LMFDB - Embedded newform 7500.2.a.n.1.12

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:	$0$
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			1.12
Root			$-2.61083$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +4.62675 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.62675 q^{7} +1.00000 q^{9} -4.94880 q^{11} -3.76277 q^{13} -2.69040 q^{17} +5.87106 q^{19} +4.62675 q^{21} +6.67829 q^{23} +1.00000 q^{27} +1.20049 q^{29} +3.30235 q^{31} -4.94880 q^{33} +1.87725 q^{37} -3.76277 q^{39} -3.03290 q^{41} -10.6626 q^{43} +0.259214 q^{47} +14.4068 q^{49} -2.69040 q^{51} +9.79659 q^{53} +5.87106 q^{57} -9.62455 q^{59} +6.27369 q^{61} +4.62675 q^{63} +2.56053 q^{67} +6.67829 q^{69} +8.67729 q^{71} +4.87481 q^{73} -22.8968 q^{77} +12.4732 q^{79} +1.00000 q^{81} -8.89025 q^{83} +1.20049 q^{87} +14.5774 q^{89} -17.4094 q^{91} +3.30235 q^{93} -3.98689 q^{97} -4.94880 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	4.62675	1.74875	0.874373	−	0.485254i	$-0.161273\pi$
$7$	4.62675	1.74875	0.874373	+	0.485254i	$0.161273\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.94880	−1.49212	−0.746059	−	0.665880i	$-0.768056\pi$
$11$	−4.94880	−1.49212	−0.746059	+	0.665880i	$0.768056\pi$
$12$	0	0
$13$	−3.76277	−1.04360	−0.521802	−	0.853066i	$-0.674740\pi$
$13$	−3.76277	−1.04360	−0.521802	+	0.853066i	$0.674740\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.69040	−0.652517	−0.326258	−	0.945281i	$-0.605788\pi$
$17$	−2.69040	−0.652517	−0.326258	+	0.945281i	$0.605788\pi$
$18$	0	0
$19$	5.87106	1.34691	0.673457	−	0.739227i	$-0.264809\pi$
$19$	5.87106	1.34691	0.673457	+	0.739227i	$0.264809\pi$
$20$	0	0
$21$	4.62675	1.00964
$22$	0	0
$23$	6.67829	1.39252	0.696260	−	0.717790i	$-0.254846\pi$
$23$	6.67829	1.39252	0.696260	+	0.717790i	$0.254846\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.20049	0.222926	0.111463	−	0.993769i	$-0.464446\pi$
$29$	1.20049	0.222926	0.111463	+	0.993769i	$0.464446\pi$
$30$	0	0
$31$	3.30235	0.593119	0.296560	−	0.955014i	$-0.404161\pi$
$31$	3.30235	0.593119	0.296560	+	0.955014i	$0.404161\pi$
$32$	0	0
$33$	−4.94880	−0.861475
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.87725	0.308618	0.154309	−	0.988023i	$-0.450685\pi$
$37$	1.87725	0.308618	0.154309	+	0.988023i	$0.450685\pi$
$38$	0	0
$39$	−3.76277	−0.602525
$40$	0	0
$41$	−3.03290	−0.473659	−0.236829	−	0.971551i	$-0.576108\pi$
$41$	−3.03290	−0.473659	−0.236829	+	0.971551i	$0.576108\pi$
$42$	0	0
$43$	−10.6626	−1.62603	−0.813014	−	0.582244i	$-0.802175\pi$
$43$	−10.6626	−1.62603	−0.813014	+	0.582244i	$0.802175\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.259214	0.0378103	0.0189051	−	0.999821i	$-0.493982\pi$
$47$	0.259214	0.0378103	0.0189051	+	0.999821i	$0.493982\pi$
$48$	0	0
$49$	14.4068	2.05811
$50$	0	0
$51$	−2.69040	−0.376731
$52$	0	0
$53$	9.79659	1.34566	0.672832	−	0.739795i	$-0.265078\pi$
$53$	9.79659	1.34566	0.672832	+	0.739795i	$0.265078\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.87106	0.777641
$58$	0	0
$59$	−9.62455	−1.25301	−0.626505	−	0.779417i	$-0.715515\pi$
$59$	−9.62455	−1.25301	−0.626505	+	0.779417i	$0.715515\pi$
$60$	0	0
$61$	6.27369	0.803264	0.401632	−	0.915801i	$-0.368443\pi$
$61$	6.27369	0.803264	0.401632	+	0.915801i	$0.368443\pi$
$62$	0	0
$63$	4.62675	0.582915
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.56053	0.312818	0.156409	−	0.987692i	$-0.450008\pi$
$67$	2.56053	0.312818	0.156409	+	0.987692i	$0.450008\pi$
$68$	0	0
$69$	6.67829	0.803971
$70$	0	0
$71$	8.67729	1.02980	0.514902	−	0.857249i	$-0.327828\pi$
$71$	8.67729	1.02980	0.514902	+	0.857249i	$0.327828\pi$
$72$	0	0
$73$	4.87481	0.570554	0.285277	−	0.958445i	$-0.407914\pi$
$73$	4.87481	0.570554	0.285277	+	0.958445i	$0.407914\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−22.8968	−2.60934
$78$	0	0
$79$	12.4732	1.40334	0.701672	−	0.712500i	$-0.252437\pi$
$79$	12.4732	1.40334	0.701672	+	0.712500i	$0.252437\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.89025	−0.975832	−0.487916	−	0.872891i	$-0.662243\pi$
$83$	−8.89025	−0.975832	−0.487916	+	0.872891i	$0.662243\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.20049	0.128706
$88$	0	0
$89$	14.5774	1.54521	0.772603	−	0.634889i	$-0.218954\pi$
$89$	14.5774	1.54521	0.772603	+	0.634889i	$0.218954\pi$
$90$	0	0
$91$	−17.4094	−1.82500
$92$	0	0
$93$	3.30235	0.342437
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.98689	−0.404808	−0.202404	−	0.979302i	$-0.564875\pi$
