LMFDB - Embedded newform 2400.2.m.e.1199.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(1199,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2400.1199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1199.20
Character	$\chi$	$=$	2400.1199
Dual form			2400.2.m.e.1199.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.12950 + 1.31310i) q^{3} +4.34495 q^{7} +(-0.448458 + 2.96629i) q^{9} +O(q^{10})$	$q+(1.12950 + 1.31310i) q^{3} +4.34495 q^{7} +(-0.448458 + 2.96629i) q^{9} -1.83679i q^{11} -0.588129 q^{13} +5.37818 q^{17} -5.38776 q^{19} +(4.90762 + 5.70535i) q^{21} +2.40885i q^{23} +(-4.40157 + 2.76156i) q^{27} +7.98077 q^{29} +7.06575i q^{31} +(2.41189 - 2.07466i) q^{33} +2.72080 q^{37} +(-0.664291 - 0.772271i) q^{39} -3.42496i q^{41} +2.96772i q^{43} -9.81525i q^{47} +11.8786 q^{49} +(6.07466 + 7.06208i) q^{51} +6.65218i q^{53} +(-6.08547 - 7.07466i) q^{57} -10.7564i q^{59} +9.27803i q^{61} +(-1.94853 + 12.8884i) q^{63} -4.13536i q^{67} +(-3.16306 + 2.72080i) q^{69} +12.2241 q^{71} -4.42003i q^{73} -7.98077i q^{77} +12.5870i q^{79} +(-8.59777 - 2.66052i) q^{81} -11.5594 q^{83} +(9.01428 + 10.4795i) q^{87} -4.21222i q^{89} -2.55539 q^{91} +(-9.27803 + 7.98077i) q^{93} +2.16763i q^{97} +(5.44846 + 0.823724i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{9}+O(q^{10}) $ 24 * q + 4 * q^9	$ 24 q + 4 q^{9} - 8 q^{19} + 72 q^{49} + 60 q^{51} - 20 q^{81} + 48 q^{91} + 116 q^{99}+O(q^{100}) $ 24 * q + 4 * q^9 - 8 * q^19 + 72 * q^49 + 60 * q^51 - 20 * q^81 + 48 * q^91 + 116 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$3$	1.12950	+	1.31310i	0.652117	+	0.758118i
$3$	1.12950	+	1.31310i	0.652117	+	0.758118i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	4.34495			1.64224			0.821118	−	0.570758i	$-0.193350\pi$
$7$	4.34495			1.64224			0.821118	+	0.570758i	$0.193350\pi$
$8$		0			0
$9$	−0.448458	+	2.96629i	−0.149486	+	0.988764i
$10$		0			0
$11$		−	1.83679i		−	0.553813i	−0.960897	−	0.276907i	$-0.910691\pi$
$11$		−	1.83679i		−	0.553813i	0.960897	−	0.276907i	$-0.0893093\pi$
$12$		0			0
$13$	−0.588129			−0.163118			−0.0815588	−	0.996669i	$-0.525990\pi$
$13$	−0.588129			−0.163118			−0.0815588	+	0.996669i	$0.525990\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.37818			1.30440			0.652200	−	0.758047i	$-0.273846\pi$
$17$	5.37818			1.30440			0.652200	+	0.758047i	$0.273846\pi$
$18$		0			0
$19$	−5.38776			−1.23604			−0.618018	−	0.786164i	$-0.712064\pi$
$19$	−5.38776			−1.23604			−0.618018	+	0.786164i	$0.712064\pi$
$20$		0			0
$21$	4.90762	+	5.70535i	1.07093	+	1.24501i
$22$		0			0
$23$			2.40885i			0.502280i	0.967951	+	0.251140i	$0.0808055\pi$
$23$			2.40885i			0.502280i	−0.967951	+	0.251140i	$0.919194\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.40157	+	2.76156i	−0.847082	+	0.531462i
$28$		0			0
$29$	7.98077			1.48199			0.740996	−	0.671510i	$-0.234354\pi$
$29$	7.98077			1.48199			0.740996	+	0.671510i	$0.234354\pi$
$30$		0			0
$31$			7.06575i			1.26905i	0.772904	+	0.634523i	$0.218803\pi$
$31$			7.06575i			1.26905i	−0.772904	+	0.634523i	$0.781197\pi$
$32$		0			0
$33$	2.41189	−	2.07466i	0.419856	−	0.361151i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.72080			0.447297			0.223648	−	0.974670i	$-0.428203\pi$
$37$	2.72080			0.447297			0.223648	+	0.974670i	$0.428203\pi$
$38$		0			0
$39$	−0.664291	−	0.772271i	−0.106372	−	0.123662i
$40$		0			0
$41$		−	3.42496i		−	0.534888i	−0.963573	−	0.267444i	$-0.913821\pi$
$41$		−	3.42496i		−	0.534888i	0.963573	−	0.267444i	$-0.0861791\pi$
$42$		0			0
$43$			2.96772i			0.452574i	0.974061	+	0.226287i	$0.0726587\pi$
$43$			2.96772i			0.452574i	−0.974061	+	0.226287i	$0.927341\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	9.81525i		−	1.43170i	−0.698254	−	0.715850i	$-0.746039\pi$
$47$		−	9.81525i		−	1.43170i	0.698254	−	0.715850i	$-0.253961\pi$
$48$		0			0
$49$	11.8786			1.69694
$50$		0			0
$51$	6.07466	+	7.06208i	0.850622	+	0.988889i
$52$		0			0
$53$			6.65218i			0.913748i	0.889531	+	0.456874i	$0.151031\pi$
$53$			6.65218i			0.913748i	−0.889531	+	0.456874i	$0.848969\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.08547	−	7.07466i	−0.806040	−	0.937061i
$58$		0			0
$59$		−	10.7564i		−	1.40036i	−0.713967	−	0.700179i	$-0.753103\pi$
$59$		−	10.7564i		−	1.40036i	0.713967	−	0.700179i	$-0.246897\pi$
$60$		0			0
$61$			9.27803i			1.18793i	0.804491	+	0.593965i	$0.202438\pi$
$61$			9.27803i			1.18793i	−0.804491	+	0.593965i	$0.797562\pi$
$62$		0			0
$63$	−1.94853	+	12.8884i	−0.245492	+	1.62378i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	4.13536i		−	0.505215i	−0.967569	−	0.252607i	$-0.918712\pi$
$67$		−	4.13536i		−	0.505215i	0.967569	−	0.252607i	$-0.0812881\pi$
$68$		0			0
$69$	−3.16306	+	2.72080i	−0.380788	+	0.327546i
$70$		0			0
$71$	12.2241			1.45073			0.725367	−	0.688363i	$-0.241670\pi$
$71$	12.2241			1.45073			0.725367	+	0.688363i	$0.241670\pi$
$72$		0			0
$73$		−	4.42003i		−	0.517325i	−0.965968	−	0.258663i	$-0.916718\pi$
$73$		−	4.42003i		−	0.517325i	0.965968	−	0.258663i	$-0.0832818\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	7.98077i		−	0.909493i
$78$		0			0
$79$			12.5870i			1.41614i	0.706141	+	0.708072i	$0.250435\pi$
$79$			12.5870i			1.41614i	−0.706141	+	0.708072i	$0.749565\pi$
$80$		0			0
$81$	−8.59777	−	2.66052i	−0.955308	−	0.295613i
$82$		0			0
$83$	−11.5594			−1.26881			−0.634404	−	0.773002i	$-0.718754\pi$
$83$	−11.5594			−1.26881			−0.634404	+	0.773002i	$0.718754\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	9.01428	+	10.4795i	0.966432	+	1.12352i
$88$		0			0
$89$		−	4.21222i		−	0.446495i	−0.974762	−	0.223247i	$-0.928334\pi$
