LMFDB - Embedded newform 2400.2.m.e.1199.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2400.1199");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1199.10
Character	$\chi$	$=$	2400.1199
Dual form			2400.2.m.e.1199.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.730070 - 1.57067i) q^{3} +1.25539 q^{7} +(-1.93400 + 2.29339i) q^{9} +O(q^{10})$	$q+(-0.730070 - 1.57067i) q^{3} +1.25539 q^{7} +(-1.93400 + 2.29339i) q^{9} -3.02346i q^{11} -5.65509 q^{13} -2.45546 q^{17} +1.77801 q^{19} +(-0.916519 - 1.97179i) q^{21} +8.84074i q^{23} +(5.01411 + 1.36333i) q^{27} +3.79561 q^{29} -5.19897i q^{31} +(-4.74886 + 2.20734i) q^{33} -6.45436 q^{37} +(4.12861 + 8.88227i) q^{39} +7.57276i q^{41} -4.37266i q^{43} +1.83304i q^{47} -5.42401 q^{49} +(1.79266 + 3.85672i) q^{51} +12.0528i q^{53} +(-1.29807 - 2.79266i) q^{57} +4.91093i q^{59} +8.16586i q^{61} +(-2.42791 + 2.87909i) q^{63} -8.50466i q^{67} +(13.8859 - 6.45436i) q^{69} +7.00770 q^{71} -4.59465i q^{73} -3.79561i q^{77} +7.36659i q^{79} +(-1.51932 - 8.87083i) q^{81} -15.7510 q^{83} +(-2.77106 - 5.96165i) q^{87} -3.65716i q^{89} -7.09931 q^{91} +(-8.16586 + 3.79561i) q^{93} +13.8773i q^{97} +(6.93400 + 5.84737i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{9}+O(q^{10}) $ 24 * q + 4 * q^9	$ 24 q + 4 q^{9} - 8 q^{19} + 72 q^{49} + 60 q^{51} - 20 q^{81} + 48 q^{91} + 116 q^{99}+O(q^{100}) $ 24 * q + 4 * q^9 - 8 * q^19 + 72 * q^49 + 60 * q^51 - 20 * q^81 + 48 * q^91 + 116 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.730070	−	1.57067i	−0.421506	−	0.906826i
$3$	−0.730070	−	1.57067i	−0.421506	−	0.906826i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.25539			0.474491			0.237245	−	0.971450i	$-0.423755\pi$
$7$	1.25539			0.474491			0.237245	+	0.971450i	$0.423755\pi$
$8$		0			0
$9$	−1.93400	+	2.29339i	−0.644665	+	0.764465i
$10$		0			0
$11$		−	3.02346i		−	0.911609i	−0.890080	−	0.455804i	$-0.849352\pi$
$11$		−	3.02346i		−	0.911609i	0.890080	−	0.455804i	$-0.150648\pi$
$12$		0			0
$13$	−5.65509			−1.56844			−0.784220	−	0.620483i	$-0.786936\pi$
$13$	−5.65509			−1.56844			−0.784220	+	0.620483i	$0.786936\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.45546			−0.595538			−0.297769	−	0.954638i	$-0.596242\pi$
$17$	−2.45546			−0.595538			−0.297769	+	0.954638i	$0.596242\pi$
$18$		0			0
$19$	1.77801			0.407903			0.203951	−	0.978981i	$-0.434622\pi$
$19$	1.77801			0.407903			0.203951	+	0.978981i	$0.434622\pi$
$20$		0			0
$21$	−0.916519	−	1.97179i	−0.200001	−	0.430281i
$22$		0			0
$23$			8.84074i			1.84342i	0.387878	+	0.921711i	$0.373208\pi$
$23$			8.84074i			1.84342i	−0.387878	+	0.921711i	$0.626792\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	5.01411	+	1.36333i	0.964967	+	0.262373i
$28$		0			0
$29$	3.79561			0.704827			0.352414	−	0.935844i	$-0.385361\pi$
$29$	3.79561			0.704827			0.352414	+	0.935844i	$0.385361\pi$
$30$		0			0
$31$		−	5.19897i		−	0.933763i	−0.884320	−	0.466881i	$-0.845377\pi$
$31$		−	5.19897i		−	0.933763i	0.884320	−	0.466881i	$-0.154623\pi$
$32$		0			0
$33$	−4.74886	+	2.20734i	−0.826670	+	0.384249i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.45436			−1.06109			−0.530545	−	0.847657i	$-0.678013\pi$
$37$	−6.45436			−1.06109			−0.530545	+	0.847657i	$0.678013\pi$
$38$		0			0
$39$	4.12861	+	8.88227i	0.661107	+	1.42230i
$40$		0			0
$41$			7.57276i			1.18267i	0.806427	+	0.591333i	$0.201398\pi$
$41$			7.57276i			1.18267i	−0.806427	+	0.591333i	$0.798602\pi$
$42$		0			0
$43$		−	4.37266i		−	0.666824i	−0.942781	−	0.333412i	$-0.891800\pi$
$43$		−	4.37266i		−	0.666824i	0.942781	−	0.333412i	$-0.108200\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.83304i			0.267376i	0.991023	+	0.133688i	$0.0426820\pi$
$47$			1.83304i			0.267376i	−0.991023	+	0.133688i	$0.957318\pi$
$48$		0			0
$49$	−5.42401			−0.774858
$50$		0			0
$51$	1.79266	+	3.85672i	0.251023	+	0.540049i
$52$		0			0
$53$			12.0528i			1.65558i	0.561035	+	0.827792i	$0.310403\pi$
$53$			12.0528i			1.65558i	−0.561035	+	0.827792i	$0.689597\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.29807	−	2.79266i	−0.171934	−	0.369897i
$58$		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	$0.175100\pi$
$61$			8.16586i			1.04553i	−0.852477	+	0.522766i	$0.824900\pi$
$62$		0			0
$63$	−2.42791	+	2.87909i	−0.305888	+	0.362732i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	8.50466i		−	1.03901i	−0.854467	−	0.519505i	$-0.826116\pi$
$67$		−	8.50466i		−	1.03901i	0.854467	−	0.519505i	$-0.173884\pi$
$68$		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
$73$		−	4.59465i		−	0.537763i	−0.963173	−	0.268881i	$-0.913346\pi$
$73$		−	4.59465i		−	0.537763i	0.963173	−	0.268881i	$-0.0866540\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.79561i		−	0.432550i
