LMFDB - Embedded newform 7500.2.a.m.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1,7500)]

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, 0]))

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

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

chi := DirichletCharacter("7500.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.a (trivial)

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:	yes
Analytic conductor:	$59.8878015160$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} + \cdots - 4 $ x^12 - 6x^11 - 11x^10 + 94x^9 + 27x^8 - 460x^7 + 55x^6 + 812x^5 - 127x^4 - 512x^3 + 40x^2 + 72*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{3} $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.83182$ of defining polynomial
Character	$\chi$	$=$	7500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.595901 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.595901 q^{7} +1.00000 q^{9} +3.35561 q^{11} -4.76906 q^{13} -7.47993 q^{17} +6.18873 q^{19} -0.595901 q^{21} +4.40898 q^{23} -1.00000 q^{27} -2.76048 q^{29} +4.48868 q^{31} -3.35561 q^{33} +1.30067 q^{37} +4.76906 q^{39} -9.40875 q^{41} -7.59854 q^{43} +4.40189 q^{47} -6.64490 q^{49} +7.47993 q^{51} -8.20731 q^{53} -6.18873 q^{57} -2.22874 q^{59} +12.6096 q^{61} +0.595901 q^{63} +8.35839 q^{67} -4.40898 q^{69} +6.79530 q^{71} -7.31221 q^{73} +1.99961 q^{77} -10.5672 q^{79} +1.00000 q^{81} +4.18160 q^{83} +2.76048 q^{87} +4.25036 q^{89} -2.84189 q^{91} -4.48868 q^{93} +12.2752 q^{97} +3.35561 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9} + 2 q^{11} - 8 q^{17} + 10 q^{19} + 8 q^{21} - 18 q^{23} - 12 q^{27} + 8 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{41} - 28 q^{43} - 22 q^{47} + 28 q^{49} + 8 q^{51} - 16 q^{53} - 10 q^{57} - 2 q^{59} + 34 q^{61} - 8 q^{63} - 32 q^{67} + 18 q^{69} - 24 q^{73} - 18 q^{77} + 6 q^{79} + 12 q^{81} - 28 q^{83} - 8 q^{87} + 10 q^{89} + 20 q^{91} + 2 q^{93} - 16 q^{97} + 2 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9 + 2 * q^11 - 8 * q^17 + 10 * q^19 + 8 * q^21 - 18 * q^23 - 12 * q^27 + 8 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^37 + 10 * q^41 - 28 * q^43 - 22 * q^47 + 28 * q^49 + 8 * q^51 - 16 * q^53 - 10 * q^57 - 2 * q^59 + 34 * q^61 - 8 * q^63 - 32 * q^67 + 18 * q^69 - 24 * q^73 - 18 * q^77 + 6 * q^79 + 12 * q^81 - 28 * q^83 - 8 * q^87 + 10 * q^89 + 20 * q^91 + 2 * q^93 - 16 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.595901	0.225229	0.112615	−	0.993639i	$-0.464077\pi$
$7$	0.595901	0.225229	0.112615	+	0.993639i	$0.464077\pi$
$8$	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
$13$	−4.76906	−1.32270	−0.661350	−	0.750077i	$-0.730016\pi$
$13$	−4.76906	−1.32270	−0.661350	+	0.750077i	$0.730016\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.47993	−1.81415	−0.907075	−	0.420969i	$-0.861690\pi$
$17$	−7.47993	−1.81415	−0.907075	+	0.420969i	$0.861690\pi$
$18$	0	0
$19$	6.18873	1.41979	0.709896	−	0.704307i	$-0.248742\pi$
$19$	6.18873	1.41979	0.709896	+	0.704307i	$0.248742\pi$
$20$	0	0
$21$	−0.595901	−0.130036
$22$	0	0
$23$	4.40898	0.919337	0.459668	−	0.888091i	$-0.347968\pi$
$23$	4.40898	0.919337	0.459668	+	0.888091i	$0.347968\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.76048	−0.512608	−0.256304	−	0.966596i	$-0.582505\pi$
$29$	−2.76048	−0.512608	−0.256304	+	0.966596i	$0.582505\pi$
$30$	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
$33$	−3.35561	−0.584137
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.30067	0.213828	0.106914	−	0.994268i	$-0.465903\pi$
$37$	1.30067	0.213828	0.106914	+	0.994268i	$0.465903\pi$
$38$	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
$43$	−7.59854	−1.15877	−0.579383	−	0.815055i	$-0.696706\pi$
$43$	−7.59854	−1.15877	−0.579383	+	0.815055i	$0.696706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.40189	0.642082	0.321041	−	0.947065i	$-0.395967\pi$
$47$	4.40189	0.642082	0.321041	+	0.947065i	$0.395967\pi$
$48$	0	0
$49$	−6.64490	−0.949272
$50$	0	0
$51$	7.47993	1.04740
$52$	0	0
$53$	−8.20731	−1.12736	−0.563680	−	0.825993i	$-0.690615\pi$
$53$	−8.20731	−1.12736	−0.563680	+	0.825993i	$0.690615\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.18873	−0.819717
$58$	0	0
