LMFDB - Embedded newform 6975.2.a.bl.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,8,0,0,-4,0,0,0,0,0,0,0,0,-4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29952.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 13 $ x^4 - 8*x^2 + 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.39417$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.39417 q^{2} +3.73205 q^{4} +0.732051 q^{7} +4.14682 q^{8} +6.54099 q^{11} -3.46410 q^{13} +1.75265 q^{14} +2.46410 q^{16} +0.641516 q^{17} -6.73205 q^{19} +15.6603 q^{22} +3.50531 q^{23} -8.29365 q^{26} +2.73205 q^{28} +7.18251 q^{29} +1.00000 q^{31} -2.39417 q^{32} +1.53590 q^{34} +2.73205 q^{37} -16.1177 q^{38} +11.3293 q^{41} +6.46410 q^{43} +24.4113 q^{44} +8.39230 q^{46} +10.6878 q^{47} -6.46410 q^{49} -12.9282 q^{52} +11.3293 q^{53} +3.03569 q^{56} +17.1962 q^{58} -6.54099 q^{59} -4.19615 q^{61} +2.39417 q^{62} -10.6603 q^{64} -7.66025 q^{67} +2.39417 q^{68} +4.14682 q^{71} +10.0000 q^{73} +6.54099 q^{74} -25.1244 q^{76} +4.78834 q^{77} -5.53590 q^{79} +27.1244 q^{82} -13.0820 q^{83} +15.4762 q^{86} +27.1244 q^{88} -7.65213 q^{89} -2.53590 q^{91} +13.0820 q^{92} +25.5885 q^{94} +13.7321 q^{97} -15.4762 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 4 q^{7} - 4 q^{16} - 20 q^{19} + 28 q^{22} + 4 q^{28} + 4 q^{31} + 20 q^{34} + 4 q^{37} + 12 q^{43} - 8 q^{46} - 12 q^{49} - 24 q^{52} + 48 q^{58} + 4 q^{61} - 8 q^{64} + 4 q^{67} + 40 q^{73}+ \cdots + 48 q^{97}+O(q^{100}) $ 4 * q + 8 * q^4 - 4 * q^7 - 4 * q^16 - 20 * q^19 + 28 * q^22 + 4 * q^28 + 4 * q^31 + 20 * q^34 + 4 * q^37 + 12 * q^43 - 8 * q^46 - 12 * q^49 - 24 * q^52 + 48 * q^58 + 4 * q^61 - 8 * q^64 + 4 * q^67 + 40 * q^73 - 52 * q^76 - 36 * q^79 + 60 * q^82 + 60 * q^88 - 24 * q^91 + 40 * q^94 + 48 * q^97

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 $

$2$	2.39417	1.69293	0.846467	−	0.532441i	$-0.178725\pi$
$2$	2.39417	1.69293	0.846467	+	0.532441i	$0.178725\pi$
$3$	0	0
$3$	0	0
$4$	3.73205	1.86603
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.732051	0.276689	0.138345	−	0.990384i	$-0.455822\pi$
$7$	0.732051	0.276689	0.138345	+	0.990384i	$0.455822\pi$
$8$	4.14682	1.46612
$9$	0	0
$10$	0	0
$11$	6.54099	1.97218	0.986092	−	0.166200i	$-0.0531498\pi$
$11$	6.54099	1.97218	0.986092	+	0.166200i	$0.0531498\pi$
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	1.75265	0.468417
$15$	0	0
$16$	2.46410	0.616025
$17$	0.641516	0.155590	0.0777952	−	0.996969i	$-0.475212\pi$
$17$	0.641516	0.155590	0.0777952	+	0.996969i	$0.475212\pi$
$18$	0	0
$19$	−6.73205	−1.54444	−0.772219	−	0.635356i	$-0.780853\pi$
$19$	−6.73205	−1.54444	−0.772219	+	0.635356i	$0.780853\pi$
$20$	0	0
$21$	0	0
$22$	15.6603	3.33878
$23$	3.50531	0.730907	0.365454	−	0.930830i	$-0.380914\pi$
$23$	3.50531	0.730907	0.365454	+	0.930830i	$0.380914\pi$
$24$	0	0
$25$	0	0
$26$	−8.29365	−1.62652
$27$	0	0
$28$	2.73205	0.516309
$29$	7.18251	1.33376	0.666879	−	0.745166i	$-0.267630\pi$
$29$	7.18251	1.33376	0.666879	+	0.745166i	$0.267630\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−2.39417	−0.423233
$33$	0	0
$34$	1.53590	0.263404
$35$	0	0
$36$	0	0
$37$	2.73205	0.449146	0.224573	−	0.974457i	$-0.427901\pi$
$37$	2.73205	0.449146	0.224573	+	0.974457i	$0.427901\pi$
$38$	−16.1177	−2.61463
$39$	0	0
$40$	0	0
$41$	11.3293	1.76934	0.884672	−	0.466213i	$-0.154382\pi$
$41$	11.3293	1.76934	0.884672	+	0.466213i	$0.154382\pi$
$42$	0	0
$43$	6.46410	0.985766	0.492883	−	0.870096i	$-0.335943\pi$
$43$	6.46410	0.985766	0.492883	+	0.870096i	$0.335943\pi$
$44$	24.4113	3.68015
$45$	0	0
$46$	8.39230	1.23738
$47$	10.6878	1.55898	0.779489	−	0.626416i	$-0.215479\pi$
$47$	10.6878	1.55898	0.779489	+	0.626416i	$0.215479\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	0	0
$52$	−12.9282	−1.79282
$53$	11.3293	1.55620	0.778102	−	0.628138i	$-0.216183\pi$
$53$	11.3293	1.55620	0.778102	+	0.628138i	$0.216183\pi$
$54$	0	0
$55$	0	0
$56$	3.03569	0.405661
$57$	0	0
$58$	17.1962	2.25797
$59$	−6.54099	−0.851565	−0.425782	−	0.904826i	$-0.640001\pi$
$59$	−6.54099	−0.851565	−0.425782	+	0.904826i	$0.640001\pi$
$60$	0	0
$61$	−4.19615	−0.537262	−0.268631	−	0.963243i	$-0.586571\pi$
$61$	−4.19615	−0.537262	−0.268631	+	0.963243i	$0.586571\pi$
$62$	2.39417	0.304060
$63$	0	0
$64$	−10.6603	−1.33253
$65$	0	0
$66$	0	0
$67$	−7.66025	−0.935849	−0.467924	−	0.883768i	$-0.654998\pi$
$67$	−7.66025	−0.935849	−0.467924	+	0.883768i	$0.654998\pi$
$68$	2.39417	0.290336
$69$	0	0
$70$	0	0
$71$	4.14682	0.492138	0.246069	−	0.969252i	$-0.420861\pi$
$71$	4.14682	0.492138	0.246069	+	0.969252i	$0.420861\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	6.54099	0.760375
