LMFDB - Embedded newform 1512.2.bl.b.1025.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1512,2,Mod(593,1512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1512, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))

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

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

chi := DirichletCharacter("1512.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.bl (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{14} + 84x^{12} - 208x^{10} + 882x^{8} + 4424x^{6} + 10340x^{4} + 20412x^{2} + 17689 $ x^16 - 7x^14 + 84x^12 - 208x^10 + 882x^8 + 4424x^6 + 10340x^4 + 20412*x^2 + 17689
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1025.8
Root			$2.23600 - 1.01798i$ of defining polynomial
Character	$\chi$	$=$	1512.1025
Dual form			1512.2.bl.b.593.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.99959 + 3.46340i) q^{5} +(0.605908 + 2.57544i) q^{7} +(4.23559 + 2.44542i) q^{11} +0.678883i q^{13} +(3.57564 - 6.19319i) q^{17} +(4.19700 - 2.42314i) q^{19} +(0.556910 - 0.321532i) q^{23} +(-5.49674 + 9.52064i) q^{25} -4.89084i q^{29} +(4.60646 + 2.65954i) q^{31} +(-7.70819 + 7.24833i) q^{35} +(-0.0905386 - 0.156817i) q^{37} -9.52435 q^{41} -2.28373 q^{43} +(1.62531 + 2.81511i) q^{47} +(-6.26575 + 3.12096i) q^{49} +(-5.96312 - 3.44281i) q^{53} +19.5594i q^{55} +(4.94932 - 8.57247i) q^{59} +(-9.15398 + 5.28505i) q^{61} +(-2.35124 + 1.35749i) q^{65} +(5.39345 - 9.34172i) q^{67} -1.03596i q^{71} +(0.409461 + 0.236403i) q^{73} +(-3.73164 + 12.3902i) q^{77} +(4.08467 + 7.07486i) q^{79} -2.53549 q^{83} +28.5993 q^{85} +(-8.90995 - 15.4325i) q^{89} +(-1.74842 + 0.411340i) q^{91} +(16.7846 + 9.69058i) q^{95} -3.07058i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{7} + 6 q^{19} - 6 q^{25} + 6 q^{31} - 8 q^{37} + 44 q^{43} - 34 q^{49} - 6 q^{61} + 20 q^{67} - 14 q^{79} + 148 q^{85} - 54 q^{91}+O(q^{100}) $ 16 * q - 2 * q^7 + 6 * q^19 - 6 * q^25 + 6 * q^31 - 8 * q^37 + 44 * q^43 - 34 * q^49 - 6 * q^61 + 20 * q^67 - 14 * q^79 + 148 * q^85 - 54 * q^91

Character values

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

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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
$5$	1.99959	+	3.46340i	0.894245	+	1.54888i	0.834736	+	0.550650i	$0.185620\pi$
$5$	1.99959	+	3.46340i	0.894245	+	1.54888i	0.0595092	+	0.998228i	$0.481046\pi$
$6$		0			0
$7$	0.605908	+	2.57544i	0.229012	+	0.973424i
$7$	0.605908	+	2.57544i	0.229012	+	0.973424i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.23559	+	2.44542i	1.27708	+	0.737321i	0.976310	−	0.216376i	$-0.0694237\pi$
$11$	4.23559	+	2.44542i	1.27708	+	0.737321i	0.300768	+	0.953697i	$0.402757\pi$
$12$		0			0
$13$			0.678883i			0.188288i	0.995559	+	0.0941441i	$0.0300114\pi$
$13$			0.678883i			0.188288i	−0.995559	+	0.0941441i	$0.969989\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.57564	−	6.19319i	0.867220	−	1.50207i	0.00239456	−	0.999997i	$-0.499238\pi$
$17$	3.57564	−	6.19319i	0.867220	−	1.50207i	0.864826	−	0.502072i	$-0.167429\pi$
$18$		0			0
$19$	4.19700	−	2.42314i	0.962858	−	0.555906i	0.0658063	−	0.997832i	$-0.479038\pi$
$19$	4.19700	−	2.42314i	0.962858	−	0.555906i	0.897051	+	0.441926i	$0.145705\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.556910	−	0.321532i	0.116124	−	0.0670440i	−0.440813	−	0.897599i	$-0.645310\pi$
$23$	0.556910	−	0.321532i	0.116124	−	0.0670440i	0.556937	+	0.830555i	$0.311977\pi$
$24$		0			0
$25$	−5.49674	+	9.52064i	−1.09935	+	1.90413i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.89084i		−	0.908206i	−0.890949	−	0.454103i	$-0.849960\pi$
$29$		−	4.89084i		−	0.908206i	0.890949	−	0.454103i	$-0.150040\pi$
$30$		0			0
$31$	4.60646	+	2.65954i	0.827345	+	0.477668i	0.852943	−	0.522005i	$-0.174816\pi$
$31$	4.60646	+	2.65954i	0.827345	+	0.477668i	−0.0255979	+	0.999672i	$0.508149\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.70819	+	7.24833i	−1.30292	+	1.22519i
$36$		0			0
$37$	−0.0905386	−	0.156817i	−0.0148844	−	0.0257806i	0.858487	−	0.512835i	$-0.171405\pi$
$37$	−0.0905386	−	0.156817i	−0.0148844	−	0.0257806i	−0.873372	+	0.487054i	$0.838071\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−9.52435			−1.48745			−0.743726	−	0.668484i	$-0.766943\pi$
$41$	−9.52435			−1.48745			−0.743726	+	0.668484i	$0.766943\pi$
$42$		0			0
$43$	−2.28373			−0.348265			−0.174133	−	0.984722i	$-0.555712\pi$
$43$	−2.28373			−0.348265			−0.174133	+	0.984722i	$0.555712\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.62531	+	2.81511i	0.237075	+	0.410627i	0.959874	−	0.280432i	$-0.0904778\pi$
$47$	1.62531	+	2.81511i	0.237075	+	0.410627i	−0.722798	+	0.691059i	$0.757144\pi$
$48$		0			0
$49$	−6.26575	+	3.12096i	−0.895107	+	0.445851i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.96312	−	3.44281i	−0.819098	−	0.472906i	0.0310076	−	0.999519i	$-0.490128\pi$
$53$	−5.96312	−	3.44281i	−0.819098	−	0.472906i	−0.850105	+	0.526613i	$0.823462\pi$
$54$		0			0
$55$			19.5594i			2.63738i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.94932	−	8.57247i	0.644346	−	1.11604i	−0.340106	−	0.940387i	$-0.610463\pi$
$59$	4.94932	−	8.57247i	0.644346	−	1.11604i	0.984452	−	0.175653i	$-0.0562037\pi$
$60$		0			0
$61$	−9.15398	+	5.28505i	−1.17205	+	0.676682i	−0.954161	−	0.299293i	$-0.903249\pi$
$61$	−9.15398	+	5.28505i	−1.17205	+	0.676682i	−0.217886	+	0.975974i	$0.569916\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.35124	+	1.35749i	−0.291635	+	0.168376i
$66$		0			0
$67$	5.39345	−	9.34172i	0.658914	−	1.14127i	−0.321983	−	0.946746i	$-0.604349\pi$
$67$	5.39345	−	9.34172i	0.658914	−	1.14127i	0.980897	−	0.194527i	$-0.0623173\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.03596i		−	0.122945i	−0.998109	−	0.0614727i	$-0.980420\pi$
$71$		−	1.03596i		−	0.122945i	0.998109	−	0.0614727i	$-0.0195797\pi$
