LMFDB - Embedded newform 7500.2.d.g.1249.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1249,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("7500.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.d (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:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.19
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.g.1249.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +0.595901i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +0.595901i q^{7} -1.00000 q^{9} +3.35561 q^{11} +4.76906i q^{13} -7.47993i q^{17} -6.18873 q^{19} -0.595901 q^{21} -4.40898i q^{23} -1.00000i q^{27} +2.76048 q^{29} +4.48868 q^{31} +3.35561i q^{33} +1.30067i q^{37} -4.76906 q^{39} -9.40875 q^{41} +7.59854i q^{43} +4.40189i q^{47} +6.64490 q^{49} +7.47993 q^{51} +8.20731i q^{53} -6.18873i q^{57} +2.22874 q^{59} +12.6096 q^{61} -0.595901i q^{63} +8.35839i q^{67} +4.40898 q^{69} +6.79530 q^{71} +7.31221i q^{73} +1.99961i q^{77} +10.5672 q^{79} +1.00000 q^{81} -4.18160i q^{83} +2.76048i q^{87} -4.25036 q^{89} -2.84189 q^{91} +4.48868i q^{93} +12.2752i q^{97} -3.35561 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{9}+O(q^{10}) $ 24 * q - 24 * q^9	$ 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) $ 24 * q - 24 * q^9 + 4 * q^11 - 20 * q^19 + 16 * q^21 - 16 * q^29 - 4 * q^31 + 20 * q^41 - 56 * q^49 + 16 * q^51 + 4 * q^59 + 68 * q^61 - 36 * q^69 - 12 * q^79 + 24 * q^81 - 20 * q^89 + 40 * q^91 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times$.

$n$	$2501$	$3751$	$6877$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.595901i	0.225229i	0.993639	+	0.112615i	$0.0359225\pi$
$7$	0.595901i	0.225229i	−0.993639	+	0.112615i	$0.964077\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.35561	1.01175	0.505877	−	0.862606i	$-0.331169\pi$
$11$	3.35561	1.01175	0.505877	+	0.862606i	$0.331169\pi$
$12$	0	0
$13$	4.76906i	1.32270i	0.750077	+	0.661350i	$0.230016\pi$
$13$	4.76906i	1.32270i	−0.750077	+	0.661350i	$0.769984\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.47993i	− 1.81415i	−0.420969	−	0.907075i	$-0.638310\pi$
$17$	− 7.47993i	− 1.81415i	0.420969	−	0.907075i	$-0.361690\pi$
$18$	0	0
$19$	−6.18873	−1.41979	−0.709896	−	0.704307i	$-0.751258\pi$
$19$	−6.18873	−1.41979	−0.709896	+	0.704307i	$0.751258\pi$
$20$	0	0
$21$	−0.595901	−0.130036
$22$	0	0
$23$	− 4.40898i	− 0.919337i	−0.888091	−	0.459668i	$-0.847968\pi$
$23$	− 4.40898i	− 0.919337i	0.888091	−	0.459668i	$-0.152032\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	2.76048	0.512608	0.256304	−	0.966596i	$-0.417495\pi$
$29$	2.76048	0.512608	0.256304	+	0.966596i	$0.417495\pi$
$30$	0	0
$31$	4.48868	0.806191	0.403096	−	0.915158i	$-0.367934\pi$
$31$	4.48868	0.806191	0.403096	+	0.915158i	$0.367934\pi$
$32$	0	0
$33$	3.35561i	0.584137i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.30067i	0.213828i	0.994268	+	0.106914i	$0.0340970\pi$
$37$	1.30067i	0.213828i	−0.994268	+	0.106914i	$0.965903\pi$
$38$	0	0
$39$	−4.76906	−0.763661
$40$	0	0
$41$	−9.40875	−1.46940	−0.734700	−	0.678392i	$-0.762677\pi$
$41$	−9.40875	−1.46940	−0.734700	+	0.678392i	$0.762677\pi$
$42$	0	0
$43$	7.59854i	1.15877i	0.815055	+	0.579383i	$0.196706\pi$
$43$	7.59854i	1.15877i	−0.815055	+	0.579383i	$0.803294\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.40189i	0.642082i	0.947065	+	0.321041i	$0.104033\pi$
$47$	4.40189i	0.642082i	−0.947065	+	0.321041i	$0.895967\pi$
$48$	0	0
$49$	6.64490	0.949272
$50$	0	0
$51$	7.47993	1.04740
$52$	0	0
$53$	8.20731i	1.12736i	0.825993	+	0.563680i	$0.190615\pi$
$53$	8.20731i	1.12736i	−0.825993	+	0.563680i	$0.809385\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6.18873i	− 0.819717i
$58$	0	0
$59$	2.22874	0.290158	0.145079	−	0.989420i	$-0.453656\pi$
$59$	2.22874	0.290158	0.145079	+	0.989420i	$0.453656\pi$
$60$	0	0
$61$	12.6096	1.61450	0.807248	−	0.590213i	$-0.200956\pi$
$61$	12.6096	1.61450	0.807248	+	0.590213i	$0.200956\pi$
$62$	0	0
$63$	− 0.595901i	− 0.0750764i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.35839i	1.02114i	0.859836	+	0.510570i	$0.170565\pi$
$67$	8.35839i	1.02114i	−0.859836	+	0.510570i	$0.829435\pi$
$68$	0	0
$69$	4.40898	0.530779
$70$	0	0
$71$	6.79530	0.806454	0.403227	−	0.915100i	$-0.367889\pi$
$71$	6.79530	0.806454	0.403227	+	0.915100i	$0.367889\pi$
$72$	0	0
$73$	7.31221i	0.855830i	0.903819	+	0.427915i	$0.140752\pi$
$73$	7.31221i	0.855830i	−0.903819	+	0.427915i	$0.859248\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.99961i	0.227877i
$78$	0	0
$79$	10.5672	1.18891	0.594454	−	0.804130i	$-0.297368\pi$
$79$	10.5672	1.18891	0.594454	+	0.804130i	$0.297368\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.18160i	− 0.458991i	−0.973310	−	0.229495i	$-0.926292\pi$
