LMFDB - Embedded newform 1456.2.k.f.337.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(337,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1456.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.k (of order $2$, degree $1$, not 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:	$11.6262185343$
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} + 21x^{10} + 152x^{8} + 456x^{6} + 532x^{4} + 192x^{2} + 16 $ x^12 + 21x^10 + 152x^8 + 456x^6 + 532x^4 + 192*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.10
Root			$1.25097i$ of defining polynomial
Character	$\chi$	$=$	1456.337
Dual form			1456.2.k.f.337.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.25097 q^{3} +2.55830i q^{5} -1.00000i q^{7} -1.43508 q^{9} +O(q^{10})$	$q+1.25097 q^{3} +2.55830i q^{5} -1.00000i q^{7} -1.43508 q^{9} -0.347794i q^{11} +(-3.40273 + 1.19223i) q^{13} +3.20035i q^{15} +2.78287 q^{17} +6.54152i q^{19} -1.25097i q^{21} +6.22674 q^{23} -1.54488 q^{25} -5.54814 q^{27} -7.72563 q^{29} +7.29337i q^{31} -0.435079i q^{33} +2.55830 q^{35} +6.07321i q^{37} +(-4.25671 + 1.49144i) q^{39} +5.60092i q^{41} +6.77454 q^{43} -3.67136i q^{45} +8.92868i q^{47} -1.00000 q^{49} +3.48129 q^{51} -5.64117 q^{53} +0.889761 q^{55} +8.18323i q^{57} +0.320308i q^{59} -12.4873 q^{61} +1.43508i q^{63} +(-3.05008 - 8.70520i) q^{65} -8.16280i q^{67} +7.78945 q^{69} -2.83212i q^{71} -10.7950i q^{73} -1.93260 q^{75} -0.347794 q^{77} +3.70802 q^{79} -2.63531 q^{81} -13.5737i q^{83} +7.11942i q^{85} -9.66451 q^{87} +7.10815i q^{89} +(1.19223 + 3.40273i) q^{91} +9.12378i q^{93} -16.7351 q^{95} +5.45877i q^{97} +0.499112i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{3} + 6 q^{9}+O(q^{10}) $ 12 * q - 2 * q^3 + 6 * q^9	$ 12 q - 2 q^{3} + 6 q^{9} + 6 q^{13} + 4 q^{17} + 20 q^{23} - 26 q^{25} - 26 q^{27} + 10 q^{29} - 2 q^{35} - 18 q^{39} + 6 q^{43} - 12 q^{49} + 12 q^{51} + 6 q^{53} + 40 q^{55} + 6 q^{61} + 38 q^{65} + 30 q^{69} + 58 q^{75} + 2 q^{77} - 40 q^{79} + 4 q^{81} - 8 q^{87} - 6 q^{91} - 54 q^{95}+O(q^{100}) $ 12 * q - 2 * q^3 + 6 * q^9 + 6 * q^13 + 4 * q^17 + 20 * q^23 - 26 * q^25 - 26 * q^27 + 10 * q^29 - 2 * q^35 - 18 * q^39 + 6 * q^43 - 12 * q^49 + 12 * q^51 + 6 * q^53 + 40 * q^55 + 6 * q^61 + 38 * q^65 + 30 * q^69 + 58 * q^75 + 2 * q^77 - 40 * q^79 + 4 * q^81 - 8 * q^87 - 6 * q^91 - 54 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$.

$n$	$561$	$911$	$1093$	$1249$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	1.25097			0.722247			0.361123	−	0.932518i	$-0.382393\pi$
$3$	1.25097			0.722247			0.361123	+	0.932518i	$0.382393\pi$
$4$		0			0
$5$			2.55830i			1.14411i	0.820217	+	0.572053i	$0.193853\pi$
$5$			2.55830i			1.14411i	−0.820217	+	0.572053i	$0.806147\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−1.43508			−0.478360
$10$		0			0
$11$		−	0.347794i		−	0.104864i	−0.998624	−	0.0524320i	$-0.983303\pi$
$11$		−	0.347794i		−	0.104864i	0.998624	−	0.0524320i	$-0.0166973\pi$
$12$		0			0
$13$	−3.40273	+	1.19223i	−0.943748	+	0.330665i
$13$	−3.40273	+	1.19223i	−0.943748	+	0.330665i
$14$		0			0
$15$			3.20035i			0.826326i
$16$		0			0
$17$	2.78287			0.674946			0.337473	−	0.941335i	$-0.390428\pi$
$17$	2.78287			0.674946			0.337473	+	0.941335i	$0.390428\pi$
$18$		0			0
$19$			6.54152i			1.50073i	0.661025	+	0.750364i	$0.270122\pi$
$19$			6.54152i			1.50073i	−0.661025	+	0.750364i	$0.729878\pi$
$20$		0			0
$21$		−	1.25097i		−	0.272984i
$22$		0			0
$23$	6.22674			1.29836			0.649182	−	0.760633i	$-0.275111\pi$
$23$	6.22674			1.29836			0.649182	+	0.760633i	$0.275111\pi$
$24$		0			0
$25$	−1.54488			−0.308977
$26$		0			0
$27$	−5.54814			−1.06774
$28$		0			0
$29$	−7.72563			−1.43461			−0.717307	−	0.696758i	$-0.754625\pi$
$29$	−7.72563			−1.43461			−0.717307	+	0.696758i	$0.754625\pi$
$30$		0			0
$31$			7.29337i			1.30993i	0.755660	+	0.654964i	$0.227316\pi$
$31$			7.29337i			1.30993i	−0.755660	+	0.654964i	$0.772684\pi$
$32$		0			0
$33$		−	0.435079i		−	0.0757376i
$34$		0			0
$35$	2.55830			0.432431
$36$		0			0
$37$			6.07321i			0.998430i	0.866478	+	0.499215i	$0.166378\pi$
$37$			6.07321i			0.998430i	−0.866478	+	0.499215i	$0.833622\pi$
$38$		0			0
$39$	−4.25671	+	1.49144i	−0.681619	+	0.238822i
$40$		0			0
$41$			5.60092i			0.874717i	0.899287	+	0.437359i	$0.144086\pi$
$41$			5.60092i			0.874717i	−0.899287	+	0.437359i	$0.855914\pi$
$42$		0			0
$43$	6.77454			1.03311			0.516554	−	0.856255i	$-0.327215\pi$
$43$	6.77454			1.03311			0.516554	+	0.856255i	$0.327215\pi$
$44$		0			0
$45$		−	3.67136i		−	0.547294i
$46$		0			0
$47$			8.92868i			1.30238i	0.758914	+	0.651191i	$0.225730\pi$
$47$			8.92868i			1.30238i	−0.758914	+	0.651191i	$0.774270\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	3.48129			0.487477
$52$		0			0
$53$	−5.64117			−0.774874			−0.387437	−	0.921896i	$-0.626640\pi$
$53$	−5.64117			−0.774874			−0.387437	+	0.921896i	$0.626640\pi$
$54$		0			0
$55$	0.889761			0.119975
$56$		0			0
$57$			8.18323i			1.08390i
$58$		0			0
$59$			0.320308i			0.0417006i	0.999783	+	0.0208503i	$0.00663733\pi$
$59$			0.320308i			0.0417006i	−0.999783	+	0.0208503i	$0.993363\pi$
$60$		0			0
$61$	−12.4873			−1.59884			−0.799419	−	0.600774i	$-0.794859\pi$
$61$	−12.4873			−1.59884			−0.799419	+	0.600774i	$0.794859\pi$
$62$		0			0
$63$			1.43508i			0.180803i
$64$		0			0
$65$	−3.05008	−	8.70520i	−0.378316	−	1.07975i
$66$		0			0