$89$		−	4.21222i		−	0.446495i	0.974762	−	0.223247i	$-0.0716658\pi$
$90$		0			0
$91$	−2.55539			−0.267878
$92$		0			0
$93$	−9.27803	+	7.98077i	−0.962087	+	0.827567i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			2.16763i			0.220090i	0.993927	+	0.110045i	$0.0350995\pi$
$97$			2.16763i			0.220090i	−0.993927	+	0.110045i	$0.964901\pi$
$98$		0			0
$99$	5.44846	+	0.823724i	0.547591	+	0.0827874i
$100$		0			0
$101$	−3.16306			−0.314736			−0.157368	−	0.987540i	$-0.550301\pi$
$101$	−3.16306			−0.314736			−0.157368	+	0.987540i	$0.550301\pi$
$102$		0			0
$103$	−12.5870			−1.24023			−0.620115	−	0.784511i	$-0.712914\pi$
$103$	−12.5870			−1.24023			−0.620115	+	0.784511i	$0.712914\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.79002			−0.366395			−0.183197	−	0.983076i	$-0.558645\pi$
$107$	−3.79002			−0.366395			−0.183197	+	0.983076i	$0.558645\pi$
$108$		0			0
$109$			0.588129i			0.0563325i	0.999603	+	0.0281663i	$0.00896678\pi$
$109$			0.588129i			0.0563325i	−0.999603	+	0.0281663i	$0.991033\pi$
$110$		0			0
$111$	3.07314	+	3.57268i	0.291690	+	0.339104i
$112$		0			0
$113$	11.0621			1.04064			0.520319	−	0.853972i	$-0.325813\pi$
$113$	11.0621			1.04064			0.520319	+	0.853972i	$0.325813\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.263751	−	1.74456i	0.0243838	−	0.161285i
$118$		0			0
$119$	23.3679			2.14213
$120$		0			0
$121$	7.62620			0.693291
$122$		0			0
$123$	4.49731	−	3.86849i	0.405508	−	0.348810i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−12.9552			−1.14959			−0.574796	−	0.818297i	$-0.694919\pi$
$127$	−12.9552			−1.14959			−0.574796	+	0.818297i	$0.694919\pi$
$128$		0			0
$129$	−3.89692	+	3.35205i	−0.343104	+	0.295131i
$130$		0			0
$131$		−	2.98699i		−	0.260974i	−0.991450	−	0.130487i	$-0.958346\pi$
$131$		−	2.98699i		−	0.260974i	0.991450	−	0.130487i	$-0.0416541\pi$
$132$		0			0
$133$	−23.4095			−2.02986
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.66820			−0.484267			−0.242133	−	0.970243i	$-0.577847\pi$
$137$	−5.66820			−0.484267			−0.242133	+	0.970243i	$0.577847\pi$
$138$		0			0
$139$	−2.69075			−0.228226			−0.114113	−	0.993468i	$-0.536403\pi$
$139$	−2.69075			−0.228226			−0.114113	+	0.993468i	$0.536403\pi$
$140$		0			0
$141$	12.8884	−	11.0863i	1.08540	−	0.933637i
$142$		0			0
$143$			1.08027i			0.0903367i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	13.4169	+	15.5978i	1.10661	+	1.28648i
$148$		0			0
$149$	20.9591			1.71703			0.858517	−	0.512785i	$-0.171386\pi$
$149$	20.9591			1.71703			0.858517	+	0.512785i	$0.171386\pi$
$150$		0			0
$151$		−	3.16869i		−	0.257865i	−0.991653	−	0.128932i	$-0.958845\pi$
$151$		−	3.16869i		−	0.257865i	0.991653	−	0.128932i	$-0.0411550\pi$
$152$		0			0
$153$	−2.41189	+	15.9533i	−0.194990	+	1.28974i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	11.9988			0.957611			0.478805	−	0.877921i	$-0.341070\pi$
$157$	11.9988			0.957611			0.478805	+	0.877921i	$0.341070\pi$
$158$		0			0
$159$	−8.73498	+	7.51364i	−0.692729	+	0.595871i
$160$		0			0
$161$			10.4663i			0.824864i
$162$		0			0
$163$			13.1816i			1.03246i	0.856449	+	0.516231i	$0.172665\pi$
$163$			13.1816i			1.03246i	−0.856449	+	0.516231i	$0.827335\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	3.73744i		−	0.289211i	−0.989489	−	0.144606i	$-0.953809\pi$
$167$		−	3.73744i		−	0.289211i	0.989489	−	0.144606i	$-0.0461914\pi$
$168$		0			0
$169$	−12.6541			−0.973393
$170$		0			0
$171$	2.41618	−	15.9817i	0.184770	−	1.22215i
$172$		0			0
$173$			6.65218i			0.505756i	0.967498	+	0.252878i	$0.0813772\pi$
$173$			6.65218i			0.505756i	−0.967498	+	0.252878i	$0.918623\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	14.1242	−	12.1493i	1.06164	−	0.913198i
$178$		0			0
$179$			18.4093i			1.37598i	0.725722	+	0.687988i	$0.241506\pi$
$179$			18.4093i			1.37598i	−0.725722	+	0.687988i	$0.758494\pi$
$180$		0			0
$181$		−	19.5125i		−	1.45035i	−0.688564	−	0.725175i	$-0.741759\pi$
$181$		−	19.5125i		−	1.45035i	0.688564	−	0.725175i	$-0.258241\pi$
$182$		0			0
$183$	−12.1830	+	10.4795i	−0.900591	+	0.774670i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	9.87859i		−	0.722394i
$188$		0			0
$189$	−19.1246	+	11.9988i	−1.39111	+	0.872786i
$190$		0			0
$191$	−8.73498			−0.632041			−0.316020	−	0.948752i	$-0.602347\pi$
$191$	−8.73498			−0.632041			−0.316020	+	0.948752i	$0.602347\pi$
$192$		0			0
$193$			1.47689i			0.106309i	0.998586	+	0.0531543i	$0.0169275\pi$
$193$			1.47689i			0.106309i	−0.998586	+	0.0531543i	$0.983072\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	11.1438i		−	0.793965i	−0.917826	−	0.396982i	$-0.870057\pi$
$197$		−	11.1438i		−	0.793965i	0.917826	−	0.396982i	$-0.129943\pi$
$198$		0			0
$199$		−	2.80041i		−	0.198516i	−0.995062	−	0.0992578i	$-0.968353\pi$
$199$		−	2.80041i		−	0.198516i	0.995062	−	0.0992578i	$-0.0316469\pi$
$200$		0			0
$201$	5.43014	−	4.67089i	0.383012	−	0.329459i
$202$		0			0
$203$	34.6760			2.43378
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−7.14536	−	1.08027i	−0.496637	−	0.0750839i
$208$		0			0
$209$			9.89618i			0.684533i
$210$		0			0
$211$	−9.86464			−0.679110			−0.339555	−	0.940586i	$-0.610276\pi$
$211$	−9.86464			−0.679110			−0.339555	+	0.940586i	$0.610276\pi$
$212$		0			0
$213$	13.8071	+	16.0515i	0.946048	+	1.09983i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			30.7003i			2.08407i
$218$		0			0
$219$	5.80394	−	4.99243i	0.392194	−	0.337357i
$220$		0			0
$221$	−3.16306			−0.212771
$222$		0			0
$223$	−1.62415			−0.108761			−0.0543806	−	0.998520i	$-0.517318\pi$
