LMFDB - Embedded newform 2400.2.b.g.2351.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,2,Mod(2351,2400)] 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("2400.2351"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [12,0,-2,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	no
Analytic conductor:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.537291533250985984.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5x^{10} + 14x^{8} - 30x^{6} + 56x^{4} - 80x^{2} + 64 $ x^12 - 5x^10 + 14x^8 - 30x^6 + 56x^4 - 80*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2351.10
Root			$1.39298 - 0.244153i$ of defining polynomial
Character	$\chi$	$=$	2400.2351
Dual form			2400.2.b.g.2351.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.31310 - 1.12950i) q^{3} +4.34495i q^{7} +(0.448458 - 2.96629i) q^{9} -1.83679i q^{11} +0.588129i q^{13} +5.37818i q^{17} +5.38776 q^{19} +(4.90762 + 5.70535i) q^{21} +2.40885 q^{23} +(-2.76156 - 4.40157i) q^{27} -7.98077 q^{29} +7.06575i q^{31} +(-2.07466 - 2.41189i) q^{33} +2.72080i q^{37} +(0.664291 + 0.772271i) q^{39} -3.42496i q^{41} +2.96772 q^{43} +9.81525 q^{47} -11.8786 q^{49} +(6.07466 + 7.06208i) q^{51} +6.65218 q^{53} +(7.07466 - 6.08547i) q^{57} +10.7564i q^{59} +9.27803i q^{61} +(12.8884 + 1.94853i) q^{63} +4.13536 q^{67} +(3.16306 - 2.72080i) q^{69} +12.2241 q^{71} -4.42003 q^{73} +7.98077 q^{77} -12.5870i q^{79} +(-8.59777 - 2.66052i) q^{81} +11.5594i q^{83} +(-10.4795 + 9.01428i) q^{87} +4.21222i q^{89} -2.55539 q^{91} +(7.98077 + 9.27803i) q^{93} -2.16763 q^{97} +(-5.44846 - 0.823724i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{3} - 2 q^{9} + 4 q^{19} - 8 q^{27} + 18 q^{33} + 40 q^{43} - 36 q^{49} + 30 q^{51} + 42 q^{57} + 60 q^{67} + 12 q^{73} - 10 q^{81} + 24 q^{91} - 32 q^{97} - 58 q^{99}+O(q^{100}) $ 12 * q - 2 * q^3 - 2 * q^9 + 4 * q^19 - 8 * q^27 + 18 * q^33 + 40 * q^43 - 36 * q^49 + 30 * q^51 + 42 * q^57 + 60 * q^67 + 12 * q^73 - 10 * q^81 + 24 * q^91 - 32 * q^97 - 58 * q^99

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
$3$	1.31310	−	1.12950i	0.758118	−	0.652117i
$3$	1.31310	−	1.12950i	0.758118	−	0.652117i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			4.34495i			1.64224i	0.570758	+	0.821118i	$0.306650\pi$
$7$			4.34495i			1.64224i	−0.570758	+	0.821118i	$0.693350\pi$
$8$		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
$13$			0.588129i			0.163118i	0.996669	+	0.0815588i	$0.0259898\pi$
$13$			0.588129i			0.163118i	−0.996669	+	0.0815588i	$0.974010\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			5.37818i			1.30440i	0.758047	+	0.652200i	$0.226154\pi$
$17$			5.37818i			1.30440i	−0.758047	+	0.652200i	$0.773846\pi$
$18$		0			0
$19$	5.38776			1.23604			0.618018	−	0.786164i	$-0.287936\pi$
$19$	5.38776			1.23604			0.618018	+	0.786164i	$0.287936\pi$
$20$		0			0
$21$	4.90762	+	5.70535i	1.07093	+	1.24501i
$22$		0			0
$23$	2.40885			0.502280			0.251140	−	0.967951i	$-0.419194\pi$
$23$	2.40885			0.502280			0.251140	+	0.967951i	$0.419194\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−2.76156	−	4.40157i	−0.531462	−	0.847082i
$28$		0			0
$29$	−7.98077			−1.48199			−0.740996	−	0.671510i	$-0.765646\pi$
$29$	−7.98077			−1.48199			−0.740996	+	0.671510i	$0.765646\pi$
$30$		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
$33$	−2.07466	−	2.41189i	−0.361151	−	0.419856i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			2.72080i			0.447297i	0.974670	+	0.223648i	$0.0717967\pi$
$37$			2.72080i			0.447297i	−0.974670	+	0.223648i	$0.928203\pi$
$38$		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
$43$	2.96772			0.452574			0.226287	−	0.974061i	$-0.427341\pi$
$43$	2.96772			0.452574			0.226287	+	0.974061i	$0.427341\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.81525			1.43170			0.715850	−	0.698254i	$-0.246039\pi$
$47$	9.81525			1.43170			0.715850	+	0.698254i	$0.246039\pi$
$48$		0			0
$49$	−11.8786			−1.69694
$50$		0			0
$51$	6.07466	+	7.06208i	0.850622	+	0.988889i
$52$		0			0
$53$	6.65218			0.913748			0.456874	−	0.889531i	$-0.348969\pi$
$53$	6.65218			0.913748			0.456874	+	0.889531i	$0.348969\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	7.07466	−	6.08547i	0.937061	−	0.806040i
$58$		0			0
$59$			10.7564i			1.40036i	0.713967	+	0.700179i	$0.246897\pi$
$59$			10.7564i			1.40036i	−0.713967	+	0.700179i	$0.753103\pi$
$60$		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
$63$	12.8884	+	1.94853i	1.62378	+	0.245492i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.13536			0.505215			0.252607	−	0.967569i	$-0.418712\pi$
$67$	4.13536			0.505215			0.252607	+	0.967569i	$0.418712\pi$
$68$		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
$73$	−4.42003			−0.517325			−0.258663	−	0.965968i	$-0.583282\pi$
