LMFDB - Embedded newform 3249.2.a.ba.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3249,2,Mod(1,3249)] 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("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3249, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,2,-2,0,-2,0,0,0,-8,0,0,0,0,-14,8,0,0,-6,0,0,24,0,-14,20, 0,-6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,-24,6,0,0,-38,0,-22,0,0,0,0,0,-6,0,0, -20,0,0,-20,-20,0,8,0,0,0,-6,0,0,0,0,-24,-40,0,0,-6,0,0,12,0,0,4,0,6,0, 0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.9433956167$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.17557$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17557 q^{2} -0.618034 q^{4} +0.618034 q^{5} +0.618034 q^{7} -3.07768 q^{8} +0.726543 q^{10} -4.23607 q^{11} +0.449028 q^{13} +0.726543 q^{14} -2.38197 q^{16} +4.23607 q^{17} -0.381966 q^{20} -4.97980 q^{22} +3.76393 q^{23} -4.61803 q^{25} +0.527864 q^{26} -0.381966 q^{28} +5.25731 q^{29} +3.35520 q^{31} +3.35520 q^{32} +4.97980 q^{34} +0.381966 q^{35} -8.50651 q^{37} -1.90211 q^{40} -5.70634 q^{41} -3.76393 q^{43} +2.61803 q^{44} +4.42477 q^{46} -10.6180 q^{47} -6.61803 q^{49} -5.42882 q^{50} -0.277515 q^{52} -10.1311 q^{53} -2.61803 q^{55} -1.90211 q^{56} +6.18034 q^{58} -12.8658 q^{59} -7.23607 q^{61} +3.94427 q^{62} +8.70820 q^{64} +0.277515 q^{65} +10.2371 q^{67} -2.61803 q^{68} +0.449028 q^{70} -2.80017 q^{71} -12.7082 q^{73} -10.0000 q^{74} -2.61803 q^{77} -13.0373 q^{79} -1.47214 q^{80} -6.70820 q^{82} -7.94427 q^{83} +2.61803 q^{85} -4.42477 q^{86} +13.0373 q^{88} +9.95959 q^{89} +0.277515 q^{91} -2.32624 q^{92} -12.4822 q^{94} +2.90617 q^{97} -7.77997 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{5} - 2 q^{7} - 8 q^{11} - 14 q^{16} + 8 q^{17} - 6 q^{20} + 24 q^{23} - 14 q^{25} + 20 q^{26} - 6 q^{28} + 6 q^{35} - 24 q^{43} + 6 q^{44} - 38 q^{47} - 22 q^{49} - 6 q^{55} - 20 q^{58}+ \cdots + 22 q^{92}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^5 - 2 * q^7 - 8 * q^11 - 14 * q^16 + 8 * q^17 - 6 * q^20 + 24 * q^23 - 14 * q^25 + 20 * q^26 - 6 * q^28 + 6 * q^35 - 24 * q^43 + 6 * q^44 - 38 * q^47 - 22 * q^49 - 6 * q^55 - 20 * q^58 - 20 * q^61 - 20 * q^62 + 8 * q^64 - 6 * q^68 - 24 * q^73 - 40 * q^74 - 6 * q^77 + 12 * q^80 + 4 * q^83 + 6 * q^85 + 22 * q^92

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 $

$2$	1.17557	0.831254	0.415627	−	0.909535i	$-0.363562\pi$
$2$	1.17557	0.831254	0.415627	+	0.909535i	$0.363562\pi$
$3$	0	0
$3$	0	0
$4$	−0.618034	−0.309017
$5$	0.618034	0.276393	0.138197	−	0.990405i	$-0.455869\pi$
$5$	0.618034	0.276393	0.138197	+	0.990405i	$0.455869\pi$
$6$	0	0
$7$	0.618034	0.233595	0.116797	−	0.993156i	$-0.462737\pi$
$7$	0.618034	0.233595	0.116797	+	0.993156i	$0.462737\pi$
$8$	−3.07768	−1.08813
$9$	0	0
$10$	0.726543	0.229753
$11$	−4.23607	−1.27722	−0.638611	−	0.769529i	$-0.720491\pi$
$11$	−4.23607	−1.27722	−0.638611	+	0.769529i	$0.720491\pi$
$12$	0	0
$13$	0.449028	0.124538	0.0622690	−	0.998059i	$-0.480166\pi$
$13$	0.449028	0.124538	0.0622690	+	0.998059i	$0.480166\pi$
$14$	0.726543	0.194177
$15$	0	0
$16$	−2.38197	−0.595492
$17$	4.23607	1.02740	0.513699	−	0.857971i	$-0.328275\pi$
$17$	4.23607	1.02740	0.513699	+	0.857971i	$0.328275\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−0.381966	−0.0854102
$21$	0	0
$22$	−4.97980	−1.06170
$23$	3.76393	0.784834	0.392417	−	0.919787i	$-0.371639\pi$
$23$	3.76393	0.784834	0.392417	+	0.919787i	$0.371639\pi$
$24$	0	0
$25$	−4.61803	−0.923607
$26$	0.527864	0.103523
$27$	0	0
$28$	−0.381966	−0.0721848
$29$	5.25731	0.976258	0.488129	−	0.872771i	$-0.337680\pi$
$29$	5.25731	0.976258	0.488129	+	0.872771i	$0.337680\pi$
$30$	0	0
$31$	3.35520	0.602611	0.301306	−	0.953528i	$-0.402578\pi$
$31$	3.35520	0.602611	0.301306	+	0.953528i	$0.402578\pi$
$32$	3.35520	0.593121
$33$	0	0
$34$	4.97980	0.854028
$35$	0.381966	0.0645640
$36$	0	0
$37$	−8.50651	−1.39846	−0.699231	−	0.714896i	$-0.746474\pi$
$37$	−8.50651	−1.39846	−0.699231	+	0.714896i	$0.746474\pi$
$38$	0	0
$39$	0	0
$40$	−1.90211	−0.300750
$41$	−5.70634	−0.891180	−0.445590	−	0.895237i	$-0.647006\pi$
$41$	−5.70634	−0.891180	−0.445590	+	0.895237i	$0.647006\pi$
$42$	0	0
$43$	−3.76393	−0.573994	−0.286997	−	0.957931i	$-0.592657\pi$
$43$	−3.76393	−0.573994	−0.286997	+	0.957931i	$0.592657\pi$
$44$	2.61803	0.394683
$45$	0	0
$46$	4.42477	0.652396
$47$	−10.6180	−1.54880	−0.774400	−	0.632697i	$-0.781948\pi$
$47$	−10.6180	−1.54880	−0.774400	+	0.632697i	$0.781948\pi$
$48$	0	0
$49$	−6.61803	−0.945433
$50$	−5.42882	−0.767752
$51$	0	0
$52$	−0.277515	−0.0384843
$53$	−10.1311	−1.39161	−0.695807	−	0.718229i	$-0.744953\pi$
$53$	−10.1311	−1.39161	−0.695807	+	0.718229i	$0.744953\pi$
$54$	0	0
$55$	−2.61803	−0.353016
$56$	−1.90211	−0.254181
$57$	0	0
$58$	6.18034	0.811518
$59$	−12.8658	−1.67498	−0.837490	−	0.546453i	$-0.815978\pi$
$59$	−12.8658	−1.67498	−0.837490	+	0.546453i	$0.815978\pi$
$60$	0	0
