LMFDB - Embedded newform 2816.2.e.o.2815.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(2815,2816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2816.2815");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.e (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:	$22.4858732092$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.1622647227216566419456.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 8x^{12} - 44x^{10} + 161x^{8} - 88x^{6} + 32x^{4} - 32x^{2} + 16 $ x^16 - 4x^14 + 8x^12 - 44x^10 + 161x^8 - 88x^6 + 32x^4 - 32*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{26} $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2815.13
Root			$-1.00237 - 1.55225i$ of defining polynomial
Character	$\chi$	$=$	2816.2815
Dual form			2816.2.e.o.2815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421i q^{3} -2.64575 q^{5} -2.27411 q^{7} -2.82843 q^{9} +(3.29066 - 0.414214i) q^{11} -5.49019i q^{13} -6.38741i q^{15} -1.36303i q^{17} +3.29066 q^{19} -5.49019i q^{21} -1.09591i q^{23} +2.00000 q^{25} +0.414214i q^{27} +4.54822i q^{29} +6.38741i q^{31} +(1.00000 + 7.94435i) q^{33} +6.01673 q^{35} +10.1291 q^{37} +13.2545 q^{39} +5.21828i q^{41} +9.30739 q^{43} +7.48331 q^{45} -1.82843 q^{49} +3.29066 q^{51} -7.48331 q^{53} +(-8.70626 + 1.09591i) q^{55} +7.94435i q^{57} -5.24264i q^{59} -4.54822i q^{61} +6.43215 q^{63} +14.5257i q^{65} -8.07107i q^{67} +2.64575 q^{69} +1.09591i q^{71} +7.94435i q^{73} +4.82843i q^{75} +(-7.48331 + 0.941967i) q^{77} -6.43215 q^{79} -9.48528 q^{81} +6.58132 q^{83} +3.60625i q^{85} -10.9804 q^{87} -0.656854 q^{89} +12.4853i q^{91} -15.4206 q^{93} -8.70626 q^{95} -9.48528 q^{97} +(-9.30739 + 1.17157i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{25} + 16 q^{33} + 16 q^{49} - 16 q^{81} + 80 q^{89} - 16 q^{97}+O(q^{100}) $ 16 * q + 32 * q^25 + 16 * q^33 + 16 * q^49 - 16 * q^81 + 80 * q^89 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$			2.41421i			1.39385i	0.717146	+	0.696923i	$0.245448\pi$
$3$			2.41421i			1.39385i	−0.717146	+	0.696923i	$0.754552\pi$
$4$		0			0
$5$	−2.64575			−1.18322			−0.591608	−	0.806226i	$-0.701507\pi$
$5$	−2.64575			−1.18322			−0.591608	+	0.806226i	$0.701507\pi$
$6$		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
$9$	−2.82843			−0.942809
$10$		0			0
$11$	3.29066	−	0.414214i	0.992171	−	0.124890i
$11$	3.29066	−	0.414214i	0.992171	−	0.124890i
$12$		0			0
$13$		−	5.49019i		−	1.52270i	−0.648339	−	0.761352i	$-0.724536\pi$
$13$		−	5.49019i		−	1.52270i	0.648339	−	0.761352i	$-0.275464\pi$
$14$		0			0
$15$		−	6.38741i		−	1.64922i
$16$		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
$19$	3.29066			0.754929			0.377464	−	0.926024i	$-0.376796\pi$
$19$	3.29066			0.754929			0.377464	+	0.926024i	$0.376796\pi$
$20$		0			0
$21$		−	5.49019i		−	1.19806i
$22$		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
$27$			0.414214i			0.0797154i
$28$		0			0
$29$			4.54822i			0.844583i	0.906460	+	0.422291i	$0.138774\pi$
$29$			4.54822i			0.844583i	−0.906460	+	0.422291i	$0.861226\pi$
$30$		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
$33$	1.00000	+	7.94435i	0.174078	+	1.38293i
$34$		0			0
$35$	6.01673			1.01701
$36$		0			0
$37$	10.1291			1.66521			0.832604	−	0.553869i	$-0.186849\pi$
$37$	10.1291			1.66521			0.832604	+	0.553869i	$0.186849\pi$
$38$		0			0
$39$	13.2545			2.12242
$40$		0			0
$41$			5.21828i			0.814958i	0.913214	+	0.407479i	$0.133592\pi$
$41$			5.21828i			0.814958i	−0.913214	+	0.407479i	$0.866408\pi$
$42$		0			0
$43$	9.30739			1.41936			0.709681	−	0.704523i	$-0.248839\pi$
$43$	9.30739			1.41936			0.709681	+	0.704523i	$0.248839\pi$
$44$		0			0
$45$	7.48331			1.11555
$46$		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
$51$	3.29066			0.460784
$52$		0			0
$53$	−7.48331			−1.02791			−0.513956	−	0.857816i	$-0.671821\pi$
$53$	−7.48331			−1.02791			−0.513956	+	0.857816i	$0.671821\pi$
$54$		0			0
$55$	−8.70626	+	1.09591i	−1.17395	+	0.147772i
$56$		0			0
$57$			7.94435i			1.05225i
$58$		0			0
$59$		−	5.24264i		−	0.682534i	−0.939966	−	0.341267i	$-0.889144\pi$
$59$		−	5.24264i		−	0.682534i	0.939966	−	0.341267i	$-0.110856\pi$
$60$		0			0
$61$		−	4.54822i		−	0.582340i	−0.956671	−	0.291170i	$-0.905956\pi$
$61$		−	4.54822i		−	0.582340i	0.956671	−	0.291170i	$-0.0940445\pi$
$62$		0			0
$63$	6.43215			0.810375
$64$		0			0
$65$			14.5257i			1.80169i
$66$		0			0
$67$		−	8.07107i		−	0.986038i	−0.870019	−	0.493019i	$-0.835893\pi$
$67$		−	8.07107i		−	0.986038i	0.870019	−	0.493019i	$-0.164107\pi$
$68$		0			0
$69$	2.64575			0.318511
$70$		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
$75$			4.82843i			0.557539i
$76$		0			0
$77$	−7.48331	+	0.941967i	−0.852803	+	0.107347i
$78$		0			0
$79$	−6.43215			−0.723674			−0.361837	−	0.932241i	$-0.617850\pi$
$79$	−6.43215			−0.723674			−0.361837	+	0.932241i	$0.617850\pi$
$80$		0			0
$81$	−9.48528			−1.05392
$82$		0			0
$83$	6.58132			0.722393			0.361197	−	0.932490i	$-0.382368\pi$
$83$	6.58132			0.722393			0.361197	+	0.932490i	$0.382368\pi$
$84$		0			0