$67$		−	8.16280i		−	0.997245i	−0.866819	−	0.498622i	$-0.833839\pi$
$67$		−	8.16280i		−	0.997245i	0.866819	−	0.498622i	$-0.166161\pi$
$68$		0			0
$69$	7.78945			0.937740
$70$		0			0
$71$		−	2.83212i		−	0.336111i	−0.985778	−	0.168056i	$-0.946251\pi$
$71$		−	2.83212i		−	0.336111i	0.985778	−	0.168056i	$-0.0537488\pi$
$72$		0			0
$73$		−	10.7950i		−	1.26346i	−0.775190	−	0.631728i	$-0.782346\pi$
$73$		−	10.7950i		−	1.26346i	0.775190	−	0.631728i	$-0.217654\pi$
$74$		0			0
$75$	−1.93260			−0.223158
$76$		0			0
$77$	−0.347794			−0.0396348
$78$		0			0
$79$	3.70802			0.417185			0.208593	−	0.978003i	$-0.433112\pi$
$79$	3.70802			0.417185			0.208593	+	0.978003i	$0.433112\pi$
$80$		0			0
$81$	−2.63531			−0.292812
$82$		0			0
$83$		−	13.5737i		−	1.48990i	−0.667119	−	0.744951i	$-0.732473\pi$
$83$		−	13.5737i		−	1.48990i	0.667119	−	0.744951i	$-0.267527\pi$
$84$		0			0
$85$			7.11942i			0.772209i
$86$		0			0
$87$	−9.66451			−1.03614
$88$		0			0
$89$			7.10815i			0.753462i	0.926323	+	0.376731i	$0.122952\pi$
$89$			7.10815i			0.753462i	−0.926323	+	0.376731i	$0.877048\pi$
$90$		0			0
$91$	1.19223	+	3.40273i	0.124980	+	0.356703i
$92$		0			0
$93$			9.12378i			0.946092i
$94$		0			0
$95$	−16.7351			−1.71699
$96$		0			0
$97$			5.45877i			0.554254i	0.960833	+	0.277127i	$0.0893822\pi$
$97$			5.45877i			0.554254i	−0.960833	+	0.277127i	$0.910618\pi$
$98$		0			0
$99$			0.499112i			0.0501627i
$100$		0			0
$101$	−4.63631			−0.461330			−0.230665	−	0.973033i	$-0.574090\pi$
$101$	−4.63631			−0.461330			−0.230665	+	0.973033i	$0.574090\pi$
$102$		0			0
$103$	2.19365			0.216147			0.108074	−	0.994143i	$-0.465532\pi$
$103$	2.19365			0.216147			0.108074	+	0.994143i	$0.465532\pi$
$104$		0			0
$105$	3.20035			0.312322
$106$		0			0
$107$	−9.95507			−0.962393			−0.481197	−	0.876613i	$-0.659798\pi$
$107$	−9.95507			−0.962393			−0.481197	+	0.876613i	$0.659798\pi$
$108$		0			0
$109$		−	11.1343i		−	1.06647i	−0.845966	−	0.533237i	$-0.820975\pi$
$109$		−	11.1343i		−	1.06647i	0.845966	−	0.533237i	$-0.179025\pi$
$110$		0			0
$111$			7.59739i			0.721113i
$112$		0			0
$113$	12.3262			1.15955			0.579776	−	0.814776i	$-0.303140\pi$
$113$	12.3262			1.15955			0.579776	+	0.814776i	$0.303140\pi$
$114$		0			0
$115$			15.9298i			1.48547i
$116$		0			0
$117$	4.88319	−	1.71094i	0.451451	−	0.158177i
$118$		0			0
$119$		−	2.78287i		−	0.255106i
$120$		0			0
$121$	10.8790			0.989004
$122$		0			0
$123$			7.00658i			0.631762i
$124$		0			0
$125$			8.83921i			0.790603i
$126$		0			0
$127$	17.6194			1.56347			0.781733	−	0.623614i	$-0.214336\pi$
$127$	17.6194			1.56347			0.781733	+	0.623614i	$0.214336\pi$
$128$		0			0
$129$	8.47473			0.746159
$130$		0			0
$131$	−17.2871			−1.51038			−0.755190	−	0.655506i	$-0.772455\pi$
$131$	−17.2871			−1.51038			−0.755190	+	0.655506i	$0.772455\pi$
$132$		0			0
$133$	6.54152			0.567222
$134$		0			0
$135$		−	14.1938i		−	1.22161i
$136$		0			0
$137$		−	7.32081i		−	0.625459i	−0.949842	−	0.312730i	$-0.898757\pi$
$137$		−	7.32081i		−	0.625459i	0.949842	−	0.312730i	$-0.101243\pi$
$138$		0			0
$139$	11.9834			1.01642			0.508209	−	0.861234i	$-0.330308\pi$
$139$	11.9834			1.01642			0.508209	+	0.861234i	$0.330308\pi$
$140$		0			0
$141$			11.1695i			0.940641i
$142$		0			0
$143$	0.414651	+	1.18345i	0.0346748	+	0.0989651i
$144$		0			0
$145$		−	19.7645i		−	1.64135i
$146$		0			0
$147$	−1.25097			−0.103178
$148$		0			0
$149$			19.8575i			1.62679i	0.581713	+	0.813394i	$0.302383\pi$
$149$			19.8575i			1.62679i	−0.581713	+	0.813394i	$0.697617\pi$
$150$		0			0
$151$		−	7.93679i		−	0.645887i	−0.946418	−	0.322944i	$-0.895328\pi$
$151$		−	7.93679i		−	0.645887i	0.946418	−	0.322944i	$-0.104672\pi$
$152$		0			0
$153$	−3.99364			−0.322867
$154$		0			0
$155$	−18.6586			−1.49870
$156$		0			0
$157$	23.0556			1.84003			0.920017	−	0.391879i	$-0.128175\pi$
$157$	23.0556			1.84003			0.920017	+	0.391879i	$0.128175\pi$
$158$		0			0
$159$	−7.05692			−0.559650
$160$		0			0
$161$		−	6.22674i		−	0.490736i
$162$		0			0
$163$		−	13.0491i		−	1.02208i	−0.859557	−	0.511040i	$-0.829260\pi$
$163$		−	13.0491i		−	1.02208i	0.859557	−	0.511040i	$-0.170740\pi$
$164$		0			0
$165$	1.11306			0.0866518
$166$		0			0
$167$			12.5463i			0.970860i	0.874276	+	0.485430i	$0.161337\pi$
$167$			12.5463i			0.970860i	−0.874276	+	0.485430i	$0.838663\pi$
$168$		0			0
$169$	10.1572	−	8.11368i	0.781321	−	0.624129i
$170$		0			0
$171$		−	9.38760i		−	0.717888i
$172$		0			0
$173$	23.9025			1.81727			0.908635	−	0.417591i	$-0.137126\pi$
$173$	23.9025			1.81727			0.908635	+	0.417591i	$0.137126\pi$
$174$		0			0
$175$			1.54488i			0.116782i
$176$		0			0
$177$			0.400695i			0.0301181i
$178$		0			0
$179$	18.1339			1.35539			0.677697	−	0.735342i	$-0.262978\pi$
$179$	18.1339			1.35539			0.677697	+	0.735342i	$0.262978\pi$
$180$		0			0
$181$	−18.7473			−1.39347			−0.696737	−	0.717327i	$-0.745365\pi$
$181$	−18.7473			−1.39347			−0.696737	+	0.717327i	$0.745365\pi$
$182$		0			0
$183$	−15.6212			−1.15475
$184$		0			0
$185$	−15.5371			−1.14231
$186$		0			0
$187$		−	0.967867i		−	0.0707775i
$188$		0			0
$189$			5.54814i			0.403568i
$190$		0			0
$191$	8.40423			0.608109			0.304054	−	0.952655i	$-0.401660\pi$
$191$	8.40423			0.608109			0.304054	+	0.952655i	$0.401660\pi$
$192$		0			0
$193$			8.38357i			0.603462i	0.953393	+	0.301731i	$0.0975645\pi$
$193$			8.38357i			0.603462i	−0.953393	+	0.301731i	$0.902436\pi$
$194$		0			0
$195$	−3.81555	−	10.8899i	−0.273237	−	0.779844i
$196$		0			0
