LMFDB - Embedded newform 8624.2.a.cd.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8624,2,Mod(1,8624)] 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("8624.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,0,0,0,0,-2,0,2,0,4,0,4,0,-8,0,0,0,-8,0,-6,0,2,0, 2,0,-8,0,-2,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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

Self dual:	yes
Analytic conductor:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{3} +1.41421 q^{5} +2.82843 q^{9} -1.00000 q^{11} +3.82843 q^{13} +3.41421 q^{15} +2.00000 q^{17} -5.41421 q^{19} -2.58579 q^{23} -3.00000 q^{25} -0.414214 q^{27} +1.00000 q^{29} +1.65685 q^{31} -2.41421 q^{33} +5.07107 q^{37} +9.24264 q^{39} +2.24264 q^{41} +8.00000 q^{43} +4.00000 q^{45} -0.828427 q^{47} +4.82843 q^{51} +5.41421 q^{53} -1.41421 q^{55} -13.0711 q^{57} +1.58579 q^{59} +13.8284 q^{61} +5.41421 q^{65} +6.07107 q^{67} -6.24264 q^{69} +14.2426 q^{71} +0.585786 q^{73} -7.24264 q^{75} +6.07107 q^{79} -9.48528 q^{81} +6.48528 q^{83} +2.82843 q^{85} +2.41421 q^{87} +14.8284 q^{89} +4.00000 q^{93} -7.65685 q^{95} +10.1716 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{11} + 2 q^{13} + 4 q^{15} + 4 q^{17} - 8 q^{19} - 8 q^{23} - 6 q^{25} + 2 q^{27} + 2 q^{29} - 8 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{39} - 4 q^{41} + 16 q^{43} + 8 q^{45} + 4 q^{47}+ \cdots + 26 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^11 + 2 * q^13 + 4 * q^15 + 4 * q^17 - 8 * q^19 - 8 * q^23 - 6 * q^25 + 2 * q^27 + 2 * q^29 - 8 * q^31 - 2 * q^33 - 4 * q^37 + 10 * q^39 - 4 * q^41 + 16 * q^43 + 8 * q^45 + 4 * q^47 + 4 * q^51 + 8 * q^53 - 12 * q^57 + 6 * q^59 + 22 * q^61 + 8 * q^65 - 2 * q^67 - 4 * q^69 + 20 * q^71 + 4 * q^73 - 6 * q^75 - 2 * q^79 - 2 * q^81 - 4 * q^83 + 2 * q^87 + 24 * q^89 + 8 * q^93 - 4 * q^95 + 26 * q^97

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
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.82843	1.06181	0.530907	−	0.847430i	$-0.321851\pi$
$13$	3.82843	1.06181	0.530907	+	0.847430i	$0.321851\pi$
$14$	0	0
$15$	3.41421	0.881546
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−5.41421	−1.24211	−0.621053	−	0.783769i	$-0.713295\pi$
$19$	−5.41421	−1.24211	−0.621053	+	0.783769i	$0.713295\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.58579	−0.539174	−0.269587	−	0.962976i	$-0.586887\pi$
$23$	−2.58579	−0.539174	−0.269587	+	0.962976i	$0.586887\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	−0.414214	−0.0797154
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	1.65685	0.297580	0.148790	−	0.988869i	$-0.452462\pi$
$31$	1.65685	0.297580	0.148790	+	0.988869i	$0.452462\pi$
$32$	0	0
$33$	−2.41421	−0.420261
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.07107	0.833678	0.416839	−	0.908980i	$-0.363138\pi$
$37$	5.07107	0.833678	0.416839	+	0.908980i	$0.363138\pi$
$38$	0	0
$39$	9.24264	1.48001
$40$	0	0
$41$	2.24264	0.350242	0.175121	−	0.984547i	$-0.443968\pi$
$41$	2.24264	0.350242	0.175121	+	0.984547i	$0.443968\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	−0.828427	−0.120839	−0.0604193	−	0.998173i	$-0.519244\pi$
$47$	−0.828427	−0.120839	−0.0604193	+	0.998173i	$0.519244\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.82843	0.676115
$52$	0	0
$53$	5.41421	0.743699	0.371850	−	0.928293i	$-0.378724\pi$
$53$	5.41421	0.743699	0.371850	+	0.928293i	$0.378724\pi$
$54$	0	0
$55$	−1.41421	−0.190693
$56$	0	0
$57$	−13.0711	−1.73131
$58$	0	0
$59$	1.58579	0.206452	0.103226	−	0.994658i	$-0.467084\pi$
$59$	1.58579	0.206452	0.103226	+	0.994658i	$0.467084\pi$
$60$	0	0
$61$	13.8284	1.77055	0.885274	−	0.465069i	$-0.153971\pi$
$61$	13.8284	1.77055	0.885274	+	0.465069i	$0.153971\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.41421	0.671551
$66$	0	0
$67$	6.07107	0.741699	0.370849	−	0.928693i	$-0.379067\pi$
$67$	6.07107	0.741699	0.370849	+	0.928693i	$0.379067\pi$
$68$	0	0
$69$	−6.24264	−0.751526
$70$	0	0
$71$	14.2426	1.69029	0.845145	−	0.534537i	$-0.179514\pi$
$71$	14.2426	1.69029	0.845145	+	0.534537i	$0.179514\pi$
$72$	0	0
$73$	0.585786	0.0685611	0.0342806	−	0.999412i	$-0.489086\pi$
$73$	0.585786	0.0685611	0.0342806	+	0.999412i	$0.489086\pi$
$74$	0	0
$75$	−7.24264	−0.836308
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.07107	0.683048	0.341524	−	0.939873i	$-0.389057\pi$
$79$	6.07107	0.683048	0.341524	+	0.939873i	$0.389057\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	6.48528	0.711852	0.355926	−	0.934514i	$-0.384165\pi$
