LMFDB - Embedded newform 3400.2.e.n.2449.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3400,2,Mod(2449,3400)] 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("3400.2449"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3400 = 2^{3} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3400.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0,0,0,0,-12,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$27.1491366872$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.251193590431744.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} + 8x^{8} - 2x^{7} + 12x^{6} - 50x^{5} + 106x^{4} - 20x^{3} + 4x^{2} - 28x + 98 $ x^10 - 4x^9 + 8x^8 - 2x^7 + 12x^6 - 50x^5 + 106x^4 - 20x^3 + 4x^2 - 28*x + 98
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.7
Root			$-0.670303 + 0.670303i$ of defining polynomial
Character	$\chi$	$=$	3400.2449
Dual form			3400.2.e.n.2449.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.50269i q^{3} -4.25913i q^{7} +0.741913 q^{9} -5.60408 q^{11} -5.09704i q^{13} -1.00000i q^{17} +3.67930 q^{19} +6.40016 q^{21} +2.08252i q^{23} +5.62295i q^{27} +5.88207 q^{29} -8.50808 q^{31} -8.42121i q^{33} +2.48191i q^{37} +7.65929 q^{39} -10.5633 q^{41} +10.0848i q^{43} +1.28105i q^{47} -11.1402 q^{49} +1.50269 q^{51} -1.32070i q^{53} +5.52886i q^{57} -11.7607 q^{59} -12.0814 q^{61} -3.15990i q^{63} -4.52695i q^{67} -3.12939 q^{69} -0.0273119 q^{71} -7.67930i q^{73} +23.8685i q^{77} -7.94712 q^{79} -6.22383 q^{81} -12.8840i q^{83} +8.83895i q^{87} -12.3228 q^{89} -21.7089 q^{91} -12.7850i q^{93} -5.70487i q^{97} -4.15774 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 12 q^{9} - 12 q^{11} + 12 q^{19} + 14 q^{21} - 12 q^{29} - 22 q^{31} - 18 q^{39} + 8 q^{41} - 60 q^{49} - 6 q^{51} - 28 q^{61} - 64 q^{69} - 10 q^{71} - 54 q^{79} - 22 q^{81} - 20 q^{89} - 126 q^{91}+ \cdots - 20 q^{99}+O(q^{100}) $ 10 * q - 12 * q^9 - 12 * q^11 + 12 * q^19 + 14 * q^21 - 12 * q^29 - 22 * q^31 - 18 * q^39 + 8 * q^41 - 60 * q^49 - 6 * q^51 - 28 * q^61 - 64 * q^69 - 10 * q^71 - 54 * q^79 - 22 * q^81 - 20 * q^89 - 126 * q^91 - 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3400\mathbb{Z}\right)^\times$.

$n$	$1601$	$1701$	$2177$	$2551$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.50269i	0.867580i	0.901014	+	0.433790i	$0.142824\pi$
$3$	1.50269i	0.867580i	−0.901014	+	0.433790i	$0.857176\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.25913i	− 1.60980i	−0.593411	−	0.804899i	$-0.702219\pi$
$7$	− 4.25913i	− 1.60980i	0.593411	−	0.804899i	$-0.297781\pi$
$8$	0	0
$9$	0.741913	0.247304
$10$	0	0
$11$	−5.60408	−1.68969	−0.844847	−	0.535008i	$-0.820309\pi$
$11$	−5.60408	−1.68969	−0.844847	+	0.535008i	$0.820309\pi$
$12$	0	0
$13$	− 5.09704i	− 1.41366i	−0.707381	−	0.706832i	$-0.750124\pi$
$13$	− 5.09704i	− 1.41366i	0.707381	−	0.706832i	$-0.249876\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.00000i	− 0.242536i
$17$	− 1.00000i	− 0.242536i
$18$	0	0
$19$	3.67930	0.844089	0.422045	−	0.906575i	$-0.361313\pi$
$19$	3.67930	0.844089	0.422045	+	0.906575i	$0.361313\pi$
$20$	0	0
$21$	6.40016	1.39663
$22$	0	0
$23$	2.08252i	0.434235i	0.976145	+	0.217118i	$0.0696655\pi$
$23$	2.08252i	0.434235i	−0.976145	+	0.217118i	$0.930334\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.62295i	1.08214i
$28$	0	0
$29$	5.88207	1.09227	0.546137	−	0.837696i	$-0.316098\pi$
$29$	5.88207	1.09227	0.546137	+	0.837696i	$0.316098\pi$
$30$	0	0
$31$	−8.50808	−1.52810	−0.764048	−	0.645159i	$-0.776791\pi$
$31$	−8.50808	−1.52810	−0.764048	+	0.645159i	$0.776791\pi$
$32$	0	0
$33$	− 8.42121i	− 1.46595i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.48191i	0.408024i	0.978968	+	0.204012i	$0.0653982\pi$
$37$	2.48191i	0.408024i	−0.978968	+	0.204012i	$0.934602\pi$
$38$	0	0
$39$	7.65929	1.22647
$40$	0	0
$41$	−10.5633	−1.64971	−0.824854	−	0.565346i	$-0.808743\pi$
$41$	−10.5633	−1.64971	−0.824854	+	0.565346i	$0.808743\pi$
$42$	0	0
$43$	10.0848i	1.53792i	0.639294	+	0.768962i	$0.279227\pi$
$43$	10.0848i	1.53792i	−0.639294	+	0.768962i	$0.720773\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.28105i	0.186861i	0.995626	+	0.0934303i	$0.0297832\pi$
$47$	1.28105i	0.186861i	−0.995626	+	0.0934303i	$0.970217\pi$
$48$	0	0
$49$	−11.1402	−1.59145
$50$	0	0
$51$	1.50269	0.210419
$52$	0	0
$53$	− 1.32070i	− 0.181412i	−0.995878	−	0.0907060i	$-0.971088\pi$
$53$	− 1.32070i	− 0.181412i	0.995878	−	0.0907060i	$-0.0289124\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.52886i	0.732315i
$58$	0	0
$59$	−11.7607	−1.53111	−0.765555	−	0.643371i	$-0.777535\pi$
$59$	−11.7607	−1.53111	−0.765555	+	0.643371i	$0.777535\pi$
$60$	0	0
$61$	−12.0814	−1.54686	−0.773431	−	0.633881i	$-0.781461\pi$
$61$	−12.0814	−1.54686	−0.773431	+	0.633881i	$0.781461\pi$
$62$	0	0
