.0162051i	$-0.994842\pi$
$277$	−8.55414	−	14.8162i	−0.513968	−	0.890219i	0.485900	−	0.874014i	$-0.338492\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.49007	−	12.9732i	−0.446820	−	0.773916i	0.551357	−	0.834270i	$-0.314110\pi$
$281$	−7.49007	−	12.9732i	−0.446820	−	0.773916i	−0.998177	+	0.0603541i	$0.980777\pi$
$282$		0			0
$283$	9.00580	−	15.5985i	0.535339	−	0.927235i	−0.463807	−	0.885936i	$-0.653517\pi$
$283$	9.00580	−	15.5985i	0.535339	−	0.927235i	0.999147	−	0.0412990i	$-0.0131496\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	11.4741			0.677294
$288$		0			0
$289$	10.2950			0.605588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−13.8362	+	23.9650i	−0.808320	+	1.40005i	0.105707	+	0.994397i	$0.466290\pi$
$293$	−13.8362	+	23.9650i	−0.808320	+	1.40005i	−0.914027	+	0.405654i	$0.867044\pi$
$294$		0			0
$295$	−2.28456	−	3.95698i	−0.133012	−	0.230384i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.24762	−	2.16095i	−0.0721519	−	0.124971i
$300$		0			0
$301$	−4.80184	+	8.31704i	−0.276774	+	0.479386i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.85056			0.163222
$306$		0			0
$307$	−6.10040			−0.348168			−0.174084	−	0.984731i	$-0.555696\pi$
$307$	−6.10040			−0.348168			−0.174084	+	0.984731i	$0.555696\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.3277	−	23.0842i	0.755743	−	1.30898i	−0.189262	−	0.981927i	$-0.560609\pi$
$311$	13.3277	−	23.0842i	0.755743	−	1.30898i	0.945004	−	0.327058i	$-0.106057\pi$
$312$		0			0
$313$	−12.0681	−	20.9026i	−0.682130	−	1.18148i	−0.974330	−	0.225126i	$-0.927721\pi$
$313$	−12.0681	−	20.9026i	−0.682130	−	1.18148i	0.292200	−	0.956357i	$-0.405613\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.80046	−	15.2428i	−0.494283	−	0.856124i	0.505695	−	0.862712i	$-0.331236\pi$
$317$	−8.80046	−	15.2428i	−0.494283	−	0.856124i	−0.999978	+	0.00658868i	$0.997903\pi$
$318$		0			0
$319$	−17.1097	+	29.6348i	−0.957957	+	1.65923i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−27.0952			−1.50762
$324$		0			0
$325$	−5.13006			−0.284565
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.984753	+	1.70564i	−0.0542912	+	0.0940351i
$330$		0			0
$331$	−12.1497	−	21.0440i	−0.667810	−	1.15668i	−0.978515	−	0.206175i	$-0.933898\pi$
$331$	−12.1497	−	21.0440i	−0.667810	−	1.15668i	0.310705	−	0.950506i	$-0.399435\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.537591	−	0.931135i	−0.0293717	−	0.0508733i
$336$		0			0
$337$	−11.4722	+	19.8704i	−0.624929	+	1.08241i	0.363626	+	0.931545i	$0.381539\pi$
$337$	−11.4722	+	19.8704i	−0.624929	+	1.08241i	−0.988555	+	0.150863i	$0.951795\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	28.5083			1.54381
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.36930	+	11.0319i	−0.341922	+	0.592226i	−0.984790	−	0.173751i	$-0.944411\pi$
$347$	−6.36930	+	11.0319i	−0.341922	+	0.592226i	0.642868	+	0.765977i	$0.277744\pi$
$348$		0			0
$349$	3.99468	+	6.91899i	0.213830	+	0.370365i	0.952910	−	0.303253i	$-0.0980727\pi$
$349$	3.99468	+	6.91899i	0.213830	+	0.370365i	−0.739080	+	0.673618i	$0.764739\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10.9781	−	19.0147i	−0.584307	−	1.01205i	−0.994961	−	0.100259i	$-0.968033\pi$
$353$	−10.9781	−	19.0147i	−0.584307	−	1.01205i	0.410654	−	0.911791i	$-0.365300\pi$
$354$		0			0
$355$	4.13854	−	7.16815i	0.219651	−	0.380446i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−20.4673			−1.08022			−0.540112	−	0.841593i	$-0.681618\pi$
$359$	−20.4673			−1.08022			−0.540112	+	0.841593i	$0.681618\pi$
$360$		0			0
$361$	7.89688			0.415625
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.85985	−	4.95341i	0.149692	−	0.259273i
$366$		0			0
$367$	9.94174	+	17.2196i	0.518955	+	0.898856i	0.999757	+	0.0220269i	$0.00701195\pi$
$367$	9.94174	+	17.2196i	0.518955	+	0.898856i	−0.480803	+	0.876829i	$0.659655\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.81709	−	6.61139i	−0.198173	−	0.343246i
$372$		0			0
$373$	−17.1438	+	29.6939i	−0.887672	+	1.53749i	−0.0450513	+	0.998985i	$0.514345\pi$