$83$	6.48528	0.711852	0.355926	+	0.934514i	$0.384165\pi$
$84$	0	0
$85$	2.82843	0.306786
$86$	0	0
$87$	2.41421	0.258831
$88$	0	0
$89$	14.8284	1.57181	0.785905	−	0.618347i	$-0.212197\pi$
$89$	14.8284	1.57181	0.785905	+	0.618347i	$0.212197\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	−7.65685	−0.785577
$96$	0	0
$97$	10.1716	1.03277	0.516383	−	0.856358i	$-0.327278\pi$
$97$	10.1716	1.03277	0.516383	+	0.856358i	$0.327278\pi$
$98$	0	0
$99$	−2.82843	−0.284268
$100$	0	0
$101$	−9.48528	−0.943821	−0.471910	−	0.881647i	$-0.656435\pi$
$101$	−9.48528	−0.943821	−0.471910	+	0.881647i	$0.656435\pi$
$102$	0	0
$103$	3.89949	0.384229	0.192114	−	0.981373i	$-0.438466\pi$
$103$	3.89949	0.384229	0.192114	+	0.981373i	$0.438466\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.5563	1.69724	0.848618	−	0.529006i	$-0.177435\pi$
$107$	17.5563	1.69724	0.848618	+	0.529006i	$0.177435\pi$
$108$	0	0
$109$	8.48528	0.812743	0.406371	−	0.913708i	$-0.366794\pi$
$109$	8.48528	0.812743	0.406371	+	0.913708i	$0.366794\pi$
$110$	0	0
$111$	12.2426	1.16202
$112$	0	0
$113$	9.48528	0.892300	0.446150	−	0.894958i	$-0.352795\pi$
$113$	9.48528	0.892300	0.446150	+	0.894958i	$0.352795\pi$
$114$	0	0
$115$	−3.65685	−0.341003
$116$	0	0
$117$	10.8284	1.00109
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.41421	0.488183
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−3.58579	−0.318187	−0.159094	−	0.987264i	$-0.550857\pi$
$127$	−3.58579	−0.318187	−0.159094	+	0.987264i	$0.550857\pi$
$128$	0	0
$129$	19.3137	1.70048
$130$	0	0
$131$	−19.0711	−1.66625	−0.833123	−	0.553087i	$-0.813450\pi$
$131$	−19.0711	−1.66625	−0.833123	+	0.553087i	$0.813450\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.585786	−0.0504165
$136$	0	0
$137$	17.0000	1.45241	0.726204	−	0.687479i	$-0.241283\pi$
$137$	17.0000	1.45241	0.726204	+	0.687479i	$0.241283\pi$
$138$	0	0
$139$	19.3137	1.63817	0.819084	−	0.573674i	$-0.194482\pi$
$139$	19.3137	1.63817	0.819084	+	0.573674i	$0.194482\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	−3.82843	−0.320149
$144$	0	0
$145$	1.41421	0.117444
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	−8.41421	−0.684739	−0.342369	−	0.939565i	$-0.611229\pi$
$151$	−8.41421	−0.684739	−0.342369	+	0.939565i	$0.611229\pi$
$152$	0	0
$153$	5.65685	0.457330
$154$	0	0
$155$	2.34315	0.188206
$156$	0	0
$157$	−15.3137	−1.22217	−0.611083	−	0.791566i	$-0.709266\pi$
$157$	−15.3137	−1.22217	−0.611083	+	0.791566i	$0.709266\pi$
$158$	0	0
$159$	13.0711	1.03660
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−24.8995	−1.95028	−0.975139	−	0.221592i	$-0.928875\pi$
$163$	−24.8995	−1.95028	−0.975139	+	0.221592i	$0.928875\pi$
$164$	0	0
$165$	−3.41421	−0.265796
$166$	0	0
$167$	−25.7279	−1.99089	−0.995443	−	0.0953565i	$-0.969601\pi$
$167$	−25.7279	−1.99089	−0.995443	+	0.0953565i	$0.969601\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	0	0
$171$	−15.3137	−1.17107
$172$	0	0
$173$	−7.82843	−0.595184	−0.297592	−	0.954693i	$-0.596184\pi$
$173$	−7.82843	−0.595184	−0.297592	+	0.954693i	$0.596184\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.82843	0.287762
$178$	0	0
$179$	0.899495	0.0672314	0.0336157	−	0.999435i	$-0.489298\pi$
$179$	0.899495	0.0672314	0.0336157	+	0.999435i	$0.489298\pi$
$180$	0	0
$181$	−9.31371	−0.692283	−0.346141	−	0.938182i	$-0.612508\pi$
$181$	−9.31371	−0.692283	−0.346141	+	0.938182i	$0.612508\pi$
$182$	0	0
$183$	33.3848	2.46787
$184$	0	0
$185$	7.17157	0.527265
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.4853	−0.758688	−0.379344	−	0.925256i	$-0.623850\pi$
$191$	−10.4853	−0.758688	−0.379344	+	0.925256i	$0.623850\pi$
$192$	0	0
$193$	−19.2132	−1.38300	−0.691498	−	0.722378i	$-0.743049\pi$
$193$	−19.2132	−1.38300	−0.691498	+	0.722378i	$0.743049\pi$
$194$	0	0
$195$	13.0711	0.936039
$196$	0	0
$197$	17.1421	1.22133	0.610663	−	0.791890i	$-0.290903\pi$
$197$	17.1421	1.22133	0.610663	+	0.791890i	$0.290903\pi$
$198$	0	0
$199$	−24.0416	−1.70427	−0.852133	−	0.523325i	$-0.824691\pi$
$199$	−24.0416	−1.70427	−0.852133	+	0.523325i	$0.824691\pi$
$200$	0	0
$201$	14.6569	1.03381
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.17157	0.221512
$206$	0	0
$207$	−7.31371	−0.508338
$208$	0	0
$209$	5.41421	0.374509
$210$	0	0
$211$	19.0711	1.31291	0.656453	−	0.754367i	$-0.272056\pi$
$211$	19.0711	1.31291	0.656453	+	0.754367i	$0.272056\pi$
$212$	0	0
$213$	34.3848	2.35601
$214$	0	0
$215$	11.3137	0.771589