$59$	−2.22874	−0.290158	−0.145079	−	0.989420i	$-0.546344\pi$
$59$	−2.22874	−0.290158	−0.145079	+	0.989420i	$0.546344\pi$
$60$	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
$63$	0.595901	0.0750764
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.35839	1.02114	0.510570	−	0.859836i	$-0.329435\pi$
$67$	8.35839	1.02114	0.510570	+	0.859836i	$0.329435\pi$
$68$	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
$73$	−7.31221	−0.855830	−0.427915	−	0.903819i	$-0.640752\pi$
$73$	−7.31221	−0.855830	−0.427915	+	0.903819i	$0.640752\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.99961	0.227877
$78$	0	0
$79$	−10.5672	−1.18891	−0.594454	−	0.804130i	$-0.702632\pi$
$79$	−10.5672	−1.18891	−0.594454	+	0.804130i	$0.702632\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.18160	0.458991	0.229495	−	0.973310i	$-0.426292\pi$
$83$	4.18160	0.458991	0.229495	+	0.973310i	$0.426292\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.76048	0.295955
$88$	0	0
$89$	4.25036	0.450538	0.225269	−	0.974297i	$-0.427674\pi$
$89$	4.25036	0.450538	0.225269	+	0.974297i	$0.427674\pi$
$90$	0	0
$91$	−2.84189	−0.297911
$92$	0	0
$93$	−4.48868	−0.465455
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.2752	1.24636	0.623180	−	0.782078i	$-0.285840\pi$
$97$	12.2752	1.24636	0.623180	+	0.782078i	$0.285840\pi$
$98$	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
$103$	−5.62521	−0.554269	−0.277134	−	0.960831i	$-0.589385\pi$
$103$	−5.62521	−0.554269	−0.277134	+	0.960831i	$0.589385\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.7701	−1.33120	−0.665601	−	0.746308i	$-0.731825\pi$
$107$	−13.7701	−1.33120	−0.665601	+	0.746308i	$0.731825\pi$
$108$	0	0
$109$	−12.8014	−1.22616	−0.613078	−	0.790023i	$-0.710069\pi$
$109$	−12.8014	−1.22616	−0.613078	+	0.790023i	$0.710069\pi$
$110$	0	0
$111$	−1.30067	−0.123454
$112$	0	0
$113$	11.6394	1.09494	0.547471	−	0.836825i	$-0.315591\pi$
$113$	11.6394	1.09494	0.547471	+	0.836825i	$0.315591\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.76906	−0.440900
$118$	0	0
$119$	−4.45730	−0.408600
$120$	0	0
$121$	0.260113	0.0236466
$122$	0	0
$123$	9.40875	0.848358
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.66172	0.236189	0.118095	−	0.993002i	$-0.462321\pi$
$127$	2.66172	0.236189	0.118095	+	0.993002i	$0.462321\pi$
$128$	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
$133$	3.68787	0.319779
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.84194	0.328239	0.164119	−	0.986440i	$-0.447522\pi$
$137$	3.84194	0.328239	0.164119	+	0.986440i	$0.447522\pi$
$138$	0	0
$139$	−4.92765	−0.417958	−0.208979	−	0.977920i	$-0.567014\pi$
$139$	−4.92765	−0.417958	−0.208979	+	0.977920i	$0.567014\pi$
$140$	0	0
$141$	−4.40189	−0.370706
$142$	0	0
$143$	−16.0031	−1.33825
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.64490	0.548062
$148$	0	0
$149$	−6.59040	−0.539907	−0.269953	−	0.962873i	$-0.587008\pi$
$149$	−6.59040	−0.539907	−0.269953	+	0.962873i	$0.587008\pi$
$150$	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
$153$	−7.47993	−0.604717
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.341995	−0.0272942	−0.0136471	−	0.999907i	$-0.504344\pi$
$157$	−0.341995	−0.0272942	−0.0136471	+	0.999907i	$0.504344\pi$
$158$	0	0
$159$	8.20731	0.650882
$160$	0	0
$161$	2.62732	0.207062
$162$	0	0
$163$	−22.5163	−1.76361	−0.881806	−	0.471612i	$-0.843672\pi$
$163$	−22.5163	−1.76361	−0.881806	+	0.471612i	$0.843672\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.31493	0.333900	0.166950	−	0.985965i	$-0.446608\pi$
$167$	4.31493	0.333900	0.166950	+	0.985965i	$0.446608\pi$
$168$	0	0
$169$	9.74397	0.749536
$170$	0	0
$171$	6.18873	0.473264
$172$	0	0
$173$	−17.1688	−1.30532	−0.652659	−	0.757652i	$-0.726347\pi$
$173$	−17.1688	−1.30532	−0.652659	+	0.757652i	$0.726347\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.22874	0.167523
$178$	0	0
$179$	1.06360	0.0794970	0.0397485	−	0.999210i	$-0.487344\pi$
$179$	1.06360	0.0794970	0.0397485	+	0.999210i	$0.487344\pi$
$180$	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
$183$	−12.6096	−0.932130
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−25.0997	−1.83547
$188$	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
$193$	20.4002	1.46844	0.734220	−	0.678912i	$-0.237548\pi$