$197$			12.1102i			0.862813i	0.902158	+	0.431407i	$0.141983\pi$
$197$			12.1102i			0.862813i	−0.902158	+	0.431407i	$0.858017\pi$
$198$		0			0
$199$	4.62101			0.327575			0.163788	−	0.986496i	$-0.447629\pi$
$199$	4.62101			0.327575			0.163788	+	0.986496i	$0.447629\pi$
$200$		0			0
$201$		−	10.2114i		−	0.720257i
$202$		0			0
$203$			7.72563i			0.542233i
$204$		0			0
$205$	−14.3288			−1.00077
$206$		0			0
$207$	−8.93586			−0.621086
$208$		0			0
$209$	2.27510			0.157372
$210$		0			0
$211$	23.7132			1.63248			0.816241	−	0.577711i	$-0.196054\pi$
$211$	23.7132			1.63248			0.816241	+	0.577711i	$0.196054\pi$
$212$		0			0
$213$		−	3.54290i		−	0.242755i
$214$		0			0
$215$			17.3313i			1.18198i
$216$		0			0
$217$	7.29337			0.495106
$218$		0			0
$219$		−	13.5042i		−	0.912527i
$220$		0			0
$221$	−9.46937	+	3.31782i	−0.636979	+	0.223181i
$222$		0			0
$223$			0.607039i			0.0406503i	0.999793	+	0.0203252i	$0.00647015\pi$
$223$			0.607039i			0.0406503i	−0.999793	+	0.0203252i	$0.993530\pi$
$224$		0			0
$225$	2.21703			0.147802
$226$		0			0
$227$			2.81871i			0.187084i	0.995615	+	0.0935422i	$0.0298190\pi$
$227$			2.81871i			0.187084i	−0.995615	+	0.0935422i	$0.970181\pi$
$228$		0			0
$229$			10.3471i			0.683758i	0.939744	+	0.341879i	$0.111063\pi$
$229$			10.3471i			0.683758i	−0.939744	+	0.341879i	$0.888937\pi$
$230$		0			0
$231$	−0.435079			−0.0286261
$232$		0			0
$233$	−9.90168			−0.648680			−0.324340	−	0.945940i	$-0.605142\pi$
$233$	−9.90168			−0.648680			−0.324340	+	0.945940i	$0.605142\pi$
$234$		0			0
$235$	−22.8422			−1.49006
$236$		0			0
$237$	4.63862			0.301311
$238$		0			0
$239$		−	14.3760i		−	0.929904i	−0.885336	−	0.464952i	$-0.846072\pi$
$239$		−	14.3760i		−	0.929904i	0.885336	−	0.464952i	$-0.153928\pi$
$240$		0			0
$241$		−	6.56693i		−	0.423013i	−0.977377	−	0.211507i	$-0.932163\pi$
$241$		−	6.56693i		−	0.423013i	0.977377	−	0.211507i	$-0.0678370\pi$
$242$		0			0
$243$	13.3477			0.856258
$244$		0			0
$245$		−	2.55830i		−	0.163444i
$246$		0			0
$247$	−7.79899	−	22.2590i	−0.496238	−	1.41631i
$248$		0			0
$249$		−	16.9802i		−	1.07608i
$250$		0			0
$251$	4.52182			0.285415			0.142707	−	0.989765i	$-0.454419\pi$
$251$	4.52182			0.285415			0.142707	+	0.989765i	$0.454419\pi$
$252$		0			0
$253$		−	2.16562i		−	0.136152i
$254$		0			0
$255$			8.90616i			0.557726i
$256$		0			0
$257$	3.69699			0.230612			0.115306	−	0.993330i	$-0.463215\pi$
$257$	3.69699			0.230612			0.115306	+	0.993330i	$0.463215\pi$
$258$		0			0
$259$	6.07321			0.377371
$260$		0			0
$261$	11.0869			0.686261
$262$		0			0
$263$	−7.30895			−0.450689			−0.225345	−	0.974279i	$-0.572351\pi$
$263$	−7.30895			−0.450689			−0.225345	+	0.974279i	$0.572351\pi$
$264$		0			0
$265$		−	14.4318i		−	0.886537i
$266$		0			0
$267$			8.89206i			0.544186i
$268$		0			0
$269$	7.61716			0.464426			0.232213	−	0.972665i	$-0.425403\pi$
$269$	7.61716			0.464426			0.232213	+	0.972665i	$0.425403\pi$
$270$		0			0
$271$		−	3.46721i		−	0.210618i	−0.994440	−	0.105309i	$-0.966417\pi$
$271$		−	3.46721i		−	0.210618i	0.994440	−	0.105309i	$-0.0335832\pi$
$272$		0			0
$273$	1.49144	+	4.25671i	0.0902661	+	0.257628i
$274$		0			0
$275$			0.537302i			0.0324005i
$276$		0			0
$277$	−22.0321			−1.32378			−0.661892	−	0.749599i	$-0.730246\pi$
$277$	−22.0321			−1.32378			−0.661892	+	0.749599i	$0.730246\pi$
$278$		0			0
$279$		−	10.4666i		−	0.626617i
$280$		0			0
$281$		−	13.9693i		−	0.833337i	−0.909059	−	0.416668i	$-0.863198\pi$
$281$		−	13.9693i		−	0.833337i	0.909059	−	0.416668i	$-0.136802\pi$
$282$		0			0
$283$	17.7311			1.05401			0.527003	−	0.849864i	$-0.323316\pi$
$283$	17.7311			1.05401			0.527003	+	0.849864i	$0.323316\pi$
$284$		0			0
$285$	−20.9351			−1.24009
$286$		0			0
$287$	5.60092			0.330612
$288$		0			0
$289$	−9.25561			−0.544448
$290$		0			0
$291$			6.82874i			0.400308i
$292$		0			0
$293$			15.9193i			0.930013i	0.885307	+	0.465007i	$0.153948\pi$
$293$			15.9193i			0.930013i	−0.885307	+	0.465007i	$0.846052\pi$
$294$		0			0
$295$	−0.819443			−0.0477098
$296$		0			0
$297$			1.92961i			0.111967i
$298$		0			0
$299$	−21.1879	+	7.42370i	−1.22533	+	0.429324i
$300$		0			0
$301$		−	6.77454i		−	0.390478i
$302$		0			0
$303$	−5.79987			−0.333194
$304$		0			0
$305$		−	31.9463i		−	1.82924i
$306$		0			0
$307$		−	20.2235i		−	1.15422i	−0.816667	−	0.577109i	$-0.804181\pi$
$307$		−	20.2235i		−	1.15422i	0.816667	−	0.577109i	$-0.195819\pi$
$308$		0			0
$309$	2.74419			0.156111
$310$		0			0
$311$	2.14723			0.121758			0.0608790	−	0.998145i	$-0.480610\pi$
$311$	2.14723			0.121758			0.0608790	+	0.998145i	$0.480610\pi$
$312$		0			0
$313$	−14.9070			−0.842593			−0.421296	−	0.906923i	$-0.638425\pi$
$313$	−14.9070			−0.842593			−0.421296	+	0.906923i	$0.638425\pi$
$314$		0			0
$315$	−3.67136			−0.206858
$316$		0			0
$317$			5.37304i			0.301780i	0.988551	+	0.150890i	$0.0482139\pi$
$317$			5.37304i			0.301780i	−0.988551	+	0.150890i	$0.951786\pi$
$318$		0			0
$319$			2.68693i			0.150439i
$320$		0			0
$321$	−12.4535			−0.695085
$322$		0			0
$323$			18.2042i			1.01291i
$324$		0			0
$325$	5.25683	−	1.84186i	0.291596	−	0.102168i
$326$		0			0
$327$		−	13.9287i		−	0.770257i
$328$		0			0
$329$	8.92868			0.492254
$330$		0			0
$331$			13.0821i			0.719057i	0.933134	+	0.359529i	$0.117062\pi$
$331$			13.0821i			0.719057i	−0.933134	+	0.359529i	$0.882938\pi$
$332$		0			0
$333$		−	8.71554i		−	0.477609i
$334$		0			0
$335$	20.8829			1.14095
$336$		0			0