$63$	− 3.15990i	− 0.398110i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.52695i	− 0.553055i	−0.961006	−	0.276527i	$-0.910816\pi$
$67$	− 4.52695i	− 0.553055i	0.961006	−	0.276527i	$-0.0891836\pi$
$68$	0	0
$69$	−3.12939	−0.376734
$70$	0	0
$71$	−0.0273119	−0.00324132	−0.00162066	−	0.999999i	$-0.500516\pi$
$71$	−0.0273119	−0.00324132	−0.00162066	+	0.999999i	$0.500516\pi$
$72$	0	0
$73$	− 7.67930i	− 0.898794i	−0.893332	−	0.449397i	$-0.851639\pi$
$73$	− 7.67930i	− 0.898794i	0.893332	−	0.449397i	$-0.148361\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	23.8685i	2.72007i
$78$	0	0
$79$	−7.94712	−0.894121	−0.447060	−	0.894504i	$-0.647529\pi$
$79$	−7.94712	−0.894121	−0.447060	+	0.894504i	$0.647529\pi$
$80$	0	0
$81$	−6.22383	−0.691536
$82$	0	0
$83$	− 12.8840i	− 1.41420i	−0.707113	−	0.707101i	$-0.750003\pi$
$83$	− 12.8840i	− 1.41420i	0.707113	−	0.707101i	$-0.249997\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.83895i	0.947635i
$88$	0	0
$89$	−12.3228	−1.30621	−0.653106	−	0.757267i	$-0.726534\pi$
$89$	−12.3228	−1.30621	−0.653106	+	0.757267i	$0.726534\pi$
$90$	0	0
$91$	−21.7089	−2.27572
$92$	0	0
$93$	− 12.7850i	− 1.32575i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 5.70487i	− 0.579242i	−0.957141	−	0.289621i	$-0.906471\pi$
$97$	− 5.70487i	− 0.579242i	0.957141	−	0.289621i	$-0.0935292\pi$
$98$	0	0
$99$	−4.15774	−0.417869
$100$	0	0
$101$	5.06800	0.504285	0.252142	−	0.967690i	$-0.418865\pi$
$101$	5.06800	0.504285	0.252142	+	0.967690i	$0.418865\pi$
$102$	0	0
$103$	5.35860i	0.527999i	0.964523	+	0.263999i	$0.0850417\pi$
$103$	5.35860i	0.527999i	−0.964523	+	0.263999i	$0.914958\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.2955i	1.76869i	0.466834	+	0.884345i	$0.345394\pi$
$107$	18.2955i	1.76869i	−0.466834	+	0.884345i	$0.654606\pi$
$108$	0	0
$109$	10.4090	0.997003	0.498502	−	0.866889i	$-0.333884\pi$
$109$	10.4090	0.997003	0.498502	+	0.866889i	$0.333884\pi$
$110$	0	0
$111$	−3.72955	−0.353994
$112$	0	0
$113$	13.7555i	1.29400i	0.762488	+	0.647002i	$0.223978\pi$
$113$	13.7555i	1.29400i	−0.762488	+	0.647002i	$0.776022\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.78156i	− 0.349605i
$118$	0	0
$119$	−4.25913	−0.390434
$120$	0	0
$121$	20.4057	1.85506
$122$	0	0
$123$	− 15.8734i	− 1.43125i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 10.2014i	− 0.905226i	−0.891707	−	0.452613i	$-0.850492\pi$
$127$	− 10.2014i	− 0.905226i	0.891707	−	0.452613i	$-0.149508\pi$
$128$	0	0
$129$	−15.1544	−1.33427
$130$	0	0
$131$	9.30955	0.813379	0.406689	−	0.913566i	$-0.366683\pi$
$131$	9.30955	0.813379	0.406689	+	0.913566i	$0.366683\pi$
$132$	0	0
$133$	− 15.6706i	− 1.35881i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.49181i	− 0.298326i	−0.988813	−	0.149163i	$-0.952342\pi$
$137$	− 3.49181i	− 0.298326i	0.988813	−	0.149163i	$-0.0476579\pi$
$138$	0	0
$139$	−12.0244	−1.01990	−0.509949	−	0.860205i	$-0.670336\pi$
$139$	−12.0244	−1.01990	−0.509949	+	0.860205i	$0.670336\pi$
$140$	0	0
$141$	−1.92503	−0.162117
$142$	0	0
$143$	28.5642i	2.38866i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 16.7402i	− 1.38071i
$148$	0	0
$149$	9.01077	0.738191	0.369096	−	0.929391i	$-0.379667\pi$
$149$	9.01077	0.738191	0.369096	+	0.929391i	$0.379667\pi$
$150$	0	0
$151$	7.60293	0.618718	0.309359	−	0.950945i	$-0.399886\pi$
$151$	7.60293	0.618718	0.309359	+	0.950945i	$0.399886\pi$
$152$	0	0
$153$	− 0.741913i	− 0.0599801i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 16.0989i	− 1.28483i	−0.766355	−	0.642417i	$-0.777932\pi$
$157$	− 16.0989i	− 1.28483i	0.766355	−	0.642417i	$-0.222068\pi$
$158$	0	0
$159$	1.98461	0.157390
$160$	0	0
$161$	8.86972	0.699031
$162$	0	0
$163$	− 13.0190i	− 1.01973i	−0.860255	−	0.509865i	$-0.829695\pi$
$163$	− 13.0190i	− 1.01973i	0.860255	−	0.509865i	$-0.170305\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.8272i	1.68904i	0.535524	+	0.844520i	$0.320114\pi$
$167$	21.8272i	1.68904i	−0.535524	+	0.844520i	$0.679886\pi$
$168$	0	0
$169$	−12.9798	−0.998448
$170$	0	0
$171$	2.72972	0.208747
$172$	0	0
$173$	12.7245i	0.967426i	0.875227	+	0.483713i	$0.160712\pi$
$173$	12.7245i	0.967426i	−0.875227	+	0.483713i	$0.839288\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 17.6727i	− 1.32836i
$178$	0	0
$179$	15.1085	1.12926	0.564631	−	0.825343i	$-0.309018\pi$
$179$	15.1085	1.12926	0.564631	+	0.825343i	$0.309018\pi$
$180$	0	0
$181$	25.6085	1.90346	0.951732	−	0.306930i	$-0.0993019\pi$
$181$	25.6085	1.90346	0.951732	+	0.306930i	$0.0993019\pi$
$182$	0	0
$183$	− 18.1546i	− 1.34203i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.60408i	0.409811i
$188$	0	0
$189$	23.9488	1.74202
$190$	0	0
$191$	−13.0708	−0.945768	−0.472884	−	0.881125i	$-0.656787\pi$
$191$	−13.0708	−0.945768	−0.472884	+	0.881125i	$0.656787\pi$
$192$	0	0