$193$	20.4002	1.46844	0.734220	+	0.678912i	$0.237548\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.10501	0.292470	0.146235	−	0.989250i	$-0.453284\pi$
$197$	4.10501	0.292470	0.146235	+	0.989250i	$0.453284\pi$
$198$	0	0
$199$	25.5940	1.81431	0.907156	−	0.420795i	$-0.138249\pi$
$199$	25.5940	1.81431	0.907156	+	0.420795i	$0.138249\pi$
$200$	0	0
$201$	−8.35839	−0.589555
$202$	0	0
$203$	−1.64497	−0.115454
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.40898	0.306446
$208$	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
$213$	−6.79530	−0.465606
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.67481	0.181578
$218$	0	0
$219$	7.31221	0.494113
$220$	0	0
$221$	35.6723	2.39958
$222$	0	0
$223$	−2.22827	−0.149216	−0.0746081	−	0.997213i	$-0.523771\pi$
$223$	−2.22827	−0.149216	−0.0746081	+	0.997213i	$0.523771\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.54796	−0.301859	−0.150929	−	0.988545i	$-0.548227\pi$
$227$	−4.54796	−0.301859	−0.150929	+	0.988545i	$0.548227\pi$
$228$	0	0
$229$	3.41511	0.225677	0.112838	−	0.993613i	$-0.464006\pi$
$229$	3.41511	0.225677	0.112838	+	0.993613i	$0.464006\pi$
$230$	0	0
$231$	−1.99961	−0.131565
$232$	0	0
$233$	−16.7395	−1.09664	−0.548320	−	0.836268i	$-0.684733\pi$
$233$	−16.7395	−1.09664	−0.548320	+	0.836268i	$0.684733\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.5672	0.686417
$238$	0	0
$239$	−22.3643	−1.44663	−0.723313	−	0.690520i	$-0.757382\pi$
$239$	−22.3643	−1.44663	−0.723313	+	0.690520i	$0.757382\pi$
$240$	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
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−29.5144	−1.87796
$248$	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
$253$	14.7948	0.930143
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.05952	0.440361	0.220180	−	0.975459i	$-0.429335\pi$
$257$	7.05952	0.440361	0.220180	+	0.975459i	$0.429335\pi$
$258$	0	0
$259$	0.775067	0.0481603
$260$	0	0
$261$	−2.76048	−0.170869
$262$	0	0
$263$	−1.38649	−0.0854949	−0.0427475	−	0.999086i	$-0.513611\pi$
$263$	−1.38649	−0.0854949	−0.0427475	+	0.999086i	$0.513611\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.25036	−0.260118
$268$	0	0
$269$	−14.1752	−0.864280	−0.432140	−	0.901807i	$-0.642241\pi$
$269$	−14.1752	−0.864280	−0.432140	+	0.901807i	$0.642241\pi$
$270$	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
$273$	2.84189	0.171999
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−12.7933	−0.768677	−0.384338	−	0.923192i	$-0.625570\pi$
$277$	−12.7933	−0.768677	−0.384338	+	0.923192i	$0.625570\pi$
$278$	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
$283$	−23.3502	−1.38802	−0.694011	−	0.719964i	$-0.744158\pi$
$283$	−23.3502	−1.38802	−0.694011	+	0.719964i	$0.744158\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.60668	−0.330952
$288$	0	0
$289$	38.9494	2.29114
$290$	0	0
$291$	−12.2752	−0.719587
$292$	0	0
$293$	−6.36651	−0.371936	−0.185968	−	0.982556i	$-0.559542\pi$
$293$	−6.36651	−0.371936	−0.185968	+	0.982556i	$0.559542\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.35561	−0.194712
$298$	0	0
$299$	−21.0267	−1.21601
$300$	0	0
$301$	−4.52797	−0.260988
$302$	0	0
$303$	8.00835	0.460068
$304$	0	0
$305$	0	0
$306$	0	0
$307$	27.2317	1.55419	0.777096	−	0.629382i	$-0.216692\pi$
$307$	27.2317	1.55419	0.777096	+	0.629382i	$0.216692\pi$
$308$	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
$313$	−20.2659	−1.14550	−0.572749	−	0.819731i	$-0.694123\pi$
$313$	−20.2659	−1.14550	−0.572749	+	0.819731i	$0.694123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.0153	1.18034	0.590169	−	0.807279i	$-0.299061\pi$
$317$	21.0153	1.18034	0.590169	+	0.807279i	$0.299061\pi$
$318$	0	0
$319$	−9.26309	−0.518634
$320$	0	0
$321$	13.7701	0.768570
$322$	0	0
$323$	−46.2913	−2.57572
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.8014	0.707921
$328$	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
$333$	1.30067	0.0712760
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.3467	−0.672567	−0.336284	−	0.941761i	$-0.609170\pi$
$337$	−12.3467	−0.672567	−0.336284	+	0.941761i	$0.609170\pi$
$338$	0	0
$339$	−11.6394	−0.632165
$340$	0	0
$341$	15.0623	0.815668
$342$	0	0
$343$	−8.13100	−0.439033