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.41421	0.0955637
$220$	0	0
$221$	7.65685	0.515056
$222$	0	0
$223$	−11.5563	−0.773870	−0.386935	−	0.922107i	$-0.626466\pi$
$223$	−11.5563	−0.773870	−0.386935	+	0.922107i	$0.626466\pi$
$224$	0	0
$225$	−8.48528	−0.565685
$226$	0	0
$227$	0.142136	0.00943387	0.00471694	−	0.999989i	$-0.498499\pi$
$227$	0.142136	0.00943387	0.00471694	+	0.999989i	$0.498499\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.3848	−1.07340	−0.536701	−	0.843772i	$-0.680330\pi$
$233$	−16.3848	−1.07340	−0.536701	+	0.843772i	$0.680330\pi$
$234$	0	0
$235$	−1.17157	−0.0764250
$236$	0	0
$237$	14.6569	0.952065
$238$	0	0
$239$	−13.7279	−0.887985	−0.443993	−	0.896030i	$-0.646438\pi$
$239$	−13.7279	−0.887985	−0.443993	+	0.896030i	$0.646438\pi$
$240$	0	0
$241$	−2.72792	−0.175721	−0.0878605	−	0.996133i	$-0.528003\pi$
$241$	−2.72792	−0.175721	−0.0878605	+	0.996133i	$0.528003\pi$
$242$	0	0
$243$	−21.6569	−1.38929
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−20.7279	−1.31889
$248$	0	0
$249$	15.6569	0.992213
$250$	0	0
$251$	−16.4853	−1.04054	−0.520271	−	0.854001i	$-0.674169\pi$
$251$	−16.4853	−1.04054	−0.520271	+	0.854001i	$0.674169\pi$
$252$	0	0
$253$	2.58579	0.162567
$254$	0	0
$255$	6.82843	0.427613
$256$	0	0
$257$	2.51472	0.156864	0.0784319	−	0.996919i	$-0.475009\pi$
$257$	2.51472	0.156864	0.0784319	+	0.996919i	$0.475009\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.82843	0.175075
$262$	0	0
$263$	−25.0416	−1.54413	−0.772067	−	0.635542i	$-0.780777\pi$
$263$	−25.0416	−1.54413	−0.772067	+	0.635542i	$0.780777\pi$
$264$	0	0
$265$	7.65685	0.470357
$266$	0	0
$267$	35.7990	2.19086
$268$	0	0
$269$	2.34315	0.142864	0.0714321	−	0.997445i	$-0.477243\pi$
$269$	2.34315	0.142864	0.0714321	+	0.997445i	$0.477243\pi$
$270$	0	0
$271$	14.0711	0.854756	0.427378	−	0.904073i	$-0.359437\pi$
$271$	14.0711	0.854756	0.427378	+	0.904073i	$0.359437\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.00000	0.180907
$276$	0	0
$277$	14.7990	0.889185	0.444593	−	0.895733i	$-0.353348\pi$
$277$	14.7990	0.889185	0.444593	+	0.895733i	$0.353348\pi$
$278$	0	0
$279$	4.68629	0.280561
$280$	0	0
$281$	7.89949	0.471244	0.235622	−	0.971845i	$-0.424287\pi$
$281$	7.89949	0.471244	0.235622	+	0.971845i	$0.424287\pi$
$282$	0	0
$283$	−3.55635	−0.211403	−0.105702	−	0.994398i	$-0.533709\pi$
$283$	−3.55635	−0.211403	−0.105702	+	0.994398i	$0.533709\pi$
$284$	0	0
$285$	−18.4853	−1.09497
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	24.5563	1.43952
$292$	0	0
$293$	18.8284	1.09997	0.549984	−	0.835175i	$-0.314634\pi$
$293$	18.8284	1.09997	0.549984	+	0.835175i	$0.314634\pi$
$294$	0	0
$295$	2.24264	0.130572
$296$	0	0
$297$	0.414214	0.0240351
$298$	0	0
$299$	−9.89949	−0.572503
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−22.8995	−1.31554
$304$	0	0
$305$	19.5563	1.11979
$306$	0	0
$307$	11.4142	0.651444	0.325722	−	0.945466i	$-0.394393\pi$
$307$	11.4142	0.651444	0.325722	+	0.945466i	$0.394393\pi$
$308$	0	0
$309$	9.41421	0.535556
$310$	0	0
$311$	15.2132	0.862662	0.431331	−	0.902194i	$-0.358044\pi$
$311$	15.2132	0.862662	0.431331	+	0.902194i	$0.358044\pi$
$312$	0	0
$313$	20.6569	1.16759	0.583797	−	0.811900i	$-0.301566\pi$
$313$	20.6569	1.16759	0.583797	+	0.811900i	$0.301566\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.9706	−0.728499	−0.364250	−	0.931301i	$-0.618675\pi$
$317$	−12.9706	−0.728499	−0.364250	+	0.931301i	$0.618675\pi$
$318$	0	0
$319$	−1.00000	−0.0559893
$320$	0	0
$321$	42.3848	2.36569
$322$	0	0
$323$	−10.8284	−0.602510
$324$	0	0
$325$	−11.4853	−0.637089
$326$	0	0
$327$	20.4853	1.13284
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−31.3848	−1.72506	−0.862532	−	0.506003i	$-0.831122\pi$
$331$	−31.3848	−1.72506	−0.862532	+	0.506003i	$0.831122\pi$
$332$	0	0
$333$	14.3431	0.786000
$334$	0	0
$335$	8.58579	0.469092
$336$	0	0
$337$	6.24264	0.340058	0.170029	−	0.985439i	$-0.445614\pi$
$337$	6.24264	0.340058	0.170029	+	0.985439i	$0.445614\pi$
$338$	0	0
$339$	22.8995	1.24373
$340$	0	0
$341$	−1.65685	−0.0897237
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−8.82843	−0.475307
$346$	0	0
$347$	11.4142	0.612747	0.306374	−	0.951911i	$-0.400884\pi$
$347$	11.4142	0.612747	0.306374	+	0.951911i	$0.400884\pi$
$348$	0	0
$349$	32.2843	1.72814	0.864069	−	0.503374i	$-0.167908\pi$
$349$	32.2843	1.72814	0.864069	+	0.503374i	$0.167908\pi$
$350$	0	0
$351$	−1.58579	−0.0846430
$352$	0	0