$373$	−17.1438	+	29.6939i	−0.887672	+	1.53749i	−0.842620	+	0.538508i	$0.818988\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.55900			−0.440811
$378$		0			0
$379$	18.6248			0.956694			0.478347	−	0.878171i	$-0.341236\pi$
$379$	18.6248			0.956694			0.478347	+	0.878171i	$0.341236\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	16.4360	−	28.4680i	0.839842	−	1.45465i	−0.0501852	−	0.998740i	$-0.515981\pi$
$383$	16.4360	−	28.4680i	0.839842	−	1.45465i	0.890027	−	0.455908i	$-0.150686\pi$
$384$		0			0
$385$	−2.32968	−	4.03512i	−0.118731	−	0.205649i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.8250	−	18.7495i	−0.548850	−	0.950635i	−0.998354	−	0.0573571i	$-0.981733\pi$
$389$	−10.8250	−	18.7495i	−0.548850	−	0.950635i	0.449504	−	0.893278i	$-0.351601\pi$
$390$		0			0
$391$	5.23836	−	9.07311i	0.264915	−	0.458847i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	11.3404			0.570600
$396$		0			0
$397$	16.2721			0.816673			0.408336	−	0.912832i	$-0.366109\pi$
$397$	16.2721			0.816673			0.408336	+	0.912832i	$0.366109\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.57133	+	7.91777i	−0.228281	+	0.395395i	−0.957299	−	0.289100i	$-0.906644\pi$
$401$	−4.57133	+	7.91777i	−0.228281	+	0.395395i	0.729018	+	0.684495i	$0.239977\pi$
$402$		0			0
$403$	3.56528	+	6.17524i	0.177599	+	0.307611i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	24.2038	+	41.9222i	1.19974	+	2.07801i
$408$		0			0
$409$	11.0976	−	19.2217i	0.548743	−	0.950450i	−0.449618	−	0.893221i	$-0.648440\pi$
$409$	11.0976	−	19.2217i	0.548743	−	0.950450i	0.998361	−	0.0572294i	$-0.0182266\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.87849			−0.240055
$414$		0			0
$415$	−0.807958			−0.0396611
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.9601	−	24.1795i	0.681994	−	1.18125i	−0.292378	−	0.956303i	$-0.594447\pi$
$419$	13.9601	−	24.1795i	0.681994	−	1.18125i	0.974372	−	0.224945i	$-0.0722202\pi$
$420$		0			0
$421$	−12.0441	−	20.8611i	−0.586996	−	1.01671i	−0.994623	−	0.103558i	$-0.966977\pi$
$421$	−12.0441	−	20.8611i	−0.586996	−	1.01671i	0.407628	−	0.913148i	$-0.366356\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.7697	−	18.6537i	−0.522408	−	0.904838i
$426$		0			0
$427$	1.52178	−	2.63580i	0.0736442	−	0.127555i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	26.1623			1.26020			0.630098	−	0.776516i	$-0.283015\pi$
$431$	26.1623			1.26020			0.630098	+	0.776516i	$0.283015\pi$
$432$		0			0
$433$	−18.5630			−0.892082			−0.446041	−	0.895013i	$-0.647166\pi$
$433$	−18.5630			−0.892082			−0.446041	+	0.895013i	$0.647166\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.20002	+	9.00670i	−0.248751	+	0.430849i
$438$		0			0
$439$	−3.85395	−	6.67524i	−0.183939	−	0.318592i	0.759279	−	0.650765i	$-0.225552\pi$
$439$	−3.85395	−	6.67524i	−0.183939	−	0.318592i	−0.943218	+	0.332173i	$0.892218\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.5204	+	28.6142i	0.784909	+	1.35950i	0.929053	+	0.369945i	$0.120624\pi$
$443$	16.5204	+	28.6142i	0.784909	+	1.35950i	−0.144145	+	0.989557i	$0.546043\pi$
$444$		0			0
$445$	5.08004	−	8.79889i	0.240817	−	0.417107i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−19.9283			−0.940477			−0.470238	−	0.882540i	$-0.655832\pi$
$449$	−19.9283			−0.940477			−0.470238	+	0.882540i	$0.655832\pi$
$450$		0			0
$451$	−57.0816			−2.68787
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.582703	−	1.00927i	0.0273175	−	0.0473154i
$456$		0			0
$457$	16.9326	+	29.3282i	0.792075	+	1.37191i	0.924680	+	0.380745i	$0.124333\pi$
$457$	16.9326	+	29.3282i	0.792075	+	1.37191i	−0.132605	+	0.991169i	$0.542334\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.25351	−	12.5634i	−0.337830	−	0.585138i	0.646195	−	0.763173i	$-0.276359\pi$
$461$	−7.25351	−	12.5634i	−0.337830	−	0.585138i	−0.984024	+	0.178035i	$0.943026\pi$
$462$		0			0
$463$	−12.5348	+	21.7110i	−0.582544	+	1.00900i	0.412633	+	0.910897i	$0.364609\pi$