$75$	0	0
$76$	−25.1244	−2.88196
$77$	4.78834	0.545682
$78$	0	0
$79$	−5.53590	−0.622837	−0.311419	−	0.950273i	$-0.600804\pi$
$79$	−5.53590	−0.622837	−0.311419	+	0.950273i	$0.600804\pi$
$80$	0	0
$81$	0	0
$82$	27.1244	2.99538
$83$	−13.0820	−1.43593	−0.717967	−	0.696077i	$-0.754927\pi$
$83$	−13.0820	−1.43593	−0.717967	+	0.696077i	$0.754927\pi$
$84$	0	0
$85$	0	0
$86$	15.4762	1.66884
$87$	0	0
$88$	27.1244	2.89147
$89$	−7.65213	−0.811124	−0.405562	−	0.914067i	$-0.632924\pi$
$89$	−7.65213	−0.811124	−0.405562	+	0.914067i	$0.632924\pi$
$90$	0	0
$91$	−2.53590	−0.265834
$92$	13.0820	1.36389
$93$	0	0
$94$	25.5885	2.63925
$95$	0	0
$96$	0	0
$97$	13.7321	1.39428	0.697139	−	0.716936i	$-0.254456\pi$
$97$	13.7321	1.39428	0.697139	+	0.716936i	$0.254456\pi$
$98$	−15.4762	−1.56333
$99$	0	0
$100$	0	0
$101$	1.28303	0.127666	0.0638332	−	0.997961i	$-0.479667\pi$
$101$	1.28303	0.127666	0.0638332	+	0.997961i	$0.479667\pi$
$102$	0	0
$103$	14.3923	1.41812	0.709058	−	0.705150i	$-0.249120\pi$
$103$	14.3923	1.41812	0.709058	+	0.705150i	$0.249120\pi$
$104$	−14.3650	−1.40861
$105$	0	0
$106$	27.1244	2.63455
$107$	5.42986	0.524924	0.262462	−	0.964942i	$-0.415466\pi$
$107$	5.42986	0.524924	0.262462	+	0.964942i	$0.415466\pi$
$108$	0	0
$109$	−20.1244	−1.92756	−0.963782	−	0.266692i	$-0.914069\pi$
$109$	−20.1244	−1.92756	−0.963782	+	0.266692i	$0.914069\pi$
$110$	0	0
$111$	0	0
$112$	1.80385	0.170448
$113$	11.3293	1.06577	0.532887	−	0.846186i	$-0.321107\pi$
$113$	11.3293	1.06577	0.532887	+	0.846186i	$0.321107\pi$
$114$	0	0
$115$	0	0
$116$	26.8055	2.48883
$117$	0	0
$118$	−15.6603	−1.44164
$119$	0.469622	0.0430502
$120$	0	0
$121$	31.7846	2.88951
$122$	−10.0463	−0.909550
$123$	0	0
$124$	3.73205	0.335148
$125$	0	0
$126$	0	0
$127$	−8.12436	−0.720920	−0.360460	−	0.932775i	$-0.617380\pi$
$127$	−8.12436	−0.720920	−0.360460	+	0.932775i	$0.617380\pi$
$128$	−20.7341	−1.83265
$129$	0	0
$130$	0	0
$131$	−13.7235	−1.19903	−0.599514	−	0.800364i	$-0.704639\pi$
$131$	−13.7235	−1.19903	−0.599514	+	0.800364i	$0.704639\pi$
$132$	0	0
$133$	−4.92820	−0.427329
$134$	−18.3400	−1.58433
$135$	0	0
$136$	2.66025	0.228115
$137$	−1.11114	−0.0949309	−0.0474655	−	0.998873i	$-0.515114\pi$
$137$	−1.11114	−0.0949309	−0.0474655	+	0.998873i	$0.515114\pi$
$138$	0	0
$139$	11.5359	0.978462	0.489231	−	0.872154i	$-0.337277\pi$
$139$	11.5359	0.978462	0.489231	+	0.872154i	$0.337277\pi$
$140$	0	0
$141$	0	0
$142$	9.92820	0.833156
$143$	−22.6587	−1.89481
$144$	0	0
$145$	0	0
$146$	23.9417	1.98143
$147$	0	0
$148$	10.1962	0.838119
$149$	−19.6230	−1.60758	−0.803789	−	0.594915i	$-0.797186\pi$
$149$	−19.6230	−1.60758	−0.803789	+	0.594915i	$0.797186\pi$
$150$	0	0
$151$	4.26795	0.347321	0.173660	−	0.984806i	$-0.444440\pi$
$151$	4.26795	0.347321	0.173660	+	0.984806i	$0.444440\pi$
$152$	−27.9166	−2.26434
$153$	0	0
$154$	11.4641	0.923804
$155$	0	0
$156$	0	0
$157$	−2.12436	−0.169542	−0.0847710	−	0.996400i	$-0.527016\pi$
$157$	−2.12436	−0.169542	−0.0847710	+	0.996400i	$0.527016\pi$
$158$	−13.2539	−1.05442
$159$	0	0
$160$	0	0
$161$	2.56606	0.202234
$162$	0	0
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	42.2817	3.30164
$165$	0	0
$166$	−31.3205	−2.43094
$167$	−1.75265	−0.135624	−0.0678122	−	0.997698i	$-0.521602\pi$
$167$	−1.75265	−0.135624	−0.0678122	+	0.997698i	$0.521602\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	24.1244	1.83946
$173$	−22.6587	−1.72271	−0.861353	−	0.508006i	$-0.830383\pi$
$173$	−22.6587	−1.72271	−0.861353	+	0.508006i	$0.830383\pi$
$174$	0	0
$175$	0	0
$176$	16.1177	1.21492
$177$	0	0
$178$	−18.3205	−1.37318
$179$	4.31872	0.322796	0.161398	−	0.986889i	$-0.448400\pi$
$179$	4.31872	0.322796	0.161398	+	0.986889i	$0.448400\pi$
$180$	0	0
$181$	−21.1244	−1.57016	−0.785080	−	0.619394i	$-0.787378\pi$
$181$	−21.1244	−1.57016	−0.785080	+	0.619394i	$0.787378\pi$
$182$	−6.07137	−0.450040
$183$	0	0
$184$	14.5359	1.07160
$185$	0	0
$186$	0	0
$187$	4.19615	0.306853
$188$	39.8875	2.90909
$189$	0	0
$190$	0	0
$191$	5.89948	0.426871	0.213436	−	0.976957i	$-0.431535\pi$
$191$	5.89948	0.426871	0.213436	+	0.976957i	$0.431535\pi$
$192$	0	0
$193$	17.3205	1.24676	0.623379	−	0.781920i	$-0.285760\pi$
$193$	17.3205	1.24676	0.623379	+	0.781920i	$0.285760\pi$
$194$	32.8769	2.36042
$195$	0	0
$196$	−24.1244	−1.72317
$197$	3.67720	0.261990	0.130995	−	0.991383i	$-0.458183\pi$
$197$	3.67720	0.261990	0.130995	+	0.991383i	$0.458183\pi$
$198$	0	0
$199$	17.8564	1.26581	0.632904	−	0.774231i	$-0.281863\pi$
$199$	17.8564	1.26581	0.632904	+	0.774231i	$0.281863\pi$
$200$	0	0
$201$	0	0
$202$	3.07180	0.216131
$203$	5.25796	0.369037
$204$	0	0
$205$	0	0