$223$	−1.62415			−0.108761			−0.0543806	+	0.998520i	$0.517318\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.94021			−0.327893			−0.163947	−	0.986469i	$-0.552422\pi$
$227$	−4.94021			−0.327893			−0.163947	+	0.986469i	$0.552422\pi$
$228$		0			0
$229$			15.2471i			1.00756i	0.863832	+	0.503779i	$0.168058\pi$
$229$			15.2471i			1.00756i	−0.863832	+	0.503779i	$0.831942\pi$
$230$		0			0
$231$	10.4795	−	9.01428i	0.689503	−	0.593096i
$232$		0			0
$233$	−22.3000			−1.46092			−0.730460	−	0.682955i	$-0.760694\pi$
$233$	−22.3000			−1.46092			−0.730460	+	0.682955i	$0.760694\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−16.5279	+	14.2170i	−1.07360	+	0.923492i
$238$		0			0
$239$	14.8813			0.962589			0.481294	−	0.876559i	$-0.340167\pi$
$239$	14.8813			0.962589			0.481294	+	0.876559i	$0.340167\pi$
$240$		0			0
$241$	−0.523114			−0.0336968			−0.0168484	−	0.999858i	$-0.505363\pi$
$241$	−0.523114			−0.0336968			−0.0168484	+	0.999858i	$0.505363\pi$
$242$		0			0
$243$	−6.21766	−	14.2948i	−0.398863	−	0.917010i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.16869			0.201619
$248$		0			0
$249$	−13.0563	−	15.1786i	−0.827412	−	0.961906i
$250$		0			0
$251$			4.82378i			0.304474i	0.988344	+	0.152237i	$0.0486477\pi$
$251$			4.82378i			0.304474i	−0.988344	+	0.152237i	$0.951352\pi$
$252$		0			0
$253$	4.42456			0.278170
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−22.8859			−1.42758			−0.713792	−	0.700358i	$-0.753024\pi$
$257$	−22.8859			−1.42758			−0.713792	+	0.700358i	$0.753024\pi$
$258$		0			0
$259$	11.8217			0.734567
$260$		0			0
$261$	−3.57904	+	23.6733i	−0.221537	+	1.46534i
$262$		0			0
$263$		−	13.5527i		−	0.835694i	−0.908517	−	0.417847i	$-0.862785\pi$
$263$		−	13.5527i		−	0.835694i	0.908517	−	0.417847i	$-0.137215\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	5.53107	−	4.75771i	0.338496	−	0.291167i
$268$		0			0
$269$	12.9783			0.791301			0.395651	−	0.918401i	$-0.370519\pi$
$269$	12.9783			0.791301			0.395651	+	0.918401i	$0.370519\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−2.88631	−	3.35548i	−0.174688	−	0.203083i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−16.7917			−1.00891			−0.504457	−	0.863437i	$-0.668307\pi$
$277$	−16.7917			−1.00891			−0.504457	+	0.863437i	$0.668307\pi$
$278$		0			0
$279$	−20.9591	−	3.16869i	−1.25479	−	0.189705i
$280$		0			0
$281$			11.0464i			0.658972i	0.944161	+	0.329486i	$0.106875\pi$
$281$			11.0464i			0.658972i	−0.944161	+	0.329486i	$0.893125\pi$
$282$		0			0
$283$		−	18.7047i		−	1.11188i	−0.831223	−	0.555940i	$-0.812359\pi$
$283$		−	18.7047i		−	1.11188i	0.831223	−	0.555940i	$-0.187641\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	14.8813i		−	0.878413i
$288$		0			0
$289$	11.9248			0.701460
$290$		0			0
$291$	−2.84632	+	2.44834i	−0.166854	+	0.143524i
$292$		0			0
$293$		−	29.4457i		−	1.72024i	−0.510093	−	0.860119i	$-0.670389\pi$
$293$		−	29.4457i		−	1.72024i	0.510093	−	0.860119i	$-0.329611\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	5.07240	+	8.08476i	0.294331	+	0.469125i
$298$		0			0
$299$		−	1.41672i		−	0.0819308i
$300$		0			0
$301$			12.8946i			0.743233i
$302$		0			0
$303$	−3.57268	−	4.15341i	−0.205245	−	0.238607i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	33.2095i		−	1.89537i	−0.319213	−	0.947683i	$-0.603419\pi$
$307$		−	33.2095i		−	1.89537i	0.319213	−	0.947683i	$-0.396581\pi$
$308$		0			0
$309$	−14.2170	−	16.5279i	−0.808775	−	0.940241i
$310$		0			0
$311$	−25.7768			−1.46167			−0.730834	−	0.682556i	$-0.760868\pi$
$311$	−25.7768			−1.46167			−0.730834	+	0.682556i	$0.760868\pi$
$312$		0			0
$313$			9.65410i			0.545682i	0.962059	+	0.272841i	$0.0879633\pi$
$313$			9.65410i			0.545682i	−0.962059	+	0.272841i	$0.912037\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			4.81770i			0.270589i	0.990805	+	0.135295i	$0.0431981\pi$
$317$			4.81770i			0.270589i	−0.990805	+	0.135295i	$0.956802\pi$
$318$		0			0
$319$		−	14.6590i		−	0.820747i
$320$		0			0
$321$	−4.28082	−	4.97667i	−0.238932	−	0.277770i
$322$		0			0
$323$	−28.9763			−1.61229
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−0.772271	+	0.664291i	−0.0427067	+	0.0367354i
$328$		0			0
$329$		−	42.6468i		−	2.35119i
$330$		0			0
$331$	1.37380			0.0755110			0.0377555	−	0.999287i	$-0.487979\pi$
$331$	1.37380			0.0755110			0.0377555	+	0.999287i	$0.487979\pi$
$332$		0			0
$333$	−1.22016	+	8.07068i	−0.0668646	+	0.442271i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	13.0925i		−	0.713192i	−0.934259	−	0.356596i	$-0.883937\pi$
$337$		−	13.0925i		−	0.713192i	0.934259	−	0.356596i	$-0.116063\pi$
$338$		0			0
$339$	12.4947	+	14.5257i	0.678618	+	0.788927i
$340$		0			0
$341$	12.9783			0.702815
$342$		0			0
$343$	21.1972			1.14454
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.97936			−0.213623			−0.106812	−	0.994279i	$-0.534064\pi$
$347$	−3.97936			−0.213623			−0.106812	+	0.994279i	$0.534064\pi$
$348$		0			0
$349$		−	12.5870i		−	0.673764i	−0.941547	−	0.336882i	$-0.890628\pi$
$349$		−	12.5870i		−	0.673764i	0.941547	−	0.336882i	$-0.109372\pi$
$350$		0			0
$351$	2.58869	−	1.62415i	0.138174	−	0.0866908i
$352$		0			0
$353$	0.787269			0.0419021			0.0209510	−	0.999781i	$-0.493331\pi$
$353$	0.787269			0.0419021			0.0209510	+	0.999781i	$0.493331\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	26.3941	+	30.6844i	1.39692	+	1.62399i
$358$		0			0
$359$	−1.08027			−0.0570144			−0.0285072	−	0.999594i	$-0.509075\pi$
$359$	−1.08027			−0.0570144			−0.0285072	+	0.999594i	$0.509075\pi$
$360$		0			0
$361$	10.0279			0.527785
$362$		0			0