$85$			3.60625i			0.391153i
$86$		0			0
$87$	−10.9804			−1.17722
$88$		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
$91$			12.4853i			1.30881i
$92$		0			0
$93$	−15.4206			−1.59904
$94$		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
$99$	−9.30739	+	1.17157i	−0.935427	+	0.117748i
$100$		0			0
$101$			7.37412i			0.733752i	0.930270	+	0.366876i	$0.119573\pi$
$101$			7.37412i			0.733752i	−0.930270	+	0.366876i	$0.880427\pi$
$102$		0			0
$103$		−	3.09969i		−	0.305422i	−0.988271	−	0.152711i	$-0.951200\pi$
$103$		−	3.09969i		−	0.305422i	0.988271	−	0.152711i	$-0.0488003\pi$
$104$		0			0
$105$			14.5257i			1.41756i
$106$		0			0
$107$	6.01673			0.581659			0.290830	−	0.956775i	$-0.406069\pi$
$107$	6.01673			0.581659			0.290830	+	0.956775i	$0.406069\pi$
$108$		0			0
$109$			10.9804i			1.05173i	0.850568	+	0.525864i	$0.176258\pi$
$109$			10.9804i			1.05173i	−0.850568	+	0.525864i	$0.823742\pi$
$110$		0			0
$111$			24.4537i			2.32105i
$112$		0			0
$113$	11.8284			1.11272			0.556362	−	0.830940i	$-0.312197\pi$
$113$	11.8284			1.11272			0.556362	+	0.830940i	$0.312197\pi$
$114$		0			0
$115$			2.89949i			0.270379i
$116$		0			0
$117$			15.5286i			1.43562i
$118$		0			0
$119$			3.09969i			0.284148i
$120$		0			0
$121$	10.6569	−	2.72607i	0.968805	−	0.247825i
$122$		0			0
$123$	−12.5980			−1.13593
$124$		0			0
$125$	7.93725			0.709930
$126$		0			0
$127$	13.2545			1.17614			0.588072	−	0.808808i	$-0.299887\pi$
$127$	13.2545			1.17614			0.588072	+	0.808808i	$0.299887\pi$
$128$		0			0
$129$			22.4700i			1.97837i
$130$		0			0
$131$	−9.87197			−0.862518			−0.431259	−	0.902228i	$-0.641931\pi$
$131$	−9.87197			−0.862518			−0.431259	+	0.902228i	$0.641931\pi$
$132$		0			0
$133$	−7.48331			−0.648886
$134$		0			0
$135$		−	1.09591i		−	0.0943206i
$136$		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
$139$	−8.74280			−0.741554			−0.370777	−	0.928722i	$-0.620909\pi$
$139$	−8.74280			−0.741554			−0.370777	+	0.928722i	$0.620909\pi$
$140$		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
$147$		−	4.41421i		−	0.364078i
$148$		0			0
$149$		−	8.15447i		−	0.668040i	−0.942566	−	0.334020i	$-0.891595\pi$
$149$		−	8.15447i		−	0.668040i	0.942566	−	0.334020i	$-0.108405\pi$
$150$		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
$153$			3.85525i			0.311678i
$154$		0			0
$155$		−	16.8995i		−	1.35740i
$156$		0			0
$157$	15.4206			1.23070			0.615348	−	0.788256i	$-0.289016\pi$
$157$	15.4206			1.23070			0.615348	+	0.788256i	$0.289016\pi$
$158$		0			0
$159$		−	18.0663i		−	1.43275i
$160$		0			0
$161$			2.49221i			0.196414i
$162$		0			0
$163$			7.17157i			0.561721i	0.959749	+	0.280860i	$0.0906198\pi$
$163$			7.17157i			0.561721i	−0.959749	+	0.280860i	$0.909380\pi$
$164$		0			0
$165$	−2.64575	−	21.0188i	−0.205971	−	1.63631i
$166$		0			0
$167$	10.5902			0.819494			0.409747	−	0.912199i	$-0.365617\pi$
$167$	10.5902			0.819494			0.409747	+	0.912199i	$0.365617\pi$
$168$		0			0
$169$	−17.1421			−1.31863
$170$		0			0
$171$	−9.30739			−0.711754
$172$		0			0
$173$			21.9607i			1.66964i	0.550520	+	0.834822i	$0.314430\pi$
$173$			21.9607i			1.66964i	−0.550520	+	0.834822i	$0.685570\pi$
$174$		0			0
$175$	−4.54822			−0.343813
$176$		0			0
$177$	12.6569			0.951347
$178$		0			0
$179$			15.7279i			1.17556i	0.809021	+	0.587780i	$0.199998\pi$
$179$			15.7279i			1.17556i	−0.809021	+	0.587780i	$0.800002\pi$
$180$		0			0
$181$	2.64575			0.196657			0.0983286	−	0.995154i	$-0.468650\pi$
$181$	2.64575			0.196657			0.0983286	+	0.995154i	$0.468650\pi$
$182$		0			0
$183$	10.9804			0.811692
$184$		0			0
$185$	−26.7990			−1.97030
$186$		0			0
$187$	−0.564588	−	4.48528i	−0.0412867	−	0.327996i
$188$		0			0
$189$		−	0.941967i		−	0.0685180i
$190$		0			0
$191$		−	9.48710i		−	0.686462i	−0.939251	−	0.343231i	$-0.888479\pi$
$191$		−	9.48710i		−	0.686462i	0.939251	−	0.343231i	$-0.111521\pi$
$192$		0			0
$193$			18.6148i			1.33992i	0.742397	+	0.669960i	$0.233689\pi$
$193$			18.6148i			1.33992i	−0.742397	+	0.669960i	$0.766311\pi$
$194$		0			0
$195$	−35.0681			−2.51128
$196$		0			0
$197$			20.0768i			1.43041i	0.698913	+	0.715207i	$0.253667\pi$
$197$			20.0768i			1.43041i	−0.698913	+	0.715207i	$0.746333\pi$
$198$		0			0
$199$			21.1660i			1.50042i	0.661200	+	0.750209i	$0.270047\pi$
$199$			21.1660i			1.50042i	−0.661200	+	0.750209i	$0.729953\pi$
$200$		0			0
$201$	19.4853			1.37439
$202$		0			0
$203$		−	10.3431i		−	0.725947i
$204$		0			0
$205$		−	13.8063i		−	0.964272i
$206$		0			0
$207$			3.09969i			0.215443i
$208$		0			0
$209$	10.8284	−	1.36303i	0.749018	−	0.0942831i
$210$		0			0
$211$	13.1626			0.906153			0.453076	−	0.891472i	$-0.350326\pi$
$211$	13.1626			0.906153			0.453076	+	0.891472i	$0.350326\pi$
$212$		0			0
$213$	−2.64575			−0.181284
$214$		0			0
$215$	−24.6250			−1.67941
$216$		0			0
$217$		−	14.5257i		−	0.986066i
$218$		0			0
$219$	−19.1794			−1.29602
$220$		0			0
$221$	−7.48331			−0.503382
$222$		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
$225$	−5.65685			−0.377124
$226$		0			0
$227$	−0.564588			−0.0374730			−0.0187365	−	0.999824i	$-0.505964\pi$