$193$	− 3.24242i	− 0.233395i	−0.993168	−	0.116697i	$-0.962769\pi$
$193$	− 3.24242i	− 0.233395i	0.993168	−	0.116697i	$-0.0372307\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.32956i	0.379715i	0.981812	+	0.189858i	$0.0608027\pi$
$197$	5.32956i	0.379715i	−0.981812	+	0.189858i	$0.939197\pi$
$198$	0	0
$199$	−23.8505	−1.69072	−0.845358	−	0.534200i	$-0.820613\pi$
$199$	−23.8505	−1.69072	−0.845358	+	0.534200i	$0.820613\pi$
$200$	0	0
$201$	6.80261	0.479819
$202$	0	0
$203$	− 25.0525i	− 1.75834i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.54505i	0.107388i
$208$	0	0
$209$	−20.6191	−1.42625
$210$	0	0
$211$	−24.6612	−1.69775	−0.848874	−	0.528596i	$-0.822719\pi$
$211$	−24.6612	−1.69775	−0.848874	+	0.528596i	$0.822719\pi$
$212$	0	0
$213$	− 0.0410414i	− 0.00281211i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	36.2370i	2.45993i
$218$	0	0
$219$	11.5396	0.779776
$220$	0	0
$221$	−5.09704	−0.342864
$222$	0	0
$223$	− 3.05389i	− 0.204504i	−0.994759	−	0.102252i	$-0.967395\pi$
$223$	− 3.05389i	− 0.204504i	0.994759	−	0.102252i	$-0.0326048\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 7.46614i	− 0.495545i	−0.968818	−	0.247773i	$-0.920301\pi$
$227$	− 7.46614i	− 0.495545i	0.968818	−	0.247773i	$-0.0796986\pi$
$228$	0	0
$229$	−14.7096	−0.972035	−0.486018	−	0.873949i	$-0.661551\pi$
$229$	−14.7096	−0.972035	−0.486018	+	0.873949i	$0.661551\pi$
$230$	0	0
$231$	−35.8670	−2.35988
$232$	0	0
$233$	− 15.1718i	− 0.993939i	−0.867768	−	0.496969i	$-0.834446\pi$
$233$	− 15.1718i	− 0.993939i	0.867768	−	0.496969i	$-0.165554\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 11.9421i	− 0.775722i
$238$	0	0
$239$	7.88068	0.509759	0.254879	−	0.966973i	$-0.417964\pi$
$239$	7.88068	0.509759	0.254879	+	0.966973i	$0.417964\pi$
$240$	0	0
$241$	−25.9292	−1.67025	−0.835123	−	0.550063i	$-0.814604\pi$
$241$	−25.9292	−1.67025	−0.835123	+	0.550063i	$0.814604\pi$
$242$	0	0
$243$	7.51634i	0.482173i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 18.7535i	− 1.19326i
$248$	0	0
$249$	19.3607	1.22693
$250$	0	0
$251$	−26.3969	−1.66615	−0.833077	−	0.553157i	$-0.813423\pi$
$251$	−26.3969	−1.66615	−0.833077	+	0.553157i	$0.813423\pi$
$252$	0	0
$253$	− 11.6706i	− 0.733725i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.58765i	0.473304i	0.971594	+	0.236652i	$0.0760502\pi$
$257$	7.58765i	0.473304i	−0.971594	+	0.236652i	$0.923950\pi$
$258$	0	0
$259$	10.5708	0.656836
$260$	0	0
$261$	4.36399	0.270124
$262$	0	0
$263$	− 12.7210i	− 0.784412i	−0.919877	−	0.392206i	$-0.871712\pi$
$263$	− 12.7210i	− 0.784412i	0.919877	−	0.392206i	$-0.128288\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 18.5174i	− 1.13324i
$268$	0	0
$269$	−4.20764	−0.256544	−0.128272	−	0.991739i	$-0.540943\pi$
$269$	−4.20764	−0.256544	−0.128272	+	0.991739i	$0.540943\pi$
$270$	0	0
$271$	18.4090	1.11827	0.559134	−	0.829077i	$-0.311134\pi$
$271$	18.4090	1.11827	0.559134	+	0.829077i	$0.311134\pi$
$272$	0	0
$273$	− 32.6219i	− 1.97437i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 23.7386i	− 1.42631i	−0.701005	−	0.713157i	$-0.747265\pi$
$277$	− 23.7386i	− 1.42631i	0.701005	−	0.713157i	$-0.252735\pi$
$278$	0	0
$279$	−6.31225	−0.377905
$280$	0	0
$281$	5.92784	0.353625	0.176813	−	0.984245i	$-0.443421\pi$
$281$	5.92784	0.353625	0.176813	+	0.984245i	$0.443421\pi$
$282$	0	0
$283$	− 11.2070i	− 0.666188i	−0.942894	−	0.333094i	$-0.891907\pi$
$283$	− 11.2070i	− 0.666188i	0.942894	−	0.333094i	$-0.108093\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	44.9904i	2.65570i
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	8.57267	0.502539
$292$	0	0
$293$	− 13.0722i	− 0.763684i	−0.924228	−	0.381842i	$-0.875290\pi$
$293$	− 13.0722i	− 0.763684i	0.924228	−	0.381842i	$-0.124710\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 31.5114i	− 1.82848i
$298$	0	0
$299$	10.6147	0.613863
$300$	0	0
$301$	42.9526	2.47575
$302$	0	0
$303$	7.61565i	0.437508i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 17.0804i	− 0.974828i	−0.873171	−	0.487414i	$-0.837940\pi$
$307$	− 17.0804i	− 0.974828i	0.873171	−	0.487414i	$-0.162060\pi$
$308$	0	0
$309$	−8.05233	−0.458081
$310$	0	0
$311$	−10.2710	−0.582418	−0.291209	−	0.956660i	$-0.594057\pi$
$311$	−10.2710	−0.582418	−0.291209	+	0.956660i	$0.594057\pi$
$312$	0	0
$313$	− 4.09096i	− 0.231235i	−0.993294	−	0.115617i	$-0.963115\pi$
$313$	− 4.09096i	− 0.231235i	0.993294	−	0.115617i	$-0.0368847\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.82279i	0.327041i	0.986540	+	0.163520i	$0.0522849\pi$
$317$	5.82279i	0.327041i	−0.986540	+	0.163520i	$0.947715\pi$
$318$	0	0
$319$	−32.9636	−1.84561
$320$	0	0
$321$	−27.4925	−1.53448
$322$	0	0
$323$	− 3.67930i	− 0.204722i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.6416i	0.864981i
$328$	0	0
$329$	5.45616	0.300808
$330$	0	0