$353$	25.3137	1.34731	0.673656	−	0.739045i	$-0.264723\pi$
$353$	25.3137	1.34731	0.673656	+	0.739045i	$0.264723\pi$
$354$	0	0
$355$	20.1421	1.06903
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.0711	−1.48153	−0.740767	−	0.671762i	$-0.765538\pi$
$359$	−28.0711	−1.48153	−0.740767	+	0.671762i	$0.765538\pi$
$360$	0	0
$361$	10.3137	0.542827
$362$	0	0
$363$	2.41421	0.126713
$364$	0	0
$365$	0.828427	0.0433619
$366$	0	0
$367$	−0.242641	−0.0126657	−0.00633287	−	0.999980i	$-0.502016\pi$
$367$	−0.242641	−0.0126657	−0.00633287	+	0.999980i	$0.502016\pi$
$368$	0	0
$369$	6.34315	0.330211
$370$	0	0
$371$	0	0
$372$	0	0
$373$	31.6274	1.63761	0.818803	−	0.574075i	$-0.194638\pi$
$373$	31.6274	1.63761	0.818803	+	0.574075i	$0.194638\pi$
$374$	0	0
$375$	−27.3137	−1.41047
$376$	0	0
$377$	3.82843	0.197174
$378$	0	0
$379$	−29.3848	−1.50939	−0.754697	−	0.656073i	$-0.772216\pi$
$379$	−29.3848	−1.50939	−0.754697	+	0.656073i	$0.772216\pi$
$380$	0	0
$381$	−8.65685	−0.443504
$382$	0	0
$383$	−27.8995	−1.42560	−0.712799	−	0.701369i	$-0.752573\pi$
$383$	−27.8995	−1.42560	−0.712799	+	0.701369i	$0.752573\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	22.6274	1.15022
$388$	0	0
$389$	−9.21320	−0.467128	−0.233564	−	0.972341i	$-0.575039\pi$
$389$	−9.21320	−0.467128	−0.233564	+	0.972341i	$0.575039\pi$
$390$	0	0
$391$	−5.17157	−0.261538
$392$	0	0
$393$	−46.0416	−2.32249
$394$	0	0
$395$	8.58579	0.431998
$396$	0	0
$397$	−24.3431	−1.22175	−0.610874	−	0.791728i	$-0.709182\pi$
$397$	−24.3431	−1.22175	−0.610874	+	0.791728i	$0.709182\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−25.0000	−1.24844	−0.624220	−	0.781248i	$-0.714583\pi$
$401$	−25.0000	−1.24844	−0.624220	+	0.781248i	$0.714583\pi$
$402$	0	0
$403$	6.34315	0.315975
$404$	0	0
$405$	−13.4142	−0.666558
$406$	0	0
$407$	−5.07107	−0.251363
$408$	0	0
$409$	14.1005	0.697225	0.348613	−	0.937267i	$-0.386653\pi$
$409$	14.1005	0.697225	0.348613	+	0.937267i	$0.386653\pi$
$410$	0	0
$411$	41.0416	2.02443
$412$	0	0
$413$	0	0
$414$	0	0
$415$	9.17157	0.450215
$416$	0	0
$417$	46.6274	2.28335
$418$	0	0
$419$	25.4558	1.24360	0.621800	−	0.783176i	$-0.286402\pi$
$419$	25.4558	1.24360	0.621800	+	0.783176i	$0.286402\pi$
$420$	0	0
$421$	−1.65685	−0.0807501	−0.0403751	−	0.999185i	$-0.512855\pi$
$421$	−1.65685	−0.0807501	−0.0403751	+	0.999185i	$0.512855\pi$
$422$	0	0
$423$	−2.34315	−0.113928
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−9.24264	−0.446239
$430$	0	0
$431$	17.7279	0.853924	0.426962	−	0.904270i	$-0.359584\pi$
$431$	17.7279	0.853924	0.426962	+	0.904270i	$0.359584\pi$
$432$	0	0
$433$	−37.4558	−1.80001	−0.900006	−	0.435876i	$-0.856438\pi$
$433$	−37.4558	−1.80001	−0.900006	+	0.435876i	$0.856438\pi$
$434$	0	0
$435$	3.41421	0.163699
$436$	0	0
$437$	14.0000	0.669711
$438$	0	0
$439$	16.0711	0.767030	0.383515	−	0.923535i	$-0.374713\pi$
$439$	16.0711	0.767030	0.383515	+	0.923535i	$0.374713\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.3137	0.822599	0.411300	−	0.911500i	$-0.365075\pi$
$443$	17.3137	0.822599	0.411300	+	0.911500i	$0.365075\pi$
$444$	0	0
$445$	20.9706	0.994100
$446$	0	0
$447$	−38.6274	−1.82701
$448$	0	0
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	0	0
$451$	−2.24264	−0.105602
$452$	0	0
$453$	−20.3137	−0.954421
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.68629	0.312772	0.156386	−	0.987696i	$-0.450016\pi$
$457$	6.68629	0.312772	0.156386	+	0.987696i	$0.450016\pi$
$458$	0	0
$459$	−0.828427	−0.0386677
$460$	0	0
$461$	13.0000	0.605470	0.302735	−	0.953075i	$-0.402100\pi$
$461$	13.0000	0.605470	0.302735	+	0.953075i	$0.402100\pi$
$462$	0	0
$463$	−25.7990	−1.19898	−0.599490	−	0.800382i	$-0.704630\pi$
$463$	−25.7990	−1.19898	−0.599490	+	0.800382i	$0.704630\pi$
$464$	0	0
$465$	5.65685	0.262330
$466$	0	0
$467$	−31.9411	−1.47806	−0.739030	−	0.673673i	$-0.764716\pi$
$467$	−31.9411	−1.47806	−0.739030	+	0.673673i	$0.764716\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−36.9706	−1.70351
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	16.2426	0.745263
$476$	0	0
$477$	15.3137	0.701167
$478$	0	0
$479$	−0.0710678	−0.00324717	−0.00162359	−	0.999999i	$-0.500517\pi$
$479$	−0.0710678	−0.00324717	−0.00162359	+	0.999999i	$0.500517\pi$
$480$	0	0
$481$	19.4142	0.885212
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.3848	0.653179
$486$	0	0
$487$	0.686292	0.0310988	0.0155494	−	0.999879i	$-0.495050\pi$
$487$	0.686292	0.0310988	0.0155494	+	0.999879i	$0.495050\pi$
$488$	0	0
$489$	−60.1127	−2.71839