$206$	34.4576	2.40078
$207$	0	0
$208$	−8.53590	−0.591858
$209$	−44.0343	−3.04592
$210$	0	0
$211$	−3.07180	−0.211471	−0.105736	−	0.994394i	$-0.533720\pi$
$211$	−3.07180	−0.211471	−0.105736	+	0.994394i	$0.533720\pi$
$212$	42.2817	2.90392
$213$	0	0
$214$	13.0000	0.888662
$215$	0	0
$216$	0	0
$217$	0.732051	0.0496948
$218$	−48.1811	−3.26324
$219$	0	0
$220$	0	0
$221$	−2.22228	−0.149486
$222$	0	0
$223$	−1.92820	−0.129122	−0.0645610	−	0.997914i	$-0.520565\pi$
$223$	−1.92820	−0.129122	−0.0645610	+	0.997914i	$0.520565\pi$
$224$	−1.75265	−0.117104
$225$	0	0
$226$	27.1244	1.80429
$227$	0.171894	0.0114090	0.00570449	−	0.999984i	$-0.498184\pi$
$227$	0.171894	0.0114090	0.00570449	+	0.999984i	$0.498184\pi$
$228$	0	0
$229$	12.9282	0.854320	0.427160	−	0.904176i	$-0.359514\pi$
$229$	12.9282	0.854320	0.427160	+	0.904176i	$0.359514\pi$
$230$	0	0
$231$	0	0
$232$	29.7846	1.95546
$233$	−9.10706	−0.596623	−0.298312	−	0.954469i	$-0.596423\pi$
$233$	−9.10706	−0.596623	−0.298312	+	0.954469i	$0.596423\pi$
$234$	0	0
$235$	0	0
$236$	−24.4113	−1.58904
$237$	0	0
$238$	1.12436	0.0728812
$239$	10.0463	0.649841	0.324921	−	0.945741i	$-0.394662\pi$
$239$	10.0463	0.649841	0.324921	+	0.945741i	$0.394662\pi$
$240$	0	0
$241$	−20.0526	−1.29170	−0.645849	−	0.763465i	$-0.723497\pi$
$241$	−20.0526	−1.29170	−0.645849	+	0.763465i	$0.723497\pi$
$242$	76.0978	4.89175
$243$	0	0
$244$	−15.6603	−1.00255
$245$	0	0
$246$	0	0
$247$	23.3205	1.48385
$248$	4.14682	0.263324
$249$	0	0
$250$	0	0
$251$	−3.97493	−0.250895	−0.125448	−	0.992100i	$-0.540037\pi$
$251$	−3.97493	−0.250895	−0.125448	+	0.992100i	$0.540037\pi$
$252$	0	0
$253$	22.9282	1.44148
$254$	−19.4511	−1.22047
$255$	0	0
$256$	−28.3205	−1.77003
$257$	10.5159	0.655965	0.327983	−	0.944684i	$-0.393631\pi$
$257$	10.5159	0.655965	0.327983	+	0.944684i	$0.393631\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	−32.8564	−2.02988
$263$	16.5873	1.02282	0.511408	−	0.859338i	$-0.329124\pi$
$263$	16.5873	1.02282	0.511408	+	0.859338i	$0.329124\pi$
$264$	0	0
$265$	0	0
$266$	−11.7990	−0.723440
$267$	0	0
$268$	−28.5885	−1.74632
$269$	−4.14682	−0.252836	−0.126418	−	0.991977i	$-0.540348\pi$
$269$	−4.14682	−0.252836	−0.126418	+	0.991977i	$0.540348\pi$
$270$	0	0
$271$	3.73205	0.226706	0.113353	−	0.993555i	$-0.463841\pi$
$271$	3.73205	0.226706	0.113353	+	0.993555i	$0.463841\pi$
$272$	1.58076	0.0958477
$273$	0	0
$274$	−2.66025	−0.160712
$275$	0	0
$276$	0	0
$277$	−20.2487	−1.21663	−0.608314	−	0.793697i	$-0.708154\pi$
$277$	−20.2487	−1.21663	−0.608314	+	0.793697i	$0.708154\pi$
$278$	27.6189	1.65647
$279$	0	0
$280$	0	0
$281$	5.25796	0.313664	0.156832	−	0.987625i	$-0.449872\pi$
$281$	5.25796	0.313664	0.156832	+	0.987625i	$0.449872\pi$
$282$	0	0
$283$	3.60770	0.214455	0.107228	−	0.994234i	$-0.465803\pi$
$283$	3.60770	0.214455	0.107228	+	0.994234i	$0.465803\pi$
$284$	15.4762	0.918341
$285$	0	0
$286$	−54.2487	−3.20779
$287$	8.29365	0.489559
$288$	0	0
$289$	−16.5885	−0.975792
$290$	0	0
$291$	0	0
$292$	37.3205	2.18402
$293$	−18.6837	−1.09152	−0.545758	−	0.837943i	$-0.683758\pi$
$293$	−18.6837	−1.09152	−0.545758	+	0.837943i	$0.683758\pi$
$294$	0	0
$295$	0	0
$296$	11.3293	0.658504
$297$	0	0
$298$	−46.9808	−2.72152
$299$	−12.1427	−0.702233
$300$	0	0
$301$	4.73205	0.272751
$302$	10.2182	0.587991
$303$	0	0
$304$	−16.5885	−0.951413
$305$	0	0
$306$	0	0
$307$	−12.7846	−0.729656	−0.364828	−	0.931075i	$-0.618872\pi$
$307$	−12.7846	−0.729656	−0.364828	+	0.931075i	$0.618872\pi$
$308$	17.8703	1.01826
$309$	0	0
$310$	0	0
$311$	−15.4762	−0.877572	−0.438786	−	0.898591i	$-0.644591\pi$
$311$	−15.4762	−0.877572	−0.438786	+	0.898591i	$0.644591\pi$
$312$	0	0
$313$	7.07180	0.399722	0.199861	−	0.979824i	$-0.435951\pi$
$313$	7.07180	0.399722	0.199861	+	0.979824i	$0.435951\pi$
$314$	−5.08607	−0.287023
$315$	0	0
$316$	−20.6603	−1.16223
$317$	17.4007	0.977321	0.488661	−	0.872474i	$-0.337486\pi$
$317$	17.4007	0.977321	0.488661	+	0.872474i	$0.337486\pi$
$318$	0	0
$319$	46.9808	2.63042
$320$	0	0
$321$	0	0
$322$	6.14359	0.342369
$323$	−4.31872	−0.240300
$324$	0	0
$325$	0	0
$326$	−33.5184	−1.85641
$327$	0	0
$328$	46.9808	2.59408
$329$	7.82403	0.431353
$330$	0	0
$331$	−16.9282	−0.930458	−0.465229	−	0.885190i	$-0.654028\pi$
$331$	−16.9282	−0.930458	−0.465229	+	0.885190i	$0.654028\pi$
$332$	−48.8226	−2.67949
$333$	0	0
$334$	−4.19615	−0.229603
$335$	0	0
$336$	0	0
$337$	15.2679	0.831698	0.415849	−	0.909434i	$-0.363484\pi$
$337$	15.2679	0.831698	0.415849	+	0.909434i	$0.363484\pi$
$338$	−2.39417	−0.130226
$339$	0	0
$340$	0	0
$341$	6.54099	0.354215
$342$	0	0
$343$	−9.85641	−0.532196
$344$	26.8055	1.44526
$345$	0	0
$346$	−54.2487	−2.91643