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.1617	−0.706555	−0.353277	−	0.935519i	$-0.614933\pi$
$347$	−13.1617	−0.706555	−0.353277	+	0.935519i	$0.614933\pi$
$348$	0	0
$349$	20.8060	1.11372	0.556861	−	0.830606i	$-0.312006\pi$
$349$	20.8060	1.11372	0.556861	+	0.830606i	$0.312006\pi$
$350$	0	0
$351$	4.76906	0.254554
$352$	0	0
$353$	−28.7945	−1.53258	−0.766289	−	0.642496i	$-0.777899\pi$
$353$	−28.7945	−1.53258	−0.766289	+	0.642496i	$0.777899\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.45730	0.235905
$358$	0	0
$359$	−33.3278	−1.75897	−0.879487	−	0.475923i	$-0.842114\pi$
$359$	−33.3278	−1.75897	−0.879487	+	0.475923i	$0.842114\pi$
$360$	0	0
$361$	19.3004	1.01581
$362$	0	0
$363$	−0.260113	−0.0136524
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.89727	0.0990366	0.0495183	−	0.998773i	$-0.484231\pi$
$367$	1.89727	0.0990366	0.0495183	+	0.998773i	$0.484231\pi$
$368$	0	0
$369$	−9.40875	−0.489800
$370$	0	0
$371$	−4.89074	−0.253914
$372$	0	0
$373$	−30.8908	−1.59947	−0.799733	−	0.600355i	$-0.795026\pi$
$373$	−30.8908	−1.59947	−0.799733	+	0.600355i	$0.795026\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.1649	0.678027
$378$	0	0
$379$	22.5547	1.15856	0.579279	−	0.815129i	$-0.303334\pi$
$379$	22.5547	1.15856	0.579279	+	0.815129i	$0.303334\pi$
$380$	0	0
$381$	−2.66172	−0.136364
$382$	0	0
$383$	4.34373	0.221954	0.110977	−	0.993823i	$-0.464602\pi$
$383$	4.34373	0.221954	0.110977	+	0.993823i	$0.464602\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.59854	−0.386255
$388$	0	0
$389$	−18.5459	−0.940313	−0.470156	−	0.882583i	$-0.655802\pi$
$389$	−18.5459	−0.940313	−0.470156	+	0.882583i	$0.655802\pi$
$390$	0	0
$391$	−32.9789	−1.66782
$392$	0	0
$393$	−9.71046	−0.489828
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−0.706994	−0.0354830	−0.0177415	−	0.999843i	$-0.505648\pi$
$397$	−0.706994	−0.0354830	−0.0177415	+	0.999843i	$0.505648\pi$
$398$	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
$403$	−21.4068	−1.06635
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.36453	0.216341
$408$	0	0
$409$	−0.730007	−0.0360965	−0.0180483	−	0.999837i	$-0.505745\pi$
$409$	−0.730007	−0.0360965	−0.0180483	+	0.999837i	$0.505745\pi$
$410$	0	0
$411$	−3.84194	−0.189509
$412$	0	0
$413$	−1.32811	−0.0653520
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.92765	0.241308
$418$	0	0
$419$	−12.9094	−0.630666	−0.315333	−	0.948981i	$-0.602116\pi$
$419$	−12.9094	−0.630666	−0.315333	+	0.948981i	$0.602116\pi$
$420$	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
$423$	4.40189	0.214027
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.51408	0.363632
$428$	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
$433$	−21.7487	−1.04518	−0.522588	−	0.852586i	$-0.675033\pi$
$433$	−21.7487	−1.04518	−0.522588	+	0.852586i	$0.675033\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	27.2860	1.30527
$438$	0	0
$439$	−39.3586	−1.87848	−0.939242	−	0.343255i	$-0.888471\pi$
$439$	−39.3586	−1.87848	−0.939242	+	0.343255i	$0.888471\pi$
$440$	0	0
$441$	−6.64490	−0.316424
$442$	0	0
$443$	−27.4919	−1.30618	−0.653089	−	0.757281i	$-0.726527\pi$
$443$	−27.4919	−1.30618	−0.653089	+	0.757281i	$0.726527\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.59040	0.311715
$448$	0	0
$449$	3.51087	0.165688	0.0828441	−	0.996563i	$-0.473600\pi$
$449$	3.51087	0.165688	0.0828441	+	0.996563i	$0.473600\pi$
$450$	0	0
$451$	−31.5721	−1.48667
$452$	0	0
$453$	19.5433	0.918226
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.1677	−0.475624	−0.237812	−	0.971311i	$-0.576430\pi$
$457$	−10.1677	−0.475624	−0.237812	+	0.971311i	$0.576430\pi$
$458$	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
$463$	4.88586	0.227065	0.113533	−	0.993534i	$-0.463783\pi$
$463$	4.88586	0.227065	0.113533	+	0.993534i	$0.463783\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−25.8046	−1.19410	−0.597048	−	0.802206i	$-0.703660\pi$
$467$	−25.8046	−1.19410	−0.597048	+	0.802206i	$0.703660\pi$
$468$	0	0
$469$	4.98077	0.229990
$470$	0	0
$471$	0.341995	0.0157583
$472$	0	0
$473$	−25.4977	−1.17239
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−8.20731	−0.375787
$478$	0	0
$479$	−16.9156	−0.772893	−0.386447	−	0.922312i	$-0.626298\pi$