$490$	0	0
$491$	−4.82843	−0.217904	−0.108952	−	0.994047i	$-0.534749\pi$
$491$	−4.82843	−0.217904	−0.108952	+	0.994047i	$0.534749\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−15.7990	−0.707260	−0.353630	−	0.935385i	$-0.615053\pi$
$499$	−15.7990	−0.707260	−0.353630	+	0.935385i	$0.615053\pi$
$500$	0	0
$501$	−62.1127	−2.77499
$502$	0	0
$503$	5.72792	0.255395	0.127698	−	0.991813i	$-0.459241\pi$
$503$	5.72792	0.255395	0.127698	+	0.991813i	$0.459241\pi$
$504$	0	0
$505$	−13.4142	−0.596925
$506$	0	0
$507$	4.00000	0.177646
$508$	0	0
$509$	−24.3431	−1.07899	−0.539495	−	0.841988i	$-0.681385\pi$
$509$	−24.3431	−1.07899	−0.539495	+	0.841988i	$0.681385\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.24264	0.0990150
$514$	0	0
$515$	5.51472	0.243008
$516$	0	0
$517$	0.828427	0.0364342
$518$	0	0
$519$	−18.8995	−0.829596
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	1.07107	0.0468345	0.0234173	−	0.999726i	$-0.492545\pi$
$523$	1.07107	0.0468345	0.0234173	+	0.999726i	$0.492545\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.31371	0.144347
$528$	0	0
$529$	−16.3137	−0.709292
$530$	0	0
$531$	4.48528	0.194645
$532$	0	0
$533$	8.58579	0.371892
$534$	0	0
$535$	24.8284	1.07343
$536$	0	0
$537$	2.17157	0.0937103
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−37.1421	−1.59687	−0.798433	−	0.602084i	$-0.794337\pi$
$541$	−37.1421	−1.59687	−0.798433	+	0.602084i	$0.794337\pi$
$542$	0	0
$543$	−22.4853	−0.964936
$544$	0	0
$545$	12.0000	0.514024
$546$	0	0
$547$	−15.6985	−0.671219	−0.335609	−	0.942001i	$-0.608942\pi$
$547$	−15.6985	−0.671219	−0.335609	+	0.942001i	$0.608942\pi$
$548$	0	0
$549$	39.1127	1.66929
$550$	0	0
$551$	−5.41421	−0.230653
$552$	0	0
$553$	0	0
$554$	0	0
$555$	17.3137	0.734926
$556$	0	0
$557$	−13.1716	−0.558097	−0.279049	−	0.960277i	$-0.590019\pi$
$557$	−13.1716	−0.558097	−0.279049	+	0.960277i	$0.590019\pi$
$558$	0	0
$559$	30.6274	1.29540
$560$	0	0
$561$	−4.82843	−0.203856
$562$	0	0
$563$	45.2132	1.90551	0.952755	−	0.303741i	$-0.0982357\pi$
$563$	45.2132	1.90551	0.952755	+	0.303741i	$0.0982357\pi$
$564$	0	0
$565$	13.4142	0.564340
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.9706	0.879132	0.439566	−	0.898210i	$-0.355132\pi$
$569$	20.9706	0.879132	0.439566	+	0.898210i	$0.355132\pi$
$570$	0	0
$571$	−4.58579	−0.191909	−0.0959546	−	0.995386i	$-0.530590\pi$
$571$	−4.58579	−0.191909	−0.0959546	+	0.995386i	$0.530590\pi$
$572$	0	0
$573$	−25.3137	−1.05750
$574$	0	0
$575$	7.75736	0.323504
$576$	0	0
$577$	17.9706	0.748124	0.374062	−	0.927404i	$-0.377965\pi$
$577$	17.9706	0.748124	0.374062	+	0.927404i	$0.377965\pi$
$578$	0	0
$579$	−46.3848	−1.92769
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.41421	−0.224234
$584$	0	0
$585$	15.3137	0.633144
$586$	0	0
$587$	−26.2132	−1.08193	−0.540967	−	0.841044i	$-0.681942\pi$
$587$	−26.2132	−1.08193	−0.540967	+	0.841044i	$0.681942\pi$
$588$	0	0
$589$	−8.97056	−0.369626
$590$	0	0
$591$	41.3848	1.70234
$592$	0	0
$593$	13.4142	0.550856	0.275428	−	0.961322i	$-0.411180\pi$
$593$	13.4142	0.550856	0.275428	+	0.961322i	$0.411180\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−58.0416	−2.37549
$598$	0	0
$599$	−11.0294	−0.450651	−0.225325	−	0.974284i	$-0.572344\pi$
$599$	−11.0294	−0.450651	−0.225325	+	0.974284i	$0.572344\pi$
$600$	0	0
$601$	12.3431	0.503487	0.251744	−	0.967794i	$-0.418996\pi$
$601$	12.3431	0.503487	0.251744	+	0.967794i	$0.418996\pi$
$602$	0	0
$603$	17.1716	0.699281
$604$	0	0
$605$	1.41421	0.0574960
$606$	0	0
$607$	−34.2843	−1.39156	−0.695778	−	0.718257i	$-0.744940\pi$
$607$	−34.2843	−1.39156	−0.695778	+	0.718257i	$0.744940\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.17157	−0.128308
$612$	0	0
$613$	−16.6274	−0.671575	−0.335788	−	0.941938i	$-0.609002\pi$
$613$	−16.6274	−0.671575	−0.335788	+	0.941938i	$0.609002\pi$
$614$	0	0
$615$	7.65685	0.308754
$616$	0	0
$617$	−11.6274	−0.468102	−0.234051	−	0.972224i	$-0.575198\pi$
$617$	−11.6274	−0.468102	−0.234051	+	0.972224i	$0.575198\pi$
$618$	0	0
$619$	23.3137	0.937057	0.468529	−	0.883448i	$-0.344784\pi$
$619$	23.3137	0.937057	0.468529	+	0.883448i	$0.344784\pi$
$620$	0	0
$621$	1.07107	0.0429805
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	13.0711	0.522008
$628$	0	0
$629$	10.1421	0.404393
$630$	0	0
$631$	−32.3848	−1.28922	−0.644609	−	0.764513i	$-0.722980\pi$
$631$	−32.3848	−1.28922	−0.644609	+	0.764513i	$0.722980\pi$
$632$	0	0
$633$	46.0416	1.82999
$634$	0	0
$635$	−5.07107	−0.201239