$227$	−0.564588			−0.0374730			−0.0187365	+	0.999824i	$0.505964\pi$
$228$		0			0
$229$	4.83756			0.319675			0.159838	−	0.987143i	$-0.448903\pi$
$229$	4.83756			0.319675			0.159838	+	0.987143i	$0.448903\pi$
$230$		0			0
$231$	−2.27411	−	18.0663i	−0.149625	−	1.18868i
$232$		0			0
$233$			25.1961i			1.65065i	0.564658	+	0.825325i	$0.309008\pi$
$233$			25.1961i			1.65065i	−0.564658	+	0.825325i	$0.690992\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	15.5286i		−	1.00869i
$238$		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	$-0.932012\pi$
$241$		−	6.58132i		−	0.423940i	0.977276	−	0.211970i	$-0.0679879\pi$
$242$		0			0
$243$		−	21.6569i		−	1.38929i
$244$		0			0
$245$	4.83756			0.309061
$246$		0			0
$247$		−	18.0663i		−	1.14953i
$248$		0			0
$249$			15.8887i			1.00691i
$250$		0			0
$251$			15.5858i			0.983766i	0.870661	+	0.491883i	$0.163691\pi$
$251$			15.5858i			0.983766i	−0.870661	+	0.491883i	$0.836309\pi$
$252$		0			0
$253$	−0.453939	−	3.60625i	−0.0285389	−	0.226723i
$254$		0			0
$255$	−8.70626			−0.545207
$256$		0			0
$257$	−10.4853			−0.654054			−0.327027	−	0.945015i	$-0.606047\pi$
$257$	−10.4853			−0.654054			−0.327027	+	0.945015i	$0.606047\pi$
$258$		0			0
$259$	−23.0346			−1.43130
$260$		0			0
$261$		−	12.8643i		−	0.796280i
$262$		0			0
$263$	0.390175			0.0240592			0.0120296	−	0.999928i	$-0.496171\pi$
$263$	0.390175			0.0240592			0.0120296	+	0.999928i	$0.496171\pi$
$264$		0			0
$265$	19.7990			1.21624
$266$		0			0
$267$		−	1.58579i		−	0.0970486i
$268$		0			0
$269$	28.6493			1.74678			0.873390	−	0.487021i	$-0.161917\pi$
$269$	28.6493			1.74678			0.873390	+	0.487021i	$0.161917\pi$
$270$		0			0
$271$	−4.15804			−0.252583			−0.126292	−	0.991993i	$-0.540307\pi$
$271$	−4.15804			−0.252583			−0.126292	+	0.991993i	$0.540307\pi$
$272$		0			0
$273$	−30.1421			−1.82429
$274$		0			0
$275$	6.58132	−	0.828427i	0.396868	−	0.0499560i
$276$		0			0
$277$		−	15.5286i		−	0.933023i	−0.884515	−	0.466511i	$-0.845511\pi$
$277$		−	15.5286i		−	0.933023i	0.884515	−	0.466511i	$-0.154489\pi$
$278$		0			0
$279$		−	18.0663i		−	1.08160i
$280$		0			0
$281$		−	9.30739i		−	0.555232i	−0.960692	−	0.277616i	$-0.910456\pi$
$281$		−	9.30739i		−	0.555232i	0.960692	−	0.277616i	$-0.0895443\pi$
$282$		0			0
$283$	15.8887			0.944485			0.472243	−	0.881469i	$-0.343445\pi$
$283$	15.8887			0.944485			0.472243	+	0.881469i	$0.343445\pi$
$284$		0			0
$285$		−	21.0188i		−	1.24504i
$286$		0			0
$287$		−	11.8669i		−	0.700483i
$288$		0			0
$289$	15.1421			0.890714
$290$		0			0
$291$		−	22.8995i		−	1.34239i
$292$		0			0
$293$			3.60625i			0.210679i	0.994436	+	0.105340i	$0.0335930\pi$
$293$			3.60625i			0.210679i	−0.994436	+	0.105340i	$0.966407\pi$
$294$		0			0
$295$			13.8707i			0.807585i
$296$		0			0
$297$	0.171573	+	1.36303i	0.00995567	+	0.0790913i
$298$		0			0
$299$	−6.01673			−0.347956
$300$		0			0
$301$	−21.1660			−1.21999
$302$		0			0
$303$	−17.8027			−1.02274
$304$		0			0
$305$			12.0335i			0.689034i
$306$		0			0
$307$	13.7272			0.783454			0.391727	−	0.920082i	$-0.371878\pi$
$307$	13.7272			0.783454			0.391727	+	0.920082i	$0.371878\pi$
$308$		0			0
$309$	7.48331			0.425711
$310$		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	$-0.519548\pi$
$313$	−2.17157			−0.122745			−0.0613723	+	0.998115i	$0.519548\pi$
$314$		0			0
$315$	−17.0179			−0.958849
$316$		0			0
$317$	−0.453939			−0.0254958			−0.0127479	−	0.999919i	$-0.504058\pi$
$317$	−0.453939			−0.0254958			−0.0127479	+	0.999919i	$0.504058\pi$
$318$		0			0
$319$	1.88393	+	14.9666i	0.105480	+	0.837970i
$320$		0			0
$321$			14.5257i			0.810744i
$322$		0			0
$323$		−	4.48528i		−	0.249568i
$324$		0			0
$325$		−	10.9804i		−	0.609081i
$326$		0			0
$327$	−26.5090			−1.46595
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	3.92893i		−	0.215954i	−0.994153	−	0.107977i	$-0.965563\pi$
$331$		−	3.92893i		−	0.215954i	0.994153	−	0.107977i	$-0.0344372\pi$
$332$		0			0
$333$	−28.6493			−1.56997
$334$		0			0
$335$			21.3540i			1.16670i
$336$		0			0
$337$		−	18.3809i		−	1.00127i	−0.865658	−	0.500636i	$-0.833100\pi$
$337$		−	18.3809i		−	1.00127i	0.865658	−	0.500636i	$-0.166900\pi$
$338$		0			0
$339$			28.5563i			1.55097i
$340$		0			0
$341$	2.64575	+	21.0188i	0.143275	+	1.13823i
$342$		0			0
$343$	20.0768			1.08405
$344$		0			0
$345$	−7.00000			−0.376867
$346$		0			0
$347$	12.5980			0.676298			0.338149	−	0.941093i	$-0.390199\pi$
$347$	12.5980			0.676298			0.338149	+	0.941093i	$0.390199\pi$
$348$		0			0
$349$			3.60625i			0.193038i	0.995331	+	0.0965191i	$0.0307709\pi$
$349$			3.60625i			0.193038i	−0.995331	+	0.0965191i	$0.969229\pi$
$350$		0			0
$351$	2.27411			0.121383
$352$		0			0
$353$	−3.82843			−0.203767			−0.101883	−	0.994796i	$-0.532487\pi$
$353$	−3.82843			−0.203767			−0.101883	+	0.994796i	$0.532487\pi$
$354$		0			0
$355$		−	2.89949i		−	0.153889i
$356$		0			0
$357$	−7.48331			−0.396059
$358$		0			0
$359$	2.27411			0.120023			0.0600114	−	0.998198i	$-0.480886\pi$
$359$	2.27411			0.120023			0.0600114	+	0.998198i	$0.480886\pi$
$360$		0			0
$361$	−8.17157			−0.430083