$363$	8.61379	+	10.0140i	0.452107	+	0.525596i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	14.5794			0.761038			0.380519	−	0.924773i	$-0.375745\pi$
$367$	14.5794			0.761038			0.380519	+	0.924773i	$0.375745\pi$
$368$		0			0
$369$	10.1594	+	1.53595i	0.528878	+	0.0799583i
$370$		0			0
$371$			28.9034i			1.50059i
$372$		0			0
$373$	31.5719			1.63473			0.817366	−	0.576118i	$-0.195433\pi$
$373$	31.5719			1.63473			0.817366	+	0.576118i	$0.195433\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.69372			−0.241739
$378$		0			0
$379$	4.61224			0.236915			0.118458	−	0.992959i	$-0.462205\pi$
$379$	4.61224			0.236915			0.118458	+	0.992959i	$0.462205\pi$
$380$		0			0
$381$	−14.6330	−	17.0115i	−0.749669	−	0.871526i
$382$		0			0
$383$		−	14.6330i		−	0.747709i	−0.927487	−	0.373854i	$-0.878036\pi$
$383$		−	14.6330i		−	0.747709i	0.927487	−	0.373854i	$-0.121964\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−8.80314	−	1.33090i	−0.447489	−	0.0676535i
$388$		0			0
$389$	−15.1388			−0.767570			−0.383785	−	0.923422i	$-0.625380\pi$
$389$	−15.1388			−0.767570			−0.383785	+	0.923422i	$0.625380\pi$
$390$		0			0
$391$			12.9552i			0.655175i
$392$		0			0
$393$	3.92221	−	3.37380i	0.197849	−	0.170186i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	18.3362			0.920268			0.460134	−	0.887849i	$-0.347801\pi$
$397$	18.3362			0.920268			0.460134	+	0.887849i	$0.347801\pi$
$398$		0			0
$399$	−26.4411	−	30.7390i	−1.32371	−	1.53888i
$400$		0			0
$401$		−	34.2381i		−	1.70977i	−0.518820	−	0.854884i	$-0.673628\pi$
$401$		−	34.2381i		−	1.70977i	0.518820	−	0.854884i	$-0.326372\pi$
$402$		0			0
$403$		−	4.15557i		−	0.207004i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4.99754i		−	0.247719i
$408$		0			0
$409$	−18.6926			−0.924291			−0.462146	−	0.886804i	$-0.652920\pi$
$409$	−18.6926			−0.924291			−0.462146	+	0.886804i	$0.652920\pi$
$410$		0			0
$411$	−6.40223	−	7.44290i	−0.315799	−	0.367131i
$412$		0			0
$413$		−	46.7359i		−	2.29972i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.03920	−	3.53322i	−0.148830	−	0.173023i
$418$		0			0
$419$		−	5.24599i		−	0.256283i	−0.991756	−	0.128142i	$-0.959099\pi$
$419$		−	5.24599i		−	0.256283i	0.991756	−	0.128142i	$-0.0409012\pi$
$420$		0			0
$421$		−	4.42456i		−	0.215640i	−0.994170	−	0.107820i	$-0.965613\pi$
$421$		−	4.42456i		−	0.215640i	0.994170	−	0.107820i	$-0.0343870\pi$
$422$		0			0
$423$	29.1149	+	4.40173i	1.41561	+	0.214019i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			40.3126i			1.95086i
$428$		0			0
$429$	−1.41850	+	1.22016i	−0.0684859	+	0.0589101i
$430$		0			0
$431$	−5.89797			−0.284095			−0.142048	−	0.989860i	$-0.545369\pi$
$431$	−5.89797			−0.284095			−0.142048	+	0.989860i	$0.545369\pi$
$432$		0			0
$433$		−	32.2158i		−	1.54819i	−0.633069	−	0.774095i	$-0.718205\pi$
$433$		−	32.2158i		−	1.54819i	0.633069	−	0.774095i	$-0.281795\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	12.9783i		−	0.620837i
$438$		0			0
$439$		−	2.27291i		−	0.108480i	−0.998528	−	0.0542399i	$-0.982726\pi$
$439$		−	2.27291i		−	0.108480i	0.998528	−	0.0542399i	$-0.0172736\pi$
$440$		0			0
$441$	−5.32705	+	35.2354i	−0.253669	+	1.67787i
$442$		0			0
$443$	−4.59091			−0.218121			−0.109060	−	0.994035i	$-0.534784\pi$
$443$	−4.59091			−0.218121			−0.109060	+	0.994035i	$0.534784\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	23.6733	+	27.5213i	1.11971	+	1.30171i
$448$		0			0
$449$			39.0461i			1.84270i	0.388736	+	0.921349i	$0.372912\pi$
$449$			39.0461i			1.84270i	−0.388736	+	0.921349i	$0.627088\pi$
$450$		0			0
$451$	−6.29093			−0.296228
$452$		0			0
$453$	4.16081	−	3.57904i	0.195492	−	0.168158i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	4.96147i		−	0.232088i	−0.993244	−	0.116044i	$-0.962979\pi$
$457$		−	4.96147i		−	0.232088i	0.993244	−	0.116044i	$-0.0370213\pi$
$458$		0			0
$459$	−23.6724	+	14.8522i	−1.10493	+	0.693239i
$460$		0			0
$461$	−14.1271			−0.657963			−0.328981	−	0.944336i	$-0.606705\pi$
$461$	−14.1271			−0.657963			−0.328981	+	0.944336i	$0.606705\pi$
$462$		0			0
$463$	14.2907			0.664146			0.332073	−	0.943254i	$-0.392252\pi$
$463$	14.2907			0.664146			0.332073	+	0.943254i	$0.392252\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	31.8469			1.47370			0.736849	−	0.676058i	$-0.236313\pi$
$467$	31.8469			1.47370			0.736849	+	0.676058i	$0.236313\pi$
$468$		0			0
$469$		−	17.9679i		−	0.829682i
$470$		0			0
$471$	13.5527	+	15.7557i	0.624475	+	0.725982i
$472$		0			0
$473$	5.45109			0.250641
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−19.7323	−	2.98323i	−0.903481	−	0.136593i
$478$		0			0
$479$	25.9566			1.18599			0.592994	−	0.805207i	$-0.297946\pi$
$479$	25.9566			1.18599			0.592994	+	0.805207i	$0.297946\pi$
$480$		0			0
$481$	−1.60018			−0.0729619
$482$		0			0
$483$	−13.7433	+	11.8217i	−0.625344	+	0.537908i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−24.4456			−1.10773			−0.553867	−	0.832605i	$-0.686848\pi$
$487$	−24.4456			−1.10773			−0.553867	+	0.832605i	$0.686848\pi$
$488$		0			0
$489$	−17.3087	+	14.8886i	−0.782728	+	0.673286i
$490$		0			0
$491$		−	37.6630i		−	1.69971i	−0.527018	−	0.849854i	$-0.676690\pi$
$491$		−	37.6630i		−	1.69971i	0.527018	−	0.849854i	$-0.323310\pi$
$492$		0			0
$493$	42.9220			1.93311
$494$		0			0
$495$		0			0
$496$		0			0
$497$	53.1131			2.38245
$498$		0			0
$499$	−36.7249			−1.64403			−0.822016	−	0.569464i	$-0.807151\pi$
$499$	−36.7249			−1.64403			−0.822016	+	0.569464i	$0.807151\pi$
$500$		0			0
$501$	4.90762	−	4.22143i	0.219256	−	0.188600i
$502$		0			0
$503$		−	29.5142i		−	1.31597i	−0.753029	−	0.657987i	$-0.771408\pi$