$72$		0			0
$73$	0.409461	+	0.236403i	0.0479238	+	0.0276688i	0.523770	−	0.851859i	$-0.324525\pi$
$73$	0.409461	+	0.236403i	0.0479238	+	0.0276688i	−0.475847	+	0.879528i	$0.657858\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.73164	+	12.3902i	−0.425260	+	1.41199i
$78$		0			0
$79$	4.08467	+	7.07486i	0.459562	+	0.795984i	0.998938	−	0.0460808i	$-0.0146732\pi$
$79$	4.08467	+	7.07486i	0.459562	+	0.795984i	−0.539376	+	0.842065i	$0.681340\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.53549			−0.278306			−0.139153	−	0.990271i	$-0.544438\pi$
$83$	−2.53549			−0.278306			−0.139153	+	0.990271i	$0.544438\pi$
$84$		0			0
$85$	28.5993			3.10203
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.90995	−	15.4325i	−0.944453	−	1.63584i	−0.756843	−	0.653597i	$-0.773259\pi$
$89$	−8.90995	−	15.4325i	−0.944453	−	1.63584i	−0.187610	−	0.982244i	$-0.560074\pi$
$90$		0			0
$91$	−1.74842	+	0.411340i	−0.183284	+	0.0431202i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	16.7846	+	9.69058i	1.72206	+	0.994233i
$96$		0			0
$97$		−	3.07058i		−	0.311770i	−0.987775	−	0.155885i	$-0.950177\pi$
$97$		−	3.07058i		−	0.311770i	0.987775	−	0.155885i	$-0.0498229\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.26652	−	5.65779i	0.325031	−	0.562971i	−0.656487	−	0.754337i	$-0.727958\pi$
$101$	3.26652	−	5.65779i	0.325031	−	0.562971i	0.981519	+	0.191366i	$0.0612918\pi$
$102$		0			0
$103$	−15.7238	+	9.07816i	−1.54932	+	0.894498i	−0.551122	+	0.834425i	$0.685800\pi$
$103$	−15.7238	+	9.07816i	−1.54932	+	0.894498i	−0.998194	+	0.0600728i	$0.980867\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.51173	+	4.33690i	−0.726187	+	0.419264i	−0.817026	−	0.576602i	$-0.804379\pi$
$107$	−7.51173	+	4.33690i	−0.726187	+	0.419264i	0.0908388	+	0.995866i	$0.471045\pi$
$108$		0			0
$109$	−0.211816	+	0.366876i	−0.0202883	+	0.0351403i	−0.875991	−	0.482327i	$-0.839792\pi$
$109$	−0.211816	+	0.366876i	−0.0202883	+	0.0351403i	0.855703	+	0.517467i	$0.173125\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	10.1027i		−	0.950378i	−0.879884	−	0.475189i	$-0.842380\pi$
$113$		−	10.1027i		−	0.950378i	0.879884	−	0.475189i	$-0.157620\pi$
$114$		0			0
$115$	2.22719	+	1.28587i	0.207686	+	0.119908i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	18.1167	+	5.45633i	1.66075	+	0.500181i
$120$		0			0
$121$	6.46014	+	11.1893i	0.587286	+	1.01721i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−23.9691			−2.14386
$126$		0			0
$127$	−12.5993			−1.11801			−0.559004	−	0.829165i	$-0.688816\pi$
$127$	−12.5993			−1.11801			−0.559004	+	0.829165i	$0.688816\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.36188	−	5.82294i	−0.293729	−	0.508753i	0.680960	−	0.732321i	$-0.261563\pi$
$131$	−3.36188	−	5.82294i	−0.293729	−	0.508753i	−0.974688	+	0.223568i	$0.928230\pi$
$132$		0			0
$133$	8.78364	+	9.34091i	0.761638	+	0.809959i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.8085	+	6.81763i	1.00887	+	0.582469i	0.910860	−	0.412716i	$-0.135420\pi$
$137$	11.8085	+	6.81763i	1.00887	+	0.582469i	0.0980070	+	0.995186i	$0.468753\pi$
$138$		0			0
$139$			19.0696i			1.61746i	0.588181	+	0.808729i	$0.299844\pi$
$139$			19.0696i			1.61746i	−0.588181	+	0.808729i	$0.700156\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.66015	+	2.87547i	−0.138829	+	0.240459i
$144$		0			0
$145$	16.9389	−	9.77968i	1.40670	−	0.812158i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	14.3433	−	8.28112i	1.17505	−	0.678416i	0.220186	−	0.975458i	$-0.429333\pi$
$149$	14.3433	−	8.28112i	1.17505	−	0.678416i	0.954864	+	0.297042i	$0.0960001\pi$
$150$		0			0
$151$	4.24897	−	7.35944i	0.345776	−	0.598902i	−0.639718	−	0.768610i	$-0.720949\pi$
$151$	4.24897	−	7.35944i	0.345776	−	0.598902i	0.985495	+	0.169707i	$0.0542822\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			21.2720i			1.70861i
$156$		0			0
$157$	−14.8574	−	8.57792i	−1.18575	−	0.684593i	−0.228412	−	0.973565i	$-0.573353\pi$
$157$	−14.8574	−	8.57792i	−1.18575	−	0.684593i	−0.957338	+	0.288972i	$0.906687\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.16552	+	1.23947i	0.0918559	+	0.0976837i
$162$		0			0
$163$	−11.6206	−	20.1275i	−0.910198	−	1.57651i	−0.813784	−	0.581168i	$-0.802596\pi$
$163$	−11.6206	−	20.1275i	−0.910198	−	1.57651i	−0.0964145	−	0.995341i	$-0.530737\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.5685			0.817813			0.408907	−	0.912576i	$-0.365910\pi$
$167$	10.5685			0.817813			0.408907	+	0.912576i	$0.365910\pi$
$168$		0			0
$169$	12.5391			0.964548
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.0926046	−	0.160396i	−0.00704060	−	0.0121947i	0.862484	−	0.506085i	$-0.168908\pi$
$173$	−0.0926046	−	0.160396i	−0.00704060	−	0.0121947i	−0.869524	+	0.493890i	$0.835574\pi$
$174$		0			0
$175$	−27.8503	−	8.38789i	−2.10529	−	0.634065i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.1115	+	6.41523i	0.830513	+	0.479497i	0.854028	−	0.520227i	$-0.174152\pi$
$179$	11.1115	+	6.41523i	0.830513	+	0.479497i	−0.0235152	+	0.999723i	$0.507486\pi$
$180$		0			0
$181$			19.0696i			1.41743i	0.705495	+	0.708715i	$0.250725\pi$
$181$			19.0696i			1.41743i	−0.705495	+	0.708715i	$0.749275\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.362081	−	0.627142i	0.0266207	−	0.0461084i
$186$		0			0
$187$	30.2899	−	17.4879i	2.21502	−	1.27884i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.97410	−	4.60385i	0.576986	−	0.333123i	−0.182949	−	0.983122i	$-0.558564\pi$
$191$	7.97410	−	4.60385i	0.576986	−	0.333123i	0.759935	+	0.649999i	$0.225231\pi$
$192$		0			0
$193$	3.19384	−	5.53189i	0.229897	−	0.398194i	−0.727880	−	0.685704i	$-0.759494\pi$
$193$	3.19384	−	5.53189i	0.229897	−	0.398194i	0.957778	+	0.287510i	$0.0928276\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2.95121i		−	0.210265i	−0.994458	−	0.105132i	$-0.966473\pi$