$83$	− 4.18160i	− 0.458991i	0.973310	−	0.229495i	$-0.0737076\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.76048i	0.295955i
$88$	0	0
$89$	−4.25036	−0.450538	−0.225269	−	0.974297i	$-0.572326\pi$
$89$	−4.25036	−0.450538	−0.225269	+	0.974297i	$0.572326\pi$
$90$	0	0
$91$	−2.84189	−0.297911
$92$	0	0
$93$	4.48868i	0.465455i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.2752i	1.24636i	0.782078	+	0.623180i	$0.214160\pi$
$97$	12.2752i	1.24636i	−0.782078	+	0.623180i	$0.785840\pi$
$98$	0	0
$99$	−3.35561	−0.337251
$100$	0	0
$101$	−8.00835	−0.796860	−0.398430	−	0.917199i	$-0.630445\pi$
$101$	−8.00835	−0.796860	−0.398430	+	0.917199i	$0.630445\pi$
$102$	0	0
$103$	5.62521i	0.554269i	0.960831	+	0.277134i	$0.0893847\pi$
$103$	5.62521i	0.554269i	−0.960831	+	0.277134i	$0.910615\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.7701i	− 1.33120i	−0.746308	−	0.665601i	$-0.768175\pi$
$107$	− 13.7701i	− 1.33120i	0.746308	−	0.665601i	$-0.231825\pi$
$108$	0	0
$109$	12.8014	1.22616	0.613078	−	0.790023i	$-0.289931\pi$
$109$	12.8014	1.22616	0.613078	+	0.790023i	$0.289931\pi$
$110$	0	0
$111$	−1.30067	−0.123454
$112$	0	0
$113$	− 11.6394i	− 1.09494i	−0.836825	−	0.547471i	$-0.815591\pi$
$113$	− 11.6394i	− 1.09494i	0.836825	−	0.547471i	$-0.184409\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.76906i	− 0.440900i
$118$	0	0
$119$	4.45730	0.408600
$120$	0	0
$121$	0.260113	0.0236466
$122$	0	0
$123$	− 9.40875i	− 0.848358i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.66172i	0.236189i	0.993002	+	0.118095i	$0.0376786\pi$
$127$	2.66172i	0.236189i	−0.993002	+	0.118095i	$0.962321\pi$
$128$	0	0
$129$	−7.59854	−0.669014
$130$	0	0
$131$	9.71046	0.848407	0.424203	−	0.905567i	$-0.360554\pi$
$131$	9.71046	0.848407	0.424203	+	0.905567i	$0.360554\pi$
$132$	0	0
$133$	− 3.68787i	− 0.319779i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.84194i	0.328239i	0.986440	+	0.164119i	$0.0524783\pi$
$137$	3.84194i	0.328239i	−0.986440	+	0.164119i	$0.947522\pi$
$138$	0	0
$139$	4.92765	0.417958	0.208979	−	0.977920i	$-0.432986\pi$
$139$	4.92765	0.417958	0.208979	+	0.977920i	$0.432986\pi$
$140$	0	0
$141$	−4.40189	−0.370706
$142$	0	0
$143$	16.0031i	1.33825i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.64490i	0.548062i
$148$	0	0
$149$	6.59040	0.539907	0.269953	−	0.962873i	$-0.412992\pi$
$149$	6.59040	0.539907	0.269953	+	0.962873i	$0.412992\pi$
$150$	0	0
$151$	−19.5433	−1.59041	−0.795207	−	0.606338i	$-0.792638\pi$
$151$	−19.5433	−1.59041	−0.795207	+	0.606338i	$0.792638\pi$
$152$	0	0
$153$	7.47993i	0.604717i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 0.341995i	− 0.0272942i	−0.999907	−	0.0136471i	$-0.995656\pi$
$157$	− 0.341995i	− 0.0272942i	0.999907	−	0.0136471i	$-0.00434414\pi$
$158$	0	0
$159$	−8.20731	−0.650882
$160$	0	0
$161$	2.62732	0.207062
$162$	0	0
$163$	22.5163i	1.76361i	0.471612	+	0.881806i	$0.343672\pi$
$163$	22.5163i	1.76361i	−0.471612	+	0.881806i	$0.656328\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.31493i	0.333900i	0.985965	+	0.166950i	$0.0533918\pi$
$167$	4.31493i	0.333900i	−0.985965	+	0.166950i	$0.946608\pi$
$168$	0	0
$169$	−9.74397	−0.749536
$170$	0	0
$171$	6.18873	0.473264
$172$	0	0
$173$	17.1688i	1.30532i	0.757652	+	0.652659i	$0.226347\pi$
$173$	17.1688i	1.30532i	−0.757652	+	0.652659i	$0.773653\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.22874i	0.167523i
$178$	0	0
$179$	−1.06360	−0.0794970	−0.0397485	−	0.999210i	$-0.512656\pi$
$179$	−1.06360	−0.0794970	−0.0397485	+	0.999210i	$0.512656\pi$
$180$	0	0
$181$	−12.0267	−0.893936	−0.446968	−	0.894550i	$-0.647496\pi$
$181$	−12.0267	−0.893936	−0.446968	+	0.894550i	$0.647496\pi$
$182$	0	0
$183$	12.6096i	0.932130i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 25.0997i	− 1.83547i
$188$	0	0
$189$	0.595901	0.0433454
$190$	0	0
$191$	−21.9695	−1.58966	−0.794830	−	0.606832i	$-0.792440\pi$
$191$	−21.9695	−1.58966	−0.794830	+	0.606832i	$0.792440\pi$
$192$	0	0
$193$	− 20.4002i	− 1.46844i	−0.678912	−	0.734220i	$-0.737548\pi$
$193$	− 20.4002i	− 1.46844i	0.678912	−	0.734220i	$-0.262452\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.10501i	0.292470i	0.989250	+	0.146235i	$0.0467155\pi$
$197$	4.10501i	0.292470i	−0.989250	+	0.146235i	$0.953284\pi$
$198$	0	0
$199$	−25.5940	−1.81431	−0.907156	−	0.420795i	$-0.861751\pi$
$199$	−25.5940	−1.81431	−0.907156	+	0.420795i	$0.861751\pi$
$200$	0	0
$201$	−8.35839	−0.589555
$202$	0	0
$203$	1.64497i	0.115454i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.40898i	0.306446i
$208$	0	0
$209$	−20.7670	−1.43648
$210$	0	0
$211$	19.5084	1.34301	0.671505	−	0.741000i	$-0.265648\pi$
$211$	19.5084	1.34301	0.671505	+	0.741000i	$0.265648\pi$