$78$		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
$83$	−15.7510			−1.72890			−0.864449	−	0.502720i	$-0.832333\pi$
$83$	−15.7510			−1.72890			−0.864449	+	0.502720i	$0.832333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−2.77106	−	5.96165i	−0.297089	−	0.639156i
$88$		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
$93$	−8.16586	+	3.79561i	−0.846760	+	0.393587i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			13.8773i			1.40903i	0.709690	+	0.704514i	$0.248835\pi$
$97$			13.8773i			1.40903i	−0.709690	+	0.704514i	$0.751165\pi$
$98$		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
$103$	−7.36659			−0.725852			−0.362926	−	0.931818i	$-0.618222\pi$
$103$	−7.36659			−0.725852			−0.362926	+	0.931818i	$0.618222\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.14076			−0.786997			−0.393499	−	0.919325i	$-0.628735\pi$
$107$	−8.14076			−0.786997			−0.393499	+	0.919325i	$0.628735\pi$
$108$		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
$113$	−11.4884			−1.08073			−0.540367	−	0.841429i	$-0.681715\pi$
$113$	−11.4884			−1.08073			−0.540367	+	0.841429i	$0.681715\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	10.9369	−	12.9693i	1.01112	−	1.19902i
$118$		0			0
$119$	−3.08255			−0.282577
$120$		0			0
$121$	1.85866			0.168969
$122$		0			0
$123$	11.8943	−	5.52865i	1.07247	−	0.498501i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	21.7081			1.92628			0.963142	−	0.268993i	$-0.0866909\pi$
$127$	21.7081			1.92628			0.963142	+	0.268993i	$0.0866909\pi$
$128$		0			0
$129$	−6.86799	+	3.19234i	−0.604693	+	0.281070i
$130$		0			0
$131$			12.5212i			1.09398i	0.837139	+	0.546990i	$0.184227\pi$
$131$			12.5212i			1.09398i	−0.837139	+	0.546990i	$0.815773\pi$
$132$		0			0
$133$	2.23208			0.193546
$134$		0			0
$135$		0			0
$136$		0			0
$137$	18.4649			1.57757			0.788783	−	0.614672i	$-0.210712\pi$
$137$	18.4649			1.57757			0.788783	+	0.614672i	$0.210712\pi$
$138$		0			0
$139$	−11.6040			−0.984237			−0.492118	−	0.870528i	$-0.663777\pi$
$139$	−11.6040			−0.984237			−0.492118	+	0.870528i	$0.663777\pi$
$140$		0			0
$141$	2.87909	−	1.33825i	0.242463	−	0.112701i
$142$		0			0
$143$			17.0980i			1.42980i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	3.95990	+	8.51932i	0.326607	+	0.702661i
$148$		0			0
$149$	−11.9233			−0.976794			−0.488397	−	0.872621i	$-0.662418\pi$
$149$	−11.9233			−0.976794			−0.488397	+	0.872621i	$0.662418\pi$
$150$		0			0
$151$			10.0548i			0.818247i	0.912479	+	0.409124i	$0.134166\pi$
$151$			10.0548i			0.818247i	−0.912479	+	0.409124i	$0.865834\pi$
$152$		0			0
$153$	4.74886	−	5.63135i	0.383922	−	0.455268i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.71150			0.136593			0.0682964	−	0.997665i	$-0.478244\pi$
$157$	1.71150			0.136593			0.0682964	+	0.997665i	$0.478244\pi$
$158$		0			0
$159$	18.9310	−	8.79941i	1.50133	−	0.697838i
$160$		0			0
$161$			11.0985i			0.874687i
$162$		0			0
$163$			11.9580i			0.936621i	0.883564	+	0.468311i	$0.155137\pi$
$163$			11.9580i			0.936621i	−0.883564	+	0.468311i	$0.844863\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	0.583522i		−	0.0451543i	−0.999745	−	0.0225771i	$-0.992813\pi$
$167$		−	0.583522i		−	0.0451543i	0.999745	−	0.0225771i	$-0.00718713\pi$
$168$		0			0
$169$	18.9800			1.46000
$170$		0			0
$171$	−3.43866	+	4.07767i	−0.262961	+	0.311827i
$172$		0			0
$173$			12.0528i			0.916360i	0.888860	+	0.458180i	$0.151498\pi$
$173$			12.0528i			0.916360i	−0.888860	+	0.458180i	$0.848502\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	7.71344	−	3.58532i	0.579778	−	0.269489i
$178$		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	$0.0848505\pi$
$181$			7.08790i			0.526840i	−0.964681	+	0.263420i	$0.915150\pi$
$182$		0			0
$183$	12.8259	−	5.96165i	0.948114	−	0.440698i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			7.42401i			0.542897i
$188$		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
$193$			4.27334i			0.307602i	0.988102	+	0.153801i	$0.0491515\pi$
$193$			4.27334i			0.307602i	−0.988102	+	0.153801i	$0.950849\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			10.0903i			0.718901i	0.933164	+	0.359450i	$0.117036\pi$
$197$			10.0903i			0.718901i	−0.933164	+	0.359450i	$0.882964\pi$
$198$		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
$203$	4.76495			0.334434
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−20.2753	−	17.0980i	−1.40923	−	1.18839i
$208$		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
$213$	−5.11611	−	11.0068i	−0.350550	−	0.754172i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	6.52671i		−	0.443062i
$218$		0			0
$219$	−7.21667	+	3.35441i	−0.487657	+	0.226670i
$220$		0			0
$221$	13.8859			0.934065
$222$		0			0
$223$	−7.70974			−0.516282			−0.258141	−	0.966107i	$-0.583110\pi$
