LMFDB - Embedded newform 2576.2.e.a.1471.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2576,2,Mod(1471,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("2576.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2576.e (of order $2$, degree $1$, 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:	$20.5694635607$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 22x^{10} + 165x^{8} + 508x^{6} + 582x^{4} + 136x^{2} + 2 $ x^12 + 22x^10 + 165x^8 + 508x^6 + 582x^4 + 136*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.12
Root			$3.17144i$ of defining polynomial
Character	$\chi$	$=$	2576.1471
Dual form			2576.2.e.a.1471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.17144i q^{3} +2.64852i q^{5} -1.00000 q^{7} -7.05804 q^{9} +O(q^{10})$	$q+3.17144i q^{3} +2.64852i q^{5} -1.00000 q^{7} -7.05804 q^{9} -1.54103 q^{11} -4.59567 q^{13} -8.39962 q^{15} +5.61697i q^{17} -5.00340 q^{19} -3.17144i q^{21} +(1.34158 + 4.60436i) q^{23} -2.01465 q^{25} -12.8699i q^{27} +10.2148 q^{29} +1.32522i q^{31} -4.88728i q^{33} -2.64852i q^{35} -11.8894i q^{37} -14.5749i q^{39} +5.27882 q^{41} +6.41087 q^{43} -18.6934i q^{45} +7.08691i q^{47} +1.00000 q^{49} -17.8139 q^{51} +10.9176i q^{53} -4.08144i q^{55} -15.8680i q^{57} +2.40454i q^{59} -6.48598i q^{61} +7.05804 q^{63} -12.1717i q^{65} -10.4969 q^{67} +(-14.6025 + 4.25473i) q^{69} +3.90680i q^{71} +15.2856 q^{73} -6.38934i q^{75} +1.54103 q^{77} -10.1748 q^{79} +19.6419 q^{81} -12.9540 q^{83} -14.8766 q^{85} +32.3957i q^{87} -2.37994i q^{89} +4.59567 q^{91} -4.20285 q^{93} -13.2516i q^{95} +3.86958i q^{97} +10.8766 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} - 8 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 - 8 * q^9	$ 12 q - 12 q^{7} - 8 q^{9} - 12 q^{13} - 8 q^{15} - 8 q^{19} - 4 q^{25} + 12 q^{29} - 12 q^{41} + 16 q^{43} + 12 q^{49} - 8 q^{51} + 8 q^{63} - 40 q^{67} - 28 q^{69} + 4 q^{73} - 16 q^{79} + 20 q^{81} - 48 q^{83} - 32 q^{85} + 12 q^{91} + 4 q^{93} - 16 q^{99}+O(q^{100}) $ 12 * q - 12 * q^7 - 8 * q^9 - 12 * q^13 - 8 * q^15 - 8 * q^19 - 4 * q^25 + 12 * q^29 - 12 * q^41 + 16 * q^43 + 12 * q^49 - 8 * q^51 + 8 * q^63 - 40 * q^67 - 28 * q^69 + 4 * q^73 - 16 * q^79 + 20 * q^81 - 48 * q^83 - 32 * q^85 + 12 * q^91 + 4 * q^93 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2576\mathbb{Z}\right)^\times$.

$n$	$645$	$1473$	$1569$	$2255$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$			3.17144i			1.83103i	0.402281	+	0.915516i	$0.368218\pi$
$3$			3.17144i			1.83103i	−0.402281	+	0.915516i	$0.631782\pi$
$4$		0			0
$5$			2.64852i			1.18445i	0.805771	+	0.592227i	$0.201751\pi$
$5$			2.64852i			1.18445i	−0.805771	+	0.592227i	$0.798249\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$	−7.05804			−2.35268
$10$		0			0
$11$	−1.54103			−0.464637			−0.232319	−	0.972640i	$-0.574631\pi$
$11$	−1.54103			−0.464637			−0.232319	+	0.972640i	$0.574631\pi$
$12$		0			0
$13$	−4.59567			−1.27461			−0.637305	−	0.770612i	$-0.719951\pi$
$13$	−4.59567			−1.27461			−0.637305	+	0.770612i	$0.719951\pi$
$14$		0			0
$15$	−8.39962			−2.16877
$16$		0			0
$17$			5.61697i			1.36232i	0.732137	+	0.681158i	$0.238523\pi$
$17$			5.61697i			1.36232i	−0.732137	+	0.681158i	$0.761477\pi$
$18$		0			0
$19$	−5.00340			−1.14786			−0.573930	−	0.818905i	$-0.694582\pi$
$19$	−5.00340			−1.14786			−0.573930	+	0.818905i	$0.694582\pi$
$20$		0			0
$21$		−	3.17144i		−	0.692065i
$22$		0			0
$23$	1.34158	+	4.60436i	0.279738	+	0.960076i
$23$	1.34158	+	4.60436i	0.279738	+	0.960076i
$24$		0			0
$25$	−2.01465			−0.402930
$26$		0			0
$27$		−	12.8699i		−	2.47680i
$28$		0			0
$29$	10.2148			1.89684			0.948422	−	0.317010i	$-0.102679\pi$
$29$	10.2148			1.89684			0.948422	+	0.317010i	$0.102679\pi$
$30$		0			0
$31$			1.32522i			0.238016i	0.992893	+	0.119008i	$0.0379715\pi$
$31$			1.32522i			0.238016i	−0.992893	+	0.119008i	$0.962029\pi$
$32$		0			0
$33$		−	4.88728i		−	0.850766i
$34$		0			0
$35$		−	2.64852i		−	0.447681i
$36$		0			0
$37$		−	11.8894i		−	1.95460i	−0.211861	−	0.977300i	$-0.567952\pi$
$37$		−	11.8894i		−	1.95460i	0.211861	−	0.977300i	$-0.432048\pi$
$38$		0			0
$39$		−	14.5749i		−	2.33385i
$40$		0			0
$41$	5.27882			0.824414			0.412207	−	0.911090i	$-0.364758\pi$
$41$	5.27882			0.824414			0.412207	+	0.911090i	$0.364758\pi$
$42$		0			0
$43$	6.41087			0.977648			0.488824	−	0.872382i	$-0.337426\pi$
$43$	6.41087			0.977648			0.488824	+	0.872382i	$0.337426\pi$
$44$		0			0
$45$		−	18.6934i		−	2.78664i
$46$		0			0
$47$			7.08691i			1.03373i	0.856066	+	0.516866i	$0.172901\pi$
$47$			7.08691i			1.03373i	−0.856066	+	0.516866i	$0.827099\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	−17.8139			−2.49444
$52$		0			0
$53$			10.9176i			1.49964i	0.661641	+	0.749820i	$0.269860\pi$
$53$			10.9176i			1.49964i	−0.661641	+	0.749820i	$0.730140\pi$
$54$		0			0
$55$		−	4.08144i		−	0.550341i
$56$		0			0
$57$		−	15.8680i		−	2.10177i
$58$		0			0
$59$			2.40454i			0.313044i	0.987674	+	0.156522i	$0.0500282\pi$
$59$			2.40454i			0.313044i	−0.987674	+	0.156522i	$0.949972\pi$
$60$		0			0
$61$		−	6.48598i		−	0.830444i	−0.909720	−	0.415222i	$-0.863704\pi$
$61$		−	6.48598i		−	0.830444i	0.909720	−	0.415222i	$-0.136296\pi$
$62$		0			0
$63$	7.05804			0.889230
$64$		0			0
$65$		−	12.1717i		−	1.50972i
$66$		0			0
$67$	−10.4969			−1.28241			−0.641203	−	0.767372i	$-0.721564\pi$