$197$		−	2.95121i		−	0.210265i	0.994458	−	0.105132i	$-0.0335267\pi$
$198$		0			0
$199$	2.42679	+	1.40111i	0.172031	+	0.0993220i	0.583543	−	0.812082i	$-0.301666\pi$
$199$	2.42679	+	1.40111i	0.172031	+	0.0993220i	−0.411512	+	0.911404i	$0.634999\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.5960	−	2.96340i	0.884069	−	0.207990i
$204$		0			0
$205$	−19.0448	−	32.9866i	−1.33015	−	2.30388i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	23.7023			1.63953
$210$		0			0
$211$	−5.60060			−0.385561			−0.192780	−	0.981242i	$-0.561751\pi$
$211$	−5.60060			−0.385561			−0.192780	+	0.981242i	$0.561751\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.56653	−	7.90946i	−0.311435	−	0.539421i
$216$		0			0
$217$	−4.05839	+	13.4751i	−0.275501	+	0.914748i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.20445	+	2.42744i	0.282822	+	0.163287i
$222$		0			0
$223$			2.76985i			0.185483i	0.995690	+	0.0927413i	$0.0295630\pi$
$223$			2.76985i			0.185483i	−0.995690	+	0.0927413i	$0.970437\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.71883	+	8.17325i	−0.313200	+	0.542478i	−0.979053	−	0.203605i	$-0.934734\pi$
$227$	−4.71883	+	8.17325i	−0.313200	+	0.542478i	0.665854	+	0.746082i	$0.268068\pi$
$228$		0			0
$229$	−23.3395	+	13.4751i	−1.54232	+	0.890459i	−0.543628	+	0.839326i	$0.682950\pi$
$229$	−23.3395	+	13.4751i	−1.54232	+	0.890459i	−0.998692	+	0.0511324i	$0.983717\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−18.7010	+	10.7970i	−1.22514	+	0.707338i	−0.966010	−	0.258503i	$-0.916771\pi$
$233$	−18.7010	+	10.7970i	−1.22514	+	0.707338i	−0.259135	+	0.965841i	$0.583437\pi$
$234$		0			0
$235$	−6.49991	+	11.2582i	−0.424007	+	0.734402i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			15.9859i			1.03404i	0.855973	+	0.517020i	$0.172959\pi$
$239$			15.9859i			1.03404i	−0.855973	+	0.517020i	$0.827041\pi$
$240$		0			0
$241$	−17.0570	−	9.84788i	−1.09874	−	0.634358i	−0.162850	−	0.986651i	$-0.552069\pi$
$241$	−17.0570	−	9.84788i	−1.09874	−	0.634358i	−0.935890	+	0.352293i	$0.885402\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−23.3381	−	15.4601i	−1.49101	−	0.987712i
$246$		0			0
$247$	1.64503	+	2.84927i	0.104671	+	0.181295i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−24.3966			−1.53990			−0.769950	−	0.638104i	$-0.779719\pi$
$251$	−24.3966			−1.53990			−0.769950	+	0.638104i	$0.779719\pi$
$252$		0			0
$253$	3.14512			0.197732
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.71999	−	9.90731i	−0.356803	−	0.618001i	0.630622	−	0.776090i	$-0.282800\pi$
$257$	−5.71999	−	9.90731i	−0.356803	−	0.618001i	−0.987425	+	0.158089i	$0.949467\pi$
$258$		0			0
$259$	0.349015	−	0.328193i	0.0216868	−	0.0203929i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.01050	−	5.20222i	−0.555612	−	0.320782i	0.195771	−	0.980650i	$-0.437279\pi$
$263$	−9.01050	−	5.20222i	−0.555612	−	0.320782i	−0.751382	+	0.659867i	$0.770613\pi$
$264$		0			0
$265$		−	27.5369i		−	1.69158i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.85382	−	8.40706i	0.295943	−	0.512588i	−0.679261	−	0.733897i	$-0.737700\pi$
$269$	4.85382	−	8.40706i	0.295943	−	0.512588i	0.975204	+	0.221309i	$0.0710329\pi$
$270$		0			0
$271$	24.9184	−	14.3867i	1.51369	−	0.873928i	0.513816	−	0.857901i	$-0.328232\pi$
$271$	24.9184	−	14.3867i	1.51369	−	0.873928i	0.999872	−	0.0160271i	$-0.00510181\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−46.5639	+	26.8837i	−2.80791	+	1.62115i
$276$		0			0
$277$	8.56550	−	14.8359i	0.514651	−	0.891401i	−0.485205	−	0.874401i	$-0.661255\pi$
$277$	8.56550	−	14.8359i	0.514651	−	0.891401i	0.999855	−	0.0170007i	$-0.00541176\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	2.61233i		−	0.155838i	−0.996960	−	0.0779191i	$-0.975172\pi$
$281$		−	2.61233i		−	0.155838i	0.996960	−	0.0779191i	$-0.0248276\pi$
$282$		0			0
$283$	2.20363	+	1.27226i	0.130992	+	0.0756283i	0.564064	−	0.825731i	$-0.309237\pi$
$283$	2.20363	+	1.27226i	0.130992	+	0.0756283i	−0.433072	+	0.901359i	$0.642570\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.77088	−	24.5294i	−0.340644	−	1.44792i
$288$		0			0
$289$	−17.0704	−	29.5668i	−1.00414	−	1.73922i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.98621			0.291297			0.145649	−	0.989336i	$-0.453473\pi$
$293$	4.98621			0.291297			0.145649	+	0.989336i	$0.453473\pi$
$294$		0			0
$295$	39.5865			2.30481
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.218282	+	0.378076i	0.0126236	+	0.0218647i
$300$		0			0
$301$	−1.38373	−	5.88160i	−0.0797569	−	0.339010i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−36.6085	−	21.1359i	−2.09619	−	1.21024i
$306$		0			0
$307$		−	0.688473i		−	0.0392932i	−0.999807	−	0.0196466i	$-0.993746\pi$
$307$		−	0.688473i		−	0.0392932i	0.999807	−	0.0196466i	$-0.00625412\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.61579	−	7.99478i	0.261737	−	0.453342i	−0.704966	−	0.709241i	$-0.749038\pi$
$311$	4.61579	−	7.99478i	0.261737	−	0.453342i	0.966704	+	0.255898i	$0.0823712\pi$
$312$		0			0
$313$	−6.79944	+	3.92566i	−0.384327	+	0.221891i	−0.679699	−	0.733491i	$-0.737890\pi$
$313$	−6.79944	+	3.92566i	−0.384327	+	0.221891i	0.295372	+	0.955382i	$0.404556\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	14.7913	−	8.53976i	0.830763	−	0.479641i	−0.0233511	−	0.999727i	$-0.507434\pi$
$317$	14.7913	−	8.53976i	0.830763	−	0.479641i	0.854114	+	0.520086i	$0.174100\pi$
$318$		0			0
$319$	11.9601	−	20.7156i	0.669639	−	1.15985i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	34.6571i		−	1.92837i
$324$		0			0
$325$	−6.46340	−	3.73164i	−0.358525	−	0.206994i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.26536	+	5.89158i	−0.345421	+	0.324813i
$330$		0			0
$331$	−15.9973	−	27.7081i	−0.879291	−	1.52298i	−0.852120	−	0.523346i	$-0.824684\pi$
$331$	−15.9973	−	27.7081i	−0.879291	−	1.52298i	−0.0271706	−	0.999631i	$-0.508650\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	43.1388			2.35692