$479$	−16.9156	−0.772893	−0.386447	+	0.922312i	$0.626298\pi$
$480$	0	0
$481$	−6.20296	−0.282830
$482$	0	0
$483$	−2.62732	−0.119547
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.3876	0.742592	0.371296	−	0.928515i	$-0.378914\pi$
$487$	16.3876	0.742592	0.371296	+	0.928515i	$0.378914\pi$
$488$	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
$493$	20.6482	0.929948
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.04932	0.181637
$498$	0	0
$499$	−16.5015	−0.738707	−0.369354	−	0.929289i	$-0.620421\pi$
$499$	−16.5015	−0.738707	−0.369354	+	0.929289i	$0.620421\pi$
$500$	0	0
$501$	−4.31493	−0.192777
$502$	0	0
$503$	−7.60033	−0.338882	−0.169441	−	0.985540i	$-0.554196\pi$
$503$	−7.60033	−0.338882	−0.169441	+	0.985540i	$0.554196\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9.74397	−0.432745
$508$	0	0
$509$	−21.9513	−0.972976	−0.486488	−	0.873687i	$-0.661722\pi$
$509$	−21.9513	−0.972976	−0.486488	+	0.873687i	$0.661722\pi$
$510$	0	0
$511$	−4.35735	−0.192758
$512$	0	0
$513$	−6.18873	−0.273239
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.7710	0.649629
$518$	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
$523$	41.5997	1.81903	0.909513	−	0.415675i	$-0.136455\pi$
$523$	41.5997	1.81903	0.909513	+	0.415675i	$0.136455\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.5751	−1.46255
$528$	0	0
$529$	−3.56085	−0.154820
$530$	0	0
$531$	−2.22874	−0.0967193
$532$	0	0
$533$	44.8709	1.94358
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.06360	−0.0458976
$538$	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
$543$	12.0267	0.516114
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−23.3039	−0.996403	−0.498202	−	0.867061i	$-0.666006\pi$
$547$	−23.3039	−0.996403	−0.498202	+	0.867061i	$0.666006\pi$
$548$	0	0
$549$	12.6096	0.538165
$550$	0	0
$551$	−17.0839	−0.727797
$552$	0	0
$553$	−6.29703	−0.267777
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.0914	0.469959	0.234979	−	0.972000i	$-0.424498\pi$
$557$	11.0914	0.469959	0.234979	+	0.972000i	$0.424498\pi$
$558$	0	0
$559$	36.2379	1.53270
$560$	0	0
$561$	25.0997	1.05971
$562$	0	0
$563$	14.3052	0.602890	0.301445	−	0.953484i	$-0.402531\pi$
$563$	14.3052	0.602890	0.301445	+	0.953484i	$0.402531\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.595901	0.0250255
$568$	0	0
$569$	−28.7743	−1.20628	−0.603141	−	0.797634i	$-0.706085\pi$
$569$	−28.7743	−1.20628	−0.603141	+	0.797634i	$0.706085\pi$
$570$	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
$573$	21.9695	0.917791
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.79973	0.0749239	0.0374619	−	0.999298i	$-0.488073\pi$
$577$	1.79973	0.0749239	0.0374619	+	0.999298i	$0.488073\pi$
$578$	0	0
$579$	−20.4002	−0.847804
$580$	0	0
$581$	2.49182	0.103378
$582$	0	0
$583$	−27.5405	−1.14061
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−40.5964	−1.67559	−0.837797	−	0.545982i	$-0.816157\pi$
$587$	−40.5964	−1.67559	−0.837797	+	0.545982i	$0.816157\pi$
$588$	0	0
$589$	27.7792	1.14462
$590$	0	0
$591$	−4.10501	−0.168857
$592$	0	0
$593$	−29.4991	−1.21138	−0.605690	−	0.795700i	$-0.707103\pi$
$593$	−29.4991	−1.21138	−0.605690	+	0.795700i	$0.707103\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−25.5940	−1.04749
$598$	0	0
$599$	−8.91418	−0.364224	−0.182112	−	0.983278i	$-0.558293\pi$
$599$	−8.91418	−0.364224	−0.182112	+	0.983278i	$0.558293\pi$
$600$	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
$603$	8.35839	0.340380
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−26.9626	−1.09438	−0.547190	−	0.837008i	$-0.684302\pi$
$607$	−26.9626	−1.09438	−0.547190	+	0.837008i	$0.684302\pi$
$608$	0	0
$609$	1.64497	0.0666576
$610$	0	0
$611$	−20.9929	−0.849282
$612$	0	0
$613$	−16.5230	−0.667356	−0.333678	−	0.942687i	$-0.608290\pi$
$613$	−16.5230	−0.667356	−0.333678	+	0.942687i	$0.608290\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.9879	−0.925456	−0.462728	−	0.886500i	$-0.653129\pi$
$617$	−22.9879	−0.925456	−0.462728	+	0.886500i	$0.653129\pi$
$618$	0	0
$619$	49.3165	1.98220	0.991099	−	0.133124i	$-0.0425008\pi$
$619$	49.3165	1.98220	0.991099	+	0.133124i	$0.0425008\pi$
$620$	0	0
$621$	−4.40898	−0.176926
$622$	0	0
$623$	2.53279	0.101474
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−20.7670	−0.829352
