LMFDB - Embedded newform 3675.2.a.bu.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, 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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 735)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.93185 q^{2} +1.00000 q^{3} +1.73205 q^{4} +1.93185 q^{6} -0.517638 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.93185 q^{2} +1.00000 q^{3} +1.73205 q^{4} +1.93185 q^{6} -0.517638 q^{8} +1.00000 q^{9} -3.46410 q^{11} +1.73205 q^{12} -4.00000 q^{13} -4.46410 q^{16} -4.00000 q^{17} +1.93185 q^{18} -0.378937 q^{19} -6.69213 q^{22} -6.31319 q^{23} -0.517638 q^{24} -7.72741 q^{26} +1.00000 q^{27} +8.92820 q^{29} -7.34847 q^{31} -7.58871 q^{32} -3.46410 q^{33} -7.72741 q^{34} +1.73205 q^{36} +0.757875 q^{37} -0.732051 q^{38} -4.00000 q^{39} -8.48528 q^{41} -6.00000 q^{44} -12.1962 q^{46} -6.00000 q^{47} -4.46410 q^{48} -4.00000 q^{51} -6.92820 q^{52} +7.34847 q^{53} +1.93185 q^{54} -0.378937 q^{57} +17.2480 q^{58} +10.5558 q^{59} +9.14162 q^{61} -14.1962 q^{62} -5.73205 q^{64} -6.69213 q^{66} +6.96953 q^{67} -6.92820 q^{68} -6.31319 q^{69} +14.3923 q^{71} -0.517638 q^{72} -10.9282 q^{73} +1.46410 q^{74} -0.656339 q^{76} -7.72741 q^{78} -11.4641 q^{79} +1.00000 q^{81} -16.3923 q^{82} -6.00000 q^{83} +8.92820 q^{87} +1.79315 q^{88} -4.14110 q^{89} -10.9348 q^{92} -7.34847 q^{93} -11.5911 q^{94} -7.58871 q^{96} -5.07180 q^{97} -3.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} - 16 q^{13} - 4 q^{16} - 16 q^{17} + 4 q^{27} + 8 q^{29} + 4 q^{38} - 16 q^{39} - 24 q^{44} - 28 q^{46} - 24 q^{47} - 4 q^{48} - 16 q^{51} - 36 q^{62} - 16 q^{64} + 16 q^{71} - 16 q^{73} - 8 q^{74} - 32 q^{79} + 4 q^{81} - 24 q^{82} - 24 q^{83} + 8 q^{87} - 48 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 - 16 * q^13 - 4 * q^16 - 16 * q^17 + 4 * q^27 + 8 * q^29 + 4 * q^38 - 16 * q^39 - 24 * q^44 - 28 * q^46 - 24 * q^47 - 4 * q^48 - 16 * q^51 - 36 * q^62 - 16 * q^64 + 16 * q^71 - 16 * q^73 - 8 * q^74 - 32 * q^79 + 4 * q^81 - 24 * q^82 - 24 * q^83 + 8 * q^87 - 48 * q^97

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 $

$2$	1.93185	1.36603	0.683013	−	0.730406i	$-0.260669\pi$
$2$	1.93185	1.36603	0.683013	+	0.730406i	$0.260669\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.73205	0.866025
$5$	0	0
$5$	0	0
$6$	1.93185	0.788675
$7$	0	0
$7$	0	0
$8$	−0.517638	−0.183013
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	1.73205	0.500000
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	−4.46410	−1.11603
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	1.93185	0.455342
$19$	−0.378937	−0.0869342	−0.0434671	−	0.999055i	$-0.513840\pi$
$19$	−0.378937	−0.0869342	−0.0434671	+	0.999055i	$0.513840\pi$
$20$	0	0
$21$	0	0
$22$	−6.69213	−1.42677
$23$	−6.31319	−1.31639	−0.658196	−	0.752847i	$-0.728680\pi$
$23$	−6.31319	−1.31639	−0.658196	+	0.752847i	$0.728680\pi$
$24$	−0.517638	−0.105662
$25$	0	0
$26$	−7.72741	−1.51547
$27$	1.00000	0.192450
$28$	0	0
$29$	8.92820	1.65793	0.828963	−	0.559304i	$-0.188931\pi$
$29$	8.92820	1.65793	0.828963	+	0.559304i	$0.188931\pi$
$30$	0	0