$73$	−4.42003			−0.517325			−0.258663	+	0.965968i	$0.583282\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.98077			0.909493
$78$		0			0
$79$		−	12.5870i		−	1.41614i	−0.706141	−	0.708072i	$-0.749565\pi$
$79$		−	12.5870i		−	1.41614i	0.706141	−	0.708072i	$-0.250435\pi$
$80$		0			0
$81$	−8.59777	−	2.66052i	−0.955308	−	0.295613i
$82$		0			0
$83$			11.5594i			1.26881i	0.773002	+	0.634404i	$0.218754\pi$
$83$			11.5594i			1.26881i	−0.773002	+	0.634404i	$0.781246\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−10.4795	+	9.01428i	−1.12352	+	0.966432i
$88$		0			0
$89$			4.21222i			0.446495i	0.974762	+	0.223247i	$0.0716658\pi$
$89$			4.21222i			0.446495i	−0.974762	+	0.223247i	$0.928334\pi$
$90$		0			0
$91$	−2.55539			−0.267878
$92$		0			0
$93$	7.98077	+	9.27803i	0.827567	+	0.962087i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.16763			−0.220090			−0.110045	−	0.993927i	$-0.535099\pi$
$97$	−2.16763			−0.220090			−0.110045	+	0.993927i	$0.535099\pi$
$98$		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
$103$			12.5870i			1.24023i	0.784511	+	0.620115i	$0.212914\pi$
$103$			12.5870i			1.24023i	−0.784511	+	0.620115i	$0.787086\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	3.79002i		−	0.366395i	−0.983076	−	0.183197i	$-0.941355\pi$
$107$		−	3.79002i		−	0.366395i	0.983076	−	0.183197i	$-0.0586447\pi$
$108$		0			0
$109$		−	0.588129i		−	0.0563325i	−0.999603	−	0.0281663i	$-0.991033\pi$
$109$		−	0.588129i		−	0.0563325i	0.999603	−	0.0281663i	$-0.00896678\pi$
$110$		0			0
$111$	3.07314	+	3.57268i	0.291690	+	0.339104i
$112$		0			0
$113$		−	11.0621i		−	1.04064i	−0.853972	−	0.520319i	$-0.825813\pi$
$113$		−	11.0621i		−	1.04064i	0.853972	−	0.520319i	$-0.174187\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.74456	+	0.263751i	0.161285	+	0.0243838i
$118$		0			0
$119$	−23.3679			−2.14213
$120$		0			0
$121$	7.62620			0.693291
$122$		0			0
$123$	−3.86849	−	4.49731i	−0.348810	−	0.405508i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	12.9552i		−	1.14959i	−0.818297	−	0.574796i	$-0.805081\pi$
$127$		−	12.9552i		−	1.14959i	0.818297	−	0.574796i	$-0.194919\pi$
$128$		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
$133$			23.4095i			2.02986i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	5.66820i		−	0.484267i	−0.970243	−	0.242133i	$-0.922153\pi$
$137$		−	5.66820i		−	0.484267i	0.970243	−	0.242133i	$-0.0778471\pi$
$138$		0			0
$139$	2.69075			0.228226			0.114113	−	0.993468i	$-0.463597\pi$
$139$	2.69075			0.228226			0.114113	+	0.993468i	$0.463597\pi$
$140$		0			0
$141$	12.8884	−	11.0863i	1.08540	−	0.933637i
$142$		0			0
$143$	1.08027			0.0903367
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−15.5978	+	13.4169i	−1.28648	+	1.10661i
$148$		0			0
$149$	−20.9591			−1.71703			−0.858517	−	0.512785i	$-0.828614\pi$
$149$	−20.9591			−1.71703			−0.858517	+	0.512785i	$0.828614\pi$
$150$		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
$153$	15.9533	+	2.41189i	1.28974	+	0.194990i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			11.9988i			0.957611i	0.877921	+	0.478805i	$0.158930\pi$
$157$			11.9988i			0.957611i	−0.877921	+	0.478805i	$0.841070\pi$
$158$		0			0
$159$	8.73498	−	7.51364i	0.692729	−	0.595871i
$160$		0			0
$161$			10.4663i			0.824864i
$162$		0			0
$163$	13.1816			1.03246			0.516231	−	0.856449i	$-0.327335\pi$
$163$	13.1816			1.03246			0.516231	+	0.856449i	$0.327335\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.73744			0.289211			0.144606	−	0.989489i	$-0.453809\pi$
$167$	3.73744			0.289211			0.144606	+	0.989489i	$0.453809\pi$
$168$		0			0
$169$	12.6541			0.973393
$170$		0			0
$171$	2.41618	−	15.9817i	0.184770	−	1.22215i
$172$		0			0
$173$	6.65218			0.505756			0.252878	−	0.967498i	$-0.418623\pi$
$173$	6.65218			0.505756			0.252878	+	0.967498i	$0.418623\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	12.1493	+	14.1242i	0.913198	+	1.06164i
$178$		0			0
$179$		−	18.4093i		−	1.37598i	−0.725722	−	0.687988i	$-0.758494\pi$
$179$		−	18.4093i		−	1.37598i	0.725722	−	0.687988i	$-0.241506\pi$
$180$		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
$183$	10.4795	+	12.1830i	0.774670	+	0.900591i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	9.87859			0.722394
$188$		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
$193$	1.47689			0.106309			0.0531543	−	0.998586i	$-0.483072\pi$
$193$	1.47689			0.106309			0.0531543	+	0.998586i	$0.483072\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	11.1438			0.793965			0.396982	−	0.917826i	$-0.370057\pi$
$197$	11.1438			0.793965			0.396982	+	0.917826i	$0.370057\pi$
$198$		0			0
$199$			2.80041i			0.198516i	0.995062	+	0.0992578i	$0.0316469\pi$
$199$			2.80041i			0.198516i	−0.995062	+	0.0992578i	$0.968353\pi$
$200$		0			0
$201$	5.43014	−	4.67089i	0.383012	−	0.329459i
$202$		0			0