$97$	−3.98689	−0.404808	−0.202404	+	0.979302i	$0.564875\pi$
$98$	0	0
$99$	−4.94880	−0.497373
$100$	0	0
$101$	9.36896	0.932246	0.466123	−	0.884720i	$-0.345650\pi$
$101$	9.36896	0.932246	0.466123	+	0.884720i	$0.345650\pi$
$102$	0	0
$103$	10.4241	1.02712	0.513559	−	0.858054i	$-0.328327\pi$
$103$	10.4241	1.02712	0.513559	+	0.858054i	$0.328327\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.220683	−0.0213342	−0.0106671	−	0.999943i	$-0.503396\pi$
$107$	−0.220683	−0.0213342	−0.0106671	+	0.999943i	$0.503396\pi$
$108$	0	0
$109$	6.68640	0.640441	0.320220	−	0.947343i	$-0.396243\pi$
$109$	6.68640	0.640441	0.320220	+	0.947343i	$0.396243\pi$
$110$	0	0
$111$	1.87725	0.178181
$112$	0	0
$113$	9.60864	0.903905	0.451952	−	0.892042i	$-0.350728\pi$
$113$	9.60864	0.903905	0.451952	+	0.892042i	$0.350728\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.76277	−0.347868
$118$	0	0
$119$	−12.4478	−1.14109
$120$	0	0
$121$	13.4906	1.22642
$122$	0	0
$123$	−3.03290	−0.273467
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.9923	1.41909	0.709543	−	0.704662i	$-0.248902\pi$
$127$	15.9923	1.41909	0.709543	+	0.704662i	$0.248902\pi$
$128$	0	0
$129$	−10.6626	−0.938788
$130$	0	0
$131$	−12.3228	−1.07665	−0.538324	−	0.842738i	$-0.680942\pi$
$131$	−12.3228	−1.07665	−0.538324	+	0.842738i	$0.680942\pi$
$132$	0	0
$133$	27.1639	2.35541
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.69454	−0.486518	−0.243259	−	0.969961i	$-0.578216\pi$
$137$	−5.69454	−0.486518	−0.243259	+	0.969961i	$0.578216\pi$
$138$	0	0
$139$	−17.9969	−1.52648	−0.763240	−	0.646115i	$-0.776393\pi$
$139$	−17.9969	−1.52648	−0.763240	+	0.646115i	$0.776393\pi$
$140$	0	0
$141$	0.259214	0.0218298
$142$	0	0
$143$	18.6212	1.55718
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.4068	1.18825
$148$	0	0
$149$	−1.09001	−0.0892972	−0.0446486	−	0.999003i	$-0.514217\pi$
$149$	−1.09001	−0.0892972	−0.0446486	+	0.999003i	$0.514217\pi$
$150$	0	0
$151$	11.3789	0.926004	0.463002	−	0.886357i	$-0.346772\pi$
$151$	11.3789	0.926004	0.463002	+	0.886357i	$0.346772\pi$
$152$	0	0
$153$	−2.69040	−0.217506
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.98415	−0.317970	−0.158985	−	0.987281i	$-0.550822\pi$
$157$	−3.98415	−0.317970	−0.158985	+	0.987281i	$0.550822\pi$
$158$	0	0
$159$	9.79659	0.776920
$160$	0	0
$161$	30.8988	2.43516
$162$	0	0
$163$	21.8314	1.70997	0.854983	−	0.518657i	$-0.173568\pi$
$163$	21.8314	1.70997	0.854983	+	0.518657i	$0.173568\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.4689	−1.19702	−0.598509	−	0.801116i	$-0.704240\pi$
$167$	−15.4689	−1.19702	−0.598509	+	0.801116i	$0.704240\pi$
$168$	0	0
$169$	1.15844	0.0891104
$170$	0	0
$171$	5.87106	0.448971
$172$	0	0
$173$	17.1704	1.30544	0.652719	−	0.757600i	$-0.273628\pi$
$173$	17.1704	1.30544	0.652719	+	0.757600i	$0.273628\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.62455	−0.723426
$178$	0	0
$179$	−14.4456	−1.07971	−0.539856	−	0.841757i	$-0.681521\pi$
$179$	−14.4456	−1.07971	−0.539856	+	0.841757i	$0.681521\pi$
$180$	0	0
$181$	12.3964	0.921420	0.460710	−	0.887551i	$-0.347595\pi$
$181$	12.3964	0.921420	0.460710	+	0.887551i	$0.347595\pi$
$182$	0	0
$183$	6.27369	0.463765
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.3142	0.973632
$188$	0	0
$189$	4.62675	0.336546
$190$	0	0
$191$	−1.65546	−0.119785	−0.0598925	−	0.998205i	$-0.519076\pi$
$191$	−1.65546	−0.119785	−0.0598925	+	0.998205i	$0.519076\pi$
$192$	0	0
$193$	16.3253	1.17512	0.587560	−	0.809181i	$-0.300089\pi$
$193$	16.3253	1.17512	0.587560	+	0.809181i	$0.300089\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.6324	−0.971267	−0.485634	−	0.874162i	$-0.661411\pi$
$197$	−13.6324	−0.971267	−0.485634	+	0.874162i	$0.661411\pi$
$198$	0	0
$199$	−6.07817	−0.430870	−0.215435	−	0.976518i	$-0.569117\pi$
$199$	−6.07817	−0.430870	−0.215435	+	0.976518i	$0.569117\pi$
$200$	0	0
$201$	2.56053	0.180606
$202$	0	0
$203$	5.55437	0.389840
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.67829	0.464173
$208$	0	0
$209$	−29.0547	−2.00975
$210$	0	0
$211$	17.0825	1.17601	0.588003	−	0.808859i	$-0.299914\pi$
$211$	17.0825	1.17601	0.588003	+	0.808859i	$0.299914\pi$
$212$	0	0
$213$	8.67729	0.594558
$214$	0	0
$215$	0	0
$216$	0	0
$217$	15.2791	1.03721
$218$	0	0
$219$	4.87481	0.329409
$220$	0	0
$221$	10.1233	0.680969
$222$	0	0
$223$	0.783790	0.0524865	0.0262432	−	0.999656i	$-0.491646\pi$
$223$	0.783790	0.0524865	0.0262432	+	0.999656i	$0.491646\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.9494	1.58958	0.794789	−	0.606886i	$-0.207581\pi$