$212$	0	0
$213$	6.79530i	0.465606i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.67481i	0.181578i
$218$	0	0
$219$	−7.31221	−0.494113
$220$	0	0
$221$	35.6723	2.39958
$222$	0	0
$223$	2.22827i	0.149216i	0.997213	+	0.0746081i	$0.0237706\pi$
$223$	2.22827i	0.149216i	−0.997213	+	0.0746081i	$0.976229\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4.54796i	− 0.301859i	−0.988545	−	0.150929i	$-0.951773\pi$
$227$	− 4.54796i	− 0.301859i	0.988545	−	0.150929i	$-0.0482266\pi$
$228$	0	0
$229$	−3.41511	−0.225677	−0.112838	−	0.993613i	$-0.535994\pi$
$229$	−3.41511	−0.225677	−0.112838	+	0.993613i	$0.535994\pi$
$230$	0	0
$231$	−1.99961	−0.131565
$232$	0	0
$233$	16.7395i	1.09664i	0.836268	+	0.548320i	$0.184733\pi$
$233$	16.7395i	1.09664i	−0.836268	+	0.548320i	$0.815267\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.5672i	0.686417i
$238$	0	0
$239$	22.3643	1.44663	0.723313	−	0.690520i	$-0.242618\pi$
$239$	22.3643	1.44663	0.723313	+	0.690520i	$0.242618\pi$
$240$	0	0
$241$	0.265356	0.0170931	0.00854655	−	0.999963i	$-0.497280\pi$
$241$	0.265356	0.0170931	0.00854655	+	0.999963i	$0.497280\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 29.5144i	− 1.87796i
$248$	0	0
$249$	4.18160	0.264998
$250$	0	0
$251$	29.0694	1.83484	0.917422	−	0.397915i	$-0.130266\pi$
$251$	29.0694	1.83484	0.917422	+	0.397915i	$0.130266\pi$
$252$	0	0
$253$	− 14.7948i	− 0.930143i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.05952i	0.440361i	0.975459	+	0.220180i	$0.0706646\pi$
$257$	7.05952i	0.440361i	−0.975459	+	0.220180i	$0.929335\pi$
$258$	0	0
$259$	−0.775067	−0.0481603
$260$	0	0
$261$	−2.76048	−0.170869
$262$	0	0
$263$	1.38649i	0.0854949i	0.999086	+	0.0427475i	$0.0136111\pi$
$263$	1.38649i	0.0854949i	−0.999086	+	0.0427475i	$0.986389\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.25036i	− 0.260118i
$268$	0	0
$269$	14.1752	0.864280	0.432140	−	0.901807i	$-0.357759\pi$
$269$	14.1752	0.864280	0.432140	+	0.901807i	$0.357759\pi$
$270$	0	0
$271$	15.0411	0.913681	0.456841	−	0.889549i	$-0.348981\pi$
$271$	15.0411	0.913681	0.456841	+	0.889549i	$0.348981\pi$
$272$	0	0
$273$	− 2.84189i	− 0.171999i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 12.7933i	− 0.768677i	−0.923192	−	0.384338i	$-0.874430\pi$
$277$	− 12.7933i	− 0.768677i	0.923192	−	0.384338i	$-0.125570\pi$
$278$	0	0
$279$	−4.48868	−0.268730
$280$	0	0
$281$	−15.3058	−0.913068	−0.456534	−	0.889706i	$-0.650909\pi$
$281$	−15.3058	−0.913068	−0.456534	+	0.889706i	$0.650909\pi$
$282$	0	0
$283$	23.3502i	1.38802i	0.719964	+	0.694011i	$0.244158\pi$
$283$	23.3502i	1.38802i	−0.719964	+	0.694011i	$0.755842\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 5.60668i	− 0.330952i
$288$	0	0
$289$	−38.9494	−2.29114
$290$	0	0
$291$	−12.2752	−0.719587
$292$	0	0
$293$	6.36651i	0.371936i	0.982556	+	0.185968i	$0.0595420\pi$
$293$	6.36651i	0.371936i	−0.982556	+	0.185968i	$0.940458\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3.35561i	− 0.194712i
$298$	0	0
$299$	21.0267	1.21601
$300$	0	0
$301$	−4.52797	−0.260988
$302$	0	0
$303$	− 8.00835i	− 0.460068i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	27.2317i	1.55419i	0.629382	+	0.777096i	$0.283308\pi$
$307$	27.2317i	1.55419i	−0.629382	+	0.777096i	$0.716692\pi$
$308$	0	0
$309$	−5.62521	−0.320007
$310$	0	0
$311$	17.1583	0.972961	0.486480	−	0.873692i	$-0.338281\pi$
$311$	17.1583	0.972961	0.486480	+	0.873692i	$0.338281\pi$
$312$	0	0
$313$	20.2659i	1.14550i	0.819731	+	0.572749i	$0.194123\pi$
$313$	20.2659i	1.14550i	−0.819731	+	0.572749i	$0.805877\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.0153i	1.18034i	0.807279	+	0.590169i	$0.200939\pi$
$317$	21.0153i	1.18034i	−0.807279	+	0.590169i	$0.799061\pi$
$318$	0	0
$319$	9.26309	0.518634
$320$	0	0
$321$	13.7701	0.768570
$322$	0	0
$323$	46.2913i	2.57572i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.8014i	0.707921i
$328$	0	0
$329$	−2.62309	−0.144616
$330$	0	0
$331$	−12.7425	−0.700391	−0.350195	−	0.936677i	$-0.613885\pi$
$331$	−12.7425	−0.700391	−0.350195	+	0.936677i	$0.613885\pi$
$332$	0	0
$333$	− 1.30067i	− 0.0712760i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 12.3467i	− 0.672567i	−0.941761	−	0.336284i	$-0.890830\pi$
$337$	− 12.3467i	− 0.672567i	0.941761	−	0.336284i	$-0.109170\pi$
$338$	0	0
$339$	11.6394	0.632165
$340$	0	0
$341$	15.0623	0.815668
$342$	0	0
$343$	8.13100i	0.439033i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 13.1617i	− 0.706555i	−0.935519	−	0.353277i	$-0.885067\pi$
$347$	− 13.1617i	− 0.706555i	0.935519	−	0.353277i	$-0.114933\pi$
$348$	0	0
$349$	−20.8060	−1.11372	−0.556861	−	0.830606i	$-0.687994\pi$
$349$	−20.8060	−1.11372	−0.556861	+	0.830606i	$0.687994\pi$
$350$	0	0