$203$		−	34.6760i		−	2.43378i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.08027	−	7.14536i	0.0750839	−	0.496637i
$208$		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
$213$	16.0515	−	13.8071i	1.09983	−	0.946048i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−30.7003			−2.08407
$218$		0			0
$219$	−5.80394	+	4.99243i	−0.392194	+	0.337357i
$220$		0			0
$221$	−3.16306			−0.212771
$222$		0			0
$223$			1.62415i			0.108761i	0.998520	+	0.0543806i	$0.0173184\pi$
$223$			1.62415i			0.108761i	−0.998520	+	0.0543806i	$0.982682\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	4.94021i		−	0.327893i	−0.986469	−	0.163947i	$-0.947578\pi$
$227$		−	4.94021i		−	0.327893i	0.986469	−	0.163947i	$-0.0524225\pi$
$228$		0			0
$229$		−	15.2471i		−	1.00756i	−0.863832	−	0.503779i	$-0.831942\pi$
$229$		−	15.2471i		−	1.00756i	0.863832	−	0.503779i	$-0.168058\pi$
$230$		0			0
$231$	10.4795	−	9.01428i	0.689503	−	0.593096i
$232$		0			0
$233$			22.3000i			1.46092i	0.682955	+	0.730460i	$0.260694\pi$
$233$			22.3000i			1.46092i	−0.682955	+	0.730460i	$0.739306\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−14.2170	−	16.5279i	−0.923492	−	1.07360i
$238$		0			0
$239$	−14.8813			−0.962589			−0.481294	−	0.876559i	$-0.659833\pi$
$239$	−14.8813			−0.962589			−0.481294	+	0.876559i	$0.659833\pi$
$240$		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
$243$	−14.2948	+	6.21766i	−0.917010	+	0.398863i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			3.16869i			0.201619i
$248$		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
$253$		−	4.42456i		−	0.278170i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	22.8859i		−	1.42758i	−0.700358	−	0.713792i	$-0.746976\pi$
$257$		−	22.8859i		−	1.42758i	0.700358	−	0.713792i	$-0.253024\pi$
$258$		0			0
$259$	−11.8217			−0.734567
$260$		0			0
$261$	−3.57904	+	23.6733i	−0.221537	+	1.46534i
$262$		0			0
$263$	−13.5527			−0.835694			−0.417847	−	0.908517i	$-0.637215\pi$
$263$	−13.5527			−0.835694			−0.417847	+	0.908517i	$0.637215\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	4.75771	+	5.53107i	0.291167	+	0.338496i
$268$		0			0
$269$	−12.9783			−0.791301			−0.395651	−	0.918401i	$-0.629481\pi$
$269$	−12.9783			−0.791301			−0.395651	+	0.918401i	$0.629481\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−3.35548	+	2.88631i	−0.203083	+	0.174688i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	16.7917i		−	1.00891i	−0.863437	−	0.504457i	$-0.831693\pi$
$277$		−	16.7917i		−	1.00891i	0.863437	−	0.504457i	$-0.168307\pi$
$278$		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
$283$	−18.7047			−1.11188			−0.555940	−	0.831223i	$-0.687641\pi$
$283$	−18.7047			−1.11188			−0.555940	+	0.831223i	$0.687641\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	14.8813			0.878413
$288$		0			0
$289$	−11.9248			−0.701460
$290$		0			0
$291$	−2.84632	+	2.44834i	−0.166854	+	0.143524i
$292$		0			0
$293$	−29.4457			−1.72024			−0.860119	−	0.510093i	$-0.829611\pi$
$293$	−29.4457			−1.72024			−0.860119	+	0.510093i	$0.829611\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−8.08476	+	5.07240i	−0.469125	+	0.294331i
$298$		0			0
$299$			1.41672i			0.0819308i
$300$		0			0
$301$			12.8946i			0.743233i
$302$		0			0
$303$	−4.15341	+	3.57268i	−0.238607	+	0.205245i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	33.2095			1.89537			0.947683	−	0.319213i	$-0.103419\pi$
$307$	33.2095			1.89537			0.947683	+	0.319213i	$0.103419\pi$
$308$		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
$313$	9.65410			0.545682			0.272841	−	0.962059i	$-0.412037\pi$
$313$	9.65410			0.545682			0.272841	+	0.962059i	$0.412037\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.81770			−0.270589			−0.135295	−	0.990805i	$-0.543198\pi$
$317$	−4.81770			−0.270589			−0.135295	+	0.990805i	$0.543198\pi$
$318$		0			0
$319$			14.6590i			0.820747i
$320$		0			0
$321$	−4.28082	−	4.97667i	−0.238932	−	0.277770i
$322$		0			0
$323$			28.9763i			1.61229i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−0.664291	−	0.772271i	−0.0367354	−	0.0427067i
$328$		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
$333$	8.07068	+	1.22016i	0.442271	+	0.0668646i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	13.0925			0.713192			0.356596	−	0.934259i	$-0.383937\pi$
$337$	13.0925			0.713192			0.356596	+	0.934259i	$0.383937\pi$
$338$		0			0
$339$	−12.4947	−	14.5257i	−0.678618	−	0.788927i
$340$		0			0
$341$	12.9783			0.702815
$342$		0			0
$343$		−	21.1972i		−	1.14454i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	3.97936i		−	0.213623i	−0.994279	−	0.106812i	$-0.965936\pi$
$347$		−	3.97936i		−	0.213623i	0.994279	−	0.106812i	$-0.0340642\pi$
$348$		0			0
$349$			12.5870i			0.673764i	0.941547	+	0.336882i	$0.109372\pi$
$349$			12.5870i			0.673764i	−0.941547	+	0.336882i	$0.890628\pi$