$636$	0	0
$637$	0	0
$638$	0	0
$639$	40.2843	1.59362
$640$	0	0
$641$	−46.2548	−1.82696	−0.913478	−	0.406888i	$-0.866614\pi$
$641$	−46.2548	−1.82696	−0.913478	+	0.406888i	$0.866614\pi$
$642$	0	0
$643$	38.4142	1.51491	0.757454	−	0.652888i	$-0.226443\pi$
$643$	38.4142	1.51491	0.757454	+	0.652888i	$0.226443\pi$
$644$	0	0
$645$	27.3137	1.07548
$646$	0	0
$647$	1.07107	0.0421080	0.0210540	−	0.999778i	$-0.493298\pi$
$647$	1.07107	0.0421080	0.0210540	+	0.999778i	$0.493298\pi$
$648$	0	0
$649$	−1.58579	−0.0622476
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−34.9289	−1.36687	−0.683437	−	0.730009i	$-0.739516\pi$
$653$	−34.9289	−1.36687	−0.683437	+	0.730009i	$0.739516\pi$
$654$	0	0
$655$	−26.9706	−1.05383
$656$	0	0
$657$	1.65685	0.0646400
$658$	0	0
$659$	−8.68629	−0.338370	−0.169185	−	0.985584i	$-0.554114\pi$
$659$	−8.68629	−0.338370	−0.169185	+	0.985584i	$0.554114\pi$
$660$	0	0
$661$	−40.6274	−1.58022	−0.790112	−	0.612963i	$-0.789978\pi$
$661$	−40.6274	−1.58022	−0.790112	+	0.612963i	$0.789978\pi$
$662$	0	0
$663$	18.4853	0.717909
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.58579	−0.100122
$668$	0	0
$669$	−27.8995	−1.07866
$670$	0	0
$671$	−13.8284	−0.533841
$672$	0	0
$673$	7.75736	0.299024	0.149512	−	0.988760i	$-0.452230\pi$
$673$	7.75736	0.299024	0.149512	+	0.988760i	$0.452230\pi$
$674$	0	0
$675$	1.24264	0.0478293
$676$	0	0
$677$	48.2843	1.85572	0.927858	−	0.372935i	$-0.121648\pi$
$677$	48.2843	1.85572	0.927858	+	0.372935i	$0.121648\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0.343146	0.0131494
$682$	0	0
$683$	4.89949	0.187474	0.0937370	−	0.995597i	$-0.470119\pi$
$683$	4.89949	0.187474	0.0937370	+	0.995597i	$0.470119\pi$
$684$	0	0
$685$	24.0416	0.918583
$686$	0	0
$687$	9.65685	0.368432
$688$	0	0
$689$	20.7279	0.789671
$690$	0	0
$691$	−7.38478	−0.280930	−0.140465	−	0.990086i	$-0.544860\pi$
$691$	−7.38478	−0.280930	−0.140465	+	0.990086i	$0.544860\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	27.3137	1.03607
$696$	0	0
$697$	4.48528	0.169892
$698$	0	0
$699$	−39.5563	−1.49616
$700$	0	0
$701$	−8.45584	−0.319373	−0.159686	−	0.987168i	$-0.551048\pi$
$701$	−8.45584	−0.319373	−0.159686	+	0.987168i	$0.551048\pi$
$702$	0	0
$703$	−27.4558	−1.03552
$704$	0	0
$705$	−2.82843	−0.106525
$706$	0	0
$707$	0	0
$708$	0	0
$709$	33.0711	1.24201	0.621005	−	0.783807i	$-0.286725\pi$
$709$	33.0711	1.24201	0.621005	+	0.783807i	$0.286725\pi$
$710$	0	0
$711$	17.1716	0.643984
$712$	0	0
$713$	−4.28427	−0.160447
$714$	0	0
$715$	−5.41421	−0.202480
$716$	0	0
$717$	−33.1421	−1.23772
$718$	0	0
$719$	23.1716	0.864154	0.432077	−	0.901837i	$-0.357781\pi$
$719$	23.1716	0.864154	0.432077	+	0.901837i	$0.357781\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−6.58579	−0.244928
$724$	0	0
$725$	−3.00000	−0.111417
$726$	0	0
$727$	25.7990	0.956832	0.478416	−	0.878133i	$-0.341211\pi$
$727$	25.7990	0.956832	0.478416	+	0.878133i	$0.341211\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	−17.3431	−0.640584	−0.320292	−	0.947319i	$-0.603781\pi$
$733$	−17.3431	−0.640584	−0.320292	+	0.947319i	$0.603781\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.07107	−0.223631
$738$	0	0
$739$	48.4264	1.78139	0.890697	−	0.454597i	$-0.150217\pi$
$739$	48.4264	1.78139	0.890697	+	0.454597i	$0.150217\pi$
$740$	0	0
$741$	−50.0416	−1.83833
$742$	0	0
$743$	−34.0000	−1.24734	−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000	−1.24734	−0.623670	+	0.781688i	$0.714359\pi$
$744$	0	0
$745$	−22.6274	−0.829004
$746$	0	0
$747$	18.3431	0.671141
$748$	0	0
$749$	0	0
$750$	0	0
$751$	11.3137	0.412843	0.206422	−	0.978463i	$-0.433818\pi$
$751$	11.3137	0.412843	0.206422	+	0.978463i	$0.433818\pi$
$752$	0	0
$753$	−39.7990	−1.45036
$754$	0	0
$755$	−11.8995	−0.433067
$756$	0	0
$757$	−5.31371	−0.193130	−0.0965650	−	0.995327i	$-0.530786\pi$
$757$	−5.31371	−0.193130	−0.0965650	+	0.995327i	$0.530786\pi$
$758$	0	0
$759$	6.24264	0.226594
$760$	0	0
$761$	−39.9411	−1.44786	−0.723932	−	0.689871i	$-0.757667\pi$
$761$	−39.9411	−1.44786	−0.723932	+	0.689871i	$0.757667\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.00000	0.289241
$766$	0	0
$767$	6.07107	0.219214
$768$	0	0
$769$	2.97056	0.107121	0.0535606	−	0.998565i	$-0.482943\pi$
$769$	2.97056	0.107121	0.0535606	+	0.998565i	$0.482943\pi$
$770$	0	0
$771$	6.07107	0.218644
$772$	0	0
$773$	10.6274	0.382242	0.191121	−	0.981567i	$-0.438788\pi$
$773$	10.6274	0.382242	0.191121	+	0.981567i	$0.438788\pi$
$774$	0	0