$351$	4.76906	0.254554
$352$	0	0
$353$	28.7945i	1.53258i	0.642496	+	0.766289i	$0.277899\pi$
$353$	28.7945i	1.53258i	−0.642496	+	0.766289i	$0.722101\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.45730i	0.235905i
$358$	0	0
$359$	33.3278	1.75897	0.879487	−	0.475923i	$-0.157886\pi$
$359$	33.3278	1.75897	0.879487	+	0.475923i	$0.157886\pi$
$360$	0	0
$361$	19.3004	1.01581
$362$	0	0
$363$	0.260113i	0.0136524i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.89727i	0.0990366i	0.998773	+	0.0495183i	$0.0157686\pi$
$367$	1.89727i	0.0990366i	−0.998773	+	0.0495183i	$0.984231\pi$
$368$	0	0
$369$	9.40875	0.489800
$370$	0	0
$371$	−4.89074	−0.253914
$372$	0	0
$373$	30.8908i	1.59947i	0.600355	+	0.799733i	$0.295026\pi$
$373$	30.8908i	1.59947i	−0.600355	+	0.799733i	$0.704974\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.1649i	0.678027i
$378$	0	0
$379$	−22.5547	−1.15856	−0.579279	−	0.815129i	$-0.696666\pi$
$379$	−22.5547	−1.15856	−0.579279	+	0.815129i	$0.696666\pi$
$380$	0	0
$381$	−2.66172	−0.136364
$382$	0	0
$383$	− 4.34373i	− 0.221954i	−0.993823	−	0.110977i	$-0.964602\pi$
$383$	− 4.34373i	− 0.221954i	0.993823	−	0.110977i	$-0.0353980\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 7.59854i	− 0.386255i
$388$	0	0
$389$	18.5459	0.940313	0.470156	−	0.882583i	$-0.344198\pi$
$389$	18.5459	0.940313	0.470156	+	0.882583i	$0.344198\pi$
$390$	0	0
$391$	−32.9789	−1.66782
$392$	0	0
$393$	9.71046i	0.489828i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 0.706994i	− 0.0354830i	−0.999843	−	0.0177415i	$-0.994352\pi$
$397$	− 0.706994i	− 0.0354830i	0.999843	−	0.0177415i	$-0.00564759\pi$
$398$	0	0
$399$	3.68787	0.184624
$400$	0	0
$401$	−24.8304	−1.23997	−0.619986	−	0.784613i	$-0.712862\pi$
$401$	−24.8304	−1.23997	−0.619986	+	0.784613i	$0.712862\pi$
$402$	0	0
$403$	21.4068i	1.06635i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.36453i	0.216341i
$408$	0	0
$409$	0.730007	0.0360965	0.0180483	−	0.999837i	$-0.494255\pi$
$409$	0.730007	0.0360965	0.0180483	+	0.999837i	$0.494255\pi$
$410$	0	0
$411$	−3.84194	−0.189509
$412$	0	0
$413$	1.32811i	0.0653520i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.92765i	0.241308i
$418$	0	0
$419$	12.9094	0.630666	0.315333	−	0.948981i	$-0.397884\pi$
$419$	12.9094	0.630666	0.315333	+	0.948981i	$0.397884\pi$
$420$	0	0
$421$	38.1301	1.85835	0.929173	−	0.369644i	$-0.120520\pi$
$421$	38.1301	1.85835	0.929173	+	0.369644i	$0.120520\pi$
$422$	0	0
$423$	− 4.40189i	− 0.214027i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.51408i	0.363632i
$428$	0	0
$429$	−16.0031	−0.772638
$430$	0	0
$431$	−36.1887	−1.74315	−0.871575	−	0.490262i	$-0.836901\pi$
$431$	−36.1887	−1.74315	−0.871575	+	0.490262i	$0.836901\pi$
$432$	0	0
$433$	21.7487i	1.04518i	0.852586	+	0.522588i	$0.175033\pi$
$433$	21.7487i	1.04518i	−0.852586	+	0.522588i	$0.824967\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	27.2860i	1.30527i
$438$	0	0
$439$	39.3586	1.87848	0.939242	−	0.343255i	$-0.111529\pi$
$439$	39.3586	1.87848	0.939242	+	0.343255i	$0.111529\pi$
$440$	0	0
$441$	−6.64490	−0.316424
$442$	0	0
$443$	27.4919i	1.30618i	0.757281	+	0.653089i	$0.226527\pi$
$443$	27.4919i	1.30618i	−0.757281	+	0.653089i	$0.773473\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.59040i	0.311715i
$448$	0	0
$449$	−3.51087	−0.165688	−0.0828441	−	0.996563i	$-0.526400\pi$
$449$	−3.51087	−0.165688	−0.0828441	+	0.996563i	$0.526400\pi$
$450$	0	0
$451$	−31.5721	−1.48667
$452$	0	0
$453$	− 19.5433i	− 0.918226i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 10.1677i	− 0.475624i	−0.971311	−	0.237812i	$-0.923570\pi$
$457$	− 10.1677i	− 0.475624i	0.971311	−	0.237812i	$-0.0764302\pi$
$458$	0	0
$459$	−7.47993	−0.349133
$460$	0	0
$461$	0.934168	0.0435085	0.0217543	−	0.999763i	$-0.493075\pi$
$461$	0.934168	0.0435085	0.0217543	+	0.999763i	$0.493075\pi$
$462$	0	0
$463$	− 4.88586i	− 0.227065i	−0.993534	−	0.113533i	$-0.963783\pi$
$463$	− 4.88586i	− 0.227065i	0.993534	−	0.113533i	$-0.0362166\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 25.8046i	− 1.19410i	−0.802206	−	0.597048i	$-0.796340\pi$
$467$	− 25.8046i	− 1.19410i	0.802206	−	0.597048i	$-0.203660\pi$
$468$	0	0
$469$	−4.98077	−0.229990
$470$	0	0
$471$	0.341995	0.0157583
$472$	0	0
$473$	25.4977i	1.17239i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 8.20731i	− 0.375787i
$478$	0	0
$479$	16.9156	0.772893	0.386447	−	0.922312i	$-0.373702\pi$
$479$	16.9156	0.772893	0.386447	+	0.922312i	$0.373702\pi$
$480$	0	0
$481$	−6.20296	−0.282830
$482$	0	0
$483$	2.62732i	0.119547i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.3876i	0.742592i	0.928515	+	0.371296i	$0.121086\pi$
$487$	16.3876i	0.742592i	−0.928515	+	0.371296i	$0.878914\pi$
$488$	0	0
