LMFDB - Embedded newform 2400.2.b.g.2351.1

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			2351.1
Root			$-1.26128 + 0.639662i$ of defining polynomial
Character	$\chi$	$=$	2400.2351
Dual form			2400.2.b.g.2351.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.57067 - 0.730070i) q^{3} -1.25539i q^{7} +(1.93400 + 2.29339i) q^{9} +3.02346i q^{11} -5.65509i q^{13} +2.45546i q^{17} -1.77801 q^{19} +(-0.916519 + 1.97179i) q^{21} +8.84074 q^{23} +(-1.36333 - 5.01411i) q^{27} -3.79561 q^{29} +5.19897i q^{31} +(2.20734 - 4.74886i) q^{33} +6.45436i q^{37} +(-4.12861 + 8.88227i) q^{39} -7.57276i q^{41} -4.37266 q^{43} -1.83304 q^{47} +5.42401 q^{49} +(1.79266 - 3.85672i) q^{51} +12.0528 q^{53} +(2.79266 + 1.29807i) q^{57} +4.91093i q^{59} -8.16586i q^{61} +(2.87909 - 2.42791i) q^{63} +8.50466 q^{67} +(-13.8859 - 6.45436i) q^{69} +7.00770 q^{71} -4.59465 q^{73} +3.79561 q^{77} +7.36659i q^{79} +(-1.51932 + 8.87083i) q^{81} -15.7510i q^{83} +(5.96165 + 2.77106i) q^{87} -3.65716i q^{89} -7.09931 q^{91} +(3.79561 - 8.16586i) q^{93} -13.8773 q^{97} +(-6.93400 + 5.84737i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.57067	−	0.730070i	−0.906826	−	0.421506i
$3$	−1.57067	−	0.730070i	−0.906826	−	0.421506i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	1.25539i		−	0.474491i	−0.971450	−	0.237245i	$-0.923755\pi$
$7$		−	1.25539i		−	0.474491i	0.971450	−	0.237245i	$-0.0762446\pi$
$8$		0			0
$9$	1.93400	+	2.29339i	0.644665	+	0.764465i
$10$		0			0
$11$			3.02346i			0.911609i	0.890080	+	0.455804i	$0.150648\pi$
$11$			3.02346i			0.911609i	−0.890080	+	0.455804i	$0.849352\pi$
$12$		0			0
$13$		−	5.65509i		−	1.56844i	−0.620483	−	0.784220i	$-0.713064\pi$
$13$		−	5.65509i		−	1.56844i	0.620483	−	0.784220i	$-0.286936\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.45546i			0.595538i	0.954638	+	0.297769i	$0.0962424\pi$
$17$			2.45546i			0.595538i	−0.954638	+	0.297769i	$0.903758\pi$
$18$		0			0
$19$	−1.77801			−0.407903			−0.203951	−	0.978981i	$-0.565378\pi$
$19$	−1.77801			−0.407903			−0.203951	+	0.978981i	$0.565378\pi$
$20$		0			0
$21$	−0.916519	+	1.97179i	−0.200001	+	0.430281i
$22$		0			0
$23$	8.84074			1.84342			0.921711	−	0.387878i	$-0.126792\pi$
$23$	8.84074			1.84342			0.921711	+	0.387878i	$0.126792\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−1.36333	−	5.01411i	−0.262373	−	0.964967i
$28$		0			0
$29$	−3.79561			−0.704827			−0.352414	−	0.935844i	$-0.614639\pi$
$29$	−3.79561			−0.704827			−0.352414	+	0.935844i	$0.614639\pi$
$30$		0			0
$31$			5.19897i			0.933763i	0.884320	+	0.466881i	$0.154623\pi$
$31$			5.19897i			0.933763i	−0.884320	+	0.466881i	$0.845377\pi$
$32$		0			0
$33$	2.20734	−	4.74886i	0.384249	−	0.826670i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			6.45436i			1.06109i	0.847657	+	0.530545i	$0.178013\pi$
$37$			6.45436i			1.06109i	−0.847657	+	0.530545i	$0.821987\pi$
$38$		0			0
$39$	−4.12861	+	8.88227i	−0.661107	+	1.42230i
$40$		0			0
$41$		−	7.57276i		−	1.18267i	−0.806427	−	0.591333i	$-0.798602\pi$
$41$		−	7.57276i		−	1.18267i	0.806427	−	0.591333i	$-0.201398\pi$
$42$		0			0
$43$	−4.37266			−0.666824			−0.333412	−	0.942781i	$-0.608200\pi$
$43$	−4.37266			−0.666824			−0.333412	+	0.942781i	$0.608200\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.83304			−0.267376			−0.133688	−	0.991023i	$-0.542682\pi$
$47$	−1.83304			−0.267376			−0.133688	+	0.991023i	$0.542682\pi$
$48$		0			0
$49$	5.42401			0.774858
$50$		0			0
$51$	1.79266	−	3.85672i	0.251023	−	0.540049i
$52$		0			0
$53$	12.0528			1.65558			0.827792	−	0.561035i	$-0.189597\pi$
$53$	12.0528			1.65558			0.827792	+	0.561035i	$0.189597\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.79266	+	1.29807i	0.369897	+	0.171934i
$58$		0			0
$59$			4.91093i			0.639348i	0.947528	+	0.319674i	$0.103573\pi$
$59$			4.91093i			0.639348i	−0.947528	+	0.319674i	$0.896427\pi$
$60$		0			0
$61$		−	8.16586i		−	1.04553i	−0.852477	−	0.522766i	$-0.824900\pi$
$61$		−	8.16586i		−	1.04553i	0.852477	−	0.522766i	$-0.175100\pi$
$62$		0			0
$63$	2.87909	−	2.42791i	0.362732	−	0.305888i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	8.50466			1.03901			0.519505	−	0.854467i	$-0.326116\pi$
$67$	8.50466			1.03901			0.519505	+	0.854467i	$0.326116\pi$
$68$		0			0
$69$	−13.8859	−	6.45436i	−1.67166	−	0.777013i
$70$		0			0
$71$	7.00770			0.831661			0.415831	−	0.909442i	$-0.363491\pi$
$71$	7.00770			0.831661			0.415831	+	0.909442i	$0.363491\pi$
$72$		0			0
$73$	−4.59465			−0.537763			−0.268881	−	0.963173i	$-0.586654\pi$
$73$	−4.59465			−0.537763			−0.268881	+	0.963173i	$0.586654\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.79561			0.432550
$78$		0			0
$79$			7.36659i			0.828806i	0.910093	+	0.414403i	$0.136010\pi$
$79$			7.36659i			0.828806i	−0.910093	+	0.414403i	$0.863990\pi$
$80$		0			0
$81$	−1.51932	+	8.87083i	−0.168813	+	0.985648i
$82$		0			0
$83$		−	15.7510i		−	1.72890i	−0.502720	−	0.864449i	$-0.667667\pi$
$83$		−	15.7510i		−	1.72890i	0.502720	−	0.864449i	$-0.332333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	5.96165	+	2.77106i	0.639156	+	0.297089i
$88$		0			0
$89$		−	3.65716i		−	0.387658i	−0.981035	−	0.193829i	$-0.937909\pi$
$89$		−	3.65716i		−	0.387658i	0.981035	−	0.193829i	$-0.0620907\pi$
$90$		0			0
$91$	−7.09931			−0.744210
$92$		0			0
$93$	3.79561	−	8.16586i	0.393587	−	0.846760i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−13.8773			−1.40903			−0.704514	−	0.709690i	$-0.748835\pi$
$97$	−13.8773			−1.40903			−0.704514	+	0.709690i	$0.748835\pi$