$223$	−7.70974			−0.516282			−0.258141	+	0.966107i	$0.583110\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.40388			0.491413			0.245706	−	0.969344i	$-0.420980\pi$
$227$	7.40388			0.491413			0.245706	+	0.969344i	$0.420980\pi$
$228$		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
$233$	−1.40807			−0.0922454			−0.0461227	−	0.998936i	$-0.514687\pi$
$233$	−1.40807			−0.0922454			−0.0461227	+	0.998936i	$0.514687\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	11.5705	−	5.37812i	0.751583	−	0.349347i
$238$		0			0
$239$	−9.50673			−0.614939			−0.307470	−	0.951558i	$-0.599482\pi$
$239$	−9.50673			−0.614939			−0.307470	+	0.951558i	$0.599482\pi$
$240$		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
$243$	−12.8239	+	8.86267i	−0.822655	+	0.568540i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−10.0548			−0.639771
$248$		0			0
$249$	11.4993	+	24.7396i	0.728741	+	1.56781i
$250$		0			0
$251$		−	9.49772i		−	0.599491i	−0.954019	−	0.299745i	$-0.903098\pi$
$251$		−	9.49772i		−	0.599491i	0.954019	−	0.299745i	$-0.0969017\pi$
$252$		0			0
$253$	26.7297			1.68048
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−27.3144			−1.70382			−0.851912	−	0.523686i	$-0.824557\pi$
$257$	−27.3144			−1.70382			−0.851912	+	0.523686i	$0.824557\pi$
$258$		0			0
$259$	−8.10270			−0.503477
$260$		0			0
$261$	−7.34070	+	8.70484i	−0.454378	+	0.538816i
$262$		0			0
$263$			1.24952i			0.0770484i	0.999258	+	0.0385242i	$0.0122657\pi$
$263$			1.24952i			0.0770484i	−0.999258	+	0.0385242i	$0.987734\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−5.74418	+	2.66998i	−0.351538	+	0.163400i
$268$		0			0
$269$	−15.7189			−0.958399			−0.479199	−	0.877706i	$-0.659073\pi$
$269$	−15.7189			−0.958399			−0.479199	+	0.877706i	$0.659073\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	5.18299	+	11.1507i	0.313689	+	0.674869i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.633548			0.0380662			0.0190331	−	0.999819i	$-0.493941\pi$
$277$	0.633548			0.0380662			0.0190331	+	0.999819i	$0.493941\pi$
$278$		0			0
$279$	11.9233	+	10.0548i	0.713829	+	0.601965i
$280$		0			0
$281$		−	20.9204i		−	1.24801i	−0.781422	−	0.624003i	$-0.785505\pi$
$281$		−	20.9204i		−	1.24801i	0.781422	−	0.624003i	$-0.214495\pi$
$282$		0			0
$283$		−	14.6846i		−	0.872911i	−0.899726	−	0.436455i	$-0.856234\pi$
$283$		−	14.6846i		−	0.872911i	0.899726	−	0.436455i	$-0.143766\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			9.50673i			0.561165i
$288$		0			0
$289$	−10.9707			−0.645335
$290$		0			0
$291$	21.7967	−	10.1314i	1.27774	−	0.593914i
$292$		0			0
$293$			5.49911i			0.321262i	0.987015	+	0.160631i	$0.0513528\pi$
$293$			5.49911i			0.321262i	−0.987015	+	0.160631i	$0.948647\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	4.12197	−	15.1600i	0.239181	−	0.879672i
$298$		0			0
$299$		−	49.9952i		−	2.89129i
$300$		0			0
$301$		−	5.48937i		−	0.316402i
$302$		0			0
$303$	−10.1377	−	21.8101i	−0.582393	−	1.25296i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	6.11929i		−	0.349246i	−0.984635	−	0.174623i	$-0.944129\pi$
$307$		−	6.11929i		−	0.349246i	0.984635	−	0.174623i	$-0.0558707\pi$
$308$		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
$313$		−	21.9800i		−	1.24238i	−0.783658	−	0.621192i	$-0.786649\pi$
$313$		−	21.9800i		−	1.24238i	0.783658	−	0.621192i	$-0.213351\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			17.6815i			0.993091i	0.868011	+	0.496545i	$0.165398\pi$
$317$			17.6815i			0.993091i	−0.868011	+	0.496545i	$0.834602\pi$
$318$		0			0
$319$		−	11.4759i		−	0.642527i
$320$		0			0
$321$	5.94333	+	12.7864i	0.331724	+	0.713669i
$322$		0			0
$323$	−4.36583			−0.242922
$324$		0			0
$325$		0			0
$326$		0			0
$327$	8.88227	−	4.12861i	0.491190	−	0.228313i
$328$		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
$333$	12.4827	−	14.8024i	0.684048	−	0.811166i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	1.90663i		−	0.103861i	−0.998651	−	0.0519303i	$-0.983463\pi$
$337$		−	1.90663i		−	0.103861i	0.998651	−	0.0519303i	$-0.0165374\pi$
$338$		0			0
$339$	8.38731	+	18.0444i	0.455536	+	0.980038i
$340$		0			0
$341$	−15.7189			−0.851226
$342$		0			0
$343$	−15.5969			−0.842154
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.530510			0.0284793			0.0142396	−	0.999899i	$-0.495467\pi$
$347$	0.530510			0.0284793			0.0142396	+	0.999899i	$0.495467\pi$
$348$		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
$353$	11.2299			0.597708			0.298854	−	0.954299i	$-0.403396\pi$
$353$	11.2299			0.597708			0.298854	+	0.954299i	$0.403396\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	2.25048	+	4.84167i	0.119108	+	0.256248i
$358$		0			0
$359$	−17.0980			−0.902396			−0.451198	−	0.892424i	$-0.649003\pi$