$489$	−22.5163	−1.01822
$490$	0	0
$491$	−10.2714	−0.463541	−0.231771	−	0.972770i	$-0.574452\pi$
$491$	−10.2714	−0.463541	−0.231771	+	0.972770i	$0.574452\pi$
$492$	0	0
$493$	− 20.6482i	− 0.929948i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.04932i	0.181637i
$498$	0	0
$499$	16.5015	0.738707	0.369354	−	0.929289i	$-0.379579\pi$
$499$	16.5015	0.738707	0.369354	+	0.929289i	$0.379579\pi$
$500$	0	0
$501$	−4.31493	−0.192777
$502$	0	0
$503$	7.60033i	0.338882i	0.985540	+	0.169441i	$0.0541962\pi$
$503$	7.60033i	0.338882i	−0.985540	+	0.169441i	$0.945804\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.74397i	− 0.432745i
$508$	0	0
$509$	21.9513	0.972976	0.486488	−	0.873687i	$-0.338278\pi$
$509$	21.9513	0.972976	0.486488	+	0.873687i	$0.338278\pi$
$510$	0	0
$511$	−4.35735	−0.192758
$512$	0	0
$513$	6.18873i	0.273239i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.7710i	0.649629i
$518$	0	0
$519$	−17.1688	−0.753626
$520$	0	0
$521$	−36.8379	−1.61390	−0.806949	−	0.590621i	$-0.798883\pi$
$521$	−36.8379	−1.61390	−0.806949	+	0.590621i	$0.798883\pi$
$522$	0	0
$523$	− 41.5997i	− 1.81903i	−0.415675	−	0.909513i	$-0.636455\pi$
$523$	− 41.5997i	− 1.81903i	0.415675	−	0.909513i	$-0.363545\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 33.5751i	− 1.46255i
$528$	0	0
$529$	3.56085	0.154820
$530$	0	0
$531$	−2.22874	−0.0967193
$532$	0	0
$533$	− 44.8709i	− 1.94358i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.06360i	− 0.0458976i
$538$	0	0
$539$	22.2977	0.960430
$540$	0	0
$541$	−15.5213	−0.667313	−0.333656	−	0.942695i	$-0.608283\pi$
$541$	−15.5213	−0.667313	−0.333656	+	0.942695i	$0.608283\pi$
$542$	0	0
$543$	− 12.0267i	− 0.516114i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 23.3039i	− 0.996403i	−0.867061	−	0.498202i	$-0.833994\pi$
$547$	− 23.3039i	− 0.996403i	0.867061	−	0.498202i	$-0.166006\pi$
$548$	0	0
$549$	−12.6096	−0.538165
$550$	0	0
$551$	−17.0839	−0.727797
$552$	0	0
$553$	6.29703i	0.267777i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.0914i	0.469959i	0.972000	+	0.234979i	$0.0755023\pi$
$557$	11.0914i	0.469959i	−0.972000	+	0.234979i	$0.924498\pi$
$558$	0	0
$559$	−36.2379	−1.53270
$560$	0	0
$561$	25.0997	1.05971
$562$	0	0
$563$	− 14.3052i	− 0.602890i	−0.953484	−	0.301445i	$-0.902531\pi$
$563$	− 14.3052i	− 0.602890i	0.953484	−	0.301445i	$-0.0974690\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.595901i	0.0250255i
$568$	0	0
$569$	28.7743	1.20628	0.603141	−	0.797634i	$-0.293915\pi$
$569$	28.7743	1.20628	0.603141	+	0.797634i	$0.293915\pi$
$570$	0	0
$571$	−13.0394	−0.545683	−0.272842	−	0.962059i	$-0.587963\pi$
$571$	−13.0394	−0.545683	−0.272842	+	0.962059i	$0.587963\pi$
$572$	0	0
$573$	− 21.9695i	− 0.917791i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.79973i	0.0749239i	0.999298	+	0.0374619i	$0.0119273\pi$
$577$	1.79973i	0.0749239i	−0.999298	+	0.0374619i	$0.988073\pi$
$578$	0	0
$579$	20.4002	0.847804
$580$	0	0
$581$	2.49182	0.103378
$582$	0	0
$583$	27.5405i	1.14061i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 40.5964i	− 1.67559i	−0.545982	−	0.837797i	$-0.683843\pi$
$587$	− 40.5964i	− 1.67559i	0.545982	−	0.837797i	$-0.316157\pi$
$588$	0	0
$589$	−27.7792	−1.14462
$590$	0	0
$591$	−4.10501	−0.168857
$592$	0	0
$593$	29.4991i	1.21138i	0.795700	+	0.605690i	$0.207103\pi$
$593$	29.4991i	1.21138i	−0.795700	+	0.605690i	$0.792897\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 25.5940i	− 1.04749i
$598$	0	0
$599$	8.91418	0.364224	0.182112	−	0.983278i	$-0.441707\pi$
$599$	8.91418	0.364224	0.182112	+	0.983278i	$0.441707\pi$
$600$	0	0
$601$	16.6757	0.680217	0.340109	−	0.940386i	$-0.389536\pi$
$601$	16.6757	0.680217	0.340109	+	0.940386i	$0.389536\pi$
$602$	0	0
$603$	− 8.35839i	− 0.340380i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 26.9626i	− 1.09438i	−0.837008	−	0.547190i	$-0.815698\pi$
$607$	− 26.9626i	− 1.09438i	0.837008	−	0.547190i	$-0.184302\pi$
$608$	0	0
$609$	−1.64497	−0.0666576
$610$	0	0
$611$	−20.9929	−0.849282
$612$	0	0
$613$	16.5230i	0.667356i	0.942687	+	0.333678i	$0.108290\pi$
$613$	16.5230i	0.667356i	−0.942687	+	0.333678i	$0.891710\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 22.9879i	− 0.925456i	−0.886500	−	0.462728i	$-0.846871\pi$
$617$	− 22.9879i	− 0.925456i	0.886500	−	0.462728i	$-0.153129\pi$
$618$	0	0
$619$	−49.3165	−1.98220	−0.991099	−	0.133124i	$-0.957499\pi$
$619$	−49.3165	−1.98220	−0.991099	+	0.133124i	$0.957499\pi$
$620$	0	0
$621$	−4.40898	−0.176926
$622$	0	0
$623$	− 2.53279i	− 0.101474i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 20.7670i	− 0.829352i
$628$	0	0
$629$	9.72889	0.387916
$630$	0	0
$631$	24.1782	0.962519	0.481260	−	0.876578i	$-0.340179\pi$
$631$	24.1782	0.962519	0.481260	+	0.876578i	$0.340179\pi$