$336$		0			0
$337$	25.6354			1.39645			0.698226	−	0.715877i	$-0.253973\pi$
$337$	25.6354			1.39645			0.698226	+	0.715877i	$0.253973\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	13.0074	+	22.5294i	0.704389	+	1.22004i
$342$		0			0
$343$	−11.8343	−	14.2460i	−0.638992	−	0.769214i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.60640	+	3.23686i	0.300967	+	0.173764i	0.642877	−	0.765969i	$-0.277740\pi$
$347$	5.60640	+	3.23686i	0.300967	+	0.173764i	−0.341910	+	0.939733i	$0.611074\pi$
$348$		0			0
$349$			27.0147i			1.44606i	0.690815	+	0.723032i	$0.257252\pi$
$349$			27.0147i			1.44606i	−0.690815	+	0.723032i	$0.742748\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15.7745	+	27.3222i	−0.839591	+	1.45421i	0.0506462	+	0.998717i	$0.483872\pi$
$353$	−15.7745	+	27.3222i	−0.839591	+	1.45421i	−0.890237	+	0.455497i	$0.849461\pi$
$354$		0			0
$355$	3.58793	−	2.07149i	0.190428	−	0.109943i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3.82534	−	2.20856i	0.201894	−	0.116563i	−0.395645	−	0.918404i	$-0.629479\pi$
$359$	3.82534	−	2.20856i	0.201894	−	0.116563i	0.597538	+	0.801840i	$0.296146\pi$
$360$		0			0
$361$	2.24320	−	3.88534i	0.118063	−	0.204491i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1.89084i			0.0989709i
$366$		0			0
$367$	−2.36455	−	1.36517i	−0.123429	−	0.0712616i	0.437015	−	0.899454i	$-0.356036\pi$
$367$	−2.36455	−	1.36517i	−0.123429	−	0.0712616i	−0.560443	+	0.828193i	$0.689369\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.25364	−	17.4437i	0.272755	−	0.905630i
$372$		0			0
$373$	2.99871	+	5.19391i	0.155267	+	0.268931i	0.933156	−	0.359471i	$-0.117043\pi$
$373$	2.99871	+	5.19391i	0.155267	+	0.268931i	−0.777889	+	0.628402i	$0.783709\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.32030			0.171004
$378$		0			0
$379$	16.7830			0.862083			0.431042	−	0.902332i	$-0.358146\pi$
$379$	16.7830			0.862083			0.431042	+	0.902332i	$0.358146\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.53825	+	14.7887i	0.436284	+	0.755667i	0.997399	−	0.0720712i	$-0.0229609\pi$
$383$	8.53825	+	14.7887i	0.436284	+	0.755667i	−0.561115	+	0.827738i	$0.689628\pi$
$384$		0			0
$385$	−50.3739	+	11.8512i	−2.56729	+	0.603992i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−30.4364	−	17.5725i	−1.54319	−	0.890959i	−0.998635	−	0.0522323i	$-0.983366\pi$
$389$	−30.4364	−	17.5725i	−1.54319	−	0.890959i	−0.544552	−	0.838727i	$-0.683300\pi$
$390$		0			0
$391$		−	4.59873i		−	0.232568i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.3354	+	28.2937i	−0.821922	+	1.42361i
$396$		0			0
$397$	−3.74191	+	2.16039i	−0.187801	+	0.108427i	−0.590953	−	0.806706i	$-0.701248\pi$
$397$	−3.74191	+	2.16039i	−0.187801	+	0.108427i	0.403152	+	0.915133i	$0.367915\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	25.6036	−	14.7822i	1.27858	−	0.738189i	0.301994	−	0.953310i	$-0.402348\pi$
$401$	25.6036	−	14.7822i	1.27858	−	0.738189i	0.976587	+	0.215121i	$0.0690145\pi$
$402$		0			0
$403$	−1.80552	+	3.12725i	−0.0899392	+	0.155779i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.885619i		−	0.0438985i
$408$		0			0
$409$	27.1261	+	15.6613i	1.34130	+	0.774401i	0.986998	−	0.160730i	$-0.0513849\pi$
$409$	27.1261	+	15.6613i	1.34130	+	0.774401i	0.354303	+	0.935131i	$0.384718\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	25.0767	+	7.55253i	1.23394	+	0.371635i
$414$		0			0
$415$	−5.06995	−	8.78141i	−0.248874	−	0.431063i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.552696			−0.0270010			−0.0135005	−	0.999909i	$-0.504297\pi$
$419$	−0.552696			−0.0270010			−0.0135005	+	0.999909i	$0.504297\pi$
$420$		0			0
$421$	−23.2742			−1.13432			−0.567158	−	0.823609i	$-0.691957\pi$
$421$	−23.2742			−1.13432			−0.567158	+	0.823609i	$0.691957\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	39.3088	+	68.0848i	1.90676	+	3.30260i
$426$		0			0
$427$	−19.1578	−	20.3732i	−0.927110	−	0.985930i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	21.7375	+	12.5501i	1.04706	+	0.604518i	0.921824	−	0.387609i	$-0.126699\pi$
$431$	21.7375	+	12.5501i	1.04706	+	0.604518i	0.125233	+	0.992127i	$0.460032\pi$
$432$		0			0
$433$		−	17.3795i		−	0.835204i	−0.908630	−	0.417602i	$-0.862871\pi$
$433$		−	17.3795i		−	0.835204i	0.908630	−	0.417602i	$-0.137129\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.55823	−	2.69894i	0.0745404	−	0.129108i
$438$		0			0
$439$	−3.78895	+	2.18755i	−0.180837	+	0.104406i	−0.587686	−	0.809089i	$-0.699961\pi$
$439$	−3.78895	+	2.18755i	−0.180837	+	0.104406i	0.406849	+	0.913495i	$0.366627\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.1887	−	11.6559i	0.959193	−	0.553790i	0.0632686	−	0.997997i	$-0.479848\pi$
$443$	20.1887	−	11.6559i	0.959193	−	0.553790i	0.895925	+	0.444206i	$0.146514\pi$
$444$		0			0
$445$	35.6326	−	61.7174i	1.68915	−	2.92569i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			24.0524i			1.13510i	0.823338	+	0.567551i	$0.192109\pi$
$449$			24.0524i			1.13510i	−0.823338	+	0.567551i	$0.807891\pi$
$450$		0			0
$451$	−40.3412	−	23.2910i	−1.89959	−	1.09673i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.92076	−	5.23296i	−0.230689	−	0.245325i
$456$		0			0
$457$	0.467757	+	0.810180i	0.0218808	+	0.0378986i	0.876758	−	0.480931i	$-0.159701\pi$
$457$	0.467757	+	0.810180i	0.0218808	+	0.0378986i	−0.854878	+	0.518830i	$0.826368\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.93609			0.369621			0.184810	−	0.982774i	$-0.440833\pi$
$461$	7.93609			0.369621			0.184810	+	0.982774i	$0.440833\pi$
$462$		0			0
$463$	−0.0963291			−0.00447679			−0.00223840	−	0.999997i	$-0.500713\pi$
$463$	−0.0963291			−0.00447679			−0.00223840	+	0.999997i	$0.500713\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.6205	−	25.3234i	−0.676555	−	1.17183i	−0.976012	−	0.217718i	$-0.930139\pi$