$359$	−17.0980			−0.902396			−0.451198	+	0.892424i	$0.649003\pi$
$360$		0			0
$361$	−15.8387			−0.833615
$362$		0			0
$363$	−1.35695	−	2.91934i	−0.0712216	−	0.153226i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−13.9984			−0.730709			−0.365355	−	0.930868i	$-0.619052\pi$
$367$	−13.9984			−0.730709			−0.365355	+	0.930868i	$0.619052\pi$
$368$		0			0
$369$	−17.3673	−	14.6457i	−0.904107	−	0.762424i
$370$		0			0
$371$			15.1309i			0.785559i
$372$		0			0
$373$	−21.5952			−1.11815			−0.559077	−	0.829116i	$-0.688845\pi$
$373$	−21.5952			−1.11815			−0.559077	+	0.829116i	$0.688845\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−21.4645			−1.10548
$378$		0			0
$379$	11.7780			0.604996			0.302498	−	0.953150i	$-0.402180\pi$
$379$	11.7780			0.604996			0.302498	+	0.953150i	$0.402180\pi$
$380$		0			0
$381$	−15.8484	−	34.0963i	−0.811940	−	1.74680i
$382$		0			0
$383$		−	15.8484i		−	0.809818i	−0.914357	−	0.404909i	$-0.867303\pi$
$383$		−	15.8484i		−	0.809818i	0.914357	−	0.404909i	$-0.132697\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	10.0282	+	8.45670i	0.509763	+	0.429878i
$388$		0			0
$389$	−18.4770			−0.936822			−0.468411	−	0.883511i	$-0.655173\pi$
$389$	−18.4770			−0.936822			−0.468411	+	0.883511i	$0.655173\pi$
$390$		0			0
$391$		−	21.7081i		−	1.09783i
$392$		0			0
$393$	19.6666	−	9.14134i	0.992050	−	0.461119i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−18.3981			−0.923373			−0.461687	−	0.887043i	$-0.652756\pi$
$397$	−18.3981			−0.923373			−0.461687	+	0.887043i	$0.652756\pi$
$398$		0			0
$399$	−1.62958	−	3.50586i	−0.0815809	−	0.175513i
$400$		0			0
$401$			0.183464i			0.00916176i	0.999990	+	0.00458088i	$0.00145814\pi$
$401$			0.183464i			0.00916176i	−0.999990	+	0.00458088i	$0.998542\pi$
$402$		0			0
$403$			29.4006i			1.46455i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			19.5145i			0.967299i
$408$		0			0
$409$	30.5933			1.51274			0.756371	−	0.654142i	$-0.226970\pi$
$409$	30.5933			1.51274			0.756371	+	0.654142i	$0.226970\pi$
$410$		0			0
$411$	−13.4807	−	29.0023i	−0.664953	−	1.43058i
$412$		0			0
$413$			6.16511i			0.303365i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	8.47171	+	18.2260i	0.414862	+	0.892531i
$418$		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	$-0.774198\pi$
$421$		−	26.7297i		−	1.30272i	0.758767	−	0.651362i	$-0.225802\pi$
$422$		0			0
$423$	−4.20388	−	3.54509i	−0.204400	−	0.172368i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			10.2513i			0.496095i
$428$		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
$433$			19.8667i			0.954731i	0.878705	+	0.477366i	$0.158408\pi$
$433$			19.8667i			0.954731i	−0.878705	+	0.477366i	$0.841592\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			15.7189i			0.751937i
$438$		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
$443$	13.6854			0.650212			0.325106	−	0.945678i	$-0.394600\pi$
$443$	13.6854			0.650212			0.325106	+	0.945678i	$0.394600\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	8.70484	+	18.7275i	0.411725	+	0.885782i
$448$		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
$453$	15.7927	−	7.34070i	0.742008	−	0.344896i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	22.6133i		−	1.05781i	−0.848682	−	0.528903i	$-0.822604\pi$
$457$		−	22.6133i		−	1.05781i	0.848682	−	0.528903i	$-0.177396\pi$
$458$		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
$463$	40.5506			1.88455			0.942273	−	0.334845i	$-0.108684\pi$
$463$	40.5506			1.88455			0.942273	+	0.334845i	$0.108684\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.2492			−0.474276			−0.237138	−	0.971476i	$-0.576209\pi$
$467$	−10.2492			−0.474276			−0.237138	+	0.971476i	$0.576209\pi$
$468$		0			0
$469$		−	10.6766i		−	0.493001i
$470$		0			0
$471$	−1.24952	−	2.68820i	−0.0575746	−	0.123866i
$472$		0			0
$473$	−13.2206			−0.607883
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−27.6419	−	23.3101i	−1.26564	−	1.06730i
$478$		0			0
$479$	−31.4378			−1.43643			−0.718215	−	0.695821i	$-0.755041\pi$
$479$	−31.4378			−1.43643			−0.718215	+	0.695821i	$0.755041\pi$
$480$		0			0
$481$	36.5000			1.66425
$482$		0			0
$483$	17.4321	−	8.10270i	0.793188	−	0.368686i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	0.177431			0.00804018			0.00402009	−	0.999992i	$-0.498720\pi$
$487$	0.177431			0.00804018			0.00402009	+	0.999992i	$0.498720\pi$
$488$		0			0
$489$	18.7820	−	8.73016i	0.849352	−	0.394791i
$490$		0			0
$491$			7.75623i			0.350034i	0.984565	+	0.175017i	$0.0559980\pi$
$491$			7.75623i			0.350034i	−0.984565	+	0.175017i	$0.944002\pi$
$492$		0			0
$493$	−9.31999			−0.419751
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.79736			0.394616
$498$		0			0
$499$	5.22067			0.233709			0.116855	−	0.993149i	$-0.462719\pi$