$337$	1.93700			0.105515			0.0527576	−	0.998607i	$-0.483199\pi$
$337$	1.93700			0.105515			0.0527576	+	0.998607i	$0.483199\pi$
$338$		0			0
$339$	15.4197			0.837483
$340$		0			0
$341$	2.53659			0.137364
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			19.9277i			1.07287i
$346$		0			0
$347$	−19.7442			−1.05992			−0.529962	−	0.848021i	$-0.677794\pi$
$347$	−19.7442			−1.05992			−0.529962	+	0.848021i	$0.677794\pi$
$348$		0			0
$349$		−	10.2818i		−	0.550374i	−0.961391	−	0.275187i	$-0.911260\pi$
$349$		−	10.2818i		−	0.550374i	0.961391	−	0.275187i	$-0.0887398\pi$
$350$		0			0
$351$	18.8788	−	6.61466i	1.00768	−	0.353064i
$352$		0			0
$353$		−	27.5367i		−	1.46563i	−0.680429	−	0.732814i	$-0.738206\pi$
$353$		−	27.5367i		−	1.46563i	0.680429	−	0.732814i	$-0.261794\pi$
$354$		0			0
$355$	7.24542			0.384547
$356$		0			0
$357$		−	3.48129i		−	0.184249i
$358$		0			0
$359$		−	23.3245i		−	1.23102i	−0.788130	−	0.615509i	$-0.788950\pi$
$359$		−	23.3245i		−	1.23102i	0.788130	−	0.615509i	$-0.211050\pi$
$360$		0			0
$361$	−23.7915			−1.25218
$362$		0			0
$363$	13.6093			0.714304
$364$		0			0
$365$	27.6167			1.44553
$366$		0			0
$367$	0.0606486			0.00316583			0.00158292	−	0.999999i	$-0.499496\pi$
$367$	0.0606486			0.00316583			0.00158292	+	0.999999i	$0.499496\pi$
$368$		0			0
$369$		−	8.03777i		−	0.418430i
$370$		0			0
$371$			5.64117i			0.292875i
$372$		0			0
$373$	−7.22028			−0.373852			−0.186926	−	0.982374i	$-0.559852\pi$
$373$	−7.22028			−0.373852			−0.186926	+	0.982374i	$0.559852\pi$
$374$		0			0
$375$			11.0576i			0.571010i
$376$		0			0
$377$	26.2883	−	9.21072i	1.35391	−	0.474376i
$378$		0			0
$379$			3.82649i			0.196554i	0.995159	+	0.0982768i	$0.0313330\pi$
$379$			3.82649i			0.196554i	−0.995159	+	0.0982768i	$0.968667\pi$
$380$		0			0
$381$	22.0412			1.12921
$382$		0			0
$383$		−	23.8886i		−	1.22065i	−0.792150	−	0.610326i	$-0.791038\pi$
$383$		−	23.8886i		−	1.22065i	0.792150	−	0.610326i	$-0.208962\pi$
$384$		0			0
$385$		−	0.889761i		−	0.0453464i
$386$		0			0
$387$	−9.72200			−0.494197
$388$		0			0
$389$	−14.4570			−0.733000			−0.366500	−	0.930418i	$-0.619444\pi$
$389$	−14.4570			−0.733000			−0.366500	+	0.930418i	$0.619444\pi$
$390$		0			0
$391$	17.3282			0.876326
$392$		0			0
$393$	−21.6256			−1.09087
$394$		0			0
$395$			9.48623i			0.477304i
$396$		0			0
$397$		−	24.7810i		−	1.24372i	−0.783127	−	0.621862i	$-0.786376\pi$
$397$		−	24.7810i		−	1.24372i	0.783127	−	0.621862i	$-0.213624\pi$
$398$		0			0
$399$	8.18323			0.409674
$400$		0			0
$401$			12.9115i			0.644768i	0.946609	+	0.322384i	$0.104484\pi$
$401$			12.9115i			0.644768i	−0.946609	+	0.322384i	$0.895516\pi$
$402$		0			0
$403$	−8.69538	−	24.8174i	−0.433148	−	1.23624i
$404$		0			0
$405$		−	6.74190i		−	0.335008i
$406$		0			0
$407$	2.11223			0.104699
$408$		0			0
$409$			20.1151i			0.994628i	0.867570	+	0.497314i	$0.165680\pi$
$409$			20.1151i			0.994628i	−0.867570	+	0.497314i	$0.834320\pi$
$410$		0			0
$411$		−	9.15810i		−	0.451736i
$412$		0			0
$413$	0.320308			0.0157613
$414$		0			0
$415$	34.7254			1.70460
$416$		0			0
$417$	14.9908			0.734104
$418$		0			0
$419$	−2.46880			−0.120609			−0.0603045	−	0.998180i	$-0.519207\pi$
$419$	−2.46880			−0.120609			−0.0603045	+	0.998180i	$0.519207\pi$
$420$		0			0
$421$		−	29.0009i		−	1.41342i	−0.707505	−	0.706709i	$-0.750179\pi$
$421$		−	29.0009i		−	1.41342i	0.707505	−	0.706709i	$-0.249821\pi$
$422$		0			0
$423$		−	12.8134i		−	0.623007i
$424$		0			0
$425$	−4.29922			−0.208543
$426$		0			0
$427$			12.4873i			0.604304i
$428$		0			0
$429$	0.518715	+	1.48046i	0.0250438	+	0.0714772i
$430$		0			0
$431$			19.1436i			0.922113i	0.887371	+	0.461056i	$0.152529\pi$
$431$			19.1436i			0.922113i	−0.887371	+	0.461056i	$0.847471\pi$
$432$		0			0
$433$	25.5304			1.22691			0.613457	−	0.789728i	$-0.289778\pi$
$433$	25.5304			1.22691			0.613457	+	0.789728i	$0.289778\pi$
$434$		0			0
$435$		−	24.7247i		−	1.18546i
$436$		0			0
$437$			40.7323i			1.94849i
$438$		0			0
$439$	5.04350			0.240713			0.120357	−	0.992731i	$-0.461596\pi$
$439$	5.04350			0.240713			0.120357	+	0.992731i	$0.461596\pi$
$440$		0			0
$441$	1.43508			0.0683371
$442$		0			0
$443$	−3.32705			−0.158073			−0.0790364	−	0.996872i	$-0.525184\pi$
$443$	−3.32705			−0.158073			−0.0790364	+	0.996872i	$0.525184\pi$
$444$		0			0
$445$	−18.1848			−0.862040
$446$		0			0
$447$			24.8411i			1.17494i
$448$		0			0
$449$			4.82389i			0.227653i	0.993501	+	0.113827i	$0.0363109\pi$
$449$			4.82389i			0.227653i	−0.993501	+	0.113827i	$0.963689\pi$
$450$		0			0
$451$	1.94797			0.0917263
$452$		0			0
$453$		−	9.92867i		−	0.466490i
$454$		0			0
$455$	−8.70520	+	3.05008i	−0.408106	+	0.142990i
$456$		0			0
$457$			30.6503i			1.43376i	0.697197	+	0.716880i	$0.254430\pi$
$457$			30.6503i			1.43376i	−0.697197	+	0.716880i	$0.745570\pi$
$458$		0			0
$459$	−15.4398			−0.720667
$460$		0			0
$461$			34.5594i			1.60959i	0.593553	+	0.804795i	$0.297725\pi$
$461$			34.5594i			1.60959i	−0.593553	+	0.804795i	$0.702275\pi$
$462$		0			0
$463$		−	22.9435i		−	1.06628i	−0.846028	−	0.533138i	$-0.821013\pi$
$463$		−	22.9435i		−	1.06628i	0.846028	−	0.533138i	$-0.178987\pi$
$464$		0			0
$465$	−23.3413			−1.08243
$466$		0			0
$467$	13.0868			0.605587			0.302794	−	0.953056i	$-0.402081\pi$
$467$	13.0868			0.605587			0.302794	+	0.953056i	$0.402081\pi$
$468$		0			0
$469$	−8.16280			−0.376923
$470$		0			0
$471$	28.8418			1.32896
$472$		0			0