$98$		0			0
$99$	−6.93400	+	5.84737i	−0.696893	+	0.587683i
$100$		0			0
$101$	13.8859			1.38170			0.690848	−	0.723000i	$-0.257237\pi$
$101$	13.8859			1.38170			0.690848	+	0.723000i	$0.257237\pi$
$102$		0			0
$103$		−	7.36659i		−	0.725852i	−0.931818	−	0.362926i	$-0.881778\pi$
$103$		−	7.36659i		−	0.725852i	0.931818	−	0.362926i	$-0.118222\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.14076i			0.786997i	0.919325	+	0.393499i	$0.128735\pi$
$107$			8.14076i			0.786997i	−0.919325	+	0.393499i	$0.871265\pi$
$108$		0			0
$109$			5.65509i			0.541659i	0.962627	+	0.270830i	$0.0872980\pi$
$109$			5.65509i			0.541659i	−0.962627	+	0.270830i	$0.912702\pi$
$110$		0			0
$111$	4.71213	−	10.1377i	0.447256	−	0.962223i
$112$		0			0
$113$		−	11.4884i		−	1.08073i	−0.841429	−	0.540367i	$-0.818285\pi$
$113$		−	11.4884i		−	1.08073i	0.841429	−	0.540367i	$-0.181715\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	12.9693	−	10.9369i	1.19902	−	1.01112i
$118$		0			0
$119$	3.08255			0.282577
$120$		0			0
$121$	1.85866			0.168969
$122$		0			0
$123$	−5.52865	+	11.8943i	−0.498501	+	1.07247i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	21.7081i		−	1.92628i	−0.268993	−	0.963142i	$-0.586691\pi$
$127$		−	21.7081i		−	1.92628i	0.268993	−	0.963142i	$-0.413309\pi$
$128$		0			0
$129$	6.86799	+	3.19234i	0.604693	+	0.281070i
$130$		0			0
$131$		−	12.5212i		−	1.09398i	−0.837139	−	0.546990i	$-0.815773\pi$
$131$		−	12.5212i		−	1.09398i	0.837139	−	0.546990i	$-0.184227\pi$
$132$		0			0
$133$			2.23208i			0.193546i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	18.4649i		−	1.57757i	−0.614672	−	0.788783i	$-0.710712\pi$
$137$		−	18.4649i		−	1.57757i	0.614672	−	0.788783i	$-0.289288\pi$
$138$		0			0
$139$	11.6040			0.984237			0.492118	−	0.870528i	$-0.336223\pi$
$139$	11.6040			0.984237			0.492118	+	0.870528i	$0.336223\pi$
$140$		0			0
$141$	2.87909	+	1.33825i	0.242463	+	0.112701i
$142$		0			0
$143$	17.0980			1.42980
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−8.51932	−	3.95990i	−0.702661	−	0.326607i
$148$		0			0
$149$	11.9233			0.976794			0.488397	−	0.872621i	$-0.337582\pi$
$149$	11.9233			0.976794			0.488397	+	0.872621i	$0.337582\pi$
$150$		0			0
$151$		−	10.0548i		−	0.818247i	−0.912479	−	0.409124i	$-0.865834\pi$
$151$		−	10.0548i		−	0.818247i	0.912479	−	0.409124i	$-0.134166\pi$
$152$		0			0
$153$	−5.63135	+	4.74886i	−0.455268	+	0.383922i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	1.71150i		−	0.136593i	−0.997665	−	0.0682964i	$-0.978244\pi$
$157$		−	1.71150i		−	0.136593i	0.997665	−	0.0682964i	$-0.0217563\pi$
$158$		0			0
$159$	−18.9310	−	8.79941i	−1.50133	−	0.697838i
$160$		0			0
$161$		−	11.0985i		−	0.874687i
$162$		0			0
$163$	11.9580			0.936621			0.468311	−	0.883564i	$-0.344863\pi$
$163$	11.9580			0.936621			0.468311	+	0.883564i	$0.344863\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.583522			0.0451543			0.0225771	−	0.999745i	$-0.492813\pi$
$167$	0.583522			0.0451543			0.0225771	+	0.999745i	$0.492813\pi$
$168$		0			0
$169$	−18.9800			−1.46000
$170$		0			0
$171$	−3.43866	−	4.07767i	−0.262961	−	0.311827i
$172$		0			0
$173$	12.0528			0.916360			0.458180	−	0.888860i	$-0.348502\pi$
$173$	12.0528			0.916360			0.458180	+	0.888860i	$0.348502\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	3.58532	−	7.71344i	0.269489	−	0.579778i
$178$		0			0
$179$			0.605490i			0.0452565i	0.999744	+	0.0226282i	$0.00720340\pi$
$179$			0.605490i			0.0452565i	−0.999744	+	0.0226282i	$0.992797\pi$
$180$		0			0
$181$		−	7.08790i		−	0.526840i	−0.964681	−	0.263420i	$-0.915150\pi$
$181$		−	7.08790i		−	0.526840i	0.964681	−	0.263420i	$-0.0848505\pi$
$182$		0			0
$183$	−5.96165	+	12.8259i	−0.440698	+	0.948114i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−7.42401			−0.542897
$188$		0			0
$189$	−6.29464	+	1.71150i	−0.457868	+	0.124493i
$190$		0			0
$191$	18.9310			1.36980			0.684899	−	0.728638i	$-0.259846\pi$
$191$	18.9310			1.36980			0.684899	+	0.728638i	$0.259846\pi$
$192$		0			0
$193$	4.27334			0.307602			0.153801	−	0.988102i	$-0.450849\pi$
$193$	4.27334			0.307602			0.153801	+	0.988102i	$0.450849\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.0903			−0.718901			−0.359450	−	0.933164i	$-0.617036\pi$
$197$	−10.0903			−0.718901			−0.359450	+	0.933164i	$0.617036\pi$
$198$		0			0
$199$		−	19.0199i		−	1.34829i	−0.738601	−	0.674143i	$-0.764513\pi$
$199$		−	19.0199i		−	1.34829i	0.738601	−	0.674143i	$-0.235487\pi$
$200$		0			0
$201$	−13.3580	−	6.20900i	−0.942201	−	0.437949i
$202$		0			0
$203$			4.76495i			0.334434i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	17.0980	+	20.2753i	1.18839	+	1.40923i
$208$		0			0
$209$		−	5.37574i		−	0.371848i
$210$		0			0
$211$	−5.49534			−0.378315			−0.189157	−	0.981947i	$-0.560576\pi$
$211$	−5.49534			−0.378315			−0.189157	+	0.981947i	$0.560576\pi$
$212$		0			0
$213$	−11.0068	−	5.11611i	−0.754172	−	0.350550i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	6.52671			0.443062
$218$		0			0
$219$	7.21667	+	3.35441i	0.487657	+	0.226670i
$220$		0			0
$221$	13.8859			0.934065
$222$		0			0
$223$		−	7.70974i		−	0.516282i	−0.966107	−	0.258141i	$-0.916890\pi$
$223$		−	7.70974i		−	0.516282i	0.966107	−	0.258141i	$-0.0831100\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	7.40388i		−	0.491413i	−0.969344	−	0.245706i	$-0.920980\pi$
$227$		−	7.40388i		−	0.491413i	0.969344	−	0.245706i	$-0.0790199\pi$
$228$		0			0
$229$			17.1310i			1.13205i	0.824389	+	0.566024i	$0.191519\pi$
