LMFDB - Embedded newform 3168.2.h.g.2287.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3168,2,Mod(2287,3168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3168, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("3168.2287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.h (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:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.10070523904.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 6x^{4} + 16 $ x^8 + 6*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2287.6
Root			$-0.804019 + 1.16342i$ of defining polynomial
Character	$\chi$	$=$	3168.2287
Dual form			3168.2.h.g.2287.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575i q^{5} -2.27411 q^{7} +(-0.414214 + 3.29066i) q^{11} +5.49019 q^{13} -1.36303i q^{17} +3.29066i q^{19} -1.09591i q^{23} -2.00000 q^{25} +4.54822 q^{29} +6.38741i q^{31} -6.01673i q^{35} +10.1291i q^{37} -5.21828i q^{41} -9.30739i q^{43} -1.82843 q^{49} +7.48331i q^{53} +(-8.70626 - 1.09591i) q^{55} -5.24264 q^{59} +4.54822 q^{61} +14.5257i q^{65} -8.07107 q^{67} +1.09591i q^{71} +7.94435i q^{73} +(0.941967 - 7.48331i) q^{77} +6.43215 q^{79} -6.58132i q^{83} +3.60625 q^{85} -0.656854 q^{89} -12.4853 q^{91} -8.70626 q^{95} -9.48528 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{11} - 16 q^{25} + 8 q^{49} - 8 q^{59} - 8 q^{67} + 40 q^{89} - 32 q^{91} - 8 q^{97}+O(q^{100}) $ 8 * q + 8 * q^11 - 16 * q^25 + 8 * q^49 - 8 * q^59 - 8 * q^67 + 40 * q^89 - 32 * q^91 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$			2.64575i			1.18322i	0.806226	+	0.591608i	$0.201507\pi$
$5$			2.64575i			1.18322i	−0.806226	+	0.591608i	$0.798493\pi$
$6$		0			0
$7$	−2.27411			−0.859533			−0.429766	−	0.902940i	$-0.641404\pi$
$7$	−2.27411			−0.859533			−0.429766	+	0.902940i	$0.641404\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.414214	+	3.29066i	−0.124890	+	0.992171i
$11$	−0.414214	+	3.29066i	−0.124890	+	0.992171i
$12$		0			0
$13$	5.49019			1.52270			0.761352	−	0.648339i	$-0.224536\pi$
$13$	5.49019			1.52270			0.761352	+	0.648339i	$0.224536\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.36303i		−	0.330585i	−0.986245	−	0.165292i	$-0.947143\pi$
$17$		−	1.36303i		−	0.330585i	0.986245	−	0.165292i	$-0.0528567\pi$
$18$		0			0
$19$			3.29066i			0.754929i	0.926024	+	0.377464i	$0.123204\pi$
$19$			3.29066i			0.754929i	−0.926024	+	0.377464i	$0.876796\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.09591i		−	0.228512i	−0.993451	−	0.114256i	$-0.963552\pi$
$23$		−	1.09591i		−	0.228512i	0.993451	−	0.114256i	$-0.0364484\pi$
$24$		0			0
$25$	−2.00000			−0.400000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.54822			0.844583			0.422291	−	0.906460i	$-0.361226\pi$
$29$	4.54822			0.844583			0.422291	+	0.906460i	$0.361226\pi$
$30$		0			0
$31$			6.38741i			1.14721i	0.819131	+	0.573606i	$0.194456\pi$
$31$			6.38741i			1.14721i	−0.819131	+	0.573606i	$0.805544\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	6.01673i		−	1.01701i
$36$		0			0
$37$			10.1291i			1.66521i	0.553869	+	0.832604i	$0.313151\pi$
$37$			10.1291i			1.66521i	−0.553869	+	0.832604i	$0.686849\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.21828i		−	0.814958i	−0.913214	−	0.407479i	$-0.866408\pi$
$41$		−	5.21828i		−	0.814958i	0.913214	−	0.407479i	$-0.133592\pi$
$42$		0			0
$43$		−	9.30739i		−	1.41936i	−0.704523	−	0.709681i	$-0.748839\pi$
$43$		−	9.30739i		−	1.41936i	0.704523	−	0.709681i	$-0.251161\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.82843			−0.261204
$50$		0			0
$51$		0			0
$52$		0			0
$53$			7.48331i			1.02791i	0.857816	+	0.513956i	$0.171821\pi$
$53$			7.48331i			1.02791i	−0.857816	+	0.513956i	$0.828179\pi$
$54$		0			0
$55$	−8.70626	−	1.09591i	−1.17395	−	0.147772i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.24264			−0.682534			−0.341267	−	0.939966i	$-0.610856\pi$
$59$	−5.24264			−0.682534			−0.341267	+	0.939966i	$0.610856\pi$
$60$		0			0
$61$	4.54822			0.582340			0.291170	−	0.956671i	$-0.405956\pi$
$61$	4.54822			0.582340			0.291170	+	0.956671i	$0.405956\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			14.5257i			1.80169i
$66$		0			0
$67$	−8.07107			−0.986038			−0.493019	−	0.870019i	$-0.664107\pi$
$67$	−8.07107			−0.986038			−0.493019	+	0.870019i	$0.664107\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			1.09591i			0.130060i	0.997883	+	0.0650301i	$0.0207143\pi$
$71$			1.09591i			0.130060i	−0.997883	+	0.0650301i	$0.979286\pi$
$72$		0			0
$73$			7.94435i			0.929816i	0.885359	+	0.464908i	$0.153913\pi$
$73$			7.94435i			0.929816i	−0.885359	+	0.464908i	$0.846087\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.941967	−	7.48331i	0.107347	−	0.852803i
$78$		0			0
$79$	6.43215			0.723674			0.361837	−	0.932241i	$-0.382150\pi$
$79$	6.43215			0.723674			0.361837	+	0.932241i	$0.382150\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	6.58132i		−	0.722393i	−0.932490	−	0.361197i	$-0.882368\pi$
$83$		−	6.58132i		−	0.722393i	0.932490	−	0.361197i	$-0.117632\pi$
$84$		0			0
$85$	3.60625			0.391153
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.656854			−0.0696264			−0.0348132	−	0.999394i	$-0.511084\pi$
