LMFDB - Embedded newform 208.2.f.b.129.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,2,Mod(129,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.129"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	208.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$1.66088836204$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.4
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.2.f.b.129.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} +2.56155i q^{5} -4.56155i q^{7} +3.56155 q^{9} +3.12311i q^{11} +(-3.56155 + 0.561553i) q^{13} +6.56155i q^{15} +0.561553 q^{17} -2.00000i q^{19} -11.6847i q^{21} -5.12311 q^{23} -1.56155 q^{25} +1.43845 q^{27} +3.12311 q^{29} -3.12311i q^{31} +8.00000i q^{33} +11.6847 q^{35} +5.43845i q^{37} +(-9.12311 + 1.43845i) q^{39} -8.00000i q^{41} -5.43845 q^{43} +9.12311i q^{45} +3.43845i q^{47} -13.8078 q^{49} +1.43845 q^{51} -4.24621 q^{53} -8.00000 q^{55} -5.12311i q^{57} -10.0000i q^{59} +11.1231 q^{61} -16.2462i q^{63} +(-1.43845 - 9.12311i) q^{65} +0.876894i q^{67} -13.1231 q^{69} +5.68466i q^{71} +15.3693i q^{73} -4.00000 q^{75} +14.2462 q^{77} +2.87689 q^{79} -7.00000 q^{81} +8.24621i q^{83} +1.43845i q^{85} +8.00000 q^{87} +5.12311i q^{89} +(2.56155 + 16.2462i) q^{91} -8.00000i q^{93} +5.12311 q^{95} -10.2462i q^{97} +11.1231i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 6 q^{9} - 6 q^{13} - 6 q^{17} - 4 q^{23} + 2 q^{25} + 14 q^{27} - 4 q^{29} + 22 q^{35} - 20 q^{39} - 30 q^{43} - 14 q^{49} + 14 q^{51} + 16 q^{53} - 32 q^{55} + 28 q^{61} - 14 q^{65} - 36 q^{69}+ \cdots + 4 q^{95}+O(q^{100}) $ 4 * q + 2 * q^3 + 6 * q^9 - 6 * q^13 - 6 * q^17 - 4 * q^23 + 2 * q^25 + 14 * q^27 - 4 * q^29 + 22 * q^35 - 20 * q^39 - 30 * q^43 - 14 * q^49 + 14 * q^51 + 16 * q^53 - 32 * q^55 + 28 * q^61 - 14 * q^65 - 36 * q^69 - 16 * q^75 + 24 * q^77 + 28 * q^79 - 28 * q^81 + 32 * q^87 + 2 * q^91 + 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.56155			1.47891			0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155			1.47891			0.739457	+	0.673204i	$0.235083\pi$
$4$		0			0
$5$			2.56155i			1.14556i	0.819709	+	0.572781i	$0.194135\pi$
$5$			2.56155i			1.14556i	−0.819709	+	0.572781i	$0.805865\pi$
$6$		0			0
$7$		−	4.56155i		−	1.72410i	−0.506819	−	0.862052i	$-0.669179\pi$
$7$		−	4.56155i		−	1.72410i	0.506819	−	0.862052i	$-0.330821\pi$
$8$		0			0
$9$	3.56155			1.18718
$10$		0			0
$11$			3.12311i			0.941652i	0.882226	+	0.470826i	$0.156044\pi$
$11$			3.12311i			0.941652i	−0.882226	+	0.470826i	$0.843956\pi$
$12$		0			0
$13$	−3.56155	+	0.561553i	−0.987797	+	0.155747i
$13$	−3.56155	+	0.561553i	−0.987797	+	0.155747i
$14$		0			0
$15$			6.56155i			1.69419i
$16$		0			0
$17$	0.561553			0.136197			0.0680983	−	0.997679i	$-0.478307\pi$
$17$	0.561553			0.136197			0.0680983	+	0.997679i	$0.478307\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$		−	11.6847i		−	2.54980i
$22$		0			0
$23$	−5.12311			−1.06824			−0.534121	−	0.845408i	$-0.679357\pi$
$23$	−5.12311			−1.06824			−0.534121	+	0.845408i	$0.679357\pi$
$24$		0			0
$25$	−1.56155			−0.312311
$26$		0			0
$27$	1.43845			0.276829
$28$		0			0
$29$	3.12311			0.579946			0.289973	−	0.957035i	$-0.406354\pi$
$29$	3.12311			0.579946			0.289973	+	0.957035i	$0.406354\pi$
$30$		0			0
$31$		−	3.12311i		−	0.560926i	−0.959865	−	0.280463i	$-0.909512\pi$
$31$		−	3.12311i		−	0.560926i	0.959865	−	0.280463i	$-0.0904881\pi$
$32$		0			0
$33$			8.00000i			1.39262i
$34$		0			0
$35$	11.6847			1.97507
$36$		0			0
$37$			5.43845i			0.894075i	0.894515	+	0.447038i	$0.147521\pi$
$37$			5.43845i			0.894075i	−0.894515	+	0.447038i	$0.852479\pi$
$38$		0			0
$39$	−9.12311	+	1.43845i	−1.46087	+	0.230336i
$40$		0			0
$41$		−	8.00000i		−	1.24939i	−0.780869	−	0.624695i	$-0.785223\pi$
$41$		−	8.00000i		−	1.24939i	0.780869	−	0.624695i	$-0.214777\pi$
$42$		0			0
$43$	−5.43845			−0.829355			−0.414678	−	0.909968i	$-0.636106\pi$
$43$	−5.43845			−0.829355			−0.414678	+	0.909968i	$0.636106\pi$
$44$		0			0
$45$			9.12311i			1.35999i
$46$		0			0
$47$			3.43845i			0.501549i	0.968046	+	0.250775i	$0.0806853\pi$
$47$			3.43845i			0.501549i	−0.968046	+	0.250775i	$0.919315\pi$
$48$		0			0
$49$	−13.8078			−1.97254
$50$		0			0
$51$	1.43845			0.201423
$52$		0			0
$53$	−4.24621			−0.583262			−0.291631	−	0.956531i	$-0.594198\pi$
$53$	−4.24621			−0.583262			−0.291631	+	0.956531i	$0.594198\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$		−	5.12311i		−	0.678572i
$58$		0			0
$59$		−	10.0000i		−	1.30189i	−0.759125	−	0.650945i	$-0.774373\pi$
$59$		−	10.0000i		−	1.30189i	0.759125	−	0.650945i	$-0.225627\pi$
$60$		0			0
$61$	11.1231			1.42417			0.712084	−	0.702094i	$-0.247752\pi$
$61$	11.1231			1.42417			0.712084	+	0.702094i	$0.247752\pi$
$62$		0			0
$63$		−	16.2462i		−	2.04683i
$64$		0			0
$65$	−1.43845	−	9.12311i	−0.178417	−	1.13158i
$66$		0			0
$67$			0.876894i			0.107130i	0.998564	+	0.0535648i	$0.0170584\pi$
$67$			0.876894i			0.107130i	−0.998564	+	0.0535648i	$0.982942\pi$
$68$		0			0
$69$	−13.1231			−1.57984
$70$		0			0
$71$			5.68466i			0.674645i	0.941389	+	0.337322i	$0.109521\pi$
$71$			5.68466i			0.674645i	−0.941389	+	0.337322i	$0.890479\pi$
$72$		0			0
$73$			15.3693i			1.79884i	0.437083	+	0.899421i	$0.356012\pi$
$73$			15.3693i			1.79884i	−0.437083	+	0.899421i	$0.643988\pi$
$74$		0			0
$75$	−4.00000			−0.461880
$76$		0			0
$77$	14.2462			1.62351
$78$		0			0
$79$	2.87689			0.323676			0.161838	−	0.986817i	$-0.448258\pi$
$79$	2.87689			0.323676			0.161838	+	0.986817i	$0.448258\pi$