$67$	−10.4969			−1.28241			−0.641203	+	0.767372i	$0.721564\pi$
$68$		0			0
$69$	−14.6025	+	4.25473i	−1.75793	+	0.512210i
$70$		0			0
$71$			3.90680i			0.463651i	0.972757	+	0.231826i	$0.0744699\pi$
$71$			3.90680i			0.463651i	−0.972757	+	0.231826i	$0.925530\pi$
$72$		0			0
$73$	15.2856			1.78905			0.894524	−	0.447020i	$-0.147515\pi$
$73$	15.2856			1.78905			0.894524	+	0.447020i	$0.147515\pi$
$74$		0			0
$75$		−	6.38934i		−	0.737777i
$76$		0			0
$77$	1.54103			0.175616
$78$		0			0
$79$	−10.1748			−1.14476			−0.572379	−	0.819989i	$-0.693979\pi$
$79$	−10.1748			−1.14476			−0.572379	+	0.819989i	$0.693979\pi$
$80$		0			0
$81$	19.6419			2.18243
$82$		0			0
$83$	−12.9540			−1.42189			−0.710946	−	0.703247i	$-0.751733\pi$
$83$	−12.9540			−1.42189			−0.710946	+	0.703247i	$0.751733\pi$
$84$		0			0
$85$	−14.8766			−1.61360
$86$		0			0
$87$			32.3957i			3.47318i
$88$		0			0
$89$		−	2.37994i		−	0.252273i	−0.992013	−	0.126136i	$-0.959742\pi$
$89$		−	2.37994i		−	0.252273i	0.992013	−	0.126136i	$-0.0402577\pi$
$90$		0			0
$91$	4.59567			0.481757
$92$		0			0
$93$	−4.20285			−0.435816
$94$		0			0
$95$		−	13.2516i		−	1.35959i
$96$		0			0
$97$			3.86958i			0.392896i	0.980514	+	0.196448i	$0.0629407\pi$
$97$			3.86958i			0.392896i	−0.980514	+	0.196448i	$0.937059\pi$
$98$		0			0
$99$	10.8766			1.09314
$100$		0			0
$101$	5.08206			0.505683			0.252842	−	0.967508i	$-0.418635\pi$
$101$	5.08206			0.505683			0.252842	+	0.967508i	$0.418635\pi$
$102$		0			0
$103$	11.9282			1.17532			0.587658	−	0.809110i	$-0.300050\pi$
$103$	11.9282			1.17532			0.587658	+	0.809110i	$0.300050\pi$
$104$		0			0
$105$	8.39962			0.819719
$106$		0			0
$107$	−14.5763			−1.40915			−0.704573	−	0.709632i	$-0.748861\pi$
$107$	−14.5763			−1.40915			−0.704573	+	0.709632i	$0.748861\pi$
$108$		0			0
$109$		−	10.6012i		−	1.01541i	−0.861532	−	0.507704i	$-0.830494\pi$
$109$		−	10.6012i		−	1.01541i	0.861532	−	0.507704i	$-0.169506\pi$
$110$		0			0
$111$	37.7064			3.57894
$112$		0			0
$113$		−	0.781241i		−	0.0734930i	−0.999325	−	0.0367465i	$-0.988301\pi$
$113$		−	0.781241i		−	0.0734930i	0.999325	−	0.0367465i	$-0.0116994\pi$
$114$		0			0
$115$	−12.1947	+	3.55319i	−1.13717	+	0.331337i
$116$		0			0
$117$	32.4364			2.99875
$118$		0			0
$119$		−	5.61697i		−	0.514907i
$120$		0			0
$121$	−8.62523			−0.784112
$122$		0			0
$123$			16.7415i			1.50953i
$124$		0			0
$125$			7.90676i			0.707202i
$126$		0			0
$127$		−	8.87188i		−	0.787252i	−0.919271	−	0.393626i	$-0.871220\pi$
$127$		−	8.87188i		−	0.787252i	0.919271	−	0.393626i	$-0.128780\pi$
$128$		0			0
$129$			20.3317i			1.79011i
$130$		0			0
$131$		−	3.58120i		−	0.312891i	−0.987687	−	0.156445i	$-0.949996\pi$
$131$		−	3.58120i		−	0.312891i	0.987687	−	0.156445i	$-0.0500035\pi$
$132$		0			0
$133$	5.00340			0.433850
$134$		0			0
$135$	34.0860			2.93366
$136$		0			0
$137$		−	4.63406i		−	0.395915i	−0.980211	−	0.197957i	$-0.936569\pi$
$137$		−	4.63406i		−	0.395915i	0.980211	−	0.197957i	$-0.0634307\pi$
$138$		0			0
$139$		−	16.8714i		−	1.43101i	−0.698607	−	0.715505i	$-0.746197\pi$
$139$		−	16.8714i		−	1.43101i	0.698607	−	0.715505i	$-0.253803\pi$
$140$		0			0
$141$	−22.4757			−1.89280
$142$		0			0
$143$	7.08206			0.592231
$144$		0			0
$145$			27.0541i			2.24672i
$146$		0			0
$147$			3.17144i			0.261576i
$148$		0			0
$149$		−	8.19308i		−	0.671203i	−0.942004	−	0.335602i	$-0.891060\pi$
$149$		−	8.19308i		−	0.671203i	0.942004	−	0.335602i	$-0.108940\pi$
$150$		0			0
$151$			8.86095i			0.721094i	0.932741	+	0.360547i	$0.117410\pi$
$151$			8.86095i			0.721094i	−0.932741	+	0.360547i	$0.882590\pi$
$152$		0			0
$153$		−	39.6448i		−	3.20509i
$154$		0			0
$155$	−3.50986			−0.281919
$156$		0			0
$157$			1.92068i			0.153287i	0.997059	+	0.0766437i	$0.0244204\pi$
$157$			1.92068i			0.153287i	−0.997059	+	0.0766437i	$0.975580\pi$
$158$		0			0
$159$	−34.6244			−2.74589
$160$		0			0
$161$	−1.34158	−	4.60436i	−0.105731	−	0.362875i
$162$		0			0
$163$			5.30193i			0.415279i	0.978205	+	0.207640i	$0.0665782\pi$
$163$			5.30193i			0.415279i	−0.978205	+	0.207640i	$0.933422\pi$
$164$		0			0
$165$	12.9441			1.00769
$166$		0			0
$167$			10.6262i			0.822279i	0.911572	+	0.411140i	$0.134869\pi$
$167$			10.6262i			0.822279i	−0.911572	+	0.411140i	$0.865131\pi$
$168$		0			0
$169$	8.12018			0.624629
$170$		0			0
$171$	35.3142			2.70055
$172$		0			0
$173$	1.45166			0.110368			0.0551839	−	0.998476i	$-0.482425\pi$
$173$	1.45166			0.110368			0.0551839	+	0.998476i	$0.482425\pi$
$174$		0			0
$175$	2.01465			0.152293
$176$		0			0
$177$	−7.62585			−0.573194
$178$		0			0
$179$		−	17.1546i		−	1.28219i	−0.767461	−	0.641096i	$-0.778480\pi$
$179$		−	17.1546i		−	1.28219i	0.767461	−	0.641096i	$-0.221520\pi$
$180$		0			0
$181$			12.4822i			0.927795i	0.885889	+	0.463898i	$0.153549\pi$
$181$			12.4822i			0.927795i	−0.885889	+	0.463898i	$0.846451\pi$
$182$		0			0
$183$	20.5699			1.52057
$184$		0			0
$185$	31.4892			2.31513
$186$		0			0
$187$		−	8.65591i		−	0.632983i
$188$		0			0
$189$			12.8699i			0.936144i
$190$		0			0
$191$	9.65586			0.698674			0.349337	−	0.936997i	$-0.386407\pi$
$191$	9.65586			0.698674			0.349337	+	0.936997i	$0.386407\pi$
$192$		0			0
$193$	−16.5358			−1.19027			−0.595136	−	0.803625i	$-0.702902\pi$
$193$	−16.5358			−1.19027			−0.595136	+	0.803625i	$0.702902\pi$
$194$		0			0
$195$	38.6019			2.76434