$331$	−21.9561	−1.20682	−0.603410	−	0.797431i	$-0.706192\pi$
$331$	−21.9561	−1.20682	−0.603410	+	0.797431i	$0.706192\pi$
$332$	0	0
$333$	1.84136i	0.100906i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 7.15965i	− 0.390011i	−0.980802	−	0.195006i	$-0.937528\pi$
$337$	− 7.15965i	− 0.390011i	0.980802	−	0.195006i	$-0.0624725\pi$
$338$	0	0
$339$	−20.6702	−1.12265
$340$	0	0
$341$	47.6800	2.58201
$342$	0	0
$343$	17.6335i	0.952118i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.3561i	1.09277i	0.837533	+	0.546387i	$0.183997\pi$
$347$	20.3561i	1.09277i	−0.837533	+	0.546387i	$0.816003\pi$
$348$	0	0
$349$	18.7735	1.00492	0.502462	−	0.864599i	$-0.332428\pi$
$349$	18.7735	1.00492	0.502462	+	0.864599i	$0.332428\pi$
$350$	0	0
$351$	28.6604	1.52978
$352$	0	0
$353$	− 31.7716i	− 1.69103i	−0.533950	−	0.845516i	$-0.679293\pi$
$353$	− 31.7716i	− 1.69103i	0.533950	−	0.845516i	$-0.320707\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6.40016i	− 0.338732i
$358$	0	0
$359$	6.52347	0.344296	0.172148	−	0.985071i	$-0.444929\pi$
$359$	6.52347	0.344296	0.172148	+	0.985071i	$0.444929\pi$
$360$	0	0
$361$	−5.46275	−0.287513
$362$	0	0
$363$	30.6635i	1.60942i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.3578i	0.906070i	0.891493	+	0.453035i	$0.149659\pi$
$367$	17.3578i	0.906070i	−0.891493	+	0.453035i	$0.850341\pi$
$368$	0	0
$369$	−7.83704	−0.407980
$370$	0	0
$371$	−5.62503	−0.292037
$372$	0	0
$373$	− 7.63878i	− 0.395521i	−0.980250	−	0.197761i	$-0.936633\pi$
$373$	− 7.63878i	− 0.395521i	0.980250	−	0.197761i	$-0.0633669\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 29.9812i	− 1.54411i
$378$	0	0
$379$	22.0927	1.13483	0.567413	−	0.823434i	$-0.307944\pi$
$379$	22.0927	1.13483	0.567413	+	0.823434i	$0.307944\pi$
$380$	0	0
$381$	15.3295	0.785356
$382$	0	0
$383$	− 12.9203i	− 0.660198i	−0.943946	−	0.330099i	$-0.892918\pi$
$383$	− 12.9203i	− 0.660198i	0.943946	−	0.330099i	$-0.107082\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.48208i	0.380335i
$388$	0	0
$389$	−1.35513	−0.0687077	−0.0343538	−	0.999410i	$-0.510937\pi$
$389$	−1.35513	−0.0687077	−0.0343538	+	0.999410i	$0.510937\pi$
$390$	0	0
$391$	2.08252	0.105318
$392$	0	0
$393$	13.9894i	0.705672i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.68991i	0.235380i	0.993050	+	0.117690i	$0.0375489\pi$
$397$	4.68991i	0.235380i	−0.993050	+	0.117690i	$0.962451\pi$
$398$	0	0
$399$	23.5481	1.17888
$400$	0	0
$401$	18.6534	0.931504	0.465752	−	0.884915i	$-0.345784\pi$
$401$	18.6534	0.931504	0.465752	+	0.884915i	$0.345784\pi$
$402$	0	0
$403$	43.3660i	2.16022i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 13.9088i	− 0.689436i
$408$	0	0
$409$	−21.0675	−1.04172	−0.520859	−	0.853642i	$-0.674388\pi$
$409$	−21.0675	−1.04172	−0.520859	+	0.853642i	$0.674388\pi$
$410$	0	0
$411$	5.24713	0.258822
$412$	0	0
$413$	50.0902i	2.46478i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 18.0690i	− 0.884843i
$418$	0	0
$419$	33.9507	1.65860	0.829300	−	0.558804i	$-0.188740\pi$
$419$	33.9507	1.65860	0.829300	+	0.558804i	$0.188740\pi$
$420$	0	0
$421$	−20.6895	−1.00835	−0.504173	−	0.863603i	$-0.668203\pi$
$421$	−20.6895	−1.00835	−0.504173	+	0.863603i	$0.668203\pi$
$422$	0	0
$423$	0.950429i	0.0462115i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	51.4561i	2.49014i
$428$	0	0
$429$	−42.9233	−2.07235
$430$	0	0
$431$	−21.2599	−1.02405	−0.512027	−	0.858970i	$-0.671105\pi$
$431$	−21.2599	−1.02405	−0.512027	+	0.858970i	$0.671105\pi$
$432$	0	0
$433$	22.7709i	1.09430i	0.837034	+	0.547151i	$0.184287\pi$
$433$	22.7709i	1.09430i	−0.837034	+	0.547151i	$0.815713\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.66222i	0.366533i
$438$	0	0
$439$	−24.6182	−1.17496	−0.587482	−	0.809237i	$-0.699881\pi$
$439$	−24.6182	−1.17496	−0.587482	+	0.809237i	$0.699881\pi$
$440$	0	0
$441$	−8.26503	−0.393573
$442$	0	0
$443$	9.80661i	0.465926i	0.972486	+	0.232963i	$0.0748421\pi$
$443$	9.80661i	0.465926i	−0.972486	+	0.232963i	$0.925158\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.5404i	0.640440i
$448$	0	0
$449$	−7.96832	−0.376048	−0.188024	−	0.982164i	$-0.560208\pi$
$449$	−7.96832	−0.376048	−0.188024	+	0.982164i	$0.560208\pi$
$450$	0	0
$451$	59.1975	2.78750
$452$	0	0
$453$	11.4249i	0.536788i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.6640i	1.06018i	0.847943	+	0.530088i	$0.177841\pi$
$457$	22.6640i	1.06018i	−0.847943	+	0.530088i	$0.822159\pi$
$458$	0	0
$459$	5.62295	0.262457
$460$	0	0
$461$	−7.03235	−0.327529	−0.163765	−	0.986499i	$-0.552364\pi$
$461$	−7.03235	−0.327529	−0.163765	+	0.986499i	$0.552364\pi$
$462$	0	0
$463$	15.3351i	0.712683i	0.934356	+	0.356342i	$0.115976\pi$
$463$	15.3351i	0.712683i	−0.934356	+	0.356342i	$0.884024\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 11.0414i	− 0.510934i	−0.966818	−	0.255467i	$-0.917771\pi$