$503$		−	29.5142i		−	1.31597i	0.753029	−	0.657987i	$-0.228592\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−14.2928	−	16.6161i	−0.634766	−	0.737947i
$508$		0			0
$509$	−2.16054			−0.0957642			−0.0478821	−	0.998853i	$-0.515247\pi$
$509$	−2.16054			−0.0957642			−0.0478821	+	0.998853i	$0.515247\pi$
$510$		0			0
$511$		−	19.2048i		−	0.849571i
$512$		0			0
$513$	23.7146	−	14.8786i	1.04702	−	0.656906i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−18.0286			−0.792895
$518$		0			0
$519$	−8.73498	+	7.51364i	−0.383423	+	0.329812i
$520$		0			0
$521$			9.39893i			0.411775i	0.978576	+	0.205887i	$0.0660080\pi$
$521$			9.39893i			0.411775i	−0.978576	+	0.205887i	$0.933992\pi$
$522$		0			0
$523$			9.68638i			0.423556i	0.977318	+	0.211778i	$0.0679253\pi$
$523$			9.68638i			0.423556i	−0.977318	+	0.211778i	$0.932075\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			38.0009i			1.65534i
$528$		0			0
$529$	17.1974			0.747714
$530$		0			0
$531$	31.9065	+	4.82378i	1.38462	+	0.209334i
$532$		0			0
$533$			2.01431i			0.0872497i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−24.1732	+	20.7933i	−1.04315	+	0.897298i
$538$		0			0
$539$		−	21.8185i		−	0.939789i
$540$		0			0
$541$			13.1751i			0.566441i	0.959055	+	0.283221i	$0.0914029\pi$
$541$			13.1751i			0.566441i	−0.959055	+	0.283221i	$0.908597\pi$
$542$		0			0
$543$	25.6218	−	22.0393i	1.09954	−	0.945799i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			28.4113i			1.21478i	0.794404	+	0.607390i	$0.207783\pi$
$547$			28.4113i			1.21478i	−0.794404	+	0.607390i	$0.792217\pi$
$548$		0			0
$549$	−27.5213	−	4.16081i	−1.17458	−	0.177579i
$550$		0			0
$551$	−42.9984			−1.83179
$552$		0			0
$553$			54.6897i			2.32564i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	32.1029i		−	1.36024i	−0.733099	−	0.680122i	$-0.761927\pi$
$557$		−	32.1029i		−	1.36024i	0.733099	−	0.680122i	$-0.238073\pi$
$558$		0			0
$559$		−	1.74540i		−	0.0738227i
$560$		0			0
$561$	12.9716	−	11.1579i	0.547660	−	0.471086i
$562$		0			0
$563$	14.2406			0.600170			0.300085	−	0.953913i	$-0.402985\pi$
$563$	14.2406			0.600170			0.300085	+	0.953913i	$0.402985\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−37.3569	−	11.5598i	−1.56884	−	0.485466i
$568$		0			0
$569$			9.08328i			0.380791i	0.981707	+	0.190396i	$0.0609770\pi$
$569$			9.08328i			0.380791i	−0.981707	+	0.190396i	$0.939023\pi$
$570$		0			0
$571$	15.7432			0.658834			0.329417	−	0.944185i	$-0.393148\pi$
$571$	15.7432			0.658834			0.329417	+	0.944185i	$0.393148\pi$
$572$		0			0
$573$	−9.86616	−	11.4699i	−0.412165	−	0.479161i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	33.1974i		−	1.38203i	−0.722842	−	0.691014i	$-0.757164\pi$
$577$		−	33.1974i		−	1.38203i	0.722842	−	0.691014i	$-0.242836\pi$
$578$		0			0
$579$	−1.93930	+	1.66814i	−0.0805944	+	0.0693256i
$580$		0			0
$581$	−50.2250			−2.08368
$582$		0			0
$583$	12.2187			0.506046
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.613686			−0.0253295			−0.0126648	−	0.999920i	$-0.504031\pi$
$587$	−0.613686			−0.0253295			−0.0126648	+	0.999920i	$0.504031\pi$
$588$		0			0
$589$		−	38.0685i		−	1.56859i
$590$		0			0
$591$	14.6330	−	12.5870i	0.601919	−	0.517758i
$592$		0			0
$593$	−15.5545			−0.638747			−0.319374	−	0.947629i	$-0.603472\pi$
$593$	−15.5545			−0.638747			−0.319374	+	0.947629i	$0.603472\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	3.67721	−	3.16306i	0.150498	−	0.129456i
$598$		0			0
$599$	−39.0811			−1.59681			−0.798406	−	0.602119i	$-0.794323\pi$
$599$	−39.0811			−1.59681			−0.798406	+	0.602119i	$0.794323\pi$
$600$		0			0
$601$	−20.5231			−0.837155			−0.418578	−	0.908181i	$-0.637471\pi$
$601$	−20.5231			−0.837155			−0.418578	+	0.908181i	$0.637471\pi$
$602$		0			0
$603$	12.2667	+	1.85454i	0.499538	+	0.0755225i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	10.8832			0.441735			0.220868	−	0.975304i	$-0.429111\pi$
$607$	10.8832			0.441735			0.220868	+	0.975304i	$0.429111\pi$
$608$		0			0
$609$	39.1666	+	45.5331i	1.58711	+	1.84509i
$610$		0			0
$611$			5.77263i			0.233535i
$612$		0			0
$613$	−30.3350			−1.22522			−0.612610	−	0.790385i	$-0.709881\pi$
$613$	−30.3350			−1.22522			−0.612610	+	0.790385i	$0.709881\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8.71447			−0.350831			−0.175416	−	0.984494i	$-0.556127\pi$
$617$	−8.71447			−0.350831			−0.175416	+	0.984494i	$0.556127\pi$
$618$		0			0
$619$	13.0323			0.523811			0.261906	−	0.965093i	$-0.415649\pi$
$619$	13.0323			0.523811			0.261906	+	0.965093i	$0.415649\pi$
$620$		0			0
$621$	−6.65218	−	10.6027i	−0.266943	−	0.425473i
$622$		0			0
$623$		−	18.3019i		−	0.733250i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−12.9947	+	11.1777i	−0.518957	+	0.446396i
$628$		0			0
$629$	14.6330			0.583454
$630$		0			0
$631$		−	14.8599i		−	0.591562i	−0.955256	−	0.295781i	$-0.904420\pi$
$631$		−	14.8599i		−	0.591562i	0.955256	−	0.295781i	$-0.0955798\pi$
$632$		0			0
$633$	−11.1421	−	12.9533i	−0.442859	−	0.514845i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−6.98614			−0.276801
$638$		0			0
$639$	−5.48200	+	36.2602i	−0.216864	+	1.43443i
$640$		0			0
$641$		−	5.97397i		−	0.235958i	−0.993016	−	0.117979i	$-0.962358\pi$
$641$		−	5.97397i		−	0.235958i	0.993016	−	0.117979i	$-0.0376415\pi$
$642$		0			0
$643$		−	15.7938i		−	0.622848i	−0.950271	−	0.311424i	$-0.899194\pi$
$643$		−	15.7938i		−	0.622848i	0.950271	−	0.311424i	$-0.100806\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	14.8128i		−	0.582351i	−0.956670	−	0.291175i	$-0.905954\pi$
$647$		−	14.8128i		−	0.582351i	0.956670	−	0.291175i	$-0.0940463\pi$
$648$		0			0
$649$	−19.7572			−0.775537