$89$	−0.656854			−0.0696264			−0.0348132	+	0.999394i	$0.511084\pi$
$90$		0			0
$91$	−12.4853			−1.30881
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.70626			−0.893244
$96$		0			0
$97$	−9.48528			−0.963084			−0.481542	−	0.876423i	$-0.659923\pi$
$97$	−9.48528			−0.963084			−0.481542	+	0.876423i	$0.659923\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.37412			−0.733752			−0.366876	−	0.930270i	$-0.619573\pi$
$101$	−7.37412			−0.733752			−0.366876	+	0.930270i	$0.619573\pi$
$102$		0			0
$103$			3.09969i			0.305422i	0.988271	+	0.152711i	$0.0488003\pi$
$103$			3.09969i			0.305422i	−0.988271	+	0.152711i	$0.951200\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			6.01673i			0.581659i	0.956775	+	0.290830i	$0.0939312\pi$
$107$			6.01673i			0.581659i	−0.956775	+	0.290830i	$0.906069\pi$
$108$		0			0
$109$	−10.9804			−1.05173			−0.525864	−	0.850568i	$-0.676258\pi$
$109$	−10.9804			−1.05173			−0.525864	+	0.850568i	$0.676258\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.8284			−1.11272			−0.556362	−	0.830940i	$-0.687803\pi$
$113$	−11.8284			−1.11272			−0.556362	+	0.830940i	$0.687803\pi$
$114$		0			0
$115$	2.89949			0.270379
$116$		0			0
$117$		0			0
$118$		0			0
$119$			3.09969i			0.284148i
$120$		0			0
$121$	−10.6569	−	2.72607i	−0.968805	−	0.247825i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			7.93725i			0.709930i
$126$		0			0
$127$	−13.2545			−1.17614			−0.588072	−	0.808808i	$-0.700113\pi$
$127$	−13.2545			−1.17614			−0.588072	+	0.808808i	$0.700113\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			9.87197i			0.862518i	0.902228	+	0.431259i	$0.141931\pi$
$131$			9.87197i			0.862518i	−0.902228	+	0.431259i	$0.858069\pi$
$132$		0			0
$133$		−	7.48331i		−	0.648886i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.65685			−0.226990			−0.113495	−	0.993539i	$-0.536205\pi$
$137$	−2.65685			−0.226990			−0.113495	+	0.993539i	$0.536205\pi$
$138$		0			0
$139$			8.74280i			0.741554i	0.928722	+	0.370777i	$0.120909\pi$
$139$			8.74280i			0.741554i	−0.928722	+	0.370777i	$0.879091\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.27411	+	18.0663i	−0.190171	+	1.51078i
$144$		0			0
$145$			12.0335i			0.999324i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.15447			0.668040			0.334020	−	0.942566i	$-0.391595\pi$
$149$	8.15447			0.668040			0.334020	+	0.942566i	$0.391595\pi$
$150$		0			0
$151$	9.09644			0.740258			0.370129	−	0.928980i	$-0.379314\pi$
$151$	9.09644			0.740258			0.370129	+	0.928980i	$0.379314\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−16.8995			−1.35740
$156$		0			0
$157$		−	15.4206i		−	1.23070i	−0.788256	−	0.615348i	$-0.789016\pi$
$157$		−	15.4206i		−	1.23070i	0.788256	−	0.615348i	$-0.210984\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2.49221i			0.196414i
$162$		0			0
$163$	7.17157			0.561721			0.280860	−	0.959749i	$-0.409380\pi$
$163$	7.17157			0.561721			0.280860	+	0.959749i	$0.409380\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.5902			−0.819494			−0.409747	−	0.912199i	$-0.634383\pi$
$167$	−10.5902			−0.819494			−0.409747	+	0.912199i	$0.634383\pi$
$168$		0			0
$169$	17.1421			1.31863
$170$		0			0
$171$		0			0
$172$		0			0
$173$	21.9607			1.66964			0.834822	−	0.550520i	$-0.185570\pi$
$173$	21.9607			1.66964			0.834822	+	0.550520i	$0.185570\pi$
$174$		0			0
$175$	4.54822			0.343813
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.7279			−1.17556			−0.587780	−	0.809021i	$-0.699998\pi$
$179$	−15.7279			−1.17556			−0.587780	+	0.809021i	$0.699998\pi$
$180$		0			0
$181$			2.64575i			0.196657i	0.995154	+	0.0983286i	$0.0313496\pi$
$181$			2.64575i			0.196657i	−0.995154	+	0.0983286i	$0.968650\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−26.7990			−1.97030
$186$		0			0
$187$	4.48528	+	0.564588i	0.327996	+	0.0412867i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			9.48710i			0.686462i	0.939251	+	0.343231i	$0.111521\pi$
$191$			9.48710i			0.686462i	−0.939251	+	0.343231i	$0.888479\pi$
$192$		0			0
$193$		−	18.6148i		−	1.33992i	−0.742397	−	0.669960i	$-0.766311\pi$
$193$		−	18.6148i		−	1.33992i	0.742397	−	0.669960i	$-0.233689\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−20.0768			−1.43041			−0.715207	−	0.698913i	$-0.753667\pi$
$197$	−20.0768			−1.43041			−0.715207	+	0.698913i	$0.753667\pi$
$198$		0			0
$199$		−	21.1660i		−	1.50042i	−0.661200	−	0.750209i	$-0.729953\pi$
$199$		−	21.1660i		−	1.50042i	0.661200	−	0.750209i	$-0.270047\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−10.3431			−0.725947
$204$		0			0
$205$	13.8063			0.964272
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.8284	−	1.36303i	−0.749018	−	0.0942831i
$210$		0			0
$211$			13.1626i			0.906153i	0.891472	+	0.453076i	$0.149674\pi$
$211$			13.1626i			0.906153i	−0.891472	+	0.453076i	$0.850326\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	24.6250			1.67941
$216$		0			0
$217$		−	14.5257i		−	0.986066i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	7.48331i		−	0.503382i
$222$		0			0
$223$			11.6789i			0.782078i	0.920374	+	0.391039i	$0.127884\pi$