$362$		0			0
$363$	6.58132	+	25.7279i	0.345429	+	1.35037i
$364$		0			0
$365$		−	21.0188i		−	1.10017i
$366$		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
$369$		−	14.7595i		−	0.768350i
$370$		0			0
$371$	17.0179			0.883524
$372$		0			0
$373$		−	31.9991i		−	1.65685i	−0.560098	−	0.828426i	$-0.689236\pi$
$373$		−	31.9991i		−	1.65685i	0.560098	−	0.828426i	$-0.310764\pi$
$374$		0			0
$375$			19.1622i			0.989533i
$376$		0			0
$377$	24.9706			1.28605
$378$		0			0
$379$		−	1.92893i		−	0.0990826i	−0.998772	−	0.0495413i	$-0.984224\pi$
$379$		−	1.92893i		−	0.0990826i	0.998772	−	0.0495413i	$-0.0157759\pi$
$380$		0			0
$381$			31.9991i			1.63937i
$382$		0			0
$383$			29.7452i			1.51991i	0.649976	+	0.759955i	$0.274779\pi$
$383$			29.7452i			1.51991i	−0.649976	+	0.759955i	$0.725221\pi$
$384$		0			0
$385$	19.7990	−	2.49221i	1.00905	−	0.127015i
$386$		0			0
$387$	−26.3253			−1.33819
$388$		0			0
$389$	−27.2875			−1.38353			−0.691766	−	0.722122i	$-0.743167\pi$
$389$	−27.2875			−1.38353			−0.691766	+	0.722122i	$0.743167\pi$
$390$		0			0
$391$	−1.49376			−0.0755426
$392$		0			0
$393$		−	23.8331i		−	1.20222i
$394$		0			0
$395$	17.0179			0.856262
$396$		0			0
$397$	−22.4499			−1.12673			−0.563365	−	0.826208i	$-0.690494\pi$
$397$	−22.4499			−1.12673			−0.563365	+	0.826208i	$0.690494\pi$
$398$		0			0
$399$		−	18.0663i		−	0.904447i
$400$		0			0
$401$	0.828427			0.0413697			0.0206848	−	0.999786i	$-0.493415\pi$
$401$	0.828427			0.0413697			0.0206848	+	0.999786i	$0.493415\pi$
$402$		0			0
$403$	35.0681			1.74686
$404$		0			0
$405$	25.0957			1.24702
$406$		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
$411$		−	6.41421i		−	0.316390i
$412$		0			0
$413$			11.9223i			0.586660i
$414$		0			0
$415$	−17.4125			−0.854747
$416$		0			0
$417$		−	21.1070i		−	1.03361i
$418$		0			0
$419$			12.1421i			0.593182i	0.955005	+	0.296591i	$0.0958498\pi$
$419$			12.1421i			0.593182i	−0.955005	+	0.296591i	$0.904150\pi$
$420$		0			0
$421$	−7.48331			−0.364714			−0.182357	−	0.983232i	$-0.558373\pi$
$421$	−7.48331			−0.364714			−0.182357	+	0.983232i	$0.558373\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	2.72607i		−	0.132234i
$426$		0			0
$427$			10.3431i			0.500540i
$428$		0			0
$429$	43.6160	−	5.49019i	2.10580	−	0.265069i
$430$		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
$435$	29.0513			1.39290
$436$		0			0
$437$		−	3.60625i		−	0.172510i
$438$		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
$441$	5.17157			0.246265
$442$		0			0
$443$			23.5858i			1.12059i	0.828292	+	0.560297i	$0.189313\pi$
$443$			23.5858i			1.12059i	−0.828292	+	0.560297i	$0.810687\pi$
$444$		0			0
$445$	1.73787			0.0823831
$446$		0			0
$447$	19.6866			0.931146
$448$		0			0
$449$	−17.9706			−0.848083			−0.424042	−	0.905643i	$-0.639389\pi$
$449$	−17.9706			−0.848083			−0.424042	+	0.905643i	$0.639389\pi$
$450$		0			0
$451$	2.16148	+	17.1716i	0.101780	+	0.808578i
$452$		0			0
$453$			21.9607i			1.03181i
$454$		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
$459$	0.564588			0.0263527
$460$		0			0
$461$			15.5286i			0.723239i	0.932326	+	0.361619i	$0.117776\pi$
$461$			15.5286i			0.723239i	−0.932326	+	0.361619i	$0.882224\pi$
$462$		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
$465$	40.7990			1.89201
$466$		0			0
$467$		−	21.9289i		−	1.01475i	−0.861725	−	0.507375i	$-0.830616\pi$
$467$		−	21.9289i		−	1.01475i	0.861725	−	0.507375i	$-0.169384\pi$
$468$		0			0
$469$			18.3545i			0.847532i
$470$		0			0
$471$			37.2285i			1.71540i
$472$		0			0
$473$	30.6274	−	3.85525i	1.40825	−	0.177264i
$474$		0			0
$475$	6.58132			0.301971
$476$		0			0
$477$	21.1660			0.969125
$478$		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
$483$	−6.01673			−0.273771
$484$		0			0
$485$	25.0957			1.13954
$486$		0			0
$487$		−	10.7710i		−	0.488082i	−0.969765	−	0.244041i	$-0.921527\pi$
$487$		−	10.7710i		−	0.488082i	0.969765	−	0.244041i	$-0.0784732\pi$
$488$		0			0
$489$	−17.3137			−0.782953
$490$		0			0
$491$	37.2295			1.68015			0.840073	−	0.542474i	$-0.182512\pi$
$491$	37.2295			1.68015			0.840073	+	0.542474i	$0.182512\pi$
$492$		0			0
$493$	6.19938			0.279206
$494$		0			0
$495$	24.6250	−	3.09969i	1.10681	−	0.139321i
$496$		0			0
$497$		−	2.49221i		−	0.111791i
$498$		0			0
$499$		−	1.51472i		−	0.0678081i	−0.999425	−	0.0339041i	$-0.989206\pi$
$499$		−	1.51472i		−	0.0678081i	0.999425	−	0.0339041i	$-0.0107941\pi$
$500$		0			0
$501$			25.5670i			1.14225i
$502$		0			0
$503$	−15.5286			−0.692386			−0.346193	−	0.938163i	$-0.612526\pi$
$503$	−15.5286			−0.692386			−0.346193	+	0.938163i	$0.612526\pi$
$504$		0			0
$505$		−	19.5101i		−	0.868187i
$506$		0			0
$507$		−	41.3848i		−	1.83796i
$508$		0			0
$509$	−9.22119			−0.408722			−0.204361	−	0.978896i	$-0.565512\pi$
$509$	−9.22119			−0.408722			−0.204361	+	0.978896i	$0.565512\pi$
$510$		0			0
$511$		−	18.0663i		−	0.799207i
$512$		0			0
$513$			1.36303i			0.0601795i
$514$		0			0
$515$			8.20101i			0.361380i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−53.0179			−2.32723
$520$		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