$650$		0			0
$651$	−40.3126	+	34.6760i	−1.57997	+	1.35906i
$652$		0			0
$653$		−	3.48912i		−	0.136540i	−0.997667	−	0.0682699i	$-0.978252\pi$
$653$		−	3.48912i		−	0.136540i	0.997667	−	0.0682699i	$-0.0217479\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	13.1111	+	1.98220i	0.511513	+	0.0773329i
$658$		0			0
$659$		−	36.5563i		−	1.42403i	−0.702163	−	0.712017i	$-0.747782\pi$
$659$		−	36.5563i		−	1.42403i	0.702163	−	0.712017i	$-0.252218\pi$
$660$		0			0
$661$		−	42.9220i		−	1.66947i	−0.550650	−	0.834736i	$-0.685620\pi$
$661$		−	42.9220i		−	1.66947i	0.550650	−	0.834736i	$-0.314380\pi$
$662$		0			0
$663$	−3.57268	−	4.15341i	−0.138751	−	0.161305i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			19.2245i			0.744375i
$668$		0			0
$669$	−1.83448	−	2.13267i	−0.0709251	−	0.0824538i
$670$		0			0
$671$	17.0418			0.657892
$672$		0			0
$673$		−	46.2899i		−	1.78434i	−0.451696	−	0.892172i	$-0.649181\pi$
$673$		−	46.2899i		−	1.78434i	0.451696	−	0.892172i	$-0.350819\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			27.7911i			1.06810i	0.845453	+	0.534049i	$0.179330\pi$
$677$			27.7911i			1.06810i	−0.845453	+	0.534049i	$0.820670\pi$
$678$		0			0
$679$			9.41826i			0.361440i
$680$		0			0
$681$	−5.57997	−	6.48699i	−0.213825	−	0.248582i
$682$		0			0
$683$	−33.6521			−1.28766			−0.643832	−	0.765167i	$-0.722656\pi$
$683$	−33.6521			−1.28766			−0.643832	+	0.765167i	$0.722656\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−20.0210	+	17.2216i	−0.763849	+	0.657047i
$688$		0			0
$689$		−	3.91234i		−	0.149048i
$690$		0			0
$691$	22.6820			0.862864			0.431432	−	0.902145i	$-0.358008\pi$
$691$	22.6820			0.862864			0.431432	+	0.902145i	$0.358008\pi$
$692$		0			0
$693$	23.6733	+	3.57904i	0.899274	+	0.135956i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	18.4200i		−	0.697708i
$698$		0			0
$699$	−25.1878	−	29.2821i	−0.952692	−	1.10755i
$700$		0			0
$701$	10.6379			0.401789			0.200895	−	0.979613i	$-0.435615\pi$
$701$	10.6379			0.401789			0.200895	+	0.979613i	$0.435615\pi$
$702$		0			0
$703$	−14.6590			−0.552875
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−13.7433			−0.516872
$708$		0			0
$709$			20.5295i			0.771002i	0.922707	+	0.385501i	$0.125971\pi$
$709$			20.5295i			0.771002i	−0.922707	+	0.385501i	$0.874029\pi$
$710$		0			0
$711$	−37.3366	−	5.64472i	−1.40023	−	0.211694i
$712$		0			0
$713$	−17.0203			−0.637417
$714$		0			0
$715$		0			0
$716$		0			0
$717$	16.8084	+	19.5406i	0.627721	+	0.729756i
$718$		0			0
$719$	−14.8813			−0.554977			−0.277489	−	0.960729i	$-0.589502\pi$
$719$	−14.8813			−0.554977			−0.277489	+	0.960729i	$0.589502\pi$
$720$		0			0
$721$	−54.6897			−2.03675
$722$		0			0
$723$	−0.590858	−	0.686901i	−0.0219742	−	0.0255461i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−16.1239			−0.598004			−0.299002	−	0.954253i	$-0.596654\pi$
$727$	−16.1239			−0.598004			−0.299002	+	0.954253i	$0.596654\pi$
$728$		0			0
$729$	11.7476	−	24.3104i	0.435096	−	0.900384i
$730$		0			0
$731$			15.9610i			0.590337i
$732$		0			0
$733$	−16.7310			−0.617975			−0.308988	−	0.951066i	$-0.599990\pi$
$733$	−16.7310			−0.617975			−0.308988	+	0.951066i	$0.599990\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7.59579			−0.279795
$738$		0			0
$739$	12.5693			0.462371			0.231185	−	0.972910i	$-0.425740\pi$
$739$	12.5693			0.462371			0.231185	+	0.972910i	$0.425740\pi$
$740$		0			0
$741$	3.57904	+	4.16081i	0.131479	+	0.152851i
$742$		0			0
$743$			22.2877i			0.817655i	0.912612	+	0.408827i	$0.134062\pi$
$743$			22.2877i			0.817655i	−0.912612	+	0.408827i	$0.865938\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	5.18390	−	34.2885i	0.189669	−	1.25455i
$748$		0			0
$749$	−16.4674			−0.601707
$750$		0			0
$751$		−	35.1279i		−	1.28183i	−0.767610	−	0.640917i	$-0.778554\pi$
$751$		−	35.1279i		−	1.28183i	0.767610	−	0.640917i	$-0.221446\pi$
$752$		0			0
$753$	−6.33410	+	5.44846i	−0.230827	+	0.198553i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−20.6887			−0.751945			−0.375972	−	0.926631i	$-0.622691\pi$
$757$	−20.6887			−0.751945			−0.375972	+	0.926631i	$0.622691\pi$
$758$		0			0
$759$	4.99754	+	5.80988i	0.181399	+	0.210885i
$760$		0			0
$761$			22.8130i			0.826971i	0.910511	+	0.413485i	$0.135689\pi$
$761$			22.8130i			0.826971i	−0.910511	+	0.413485i	$0.864311\pi$
$762$		0			0
$763$			2.55539i			0.0925113i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			6.32612i			0.228423i
$768$		0			0
$769$	46.3082			1.66992			0.834958	−	0.550313i	$-0.185492\pi$
$769$	46.3082			1.66992			0.834958	+	0.550313i	$0.185492\pi$
$770$		0			0
$771$	−25.8496	−	30.0515i	−0.930952	−	1.08228i
$772$		0			0
$773$		−	30.2684i		−	1.08868i	−0.838865	−	0.544340i	$-0.816780\pi$
$773$		−	30.2684i		−	1.08868i	0.838865	−	0.544340i	$-0.183220\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	13.3527	+	15.5231i	0.479024	+	0.556889i
$778$		0			0
$779$			18.4528i			0.661141i
$780$		0			0
$781$		−	22.4531i		−	0.803436i
$782$		0			0
$783$	−35.1279	+	22.0393i	−1.25537	+	0.787622i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	19.4080i		−	0.691819i	−0.938268	−	0.345910i	$-0.887570\pi$
$787$		−	19.4080i		−	0.691819i	0.938268	−	0.345910i	$-0.112430\pi$
$788$		0			0
$789$	17.7960	−	15.3078i	0.633555	−	0.544971i
$790$		0			0
$791$	48.0644			1.70897
$792$		0			0
$793$		−	5.45667i		−	0.193772i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			30.7743i			1.09008i	0.838409	+	0.545041i	$0.183486\pi$
$797$			30.7743i			1.09008i	−0.838409	+	0.545041i	$0.816514\pi$
$798$		0			0
$799$		−	52.7882i		−	1.86751i
$800$		0			0