$223$			11.6789i			0.782078i	−0.920374	+	0.391039i	$0.872116\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			0.564588i			0.0374730i	0.999824	+	0.0187365i	$0.00596436\pi$
$227$			0.564588i			0.0374730i	−0.999824	+	0.0187365i	$0.994036\pi$
$228$		0			0
$229$			4.83756i			0.319675i	0.987143	+	0.159838i	$0.0510971\pi$
$229$			4.83756i			0.319675i	−0.987143	+	0.159838i	$0.948903\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	25.1961i		−	1.65065i	−0.564658	−	0.825325i	$-0.690992\pi$
$233$		−	25.1961i		−	1.65065i	0.564658	−	0.825325i	$-0.309008\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	21.9607			1.42052			0.710261	−	0.703938i	$-0.248577\pi$
$239$	21.9607			1.42052			0.710261	+	0.703938i	$0.248577\pi$
$240$		0			0
$241$			6.58132i			0.423940i	0.977276	+	0.211970i	$0.0679879\pi$
$241$			6.58132i			0.423940i	−0.977276	+	0.211970i	$0.932012\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	4.83756i		−	0.309061i
$246$		0			0
$247$			18.0663i			1.14953i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.5858			0.983766			0.491883	−	0.870661i	$-0.336309\pi$
$251$	15.5858			0.983766			0.491883	+	0.870661i	$0.336309\pi$
$252$		0			0
$253$	3.60625	+	0.453939i	0.226723	+	0.0285389i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.4853			0.654054			0.327027	−	0.945015i	$-0.393953\pi$
$257$	10.4853			0.654054			0.327027	+	0.945015i	$0.393953\pi$
$258$		0			0
$259$		−	23.0346i		−	1.43130i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.390175			−0.0240592			−0.0120296	−	0.999928i	$-0.503829\pi$
$263$	−0.390175			−0.0240592			−0.0120296	+	0.999928i	$0.503829\pi$
$264$		0			0
$265$	−19.7990			−1.21624
$266$		0			0
$267$		0			0
$268$		0			0
$269$			28.6493i			1.74678i	0.487021	+	0.873390i	$0.338083\pi$
$269$			28.6493i			1.74678i	−0.487021	+	0.873390i	$0.661917\pi$
$270$		0			0
$271$	4.15804			0.252583			0.126292	−	0.991993i	$-0.459693\pi$
$271$	4.15804			0.252583			0.126292	+	0.991993i	$0.459693\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.828427	−	6.58132i	0.0499560	−	0.396868i
$276$		0			0
$277$	−15.5286			−0.933023			−0.466511	−	0.884515i	$-0.654489\pi$
$277$	−15.5286			−0.933023			−0.466511	+	0.884515i	$0.654489\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			9.30739i			0.555232i	0.960692	+	0.277616i	$0.0895443\pi$
$281$			9.30739i			0.555232i	−0.960692	+	0.277616i	$0.910456\pi$
$282$		0			0
$283$		−	15.8887i		−	0.944485i	−0.881469	−	0.472243i	$-0.843445\pi$
$283$		−	15.8887i		−	0.944485i	0.881469	−	0.472243i	$-0.156555\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			11.8669i			0.700483i
$288$		0			0
$289$	15.1421			0.890714
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.60625			−0.210679			−0.105340	−	0.994436i	$-0.533593\pi$
$293$	−3.60625			−0.210679			−0.105340	+	0.994436i	$0.533593\pi$
$294$		0			0
$295$		−	13.8707i		−	0.807585i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	6.01673i		−	0.347956i
$300$		0			0
$301$			21.1660i			1.21999i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			12.0335i			0.689034i
$306$		0			0
$307$			13.7272i			0.783454i	0.920082	+	0.391727i	$0.128122\pi$
$307$			13.7272i			0.783454i	−0.920082	+	0.391727i	$0.871878\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			14.9666i			0.848680i	0.905503	+	0.424340i	$0.139494\pi$
$311$			14.9666i			0.848680i	−0.905503	+	0.424340i	$0.860506\pi$
$312$		0			0
$313$	2.17157			0.122745			0.0613723	−	0.998115i	$-0.480452\pi$
$313$	2.17157			0.122745			0.0613723	+	0.998115i	$0.480452\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	0.453939i		−	0.0254958i	−0.999919	−	0.0127479i	$-0.995942\pi$
$317$		−	0.453939i		−	0.0254958i	0.999919	−	0.0127479i	$-0.00405789\pi$
$318$		0			0
$319$	−1.88393	+	14.9666i	−0.105480	+	0.837970i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.48528			0.249568
$324$		0			0
$325$	−10.9804			−0.609081
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	3.92893			0.215954			0.107977	−	0.994153i	$-0.465563\pi$
$331$	3.92893			0.215954			0.107977	+	0.994153i	$0.465563\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	21.3540i		−	1.16670i
$336$		0			0
$337$			18.3809i			1.00127i	0.865658	+	0.500636i	$0.166900\pi$
$337$			18.3809i			1.00127i	−0.865658	+	0.500636i	$0.833100\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−21.0188	−	2.64575i	−1.13823	−	0.143275i
$342$		0			0
$343$	20.0768			1.08405
$344$		0			0
$345$		0			0
$346$		0			0
$347$			12.5980i			0.676298i	0.941093	+	0.338149i	$0.109801\pi$
$347$			12.5980i			0.676298i	−0.941093	+	0.338149i	$0.890199\pi$
$348$		0			0
$349$	−3.60625			−0.193038			−0.0965191	−	0.995331i	$-0.530771\pi$
$349$	−3.60625			−0.193038			−0.0965191	+	0.995331i	$0.530771\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.82843			0.203767			0.101883	−	0.994796i	$-0.467513\pi$
$353$	3.82843			0.203767			0.101883	+	0.994796i	$0.467513\pi$
$354$		0			0
$355$	−2.89949			−0.153889
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.27411			−0.120023			−0.0600114	−	0.998198i	$-0.519114\pi$
$359$	−2.27411			−0.120023			−0.0600114	+	0.998198i	$0.519114\pi$
$360$		0			0
$361$	8.17157			0.430083
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−21.0188			−1.10017