$61$	−7.23607	−0.926484	−0.463242	−	0.886232i	$-0.653314\pi$
$61$	−7.23607	−0.926484	−0.463242	+	0.886232i	$0.653314\pi$
$62$	3.94427	0.500923
$63$	0	0
$64$	8.70820	1.08853
$65$	0.277515	0.0344214
$66$	0	0
$67$	10.2371	1.25066	0.625331	−	0.780360i	$-0.284964\pi$
$67$	10.2371	1.25066	0.625331	+	0.780360i	$0.284964\pi$
$68$	−2.61803	−0.317483
$69$	0	0
$70$	0.449028	0.0536691
$71$	−2.80017	−0.332319	−0.166159	−	0.986099i	$-0.553137\pi$
$71$	−2.80017	−0.332319	−0.166159	+	0.986099i	$0.553137\pi$
$72$	0	0
$73$	−12.7082	−1.48738	−0.743691	−	0.668523i	$-0.766927\pi$
$73$	−12.7082	−1.48738	−0.743691	+	0.668523i	$0.766927\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	0	0
$77$	−2.61803	−0.298353
$78$	0	0
$79$	−13.0373	−1.46681	−0.733404	−	0.679793i	$-0.762070\pi$
$79$	−13.0373	−1.46681	−0.733404	+	0.679793i	$0.762070\pi$
$80$	−1.47214	−0.164590
$81$	0	0
$82$	−6.70820	−0.740797
$83$	−7.94427	−0.871997	−0.435999	−	0.899947i	$-0.643605\pi$
$83$	−7.94427	−0.871997	−0.435999	+	0.899947i	$0.643605\pi$
$84$	0	0
$85$	2.61803	0.283966
$86$	−4.42477	−0.477135
$87$	0	0
$88$	13.0373	1.38978
$89$	9.95959	1.05571	0.527857	−	0.849333i	$-0.322996\pi$
$89$	9.95959	1.05571	0.527857	+	0.849333i	$0.322996\pi$
$90$	0	0
$91$	0.277515	0.0290914
$92$	−2.32624	−0.242527
$93$	0	0
$94$	−12.4822	−1.28745
$95$	0	0
$96$	0	0
$97$	2.90617	0.295077	0.147538	−	0.989056i	$-0.452865\pi$
$97$	2.90617	0.295077	0.147538	+	0.989056i	$0.452865\pi$
$98$	−7.77997	−0.785895
$99$	0	0
$100$	2.85410	0.285410
$101$	17.6525	1.75649	0.878243	−	0.478214i	$-0.158716\pi$
$101$	17.6525	1.75649	0.878243	+	0.478214i	$0.158716\pi$
$102$	0	0
$103$	12.5882	1.24036	0.620179	−	0.784461i	$-0.287060\pi$
$103$	12.5882	1.24036	0.620179	+	0.784461i	$0.287060\pi$
$104$	−1.38197	−0.135513
$105$	0	0
$106$	−11.9098	−1.15678
$107$	6.88191	0.665299	0.332650	−	0.943050i	$-0.392057\pi$
$107$	6.88191	0.665299	0.332650	+	0.943050i	$0.392057\pi$
$108$	0	0
$109$	−3.63271	−0.347951	−0.173975	−	0.984750i	$-0.555661\pi$
$109$	−3.63271	−0.347951	−0.173975	+	0.984750i	$0.555661\pi$
$110$	−3.07768	−0.293446
$111$	0	0
$112$	−1.47214	−0.139104
$113$	−7.88597	−0.741849	−0.370925	−	0.928663i	$-0.620959\pi$
$113$	−7.88597	−0.741849	−0.370925	+	0.928663i	$0.620959\pi$
$114$	0	0
$115$	2.32624	0.216923
$116$	−3.24920	−0.301680
$117$	0	0
$118$	−15.1246	−1.39233
$119$	2.61803	0.239995
$120$	0	0
$121$	6.94427	0.631297
$122$	−8.50651	−0.770143
$123$	0	0
$124$	−2.07363	−0.186217
$125$	−5.94427	−0.531672
$126$	0	0
$127$	4.70228	0.417260	0.208630	−	0.977995i	$-0.433100\pi$
$127$	4.70228	0.417260	0.208630	+	0.977995i	$0.433100\pi$
$128$	3.52671	0.311720
$129$	0	0
$130$	0.326238	0.0286130
$131$	12.2361	1.06907	0.534535	−	0.845146i	$-0.320487\pi$
$131$	12.2361	1.06907	0.534535	+	0.845146i	$0.320487\pi$
$132$	0	0
$133$	0	0
$134$	12.0344	1.03962
$135$	0	0
$136$	−13.0373	−1.11794
$137$	8.85410	0.756457	0.378228	−	0.925712i	$-0.376533\pi$
$137$	8.85410	0.756457	0.378228	+	0.925712i	$0.376533\pi$
$138$	0	0
$139$	−22.1803	−1.88131	−0.940656	−	0.339362i	$-0.889789\pi$
$139$	−22.1803	−1.88131	−0.940656	+	0.339362i	$0.889789\pi$
$140$	−0.236068	−0.0199514
$141$	0	0
$142$	−3.29180	−0.276241
$143$	−1.90211	−0.159063
$144$	0	0
$145$	3.24920	0.269831
$146$	−14.9394	−1.23639
$147$	0	0
$148$	5.25731	0.432148
$149$	3.61803	0.296401	0.148200	−	0.988957i	$-0.452652\pi$
$149$	3.61803	0.296401	0.148200	+	0.988957i	$0.452652\pi$
$150$	0	0
$151$	−19.0211	−1.54792	−0.773959	−	0.633236i	$-0.781726\pi$
$151$	−19.0211	−1.54792	−0.773959	+	0.633236i	$0.781726\pi$
$152$	0	0
$153$	0	0
$154$	−3.07768	−0.248007
$155$	2.07363	0.166558
$156$	0	0
$157$	−6.41641	−0.512085	−0.256043	−	0.966666i	$-0.582419\pi$
$157$	−6.41641	−0.512085	−0.256043	+	0.966666i	$0.582419\pi$
$158$	−15.3262	−1.21929
$159$	0	0
$160$	2.07363	0.163935
$161$	2.32624	0.183333
$162$	0	0
$163$	−16.3262	−1.27877	−0.639385	−	0.768887i	$-0.720811\pi$
$163$	−16.3262	−1.27877	−0.639385	+	0.768887i	$0.720811\pi$
$164$	3.52671	0.275390
$165$	0	0
$166$	−9.33905	−0.724851
$167$	9.06154	0.701203	0.350601	−	0.936525i	$-0.385977\pi$
$167$	9.06154	0.701203	0.350601	+	0.936525i	$0.385977\pi$
$168$	0	0
$169$	−12.7984	−0.984490
$170$	3.07768	0.236048
$171$	0	0
$172$	2.32624	0.177374
$173$	15.5599	1.18300	0.591500	−	0.806305i	$-0.298536\pi$
$173$	15.5599	1.18300	0.591500	+	0.806305i	$0.298536\pi$
$174$	0	0
$175$	−2.85410	−0.215750
$176$	10.0902	0.760575
$177$	0	0
$178$	11.7082	0.877567
$179$	−1.17557	−0.0878663	−0.0439331	−	0.999034i	$-0.513989\pi$
$179$	−1.17557	−0.0878663	−0.0439331	+	0.999034i	$0.513989\pi$
$180$	0	0
$181$	18.0171	1.33920	0.669599	−	0.742723i	$-0.266466\pi$
$181$	18.0171	1.33920	0.669599	+	0.742723i	$0.266466\pi$
$182$	0.326238	0.0241824
$183$	0	0
$184$	−11.5842	−0.853998
$185$	−5.25731	−0.386525
$186$	0	0
$187$	−17.9443	−1.31222
$188$	6.56231	0.478605
$189$	0	0
$190$	0	0
$191$	−8.41641	−0.608990	−0.304495	−	0.952514i	$-0.598488\pi$