$80$		0			0
$81$	−7.00000			−0.777778
$82$		0			0
$83$			8.24621i			0.905139i	0.891729	+	0.452570i	$0.149493\pi$
$83$			8.24621i			0.905139i	−0.891729	+	0.452570i	$0.850507\pi$
$84$		0			0
$85$			1.43845i			0.156022i
$86$		0			0
$87$	8.00000			0.857690
$88$		0			0
$89$			5.12311i			0.543048i	0.962432	+	0.271524i	$0.0875277\pi$
$89$			5.12311i			0.543048i	−0.962432	+	0.271524i	$0.912472\pi$
$90$		0			0
$91$	2.56155	+	16.2462i	0.268524	+	1.70307i
$92$		0			0
$93$		−	8.00000i		−	0.829561i
$94$		0			0
$95$	5.12311			0.525620
$96$		0			0
$97$		−	10.2462i		−	1.04035i	−0.854061	−	0.520173i	$-0.825868\pi$
$97$		−	10.2462i		−	1.04035i	0.854061	−	0.520173i	$-0.174132\pi$
$98$		0			0
$99$			11.1231i			1.11791i
$100$		0			0
$101$	13.3693			1.33030			0.665148	−	0.746711i	$-0.268368\pi$
$101$	13.3693			1.33030			0.665148	+	0.746711i	$0.268368\pi$
$102$		0			0
$103$	5.12311			0.504795			0.252397	−	0.967624i	$-0.418781\pi$
$103$	5.12311			0.504795			0.252397	+	0.967624i	$0.418781\pi$
$104$		0			0
$105$	29.9309			2.92095
$106$		0			0
$107$	4.00000			0.386695			0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000			0.386695			0.193347	+	0.981130i	$0.438066\pi$
$108$		0			0
$109$		−	15.6847i		−	1.50232i	−0.660121	−	0.751159i	$-0.729495\pi$
$109$		−	15.6847i		−	1.50232i	0.660121	−	0.751159i	$-0.270505\pi$
$110$		0			0
$111$			13.9309i			1.32226i
$112$		0			0
$113$	10.0000			0.940721			0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000			0.940721			0.470360	+	0.882474i	$0.344124\pi$
$114$		0			0
$115$		−	13.1231i		−	1.22374i
$116$		0			0
$117$	−12.6847	+	2.00000i	−1.17270	+	0.184900i
$118$		0			0
$119$		−	2.56155i		−	0.234817i
$120$		0			0
$121$	1.24621			0.113292
$122$		0			0
$123$		−	20.4924i		−	1.84774i
$124$		0			0
$125$			8.80776i			0.787790i
$126$		0			0
$127$	13.1231			1.16449			0.582244	−	0.813014i	$-0.302175\pi$
$127$	13.1231			1.16449			0.582244	+	0.813014i	$0.302175\pi$
$128$		0			0
$129$	−13.9309			−1.22654
$130$		0			0
$131$	15.6847			1.37037			0.685187	−	0.728367i	$-0.259720\pi$
$131$	15.6847			1.37037			0.685187	+	0.728367i	$0.259720\pi$
$132$		0			0
$133$	−9.12311			−0.791074
$134$		0			0
$135$			3.68466i			0.317125i
$136$		0			0
$137$			5.12311i			0.437696i	0.975759	+	0.218848i	$0.0702300\pi$
$137$			5.12311i			0.437696i	−0.975759	+	0.218848i	$0.929770\pi$
$138$		0			0
$139$	−12.8078			−1.08634			−0.543170	−	0.839623i	$-0.682776\pi$
$139$	−12.8078			−1.08634			−0.543170	+	0.839623i	$0.682776\pi$
$140$		0			0
$141$			8.80776i			0.741748i
$142$		0			0
$143$	−1.75379	−	11.1231i	−0.146659	−	0.930161i
$144$		0			0
$145$			8.00000i			0.664364i
$146$		0			0
$147$	−35.3693			−2.91721
$148$		0			0
$149$		−	1.12311i		−	0.0920084i	−0.998941	−	0.0460042i	$-0.985351\pi$
$149$		−	1.12311i		−	0.0920084i	0.998941	−	0.0460042i	$-0.0146488\pi$
$150$		0			0
$151$		−	7.43845i		−	0.605332i	−0.953097	−	0.302666i	$-0.902123\pi$
$151$		−	7.43845i		−	0.605332i	0.953097	−	0.302666i	$-0.0978767\pi$
$152$		0			0
$153$	2.00000			0.161690
$154$		0			0
$155$	8.00000			0.642575
$156$		0			0
$157$	−4.24621			−0.338885			−0.169442	−	0.985540i	$-0.554197\pi$
$157$	−4.24621			−0.338885			−0.169442	+	0.985540i	$0.554197\pi$
$158$		0			0
$159$	−10.8769			−0.862594
$160$		0			0
$161$			23.3693i			1.84176i
$162$		0			0
$163$			3.75379i			0.294019i	0.989135	+	0.147010i	$0.0469649\pi$
$163$			3.75379i			0.294019i	−0.989135	+	0.147010i	$0.953035\pi$
$164$		0			0
$165$	−20.4924			−1.59533
$166$		0			0
$167$			15.1231i			1.17026i	0.810939	+	0.585131i	$0.198957\pi$
$167$			15.1231i			1.17026i	−0.810939	+	0.585131i	$0.801043\pi$
$168$		0			0
$169$	12.3693	−	4.00000i	0.951486	−	0.307692i
$170$		0			0
$171$		−	7.12311i		−	0.544718i
$172$		0			0
$173$	−12.2462			−0.931062			−0.465531	−	0.885032i	$-0.654137\pi$
$173$	−12.2462			−0.931062			−0.465531	+	0.885032i	$0.654137\pi$
$174$		0			0
$175$			7.12311i			0.538456i
$176$		0			0
$177$		−	25.6155i		−	1.92538i
$178$		0			0
$179$	−13.4384			−1.00444			−0.502218	−	0.864741i	$-0.667483\pi$
$179$	−13.4384			−1.00444			−0.502218	+	0.864741i	$0.667483\pi$
$180$		0			0
$181$	8.24621			0.612936			0.306468	−	0.951881i	$-0.400853\pi$
$181$	8.24621			0.612936			0.306468	+	0.951881i	$0.400853\pi$
$182$		0			0
$183$	28.4924			2.10622
$184$		0			0
$185$	−13.9309			−1.02422
$186$		0			0
$187$			1.75379i			0.128250i
$188$		0			0
$189$		−	6.56155i		−	0.477283i
$190$		0			0
$191$	5.75379			0.416330			0.208165	−	0.978094i	$-0.433251\pi$
$191$	5.75379			0.416330			0.208165	+	0.978094i	$0.433251\pi$
$192$		0			0
$193$		−	17.6155i		−	1.26799i	−0.773336	−	0.633997i	$-0.781413\pi$
$193$		−	17.6155i		−	1.26799i	0.773336	−	0.633997i	$-0.218587\pi$
$194$		0			0
$195$	−3.68466	−	23.3693i	−0.263864	−	1.67351i
$196$		0			0
$197$			5.43845i			0.387473i	0.981054	+	0.193737i	$0.0620607\pi$
$197$			5.43845i			0.387473i	−0.981054	+	0.193737i	$0.937939\pi$
$198$		0			0
$199$	−2.24621			−0.159230			−0.0796148	−	0.996826i	$-0.525369\pi$
$199$	−2.24621			−0.159230			−0.0796148	+	0.996826i	$0.525369\pi$
$200$		0			0
$201$			2.24621i			0.158436i
$202$		0			0
$203$		−	14.2462i		−	0.999888i
$204$		0			0
$205$	20.4924			1.43125
$206$		0			0
$207$	−18.2462			−1.26820
$208$		0			0
$209$	6.24621			0.432059
$210$		0			0
$211$	−15.0540			−1.03636			−0.518179	−	0.855272i	$-0.673390\pi$
$211$	−15.0540			−1.03636			−0.518179	+	0.855272i	$0.673390\pi$
$212$		0			0
$213$			14.5616i			0.997741i
$214$		0			0