$366$		0			0
$367$		−	14.7786i		−	0.771437i	−0.922617	−	0.385718i	$-0.873954\pi$
$367$		−	14.7786i		−	0.771437i	0.922617	−	0.385718i	$-0.126046\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	17.0179i		−	0.883524i
$372$		0			0
$373$	−31.9991			−1.65685			−0.828426	−	0.560098i	$-0.810764\pi$
$373$	−31.9991			−1.65685			−0.828426	+	0.560098i	$0.810764\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	24.9706			1.28605
$378$		0			0
$379$	1.92893			0.0990826			0.0495413	−	0.998772i	$-0.484224\pi$
$379$	1.92893			0.0990826			0.0495413	+	0.998772i	$0.484224\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	29.7452i		−	1.51991i	−0.649976	−	0.759955i	$-0.725221\pi$
$383$		−	29.7452i		−	1.51991i	0.649976	−	0.759955i	$-0.274779\pi$
$384$		0			0
$385$	19.7990	+	2.49221i	1.00905	+	0.127015i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			27.2875i			1.38353i	0.722122	+	0.691766i	$0.243167\pi$
$389$			27.2875i			1.38353i	−0.722122	+	0.691766i	$0.756833\pi$
$390$		0			0
$391$	−1.49376			−0.0755426
$392$		0			0
$393$		0			0
$394$		0			0
$395$			17.0179i			0.856262i
$396$		0			0
$397$			22.4499i			1.12673i	0.826208	+	0.563365i	$0.190494\pi$
$397$			22.4499i			1.12673i	−0.826208	+	0.563365i	$0.809506\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.828427			−0.0413697			−0.0206848	−	0.999786i	$-0.506585\pi$
$401$	−0.828427			−0.0413697			−0.0206848	+	0.999786i	$0.506585\pi$
$402$		0			0
$403$			35.0681i			1.74686i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−33.3313	−	4.19560i	−1.65217	−	0.207968i
$408$		0			0
$409$		−	15.8887i		−	0.785646i	−0.919614	−	0.392823i	$-0.871499\pi$
$409$		−	15.8887i		−	0.785646i	0.919614	−	0.392823i	$-0.128501\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	11.9223			0.586660
$414$		0			0
$415$	17.4125			0.854747
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.1421			−0.593182			−0.296591	−	0.955005i	$-0.595850\pi$
$419$	−12.1421			−0.593182			−0.296591	+	0.955005i	$0.595850\pi$
$420$		0			0
$421$		−	7.48331i		−	0.364714i	−0.983232	−	0.182357i	$-0.941627\pi$
$421$		−	7.48331i		−	0.364714i	0.983232	−	0.182357i	$-0.0583727\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			2.72607i			0.132234i
$426$		0			0
$427$	−10.3431			−0.500540
$428$		0			0
$429$		0			0
$430$		0			0
$431$	24.2349			1.16735			0.583676	−	0.811987i	$-0.301614\pi$
$431$	24.2349			1.16735			0.583676	+	0.811987i	$0.301614\pi$
$432$		0			0
$433$	−28.6569			−1.37716			−0.688580	−	0.725160i	$-0.741766\pi$
$433$	−28.6569			−1.37716			−0.688580	+	0.725160i	$0.741766\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.60625			0.172510
$438$		0			0
$439$	14.7482			0.703895			0.351948	−	0.936020i	$-0.385520\pi$
$439$	14.7482			0.703895			0.351948	+	0.936020i	$0.385520\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	23.5858			1.12059			0.560297	−	0.828292i	$-0.310687\pi$
$443$	23.5858			1.12059			0.560297	+	0.828292i	$0.310687\pi$
$444$		0			0
$445$		−	1.73787i		−	0.0823831i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	17.9706			0.848083			0.424042	−	0.905643i	$-0.360611\pi$
$449$	17.9706			0.848083			0.424042	+	0.905643i	$0.360611\pi$
$450$		0			0
$451$	17.1716	+	2.16148i	0.808578	+	0.101780i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	33.0329i		−	1.54861i
$456$		0			0
$457$		−	32.9066i		−	1.53930i	−0.638463	−	0.769652i	$-0.720430\pi$
$457$		−	32.9066i		−	1.53930i	0.638463	−	0.769652i	$-0.279570\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.5286			0.723239			0.361619	−	0.932326i	$-0.382224\pi$
$461$	15.5286			0.723239			0.361619	+	0.932326i	$0.382224\pi$
$462$		0			0
$463$			14.7786i			0.686820i	0.939186	+	0.343410i	$0.111582\pi$
$463$			14.7786i			0.686820i	−0.939186	+	0.343410i	$0.888418\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	21.9289			1.01475			0.507375	−	0.861725i	$-0.330616\pi$
$467$	21.9289			1.01475			0.507375	+	0.861725i	$0.330616\pi$
$468$		0			0
$469$	18.3545			0.847532
$470$		0			0
$471$		0			0
$472$		0			0
$473$	30.6274	+	3.85525i	1.40825	+	0.177264i
$474$		0			0
$475$		−	6.58132i		−	0.301971i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−32.5509			−1.48729			−0.743645	−	0.668575i	$-0.766905\pi$
$479$	−32.5509			−1.48729			−0.743645	+	0.668575i	$0.766905\pi$
$480$		0			0
$481$			55.6105i			2.53562i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	25.0957i		−	1.13954i
$486$		0			0
$487$			10.7710i			0.488082i	0.969765	+	0.244041i	$0.0784732\pi$
$487$			10.7710i			0.488082i	−0.969765	+	0.244041i	$0.921527\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			37.2295i			1.68015i	0.542474	+	0.840073i	$0.317488\pi$
$491$			37.2295i			1.68015i	−0.542474	+	0.840073i	$0.682512\pi$
$492$		0			0
$493$		−	6.19938i		−	0.279206i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	2.49221i		−	0.111791i
$498$		0			0
$499$	−1.51472			−0.0678081			−0.0339041	−	0.999425i	$-0.510794\pi$
$499$	−1.51472			−0.0678081			−0.0339041	+	0.999425i	$0.510794\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	15.5286			0.692386			0.346193	−	0.938163i	$-0.387474\pi$