$350$		0			0
$351$	2.58869	−	1.62415i	0.138174	−	0.0866908i
$352$		0			0
$353$		−	0.787269i		−	0.0419021i	−0.999781	−	0.0209510i	$-0.993331\pi$
$353$		−	0.787269i		−	0.0419021i	0.999781	−	0.0209510i	$-0.00666941\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−30.6844	+	26.3941i	−1.62399	+	1.39692i
$358$		0			0
$359$	1.08027			0.0570144			0.0285072	−	0.999594i	$-0.490925\pi$
$359$	1.08027			0.0570144			0.0285072	+	0.999594i	$0.490925\pi$
$360$		0			0
$361$	10.0279			0.527785
$362$		0			0
$363$	10.0140	−	8.61379i	0.525596	−	0.452107i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			14.5794i			0.761038i	0.924773	+	0.380519i	$0.124255\pi$
$367$			14.5794i			0.761038i	−0.924773	+	0.380519i	$0.875745\pi$
$368$		0			0
$369$	−10.1594	−	1.53595i	−0.528878	−	0.0799583i
$370$		0			0
$371$			28.9034i			1.50059i
$372$		0			0
$373$		−	31.5719i		−	1.63473i	−0.576118	−	0.817366i	$-0.695433\pi$
$373$		−	31.5719i		−	1.63473i	0.576118	−	0.817366i	$-0.304567\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	4.69372i		−	0.241739i
$378$		0			0
$379$	−4.61224			−0.236915			−0.118458	−	0.992959i	$-0.537795\pi$
$379$	−4.61224			−0.236915			−0.118458	+	0.992959i	$0.537795\pi$
$380$		0			0
$381$	−14.6330	−	17.0115i	−0.749669	−	0.871526i
$382$		0			0
$383$	−14.6330			−0.747709			−0.373854	−	0.927487i	$-0.621964\pi$
$383$	−14.6330			−0.747709			−0.373854	+	0.927487i	$0.621964\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.33090	−	8.80314i	0.0676535	−	0.447489i
$388$		0			0
$389$	15.1388			0.767570			0.383785	−	0.923422i	$-0.374620\pi$
$389$	15.1388			0.767570			0.383785	+	0.923422i	$0.374620\pi$
$390$		0			0
$391$			12.9552i			0.655175i
$392$		0			0
$393$	−3.37380	−	3.92221i	−0.170186	−	0.197849i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			18.3362i			0.920268i	0.887849	+	0.460134i	$0.152199\pi$
$397$			18.3362i			0.920268i	−0.887849	+	0.460134i	$0.847801\pi$
$398$		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
$403$	−4.15557			−0.207004
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.99754			0.247719
$408$		0			0
$409$	18.6926			0.924291			0.462146	−	0.886804i	$-0.347080\pi$
$409$	18.6926			0.924291			0.462146	+	0.886804i	$0.347080\pi$
$410$		0			0
$411$	−6.40223	−	7.44290i	−0.315799	−	0.367131i
$412$		0			0
$413$	−46.7359			−2.29972
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.53322	−	3.03920i	0.173023	−	0.148830i
$418$		0			0
$419$			5.24599i			0.256283i	0.991756	+	0.128142i	$0.0409012\pi$
$419$			5.24599i			0.256283i	−0.991756	+	0.128142i	$0.959099\pi$
$420$		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
$423$	4.40173	−	29.1149i	0.214019	−	1.41561i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−40.3126			−1.95086
$428$		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
$433$	−32.2158			−1.54819			−0.774095	−	0.633069i	$-0.781795\pi$
$433$	−32.2158			−1.54819			−0.774095	+	0.633069i	$0.781795\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.9783			0.620837
$438$		0			0
$439$			2.27291i			0.108480i	0.998528	+	0.0542399i	$0.0172736\pi$
$439$			2.27291i			0.108480i	−0.998528	+	0.0542399i	$0.982726\pi$
$440$		0			0
$441$	−5.32705	+	35.2354i	−0.253669	+	1.67787i
$442$		0			0
$443$			4.59091i			0.218121i	0.994035	+	0.109060i	$0.0347842\pi$
$443$			4.59091i			0.218121i	−0.994035	+	0.109060i	$0.965216\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−27.5213	+	23.6733i	−1.30171	+	1.11971i
$448$		0			0
$449$		−	39.0461i		−	1.84270i	−0.388736	−	0.921349i	$-0.627088\pi$
$449$		−	39.0461i		−	1.84270i	0.388736	−	0.921349i	$-0.372912\pi$
$450$		0			0
$451$	−6.29093			−0.296228
$452$		0			0
$453$	−3.57904	−	4.16081i	−0.168158	−	0.195492i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.96147			0.232088			0.116044	−	0.993244i	$-0.462979\pi$
$457$	4.96147			0.232088			0.116044	+	0.993244i	$0.462979\pi$
$458$		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
$463$		−	14.2907i		−	0.664146i	−0.943254	−	0.332073i	$-0.892252\pi$
$463$		−	14.2907i		−	0.664146i	0.943254	−	0.332073i	$-0.107748\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			31.8469i			1.47370i	0.676058	+	0.736849i	$0.263687\pi$
$467$			31.8469i			1.47370i	−0.676058	+	0.736849i	$0.736313\pi$
$468$		0			0
$469$			17.9679i			0.829682i
$470$		0			0
$471$	13.5527	+	15.7557i	0.624475	+	0.725982i
$472$		0			0
$473$		−	5.45109i		−	0.250641i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	2.98323	−	19.7323i	0.136593	−	0.903481i
$478$		0			0
$479$	−25.9566			−1.18599			−0.592994	−	0.805207i	$-0.702054\pi$
$479$	−25.9566			−1.18599			−0.592994	+	0.805207i	$0.702054\pi$
$480$		0			0
$481$	−1.60018			−0.0729619
$482$		0			0