$523$	31.2128			1.36484			0.682421	−	0.730960i	$-0.260927\pi$
$523$	31.2128			1.36484			0.682421	+	0.730960i	$0.260927\pi$
$524$		0			0
$525$		−	10.9804i		−	0.479223i
$526$		0			0
$527$	8.70626			0.379251
$528$		0			0
$529$	21.7990			0.947782
$530$		0			0
$531$			14.8284i			0.643499i
$532$		0			0
$533$	28.6493			1.24094
$534$		0			0
$535$	−15.9188			−0.688228
$536$		0			0
$537$	−37.9706			−1.63855
$538$		0			0
$539$	−6.01673	+	0.757359i	−0.259159	+	0.0326218i
$540$		0			0
$541$			7.37412i			0.317038i	0.987356	+	0.158519i	$0.0506719\pi$
$541$			7.37412i			0.317038i	−0.987356	+	0.158519i	$0.949328\pi$
$542$		0			0
$543$			6.38741i			0.274110i
$544$		0			0
$545$		−	29.0513i		−	1.24442i
$546$		0			0
$547$	33.3743			1.42698			0.713491	−	0.700664i	$-0.247113\pi$
$547$	33.3743			1.42698			0.713491	+	0.700664i	$0.247113\pi$
$548$		0			0
$549$			12.8643i			0.549035i
$550$		0			0
$551$			14.9666i			0.637600i
$552$		0			0
$553$	14.6274			0.622021
$554$		0			0
$555$		−	64.6985i		−	2.74630i
$556$		0			0
$557$			2.82590i			0.119737i	0.998206	+	0.0598686i	$0.0190682\pi$
$557$			2.82590i			0.119737i	−0.998206	+	0.0598686i	$0.980932\pi$
$558$		0			0
$559$		−	51.0993i		−	2.16127i
$560$		0			0
$561$	10.8284	−	1.36303i	0.457177	−	0.0575474i
$562$		0			0
$563$	35.6326			1.50174			0.750868	−	0.660452i	$-0.229635\pi$
$563$	35.6326			1.50174			0.750868	+	0.660452i	$0.229635\pi$
$564$		0			0
$565$	−31.2951			−1.31659
$566$		0			0
$567$	21.5706			0.905879
$568$		0			0
$569$			22.4700i			0.941992i	0.882135	+	0.470996i	$0.156105\pi$
$569$			22.4700i			0.941992i	−0.882135	+	0.470996i	$0.843895\pi$
$570$		0			0
$571$	5.45214			0.228165			0.114082	−	0.993471i	$-0.463607\pi$
$571$	5.45214			0.228165			0.114082	+	0.993471i	$0.463607\pi$
$572$		0			0
$573$	22.9039			0.956823
$574$		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
$579$	−44.9400			−1.86764
$580$		0			0
$581$	−14.9666			−0.620920
$582$		0			0
$583$	−24.6250	+	3.09969i	−1.01986	+	0.128376i
$584$		0			0
$585$		−	41.0848i		−	1.69865i
$586$		0			0
$587$			26.2843i			1.08487i	0.840098	+	0.542434i	$0.182497\pi$
$587$			26.2843i			1.08487i	−0.840098	+	0.542434i	$0.817503\pi$
$588$		0			0
$589$			21.0188i			0.866064i
$590$		0			0
$591$	−48.4697			−1.99378
$592$		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
$595$		−	8.20101i		−	0.336209i
$596$		0			0
$597$	−51.0993			−2.09135
$598$		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
$603$			22.8284i			0.929645i
$604$		0			0
$605$	−28.1954	+	7.21250i	−1.14631	+	0.293230i
$606$		0			0
$607$	−7.21250			−0.292746			−0.146373	−	0.989229i	$-0.546760\pi$
$607$	−7.21250			−0.292746			−0.146373	+	0.989229i	$0.546760\pi$
$608$		0			0
$609$	24.9706			1.01186
$610$		0			0
$611$		0			0
$612$		0			0
$613$			45.8054i			1.85006i	0.379890	+	0.925032i	$0.375962\pi$
$613$			45.8054i			1.85006i	−0.379890	+	0.925032i	$0.624038\pi$
$614$		0			0
$615$	33.3313			1.34405
$616$		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
$619$		−	33.5269i		−	1.34756i	−0.738932	−	0.673780i	$-0.764669\pi$
$619$		−	33.5269i		−	1.34756i	0.738932	−	0.673780i	$-0.235331\pi$
$620$		0			0
$621$	0.453939			0.0182160
$622$		0			0
$623$	1.49376			0.0598462
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$	3.29066	+	26.1421i	0.131416	+	1.04402i
$628$		0			0
$629$		−	13.8063i		−	0.550492i
$630$		0			0
$631$		−	4.19560i		−	0.167024i	−0.996507	−	0.0835120i	$-0.973386\pi$
$631$		−	4.19560i		−	0.167024i	0.996507	−	0.0835120i	$-0.0266137\pi$
$632$		0			0
$633$			31.7774i			1.26304i
$634$		0			0
$635$	−35.0681			−1.39163
$636$		0			0
$637$			10.0384i			0.397736i
$638$		0			0
$639$		−	3.09969i		−	0.122622i
$640$		0			0
$641$	−12.1716			−0.480748			−0.240374	−	0.970680i	$-0.577270\pi$
$641$	−12.1716			−0.480748			−0.240374	+	0.970680i	$0.577270\pi$
$642$		0			0
$643$		−	37.3848i		−	1.47431i	−0.675723	−	0.737156i	$-0.736168\pi$
$643$		−	37.3848i		−	1.47431i	0.675723	−	0.737156i	$-0.263832\pi$
$644$		0			0
$645$		−	59.4501i		−	2.34084i
$646$		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
$651$	35.0681			1.37443
$652$		0			0
$653$	−21.6199			−0.846054			−0.423027	−	0.906117i	$-0.639032\pi$
$653$	−21.6199			−0.846054			−0.423027	+	0.906117i	$0.639032\pi$
$654$		0			0
$655$	26.1188			1.02055
$656$		0			0
$657$		−	22.4700i		−	0.876639i
$658$		0			0
$659$	2.72607			0.106193			0.0530963	−	0.998589i	$-0.483091\pi$
$659$	2.72607			0.106193			0.0530963	+	0.998589i	$0.483091\pi$
$660$		0			0
$661$	−17.6124			−0.685042			−0.342521	−	0.939510i	$-0.611281\pi$
$661$	−17.6124			−0.685042			−0.342521	+	0.939510i	$0.611281\pi$
$662$		0			0
$663$		−	18.0663i		−	0.701638i
$664$		0			0
$665$	19.7990			0.767772
$666$		0			0
$667$	4.98442			0.192998
$668$		0			0
$669$	−28.1954			−1.09010
$670$		0			0
$671$	−1.88393	−	14.9666i	−0.0727285	−	0.577780i
$672$		0			0
$673$		−	36.7618i		−	1.41706i	−0.705679	−	0.708532i	$-0.749358\pi$
$673$		−	36.7618i		−	1.41706i	0.705679	−	0.708532i	$-0.250642\pi$
$674$		0			0
$675$			0.828427i			0.0318862i