$473$		−	2.35615i		−	0.108336i
$474$		0			0
$475$		−	10.1059i		−	0.463690i
$476$		0			0
$477$	8.09552			0.370669
$478$		0			0
$479$			18.2224i			0.832604i	0.909226	+	0.416302i	$0.136674\pi$
$479$			18.2224i			0.832604i	−0.909226	+	0.416302i	$0.863326\pi$
$480$		0			0
$481$	−7.24066	−	20.6655i	−0.330146	−	0.942266i
$482$		0			0
$483$		−	7.78945i		−	0.354432i
$484$		0			0
$485$	−13.9651			−0.634125
$486$		0			0
$487$			9.00582i			0.408093i	0.978961	+	0.204046i	$0.0654094\pi$
$487$			9.00582i			0.408093i	−0.978961	+	0.204046i	$0.934591\pi$
$488$		0			0
$489$		−	16.3239i		−	0.738194i
$490$		0			0
$491$	27.6827			1.24930			0.624651	−	0.780904i	$-0.285241\pi$
$491$	27.6827			1.24930			0.624651	+	0.780904i	$0.285241\pi$
$492$		0			0
$493$	−21.4995			−0.968287
$494$		0			0
$495$	−1.27688			−0.0573914
$496$		0			0
$497$	−2.83212			−0.127038
$498$		0			0
$499$			26.0910i			1.16799i	0.811756	+	0.583996i	$0.198512\pi$
$499$			26.0910i			1.16799i	−0.811756	+	0.583996i	$0.801488\pi$
$500$		0			0
$501$			15.6950i			0.701200i
$502$		0			0
$503$	7.02032			0.313021			0.156510	−	0.987676i	$-0.449976\pi$
$503$	7.02032			0.313021			0.156510	+	0.987676i	$0.449976\pi$
$504$		0			0
$505$		−	11.8611i		−	0.527810i
$506$		0			0
$507$	12.7063	−	10.1499i	0.564307	−	0.450775i
$508$		0			0
$509$		−	37.2033i		−	1.64901i	−0.565856	−	0.824504i	$-0.691454\pi$
$509$		−	37.2033i		−	1.64901i	0.565856	−	0.824504i	$-0.308546\pi$
$510$		0			0
$511$	−10.7950			−0.477541
$512$		0			0
$513$		−	36.2933i		−	1.60239i
$514$		0			0
$515$			5.61202i			0.247295i
$516$		0			0
$517$	3.10534			0.136573
$518$		0			0
$519$	29.9012			1.31252
$520$		0			0
$521$	28.8161			1.26245			0.631227	−	0.775598i	$-0.282552\pi$
$521$	28.8161			1.26245			0.631227	+	0.775598i	$0.282552\pi$
$522$		0			0
$523$	7.19007			0.314400			0.157200	−	0.987567i	$-0.449753\pi$
$523$	7.19007			0.314400			0.157200	+	0.987567i	$0.449753\pi$
$524$		0			0
$525$			1.93260i			0.0843456i
$526$		0			0
$527$			20.2965i			0.884131i
$528$		0			0
$529$	15.7723			0.685751
$530$		0			0
$531$		−	0.459668i		−	0.0199479i
$532$		0			0
$533$	−6.67759	−	19.0584i	−0.289238	−	0.825513i
$534$		0			0
$535$		−	25.4680i		−	1.10108i
$536$		0			0
$537$	22.6850			0.978928
$538$		0			0
$539$			0.347794i			0.0149806i
$540$		0			0
$541$		−	27.3190i		−	1.17454i	−0.809392	−	0.587269i	$-0.800203\pi$
$541$		−	27.3190i		−	1.17454i	0.809392	−	0.587269i	$-0.199797\pi$
$542$		0			0
$543$	−23.4522			−1.00643
$544$		0			0
$545$	28.4849			1.22016
$546$		0			0
$547$	−34.4349			−1.47233			−0.736165	−	0.676802i	$-0.763365\pi$
$547$	−34.4349			−1.47233			−0.736165	+	0.676802i	$0.763365\pi$
$548$		0			0
$549$	17.9203			0.764820
$550$		0			0
$551$		−	50.5374i		−	2.15296i
$552$		0			0
$553$		−	3.70802i		−	0.157681i
$554$		0			0
$555$	−19.4364			−0.825029
$556$		0			0
$557$			44.8311i			1.89955i	0.312929	+	0.949777i	$0.398690\pi$
$557$			44.8311i			1.89955i	−0.312929	+	0.949777i	$0.601310\pi$
$558$		0			0
$559$	−23.0520	+	8.07681i	−0.974994	+	0.341613i
$560$		0			0
$561$		−	1.21077i		−	0.0511188i
$562$		0			0
$563$	6.83500			0.288061			0.144030	−	0.989573i	$-0.453994\pi$
$563$	6.83500			0.288061			0.144030	+	0.989573i	$0.453994\pi$
$564$		0			0
$565$			31.5341i			1.32665i
$566$		0			0
$567$			2.63531i			0.110673i
$568$		0			0
$569$	−32.2229			−1.35085			−0.675426	−	0.737428i	$-0.736040\pi$
$569$	−32.2229			−1.35085			−0.675426	+	0.737428i	$0.736040\pi$
$570$		0			0
$571$	16.3337			0.683542			0.341771	−	0.939783i	$-0.388973\pi$
$571$	16.3337			0.683542			0.341771	+	0.939783i	$0.388973\pi$
$572$		0			0
$573$	10.5134			0.439204
$574$		0			0
$575$	−9.61959			−0.401165
$576$		0			0
$577$		−	6.62932i		−	0.275982i	−0.990433	−	0.137991i	$-0.955935\pi$
$577$		−	6.62932i		−	0.275982i	0.990433	−	0.137991i	$-0.0440646\pi$
$578$		0			0
$579$			10.4876i			0.435849i
$580$		0			0
$581$	−13.5737			−0.563130
$582$		0			0
$583$			1.96197i			0.0812563i
$584$		0			0
$585$	4.37710	+	12.4927i	0.180971	+	0.516508i
$586$		0			0
$587$			12.1333i			0.500796i	0.968143	+	0.250398i	$0.0805615\pi$
$587$			12.1333i			0.500796i	−0.968143	+	0.250398i	$0.919438\pi$
$588$		0			0
$589$	−47.7097			−1.96585
$590$		0			0
$591$			15.1494i			0.623164i
$592$		0			0
$593$		−	1.64576i		−	0.0675831i	−0.999429	−	0.0337915i	$-0.989242\pi$
$593$		−	1.64576i		−	0.0675831i	0.999429	−	0.0337915i	$-0.0107582\pi$
$594$		0			0
$595$	7.11942			0.291868
$596$		0			0
$597$	5.78074			0.236590
$598$		0			0
$599$	−28.1742			−1.15117			−0.575584	−	0.817743i	$-0.695225\pi$
$599$	−28.1742			−1.15117			−0.575584	+	0.817743i	$0.695225\pi$
$600$		0			0
$601$	26.9184			1.09803			0.549013	−	0.835814i	$-0.315004\pi$
$601$	26.9184			1.09803			0.549013	+	0.835814i	$0.315004\pi$
$602$		0			0
$603$			11.7143i			0.477042i
$604$		0			0
$605$			27.8318i			1.13152i
$606$		0			0
$607$	−46.4660			−1.88600			−0.942999	−	0.332795i	$-0.892008\pi$
$607$	−46.4660			−1.88600			−0.942999	+	0.332795i	$0.892008\pi$
$608$		0			0
$609$			9.66451i			0.391626i
$610$		0			0
$611$	−10.6450	−	30.3819i	−0.430652	−	1.22912i
$612$		0			0
$613$		−	43.2218i		−	1.74571i	−0.487978	−	0.872856i	$-0.662265\pi$
$613$		−	43.2218i		−	1.74571i	0.487978	−	0.872856i	$-0.337735\pi$
$614$		0			0
$615$	−17.9249			−0.722802
$616$		0			0
$617$			28.8669i			1.16214i	0.813855	+	0.581068i	$0.197365\pi$
$617$			28.8669i			1.16214i	−0.813855	+	0.581068i	$0.802635\pi$