$196$		0			0
$197$	−18.3115			−1.30464			−0.652321	−	0.757943i	$-0.726205\pi$
$197$	−18.3115			−1.30464			−0.652321	+	0.757943i	$0.726205\pi$
$198$		0			0
$199$	−12.3509			−0.875531			−0.437765	−	0.899089i	$-0.644230\pi$
$199$	−12.3509			−0.875531			−0.437765	+	0.899089i	$0.644230\pi$
$200$		0			0
$201$		−	33.2904i		−	2.34813i
$202$		0			0
$203$	−10.2148			−0.716940
$204$		0			0
$205$			13.9811i			0.976480i
$206$		0			0
$207$	−9.46891	−	32.4978i	−0.658135	−	2.25875i
$208$		0			0
$209$	7.71038			0.533338
$210$		0			0
$211$		−	2.35716i		−	0.162274i	−0.996703	−	0.0811368i	$-0.974145\pi$
$211$		−	2.35716i		−	0.162274i	0.996703	−	0.0811368i	$-0.0258551\pi$
$212$		0			0
$213$	−12.3902			−0.848961
$214$		0			0
$215$			16.9793i			1.15798i
$216$		0			0
$217$		−	1.32522i		−	0.0899617i
$218$		0			0
$219$			48.4775i			3.27580i
$220$		0			0
$221$		−	25.8137i		−	1.73642i
$222$		0			0
$223$			14.1581i			0.948098i	0.880498	+	0.474049i	$0.157208\pi$
$223$			14.1581i			0.948098i	−0.880498	+	0.474049i	$0.842792\pi$
$224$		0			0
$225$	14.2195			0.947965
$226$		0			0
$227$	−0.445859			−0.0295927			−0.0147963	−	0.999891i	$-0.504710\pi$
$227$	−0.445859			−0.0295927			−0.0147963	+	0.999891i	$0.504710\pi$
$228$		0			0
$229$			21.2204i			1.40228i	0.713023	+	0.701140i	$0.247325\pi$
$229$			21.2204i			1.40228i	−0.713023	+	0.701140i	$0.752675\pi$
$230$		0			0
$231$			4.88728i			0.321559i
$232$		0			0
$233$	−21.0994			−1.38227			−0.691134	−	0.722726i	$-0.742889\pi$
$233$	−21.0994			−1.38227			−0.691134	+	0.722726i	$0.742889\pi$
$234$		0			0
$235$	−18.7698			−1.22441
$236$		0			0
$237$		−	32.2689i		−	2.09609i
$238$		0			0
$239$			19.1265i			1.23719i	0.785711	+	0.618594i	$0.212297\pi$
$239$			19.1265i			1.23719i	−0.785711	+	0.618594i	$0.787703\pi$
$240$		0			0
$241$		−	18.9426i		−	1.22020i	−0.792324	−	0.610100i	$-0.791129\pi$
$241$		−	18.9426i		−	1.22020i	0.792324	−	0.610100i	$-0.208871\pi$
$242$		0			0
$243$			23.6834i			1.51929i
$244$		0			0
$245$			2.64852i			0.169208i
$246$		0			0
$247$	22.9940			1.46307
$248$		0			0
$249$		−	41.0830i		−	2.60353i
$250$		0			0
$251$	−12.9084			−0.814772			−0.407386	−	0.913256i	$-0.633560\pi$
$251$	−12.9084			−0.814772			−0.407386	+	0.913256i	$0.633560\pi$
$252$		0			0
$253$	−2.06741	−	7.09545i	−0.129977	−	0.446087i
$254$		0			0
$255$		−	47.1804i		−	2.95455i
$256$		0			0
$257$	11.9688			0.746592			0.373296	−	0.927712i	$-0.378228\pi$
$257$	11.9688			0.746592			0.373296	+	0.927712i	$0.378228\pi$
$258$		0			0
$259$			11.8894i			0.738769i
$260$		0			0
$261$	−72.0966			−4.46267
$262$		0			0
$263$	27.4218			1.69090			0.845450	−	0.534054i	$-0.179332\pi$
$263$	27.4218			1.69090			0.845450	+	0.534054i	$0.179332\pi$
$264$		0			0
$265$	−28.9153			−1.77625
$266$		0			0
$267$	7.54783			0.461920
$268$		0			0
$269$	16.6318			1.01406			0.507028	−	0.861929i	$-0.330744\pi$
$269$	16.6318			1.01406			0.507028	+	0.861929i	$0.330744\pi$
$270$		0			0
$271$			21.9673i			1.33442i	0.744870	+	0.667210i	$0.232512\pi$
$271$			21.9673i			1.33442i	−0.744870	+	0.667210i	$0.767488\pi$
$272$		0			0
$273$			14.5749i			0.882113i
$274$		0			0
$275$	3.10463			0.187216
$276$		0			0
$277$	7.92323			0.476061			0.238030	−	0.971258i	$-0.423498\pi$
$277$	7.92323			0.476061			0.238030	+	0.971258i	$0.423498\pi$
$278$		0			0
$279$		−	9.35345i		−	0.559976i
$280$		0			0
$281$		−	13.1474i		−	0.784306i	−0.919900	−	0.392153i	$-0.871730\pi$
$281$		−	13.1474i		−	0.784306i	0.919900	−	0.392153i	$-0.128270\pi$
$282$		0			0
$283$	1.40364			0.0834376			0.0417188	−	0.999129i	$-0.486717\pi$
$283$	1.40364			0.0834376			0.0417188	+	0.999129i	$0.486717\pi$
$284$		0			0
$285$	42.0267			2.48945
$286$		0			0
$287$	−5.27882			−0.311599
$288$		0			0
$289$	−14.5503			−0.855902
$290$		0			0
$291$	−12.2721			−0.719406
$292$		0			0
$293$			1.92635i			0.112539i	0.998416	+	0.0562693i	$0.0179205\pi$
$293$			1.92635i			0.112539i	−0.998416	+	0.0562693i	$0.982079\pi$
$294$		0			0
$295$	−6.36846			−0.370786
$296$		0			0
$297$			19.8328i			1.15082i
$298$		0			0
$299$	−6.16545	−	21.1601i	−0.356557	−	1.22372i
$300$		0			0
$301$	−6.41087			−0.369516
$302$		0			0
$303$			16.1174i			0.925923i
$304$		0			0
$305$	17.1782			0.983622
$306$		0			0
$307$		−	19.2308i		−	1.09756i	−0.835967	−	0.548780i	$-0.815092\pi$
$307$		−	19.2308i		−	1.09756i	0.835967	−	0.548780i	$-0.184908\pi$
$308$		0			0
$309$			37.8294i			2.15204i
$310$		0			0
$311$		−	22.8615i		−	1.29636i	−0.761488	−	0.648179i	$-0.775531\pi$
$311$		−	22.8615i		−	1.29636i	0.761488	−	0.648179i	$-0.224469\pi$
$312$		0			0
$313$			4.21509i			0.238251i	0.992879	+	0.119126i	$0.0380091\pi$
$313$			4.21509i			0.238251i	−0.992879	+	0.119126i	$0.961991\pi$
$314$		0			0
$315$			18.6934i			1.05325i
$316$		0			0
$317$	8.60006			0.483027			0.241514	−	0.970397i	$-0.422356\pi$
$317$	8.60006			0.483027			0.241514	+	0.970397i	$0.422356\pi$
$318$		0			0
$319$	−15.7413			−0.881345
$320$		0			0
$321$		−	46.2279i		−	2.58019i
$322$		0			0
$323$		−	28.1040i		−	1.56375i
$324$		0			0
$325$	9.25866			0.513578
$326$		0			0
$327$	33.6210			1.85924
$328$		0			0
$329$		−	7.08691i		−	0.390714i
$330$		0			0
$331$		−	22.9430i		−	1.26106i	−0.776164	−	0.630530i	$-0.782837\pi$
$331$		−	22.9430i		−	1.26106i	0.776164	−	0.630530i	$-0.217163\pi$
$332$		0			0
$333$			83.9157i			4.59855i
$334$		0			0
$335$		−	27.8013i		−	1.51895i