$227$	23.9494	1.58958	0.794789	+	0.606886i	$0.207581\pi$
$228$	0	0
$229$	−5.53383	−0.365686	−0.182843	−	0.983142i	$-0.558530\pi$
$229$	−5.53383	−0.365686	−0.182843	+	0.983142i	$0.558530\pi$
$230$	0	0
$231$	−22.8968	−1.50650
$232$	0	0
$233$	0.907742	0.0594682	0.0297341	−	0.999558i	$-0.490534\pi$
$233$	0.907742	0.0594682	0.0297341	+	0.999558i	$0.490534\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.4732	0.810221
$238$	0	0
$239$	−11.2387	−0.726969	−0.363484	−	0.931600i	$-0.618413\pi$
$239$	−11.2387	−0.726969	−0.363484	+	0.931600i	$0.618413\pi$
$240$	0	0
$241$	−20.2517	−1.30453	−0.652264	−	0.757992i	$-0.726181\pi$
$241$	−20.2517	−1.30453	−0.652264	+	0.757992i	$0.726181\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−22.0914	−1.40564
$248$	0	0
$249$	−8.89025	−0.563397
$250$	0	0
$251$	−30.6919	−1.93725	−0.968627	−	0.248520i	$-0.920056\pi$
$251$	−30.6919	−1.93725	−0.968627	+	0.248520i	$0.920056\pi$
$252$	0	0
$253$	−33.0495	−2.07780
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.77543	0.297883	0.148941	−	0.988846i	$-0.452413\pi$
$257$	4.77543	0.297883	0.148941	+	0.988846i	$0.452413\pi$
$258$	0	0
$259$	8.68556	0.539694
$260$	0	0
$261$	1.20049	0.0743085
$262$	0	0
$263$	1.20966	0.0745910	0.0372955	−	0.999304i	$-0.488126\pi$
$263$	1.20966	0.0745910	0.0372955	+	0.999304i	$0.488126\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	14.5774	0.892125
$268$	0	0
$269$	−25.5249	−1.55628	−0.778140	−	0.628090i	$-0.783837\pi$
$269$	−25.5249	−1.55628	−0.778140	+	0.628090i	$0.783837\pi$
$270$	0	0
$271$	−5.87342	−0.356785	−0.178393	−	0.983959i	$-0.557090\pi$
$271$	−5.87342	−0.356785	−0.178393	+	0.983959i	$0.557090\pi$
$272$	0	0
$273$	−17.4094	−1.05366
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4.90539	−0.294737	−0.147368	−	0.989082i	$-0.547080\pi$
$277$	−4.90539	−0.294737	−0.147368	+	0.989082i	$0.547080\pi$
$278$	0	0
$279$	3.30235	0.197706
$280$	0	0
$281$	−1.29526	−0.0772686	−0.0386343	−	0.999253i	$-0.512301\pi$
$281$	−1.29526	−0.0772686	−0.0386343	+	0.999253i	$0.512301\pi$
$282$	0	0
$283$	30.2297	1.79697	0.898484	−	0.439006i	$-0.144669\pi$
$283$	30.2297	1.79697	0.898484	+	0.439006i	$0.144669\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.0324	−0.828309
$288$	0	0
$289$	−9.76177	−0.574222
$290$	0	0
$291$	−3.98689	−0.233716
$292$	0	0
$293$	8.06831	0.471356	0.235678	−	0.971831i	$-0.424269\pi$
$293$	8.06831	0.471356	0.235678	+	0.971831i	$0.424269\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.94880	−0.287158
$298$	0	0
$299$	−25.1289	−1.45324
$300$	0	0
$301$	−49.3331	−2.84351
$302$	0	0
$303$	9.36896	0.538233
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−14.7750	−0.843255	−0.421628	−	0.906769i	$-0.638541\pi$
$307$	−14.7750	−0.843255	−0.421628	+	0.906769i	$0.638541\pi$
$308$	0	0
$309$	10.4241	0.593007
$310$	0	0
$311$	−9.15373	−0.519060	−0.259530	−	0.965735i	$-0.583568\pi$
$311$	−9.15373	−0.519060	−0.259530	+	0.965735i	$0.583568\pi$
$312$	0	0
$313$	−13.2553	−0.749235	−0.374618	−	0.927179i	$-0.622226\pi$
$313$	−13.2553	−0.749235	−0.374618	+	0.927179i	$0.622226\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.790550	−0.0444017	−0.0222009	−	0.999754i	$-0.507067\pi$
$317$	−0.790550	−0.0444017	−0.0222009	+	0.999754i	$0.507067\pi$
$318$	0	0
$319$	−5.94099	−0.332631
$320$	0	0
$321$	−0.220683	−0.0123173
$322$	0	0
$323$	−15.7955	−0.878883
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.68640	0.369759
$328$	0	0
$329$	1.19932	0.0661205
$330$	0	0
$331$	−3.37370	−0.185435	−0.0927176	−	0.995692i	$-0.529555\pi$
$331$	−3.37370	−0.185435	−0.0927176	+	0.995692i	$0.529555\pi$
$332$	0	0
$333$	1.87725	0.102873
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−18.0660	−0.984116	−0.492058	−	0.870563i	$-0.663755\pi$
$337$	−18.0660	−0.984116	−0.492058	+	0.870563i	$0.663755\pi$
$338$	0	0
$339$	9.60864	0.521870
$340$	0	0
$341$	−16.3426	−0.885004
$342$	0	0
$343$	34.2694	1.85037
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.05223	−0.163852	−0.0819262	−	0.996638i	$-0.526107\pi$
$347$	−3.05223	−0.163852	−0.0819262	+	0.996638i	$0.526107\pi$
$348$	0	0
$349$	−0.628744	−0.0336559	−0.0168280	−	0.999858i	$-0.505357\pi$
$349$	−0.628744	−0.0336559	−0.0168280	+	0.999858i	$0.505357\pi$
$350$	0	0
$351$	−3.76277	−0.200842
$352$	0	0
$353$	18.8896	1.00539	0.502696	−	0.864463i	$-0.332342\pi$
$353$	18.8896	1.00539	0.502696	+	0.864463i	$0.332342\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−12.4478	−0.658807