$229$			17.1310i			1.13205i	−0.824389	+	0.566024i	$0.808481\pi$
$230$		0			0
$231$	−5.96165	−	2.77106i	−0.392248	−	0.182322i
$232$		0			0
$233$		−	1.40807i		−	0.0922454i	−0.998936	−	0.0461227i	$-0.985313\pi$
$233$		−	1.40807i		−	0.0922454i	0.998936	−	0.0461227i	$-0.0146865\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	5.37812	−	11.5705i	0.349347	−	0.751583i
$238$		0			0
$239$	9.50673			0.614939			0.307470	−	0.951558i	$-0.400518\pi$
$239$	9.50673			0.614939			0.307470	+	0.951558i	$0.400518\pi$
$240$		0			0
$241$	2.27334			0.146439			0.0732195	−	0.997316i	$-0.476673\pi$
$241$	2.27334			0.146439			0.0732195	+	0.997316i	$0.476673\pi$
$242$		0			0
$243$	8.86267	−	12.8239i	0.568540	−	0.822655i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			10.0548i			0.639771i
$248$		0			0
$249$	−11.4993	+	24.7396i	−0.728741	+	1.56781i
$250$		0			0
$251$			9.49772i			0.599491i	0.954019	+	0.299745i	$0.0969017\pi$
$251$			9.49772i			0.599491i	−0.954019	+	0.299745i	$0.903098\pi$
$252$		0			0
$253$			26.7297i			1.68048i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			27.3144i			1.70382i	0.523686	+	0.851912i	$0.324557\pi$
$257$			27.3144i			1.70382i	−0.523686	+	0.851912i	$0.675443\pi$
$258$		0			0
$259$	8.10270			0.503477
$260$		0			0
$261$	−7.34070	−	8.70484i	−0.454378	−	0.538816i
$262$		0			0
$263$	1.24952			0.0770484			0.0385242	−	0.999258i	$-0.487734\pi$
$263$	1.24952			0.0770484			0.0385242	+	0.999258i	$0.487734\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−2.66998	+	5.74418i	−0.163400	+	0.351538i
$268$		0			0
$269$	15.7189			0.958399			0.479199	−	0.877706i	$-0.340927\pi$
$269$	15.7189			0.958399			0.479199	+	0.877706i	$0.340927\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	11.1507	+	5.18299i	0.674869	+	0.313689i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	0.633548i		−	0.0380662i	−0.999819	−	0.0190331i	$-0.993941\pi$
$277$		−	0.633548i		−	0.0380662i	0.999819	−	0.0190331i	$-0.00605879\pi$
$278$		0			0
$279$	−11.9233	+	10.0548i	−0.713829	+	0.601965i
$280$		0			0
$281$			20.9204i			1.24801i	0.781422	+	0.624003i	$0.214495\pi$
$281$			20.9204i			1.24801i	−0.781422	+	0.624003i	$0.785505\pi$
$282$		0			0
$283$	−14.6846			−0.872911			−0.436455	−	0.899726i	$-0.643766\pi$
$283$	−14.6846			−0.872911			−0.436455	+	0.899726i	$0.643766\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.50673			−0.561165
$288$		0			0
$289$	10.9707			0.645335
$290$		0			0
$291$	21.7967	+	10.1314i	1.27774	+	0.593914i
$292$		0			0
$293$	5.49911			0.321262			0.160631	−	0.987015i	$-0.448647\pi$
$293$	5.49911			0.321262			0.160631	+	0.987015i	$0.448647\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	15.1600	−	4.12197i	0.879672	−	0.239181i
$298$		0			0
$299$		−	49.9952i		−	2.89129i
$300$		0			0
$301$			5.48937i			0.316402i
$302$		0			0
$303$	−21.8101	−	10.1377i	−1.25296	−	0.582393i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	6.11929			0.349246			0.174623	−	0.984635i	$-0.444129\pi$
$307$	6.11929			0.349246			0.174623	+	0.984635i	$0.444129\pi$
$308$		0			0
$309$	−5.37812	+	11.5705i	−0.305951	+	0.658221i
$310$		0			0
$311$	−5.75819			−0.326517			−0.163258	−	0.986583i	$-0.552200\pi$
$311$	−5.75819			−0.326517			−0.163258	+	0.986583i	$0.552200\pi$
$312$		0			0
$313$	−21.9800			−1.24238			−0.621192	−	0.783658i	$-0.713351\pi$
$313$	−21.9800			−1.24238			−0.621192	+	0.783658i	$0.713351\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−17.6815			−0.993091			−0.496545	−	0.868011i	$-0.665398\pi$
$317$	−17.6815			−0.993091			−0.496545	+	0.868011i	$0.665398\pi$
$318$		0			0
$319$		−	11.4759i		−	0.642527i
$320$		0			0
$321$	5.94333	−	12.7864i	0.331724	−	0.713669i
$322$		0			0
$323$		−	4.36583i		−	0.242922i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	4.12861	−	8.88227i	0.228313	−	0.491190i
$328$		0			0
$329$			2.30117i			0.126867i
$330$		0			0
$331$	7.14134			0.392523			0.196262	−	0.980552i	$-0.437120\pi$
$331$	7.14134			0.392523			0.196262	+	0.980552i	$0.437120\pi$
$332$		0			0
$333$	−14.8024	+	12.4827i	−0.811166	+	0.684048i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.90663			0.103861			0.0519303	−	0.998651i	$-0.483463\pi$
$337$	1.90663			0.103861			0.0519303	+	0.998651i	$0.483463\pi$
$338$		0			0
$339$	−8.38731	+	18.0444i	−0.455536	+	0.980038i
$340$		0			0
$341$	−15.7189			−0.851226
$342$		0			0
$343$		−	15.5969i		−	0.842154i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	0.530510i		−	0.0284793i	−0.999899	−	0.0142396i	$-0.995467\pi$
$347$		−	0.530510i		−	0.0284793i	0.999899	−	0.0142396i	$-0.00453277\pi$
$348$		0			0
$349$		−	7.36659i		−	0.394325i	−0.980371	−	0.197162i	$-0.936827\pi$
$349$		−	7.36659i		−	0.394325i	0.980371	−	0.197162i	$-0.0631726\pi$
$350$		0			0
$351$	−28.3553	+	7.70974i	−1.51349	+	0.411516i
$352$		0			0
$353$			11.2299i			0.597708i	0.954299	+	0.298854i	$0.0966044\pi$
$353$			11.2299i			0.597708i	−0.954299	+	0.298854i	$0.903396\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−4.84167	−	2.25048i	−0.256248	−	0.119108i
$358$		0			0
$359$	17.0980			0.902396			0.451198	−	0.892424i	$-0.350997\pi$
$359$	17.0980			0.902396			0.451198	+	0.892424i	$0.350997\pi$
$360$		0			0
$361$	−15.8387			−0.833615
$362$		0			0
$363$	−2.91934	−	1.35695i	−0.153226	−	0.0712216i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			13.9984i			0.730709i	0.930868	+	0.365355i	$0.119052\pi$
$367$			13.9984i			0.730709i	−0.930868	+	0.365355i	$0.880948\pi$
$368$		0			0
$369$	17.3673	−	14.6457i	0.904107	−	0.762424i