$503$	15.5286			0.692386			0.346193	+	0.938163i	$0.387474\pi$
$504$		0			0
$505$		−	19.5101i		−	0.868187i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	9.22119i		−	0.408722i	−0.978896	−	0.204361i	$-0.934488\pi$
$509$		−	9.22119i		−	0.408722i	0.978896	−	0.204361i	$-0.0655116\pi$
$510$		0			0
$511$		−	18.0663i		−	0.799207i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.20101			−0.361380
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−42.1127			−1.84499			−0.922495	−	0.386008i	$-0.873854\pi$
$521$	−42.1127			−1.84499			−0.922495	+	0.386008i	$0.873854\pi$
$522$		0			0
$523$		−	31.2128i		−	1.36484i	−0.730960	−	0.682421i	$-0.760927\pi$
$523$		−	31.2128i		−	1.36484i	0.730960	−	0.682421i	$-0.239073\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.70626			0.379251
$528$		0			0
$529$	21.7990			0.947782
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	28.6493i		−	1.24094i
$534$		0			0
$535$	−15.9188			−0.688228
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.757359	−	6.01673i	0.0326218	−	0.259159i
$540$		0			0
$541$	−7.37412			−0.317038			−0.158519	−	0.987356i	$-0.550672\pi$
$541$	−7.37412			−0.317038			−0.158519	+	0.987356i	$0.550672\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	29.0513i		−	1.24442i
$546$		0			0
$547$			33.3743i			1.42698i	0.700664	+	0.713491i	$0.252887\pi$
$547$			33.3743i			1.42698i	−0.700664	+	0.713491i	$0.747113\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			14.9666i			0.637600i
$552$		0			0
$553$	−14.6274			−0.622021
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.82590			0.119737			0.0598686	−	0.998206i	$-0.480932\pi$
$557$	2.82590			0.119737			0.0598686	+	0.998206i	$0.480932\pi$
$558$		0			0
$559$		−	51.0993i		−	2.16127i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	35.6326i		−	1.50174i	−0.660452	−	0.750868i	$-0.729635\pi$
$563$		−	35.6326i		−	1.50174i	0.660452	−	0.750868i	$-0.270365\pi$
$564$		0			0
$565$		−	31.2951i		−	1.31659i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	22.4700i		−	0.941992i	−0.882135	−	0.470996i	$-0.843895\pi$
$569$		−	22.4700i		−	0.941992i	0.882135	−	0.470996i	$-0.156105\pi$
$570$		0			0
$571$		−	5.45214i		−	0.228165i	−0.993471	−	0.114082i	$-0.963607\pi$
$571$		−	5.45214i		−	0.228165i	0.993471	−	0.114082i	$-0.0363928\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			2.19181i			0.0914049i
$576$		0			0
$577$	13.6274			0.567317			0.283658	−	0.958925i	$-0.408452\pi$
$577$	13.6274			0.567317			0.283658	+	0.958925i	$0.408452\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			14.9666i			0.620920i
$582$		0			0
$583$	−24.6250	−	3.09969i	−1.01986	−	0.128376i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	26.2843			1.08487			0.542434	−	0.840098i	$-0.317503\pi$
$587$	26.2843			1.08487			0.542434	+	0.840098i	$0.317503\pi$
$588$		0			0
$589$	−21.0188			−0.866064
$590$		0			0
$591$		0			0
$592$		0			0
$593$			15.8887i			0.652471i	0.945289	+	0.326235i	$0.105780\pi$
$593$			15.8887i			0.652471i	−0.945289	+	0.326235i	$0.894220\pi$
$594$		0			0
$595$	−8.20101			−0.336209
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		−	38.1249i		−	1.55515i	−0.628793	−	0.777573i	$-0.716451\pi$
$601$		−	38.1249i		−	1.55515i	0.628793	−	0.777573i	$-0.283549\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.21250	−	28.1954i	0.293230	−	1.14631i
$606$		0			0
$607$	7.21250			0.292746			0.146373	−	0.989229i	$-0.453240\pi$
$607$	7.21250			0.292746			0.146373	+	0.989229i	$0.453240\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	45.8054			1.85006			0.925032	−	0.379890i	$-0.124038\pi$
$613$	45.8054			1.85006			0.925032	+	0.379890i	$0.124038\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	29.1127			1.17203			0.586017	−	0.810299i	$-0.300695\pi$
$617$	29.1127			1.17203			0.586017	+	0.810299i	$0.300695\pi$
$618$		0			0
$619$	33.5269			1.34756			0.673780	−	0.738932i	$-0.264669\pi$
$619$	33.5269			1.34756			0.673780	+	0.738932i	$0.264669\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.49376			0.0598462
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13.8063			0.550492
$630$		0			0
$631$			4.19560i			0.167024i	0.996507	+	0.0835120i	$0.0266137\pi$
$631$			4.19560i			0.167024i	−0.996507	+	0.0835120i	$0.973386\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	35.0681i		−	1.39163i
$636$		0			0
$637$	−10.0384			−0.397736
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12.1716			0.480748			0.240374	−	0.970680i	$-0.422730\pi$
$641$	12.1716			0.480748			0.240374	+	0.970680i	$0.422730\pi$
$642$		0			0
$643$	−37.3848			−1.47431			−0.737156	−	0.675723i	$-0.763832\pi$
$643$	−37.3848			−1.47431			−0.737156	+	0.675723i	$0.763832\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	28.8374i		−	1.13371i	−0.823816	−	0.566857i	$-0.808159\pi$
$647$		−	28.8374i		−	1.13371i	0.823816	−	0.566857i	$-0.191841\pi$
$648$		0			0
$649$	2.17157	−	17.2517i	0.0852417	−	0.677190i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	21.6199i		−	0.846054i	−0.906117	−	0.423027i	$-0.860968\pi$
$653$		−	21.6199i		−	0.846054i	0.906117	−	0.423027i	$-0.139032\pi$
$654$		0			0
$655$	−26.1188			−1.02055