$801$	12.4947	+	1.88901i	0.441478	+	0.0667448i
$802$		0			0
$803$	−8.11867			−0.286502
$804$		0			0
$805$		0			0
$806$		0			0
$807$	14.6590	+	17.0418i	0.516021	+	0.599900i
$808$		0			0
$809$		−	37.5744i		−	1.32104i	−0.750807	−	0.660522i	$-0.770335\pi$
$809$		−	37.5744i		−	1.32104i	0.750807	−	0.660522i	$-0.229665\pi$
$810$		0			0
$811$	30.3492			1.06571			0.532853	−	0.846208i	$-0.321120\pi$
$811$	30.3492			1.06571			0.532853	+	0.846208i	$0.321120\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	15.9894i		−	0.559397i
$818$		0			0
$819$	1.14599	−	7.58003i	0.0400440	−	0.264868i
$820$		0			0
$821$	6.97824			0.243542			0.121771	−	0.992558i	$-0.461143\pi$
$821$	6.97824			0.243542			0.121771	+	0.992558i	$0.461143\pi$
$822$		0			0
$823$	−0.569147			−0.0198392			−0.00991960	−	0.999951i	$-0.503158\pi$
$823$	−0.569147			−0.0198392			−0.00991960	+	0.999951i	$0.503158\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−23.0851			−0.802748			−0.401374	−	0.915914i	$-0.631467\pi$
$827$	−23.0851			−0.802748			−0.401374	+	0.915914i	$0.631467\pi$
$828$		0			0
$829$			48.3636i			1.67974i	0.542790	+	0.839869i	$0.317368\pi$
$829$			48.3636i			1.67974i	−0.542790	+	0.839869i	$0.682632\pi$
$830$		0			0
$831$	−18.9662	−	22.0491i	−0.657930	−	0.764876i
$832$		0			0
$833$	63.8852			2.21349
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−19.5125	−	31.1004i	−0.674450	−	1.07499i
$838$		0			0
$839$	22.0393			0.760883			0.380441	−	0.924805i	$-0.375772\pi$
$839$	22.0393			0.760883			0.380441	+	0.924805i	$0.375772\pi$
$840$		0			0
$841$	34.6926			1.19630
$842$		0			0
$843$	−14.5050	+	12.4769i	−0.499578	+	0.429727i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	33.1355			1.13855
$848$		0			0
$849$	24.5611	−	21.1270i	0.842936	−	0.725076i
$850$		0			0
$851$			6.55400i			0.224668i
$852$		0			0
$853$	−6.39801			−0.219064			−0.109532	−	0.993983i	$-0.534935\pi$
$853$	−6.39801			−0.219064			−0.109532	+	0.993983i	$0.534935\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−27.2124			−0.929559			−0.464780	−	0.885426i	$-0.653866\pi$
$857$	−27.2124			−0.929559			−0.464780	+	0.885426i	$0.653866\pi$
$858$		0			0
$859$	23.4663			0.800658			0.400329	−	0.916371i	$-0.368896\pi$
$859$	23.4663			0.800658			0.400329	+	0.916371i	$0.368896\pi$
$860$		0			0
$861$	19.5406	−	16.8084i	0.665941	−	0.572828i
$862$		0			0
$863$		−	12.4724i		−	0.424566i	−0.977208	−	0.212283i	$-0.931910\pi$
$863$		−	12.4724i		−	0.424566i	0.977208	−	0.212283i	$-0.0680898\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	13.4691	+	15.6585i	0.457434	+	0.531790i
$868$		0			0
$869$	23.1196			0.784279
$870$		0			0
$871$			2.43212i			0.0824093i
$872$		0			0
$873$	−6.42984	−	0.972093i	−0.217617	−	0.0329004i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−48.9517			−1.65298			−0.826491	−	0.562950i	$-0.809667\pi$
$877$	−48.9517			−1.65298			−0.826491	+	0.562950i	$0.809667\pi$
$878$		0			0
$879$	38.6652	−	33.2590i	1.30414	−	1.12180i
$880$		0			0
$881$		−	9.76612i		−	0.329029i	−0.986375	−	0.164514i	$-0.947394\pi$
$881$		−	9.76612i		−	0.329029i	0.986375	−	0.164514i	$-0.0526057\pi$
$882$		0			0
$883$			37.1170i			1.24909i	0.780990	+	0.624544i	$0.214715\pi$
$883$			37.1170i			1.24909i	−0.780990	+	0.624544i	$0.785285\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	20.7792i		−	0.697699i	−0.937179	−	0.348849i	$-0.886573\pi$
$887$		−	20.7792i		−	0.697699i	0.937179	−	0.348849i	$-0.113427\pi$
$888$		0			0
$889$	−56.2899			−1.88790
$890$		0			0
$891$	−4.88681	+	15.7923i	−0.163714	+	0.529062i
$892$		0			0
$893$			52.8821i			1.76963i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	1.86029	−	1.60018i	0.0621132	−	0.0534285i
$898$		0			0
$899$			56.3901i			1.88072i
$900$		0			0
$901$			35.7766i			1.19189i
$902$		0			0
$903$	−16.9319	+	14.5645i	−0.563459	+	0.484675i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			25.5231i			0.847481i	0.905784	+	0.423741i	$0.139283\pi$
$907$			25.5231i			0.847481i	−0.905784	+	0.423741i	$0.860717\pi$
$908$		0			0
$909$	1.41850	−	9.38256i	0.0470487	−	0.311200i
$910$		0			0
$911$	−18.6187			−0.616865			−0.308433	−	0.951246i	$-0.599804\pi$
$911$	−18.6187			−0.616865			−0.308433	+	0.951246i	$0.599804\pi$
$912$		0			0
$913$			21.2322i			0.702683i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	12.9783i		−	0.428581i
$918$		0			0
$919$			27.9743i			0.922788i	0.887195	+	0.461394i	$0.152650\pi$
$919$			27.9743i			0.922788i	−0.887195	+	0.461394i	$0.847350\pi$
$920$		0			0
$921$	43.6074	−	37.5101i	1.43691	−	1.23600i
$922$		0			0
$923$	−7.18934			−0.236640
$924$		0			0
$925$		0			0
$926$		0			0
$927$	5.64472	−	37.3366i	0.185397	−	1.22629i
$928$		0			0
$929$			46.7604i			1.53416i	0.641551	+	0.767080i	$0.278291\pi$
$929$			46.7604i			1.53416i	−0.641551	+	0.767080i	$0.721709\pi$
$930$		0			0
$931$	−63.9990			−2.09748
$932$		0			0
$933$	−29.1149	−	33.8475i	−0.953178	−	1.10812i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			41.2253i			1.34677i	0.739291	+	0.673387i	$0.235161\pi$
$937$			41.2253i			1.34677i	−0.739291	+	0.673387i	$0.764839\pi$
$938$		0			0
$939$	−12.6768	+	10.9043i	−0.413692	+	0.355849i
$940$		0			0
$941$	−0.179836			−0.00586248			−0.00293124	−	0.999996i	$-0.500933\pi$
$941$	−0.179836			−0.00586248			−0.00293124	+	0.999996i	$0.500933\pi$
$942$		0			0
$943$	8.25021			0.268664
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21.3864			−0.694965			−0.347483	−	0.937686i	$-0.612963\pi$
$947$	−21.3864			−0.694965			−0.347483	+	0.937686i	$0.612963\pi$
$948$		0			0
$949$			2.59955i			0.0843849i
$950$		0			0