$463$	−12.5348	+	21.7110i	−0.582544	+	1.00900i	−0.995177	+	0.0980981i	$0.968724\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.102955			0.00476419			0.00238209	−	0.999997i	$-0.499242\pi$
$467$	0.102955			0.00476419			0.00238209	+	0.999997i	$0.499242\pi$
$468$		0			0
$469$	−1.14798			−0.0530088
$470$		0			0
$471$		0			0
$472$		0			0
$473$	23.8883	−	41.3758i	1.09839	−	1.90246i
$474$		0			0
$475$	10.6909	+	18.5172i	0.490532	+	0.849626i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.62668	+	4.54954i	0.120016	+	0.207874i	0.919774	−	0.392449i	$-0.128372\pi$
$479$	2.62668	+	4.54954i	0.120016	+	0.207874i	−0.799758	+	0.600323i	$0.795039\pi$
$480$		0			0
$481$	−6.05390	+	10.4857i	−0.276034	+	0.478105i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.09036			−0.321957
$486$		0			0
$487$	−30.7319			−1.39260			−0.696299	−	0.717752i	$-0.745171\pi$
$487$	−30.7319			−1.39260			−0.696299	+	0.717752i	$0.745171\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.35920	+	4.08626i	−0.106469	+	0.184410i	−0.914338	−	0.404953i	$-0.867288\pi$
$491$	−2.35920	+	4.08626i	−0.106469	+	0.184410i	0.807868	+	0.589363i	$0.200621\pi$
$492$		0			0
$493$	−17.9682	−	31.1219i	−0.809248	−	1.40166i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.41875	−	7.65349i	−0.198208	−	0.343306i
$498$		0			0
$499$	−12.3566	+	21.4023i	−0.553158	+	0.958098i	0.444886	+	0.895587i	$0.353244\pi$
$499$	−12.3566	+	21.4023i	−0.553158	+	0.958098i	−0.998044	+	0.0625105i	$0.980089\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	19.4083			0.865371			0.432686	−	0.901545i	$-0.357566\pi$
$503$	19.4083			0.865371			0.432686	+	0.901545i	$0.357566\pi$
$504$		0			0
$505$	4.73856			0.210863
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.72850	−	4.72591i	0.120939	−	0.209472i	−0.799199	−	0.601066i	$-0.794743\pi$
$509$	2.72850	−	4.72591i	0.120939	−	0.209472i	0.920138	+	0.391594i	$0.128076\pi$
$510$		0			0
$511$	−3.05349	−	5.28880i	−0.135078	−	0.233963i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.111918	+	0.193848i	0.00493170	+	0.00854196i
$516$		0			0
$517$	4.89897	−	8.48527i	0.215457	−	0.373182i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	24.7683			1.08512			0.542558	−	0.840018i	$-0.317456\pi$
$521$	24.7683			1.08512			0.542558	+	0.840018i	$0.317456\pi$
$522$		0			0
$523$	−32.7530			−1.43219			−0.716094	−	0.698004i	$-0.754072\pi$
$523$	−32.7530			−1.43219			−0.716094	+	0.698004i	$0.754072\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14.9694	+	25.9278i	−0.652078	+	1.12943i
$528$		0			0
$529$	9.48934	+	16.4360i	0.412580	+	0.714610i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.13868	−	12.3646i	−0.309211	−	0.535569i
$534$		0			0
$535$	−4.33403	+	7.50676i	−0.187377	+	0.324546i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.97483			−0.214281
$540$		0			0
$541$	37.8575			1.62762			0.813811	−	0.581130i	$-0.197389\pi$
$541$	37.8575			1.62762			0.813811	+	0.581130i	$0.197389\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.93827	+	8.55333i	−0.211532	+	0.366385i
$546$		0			0
$547$	−1.74835	−	3.02824i	−0.0747543	−	0.129478i	0.826225	−	0.563340i	$-0.190484\pi$
$547$	−1.74835	−	3.02824i	−0.0747543	−	0.129478i	−0.900979	+	0.433862i	$0.857151\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	17.8367	+	30.8941i	0.759869	+	1.31613i
$552$		0			0
$553$	6.05414	−	10.4861i	0.257448	−	0.445913i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	34.1226			1.44582			0.722911	−	0.690941i	$-0.242803\pi$
$557$	34.1226			1.44582			0.722911	+	0.690941i	$0.242803\pi$
$558$		0			0
$559$	11.9500			0.505431
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.65071	−	13.2514i	0.322439	−	0.558481i	−0.658552	−	0.752535i	$-0.728831\pi$
$563$	7.65071	−	13.2514i	0.322439	−	0.558481i	0.980991	+	0.194055i	$0.0621640\pi$
$564$		0			0
$565$	4.74293	+	8.21500i	0.199537	+	0.345608i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	16.4953	+	28.5707i	0.691520	+	1.19775i	0.971340	+	0.237695i	$0.0763920\pi$