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	2.72607i		−	0.106193i	−0.998589	−	0.0530963i	$-0.983091\pi$
$659$		−	2.72607i		−	0.106193i	0.998589	−	0.0530963i	$-0.0169090\pi$
$660$		0			0
$661$		−	17.6124i		−	0.685042i	−0.939510	−	0.342521i	$-0.888719\pi$
$661$		−	17.6124i		−	0.685042i	0.939510	−	0.342521i	$-0.111281\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	19.7990			0.767772
$666$		0			0
$667$		−	4.98442i		−	0.192998i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.88393	+	14.9666i	−0.0727285	+	0.577780i
$672$		0			0
$673$			36.7618i			1.41706i	0.705679	+	0.708532i	$0.250642\pi$
$673$			36.7618i			1.41706i	−0.705679	+	0.708532i	$0.749358\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21.0188			0.807817			0.403909	−	0.914799i	$-0.367651\pi$
$677$	21.0188			0.807817			0.403909	+	0.914799i	$0.367651\pi$
$678$		0			0
$679$	21.5706			0.827802
$680$		0			0
$681$		0			0
$682$		0			0
$683$	48.4264			1.85298			0.926492	−	0.376313i	$-0.122808\pi$
$683$	48.4264			1.85298			0.926492	+	0.376313i	$0.122808\pi$
$684$		0			0
$685$		−	7.02938i		−	0.268579i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			41.0848i			1.56521i
$690$		0			0
$691$	17.2426			0.655941			0.327970	−	0.944688i	$-0.393635\pi$
$691$	17.2426			0.655941			0.327970	+	0.944688i	$0.393635\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−23.1313			−0.877419
$696$		0			0
$697$	−7.11270			−0.269413
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−50.1920			−1.89573			−0.947863	−	0.318677i	$-0.896761\pi$
$701$	−50.1920			−1.89573			−0.947863	+	0.318677i	$0.896761\pi$
$702$		0			0
$703$	−33.3313			−1.25711
$704$		0			0
$705$		0			0
$706$		0			0
$707$	16.7696			0.630684
$708$		0			0
$709$		−	1.36182i		−	0.0511441i	−0.999673	−	0.0255721i	$-0.991859\pi$
$709$		−	1.36182i		−	0.0511441i	0.999673	−	0.0255721i	$-0.00814073\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	7.00000			0.262152
$714$		0			0
$715$	−47.7990	−	6.01673i	−1.78758	−	0.225013i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	42.5200i		−	1.58573i	−0.609397	−	0.792865i	$-0.708588\pi$
$719$		−	42.5200i		−	1.58573i	0.609397	−	0.792865i	$-0.291412\pi$
$720$		0			0
$721$		−	7.04903i		−	0.262520i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.09644			−0.337833
$726$		0			0
$727$			13.8707i			0.514437i	0.966353	+	0.257218i	$0.0828060\pi$
$727$			13.8707i			0.514437i	−0.966353	+	0.257218i	$0.917194\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.6863			−0.469219
$732$		0			0
$733$	44.7018			1.65110			0.825550	−	0.564329i	$-0.190865\pi$
$733$	44.7018			1.65110			0.825550	+	0.564329i	$0.190865\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.34315	−	26.5591i	0.123146	−	0.978318i
$738$		0			0
$739$		−	42.6817i		−	1.57007i	−0.619451	−	0.785035i	$-0.712645\pi$
$739$		−	42.6817i		−	1.57007i	0.619451	−	0.785035i	$-0.287355\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.31609			0.305088			0.152544	−	0.988297i	$-0.451253\pi$
$743$	8.31609			0.305088			0.152544	+	0.988297i	$0.451253\pi$
$744$		0			0
$745$			21.5747i			0.790436i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	13.6827i		−	0.499955i
$750$		0			0
$751$			29.7452i			1.08542i	0.839921	+	0.542709i	$0.182602\pi$
$751$			29.7452i			1.08542i	−0.839921	+	0.542709i	$0.817398\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			24.0669i			0.875885i
$756$		0			0
$757$			13.6827i			0.497306i	0.968593	+	0.248653i	$0.0799879\pi$
$757$			13.6827i			0.497306i	−0.968593	+	0.248653i	$0.920012\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			17.2517i			0.625375i	0.949856	+	0.312687i	$0.101229\pi$
$761$			17.2517i			0.625375i	−0.949856	+	0.312687i	$0.898771\pi$
$762$		0			0
$763$	24.9706			0.903995
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−28.7831			−1.03930
$768$		0			0
$769$			23.8331i			0.859442i	0.902962	+	0.429721i	$0.141388\pi$
$769$			23.8331i			0.859442i	−0.902962	+	0.429721i	$0.858612\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	51.0993i		−	1.83791i	−0.394359	−	0.918956i	$-0.629033\pi$
$773$		−	51.0993i		−	1.83791i	0.394359	−	0.918956i	$-0.370967\pi$
$774$		0			0
$775$		−	12.7748i		−	0.458885i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	17.1716			0.615235
$780$		0			0
$781$	−3.60625	−	0.453939i	−0.129042	−	0.0162432i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	40.7990			1.45618
$786$		0			0
$787$		−	8.17821i		−	0.291522i	−0.989320	−	0.145761i	$-0.953437\pi$
$787$		−	8.17821i		−	0.291522i	0.989320	−	0.145761i	$-0.0465630\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	26.8991			0.956423
$792$		0			0
$793$	24.9706			0.886731
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	14.1366i		−	0.500745i	−0.968150	−	0.250373i	$-0.919447\pi$
$797$		−	14.1366i		−	0.500745i	0.968150	−	0.250373i	$-0.0805531\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−26.1421	−	3.29066i	−0.922536	−	0.116125i
$804$		0			0
$805$	−6.59377			−0.232400
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	7.94435i		−	0.279308i	−0.990200	−	0.139654i	$-0.955401\pi$
$809$		−	7.94435i		−	0.279308i	0.990200	−	0.139654i	$-0.0445991\pi$
$810$		0			0