$951$	−6.32612	+	5.44160i	−0.205139	+	0.176456i
$952$		0			0
$953$	−11.0306			−0.357317			−0.178658	−	0.983911i	$-0.557176\pi$
$953$	−11.0306			−0.357317			−0.178658	+	0.983911i	$0.557176\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	19.2487	−	16.5573i	0.622223	−	0.535223i
$958$		0			0
$959$	−24.6280			−0.795281
$960$		0			0
$961$	−18.9248			−0.610478
$962$		0			0
$963$	1.69966	−	11.2423i	0.0547709	−	0.362278i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	10.8832			0.349980			0.174990	−	0.984570i	$-0.444011\pi$
$967$	10.8832			0.349980			0.174990	+	0.984570i	$0.444011\pi$
$968$		0			0
$969$	−32.7288	−	38.0488i	−1.05140	−	1.22230i
$970$		0			0
$971$			2.40482i			0.0771744i	0.999255	+	0.0385872i	$0.0122857\pi$
$971$			2.40482i			0.0771744i	−0.999255	+	0.0385872i	$0.987714\pi$
$972$		0			0
$973$	−11.6912			−0.374802
$974$		0			0
$975$		0			0
$976$		0			0
$977$	41.7609			1.33605			0.668026	−	0.744138i	$-0.267140\pi$
$977$	41.7609			1.33605			0.668026	+	0.744138i	$0.267140\pi$
$978$		0			0
$979$	−7.73698			−0.247275
$980$		0			0
$981$	−1.74456	−	0.263751i	−0.0556995	−	0.00842092i
$982$		0			0
$983$			33.2516i			1.06056i	0.847822	+	0.530281i	$0.177914\pi$
$983$			33.2516i			1.06056i	−0.847822	+	0.530281i	$0.822086\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	55.9994	−	48.1695i	1.78248	−	1.53325i
$988$		0			0
$989$	−7.14881			−0.227319
$990$		0			0
$991$			33.9434i			1.07825i	0.842226	+	0.539124i	$0.181245\pi$
$991$			33.9434i			1.07825i	−0.842226	+	0.539124i	$0.818755\pi$
$992$		0			0
$993$	1.55171	+	1.80394i	0.0492420	+	0.0572462i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	45.6428			1.44552			0.722761	−	0.691098i	$-0.242873\pi$
$997$	45.6428			1.44552			0.722761	+	0.691098i	$0.242873\pi$
$998$		0			0
$999$	−11.9758	+	7.51364i	−0.378897	+	0.237721i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.m.e.1199.20		24
3.2	odd	2	inner	2400.2.m.e.1199.8		24
4.3	odd	2		600.2.m.e.299.11		24
5.2	odd	4		2400.2.b.g.2351.10		12
5.3	odd	4		2400.2.b.h.2351.3		12
5.4	even	2	inner	2400.2.m.e.1199.5		24
8.3	odd	2	inner	2400.2.m.e.1199.19		24
8.5	even	2		600.2.m.e.299.9		24
12.11	even	2		600.2.m.e.299.13		24
15.2	even	4		2400.2.b.g.2351.12		12
15.8	even	4		2400.2.b.h.2351.1		12
15.14	odd	2	inner	2400.2.m.e.1199.17		24
20.3	even	4		600.2.b.g.251.12	yes	12
20.7	even	4		600.2.b.h.251.1	yes	12
20.19	odd	2		600.2.m.e.299.14		24
24.5	odd	2		600.2.m.e.299.15		24
24.11	even	2	inner	2400.2.m.e.1199.7		24
40.3	even	4		2400.2.b.h.2351.4		12
40.13	odd	4		600.2.b.g.251.2	yes	12
40.19	odd	2	inner	2400.2.m.e.1199.6		24
40.27	even	4		2400.2.b.g.2351.9		12
40.29	even	2		600.2.m.e.299.16		24
40.37	odd	4		600.2.b.h.251.11	yes	12
60.23	odd	4		600.2.b.g.251.1	✓	12
60.47	odd	4		600.2.b.h.251.12	yes	12
60.59	even	2		600.2.m.e.299.12		24
120.29	odd	2		600.2.m.e.299.10		24
120.53	even	4		600.2.b.g.251.11	yes	12
120.59	even	2	inner	2400.2.m.e.1199.18		24
120.77	even	4		600.2.b.h.251.2	yes	12
120.83	odd	4		2400.2.b.h.2351.2		12
120.107	odd	4		2400.2.b.g.2351.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.b.g.251.1	✓	12	60.23	odd	4
600.2.b.g.251.2	yes	12	40.13	odd	4
600.2.b.g.251.11	yes	12	120.53	even	4
600.2.b.g.251.12	yes	12	20.3	even	4
600.2.b.h.251.1	yes	12	20.7	even	4
600.2.b.h.251.2	yes	12	120.77	even	4
600.2.b.h.251.11	yes	12	40.37	odd	4
600.2.b.h.251.12	yes	12	60.47	odd	4
600.2.m.e.299.9		24	8.5	even	2
600.2.m.e.299.10		24	120.29	odd	2
600.2.m.e.299.11		24	4.3	odd	2
600.2.m.e.299.12		24	60.59	even	2
600.2.m.e.299.13		24	12.11	even	2
600.2.m.e.299.14		24	20.19	odd	2
600.2.m.e.299.15		24	24.5	odd	2
600.2.m.e.299.16		24	40.29	even	2
2400.2.b.g.2351.9		12	40.27	even	4
2400.2.b.g.2351.10		12	5.2	odd	4
2400.2.b.g.2351.11		12	120.107	odd	4
2400.2.b.g.2351.12		12	15.2	even	4
2400.2.b.h.2351.1		12	15.8	even	4
2400.2.b.h.2351.2		12	120.83	odd	4
2400.2.b.h.2351.3		12	5.3	odd	4
2400.2.b.h.2351.4		12	40.3	even	4
2400.2.m.e.1199.5		24	5.4	even	2	inner
2400.2.m.e.1199.6		24	40.19	odd	2	inner
2400.2.m.e.1199.7		24	24.11	even	2	inner
2400.2.m.e.1199.8		24	3.2	odd	2	inner
2400.2.m.e.1199.17		24	15.14	odd	2	inner
2400.2.m.e.1199.18		24	120.59	even	2	inner
2400.2.m.e.1199.19		24	8.3	odd	2	inner
2400.2.m.e.1199.20		24	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.12950 + 1.31310i) q^{3} +4.34495 q^{7} +(-0.448458 + 2.96629i) q^{9} +O(q^{10})\)	\(q+(1.12950 + 1.31310i) q^{3} +4.34495 q^{7} +(-0.448458 + 2.96629i) q^{9} -1.83679i q^{11} -0.588129 q^{13} +5.37818 q^{17} -5.38776 q^{19} +(4.90762 + 5.70535i) q^{21} +2.40885i q^{23} +(-4.40157 + 2.76156i) q^{27} +7.98077 q^{29} +7.06575i q^{31} +(2.41189 - 2.07466i) q^{33} +2.72080 q^{37} +(-0.664291 - 0.772271i) q^{39} -3.42496i q^{41} +2.96772i q^{43} -9.81525i q^{47} +11.8786 q^{49} +(6.07466 + 7.06208i) q^{51} +6.65218i q^{53} +(-6.08547 - 7.07466i) q^{57} -10.7564i q^{59} +9.27803i q^{61} +(-1.94853 + 12.8884i) q^{63} -4.13536i q^{67} +(-3.16306 + 2.72080i) q^{69} +12.2241 q^{71} -4.42003i q^{73} -7.98077i q^{77} +12.5870i q^{79} +(-8.59777 - 2.66052i) q^{81} -11.5594 q^{83} +(9.01428 + 10.4795i) q^{87} -4.21222i q^{89} -2.55539 q^{91} +(-9.27803 + 7.98077i) q^{93} +2.16763i q^{97} +(5.44846 + 0.823724i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9}+O(q^{10}) \) 24 * q + 4 * q^9	\( 24 q + 4 q^{9} - 8 q^{19} + 72 q^{49} + 60 q^{51} - 20 q^{81} + 48 q^{91} + 116 q^{99}+O(q^{100}) \) 24 * q + 4 * q^9 - 8 * q^19 + 72 * q^49 + 60 * q^51 - 20 * q^81 + 48 * q^91 + 116 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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