$191$	−8.41641	−0.608990	−0.304495	+	0.952514i	$0.598488\pi$
$192$	0	0
$193$	−5.25731	−0.378430	−0.189215	−	0.981936i	$-0.560594\pi$
$193$	−5.25731	−0.378430	−0.189215	+	0.981936i	$0.560594\pi$
$194$	3.41641	0.245284
$195$	0	0
$196$	4.09017	0.292155
$197$	15.7426	1.12162	0.560809	−	0.827946i	$-0.310490\pi$
$197$	15.7426	1.12162	0.560809	+	0.827946i	$0.310490\pi$
$198$	0	0
$199$	1.18034	0.0836721	0.0418360	−	0.999124i	$-0.486679\pi$
$199$	1.18034	0.0836721	0.0418360	+	0.999124i	$0.486679\pi$
$200$	14.2128	1.00500
$201$	0	0
$202$	20.7517	1.46009
$203$	3.24920	0.228049
$204$	0	0
$205$	−3.52671	−0.246316
$206$	14.7984	1.03105
$207$	0	0
$208$	−1.06957	−0.0741613
$209$	0	0
$210$	0	0
$211$	8.33499	0.573805	0.286902	−	0.957960i	$-0.407374\pi$
$211$	8.33499	0.573805	0.286902	+	0.957960i	$0.407374\pi$
$212$	6.26137	0.430033
$213$	0	0
$214$	8.09017	0.553033
$215$	−2.32624	−0.158648
$216$	0	0
$217$	2.07363	0.140767
$218$	−4.27051	−0.289235
$219$	0	0
$220$	1.61803	0.109088
$221$	1.90211	0.127950
$222$	0	0
$223$	1.34708	0.0902074	0.0451037	−	0.998982i	$-0.485638\pi$
$223$	1.34708	0.0902074	0.0451037	+	0.998982i	$0.485638\pi$
$224$	2.07363	0.138550
$225$	0	0
$226$	−9.27051	−0.616665
$227$	18.2946	1.21425	0.607127	−	0.794605i	$-0.292322\pi$
$227$	18.2946	1.21425	0.607127	+	0.794605i	$0.292322\pi$
$228$	0	0
$229$	9.65248	0.637854	0.318927	−	0.947779i	$-0.396678\pi$
$229$	9.65248	0.637854	0.318927	+	0.947779i	$0.396678\pi$
$230$	2.73466	0.180318
$231$	0	0
$232$	−16.1803	−1.06229
$233$	−23.8885	−1.56499	−0.782495	−	0.622657i	$-0.786053\pi$
$233$	−23.8885	−1.56499	−0.782495	+	0.622657i	$0.786053\pi$
$234$	0	0
$235$	−6.56231	−0.428078
$236$	7.95148	0.517597
$237$	0	0
$238$	3.07768	0.199497
$239$	0.145898	0.00943736	0.00471868	−	0.999989i	$-0.498498\pi$
$239$	0.145898	0.00943736	0.00471868	+	0.999989i	$0.498498\pi$
$240$	0	0
$241$	10.4086	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$241$	10.4086	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$242$	8.16348	0.524768
$243$	0	0
$244$	4.47214	0.286299
$245$	−4.09017	−0.261311
$246$	0	0
$247$	0	0
$248$	−10.3262	−0.655717
$249$	0	0
$250$	−6.98791	−0.441954
$251$	−28.4164	−1.79363	−0.896814	−	0.442408i	$-0.854124\pi$
$251$	−28.4164	−1.79363	−0.896814	+	0.442408i	$0.854124\pi$
$252$	0	0
$253$	−15.9443	−1.00241
$254$	5.52786	0.346849
$255$	0	0
$256$	−13.2705	−0.829407
$257$	−11.2412	−0.701205	−0.350602	−	0.936524i	$-0.614023\pi$
$257$	−11.2412	−0.701205	−0.350602	+	0.936524i	$0.614023\pi$
$258$	0	0
$259$	−5.25731	−0.326673
$260$	−0.171513	−0.0106368
$261$	0	0
$262$	14.3844	0.888669
$263$	−28.4721	−1.75567	−0.877834	−	0.478966i	$-0.841012\pi$
$263$	−28.4721	−1.75567	−0.877834	+	0.478966i	$0.841012\pi$
$264$	0	0
$265$	−6.26137	−0.384633
$266$	0	0
$267$	0	0
$268$	−6.32688	−0.386476
$269$	20.9888	1.27971	0.639854	−	0.768497i	$-0.278995\pi$
$269$	20.9888	1.27971	0.639854	+	0.768497i	$0.278995\pi$
$270$	0	0
$271$	−20.8541	−1.26680	−0.633398	−	0.773826i	$-0.718340\pi$
$271$	−20.8541	−1.26680	−0.633398	+	0.773826i	$0.718340\pi$
$272$	−10.0902	−0.611806
$273$	0	0
$274$	10.4086	0.628808
$275$	19.5623	1.17965
$276$	0	0
$277$	−12.7984	−0.768980	−0.384490	−	0.923129i	$-0.625623\pi$
$277$	−12.7984	−0.768980	−0.384490	+	0.923129i	$0.625623\pi$
$278$	−26.0746	−1.56385
$279$	0	0
$280$	−1.17557	−0.0702538
$281$	26.4176	1.57594	0.787970	−	0.615713i	$-0.211132\pi$
$281$	26.4176	1.57594	0.787970	+	0.615713i	$0.211132\pi$
$282$	0	0
$283$	19.2148	1.14220	0.571100	−	0.820880i	$-0.306517\pi$
$283$	19.2148	1.14220	0.571100	+	0.820880i	$0.306517\pi$
$284$	1.73060	0.102692
$285$	0	0
$286$	−2.23607	−0.132221
$287$	−3.52671	−0.208175
$288$	0	0
$289$	0.944272	0.0555454
$290$	3.81966	0.224298
$291$	0	0
$292$	7.85410	0.459627
$293$	16.5640	0.967679	0.483839	−	0.875157i	$-0.339242\pi$
$293$	16.5640	0.967679	0.483839	+	0.875157i	$0.339242\pi$
$294$	0	0
$295$	−7.95148	−0.462953
$296$	26.1803	1.52170
$297$	0	0
$298$	4.25325	0.246384
$299$	1.69011	0.0977416
$300$	0	0
$301$	−2.32624	−0.134082
$302$	−22.3607	−1.28671
$303$	0	0
$304$	0	0
$305$	−4.47214	−0.256074
$306$	0	0
$307$	−3.42071	−0.195230	−0.0976151	−	0.995224i	$-0.531121\pi$
$307$	−3.42071	−0.195230	−0.0976151	+	0.995224i	$0.531121\pi$
$308$	1.61803	0.0921960
$309$	0	0
$310$	2.43769	0.138452
$311$	−8.05573	−0.456798	−0.228399	−	0.973568i	$-0.573349\pi$
$311$	−8.05573	−0.456798	−0.228399	+	0.973568i	$0.573349\pi$
$312$	0	0
$313$	22.1803	1.25371	0.626853	−	0.779137i	$-0.284342\pi$
$313$	22.1803	1.25371	0.626853	+	0.779137i	$0.284342\pi$
$314$	−7.54294	−0.425673
$315$	0	0
$316$	8.05748	0.453269
$317$	4.42477	0.248520	0.124260	−	0.992250i	$-0.460344\pi$
$317$	4.42477	0.248520	0.124260	+	0.992250i	$0.460344\pi$
$318$	0	0
$319$	−22.2703	−1.24690
$320$	5.38197	0.300861
$321$	0	0
$322$	2.73466	0.152396
$323$	0	0
$324$	0	0
$325$	−2.07363	−0.115024
$326$	−19.1926	−1.06298
$327$	0	0
$328$	17.5623	0.969716
$329$	−6.56231	−0.361792
$330$	0	0