$347$	27.9166	1.49864	0.749322	−	0.662206i	$-0.230380\pi$
$347$	27.9166	1.49864	0.749322	+	0.662206i	$0.230380\pi$
$348$	0	0
$349$	8.26795	0.442573	0.221287	−	0.975209i	$-0.428974\pi$
$349$	8.26795	0.442573	0.221287	+	0.975209i	$0.428974\pi$
$350$	0	0
$351$	0	0
$352$	−15.6603	−0.834694
$353$	−34.9272	−1.85899	−0.929495	−	0.368835i	$-0.879757\pi$
$353$	−34.9272	−1.85899	−0.929495	+	0.368835i	$0.879757\pi$
$354$	0	0
$355$	0	0
$356$	−28.5581	−1.51358
$357$	0	0
$358$	10.3397	0.546473
$359$	33.6442	1.77567	0.887837	−	0.460158i	$-0.152207\pi$
$359$	33.6442	1.77567	0.887837	+	0.460158i	$0.152207\pi$
$360$	0	0
$361$	26.3205	1.38529
$362$	−50.5753	−2.65818
$363$	0	0
$364$	−9.46410	−0.496054
$365$	0	0
$366$	0	0
$367$	−29.9808	−1.56498	−0.782492	−	0.622661i	$-0.786051\pi$
$367$	−29.9808	−1.56498	−0.782492	+	0.622661i	$0.786051\pi$
$368$	8.63744	0.450257
$369$	0	0
$370$	0	0
$371$	8.29365	0.430585
$372$	0	0
$373$	−4.07180	−0.210830	−0.105415	−	0.994428i	$-0.533617\pi$
$373$	−4.07180	−0.210830	−0.105415	+	0.994428i	$0.533617\pi$
$374$	10.0463	0.519482
$375$	0	0
$376$	44.3205	2.28566
$377$	−24.8809	−1.28143
$378$	0	0
$379$	21.3205	1.09516	0.547580	−	0.836753i	$-0.315549\pi$
$379$	21.3205	1.09516	0.547580	+	0.836753i	$0.315549\pi$
$380$	0	0
$381$	0	0
$382$	14.1244	0.722665
$383$	3.84910	0.196680	0.0983398	−	0.995153i	$-0.468647\pi$
$383$	3.84910	0.196680	0.0983398	+	0.995153i	$0.468647\pi$
$384$	0	0
$385$	0	0
$386$	41.4682	2.11068
$387$	0	0
$388$	51.2487	2.60176
$389$	12.2686	0.622042	0.311021	−	0.950403i	$-0.399329\pi$
$389$	12.2686	0.622042	0.311021	+	0.950403i	$0.399329\pi$
$390$	0	0
$391$	2.24871	0.113722
$392$	−26.8055	−1.35388
$393$	0	0
$394$	8.80385	0.443531
$395$	0	0
$396$	0	0
$397$	−10.5359	−0.528782	−0.264391	−	0.964416i	$-0.585171\pi$
$397$	−10.5359	−0.528782	−0.264391	+	0.964416i	$0.585171\pi$
$398$	42.7513	2.14293
$399$	0	0
$400$	0	0
$401$	13.8954	0.693903	0.346952	−	0.937883i	$-0.387217\pi$
$401$	13.8954	0.693903	0.346952	+	0.937883i	$0.387217\pi$
$402$	0	0
$403$	−3.46410	−0.172559
$404$	4.78834	0.238229
$405$	0	0
$406$	12.5885	0.624755
$407$	17.8703	0.885799
$408$	0	0
$409$	23.6603	1.16992	0.584962	−	0.811061i	$-0.301109\pi$
$409$	23.6603	1.16992	0.584962	+	0.811061i	$0.301109\pi$
$410$	0	0
$411$	0	0
$412$	53.7128	2.64624
$413$	−4.78834	−0.235619
$414$	0	0
$415$	0	0
$416$	8.29365	0.406630
$417$	0	0
$418$	−105.426	−5.15654
$419$	−0.641516	−0.0313401	−0.0156701	−	0.999877i	$-0.504988\pi$
$419$	−0.641516	−0.0313401	−0.0156701	+	0.999877i	$0.504988\pi$
$420$	0	0
$421$	−9.00000	−0.438633	−0.219317	−	0.975654i	$-0.570383\pi$
$421$	−9.00000	−0.438633	−0.219317	+	0.975654i	$0.570383\pi$
$422$	−7.35440	−0.358007
$423$	0	0
$424$	46.9808	2.28159
$425$	0	0
$426$	0	0
$427$	−3.07180	−0.148655
$428$	20.2645	0.979522
$429$	0	0
$430$	0	0
$431$	10.9855	0.529155	0.264578	−	0.964364i	$-0.414767\pi$
$431$	10.9855	0.529155	0.264578	+	0.964364i	$0.414767\pi$
$432$	0	0
$433$	−26.1962	−1.25891	−0.629453	−	0.777038i	$-0.716721\pi$
$433$	−26.1962	−1.25891	−0.629453	+	0.777038i	$0.716721\pi$
$434$	1.75265	0.0841301
$435$	0	0
$436$	−75.1051	−3.59688
$437$	−23.5979	−1.12884
$438$	0	0
$439$	−36.5885	−1.74627	−0.873136	−	0.487477i	$-0.837917\pi$
$439$	−36.5885	−1.74627	−0.873136	+	0.487477i	$0.837917\pi$
$440$	0	0
$441$	0	0
$442$	−5.32051	−0.253071
$443$	24.2394	1.15165	0.575825	−	0.817573i	$-0.304681\pi$
$443$	24.2394	1.15165	0.575825	+	0.817573i	$0.304681\pi$
$444$	0	0
$445$	0	0
$446$	−4.61645	−0.218595
$447$	0	0
$448$	−7.80385	−0.368697
$449$	−33.3465	−1.57372	−0.786859	−	0.617133i	$-0.788294\pi$
$449$	−33.3465	−1.57372	−0.786859	+	0.617133i	$0.788294\pi$
$450$	0	0
$451$	74.1051	3.48947
$452$	42.2817	1.98876
$453$	0	0
$454$	0.411543	0.0193147
$455$	0	0
$456$	0	0
$457$	38.2487	1.78920	0.894600	−	0.446869i	$-0.147461\pi$
$457$	38.2487	1.78920	0.894600	+	0.446869i	$0.147461\pi$
$458$	30.9523	1.44631
$459$	0	0
$460$	0	0
$461$	−6.88478	−0.320656	−0.160328	−	0.987064i	$-0.551255\pi$
$461$	−6.88478	−0.320656	−0.160328	+	0.987064i	$0.551255\pi$
$462$	0	0
$463$	−29.9282	−1.39088	−0.695441	−	0.718583i	$-0.744791\pi$
$463$	−29.9282	−1.39088	−0.695441	+	0.718583i	$0.744791\pi$
$464$	17.6984	0.821629
$465$	0	0
$466$	−21.8038	−1.01004
$467$	−0.641516	−0.0296858	−0.0148429	−	0.999890i	$-0.504725\pi$
$467$	−0.641516	−0.0296858	−0.0148429	+	0.999890i	$0.504725\pi$
$468$	0	0
$469$	−5.60770	−0.258939
$470$	0	0
$471$	0	0
$472$	−27.1244	−1.24850
$473$	42.2817	1.94411
$474$	0	0
$475$	0	0
$476$	1.75265	0.0803328
$477$	0	0
$478$	24.0526	1.10014
$479$	4.61645	0.210931	0.105465	−	0.994423i	$-0.466367\pi$