$370$		0			0
$371$		−	15.1309i		−	0.785559i
$372$		0			0
$373$		−	21.5952i		−	1.11815i	−0.829116	−	0.559077i	$-0.811155\pi$
$373$		−	21.5952i		−	1.11815i	0.829116	−	0.559077i	$-0.188845\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			21.4645i			1.10548i
$378$		0			0
$379$	−11.7780			−0.604996			−0.302498	−	0.953150i	$-0.597820\pi$
$379$	−11.7780			−0.604996			−0.302498	+	0.953150i	$0.597820\pi$
$380$		0			0
$381$	−15.8484	+	34.0963i	−0.811940	+	1.74680i
$382$		0			0
$383$	−15.8484			−0.809818			−0.404909	−	0.914357i	$-0.632697\pi$
$383$	−15.8484			−0.809818			−0.404909	+	0.914357i	$0.632697\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−8.45670	−	10.0282i	−0.429878	−	0.509763i
$388$		0			0
$389$	18.4770			0.936822			0.468411	−	0.883511i	$-0.344827\pi$
$389$	18.4770			0.936822			0.468411	+	0.883511i	$0.344827\pi$
$390$		0			0
$391$			21.7081i			1.09783i
$392$		0			0
$393$	−9.14134	+	19.6666i	−0.461119	+	0.992050i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			18.3981i			0.923373i	0.887043	+	0.461687i	$0.152756\pi$
$397$			18.3981i			0.923373i	−0.887043	+	0.461687i	$0.847244\pi$
$398$		0			0
$399$	1.62958	−	3.50586i	0.0815809	−	0.175513i
$400$		0			0
$401$		−	0.183464i		−	0.00916176i	−0.999990	−	0.00458088i	$-0.998542\pi$
$401$		−	0.183464i		−	0.00916176i	0.999990	−	0.00458088i	$-0.00145814\pi$
$402$		0			0
$403$	29.4006			1.46455
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−19.5145			−0.967299
$408$		0			0
$409$	−30.5933			−1.51274			−0.756371	−	0.654142i	$-0.773030\pi$
$409$	−30.5933			−1.51274			−0.756371	+	0.654142i	$0.773030\pi$
$410$		0			0
$411$	−13.4807	+	29.0023i	−0.664953	+	1.43058i
$412$		0			0
$413$	6.16511			0.303365
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−18.2260	−	8.47171i	−0.892531	−	0.414862i
$418$		0			0
$419$			13.9813i			0.683032i	0.939876	+	0.341516i	$0.110940\pi$
$419$			13.9813i			0.683032i	−0.939876	+	0.341516i	$0.889060\pi$
$420$		0			0
$421$			26.7297i			1.30272i	0.758767	+	0.651362i	$0.225802\pi$
$421$			26.7297i			1.30272i	−0.758767	+	0.651362i	$0.774198\pi$
$422$		0			0
$423$	−3.54509	−	4.20388i	−0.172368	−	0.204400i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−10.2513			−0.496095
$428$		0			0
$429$	−26.8552	−	12.4827i	−1.29658	−	0.602671i
$430$		0			0
$431$	−34.7794			−1.67527			−0.837633	−	0.546233i	$-0.816061\pi$
$431$	−34.7794			−1.67527			−0.837633	+	0.546233i	$0.816061\pi$
$432$		0			0
$433$	19.8667			0.954731			0.477366	−	0.878705i	$-0.341592\pi$
$433$	19.8667			0.954731			0.477366	+	0.878705i	$0.341592\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−15.7189			−0.751937
$438$		0			0
$439$			2.85392i			0.136210i	0.997678	+	0.0681051i	$0.0216953\pi$
$439$			2.85392i			0.136210i	−0.997678	+	0.0681051i	$0.978305\pi$
$440$		0			0
$441$	10.4900	+	12.4394i	0.499524	+	0.592352i
$442$		0			0
$443$			13.6854i			0.650212i	0.945678	+	0.325106i	$0.105400\pi$
$443$			13.6854i			0.650212i	−0.945678	+	0.325106i	$0.894600\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−18.7275	−	8.70484i	−0.885782	−	0.411725i
$448$		0			0
$449$		−	19.1132i		−	0.902008i	−0.892522	−	0.451004i	$-0.851066\pi$
$449$		−	19.1132i		−	0.902008i	0.892522	−	0.451004i	$-0.148934\pi$
$450$		0			0
$451$	22.8960			1.07813
$452$		0			0
$453$	−7.34070	+	15.7927i	−0.344896	+	0.742008i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	22.6133			1.05781			0.528903	−	0.848682i	$-0.322604\pi$
$457$	22.6133			1.05781			0.528903	+	0.848682i	$0.322604\pi$
$458$		0			0
$459$	12.3120	−	3.34760i	0.574674	−	0.156253i
$460$		0			0
$461$	−13.2199			−0.615711			−0.307855	−	0.951433i	$-0.599611\pi$
$461$	−13.2199			−0.615711			−0.307855	+	0.951433i	$0.599611\pi$
$462$		0			0
$463$			40.5506i			1.88455i	0.334845	+	0.942273i	$0.391316\pi$
$463$			40.5506i			1.88455i	−0.334845	+	0.942273i	$0.608684\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			10.2492i			0.474276i	0.971476	+	0.237138i	$0.0762093\pi$
$467$			10.2492i			0.474276i	−0.971476	+	0.237138i	$0.923791\pi$
$468$		0			0
$469$		−	10.6766i		−	0.493001i
$470$		0			0
$471$	−1.24952	+	2.68820i	−0.0575746	+	0.123866i
$472$		0			0
$473$		−	13.2206i		−	0.607883i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	23.3101	+	27.6419i	1.06730	+	1.26564i
$478$		0			0
$479$	31.4378			1.43643			0.718215	−	0.695821i	$-0.244959\pi$
$479$	31.4378			1.43643			0.718215	+	0.695821i	$0.244959\pi$
$480$		0			0
$481$	36.5000			1.66425
$482$		0			0
$483$	−8.10270	+	17.4321i	−0.368686	+	0.793188i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	0.177431i		−	0.00804018i	−0.999992	−	0.00402009i	$-0.998720\pi$
$487$		−	0.177431i		−	0.00804018i	0.999992	−	0.00402009i	$-0.00127964\pi$
$488$		0			0
$489$	−18.7820	−	8.73016i	−0.849352	−	0.394791i
$490$		0			0
$491$		−	7.75623i		−	0.350034i	−0.984565	−	0.175017i	$-0.944002\pi$
$491$		−	7.75623i		−	0.350034i	0.984565	−	0.175017i	$-0.0559980\pi$
$492$		0			0
$493$		−	9.31999i		−	0.419751i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	8.79736i		−	0.394616i
$498$		0			0
$499$	−5.22067			−0.233709			−0.116855	−	0.993149i	$-0.537281\pi$
$499$	−5.22067			−0.233709			−0.116855	+	0.993149i	$0.537281\pi$
$500$		0			0
$501$	−0.916519	−	0.426011i	−0.0409470	−	0.0190328i
$502$		0			0
$503$	−6.34171			−0.282763			−0.141381	−	0.989955i	$-0.545154\pi$
$503$	−6.34171			−0.282763			−0.141381	+	0.989955i	$0.545154\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	29.8113	+	13.8567i	1.32397	+	0.615399i