$331$	−10.6861	−0.587363	−0.293682	−	0.955903i	$-0.594881\pi$
$331$	−10.6861	−0.587363	−0.293682	+	0.955903i	$0.594881\pi$
$332$	4.90983	0.269462
$333$	0	0
$334$	10.6525	0.582878
$335$	6.32688	0.345674
$336$	0	0
$337$	−10.4086	−0.566994	−0.283497	−	0.958973i	$-0.591495\pi$
$337$	−10.4086	−0.566994	−0.283497	+	0.958973i	$0.591495\pi$
$338$	−15.0454	−0.818361
$339$	0	0
$340$	−1.61803	−0.0877502
$341$	−14.2128	−0.769669
$342$	0	0
$343$	−8.41641	−0.454443
$344$	11.5842	0.624578
$345$	0	0
$346$	18.2918	0.983373
$347$	−15.4164	−0.827596	−0.413798	−	0.910369i	$-0.635798\pi$
$347$	−15.4164	−0.827596	−0.413798	+	0.910369i	$0.635798\pi$
$348$	0	0
$349$	−5.20163	−0.278437	−0.139218	−	0.990262i	$-0.544459\pi$
$349$	−5.20163	−0.278437	−0.139218	+	0.990262i	$0.544459\pi$
$350$	−3.35520	−0.179343
$351$	0	0
$352$	−14.2128	−0.757547
$353$	21.3607	1.13691	0.568457	−	0.822713i	$-0.307541\pi$
$353$	21.3607	1.13691	0.568457	+	0.822713i	$0.307541\pi$
$354$	0	0
$355$	−1.73060	−0.0918507
$356$	−6.15537	−0.326234
$357$	0	0
$358$	−1.38197	−0.0730392
$359$	−6.50658	−0.343404	−0.171702	−	0.985149i	$-0.554927\pi$
$359$	−6.50658	−0.343404	−0.171702	+	0.985149i	$0.554927\pi$
$360$	0	0
$361$	0	0
$362$	21.1803	1.11321
$363$	0	0
$364$	−0.171513	−0.00898975
$365$	−7.85410	−0.411102
$366$	0	0
$367$	4.88854	0.255180	0.127590	−	0.991827i	$-0.459276\pi$
$367$	4.88854	0.255180	0.127590	+	0.991827i	$0.459276\pi$
$368$	−8.96556	−0.467362
$369$	0	0
$370$	−6.18034	−0.321301
$371$	−6.26137	−0.325074
$372$	0	0
$373$	−20.0907	−1.04026	−0.520128	−	0.854088i	$-0.674116\pi$
$373$	−20.0907	−1.04026	−0.520128	+	0.854088i	$0.674116\pi$
$374$	−21.0948	−1.09078
$375$	0	0
$376$	32.6789	1.68529
$377$	2.36068	0.121581
$378$	0	0
$379$	6.15537	0.316180	0.158090	−	0.987425i	$-0.449466\pi$
$379$	6.15537	0.316180	0.158090	+	0.987425i	$0.449466\pi$
$380$	0	0
$381$	0	0
$382$	−9.89408	−0.506225
$383$	31.6094	1.61516	0.807582	−	0.589755i	$-0.200776\pi$
$383$	31.6094	1.61516	0.807582	+	0.589755i	$0.200776\pi$
$384$	0	0
$385$	−1.61803	−0.0824626
$386$	−6.18034	−0.314571
$387$	0	0
$388$	−1.79611	−0.0911838
$389$	35.1803	1.78371	0.891857	−	0.452317i	$-0.149403\pi$
$389$	35.1803	1.78371	0.891857	+	0.452317i	$0.149403\pi$
$390$	0	0
$391$	15.9443	0.806336
$392$	20.3682	1.02875
$393$	0	0
$394$	18.5066	0.932349
$395$	−8.05748	−0.405416
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	1.38757	0.0695527
$399$	0	0
$400$	11.0000	0.550000
$401$	5.42882	0.271103	0.135551	−	0.990770i	$-0.456719\pi$
$401$	5.42882	0.271103	0.135551	+	0.990770i	$0.456719\pi$
$402$	0	0
$403$	1.50658	0.0750480
$404$	−10.9098	−0.542784
$405$	0	0
$406$	3.81966	0.189567
$407$	36.0341	1.78615
$408$	0	0
$409$	31.6749	1.56622	0.783111	−	0.621882i	$-0.213631\pi$
$409$	31.6749	1.56622	0.783111	+	0.621882i	$0.213631\pi$
$410$	−4.14590	−0.204751
$411$	0	0
$412$	−7.77997	−0.383291
$413$	−7.95148	−0.391267
$414$	0	0
$415$	−4.90983	−0.241014
$416$	1.50658	0.0738661
$417$	0	0
$418$	0	0
$419$	7.29180	0.356228	0.178114	−	0.984010i	$-0.443000\pi$
$419$	7.29180	0.356228	0.178114	+	0.984010i	$0.443000\pi$
$420$	0	0
$421$	30.1563	1.46973	0.734864	−	0.678215i	$-0.237246\pi$
$421$	30.1563	1.46973	0.734864	+	0.678215i	$0.237246\pi$
$422$	9.79837	0.476977
$423$	0	0
$424$	31.1803	1.51425
$425$	−19.5623	−0.948911
$426$	0	0
$427$	−4.47214	−0.216422
$428$	−4.25325	−0.205589
$429$	0	0
$430$	−2.73466	−0.131877
$431$	−24.0009	−1.15608	−0.578042	−	0.816007i	$-0.696183\pi$
$431$	−24.0009	−1.15608	−0.578042	+	0.816007i	$0.696183\pi$
$432$	0	0
$433$	−22.6134	−1.08673	−0.543364	−	0.839497i	$-0.682850\pi$
$433$	−22.6134	−1.08673	−0.543364	+	0.839497i	$0.682850\pi$
$434$	2.43769	0.117013
$435$	0	0
$436$	2.24514	0.107523
$437$	0	0
$438$	0	0
$439$	20.9888	1.00174	0.500869	−	0.865523i	$-0.333014\pi$
$439$	20.9888	1.00174	0.500869	+	0.865523i	$0.333014\pi$
$440$	8.05748	0.384125
$441$	0	0
$442$	2.23607	0.106359
$443$	−4.81966	−0.228989	−0.114494	−	0.993424i	$-0.536525\pi$
$443$	−4.81966	−0.228989	−0.114494	+	0.993424i	$0.536525\pi$
$444$	0	0
$445$	6.15537	0.291792
$446$	1.58359	0.0749853
$447$	0	0
$448$	5.38197	0.254274
$449$	29.2582	1.38078	0.690391	−	0.723437i	$-0.257439\pi$
$449$	29.2582	1.38078	0.690391	+	0.723437i	$0.257439\pi$
$450$	0	0
$451$	24.1724	1.13824
$452$	4.87380	0.229244
$453$	0	0
$454$	21.5066	1.00935
$455$	0.171513	0.00804067
$456$	0	0
$457$	−4.56231	−0.213416	−0.106708	−	0.994290i	$-0.534031\pi$
$457$	−4.56231	−0.213416	−0.106708	+	0.994290i	$0.534031\pi$
$458$	11.3472	0.530218
$459$	0	0
$460$	−1.43769	−0.0670328
$461$	21.9098	1.02044	0.510221	−	0.860043i	$-0.329564\pi$
$461$	21.9098	1.02044	0.510221	+	0.860043i	$0.329564\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−12.5227	−0.581353
$465$	0	0
$466$	−28.0827	−1.30090
$467$	6.67376	0.308825	0.154412	−	0.988006i	$-0.450652\pi$
$467$	6.67376	0.308825	0.154412	+	0.988006i	$0.450652\pi$
$468$	0	0
$469$	6.32688	0.292148
$470$	−7.71445	−0.355841