$467$	−14.6205	−	25.3234i	−0.676555	−	1.17183i	0.299456	−	0.954110i	$-0.403195\pi$
$468$		0			0
$469$	27.3269	+	8.23026i	1.26184	+	0.380038i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−9.67294	−	5.58467i	−0.444762	−	0.256784i
$474$		0			0
$475$			53.2775i			2.44454i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.93986	+	8.55609i	−0.225708	+	0.390938i	−0.956532	−	0.291629i	$-0.905803\pi$
$479$	−4.93986	+	8.55609i	−0.225708	+	0.390938i	0.730824	+	0.682566i	$0.239136\pi$
$480$		0			0
$481$	0.106461	−	0.0614650i	0.00485418	−	0.00280256i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.6346	−	6.13990i	0.482893	−	0.278799i
$486$		0			0
$487$	4.10907	−	7.11712i	0.186200	−	0.322507i	−0.757780	−	0.652510i	$-0.773716\pi$
$487$	4.10907	−	7.11712i	0.186200	−	0.322507i	0.943980	+	0.330002i	$0.107050\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			14.0806i			0.635450i	0.948183	+	0.317725i	$0.102919\pi$
$491$			14.0806i			0.635450i	−0.948183	+	0.317725i	$0.897081\pi$
$492$		0			0
$493$	−30.2899	−	17.4879i	−1.36419	−	0.787614i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.66804	−	0.627694i	0.119678	−	0.0281559i
$498$		0			0
$499$	8.08403	+	14.0019i	0.361891	+	0.626813i	0.988272	−	0.152704i	$-0.0487980\pi$
$499$	8.08403	+	14.0019i	0.361891	+	0.626813i	−0.626381	+	0.779517i	$0.715465\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	36.8914			1.64491			0.822453	−	0.568834i	$-0.192605\pi$
$503$	36.8914			1.64491			0.822453	+	0.568834i	$0.192605\pi$
$504$		0			0
$505$	26.1269			1.16263
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.90414	+	13.6904i	0.350345	+	0.606815i	0.986310	−	0.164903i	$-0.0527311\pi$
$509$	7.90414	+	13.6904i	0.350345	+	0.606815i	−0.635965	+	0.771718i	$0.719398\pi$
$510$		0			0
$511$	−0.360744	+	1.19778i	−0.0159584	+	0.0529867i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−62.8825	−	36.3053i	−2.77094	−	1.59980i
$516$		0			0
$517$			15.8982i			0.699203i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	7.71479	−	13.3624i	0.337991	−	0.585417i	−0.646064	−	0.763283i	$-0.723586\pi$
$521$	7.71479	−	13.3624i	0.337991	−	0.585417i	0.984055	+	0.177866i	$0.0569193\pi$
$522$		0			0
$523$	−27.0967	+	15.6443i	−1.18486	+	0.684077i	−0.957133	−	0.289649i	$-0.906461\pi$
$523$	−27.0967	+	15.6443i	−1.18486	+	0.684077i	−0.227723	+	0.973726i	$0.573128\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	32.9421	−	19.0191i	1.43498	−	0.828486i
$528$		0			0
$529$	−11.2932	+	19.5605i	−0.491010	+	0.850455i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	6.46591i		−	0.280070i
$534$		0			0
$535$	−30.0408	−	17.3441i	−1.29878	−	0.749850i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−34.1712	−	2.10330i	−1.47186	−	0.0905955i
$540$		0			0
$541$	−2.65929	−	4.60603i	−0.114332	−	0.198029i	0.803181	−	0.595736i	$-0.203139\pi$
$541$	−2.65929	−	4.60603i	−0.114332	−	0.198029i	−0.917512	+	0.397707i	$0.869806\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.69418			−0.0725708
$546$		0			0
$547$	14.0296			0.599864			0.299932	−	0.953961i	$-0.403036\pi$
$547$	14.0296			0.599864			0.299932	+	0.953961i	$0.403036\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−11.8512	−	20.5268i	−0.504877	−	0.874473i
$552$		0			0
$553$	−15.7459	+	14.8065i	−0.669585	+	0.629638i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.06802	−	3.50337i	−0.257110	−	0.148443i	0.365905	−	0.930652i	$-0.380759\pi$
$557$	−6.06802	−	3.50337i	−0.257110	−	0.148443i	−0.623016	+	0.782209i	$0.714093\pi$
$558$		0			0
$559$		−	1.55038i		−	0.0655743i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.5937	+	18.3488i	−0.446471	+	0.773311i	−0.998153	−	0.0607435i	$-0.980653\pi$
$563$	−10.5937	+	18.3488i	−0.446471	+	0.773311i	0.551682	+	0.834054i	$0.313986\pi$
$564$		0			0
$565$	34.9895	−	20.2012i	1.47202	−	0.849871i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	27.1180	−	15.6566i	1.13685	−	0.656358i	0.191198	−	0.981551i	$-0.438763\pi$
$569$	27.1180	−	15.6566i	1.13685	−	0.656358i	0.945648	+	0.325193i	$0.105429\pi$
$570$		0			0
$571$	−11.5820	+	20.0606i	−0.484690	+	0.839508i	−0.999845	−	0.0175887i	$-0.994401\pi$
$571$	−11.5820	+	20.0606i	−0.484690	+	0.839508i	0.515155	+	0.857097i	$0.327734\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			7.06952i			0.294819i
$576$		0			0
$577$	17.2908	+	9.98287i	0.719827	+	0.415592i	0.814689	−	0.579898i	$-0.196908\pi$
$577$	17.2908	+	9.98287i	0.719827	+	0.415592i	−0.0948620	+	0.995490i	$0.530241\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.53627	−	6.53000i	−0.0637354	−	0.270910i
$582$		0			0
$583$	−16.8382	−	29.1646i	−0.697368	−	1.20788i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−22.7290			−0.938127			−0.469063	−	0.883164i	$-0.655408\pi$
$587$	−22.7290			−0.938127			−0.469063	+	0.883164i	$0.655408\pi$
$588$		0			0
$589$	25.7777			1.06215
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.90166	+	17.1502i	0.406612	+	0.704273i	0.994508	−	0.104664i	$-0.0333767\pi$
$593$	9.90166	+	17.1502i	0.406612	+	0.704273i	−0.587895	+	0.808937i	$0.700043\pi$
$594$		0			0
$595$	17.3285	+	73.6557i	0.710401	+	3.01959i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−13.6917	−	7.90491i	−0.559428	−	0.322986i	0.193488	−	0.981103i	$-0.438020\pi$
$599$	−13.6917	−	7.90491i	−0.559428	−	0.322986i	−0.752916	+	0.658117i	$0.771353\pi$
$600$		0			0
$601$			32.2987i			1.31749i	0.752366	+	0.658746i	$0.228913\pi$
$601$			32.2987i			1.31749i	−0.752366	+	0.658746i	$0.771087\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−25.8353	+	44.7481i	−1.05035	+	1.81927i
$606$		0			0
$607$	1.26070	−	0.727866i	0.0511703	−	0.0295432i	−0.474197	−	0.880419i	$-0.657261\pi$
$607$	1.26070	−	0.727866i	0.0511703	−	0.0295432i	0.525367	+	0.850876i	$0.323928\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.91113	+	1.10339i	−0.0773161	+	0.0446385i