$499$	5.22067			0.233709			0.116855	+	0.993149i	$0.462719\pi$
$500$		0			0
$501$	−0.916519	+	0.426011i	−0.0409470	+	0.0190328i
$502$		0			0
$503$		−	6.34171i		−	0.282763i	−0.989955	−	0.141381i	$-0.954846\pi$
$503$		−	6.34171i		−	0.282763i	0.989955	−	0.141381i	$-0.0451544\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−13.8567	−	29.8113i	−0.615399	−	1.32397i
$508$		0			0
$509$	−34.1959			−1.51571			−0.757854	−	0.652425i	$-0.773752\pi$
$509$	−34.1959			−1.51571			−0.757854	+	0.652425i	$0.773752\pi$
$510$		0			0
$511$		−	5.76805i		−	0.255164i
$512$		0			0
$513$	8.91513	+	2.42401i	0.393613	+	0.107023i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	5.54212			0.243742
$518$		0			0
$519$	18.9310	−	8.79941i	0.830978	−	0.386251i
$520$		0			0
$521$		−	32.6151i		−	1.42889i	−0.699689	−	0.714447i	$-0.746678\pi$
$521$		−	32.6151i		−	1.42889i	0.699689	−	0.714447i	$-0.253322\pi$
$522$		0			0
$523$		−	14.6074i		−	0.638736i	−0.947631	−	0.319368i	$-0.896529\pi$
$523$		−	14.6074i		−	0.638736i	0.947631	−	0.319368i	$-0.103471\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			12.7659i			0.556091i
$528$		0			0
$529$	−55.1587			−2.39820
$530$		0			0
$531$	−11.2627	−	9.49772i	−0.488759	−	0.412166i
$532$		0			0
$533$		−	42.8246i		−	1.85494i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.951024	−	0.442050i	0.0410397	−	0.0190759i
$538$		0			0
$539$			16.3993i			0.706368i
$540$		0			0
$541$			13.0217i			0.559846i	0.960023	+	0.279923i	$0.0903089\pi$
$541$			13.0217i			0.559846i	−0.960023	+	0.279923i	$0.909691\pi$
$542$		0			0
$543$	11.1327	−	5.17466i	0.477752	−	0.222066i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	37.8280i		−	1.61741i	−0.588214	−	0.808705i	$-0.700169\pi$
$547$		−	37.8280i		−	1.61741i	0.588214	−	0.808705i	$-0.299831\pi$
$548$		0			0
$549$	−18.7275	−	15.7927i	−0.799272	−	0.674018i
$550$		0			0
$551$	6.74863			0.287501
$552$		0			0
$553$			9.24791i			0.393261i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			22.0135i			0.932744i	0.884589	+	0.466372i	$0.154439\pi$
$557$			22.0135i			0.932744i	−0.884589	+	0.466372i	$0.845561\pi$
$558$		0			0
$559$			24.7278i			1.04587i
$560$		0			0
$561$	11.6607	−	5.42004i	0.492313	−	0.228834i
$562$		0			0
$563$	9.80727			0.413327			0.206664	−	0.978412i	$-0.433739\pi$
$563$	9.80727			0.413327			0.206664	+	0.978412i	$0.433739\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.90733	−	11.1363i	−0.0801002	−	0.467681i
$568$		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
$573$	−13.8209	−	29.7343i	−0.577378	−	1.24217i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			39.1587i			1.63020i	0.579322	+	0.815098i	$0.303317\pi$
$577$			39.1587i			1.63020i	−0.579322	+	0.815098i	$0.696683\pi$
$578$		0			0
$579$	6.71200	−	3.11984i	0.278941	−	0.129656i
$580$		0			0
$581$	−19.7736			−0.820347
$582$		0			0
$583$	36.4413			1.50924
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−29.3332			−1.21071			−0.605356	−	0.795955i	$-0.706969\pi$
$587$	−29.3332			−1.21071			−0.605356	+	0.795955i	$0.706969\pi$
$588$		0			0
$589$		−	9.24381i		−	0.380885i
$590$		0			0
$591$	15.8484	−	7.36659i	0.651918	−	0.303021i
$592$		0			0
$593$	−24.6525			−1.01236			−0.506179	−	0.862428i	$-0.668942\pi$
$593$	−24.6525			−1.01236			−0.506179	+	0.862428i	$0.668942\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−29.8740	+	13.8859i	−1.22266	+	0.568311i
$598$		0			0
$599$	−29.8638			−1.22020			−0.610102	−	0.792323i	$-0.708871\pi$
$599$	−29.8638			−1.22020			−0.610102	+	0.792323i	$0.708871\pi$
$600$		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
$603$	19.5046	+	16.4480i	0.794287	+	0.669814i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−25.8174			−1.04790			−0.523949	−	0.851750i	$-0.675542\pi$
$607$	−25.8174			−1.04790			−0.523949	+	0.851750i	$0.675542\pi$
$608$		0			0
$609$	−3.47875	−	7.48416i	−0.140966	−	0.303274i
$610$		0			0
$611$		−	10.3660i		−	0.419363i
$612$		0			0
$613$	16.6866			0.673965			0.336982	−	0.941511i	$-0.390594\pi$
$613$	16.6866			0.673965			0.336982	+	0.941511i	$0.390594\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.69514			0.350053			0.175027	−	0.984564i	$-0.443999\pi$
$617$	8.69514			0.350053			0.175027	+	0.984564i	$0.443999\pi$
$618$		0			0
$619$	20.3727			0.818846			0.409423	−	0.912345i	$-0.365730\pi$
$619$	20.3727			0.818846			0.409423	+	0.912345i	$0.365730\pi$
$620$		0			0
$621$	−12.0528	+	44.3285i	−0.483663	+	1.77884i
$622$		0			0
$623$		−	4.59114i		−	0.183940i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−8.44351	+	3.92467i	−0.337201	+	0.156736i
$628$		0			0
$629$	15.8484			0.631919
$630$		0			0
$631$		−	4.51267i		−	0.179647i	−0.995958	−	0.0898233i	$-0.971370\pi$
$631$		−	4.51267i		−	0.179647i	0.995958	−	0.0898233i	$-0.0286302\pi$