$471$	0	0
$472$	39.5967	1.82259
$473$	15.9443	0.733118
$474$	0	0
$475$	0	0
$476$	−1.61803	−0.0741625
$477$	0	0
$478$	0.171513	0.00784484
$479$	23.2148	1.06071	0.530355	−	0.847776i	$-0.322059\pi$
$479$	23.2148	1.06071	0.530355	+	0.847776i	$0.322059\pi$
$480$	0	0
$481$	−3.81966	−0.174162
$482$	12.2361	0.557338
$483$	0	0
$484$	−4.29180	−0.195082
$485$	1.79611	0.0815572
$486$	0	0
$487$	−33.2995	−1.50894	−0.754472	−	0.656332i	$-0.772107\pi$
$487$	−33.2995	−1.50894	−0.754472	+	0.656332i	$0.772107\pi$
$488$	22.2703	1.00813
$489$	0	0
$490$	−4.80828	−0.217216
$491$	1.70820	0.0770902	0.0385451	−	0.999257i	$-0.487728\pi$
$491$	1.70820	0.0770902	0.0385451	+	0.999257i	$0.487728\pi$
$492$	0	0
$493$	22.2703	1.00301
$494$	0	0
$495$	0	0
$496$	−7.99197	−0.358850
$497$	−1.73060	−0.0776280
$498$	0	0
$499$	−13.7639	−0.616158	−0.308079	−	0.951361i	$-0.599686\pi$
$499$	−13.7639	−0.616158	−0.308079	+	0.951361i	$0.599686\pi$
$500$	3.67376	0.164296
$501$	0	0
$502$	−33.4055	−1.49096
$503$	−3.96556	−0.176815	−0.0884077	−	0.996084i	$-0.528178\pi$
$503$	−3.96556	−0.176815	−0.0884077	+	0.996084i	$0.528178\pi$
$504$	0	0
$505$	10.9098	0.485481
$506$	−18.7436	−0.833255
$507$	0	0
$508$	−2.90617	−0.128940
$509$	10.4086	0.461354	0.230677	−	0.973030i	$-0.425906\pi$
$509$	10.4086	0.461354	0.230677	+	0.973030i	$0.425906\pi$
$510$	0	0
$511$	−7.85410	−0.347445
$512$	−22.6538	−1.00117
$513$	0	0
$514$	−13.2148	−0.582879
$515$	7.77997	0.342826
$516$	0	0
$517$	44.9787	1.97816
$518$	−6.18034	−0.271549
$519$	0	0
$520$	−0.854102	−0.0374548
$521$	−27.1441	−1.18921	−0.594603	−	0.804020i	$-0.702691\pi$
$521$	−27.1441	−1.18921	−0.594603	+	0.804020i	$0.702691\pi$
$522$	0	0
$523$	−26.6296	−1.16443	−0.582215	−	0.813035i	$-0.697814\pi$
$523$	−26.6296	−1.16443	−0.582215	+	0.813035i	$0.697814\pi$
$524$	−7.56231	−0.330361
$525$	0	0
$526$	−33.4710	−1.45941
$527$	14.2128	0.619121
$528$	0	0
$529$	−8.83282	−0.384035
$530$	−7.36068	−0.319727
$531$	0	0
$532$	0	0
$533$	−2.56231	−0.110986
$534$	0	0
$535$	4.25325	0.183884
$536$	−31.5066	−1.36088
$537$	0	0
$538$	24.6738	1.06376
$539$	28.0344	1.20753
$540$	0	0
$541$	13.4164	0.576816	0.288408	−	0.957508i	$-0.406874\pi$
$541$	13.4164	0.576816	0.288408	+	0.957508i	$0.406874\pi$
$542$	−24.5155	−1.05303
$543$	0	0
$544$	14.2128	0.609371
$545$	−2.24514	−0.0961712
$546$	0	0
$547$	−7.99197	−0.341712	−0.170856	−	0.985296i	$-0.554653\pi$
$547$	−7.99197	−0.341712	−0.170856	+	0.985296i	$0.554653\pi$
$548$	−5.47214	−0.233758
$549$	0	0
$550$	22.9969	0.980590
$551$	0	0
$552$	0	0
$553$	−8.05748	−0.342639
$554$	−15.0454	−0.639217
$555$	0	0
$556$	13.7082	0.581357
$557$	−0.944272	−0.0400101	−0.0200050	−	0.999800i	$-0.506368\pi$
$557$	−0.944272	−0.0400101	−0.0200050	+	0.999800i	$0.506368\pi$
$558$	0	0
$559$	−1.69011	−0.0714841
$560$	−0.909830	−0.0384473
$561$	0	0
$562$	31.0557	1.31001
$563$	−24.0009	−1.01152	−0.505759	−	0.862675i	$-0.668788\pi$
$563$	−24.0009	−1.01152	−0.505759	+	0.862675i	$0.668788\pi$
$564$	0	0
$565$	−4.87380	−0.205042
$566$	22.5883	0.949458
$567$	0	0
$568$	8.61803	0.361605
$569$	41.9120	1.75704	0.878521	−	0.477703i	$-0.158531\pi$
$569$	41.9120	1.75704	0.878521	+	0.477703i	$0.158531\pi$
$570$	0	0
$571$	−23.4164	−0.979946	−0.489973	−	0.871738i	$-0.662993\pi$
$571$	−23.4164	−0.979946	−0.489973	+	0.871738i	$0.662993\pi$
$572$	1.17557	0.0491531
$573$	0	0
$574$	−4.14590	−0.173046
$575$	−17.3820	−0.724878
$576$	0	0
$577$	−18.1246	−0.754537	−0.377269	−	0.926104i	$-0.623137\pi$
$577$	−18.1246	−0.754537	−0.377269	+	0.926104i	$0.623137\pi$
$578$	1.11006	0.0461723
$579$	0	0
$580$	−2.00811	−0.0833824
$581$	−4.90983	−0.203694
$582$	0	0
$583$	42.9161	1.77740
$584$	39.1118	1.61846
$585$	0	0
$586$	19.4721	0.804387
$587$	4.50658	0.186006	0.0930032	−	0.995666i	$-0.470353\pi$
$587$	4.50658	0.186006	0.0930032	+	0.995666i	$0.470353\pi$
$588$	0	0
$589$	0	0
$590$	−9.34752	−0.384831
$591$	0	0
$592$	20.2622	0.832772
$593$	−7.41641	−0.304555	−0.152278	−	0.988338i	$-0.548661\pi$
$593$	−7.41641	−0.304555	−0.152278	+	0.988338i	$0.548661\pi$
$594$	0	0
$595$	1.61803	0.0663329
$596$	−2.23607	−0.0915929
$597$	0	0
$598$	1.98684	0.0812481
$599$	35.5851	1.45397	0.726984	−	0.686654i	$-0.240921\pi$
$599$	35.5851	1.45397	0.726984	+	0.686654i	$0.240921\pi$
$600$	0	0
$601$	−37.7647	−1.54046	−0.770228	−	0.637769i	$-0.779858\pi$
$601$	−37.7647	−1.54046	−0.770228	+	0.637769i	$0.779858\pi$
$602$	−2.73466	−0.111456
$603$	0	0
$604$	11.7557	0.478333
$605$	4.29180	0.174486
$606$	0	0
$607$	29.3893	1.19287	0.596437	−	0.802660i	$-0.296583\pi$
$607$	29.3893	1.19287	0.596437	+	0.802660i	$0.296583\pi$
$608$	0	0
$609$	0	0
$610$	−5.25731	−0.212862
$611$	−4.76779	−0.192884
$612$	0	0
$613$	23.4721	0.948031	0.474015	−	0.880517i	$-0.342804\pi$
$613$	23.4721	0.948031	0.474015	+	0.880517i	$0.342804\pi$
$614$	−4.02129	−0.162286
$615$	0	0
$616$	8.05748	0.324645
$617$	−43.1246	−1.73613	−0.868066	−	0.496449i	$-0.834637\pi$
$617$	−43.1246	−1.73613	−0.868066	+	0.496449i	$0.834637\pi$