$811$			47.6661i			1.67378i	0.547369	+	0.836892i	$0.315630\pi$
$811$			47.6661i			1.67378i	−0.547369	+	0.836892i	$0.684370\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			18.9742i			0.664637i
$816$		0			0
$817$	30.6274			1.07152
$818$		0			0
$819$		0			0
$820$		0			0
$821$	23.8447			0.832185			0.416092	−	0.909322i	$-0.363399\pi$
$821$	23.8447			0.832185			0.416092	+	0.909322i	$0.363399\pi$
$822$		0			0
$823$			43.4279i			1.51380i	0.653529	+	0.756901i	$0.273288\pi$
$823$			43.4279i			1.51380i	−0.653529	+	0.756901i	$0.726712\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1.03231i			0.0358968i	0.999839	+	0.0179484i	$0.00571347\pi$
$827$			1.03231i			0.0358968i	−0.999839	+	0.0179484i	$0.994287\pi$
$828$		0			0
$829$		−	26.9115i		−	0.934673i	−0.884079	−	0.467337i	$-0.845214\pi$
$829$		−	26.9115i		−	0.934673i	0.884079	−	0.467337i	$-0.154786\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			2.49221i			0.0863500i
$834$		0			0
$835$		−	28.0190i		−	0.969638i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			27.9295i			0.964233i	0.876107	+	0.482116i	$0.160132\pi$
$839$			27.9295i			0.964233i	−0.876107	+	0.482116i	$0.839868\pi$
$840$		0			0
$841$	−8.31371			−0.286680
$842$		0			0
$843$		0			0
$844$		0			0
$845$			45.3538i			1.56022i
$846$		0			0
$847$	24.2349	+	6.19938i	0.832719	+	0.213013i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11.1005			0.380520
$852$		0			0
$853$	14.4250			0.493903			0.246951	−	0.969028i	$-0.420571\pi$
$853$	14.4250			0.493903			0.246951	+	0.969028i	$0.420571\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			5.45214i			0.186242i	0.995655	+	0.0931208i	$0.0296843\pi$
$857$			5.45214i			0.186242i	−0.995655	+	0.0931208i	$0.970316\pi$
$858$		0			0
$859$	40.7574			1.39062			0.695312	−	0.718708i	$-0.255266\pi$
$859$	40.7574			1.39062			0.695312	+	0.718708i	$0.255266\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			14.9666i			0.509470i	0.967011	+	0.254735i	$0.0819882\pi$
$863$			14.9666i			0.509470i	−0.967011	+	0.254735i	$0.918012\pi$
$864$		0			0
$865$			58.1027i			1.97555i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.66428	+	21.1660i	−0.0903797	+	0.718008i
$870$		0			0
$871$	−44.3117			−1.50144
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	18.0502i		−	0.610208i
$876$		0			0
$877$	24.7866			0.836985			0.418493	−	0.908220i	$-0.362559\pi$
$877$	24.7866			0.836985			0.418493	+	0.908220i	$0.362559\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−23.4853			−0.791239			−0.395620	−	0.918414i	$-0.629470\pi$
$881$	−23.4853			−0.791239			−0.395620	+	0.918414i	$0.629470\pi$
$882$		0			0
$883$	15.6569			0.526895			0.263448	−	0.964674i	$-0.415140\pi$
$883$	15.6569			0.526895			0.263448	+	0.964674i	$0.415140\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	32.1608			1.07985			0.539926	−	0.841712i	$-0.318452\pi$
$887$	32.1608			1.07985			0.539926	+	0.841712i	$0.318452\pi$
$888$		0			0
$889$	30.1421			1.01093
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	41.6122i		−	1.39094i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			29.0513i			0.968916i
$900$		0			0
$901$	10.2000			0.339812
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.00000			−0.232688
$906$		0			0
$907$	50.4853			1.67634			0.838168	−	0.545412i	$-0.183627\pi$
$907$	50.4853			1.67634			0.838168	+	0.545412i	$0.183627\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	44.8999i		−	1.48760i	−0.668402	−	0.743800i	$-0.733022\pi$
$911$		−	44.8999i		−	1.48760i	0.668402	−	0.743800i	$-0.266978\pi$
$912$		0			0
$913$	21.6569	+	2.72607i	0.716737	+	0.0902197i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	22.4499i		−	0.741362i
$918$		0			0
$919$	−29.1732			−0.962337			−0.481168	−	0.876628i	$-0.659787\pi$
$919$	−29.1732			−0.962337			−0.481168	+	0.876628i	$0.659787\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			6.01673i			0.198043i
$924$		0			0
$925$		−	20.2581i		−	0.666083i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17.5147			−0.574639			−0.287320	−	0.957835i	$-0.592764\pi$
$929$	−17.5147			−0.574639			−0.287320	+	0.957835i	$0.592764\pi$
$930$		0			0
$931$		−	6.01673i		−	0.197190i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.49376	+	11.8669i	−0.0488511	+	0.388090i
$936$		0			0
$937$		−	40.8509i		−	1.33454i	−0.744815	−	0.667271i	$-0.767462\pi$
$937$		−	40.8509i		−	1.33454i	0.744815	−	0.667271i	$-0.232538\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−28.3929			−0.925582			−0.462791	−	0.886467i	$-0.653152\pi$
$941$	−28.3929			−0.925582			−0.462791	+	0.886467i	$0.653152\pi$
$942$		0			0
$943$	−5.71874			−0.186228
$944$		0			0
$945$		0			0
$946$		0			0
$947$	21.5858			0.701444			0.350722	−	0.936480i	$-0.385936\pi$
$947$	21.5858			0.701444			0.350722	+	0.936480i	$0.385936\pi$
$948$		0			0
$949$			43.6160i			1.41583i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	6.34746i		−	0.205614i	−0.994701	−	0.102807i	$-0.967218\pi$
$953$		−	6.34746i		−	0.205614i	0.994701	−	0.102807i	$-0.0327824\pi$
$954$		0			0
$955$	−25.1005			−0.812233
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.04198			0.195106
$960$		0			0