$632$		0			0
$633$	4.01198	+	8.63135i	0.159462	+	0.343065i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	30.6732			1.21532
$638$		0			0
$639$	−13.5529	+	16.0714i	−0.536143	+	0.635776i
$640$		0			0
$641$			25.0424i			0.989114i	0.869145	+	0.494557i	$0.164670\pi$
$641$			25.0424i			0.989114i	−0.869145	+	0.494557i	$0.835330\pi$
$642$		0			0
$643$		−	21.7360i		−	0.857184i	−0.903498	−	0.428592i	$-0.859010\pi$
$643$		−	21.7360i		−	0.857184i	0.903498	−	0.428592i	$-0.140990\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			21.3476i			0.839259i	0.907695	+	0.419629i	$0.137840\pi$
$647$			21.3476i			0.839259i	−0.907695	+	0.419629i	$0.862160\pi$
$648$		0			0
$649$	14.8480			0.582836
$650$		0			0
$651$	−10.2513	+	4.76495i	−0.401780	+	0.186753i
$652$		0			0
$653$		−	25.9387i		−	1.01506i	−0.861634	−	0.507530i	$-0.830559\pi$
$653$		−	25.9387i		−	1.01506i	0.861634	−	0.507530i	$-0.169441\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	10.5373	+	8.88603i	0.411101	+	0.346677i
$658$		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	$0.0580152\pi$
$661$			9.31999i			0.362506i	−0.983437	+	0.181253i	$0.941985\pi$
$662$		0			0
$663$	−10.1377	−	21.8101i	−0.393714	−	0.847034i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			33.5560i			1.29929i
$668$		0			0
$669$	5.62865	+	12.1094i	0.217616	+	0.468178i
$670$		0			0
$671$	24.6892			0.953115
$672$		0			0
$673$			37.2520i			1.43596i	0.696063	+	0.717980i	$0.254933\pi$
$673$			37.2520i			1.43596i	−0.696063	+	0.717980i	$0.745067\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	37.0665i		−	1.42458i	−0.701885	−	0.712290i	$-0.747658\pi$
$677$		−	37.0665i		−	1.42458i	0.701885	−	0.712290i	$-0.252342\pi$
$678$		0			0
$679$			17.4214i			0.668571i
$680$		0			0
$681$	−5.40535	−	11.6290i	−0.207133	−	0.445626i
$682$		0			0
$683$	26.0898			0.998297			0.499149	−	0.866516i	$-0.333646\pi$
$683$	26.0898			0.998297			0.499149	+	0.866516i	$0.333646\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	26.9071	−	12.5068i	1.02657	−	0.477165i
$688$		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
$693$	8.70484	+	7.34070i	0.330669	+	0.278850i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	18.5946i		−	0.704323i
$698$		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
$703$	−11.4759			−0.432822
$704$		0			0
$705$		0			0
$706$		0			0
$707$	17.4321			0.655602
$708$		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
$713$	45.9628			1.72132
$714$		0			0
$715$		0			0
$716$		0			0
$717$	6.94058	+	14.9319i	0.259201	+	0.557643i
$718$		0			0
$719$	9.50673			0.354541			0.177271	−	0.984162i	$-0.443273\pi$
$719$	9.50673			0.354541			0.177271	+	0.984162i	$0.443273\pi$
$720$		0			0
$721$	−9.24791			−0.344410
$722$		0			0
$723$	−1.65970	−	3.57067i	−0.0617249	−	0.132795i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	31.7629			1.17802			0.589011	−	0.808125i	$-0.299518\pi$
$727$	31.7629			1.17802			0.589011	+	0.808125i	$0.299518\pi$
$728$		0			0
$729$	23.2827	+	13.6718i	0.862321	+	0.506362i
$730$		0			0
$731$			10.7369i			0.397119i
$732$		0			0
$733$	−15.5852			−0.575653			−0.287826	−	0.957683i	$-0.592933\pi$
$733$	−15.5852			−0.575653			−0.287826	+	0.957683i	$0.592933\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−25.7135			−0.947171
$738$		0			0
$739$	4.17997			0.153763			0.0768813	−	0.997040i	$-0.475504\pi$
$739$	4.17997			0.153763			0.0768813	+	0.997040i	$0.475504\pi$
$740$		0			0
$741$	7.34070	+	15.7927i	0.269667	+	0.580161i
$742$		0			0
$743$		−	20.1805i		−	0.740351i	−0.928962	−	0.370176i	$-0.879298\pi$
$743$		−	20.1805i		−	0.740351i	0.928962	−	0.370176i	$-0.120702\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	30.4624	−	36.1233i	1.11456	−	1.32168i
$748$		0			0
$749$	−10.2198			−0.373423
$750$		0			0
$751$			19.0316i			0.694474i	0.937777	+	0.347237i	$0.112880\pi$
$751$			19.0316i			0.694474i	−0.937777	+	0.347237i	$0.887120\pi$
$752$		0			0
$753$	−14.9178	+	6.93400i	−0.543633	+	0.252689i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−4.22227			−0.153461			−0.0767305	−	0.997052i	$-0.524448\pi$
$757$	−4.22227			−0.153461			−0.0767305	+	0.997052i	$0.524448\pi$
$758$		0			0
$759$	−19.5145	−	41.9834i	−0.708332	−	1.52390i
$760$		0			0
$761$			38.0795i			1.38038i	0.723628	+	0.690190i	$0.242473\pi$
$761$			38.0795i			1.38038i	−0.723628	+	0.690190i	$0.757527\pi$
$762$		0			0
$763$			7.09931i			0.257012i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	27.7717i		−	1.00278i
$768$		0			0
$769$	−16.9600			−0.611595			−0.305797	−	0.952097i	$-0.598923\pi$
$769$	−16.9600			−0.611595			−0.305797	+	0.952097i	$0.598923\pi$
$770$		0			0
$771$	19.9414	+	42.9018i	0.718172	+	1.54507i
$772$		0			0
$773$			16.3849i			0.589324i	0.955602	+	0.294662i	$0.0952070\pi$