$618$	0	0
$619$	2.09017	0.0840110	0.0420055	−	0.999117i	$-0.486625\pi$
$619$	2.09017	0.0840110	0.0420055	+	0.999117i	$0.486625\pi$
$620$	−1.28157	−0.0514692
$621$	0	0
$622$	−9.47008	−0.379715
$623$	6.15537	0.246610
$624$	0	0
$625$	19.4164	0.776656
$626$	26.0746	1.04215
$627$	0	0
$628$	3.96556	0.158243
$629$	−36.0341	−1.43678
$630$	0	0
$631$	16.8328	0.670104	0.335052	−	0.942200i	$-0.391246\pi$
$631$	16.8328	0.670104	0.335052	+	0.942200i	$0.391246\pi$
$632$	40.1246	1.59607
$633$	0	0
$634$	5.20163	0.206583
$635$	2.90617	0.115328
$636$	0	0
$637$	−2.97168	−0.117742
$638$	−26.1803	−1.03649
$639$	0	0
$640$	2.17963	0.0861573
$641$	−37.3157	−1.47388	−0.736941	−	0.675957i	$-0.763731\pi$
$641$	−37.3157	−1.47388	−0.736941	+	0.675957i	$0.763731\pi$
$642$	0	0
$643$	33.5623	1.32357	0.661784	−	0.749695i	$-0.269800\pi$
$643$	33.5623	1.32357	0.661784	+	0.749695i	$0.269800\pi$
$644$	−1.43769	−0.0566531
$645$	0	0
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	54.5002	2.13932
$650$	−2.43769	−0.0956142
$651$	0	0
$652$	10.0902	0.395162
$653$	17.9443	0.702214	0.351107	−	0.936335i	$-0.385805\pi$
$653$	17.9443	0.702214	0.351107	+	0.936335i	$0.385805\pi$
$654$	0	0
$655$	7.56231	0.295484
$656$	13.5923	0.530690
$657$	0	0
$658$	−7.71445	−0.300741
$659$	36.2057	1.41037	0.705186	−	0.709022i	$-0.250863\pi$
$659$	36.2057	1.41037	0.705186	+	0.709022i	$0.250863\pi$
$660$	0	0
$661$	−29.3893	−1.14311	−0.571555	−	0.820564i	$-0.693660\pi$
$661$	−29.3893	−1.14311	−0.571555	+	0.820564i	$0.693660\pi$
$662$	−12.5623	−0.488248
$663$	0	0
$664$	24.4500	0.948842
$665$	0	0
$666$	0	0
$667$	19.7882	0.766201
$668$	−5.60034	−0.216684
$669$	0	0
$670$	7.43769	0.287343
$671$	30.6525	1.18333
$672$	0	0
$673$	−47.2348	−1.82077	−0.910384	−	0.413764i	$-0.864214\pi$
$673$	−47.2348	−1.82077	−0.910384	+	0.413764i	$0.864214\pi$
$674$	−12.2361	−0.471316
$675$	0	0
$676$	7.90983	0.304224
$677$	−7.60845	−0.292417	−0.146208	−	0.989254i	$-0.546707\pi$
$677$	−7.60845	−0.292417	−0.146208	+	0.989254i	$0.546707\pi$
$678$	0	0
$679$	1.79611	0.0689284
$680$	−8.05748	−0.308990
$681$	0	0
$682$	−16.7082	−0.639790
$683$	−30.8828	−1.18170	−0.590849	−	0.806782i	$-0.701207\pi$
$683$	−30.8828	−1.18170	−0.590849	+	0.806782i	$0.701207\pi$
$684$	0	0
$685$	5.47214	0.209080
$686$	−9.89408	−0.377758
$687$	0	0
$688$	8.96556	0.341809
$689$	−4.54915	−0.173309
$690$	0	0
$691$	−50.7771	−1.93165	−0.965826	−	0.259192i	$-0.916544\pi$
$691$	−50.7771	−1.93165	−0.965826	+	0.259192i	$0.916544\pi$
$692$	−9.61657	−0.365567
$693$	0	0
$694$	−18.1231	−0.687942
$695$	−13.7082	−0.519982
$696$	0	0
$697$	−24.1724	−0.915596
$698$	−6.11488	−0.231452
$699$	0	0
$700$	1.76393	0.0666704
$701$	19.1803	0.724431	0.362216	−	0.932094i	$-0.382020\pi$
$701$	19.1803	0.724431	0.362216	+	0.932094i	$0.382020\pi$
$702$	0	0
$703$	0	0
$704$	−36.8885	−1.39029
$705$	0	0
$706$	25.1110	0.945064
$707$	10.9098	0.410306
$708$	0	0
$709$	37.6869	1.41536	0.707681	−	0.706532i	$-0.249741\pi$
$709$	37.6869	1.41536	0.707681	+	0.706532i	$0.249741\pi$
$710$	−2.03444	−0.0763512
$711$	0	0
$712$	−30.6525	−1.14875
$713$	12.6287	0.472950
$714$	0	0
$715$	−1.17557	−0.0439638
$716$	0.726543	0.0271522
$717$	0	0
$718$	−7.64894	−0.285456
$719$	45.3262	1.69038	0.845192	−	0.534463i	$-0.179486\pi$
$719$	45.3262	1.69038	0.845192	+	0.534463i	$0.179486\pi$
$720$	0	0
$721$	7.77997	0.289741
$722$	0	0
$723$	0	0
$724$	−11.1352	−0.413835
$725$	−24.2784	−0.901679
$726$	0	0
$727$	−40.1591	−1.48942	−0.744708	−	0.667390i	$-0.767411\pi$
$727$	−40.1591	−1.48942	−0.744708	+	0.667390i	$0.767411\pi$
$728$	−0.854102	−0.0316551
$729$	0	0
$730$	−9.23305	−0.341731
$731$	−15.9443	−0.589720
$732$	0	0
$733$	35.1803	1.29942	0.649708	−	0.760184i	$-0.274891\pi$
$733$	35.1803	1.29942	0.649708	+	0.760184i	$0.274891\pi$
$734$	5.74683	0.212119
$735$	0	0
$736$	12.6287	0.465501
$737$	−43.3651	−1.59737
$738$	0	0
$739$	19.7984	0.728295	0.364147	−	0.931341i	$-0.381360\pi$
$739$	19.7984	0.728295	0.364147	+	0.931341i	$0.381360\pi$
$740$	3.24920	0.119443
$741$	0	0
$742$	−7.36068	−0.270219
$743$	−1.62460	−0.0596007	−0.0298004	−	0.999556i	$-0.509487\pi$
$743$	−1.62460	−0.0596007	−0.0298004	+	0.999556i	$0.509487\pi$
$744$	0	0
$745$	2.23607	0.0819232
$746$	−23.6180	−0.864718
$747$	0	0
$748$	11.0902	0.405497
$749$	4.25325	0.155411
$750$	0	0
$751$	17.0785	0.623204	0.311602	−	0.950213i	$-0.399134\pi$
$751$	17.0785	0.623204	0.311602	+	0.950213i	$0.399134\pi$
$752$	25.2918	0.922297
$753$	0	0
$754$	2.77515	0.101065
$755$	−11.7557	−0.427834
$756$	0	0
$757$	−12.1246	−0.440677	−0.220338	−	0.975424i	$-0.570716\pi$
$757$	−12.1246	−0.440677	−0.220338	+	0.975424i	$0.570716\pi$
$758$	7.23607	0.262826
$759$	0	0
$760$	0	0
$761$	43.3050	1.56980	0.784902	−	0.619620i	$-0.212713\pi$
$761$	43.3050	1.56980	0.784902	+	0.619620i	$0.212713\pi$
$762$	0	0
$763$	−2.24514	−0.0812795
$764$	5.20163	0.188188
$765$	0	0
$766$	37.1591	1.34261
$767$	−5.77709	−0.208599