$508$		0			0
$509$	34.1959			1.51571			0.757854	−	0.652425i	$-0.226248\pi$
$509$	34.1959			1.51571			0.757854	+	0.652425i	$0.226248\pi$
$510$		0			0
$511$			5.76805i			0.255164i
$512$		0			0
$513$	2.42401	+	8.91513i	0.107023	+	0.393613i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	5.54212i		−	0.243742i
$518$		0			0
$519$	−18.9310	−	8.79941i	−0.830978	−	0.386251i
$520$		0			0
$521$			32.6151i			1.42889i	0.699689	+	0.714447i	$0.253322\pi$
$521$			32.6151i			1.42889i	−0.699689	+	0.714447i	$0.746678\pi$
$522$		0			0
$523$	−14.6074			−0.638736			−0.319368	−	0.947631i	$-0.603471\pi$
$523$	−14.6074			−0.638736			−0.319368	+	0.947631i	$0.603471\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.7659			−0.556091
$528$		0			0
$529$	55.1587			2.39820
$530$		0			0
$531$	−11.2627	+	9.49772i	−0.488759	+	0.412166i
$532$		0			0
$533$	−42.8246			−1.85494
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.442050	−	0.951024i	0.0190759	−	0.0410397i
$538$		0			0
$539$			16.3993i			0.706368i
$540$		0			0
$541$		−	13.0217i		−	0.559846i	−0.960023	−	0.279923i	$-0.909691\pi$
$541$		−	13.0217i		−	0.559846i	0.960023	−	0.279923i	$-0.0903089\pi$
$542$		0			0
$543$	−5.17466	+	11.1327i	−0.222066	+	0.477752i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	37.8280			1.61741			0.808705	−	0.588214i	$-0.200169\pi$
$547$	37.8280			1.61741			0.808705	+	0.588214i	$0.200169\pi$
$548$		0			0
$549$	18.7275	−	15.7927i	0.799272	−	0.674018i
$550$		0			0
$551$	6.74863			0.287501
$552$		0			0
$553$	9.24791			0.393261
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.0135			−0.932744			−0.466372	−	0.884589i	$-0.654439\pi$
$557$	−22.0135			−0.932744			−0.466372	+	0.884589i	$0.654439\pi$
$558$		0			0
$559$			24.7278i			1.04587i
$560$		0			0
$561$	11.6607	+	5.42004i	0.492313	+	0.228834i
$562$		0			0
$563$			9.80727i			0.413327i	0.978412	+	0.206664i	$0.0662606\pi$
$563$			9.80727i			0.413327i	−0.978412	+	0.206664i	$0.933739\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	11.1363	+	1.90733i	0.467681	+	0.0801002i
$568$		0			0
$569$			22.4555i			0.941384i	0.882298	+	0.470692i	$0.155996\pi$
$569$			22.4555i			0.941384i	−0.882298	+	0.470692i	$0.844004\pi$
$570$		0			0
$571$	−5.92867			−0.248107			−0.124054	−	0.992276i	$-0.539589\pi$
$571$	−5.92867			−0.248107			−0.124054	+	0.992276i	$0.539589\pi$
$572$		0			0
$573$	−29.7343	−	13.8209i	−1.24217	−	0.577378i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−39.1587			−1.63020			−0.815098	−	0.579322i	$-0.803317\pi$
$577$	−39.1587			−1.63020			−0.815098	+	0.579322i	$0.803317\pi$
$578$		0			0
$579$	−6.71200	−	3.11984i	−0.278941	−	0.129656i
$580$		0			0
$581$	−19.7736			−0.820347
$582$		0			0
$583$			36.4413i			1.50924i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			29.3332i			1.21071i	0.795955	+	0.605356i	$0.206969\pi$
$587$			29.3332i			1.21071i	−0.795955	+	0.605356i	$0.793031\pi$
$588$		0			0
$589$		−	9.24381i		−	0.380885i
$590$		0			0
$591$	15.8484	+	7.36659i	0.651918	+	0.303021i
$592$		0			0
$593$		−	24.6525i		−	1.01236i	−0.862428	−	0.506179i	$-0.831058\pi$
$593$		−	24.6525i		−	1.01236i	0.862428	−	0.506179i	$-0.168942\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−13.8859	+	29.8740i	−0.568311	+	1.22266i
$598$		0			0
$599$	29.8638			1.22020			0.610102	−	0.792323i	$-0.291129\pi$
$599$	29.8638			1.22020			0.610102	+	0.792323i	$0.291129\pi$
$600$		0			0
$601$	−17.7267			−0.723085			−0.361543	−	0.932356i	$-0.617750\pi$
$601$	−17.7267			−0.723085			−0.361543	+	0.932356i	$0.617750\pi$
$602$		0			0
$603$	16.4480	+	19.5046i	0.669814	+	0.794287i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			25.8174i			1.04790i	0.851750	+	0.523949i	$0.175542\pi$
$607$			25.8174i			1.04790i	−0.851750	+	0.523949i	$0.824458\pi$
$608$		0			0
$609$	3.47875	−	7.48416i	0.140966	−	0.303274i
$610$		0			0
$611$			10.3660i			0.419363i
$612$		0			0
$613$			16.6866i			0.673965i	0.941511	+	0.336982i	$0.109406\pi$
$613$			16.6866i			0.673965i	−0.941511	+	0.336982i	$0.890594\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	8.69514i		−	0.350053i	−0.984564	−	0.175027i	$-0.943999\pi$
$617$		−	8.69514i		−	0.350053i	0.984564	−	0.175027i	$-0.0560012\pi$
$618$		0			0
$619$	−20.3727			−0.818846			−0.409423	−	0.912345i	$-0.634270\pi$
$619$	−20.3727			−0.818846			−0.409423	+	0.912345i	$0.634270\pi$
$620$		0			0
$621$	−12.0528	−	44.3285i	−0.483663	−	1.77884i
$622$		0			0
$623$	−4.59114			−0.183940
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−3.92467	+	8.44351i	−0.156736	+	0.337201i
$628$		0			0
$629$	−15.8484			−0.631919
$630$		0			0
$631$			4.51267i			0.179647i	0.995958	+	0.0898233i	$0.0286302\pi$
$631$			4.51267i			0.179647i	−0.995958	+	0.0898233i	$0.971370\pi$
$632$		0			0
$633$	8.63135	+	4.01198i	0.343065	+	0.159462i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	30.6732i		−	1.21532i
$638$		0			0
$639$	13.5529	+	16.0714i	0.536143	+	0.635776i
$640$		0			0
$641$		−	25.0424i		−	0.989114i	−0.869145	−	0.494557i	$-0.835330\pi$
$641$		−	25.0424i		−	0.989114i	0.869145	−	0.494557i	$-0.164670\pi$
$642$		0			0
$643$	−21.7360			−0.857184			−0.428592	−	0.903498i	$-0.640990\pi$
$643$	−21.7360			−0.857184			−0.428592	+	0.903498i	$0.640990\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21.3476			−0.839259			−0.419629	−	0.907695i	$-0.637840\pi$
$647$	−21.3476			−0.839259			−0.419629	+	0.907695i	$0.637840\pi$
$648$		0			0
$649$	−14.8480			−0.582836
$650$		0			0
$651$	−10.2513	−	4.76495i	−0.401780	−	0.186753i