$358$	0	0
$359$	28.9347	1.52712	0.763558	−	0.645739i	$-0.223450\pi$
$359$	28.9347	1.52712	0.763558	+	0.645739i	$0.223450\pi$
$360$	0	0
$361$	15.4693	0.814175
$362$	0	0
$363$	13.4906	0.708073
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.38295	−0.280988	−0.140494	−	0.990082i	$-0.544869\pi$
$367$	−5.38295	−0.280988	−0.140494	+	0.990082i	$0.544869\pi$
$368$	0	0
$369$	−3.03290	−0.157886
$370$	0	0
$371$	45.3263	2.35323
$372$	0	0
$373$	10.9975	0.569428	0.284714	−	0.958613i	$-0.408101\pi$
$373$	10.9975	0.569428	0.284714	+	0.958613i	$0.408101\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.51717	−0.232646
$378$	0	0
$379$	7.06775	0.363046	0.181523	−	0.983387i	$-0.441897\pi$
$379$	7.06775	0.363046	0.181523	+	0.983387i	$0.441897\pi$
$380$	0	0
$381$	15.9923	0.819309
$382$	0	0
$383$	−8.10135	−0.413960	−0.206980	−	0.978345i	$-0.566363\pi$
$383$	−8.10135	−0.413960	−0.206980	+	0.978345i	$0.566363\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.6626	−0.542009
$388$	0	0
$389$	33.0432	1.67536	0.837678	−	0.546164i	$-0.183912\pi$
$389$	33.0432	1.67536	0.837678	+	0.546164i	$0.183912\pi$
$390$	0	0
$391$	−17.9672	−0.908642
$392$	0	0
$393$	−12.3228	−0.621603
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.3668	0.620671	0.310336	−	0.950627i	$-0.399559\pi$
$397$	12.3668	0.620671	0.310336	+	0.950627i	$0.399559\pi$
$398$	0	0
$399$	27.1639	1.35990
$400$	0	0
$401$	−14.7983	−0.738993	−0.369496	−	0.929232i	$-0.620470\pi$
$401$	−14.7983	−0.738993	−0.369496	+	0.929232i	$0.620470\pi$
$402$	0	0
$403$	−12.4260	−0.618982
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.29012	−0.460494
$408$	0	0
$409$	−23.6853	−1.17116	−0.585581	−	0.810614i	$-0.699134\pi$
$409$	−23.6853	−1.17116	−0.585581	+	0.810614i	$0.699134\pi$
$410$	0	0
$411$	−5.69454	−0.280891
$412$	0	0
$413$	−44.5304	−2.19120
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−17.9969	−0.881314
$418$	0	0
$419$	1.54379	0.0754193	0.0377096	−	0.999289i	$-0.487994\pi$
$419$	1.54379	0.0754193	0.0377096	+	0.999289i	$0.487994\pi$
$420$	0	0
$421$	−17.5965	−0.857599	−0.428799	−	0.903400i	$-0.641063\pi$
$421$	−17.5965	−0.857599	−0.428799	+	0.903400i	$0.641063\pi$
$422$	0	0
$423$	0.259214	0.0126034
$424$	0	0
$425$	0	0
$426$	0	0
$427$	29.0268	1.40471
$428$	0	0
$429$	18.6212	0.899039
$430$	0	0
$431$	15.6473	0.753702	0.376851	−	0.926274i	$-0.377007\pi$
$431$	15.6473	0.753702	0.376851	+	0.926274i	$0.377007\pi$
$432$	0	0
$433$	0.228980	0.0110041	0.00550203	−	0.999985i	$-0.498249\pi$
$433$	0.228980	0.0110041	0.00550203	+	0.999985i	$0.498249\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	39.2086	1.87560
$438$	0	0
$439$	19.4199	0.926863	0.463431	−	0.886133i	$-0.346618\pi$
$439$	19.4199	0.926863	0.463431	+	0.886133i	$0.346618\pi$
$440$	0	0
$441$	14.4068	0.686038
$442$	0	0
$443$	−12.7980	−0.608051	−0.304026	−	0.952664i	$-0.598331\pi$
$443$	−12.7980	−0.608051	−0.304026	+	0.952664i	$0.598331\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.09001	−0.0515558
$448$	0	0
$449$	−21.1499	−0.998124	−0.499062	−	0.866566i	$-0.666322\pi$
$449$	−21.1499	−0.998124	−0.499062	+	0.866566i	$0.666322\pi$
$450$	0	0
$451$	15.0092	0.706755
$452$	0	0
$453$	11.3789	0.534629
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.97118	−0.419654	−0.209827	−	0.977739i	$-0.567290\pi$
$457$	−8.97118	−0.419654	−0.209827	+	0.977739i	$0.567290\pi$
$458$	0	0
$459$	−2.69040	−0.125577
$460$	0	0
$461$	27.5351	1.28244	0.641218	−	0.767358i	$-0.278429\pi$
$461$	27.5351	1.28244	0.641218	+	0.767358i	$0.278429\pi$
$462$	0	0
$463$	29.4011	1.36638	0.683191	−	0.730239i	$-0.260591\pi$
$463$	29.4011	1.36638	0.683191	+	0.730239i	$0.260591\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.2609	0.613643	0.306821	−	0.951767i	$-0.400735\pi$
$467$	13.2609	0.613643	0.306821	+	0.951767i	$0.400735\pi$
$468$	0	0
$469$	11.8469	0.547040
$470$	0	0
$471$	−3.98415	−0.183580
$472$	0	0
$473$	52.7669	2.42623
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.79659	0.448555
$478$	0	0
$479$	−18.0591	−0.825142	−0.412571	−	0.910925i	$-0.635369\pi$
$479$	−18.0591	−0.825142	−0.412571	+	0.910925i	$0.635369\pi$
$480$	0	0
$481$	−7.06365	−0.322075
$482$	0	0
$483$	30.8988	1.40594
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.0571	0.636990	0.318495	−	0.947925i	$-0.396823\pi$
$487$	14.0571	0.636990	0.318495	+	0.947925i	$0.396823\pi$
$488$	0	0
$489$	21.8314	0.987249
$490$	0	0
$491$	−27.2319	−1.22896	−0.614478	−	0.788934i	$-0.710634\pi$