$467$	− 11.0414i	− 0.510934i	0.966818	−	0.255467i	$-0.0822292\pi$
$468$	0	0
$469$	−19.2808	−0.890307
$470$	0	0
$471$	24.1918	1.11470
$472$	0	0
$473$	− 56.5163i	− 2.59862i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.979844i	− 0.0448640i
$478$	0	0
$479$	−1.02113	−0.0466566	−0.0233283	−	0.999728i	$-0.507426\pi$
$479$	−1.02113	−0.0466566	−0.0233283	+	0.999728i	$0.507426\pi$
$480$	0	0
$481$	12.6504	0.576809
$482$	0	0
$483$	13.3285i	0.606466i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 25.2759i	− 1.14536i	−0.819779	−	0.572680i	$-0.805904\pi$
$487$	− 25.2759i	− 1.14536i	0.819779	−	0.572680i	$-0.194096\pi$
$488$	0	0
$489$	19.5636	0.884697
$490$	0	0
$491$	4.07984	0.184121	0.0920603	−	0.995753i	$-0.470655\pi$
$491$	4.07984	0.184121	0.0920603	+	0.995753i	$0.470655\pi$
$492$	0	0
$493$	− 5.88207i	− 0.264915i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.116325i	0.00521788i
$498$	0	0
$499$	16.8340	0.753593	0.376797	−	0.926296i	$-0.377026\pi$
$499$	16.8340	0.753593	0.376797	+	0.926296i	$0.377026\pi$
$500$	0	0
$501$	−32.7996	−1.46538
$502$	0	0
$503$	− 23.1652i	− 1.03289i	−0.856321	−	0.516444i	$-0.827256\pi$
$503$	− 23.1652i	− 1.03289i	0.856321	−	0.516444i	$-0.172744\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 19.5047i	− 0.866234i
$508$	0	0
$509$	−1.05059	−0.0465665	−0.0232832	−	0.999729i	$-0.507412\pi$
$509$	−1.05059	−0.0465665	−0.0232832	+	0.999729i	$0.507412\pi$
$510$	0	0
$511$	−32.7071	−1.44688
$512$	0	0
$513$	20.6885i	0.913420i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 7.17912i	− 0.315737i
$518$	0	0
$519$	−19.1210	−0.839320
$520$	0	0
$521$	−14.9831	−0.656421	−0.328210	−	0.944605i	$-0.606445\pi$
$521$	−14.9831	−0.656421	−0.328210	+	0.944605i	$0.606445\pi$
$522$	0	0
$523$	− 2.71233i	− 0.118602i	−0.998240	−	0.0593010i	$-0.981113\pi$
$523$	− 2.71233i	− 0.118602i	0.998240	−	0.0593010i	$-0.0188872\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.50808i	0.370618i
$528$	0	0
$529$	18.6631	0.811440
$530$	0	0
$531$	−8.72540	−0.378650
$532$	0	0
$533$	53.8415i	2.33213i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	22.7034i	0.979726i
$538$	0	0
$539$	62.4304	2.68907
$540$	0	0
$541$	28.7748	1.23712	0.618562	−	0.785736i	$-0.287716\pi$
$541$	28.7748	1.23712	0.618562	+	0.785736i	$0.287716\pi$
$542$	0	0
$543$	38.4817i	1.65141i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 18.0706i	− 0.772643i	−0.922364	−	0.386322i	$-0.873746\pi$
$547$	− 18.0706i	− 0.772643i	0.922364	−	0.386322i	$-0.126254\pi$
$548$	0	0
$549$	−8.96333	−0.382546
$550$	0	0
$551$	21.6419	0.921977
$552$	0	0
$553$	33.8478i	1.43935i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.3000i	0.478797i	0.970921	+	0.239399i	$0.0769503\pi$
$557$	11.3000i	0.478797i	−0.970921	+	0.239399i	$0.923050\pi$
$558$	0	0
$559$	51.4029	2.17411
$560$	0	0
$561$	−8.42121	−0.355544
$562$	0	0
$563$	32.8640i	1.38505i	0.721394	+	0.692525i	$0.243502\pi$
$563$	32.8640i	1.38505i	−0.721394	+	0.692525i	$0.756498\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	26.5081i	1.11323i
$568$	0	0
$569$	−23.5100	−0.985592	−0.492796	−	0.870145i	$-0.664025\pi$
$569$	−23.5100	−0.985592	−0.492796	+	0.870145i	$0.664025\pi$
$570$	0	0
$571$	22.3428	0.935018	0.467509	−	0.883988i	$-0.345151\pi$
$571$	22.3428	0.935018	0.467509	+	0.883988i	$0.345151\pi$
$572$	0	0
$573$	− 19.6414i	− 0.820529i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 26.1662i	− 1.08931i	−0.838659	−	0.544657i	$-0.816660\pi$
$577$	− 26.1662i	− 1.08931i	0.838659	−	0.544657i	$-0.183340\pi$
$578$	0	0
$579$	4.87236	0.202489
$580$	0	0
$581$	−54.8745	−2.27658
$582$	0	0
$583$	7.40131i	0.306531i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.67859i	0.110557i	0.998471	+	0.0552785i	$0.0176047\pi$
$587$	2.67859i	0.110557i	−0.998471	+	0.0552785i	$0.982395\pi$
$588$	0	0
$589$	−31.3038	−1.28985
$590$	0	0
$591$	−8.00869	−0.329434
$592$	0	0
$593$	26.8528i	1.10271i	0.834270	+	0.551357i	$0.185890\pi$
$593$	26.8528i	1.10271i	−0.834270	+	0.551357i	$0.814110\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 35.8400i	− 1.46683i
$598$	0	0
$599$	−33.0060	−1.34859	−0.674295	−	0.738462i	$-0.735552\pi$
$599$	−33.0060	−1.34859	−0.674295	+	0.738462i	$0.735552\pi$
$600$	0	0
$601$	−17.1268	−0.698616	−0.349308	−	0.937008i	$-0.613583\pi$
$601$	−17.1268	−0.698616	−0.349308	+	0.937008i	$0.613583\pi$
$602$	0	0
$603$	− 3.35860i	− 0.136773i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	41.3785i	1.67950i	0.542973	+	0.839750i	$0.317298\pi$
$607$	41.3785i	1.67950i	−0.542973	+	0.839750i	$0.682702\pi$
$608$	0	0
$609$	37.6462	1.52550
$610$	0	0
$611$	6.52957	0.264158
$612$	0	0
$613$	30.1547i	1.21794i	0.793194	+	0.608969i	$0.208417\pi$
$613$	30.1547i	1.21794i	−0.793194	+	0.608969i	$0.791583\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 31.8901i	− 1.28385i	−0.766769	−	0.641923i	$-0.778137\pi$