$652$		0			0
$653$	−25.9387			−1.01506			−0.507530	−	0.861634i	$-0.669441\pi$
$653$	−25.9387			−1.01506			−0.507530	+	0.861634i	$0.669441\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−8.88603	−	10.5373i	−0.346677	−	0.411101i
$658$		0			0
$659$		−	20.6474i		−	0.804307i	−0.915572	−	0.402153i	$-0.868262\pi$
$659$		−	20.6474i		−	0.804307i	0.915572	−	0.402153i	$-0.131738\pi$
$660$		0			0
$661$		−	9.31999i		−	0.362506i	−0.983437	−	0.181253i	$-0.941985\pi$
$661$		−	9.31999i		−	0.362506i	0.983437	−	0.181253i	$-0.0580152\pi$
$662$		0			0
$663$	−21.8101	−	10.1377i	−0.847034	−	0.393714i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−33.5560			−1.29929
$668$		0			0
$669$	−5.62865	+	12.1094i	−0.217616	+	0.468178i
$670$		0			0
$671$	24.6892			0.953115
$672$		0			0
$673$	37.2520			1.43596			0.717980	−	0.696063i	$-0.245067\pi$
$673$	37.2520			1.43596			0.717980	+	0.696063i	$0.245067\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	37.0665			1.42458			0.712290	−	0.701885i	$-0.247658\pi$
$677$	37.0665			1.42458			0.712290	+	0.701885i	$0.247658\pi$
$678$		0			0
$679$			17.4214i			0.668571i
$680$		0			0
$681$	−5.40535	+	11.6290i	−0.207133	+	0.445626i
$682$		0			0
$683$			26.0898i			0.998297i	0.866516	+	0.499149i	$0.166354\pi$
$683$			26.0898i			0.998297i	−0.866516	+	0.499149i	$0.833646\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	12.5068	−	26.9071i	0.477165	−	1.02657i
$688$		0			0
$689$		−	68.1598i		−	2.59668i
$690$		0			0
$691$	−34.8187			−1.32457			−0.662283	−	0.749254i	$-0.730412\pi$
$691$	−34.8187			−1.32457			−0.662283	+	0.749254i	$0.730412\pi$
$692$		0			0
$693$	7.34070	+	8.70484i	0.278850	+	0.330669i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	18.5946			0.704323
$698$		0			0
$699$	−1.02799	+	2.21160i	−0.0388820	+	0.0836505i
$700$		0			0
$701$	−12.7188			−0.480383			−0.240192	−	0.970725i	$-0.577210\pi$
$701$	−12.7188			−0.480383			−0.240192	+	0.970725i	$0.577210\pi$
$702$		0			0
$703$		−	11.4759i		−	0.432822i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	17.4321i		−	0.655602i
$708$		0			0
$709$		−	46.7263i		−	1.75484i	−0.479721	−	0.877421i	$-0.659262\pi$
$709$		−	46.7263i		−	1.75484i	0.479721	−	0.877421i	$-0.340738\pi$
$710$		0			0
$711$	−16.8945	+	14.2470i	−0.633593	+	0.534303i
$712$		0			0
$713$			45.9628i			1.72132i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−14.9319	−	6.94058i	−0.557643	−	0.259201i
$718$		0			0
$719$	−9.50673			−0.354541			−0.177271	−	0.984162i	$-0.556727\pi$
$719$	−9.50673			−0.354541			−0.177271	+	0.984162i	$0.556727\pi$
$720$		0			0
$721$	−9.24791			−0.344410
$722$		0			0
$723$	−3.57067	−	1.65970i	−0.132795	−	0.0617249i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	31.7629i		−	1.17802i	−0.808125	−	0.589011i	$-0.799518\pi$
$727$		−	31.7629i		−	1.17802i	0.808125	−	0.589011i	$-0.200482\pi$
$728$		0			0
$729$	−23.2827	+	13.6718i	−0.862321	+	0.506362i
$730$		0			0
$731$		−	10.7369i		−	0.397119i
$732$		0			0
$733$		−	15.5852i		−	0.575653i	−0.957683	−	0.287826i	$-0.907067\pi$
$733$		−	15.5852i		−	0.575653i	0.957683	−	0.287826i	$-0.0929326\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			25.7135i			0.947171i
$738$		0			0
$739$	−4.17997			−0.153763			−0.0768813	−	0.997040i	$-0.524496\pi$
$739$	−4.17997			−0.153763			−0.0768813	+	0.997040i	$0.524496\pi$
$740$		0			0
$741$	7.34070	−	15.7927i	0.269667	−	0.580161i
$742$		0			0
$743$	−20.1805			−0.740351			−0.370176	−	0.928962i	$-0.620702\pi$
$743$	−20.1805			−0.740351			−0.370176	+	0.928962i	$0.620702\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	36.1233	−	30.4624i	1.32168	−	1.11456i
$748$		0			0
$749$	10.2198			0.373423
$750$		0			0
$751$		−	19.0316i		−	0.694474i	−0.937777	−	0.347237i	$-0.887120\pi$
$751$		−	19.0316i		−	0.694474i	0.937777	−	0.347237i	$-0.112880\pi$
$752$		0			0
$753$	6.93400	−	14.9178i	0.252689	−	0.543633i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			4.22227i			0.153461i	0.997052	+	0.0767305i	$0.0244481\pi$
$757$			4.22227i			0.153461i	−0.997052	+	0.0767305i	$0.975552\pi$
$758$		0			0
$759$	19.5145	−	41.9834i	0.708332	−	1.52390i
$760$		0			0
$761$		−	38.0795i		−	1.38038i	−0.723628	−	0.690190i	$-0.757527\pi$
$761$		−	38.0795i		−	1.38038i	0.723628	−	0.690190i	$-0.242473\pi$
$762$		0			0
$763$	7.09931			0.257012
$764$		0			0
$765$		0			0
$766$		0			0
$767$	27.7717			1.00278
$768$		0			0
$769$	16.9600			0.611595			0.305797	−	0.952097i	$-0.401077\pi$
$769$	16.9600			0.611595			0.305797	+	0.952097i	$0.401077\pi$
$770$		0			0
$771$	19.9414	−	42.9018i	0.718172	−	1.54507i
$772$		0			0
$773$	16.3849			0.589324			0.294662	−	0.955602i	$-0.404793\pi$
$773$	16.3849			0.589324			0.294662	+	0.955602i	$0.404793\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−12.7267	−	5.91554i	−0.456566	−	0.212219i
$778$		0			0
$779$			13.4644i			0.482413i
$780$		0			0
$781$			21.1875i			0.758150i
$782$		0			0
$783$	5.17466	+	19.0316i	0.184927	+	0.680135i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−25.6833			−0.915511			−0.457756	−	0.889078i	$-0.651347\pi$
$787$	−25.6833			−0.915511			−0.457756	+	0.889078i	$0.651347\pi$
$788$		0			0
$789$	−1.96257	−	0.912234i	−0.0698695	−	0.0324764i
$790$		0			0
$791$	−14.4223			−0.512799
$792$		0			0
$793$	−46.1787			−1.63985
$794$		0			0
$795$		0			0
$796$		0			0
$797$	13.7563			0.487274			0.243637	−	0.969866i	$-0.421659\pi$
$797$	13.7563			0.487274			0.243637	+	0.969866i	$0.421659\pi$
$798$		0			0
$799$		−	4.50096i		−	0.159232i
$800$		0			0