$612$		0			0
$613$	−1.57946	+	2.73570i	−0.0637936	+	0.110494i	−0.896158	−	0.443735i	$-0.853653\pi$
$613$	−1.57946	+	2.73570i	−0.0637936	+	0.110494i	0.832365	+	0.554228i	$0.186987\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	28.7984i		−	1.15938i	−0.814837	−	0.579691i	$-0.803173\pi$
$617$		−	28.7984i		−	1.15938i	0.814837	−	0.579691i	$-0.196827\pi$
$618$		0			0
$619$	20.3013	+	11.7210i	0.815979	+	0.471106i	0.849028	−	0.528348i	$-0.177188\pi$
$619$	20.3013	+	11.7210i	0.815979	+	0.471106i	−0.0330487	+	0.999454i	$0.510522\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	34.3468	−	32.2977i	1.37608	−	1.29398i
$624$		0			0
$625$	−20.4447	−	35.4112i	−0.817787	−	1.41645i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.29493			−0.0516324
$630$		0			0
$631$	41.0358			1.63361			0.816804	−	0.576915i	$-0.195744\pi$
$631$	41.0358			1.63361			0.816804	+	0.576915i	$0.195744\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−25.1935	−	43.6364i	−0.999773	−	1.73166i
$636$		0			0
$637$	−2.11876	−	4.25371i	−0.0839484	−	0.168538i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−13.2162	−	7.63040i	−0.522010	−	0.301383i	0.215747	−	0.976449i	$-0.430781\pi$
$641$	−13.2162	−	7.63040i	−0.522010	−	0.301383i	−0.737757	+	0.675067i	$0.764115\pi$
$642$		0			0
$643$			30.2829i			1.19424i	0.802151	+	0.597121i	$0.203689\pi$
$643$			30.2829i			1.19424i	−0.802151	+	0.597121i	$0.796311\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.17732	+	5.50328i	−0.124913	+	0.216356i	−0.921699	−	0.387906i	$-0.873199\pi$
$647$	−3.17732	+	5.50328i	−0.124913	+	0.216356i	0.796786	+	0.604262i	$0.206532\pi$
$648$		0			0
$649$	41.9265	−	24.2063i	1.64576	−	0.950180i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.14653	+	1.23930i	−0.0840002	+	0.0484975i	−0.541412	−	0.840758i	$-0.682110\pi$
$653$	−2.14653	+	1.23930i	−0.0840002	+	0.0484975i	0.457412	+	0.889255i	$0.348777\pi$
$654$		0			0
$655$	13.4448	−	23.2870i	0.525331	−	0.909899i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			36.8651i			1.43606i	0.696011	+	0.718031i	$0.254956\pi$
$659$			36.8651i			1.43606i	−0.696011	+	0.718031i	$0.745044\pi$
$660$		0			0
$661$	16.8816	+	9.74661i	0.656619	+	0.379099i	0.790988	−	0.611832i	$-0.209567\pi$
$661$	16.8816	+	9.74661i	0.656619	+	0.379099i	−0.134369	+	0.990931i	$0.542901\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−14.7876	+	49.0992i	−0.573437	+	1.90399i
$666$		0			0
$667$	−1.57256	−	2.72375i	−0.0608898	−	0.105464i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−51.6967			−1.99573
$672$		0			0
$673$	−19.7560			−0.761538			−0.380769	−	0.924670i	$-0.624341\pi$
$673$	−19.7560			−0.761538			−0.380769	+	0.924670i	$0.624341\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.42478	−	7.66394i	−0.170058	−	0.294549i	0.768382	−	0.639992i	$-0.221062\pi$
$677$	−4.42478	−	7.66394i	−0.170058	−	0.294549i	−0.938440	+	0.345442i	$0.887729\pi$
$678$		0			0
$679$	7.90807	−	1.86049i	0.303484	−	0.0713989i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	2.07247	+	1.19654i	0.0793008	+	0.0457843i	0.539126	−	0.842225i	$-0.318755\pi$
$683$	2.07247	+	1.19654i	0.0793008	+	0.0457843i	−0.459825	+	0.888009i	$0.652088\pi$
$684$		0			0
$685$			54.5299i			2.08348i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.33726	−	4.04826i	0.0890426	−	0.154226i
$690$		0			0
$691$	−1.87166	+	1.08060i	−0.0712013	+	0.0411081i	−0.535178	−	0.844739i	$-0.679755\pi$
$691$	−1.87166	+	1.08060i	−0.0712013	+	0.0411081i	0.463977	+	0.885847i	$0.346422\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−66.0454	+	38.1314i	−2.50525	+	1.44640i
$696$		0			0
$697$	−34.0556	+	58.9861i	−1.28995	+	2.23426i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			32.7240i			1.23597i	0.786191	+	0.617984i	$0.212050\pi$
$701$			32.7240i			1.23597i	−0.786191	+	0.617984i	$0.787950\pi$
$702$		0			0
$703$	−0.759980	−	0.438775i	−0.0286632	−	0.0165487i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	16.5505	+	4.98463i	0.622445	+	0.187466i
$708$		0			0
$709$	16.7348	+	28.9856i	0.628490	+	1.08858i	0.987855	+	0.155380i	$0.0496601\pi$
$709$	16.7348	+	28.9856i	0.628490	+	1.08858i	−0.359365	+	0.933197i	$0.617007\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.42051			0.128099
$714$		0			0
$715$	−13.2785			−0.496588
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.2023	−	26.3312i	−0.566952	−	0.981989i	−0.996865	−	0.0791190i	$-0.974789\pi$
$719$	−15.2023	−	26.3312i	−0.566952	−	0.981989i	0.429914	−	0.902870i	$-0.358544\pi$
$720$		0			0
$721$	−32.9074	−	34.9952i	−1.22554	−	1.30329i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	46.5639	+	26.8837i	1.72934	+	0.998435i
$726$		0			0
$727$		−	12.4585i		−	0.462061i	−0.972946	−	0.231031i	$-0.925790\pi$
$727$		−	12.4585i		−	0.462061i	0.972946	−	0.231031i	$-0.0742098\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.16580	+	14.1436i	−0.302023	+	0.523119i
$732$		0			0
$733$	2.77872	−	1.60430i	0.102634	−	0.0592560i	−0.447804	−	0.894132i	$-0.647794\pi$
$733$	2.77872	−	1.60430i	0.102634	−	0.0592560i	0.550439	+	0.834876i	$0.314460\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	45.6888	−	26.3785i	1.68297	−	0.971663i
$738$		0			0
$739$	−3.11747	+	5.39962i	−0.114678	+	0.198628i	−0.917651	−	0.397387i	$-0.869917\pi$
$739$	−3.11747	+	5.39962i	−0.114678	+	0.198628i	0.802973	+	0.596015i	$0.203250\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	44.5451i		−	1.63420i	−0.576495	−	0.817101i	$-0.695580\pi$
$743$		−	44.5451i		−	1.63420i	0.576495	−	0.817101i	$-0.304420\pi$
$744$		0			0
$745$	57.3616	+	33.1177i	2.10157	+	1.21334i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−15.7208	−	16.7182i	−0.574427	−	0.610871i
$750$		0			0
$751$	8.74572	+	15.1480i	0.319136	+	0.552759i	0.980308	−	0.197475i	$-0.0632741\pi$