$617$	− 31.8901i	− 1.28385i	0.766769	−	0.641923i	$-0.221863\pi$
$618$	0	0
$619$	14.8764	0.597935	0.298967	−	0.954263i	$-0.403358\pi$
$619$	14.8764	0.597935	0.298967	+	0.954263i	$0.403358\pi$
$620$	0	0
$621$	−11.7099	−0.469902
$622$	0	0
$623$	52.4843i	2.10274i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 30.9842i	− 1.23739i
$628$	0	0
$629$	2.48191	0.0989604
$630$	0	0
$631$	−9.33949	−0.371799	−0.185900	−	0.982569i	$-0.559520\pi$
$631$	−9.33949	−0.371799	−0.185900	+	0.982569i	$0.559520\pi$
$632$	0	0
$633$	− 37.0582i	− 1.47293i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	56.7818i	2.24978i
$638$	0	0
$639$	−0.0202630	−0.000801593 0
$640$	0	0
$641$	−3.28606	−0.129792	−0.0648958	−	0.997892i	$-0.520672\pi$
$641$	−3.28606	−0.129792	−0.0648958	+	0.997892i	$0.520672\pi$
$642$	0	0
$643$	− 1.78820i	− 0.0705196i	−0.999378	−	0.0352598i	$-0.988774\pi$
$643$	− 1.78820i	− 0.0705196i	0.999378	−	0.0352598i	$-0.0112259\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.04558i	0.0804200i	0.999191	+	0.0402100i	$0.0128027\pi$
$647$	2.04558i	0.0804200i	−0.999191	+	0.0402100i	$0.987197\pi$
$648$	0	0
$649$	65.9078	2.58711
$650$	0	0
$651$	−54.4531	−2.13418
$652$	0	0
$653$	2.36051i	0.0923740i	0.998933	+	0.0461870i	$0.0147070\pi$
$653$	2.36051i	0.0923740i	−0.998933	+	0.0461870i	$0.985293\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5.69737i	− 0.222276i
$658$	0	0
$659$	−3.25145	−0.126658	−0.0633292	−	0.997993i	$-0.520172\pi$
$659$	−3.25145	−0.126658	−0.0633292	+	0.997993i	$0.520172\pi$
$660$	0	0
$661$	−2.78242	−0.108223	−0.0541117	−	0.998535i	$-0.517233\pi$
$661$	−2.78242	−0.108223	−0.0541117	+	0.998535i	$0.517233\pi$
$662$	0	0
$663$	− 7.65929i	− 0.297462i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.2495i	0.474304i
$668$	0	0
$669$	4.58907	0.177424
$670$	0	0
$671$	67.7050	2.61372
$672$	0	0
$673$	− 17.4485i	− 0.672589i	−0.941757	−	0.336294i	$-0.890826\pi$
$673$	− 17.4485i	− 0.672589i	0.941757	−	0.336294i	$-0.109174\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 5.57095i	− 0.214109i	−0.994253	−	0.107054i	$-0.965858\pi$
$677$	− 5.57095i	− 0.214109i	0.994253	−	0.107054i	$-0.0341419\pi$
$678$	0	0
$679$	−24.2978	−0.932462
$680$	0	0
$681$	11.2193	0.429925
$682$	0	0
$683$	− 32.0094i	− 1.22480i	−0.790546	−	0.612402i	$-0.790203\pi$
$683$	− 32.0094i	− 1.22480i	0.790546	−	0.612402i	$-0.209797\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 22.1040i	− 0.843319i
$688$	0	0
$689$	−6.73166	−0.256456
$690$	0	0
$691$	−30.0817	−1.14436	−0.572181	−	0.820127i	$-0.693903\pi$
$691$	−30.0817	−1.14436	−0.572181	+	0.820127i	$0.693903\pi$
$692$	0	0
$693$	17.7083i	0.672684i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.5633i	0.400113i
$698$	0	0
$699$	22.7986	0.862322
$700$	0	0
$701$	8.62366	0.325711	0.162856	−	0.986650i	$-0.447930\pi$
$701$	8.62366	0.325711	0.162856	+	0.986650i	$0.447930\pi$
$702$	0	0
$703$	9.13170i	0.344409i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 21.5853i	− 0.811797i
$708$	0	0
$709$	11.9495	0.448774	0.224387	−	0.974500i	$-0.427962\pi$
$709$	11.9495	0.448774	0.224387	+	0.974500i	$0.427962\pi$
$710$	0	0
$711$	−5.89607	−0.221120
$712$	0	0
$713$	− 17.7182i	− 0.663553i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.8422i	0.442257i
$718$	0	0
$719$	−0.453808	−0.0169242	−0.00846209	−	0.999964i	$-0.502694\pi$
$719$	−0.453808	−0.0169242	−0.00846209	+	0.999964i	$0.502694\pi$
$720$	0	0
$721$	22.8230	0.849971
$722$	0	0
$723$	− 38.9636i	− 1.44907i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 33.8668i	− 1.25605i	−0.778194	−	0.628024i	$-0.783864\pi$
$727$	− 33.8668i	− 1.25605i	0.778194	−	0.628024i	$-0.216136\pi$
$728$	0	0
$729$	−29.9662	−1.10986
$730$	0	0
$731$	10.0848	0.373002
$732$	0	0
$733$	− 12.1335i	− 0.448162i	−0.974571	−	0.224081i	$-0.928062\pi$
$733$	− 12.1335i	− 0.448162i	0.974571	−	0.224081i	$-0.0719380\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.3694i	0.934493i
$738$	0	0
$739$	22.0480	0.811049	0.405525	−	0.914084i	$-0.367089\pi$
$739$	22.0480	0.811049	0.405525	+	0.914084i	$0.367089\pi$
$740$	0	0
$741$	28.1808	1.03525
$742$	0	0
$743$	− 53.5023i	− 1.96281i	−0.191954	−	0.981404i	$-0.561482\pi$
$743$	− 53.5023i	− 1.96281i	0.191954	−	0.981404i	$-0.438518\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 9.55880i	− 0.349738i
$748$	0	0
$749$	77.9227	2.84723
$750$	0	0
$751$	−27.4545	−1.00183	−0.500914	−	0.865497i	$-0.667003\pi$
$751$	−27.4545	−1.00183	−0.500914	+	0.865497i	$0.667003\pi$
$752$	0	0
$753$	− 39.6664i	− 1.44552i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 35.2758i	− 1.28212i	−0.767490	−	0.641061i	$-0.778495\pi$
$757$	− 35.2758i	− 1.28212i	0.767490	−	0.641061i	$-0.221505\pi$
$758$	0	0