$632$	0	0
$633$	19.5084i	0.775387i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	31.6900i	1.25560i
$638$	0	0
$639$	−6.79530	−0.268818
$640$	0	0
$641$	−49.3663	−1.94985	−0.974927	−	0.222526i	$-0.928570\pi$
$641$	−49.3663	−1.94985	−0.974927	+	0.222526i	$0.928570\pi$
$642$	0	0
$643$	3.97743i	0.156854i	0.996920	+	0.0784272i	$0.0249898\pi$
$643$	3.97743i	0.156854i	−0.996920	+	0.0784272i	$0.975010\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.01441i	− 0.0398807i	−0.999801	−	0.0199404i	$-0.993652\pi$
$647$	− 1.01441i	− 0.0398807i	0.999801	−	0.0199404i	$-0.00634764\pi$
$648$	0	0
$649$	7.47879	0.293568
$650$	0	0
$651$	−2.67481	−0.104834
$652$	0	0
$653$	− 8.17155i	− 0.319778i	−0.987135	−	0.159889i	$-0.948886\pi$
$653$	− 8.17155i	− 0.319778i	0.987135	−	0.159889i	$-0.0511136\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 7.31221i	− 0.285277i
$658$	0	0
$659$	−7.68859	−0.299505	−0.149752	−	0.988724i	$-0.547848\pi$
$659$	−7.68859	−0.299505	−0.149752	+	0.988724i	$0.547848\pi$
$660$	0	0
$661$	−36.2000	−1.40802	−0.704008	−	0.710192i	$-0.748608\pi$
$661$	−36.2000	−1.40802	−0.704008	+	0.710192i	$0.748608\pi$
$662$	0	0
$663$	35.6723i	1.38540i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12.1709i	− 0.471260i
$668$	0	0
$669$	−2.22827	−0.0861500
$670$	0	0
$671$	42.3129	1.63347
$672$	0	0
$673$	− 30.1819i	− 1.16343i	−0.813394	−	0.581714i	$-0.802382\pi$
$673$	− 30.1819i	− 1.16343i	0.813394	−	0.581714i	$-0.197618\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	47.0154i	1.80695i	0.428644	+	0.903473i	$0.358991\pi$
$677$	47.0154i	1.80695i	−0.428644	+	0.903473i	$0.641009\pi$
$678$	0	0
$679$	−7.31482	−0.280717
$680$	0	0
$681$	4.54796	0.174278
$682$	0	0
$683$	2.19204i	0.0838762i	0.999120	+	0.0419381i	$0.0133532\pi$
$683$	2.19204i	0.0838762i	−0.999120	+	0.0419381i	$0.986647\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.41511i	− 0.130295i
$688$	0	0
$689$	−39.1412	−1.49116
$690$	0	0
$691$	1.78521	0.0679127	0.0339563	−	0.999423i	$-0.489189\pi$
$691$	1.78521	0.0679127	0.0339563	+	0.999423i	$0.489189\pi$
$692$	0	0
$693$	− 1.99961i	− 0.0759589i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	70.3768i	2.66571i
$698$	0	0
$699$	−16.7395	−0.633146
$700$	0	0
$701$	−11.0728	−0.418215	−0.209107	−	0.977893i	$-0.567056\pi$
$701$	−11.0728	−0.418215	−0.209107	+	0.977893i	$0.567056\pi$
$702$	0	0
$703$	− 8.04947i	− 0.303591i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.77218i	− 0.179476i
$708$	0	0
$709$	−19.9129	−0.747846	−0.373923	−	0.927460i	$-0.621988\pi$
$709$	−19.9129	−0.747846	−0.373923	+	0.927460i	$0.621988\pi$
$710$	0	0
$711$	−10.5672	−0.396303
$712$	0	0
$713$	− 19.7905i	− 0.741162i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	22.3643i	0.835210i
$718$	0	0
$719$	−34.7804	−1.29709	−0.648546	−	0.761176i	$-0.724623\pi$
$719$	−34.7804	−1.29709	−0.648546	+	0.761176i	$0.724623\pi$
$720$	0	0
$721$	−3.35207	−0.124837
$722$	0	0
$723$	0.265356i	0.00986871i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 15.9677i	− 0.592209i	−0.955156	−	0.296105i	$-0.904312\pi$
$727$	− 15.9677i	− 0.592209i	0.955156	−	0.296105i	$-0.0956877\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	56.8365	2.10218
$732$	0	0
$733$	10.1425i	0.374621i	0.982301	+	0.187310i	$0.0599770\pi$
$733$	10.1425i	0.374621i	−0.982301	+	0.187310i	$0.940023\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	28.0475i	1.03314i
$738$	0	0
$739$	−3.27848	−0.120601	−0.0603004	−	0.998180i	$-0.519206\pi$
$739$	−3.27848	−0.120601	−0.0603004	+	0.998180i	$0.519206\pi$
$740$	0	0
$741$	29.5144	1.08424
$742$	0	0
$743$	− 38.7278i	− 1.42078i	−0.703806	−	0.710392i	$-0.748518\pi$
$743$	− 38.7278i	− 1.42078i	0.703806	−	0.710392i	$-0.251482\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.18160i	0.152997i
$748$	0	0
$749$	8.20559	0.299826
$750$	0	0
$751$	13.0211	0.475146	0.237573	−	0.971370i	$-0.423648\pi$
$751$	13.0211	0.475146	0.237573	+	0.971370i	$0.423648\pi$
$752$	0	0
$753$	29.0694i	1.05935i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33.4057i	1.21415i	0.794645	+	0.607075i	$0.207657\pi$
$757$	33.4057i	1.21415i	−0.794645	+	0.607075i	$0.792343\pi$
$758$	0	0
$759$	14.7948	0.537018
$760$	0	0
$761$	29.7319	1.07778	0.538890	−	0.842376i	$-0.318844\pi$
$761$	29.7319	1.07778	0.538890	+	0.842376i	$0.318844\pi$
$762$	0	0
$763$	7.62838i	0.276166i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.6290i	0.383792i
$768$	0	0
$769$	5.45427	0.196686	0.0983431	−	0.995153i	$-0.468646\pi$
$769$	5.45427	0.196686	0.0983431	+	0.995153i	$0.468646\pi$
$770$	0	0
$771$	−7.05952	−0.254242
$772$	0	0
$773$	29.0843i	1.04609i	0.852306	+	0.523044i	$0.175204\pi$