$483$	11.8217	+	13.7433i	0.537908	+	0.625344i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	24.4456i		−	1.10773i	−0.832605	−	0.553867i	$-0.813152\pi$
$487$		−	24.4456i		−	1.10773i	0.832605	−	0.553867i	$-0.186848\pi$
$488$		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
$493$		−	42.9220i		−	1.93311i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			53.1131i			2.38245i
$498$		0			0
$499$	36.7249			1.64403			0.822016	−	0.569464i	$-0.192849\pi$
$499$	36.7249			1.64403			0.822016	+	0.569464i	$0.192849\pi$
$500$		0			0
$501$	4.90762	−	4.22143i	0.219256	−	0.188600i
$502$		0			0
$503$	−29.5142			−1.31597			−0.657987	−	0.753029i	$-0.728592\pi$
$503$	−29.5142			−1.31597			−0.657987	+	0.753029i	$0.728592\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	16.6161	−	14.2928i	0.737947	−	0.634766i
$508$		0			0
$509$	2.16054			0.0957642			0.0478821	−	0.998853i	$-0.484753\pi$
$509$	2.16054			0.0957642			0.0478821	+	0.998853i	$0.484753\pi$
$510$		0			0
$511$		−	19.2048i		−	0.849571i
$512$		0			0
$513$	−14.8786	−	23.7146i	−0.656906	−	1.04702i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	18.0286i		−	0.792895i
$518$		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
$523$	9.68638			0.423556			0.211778	−	0.977318i	$-0.432075\pi$
$523$	9.68638			0.423556			0.211778	+	0.977318i	$0.432075\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−38.0009			−1.65534
$528$		0			0
$529$	−17.1974			−0.747714
$530$		0			0
$531$	31.9065	+	4.82378i	1.38462	+	0.209334i
$532$		0			0
$533$	2.01431			0.0872497
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−20.7933	−	24.1732i	−0.897298	−	1.04315i
$538$		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
$543$	−22.0393	−	25.6218i	−0.945799	−	1.09954i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−28.4113			−1.21478			−0.607390	−	0.794404i	$-0.707783\pi$
$547$	−28.4113			−1.21478			−0.607390	+	0.794404i	$0.707783\pi$
$548$		0			0
$549$	27.5213	+	4.16081i	1.17458	+	0.177579i
$550$		0			0
$551$	−42.9984			−1.83179
$552$		0			0
$553$	54.6897			2.32564
$554$		0			0
$555$		0			0
$556$		0			0
$557$	32.1029			1.36024			0.680122	−	0.733099i	$-0.261927\pi$
$557$	32.1029			1.36024			0.680122	+	0.733099i	$0.261927\pi$
$558$		0			0
$559$			1.74540i			0.0738227i
$560$		0			0
$561$	12.9716	−	11.1579i	0.547660	−	0.471086i
$562$		0			0
$563$		−	14.2406i		−	0.600170i	−0.953913	−	0.300085i	$-0.902985\pi$
$563$		−	14.2406i		−	0.600170i	0.953913	−	0.300085i	$-0.0970150\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	11.5598	−	37.3569i	0.485466	−	1.56884i
$568$		0			0
$569$		−	9.08328i		−	0.380791i	−0.981707	−	0.190396i	$-0.939023\pi$
$569$		−	9.08328i		−	0.380791i	0.981707	−	0.190396i	$-0.0609770\pi$
$570$		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
$573$	−11.4699	+	9.86616i	−0.479161	+	0.412165i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	33.1974			1.38203			0.691014	−	0.722842i	$-0.257164\pi$
$577$	33.1974			1.38203			0.691014	+	0.722842i	$0.257164\pi$
$578$		0			0
$579$	1.93930	−	1.66814i	0.0805944	−	0.0693256i
$580$		0			0
$581$	−50.2250			−2.08368
$582$		0			0
$583$		−	12.2187i		−	0.506046i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	0.613686i		−	0.0253295i	−0.999920	−	0.0126648i	$-0.995969\pi$
$587$		−	0.613686i		−	0.0253295i	0.999920	−	0.0126648i	$-0.00403143\pi$
$588$		0			0
$589$			38.0685i			1.56859i
$590$		0			0
$591$	14.6330	−	12.5870i	0.601919	−	0.517758i
$592$		0			0
$593$			15.5545i			0.638747i	0.947629	+	0.319374i	$0.103472\pi$
$593$			15.5545i			0.638747i	−0.947629	+	0.319374i	$0.896528\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	3.16306	+	3.67721i	0.129456	+	0.150498i
$598$		0			0
$599$	39.0811			1.59681			0.798406	−	0.602119i	$-0.205677\pi$
$599$	39.0811			1.59681			0.798406	+	0.602119i	$0.205677\pi$
$600$		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
$603$	1.85454	−	12.2667i	0.0755225	−	0.499538i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			10.8832i			0.441735i	0.975304	+	0.220868i	$0.0708889\pi$
$607$			10.8832i			0.441735i	−0.975304	+	0.220868i	$0.929111\pi$
$608$		0			0
$609$	−39.1666	−	45.5331i	−1.58711	−	1.84509i
$610$		0			0
$611$			5.77263i			0.233535i
$612$		0			0
$613$			30.3350i			1.22522i	0.790385	+	0.612610i	$0.209881\pi$
$613$			30.3350i			1.22522i	−0.790385	+	0.612610i	$0.790119\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	8.71447i		−	0.350831i	−0.984494	−	0.175416i	$-0.943873\pi$
$617$		−	8.71447i		−	0.350831i	0.984494	−	0.175416i	$-0.0561269\pi$
$618$		0			0
$619$	−13.0323			−0.523811			−0.261906	−	0.965093i	$-0.584351\pi$
$619$	−13.0323			−0.523811			−0.261906	+	0.965093i	$0.584351\pi$
$620$		0			0
$621$	−6.65218	−	10.6027i	−0.266943	−	0.425473i
$622$		0			0
$623$	−18.3019			−0.733250
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−11.1777	−	12.9947i	−0.446396	−	0.518957i