$491$	−27.2319	−1.22896	−0.614478	+	0.788934i	$0.710634\pi$
$492$	0	0
$493$	−3.22980	−0.145463
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.1476	1.80087
$498$	0	0
$499$	−8.17654	−0.366032	−0.183016	−	0.983110i	$-0.558586\pi$
$499$	−8.17654	−0.366032	−0.183016	+	0.983110i	$0.558586\pi$
$500$	0	0
$501$	−15.4689	−0.691099
$502$	0	0
$503$	9.94479	0.443416	0.221708	−	0.975113i	$-0.428837\pi$
$503$	9.94479	0.443416	0.221708	+	0.975113i	$0.428837\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.15844	0.0514479
$508$	0	0
$509$	17.2750	0.765700	0.382850	−	0.923811i	$-0.374943\pi$
$509$	17.2750	0.765700	0.382850	+	0.923811i	$0.374943\pi$
$510$	0	0
$511$	22.5545	0.997754
$512$	0	0
$513$	5.87106	0.259214
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.28280	−0.0564174
$518$	0	0
$519$	17.1704	0.753695
$520$	0	0
$521$	−11.1252	−0.487402	−0.243701	−	0.969850i	$-0.578362\pi$
$521$	−11.1252	−0.487402	−0.243701	+	0.969850i	$0.578362\pi$
$522$	0	0
$523$	−3.75282	−0.164099	−0.0820496	−	0.996628i	$-0.526147\pi$
$523$	−3.75282	−0.164099	−0.0820496	+	0.996628i	$0.526147\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.88462	−0.387020
$528$	0	0
$529$	21.5995	0.939109
$530$	0	0
$531$	−9.62455	−0.417670
$532$	0	0
$533$	11.4121	0.494312
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−14.4456	−0.623372
$538$	0	0
$539$	−71.2963	−3.07095
$540$	0	0
$541$	18.8812	0.811764	0.405882	−	0.913925i	$-0.366964\pi$
$541$	18.8812	0.811764	0.405882	+	0.913925i	$0.366964\pi$
$542$	0	0
$543$	12.3964	0.531982
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0.155307	0.00664046	0.00332023	−	0.999994i	$-0.498943\pi$
$547$	0.155307	0.00664046	0.00332023	+	0.999994i	$0.498943\pi$
$548$	0	0
$549$	6.27369	0.267755
$550$	0	0
$551$	7.04815	0.300261
$552$	0	0
$553$	57.7103	2.45409
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.0371	−0.806626	−0.403313	−	0.915062i	$-0.632141\pi$
$557$	−19.0371	−0.806626	−0.403313	+	0.915062i	$0.632141\pi$
$558$	0	0
$559$	40.1208	1.69693
$560$	0	0
$561$	13.3142	0.562127
$562$	0	0
$563$	−31.0709	−1.30948	−0.654741	−	0.755854i	$-0.727222\pi$
$563$	−31.0709	−1.30948	−0.654741	+	0.755854i	$0.727222\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.62675	0.194305
$568$	0	0
$569$	39.3417	1.64929	0.824646	−	0.565649i	$-0.191374\pi$
$569$	39.3417	1.64929	0.824646	+	0.565649i	$0.191374\pi$
$570$	0	0
$571$	28.4331	1.18989	0.594944	−	0.803767i	$-0.297174\pi$
$571$	28.4331	1.18989	0.594944	+	0.803767i	$0.297174\pi$
$572$	0	0
$573$	−1.65546	−0.0691579
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−22.4595	−0.935002	−0.467501	−	0.883992i	$-0.654846\pi$
$577$	−22.4595	−0.935002	−0.467501	+	0.883992i	$0.654846\pi$
$578$	0	0
$579$	16.3253	0.678456
$580$	0	0
$581$	−41.1330	−1.70648
$582$	0	0
$583$	−48.4813	−2.00789
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.0830	−0.663816	−0.331908	−	0.943312i	$-0.607692\pi$
$587$	−16.0830	−0.663816	−0.331908	+	0.943312i	$0.607692\pi$
$588$	0	0
$589$	19.3883	0.798880
$590$	0	0
$591$	−13.6324	−0.560761
$592$	0	0
$593$	20.4648	0.840389	0.420194	−	0.907434i	$-0.361962\pi$
$593$	20.4648	0.840389	0.420194	+	0.907434i	$0.361962\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.07817	−0.248763
$598$	0	0
$599$	18.5688	0.758699	0.379349	−	0.925253i	$-0.376148\pi$
$599$	18.5688	0.758699	0.379349	+	0.925253i	$0.376148\pi$
$600$	0	0
$601$	47.2047	1.92552	0.962761	−	0.270355i	$-0.0871409\pi$
$601$	47.2047	1.92552	0.962761	+	0.270355i	$0.0871409\pi$
$602$	0	0
$603$	2.56053	0.104273
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0.0786576	0.00319261	0.00159631	−	0.999999i	$-0.499492\pi$
$607$	0.0786576	0.00319261	0.00159631	+	0.999999i	$0.499492\pi$
$608$	0	0
$609$	5.55437	0.225074
$610$	0	0
$611$	−0.975363	−0.0394590
$612$	0	0
$613$	5.31081	0.214502	0.107251	−	0.994232i	$-0.465795\pi$
$613$	5.31081	0.214502	0.107251	+	0.994232i	$0.465795\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.4323	1.22516	0.612580	−	0.790409i	$-0.290132\pi$
$617$	30.4323	1.22516	0.612580	+	0.790409i	$0.290132\pi$
$618$	0	0
$619$	−31.6461	−1.27196	−0.635982	−	0.771704i	$-0.719405\pi$
$619$	−31.6461	−1.27196	−0.635982	+	0.771704i	$0.719405\pi$
$620$	0	0
$621$	6.67829	0.267990
$622$	0	0
$623$	67.4462	2.70217
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−29.0547	−1.16033
$628$	0	0
$629$	−5.05054	−0.201378
$630$	0	0
$631$	−33.8887	−1.34909	−0.674544	−	0.738234i	$-0.735660\pi$