$336$		0			0
$337$			6.88646i			0.375130i	0.982252	+	0.187565i	$0.0600595\pi$
$337$			6.88646i			0.375130i	−0.982252	+	0.187565i	$0.939941\pi$
$338$		0			0
$339$	2.47766			0.134568
$340$		0			0
$341$		−	2.04220i		−	0.110591i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$	−11.2687	−	38.6749i	−0.606689	−	2.08219i
$346$		0			0
$347$			15.1910i			0.815495i	0.913095	+	0.407748i	$0.133686\pi$
$347$			15.1910i			0.815495i	−0.913095	+	0.407748i	$0.866314\pi$
$348$		0			0
$349$	−4.77064			−0.255366			−0.127683	−	0.991815i	$-0.540754\pi$
$349$	−4.77064			−0.255366			−0.127683	+	0.991815i	$0.540754\pi$
$350$		0			0
$351$			59.1456i			3.15696i
$352$		0			0
$353$	4.12732			0.219675			0.109838	−	0.993950i	$-0.464967\pi$
$353$	4.12732			0.219675			0.109838	+	0.993950i	$0.464967\pi$
$354$		0			0
$355$	−10.3472			−0.549173
$356$		0			0
$357$	17.8139			0.942811
$358$		0			0
$359$	−36.7948			−1.94196			−0.970978	−	0.239170i	$-0.923125\pi$
$359$	−36.7948			−1.94196			−0.970978	+	0.239170i	$0.923125\pi$
$360$		0			0
$361$	6.03403			0.317581
$362$		0			0
$363$		−	27.3544i		−	1.43573i
$364$		0			0
$365$			40.4843i			2.11904i
$366$		0			0
$367$	28.9555			1.51146			0.755732	−	0.654881i	$-0.227281\pi$
$367$	28.9555			1.51146			0.755732	+	0.654881i	$0.227281\pi$
$368$		0			0
$369$	−37.2582			−1.93958
$370$		0			0
$371$		−	10.9176i		−	0.566811i
$372$		0			0
$373$			13.5933i			0.703837i	0.936031	+	0.351918i	$0.114471\pi$
$373$			13.5933i			0.703837i	−0.936031	+	0.351918i	$0.885529\pi$
$374$		0			0
$375$	−25.0758			−1.29491
$376$		0			0
$377$	−46.9439			−2.41774
$378$		0			0
$379$	−5.67398			−0.291453			−0.145726	−	0.989325i	$-0.546552\pi$
$379$	−5.67398			−0.291453			−0.145726	+	0.989325i	$0.546552\pi$
$380$		0			0
$381$	28.1367			1.44148
$382$		0			0
$383$	−37.2356			−1.90265			−0.951325	−	0.308191i	$-0.900276\pi$
$383$	−37.2356			−1.90265			−0.951325	+	0.308191i	$0.900276\pi$
$384$		0			0
$385$			4.08144i			0.208009i
$386$		0			0
$387$	−45.2482			−2.30009
$388$		0			0
$389$			25.9929i			1.31789i	0.752190	+	0.658947i	$0.228998\pi$
$389$			25.9929i			1.31789i	−0.752190	+	0.658947i	$0.771002\pi$
$390$		0			0
$391$	−25.8626	+	7.53560i	−1.30793	+	0.381092i
$392$		0			0
$393$	11.3576			0.572913
$394$		0			0
$395$		−	26.9482i		−	1.35591i
$396$		0			0
$397$	−9.06653			−0.455036			−0.227518	−	0.973774i	$-0.573061\pi$
$397$	−9.06653			−0.455036			−0.227518	+	0.973774i	$0.573061\pi$
$398$		0			0
$399$			15.8680i			0.794394i
$400$		0			0
$401$		−	3.70423i		−	0.184981i	−0.995714	−	0.0924903i	$-0.970517\pi$
$401$		−	3.70423i		−	0.184981i	0.995714	−	0.0924903i	$-0.0294827\pi$
$402$		0			0
$403$		−	6.09027i		−	0.303378i
$404$		0			0
$405$			52.0218i			2.58498i
$406$		0			0
$407$			18.3218i			0.908180i
$408$		0			0
$409$	−6.06446			−0.299868			−0.149934	−	0.988696i	$-0.547906\pi$
$409$	−6.06446			−0.299868			−0.149934	+	0.988696i	$0.547906\pi$
$410$		0			0
$411$	14.6967			0.724933
$412$		0			0
$413$		−	2.40454i		−	0.118319i
$414$		0			0
$415$		−	34.3090i		−	1.68416i
$416$		0			0
$417$	53.5066			2.62023
$418$		0			0
$419$	−23.9547			−1.17027			−0.585133	−	0.810938i	$-0.698958\pi$
$419$	−23.9547			−1.17027			−0.585133	+	0.810938i	$0.698958\pi$
$420$		0			0
$421$		−	10.6016i		−	0.516691i	−0.966053	−	0.258346i	$-0.916823\pi$
$421$		−	10.6016i		−	0.516691i	0.966053	−	0.258346i	$-0.0831773\pi$
$422$		0			0
$423$		−	50.0197i		−	2.43204i
$424$		0			0
$425$		−	11.3162i		−	0.548917i
$426$		0			0
$427$			6.48598i			0.313878i
$428$		0			0
$429$			22.4603i			1.08439i
$430$		0			0
$431$	16.8299			0.810666			0.405333	−	0.914169i	$-0.367155\pi$
$431$	16.8299			0.810666			0.405333	+	0.914169i	$0.367155\pi$
$432$		0			0
$433$			30.2591i			1.45416i	0.686552	+	0.727081i	$0.259123\pi$
$433$			30.2591i			1.45416i	−0.686552	+	0.727081i	$0.740877\pi$
$434$		0			0
$435$	−85.8006			−4.11382
$436$		0			0
$437$	−6.71245	−	23.0375i	−0.321100	−	1.10203i
$438$		0			0
$439$			2.02611i			0.0967008i	0.998830	+	0.0483504i	$0.0153964\pi$
$439$			2.02611i			0.0967008i	−0.998830	+	0.0483504i	$0.984604\pi$
$440$		0			0
$441$	−7.05804			−0.336097
$442$		0			0
$443$			14.3626i			0.682388i	0.939993	+	0.341194i	$0.110831\pi$
$443$			14.3626i			0.682388i	−0.939993	+	0.341194i	$0.889169\pi$
$444$		0			0
$445$	6.30331			0.298805
$446$		0			0
$447$	25.9839			1.22900
$448$		0			0
$449$	3.78316			0.178538			0.0892691	−	0.996008i	$-0.471547\pi$
$449$	3.78316			0.178538			0.0892691	+	0.996008i	$0.471547\pi$
$450$		0			0
$451$	−8.13482			−0.383054
$452$		0			0
$453$	−28.1020			−1.32035
$454$		0			0
$455$			12.1717i			0.570619i
$456$		0			0
$457$			3.25409i			0.152220i	0.997099	+	0.0761098i	$0.0242500\pi$
$457$			3.25409i			0.152220i	−0.997099	+	0.0761098i	$0.975750\pi$
$458$		0			0
$459$	72.2896			3.37419
$460$		0			0
$461$	20.0286			0.932825			0.466413	−	0.884567i	$-0.345546\pi$
$461$	20.0286			0.932825			0.466413	+	0.884567i	$0.345546\pi$
$462$		0			0
$463$			33.2240i			1.54405i	0.635591	+	0.772026i	$0.280757\pi$
$463$			33.2240i			1.54405i	−0.635591	+	0.772026i	$0.719243\pi$
$464$		0			0
$465$		−	11.1313i		−	0.516203i
$466$		0			0
$467$	35.8463			1.65877			0.829386	−	0.558677i	$-0.188691\pi$
$467$	35.8463			1.65877			0.829386	+	0.558677i	$0.188691\pi$
$468$		0			0
$469$	10.4969			0.484704
$470$		0			0
$471$	−6.09134			−0.280674