$479$	4.61645	0.210931	0.105465	+	0.994423i	$0.466367\pi$
$480$	0	0
$481$	−9.46410	−0.431526
$482$	−48.0092	−2.18676
$483$	0	0
$484$	118.622	5.39190
$485$	0	0
$486$	0	0
$487$	−16.6603	−0.754948	−0.377474	−	0.926020i	$-0.623207\pi$
$487$	−16.6603	−0.754948	−0.377474	+	0.926020i	$0.623207\pi$
$488$	−17.4007	−0.787693
$489$	0	0
$490$	0	0
$491$	−42.2817	−1.90814	−0.954072	−	0.299577i	$-0.903154\pi$
$491$	−42.2817	−1.90814	−0.954072	+	0.299577i	$0.903154\pi$
$492$	0	0
$493$	4.60770	0.207520
$494$	55.8333	2.51206
$495$	0	0
$496$	2.46410	0.110641
$497$	3.03569	0.136169
$498$	0	0
$499$	−19.1962	−0.859338	−0.429669	−	0.902987i	$-0.641370\pi$
$499$	−19.1962	−0.859338	−0.429669	+	0.902987i	$0.641370\pi$
$500$	0	0
$501$	0	0
$502$	−9.51666	−0.424749
$503$	−26.8055	−1.19520	−0.597599	−	0.801795i	$-0.703878\pi$
$503$	−26.8055	−1.19520	−0.597599	+	0.801795i	$0.703878\pi$
$504$	0	0
$505$	0	0
$506$	54.8940	2.44034
$507$	0	0
$508$	−30.3205	−1.34526
$509$	0.469622	0.0208156	0.0104078	−	0.999946i	$-0.496687\pi$
$509$	0.469622	0.0208156	0.0104078	+	0.999946i	$0.496687\pi$
$510$	0	0
$511$	7.32051	0.323840
$512$	−26.3359	−1.16389
$513$	0	0
$514$	25.1769	1.11051
$515$	0	0
$516$	0	0
$517$	69.9090	3.07459
$518$	4.78834	0.210388
$519$	0	0
$520$	0	0
$521$	−27.9166	−1.22305	−0.611525	−	0.791225i	$-0.709444\pi$
$521$	−27.9166	−1.22305	−0.611525	+	0.791225i	$0.709444\pi$
$522$	0	0
$523$	−10.1244	−0.442707	−0.221354	−	0.975194i	$-0.571047\pi$
$523$	−10.1244	−0.442707	−0.221354	+	0.975194i	$0.571047\pi$
$524$	−51.2168	−2.23742
$525$	0	0
$526$	39.7128	1.73156
$527$	0.641516	0.0279449
$528$	0	0
$529$	−10.7128	−0.465774
$530$	0	0
$531$	0	0
$532$	−18.3923	−0.797408
$533$	−39.2460	−1.69993
$534$	0	0
$535$	0	0
$536$	−31.7657	−1.37207
$537$	0	0
$538$	−9.92820	−0.428035
$539$	−42.2817	−1.82120
$540$	0	0
$541$	−25.7846	−1.10857	−0.554283	−	0.832328i	$-0.687008\pi$
$541$	−25.7846	−1.10857	−0.554283	+	0.832328i	$0.687008\pi$
$542$	8.93516	0.383798
$543$	0	0
$544$	−1.53590	−0.0658511
$545$	0	0
$546$	0	0
$547$	−3.60770	−0.154254	−0.0771270	−	0.997021i	$-0.524575\pi$
$547$	−3.60770	−0.154254	−0.0771270	+	0.997021i	$0.524575\pi$
$548$	−4.14682	−0.177144
$549$	0	0
$550$	0	0
$551$	−48.3530	−2.05991
$552$	0	0
$553$	−4.05256	−0.172332
$554$	−48.4789	−2.05967
$555$	0	0
$556$	43.0526	1.82584
$557$	24.2394	1.02706	0.513529	−	0.858072i	$-0.328338\pi$
$557$	24.2394	1.02706	0.513529	+	0.858072i	$0.328338\pi$
$558$	0	0
$559$	−22.3923	−0.947094
$560$	0	0
$561$	0	0
$562$	12.5885	0.531012
$563$	15.4762	0.652242	0.326121	−	0.945328i	$-0.394258\pi$
$563$	15.4762	0.652242	0.326121	+	0.945328i	$0.394258\pi$
$564$	0	0
$565$	0	0
$566$	8.63744	0.363059
$567$	0	0
$568$	17.1962	0.721535
$569$	−6.88478	−0.288625	−0.144313	−	0.989532i	$-0.546097\pi$
$569$	−6.88478	−0.288625	−0.144313	+	0.989532i	$0.546097\pi$
$570$	0	0
$571$	−0.392305	−0.0164174	−0.00820872	−	0.999966i	$-0.502613\pi$
$571$	−0.392305	−0.0164174	−0.00820872	+	0.999966i	$0.502613\pi$
$572$	−84.5633	−3.53577
$573$	0	0
$574$	19.8564	0.828790
$575$	0	0
$576$	0	0
$577$	−10.1244	−0.421482	−0.210741	−	0.977542i	$-0.567588\pi$
$577$	−10.1244	−0.421482	−0.210741	+	0.977542i	$0.567588\pi$
$578$	−39.7156	−1.65195
$579$	0	0
$580$	0	0
$581$	−9.57668	−0.397308
$582$	0	0
$583$	74.1051	3.06912
$584$	41.4682	1.71597
$585$	0	0
$586$	−44.7321	−1.84786
$587$	14.3650	0.592908	0.296454	−	0.955047i	$-0.404196\pi$
$587$	14.3650	0.592908	0.296454	+	0.955047i	$0.404196\pi$
$588$	0	0
$589$	−6.73205	−0.277389
$590$	0	0
$591$	0	0
$592$	6.73205	0.276686
$593$	−24.8809	−1.02174	−0.510869	−	0.859659i	$-0.670676\pi$
$593$	−24.8809	−1.02174	−0.510869	+	0.859659i	$0.670676\pi$
$594$	0	0
$595$	0	0
$596$	−73.2340	−2.99978
$597$	0	0
$598$	−29.0718	−1.18883
$599$	21.0779	0.861220	0.430610	−	0.902538i	$-0.358298\pi$
$599$	21.0779	0.861220	0.430610	+	0.902538i	$0.358298\pi$
$600$	0	0
$601$	−14.0526	−0.573216	−0.286608	−	0.958048i	$-0.592528\pi$
$601$	−14.0526	−0.573216	−0.286608	+	0.958048i	$0.592528\pi$
$602$	11.3293	0.461749
$603$	0	0
$604$	15.9282	0.648109
$605$	0	0
$606$	0	0
$607$	9.07180	0.368213	0.184106	−	0.982906i	$-0.441061\pi$
$607$	9.07180	0.368213	0.184106	+	0.982906i	$0.441061\pi$
$608$	16.1177	0.653658
$609$	0	0
$610$	0	0
$611$	−37.0237	−1.49782
$612$	0	0
$613$	11.1244	0.449308	0.224654	−	0.974439i	$-0.427875\pi$
$613$	11.1244	0.449308	0.224654	+	0.974439i	$0.427875\pi$
$614$	−30.6085	−1.23526
$615$	0	0
$616$	19.8564	0.800037
$617$	−39.5898	−1.59382	−0.796912	−	0.604096i	$-0.793534\pi$
$617$	−39.5898	−1.59382	−0.796912	+	0.604096i	$0.793534\pi$
$618$	0	0
$619$	−19.9282	−0.800982	−0.400491	−	0.916301i	$-0.631160\pi$