$215$		−	13.9309i		−	0.950077i
$216$		0			0
$217$	−14.2462			−0.967096
$218$		0			0
$219$			39.3693i			2.66033i
$220$		0			0
$221$	−2.00000	+	0.315342i	−0.134535	+	0.0212122i
$222$		0			0
$223$			18.1771i			1.21723i	0.793467	+	0.608614i	$0.208274\pi$
$223$			18.1771i			1.21723i	−0.793467	+	0.608614i	$0.791726\pi$
$224$		0			0
$225$	−5.56155			−0.370770
$226$		0			0
$227$			21.3693i			1.41833i	0.705042	+	0.709166i	$0.250928\pi$
$227$			21.3693i			1.41833i	−0.705042	+	0.709166i	$0.749072\pi$
$228$		0			0
$229$			8.31534i			0.549493i	0.961517	+	0.274747i	$0.0885940\pi$
$229$			8.31534i			0.549493i	−0.961517	+	0.274747i	$0.911406\pi$
$230$		0			0
$231$	36.4924			2.40103
$232$		0			0
$233$	0.561553			0.0367885			0.0183943	−	0.999831i	$-0.494145\pi$
$233$	0.561553			0.0367885			0.0183943	+	0.999831i	$0.494145\pi$
$234$		0			0
$235$	−8.80776			−0.574555
$236$		0			0
$237$	7.36932			0.478689
$238$		0			0
$239$		−	19.9309i		−	1.28922i	−0.764511	−	0.644610i	$-0.777020\pi$
$239$		−	19.9309i		−	1.28922i	0.764511	−	0.644610i	$-0.222980\pi$
$240$		0			0
$241$		−	18.8769i		−	1.21597i	−0.793949	−	0.607984i	$-0.791979\pi$
$241$		−	18.8769i		−	1.21597i	0.793949	−	0.607984i	$-0.208021\pi$
$242$		0			0
$243$	−22.2462			−1.42710
$244$		0			0
$245$		−	35.3693i		−	2.25966i
$246$		0			0
$247$	1.12311	+	7.12311i	0.0714615	+	0.453232i
$248$		0			0
$249$			21.1231i			1.33862i
$250$		0			0
$251$	8.49242			0.536037			0.268018	−	0.963414i	$-0.413631\pi$
$251$	8.49242			0.536037			0.268018	+	0.963414i	$0.413631\pi$
$252$		0			0
$253$		−	16.0000i		−	1.00591i
$254$		0			0
$255$			3.68466i			0.230742i
$256$		0			0
$257$	−27.9309			−1.74228			−0.871140	−	0.491035i	$-0.836619\pi$
$257$	−27.9309			−1.74228			−0.871140	+	0.491035i	$0.836619\pi$
$258$		0			0
$259$	24.8078			1.54148
$260$		0			0
$261$	11.1231			0.688503
$262$		0			0
$263$	−28.4924			−1.75692			−0.878459	−	0.477818i	$-0.841428\pi$
$263$	−28.4924			−1.75692			−0.878459	+	0.477818i	$0.841428\pi$
$264$		0			0
$265$		−	10.8769i		−	0.668162i
$266$		0			0
$267$			13.1231i			0.803121i
$268$		0			0
$269$	19.1231			1.16596			0.582978	−	0.812488i	$-0.301887\pi$
$269$	19.1231			1.16596			0.582978	+	0.812488i	$0.301887\pi$
$270$		0			0
$271$		−	2.31534i		−	0.140647i	−0.997524	−	0.0703235i	$-0.977597\pi$
$271$		−	2.31534i		−	0.140647i	0.997524	−	0.0703235i	$-0.0224032\pi$
$272$		0			0
$273$	6.56155	+	41.6155i	0.397123	+	2.51869i
$274$		0			0
$275$		−	4.87689i		−	0.294088i
$276$		0			0
$277$	−17.3693			−1.04362			−0.521811	−	0.853061i	$-0.674743\pi$
$277$	−17.3693			−1.04362			−0.521811	+	0.853061i	$0.674743\pi$
$278$		0			0
$279$		−	11.1231i		−	0.665923i
$280$		0			0
$281$		−	15.3693i		−	0.916857i	−0.888731	−	0.458428i	$-0.848413\pi$
$281$		−	15.3693i		−	0.916857i	0.888731	−	0.458428i	$-0.151587\pi$
$282$		0			0
$283$	−12.0000			−0.713326			−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000			−0.713326			−0.356663	+	0.934233i	$0.616086\pi$
$284$		0			0
$285$	13.1231			0.777346
$286$		0			0
$287$	−36.4924			−2.15408
$288$		0			0
$289$	−16.6847			−0.981450
$290$		0			0
$291$		−	26.2462i		−	1.53858i
$292$		0			0
$293$		−	0.315342i		−	0.0184225i	−0.999958	−	0.00921123i	$-0.997068\pi$
$293$		−	0.315342i		−	0.0184225i	0.999958	−	0.00921123i	$-0.00293207\pi$
$294$		0			0
$295$	25.6155			1.49139
$296$		0			0
$297$			4.49242i			0.260677i
$298$		0			0
$299$	18.2462	−	2.87689i	1.05521	−	0.166375i
$300$		0			0
$301$			24.8078i			1.42990i
$302$		0			0
$303$	34.2462			1.96739
$304$		0			0
$305$			28.4924i			1.63147i
$306$		0			0
$307$		−	18.0000i		−	1.02731i	−0.857996	−	0.513657i	$-0.828290\pi$
$307$		−	18.0000i		−	1.02731i	0.857996	−	0.513657i	$-0.171710\pi$
$308$		0			0
$309$	13.1231			0.746547
$310$		0			0
$311$	−10.8769			−0.616772			−0.308386	−	0.951261i	$-0.599789\pi$
$311$	−10.8769			−0.616772			−0.308386	+	0.951261i	$0.599789\pi$
$312$		0			0
$313$	8.56155			0.483928			0.241964	−	0.970285i	$-0.422208\pi$
$313$	8.56155			0.483928			0.241964	+	0.970285i	$0.422208\pi$
$314$		0			0
$315$	41.6155			2.34477
$316$		0			0
$317$			6.87689i			0.386245i	0.981175	+	0.193122i	$0.0618615\pi$
$317$			6.87689i			0.386245i	−0.981175	+	0.193122i	$0.938139\pi$
$318$		0			0
$319$			9.75379i			0.546107i
$320$		0			0
$321$	10.2462			0.571888
$322$		0			0
$323$		−	1.12311i		−	0.0624913i
$324$		0			0
$325$	5.56155	−	0.876894i	0.308499	−	0.0486413i
$326$		0			0
$327$		−	40.1771i		−	2.22180i
$328$		0			0
$329$	15.6847			0.864723
$330$		0			0
$331$			16.2462i			0.892973i	0.894790	+	0.446486i	$0.147325\pi$
$331$			16.2462i			0.892973i	−0.894790	+	0.446486i	$0.852675\pi$
$332$		0			0
$333$			19.3693i			1.06143i
$334$		0			0
$335$	−2.24621			−0.122724
$336$		0			0
$337$	13.6847			0.745451			0.372725	−	0.927942i	$-0.378423\pi$
$337$	13.6847			0.745451			0.372725	+	0.927942i	$0.378423\pi$
$338$		0			0
$339$	25.6155			1.39124
$340$		0			0
$341$	9.75379			0.528197
$342$		0			0
$343$			31.0540i			1.67676i
$344$		0			0
$345$		−	33.6155i		−	1.80980i
$346$		0			0
$347$	−2.56155			−0.137511			−0.0687557	−	0.997634i	$-0.521903\pi$
$347$	−2.56155			−0.137511			−0.0687557	+	0.997634i	$0.521903\pi$
$348$		0			0
$349$			17.9309i			0.959817i	0.877318	+	0.479909i	$0.159330\pi$
$349$			17.9309i			0.959817i	−0.877318	+	0.479909i	$0.840670\pi$
$350$		0			0
$351$	−5.12311	+	0.807764i	−0.273451	+	0.0431153i
$352$		0			0
$353$		−	33.6155i		−	1.78917i	−0.446894	−	0.894587i	$-0.647470\pi$
$353$		−	33.6155i		−	1.78917i	0.446894	−	0.894587i	$-0.352530\pi$