$676$		0			0
$677$		−	21.0188i		−	0.807817i	−0.914799	−	0.403909i	$-0.867651\pi$
$677$		−	21.0188i		−	0.807817i	0.914799	−	0.403909i	$-0.132349\pi$
$678$		0			0
$679$	21.5706			0.827802
$680$		0			0
$681$		−	1.36303i		−	0.0522316i
$682$		0			0
$683$			48.4264i			1.85298i	0.376313	+	0.926492i	$0.377192\pi$
$683$			48.4264i			1.85298i	−0.376313	+	0.926492i	$0.622808\pi$
$684$		0			0
$685$	7.02938			0.268579
$686$		0			0
$687$			11.6789i			0.445578i
$688$		0			0
$689$			41.0848i			1.56521i
$690$		0			0
$691$			17.2426i			0.655941i	0.944688	+	0.327970i	$0.106365\pi$
$691$			17.2426i			0.655941i	−0.944688	+	0.327970i	$0.893635\pi$
$692$		0			0
$693$	21.1660	−	2.66428i	0.804030	−	0.101208i
$694$		0			0
$695$	23.1313			0.877419
$696$		0			0
$697$	7.11270			0.269413
$698$		0			0
$699$	−60.8287			−2.30075
$700$		0			0
$701$		−	50.1920i		−	1.89573i	−0.318677	−	0.947863i	$-0.603239\pi$
$701$		−	50.1920i		−	1.89573i	0.318677	−	0.947863i	$-0.396761\pi$
$702$		0			0
$703$	33.3313			1.25711
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	16.7696i		−	0.630684i
$708$		0			0
$709$	−1.36182			−0.0511441			−0.0255721	−	0.999673i	$-0.508141\pi$
$709$	−1.36182			−0.0511441			−0.0255721	+	0.999673i	$0.508141\pi$
$710$		0			0
$711$	18.1929			0.682286
$712$		0			0
$713$	7.00000			0.262152
$714$		0			0
$715$	6.01673	+	47.7990i	0.225013	+	1.78758i
$716$		0			0
$717$			53.0179i			1.97999i
$718$		0			0
$719$			42.5200i			1.58573i	0.609397	+	0.792865i	$0.291412\pi$
$719$			42.5200i			1.58573i	−0.609397	+	0.792865i	$0.708588\pi$
$720$		0			0
$721$			7.04903i			0.262520i
$722$		0			0
$723$	15.8887			0.590907
$724$		0			0
$725$			9.09644i			0.337833i
$726$		0			0
$727$		−	13.8707i		−	0.514437i	−0.966353	−	0.257218i	$-0.917194\pi$
$727$		−	13.8707i		−	0.514437i	0.966353	−	0.257218i	$-0.0828060\pi$
$728$		0			0
$729$	23.8284			0.882534
$730$		0			0
$731$		−	12.6863i		−	0.469219i
$732$		0			0
$733$		−	44.7018i		−	1.65110i	−0.564329	−	0.825550i	$-0.690865\pi$
$733$		−	44.7018i		−	1.65110i	0.564329	−	0.825550i	$-0.309135\pi$
$734$		0			0
$735$			11.6789i			0.430783i
$736$		0			0
$737$	−3.34315	−	26.5591i	−0.123146	−	0.978318i
$738$		0			0
$739$	−42.6817			−1.57007			−0.785035	−	0.619451i	$-0.787355\pi$
$739$	−42.6817			−1.57007			−0.785035	+	0.619451i	$0.787355\pi$
$740$		0			0
$741$	43.6160			1.60227
$742$		0			0
$743$	−8.31609			−0.305088			−0.152544	−	0.988297i	$-0.548747\pi$
$743$	−8.31609			−0.305088			−0.152544	+	0.988297i	$0.548747\pi$
$744$		0			0
$745$			21.5747i			0.790436i
$746$		0			0
$747$	−18.6148			−0.681079
$748$		0			0
$749$	−13.6827			−0.499955
$750$		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
$753$	−37.6274			−1.37122
$754$		0			0
$755$	−24.0669			−0.875885
$756$		0			0
$757$	13.6827			0.497306			0.248653	−	0.968593i	$-0.420012\pi$
$757$	13.6827			0.497306			0.248653	+	0.968593i	$0.420012\pi$
$758$		0			0
$759$	8.70626	−	1.09591i	0.316017	−	0.0397789i
$760$		0			0
$761$		−	17.2517i		−	0.625375i	−0.949856	−	0.312687i	$-0.898771\pi$
$761$		−	17.2517i		−	0.625375i	0.949856	−	0.312687i	$-0.101229\pi$
$762$		0			0
$763$		−	24.9706i		−	0.903995i
$764$		0			0
$765$		−	10.2000i		−	0.368782i
$766$		0			0
$767$	−28.7831			−1.03930
$768$		0			0
$769$		−	23.8331i		−	0.859442i	−0.902962	−	0.429721i	$-0.858612\pi$
$769$		−	23.8331i		−	0.859442i	0.902962	−	0.429721i	$-0.141388\pi$
$770$		0			0
$771$		−	25.3137i		−	0.911651i
$772$		0			0
$773$	51.0993			1.83791			0.918956	−	0.394359i	$-0.129033\pi$
$773$	51.0993			1.83791			0.918956	+	0.394359i	$0.129033\pi$
$774$		0			0
$775$			12.7748i			0.458885i
$776$		0			0
$777$		−	55.6105i		−	1.99501i
$778$		0			0
$779$			17.1716i			0.615235i
$780$		0			0
$781$	0.453939	+	3.60625i	0.0162432	+	0.129042i
$782$		0			0
$783$	−1.88393			−0.0673263
$784$		0			0
$785$	−40.7990			−1.45618
$786$		0			0
$787$	−8.17821			−0.291522			−0.145761	−	0.989320i	$-0.546563\pi$
$787$	−8.17821			−0.291522			−0.145761	+	0.989320i	$0.546563\pi$
$788$		0			0
$789$			0.941967i			0.0335349i
$790$		0			0
$791$	−26.8991			−0.956423
$792$		0			0
$793$	−24.9706			−0.886731
$794$		0			0
$795$			47.7990i			1.69526i
$796$		0			0
$797$	−14.1366			−0.500745			−0.250373	−	0.968150i	$-0.580553\pi$
$797$	−14.1366			−0.500745			−0.250373	+	0.968150i	$0.580553\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	1.85786			0.0656444
$802$		0			0
$803$	3.29066	+	26.1421i	0.116125	+	0.922536i
$804$		0			0
$805$		−	6.59377i		−	0.232400i
$806$		0			0
$807$			69.1656i			2.43474i
$808$		0			0
$809$			7.94435i			0.279308i	0.990200	+	0.139654i	$0.0445991\pi$
$809$			7.94435i			0.279308i	−0.990200	+	0.139654i	$0.955401\pi$
$810$		0			0
$811$	−47.6661			−1.67378			−0.836892	−	0.547369i	$-0.815630\pi$
$811$	−47.6661			−1.67378			−0.836892	+	0.547369i	$0.815630\pi$
$812$		0			0
$813$		−	10.0384i		−	0.352062i
$814$		0			0
$815$		−	18.9742i		−	0.664637i
$816$		0			0
$817$	30.6274			1.07152
$818$		0			0
$819$		−	35.3137i		−	1.23396i
$820$		0			0
$821$		−	23.8447i		−	0.832185i	−0.909322	−	0.416092i	$-0.863399\pi$