$618$		0			0
$619$			32.1542i			1.29238i	0.763174	+	0.646192i	$0.223640\pi$
$619$			32.1542i			1.29238i	−0.763174	+	0.646192i	$0.776360\pi$
$620$		0			0
$621$	−34.5468			−1.38632
$622$		0			0
$623$	7.10815			0.284782
$624$		0			0
$625$	−30.3378			−1.21351
$626$		0			0
$627$	2.84608			0.113661
$628$		0			0
$629$			16.9010i			0.673886i
$630$		0			0
$631$			31.0569i			1.23636i	0.786038	+	0.618178i	$0.212129\pi$
$631$			31.0569i			1.23636i	−0.786038	+	0.618178i	$0.787871\pi$
$632$		0			0
$633$	29.6644			1.17905
$634$		0			0
$635$			45.0756i			1.78877i
$636$		0			0
$637$	3.40273	−	1.19223i	0.134821	−	0.0472379i
$638$		0			0
$639$			4.06432i			0.160782i
$640$		0			0
$641$	−28.6255			−1.13064			−0.565319	−	0.824872i	$-0.691247\pi$
$641$	−28.6255			−1.13064			−0.565319	+	0.824872i	$0.691247\pi$
$642$		0			0
$643$		−	4.57879i		−	0.180570i	−0.995916	−	0.0902849i	$-0.971222\pi$
$643$		−	4.57879i		−	0.180570i	0.995916	−	0.0902849i	$-0.0287778\pi$
$644$		0			0
$645$			21.6809i			0.853684i
$646$		0			0
$647$	−26.4784			−1.04097			−0.520486	−	0.853870i	$-0.674249\pi$
$647$	−26.4784			−1.04097			−0.520486	+	0.853870i	$0.674249\pi$
$648$		0			0
$649$	0.111401			0.00437288
$650$		0			0
$651$	9.12378			0.357589
$652$		0			0
$653$	19.4153			0.759778			0.379889	−	0.925032i	$-0.375962\pi$
$653$	19.4153			0.759778			0.379889	+	0.925032i	$0.375962\pi$
$654$		0			0
$655$		−	44.2255i		−	1.72803i
$656$		0			0
$657$			15.4916i			0.604386i
$658$		0			0
$659$	−30.1450			−1.17428			−0.587141	−	0.809485i	$-0.699747\pi$
$659$	−30.1450			−1.17428			−0.587141	+	0.809485i	$0.699747\pi$
$660$		0			0
$661$			13.1734i			0.512385i	0.966626	+	0.256193i	$0.0824681\pi$
$661$			13.1734i			0.512385i	−0.966626	+	0.256193i	$0.917532\pi$
$662$		0			0
$663$	−11.8459	+	4.15049i	−0.460056	+	0.161192i
$664$		0			0
$665$			16.7351i			0.648961i
$666$		0			0
$667$	−48.1055			−1.86265
$668$		0			0
$669$			0.759386i			0.0293596i
$670$		0			0
$671$			4.34302i			0.167660i
$672$		0			0
$673$	41.1762			1.58723			0.793613	−	0.608423i	$-0.208197\pi$
$673$	41.1762			1.58723			0.793613	+	0.608423i	$0.208197\pi$
$674$		0			0
$675$	8.57124			0.329907
$676$		0			0
$677$	3.50414			0.134675			0.0673374	−	0.997730i	$-0.478550\pi$
$677$	3.50414			0.134675			0.0673374	+	0.997730i	$0.478550\pi$
$678$		0			0
$679$	5.45877			0.209488
$680$		0			0
$681$			3.52612i			0.135121i
$682$		0			0
$683$			16.5881i			0.634727i	0.948304	+	0.317364i	$0.102798\pi$
$683$			16.5881i			0.634727i	−0.948304	+	0.317364i	$0.897202\pi$
$684$		0			0
$685$	18.7288			0.715591
$686$		0			0
$687$			12.9439i			0.493842i
$688$		0			0
$689$	19.1954	−	6.72557i	0.731286	−	0.256224i
$690$		0			0
$691$		−	16.7466i		−	0.637069i	−0.947911	−	0.318535i	$-0.896809\pi$
$691$		−	16.7466i		−	0.637069i	0.947911	−	0.318535i	$-0.103191\pi$
$692$		0			0
$693$	0.499112			0.0189597
$694$		0			0
$695$			30.6571i			1.16289i
$696$		0			0
$697$			15.5867i			0.590387i
$698$		0			0
$699$	−12.3867			−0.468507
$700$		0			0
$701$	−31.2374			−1.17982			−0.589910	−	0.807469i	$-0.700837\pi$
$701$	−31.2374			−1.17982			−0.589910	+	0.807469i	$0.700837\pi$
$702$		0			0
$703$	−39.7280			−1.49837
$704$		0			0
$705$	−28.5749			−1.07619
$706$		0			0
$707$			4.63631i			0.174366i
$708$		0			0
$709$			27.6671i			1.03906i	0.854452	+	0.519530i	$0.173893\pi$
$709$			27.6671i			1.03906i	−0.854452	+	0.519530i	$0.826107\pi$
$710$		0			0
$711$	−5.32131			−0.199565
$712$		0			0
$713$			45.4139i			1.70077i
$714$		0			0
$715$	−3.02762	+	1.06080i	−0.113227	+	0.0396716i
$716$		0			0
$717$		−	17.9839i		−	0.671620i
$718$		0			0
$719$	−22.1407			−0.825710			−0.412855	−	0.910797i	$-0.635468\pi$
$719$	−22.1407			−0.825710			−0.412855	+	0.910797i	$0.635468\pi$
$720$		0			0
$721$		−	2.19365i		−	0.0816959i
$722$		0			0
$723$		−	8.21502i		−	0.305520i
$724$		0			0
$725$	11.9352			0.443262
$726$		0			0
$727$	42.2319			1.56629			0.783147	−	0.621837i	$-0.213613\pi$
$727$	42.2319			1.56629			0.783147	+	0.621837i	$0.213613\pi$
$728$		0			0
$729$	24.6035			0.911241
$730$		0			0
$731$	18.8527			0.697292
$732$		0			0
$733$			41.2804i			1.52473i	0.647149	+	0.762363i	$0.275961\pi$
$733$			41.2804i			1.52473i	−0.647149	+	0.762363i	$0.724039\pi$
$734$		0			0
$735$		−	3.20035i		−	0.118047i
$736$		0			0
$737$	−2.83897			−0.104575
$738$		0			0
$739$		−	44.4419i		−	1.63482i	−0.576056	−	0.817410i	$-0.695409\pi$
$739$		−	44.4419i		−	1.63482i	0.576056	−	0.817410i	$-0.304591\pi$
$740$		0			0
$741$	−9.75629	−	27.8453i	−0.358406	−	1.02292i
$742$		0			0
$743$		−	18.7354i		−	0.687334i	−0.939092	−	0.343667i	$-0.888331\pi$
$743$		−	18.7354i		−	0.687334i	0.939092	−	0.343667i	$-0.111669\pi$
$744$		0			0
$745$	−50.8013			−1.86122
$746$		0			0
$747$			19.4793i			0.712709i
$748$		0			0
$749$			9.95507i			0.363750i
$750$		0			0
$751$	−21.8505			−0.797338			−0.398669	−	0.917095i	$-0.630528\pi$
$751$	−21.8505			−0.797338			−0.398669	+	0.917095i	$0.630528\pi$
$752$		0			0
$753$	5.65665			0.206140
$754$		0			0
$755$	20.3047			0.738963
$756$		0			0
$757$	33.1098			1.20340			0.601698	−	0.798724i	$-0.294491\pi$
$757$	33.1098			1.20340			0.601698	+	0.798724i	$0.294491\pi$
$758$		0			0
$759$		−	2.70913i		−	0.0983350i
$760$		0			0
$761$		−	2.14749i		−	0.0778464i	−0.999242	−	0.0389232i	$-0.987607\pi$
$761$		−	2.14749i		−	0.0778464i	0.999242	−	0.0389232i	$-0.0123928\pi$
$762$		0			0
$763$	−11.1343			−0.403089
$764$		0			0