$631$	−33.8887	−1.34909	−0.674544	+	0.738234i	$0.735660\pi$
$632$	0	0
$633$	17.0825	0.678967
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−54.2095	−2.14786
$638$	0	0
$639$	8.67729	0.343268
$640$	0	0
$641$	−49.0347	−1.93676	−0.968378	−	0.249489i	$-0.919737\pi$
$641$	−49.0347	−1.93676	−0.968378	+	0.249489i	$0.919737\pi$
$642$	0	0
$643$	−14.2509	−0.562000	−0.281000	−	0.959708i	$-0.590666\pi$
$643$	−14.2509	−0.562000	−0.281000	+	0.959708i	$0.590666\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.3343	1.66433	0.832166	−	0.554526i	$-0.187101\pi$
$647$	42.3343	1.66433	0.832166	+	0.554526i	$0.187101\pi$
$648$	0	0
$649$	47.6300	1.86964
$650$	0	0
$651$	15.2791	0.598836
$652$	0	0
$653$	47.0292	1.84039	0.920197	−	0.391456i	$-0.128029\pi$
$653$	47.0292	1.84039	0.920197	+	0.391456i	$0.128029\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.87481	0.190185
$658$	0	0
$659$	−11.1866	−0.435767	−0.217884	−	0.975975i	$-0.569915\pi$
$659$	−11.1866	−0.435767	−0.217884	+	0.975975i	$0.569915\pi$
$660$	0	0
$661$	−46.6124	−1.81301	−0.906505	−	0.422195i	$-0.861260\pi$
$661$	−46.6124	−1.81301	−0.906505	+	0.422195i	$0.861260\pi$
$662$	0	0
$663$	10.1233	0.393158
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.01722	0.310428
$668$	0	0
$669$	0.783790	0.0303031
$670$	0	0
$671$	−31.0472	−1.19857
$672$	0	0
$673$	−13.7340	−0.529407	−0.264703	−	0.964330i	$-0.585274\pi$
$673$	−13.7340	−0.529407	−0.264703	+	0.964330i	$0.585274\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.8210	1.41514	0.707572	−	0.706641i	$-0.249790\pi$
$677$	36.8210	1.41514	0.707572	+	0.706641i	$0.249790\pi$
$678$	0	0
$679$	−18.4464	−0.707906
$680$	0	0
$681$	23.9494	0.917744
$682$	0	0
$683$	−30.0906	−1.15138	−0.575692	−	0.817666i	$-0.695267\pi$
$683$	−30.0906	−1.15138	−0.575692	+	0.817666i	$0.695267\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−5.53383	−0.211129
$688$	0	0
$689$	−36.8623	−1.40434
$690$	0	0
$691$	−8.36619	−0.318265	−0.159132	−	0.987257i	$-0.550870\pi$
$691$	−8.36619	−0.318265	−0.159132	+	0.987257i	$0.550870\pi$
$692$	0	0
$693$	−22.8968	−0.869779
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.15969	0.309070
$698$	0	0
$699$	0.907742	0.0343340
$700$	0	0
$701$	−3.42495	−0.129359	−0.0646794	−	0.997906i	$-0.520602\pi$
$701$	−3.42495	−0.129359	−0.0646794	+	0.997906i	$0.520602\pi$
$702$	0	0
$703$	11.0214	0.415681
$704$	0	0
$705$	0	0
$706$	0	0
$707$	43.3478	1.63026
$708$	0	0
$709$	−49.4835	−1.85839	−0.929197	−	0.369585i	$-0.879500\pi$
$709$	−49.4835	−1.85839	−0.929197	+	0.369585i	$0.879500\pi$
$710$	0	0
$711$	12.4732	0.467781
$712$	0	0
$713$	22.0540	0.825930
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.2387	−0.419716
$718$	0	0
$719$	26.1522	0.975313	0.487656	−	0.873036i	$-0.337852\pi$
$719$	26.1522	0.975313	0.487656	+	0.873036i	$0.337852\pi$
$720$	0	0
$721$	48.2297	1.79617
$722$	0	0
$723$	−20.2517	−0.753170
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−34.6172	−1.28388	−0.641941	−	0.766754i	$-0.721871\pi$
$727$	−34.6172	−1.28388	−0.641941	+	0.766754i	$0.721871\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	28.6866	1.06101
$732$	0	0
$733$	17.4652	0.645091	0.322545	−	0.946554i	$-0.395461\pi$
$733$	17.4652	0.645091	0.322545	+	0.946554i	$0.395461\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.6715	−0.466762
$738$	0	0
$739$	20.0685	0.738233	0.369116	−	0.929383i	$-0.379660\pi$
$739$	20.0685	0.738233	0.369116	+	0.929383i	$0.379660\pi$
$740$	0	0
$741$	−22.0914	−0.811549
$742$	0	0
$743$	5.84644	0.214485	0.107243	−	0.994233i	$-0.465798\pi$
$743$	5.84644	0.214485	0.107243	+	0.994233i	$0.465798\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.89025	−0.325277
$748$	0	0
$749$	−1.02104	−0.0373081
$750$	0	0
$751$	−32.7925	−1.19662	−0.598308	−	0.801266i	$-0.704160\pi$
$751$	−32.7925	−1.19662	−0.598308	+	0.801266i	$0.704160\pi$
$752$	0	0
$753$	−30.6919	−1.11847
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.6371	0.822759	0.411379	−	0.911464i	$-0.365047\pi$
$757$	22.6371	0.822759	0.411379	+	0.911464i	$0.365047\pi$
$758$	0	0
$759$	−33.0495	−1.19962
$760$	0	0
$761$	−36.3308	−1.31699	−0.658496	−	0.752584i	$-0.728807\pi$
$761$	−36.3308	−1.31699	−0.658496	+	0.752584i	$0.728807\pi$
$762$	0	0
$763$	30.9363	1.11997
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.2150	1.30765
$768$	0	0
$769$	14.6238	0.527349	0.263675	−	0.964612i	$-0.415065\pi$
$769$	14.6238	0.527349	0.263675	+	0.964612i	$0.415065\pi$
$770$	0	0
$771$	4.77543	0.171983
$772$	0	0