$759$	17.5373	0.636565
$760$	0	0
$761$	29.2373	1.05985	0.529926	−	0.848044i	$-0.322220\pi$
$761$	29.2373	1.05985	0.529926	+	0.848044i	$0.322220\pi$
$762$	0	0
$763$	− 44.3333i	− 1.60497i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	59.9446i	2.16448i
$768$	0	0
$769$	22.0766	0.796104	0.398052	−	0.917363i	$-0.369686\pi$
$769$	22.0766	0.796104	0.398052	+	0.917363i	$0.369686\pi$
$770$	0	0
$771$	−11.4019	−0.410630
$772$	0	0
$773$	29.2667i	1.05265i	0.850283	+	0.526326i	$0.176431\pi$
$773$	29.2667i	1.05265i	−0.850283	+	0.526326i	$0.823569\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	15.8846i	0.569858i
$778$	0	0
$779$	−38.8655	−1.39250
$780$	0	0
$781$	0.153058	0.00547684
$782$	0	0
$783$	33.0746i	1.18199i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 7.40215i	− 0.263858i	−0.991259	−	0.131929i	$-0.957883\pi$
$787$	− 7.40215i	− 0.263858i	0.991259	−	0.131929i	$-0.0421171\pi$
$788$	0	0
$789$	19.1158	0.680541
$790$	0	0
$791$	58.5862	2.08309
$792$	0	0
$793$	61.5793i	2.18674i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 51.5831i	− 1.82717i	−0.406652	−	0.913583i	$-0.633304\pi$
$797$	− 51.5831i	− 1.82717i	0.406652	−	0.913583i	$-0.366696\pi$
$798$	0	0
$799$	1.28105	0.0453204
$800$	0	0
$801$	−9.14243	−0.323032
$802$	0	0
$803$	43.0354i	1.51869i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6.32279i	− 0.222573i
$808$	0	0
$809$	30.8840	1.08582	0.542912	−	0.839790i	$-0.317322\pi$
$809$	30.8840	1.08582	0.542912	+	0.839790i	$0.317322\pi$
$810$	0	0
$811$	1.66218	0.0583670	0.0291835	−	0.999574i	$-0.490709\pi$
$811$	1.66218	0.0583670	0.0291835	+	0.999574i	$0.490709\pi$
$812$	0	0
$813$	27.6631i	0.970188i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	37.1052i	1.29815i
$818$	0	0
$819$	−16.1061	−0.562794
$820$	0	0
$821$	39.9351	1.39374	0.696872	−	0.717195i	$-0.254574\pi$
$821$	39.9351	1.39374	0.696872	+	0.717195i	$0.254574\pi$
$822$	0	0
$823$	− 2.20540i	− 0.0768754i	−0.999261	−	0.0384377i	$-0.987762\pi$
$823$	− 2.20540i	− 0.0768754i	0.999261	−	0.0384377i	$-0.0122381\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 14.1644i	− 0.492546i	−0.969201	−	0.246273i	$-0.920794\pi$
$827$	− 14.1644i	− 0.492546i	0.969201	−	0.246273i	$-0.0792059\pi$
$828$	0	0
$829$	−34.9292	−1.21314	−0.606570	−	0.795030i	$-0.707455\pi$
$829$	−34.9292	−1.21314	−0.606570	+	0.795030i	$0.707455\pi$
$830$	0	0
$831$	35.6718	1.23744
$832$	0	0
$833$	11.1402i	0.385984i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 47.8405i	− 1.65361i
$838$	0	0
$839$	10.6389	0.367294	0.183647	−	0.982992i	$-0.441210\pi$
$839$	10.6389	0.367294	0.183647	+	0.982992i	$0.441210\pi$
$840$	0	0
$841$	5.59880	0.193062
$842$	0	0
$843$	8.90772i	0.306798i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 86.9105i	− 2.98628i
$848$	0	0
$849$	16.8407	0.577972
$850$	0	0
$851$	−5.16863	−0.177178
$852$	0	0
$853$	4.53305i	0.155209i	0.996984	+	0.0776043i	$0.0247271\pi$
$853$	4.53305i	0.155209i	−0.996984	+	0.0776043i	$0.975273\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 26.8119i	− 0.915878i	−0.888983	−	0.457939i	$-0.848588\pi$
$857$	− 26.8119i	− 0.915878i	0.888983	−	0.457939i	$-0.151412\pi$
$858$	0	0
$859$	−2.54555	−0.0868530	−0.0434265	−	0.999057i	$-0.513827\pi$
$859$	−2.54555	−0.0868530	−0.0434265	+	0.999057i	$0.513827\pi$
$860$	0	0
$861$	−67.6067	−2.30403
$862$	0	0
$863$	21.3673i	0.727351i	0.931526	+	0.363676i	$0.118478\pi$
$863$	21.3673i	0.727351i	−0.931526	+	0.363676i	$0.881522\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.50269i	− 0.0510341i
$868$	0	0
$869$	44.5363	1.51079
$870$	0	0
$871$	−23.0740	−0.781834
$872$	0	0
$873$	− 4.23252i	− 0.143249i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.1228i	0.645730i	0.946445	+	0.322865i	$0.104646\pi$
$877$	19.1228i	0.645730i	−0.946445	+	0.322865i	$0.895354\pi$
$878$	0	0
$879$	19.6434	0.662557
$880$	0	0
$881$	33.8416	1.14015	0.570075	−	0.821592i	$-0.306914\pi$
$881$	33.8416	1.14015	0.570075	+	0.821592i	$0.306914\pi$
$882$	0	0
$883$	− 22.7756i	− 0.766461i	−0.923653	−	0.383231i	$-0.874811\pi$
$883$	− 22.7756i	− 0.766461i	0.923653	−	0.383231i	$-0.125189\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.7568i	0.529062i	0.964377	+	0.264531i	$0.0852171\pi$
$887$	15.7568i	0.529062i	−0.964377	+	0.264531i	$0.914783\pi$
$888$	0	0
$889$	−43.4490	−1.45723
$890$	0	0
$891$	34.8788	1.16848
$892$	0	0
$893$	4.71338i	0.157727i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	15.9506i	0.532576i
$898$	0	0
$899$	−50.0452	−1.66910
$900$	0	0
$901$	−1.32070	−0.0439989
$902$	0	0
$903$	64.5447i	2.14791i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 28.3574i	− 0.941593i	−0.882242	−	0.470796i	$-0.843967\pi$
$907$	− 28.3574i	− 0.941593i	0.882242	−	0.470796i	$-0.156033\pi$
$908$	0	0
$909$	3.76001	0.124712
$910$	0	0