$775$	−4.97056	−0.178548
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.1421	−0.435037
$780$	0	0
$781$	−14.2426	−0.509642
$782$	0	0
$783$	−0.414214	−0.0148028
$784$	0	0
$785$	−21.6569	−0.772966
$786$	0	0
$787$	42.3848	1.51085	0.755427	−	0.655233i	$-0.227429\pi$
$787$	42.3848	1.51085	0.755427	+	0.655233i	$0.227429\pi$
$788$	0	0
$789$	−60.4558	−2.15229
$790$	0	0
$791$	0	0
$792$	0	0
$793$	52.9411	1.87999
$794$	0	0
$795$	18.4853	0.655605
$796$	0	0
$797$	13.3137	0.471596	0.235798	−	0.971802i	$-0.424230\pi$
$797$	13.3137	0.471596	0.235798	+	0.971802i	$0.424230\pi$
$798$	0	0
$799$	−1.65685	−0.0586153
$800$	0	0
$801$	41.9411	1.48192
$802$	0	0
$803$	−0.585786	−0.0206720
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.65685	0.199131
$808$	0	0
$809$	−8.00000	−0.281265	−0.140633	−	0.990062i	$-0.544914\pi$
$809$	−8.00000	−0.281265	−0.140633	+	0.990062i	$0.544914\pi$
$810$	0	0
$811$	38.3431	1.34641	0.673205	−	0.739456i	$-0.264917\pi$
$811$	38.3431	1.34641	0.673205	+	0.739456i	$0.264917\pi$
$812$	0	0
$813$	33.9706	1.19140
$814$	0	0
$815$	−35.2132	−1.23346
$816$	0	0
$817$	−43.3137	−1.51535
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.8284	−0.971219	−0.485609	−	0.874176i	$-0.661402\pi$
$821$	−27.8284	−0.971219	−0.485609	+	0.874176i	$0.661402\pi$
$822$	0	0
$823$	32.8284	1.14433	0.572164	−	0.820140i	$-0.306104\pi$
$823$	32.8284	1.14433	0.572164	+	0.820140i	$0.306104\pi$
$824$	0	0
$825$	7.24264	0.252156
$826$	0	0
$827$	32.5269	1.13107	0.565536	−	0.824724i	$-0.308669\pi$
$827$	32.5269	1.13107	0.565536	+	0.824724i	$0.308669\pi$
$828$	0	0
$829$	−34.5858	−1.20121	−0.600607	−	0.799544i	$-0.705074\pi$
$829$	−34.5858	−1.20121	−0.600607	+	0.799544i	$0.705074\pi$
$830$	0	0
$831$	35.7279	1.23939
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−36.3848	−1.25915
$836$	0	0
$837$	−0.686292	−0.0237217
$838$	0	0
$839$	9.51472	0.328485	0.164242	−	0.986420i	$-0.447482\pi$
$839$	9.51472	0.328485	0.164242	+	0.986420i	$0.447482\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	19.0711	0.656842
$844$	0	0
$845$	2.34315	0.0806067
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.58579	−0.294663
$850$	0	0
$851$	−13.1127	−0.449498
$852$	0	0
$853$	50.0000	1.71197	0.855984	−	0.517003i	$-0.172952\pi$
$853$	50.0000	1.71197	0.855984	+	0.517003i	$0.172952\pi$
$854$	0	0
$855$	−21.6569	−0.740649
$856$	0	0
$857$	−8.28427	−0.282985	−0.141493	−	0.989939i	$-0.545190\pi$
$857$	−8.28427	−0.282985	−0.141493	+	0.989939i	$0.545190\pi$
$858$	0	0
$859$	−9.58579	−0.327063	−0.163531	−	0.986538i	$-0.552289\pi$
$859$	−9.58579	−0.327063	−0.163531	+	0.986538i	$0.552289\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−10.3848	−0.353502	−0.176751	−	0.984256i	$-0.556559\pi$
$863$	−10.3848	−0.353502	−0.176751	+	0.984256i	$0.556559\pi$
$864$	0	0
$865$	−11.0711	−0.376428
$866$	0	0
$867$	−31.3848	−1.06588
$868$	0	0
$869$	−6.07107	−0.205947
$870$	0	0
$871$	23.2426	0.787547
$872$	0	0
$873$	28.7696	0.973702
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−43.2843	−1.46161	−0.730803	−	0.682588i	$-0.760854\pi$
$877$	−43.2843	−1.46161	−0.730803	+	0.682588i	$0.760854\pi$
$878$	0	0
$879$	45.4558	1.53319
$880$	0	0
$881$	−14.1716	−0.477452	−0.238726	−	0.971087i	$-0.576730\pi$
$881$	−14.1716	−0.477452	−0.238726	+	0.971087i	$0.576730\pi$
$882$	0	0
$883$	53.6690	1.80611	0.903054	−	0.429528i	$-0.141320\pi$
$883$	53.6690	1.80611	0.903054	+	0.429528i	$0.141320\pi$
$884$	0	0
$885$	5.41421	0.181997
$886$	0	0
$887$	−6.27208	−0.210596	−0.105298	−	0.994441i	$-0.533580\pi$
$887$	−6.27208	−0.210596	−0.105298	+	0.994441i	$0.533580\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.48528	0.317769
$892$	0	0
$893$	4.48528	0.150094
$894$	0	0
$895$	1.27208	0.0425209
$896$	0	0
$897$	−23.8995	−0.797981
$898$	0	0
$899$	1.65685	0.0552592
$900$	0	0
$901$	10.8284	0.360747
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.1716	−0.437838
$906$	0	0
$907$	−32.6274	−1.08338	−0.541688	−	0.840580i	$-0.682214\pi$
$907$	−32.6274	−1.08338	−0.541688	+	0.840580i	$0.682214\pi$
$908$	0	0
$909$	−26.8284	−0.889843
$910$	0	0
$911$	−21.5147	−0.712814	−0.356407	−	0.934331i	$-0.615998\pi$
$911$	−21.5147	−0.712814	−0.356407	+	0.934331i	$0.615998\pi$
$912$	0	0
$913$	−6.48528	−0.214631
$914$	0	0
$915$	47.2132	1.56082
$916$	0	0
$917$	0	0
$918$	0	0
$919$	45.4558	1.49945	0.749725	−	0.661750i	$-0.230186\pi$
$919$	45.4558	1.49945	0.749725	+	0.661750i	$0.230186\pi$
$920$	0	0
$921$	27.5563	0.908013