$472$		0			0
$473$	−9.87933			−0.454252
$474$		0			0
$475$	10.0801			0.462506
$476$		0			0
$477$		−	77.0566i		−	3.52818i
$478$		0			0
$479$	−29.6825			−1.35623			−0.678114	−	0.734957i	$-0.737202\pi$
$479$	−29.6825			−1.35623			−0.678114	+	0.734957i	$0.737202\pi$
$480$		0			0
$481$			54.6396i			2.49135i
$482$		0			0
$483$	14.6025	−	4.25473i	0.664436	−	0.193597i
$484$		0			0
$485$	−10.2486			−0.465367
$486$		0			0
$487$		−	25.2245i		−	1.14303i	−0.820591	−	0.571516i	$-0.806355\pi$
$487$		−	25.2245i		−	1.14303i	0.820591	−	0.571516i	$-0.193645\pi$
$488$		0			0
$489$	−16.8148			−0.760390
$490$		0			0
$491$			22.9284i			1.03474i	0.855761	+	0.517371i	$0.173089\pi$
$491$			22.9284i			1.03474i	−0.855761	+	0.517371i	$0.826911\pi$
$492$		0			0
$493$			57.3763i			2.58410i
$494$		0			0
$495$			28.8070i			1.29478i
$496$		0			0
$497$		−	3.90680i		−	0.175244i
$498$		0			0
$499$		−	38.1273i		−	1.70681i	−0.521247	−	0.853406i	$-0.674533\pi$
$499$		−	38.1273i		−	1.70681i	0.521247	−	0.853406i	$-0.325467\pi$
$500$		0			0
$501$	−33.7004			−1.50562
$502$		0			0
$503$	13.5480			0.604074			0.302037	−	0.953296i	$-0.402333\pi$
$503$	13.5480			0.604074			0.302037	+	0.953296i	$0.402333\pi$
$504$		0			0
$505$			13.4599i			0.598958i
$506$		0			0
$507$			25.7527i			1.14372i
$508$		0			0
$509$	−13.1215			−0.581601			−0.290801	−	0.956784i	$-0.593922\pi$
$509$	−13.1215			−0.581601			−0.290801	+	0.956784i	$0.593922\pi$
$510$		0			0
$511$	−15.2856			−0.676196
$512$		0			0
$513$			64.3930i			2.84302i
$514$		0			0
$515$			31.5919i			1.39211i
$516$		0			0
$517$		−	10.9211i		−	0.480310i
$518$		0			0
$519$			4.60386i			0.202087i
$520$		0			0
$521$			14.9576i			0.655306i	0.944798	+	0.327653i	$0.106258\pi$
$521$			14.9576i			0.655306i	−0.944798	+	0.327653i	$0.893742\pi$
$522$		0			0
$523$	−42.8793			−1.87498			−0.937491	−	0.348009i	$-0.886858\pi$
$523$	−42.8793			−1.87498			−0.937491	+	0.348009i	$0.886858\pi$
$524$		0			0
$525$			6.38934i			0.278854i
$526$		0			0
$527$	−7.44371			−0.324253
$528$		0			0
$529$	−19.4003	+	12.3542i	−0.843493	+	0.537140i
$530$		0			0
$531$		−	16.9713i		−	0.736493i
$532$		0			0
$533$	−24.2597			−1.05081
$534$		0			0
$535$		−	38.6056i		−	1.66907i
$536$		0			0
$537$	54.4047			2.34774
$538$		0			0
$539$	−1.54103			−0.0663768
$540$		0			0
$541$	16.0393			0.689584			0.344792	−	0.938679i	$-0.387949\pi$
$541$	16.0393			0.689584			0.344792	+	0.938679i	$0.387949\pi$
$542$		0			0
$543$	−39.5866			−1.69882
$544$		0			0
$545$	28.0774			1.20270
$546$		0			0
$547$		−	3.83236i		−	0.163860i	−0.996638	−	0.0819300i	$-0.973892\pi$
$547$		−	3.83236i		−	0.163860i	0.996638	−	0.0819300i	$-0.0261084\pi$
$548$		0			0
$549$			45.7783i			1.95377i
$550$		0			0
$551$	−51.1088			−2.17731
$552$		0			0
$553$	10.1748			0.432678
$554$		0			0
$555$			99.8662i			4.23908i
$556$		0			0
$557$		−	11.3527i		−	0.481028i	−0.970646	−	0.240514i	$-0.922684\pi$
$557$		−	11.3527i		−	0.481028i	0.970646	−	0.240514i	$-0.0773159\pi$
$558$		0			0
$559$	−29.4622			−1.24612
$560$		0			0
$561$	27.4517			1.15901
$562$		0			0
$563$	11.1900			0.471603			0.235801	−	0.971801i	$-0.424229\pi$
$563$	11.1900			0.471603			0.235801	+	0.971801i	$0.424229\pi$
$564$		0			0
$565$	2.06913			0.0870490
$566$		0			0
$567$	−19.6419			−0.824880
$568$		0			0
$569$		−	39.0593i		−	1.63745i	−0.574185	−	0.818726i	$-0.694681\pi$
$569$		−	39.0593i		−	1.63745i	0.574185	−	0.818726i	$-0.305319\pi$
$570$		0			0
$571$	−1.48495			−0.0621431			−0.0310715	−	0.999517i	$-0.509892\pi$
$571$	−1.48495			−0.0621431			−0.0310715	+	0.999517i	$0.509892\pi$
$572$		0			0
$573$			30.6230i			1.27929i
$574$		0			0
$575$	−2.70281	−	9.27617i	−0.112715	−	0.386843i
$576$		0			0
$577$	−12.5209			−0.521250			−0.260625	−	0.965440i	$-0.583929\pi$
$577$	−12.5209			−0.521250			−0.260625	+	0.965440i	$0.583929\pi$
$578$		0			0
$579$		−	52.4423i		−	2.17943i
$580$		0			0
$581$	12.9540			0.537424
$582$		0			0
$583$		−	16.8243i		−	0.696789i
$584$		0			0
$585$			85.9085i			3.55188i
$586$		0			0
$587$			3.56792i			0.147264i	0.997285	+	0.0736318i	$0.0234590\pi$
$587$			3.56792i			0.147264i	−0.997285	+	0.0736318i	$0.976541\pi$
$588$		0			0
$589$		−	6.63060i		−	0.273209i
$590$		0			0
$591$		−	58.0739i		−	2.38884i
$592$		0			0
$593$	8.31785			0.341573			0.170787	−	0.985308i	$-0.445369\pi$
$593$	8.31785			0.341573			0.170787	+	0.985308i	$0.445369\pi$
$594$		0			0
$595$	14.8766			0.609883
$596$		0			0
$597$		−	39.1701i		−	1.60313i
$598$		0			0
$599$			14.8515i			0.606816i	0.952861	+	0.303408i	$0.0981245\pi$
$599$			14.8515i			0.606816i	−0.952861	+	0.303408i	$0.901876\pi$
$600$		0			0
$601$	25.9581			1.05885			0.529426	−	0.848356i	$-0.322407\pi$
$601$	25.9581			1.05885			0.529426	+	0.848356i	$0.322407\pi$
$602$		0			0
$603$	74.0879			3.01709
$604$		0			0
$605$		−	22.8441i		−	0.928744i
$606$		0			0
$607$		−	14.3702i		−	0.583269i	−0.956530	−	0.291635i	$-0.905801\pi$
$607$		−	14.3702i		−	0.583269i	0.956530	−	0.291635i	$-0.0941991\pi$
$608$		0			0
$609$		−	32.3957i		−	1.31274i
$610$		0			0
$611$		−	32.5691i		−	1.31760i
$612$		0			0
$613$		−	36.3609i		−	1.46860i	−0.678824	−	0.734301i	$-0.737510\pi$
$613$		−	36.3609i		−	1.46860i	0.678824	−	0.734301i	$-0.262490\pi$
$614$		0			0
$615$	−44.3401			−1.78797
$616$		0			0
$617$			30.8406i			1.24160i	0.783971	+	0.620798i	$0.213191\pi$