$354$		0			0
$355$	−14.5616			−0.772847
$356$		0			0
$357$		−	6.56155i		−	0.347274i
$358$		0			0
$359$			4.87689i			0.257393i	0.991684	+	0.128696i	$0.0410792\pi$
$359$			4.87689i			0.257393i	−0.991684	+	0.128696i	$0.958921\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	3.19224			0.167549
$364$		0			0
$365$	−39.3693			−2.06068
$366$		0			0
$367$	4.49242			0.234503			0.117251	−	0.993102i	$-0.462592\pi$
$367$	4.49242			0.234503			0.117251	+	0.993102i	$0.462592\pi$
$368$		0			0
$369$		−	28.4924i		−	1.48326i
$370$		0			0
$371$			19.3693i			1.00560i
$372$		0			0
$373$	−10.6307			−0.550436			−0.275218	−	0.961382i	$-0.588750\pi$
$373$	−10.6307			−0.550436			−0.275218	+	0.961382i	$0.588750\pi$
$374$		0			0
$375$			22.5616i			1.16507i
$376$		0			0
$377$	−11.1231	+	1.75379i	−0.572869	+	0.0903247i
$378$		0			0
$379$		−	26.0000i		−	1.33553i	−0.744372	−	0.667765i	$-0.767251\pi$
$379$		−	26.0000i		−	1.33553i	0.744372	−	0.667765i	$-0.232749\pi$
$380$		0			0
$381$	33.6155			1.72218
$382$		0			0
$383$			9.19224i			0.469701i	0.972031	+	0.234851i	$0.0754601\pi$
$383$			9.19224i			0.469701i	−0.972031	+	0.234851i	$0.924540\pi$
$384$		0			0
$385$			36.4924i			1.85983i
$386$		0			0
$387$	−19.3693			−0.984598
$388$		0			0
$389$	−28.2462			−1.43214			−0.716070	−	0.698029i	$-0.754061\pi$
$389$	−28.2462			−1.43214			−0.716070	+	0.698029i	$0.754061\pi$
$390$		0			0
$391$	−2.87689			−0.145491
$392$		0			0
$393$	40.1771			2.02667
$394$		0			0
$395$			7.36932i			0.370791i
$396$		0			0
$397$		−	9.12311i		−	0.457876i	−0.973441	−	0.228938i	$-0.926475\pi$
$397$		−	9.12311i		−	0.457876i	0.973441	−	0.228938i	$-0.0735252\pi$
$398$		0			0
$399$	−23.3693			−1.16993
$400$		0			0
$401$			7.36932i			0.368006i	0.982926	+	0.184003i	$0.0589056\pi$
$401$			7.36932i			0.368006i	−0.982926	+	0.184003i	$0.941094\pi$
$402$		0			0
$403$	1.75379	+	11.1231i	0.0873624	+	0.554081i
$404$		0			0
$405$		−	17.9309i		−	0.890992i
$406$		0			0
$407$	−16.9848			−0.841908
$408$		0			0
$409$			0.630683i			0.0311853i	0.999878	+	0.0155926i	$0.00496349\pi$
$409$			0.630683i			0.0311853i	−0.999878	+	0.0155926i	$0.995037\pi$
$410$		0			0
$411$			13.1231i			0.647315i
$412$		0			0
$413$	−45.6155			−2.24459
$414$		0			0
$415$	−21.1231			−1.03689
$416$		0			0
$417$	−32.8078			−1.60660
$418$		0			0
$419$	34.5616			1.68844			0.844221	−	0.535995i	$-0.180063\pi$
$419$	34.5616			1.68844			0.844221	+	0.535995i	$0.180063\pi$
$420$		0			0
$421$		−	3.19224i		−	0.155580i	−0.996970	−	0.0777900i	$-0.975214\pi$
$421$		−	3.19224i		−	0.155580i	0.996970	−	0.0777900i	$-0.0247864\pi$
$422$		0			0
$423$			12.2462i			0.595431i
$424$		0			0
$425$	−0.876894			−0.0425356
$426$		0			0
$427$		−	50.7386i		−	2.45541i
$428$		0			0
$429$	−4.49242	−	28.4924i	−0.216896	−	1.37563i
$430$		0			0
$431$		−	24.4233i		−	1.17643i	−0.808705	−	0.588214i	$-0.799831\pi$
$431$		−	24.4233i		−	1.17643i	0.808705	−	0.588214i	$-0.200169\pi$
$432$		0			0
$433$	21.6847			1.04210			0.521049	−	0.853527i	$-0.325541\pi$
$433$	21.6847			1.04210			0.521049	+	0.853527i	$0.325541\pi$
$434$		0			0
$435$			20.4924i			0.982536i
$436$		0			0
$437$			10.2462i			0.490143i
$438$		0			0
$439$	−19.8617			−0.947949			−0.473975	−	0.880539i	$-0.657181\pi$
$439$	−19.8617			−0.947949			−0.473975	+	0.880539i	$0.657181\pi$
$440$		0			0
$441$	−49.1771			−2.34177
$442$		0			0
$443$	−36.1771			−1.71882			−0.859412	−	0.511283i	$-0.829170\pi$
$443$	−36.1771			−1.71882			−0.859412	+	0.511283i	$0.829170\pi$
$444$		0			0
$445$	−13.1231			−0.622095
$446$		0			0
$447$		−	2.87689i		−	0.136072i
$448$		0			0
$449$			16.0000i			0.755087i	0.925992	+	0.377543i	$0.123231\pi$
$449$			16.0000i			0.755087i	−0.925992	+	0.377543i	$0.876769\pi$
$450$		0			0
$451$	24.9848			1.17649
$452$		0			0
$453$		−	19.0540i		−	0.895234i
$454$		0			0
$455$	−41.6155	+	6.56155i	−1.95097	+	0.307610i
$456$		0			0
$457$			25.6155i			1.19824i	0.800658	+	0.599122i	$0.204484\pi$
$457$			25.6155i			1.19824i	−0.800658	+	0.599122i	$0.795516\pi$
$458$		0			0
$459$	0.807764			0.0377032
$460$		0			0
$461$		−	36.1771i		−	1.68493i	−0.538747	−	0.842467i	$-0.681102\pi$
$461$		−	36.1771i		−	1.68493i	0.538747	−	0.842467i	$-0.318898\pi$
$462$		0			0
$463$		−	39.6155i		−	1.84109i	−0.390637	−	0.920545i	$-0.627745\pi$
$463$		−	39.6155i		−	1.84109i	0.390637	−	0.920545i	$-0.372255\pi$
$464$		0			0
$465$	20.4924			0.950313
$466$		0			0
$467$	16.4924			0.763178			0.381589	−	0.924332i	$-0.375377\pi$
$467$	16.4924			0.763178			0.381589	+	0.924332i	$0.375377\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$		0			0
$471$	−10.8769			−0.501181
$472$		0			0
$473$		−	16.9848i		−	0.780964i
$474$		0			0
$475$			3.12311i			0.143298i
$476$		0			0
$477$	−15.1231			−0.692439
$478$		0			0
$479$			9.19224i			0.420004i	0.977701	+	0.210002i	$0.0673470\pi$
$479$			9.19224i			0.420004i	−0.977701	+	0.210002i	$0.932653\pi$
$480$		0			0
$481$	−3.05398	−	19.3693i	−0.139249	−	0.883165i
$482$		0			0
$483$			59.8617i			2.72380i
$484$		0			0
$485$	26.2462			1.19178
$486$		0			0
$487$		−	41.8617i		−	1.89694i	−0.316872	−	0.948468i	$-0.602633\pi$
$487$		−	41.8617i		−	1.89694i	0.316872	−	0.948468i	$-0.397367\pi$
$488$		0			0
$489$			9.61553i			0.434829i
$490$		0			0
$491$	−11.1922			−0.505099			−0.252549	−	0.967584i	$-0.581269\pi$
$491$	−11.1922			−0.505099			−0.252549	+	0.967584i	$0.581269\pi$
$492$		0			0
$493$	1.75379			0.0789867
$494$		0			0
$495$	−28.4924			−1.28064
$496$		0			0
$497$	25.9309			1.16316