$751$	8.74572	+	15.1480i	0.319136	+	0.552759i	−0.661172	+	0.750234i	$0.729941\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	33.9849			1.23684
$756$		0			0
$757$	−25.5174			−0.927445			−0.463722	−	0.885981i	$-0.653486\pi$
$757$	−25.5174			−0.927445			−0.463722	+	0.885981i	$0.653486\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.89391	+	15.4047i	0.322404	+	0.558420i	0.980984	−	0.194091i	$-0.0621757\pi$
$761$	8.89391	+	15.4047i	0.322404	+	0.558420i	−0.658580	+	0.752511i	$0.728842\pi$
$762$		0			0
$763$	−1.07321	−	0.323225i	−0.0388527	−	0.0117015i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.81970	+	3.36000i	0.210137	+	0.121323i
$768$		0			0
$769$			38.3951i			1.38456i	0.721627	+	0.692282i	$0.243394\pi$
$769$			38.3951i			1.38456i	−0.721627	+	0.692282i	$0.756606\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13.7588	−	23.8310i	0.494871	−	0.857142i	−0.505111	−	0.863054i	$-0.668549\pi$
$773$	13.7588	−	23.8310i	0.494871	−	0.857142i	0.999983	+	0.00591211i	$0.00188189\pi$
$774$		0			0
$775$	−50.6411	+	29.2376i	−1.81908	+	1.05025i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−39.9737	+	23.0788i	−1.43221	+	0.826884i
$780$		0			0
$781$	2.53335	−	4.38789i	0.0906503	−	0.157011i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	68.6094i		−	2.44877i
$786$		0			0
$787$	−11.6948	−	6.75200i	−0.416875	−	0.240683i	0.276864	−	0.960909i	$-0.410705\pi$
$787$	−11.6948	−	6.75200i	−0.416875	−	0.240683i	−0.693739	+	0.720226i	$0.744038\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	26.0187	−	6.12128i	0.925120	−	0.217648i
$792$		0			0
$793$	−3.58793	−	6.21448i	−0.127411	−	0.220683i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−32.1072			−1.13729			−0.568647	−	0.822582i	$-0.692533\pi$
$797$	−32.1072			−1.13729			−0.568647	+	0.822582i	$0.692533\pi$
$798$		0			0
$799$	23.2461			0.822386
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.15621	+	2.00261i	0.0408017	+	0.0706705i
$804$		0			0
$805$	−1.96220	+	6.51509i	−0.0691584	+	0.229627i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.84513	+	1.06529i	0.0648713	+	0.0374534i	0.532085	−	0.846691i	$-0.321409\pi$
$809$	1.84513	+	1.06529i	0.0648713	+	0.0374534i	−0.467214	+	0.884145i	$0.654742\pi$
$810$		0			0
$811$		−	18.7062i		−	0.656864i	−0.944528	−	0.328432i	$-0.893480\pi$
$811$		−	18.7062i		−	0.656864i	0.944528	−	0.328432i	$-0.106520\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	46.4731	−	80.4937i	1.62788	−	2.81957i
$816$		0			0
$817$	−9.58481	+	5.53379i	−0.335330	+	0.193603i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	20.6223	−	11.9063i	0.719724	−	0.415533i	−0.0949272	−	0.995484i	$-0.530262\pi$
$821$	20.6223	−	11.9063i	0.719724	−	0.415533i	0.814651	+	0.579951i	$0.196929\pi$
$822$		0			0
$823$	−3.91318	+	6.77782i	−0.136405	+	0.236260i	−0.926133	−	0.377197i	$-0.876888\pi$
$823$	−3.91318	+	6.77782i	−0.136405	+	0.236260i	0.789728	+	0.613457i	$0.210221\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			5.94710i			0.206801i	0.994640	+	0.103400i	$0.0329723\pi$
$827$			5.94710i			0.206801i	−0.994640	+	0.103400i	$0.967028\pi$
$828$		0			0
$829$	−10.5009	−	6.06269i	−0.364711	−	0.210566i	0.306435	−	0.951892i	$-0.400864\pi$
$829$	−10.5009	−	6.06269i	−0.364711	−	0.210566i	−0.671145	+	0.741326i	$0.734197\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.07540	+	49.9644i	−0.106556	+	1.73116i
$834$		0			0
$835$	21.1327	+	36.6028i	0.731326	+	1.26669i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	29.5941			1.02170			0.510851	−	0.859669i	$-0.329330\pi$
$839$	29.5941			1.02170			0.510851	+	0.859669i	$0.329330\pi$
$840$		0			0
$841$	5.07972			0.175163
$842$		0			0
$843$		0			0
$844$		0			0
$845$	25.0731	+	43.4279i	0.862542	+	1.49397i
$846$		0			0
$847$	−24.9031	+	23.4174i	−0.855680	+	0.804630i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.100844	−	0.0582221i	−0.00345687	−	0.00199583i
$852$		0			0
$853$			5.00490i			0.171364i	0.996323	+	0.0856822i	$0.0273070\pi$
$853$			5.00490i			0.171364i	−0.996323	+	0.0856822i	$0.972693\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22.3412	+	38.6961i	−0.763160	+	1.32183i	0.178053	+	0.984021i	$0.443020\pi$
$857$	−22.3412	+	38.6961i	−0.763160	+	1.32183i	−0.941214	+	0.337812i	$0.890313\pi$
$858$		0			0
$859$	17.4974	−	10.1021i	0.597003	−	0.344680i	−0.170858	−	0.985296i	$-0.554654\pi$
$859$	17.4974	−	10.1021i	0.597003	−	0.344680i	0.767862	+	0.640616i	$0.221321\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−25.2738	+	14.5919i	−0.860331	+	0.496713i	−0.864123	−	0.503280i	$-0.832126\pi$
$863$	−25.2738	+	14.5919i	−0.860331	+	0.496713i	0.00379192	+	0.999993i	$0.498793\pi$
$864$		0			0
$865$	0.370343	−	0.641453i	0.0125920	−	0.0218100i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			39.9549i			1.35538i
$870$		0			0
$871$	6.34193	+	3.66152i	0.214888	+	0.124066i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−14.5231	−	61.7308i	−0.490969	−	2.08688i
$876$		0			0
$877$	4.27851	+	7.41060i	0.144475	+	0.250238i	0.929177	−	0.369635i	$-0.120517\pi$
$877$	4.27851	+	7.41060i	0.144475	+	0.250238i	−0.784702	+	0.619873i	$0.787184\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.2255			0.883561			0.441781	−	0.897123i	$-0.354347\pi$
$881$	26.2255			0.883561			0.441781	+	0.897123i	$0.354347\pi$
$882$		0			0
$883$	49.3518			1.66082			0.830410	−	0.557153i	$-0.188106\pi$
$883$	49.3518			1.66082			0.830410	+	0.557153i	$0.188106\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.96343	+	8.59691i	0.166656	+	0.288656i	0.937242	−	0.348680i	$-0.113370\pi$
$887$	4.96343	+	8.59691i	0.166656	+	0.288656i	−0.770586	+	0.637336i	$0.780037\pi$
$888$		0			0
$889$	−7.63402	−	32.4487i	−0.256037	−	1.08829i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	13.6428	+	7.87669i	0.456540	+	0.263583i
$894$		0			0