$821$		−	23.8447i		−	0.832185i	0.909322	−	0.416092i	$-0.136601\pi$
$822$		0			0
$823$		−	43.4279i		−	1.51380i	−0.653529	−	0.756901i	$-0.726712\pi$
$823$		−	43.4279i		−	1.51380i	0.653529	−	0.756901i	$-0.273288\pi$
$824$		0			0
$825$	2.00000	+	15.8887i	0.0696311	+	0.553174i
$826$		0			0
$827$	1.03231			0.0358968			0.0179484	−	0.999839i	$-0.494287\pi$
$827$	1.03231			0.0358968			0.0179484	+	0.999839i	$0.494287\pi$
$828$		0			0
$829$	26.9115			0.934673			0.467337	−	0.884079i	$-0.345214\pi$
$829$	26.9115			0.934673			0.467337	+	0.884079i	$0.345214\pi$
$830$		0			0
$831$	37.4893			1.30049
$832$		0			0
$833$			2.49221i			0.0863500i
$834$		0			0
$835$	−28.0190			−0.969638
$836$		0			0
$837$	−2.64575			−0.0914505
$838$		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
$843$	22.4700			0.773908
$844$		0			0
$845$	45.3538			1.56022
$846$		0			0
$847$	−24.2349	+	6.19938i	−0.832719	+	0.213013i
$848$		0			0
$849$			38.3587i			1.31647i
$850$		0			0
$851$		−	11.1005i		−	0.380520i
$852$		0			0
$853$			14.4250i			0.493903i	0.969028	+	0.246951i	$0.0794288\pi$
$853$			14.4250i			0.493903i	−0.969028	+	0.246951i	$0.920571\pi$
$854$		0			0
$855$	24.6250			0.842158
$856$		0			0
$857$		−	5.45214i		−	0.186242i	−0.995655	−	0.0931208i	$-0.970316\pi$
$857$		−	5.45214i		−	0.186242i	0.995655	−	0.0931208i	$-0.0296843\pi$
$858$		0			0
$859$		−	40.7574i		−	1.39062i	−0.718708	−	0.695312i	$-0.755266\pi$
$859$		−	40.7574i		−	1.39062i	0.718708	−	0.695312i	$-0.244734\pi$
$860$		0			0
$861$	28.6493			0.976366
$862$		0			0
$863$		−	14.9666i		−	0.509470i	−0.967011	−	0.254735i	$-0.918012\pi$
$863$		−	14.9666i		−	0.509470i	0.967011	−	0.254735i	$-0.0819882\pi$
$864$		0			0
$865$		−	58.1027i		−	1.97555i
$866$		0			0
$867$			36.5563i			1.24152i
$868$		0			0
$869$	−21.1660	+	2.66428i	−0.718008	+	0.0903797i
$870$		0			0
$871$	−44.3117			−1.50144
$872$		0			0
$873$	26.8284			0.908005
$874$		0			0
$875$	−18.0502			−0.610208
$876$		0			0
$877$		−	24.7866i		−	0.836985i	−0.908220	−	0.418493i	$-0.862559\pi$
$877$		−	24.7866i		−	0.836985i	0.908220	−	0.418493i	$-0.137441\pi$
$878$		0			0
$879$	−8.70626			−0.293655
$880$		0			0
$881$	23.4853			0.791239			0.395620	−	0.918414i	$-0.370530\pi$
$881$	23.4853			0.791239			0.395620	+	0.918414i	$0.370530\pi$
$882$		0			0
$883$			15.6569i			0.526895i	0.964674	+	0.263448i	$0.0848596\pi$
$883$			15.6569i			0.526895i	−0.964674	+	0.263448i	$0.915140\pi$
$884$		0			0
$885$	−33.4869			−1.12565
$886$		0			0
$887$	−32.1608			−1.07985			−0.539926	−	0.841712i	$-0.681548\pi$
$887$	−32.1608			−1.07985			−0.539926	+	0.841712i	$0.681548\pi$
$888$		0			0
$889$	−30.1421			−1.01093
$890$		0			0
$891$	−31.2128	+	3.92893i	−1.04567	+	0.131624i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	41.6122i		−	1.39094i
$896$		0			0
$897$		−	14.5257i		−	0.484998i
$898$		0			0
$899$	−29.0513			−0.968916
$900$		0			0
$901$			10.2000i			0.339812i
$902$		0			0
$903$		−	51.0993i		−	1.70048i
$904$		0			0
$905$	−7.00000			−0.232688
$906$		0			0
$907$		−	50.4853i		−	1.67634i	−0.545412	−	0.838168i	$-0.683627\pi$
$907$		−	50.4853i		−	1.67634i	0.545412	−	0.838168i	$-0.316373\pi$
$908$		0			0
$909$		−	20.8572i		−	0.691788i
$910$		0			0
$911$			44.8999i			1.48760i	0.668402	+	0.743800i	$0.266978\pi$
$911$			44.8999i			1.48760i	−0.668402	+	0.743800i	$0.733022\pi$
$912$		0			0
$913$	21.6569	−	2.72607i	0.716737	−	0.0902197i
$914$		0			0
$915$	−29.0513			−0.960407
$916$		0			0
$917$	22.4499			0.741362
$918$		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
$921$			33.1404i			1.09201i
$922$		0			0
$923$	6.01673			0.198043
$924$		0			0
$925$	20.2581			0.666083
$926$		0			0
$927$			8.76725i			0.287954i
$928$		0			0
$929$	17.5147			0.574639			0.287320	−	0.957835i	$-0.407236\pi$
$929$	17.5147			0.574639			0.287320	+	0.957835i	$0.407236\pi$
$930$		0			0
$931$	−6.01673			−0.197190
$932$		0			0
$933$	−36.1326			−1.18293
$934$		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
$939$		−	5.24264i		−	0.171087i
$940$		0			0
$941$		−	28.3929i		−	0.925582i	−0.886467	−	0.462791i	$-0.846848\pi$
$941$		−	28.3929i		−	0.925582i	0.886467	−	0.462791i	$-0.153152\pi$
$942$		0			0
$943$	5.71874			0.186228
$944$		0			0
$945$			2.49221i			0.0810716i
$946$		0			0
$947$		−	21.5858i		−	0.701444i	−0.936480	−	0.350722i	$-0.885936\pi$
$947$		−	21.5858i		−	0.701444i	0.936480	−	0.350722i	$-0.114064\pi$
$948$		0			0
$949$	43.6160			1.41583
$950$		0			0
$951$		−	1.09591i		−	0.0355372i
$952$		0			0
$953$			6.34746i			0.205614i	0.994701	+	0.102807i	$0.0327824\pi$
$953$			6.34746i			0.205614i	−0.994701	+	0.102807i	$0.967218\pi$
$954$		0			0
$955$			25.1005i			0.812233i
$956$		0			0
$957$	−36.1326	+	4.54822i	−1.16800	+	0.147023i
$958$		0			0
$959$	6.04198			0.195106
$960$		0			0
$961$	−9.79899			−0.316096
$962$		0			0
$963$	−17.0179			−0.548393
$964$		0			0
$965$		−	49.2501i		−	1.58542i
$966$		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
$969$	10.8284			0.347859
$970$		0			0
$971$		−	54.8406i		−	1.75992i	−0.475048	−	0.879960i	$-0.657569\pi$