$498$		0			0
$499$			16.8769i			0.755514i	0.925905	+	0.377757i	$0.123304\pi$
$499$			16.8769i			0.755514i	−0.925905	+	0.377757i	$0.876696\pi$
$500$		0			0
$501$			38.7386i			1.73071i
$502$		0			0
$503$	15.3693			0.685284			0.342642	−	0.939466i	$-0.388678\pi$
$503$	15.3693			0.685284			0.342642	+	0.939466i	$0.388678\pi$
$504$		0			0
$505$			34.2462i			1.52394i
$506$		0			0
$507$	31.6847	−	10.2462i	1.40717	−	0.455050i
$508$		0			0
$509$			27.3693i			1.21312i	0.795036	+	0.606562i	$0.207452\pi$
$509$			27.3693i			1.21312i	−0.795036	+	0.606562i	$0.792548\pi$
$510$		0			0
$511$	70.1080			3.10139
$512$		0			0
$513$		−	2.87689i		−	0.127018i
$514$		0			0
$515$			13.1231i			0.578273i
$516$		0			0
$517$	−10.7386			−0.472285
$518$		0			0
$519$	−31.3693			−1.37696
$520$		0			0
$521$	−27.9309			−1.22367			−0.611837	−	0.790984i	$-0.709569\pi$
$521$	−27.9309			−1.22367			−0.611837	+	0.790984i	$0.709569\pi$
$522$		0			0
$523$	30.2462			1.32257			0.661287	−	0.750133i	$-0.270010\pi$
$523$	30.2462			1.32257			0.661287	+	0.750133i	$0.270010\pi$
$524$		0			0
$525$			18.2462i			0.796330i
$526$		0			0
$527$		−	1.75379i		−	0.0763962i
$528$		0			0
$529$	3.24621			0.141140
$530$		0			0
$531$		−	35.6155i		−	1.54558i
$532$		0			0
$533$	4.49242	+	28.4924i	0.194588	+	1.23414i
$534$		0			0
$535$			10.2462i			0.442982i
$536$		0			0
$537$	−34.4233			−1.48547
$538$		0			0
$539$		−	43.1231i		−	1.85744i
$540$		0			0
$541$		−	20.1771i		−	0.867480i	−0.901038	−	0.433740i	$-0.857194\pi$
$541$		−	20.1771i		−	0.867480i	0.901038	−	0.433740i	$-0.142806\pi$
$542$		0			0
$543$	21.1231			0.906479
$544$		0			0
$545$	40.1771			1.72100
$546$		0			0
$547$	−3.19224			−0.136490			−0.0682451	−	0.997669i	$-0.521740\pi$
$547$	−3.19224			−0.136490			−0.0682451	+	0.997669i	$0.521740\pi$
$548$		0			0
$549$	39.6155			1.69075
$550$		0			0
$551$		−	6.24621i		−	0.266098i
$552$		0			0
$553$		−	13.1231i		−	0.558051i
$554$		0			0
$555$	−35.6847			−1.51473
$556$		0			0
$557$		−	9.93087i		−	0.420784i	−0.977617	−	0.210392i	$-0.932526\pi$
$557$		−	9.93087i		−	0.420784i	0.977617	−	0.210392i	$-0.0674741\pi$
$558$		0			0
$559$	19.3693	−	3.05398i	0.819235	−	0.129169i
$560$		0			0
$561$			4.49242i			0.189670i
$562$		0			0
$563$	−1.93087			−0.0813765			−0.0406882	−	0.999172i	$-0.512955\pi$
$563$	−1.93087			−0.0813765			−0.0406882	+	0.999172i	$0.512955\pi$
$564$		0			0
$565$			25.6155i			1.07765i
$566$		0			0
$567$			31.9309i			1.34097i
$568$		0			0
$569$	−22.8078			−0.956151			−0.478076	−	0.878319i	$-0.658666\pi$
$569$	−22.8078			−0.956151			−0.478076	+	0.878319i	$0.658666\pi$
$570$		0			0
$571$	−23.0540			−0.964779			−0.482389	−	0.875957i	$-0.660231\pi$
$571$	−23.0540			−0.964779			−0.482389	+	0.875957i	$0.660231\pi$
$572$		0			0
$573$	14.7386			0.615715
$574$		0			0
$575$	8.00000			0.333623
$576$		0			0
$577$			27.8617i			1.15990i	0.814652	+	0.579950i	$0.196928\pi$
$577$			27.8617i			1.15990i	−0.814652	+	0.579950i	$0.803072\pi$
$578$		0			0
$579$		−	45.1231i		−	1.87525i
$580$		0			0
$581$	37.6155			1.56056
$582$		0			0
$583$		−	13.2614i		−	0.549230i
$584$		0			0
$585$	−5.12311	−	32.4924i	−0.211814	−	1.34340i
$586$		0			0
$587$			42.4924i			1.75385i	0.480627	+	0.876925i	$0.340409\pi$
$587$			42.4924i			1.75385i	−0.480627	+	0.876925i	$0.659591\pi$
$588$		0			0
$589$	−6.24621			−0.257371
$590$		0			0
$591$			13.9309i			0.573039i
$592$		0			0
$593$		−	42.2462i		−	1.73484i	−0.497573	−	0.867422i	$-0.665775\pi$
$593$		−	42.2462i		−	1.73484i	0.497573	−	0.867422i	$-0.334225\pi$
$594$		0			0
$595$	6.56155			0.268997
$596$		0			0
$597$	−5.75379			−0.235487
$598$		0			0
$599$	−16.6307			−0.679511			−0.339756	−	0.940514i	$-0.610344\pi$
$599$	−16.6307			−0.679511			−0.339756	+	0.940514i	$0.610344\pi$
$600$		0			0
$601$	1.19224			0.0486323			0.0243162	−	0.999704i	$-0.492259\pi$
$601$	1.19224			0.0486323			0.0243162	+	0.999704i	$0.492259\pi$
$602$		0			0
$603$			3.12311i			0.127183i
$604$		0			0
$605$			3.19224i			0.129783i
$606$		0			0
$607$	−33.6155			−1.36441			−0.682206	−	0.731160i	$-0.738979\pi$
$607$	−33.6155			−1.36441			−0.682206	+	0.731160i	$0.738979\pi$
$608$		0			0
$609$		−	36.4924i		−	1.47875i
$610$		0			0
$611$	−1.93087	−	12.2462i	−0.0781146	−	0.495429i
$612$		0			0
$613$		−	43.3693i		−	1.75167i	−0.482610	−	0.875835i	$-0.660311\pi$
$613$		−	43.3693i		−	1.75167i	0.482610	−	0.875835i	$-0.339689\pi$
$614$		0			0
$615$	52.4924			2.11670
$616$		0			0
$617$		−	18.2462i		−	0.734565i	−0.930109	−	0.367282i	$-0.880288\pi$
$617$		−	18.2462i		−	0.734565i	0.930109	−	0.367282i	$-0.119712\pi$
$618$		0			0
$619$			23.6155i			0.949188i	0.880205	+	0.474594i	$0.157405\pi$
$619$			23.6155i			0.949188i	−0.880205	+	0.474594i	$0.842595\pi$
$620$		0			0
$621$	−7.36932			−0.295720
$622$		0			0
$623$	23.3693			0.936272
$624$		0			0
$625$	−30.3693			−1.21477
$626$		0			0
$627$	16.0000			0.638978
$628$		0			0
$629$			3.05398i			0.121770i
$630$		0			0
$631$		−	7.43845i		−	0.296120i	−0.988978	−	0.148060i	$-0.952697\pi$
$631$		−	7.43845i		−	0.296120i	0.988978	−	0.148060i	$-0.0473029\pi$
$632$		0			0
$633$	−38.5616			−1.53268
$634$		0			0
$635$			33.6155i			1.33399i
$636$		0			0
$637$	49.1771	−	7.75379i	1.94847	−	0.307216i
$638$		0			0
$639$			20.2462i			0.800928i
$640$		0			0
$641$	20.2462			0.799677			0.399839	−	0.916586i	$-0.369066\pi$
$641$	20.2462			0.799677			0.399839	+	0.916586i	$0.369066\pi$
$642$		0			0
$643$			16.8769i			0.665560i	0.943005	+	0.332780i	$0.107987\pi$