$895$			51.3114i			1.71515i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	13.0074	−	22.5294i	0.433820	−	0.751399i
$900$		0			0
$901$	−42.6440	+	24.6205i	−1.42068	+	0.820228i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−66.0454	+	38.1314i	−2.19543	+	1.26753i
$906$		0			0
$907$	7.92790	−	13.7315i	0.263242	−	0.455948i	−0.703860	−	0.710339i	$-0.748542\pi$
$907$	7.92790	−	13.7315i	0.263242	−	0.455948i	0.967101	+	0.254391i	$0.0818750\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			7.46438i			0.247306i	0.992326	+	0.123653i	$0.0394610\pi$
$911$			7.46438i			0.247306i	−0.992326	+	0.123653i	$0.960539\pi$
$912$		0			0
$913$	−10.7393	−	6.20034i	−0.355419	−	0.205201i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12.9596	−	12.1865i	0.427965	−	0.402433i
$918$		0			0
$919$	1.70138	+	2.94688i	0.0561233	+	0.0972085i	0.892722	−	0.450608i	$-0.148793\pi$
$919$	1.70138	+	2.94688i	0.0561233	+	0.0972085i	−0.836599	+	0.547816i	$0.815459\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.703293			0.0231492
$924$		0			0
$925$	1.99067			0.0654528
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−23.3809	−	40.4968i	−0.767101	−	1.32866i	−0.939128	−	0.343566i	$-0.888365\pi$
$929$	−23.3809	−	40.4968i	−0.767101	−	1.32866i	0.172027	−	0.985092i	$-0.444968\pi$
$930$		0			0
$931$	−18.7348	+	28.2814i	−0.614010	+	0.926886i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	121.135	+	69.9373i	3.96153	+	2.28719i
$936$		0			0
$937$			14.2702i			0.466187i	0.972454	+	0.233093i	$0.0748848\pi$
$937$			14.2702i			0.466187i	−0.972454	+	0.233093i	$0.925115\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	2.70002	−	4.67658i	0.0880183	−	0.152452i	−0.818655	−	0.574285i	$-0.805280\pi$
$941$	2.70002	−	4.67658i	0.0880183	−	0.152452i	0.906673	+	0.421833i	$0.138613\pi$
$942$		0			0
$943$	−5.30420	+	3.06238i	−0.172729	+	0.0997249i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−40.0390	+	23.1165i	−1.30109	+	0.751186i	−0.980591	−	0.196062i	$-0.937184\pi$
$947$	−40.0390	+	23.1165i	−1.30109	+	0.751186i	−0.320501	+	0.947248i	$0.603851\pi$
$948$		0			0
$949$	−0.160490	+	0.277976i	−0.00520971	+	0.00902349i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	8.50095i		−	0.275373i	−0.990476	−	0.137686i	$-0.956033\pi$
$953$		−	8.50095i		−	0.275373i	0.990476	−	0.137686i	$-0.0439666\pi$
$954$		0			0
$955$	31.8899	+	18.4117i	1.03193	+	0.595787i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.4035	+	34.5429i	−0.335947	+	1.11545i
$960$		0			0
$961$	−1.35368	−	2.34464i	−0.0436671	−	0.0756337i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	25.5455			0.822339
$966$		0			0
$967$	19.4198			0.624497			0.312249	−	0.950000i	$-0.398918\pi$
$967$	19.4198			0.624497			0.312249	+	0.950000i	$0.398918\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.34427	−	7.52449i	−0.139414	−	0.241472i	0.787861	−	0.615853i	$-0.211189\pi$
$971$	−4.34427	−	7.52449i	−0.139414	−	0.241472i	−0.927275	+	0.374381i	$0.877855\pi$
$972$		0			0
$973$	−49.1124	+	11.5544i	−1.57447	+	0.370417i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.2821	+	12.8646i	0.712869	+	0.411575i	0.812122	−	0.583487i	$-0.198312\pi$
$977$	22.2821	+	12.8646i	0.712869	+	0.411575i	−0.0992537	+	0.995062i	$0.531646\pi$
$978$		0			0
$979$		−	87.1542i		−	2.78546i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	5.51952	−	9.56009i	0.176045	−	0.304920i	−0.764477	−	0.644651i	$-0.777003\pi$
$983$	5.51952	−	9.56009i	0.176045	−	0.304920i	0.940523	+	0.339731i	$0.110336\pi$
$984$		0			0
$985$	10.2212	−	5.90122i	0.325675	−	0.188028i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.27183	+	0.734292i	−0.0404419	+	0.0233491i
$990$		0			0
$991$	−7.68283	+	13.3070i	−0.244053	+	0.422712i	−0.961865	−	0.273525i	$-0.911810\pi$
$991$	−7.68283	+	13.3070i	−0.244053	+	0.422712i	0.717812	+	0.696237i	$0.245144\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			11.2066i			0.355273i
$996$		0			0
$997$	−54.1503	−	31.2637i	−1.71496	−	0.990132i	−0.927549	−	0.373702i	$-0.878088\pi$
$997$	−54.1503	−	31.2637i	−1.71496	−	0.990132i	−0.787410	−	0.616430i	$-0.788578\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.bl.b.1025.8	yes	16
3.2	odd	2	inner	1512.2.bl.b.1025.1	yes	16
7.5	odd	6	inner	1512.2.bl.b.593.1	✓	16
21.5	even	6	inner	1512.2.bl.b.593.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.bl.b.593.1	✓	16	7.5	odd	6	inner
1512.2.bl.b.593.8	yes	16	21.5	even	6	inner
1512.2.bl.b.1025.1	yes	16	3.2	odd	2	inner
1512.2.bl.b.1025.8	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.bl (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.99959 + 3.46340i) q^{5} +(0.605908 + 2.57544i) q^{7} +(4.23559 + 2.44542i) q^{11} +0.678883i q^{13} +(3.57564 - 6.19319i) q^{17} +(4.19700 - 2.42314i) q^{19} +(0.556910 - 0.321532i) q^{23} +(-5.49674 + 9.52064i) q^{25} -4.89084i q^{29} +(4.60646 + 2.65954i) q^{31} +(-7.70819 + 7.24833i) q^{35} +(-0.0905386 - 0.156817i) q^{37} -9.52435 q^{41} -2.28373 q^{43} +(1.62531 + 2.81511i) q^{47} +(-6.26575 + 3.12096i) q^{49} +(-5.96312 - 3.44281i) q^{53} +19.5594i q^{55} +(4.94932 - 8.57247i) q^{59} +(-9.15398 + 5.28505i) q^{61} +(-2.35124 + 1.35749i) q^{65} +(5.39345 - 9.34172i) q^{67} -1.03596i q^{71} +(0.409461 + 0.236403i) q^{73} +(-3.73164 + 12.3902i) q^{77} +(4.08467 + 7.07486i) q^{79} -2.53549 q^{83} +28.5993 q^{85} +(-8.90995 - 15.4325i) q^{89} +(-1.74842 + 0.411340i) q^{91} +(16.7846 + 9.69058i) q^{95} -3.07058i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{7} + 6 q^{19} - 6 q^{25} + 6 q^{31} - 8 q^{37} + 44 q^{43} - 34 q^{49} - 6 q^{61} + 20 q^{67} - 14 q^{79} + 148 q^{85} - 54 q^{91}+O(q^{100}) \) 16 * q - 2 * q^7 + 6 * q^19 - 6 * q^25 + 6 * q^31 - 8 * q^37 + 44 * q^43 - 34 * q^49 - 6 * q^61 + 20 * q^67 - 14 * q^79 + 148 * q^85 - 54 * q^91

Introduction

L-functions

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