$628$		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
$633$	−12.9533	+	11.1421i	−0.514845	+	0.442859i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	6.98614i		−	0.276801i
$638$		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
$643$	−15.7938			−0.622848			−0.311424	−	0.950271i	$-0.600806\pi$
$643$	−15.7938			−0.622848			−0.311424	+	0.950271i	$0.600806\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.8128			0.582351			0.291175	−	0.956670i	$-0.405954\pi$
$647$	14.8128			0.582351			0.291175	+	0.956670i	$0.405954\pi$
$648$		0			0
$649$	19.7572			0.775537
$650$		0			0
$651$	−40.3126	+	34.6760i	−1.57997	+	1.35906i
$652$		0			0
$653$	−3.48912			−0.136540			−0.0682699	−	0.997667i	$-0.521748\pi$
$653$	−3.48912			−0.136540			−0.0682699	+	0.997667i	$0.521748\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.98220	+	13.1111i	−0.0773329	+	0.511513i
$658$		0			0
$659$			36.5563i			1.42403i	0.702163	+	0.712017i	$0.252218\pi$
$659$			36.5563i			1.42403i	−0.702163	+	0.712017i	$0.747782\pi$
$660$		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
$663$	−4.15341	+	3.57268i	−0.161305	+	0.138751i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−19.2245			−0.744375
$668$		0			0
$669$	1.83448	+	2.13267i	0.0709251	+	0.0824538i
$670$		0			0
$671$	17.0418			0.657892
$672$		0			0
$673$	−46.2899			−1.78434			−0.892172	−	0.451696i	$-0.850819\pi$
$673$	−46.2899			−1.78434			−0.892172	+	0.451696i	$0.850819\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−27.7911			−1.06810			−0.534049	−	0.845453i	$-0.679330\pi$
$677$	−27.7911			−1.06810			−0.534049	+	0.845453i	$0.679330\pi$
$678$		0			0
$679$		−	9.41826i		−	0.361440i
$680$		0			0
$681$	−5.57997	−	6.48699i	−0.213825	−	0.248582i
$682$		0			0
$683$			33.6521i			1.28766i	0.765167	+	0.643832i	$0.222656\pi$
$683$			33.6521i			1.28766i	−0.765167	+	0.643832i	$0.777344\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−17.2216	−	20.0210i	−0.657047	−	0.763849i
$688$		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
$693$	3.57904	−	23.6733i	0.135956	−	0.899274i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	18.4200			0.697708
$698$		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
$703$			14.6590i			0.552875i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	13.7433i		−	0.516872i
$708$		0			0
$709$		−	20.5295i		−	0.771002i	−0.922707	−	0.385501i	$-0.874029\pi$
$709$		−	20.5295i		−	0.771002i	0.922707	−	0.385501i	$-0.125971\pi$
$710$		0			0
$711$	−37.3366	−	5.64472i	−1.40023	−	0.211694i
$712$		0			0
$713$			17.0203i			0.637417i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−19.5406	+	16.8084i	−0.729756	+	0.627721i
$718$		0			0
$719$	14.8813			0.554977			0.277489	−	0.960729i	$-0.410498\pi$
$719$	14.8813			0.554977			0.277489	+	0.960729i	$0.410498\pi$
$720$		0			0
$721$	−54.6897			−2.03675
$722$		0			0
$723$	−0.686901	+	0.590858i	−0.0255461	+	0.0219742i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	16.1239i		−	0.598004i	−0.954253	−	0.299002i	$-0.903346\pi$
$727$		−	16.1239i		−	0.598004i	0.954253	−	0.299002i	$-0.0966537\pi$
$728$		0			0
$729$	−11.7476	+	24.3104i	−0.435096	+	0.900384i
$730$		0			0
$731$			15.9610i			0.590337i
$732$		0			0
$733$			16.7310i			0.617975i	0.951066	+	0.308988i	$0.0999902\pi$
$733$			16.7310i			0.617975i	−0.951066	+	0.308988i	$0.900010\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	7.59579i		−	0.279795i
$738$		0			0
$739$	−12.5693			−0.462371			−0.231185	−	0.972910i	$-0.574260\pi$
$739$	−12.5693			−0.462371			−0.231185	+	0.972910i	$0.574260\pi$
$740$		0			0
$741$	3.57904	+	4.16081i	0.131479	+	0.152851i
$742$		0			0
$743$	22.2877			0.817655			0.408827	−	0.912612i	$-0.365938\pi$
$743$	22.2877			0.817655			0.408827	+	0.912612i	$0.365938\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	34.2885	+	5.18390i	1.25455	+	0.189669i
$748$		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
$753$	5.44846	+	6.33410i	0.198553	+	0.230827i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	20.6887i		−	0.751945i	−0.926631	−	0.375972i	$-0.877309\pi$
$757$		−	20.6887i		−	0.751945i	0.926631	−	0.375972i	$-0.122691\pi$
$758$		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
$763$	2.55539			0.0925113
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.32612			−0.228423
$768$		0			0
$769$	−46.3082			−1.66992			−0.834958	−	0.550313i	$-0.814508\pi$
$769$	−46.3082			−1.66992			−0.834958	+	0.550313i	$0.814508\pi$
$770$		0			0
$771$	−25.8496	−	30.0515i	−0.930952	−	1.08228i
$772$		0			0
$773$	−30.2684			−1.08868			−0.544340	−	0.838865i	$-0.683220\pi$
$773$	−30.2684			−1.08868			−0.544340	+	0.838865i	$0.683220\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−15.5231	+	13.3527i	−0.556889	+	0.479024i
$778$		0			0
$779$		−	18.4528i		−	0.661141i