$971$		−	54.8406i		−	1.75992i	0.475048	−	0.879960i	$-0.342431\pi$
$972$		0			0
$973$	19.8821			0.637390
$974$		0			0
$975$	26.5090			0.848966
$976$		0			0
$977$	−0.372583			−0.0119200			−0.00595999	−	0.999982i	$-0.501897\pi$
$977$	−0.372583			−0.0119200			−0.00595999	+	0.999982i	$0.501897\pi$
$978$		0			0
$979$	−2.16148	+	0.272078i	−0.0690813	+	0.00869565i
$980$		0			0
$981$		−	31.0572i		−	0.991579i
$982$		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
$989$		−	10.2000i		−	0.324342i
$990$		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
$993$	9.48528			0.301006
$994$		0			0
$995$		−	56.0000i		−	1.77532i
$996$		0			0
$997$			6.59377i			0.208827i	0.994534	+	0.104413i	$0.0332965\pi$
$997$			6.59377i			0.208827i	−0.994534	+	0.104413i	$0.966704\pi$
$998$		0			0
$999$			4.19560i			0.132743i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.e.o.2815.13		16
4.3	odd	2	inner	2816.2.e.o.2815.2		16
8.3	odd	2	inner	2816.2.e.o.2815.16		16
8.5	even	2	inner	2816.2.e.o.2815.3		16
11.10	odd	2	inner	2816.2.e.o.2815.14		16
16.3	odd	4		352.2.g.b.175.7		8
16.5	even	4		352.2.g.b.175.6		8
16.11	odd	4		88.2.g.b.43.4	yes	8
16.13	even	4		88.2.g.b.43.6	yes	8
44.43	even	2	inner	2816.2.e.o.2815.1		16
48.5	odd	4		3168.2.h.g.2287.7		8
48.11	even	4		792.2.h.g.307.5		8
48.29	odd	4		792.2.h.g.307.3		8
48.35	even	4		3168.2.h.g.2287.2		8
88.21	odd	2	inner	2816.2.e.o.2815.4		16
88.43	even	2	inner	2816.2.e.o.2815.15		16
176.13	odd	20		968.2.k.g.403.4		32
176.21	odd	4		352.2.g.b.175.5		8
176.27	odd	20		968.2.k.g.723.4		32
176.29	odd	20		968.2.k.g.699.6		32
176.43	even	4		88.2.g.b.43.5	yes	8
176.59	odd	20		968.2.k.g.699.1		32
176.61	odd	20		968.2.k.g.723.7		32
176.75	odd	20		968.2.k.g.403.7		32
176.91	odd	20		968.2.k.g.475.6		32
176.93	even	20		968.2.k.g.723.2		32
176.107	even	20		968.2.k.g.475.3		32
176.109	odd	4		88.2.g.b.43.3	✓	8
176.123	even	20		968.2.k.g.403.2		32
176.125	even	20		968.2.k.g.699.3		32
176.131	even	4		352.2.g.b.175.8		8
176.139	even	20		968.2.k.g.699.8		32
176.141	even	20		968.2.k.g.403.5		32
176.157	even	20		968.2.k.g.475.8		32
176.171	even	20		968.2.k.g.723.5		32
176.173	odd	20		968.2.k.g.475.1		32
528.131	odd	4		3168.2.h.g.2287.3		8
528.197	even	4		3168.2.h.g.2287.6		8
528.395	odd	4		792.2.h.g.307.4		8
528.461	even	4		792.2.h.g.307.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.g.b.43.3	✓	8	176.109	odd	4
88.2.g.b.43.4	yes	8	16.11	odd	4
88.2.g.b.43.5	yes	8	176.43	even	4
88.2.g.b.43.6	yes	8	16.13	even	4
352.2.g.b.175.5		8	176.21	odd	4
352.2.g.b.175.6		8	16.5	even	4
352.2.g.b.175.7		8	16.3	odd	4
352.2.g.b.175.8		8	176.131	even	4
792.2.h.g.307.3		8	48.29	odd	4
792.2.h.g.307.4		8	528.395	odd	4
792.2.h.g.307.5		8	48.11	even	4
792.2.h.g.307.6		8	528.461	even	4
968.2.k.g.403.2		32	176.123	even	20
968.2.k.g.403.4		32	176.13	odd	20
968.2.k.g.403.5		32	176.141	even	20
968.2.k.g.403.7		32	176.75	odd	20
968.2.k.g.475.1		32	176.173	odd	20
968.2.k.g.475.3		32	176.107	even	20
968.2.k.g.475.6		32	176.91	odd	20
968.2.k.g.475.8		32	176.157	even	20
968.2.k.g.699.1		32	176.59	odd	20
968.2.k.g.699.3		32	176.125	even	20
968.2.k.g.699.6		32	176.29	odd	20
968.2.k.g.699.8		32	176.139	even	20
968.2.k.g.723.2		32	176.93	even	20
968.2.k.g.723.4		32	176.27	odd	20
968.2.k.g.723.5		32	176.171	even	20
968.2.k.g.723.7		32	176.61	odd	20
2816.2.e.o.2815.1		16	44.43	even	2	inner
2816.2.e.o.2815.2		16	4.3	odd	2	inner
2816.2.e.o.2815.3		16	8.5	even	2	inner
2816.2.e.o.2815.4		16	88.21	odd	2	inner
2816.2.e.o.2815.13		16	1.1	even	1	trivial
2816.2.e.o.2815.14		16	11.10	odd	2	inner
2816.2.e.o.2815.15		16	88.43	even	2	inner
2816.2.e.o.2815.16		16	8.3	odd	2	inner
3168.2.h.g.2287.2		8	48.35	even	4
3168.2.h.g.2287.3		8	528.131	odd	4
3168.2.h.g.2287.6		8	528.197	even	4
3168.2.h.g.2287.7		8	48.5	odd	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:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.41421i q^{3} -2.64575 q^{5} -2.27411 q^{7} -2.82843 q^{9} +(3.29066 - 0.414214i) q^{11} -5.49019i q^{13} -6.38741i q^{15} -1.36303i q^{17} +3.29066 q^{19} -5.49019i q^{21} -1.09591i q^{23} +2.00000 q^{25} +0.414214i q^{27} +4.54822i q^{29} +6.38741i q^{31} +(1.00000 + 7.94435i) q^{33} +6.01673 q^{35} +10.1291 q^{37} +13.2545 q^{39} +5.21828i q^{41} +9.30739 q^{43} +7.48331 q^{45} -1.82843 q^{49} +3.29066 q^{51} -7.48331 q^{53} +(-8.70626 + 1.09591i) q^{55} +7.94435i q^{57} -5.24264i q^{59} -4.54822i q^{61} +6.43215 q^{63} +14.5257i q^{65} -8.07107i q^{67} +2.64575 q^{69} +1.09591i q^{71} +7.94435i q^{73} +4.82843i q^{75} +(-7.48331 + 0.941967i) q^{77} -6.43215 q^{79} -9.48528 q^{81} +6.58132 q^{83} +3.60625i q^{85} -10.9804 q^{87} -0.656854 q^{89} +12.4853i q^{91} -15.4206 q^{93} -8.70626 q^{95} -9.48528 q^{97} +(-9.30739 + 1.17157i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{25} + 16 q^{33} + 16 q^{49} - 16 q^{81} + 80 q^{89} - 16 q^{97}+O(q^{100}) \) 16 * q + 32 * q^25 + 16 * q^33 + 16 * q^49 - 16 * q^81 + 80 * q^89 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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