$569$	16.4953	+	28.5707i	0.691520	+	1.19775i	−0.279820	+	0.960053i	$0.590275\pi$
$570$		0			0
$571$	10.7150	−	18.5590i	0.448410	−	0.776668i	−0.549873	−	0.835248i	$-0.685324\pi$
$571$	10.7150	−	18.5590i	0.448410	−	0.776668i	0.998283	+	0.0585801i	$0.0186573\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.26755			−0.344781
$576$		0			0
$577$	−38.4188			−1.59939			−0.799697	−	0.600404i	$-0.795007\pi$
$577$	−38.4188			−1.59939			−0.799697	+	0.600404i	$0.795007\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.431332	+	0.747088i	−0.0178947	+	0.0309944i
$582$		0			0
$583$	18.9894	+	32.8905i	0.786459	+	1.36219i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.43772	−	14.6146i	−0.348262	−	0.603207i	0.637679	−	0.770302i	$-0.279895\pi$
$587$	−8.43772	−	14.6146i	−0.348262	−	0.603207i	−0.985941	+	0.167095i	$0.946561\pi$
$588$		0			0
$589$	14.8599	−	25.7380i	0.612290	−	1.06052i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−21.7817			−0.894467			−0.447233	−	0.894417i	$-0.647591\pi$
$593$	−21.7817			−0.894467			−0.447233	+	0.894417i	$0.647591\pi$
$594$		0			0
$595$	4.89316			0.200600
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−13.3778	+	23.1710i	−0.546601	+	0.946740i	0.451904	+	0.892067i	$0.350745\pi$
$599$	−13.3778	+	23.1710i	−0.546601	+	0.946740i	−0.998504	+	0.0546732i	$0.982588\pi$
$600$		0			0
$601$	12.5615	+	21.7571i	0.512393	+	0.887491i	0.999897	+	0.0143699i	$0.00457423\pi$
$601$	12.5615	+	21.7571i	0.512393	+	0.887491i	−0.487504	+	0.873121i	$0.662092\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.43852	+	11.1518i	0.261763	+	0.453387i
$606$		0			0
$607$	−4.31975	+	7.48203i	−0.175333	+	0.303686i	−0.940277	−	0.340411i	$-0.889434\pi$
$607$	−4.31975	+	7.48203i	−0.175333	+	0.303686i	0.764943	+	0.644098i	$0.222767\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.45068			0.0991440
$612$		0			0
$613$	46.1789			1.86515			0.932575	−	0.360977i	$-0.117557\pi$
$613$	46.1789			1.86515			0.932575	+	0.360977i	$0.117557\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.76222	+	9.98046i	−0.231978	+	0.401798i	−0.958390	−	0.285461i	$-0.907853\pi$
$617$	−5.76222	+	9.98046i	−0.231978	+	0.401798i	0.726412	+	0.687260i	$0.241186\pi$
$618$		0			0
$619$	−19.6325	−	34.0045i	−0.789096	−	1.36675i	−0.926521	−	0.376242i	$-0.877216\pi$
$619$	−19.6325	−	34.0045i	−0.789096	−	1.36675i	0.137425	−	0.990512i	$-0.456117\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.42400	−	9.39464i	−0.217308	−	0.376388i
$624$		0			0
$625$	−6.30578	+	10.9219i	−0.252231	+	0.436877i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−50.8367			−2.02699
$630$		0			0
$631$	22.8387			0.909193			0.454596	−	0.890698i	$-0.349784\pi$
$631$	22.8387			0.909193			0.454596	+	0.890698i	$0.349784\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.05664	−	5.29426i	0.121299	−	0.210096i
$636$		0			0
$637$	−0.622156	−	1.07761i	−0.0246507	−	0.0426963i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.04899	−	15.6733i	−0.357413	−	0.619058i	0.630114	−	0.776502i	$-0.283008\pi$
$641$	−9.04899	−	15.6733i	−0.357413	−	0.619058i	−0.987528	+	0.157444i	$0.949675\pi$
$642$		0			0
$643$	15.0416	−	26.0529i	0.593184	−	1.02742i	−0.400616	−	0.916246i	$-0.631204\pi$
$643$	15.0416	−	26.0529i	0.593184	−	1.02742i	0.993800	−	0.111179i	$-0.0354627\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	25.1323			0.988054			0.494027	−	0.869447i	$-0.335524\pi$
$647$	25.1323			0.988054			0.494027	+	0.869447i	$0.335524\pi$
$648$		0			0
$649$	24.2697			0.952668
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.24884	−	9.09125i	0.205403	−	0.355768i	−0.744858	−	0.667223i	$-0.767483\pi$
$653$	5.24884	−	9.09125i	0.205403	−	0.355768i	0.950261	+	0.311455i	$0.100816\pi$
$654$		0			0
$655$	1.65539	+	2.86722i	0.0646815	+	0.112032i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4.04138	−	6.99988i	−0.157430	−	0.272677i	0.776511	−	0.630103i	$-0.216988\pi$
$659$	−4.04138	−	6.99988i	−0.157430	−	0.272677i	−0.933941	+	0.357427i	$0.883654\pi$