$643$			16.8769i			0.665560i	−0.943005	+	0.332780i	$0.892013\pi$
$644$		0			0
$645$		−	35.6847i		−	1.40508i
$646$		0			0
$647$	5.12311			0.201410			0.100705	−	0.994916i	$-0.467890\pi$
$647$	5.12311			0.201410			0.100705	+	0.994916i	$0.467890\pi$
$648$		0			0
$649$	31.2311			1.22593
$650$		0			0
$651$	−36.4924			−1.43025
$652$		0			0
$653$	−27.6155			−1.08068			−0.540340	−	0.841447i	$-0.681704\pi$
$653$	−27.6155			−1.08068			−0.540340	+	0.841447i	$0.681704\pi$
$654$		0			0
$655$			40.1771i			1.56985i
$656$		0			0
$657$			54.7386i			2.13556i
$658$		0			0
$659$	42.7386			1.66486			0.832430	−	0.554130i	$-0.186949\pi$
$659$	42.7386			1.66486			0.832430	+	0.554130i	$0.186949\pi$
$660$		0			0
$661$			41.1231i			1.59950i	0.600331	+	0.799752i	$0.295036\pi$
$661$			41.1231i			1.59950i	−0.600331	+	0.799752i	$0.704964\pi$
$662$		0			0
$663$	−5.12311	+	0.807764i	−0.198965	+	0.0313710i
$664$		0			0
$665$		−	23.3693i		−	0.906223i
$666$		0			0
$667$	−16.0000			−0.619522
$668$		0			0
$669$			46.5616i			1.80017i
$670$		0			0
$671$			34.7386i			1.34107i
$672$		0			0
$673$	31.3002			1.20653			0.603267	−	0.797539i	$-0.293865\pi$
$673$	31.3002			1.20653			0.603267	+	0.797539i	$0.293865\pi$
$674$		0			0
$675$	−2.24621			−0.0864567
$676$		0			0
$677$	20.7386			0.797050			0.398525	−	0.917157i	$-0.369522\pi$
$677$	20.7386			0.797050			0.398525	+	0.917157i	$0.369522\pi$
$678$		0			0
$679$	−46.7386			−1.79366
$680$		0			0
$681$			54.7386i			2.09759i
$682$		0			0
$683$		−	4.24621i		−	0.162477i	−0.996695	−	0.0812384i	$-0.974112\pi$
$683$		−	4.24621i		−	0.162477i	0.996695	−	0.0812384i	$-0.0258875\pi$
$684$		0			0
$685$	−13.1231			−0.501408
$686$		0			0
$687$			21.3002i			0.812653i
$688$		0			0
$689$	15.1231	−	2.38447i	0.576144	−	0.0908411i
$690$		0			0
$691$		−	29.8617i		−	1.13599i	−0.823031	−	0.567997i	$-0.807718\pi$
$691$		−	29.8617i		−	1.13599i	0.823031	−	0.567997i	$-0.192282\pi$
$692$		0			0
$693$	50.7386			1.92740
$694$		0			0
$695$		−	32.8078i		−	1.24447i
$696$		0			0
$697$		−	4.49242i		−	0.170163i
$698$		0			0
$699$	1.43845			0.0544071
$700$		0			0
$701$	−45.2311			−1.70835			−0.854177	−	0.519983i	$-0.825938\pi$
$701$	−45.2311			−1.70835			−0.854177	+	0.519983i	$0.825938\pi$
$702$		0			0
$703$	10.8769			0.410230
$704$		0			0
$705$	−22.5616			−0.849717
$706$		0			0
$707$		−	60.9848i		−	2.29357i
$708$		0			0
$709$		−	1.12311i		−	0.0421791i	−0.999778	−	0.0210896i	$-0.993286\pi$
$709$		−	1.12311i		−	0.0421791i	0.999778	−	0.0210896i	$-0.00671351\pi$
$710$		0			0
$711$	10.2462			0.384263
$712$		0			0
$713$			16.0000i			0.599205i
$714$		0			0
$715$	28.4924	−	4.49242i	1.06556	−	0.168007i
$716$		0			0
$717$		−	51.0540i		−	1.90665i
$718$		0			0
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$		0			0
$721$		−	23.3693i		−	0.870319i
$722$		0			0
$723$		−	48.3542i		−	1.79831i
$724$		0			0
$725$	−4.87689			−0.181123
$726$		0			0
$727$	43.2311			1.60335			0.801676	−	0.597759i	$-0.203942\pi$
$727$	43.2311			1.60335			0.801676	+	0.597759i	$0.203942\pi$
$728$		0			0
$729$	−35.9848			−1.33277
$730$		0			0
$731$	−3.05398			−0.112955
$732$		0			0
$733$		−	49.3002i		−	1.82094i	−0.413571	−	0.910472i	$-0.635719\pi$
$733$		−	49.3002i		−	1.82094i	0.413571	−	0.910472i	$-0.364281\pi$
$734$		0			0
$735$		−	90.6004i		−	3.34185i
$736$		0			0
$737$	−2.73863			−0.100879
$738$		0			0
$739$			41.8617i			1.53991i	0.638099	+	0.769954i	$0.279721\pi$
$739$			41.8617i			1.53991i	−0.638099	+	0.769954i	$0.720279\pi$
$740$		0			0
$741$	2.87689	+	18.2462i	0.105685	+	0.670291i
$742$		0			0
$743$			45.0540i			1.65287i	0.563032	+	0.826435i	$0.309635\pi$
$743$			45.0540i			1.65287i	−0.563032	+	0.826435i	$0.690365\pi$
$744$		0			0
$745$	2.87689			0.105401
$746$		0			0
$747$			29.3693i			1.07457i
$748$		0			0
$749$		−	18.2462i		−	0.666702i
$750$		0			0
$751$	36.4924			1.33163			0.665814	−	0.746118i	$-0.268085\pi$
$751$	36.4924			1.33163			0.665814	+	0.746118i	$0.268085\pi$
$752$		0			0
$753$	21.7538			0.792752
$754$		0			0
$755$	19.0540			0.693445
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$		0			0
$759$		−	40.9848i		−	1.48766i
$760$		0			0
$761$			53.1231i			1.92571i	0.270017	+	0.962856i	$0.412971\pi$
$761$			53.1231i			1.92571i	−0.270017	+	0.962856i	$0.587029\pi$
$762$		0			0
$763$	−71.5464			−2.59015
$764$		0			0
$765$			5.12311i			0.185226i
$766$		0			0
$767$	5.61553	+	35.6155i	0.202765	+	1.28600i
$768$		0			0
$769$		−	4.49242i		−	0.162001i	−0.996714	−	0.0810004i	$-0.974188\pi$
$769$		−	4.49242i		−	0.162001i	0.996714	−	0.0810004i	$-0.0258115\pi$
$770$		0			0
$771$	−71.5464			−2.57668
$772$		0			0
$773$			23.0540i			0.829194i	0.910005	+	0.414597i	$0.136077\pi$
$773$			23.0540i			0.829194i	−0.910005	+	0.414597i	$0.863923\pi$
$774$		0			0
$775$			4.87689i			0.175183i
$776$		0			0
$777$	63.5464			2.27971
$778$		0			0
$779$	−16.0000			−0.573259
$780$		0			0
$781$	−17.7538			−0.635281
$782$		0			0
$783$	4.49242			0.160546
$784$		0			0
$785$		−	10.8769i		−	0.388213i
$786$		0			0
$787$			21.3693i			0.761734i	0.924630	+	0.380867i	$0.124374\pi$
$787$			21.3693i			0.761734i	−0.924630	+	0.380867i	$0.875626\pi$
$788$		0			0
$789$	−72.9848			−2.59833
$790$		0			0
$791$		−	45.6155i		−	1.62190i
$792$		0			0
$793$	−39.6155	+	6.24621i	−1.40679	+	0.221809i
$794$		0			0
$795$		−	27.8617i		−	0.988154i
$796$		0			0
$797$	−4.24621			−0.150409			−0.0752043	−	0.997168i	$-0.523961\pi$
$797$	−4.24621			−0.150409			−0.0752043	+	0.997168i	$0.523961\pi$
$798$		0			0
$799$			1.93087i			0.0683093i