$773$	29.0843i	1.04609i	−0.852306	+	0.523044i	$0.824796\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 0.775067i	− 0.0278054i
$778$	0	0
$779$	58.2282	2.08624
$780$	0	0
$781$	22.8024	0.815933
$782$	0	0
$783$	− 2.76048i	− 0.0986515i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.1141i	1.21604i	0.793923	+	0.608018i	$0.208035\pi$
$787$	34.1141i	1.21604i	−0.793923	+	0.608018i	$0.791965\pi$
$788$	0	0
$789$	−1.38649	−0.0493605
$790$	0	0
$791$	6.93592	0.246613
$792$	0	0
$793$	60.1361i	2.13549i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 24.4560i	− 0.866276i	−0.901328	−	0.433138i	$-0.857406\pi$
$797$	− 24.4560i	− 0.866276i	0.901328	−	0.433138i	$-0.142594\pi$
$798$	0	0
$799$	32.9259	1.16483
$800$	0	0
$801$	4.25036	0.150179
$802$	0	0
$803$	24.5369i	0.865889i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.1752i	0.498992i
$808$	0	0
$809$	52.2487	1.83697	0.918483	−	0.395460i	$-0.129415\pi$
$809$	52.2487	1.83697	0.918483	+	0.395460i	$0.129415\pi$
$810$	0	0
$811$	17.6487	0.619728	0.309864	−	0.950781i	$-0.399716\pi$
$811$	17.6487	0.619728	0.309864	+	0.950781i	$0.399716\pi$
$812$	0	0
$813$	15.0411i	0.527514i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 47.0253i	− 1.64521i
$818$	0	0
$819$	2.84189	0.0993036
$820$	0	0
$821$	−34.1306	−1.19117	−0.595584	−	0.803293i	$-0.703079\pi$
$821$	−34.1306	−1.19117	−0.595584	+	0.803293i	$0.703079\pi$
$822$	0	0
$823$	− 45.5282i	− 1.58701i	−0.608562	−	0.793506i	$-0.708253\pi$
$823$	− 45.5282i	− 1.58701i	0.608562	−	0.793506i	$-0.291747\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 19.0967i	− 0.664057i	−0.943269	−	0.332029i	$-0.892267\pi$
$827$	− 19.0967i	− 0.664057i	0.943269	−	0.332029i	$-0.107733\pi$
$828$	0	0
$829$	−11.3819	−0.395309	−0.197654	−	0.980272i	$-0.563332\pi$
$829$	−11.3819	−0.395309	−0.197654	+	0.980272i	$0.563332\pi$
$830$	0	0
$831$	12.7933	0.443796
$832$	0	0
$833$	− 49.7034i	− 1.72212i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4.48868i	− 0.155152i
$838$	0	0
$839$	−25.7924	−0.890452	−0.445226	−	0.895418i	$-0.646877\pi$
$839$	−25.7924	−0.890452	−0.445226	+	0.895418i	$0.646877\pi$
$840$	0	0
$841$	−21.3798	−0.737233
$842$	0	0
$843$	− 15.3058i	− 0.527160i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.155001i	0.00532591i
$848$	0	0
$849$	−23.3502	−0.801375
$850$	0	0
$851$	5.73461	0.196580
$852$	0	0
$853$	15.7410i	0.538961i	0.963006	+	0.269480i	$0.0868519\pi$
$853$	15.7410i	0.538961i	−0.963006	+	0.269480i	$0.913148\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.4948i	1.58823i	0.607767	+	0.794116i	$0.292066\pi$
$857$	46.4948i	1.58823i	−0.607767	+	0.794116i	$0.707934\pi$
$858$	0	0
$859$	−12.1958	−0.416116	−0.208058	−	0.978116i	$-0.566714\pi$
$859$	−12.1958	−0.416116	−0.208058	+	0.978116i	$0.566714\pi$
$860$	0	0
$861$	5.60668	0.191075
$862$	0	0
$863$	− 45.3926i	− 1.54518i	−0.634904	−	0.772591i	$-0.718960\pi$
$863$	− 45.3926i	− 1.54518i	0.634904	−	0.772591i	$-0.281040\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 38.9494i	− 1.32279i
$868$	0	0
$869$	35.4596	1.20288
$870$	0	0
$871$	−39.8617	−1.35066
$872$	0	0
$873$	− 12.2752i	− 0.415454i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.0305i	0.642615i	0.946975	+	0.321308i	$0.104122\pi$
$877$	19.0305i	0.642615i	−0.946975	+	0.321308i	$0.895878\pi$
$878$	0	0
$879$	−6.36651	−0.214737
$880$	0	0
$881$	−15.2097	−0.512426	−0.256213	−	0.966620i	$-0.582475\pi$
$881$	−15.2097	−0.512426	−0.256213	+	0.966620i	$0.582475\pi$
$882$	0	0
$883$	− 48.3181i	− 1.62603i	−0.582240	−	0.813017i	$-0.697824\pi$
$883$	− 48.3181i	− 1.62603i	0.582240	−	0.813017i	$-0.302176\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.4010i	1.08792i	0.839111	+	0.543960i	$0.183076\pi$
$887$	32.4010i	1.08792i	−0.839111	+	0.543960i	$0.816924\pi$
$888$	0	0
$889$	−1.58612	−0.0531968
$890$	0	0
$891$	3.35561	0.112417
$892$	0	0
$893$	− 27.2421i	− 0.911622i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	21.0267i	0.702062i
$898$	0	0
$899$	12.3909	0.413260
$900$	0	0
$901$	61.3901	2.04520
$902$	0	0
$903$	− 4.52797i	− 0.150681i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 30.8525i	− 1.02444i	−0.858854	−	0.512220i	$-0.828823\pi$
$907$	− 30.8525i	− 1.02444i	0.858854	−	0.512220i	$-0.171177\pi$
$908$	0	0
$909$	8.00835	0.265620
$910$	0	0
$911$	36.2181	1.19996	0.599980	−	0.800015i	$-0.295175\pi$
$911$	36.2181	1.19996	0.599980	+	0.800015i	$0.295175\pi$
$912$	0	0
$913$	− 14.0318i	− 0.464386i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.78647i	0.191086i
$918$	0	0
$919$	37.9977	1.25343	0.626715	−	0.779249i	$-0.284399\pi$
$919$	37.9977	1.25343	0.626715	+	0.779249i	$0.284399\pi$
$920$	0	0
$921$	−27.2317	−0.897314
$922$	0	0
$923$	32.4072i	1.06670i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 5.62521i	− 0.184756i