$660$		0			0
$661$	13.4530	−	23.3012i	0.523260	−	0.906312i	−0.476374	−	0.879243i	$-0.658049\pi$
$661$	13.4530	−	23.3012i	0.523260	−	0.906312i	0.999634	−	0.0270695i	$-0.00861755\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.85734			−0.188360
$666$		0			0
$667$	−13.7936			−0.534090
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−7.57060	+	13.1127i	−0.292260	+	0.506209i
$672$		0			0
$673$	6.11160	+	10.5856i	0.235585	+	0.408045i	0.959443	−	0.281904i	$-0.0909661\pi$
$673$	6.11160	+	10.5856i	0.235585	+	0.408045i	−0.723858	+	0.689949i	$0.757633\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.34015	+	5.78531i	0.128372	+	0.222348i	0.923046	−	0.384689i	$-0.125691\pi$
$677$	3.34015	+	5.78531i	0.128372	+	0.222348i	−0.794674	+	0.607037i	$0.792358\pi$
$678$		0			0
$679$	−3.78521	+	6.55618i	−0.145263	+	0.251603i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−19.2125			−0.735146			−0.367573	−	0.929995i	$-0.619811\pi$
$683$	−19.2125			−0.735146			−0.367573	+	0.929995i	$0.619811\pi$
$684$		0			0
$685$	4.65677			0.177926
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.74966	+	8.22664i	−0.180947	+	0.313410i
$690$		0			0
$691$	5.72706	+	9.91955i	0.217867	+	0.377357i	0.954156	−	0.299310i	$-0.0967566\pi$
$691$	5.72706	+	9.91955i	0.217867	+	0.377357i	−0.736288	+	0.676668i	$0.763423\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4.56648	+	7.90937i	0.173216	+	0.300020i
$696$		0			0
$697$	29.9730	−	51.9147i	1.13531	−	1.96641i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.29596			0.162256			0.0811281	−	0.996704i	$-0.474148\pi$
$701$	4.29596			0.162256			0.0811281	+	0.996704i	$0.474148\pi$
$702$		0			0
$703$	50.4646			1.90331
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.52970	−	4.38156i	0.0951390	−	0.164786i
$708$		0			0
$709$	−6.44506	−	11.1632i	−0.242049	−	0.419242i	0.719248	−	0.694753i	$-0.244486\pi$
$709$	−6.44506	−	11.1632i	−0.242049	−	0.419242i	−0.961298	+	0.275511i	$0.911153\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.74576	+	9.95195i	0.215180	+	0.372703i
$714$		0			0
$715$	−2.89885	+	5.02095i	−0.108411	+	0.187773i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−5.46489			−0.203806			−0.101903	−	0.994794i	$-0.532493\pi$
$719$	−5.46489			−0.203806			−0.101903	+	0.994794i	$0.532493\pi$
$720$		0			0
$721$	0.238992			0.00890051
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−14.1793	+	24.5594i	−0.526608	+	0.912111i
$726$		0			0
$727$	24.9300	+	43.1800i	0.924601	+	1.60146i	0.792201	+	0.610260i	$0.208935\pi$
$727$	24.9300	+	43.1800i	0.924601	+	1.60146i	0.132401	+	0.991196i	$0.457731\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	25.0870	+	43.4520i	0.927878	+	1.60713i
$732$		0			0
$733$	17.6123	−	30.5054i	0.650525	−	1.12674i	−0.332471	−	0.943114i	$-0.607882\pi$
$733$	17.6123	−	30.5054i	0.650525	−	1.12674i	0.982996	−	0.183629i	$-0.0587844\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.71100			0.210367
$738$		0			0
$739$	−35.7209			−1.31401			−0.657007	−	0.753885i	$-0.728178\pi$
$739$	−35.7209			−1.31401			−0.657007	+	0.753885i	$0.728178\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6.53043	−	11.3110i	0.239578	−	0.414962i	−0.721015	−	0.692919i	$-0.756324\pi$
$743$	6.53043	−	11.3110i	0.239578	−	0.414962i	0.960593	+	0.277958i	$0.0896575\pi$
$744$		0			0
$745$	5.98580	+	10.3677i	0.219303	+	0.379843i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.62748	+	8.01503i	0.169084	+	0.292863i
$750$		0			0
$751$	1.11223	−	1.92644i	0.0405859	−	0.0702968i	−0.845019	−	0.534736i	$-0.820411\pi$
$751$	1.11223	−	1.92644i	0.0405859	−	0.0702968i	0.885605	+	0.464440i	$0.153744\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.5005			−0.782484
$756$		0			0
$757$	4.27335			0.155317			0.0776587	−	0.996980i	$-0.475256\pi$
$757$	4.27335			0.155317			0.0776587	+	0.996980i	$0.475256\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	25.2206	−	43.6833i	0.914245	−	1.58352i	0.106242	−	0.994340i	$-0.466118\pi$