$768$	0	0
$769$	1.96556	0.0708798	0.0354399	−	0.999372i	$-0.488717\pi$
$769$	1.96556	0.0708798	0.0354399	+	0.999372i	$0.488717\pi$
$770$	−1.90211	−0.0685474
$771$	0	0
$772$	3.24920	0.116941
$773$	−19.0866	−0.686499	−0.343249	−	0.939244i	$-0.611528\pi$
$773$	−19.0866	−0.686499	−0.343249	+	0.939244i	$0.611528\pi$
$774$	0	0
$775$	−15.4944	−0.556576
$776$	−8.94427	−0.321081
$777$	0	0
$778$	41.3570	1.48272
$779$	0	0
$780$	0	0
$781$	11.8617	0.424445
$782$	18.7436	0.670270
$783$	0	0
$784$	15.7639	0.562998
$785$	−3.96556	−0.141537
$786$	0	0
$787$	17.8456	0.636126	0.318063	−	0.948070i	$-0.396968\pi$
$787$	17.8456	0.636126	0.318063	+	0.948070i	$0.396968\pi$
$788$	−9.72949	−0.346599
$789$	0	0
$790$	−9.47214	−0.337003
$791$	−4.87380	−0.173292
$792$	0	0
$793$	−3.24920	−0.115382
$794$	−8.22899	−0.292036
$795$	0	0
$796$	−0.729490	−0.0258561
$797$	−27.0786	−0.959174	−0.479587	−	0.877494i	$-0.659213\pi$
$797$	−27.0786	−0.959174	−0.479587	+	0.877494i	$0.659213\pi$
$798$	0	0
$799$	−44.9787	−1.59123
$800$	−15.4944	−0.547810
$801$	0	0
$802$	6.38197	0.225355
$803$	53.8328	1.89972
$804$	0	0
$805$	1.43769	0.0506721
$806$	1.77109	0.0623839
$807$	0	0
$808$	−54.3287	−1.91128
$809$	−29.5410	−1.03861	−0.519303	−	0.854590i	$-0.673809\pi$
$809$	−29.5410	−1.03861	−0.519303	+	0.854590i	$0.673809\pi$
$810$	0	0
$811$	−22.4418	−0.788040	−0.394020	−	0.919102i	$-0.628916\pi$
$811$	−22.4418	−0.788040	−0.394020	+	0.919102i	$0.628916\pi$
$812$	−2.00811	−0.0704710
$813$	0	0
$814$	42.3607	1.48474
$815$	−10.0902	−0.353443
$816$	0	0
$817$	0	0
$818$	37.2361	1.30193
$819$	0	0
$820$	2.17963	0.0761159
$821$	45.3262	1.58190	0.790948	−	0.611883i	$-0.209588\pi$
$821$	45.3262	1.58190	0.790948	+	0.611883i	$0.209588\pi$
$822$	0	0
$823$	−22.3050	−0.777502	−0.388751	−	0.921343i	$-0.627093\pi$
$823$	−22.3050	−0.777502	−0.388751	+	0.921343i	$0.627093\pi$
$824$	−38.7426	−1.34966
$825$	0	0
$826$	−9.34752	−0.325242
$827$	−13.8293	−0.480893	−0.240446	−	0.970662i	$-0.577294\pi$
$827$	−13.8293	−0.480893	−0.240446	+	0.970662i	$0.577294\pi$
$828$	0	0
$829$	−29.6418	−1.02950	−0.514750	−	0.857340i	$-0.672115\pi$
$829$	−29.6418	−1.02950	−0.514750	+	0.857340i	$0.672115\pi$
$830$	−5.77185	−0.200344
$831$	0	0
$832$	3.91023	0.135563
$833$	−28.0344	−0.971336
$834$	0	0
$835$	5.60034	0.193808
$836$	0	0
$837$	0	0
$838$	8.57202	0.296116
$839$	−57.6184	−1.98921	−0.994604	−	0.103741i	$-0.966919\pi$
$839$	−57.6184	−1.98921	−0.994604	+	0.103741i	$0.966919\pi$
$840$	0	0
$841$	−1.36068	−0.0469200
$842$	35.4508	1.22172
$843$	0	0
$844$	−5.15131	−0.177315
$845$	−7.90983	−0.272106
$846$	0	0
$847$	4.29180	0.147468
$848$	24.1320	0.828695
$849$	0	0
$850$	−22.9969	−0.788786
$851$	−32.0179	−1.09756
$852$	0	0
$853$	−15.5967	−0.534022	−0.267011	−	0.963693i	$-0.586036\pi$
$853$	−15.5967	−0.534022	−0.267011	+	0.963693i	$0.586036\pi$
$854$	−5.25731	−0.179901
$855$	0	0
$856$	−21.1803	−0.723929
$857$	−3.63271	−0.124091	−0.0620455	−	0.998073i	$-0.519762\pi$
$857$	−3.63271	−0.124091	−0.0620455	+	0.998073i	$0.519762\pi$
$858$	0	0
$859$	42.1591	1.43845	0.719224	−	0.694778i	$-0.244497\pi$
$859$	42.1591	1.43845	0.719224	+	0.694778i	$0.244497\pi$
$860$	1.43769	0.0490250
$861$	0	0
$862$	−28.2148	−0.960999
$863$	−46.1653	−1.57148	−0.785742	−	0.618555i	$-0.787719\pi$
$863$	−46.1653	−1.57148	−0.785742	+	0.618555i	$0.787719\pi$
$864$	0	0
$865$	9.61657	0.326973
$866$	−26.5836	−0.903347
$867$	0	0
$868$	−1.28157	−0.0434994
$869$	55.2268	1.87344
$870$	0	0
$871$	4.59675	0.155755
$872$	11.1803	0.378614
$873$	0	0
$874$	0	0
$875$	−3.67376	−0.124196
$876$	0	0
$877$	41.8060	1.41169	0.705844	−	0.708367i	$-0.250568\pi$
$877$	41.8060	1.41169	0.705844	+	0.708367i	$0.250568\pi$
$878$	24.6738	0.832699
$879$	0	0
$880$	6.23607	0.210218
$881$	−40.3607	−1.35979	−0.679893	−	0.733311i	$-0.737974\pi$
$881$	−40.3607	−1.35979	−0.679893	+	0.733311i	$0.737974\pi$
$882$	0	0
$883$	13.4721	0.453373	0.226687	−	0.973968i	$-0.427211\pi$
$883$	13.4721	0.453373	0.226687	+	0.973968i	$0.427211\pi$
$884$	−1.17557	−0.0395387
$885$	0	0
$886$	−5.66585	−0.190348
$887$	30.4338	1.02187	0.510934	−	0.859620i	$-0.329300\pi$
$887$	30.4338	1.02187	0.510934	+	0.859620i	$0.329300\pi$
$888$	0	0
$889$	2.90617	0.0974698
$890$	7.23607	0.242554
$891$	0	0
$892$	−0.832544	−0.0278756
$893$	0	0
$894$	0	0
$895$	−0.726543	−0.0242856
$896$	2.17963	0.0728162
$897$	0	0
$898$	34.3951	1.14778
$899$	17.6393	0.588304
$900$	0	0
$901$	−42.9161	−1.42974
$902$	28.4164	0.946163
$903$	0	0
$904$	24.2705	0.807225
$905$	11.1352	0.370145
$906$	0	0
$907$	33.2995	1.10569	0.552846	−	0.833284i	$-0.313542\pi$
$907$	33.2995	1.10569	0.552846	+	0.833284i	$0.313542\pi$
$908$	−11.3067	−0.375225
$909$	0	0
$910$	0.201626	0.00668384
$911$	−34.6466	−1.14789	−0.573946	−	0.818893i	$-0.694588\pi$
$911$	−34.6466	−1.14789	−0.573946	+	0.818893i	$0.694588\pi$
$912$	0	0
$913$	33.6525	1.11373
$914$	−5.36331	−0.177403
$915$	0	0
$916$	−5.96556	−0.197108
$917$	7.56231	0.249729
$918$	0	0