$780$		0			0
$781$		−	22.4531i		−	0.803436i
$782$		0			0
$783$	22.0393	+	35.1279i	0.787622	+	1.25537i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	19.4080			0.691819			0.345910	−	0.938268i	$-0.387570\pi$
$787$	19.4080			0.691819			0.345910	+	0.938268i	$0.387570\pi$
$788$		0			0
$789$	−17.7960	+	15.3078i	−0.633555	+	0.544971i
$790$		0			0
$791$	48.0644			1.70897
$792$		0			0
$793$	−5.45667			−0.193772
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−30.7743			−1.09008			−0.545041	−	0.838409i	$-0.683486\pi$
$797$	−30.7743			−1.09008			−0.545041	+	0.838409i	$0.683486\pi$
$798$		0			0
$799$			52.7882i			1.86751i
$800$		0			0
$801$	12.4947	+	1.88901i	0.441478	+	0.0667448i
$802$		0			0
$803$			8.11867i			0.286502i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−17.0418	+	14.6590i	−0.599900	+	0.516021i
$808$		0			0
$809$			37.5744i			1.32104i	0.750807	+	0.660522i	$0.229665\pi$
$809$			37.5744i			1.32104i	−0.750807	+	0.660522i	$0.770335\pi$
$810$		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
$817$	15.9894			0.559397
$818$		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
$823$			0.569147i			0.0198392i	0.999951	+	0.00991960i	$0.00315756\pi$
$823$			0.569147i			0.0198392i	−0.999951	+	0.00991960i	$0.996842\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	23.0851i		−	0.802748i	−0.915914	−	0.401374i	$-0.868533\pi$
$827$		−	23.0851i		−	0.802748i	0.915914	−	0.401374i	$-0.131467\pi$
$828$		0			0
$829$		−	48.3636i		−	1.67974i	−0.542790	−	0.839869i	$-0.682632\pi$
$829$		−	48.3636i		−	1.67974i	0.542790	−	0.839869i	$-0.317368\pi$
$830$		0			0
$831$	−18.9662	−	22.0491i	−0.657930	−	0.764876i
$832$		0			0
$833$		−	63.8852i		−	2.21349i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	31.1004	−	19.5125i	1.07499	−	0.674450i
$838$		0			0
$839$	−22.0393			−0.760883			−0.380441	−	0.924805i	$-0.624228\pi$
$839$	−22.0393			−0.760883			−0.380441	+	0.924805i	$0.624228\pi$
$840$		0			0
$841$	34.6926			1.19630
$842$		0			0
$843$	12.4769	+	14.5050i	0.429727	+	0.499578i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			33.1355i			1.13855i
$848$		0			0
$849$	−24.5611	+	21.1270i	−0.842936	+	0.725076i
$850$		0			0
$851$			6.55400i			0.224668i
$852$		0			0
$853$			6.39801i			0.219064i	0.993983	+	0.109532i	$0.0349352\pi$
$853$			6.39801i			0.219064i	−0.993983	+	0.109532i	$0.965065\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	27.2124i		−	0.929559i	−0.885426	−	0.464780i	$-0.846134\pi$
$857$		−	27.2124i		−	0.929559i	0.885426	−	0.464780i	$-0.153866\pi$
$858$		0			0
$859$	−23.4663			−0.800658			−0.400329	−	0.916371i	$-0.631104\pi$
$859$	−23.4663			−0.800658			−0.400329	+	0.916371i	$0.631104\pi$
$860$		0			0
$861$	19.5406	−	16.8084i	0.665941	−	0.572828i
$862$		0			0
$863$	−12.4724			−0.424566			−0.212283	−	0.977208i	$-0.568090\pi$
$863$	−12.4724			−0.424566			−0.212283	+	0.977208i	$0.568090\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−15.6585	+	13.4691i	−0.531790	+	0.457434i
$868$		0			0
$869$	−23.1196			−0.784279
$870$		0			0
$871$			2.43212i			0.0824093i
$872$		0			0
$873$	−0.972093	+	6.42984i	−0.0329004	+	0.217617i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	48.9517i		−	1.65298i	−0.562950	−	0.826491i	$-0.690333\pi$
$877$		−	48.9517i		−	1.65298i	0.562950	−	0.826491i	$-0.309667\pi$
$878$		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
$883$	37.1170			1.24909			0.624544	−	0.780990i	$-0.285285\pi$
$883$	37.1170			1.24909			0.624544	+	0.780990i	$0.285285\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.7792			0.697699			0.348849	−	0.937179i	$-0.386573\pi$
$887$	20.7792			0.697699			0.348849	+	0.937179i	$0.386573\pi$
$888$		0			0
$889$	56.2899			1.88790
$890$		0			0
$891$	−4.88681	+	15.7923i	−0.163714	+	0.529062i
$892$		0			0
$893$	52.8821			1.76963
$894$		0			0
$895$		0			0
$896$		0			0
$897$	1.60018	+	1.86029i	0.0534285	+	0.0621132i
$898$		0			0
$899$		−	56.3901i		−	1.88072i
$900$		0			0
$901$			35.7766i			1.19189i
$902$		0			0
$903$	14.5645	+	16.9319i	0.484675	+	0.563459i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−25.5231			−0.847481			−0.423741	−	0.905784i	$-0.639283\pi$
$907$	−25.5231			−0.847481			−0.423741	+	0.905784i	$0.639283\pi$
$908$		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
$913$	21.2322			0.702683
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12.9783			0.428581
$918$		0			0
$919$		−	27.9743i		−	0.922788i	−0.887195	−	0.461394i	$-0.847350\pi$
$919$		−	27.9743i		−	0.922788i	0.887195	−	0.461394i	$-0.152650\pi$
$920$		0			0
$921$	43.6074	−	37.5101i	1.43691	−	1.23600i
$922$		0			0
$923$			7.18934i			0.236640i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	37.3366	+	5.64472i	1.22629	+	0.185397i
$928$		0			0
$929$		−	46.7604i		−	1.53416i	−0.641551	−	0.767080i	$-0.721709\pi$