$761$	25.2206	−	43.6833i	0.914245	−	1.58352i	0.808003	−	0.589179i	$-0.200549\pi$
$762$		0			0
$763$	5.27263	+	9.13246i	0.190882	+	0.330617i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.03519	+	5.25710i	0.109594	+	0.189823i
$768$		0			0
$769$	1.03493	−	1.79255i	0.0373205	−	0.0646411i	−0.846762	−	0.531972i	$-0.821451\pi$
$769$	1.03493	−	1.79255i	0.0373205	−	0.0646411i	0.884082	+	0.467331i	$0.154784\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	28.3739			1.02054			0.510269	−	0.860015i	$-0.329546\pi$
$773$	28.3739			1.02054			0.510269	+	0.860015i	$0.329546\pi$
$774$		0			0
$775$	23.6258			0.848664
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−29.7536	+	51.5347i	−1.06603	+	1.84642i
$780$		0			0
$781$	21.9825	+	38.0748i	0.786595	+	1.36242i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.73141	+	13.3912i	0.275946	+	0.477952i
$786$		0			0
$787$	1.72536	−	2.98841i	0.0615025	−	0.106525i	−0.833635	−	0.552316i	$-0.813744\pi$
$787$	1.72536	−	2.98841i	0.0615025	−	0.106525i	0.895137	+	0.445791i	$0.147077\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	10.1281			0.360115
$792$		0			0
$793$	−3.78714			−0.134485
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11.2230	+	19.4388i	−0.397540	+	0.688559i	−0.993422	−	0.114513i	$-0.963469\pi$
$797$	−11.2230	+	19.4388i	−0.397540	+	0.688559i	0.595882	+	0.803072i	$0.296803\pi$
$798$		0			0
$799$	5.14480	+	8.91106i	0.182010	+	0.315251i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	15.1906	+	26.3108i	0.536064	+	0.928490i
$804$		0			0
$805$	0.939078	−	1.62653i	0.0330982	−	0.0573277i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18.2366			−0.641164			−0.320582	−	0.947221i	$-0.603878\pi$
$809$	−18.2366			−0.641164			−0.320582	+	0.947221i	$0.603878\pi$
$810$		0			0
$811$	16.8280			0.590910			0.295455	−	0.955357i	$-0.404529\pi$
$811$	16.8280			0.590910			0.295455	+	0.955357i	$0.404529\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.06610	−	13.9709i	0.282543	−	0.489379i
$816$		0			0
$817$	−24.9034	−	43.1340i	−0.871260	−	1.50907i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18.6149	−	32.2420i	−0.649665	−	1.12525i	−0.983203	−	0.182516i	$-0.941576\pi$
$821$	−18.6149	−	32.2420i	−0.649665	−	1.12525i	0.333538	−	0.942737i	$-0.391757\pi$
$822$		0			0
$823$	6.13747	−	10.6304i	0.213939	−	0.370553i	−0.739005	−	0.673700i	$-0.764704\pi$
$823$	6.13747	−	10.6304i	0.213939	−	0.370553i	0.952944	+	0.303147i	$0.0980374\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.60355			0.299175			0.149587	−	0.988749i	$-0.452205\pi$
$827$	8.60355			0.299175			0.149587	+	0.988749i	$0.452205\pi$
$828$		0			0
$829$	−44.2887			−1.53821			−0.769105	−	0.639123i	$-0.779298\pi$
$829$	−44.2887			−1.53821			−0.769105	+	0.639123i	$0.779298\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.61223	−	4.52452i	0.0905084	−	0.156765i
$834$		0			0
$835$	−6.14781	−	10.6483i	−0.212754	−	0.368500i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	23.6154	+	40.9030i	0.815293	+	1.41213i	0.909117	+	0.416540i	$0.136758\pi$
$839$	23.6154	+	40.9030i	0.815293	+	1.41213i	−0.0938240	+	0.995589i	$0.529909\pi$
$840$		0			0
$841$	−9.15684	+	15.8601i	−0.315753	+	0.546900i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.7255			0.368968
$846$		0			0
$847$	13.7489			0.472418
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−9.75640	+	16.8986i	−0.334445	+	0.579276i
$852$		0			0
$853$	11.9825	+	20.7543i	0.410273	+	0.710614i	0.994919	−	0.100674i	$-0.0321000\pi$
$853$	11.9825	+	20.7543i	0.410273	+	0.710614i	−0.584646	+	0.811288i	$0.698767\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.50903	+	11.2740i	0.222344	+	0.385111i	0.955519	−	0.294928i	$-0.0952958\pi$
$857$	6.50903	+	11.2740i	0.222344	+	0.385111i	−0.733175	+	0.680040i	$0.761962\pi$
$858$		0			0
$859$	0.0473685	−	0.0820447i	0.00161619	−	0.00279933i	−0.865216	−	0.501399i	$-0.832819\pi$