$31$	−7.34847	−1.31982	−0.659912	−	0.751343i	$-0.729406\pi$
$31$	−7.34847	−1.31982	−0.659912	+	0.751343i	$0.729406\pi$
$32$	−7.58871	−1.34151
$33$	−3.46410	−0.603023
$34$	−7.72741	−1.32524
$35$	0	0
$36$	1.73205	0.288675
$37$	0.757875	0.124594	0.0622969	−	0.998058i	$-0.480157\pi$
$37$	0.757875	0.124594	0.0622969	+	0.998058i	$0.480157\pi$
$38$	−0.732051	−0.118754
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−8.48528	−1.32518	−0.662589	−	0.748983i	$-0.730542\pi$
$41$	−8.48528	−1.32518	−0.662589	+	0.748983i	$0.730542\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	−12.1962	−1.79822
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−4.46410	−0.644338
$49$	0	0
$50$	0	0
$51$	−4.00000	−0.560112
$52$	−6.92820	−0.960769
$53$	7.34847	1.00939	0.504695	−	0.863298i	$-0.331605\pi$
$53$	7.34847	1.00939	0.504695	+	0.863298i	$0.331605\pi$
$54$	1.93185	0.262892
$55$	0	0
$56$	0	0
$57$	−0.378937	−0.0501915
$58$	17.2480	2.26477
$59$	10.5558	1.37425	0.687126	−	0.726538i	$-0.258872\pi$
$59$	10.5558	1.37425	0.687126	+	0.726538i	$0.258872\pi$
$60$	0	0
$61$	9.14162	1.17046	0.585232	−	0.810866i	$-0.301003\pi$
$61$	9.14162	1.17046	0.585232	+	0.810866i	$0.301003\pi$
$62$	−14.1962	−1.80291
$63$	0	0
$64$	−5.73205	−0.716506
$65$	0	0
$66$	−6.69213	−0.823744
$67$	6.96953	0.851464	0.425732	−	0.904849i	$-0.360017\pi$
$67$	6.96953	0.851464	0.425732	+	0.904849i	$0.360017\pi$
$68$	−6.92820	−0.840168
$69$	−6.31319	−0.760019
$70$	0	0
$71$	14.3923	1.70805	0.854026	−	0.520230i	$-0.174154\pi$
$71$	14.3923	1.70805	0.854026	+	0.520230i	$0.174154\pi$
$72$	−0.517638	−0.0610042
$73$	−10.9282	−1.27905	−0.639525	−	0.768771i	$-0.720869\pi$
$73$	−10.9282	−1.27905	−0.639525	+	0.768771i	$0.720869\pi$
$74$	1.46410	0.170198
$75$	0	0
$76$	−0.656339	−0.0752872
$77$	0	0
$78$	−7.72741	−0.874957
$79$	−11.4641	−1.28981	−0.644906	−	0.764262i	$-0.723104\pi$
$79$	−11.4641	−1.28981	−0.644906	+	0.764262i	$0.723104\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−16.3923	−1.81023
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.92820	0.957204
$88$	1.79315	0.191151
$89$	−4.14110	−0.438956	−0.219478	−	0.975617i	$-0.570435\pi$
$89$	−4.14110	−0.438956	−0.219478	+	0.975617i	$0.570435\pi$
$90$	0	0
$91$	0	0
$92$	−10.9348	−1.14003
$93$	−7.34847	−0.762001
$94$	−11.5911	−1.19553
$95$	0	0
$96$	−7.58871	−0.774519
$97$	−5.07180	−0.514963	−0.257481	−	0.966283i	$-0.582893\pi$
$97$	−5.07180	−0.514963	−0.257481	+	0.966283i	$0.582893\pi$
$98$	0	0
$99$	−3.46410	−0.348155
$100$	0	0
$101$	−12.6264	−1.25637	−0.628186	−	0.778063i	$-0.716202\pi$
$101$	−12.6264	−1.25637	−0.628186	+	0.778063i	$0.716202\pi$
$102$	−7.72741	−0.765127
$103$	13.8564	1.36531	0.682656	−	0.730740i	$-0.260825\pi$
$103$	13.8564	1.36531	0.682656	+	0.730740i	$0.260825\pi$
$104$	2.07055	0.203034
$105$	0	0
$106$	14.1962	1.37885
$107$	15.5563	1.50389	0.751945	−	0.659226i	$-0.229116\pi$
$107$	15.5563	1.50389	0.751945	+	0.659226i	$0.229116\pi$
$108$	1.73205	0.166667
$109$	−11.8564	−1.13564	−0.567819	−	0.823154i	$-0.692213\pi$
$109$	−11.8564	−1.13564	−0.567819	+	0.823154i	$0.692213\pi$
$110$	0	0