$617$			30.8406i			1.24160i	−0.783971	+	0.620798i	$0.786809\pi$
$618$		0			0
$619$	−6.21096			−0.249640			−0.124820	−	0.992179i	$-0.539835\pi$
$619$	−6.21096			−0.249640			−0.124820	+	0.992179i	$0.539835\pi$
$620$		0			0
$621$	59.2575	−	17.2659i	2.37792	−	0.692857i
$622$		0			0
$623$			2.37994i			0.0953502i
$624$		0			0
$625$	−31.0144			−1.24058
$626$		0			0
$627$			24.4530i			0.976560i
$628$		0			0
$629$	66.7822			2.66278
$630$		0			0
$631$	−21.1279			−0.841088			−0.420544	−	0.907272i	$-0.638161\pi$
$631$	−21.1279			−0.841088			−0.420544	+	0.907272i	$0.638161\pi$
$632$		0			0
$633$	7.47559			0.297128
$634$		0			0
$635$	23.4973			0.932463
$636$		0			0
$637$	−4.59567			−0.182087
$638$		0			0
$639$		−	27.5743i		−	1.09082i
$640$		0			0
$641$		−	23.3058i		−	0.920524i	−0.887783	−	0.460262i	$-0.847755\pi$
$641$		−	23.3058i		−	0.920524i	0.887783	−	0.460262i	$-0.152245\pi$
$642$		0			0
$643$	29.3195			1.15625			0.578123	−	0.815949i	$-0.303785\pi$
$643$	29.3195			1.15625			0.578123	+	0.815949i	$0.303785\pi$
$644$		0			0
$645$	−53.8489			−2.12030
$646$		0			0
$647$			5.13348i			0.201818i	0.994896	+	0.100909i	$0.0321751\pi$
$647$			5.13348i			0.201818i	−0.994896	+	0.100909i	$0.967825\pi$
$648$		0			0
$649$		−	3.70546i		−	0.145452i
$650$		0			0
$651$	4.20285			0.164723
$652$		0			0
$653$	7.54478			0.295250			0.147625	−	0.989043i	$-0.452837\pi$
$653$	7.54478			0.295250			0.147625	+	0.989043i	$0.452837\pi$
$654$		0			0
$655$	9.48487			0.370604
$656$		0			0
$657$	−107.887			−4.20906
$658$		0			0
$659$	11.0877			0.431915			0.215958	−	0.976403i	$-0.430713\pi$
$659$	11.0877			0.431915			0.215958	+	0.976403i	$0.430713\pi$
$660$		0			0
$661$			11.5997i			0.451177i	0.974223	+	0.225589i	$0.0724305\pi$
$661$			11.5997i			0.451177i	−0.974223	+	0.225589i	$0.927569\pi$
$662$		0			0
$663$	81.8668			3.17944
$664$		0			0
$665$			13.2516i			0.513875i
$666$		0			0
$667$	13.7040	+	47.0327i	0.530620	+	1.82112i
$668$		0			0
$669$	−44.9017			−1.73600
$670$		0			0
$671$			9.99507i			0.385855i
$672$		0			0
$673$	−7.43521			−0.286606			−0.143303	−	0.989679i	$-0.545772\pi$
$673$	−7.43521			−0.286606			−0.143303	+	0.989679i	$0.545772\pi$
$674$		0			0
$675$			25.9282i			0.997978i
$676$		0			0
$677$			27.4460i			1.05483i	0.849607	+	0.527417i	$0.176839\pi$
$677$			27.4460i			1.05483i	−0.849607	+	0.527417i	$0.823161\pi$
$678$		0			0
$679$		−	3.86958i		−	0.148501i
$680$		0			0
$681$		−	1.41401i		−	0.0541852i
$682$		0			0
$683$			2.51920i			0.0963947i	0.998838	+	0.0481973i	$0.0153476\pi$
$683$			2.51920i			0.0963947i	−0.998838	+	0.0481973i	$0.984652\pi$
$684$		0			0
$685$	12.2734			0.468942
$686$		0			0
$687$	−67.2991			−2.56762
$688$		0			0
$689$		−	50.1735i		−	1.91146i
$690$		0			0
$691$			46.1211i			1.75453i	0.480006	+	0.877265i	$0.340634\pi$
$691$			46.1211i			1.75453i	−0.480006	+	0.877265i	$0.659366\pi$
$692$		0			0
$693$	−10.8766			−0.413170
$694$		0			0
$695$	44.6841			1.69497
$696$		0			0
$697$			29.6510i			1.12311i
$698$		0			0
$699$		−	66.9156i		−	2.53098i
$700$		0			0
$701$		−	29.0673i		−	1.09786i	−0.835869	−	0.548928i	$-0.815036\pi$
$701$		−	29.0673i		−	1.09786i	0.835869	−	0.548928i	$-0.184964\pi$
$702$		0			0
$703$			59.4873i			2.24361i
$704$		0			0
$705$		−	59.5273i		−	2.24193i
$706$		0			0
$707$	−5.08206			−0.191130
$708$		0			0
$709$		−	13.0816i		−	0.491288i	−0.969360	−	0.245644i	$-0.921001\pi$
$709$		−	13.0816i		−	0.491288i	0.969360	−	0.245644i	$-0.0789994\pi$
$710$		0			0
$711$	71.8144			2.69325
$712$		0			0
$713$	−6.10179	+	1.77788i	−0.228514	+	0.0665822i
$714$		0			0
$715$			18.7570i			0.701470i
$716$		0			0
$717$	−60.6585			−2.26533
$718$		0			0
$719$			5.71978i			0.213312i	0.994296	+	0.106656i	$0.0340143\pi$
$719$			5.71978i			0.213312i	−0.994296	+	0.106656i	$0.965986\pi$
$720$		0			0
$721$	−11.9282			−0.444228
$722$		0			0
$723$	60.0754			2.23423
$724$		0			0
$725$	−20.5793			−0.764295
$726$		0			0
$727$	28.7262			1.06539			0.532697	−	0.846306i	$-0.321178\pi$
$727$	28.7262			1.06539			0.532697	+	0.846306i	$0.321178\pi$
$728$		0			0
$729$	−16.1851			−0.599449
$730$		0			0
$731$			36.0096i			1.33186i
$732$		0			0
$733$			1.12643i			0.0416058i	0.999784	+	0.0208029i	$0.00662224\pi$
$733$			1.12643i			0.0416058i	−0.999784	+	0.0208029i	$0.993378\pi$
$734$		0			0
$735$	−8.39962			−0.309825
$736$		0			0
$737$	16.1761			0.595853
$738$		0			0
$739$		−	10.5536i		−	0.388221i	−0.980980	−	0.194110i	$-0.937818\pi$
$739$		−	10.5536i		−	0.388221i	0.980980	−	0.194110i	$-0.0621820\pi$
$740$		0			0
$741$			72.9241i			2.67893i
$742$		0			0
$743$	16.5955			0.608829			0.304414	−	0.952540i	$-0.401539\pi$
$743$	16.5955			0.608829			0.304414	+	0.952540i	$0.401539\pi$
$744$		0			0
$745$	21.6995			0.795009
$746$		0			0
$747$	91.4302			3.34526
$748$		0			0
$749$	14.5763			0.532607
$750$		0			0
$751$	−16.9043			−0.616847			−0.308423	−	0.951249i	$-0.599801\pi$
$751$	−16.9043			−0.616847			−0.308423	+	0.951249i	$0.599801\pi$
$752$		0			0
$753$		−	40.9383i		−	1.49187i
$754$		0			0
$755$	−23.4684			−0.854102
$756$		0			0
$757$			21.1764i			0.769668i	0.922986	+	0.384834i	$0.125741\pi$
$757$			21.1764i			0.769668i	−0.922986	+	0.384834i	$0.874259\pi$
$758$		0			0
$759$	22.5028	−	6.55666i	0.816801	−	0.237992i
$760$		0			0
$761$	−37.3684			−1.35460			−0.677301	−	0.735706i	$-0.736851\pi$