$765$		−	10.2169i		−	0.369394i
$766$		0			0
$767$	−0.381881	−	1.08992i	−0.0137889	−	0.0393548i
$768$		0			0
$769$			15.5408i			0.560415i	0.959939	+	0.280208i	$0.0904033\pi$
$769$			15.5408i			0.560415i	−0.959939	+	0.280208i	$0.909597\pi$
$770$		0			0
$771$	4.62481			0.166559
$772$		0			0
$773$			22.9337i			0.824867i	0.910988	+	0.412433i	$0.135321\pi$
$773$			22.9337i			0.824867i	−0.910988	+	0.412433i	$0.864679\pi$
$774$		0			0
$775$		−	11.2674i		−	0.404738i
$776$		0			0
$777$	7.59739			0.272555
$778$		0			0
$779$	−36.6386			−1.31271
$780$		0			0
$781$	−0.984997			−0.0352459
$782$		0			0
$783$	42.8629			1.53179
$784$		0			0
$785$			58.9830i			2.10519i
$786$		0			0
$787$			11.7463i			0.418711i	0.977840	+	0.209355i	$0.0671366\pi$
$787$			11.7463i			0.418711i	−0.977840	+	0.209355i	$0.932863\pi$
$788$		0			0
$789$	−9.14326			−0.325509
$790$		0			0
$791$		−	12.3262i		−	0.438270i
$792$		0			0
$793$	42.4910	−	14.8878i	1.50890	−	0.528680i
$794$		0			0
$795$		−	18.0537i		−	0.640299i
$796$		0			0
$797$	25.9909			0.920646			0.460323	−	0.887751i	$-0.347734\pi$
$797$	25.9909			0.920646			0.460323	+	0.887751i	$0.347734\pi$
$798$		0			0
$799$			24.8474i			0.879038i
$800$		0			0
$801$		−	10.2008i		−	0.360426i
$802$		0			0
$803$	−3.75443			−0.132491
$804$		0			0
$805$	15.9298			0.561453
$806$		0			0
$807$	9.52882			0.335430
$808$		0			0
$809$	−2.57898			−0.0906722			−0.0453361	−	0.998972i	$-0.514436\pi$
$809$	−2.57898			−0.0906722			−0.0453361	+	0.998972i	$0.514436\pi$
$810$		0			0
$811$			26.5319i			0.931661i	0.884874	+	0.465830i	$0.154244\pi$
$811$			26.5319i			0.931661i	−0.884874	+	0.465830i	$0.845756\pi$
$812$		0			0
$813$		−	4.33737i		−	0.152118i
$814$		0			0
$815$	33.3834			1.16937
$816$		0			0
$817$			44.3158i			1.55041i
$818$		0			0
$819$	−1.71094	−	4.88319i	−0.0597852	−	0.170633i
$820$		0			0
$821$		−	21.6806i		−	0.756659i	−0.925671	−	0.378329i	$-0.876499\pi$
$821$		−	21.6806i		−	0.756659i	0.925671	−	0.378329i	$-0.123501\pi$
$822$		0			0
$823$	7.98703			0.278410			0.139205	−	0.990264i	$-0.455545\pi$
$823$	7.98703			0.278410			0.139205	+	0.990264i	$0.455545\pi$
$824$		0			0
$825$			0.672147i			0.0234012i
$826$		0			0
$827$			13.9617i			0.485496i	0.970089	+	0.242748i	$0.0780488\pi$
$827$			13.9617i			0.485496i	−0.970089	+	0.242748i	$0.921951\pi$
$828$		0			0
$829$	28.8547			1.00216			0.501082	−	0.865400i	$-0.332935\pi$
$829$	28.8547			1.00216			0.501082	+	0.865400i	$0.332935\pi$
$830$		0			0
$831$	−27.5615			−0.956098
$832$		0			0
$833$	−2.78287			−0.0964209
$834$		0			0
$835$	−32.0971			−1.11077
$836$		0			0
$837$		−	40.4647i		−	1.39866i
$838$		0			0
$839$			28.2321i			0.974682i	0.873212	+	0.487341i	$0.162033\pi$
$839$			28.2321i			0.974682i	−0.873212	+	0.487341i	$0.837967\pi$
$840$		0			0
$841$	30.6854			1.05812
$842$		0			0
$843$		−	17.4751i		−	0.601875i
$844$		0			0
$845$	20.7572	+	25.9851i	0.714069	+	0.893914i
$846$		0			0
$847$		−	10.8790i		−	0.373808i
$848$		0			0
$849$	22.1811			0.761252
$850$		0			0
$851$			37.8163i			1.29633i
$852$		0			0
$853$		−	32.3033i		−	1.10604i	−0.833167	−	0.553021i	$-0.813475\pi$
$853$		−	32.3033i		−	1.10604i	0.833167	−	0.553021i	$-0.186525\pi$
$854$		0			0
$855$	24.0163			0.821339
$856$		0			0
$857$	−33.6189			−1.14840			−0.574200	−	0.818715i	$-0.694687\pi$
$857$	−33.6189			−1.14840			−0.574200	+	0.818715i	$0.694687\pi$
$858$		0			0
$859$	51.3339			1.75149			0.875745	−	0.482774i	$-0.160371\pi$
$859$	51.3339			1.75149			0.875745	+	0.482774i	$0.160371\pi$
$860$		0			0
$861$	7.00658			0.238783
$862$		0			0
$863$			56.0814i			1.90903i	0.298157	+	0.954517i	$0.403628\pi$
$863$			56.0814i			1.90903i	−0.298157	+	0.954517i	$0.596372\pi$
$864$		0			0
$865$			61.1496i			2.07915i
$866$		0			0
$867$	−11.5785			−0.393226
$868$		0			0
$869$		−	1.28963i		−	0.0437477i
$870$		0			0
$871$	9.73193	+	27.7758i	0.329754	+	0.941148i
$872$		0			0
$873$		−	7.83376i		−	0.265133i
$874$		0			0
$875$	8.83921			0.298820
$876$		0			0
$877$		−	46.2698i		−	1.56242i	−0.624269	−	0.781209i	$-0.714603\pi$
$877$		−	46.2698i		−	1.56242i	0.624269	−	0.781209i	$-0.285397\pi$
$878$		0			0
$879$			19.9145i			0.671699i
$880$		0			0
$881$	−19.6096			−0.660664			−0.330332	−	0.943865i	$-0.607161\pi$
$881$	−19.6096			−0.660664			−0.330332	+	0.943865i	$0.607161\pi$
$882$		0			0
$883$	10.5960			0.356583			0.178291	−	0.983978i	$-0.442943\pi$
$883$	10.5960			0.356583			0.178291	+	0.983978i	$0.442943\pi$
$884$		0			0
$885$	−1.02510			−0.0344583
$886$		0			0
$887$	−31.6541			−1.06284			−0.531421	−	0.847108i	$-0.678342\pi$
$887$	−31.6541			−1.06284			−0.531421	+	0.847108i	$0.678342\pi$
$888$		0			0
$889$		−	17.6194i		−	0.590934i
$890$		0			0
$891$			0.916545i			0.0307054i
$892$		0			0
$893$	−58.4071			−1.95452
$894$		0			0
$895$			46.3920i			1.55071i
$896$		0			0
$897$	−26.5054	+	9.28681i	−0.884990	+	0.310078i
$898$		0			0
$899$		−	56.3459i		−	1.87924i
$900$		0			0
$901$	−15.6987			−0.522998
$902$		0			0
$903$		−	8.47473i		−	0.282021i
$904$		0			0
$905$		−	47.9611i		−	1.59428i
$906$		0			0
$907$	16.8607			0.559849			0.279924	−	0.960022i	$-0.409691\pi$
$907$	16.8607			0.559849			0.279924	+	0.960022i	$0.409691\pi$
$908$		0			0
$909$	6.65347			0.220682
$910$		0			0
$911$	−32.3573			−1.07204			−0.536022	−	0.844204i	$-0.680074\pi$
$911$	−32.3573			−1.07204			−0.536022	+	0.844204i	$0.680074\pi$
$912$		0			0
$913$	−4.72084			−0.156237
$914$		0			0