$111$	0.757875	0.0719343
$112$	0	0
$113$	−0.378937	−0.0356474	−0.0178237	−	0.999841i	$-0.505674\pi$
$113$	−0.378937	−0.0356474	−0.0178237	+	0.999841i	$0.505674\pi$
$114$	−0.732051	−0.0685628
$115$	0	0
$116$	15.4641	1.43581
$117$	−4.00000	−0.369800
$118$	20.3923	1.87726
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	17.6603	1.59888
$123$	−8.48528	−0.765092
$124$	−12.7279	−1.14300
$125$	0	0
$126$	0	0
$127$	2.82843	0.250982	0.125491	−	0.992095i	$-0.459949\pi$
$127$	2.82843	0.250982	0.125491	+	0.992095i	$0.459949\pi$
$128$	4.10394	0.362740
$129$	0	0
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	13.4641	1.16312
$135$	0	0
$136$	2.07055	0.177548
$137$	−4.52004	−0.386173	−0.193087	−	0.981182i	$-0.561850\pi$
$137$	−4.52004	−0.386173	−0.193087	+	0.981182i	$0.561850\pi$
$138$	−12.1962	−1.03821
$139$	5.27792	0.447667	0.223834	−	0.974627i	$-0.428143\pi$
$139$	5.27792	0.447667	0.223834	+	0.974627i	$0.428143\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	27.8038	2.33324
$143$	13.8564	1.15873
$144$	−4.46410	−0.372008
$145$	0	0
$146$	−21.1117	−1.74721
$147$	0	0
$148$	1.31268	0.107901
$149$	20.9282	1.71451	0.857253	−	0.514896i	$-0.172169\pi$
$149$	20.9282	1.71451	0.857253	+	0.514896i	$0.172169\pi$
$150$	0	0
$151$	−12.5359	−1.02016	−0.510078	−	0.860128i	$-0.670384\pi$
$151$	−12.5359	−1.02016	−0.510078	+	0.860128i	$0.670384\pi$
$152$	0.196152	0.0159101
$153$	−4.00000	−0.323381
$154$	0	0
$155$	0	0
$156$	−6.92820	−0.554700
$157$	−18.9282	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$157$	−18.9282	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$158$	−22.1469	−1.76192
$159$	7.34847	0.582772
$160$	0	0
$161$	0	0
$162$	1.93185	0.151781
$163$	16.7675	1.31333	0.656666	−	0.754182i	$-0.271966\pi$
$163$	16.7675	1.31333	0.656666	+	0.754182i	$0.271966\pi$
$164$	−14.6969	−1.14764
$165$	0	0
$166$	−11.5911	−0.899645
$167$	−5.07180	−0.392467	−0.196234	−	0.980557i	$-0.562871\pi$
$167$	−5.07180	−0.392467	−0.196234	+	0.980557i	$0.562871\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−0.378937	−0.0289781
$172$	0	0
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	17.2480	1.30756
$175$	0	0
$176$	15.4641	1.16565
$177$	10.5558	0.793425
$178$	−8.00000	−0.599625
$179$	−10.3923	−0.776757	−0.388379	−	0.921500i	$-0.626965\pi$
$179$	−10.3923	−0.776757	−0.388379	+	0.921500i	$0.626965\pi$
$180$	0	0
$181$	−3.48477	−0.259021	−0.129510	−	0.991578i	$-0.541341\pi$
$181$	−3.48477	−0.259021	−0.129510	+	0.991578i	$0.541341\pi$
$182$	0	0
$183$	9.14162	0.675768
$184$	3.26795	0.240916
$185$	0	0
$186$	−14.1962	−1.04091
$187$	13.8564	1.01328
$188$	−10.3923	−0.757937
$189$	0	0
$190$	0	0
$191$	−0.535898	−0.0387762	−0.0193881	−	0.999812i	$-0.506172\pi$
$191$	−0.535898	−0.0387762	−0.0193881	+	0.999812i	$0.506172\pi$
$192$	−5.73205	−0.413675
$193$	−16.2127	−1.16701	−0.583507	−	0.812108i	$-0.698320\pi$
$193$	−16.2127	−1.16701	−0.583507	+	0.812108i	$0.698320\pi$
$194$	−9.79796	−0.703452
$195$	0	0
$196$	0	0
$197$	11.4896	0.818598	0.409299	−	0.912400i	$-0.365773\pi$
$197$	11.4896	0.818598	0.409299	+	0.912400i	$0.365773\pi$
$198$	−6.69213	−0.475589