$773$			16.3849i			0.589324i	−0.955602	+	0.294662i	$0.904793\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	5.91554	+	12.7267i	0.212219	+	0.456566i
$778$		0			0
$779$			13.4644i			0.482413i
$780$		0			0
$781$		−	21.1875i		−	0.758150i
$782$		0			0
$783$	19.0316	+	5.17466i	0.680135	+	0.184927i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			25.6833i			0.915511i	0.889078	+	0.457756i	$0.151347\pi$
$787$			25.6833i			0.915511i	−0.889078	+	0.457756i	$0.848653\pi$
$788$		0			0
$789$	1.96257	−	0.912234i	0.0698695	−	0.0324764i
$790$		0			0
$791$	−14.4223			−0.512799
$792$		0			0
$793$		−	46.1787i		−	1.63985i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	13.7563i		−	0.487274i	−0.969866	−	0.243637i	$-0.921659\pi$
$797$		−	13.7563i		−	0.487274i	0.969866	−	0.243637i	$-0.0783406\pi$
$798$		0			0
$799$		−	4.50096i		−	0.159232i
$800$		0			0
$801$	8.38731	+	7.07293i	0.296351	+	0.249910i
$802$		0			0
$803$	−13.8918			−0.490229
$804$		0			0
$805$		0			0
$806$		0			0
$807$	11.4759	+	24.6892i	0.403971	+	0.869100i
$808$		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
$817$		−	7.77462i		−	0.271999i
$818$		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
$823$	36.0379			1.25620			0.628102	−	0.778131i	$-0.283832\pi$
$823$	36.0379			1.25620			0.628102	+	0.778131i	$0.283832\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	29.8501			1.03799			0.518995	−	0.854777i	$-0.326306\pi$
$827$	29.8501			1.03799			0.518995	+	0.854777i	$0.326306\pi$
$828$		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
$833$	13.3185			0.461457
$834$		0			0
$835$		0			0
$836$		0			0
$837$	7.08790	−	26.0682i	0.244994	−	0.901050i
$838$		0			0
$839$	5.17466			0.178649			0.0893246	−	0.996003i	$-0.471529\pi$
$839$	5.17466			0.178649			0.0893246	+	0.996003i	$0.471529\pi$
$840$		0			0
$841$	−14.5933			−0.503218
$842$		0			0
$843$	−32.8590	+	15.2733i	−1.13172	+	0.526042i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.33334			0.0801745
$848$		0			0
$849$	−23.0647	+	10.7208i	−0.791578	+	0.367937i
$850$		0			0
$851$		−	57.0613i		−	1.95604i
$852$		0			0
$853$	36.3283			1.24386			0.621929	−	0.783073i	$-0.286349\pi$
$853$	36.3283			1.24386			0.621929	+	0.783073i	$0.286349\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.42274			0.321875			0.160937	−	0.986965i	$-0.448548\pi$
$857$	9.42274			0.321875			0.160937	+	0.986965i	$0.448548\pi$
$858$		0			0
$859$	18.0480			0.615789			0.307894	−	0.951421i	$-0.400376\pi$
$859$	18.0480			0.615789			0.307894	+	0.951421i	$0.400376\pi$
$860$		0			0
$861$	14.9319	−	6.94058i	0.508879	−	0.236534i
$862$		0			0
$863$			18.3475i			0.624555i	0.949991	+	0.312278i	$0.101092\pi$
$863$			18.3475i			0.624555i	−0.949991	+	0.312278i	$0.898908\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	8.00937	+	17.2313i	0.272013	+	0.585206i
$868$		0			0
$869$	22.2726			0.755547
$870$		0			0
$871$			48.0946i			1.62962i
$872$		0			0
$873$	−31.8262	−	26.8387i	−1.07715	−	0.908352i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	16.5736			0.559651			0.279826	−	0.960051i	$-0.409723\pi$
$877$	16.5736			0.559651			0.279826	+	0.960051i	$0.409723\pi$
$878$		0			0
$879$	8.63728	−	4.01474i	0.291328	−	0.135414i
$880$		0			0
$881$		−	25.5865i		−	0.862031i	−0.902345	−	0.431015i	$-0.858155\pi$
$881$		−	25.5865i		−	0.862031i	0.902345	−	0.431015i	$-0.141845\pi$
$882$		0			0
$883$			21.2127i			0.713863i	0.934131	+	0.356931i	$0.116177\pi$
$883$			21.2127i			0.713863i	−0.934131	+	0.356931i	$0.883823\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	25.2727i		−	0.848574i	−0.905528	−	0.424287i	$-0.860525\pi$
$887$		−	25.2727i		−	0.848574i	0.905528	−	0.424287i	$-0.139475\pi$
$888$		0			0
$889$	27.2520			0.914004
$890$		0			0
$891$	−26.8206	+	4.59360i	−0.898525	+	0.153891i
$892$		0			0
$893$			3.25915i			0.109063i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−78.5258	+	36.5000i	−2.62190	+	1.21870i
$898$		0			0
$899$		−	19.7333i		−	0.658142i
$900$		0			0
$901$		−	29.5953i		−	0.985962i
$902$		0			0
$903$	−8.62198	+	4.00762i	−0.286921	+	0.133365i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			22.7267i			0.754626i	0.926086	+	0.377313i	$0.123152\pi$
$907$			22.7267i			0.754626i	−0.926086	+	0.377313i	$0.876848\pi$
$908$		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
$913$			47.6226i			1.57608i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			15.7189i			0.519084i
$918$		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
$923$	−39.6292			−1.30441
$924$		0			0
$925$		0			0
$926$		0			0
$927$	14.2470	−	16.8945i	0.467932	−	0.554888i
$928$		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
$933$	4.20388	+	9.04420i	0.137629	+	0.296094i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	56.9974i		−	1.86202i	−0.364990	−	0.931011i	$-0.618928\pi$