$619$	−19.9282	−0.800982	−0.400491	+	0.916301i	$0.631160\pi$
$620$	0	0
$621$	0	0
$622$	−37.0526	−1.48567
$623$	−5.60175	−0.224429
$624$	0	0
$625$	0	0
$626$	16.9311	0.676702
$627$	0	0
$628$	−7.92820	−0.316370
$629$	1.75265	0.0698829
$630$	0	0
$631$	−41.0526	−1.63428	−0.817138	−	0.576442i	$-0.804441\pi$
$631$	−41.0526	−1.63428	−0.817138	+	0.576442i	$0.804441\pi$
$632$	−22.9564	−0.913156
$633$	0	0
$634$	41.6603	1.65454
$635$	0	0
$636$	0	0
$637$	22.3923	0.887215
$638$	112.480	4.45312
$639$	0	0
$640$	0	0
$641$	6.19721	0.244775	0.122387	−	0.992482i	$-0.460945\pi$
$641$	6.19721	0.244775	0.122387	+	0.992482i	$0.460945\pi$
$642$	0	0
$643$	−23.0526	−0.909104	−0.454552	−	0.890720i	$-0.650201\pi$
$643$	−23.0526	−0.909104	−0.454552	+	0.890720i	$0.650201\pi$
$644$	9.57668	0.377374
$645$	0	0
$646$	−10.3397	−0.406812
$647$	20.5622	0.808385	0.404192	−	0.914674i	$-0.367553\pi$
$647$	20.5622	0.808385	0.404192	+	0.914674i	$0.367553\pi$
$648$	0	0
$649$	−42.7846	−1.67944
$650$	0	0
$651$	0	0
$652$	−52.2487	−2.04622
$653$	−32.7050	−1.27984	−0.639922	−	0.768440i	$-0.721033\pi$
$653$	−32.7050	−1.27984	−0.639922	+	0.768440i	$0.721033\pi$
$654$	0	0
$655$	0	0
$656$	27.9166	1.08996
$657$	0	0
$658$	18.7321	0.730251
$659$	−11.3293	−0.441328	−0.220664	−	0.975350i	$-0.570822\pi$
$659$	−11.3293	−0.441328	−0.220664	+	0.975350i	$0.570822\pi$
$660$	0	0
$661$	10.4641	0.407006	0.203503	−	0.979074i	$-0.434767\pi$
$661$	10.4641	0.407006	0.203503	+	0.979074i	$0.434767\pi$
$662$	−40.5290	−1.57520
$663$	0	0
$664$	−54.2487	−2.10526
$665$	0	0
$666$	0	0
$667$	25.1769	0.974854
$668$	−6.54099	−0.253079
$669$	0	0
$670$	0	0
$671$	−27.4470	−1.05958
$672$	0	0
$673$	23.8038	0.917571	0.458785	−	0.888547i	$-0.348285\pi$
$673$	23.8038	0.917571	0.458785	+	0.888547i	$0.348285\pi$
$674$	36.5541	1.40801
$675$	0	0
$676$	−3.73205	−0.143540
$677$	−6.71289	−0.257997	−0.128999	−	0.991645i	$-0.541176\pi$
$677$	−6.71289	−0.257997	−0.128999	+	0.991645i	$0.541176\pi$
$678$	0	0
$679$	10.0526	0.385782
$680$	0	0
$681$	0	0
$682$	15.6603	0.599662
$683$	15.8199	0.605333	0.302667	−	0.953096i	$-0.402123\pi$
$683$	15.8199	0.605333	0.302667	+	0.953096i	$0.402123\pi$
$684$	0	0
$685$	0	0
$686$	−23.5979	−0.900973
$687$	0	0
$688$	15.9282	0.607257
$689$	−39.2460	−1.49515
$690$	0	0
$691$	11.0718	0.421191	0.210595	−	0.977573i	$-0.432460\pi$
$691$	11.0718	0.421191	0.210595	+	0.977573i	$0.432460\pi$
$692$	−84.5633	−3.21461
$693$	0	0
$694$	66.8372	2.53710
$695$	0	0
$696$	0	0
$697$	7.26795	0.275293
$698$	19.7949	0.749247
$699$	0	0
$700$	0	0
$701$	33.1746	1.25299	0.626494	−	0.779427i	$-0.284489\pi$
$701$	33.1746	1.25299	0.626494	+	0.779427i	$0.284489\pi$
$702$	0	0
$703$	−18.3923	−0.693679
$704$	−69.7287	−2.62800
$705$	0	0
$706$	−83.6218	−3.14715
$707$	0.939245	0.0353239
$708$	0	0
$709$	−35.9090	−1.34859	−0.674295	−	0.738462i	$-0.735552\pi$
$709$	−35.9090	−1.34859	−0.674295	+	0.738462i	$0.735552\pi$
$710$	0	0
$711$	0	0
$712$	−31.7321	−1.18921
$713$	3.50531	0.131275
$714$	0	0
$715$	0	0
$716$	16.1177	0.602346
$717$	0	0
$718$	80.5500	3.00610
$719$	−5.72758	−0.213603	−0.106801	−	0.994280i	$-0.534061\pi$
$719$	−5.72758	−0.213603	−0.106801	+	0.994280i	$0.534061\pi$
$720$	0	0
$721$	10.5359	0.392377
$722$	63.0158	2.34520
$723$	0	0
$724$	−78.8372	−2.92996
$725$	0	0
$726$	0	0
$727$	21.7128	0.805284	0.402642	−	0.915358i	$-0.368092\pi$
$727$	21.7128	0.805284	0.402642	+	0.915358i	$0.368092\pi$
$728$	−10.5159	−0.389746
$729$	0	0
$730$	0	0
$731$	4.14682	0.153376
$732$	0	0
$733$	46.3205	1.71089	0.855444	−	0.517896i	$-0.173285\pi$
$733$	46.3205	1.71089	0.855444	+	0.517896i	$0.173285\pi$
$734$	−71.7790	−2.64941
$735$	0	0
$736$	−8.39230	−0.309344
$737$	−50.1057	−1.84567
$738$	0	0
$739$	−41.5885	−1.52986	−0.764928	−	0.644116i	$-0.777226\pi$
$739$	−41.5885	−1.52986	−0.764928	+	0.644116i	$0.777226\pi$
$740$	0	0
$741$	0	0
$742$	19.8564	0.728952
$743$	−24.0675	−0.882952	−0.441476	−	0.897273i	$-0.645545\pi$
$743$	−24.0675	−0.882952	−0.441476	+	0.897273i	$0.645545\pi$
$744$	0	0
$745$	0	0
$746$	−9.74857	−0.356921
$747$	0	0
$748$	15.6603	0.572596
$749$	3.97493	0.145241
$750$	0	0
$751$	−18.7321	−0.683542	−0.341771	−	0.939783i	$-0.611027\pi$
$751$	−18.7321	−0.683542	−0.341771	+	0.939783i	$0.611027\pi$
$752$	26.3359	0.960370
$753$	0	0
$754$	−59.5692	−2.16938
$755$	0	0
$756$	0	0
$757$	−47.7654	−1.73606	−0.868031	−	0.496510i	$-0.834615\pi$
$757$	−47.7654	−1.73606	−0.868031	+	0.496510i	$0.834615\pi$
$758$	51.0449	1.85404
$759$	0	0
$760$	0	0
$761$	−4.96023	−0.179808	−0.0899042	−	0.995950i	$-0.528656\pi$
$761$	−4.96023	−0.179808	−0.0899042	+	0.995950i	$0.528656\pi$