$628$	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
$633$	−19.5084	−0.775387
$634$	0	0
$635$	0	0
$636$	0	0
$637$	31.6900	1.25560
$638$	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
$643$	−3.97743	−0.156854	−0.0784272	−	0.996920i	$-0.524990\pi$
$643$	−3.97743	−0.156854	−0.0784272	+	0.996920i	$0.524990\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.01441	−0.0398807	−0.0199404	−	0.999801i	$-0.506348\pi$
$647$	−1.01441	−0.0398807	−0.0199404	+	0.999801i	$0.506348\pi$
$648$	0	0
$649$	−7.47879	−0.293568
$650$	0	0
$651$	−2.67481	−0.104834
$652$	0	0
$653$	8.17155	0.319778	0.159889	−	0.987135i	$-0.448886\pi$
$653$	8.17155	0.319778	0.159889	+	0.987135i	$0.448886\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.31221	−0.285277
$658$	0	0
$659$	7.68859	0.299505	0.149752	−	0.988724i	$-0.452152\pi$
$659$	7.68859	0.299505	0.149752	+	0.988724i	$0.452152\pi$
$660$	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
$663$	−35.6723	−1.38540
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.1709	−0.471260
$668$	0	0
$669$	2.22827	0.0861500
$670$	0	0
$671$	42.3129	1.63347
$672$	0	0
$673$	30.1819	1.16343	0.581714	−	0.813394i	$-0.302382\pi$
$673$	30.1819	1.16343	0.581714	+	0.813394i	$0.302382\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	47.0154	1.80695	0.903473	−	0.428644i	$-0.141009\pi$
$677$	47.0154	1.80695	0.903473	+	0.428644i	$0.141009\pi$
$678$	0	0
$679$	7.31482	0.280717
$680$	0	0
$681$	4.54796	0.174278
$682$	0	0
$683$	−2.19204	−0.0838762	−0.0419381	−	0.999120i	$-0.513353\pi$
$683$	−2.19204	−0.0838762	−0.0419381	+	0.999120i	$0.513353\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3.41511	−0.130295
$688$	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
$693$	1.99961	0.0759589
$694$	0	0
$695$	0	0
$696$	0	0
$697$	70.3768	2.66571
$698$	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
$703$	8.04947	0.303591
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.77218	−0.179476
$708$	0	0
$709$	19.9129	0.747846	0.373923	−	0.927460i	$-0.378012\pi$
$709$	19.9129	0.747846	0.373923	+	0.927460i	$0.378012\pi$
$710$	0	0
$711$	−10.5672	−0.396303
$712$	0	0
$713$	19.7905	0.741162
$714$	0	0
$715$	0	0
$716$	0	0
$717$	22.3643	0.835210
$718$	0	0
$719$	34.7804	1.29709	0.648546	−	0.761176i	$-0.275377\pi$
$719$	34.7804	1.29709	0.648546	+	0.761176i	$0.275377\pi$
$720$	0	0
$721$	−3.35207	−0.124837
$722$	0	0
$723$	−0.265356	−0.00986871
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−15.9677	−0.592209	−0.296105	−	0.955156i	$-0.595688\pi$
$727$	−15.9677	−0.592209	−0.296105	+	0.955156i	$0.595688\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	56.8365	2.10218
$732$	0	0
$733$	−10.1425	−0.374621	−0.187310	−	0.982301i	$-0.559977\pi$
$733$	−10.1425	−0.374621	−0.187310	+	0.982301i	$0.559977\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	28.0475	1.03314
$738$	0	0
$739$	3.27848	0.120601	0.0603004	−	0.998180i	$-0.480794\pi$
$739$	3.27848	0.120601	0.0603004	+	0.998180i	$0.480794\pi$
$740$	0	0
$741$	29.5144	1.08424
$742$	0	0
$743$	38.7278	1.42078	0.710392	−	0.703806i	$-0.248518\pi$
$743$	38.7278	1.42078	0.710392	+	0.703806i	$0.248518\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.18160	0.152997
$748$	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
$753$	−29.0694	−1.05935
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33.4057	1.21415	0.607075	−	0.794645i	$-0.292343\pi$
$757$	33.4057	1.21415	0.607075	+	0.794645i	$0.292343\pi$
$758$	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
$763$	−7.62838	−0.276166
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.6290	0.383792
$768$	0	0
$769$	−5.45427	−0.196686	−0.0983431	−	0.995153i	$-0.531354\pi$
$769$	−5.45427	−0.196686	−0.0983431	+	0.995153i	$0.531354\pi$
$770$	0	0
$771$	−7.05952	−0.254242
$772$	0	0
$773$	−29.0843	−1.04609	−0.523044	−	0.852306i	$-0.675204\pi$
$773$	−29.0843	−1.04609	−0.523044	+	0.852306i	$0.675204\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.775067	−0.0278054
$778$	0	0
$779$	−58.2282	−2.08624
$780$	0	0
$781$	22.8024	0.815933
$782$	0	0
$783$	2.76048	0.0986515
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.1141	1.21604	0.608018	−	0.793923i	$-0.291965\pi$