$761$	−37.3684			−1.35460			−0.677301	+	0.735706i	$0.736851\pi$
$762$		0			0
$763$			10.6012i			0.383788i
$764$		0			0
$765$	105.000			3.79628
$766$		0			0
$767$		−	11.0505i		−	0.399009i
$768$		0			0
$769$			28.8200i			1.03927i	0.854387	+	0.519637i	$0.173933\pi$
$769$			28.8200i			1.03927i	−0.854387	+	0.519637i	$0.826067\pi$
$770$		0			0
$771$			37.9583i			1.36703i
$772$		0			0
$773$		−	2.09912i		−	0.0755001i	−0.999287	−	0.0377500i	$-0.987981\pi$
$773$		−	2.09912i		−	0.0755001i	0.999287	−	0.0377500i	$-0.0120191\pi$
$774$		0			0
$775$		−	2.66985i		−	0.0959038i
$776$		0			0
$777$	−37.7064			−1.35271
$778$		0			0
$779$	−26.4121			−0.946311
$780$		0			0
$781$		−	6.02048i		−	0.215430i
$782$		0			0
$783$		−	131.463i		−	4.69811i
$784$		0			0
$785$	−5.08697			−0.181562
$786$		0			0
$787$	−31.5567			−1.12488			−0.562438	−	0.826839i	$-0.690136\pi$
$787$	−31.5567			−1.12488			−0.562438	+	0.826839i	$0.690136\pi$
$788$		0			0
$789$			86.9666i			3.09609i
$790$		0			0
$791$			0.781241i			0.0277777i
$792$		0			0
$793$			29.8074i			1.05849i
$794$		0			0
$795$		−	91.7033i		−	3.25238i
$796$		0			0
$797$		−	22.3998i		−	0.793442i	−0.917939	−	0.396721i	$-0.870148\pi$
$797$		−	22.3998i		−	0.793442i	0.917939	−	0.396721i	$-0.129852\pi$
$798$		0			0
$799$	−39.8069			−1.40827
$800$		0			0
$801$			16.7977i			0.593518i
$802$		0			0
$803$	−23.5556			−0.831258
$804$		0			0
$805$	12.1947	−	3.55319i	0.429808	−	0.125234i
$806$		0			0
$807$			52.7467i			1.85677i
$808$		0			0
$809$	−23.5865			−0.829258			−0.414629	−	0.909991i	$-0.636089\pi$
$809$	−23.5865			−0.829258			−0.414629	+	0.909991i	$0.636089\pi$
$810$		0			0
$811$		−	16.6418i		−	0.584372i	−0.956362	−	0.292186i	$-0.905617\pi$
$811$		−	16.6418i		−	0.584372i	0.956362	−	0.292186i	$-0.0943826\pi$
$812$		0			0
$813$	−69.6681			−2.44337
$814$		0			0
$815$	−14.0423			−0.491879
$816$		0			0
$817$	−32.0761			−1.12220
$818$		0			0
$819$	−32.4364			−1.13342
$820$		0			0
$821$	−6.89037			−0.240476			−0.120238	−	0.992745i	$-0.538366\pi$
$821$	−6.89037			−0.240476			−0.120238	+	0.992745i	$0.538366\pi$
$822$		0			0
$823$			31.8702i			1.11093i	0.831541	+	0.555463i	$0.187459\pi$
$823$			31.8702i			1.11093i	−0.831541	+	0.555463i	$0.812541\pi$
$824$		0			0
$825$			9.84615i			0.342799i
$826$		0			0
$827$	3.41970			0.118915			0.0594573	−	0.998231i	$-0.481063\pi$
$827$	3.41970			0.118915			0.0594573	+	0.998231i	$0.481063\pi$
$828$		0			0
$829$	−54.4244			−1.89024			−0.945118	−	0.326729i	$-0.894053\pi$
$829$	−54.4244			−1.89024			−0.945118	+	0.326729i	$0.894053\pi$
$830$		0			0
$831$			25.1281i			0.871683i
$832$		0			0
$833$			5.61697i			0.194616i
$834$		0			0
$835$	−28.1437			−0.973952
$836$		0			0
$837$	17.0554			0.589520
$838$		0			0
$839$	0.927467			0.0320197			0.0160099	−	0.999872i	$-0.494904\pi$
$839$	0.927467			0.0320197			0.0160099	+	0.999872i	$0.494904\pi$
$840$		0			0
$841$	75.3425			2.59802
$842$		0			0
$843$	41.6961			1.43609
$844$		0			0
$845$			21.5064i			0.739844i
$846$		0			0
$847$	8.62523			0.296367
$848$		0			0
$849$			4.45156i			0.152777i
$850$		0			0
$851$	54.7430	−	15.9505i	1.87656	−	0.546776i
$852$		0			0
$853$	4.64837			0.159157			0.0795785	−	0.996829i	$-0.474643\pi$
$853$	4.64837			0.159157			0.0795785	+	0.996829i	$0.474643\pi$
$854$		0			0
$855$			93.5304i			3.19867i
$856$		0			0
$857$	−29.9069			−1.02160			−0.510799	−	0.859700i	$-0.670651\pi$
$857$	−29.9069			−1.02160			−0.510799	+	0.859700i	$0.670651\pi$
$858$		0			0
$859$			41.2763i			1.40833i	0.710037	+	0.704164i	$0.248678\pi$
$859$			41.2763i			1.40833i	−0.710037	+	0.704164i	$0.751322\pi$
$860$		0			0
$861$		−	16.7415i		−	0.570548i
$862$		0			0
$863$			50.9223i			1.73342i	0.498816	+	0.866708i	$0.333768\pi$
$863$			50.9223i			1.73342i	−0.498816	+	0.866708i	$0.666232\pi$
$864$		0			0
$865$			3.84475i			0.130725i
$866$		0			0
$867$		−	46.1456i		−	1.56719i
$868$		0			0
$869$	15.6797			0.531897
$870$		0			0
$871$	48.2405			1.63457
$872$		0			0
$873$		−	27.3117i		−	0.924359i
$874$		0			0
$875$		−	7.90676i		−	0.267297i
$876$		0			0
$877$	−27.5386			−0.929912			−0.464956	−	0.885334i	$-0.653930\pi$
$877$	−27.5386			−0.929912			−0.464956	+	0.885334i	$0.653930\pi$
$878$		0			0
$879$	−6.10931			−0.206062
$880$		0			0
$881$			27.8340i			0.937750i	0.883264	+	0.468875i	$0.155341\pi$
$881$			27.8340i			0.937750i	−0.883264	+	0.468875i	$0.844659\pi$
$882$		0			0
$883$		−	27.9735i		−	0.941383i	−0.882298	−	0.470692i	$-0.844004\pi$
$883$		−	27.9735i		−	0.941383i	0.882298	−	0.470692i	$-0.155996\pi$
$884$		0			0
$885$		−	20.1972i		−	0.678921i
$886$		0			0
$887$		−	39.7535i		−	1.33479i	−0.744704	−	0.667395i	$-0.767409\pi$
$887$		−	39.7535i		−	1.33479i	0.744704	−	0.667395i	$-0.232591\pi$
$888$		0			0
$889$			8.87188i			0.297553i
$890$		0			0
$891$	−30.2686			−1.01404
$892$		0			0
$893$		−	35.4586i		−	1.18658i
$894$		0			0
$895$	45.4342			1.51870
$896$		0			0
$897$	67.1082	−	19.5534i	2.24068	−	0.652868i
$898$		0			0
$899$			13.5369i			0.451480i
$900$		0			0
$901$	−61.3235			−2.04298
$902$		0			0
$903$		−	20.3317i		−	0.676596i
$904$		0			0
$905$	−33.0594			−1.09893
$906$		0			0
$907$	35.7827			1.18814			0.594072	−	0.804412i	$-0.297519\pi$
$907$	35.7827			1.18814			0.594072	+	0.804412i	$0.297519\pi$
$908$		0			0
$909$	−35.8694			−1.18971
$910$		0			0
$911$	9.90802			0.328267			0.164134	−	0.986438i	$-0.447517\pi$