$762$	0	0
$763$	−14.7321	−0.533336
$764$	22.0172	0.796553
$765$	0	0
$766$	9.21539	0.332966
$767$	22.6587	0.818157
$768$	0	0
$769$	4.12436	0.148728	0.0743640	−	0.997231i	$-0.476307\pi$
$769$	4.12436	0.148728	0.0743640	+	0.997231i	$0.476307\pi$
$770$	0	0
$771$	0	0
$772$	64.6410	2.32648
$773$	1.11114	0.0399649	0.0199824	−	0.999800i	$-0.493639\pi$
$773$	1.11114	0.0399649	0.0199824	+	0.999800i	$0.493639\pi$
$774$	0	0
$775$	0	0
$776$	56.9444	2.04418
$777$	0	0
$778$	29.3731	1.05308
$779$	−76.2697	−2.73264
$780$	0	0
$781$	27.1244	0.970586
$782$	5.38380	0.192524
$783$	0	0
$784$	−15.9282	−0.568864
$785$	0	0
$786$	0	0
$787$	−20.1436	−0.718042	−0.359021	−	0.933330i	$-0.616889\pi$
$787$	−20.1436	−0.718042	−0.359021	+	0.933330i	$0.616889\pi$
$788$	13.7235	0.488880
$789$	0	0
$790$	0	0
$791$	8.29365	0.294888
$792$	0	0
$793$	14.5359	0.516185
$794$	−25.2247	−0.895192
$795$	0	0
$796$	66.6410	2.36203
$797$	0.813410	0.0288124	0.0144062	−	0.999896i	$-0.495414\pi$
$797$	0.813410	0.0288124	0.0144062	+	0.999896i	$0.495414\pi$
$798$	0	0
$799$	6.85641	0.242562
$800$	0	0
$801$	0	0
$802$	33.2679	1.17473
$803$	65.4099	2.30827
$804$	0	0
$805$	0	0
$806$	−8.29365	−0.292131
$807$	0	0
$808$	5.32051	0.187175
$809$	−9.40479	−0.330655	−0.165327	−	0.986239i	$-0.552868\pi$
$809$	−9.40479	−0.330655	−0.165327	+	0.986239i	$0.552868\pi$
$810$	0	0
$811$	23.8038	0.835866	0.417933	−	0.908478i	$-0.362755\pi$
$811$	23.8038	0.835866	0.417933	+	0.908478i	$0.362755\pi$
$812$	19.6230	0.688632
$813$	0	0
$814$	42.7846	1.49960
$815$	0	0
$816$	0	0
$817$	−43.5167	−1.52246
$818$	56.6467	1.98060
$819$	0	0
$820$	0	0
$821$	−37.8371	−1.32052	−0.660262	−	0.751035i	$-0.729555\pi$
$821$	−37.8371	−1.32052	−0.660262	+	0.751035i	$0.729555\pi$
$822$	0	0
$823$	42.3923	1.47770	0.738851	−	0.673868i	$-0.235369\pi$
$823$	42.3923	1.47770	0.738851	+	0.673868i	$0.235369\pi$
$824$	59.6824	2.07913
$825$	0	0
$826$	−11.4641	−0.398887
$827$	50.5753	1.75868	0.879338	−	0.476199i	$-0.157986\pi$
$827$	50.5753	1.75868	0.879338	+	0.476199i	$0.157986\pi$
$828$	0	0
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	0	0
$831$	0	0
$832$	36.9282	1.28026
$833$	−4.14682	−0.143679
$834$	0	0
$835$	0	0
$836$	−164.338	−5.68376
$837$	0	0
$838$	−1.53590	−0.0530567
$839$	11.6271	0.401411	0.200705	−	0.979652i	$-0.435677\pi$
$839$	11.6271	0.401411	0.200705	+	0.979652i	$0.435677\pi$
$840$	0	0
$841$	22.5885	0.778912
$842$	−21.5475	−0.742577
$843$	0	0
$844$	−11.4641	−0.394611
$845$	0	0
$846$	0	0
$847$	23.2679	0.799496
$848$	27.9166	0.958661
$849$	0	0
$850$	0	0
$851$	9.57668	0.328284
$852$	0	0
$853$	38.8038	1.32862	0.664309	−	0.747458i	$-0.268726\pi$
$853$	38.8038	1.32862	0.664309	+	0.747458i	$0.268726\pi$
$854$	−7.35440	−0.251663
$855$	0	0
$856$	22.5167	0.769604
$857$	9.10706	0.311091	0.155546	−	0.987829i	$-0.450286\pi$
$857$	9.10706	0.311091	0.155546	+	0.987829i	$0.450286\pi$
$858$	0	0
$859$	−38.3731	−1.30927	−0.654636	−	0.755944i	$-0.727178\pi$
$859$	−38.3731	−1.30927	−0.654636	+	0.755944i	$0.727178\pi$
$860$	0	0
$861$	0	0
$862$	26.3013	0.895825
$863$	−19.2792	−0.656271	−0.328136	−	0.944631i	$-0.606420\pi$
$863$	−19.2792	−0.656271	−0.328136	+	0.944631i	$0.606420\pi$
$864$	0	0
$865$	0	0
$866$	−62.7180	−2.13125
$867$	0	0
$868$	2.73205	0.0927318
$869$	−36.2103	−1.22835
$870$	0	0
$871$	26.5359	0.899135
$872$	−83.4522	−2.82605
$873$	0	0
$874$	−56.4974	−1.91105
$875$	0	0
$876$	0	0
$877$	29.1962	0.985884	0.492942	−	0.870062i	$-0.335921\pi$
$877$	29.1962	0.985884	0.492942	+	0.870062i	$0.335921\pi$
$878$	−87.5990	−2.95632
$879$	0	0
$880$	0	0
$881$	−0.641516	−0.0216132	−0.0108066	−	0.999942i	$-0.503440\pi$
$881$	−0.641516	−0.0216132	−0.0108066	+	0.999942i	$0.503440\pi$
$882$	0	0
$883$	−20.6410	−0.694625	−0.347313	−	0.937749i	$-0.612906\pi$
$883$	−20.6410	−0.694625	−0.347313	+	0.937749i	$0.612906\pi$
$884$	−8.29365	−0.278946
$885$	0	0
$886$	58.0333	1.94967
$887$	21.5475	0.723495	0.361748	−	0.932276i	$-0.382180\pi$
$887$	21.5475	0.723495	0.361748	+	0.932276i	$0.382180\pi$
$888$	0	0
$889$	−5.94744	−0.199471
$890$	0	0
$891$	0	0
$892$	−7.19615	−0.240945
$893$	−71.9509	−2.40775
$894$	0	0
$895$	0	0
$896$	−15.1784	−0.507076
$897$	0	0
$898$	−79.8372	−2.66420
$899$	7.18251	0.239550
$900$	0	0
$901$	7.26795	0.242130
$902$	177.420	5.90745
$903$	0	0
$904$	46.9808	1.56256
$905$	0	0
$906$	0	0
$907$	−51.7128	−1.71710	−0.858548	−	0.512733i	$-0.828633\pi$
$907$	−51.7128	−1.71710	−0.858548	+	0.512733i	$0.828633\pi$
$908$	0.641516	0.0212895
$909$	0	0
$910$	0	0
$911$	−17.8703	−0.592070	−0.296035	−	0.955177i	$-0.595665\pi$
$911$	−17.8703	−0.592070	−0.296035	+	0.955177i	$0.595665\pi$
$912$	0	0