$773$	30.5251	1.09791	0.548956	−	0.835851i	$-0.315025\pi$
$773$	30.5251	1.09791	0.548956	+	0.835851i	$0.315025\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.68556	0.311593
$778$	0	0
$779$	−17.8063	−0.637977
$780$	0	0
$781$	−42.9421	−1.53659
$782$	0	0
$783$	1.20049	0.0429020
$784$	0	0
$785$	0	0
$786$	0	0
$787$	49.1515	1.75206	0.876031	−	0.482254i	$-0.160182\pi$
$787$	49.1515	1.75206	0.876031	+	0.482254i	$0.160182\pi$
$788$	0	0
$789$	1.20966	0.0430651
$790$	0	0
$791$	44.4567	1.58070
$792$	0	0
$793$	−23.6065	−0.838290
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−44.5325	−1.57742	−0.788711	−	0.614764i	$-0.789251\pi$
$797$	−44.5325	−1.57742	−0.788711	+	0.614764i	$0.789251\pi$
$798$	0	0
$799$	−0.697388	−0.0246718
$800$	0	0
$801$	14.5774	0.515069
$802$	0	0
$803$	−24.1245	−0.851334
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−25.5249	−0.898519
$808$	0	0
$809$	−30.2911	−1.06498	−0.532488	−	0.846437i	$-0.678743\pi$
$809$	−30.2911	−1.06498	−0.532488	+	0.846437i	$0.678743\pi$
$810$	0	0
$811$	6.24687	0.219357	0.109679	−	0.993967i	$-0.465018\pi$
$811$	6.24687	0.219357	0.109679	+	0.993967i	$0.465018\pi$
$812$	0	0
$813$	−5.87342	−0.205990
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−62.6006	−2.19012
$818$	0	0
$819$	−17.4094	−0.608333
$820$	0	0
$821$	−19.2716	−0.672582	−0.336291	−	0.941758i	$-0.609173\pi$
$821$	−19.2716	−0.672582	−0.336291	+	0.941758i	$0.609173\pi$
$822$	0	0
$823$	−7.55750	−0.263438	−0.131719	−	0.991287i	$-0.542050\pi$
$823$	−7.55750	−0.263438	−0.131719	+	0.991287i	$0.542050\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31.3740	−1.09098	−0.545490	−	0.838117i	$-0.683656\pi$
$827$	−31.3740	−1.09098	−0.545490	+	0.838117i	$0.683656\pi$
$828$	0	0
$829$	27.6886	0.961665	0.480832	−	0.876813i	$-0.340335\pi$
$829$	27.6886	0.961665	0.480832	+	0.876813i	$0.340335\pi$
$830$	0	0
$831$	−4.90539	−0.170166
$832$	0	0
$833$	−38.7600	−1.34295
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.30235	0.114146
$838$	0	0
$839$	−37.7794	−1.30429	−0.652145	−	0.758094i	$-0.726131\pi$
$839$	−37.7794	−1.30429	−0.652145	+	0.758094i	$0.726131\pi$
$840$	0	0
$841$	−27.5588	−0.950304
$842$	0	0
$843$	−1.29526	−0.0446110
$844$	0	0
$845$	0	0
$846$	0	0
$847$	62.4176	2.14469
$848$	0	0
$849$	30.2297	1.03748
$850$	0	0
$851$	12.5368	0.429756
$852$	0	0
$853$	−49.3670	−1.69029	−0.845147	−	0.534535i	$-0.820487\pi$
$853$	−49.3670	−1.69029	−0.845147	+	0.534535i	$0.820487\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.7692	−0.538667	−0.269333	−	0.963047i	$-0.586803\pi$
$857$	−15.7692	−0.538667	−0.269333	+	0.963047i	$0.586803\pi$
$858$	0	0
$859$	9.39688	0.320617	0.160309	−	0.987067i	$-0.448751\pi$
$859$	9.39688	0.320617	0.160309	+	0.987067i	$0.448751\pi$
$860$	0	0
$861$	−14.0324	−0.478225
$862$	0	0
$863$	−17.8705	−0.608318	−0.304159	−	0.952621i	$-0.598375\pi$
$863$	−17.8705	−0.608318	−0.304159	+	0.952621i	$0.598375\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−9.76177	−0.331527
$868$	0	0
$869$	−61.7273	−2.09395
$870$	0	0
$871$	−9.63468	−0.326459
$872$	0	0
$873$	−3.98689	−0.134936
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−17.7052	−0.597862	−0.298931	−	0.954275i	$-0.596630\pi$
$877$	−17.7052	−0.597862	−0.298931	+	0.954275i	$0.596630\pi$
$878$	0	0
$879$	8.06831	0.272137
$880$	0	0
$881$	−31.7784	−1.07064	−0.535321	−	0.844649i	$-0.679809\pi$
$881$	−31.7784	−1.07064	−0.535321	+	0.844649i	$0.679809\pi$
$882$	0	0
$883$	34.6098	1.16471	0.582356	−	0.812934i	$-0.302131\pi$
$883$	34.6098	1.16471	0.582356	+	0.812934i	$0.302131\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.5204	0.957622	0.478811	−	0.877918i	$-0.341068\pi$
$887$	28.5204	0.957622	0.478811	+	0.877918i	$0.341068\pi$
$888$	0	0
$889$	73.9923	2.48162
$890$	0	0
$891$	−4.94880	−0.165791
$892$	0	0
$893$	1.52186	0.0509271
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−25.1289	−0.839028
$898$	0	0
$899$	3.96444	0.132221
$900$	0	0
$901$	−26.3567	−0.878069
$902$	0	0
$903$	−49.3331	−1.64170
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7.01754	0.233013	0.116507	−	0.993190i	$-0.462830\pi$
$907$	7.01754	0.233013	0.116507	+	0.993190i	$0.462830\pi$
$908$	0	0
$909$	9.36896	0.310749
$910$	0	0
$911$	−20.9075	−0.692696	−0.346348	−	0.938106i	$-0.612578\pi$
$911$	−20.9075	−0.692696	−0.346348	+	0.938106i	$0.612578\pi$
$912$	0	0
$913$	43.9961	1.45606
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−57.0145	−1.88278
$918$	0	0
$919$	59.7567	1.97119	0.985596	−	0.169117i	$-0.0540917\pi$