$787$	34.1141	1.21604	0.608018	+	0.793923i	$0.291965\pi$
$788$	0	0
$789$	1.38649	0.0493605
$790$	0	0
$791$	6.93592	0.246613
$792$	0	0
$793$	−60.1361	−2.13549
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.4560	−0.866276	−0.433138	−	0.901328i	$-0.642594\pi$
$797$	−24.4560	−0.866276	−0.433138	+	0.901328i	$0.642594\pi$
$798$	0	0
$799$	−32.9259	−1.16483
$800$	0	0
$801$	4.25036	0.150179
$802$	0	0
$803$	−24.5369	−0.865889
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.1752	0.498992
$808$	0	0
$809$	−52.2487	−1.83697	−0.918483	−	0.395460i	$-0.870585\pi$
$809$	−52.2487	−1.83697	−0.918483	+	0.395460i	$0.870585\pi$
$810$	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
$813$	−15.0411	−0.527514
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−47.0253	−1.64521
$818$	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
$823$	45.5282	1.58701	0.793506	−	0.608562i	$-0.208253\pi$
$823$	45.5282	1.58701	0.793506	+	0.608562i	$0.208253\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−19.0967	−0.664057	−0.332029	−	0.943269i	$-0.607733\pi$
$827$	−19.0967	−0.664057	−0.332029	+	0.943269i	$0.607733\pi$
$828$	0	0
$829$	11.3819	0.395309	0.197654	−	0.980272i	$-0.436668\pi$
$829$	11.3819	0.395309	0.197654	+	0.980272i	$0.436668\pi$
$830$	0	0
$831$	12.7933	0.443796
$832$	0	0
$833$	49.7034	1.72212
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.48868	−0.155152
$838$	0	0
$839$	25.7924	0.890452	0.445226	−	0.895418i	$-0.353123\pi$
$839$	25.7924	0.890452	0.445226	+	0.895418i	$0.353123\pi$
$840$	0	0
$841$	−21.3798	−0.737233
$842$	0	0
$843$	15.3058	0.527160
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.155001	0.00532591
$848$	0	0
$849$	23.3502	0.801375
$850$	0	0
$851$	5.73461	0.196580
$852$	0	0
$853$	−15.7410	−0.538961	−0.269480	−	0.963006i	$-0.586852\pi$
$853$	−15.7410	−0.538961	−0.269480	+	0.963006i	$0.586852\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.4948	1.58823	0.794116	−	0.607767i	$-0.207934\pi$
$857$	46.4948	1.58823	0.794116	+	0.607767i	$0.207934\pi$
$858$	0	0
$859$	12.1958	0.416116	0.208058	−	0.978116i	$-0.433286\pi$
$859$	12.1958	0.416116	0.208058	+	0.978116i	$0.433286\pi$
$860$	0	0
$861$	5.60668	0.191075
$862$	0	0
$863$	45.3926	1.54518	0.772591	−	0.634904i	$-0.218960\pi$
$863$	45.3926	1.54518	0.772591	+	0.634904i	$0.218960\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−38.9494	−1.32279
$868$	0	0
$869$	−35.4596	−1.20288
$870$	0	0
$871$	−39.8617	−1.35066
$872$	0	0
$873$	12.2752	0.415454
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.0305	0.642615	0.321308	−	0.946975i	$-0.395878\pi$
$877$	19.0305	0.642615	0.321308	+	0.946975i	$0.395878\pi$
$878$	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
$883$	48.3181	1.62603	0.813017	−	0.582240i	$-0.197824\pi$
$883$	48.3181	1.62603	0.813017	+	0.582240i	$0.197824\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.4010	1.08792	0.543960	−	0.839111i	$-0.316924\pi$
$887$	32.4010	1.08792	0.543960	+	0.839111i	$0.316924\pi$
$888$	0	0
$889$	1.58612	0.0531968
$890$	0	0
$891$	3.35561	0.112417
$892$	0	0
$893$	27.2421	0.911622
$894$	0	0
$895$	0	0
$896$	0	0
$897$	21.0267	0.702062
$898$	0	0
$899$	−12.3909	−0.413260
$900$	0	0
$901$	61.3901	2.04520
$902$	0	0
$903$	4.52797	0.150681
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−30.8525	−1.02444	−0.512220	−	0.858854i	$-0.671177\pi$
$907$	−30.8525	−1.02444	−0.512220	+	0.858854i	$0.671177\pi$
$908$	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
$913$	14.0318	0.464386
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.78647	0.191086
$918$	0	0
$919$	−37.9977	−1.25343	−0.626715	−	0.779249i	$-0.715601\pi$
$919$	−37.9977	−1.25343	−0.626715	+	0.779249i	$0.715601\pi$
$920$	0	0
$921$	−27.2317	−0.897314
$922$	0	0
$923$	−32.4072	−1.06670
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−5.62521	−0.184756
$928$	0	0
$929$	−10.4600	−0.343180	−0.171590	−	0.985168i	$-0.554890\pi$
$929$	−10.4600	−0.343180	−0.171590	+	0.985168i	$0.554890\pi$
$930$	0	0
$931$	−41.1235	−1.34777
$932$	0	0
$933$	−17.1583	−0.561739
$934$	0	0
$935$	0	0
$936$	0	0
$937$	33.7638	1.10301	0.551507	−	0.834170i	$-0.314053\pi$
$937$	33.7638	1.10301	0.551507	+	0.834170i	$0.314053\pi$
$938$	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