$929$		−	46.7604i		−	1.53416i	0.641551	−	0.767080i	$-0.278291\pi$
$930$		0			0
$931$	−63.9990			−2.09748
$932$		0			0
$933$	−33.8475	+	29.1149i	−1.10812	+	0.953178i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−41.2253			−1.34677			−0.673387	−	0.739291i	$-0.735161\pi$
$937$	−41.2253			−1.34677			−0.673387	+	0.739291i	$0.735161\pi$
$938$		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
$943$		−	8.25021i		−	0.268664i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	21.3864i		−	0.694965i	−0.937686	−	0.347483i	$-0.887037\pi$
$947$		−	21.3864i		−	0.694965i	0.937686	−	0.347483i	$-0.112963\pi$
$948$		0			0
$949$		−	2.59955i		−	0.0843849i
$950$		0			0
$951$	−6.32612	+	5.44160i	−0.205139	+	0.176456i
$952$		0			0
$953$			11.0306i			0.357317i	0.983911	+	0.178658i	$0.0571757\pi$
$953$			11.0306i			0.357317i	−0.983911	+	0.178658i	$0.942824\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	16.5573	+	19.2487i	0.535223	+	0.622223i
$958$		0			0
$959$	24.6280			0.795281
$960$		0			0
$961$	−18.9248			−0.610478
$962$		0			0
$963$	−11.2423	−	1.69966i	−0.362278	−	0.0547709i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			10.8832i			0.349980i	0.984570	+	0.174990i	$0.0559893\pi$
$967$			10.8832i			0.349980i	−0.984570	+	0.174990i	$0.944011\pi$
$968$		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
$973$			11.6912i			0.374802i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			41.7609i			1.33605i	0.744138	+	0.668026i	$0.232860\pi$
$977$			41.7609i			1.33605i	−0.744138	+	0.668026i	$0.767140\pi$
$978$		0			0
$979$	7.73698			0.247275
$980$		0			0
$981$	−1.74456	−	0.263751i	−0.0556995	−	0.00842092i
$982$		0			0
$983$	33.2516			1.06056			0.530281	−	0.847822i	$-0.322086\pi$
$983$	33.2516			1.06056			0.530281	+	0.847822i	$0.322086\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	48.1695	+	55.9994i	1.53325	+	1.78248i
$988$		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
$993$	1.80394	−	1.55171i	0.0572462	−	0.0492420i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			45.6428i			1.44552i	0.691098	+	0.722761i	$0.257127\pi$
$997$			45.6428i			1.44552i	−0.691098	+	0.722761i	$0.742873\pi$
$998$		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.b.g.2351.10		12
3.2	odd	2	inner	2400.2.b.g.2351.12		12
4.3	odd	2		600.2.b.h.251.1	yes	12
5.2	odd	4		2400.2.m.e.1199.5		24
5.3	odd	4		2400.2.m.e.1199.20		24
5.4	even	2		2400.2.b.h.2351.3		12
8.3	odd	2	inner	2400.2.b.g.2351.9		12
8.5	even	2		600.2.b.h.251.11	yes	12
12.11	even	2		600.2.b.h.251.12	yes	12
15.2	even	4		2400.2.m.e.1199.17		24
15.8	even	4		2400.2.m.e.1199.8		24
15.14	odd	2		2400.2.b.h.2351.1		12
20.3	even	4		600.2.m.e.299.11		24
20.7	even	4		600.2.m.e.299.14		24
20.19	odd	2		600.2.b.g.251.12	yes	12
24.5	odd	2		600.2.b.h.251.2	yes	12
24.11	even	2	inner	2400.2.b.g.2351.11		12
40.3	even	4		2400.2.m.e.1199.19		24
40.13	odd	4		600.2.m.e.299.9		24
40.19	odd	2		2400.2.b.h.2351.4		12
40.27	even	4		2400.2.m.e.1199.6		24
40.29	even	2		600.2.b.g.251.2	yes	12
40.37	odd	4		600.2.m.e.299.16		24
60.23	odd	4		600.2.m.e.299.13		24
60.47	odd	4		600.2.m.e.299.12		24
60.59	even	2		600.2.b.g.251.1	✓	12
120.29	odd	2		600.2.b.g.251.11	yes	12
120.53	even	4		600.2.m.e.299.15		24
120.59	even	2		2400.2.b.h.2351.2		12
120.77	even	4		600.2.m.e.299.10		24
120.83	odd	4		2400.2.m.e.1199.7		24
120.107	odd	4		2400.2.m.e.1199.18		24

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(1.31310 - 1.12950i) q^{3} +4.34495i q^{7} +(0.448458 - 2.96629i) q^{9} -1.83679i q^{11} +0.588129i q^{13} +5.37818i q^{17} +5.38776 q^{19} +(4.90762 + 5.70535i) q^{21} +2.40885 q^{23} +(-2.76156 - 4.40157i) q^{27} -7.98077 q^{29} +7.06575i q^{31} +(-2.07466 - 2.41189i) q^{33} +2.72080i q^{37} +(0.664291 + 0.772271i) q^{39} -3.42496i q^{41} +2.96772 q^{43} +9.81525 q^{47} -11.8786 q^{49} +(6.07466 + 7.06208i) q^{51} +6.65218 q^{53} +(7.07466 - 6.08547i) q^{57} +10.7564i q^{59} +9.27803i q^{61} +(12.8884 + 1.94853i) q^{63} +4.13536 q^{67} +(3.16306 - 2.72080i) q^{69} +12.2241 q^{71} -4.42003 q^{73} +7.98077 q^{77} -12.5870i q^{79} +(-8.59777 - 2.66052i) q^{81} +11.5594i q^{83} +(-10.4795 + 9.01428i) q^{87} +4.21222i q^{89} -2.55539 q^{91} +(7.98077 + 9.27803i) q^{93} -2.16763 q^{97} +(-5.44846 - 0.823724i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{3} - 2 q^{9} + 4 q^{19} - 8 q^{27} + 18 q^{33} + 40 q^{43} - 36 q^{49} + 30 q^{51} + 42 q^{57} + 60 q^{67} + 12 q^{73} - 10 q^{81} + 24 q^{91} - 32 q^{97} - 58 q^{99}+O(q^{100}) \) 12 * q - 2 * q^3 - 2 * q^9 + 4 * q^19 - 8 * q^27 + 18 * q^33 + 40 * q^43 - 36 * q^49 + 30 * q^51 + 42 * q^57 + 60 * q^67 + 12 * q^73 - 10 * q^81 + 24 * q^91 - 32 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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