$913$	−85.5692	−2.83193
$914$	91.5739	3.02900
$915$	0	0
$916$	48.2487	1.59418
$917$	−10.0463	−0.331758
$918$	0	0
$919$	16.1962	0.534262	0.267131	−	0.963660i	$-0.413924\pi$
$919$	16.1962	0.534262	0.267131	+	0.963660i	$0.413924\pi$
$920$	0	0
$921$	0	0
$922$	−16.4833	−0.542850
$923$	−14.3650	−0.472830
$924$	0	0
$925$	0	0
$926$	−71.6532	−2.35467
$927$	0	0
$928$	−17.1962	−0.564491
$929$	−35.2710	−1.15721	−0.578603	−	0.815610i	$-0.696402\pi$
$929$	−35.2710	−1.15721	−0.578603	+	0.815610i	$0.696402\pi$
$930$	0	0
$931$	43.5167	1.42620
$932$	−33.9880	−1.11331
$933$	0	0
$934$	−1.53590	−0.0502561
$935$	0	0
$936$	0	0
$937$	−29.5692	−0.965984	−0.482992	−	0.875625i	$-0.660450\pi$
$937$	−29.5692	−0.965984	−0.482992	+	0.875625i	$0.660450\pi$
$938$	−13.4258	−0.438367
$939$	0	0
$940$	0	0
$941$	−24.5832	−0.801390	−0.400695	−	0.916212i	$-0.631231\pi$
$941$	−24.5832	−0.801390	−0.400695	+	0.916212i	$0.631231\pi$
$942$	0	0
$943$	39.7128	1.29323
$944$	−16.1177	−0.524586
$945$	0	0
$946$	101.229	3.29125
$947$	−45.3173	−1.47262	−0.736308	−	0.676647i	$-0.763432\pi$
$947$	−45.3173	−1.47262	−0.736308	+	0.676647i	$0.763432\pi$
$948$	0	0
$949$	−34.6410	−1.12449
$950$	0	0
$951$	0	0
$952$	1.94744	0.0631169
$953$	−15.1784	−0.491678	−0.245839	−	0.969311i	$-0.579063\pi$
$953$	−15.1784	−0.491678	−0.245839	+	0.969311i	$0.579063\pi$
$954$	0	0
$955$	0	0
$956$	37.4933	1.21262
$957$	0	0
$958$	11.0526	0.357092
$959$	−0.813410	−0.0262664
$960$	0	0
$961$	1.00000	0.0322581
$962$	−22.6587	−0.730545
$963$	0	0
$964$	−74.8372	−2.41034
$965$	0	0
$966$	0	0
$967$	20.5359	0.660390	0.330195	−	0.943913i	$-0.392885\pi$
$967$	20.5359	0.660390	0.330195	+	0.943913i	$0.392885\pi$
$968$	131.805	4.23638
$969$	0	0
$970$	0	0
$971$	2.73796	0.0878652	0.0439326	−	0.999034i	$-0.486011\pi$
$971$	2.73796	0.0878652	0.0439326	+	0.999034i	$0.486011\pi$
$972$	0	0
$973$	8.44486	0.270730
$974$	−39.8875	−1.27808
$975$	0	0
$976$	−10.3397	−0.330967
$977$	−23.4721	−0.750938	−0.375469	−	0.926835i	$-0.622518\pi$
$977$	−23.4721	−0.750938	−0.375469	+	0.926835i	$0.622518\pi$
$978$	0	0
$979$	−50.0526	−1.59969
$980$	0	0
$981$	0	0
$982$	−101.229	−3.23036
$983$	−45.4432	−1.44941	−0.724706	−	0.689058i	$-0.758024\pi$
$983$	−45.4432	−1.44941	−0.724706	+	0.689058i	$0.758024\pi$
$984$	0	0
$985$	0	0
$986$	11.0316	0.351318
$987$	0	0
$988$	87.0333	2.76890
$989$	22.6587	0.720504
$990$	0	0
$991$	−3.98076	−0.126453	−0.0632265	−	0.997999i	$-0.520139\pi$
$991$	−3.98076	−0.126453	−0.0632265	+	0.997999i	$0.520139\pi$
$992$	−2.39417	−0.0760150
$993$	0	0
$994$	7.26795	0.230525
$995$	0	0
$996$	0	0
$997$	46.6410	1.47714	0.738568	−	0.674179i	$-0.235502\pi$
$997$	46.6410	1.47714	0.738568	+	0.674179i	$0.235502\pi$
$998$	−45.9589	−1.45480
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bl.1.4	yes	4
3.2	odd	2	inner	6975.2.a.bl.1.1	✓	4
5.4	even	2		6975.2.a.bm.1.1	yes	4
15.14	odd	2		6975.2.a.bm.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6975.2.a.bl.1.1	✓	4	3.2	odd	2	inner
6975.2.a.bl.1.4	yes	4	1.1	even	1	trivial
6975.2.a.bm.1.1	yes	4	5.4	even	2
6975.2.a.bm.1.4	yes	4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q+2.39417 q^{2} +3.73205 q^{4} +0.732051 q^{7} +4.14682 q^{8} +6.54099 q^{11} -3.46410 q^{13} +1.75265 q^{14} +2.46410 q^{16} +0.641516 q^{17} -6.73205 q^{19} +15.6603 q^{22} +3.50531 q^{23} -8.29365 q^{26} +2.73205 q^{28} +7.18251 q^{29} +1.00000 q^{31} -2.39417 q^{32} +1.53590 q^{34} +2.73205 q^{37} -16.1177 q^{38} +11.3293 q^{41} +6.46410 q^{43} +24.4113 q^{44} +8.39230 q^{46} +10.6878 q^{47} -6.46410 q^{49} -12.9282 q^{52} +11.3293 q^{53} +3.03569 q^{56} +17.1962 q^{58} -6.54099 q^{59} -4.19615 q^{61} +2.39417 q^{62} -10.6603 q^{64} -7.66025 q^{67} +2.39417 q^{68} +4.14682 q^{71} +10.0000 q^{73} +6.54099 q^{74} -25.1244 q^{76} +4.78834 q^{77} -5.53590 q^{79} +27.1244 q^{82} -13.0820 q^{83} +15.4762 q^{86} +27.1244 q^{88} -7.65213 q^{89} -2.53590 q^{91} +13.0820 q^{92} +25.5885 q^{94} +13.7321 q^{97} -15.4762 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 4 q^{7} - 4 q^{16} - 20 q^{19} + 28 q^{22} + 4 q^{28} + 4 q^{31} + 20 q^{34} + 4 q^{37} + 12 q^{43} - 8 q^{46} - 12 q^{49} - 24 q^{52} + 48 q^{58} + 4 q^{61} - 8 q^{64} + 4 q^{67} + 40 q^{73}+ \cdots + 48 q^{97}+O(q^{100}) \) 4 * q + 8 * q^4 - 4 * q^7 - 4 * q^16 - 20 * q^19 + 28 * q^22 + 4 * q^28 + 4 * q^31 + 20 * q^34 + 4 * q^37 + 12 * q^43 - 8 * q^46 - 12 * q^49 - 24 * q^52 + 48 * q^58 + 4 * q^61 - 8 * q^64 + 4 * q^67 + 40 * q^73 - 52 * q^76 - 36 * q^79 + 60 * q^82 + 60 * q^88 - 24 * q^91 + 40 * q^94 + 48 * q^97

Introduction

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