$922$	0	0
$923$	54.5269	1.79478
$924$	0	0
$925$	−15.2132	−0.500207
$926$	0	0
$927$	11.0294	0.362254
$928$	0	0
$929$	−43.4264	−1.42477	−0.712387	−	0.701787i	$-0.752386\pi$
$929$	−43.4264	−1.42477	−0.712387	+	0.701787i	$0.752386\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	36.7279	1.20242
$934$	0	0
$935$	−2.82843	−0.0924995
$936$	0	0
$937$	18.5858	0.607171	0.303586	−	0.952804i	$-0.401816\pi$
$937$	18.5858	0.607171	0.303586	+	0.952804i	$0.401816\pi$
$938$	0	0
$939$	49.8701	1.62745
$940$	0	0
$941$	−16.9411	−0.552265	−0.276132	−	0.961120i	$-0.589053\pi$
$941$	−16.9411	−0.552265	−0.276132	+	0.961120i	$0.589053\pi$
$942$	0	0
$943$	−5.79899	−0.188841
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.7990	−1.29329	−0.646647	−	0.762790i	$-0.723829\pi$
$947$	−39.7990	−1.29329	−0.646647	+	0.762790i	$0.723829\pi$
$948$	0	0
$949$	2.24264	0.0727992
$950$	0	0
$951$	−31.3137	−1.01542
$952$	0	0
$953$	54.3848	1.76170	0.880848	−	0.473399i	$-0.156973\pi$
$953$	54.3848	1.76170	0.880848	+	0.473399i	$0.156973\pi$
$954$	0	0
$955$	−14.8284	−0.479837
$956$	0	0
$957$	−2.41421	−0.0780404
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.2548	−0.911446
$962$	0	0
$963$	49.6569	1.60017
$964$	0	0
$965$	−27.1716	−0.874684
$966$	0	0
$967$	29.6569	0.953700	0.476850	−	0.878985i	$-0.341778\pi$
$967$	29.6569	0.953700	0.476850	+	0.878985i	$0.341778\pi$
$968$	0	0
$969$	−26.1421	−0.839806
$970$	0	0
$971$	−12.2132	−0.391940	−0.195970	−	0.980610i	$-0.562786\pi$
$971$	−12.2132	−0.391940	−0.195970	+	0.980610i	$0.562786\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−27.7279	−0.888004
$976$	0	0
$977$	36.9706	1.18279	0.591397	−	0.806381i	$-0.298577\pi$
$977$	36.9706	1.18279	0.591397	+	0.806381i	$0.298577\pi$
$978$	0	0
$979$	−14.8284	−0.473919
$980$	0	0
$981$	24.0000	0.766261
$982$	0	0
$983$	−46.4853	−1.48265	−0.741325	−	0.671146i	$-0.765802\pi$
$983$	−46.4853	−1.48265	−0.741325	+	0.671146i	$0.765802\pi$
$984$	0	0
$985$	24.2426	0.772435
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20.6863	−0.657786
$990$	0	0
$991$	−19.8995	−0.632128	−0.316064	−	0.948738i	$-0.602361\pi$
$991$	−19.8995	−0.632128	−0.316064	+	0.948738i	$0.602361\pi$
$992$	0	0
$993$	−75.7696	−2.40447
$994$	0	0
$995$	−34.0000	−1.07787
$996$	0	0
$997$	9.45584	0.299470	0.149735	−	0.988726i	$-0.452158\pi$
$997$	9.45584	0.299470	0.149735	+	0.988726i	$0.452158\pi$
$998$	0	0
$999$	−2.10051	−0.0664570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.cd.1.2		2
4.3	odd	2		4312.2.a.m.1.1		2
7.3	odd	6		1232.2.q.h.177.2		4
7.5	odd	6		1232.2.q.h.529.2		4
7.6	odd	2		8624.2.a.bg.1.1		2
28.3	even	6		616.2.q.b.177.1	✓	4
28.19	even	6		616.2.q.b.529.1	yes	4
28.27	even	2		4312.2.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.q.b.177.1	✓	4	28.3	even	6
616.2.q.b.529.1	yes	4	28.19	even	6
1232.2.q.h.177.2		4	7.3	odd	6
1232.2.q.h.529.2		4	7.5	odd	6
4312.2.a.m.1.1		2	4.3	odd	2
4312.2.a.u.1.2		2	28.27	even	2
8624.2.a.bg.1.1		2	7.6	odd	2
8624.2.a.cd.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{3} +1.41421 q^{5} +2.82843 q^{9} -1.00000 q^{11} +3.82843 q^{13} +3.41421 q^{15} +2.00000 q^{17} -5.41421 q^{19} -2.58579 q^{23} -3.00000 q^{25} -0.414214 q^{27} +1.00000 q^{29} +1.65685 q^{31} -2.41421 q^{33} +5.07107 q^{37} +9.24264 q^{39} +2.24264 q^{41} +8.00000 q^{43} +4.00000 q^{45} -0.828427 q^{47} +4.82843 q^{51} +5.41421 q^{53} -1.41421 q^{55} -13.0711 q^{57} +1.58579 q^{59} +13.8284 q^{61} +5.41421 q^{65} +6.07107 q^{67} -6.24264 q^{69} +14.2426 q^{71} +0.585786 q^{73} -7.24264 q^{75} +6.07107 q^{79} -9.48528 q^{81} +6.48528 q^{83} +2.82843 q^{85} +2.41421 q^{87} +14.8284 q^{89} +4.00000 q^{93} -7.65685 q^{95} +10.1716 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{11} + 2 q^{13} + 4 q^{15} + 4 q^{17} - 8 q^{19} - 8 q^{23} - 6 q^{25} + 2 q^{27} + 2 q^{29} - 8 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{39} - 4 q^{41} + 16 q^{43} + 8 q^{45} + 4 q^{47}+ \cdots + 26 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^11 + 2 * q^13 + 4 * q^15 + 4 * q^17 - 8 * q^19 - 8 * q^23 - 6 * q^25 + 2 * q^27 + 2 * q^29 - 8 * q^31 - 2 * q^33 - 4 * q^37 + 10 * q^39 - 4 * q^41 + 16 * q^43 + 8 * q^45 + 4 * q^47 + 4 * q^51 + 8 * q^53 - 12 * q^57 + 6 * q^59 + 22 * q^61 + 8 * q^65 - 2 * q^67 - 4 * q^69 + 20 * q^71 + 4 * q^73 - 6 * q^75 - 2 * q^79 - 2 * q^81 - 4 * q^83 + 2 * q^87 + 24 * q^89 + 8 * q^93 - 4 * q^95 + 26 * q^97

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