$928$	0	0
$929$	10.4600	0.343180	0.171590	−	0.985168i	$-0.445110\pi$
$929$	10.4600	0.343180	0.171590	+	0.985168i	$0.445110\pi$
$930$	0	0
$931$	−41.1235	−1.34777
$932$	0	0
$933$	17.1583i	0.561739i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	33.7638i	1.10301i	0.834170	+	0.551507i	$0.185947\pi$
$937$	33.7638i	1.10301i	−0.834170	+	0.551507i	$0.814053\pi$
$938$	0	0
$939$	−20.2659	−0.661354
$940$	0	0
$941$	2.23047	0.0727113	0.0363556	−	0.999339i	$-0.488425\pi$
$941$	2.23047	0.0727113	0.0363556	+	0.999339i	$0.488425\pi$
$942$	0	0
$943$	41.4830i	1.35087i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.65150i	0.248640i	0.992242	+	0.124320i	$0.0396750\pi$
$947$	7.65150i	0.248640i	−0.992242	+	0.124320i	$0.960325\pi$
$948$	0	0
$949$	−34.8724	−1.13201
$950$	0	0
$951$	−21.0153	−0.681469
$952$	0	0
$953$	11.6526i	0.377465i	0.982029	+	0.188733i	$0.0604380\pi$
$953$	11.6526i	0.377465i	−0.982029	+	0.188733i	$0.939562\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.26309i	0.299433i
$958$	0	0
$959$	−2.28941	−0.0739290
$960$	0	0
$961$	−10.8517	−0.350055
$962$	0	0
$963$	13.7701i	0.443734i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 33.7782i	− 1.08623i	−0.839657	−	0.543117i	$-0.817244\pi$
$967$	− 33.7782i	− 1.08623i	0.839657	−	0.543117i	$-0.182756\pi$
$968$	0	0
$969$	−46.2913	−1.48709
$970$	0	0
$971$	33.7855	1.08423	0.542114	−	0.840305i	$-0.317624\pi$
$971$	33.7855	1.08423	0.542114	+	0.840305i	$0.317624\pi$
$972$	0	0
$973$	2.93639i	0.0941363i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 3.77099i	− 0.120645i	−0.998179	−	0.0603223i	$-0.980787\pi$
$977$	− 3.77099i	− 0.120645i	0.998179	−	0.0603223i	$-0.0192128\pi$
$978$	0	0
$979$	−14.2626	−0.455833
$980$	0	0
$981$	−12.8014	−0.408718
$982$	0	0
$983$	− 42.3040i	− 1.34929i	−0.738143	−	0.674644i	$-0.764297\pi$
$983$	− 42.3040i	− 1.34929i	0.738143	−	0.674644i	$-0.235703\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.62309i	− 0.0834939i
$988$	0	0
$989$	33.5018	1.06530
$990$	0	0
$991$	31.9881	1.01614	0.508068	−	0.861317i	$-0.330360\pi$
$991$	31.9881	1.01614	0.508068	+	0.861317i	$0.330360\pi$
$992$	0	0
$993$	− 12.7425i	− 0.404371i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 16.7077i	− 0.529138i	−0.964367	−	0.264569i	$-0.914770\pi$
$997$	− 16.7077i	− 0.529138i	0.964367	−	0.264569i	$-0.0852296\pi$
$998$	0	0
$999$	1.30067	0.0411512

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.g.1249.19		24
5.2	odd	4		7500.2.a.n.1.6		12
5.3	odd	4		7500.2.a.m.1.7		12
5.4	even	2	inner	7500.2.d.g.1249.6		24
25.3	odd	20		1500.2.m.d.1201.4		24
25.4	even	10		1500.2.o.c.49.5		24
25.6	even	5		1500.2.o.c.949.5		24
25.8	odd	20		1500.2.m.d.301.4		24
25.17	odd	20		1500.2.m.c.301.3		24
25.19	even	10		300.2.o.a.289.1	yes	24
25.21	even	5		300.2.o.a.109.1	✓	24
25.22	odd	20		1500.2.m.c.1201.3		24
75.44	odd	10		900.2.w.c.289.6		24
75.71	odd	10		900.2.w.c.109.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.109.1	✓	24	25.21	even	5
300.2.o.a.289.1	yes	24	25.19	even	10
900.2.w.c.109.6		24	75.71	odd	10
900.2.w.c.289.6		24	75.44	odd	10
1500.2.m.c.301.3		24	25.17	odd	20
1500.2.m.c.1201.3		24	25.22	odd	20
1500.2.m.d.301.4		24	25.8	odd	20
1500.2.m.d.1201.4		24	25.3	odd	20
1500.2.o.c.49.5		24	25.4	even	10
1500.2.o.c.949.5		24	25.6	even	5
7500.2.a.m.1.7		12	5.3	odd	4
7500.2.a.n.1.6		12	5.2	odd	4
7500.2.d.g.1249.6		24	5.4	even	2	inner
7500.2.d.g.1249.19		24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +0.595901i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +0.595901i q^{7} -1.00000 q^{9} +3.35561 q^{11} +4.76906i q^{13} -7.47993i q^{17} -6.18873 q^{19} -0.595901 q^{21} -4.40898i q^{23} -1.00000i q^{27} +2.76048 q^{29} +4.48868 q^{31} +3.35561i q^{33} +1.30067i q^{37} -4.76906 q^{39} -9.40875 q^{41} +7.59854i q^{43} +4.40189i q^{47} +6.64490 q^{49} +7.47993 q^{51} +8.20731i q^{53} -6.18873i q^{57} +2.22874 q^{59} +12.6096 q^{61} -0.595901i q^{63} +8.35839i q^{67} +4.40898 q^{69} +6.79530 q^{71} +7.31221i q^{73} +1.99961i q^{77} +10.5672 q^{79} +1.00000 q^{81} -4.18160i q^{83} +2.76048i q^{87} -4.25036 q^{89} -2.84189 q^{91} +4.48868i q^{93} +12.2752i q^{97} -3.35561 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{9}+O(q^{10}) \) 24 * q - 24 * q^9	\( 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) \) 24 * q - 24 * q^9 + 4 * q^11 - 20 * q^19 + 16 * q^21 - 16 * q^29 - 4 * q^31 + 20 * q^41 - 56 * q^49 + 16 * q^51 + 4 * q^59 + 68 * q^61 - 36 * q^69 - 12 * q^79 + 24 * q^81 - 20 * q^89 + 40 * q^91 - 4 * q^99

Introduction

L-functions

Modular forms

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Database

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