$859$	0.0473685	−	0.0820447i	0.00161619	−	0.00279933i	0.866832	+	0.498600i	$0.166152\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	25.8729			0.880723			0.440361	−	0.897821i	$-0.354850\pi$
$863$	25.8729			0.880723			0.440361	+	0.897821i	$0.354850\pi$
$864$		0			0
$865$	−17.7841			−0.604678
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−30.1183	+	52.1664i	−1.02169	+	1.76962i
$870$		0			0
$871$	0.714223	+	1.23707i	0.0242005	+	0.0419165i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.27215	−	7.39958i	−0.144425	−	0.250151i
$876$		0			0
$877$	26.9042	−	46.5994i	0.908489	−	1.57355i	0.0923254	−	0.995729i	$-0.470570\pi$
$877$	26.9042	−	46.5994i	0.908489	−	1.57355i	0.816164	−	0.577821i	$-0.196097\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	42.8689			1.44429			0.722144	−	0.691743i	$-0.243157\pi$
$881$	42.8689			1.44429			0.722144	+	0.691743i	$0.243157\pi$
$882$		0			0
$883$	34.5967			1.16427			0.582136	−	0.813092i	$-0.302217\pi$
$883$	34.5967			1.16427			0.582136	+	0.813092i	$0.302217\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.2278	+	17.7150i	−0.343415	+	0.594812i	−0.985064	−	0.172186i	$-0.944917\pi$
$887$	−10.2278	+	17.7150i	−0.343415	+	0.594812i	0.641650	+	0.766998i	$0.278250\pi$
$888$		0			0
$889$	−3.26360	−	5.65272i	−0.109458	−	0.189586i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5.10714	−	8.84583i	−0.170904	−	0.296015i
$894$		0			0
$895$	−3.56762	+	6.17930i	−0.119252	+	0.206551i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	39.4173			1.31464
$900$		0			0
$901$	−39.8845			−1.32874
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.34453	+	7.52494i	−0.144417	+	0.250137i
$906$		0			0
$907$	3.92456	+	6.79754i	0.130313	+	0.225709i	0.923797	−	0.382882i	$-0.125068\pi$
$907$	3.92456	+	6.79754i	0.130313	+	0.225709i	−0.793484	+	0.608591i	$0.791735\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.87189	−	6.70631i	−0.128282	−	0.222190i	0.794729	−	0.606964i	$-0.207613\pi$
$911$	−3.87189	−	6.70631i	−0.128282	−	0.222190i	−0.923011	+	0.384774i	$0.874279\pi$
$912$		0			0
$913$	2.14580	−	3.71663i	0.0710156	−	0.123003i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.53495			0.116734
$918$		0			0
$919$	19.3702			0.638963			0.319482	−	0.947592i	$-0.396491\pi$
$919$	19.3702			0.638963			0.319482	+	0.947592i	$0.396491\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.49830	+	9.52334i	−0.180979	+	0.313464i
$924$		0			0
$925$	20.0585	+	34.7424i	0.659520	+	1.14232i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.77989	−	3.08287i	−0.0583964	−	0.101146i	0.835349	−	0.549720i	$-0.185265\pi$
$929$	−1.77989	−	3.08287i	−0.0583964	−	0.101146i	−0.893746	+	0.448574i	$0.851932\pi$
$930$		0			0
$931$	−2.59311	+	4.49140i	−0.0849858	+	0.147200i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−24.3426			−0.796089
$936$		0			0
$937$	−34.2230			−1.11802			−0.559008	−	0.829162i	$-0.688818\pi$
$937$	−34.2230			−1.11802			−0.559008	+	0.829162i	$0.688818\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.02033	+	10.4275i	−0.196257	+	0.339928i	−0.947312	−	0.320312i	$-0.896212\pi$
$941$	−6.02033	+	10.4275i	−0.196257	+	0.339928i	0.751055	+	0.660240i	$0.229545\pi$
$942$		0			0
$943$	−11.5046	−	19.9266i	−0.374642	−	0.648899i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−11.3623	−	19.6802i	−0.369227	−	0.639519i	0.620218	−	0.784429i	$-0.287044\pi$
$947$	−11.3623	−	19.6802i	−0.369227	−	0.639519i	−0.989445	+	0.144910i	$0.953711\pi$
$948$		0			0
$949$	−3.79949	+	6.58092i	−0.123337	+	0.213626i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6.74488			0.218488			0.109244	−	0.994015i	$-0.465157\pi$
$953$	6.74488			0.218488			0.109244	+	0.994015i	$0.465157\pi$
$954$		0			0
$955$	8.99346			0.291022
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.48603	−	4.30593i	0.0802782	−	0.139046i
$960$		0			0
$961$	−0.919395	−	1.59244i	−0.0296579	−	0.0513690i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.06331	−	12.2340i	−0.227376	−	0.393827i
$966$		0			0