$919$	59.7567	1.97119	0.985596	+	0.169117i	$0.0540917\pi$
$920$	0	0
$921$	−14.7750	−0.486854
$922$	0	0
$923$	−32.6506	−1.07471
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.4241	0.342372
$928$	0	0
$929$	−40.2308	−1.31993	−0.659965	−	0.751296i	$-0.729429\pi$
$929$	−40.2308	−1.31993	−0.659965	+	0.751296i	$0.729429\pi$
$930$	0	0
$931$	84.5832	2.77210
$932$	0	0
$933$	−9.15373	−0.299680
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−49.4155	−1.61433	−0.807167	−	0.590324i	$-0.799000\pi$
$937$	−49.4155	−1.61433	−0.807167	+	0.590324i	$0.799000\pi$
$938$	0	0
$939$	−13.2553	−0.432571
$940$	0	0
$941$	22.2653	0.725828	0.362914	−	0.931823i	$-0.381782\pi$
$941$	22.2653	0.725828	0.362914	+	0.931823i	$0.381782\pi$
$942$	0	0
$943$	−20.2546	−0.659579
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−35.6778	−1.15937	−0.579687	−	0.814839i	$-0.696825\pi$
$947$	−35.6778	−1.15937	−0.579687	+	0.814839i	$0.696825\pi$
$948$	0	0
$949$	−18.3428	−0.595433
$950$	0	0
$951$	−0.790550	−0.0256354
$952$	0	0
$953$	2.80677	0.0909203	0.0454602	−	0.998966i	$-0.485525\pi$
$953$	2.80677	0.0909203	0.0454602	+	0.998966i	$0.485525\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−5.94099	−0.192045
$958$	0	0
$959$	−26.3472	−0.850796
$960$	0	0
$961$	−20.0945	−0.648210
$962$	0	0
$963$	−0.220683	−0.00711140
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−11.5283	−0.370724	−0.185362	−	0.982670i	$-0.559346\pi$
$967$	−11.5283	−0.370724	−0.185362	+	0.982670i	$0.559346\pi$
$968$	0	0
$969$	−15.7955	−0.507424
$970$	0	0
$971$	23.1566	0.743130	0.371565	−	0.928407i	$-0.378821\pi$
$971$	23.1566	0.743130	0.371565	+	0.928407i	$0.378821\pi$
$972$	0	0
$973$	−83.2673	−2.66943
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.26069	0.264283	0.132141	−	0.991231i	$-0.457815\pi$
$977$	8.26069	0.264283	0.132141	+	0.991231i	$0.457815\pi$
$978$	0	0
$979$	−72.1408	−2.30563
$980$	0	0
$981$	6.68640	0.213480
$982$	0	0
$983$	−19.7887	−0.631163	−0.315581	−	0.948899i	$-0.602200\pi$
$983$	−19.7887	−0.631163	−0.315581	+	0.948899i	$0.602200\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.19932	0.0381747
$988$	0	0
$989$	−71.2078	−2.26427
$990$	0	0
$991$	21.3884	0.679424	0.339712	−	0.940529i	$-0.389670\pi$
$991$	21.3884	0.679424	0.339712	+	0.940529i	$0.389670\pi$
$992$	0	0
$993$	−3.37370	−0.107061
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−17.5832	−0.556865	−0.278432	−	0.960456i	$-0.589815\pi$
$997$	−17.5832	−0.556865	−0.278432	+	0.960456i	$0.589815\pi$
$998$	0	0
$999$	1.87725	0.0593935

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.a.n.1.12		12
5.2	odd	4		7500.2.d.g.1249.12		24
5.3	odd	4		7500.2.d.g.1249.13		24
5.4	even	2		7500.2.a.m.1.1		12
25.3	odd	20		300.2.o.a.109.2	✓	24
25.4	even	10		1500.2.m.d.1201.1		24
25.6	even	5		1500.2.m.c.301.6		24
25.8	odd	20		1500.2.o.c.949.6		24
25.17	odd	20		300.2.o.a.289.2	yes	24
25.19	even	10		1500.2.m.d.301.1		24
25.21	even	5		1500.2.m.c.1201.6		24
25.22	odd	20		1500.2.o.c.49.6		24
75.17	even	20		900.2.w.c.289.3		24
75.53	even	20		900.2.w.c.109.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.109.2	✓	24	25.3	odd	20
300.2.o.a.289.2	yes	24	25.17	odd	20
900.2.w.c.109.3		24	75.53	even	20
900.2.w.c.289.3		24	75.17	even	20
1500.2.m.c.301.6		24	25.6	even	5
1500.2.m.c.1201.6		24	25.21	even	5
1500.2.m.d.301.1		24	25.19	even	10
1500.2.m.d.1201.1		24	25.4	even	10
1500.2.o.c.49.6		24	25.22	odd	20
1500.2.o.c.949.6		24	25.8	odd	20
7500.2.a.m.1.1		12	5.4	even	2
7500.2.a.n.1.12		12	1.1	even	1	trivial
7500.2.d.g.1249.12		24	5.2	odd	4
7500.2.d.g.1249.13		24	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.62675 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.62675 q^{7} +1.00000 q^{9} -4.94880 q^{11} -3.76277 q^{13} -2.69040 q^{17} +5.87106 q^{19} +4.62675 q^{21} +6.67829 q^{23} +1.00000 q^{27} +1.20049 q^{29} +3.30235 q^{31} -4.94880 q^{33} +1.87725 q^{37} -3.76277 q^{39} -3.03290 q^{41} -10.6626 q^{43} +0.259214 q^{47} +14.4068 q^{49} -2.69040 q^{51} +9.79659 q^{53} +5.87106 q^{57} -9.62455 q^{59} +6.27369 q^{61} +4.62675 q^{63} +2.56053 q^{67} +6.67829 q^{69} +8.67729 q^{71} +4.87481 q^{73} -22.8968 q^{77} +12.4732 q^{79} +1.00000 q^{81} -8.89025 q^{83} +1.20049 q^{87} +14.5774 q^{89} -17.4094 q^{91} +3.30235 q^{93} -3.98689 q^{97} -4.94880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 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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