$915$		−	39.9638i		−	1.32116i
$916$		0			0
$917$			17.2871i			0.570870i
$918$		0			0
$919$	22.5519			0.743919			0.371959	−	0.928249i	$-0.378686\pi$
$919$	22.5519			0.743919			0.371959	+	0.928249i	$0.378686\pi$
$920$		0			0
$921$		−	25.2990i		−	0.833630i
$922$		0			0
$923$	3.37654	+	9.63696i	0.111140	+	0.317204i
$924$		0			0
$925$		−	9.38241i		−	0.308492i
$926$		0			0
$927$	−3.14807			−0.103396
$928$		0			0
$929$			23.6163i			0.774826i	0.921906	+	0.387413i	$0.126631\pi$
$929$			23.6163i			0.774826i	−0.921906	+	0.387413i	$0.873369\pi$
$930$		0			0
$931$		−	6.54152i		−	0.214390i
$932$		0			0
$933$	2.68611			0.0879393
$934$		0			0
$935$	2.47609			0.0809769
$936$		0			0
$937$	−37.6175			−1.22891			−0.614455	−	0.788952i	$-0.710624\pi$
$937$	−37.6175			−1.22891			−0.614455	+	0.788952i	$0.710624\pi$
$938$		0			0
$939$	−18.6482			−0.608560
$940$		0			0
$941$		−	53.5875i		−	1.74690i	−0.486913	−	0.873451i	$-0.661877\pi$
$941$		−	53.5875i		−	1.74690i	0.486913	−	0.873451i	$-0.338123\pi$
$942$		0			0
$943$			34.8755i			1.13570i
$944$		0			0
$945$	−14.1938			−0.461724
$946$		0			0
$947$			39.2361i			1.27500i	0.770450	+	0.637501i	$0.220032\pi$
$947$			39.2361i			1.27500i	−0.770450	+	0.637501i	$0.779968\pi$
$948$		0			0
$949$	12.8701	+	36.7324i	0.417781	+	1.19238i
$950$		0			0
$951$			6.72149i			0.217959i
$952$		0			0
$953$	−28.3871			−0.919550			−0.459775	−	0.888036i	$-0.652070\pi$
$953$	−28.3871			−0.919550			−0.459775	+	0.888036i	$0.652070\pi$
$954$		0			0
$955$			21.5005i			0.695740i
$956$		0			0
$957$			3.36126i			0.108654i
$958$		0			0
$959$	−7.32081			−0.236401
$960$		0			0
$961$	−22.1933			−0.715913
$962$		0			0
$963$	14.2863			0.460370
$964$		0			0
$965$	−21.4477			−0.690424
$966$		0			0
$967$		−	23.1722i		−	0.745168i	−0.927999	−	0.372584i	$-0.878472\pi$
$967$		−	23.1722i		−	0.745168i	0.927999	−	0.372584i	$-0.121528\pi$
$968$		0			0
$969$			22.7729i			0.731571i
$970$		0			0
$971$	53.7799			1.72588			0.862940	−	0.505307i	$-0.168621\pi$
$971$	53.7799			1.72588			0.862940	+	0.505307i	$0.168621\pi$
$972$		0			0
$973$		−	11.9834i		−	0.384170i
$974$		0			0
$975$	6.57612	−	2.30410i	0.210605	−	0.0737904i
$976$		0			0
$977$			29.4656i			0.942687i	0.881950	+	0.471343i	$0.156231\pi$
$977$			29.4656i			0.942687i	−0.881950	+	0.471343i	$0.843769\pi$
$978$		0			0
$979$	2.47217			0.0790110
$980$		0			0
$981$			15.9786i			0.510158i
$982$		0			0
$983$		−	31.1347i		−	0.993042i	−0.868025	−	0.496521i	$-0.834611\pi$
$983$		−	31.1347i		−	0.993042i	0.868025	−	0.496521i	$-0.165389\pi$
$984$		0			0
$985$	−30.9814			−0.987149
$986$		0			0
$987$	11.1695			0.355529
$988$		0			0
$989$	42.1833			1.34135
$990$		0			0
$991$	28.6336			0.909577			0.454789	−	0.890599i	$-0.349715\pi$
$991$	28.6336			0.909577			0.454789	+	0.890599i	$0.349715\pi$
$992$		0			0
$993$			16.3653i			0.519337i
$994$		0			0
$995$			11.8219i			0.374780i
$996$		0			0
$997$	22.3431			0.707613			0.353807	−	0.935319i	$-0.384887\pi$
$997$	22.3431			0.707613			0.353807	+	0.935319i	$0.384887\pi$
$998$		0			0
$999$		−	33.6950i		−	1.06606i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.k.f.337.10		12
4.3	odd	2		728.2.k.b.337.4	yes	12
13.12	even	2	inner	1456.2.k.f.337.9		12
52.31	even	4		9464.2.a.bd.1.2		6
52.47	even	4		9464.2.a.be.1.2		6
52.51	odd	2		728.2.k.b.337.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.k.b.337.3	✓	12	52.51	odd	2
728.2.k.b.337.4	yes	12	4.3	odd	2
1456.2.k.f.337.9		12	13.12	even	2	inner
1456.2.k.f.337.10		12	1.1	even	1	trivial
9464.2.a.bd.1.2		6	52.31	even	4
9464.2.a.be.1.2		6	52.47	even	4

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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.25097 q^{3} +2.55830i q^{5} -1.00000i q^{7} -1.43508 q^{9} +O(q^{10})\)	\(q+1.25097 q^{3} +2.55830i q^{5} -1.00000i q^{7} -1.43508 q^{9} -0.347794i q^{11} +(-3.40273 + 1.19223i) q^{13} +3.20035i q^{15} +2.78287 q^{17} +6.54152i q^{19} -1.25097i q^{21} +6.22674 q^{23} -1.54488 q^{25} -5.54814 q^{27} -7.72563 q^{29} +7.29337i q^{31} -0.435079i q^{33} +2.55830 q^{35} +6.07321i q^{37} +(-4.25671 + 1.49144i) q^{39} +5.60092i q^{41} +6.77454 q^{43} -3.67136i q^{45} +8.92868i q^{47} -1.00000 q^{49} +3.48129 q^{51} -5.64117 q^{53} +0.889761 q^{55} +8.18323i q^{57} +0.320308i q^{59} -12.4873 q^{61} +1.43508i q^{63} +(-3.05008 - 8.70520i) q^{65} -8.16280i q^{67} +7.78945 q^{69} -2.83212i q^{71} -10.7950i q^{73} -1.93260 q^{75} -0.347794 q^{77} +3.70802 q^{79} -2.63531 q^{81} -13.5737i q^{83} +7.11942i q^{85} -9.66451 q^{87} +7.10815i q^{89} +(1.19223 + 3.40273i) q^{91} +9.12378i q^{93} -16.7351 q^{95} +5.45877i q^{97} +0.499112i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{3} + 6 q^{9}+O(q^{10}) \) 12 * q - 2 * q^3 + 6 * q^9	\( 12 q - 2 q^{3} + 6 q^{9} + 6 q^{13} + 4 q^{17} + 20 q^{23} - 26 q^{25} - 26 q^{27} + 10 q^{29} - 2 q^{35} - 18 q^{39} + 6 q^{43} - 12 q^{49} + 12 q^{51} + 6 q^{53} + 40 q^{55} + 6 q^{61} + 38 q^{65} + 30 q^{69} + 58 q^{75} + 2 q^{77} - 40 q^{79} + 4 q^{81} - 8 q^{87} - 6 q^{91} - 54 q^{95}+O(q^{100}) \) 12 * q - 2 * q^3 + 6 * q^9 + 6 * q^13 + 4 * q^17 + 20 * q^23 - 26 * q^25 - 26 * q^27 + 10 * q^29 - 2 * q^35 - 18 * q^39 + 6 * q^43 - 12 * q^49 + 12 * q^51 + 6 * q^53 + 40 * q^55 + 6 * q^61 + 38 * q^65 + 30 * q^69 + 58 * q^75 + 2 * q^77 - 40 * q^79 + 4 * q^81 - 8 * q^87 - 6 * q^91 - 54 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more