$801$	8.38731	−	7.07293i	0.296351	−	0.249910i
$802$		0			0
$803$		−	13.8918i		−	0.490229i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−24.6892	−	11.4759i	−0.869100	−	0.403971i
$808$		0			0
$809$			6.42314i			0.225826i	0.993605	+	0.112913i	$0.0360181\pi$
$809$			6.42314i			0.225826i	−0.993605	+	0.112913i	$0.963982\pi$
$810$		0			0
$811$	40.8353			1.43392			0.716961	−	0.697114i	$-0.245533\pi$
$811$	40.8353			1.43392			0.716961	+	0.697114i	$0.245533\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	7.77462			0.271999
$818$		0			0
$819$	−13.7300	−	16.2815i	−0.479767	−	0.568923i
$820$		0			0
$821$	51.8774			1.81053			0.905267	−	0.424844i	$-0.139671\pi$
$821$	51.8774			1.81053			0.905267	+	0.424844i	$0.139671\pi$
$822$		0			0
$823$			36.0379i			1.25620i	0.778131	+	0.628102i	$0.216168\pi$
$823$			36.0379i			1.25620i	−0.778131	+	0.628102i	$0.783832\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	29.8501i		−	1.03799i	−0.854777	−	0.518995i	$-0.826306\pi$
$827$		−	29.8501i		−	1.03799i	0.854777	−	0.518995i	$-0.173694\pi$
$828$		0			0
$829$		−	22.2287i		−	0.772035i	−0.922492	−	0.386017i	$-0.873850\pi$
$829$		−	22.2287i		−	0.772035i	0.922492	−	0.386017i	$-0.126150\pi$
$830$		0			0
$831$	−0.462534	+	0.995094i	−0.0160451	+	0.0345194i
$832$		0			0
$833$			13.3185i			0.461457i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	26.0682	−	7.08790i	0.901050	−	0.244994i
$838$		0			0
$839$	−5.17466			−0.178649			−0.0893246	−	0.996003i	$-0.528471\pi$
$839$	−5.17466			−0.178649			−0.0893246	+	0.996003i	$0.528471\pi$
$840$		0			0
$841$	−14.5933			−0.503218
$842$		0			0
$843$	15.2733	−	32.8590i	0.526042	−	1.13172i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	2.33334i		−	0.0801745i
$848$		0			0
$849$	23.0647	+	10.7208i	0.791578	+	0.367937i
$850$		0			0
$851$			57.0613i			1.95604i
$852$		0			0
$853$			36.3283i			1.24386i	0.783073	+	0.621929i	$0.213651\pi$
$853$			36.3283i			1.24386i	−0.783073	+	0.621929i	$0.786349\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	9.42274i		−	0.321875i	−0.986965	−	0.160937i	$-0.948548\pi$
$857$		−	9.42274i		−	0.321875i	0.986965	−	0.160937i	$-0.0514517\pi$
$858$		0			0
$859$	−18.0480			−0.615789			−0.307894	−	0.951421i	$-0.599624\pi$
$859$	−18.0480			−0.615789			−0.307894	+	0.951421i	$0.599624\pi$
$860$		0			0
$861$	14.9319	+	6.94058i	0.508879	+	0.236534i
$862$		0			0
$863$	18.3475			0.624555			0.312278	−	0.949991i	$-0.398908\pi$
$863$	18.3475			0.624555			0.312278	+	0.949991i	$0.398908\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−17.2313	−	8.00937i	−0.585206	−	0.272013i
$868$		0			0
$869$	−22.2726			−0.755547
$870$		0			0
$871$		−	48.0946i		−	1.62962i
$872$		0			0
$873$	−26.8387	−	31.8262i	−0.908352	−	1.07715i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	16.5736i		−	0.559651i	−0.960051	−	0.279826i	$-0.909723\pi$
$877$		−	16.5736i		−	0.559651i	0.960051	−	0.279826i	$-0.0902766\pi$
$878$		0			0
$879$	−8.63728	−	4.01474i	−0.291328	−	0.135414i
$880$		0			0
$881$			25.5865i			0.862031i	0.902345	+	0.431015i	$0.141845\pi$
$881$			25.5865i			0.862031i	−0.902345	+	0.431015i	$0.858155\pi$
$882$		0			0
$883$	21.2127			0.713863			0.356931	−	0.934131i	$-0.383823\pi$
$883$	21.2127			0.713863			0.356931	+	0.934131i	$0.383823\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	25.2727			0.848574			0.424287	−	0.905528i	$-0.360525\pi$
$887$	25.2727			0.848574			0.424287	+	0.905528i	$0.360525\pi$
$888$		0			0
$889$	−27.2520			−0.914004
$890$		0			0
$891$	−26.8206	−	4.59360i	−0.898525	−	0.153891i
$892$		0			0
$893$	3.25915			0.109063
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−36.5000	+	78.5258i	−1.21870	+	2.62190i
$898$		0			0
$899$		−	19.7333i		−	0.658142i
$900$		0			0
$901$			29.5953i			0.985962i
$902$		0			0
$903$	4.00762	−	8.62198i	0.133365	−	0.286921i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−22.7267			−0.754626			−0.377313	−	0.926086i	$-0.623152\pi$
$907$	−22.7267			−0.754626			−0.377313	+	0.926086i	$0.623152\pi$
$908$		0			0
$909$	26.8552	+	31.8458i	0.890731	+	1.05626i
$910$		0			0
$911$	8.92321			0.295639			0.147820	−	0.989014i	$-0.452774\pi$
$911$	8.92321			0.295639			0.147820	+	0.989014i	$0.452774\pi$
$912$		0			0
$913$	47.6226			1.57608
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−15.7189			−0.519084
$918$		0			0
$919$			33.7531i			1.11341i	0.830710	+	0.556706i	$0.187935\pi$
$919$			33.7531i			1.11341i	−0.830710	+	0.556706i	$0.812065\pi$
$920$		0			0
$921$	−9.61137	−	4.46751i	−0.316705	−	0.147209i
$922$		0			0
$923$		−	39.6292i		−	1.30441i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	16.8945	−	14.2470i	0.554888	−	0.467932i
$928$		0			0
$929$			51.1823i			1.67924i	0.543177	+	0.839618i	$0.317221\pi$
$929$			51.1823i			1.67924i	−0.543177	+	0.839618i	$0.682779\pi$
$930$		0			0
$931$	−9.64393			−0.316067
$932$		0			0
$933$	9.04420	+	4.20388i	0.296094	+	0.137629i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	56.9974			1.86202			0.931011	−	0.364990i	$-0.118928\pi$
$937$	56.9974			1.86202			0.931011	+	0.364990i	$0.118928\pi$
$938$		0			0
$939$	34.5233	+	16.0470i	1.12663	+	0.523672i
$940$		0			0
$941$	37.1960			1.21255			0.606277	−	0.795253i	$-0.292662\pi$
$941$	37.1960			1.21255			0.606277	+	0.795253i	$0.292662\pi$
$942$		0			0
$943$		−	66.9488i		−	2.18015i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			36.5775i			1.18861i	0.804240	+	0.594305i	$0.202573\pi$
$947$			36.5775i			1.18861i	−0.804240	+	0.594305i	$0.797427\pi$
$948$		0			0