$911$	9.90802			0.328267			0.164134	+	0.986438i	$0.447517\pi$
$912$		0			0
$913$	19.9625			0.660664
$914$		0			0
$915$			54.4797i			1.80104i
$916$		0			0
$917$			3.58120i			0.118262i
$918$		0			0
$919$	46.4022			1.53067			0.765333	−	0.643634i	$-0.222574\pi$
$919$	46.4022			1.53067			0.765333	+	0.643634i	$0.222574\pi$
$920$		0			0
$921$	60.9894			2.00967
$922$		0			0
$923$		−	17.9543i		−	0.590974i
$924$		0			0
$925$			23.9529i			0.787566i
$926$		0			0
$927$	−84.1894			−2.76514
$928$		0			0
$929$	−4.58673			−0.150486			−0.0752429	−	0.997165i	$-0.523973\pi$
$929$	−4.58673			−0.150486			−0.0752429	+	0.997165i	$0.523973\pi$
$930$		0			0
$931$	−5.00340			−0.163980
$932$		0			0
$933$	72.5040			2.37368
$934$		0			0
$935$	22.9253			0.749738
$936$		0			0
$937$		−	6.83354i		−	0.223242i	−0.993751	−	0.111621i	$-0.964396\pi$
$937$		−	6.83354i		−	0.223242i	0.993751	−	0.111621i	$-0.0356043\pi$
$938$		0			0
$939$	−13.3679			−0.436246
$940$		0			0
$941$			31.9199i			1.04056i	0.853996	+	0.520280i	$0.174172\pi$
$941$			31.9199i			1.04056i	−0.853996	+	0.520280i	$0.825828\pi$
$942$		0			0
$943$	7.08195	+	24.3056i	0.230620	+	0.791500i
$944$		0			0
$945$	−34.0860			−1.10882
$946$		0			0
$947$			34.2008i			1.11138i	0.831391	+	0.555688i	$0.187545\pi$
$947$			34.2008i			1.11138i	−0.831391	+	0.555688i	$0.812455\pi$
$948$		0			0
$949$	−70.2477			−2.28034
$950$		0			0
$951$			27.2746i			0.884439i
$952$		0			0
$953$			18.3681i			0.595000i	0.954722	+	0.297500i	$0.0961528\pi$
$953$			18.3681i			0.595000i	−0.954722	+	0.297500i	$0.903847\pi$
$954$		0			0
$955$			25.5737i			0.827546i
$956$		0			0
$957$		−	49.9227i		−	1.61377i
$958$		0			0
$959$			4.63406i			0.149642i
$960$		0			0
$961$	29.2438			0.943348
$962$		0			0
$963$	102.880			3.31527
$964$		0			0
$965$		−	43.7953i		−	1.40982i
$966$		0			0
$967$		−	59.3891i		−	1.90982i	−0.296889	−	0.954912i	$-0.595949\pi$
$967$		−	59.3891i		−	1.90982i	0.296889	−	0.954912i	$-0.404051\pi$
$968$		0			0
$969$	89.1301			2.86327
$970$		0			0
$971$	19.3139			0.619812			0.309906	−	0.950767i	$-0.399702\pi$
$971$	19.3139			0.619812			0.309906	+	0.950767i	$0.399702\pi$
$972$		0			0
$973$			16.8714i			0.540871i
$974$		0			0
$975$			29.3633i			0.940378i
$976$		0			0
$977$		−	16.3366i		−	0.522655i	−0.965250	−	0.261328i	$-0.915840\pi$
$977$		−	16.3366i		−	0.522655i	0.965250	−	0.261328i	$-0.0841603\pi$
$978$		0			0
$979$			3.66755i			0.117215i
$980$		0			0
$981$			74.8235i			2.38893i
$982$		0			0
$983$	12.8408			0.409557			0.204779	−	0.978808i	$-0.434353\pi$
$983$	12.8408			0.409557			0.204779	+	0.978808i	$0.434353\pi$
$984$		0			0
$985$		−	48.4984i		−	1.54529i
$986$		0			0
$987$	22.4757			0.715410
$988$		0			0
$989$	8.60067	+	29.5180i	0.273486	+	0.938617i
$990$		0			0
$991$		−	27.8534i		−	0.884793i	−0.896819	−	0.442397i	$-0.854128\pi$
$991$		−	27.8534i		−	0.884793i	0.896819	−	0.442397i	$-0.145872\pi$
$992$		0			0
$993$	72.7624			2.30904
$994$		0			0
$995$		−	32.7115i		−	1.03702i
$996$		0			0
$997$	52.7736			1.67136			0.835678	−	0.549219i	$-0.185075\pi$
$997$	52.7736			1.67136			0.835678	+	0.549219i	$0.185075\pi$
$998$		0			0
$999$	−153.014			−4.84116

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.e.a.1471.12	yes	12
4.3	odd	2		2576.2.e.b.1471.1	yes	12
23.22	odd	2		2576.2.e.b.1471.12	yes	12
92.91	even	2	inner	2576.2.e.a.1471.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2576.2.e.a.1471.1	✓	12	92.91	even	2	inner
2576.2.e.a.1471.12	yes	12	1.1	even	1	trivial
2576.2.e.b.1471.1	yes	12	4.3	odd	2
2576.2.e.b.1471.12	yes	12	23.22	odd	2

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:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.17144i q^{3} +2.64852i q^{5} -1.00000 q^{7} -7.05804 q^{9} +O(q^{10})\)	\(q+3.17144i q^{3} +2.64852i q^{5} -1.00000 q^{7} -7.05804 q^{9} -1.54103 q^{11} -4.59567 q^{13} -8.39962 q^{15} +5.61697i q^{17} -5.00340 q^{19} -3.17144i q^{21} +(1.34158 + 4.60436i) q^{23} -2.01465 q^{25} -12.8699i q^{27} +10.2148 q^{29} +1.32522i q^{31} -4.88728i q^{33} -2.64852i q^{35} -11.8894i q^{37} -14.5749i q^{39} +5.27882 q^{41} +6.41087 q^{43} -18.6934i q^{45} +7.08691i q^{47} +1.00000 q^{49} -17.8139 q^{51} +10.9176i q^{53} -4.08144i q^{55} -15.8680i q^{57} +2.40454i q^{59} -6.48598i q^{61} +7.05804 q^{63} -12.1717i q^{65} -10.4969 q^{67} +(-14.6025 + 4.25473i) q^{69} +3.90680i q^{71} +15.2856 q^{73} -6.38934i q^{75} +1.54103 q^{77} -10.1748 q^{79} +19.6419 q^{81} -12.9540 q^{83} -14.8766 q^{85} +32.3957i q^{87} -2.37994i q^{89} +4.59567 q^{91} -4.20285 q^{93} -13.2516i q^{95} +3.86958i q^{97} +10.8766 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 8 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 - 8 * q^9	\( 12 q - 12 q^{7} - 8 q^{9} - 12 q^{13} - 8 q^{15} - 8 q^{19} - 4 q^{25} + 12 q^{29} - 12 q^{41} + 16 q^{43} + 12 q^{49} - 8 q^{51} + 8 q^{63} - 40 q^{67} - 28 q^{69} + 4 q^{73} - 16 q^{79} + 20 q^{81} - 48 q^{83} - 32 q^{85} + 12 q^{91} + 4 q^{93} - 16 q^{99}+O(q^{100}) \) 12 * q - 12 * q^7 - 8 * q^9 - 12 * q^13 - 8 * q^15 - 8 * q^19 - 4 * q^25 + 12 * q^29 - 12 * q^41 + 16 * q^43 + 12 * q^49 - 8 * q^51 + 8 * q^63 - 40 * q^67 - 28 * q^69 + 4 * q^73 - 16 * q^79 + 20 * q^81 - 48 * q^83 - 32 * q^85 + 12 * q^91 + 4 * q^93 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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