$943$	−41.4830	−1.35087
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.65150	0.248640	0.124320	−	0.992242i	$-0.460325\pi$
$947$	7.65150	0.248640	0.124320	+	0.992242i	$0.460325\pi$
$948$	0	0
$949$	34.8724	1.13201
$950$	0	0
$951$	−21.0153	−0.681469
$952$	0	0
$953$	−11.6526	−0.377465	−0.188733	−	0.982029i	$-0.560438\pi$
$953$	−11.6526	−0.377465	−0.188733	+	0.982029i	$0.560438\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.26309	0.299433
$958$	0	0
$959$	2.28941	0.0739290
$960$	0	0
$961$	−10.8517	−0.350055
$962$	0	0
$963$	−13.7701	−0.443734
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−33.7782	−1.08623	−0.543117	−	0.839657i	$-0.682756\pi$
$967$	−33.7782	−1.08623	−0.543117	+	0.839657i	$0.682756\pi$
$968$	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
$973$	−2.93639	−0.0941363
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.77099	−0.120645	−0.0603223	−	0.998179i	$-0.519213\pi$
$977$	−3.77099	−0.120645	−0.0603223	+	0.998179i	$0.519213\pi$
$978$	0	0
$979$	14.2626	0.455833
$980$	0	0
$981$	−12.8014	−0.408718
$982$	0	0
$983$	42.3040	1.34929	0.674644	−	0.738143i	$-0.264297\pi$
$983$	42.3040	1.34929	0.674644	+	0.738143i	$0.264297\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.62309	−0.0834939
$988$	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
$993$	12.7425	0.404371
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−16.7077	−0.529138	−0.264569	−	0.964367i	$-0.585230\pi$
$997$	−16.7077	−0.529138	−0.264569	+	0.964367i	$0.585230\pi$
$998$	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.a.m.1.7		12
5.2	odd	4		7500.2.d.g.1249.19		24
5.3	odd	4		7500.2.d.g.1249.6		24
5.4	even	2		7500.2.a.n.1.6		12
25.3	odd	20		1500.2.o.c.49.5		24
25.4	even	10		1500.2.m.c.1201.3		24
25.6	even	5		1500.2.m.d.301.4		24
25.8	odd	20		300.2.o.a.289.1	yes	24
25.17	odd	20		1500.2.o.c.949.5		24
25.19	even	10		1500.2.m.c.301.3		24
25.21	even	5		1500.2.m.d.1201.4		24
25.22	odd	20		300.2.o.a.109.1	✓	24
75.8	even	20		900.2.w.c.289.6		24
75.47	even	20		900.2.w.c.109.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.109.1	✓	24	25.22	odd	20
300.2.o.a.289.1	yes	24	25.8	odd	20
900.2.w.c.109.6		24	75.47	even	20
900.2.w.c.289.6		24	75.8	even	20
1500.2.m.c.301.3		24	25.19	even	10
1500.2.m.c.1201.3		24	25.4	even	10
1500.2.m.d.301.4		24	25.6	even	5
1500.2.m.d.1201.4		24	25.21	even	5
1500.2.o.c.49.5		24	25.3	odd	20
1500.2.o.c.949.5		24	25.17	odd	20
7500.2.a.m.1.7		12	1.1	even	1	trivial
7500.2.a.n.1.6		12	5.4	even	2
7500.2.d.g.1249.6		24	5.3	odd	4
7500.2.d.g.1249.19		24	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.595901 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.595901 q^{7} +1.00000 q^{9} +3.35561 q^{11} -4.76906 q^{13} -7.47993 q^{17} +6.18873 q^{19} -0.595901 q^{21} +4.40898 q^{23} -1.00000 q^{27} -2.76048 q^{29} +4.48868 q^{31} -3.35561 q^{33} +1.30067 q^{37} +4.76906 q^{39} -9.40875 q^{41} -7.59854 q^{43} +4.40189 q^{47} -6.64490 q^{49} +7.47993 q^{51} -8.20731 q^{53} -6.18873 q^{57} -2.22874 q^{59} +12.6096 q^{61} +0.595901 q^{63} +8.35839 q^{67} -4.40898 q^{69} +6.79530 q^{71} -7.31221 q^{73} +1.99961 q^{77} -10.5672 q^{79} +1.00000 q^{81} +4.18160 q^{83} +2.76048 q^{87} +4.25036 q^{89} -2.84189 q^{91} -4.48868 q^{93} +12.2752 q^{97} +3.35561 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9} + 2 q^{11} - 8 q^{17} + 10 q^{19} + 8 q^{21} - 18 q^{23} - 12 q^{27} + 8 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{41} - 28 q^{43} - 22 q^{47} + 28 q^{49} + 8 q^{51} - 16 q^{53} - 10 q^{57} - 2 q^{59} + 34 q^{61} - 8 q^{63} - 32 q^{67} + 18 q^{69} - 24 q^{73} - 18 q^{77} + 6 q^{79} + 12 q^{81} - 28 q^{83} - 8 q^{87} + 10 q^{89} + 20 q^{91} + 2 q^{93} - 16 q^{97} + 2 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9 + 2 * q^11 - 8 * q^17 + 10 * q^19 + 8 * q^21 - 18 * q^23 - 12 * q^27 + 8 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^37 + 10 * q^41 - 28 * q^43 - 22 * q^47 + 28 * q^49 + 8 * q^51 - 16 * q^53 - 10 * q^57 - 2 * q^59 + 34 * q^61 - 8 * q^63 - 32 * q^67 + 18 * q^69 - 24 * q^73 - 18 * q^77 + 6 * q^79 + 12 * q^81 - 28 * q^83 - 8 * q^87 + 10 * q^89 + 20 * q^91 + 2 * q^93 - 16 * q^97 + 2 * q^99

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