$949$			25.9831i			0.843449i
$950$		0			0
$951$	27.7717	+	12.9087i	0.900560	+	0.418594i
$952$		0			0
$953$			30.3524i			0.983211i	0.870818	+	0.491606i	$0.163590\pi$
$953$			30.3524i			0.983211i	−0.870818	+	0.491606i	$0.836410\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−8.37821	+	18.0248i	−0.270829	+	0.582660i
$958$		0			0
$959$	−23.1806			−0.748540
$960$		0			0
$961$	3.97070			0.128087
$962$		0			0
$963$	−18.6700	+	15.7442i	−0.601632	+	0.507350i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			25.8174i			0.830232i	0.909768	+	0.415116i	$0.136259\pi$
$967$			25.8174i			0.830232i	−0.909768	+	0.415116i	$0.863741\pi$
$968$		0			0
$969$	−3.18736	+	6.85728i	−0.102393	+	0.220287i
$970$		0			0
$971$			22.9904i			0.737796i	0.929470	+	0.368898i	$0.120265\pi$
$971$			22.9904i			0.737796i	−0.929470	+	0.368898i	$0.879735\pi$
$972$		0			0
$973$		−	14.5675i		−	0.467011i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	36.2952i		−	1.16119i	−0.814194	−	0.580593i	$-0.802821\pi$
$977$		−	36.2952i		−	1.16119i	0.814194	−	0.580593i	$-0.197179\pi$
$978$		0			0
$979$	11.0573			0.353393
$980$		0			0
$981$	−12.9693	+	10.9369i	−0.414079	+	0.349189i
$982$		0			0
$983$	6.92523			0.220881			0.110440	−	0.993883i	$-0.464774\pi$
$983$	6.92523			0.220881			0.110440	+	0.993883i	$0.464774\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	1.68001	−	3.61437i	0.0534754	−	0.115047i
$988$		0			0
$989$	−38.6575			−1.22924
$990$		0			0
$991$		−	42.7182i		−	1.35699i	−0.734605	−	0.678495i	$-0.762633\pi$
$991$		−	42.7182i		−	1.35699i	0.734605	−	0.678495i	$-0.237367\pi$
$992$		0			0
$993$	−11.2167	−	5.21367i	−0.355950	−	0.165451i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			15.7743i			0.499579i	0.968300	+	0.249789i	$0.0803613\pi$
$997$			15.7743i			0.499579i	−0.968300	+	0.249789i	$0.919639\pi$
$998$		0			0
$999$	32.3629	−	8.79941i	1.02392	−	0.278401i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.b.g.2351.1		12
3.2	odd	2	inner	2400.2.b.g.2351.3		12
4.3	odd	2		600.2.b.h.251.10	yes	12
5.2	odd	4		2400.2.m.e.1199.12		24
5.3	odd	4		2400.2.m.e.1199.13		24
5.4	even	2		2400.2.b.h.2351.12		12
8.3	odd	2	inner	2400.2.b.g.2351.2		12
8.5	even	2		600.2.b.h.251.4	yes	12
12.11	even	2		600.2.b.h.251.3	yes	12
15.2	even	4		2400.2.m.e.1199.16		24
15.8	even	4		2400.2.m.e.1199.9		24
15.14	odd	2		2400.2.b.h.2351.10		12
20.3	even	4		600.2.m.e.299.18		24
20.7	even	4		600.2.m.e.299.7		24
20.19	odd	2		600.2.b.g.251.3	✓	12
24.5	odd	2		600.2.b.h.251.9	yes	12
24.11	even	2	inner	2400.2.b.g.2351.4		12
40.3	even	4		2400.2.m.e.1199.14		24
40.13	odd	4		600.2.m.e.299.20		24
40.19	odd	2		2400.2.b.h.2351.11		12
40.27	even	4		2400.2.m.e.1199.11		24
40.29	even	2		600.2.b.g.251.9	yes	12
40.37	odd	4		600.2.m.e.299.5		24
60.23	odd	4		600.2.m.e.299.8		24
60.47	odd	4		600.2.m.e.299.17		24
60.59	even	2		600.2.b.g.251.10	yes	12
120.29	odd	2		600.2.b.g.251.4	yes	12
120.53	even	4		600.2.m.e.299.6		24
120.59	even	2		2400.2.b.h.2351.9		12
120.77	even	4		600.2.m.e.299.19		24
120.83	odd	4		2400.2.m.e.1199.10		24
120.107	odd	4		2400.2.m.e.1199.15		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.b.g.251.3	✓	12	20.19	odd	2
600.2.b.g.251.4	yes	12	120.29	odd	2
600.2.b.g.251.9	yes	12	40.29	even	2
600.2.b.g.251.10	yes	12	60.59	even	2
600.2.b.h.251.3	yes	12	12.11	even	2
600.2.b.h.251.4	yes	12	8.5	even	2
600.2.b.h.251.9	yes	12	24.5	odd	2
600.2.b.h.251.10	yes	12	4.3	odd	2
600.2.m.e.299.5		24	40.37	odd	4
600.2.m.e.299.6		24	120.53	even	4
600.2.m.e.299.7		24	20.7	even	4
600.2.m.e.299.8		24	60.23	odd	4
600.2.m.e.299.17		24	60.47	odd	4
600.2.m.e.299.18		24	20.3	even	4
600.2.m.e.299.19		24	120.77	even	4
600.2.m.e.299.20		24	40.13	odd	4
2400.2.b.g.2351.1		12	1.1	even	1	trivial
2400.2.b.g.2351.2		12	8.3	odd	2	inner
2400.2.b.g.2351.3		12	3.2	odd	2	inner
2400.2.b.g.2351.4		12	24.11	even	2	inner
2400.2.b.h.2351.9		12	120.59	even	2
2400.2.b.h.2351.10		12	15.14	odd	2
2400.2.b.h.2351.11		12	40.19	odd	2
2400.2.b.h.2351.12		12	5.4	even	2
2400.2.m.e.1199.9		24	15.8	even	4
2400.2.m.e.1199.10		24	120.83	odd	4
2400.2.m.e.1199.11		24	40.27	even	4
2400.2.m.e.1199.12		24	5.2	odd	4
2400.2.m.e.1199.13		24	5.3	odd	4
2400.2.m.e.1199.14		24	40.3	even	4
2400.2.m.e.1199.15		24	120.107	odd	4
2400.2.m.e.1199.16		24	15.2	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.57067 - 0.730070i) q^{3} -1.25539i q^{7} +(1.93400 + 2.29339i) q^{9} +3.02346i q^{11} -5.65509i q^{13} +2.45546i q^{17} -1.77801 q^{19} +(-0.916519 + 1.97179i) q^{21} +8.84074 q^{23} +(-1.36333 - 5.01411i) q^{27} -3.79561 q^{29} +5.19897i q^{31} +(2.20734 - 4.74886i) q^{33} +6.45436i q^{37} +(-4.12861 + 8.88227i) q^{39} -7.57276i q^{41} -4.37266 q^{43} -1.83304 q^{47} +5.42401 q^{49} +(1.79266 - 3.85672i) q^{51} +12.0528 q^{53} +(2.79266 + 1.29807i) q^{57} +4.91093i q^{59} -8.16586i q^{61} +(2.87909 - 2.42791i) q^{63} +8.50466 q^{67} +(-13.8859 - 6.45436i) q^{69} +7.00770 q^{71} -4.59465 q^{73} +3.79561 q^{77} +7.36659i q^{79} +(-1.51932 + 8.87083i) q^{81} -15.7510i q^{83} +(5.96165 + 2.77106i) q^{87} -3.65716i q^{89} -7.09931 q^{91} +(3.79561 - 8.16586i) q^{93} -13.8773 q^{97} +(-6.93400 + 5.84737i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{3} - 2 q^{9} + 4 q^{19} - 8 q^{27} + 18 q^{33} + 40 q^{43} - 36 q^{49} + 30 q^{51} + 42 q^{57} + 60 q^{67} + 12 q^{73} - 10 q^{81} + 24 q^{91} - 32 q^{97} - 58 q^{99}+O(q^{100}) \) 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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