$800$		0			0
$801$			18.2462i			0.644698i
$802$		0			0
$803$	−48.0000			−1.69388
$804$		0			0
$805$	−59.8617			−2.10985
$806$		0			0
$807$	48.9848			1.72435
$808$		0			0
$809$	30.3153			1.06583			0.532915	−	0.846169i	$-0.321096\pi$
$809$	30.3153			1.06583			0.532915	+	0.846169i	$0.321096\pi$
$810$		0			0
$811$		−	1.36932i		−	0.0480832i	−0.999711	−	0.0240416i	$-0.992347\pi$
$811$		−	1.36932i		−	0.0480832i	0.999711	−	0.0240416i	$-0.00765342\pi$
$812$		0			0
$813$		−	5.93087i		−	0.208005i
$814$		0			0
$815$	−9.61553			−0.336817
$816$		0			0
$817$			10.8769i			0.380534i
$818$		0			0
$819$	9.12311	+	57.8617i	0.318787	+	2.02185i
$820$		0			0
$821$			27.5464i			0.961376i	0.876892	+	0.480688i	$0.159613\pi$
$821$			27.5464i			0.961376i	−0.876892	+	0.480688i	$0.840387\pi$
$822$		0			0
$823$	−40.3542			−1.40666			−0.703329	−	0.710865i	$-0.748304\pi$
$823$	−40.3542			−1.40666			−0.703329	+	0.710865i	$0.748304\pi$
$824$		0			0
$825$		−	12.4924i		−	0.434930i
$826$		0			0
$827$			22.0000i			0.765015i	0.923952	+	0.382507i	$0.124939\pi$
$827$			22.0000i			0.765015i	−0.923952	+	0.382507i	$0.875061\pi$
$828$		0			0
$829$	−15.1231			−0.525247			−0.262624	−	0.964898i	$-0.584588\pi$
$829$	−15.1231			−0.525247			−0.262624	+	0.964898i	$0.584588\pi$
$830$		0			0
$831$	−44.4924			−1.54343
$832$		0			0
$833$	−7.75379			−0.268653
$834$		0			0
$835$	−38.7386			−1.34061
$836$		0			0
$837$		−	4.49242i		−	0.155281i
$838$		0			0
$839$			13.8617i			0.478560i	0.970951	+	0.239280i	$0.0769114\pi$
$839$			13.8617i			0.478560i	−0.970951	+	0.239280i	$0.923089\pi$
$840$		0			0
$841$	−19.2462			−0.663662
$842$		0			0
$843$		−	39.3693i		−	1.35595i
$844$		0			0
$845$	10.2462	+	31.6847i	0.352480	+	1.08999i
$846$		0			0
$847$		−	5.68466i		−	0.195327i
$848$		0			0
$849$	−30.7386			−1.05495
$850$		0			0
$851$		−	27.8617i		−	0.955088i
$852$		0			0
$853$		−	7.68466i		−	0.263118i	−0.991308	−	0.131559i	$-0.958002\pi$
$853$		−	7.68466i		−	0.263118i	0.991308	−	0.131559i	$-0.0419982\pi$
$854$		0			0
$855$	18.2462			0.624007
$856$		0			0
$857$	−17.5076			−0.598047			−0.299024	−	0.954246i	$-0.596661\pi$
$857$	−17.5076			−0.598047			−0.299024	+	0.954246i	$0.596661\pi$
$858$		0			0
$859$	−6.24621			−0.213118			−0.106559	−	0.994306i	$-0.533983\pi$
$859$	−6.24621			−0.213118			−0.106559	+	0.994306i	$0.533983\pi$
$860$		0			0
$861$	−93.4773			−3.18570
$862$		0			0
$863$		−	10.9460i		−	0.372607i	−0.982492	−	0.186304i	$-0.940349\pi$
$863$		−	10.9460i		−	0.372607i	0.982492	−	0.186304i	$-0.0596508\pi$
$864$		0			0
$865$		−	31.3693i		−	1.06659i
$866$		0			0
$867$	−42.7386			−1.45148
$868$		0			0
$869$			8.98485i			0.304790i
$870$		0			0
$871$	−0.492423	−	3.12311i	−0.0166851	−	0.105822i
$872$		0			0
$873$		−	36.4924i		−	1.23508i
$874$		0			0
$875$	40.1771			1.35823
$876$		0			0
$877$			54.4233i			1.83774i	0.394556	+	0.918872i	$0.370898\pi$
$877$			54.4233i			1.83774i	−0.394556	+	0.918872i	$0.629102\pi$
$878$		0			0
$879$		−	0.807764i		−	0.0272452i
$880$		0			0
$881$	43.7926			1.47541			0.737705	−	0.675123i	$-0.235909\pi$
$881$	43.7926			1.47541			0.737705	+	0.675123i	$0.235909\pi$
$882$		0			0
$883$	11.1922			0.376649			0.188324	−	0.982107i	$-0.439694\pi$
$883$	11.1922			0.376649			0.188324	+	0.982107i	$0.439694\pi$
$884$		0			0
$885$	65.6155			2.20564
$886$		0			0
$887$	54.7386			1.83794			0.918972	−	0.394323i	$-0.129021\pi$
$887$	54.7386			1.83794			0.918972	+	0.394323i	$0.129021\pi$
$888$		0			0
$889$		−	59.8617i		−	2.00770i
$890$		0			0
$891$		−	21.8617i		−	0.732396i
$892$		0			0
$893$	6.87689			0.230126
$894$		0			0
$895$		−	34.4233i		−	1.15064i
$896$		0			0
$897$	46.7386	−	7.36932i	1.56056	−	0.246054i
$898$		0			0
$899$		−	9.75379i		−	0.325307i
$900$		0			0
$901$	−2.38447			−0.0794383
$902$		0			0
$903$			63.5464i			2.11469i
$904$		0			0
$905$			21.1231i			0.702156i
$906$		0			0
$907$	26.5616			0.881962			0.440981	−	0.897516i	$-0.354631\pi$
$907$	26.5616			0.881962			0.440981	+	0.897516i	$0.354631\pi$
$908$		0			0
$909$	47.6155			1.57931
$910$		0			0
$911$	−42.2462			−1.39968			−0.699840	−	0.714300i	$-0.746745\pi$
$911$	−42.2462			−1.39968			−0.699840	+	0.714300i	$0.746745\pi$
$912$		0			0
$913$	−25.7538			−0.852326
$914$		0			0
$915$			72.9848i			2.41280i
$916$		0			0
$917$		−	71.5464i		−	2.36267i
$918$		0			0
$919$	−18.2462			−0.601887			−0.300943	−	0.953642i	$-0.597302\pi$
$919$	−18.2462			−0.601887			−0.300943	+	0.953642i	$0.597302\pi$
$920$		0			0
$921$		−	46.1080i		−	1.51931i
$922$		0			0
$923$	−3.19224	−	20.2462i	−0.105074	−	0.666412i
$924$		0			0
$925$		−	8.49242i		−	0.279229i
$926$		0			0
$927$	18.2462			0.599284
$928$		0			0
$929$			30.7386i			1.00850i	0.863557	+	0.504251i	$0.168231\pi$
$929$			30.7386i			1.00850i	−0.863557	+	0.504251i	$0.831769\pi$
$930$		0			0
$931$			27.6155i			0.905062i
$932$		0			0
$933$	−27.8617			−0.912152
$934$		0			0
$935$	−4.49242			−0.146918
$936$		0			0
$937$	36.2462			1.18411			0.592056	−	0.805897i	$-0.298316\pi$
$937$	36.2462			1.18411			0.592056	+	0.805897i	$0.298316\pi$
$938$		0			0
$939$	21.9309			0.715687
$940$		0			0
$941$		−	27.5464i		−	0.897987i	−0.893535	−	0.448993i	$-0.851783\pi$
$941$		−	27.5464i		−	0.897987i	0.893535	−	0.448993i	$-0.148217\pi$
$942$		0			0
$943$			40.9848i			1.33465i
$944$		0			0
$945$	16.8078			0.546757
$946$		0			0
$947$		−	9.36932i		−	0.304462i	−0.988345	−	0.152231i	$-0.951354\pi$
$947$		−	9.36932i		−	0.304462i	0.988345	−	0.152231i	$-0.0486458\pi$
$948$		0			0