$961$	−9.79899			−0.316096
$962$		0			0
$963$		0			0
$964$		0			0
$965$	49.2501			1.58542
$966$		0			0
$967$	15.5286			0.499366			0.249683	−	0.968328i	$-0.419674\pi$
$967$	15.5286			0.499366			0.249683	+	0.968328i	$0.419674\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−54.8406			−1.75992			−0.879960	−	0.475048i	$-0.842431\pi$
$971$	−54.8406			−1.75992			−0.879960	+	0.475048i	$0.842431\pi$
$972$		0			0
$973$		−	19.8821i		−	0.637390i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.372583			0.0119200			0.00595999	−	0.999982i	$-0.498103\pi$
$977$	0.372583			0.0119200			0.00595999	+	0.999982i	$0.498103\pi$
$978$		0			0
$979$	0.272078	−	2.16148i	0.00869565	−	0.0690813i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			40.3282i			1.28627i	0.765752	+	0.643135i	$0.222367\pi$
$983$			40.3282i			1.28627i	−0.765752	+	0.643135i	$0.777633\pi$
$984$		0			0
$985$		−	53.1182i		−	1.69249i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−10.2000			−0.324342
$990$		0			0
$991$			29.9333i			0.950861i	0.879753	+	0.475431i	$0.157708\pi$
$991$			29.9333i			0.950861i	−0.879753	+	0.475431i	$0.842292\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	56.0000			1.77532
$996$		0			0
$997$	6.59377			0.208827			0.104413	−	0.994534i	$-0.466704\pi$
$997$	6.59377			0.208827			0.104413	+	0.994534i	$0.466704\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.h.g.2287.6		8
3.2	odd	2		352.2.g.b.175.5		8
4.3	odd	2		792.2.h.g.307.4		8
8.3	odd	2	inner	3168.2.h.g.2287.3		8
8.5	even	2		792.2.h.g.307.6		8
11.10	odd	2	inner	3168.2.h.g.2287.7		8
12.11	even	2		88.2.g.b.43.5	yes	8
24.5	odd	2		88.2.g.b.43.3	✓	8
24.11	even	2		352.2.g.b.175.8		8
33.32	even	2		352.2.g.b.175.6		8
44.43	even	2		792.2.h.g.307.5		8
48.5	odd	4		2816.2.e.o.2815.4		16
48.11	even	4		2816.2.e.o.2815.15		16
48.29	odd	4		2816.2.e.o.2815.14		16
48.35	even	4		2816.2.e.o.2815.1		16
88.21	odd	2		792.2.h.g.307.3		8
88.43	even	2	inner	3168.2.h.g.2287.2		8
132.35	odd	10		968.2.k.g.403.7		32
132.47	even	10		968.2.k.g.475.3		32
132.59	even	10		968.2.k.g.699.8		32
132.71	even	10		968.2.k.g.723.5		32
132.83	odd	10		968.2.k.g.723.4		32
132.95	odd	10		968.2.k.g.699.1		32
132.107	odd	10		968.2.k.g.475.6		32
132.119	even	10		968.2.k.g.403.2		32
132.131	odd	2		88.2.g.b.43.4	yes	8
264.5	odd	10		968.2.k.g.723.7		32
264.29	even	10		968.2.k.g.699.3		32
264.53	odd	10		968.2.k.g.403.4		32
264.101	even	10		968.2.k.g.403.5		32
264.125	odd	10		968.2.k.g.699.6		32
264.131	odd	2		352.2.g.b.175.7		8
264.149	even	10		968.2.k.g.723.2		32
264.173	even	10		968.2.k.g.475.8		32
264.197	even	2		88.2.g.b.43.6	yes	8
264.245	odd	10		968.2.k.g.475.1		32
528.131	odd	4		2816.2.e.o.2815.2		16
528.197	even	4		2816.2.e.o.2815.3		16
528.395	odd	4		2816.2.e.o.2815.16		16
528.461	even	4		2816.2.e.o.2815.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.g.b.43.3	✓	8	24.5	odd	2
88.2.g.b.43.4	yes	8	132.131	odd	2
88.2.g.b.43.5	yes	8	12.11	even	2
88.2.g.b.43.6	yes	8	264.197	even	2
352.2.g.b.175.5		8	3.2	odd	2
352.2.g.b.175.6		8	33.32	even	2
352.2.g.b.175.7		8	264.131	odd	2
352.2.g.b.175.8		8	24.11	even	2
792.2.h.g.307.3		8	88.21	odd	2
792.2.h.g.307.4		8	4.3	odd	2
792.2.h.g.307.5		8	44.43	even	2
792.2.h.g.307.6		8	8.5	even	2
968.2.k.g.403.2		32	132.119	even	10
968.2.k.g.403.4		32	264.53	odd	10
968.2.k.g.403.5		32	264.101	even	10
968.2.k.g.403.7		32	132.35	odd	10
968.2.k.g.475.1		32	264.245	odd	10
968.2.k.g.475.3		32	132.47	even	10
968.2.k.g.475.6		32	132.107	odd	10
968.2.k.g.475.8		32	264.173	even	10
968.2.k.g.699.1		32	132.95	odd	10
968.2.k.g.699.3		32	264.29	even	10
968.2.k.g.699.6		32	264.125	odd	10
968.2.k.g.699.8		32	132.59	even	10
968.2.k.g.723.2		32	264.149	even	10
968.2.k.g.723.4		32	132.83	odd	10
968.2.k.g.723.5		32	132.71	even	10
968.2.k.g.723.7		32	264.5	odd	10
2816.2.e.o.2815.1		16	48.35	even	4
2816.2.e.o.2815.2		16	528.131	odd	4
2816.2.e.o.2815.3		16	528.197	even	4
2816.2.e.o.2815.4		16	48.5	odd	4
2816.2.e.o.2815.13		16	528.461	even	4
2816.2.e.o.2815.14		16	48.29	odd	4
2816.2.e.o.2815.15		16	48.11	even	4
2816.2.e.o.2815.16		16	528.395	odd	4
3168.2.h.g.2287.2		8	88.43	even	2	inner
3168.2.h.g.2287.3		8	8.3	odd	2	inner
3168.2.h.g.2287.6		8	1.1	even	1	trivial
3168.2.h.g.2287.7		8	11.10	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.64575i q^{5} -2.27411 q^{7} +(-0.414214 + 3.29066i) q^{11} +5.49019 q^{13} -1.36303i q^{17} +3.29066i q^{19} -1.09591i q^{23} -2.00000 q^{25} +4.54822 q^{29} +6.38741i q^{31} -6.01673i q^{35} +10.1291i q^{37} -5.21828i q^{41} -9.30739i q^{43} -1.82843 q^{49} +7.48331i q^{53} +(-8.70626 - 1.09591i) q^{55} -5.24264 q^{59} +4.54822 q^{61} +14.5257i q^{65} -8.07107 q^{67} +1.09591i q^{71} +7.94435i q^{73} +(0.941967 - 7.48331i) q^{77} +6.43215 q^{79} -6.58132i q^{83} +3.60625 q^{85} -0.656854 q^{89} -12.4853 q^{91} -8.70626 q^{95} -9.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{11} - 16 q^{25} + 8 q^{49} - 8 q^{59} - 8 q^{67} + 40 q^{89} - 32 q^{91} - 8 q^{97}+O(q^{100}) \) 8 * q + 8 * q^11 - 16 * q^25 + 8 * q^49 - 8 * q^59 - 8 * q^67 + 40 * q^89 - 32 * q^91 - 8 * q^97

Introduction

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