$919$	−16.0557	−0.529630	−0.264815	−	0.964299i	$-0.585311\pi$
$919$	−16.0557	−0.529630	−0.264815	+	0.964299i	$0.585311\pi$
$920$	−7.15942	−0.236039
$921$	0	0
$922$	25.7565	0.848247
$923$	−1.25735	−0.0413863
$924$	0	0
$925$	39.2833	1.29163
$926$	−28.2137	−0.927159
$927$	0	0
$928$	17.6393	0.579039
$929$	−37.1803	−1.21985	−0.609924	−	0.792460i	$-0.708800\pi$
$929$	−37.1803	−1.21985	−0.609924	+	0.792460i	$0.708800\pi$
$930$	0	0
$931$	0	0
$932$	14.7639	0.483609
$933$	0	0
$934$	7.84548	0.256712
$935$	−11.0902	−0.362687
$936$	0	0
$937$	4.18034	0.136566	0.0682829	−	0.997666i	$-0.478248\pi$
$937$	4.18034	0.136566	0.0682829	+	0.997666i	$0.478248\pi$
$938$	7.43769	0.242849
$939$	0	0
$940$	4.05573	0.132283
$941$	−7.50245	−0.244573	−0.122286	−	0.992495i	$-0.539023\pi$
$941$	−7.50245	−0.244573	−0.122286	+	0.992495i	$0.539023\pi$
$942$	0	0
$943$	−21.4783	−0.699429
$944$	30.6458	0.997436
$945$	0	0
$946$	18.7436	0.609408
$947$	−15.8885	−0.516308	−0.258154	−	0.966104i	$-0.583114\pi$
$947$	−15.8885	−0.516308	−0.258154	+	0.966104i	$0.583114\pi$
$948$	0	0
$949$	−5.70634	−0.185236
$950$	0	0
$951$	0	0
$952$	−8.05748	−0.261144
$953$	10.0001	0.323934	0.161967	−	0.986796i	$-0.448216\pi$
$953$	10.0001	0.323934	0.161967	+	0.986796i	$0.448216\pi$
$954$	0	0
$955$	−5.20163	−0.168321
$956$	−0.0901699	−0.00291630
$957$	0	0
$958$	27.2906	0.881720
$959$	5.47214	0.176704
$960$	0	0
$961$	−19.7426	−0.636860
$962$	−4.49028	−0.144772
$963$	0	0
$964$	−6.43288	−0.207189
$965$	−3.24920	−0.104595
$966$	0	0
$967$	27.2705	0.876960	0.438480	−	0.898741i	$-0.355517\pi$
$967$	27.2705	0.876960	0.438480	+	0.898741i	$0.355517\pi$
$968$	−21.3723	−0.686931
$969$	0	0
$970$	2.11146	0.0677948
$971$	−5.42882	−0.174219	−0.0871096	−	0.996199i	$-0.527763\pi$
$971$	−5.42882	−0.174219	−0.0871096	+	0.996199i	$0.527763\pi$
$972$	0	0
$973$	−13.7082	−0.439465
$974$	−39.1459	−1.25432
$975$	0	0
$976$	17.2361	0.551713
$977$	−26.6296	−0.851956	−0.425978	−	0.904734i	$-0.640070\pi$
$977$	−26.6296	−0.851956	−0.425978	+	0.904734i	$0.640070\pi$
$978$	0	0
$979$	−42.1895	−1.34838
$980$	2.52786	0.0807497
$981$	0	0
$982$	2.00811	0.0640815
$983$	−26.4176	−0.842590	−0.421295	−	0.906924i	$-0.638424\pi$
$983$	−26.4176	−0.842590	−0.421295	+	0.906924i	$0.638424\pi$
$984$	0	0
$985$	9.72949	0.310007
$986$	26.1803	0.833752
$987$	0	0
$988$	0	0
$989$	−14.1672	−0.450490
$990$	0	0
$991$	−3.63271	−0.115397	−0.0576985	−	0.998334i	$-0.518376\pi$
$991$	−3.63271	−0.115397	−0.0576985	+	0.998334i	$0.518376\pi$
$992$	11.2574	0.357421
$993$	0	0
$994$	−2.03444	−0.0645286
$995$	0.729490	0.0231264
$996$	0	0
$997$	28.1246	0.890715	0.445358	−	0.895353i	$-0.353077\pi$
$997$	28.1246	0.890715	0.445358	+	0.895353i	$0.353077\pi$
$998$	−16.1805	−0.512184
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.ba.1.3	yes	4
3.2	odd	2		3249.2.a.bb.1.2	yes	4
19.18	odd	2	inner	3249.2.a.ba.1.2	✓	4
57.56	even	2		3249.2.a.bb.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3249.2.a.ba.1.2	✓	4	19.18	odd	2	inner
3249.2.a.ba.1.3	yes	4	1.1	even	1	trivial
3249.2.a.bb.1.2	yes	4	3.2	odd	2
3249.2.a.bb.1.3	yes	4	57.56	even	2

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:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17557 q^{2} -0.618034 q^{4} +0.618034 q^{5} +0.618034 q^{7} -3.07768 q^{8} +0.726543 q^{10} -4.23607 q^{11} +0.449028 q^{13} +0.726543 q^{14} -2.38197 q^{16} +4.23607 q^{17} -0.381966 q^{20} -4.97980 q^{22} +3.76393 q^{23} -4.61803 q^{25} +0.527864 q^{26} -0.381966 q^{28} +5.25731 q^{29} +3.35520 q^{31} +3.35520 q^{32} +4.97980 q^{34} +0.381966 q^{35} -8.50651 q^{37} -1.90211 q^{40} -5.70634 q^{41} -3.76393 q^{43} +2.61803 q^{44} +4.42477 q^{46} -10.6180 q^{47} -6.61803 q^{49} -5.42882 q^{50} -0.277515 q^{52} -10.1311 q^{53} -2.61803 q^{55} -1.90211 q^{56} +6.18034 q^{58} -12.8658 q^{59} -7.23607 q^{61} +3.94427 q^{62} +8.70820 q^{64} +0.277515 q^{65} +10.2371 q^{67} -2.61803 q^{68} +0.449028 q^{70} -2.80017 q^{71} -12.7082 q^{73} -10.0000 q^{74} -2.61803 q^{77} -13.0373 q^{79} -1.47214 q^{80} -6.70820 q^{82} -7.94427 q^{83} +2.61803 q^{85} -4.42477 q^{86} +13.0373 q^{88} +9.95959 q^{89} +0.277515 q^{91} -2.32624 q^{92} -12.4822 q^{94} +2.90617 q^{97} -7.77997 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{5} - 2 q^{7} - 8 q^{11} - 14 q^{16} + 8 q^{17} - 6 q^{20} + 24 q^{23} - 14 q^{25} + 20 q^{26} - 6 q^{28} + 6 q^{35} - 24 q^{43} + 6 q^{44} - 38 q^{47} - 22 q^{49} - 6 q^{55} - 20 q^{58}+ \cdots + 22 q^{92}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^5 - 2 * q^7 - 8 * q^11 - 14 * q^16 + 8 * q^17 - 6 * q^20 + 24 * q^23 - 14 * q^25 + 20 * q^26 - 6 * q^28 + 6 * q^35 - 24 * q^43 + 6 * q^44 - 38 * q^47 - 22 * q^49 - 6 * q^55 - 20 * q^58 - 20 * q^61 - 20 * q^62 + 8 * q^64 - 6 * q^68 - 24 * q^73 - 40 * q^74 - 6 * q^77 + 12 * q^80 + 4 * q^83 + 6 * q^85 + 22 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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