$911$	4.46190	0.147829	0.0739147	−	0.997265i	$-0.476451\pi$
$911$	4.46190	0.147829	0.0739147	+	0.997265i	$0.476451\pi$
$912$	0	0
$913$	72.2029i	2.38957i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 39.6505i	− 1.30938i
$918$	0	0
$919$	−12.2144	−0.402917	−0.201459	−	0.979497i	$-0.564568\pi$
$919$	−12.2144	−0.402917	−0.201459	+	0.979497i	$0.564568\pi$
$920$	0	0
$921$	25.6665	0.845741
$922$	0	0
$923$	0.139210i	0.00458214i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.97562i	0.130576i
$928$	0	0
$929$	24.0052	0.787586	0.393793	−	0.919199i	$-0.371163\pi$
$929$	24.0052	0.787586	0.393793	+	0.919199i	$0.371163\pi$
$930$	0	0
$931$	−40.9880	−1.34333
$932$	0	0
$933$	− 15.4342i	− 0.505294i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	46.3819i	1.51523i	0.652702	+	0.757615i	$0.273635\pi$
$937$	46.3819i	1.51523i	−0.652702	+	0.757615i	$0.726365\pi$
$938$	0	0
$939$	6.14746	0.200615
$940$	0	0
$941$	0.530210	0.0172843	0.00864217	−	0.999963i	$-0.497249\pi$
$941$	0.530210	0.0172843	0.00864217	+	0.999963i	$0.497249\pi$
$942$	0	0
$943$	− 21.9983i	− 0.716362i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.3865i	0.889942i	0.895545	+	0.444971i	$0.146786\pi$
$947$	27.3865i	0.889942i	−0.895545	+	0.444971i	$0.853214\pi$
$948$	0	0
$949$	−39.1417	−1.27059
$950$	0	0
$951$	−8.74987	−0.283734
$952$	0	0
$953$	− 39.8826i	− 1.29192i	−0.763370	−	0.645962i	$-0.776457\pi$
$953$	− 39.8826i	− 1.29192i	0.763370	−	0.645962i	$-0.223543\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 49.5342i	− 1.60121i
$958$	0	0
$959$	−14.8721	−0.480245
$960$	0	0
$961$	41.3874	1.33508
$962$	0	0
$963$	13.5736i	0.437405i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 33.1843i	− 1.06714i	−0.845757	−	0.533568i	$-0.820851\pi$
$967$	− 33.1843i	− 1.06714i	0.845757	−	0.533568i	$-0.179149\pi$
$968$	0	0
$969$	5.52886	0.177613
$970$	0	0
$971$	1.88467	0.0604818	0.0302409	−	0.999543i	$-0.490373\pi$
$971$	1.88467	0.0604818	0.0302409	+	0.999543i	$0.490373\pi$
$972$	0	0
$973$	51.2135i	1.64183i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.8166i	1.72175i	0.508819	+	0.860874i	$0.330082\pi$
$977$	53.8166i	1.72175i	−0.508819	+	0.860874i	$0.669918\pi$
$978$	0	0
$979$	69.0578	2.20710
$980$	0	0
$981$	7.72259	0.246563
$982$	0	0
$983$	41.8680i	1.33538i	0.744439	+	0.667690i	$0.232717\pi$
$983$	41.8680i	1.33538i	−0.744439	+	0.667690i	$0.767283\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.19894i	0.260975i
$988$	0	0
$989$	−21.0019	−0.667821
$990$	0	0
$991$	29.6430	0.941642	0.470821	−	0.882229i	$-0.343958\pi$
$991$	29.6430	0.941642	0.470821	+	0.882229i	$0.343958\pi$
$992$	0	0
$993$	− 32.9934i	− 1.04701i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.3132i	0.896688i	0.893861	+	0.448344i	$0.147986\pi$
$997$	28.3132i	0.896688i	−0.893861	+	0.448344i	$0.852014\pi$
$998$	0	0
$999$	−13.9557	−0.441538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3400.2.e.n.2449.7		10
5.2	odd	4		3400.2.a.r.1.4	✓	5
5.3	odd	4		3400.2.a.w.1.2	yes	5
5.4	even	2	inner	3400.2.e.n.2449.4		10
20.3	even	4		6800.2.a.bx.1.4		5
20.7	even	4		6800.2.a.cg.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3400.2.a.r.1.4	✓	5	5.2	odd	4
3400.2.a.w.1.2	yes	5	5.3	odd	4
3400.2.e.n.2449.4		10	5.4	even	2	inner
3400.2.e.n.2449.7		10	1.1	even	1	trivial
6800.2.a.bx.1.4		5	20.3	even	4
6800.2.a.cg.1.2		5	20.7	even	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:	\( N \)	\(=\)	\( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3400.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.50269i q^{3} -4.25913i q^{7} +0.741913 q^{9} -5.60408 q^{11} -5.09704i q^{13} -1.00000i q^{17} +3.67930 q^{19} +6.40016 q^{21} +2.08252i q^{23} +5.62295i q^{27} +5.88207 q^{29} -8.50808 q^{31} -8.42121i q^{33} +2.48191i q^{37} +7.65929 q^{39} -10.5633 q^{41} +10.0848i q^{43} +1.28105i q^{47} -11.1402 q^{49} +1.50269 q^{51} -1.32070i q^{53} +5.52886i q^{57} -11.7607 q^{59} -12.0814 q^{61} -3.15990i q^{63} -4.52695i q^{67} -3.12939 q^{69} -0.0273119 q^{71} -7.67930i q^{73} +23.8685i q^{77} -7.94712 q^{79} -6.22383 q^{81} -12.8840i q^{83} +8.83895i q^{87} -12.3228 q^{89} -21.7089 q^{91} -12.7850i q^{93} -5.70487i q^{97} -4.15774 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 12 q^{9} - 12 q^{11} + 12 q^{19} + 14 q^{21} - 12 q^{29} - 22 q^{31} - 18 q^{39} + 8 q^{41} - 60 q^{49} - 6 q^{51} - 28 q^{61} - 64 q^{69} - 10 q^{71} - 54 q^{79} - 22 q^{81} - 20 q^{89} - 126 q^{91}+ \cdots - 20 q^{99}+O(q^{100}) \) 10 * q - 12 * q^9 - 12 * q^11 + 12 * q^19 + 14 * q^21 - 12 * q^29 - 22 * q^31 - 18 * q^39 + 8 * q^41 - 60 * q^49 - 6 * q^51 - 28 * q^61 - 64 * q^69 - 10 * q^71 - 54 * q^79 - 22 * q^81 - 20 * q^89 - 126 * q^91 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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