$967$	12.8267	−	22.2165i	0.412479	−	0.714434i	−0.582681	−	0.812701i	$-0.697996\pi$
$967$	12.8267	−	22.2165i	0.412479	−	0.714434i	0.995160	+	0.0982664i	$0.0313297\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−40.9298			−1.31350			−0.656750	−	0.754108i	$-0.728069\pi$
$971$	−40.9298			−1.31350			−0.656750	+	0.754108i	$0.728069\pi$
$972$		0			0
$973$	9.75133			0.312613
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.2365	−	38.5147i	0.711408	−	1.23220i	−0.252920	−	0.967487i	$-0.581391\pi$
$977$	22.2365	−	38.5147i	0.711408	−	1.23220i	0.964329	−	0.264708i	$-0.0852756\pi$
$978$		0			0
$979$	26.9835	+	46.7367i	0.862395	+	1.49371i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−28.4550	−	49.2855i	−0.907573	−	1.57196i	−0.817425	−	0.576035i	$-0.804599\pi$
$983$	−28.4550	−	49.2855i	−0.907573	−	1.57196i	−0.0901483	−	0.995928i	$-0.528734\pi$
$984$		0			0
$985$	5.18455	−	8.97990i	0.165193	−	0.286123i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	19.2585			0.612384
$990$		0			0
$991$	−44.7331			−1.42099			−0.710496	−	0.703701i	$-0.751530\pi$
$991$	−44.7331			−1.42099			−0.710496	+	0.703701i	$0.751530\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.58698	−	14.8731i	0.272225	−	0.471508i
$996$		0			0
$997$	9.36681	+	16.2238i	0.296650	+	0.513812i	0.975367	−	0.220587i	$-0.0707973\pi$
$997$	9.36681	+	16.2238i	0.296650	+	0.513812i	−0.678718	+	0.734399i	$0.737464\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.r.d.1009.3		8
3.2	odd	2		504.2.r.d.337.2	yes	8
4.3	odd	2		3024.2.r.l.1009.3		8
9.2	odd	6		504.2.r.d.169.2	✓	8
9.4	even	3		4536.2.a.ba.1.2		4
9.5	odd	6		4536.2.a.x.1.3		4
9.7	even	3	inner	1512.2.r.d.505.3		8
12.11	even	2		1008.2.r.m.337.3		8
36.7	odd	6		3024.2.r.l.2017.3		8
36.11	even	6		1008.2.r.m.673.3		8
36.23	even	6		9072.2.a.ce.1.3		4
36.31	odd	6		9072.2.a.cl.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.d.169.2	✓	8	9.2	odd	6
504.2.r.d.337.2	yes	8	3.2	odd	2
1008.2.r.m.337.3		8	12.11	even	2
1008.2.r.m.673.3		8	36.11	even	6
1512.2.r.d.505.3		8	9.7	even	3	inner
1512.2.r.d.1009.3		8	1.1	even	1	trivial
3024.2.r.l.1009.3		8	4.3	odd	2
3024.2.r.l.2017.3		8	36.7	odd	6
4536.2.a.x.1.3		4	9.5	odd	6
4536.2.a.ba.1.2		4	9.4	even	3
9072.2.a.ce.1.3		4	36.23	even	6
9072.2.a.cl.1.2		4	36.31	odd	6

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:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.r (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.468293 - 0.811107i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(2.48741 + 4.30833i) q^{11} +(-0.622156 + 1.07761i) q^{13} -5.22446 q^{17} +5.18622 q^{19} +(-1.00266 + 1.73666i) q^{23} +(2.06140 + 3.57046i) q^{25} +(3.43925 + 5.95695i) q^{29} +(2.86526 - 4.96277i) q^{31} -0.936586 q^{35} +9.73051 q^{37} +(-5.73705 + 9.93686i) q^{41} +(-4.80184 - 8.31704i) q^{43} +(-0.984753 - 1.70564i) q^{47} +(-0.500000 + 0.866025i) q^{49} +7.63418 q^{53} +4.65935 q^{55} +(2.43925 - 4.22490i) q^{59} +(1.52178 + 2.63580i) q^{61} +(0.582703 + 1.00927i) q^{65} +(0.573990 - 0.994179i) q^{67} +8.83749 q^{71} +6.10698 q^{73} +(2.48741 - 4.30833i) q^{77} +(6.05414 + 10.4861i) q^{79} +(-0.431332 - 0.747088i) q^{83} +(-2.44658 + 4.23760i) q^{85} +10.8480 q^{89} +1.24431 q^{91} +(2.42867 - 4.20658i) q^{95} +(-3.78521 - 6.55618i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{5} - 4 q^{7} + 6 q^{11} - 3 q^{13} + 16 q^{17} - 4 q^{19} + 5 q^{23} - 14 q^{25} - q^{29} + 11 q^{31} + 8 q^{35} + 54 q^{37} - 2 q^{41} - 11 q^{43} - 7 q^{47} - 4 q^{49} + 8 q^{53} + 12 q^{55}+ \cdots - q^{97}+O(q^{100}) \) 8 * q - 4 * q^5 - 4 * q^7 + 6 * q^11 - 3 * q^13 + 16 * q^17 - 4 * q^19 + 5 * q^23 - 14 * q^25 - q^29 + 11 * q^31 + 8 * q^35 + 54 * q^37 - 2 * q^41 - 11 * q^43 - 7 * q^47 - 4 * q^49 + 8 * q^53 + 12 * q^55 - 9 * q^59 - 7 * q^61 + 9 * q^65 - 12 * q^67 + 24 * q^71 + 26 * q^73 + 6 * q^77 - 22 * q^79 + 6 * q^83 - 11 * q^85 + 28 * q^89 + 6 * q^91 + 23 * q^95 - q^97

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