$949$	−8.63068	−	54.7386i	−0.280164	−	1.77689i
$950$		0			0
$951$			17.6155i			0.571223i
$952$		0			0
$953$	−17.0540			−0.552432			−0.276216	−	0.961096i	$-0.589081\pi$
$953$	−17.0540			−0.552432			−0.276216	+	0.961096i	$0.589081\pi$
$954$		0			0
$955$			14.7386i			0.476931i
$956$		0			0
$957$			24.9848i			0.807645i
$958$		0			0
$959$	23.3693			0.754635
$960$		0			0
$961$	21.2462			0.685362
$962$		0			0
$963$	14.2462			0.459078
$964$		0			0
$965$	45.1231			1.45256
$966$		0			0
$967$			11.4384i			0.367836i	0.982942	+	0.183918i	$0.0588780\pi$
$967$			11.4384i			0.367836i	−0.982942	+	0.183918i	$0.941122\pi$
$968$		0			0
$969$		−	2.87689i		−	0.0924192i
$970$		0			0
$971$	11.8229			0.379416			0.189708	−	0.981841i	$-0.439246\pi$
$971$	11.8229			0.379416			0.189708	+	0.981841i	$0.439246\pi$
$972$		0			0
$973$			58.4233i			1.87296i
$974$		0			0
$975$	14.2462	−	2.24621i	0.456244	−	0.0719363i
$976$		0			0
$977$			35.2311i			1.12714i	0.826068	+	0.563571i	$0.190573\pi$
$977$			35.2311i			1.12714i	−0.826068	+	0.563571i	$0.809427\pi$
$978$		0			0
$979$	−16.0000			−0.511362
$980$		0			0
$981$		−	55.8617i		−	1.78353i
$982$		0			0
$983$		−	30.8078i		−	0.982615i	−0.870986	−	0.491308i	$-0.836519\pi$
$983$		−	30.8078i		−	0.982615i	0.870986	−	0.491308i	$-0.163481\pi$
$984$		0			0
$985$	−13.9309			−0.443874
$986$		0			0
$987$	40.1771			1.27885
$988$		0			0
$989$	27.8617			0.885952
$990$		0			0
$991$	−33.6155			−1.06783			−0.533916	−	0.845537i	$-0.679280\pi$
$991$	−33.6155			−1.06783			−0.533916	+	0.845537i	$0.679280\pi$
$992$		0			0
$993$			41.6155i			1.32063i
$994$		0			0
$995$		−	5.75379i		−	0.182407i
$996$		0			0
$997$	39.6155			1.25464			0.627318	−	0.778763i	$-0.284153\pi$
$997$	39.6155			1.25464			0.627318	+	0.778763i	$0.284153\pi$
$998$		0			0
$999$			7.82292i			0.247506i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.2.f.b.129.4		4
3.2	odd	2		1872.2.c.j.1585.1		4
4.3	odd	2		104.2.f.a.25.2	yes	4
8.3	odd	2		832.2.f.i.129.3		4
8.5	even	2		832.2.f.f.129.1		4
12.11	even	2		936.2.c.d.649.1		4
13.5	odd	4		2704.2.a.t.1.2		2
13.8	odd	4		2704.2.a.s.1.2		2
13.12	even	2	inner	208.2.f.b.129.3		4
20.3	even	4		2600.2.f.b.649.1		4
20.7	even	4		2600.2.f.a.649.4		4
20.19	odd	2		2600.2.k.a.2001.3		4
39.38	odd	2		1872.2.c.j.1585.4		4
52.3	odd	6		1352.2.o.e.1161.4		8
52.7	even	12		1352.2.i.g.1329.2		4
52.11	even	12		1352.2.i.g.529.2		4
52.15	even	12		1352.2.i.h.529.2		4
52.19	even	12		1352.2.i.h.1329.2		4
52.23	odd	6		1352.2.o.e.1161.3		8
52.31	even	4		1352.2.a.e.1.1		2
52.35	odd	6		1352.2.o.e.361.4		8
52.43	odd	6		1352.2.o.e.361.3		8
52.47	even	4		1352.2.a.d.1.1		2
52.51	odd	2		104.2.f.a.25.1	✓	4
104.51	odd	2		832.2.f.i.129.4		4
104.77	even	2		832.2.f.f.129.2		4
156.155	even	2		936.2.c.d.649.4		4
260.103	even	4		2600.2.f.a.649.1		4
260.207	even	4		2600.2.f.b.649.4		4
260.259	odd	2		2600.2.k.a.2001.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.f.a.25.1	✓	4	52.51	odd	2
104.2.f.a.25.2	yes	4	4.3	odd	2
208.2.f.b.129.3		4	13.12	even	2	inner
208.2.f.b.129.4		4	1.1	even	1	trivial
832.2.f.f.129.1		4	8.5	even	2
832.2.f.f.129.2		4	104.77	even	2
832.2.f.i.129.3		4	8.3	odd	2
832.2.f.i.129.4		4	104.51	odd	2
936.2.c.d.649.1		4	12.11	even	2
936.2.c.d.649.4		4	156.155	even	2
1352.2.a.d.1.1		2	52.47	even	4
1352.2.a.e.1.1		2	52.31	even	4
1352.2.i.g.529.2		4	52.11	even	12
1352.2.i.g.1329.2		4	52.7	even	12
1352.2.i.h.529.2		4	52.15	even	12
1352.2.i.h.1329.2		4	52.19	even	12
1352.2.o.e.361.3		8	52.43	odd	6
1352.2.o.e.361.4		8	52.35	odd	6
1352.2.o.e.1161.3		8	52.23	odd	6
1352.2.o.e.1161.4		8	52.3	odd	6
1872.2.c.j.1585.1		4	3.2	odd	2
1872.2.c.j.1585.4		4	39.38	odd	2
2600.2.f.a.649.1		4	260.103	even	4
2600.2.f.a.649.4		4	20.7	even	4
2600.2.f.b.649.1		4	20.3	even	4
2600.2.f.b.649.4		4	260.207	even	4
2600.2.k.a.2001.3		4	20.19	odd	2
2600.2.k.a.2001.4		4	260.259	odd	2
2704.2.a.s.1.2		2	13.8	odd	4
2704.2.a.t.1.2		2	13.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} +2.56155i q^{5} -4.56155i q^{7} +3.56155 q^{9} +3.12311i q^{11} +(-3.56155 + 0.561553i) q^{13} +6.56155i q^{15} +0.561553 q^{17} -2.00000i q^{19} -11.6847i q^{21} -5.12311 q^{23} -1.56155 q^{25} +1.43845 q^{27} +3.12311 q^{29} -3.12311i q^{31} +8.00000i q^{33} +11.6847 q^{35} +5.43845i q^{37} +(-9.12311 + 1.43845i) q^{39} -8.00000i q^{41} -5.43845 q^{43} +9.12311i q^{45} +3.43845i q^{47} -13.8078 q^{49} +1.43845 q^{51} -4.24621 q^{53} -8.00000 q^{55} -5.12311i q^{57} -10.0000i q^{59} +11.1231 q^{61} -16.2462i q^{63} +(-1.43845 - 9.12311i) q^{65} +0.876894i q^{67} -13.1231 q^{69} +5.68466i q^{71} +15.3693i q^{73} -4.00000 q^{75} +14.2462 q^{77} +2.87689 q^{79} -7.00000 q^{81} +8.24621i q^{83} +1.43845i q^{85} +8.00000 q^{87} +5.12311i q^{89} +(2.56155 + 16.2462i) q^{91} -8.00000i q^{93} +5.12311 q^{95} -10.2462i q^{97} +11.1231i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 6 q^{9} - 6 q^{13} - 6 q^{17} - 4 q^{23} + 2 q^{25} + 14 q^{27} - 4 q^{29} + 22 q^{35} - 20 q^{39} - 30 q^{43} - 14 q^{49} + 14 q^{51} + 16 q^{53} - 32 q^{55} + 28 q^{61} - 14 q^{65} - 36 q^{69}+ \cdots + 4 q^{95}+O(q^{100}) \) 4 * q + 2 * q^3 + 6 * q^9 - 6 * q^13 - 6 * q^17 - 4 * q^23 + 2 * q^25 + 14 * q^27 - 4 * q^29 + 22 * q^35 - 20 * q^39 - 30 * q^43 - 14 * q^49 + 14 * q^51 + 16 * q^53 - 32 * q^55 + 28 * q^61 - 14 * q^65 - 36 * q^69 - 16 * q^75 + 24 * q^77 + 28 * q^79 - 28 * q^81 + 32 * q^87 + 2 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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