$937$		−	56.9974i		−	1.86202i	0.364990	−	0.931011i	$-0.381072\pi$
$938$		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
$943$	−66.9488			−2.18015
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−36.5775			−1.18861			−0.594305	−	0.804240i	$-0.702573\pi$
$947$	−36.5775			−1.18861			−0.594305	+	0.804240i	$0.702573\pi$
$948$		0			0
$949$			25.9831i			0.843449i
$950$		0			0
$951$	27.7717	−	12.9087i	0.900560	−	0.418594i
$952$		0			0
$953$	30.3524			0.983211			0.491606	−	0.870818i	$-0.336410\pi$
$953$	30.3524			0.983211			0.491606	+	0.870818i	$0.336410\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−18.0248	+	8.37821i	−0.582660	+	0.270829i
$958$		0			0
$959$	23.1806			0.748540
$960$		0			0
$961$	3.97070			0.128087
$962$		0			0
$963$	15.7442	−	18.6700i	0.507350	−	0.601632i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−25.8174			−0.830232			−0.415116	−	0.909768i	$-0.636259\pi$
$967$	−25.8174			−0.830232			−0.415116	+	0.909768i	$0.636259\pi$
$968$		0			0
$969$	3.18736	+	6.85728i	0.102393	+	0.220287i
$970$		0			0
$971$		−	22.9904i		−	0.737796i	−0.929470	−	0.368898i	$-0.879735\pi$
$971$		−	22.9904i		−	0.737796i	0.929470	−	0.368898i	$-0.120265\pi$
$972$		0			0
$973$	−14.5675			−0.467011
$974$		0			0
$975$		0			0
$976$		0			0
$977$	36.2952			1.16119			0.580593	−	0.814194i	$-0.302821\pi$
$977$	36.2952			1.16119			0.580593	+	0.814194i	$0.302821\pi$
$978$		0			0
$979$	−11.0573			−0.353393
$980$		0			0
$981$	−12.9693	−	10.9369i	−0.414079	−	0.349189i
$982$		0			0
$983$			6.92523i			0.220881i	0.993883	+	0.110440i	$0.0352261\pi$
$983$			6.92523i			0.220881i	−0.993883	+	0.110440i	$0.964774\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	3.61437	−	1.68001i	0.115047	−	0.0534754i
$988$		0			0
$989$	38.6575			1.22924
$990$		0			0
$991$			42.7182i			1.35699i	0.734605	+	0.678495i	$0.237367\pi$
$991$			42.7182i			1.35699i	−0.734605	+	0.678495i	$0.762633\pi$
$992$		0			0
$993$	−5.21367	−	11.2167i	−0.165451	−	0.355950i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−15.7743			−0.499579			−0.249789	−	0.968300i	$-0.580361\pi$
$997$	−15.7743			−0.499579			−0.249789	+	0.968300i	$0.580361\pi$
$998$		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.m.e.1199.10		24
3.2	odd	2	inner	2400.2.m.e.1199.14		24
4.3	odd	2		600.2.m.e.299.6		24
5.2	odd	4		2400.2.b.g.2351.4		12
5.3	odd	4		2400.2.b.h.2351.9		12
5.4	even	2	inner	2400.2.m.e.1199.15		24
8.3	odd	2	inner	2400.2.m.e.1199.9		24
8.5	even	2		600.2.m.e.299.8		24
12.11	even	2		600.2.m.e.299.20		24
15.2	even	4		2400.2.b.g.2351.2		12
15.8	even	4		2400.2.b.h.2351.11		12
15.14	odd	2	inner	2400.2.m.e.1199.11		24
20.3	even	4		600.2.b.g.251.4	yes	12
20.7	even	4		600.2.b.h.251.9	yes	12
20.19	odd	2		600.2.m.e.299.19		24
24.5	odd	2		600.2.m.e.299.18		24
24.11	even	2	inner	2400.2.m.e.1199.13		24
40.3	even	4		2400.2.b.h.2351.10		12
40.13	odd	4		600.2.b.g.251.10	yes	12
40.19	odd	2	inner	2400.2.m.e.1199.16		24
40.27	even	4		2400.2.b.g.2351.3		12
40.29	even	2		600.2.m.e.299.17		24
40.37	odd	4		600.2.b.h.251.3	yes	12
60.23	odd	4		600.2.b.g.251.9	yes	12
60.47	odd	4		600.2.b.h.251.4	yes	12
60.59	even	2		600.2.m.e.299.5		24
120.29	odd	2		600.2.m.e.299.7		24
120.53	even	4		600.2.b.g.251.3	✓	12
120.59	even	2	inner	2400.2.m.e.1199.12		24
120.77	even	4		600.2.b.h.251.10	yes	12
120.83	odd	4		2400.2.b.h.2351.12		12
120.107	odd	4		2400.2.b.g.2351.1		12

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.730070 - 1.57067i) q^{3} +1.25539 q^{7} +(-1.93400 + 2.29339i) q^{9} +O(q^{10})\)	\(q+(-0.730070 - 1.57067i) q^{3} +1.25539 q^{7} +(-1.93400 + 2.29339i) q^{9} -3.02346i q^{11} -5.65509 q^{13} -2.45546 q^{17} +1.77801 q^{19} +(-0.916519 - 1.97179i) q^{21} +8.84074i q^{23} +(5.01411 + 1.36333i) q^{27} +3.79561 q^{29} -5.19897i q^{31} +(-4.74886 + 2.20734i) q^{33} -6.45436 q^{37} +(4.12861 + 8.88227i) q^{39} +7.57276i q^{41} -4.37266i q^{43} +1.83304i q^{47} -5.42401 q^{49} +(1.79266 + 3.85672i) q^{51} +12.0528i q^{53} +(-1.29807 - 2.79266i) q^{57} +4.91093i q^{59} +8.16586i q^{61} +(-2.42791 + 2.87909i) q^{63} -8.50466i q^{67} +(13.8859 - 6.45436i) q^{69} +7.00770 q^{71} -4.59465i q^{73} -3.79561i q^{77} +7.36659i q^{79} +(-1.51932 - 8.87083i) q^{81} -15.7510 q^{83} +(-2.77106 - 5.96165i) q^{87} -3.65716i q^{89} -7.09931 q^{91} +(-8.16586 + 3.79561i) q^{93} +13.8773i q^{97} +(6.93400 + 5.84737i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9}+O(q^{10}) \) 24 * q + 4 * q^9	\( 24 q + 4 q^{9} - 8 q^{19} + 72 q^{49} + 60 q^{51} - 20 q^{81} + 48 q^{91} + 116 q^{99}+O(q^{100}) \) 24 * q + 4 * q^9 - 8 * q^19 + 72 * q^49 + 60 * q^51 - 20 * q^81 + 48 * q^91 + 116 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more