$199$	−3.20736	−0.227364	−0.113682	−	0.993517i	$-0.536265\pi$
$199$	−3.20736	−0.227364	−0.113682	+	0.993517i	$0.536265\pi$
$200$	0	0
$201$	6.96953	0.491593
$202$	−24.3923	−1.71624
$203$	0	0
$204$	−6.92820	−0.485071
$205$	0	0
$206$	26.7685	1.86505
$207$	−6.31319	−0.438797
$208$	17.8564	1.23812
$209$	1.31268	0.0907998
$210$	0	0
$211$	−13.0718	−0.899900	−0.449950	−	0.893054i	$-0.648558\pi$
$211$	−13.0718	−0.899900	−0.449950	+	0.893054i	$0.648558\pi$
$212$	12.7279	0.874157
$213$	14.3923	0.986144
$214$	30.0526	2.05435
$215$	0	0
$216$	−0.517638	−0.0352208
$217$	0	0
$218$	−22.9048	−1.55131
$219$	−10.9282	−0.738460
$220$	0	0
$221$	16.0000	1.07628
$222$	1.46410	0.0982641
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	0	0
$226$	−0.732051	−0.0486953
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	−0.656339	−0.0434671
$229$	23.2838	1.53863	0.769317	−	0.638867i	$-0.220597\pi$
$229$	23.2838	1.53863	0.769317	+	0.638867i	$0.220597\pi$
$230$	0	0
$231$	0	0
$232$	−4.62158	−0.303421
$233$	−0.378937	−0.0248250	−0.0124125	−	0.999923i	$-0.503951\pi$
$233$	−0.378937	−0.0248250	−0.0124125	+	0.999923i	$0.503951\pi$
$234$	−7.72741	−0.505156
$235$	0	0
$236$	18.2832	1.19014
$237$	−11.4641	−0.744673
$238$	0	0
$239$	−15.4641	−1.00029	−0.500145	−	0.865942i	$-0.666720\pi$
$239$	−15.4641	−1.00029	−0.500145	+	0.865942i	$0.666720\pi$
$240$	0	0
$241$	5.55532	0.357850	0.178925	−	0.983863i	$-0.442738\pi$
$241$	5.55532	0.357850	0.178925	+	0.983863i	$0.442738\pi$
$242$	1.93185	0.124184
$243$	1.00000	0.0641500
$244$	15.8338	1.01365
$245$	0	0
$246$	−16.3923	−1.04514
$247$	1.51575	0.0964448
$248$	3.80385	0.241545
$249$	−6.00000	−0.380235
$250$	0	0
$251$	9.04008	0.570605	0.285303	−	0.958438i	$-0.407906\pi$
$251$	9.04008	0.570605	0.285303	+	0.958438i	$0.407906\pi$
$252$	0	0
$253$	21.8695	1.37493
$254$	5.46410	0.342848
$255$	0	0
$256$	19.3923	1.21202
$257$	−3.46410	−0.216085	−0.108042	−	0.994146i	$-0.534458\pi$
$257$	−3.46410	−0.216085	−0.108042	+	0.994146i	$0.534458\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.92820	0.552642
$262$	10.9282	0.675147
$263$	−21.7680	−1.34227	−0.671136	−	0.741334i	$-0.734194\pi$
$263$	−21.7680	−1.34227	−0.671136	+	0.741334i	$0.734194\pi$
$264$	1.79315	0.110361
$265$	0	0
$266$	0	0
$267$	−4.14110	−0.253431
$268$	12.0716	0.737389
$269$	8.28221	0.504975	0.252488	−	0.967600i	$-0.418751\pi$
$269$	8.28221	0.504975	0.252488	+	0.967600i	$0.418751\pi$
$270$	0	0
$271$	−13.0053	−0.790017	−0.395009	−	0.918677i	$-0.629258\pi$
$271$	−13.0053	−0.790017	−0.395009	+	0.918677i	$0.629258\pi$
$272$	17.8564	1.08270
$273$	0	0
$274$	−8.73205	−0.527522
$275$	0	0
$276$	−10.9348	−0.658196
$277$	−11.3137	−0.679775	−0.339887	−	0.940466i	$-0.610389\pi$
$277$	−11.3137	−0.679775	−0.339887	+	0.940466i	$0.610389\pi$
$278$	10.1962	0.611525
$279$	−7.34847	−0.439941
$280$	0	0
$281$	0.143594	0.00856607	0.00428304	−	0.999991i	$-0.498637\pi$
$281$	0.143594	0.00856607	0.00428304	+	0.999991i	$0.498637\pi$
$282$	−11.5911	−0.690241
$283$	−29.8564	−1.77478	−0.887390	−	0.461020i	$-0.847484\pi$
$283$	−29.8564	−1.77478	−0.887390	+	0.461020i	$0.847484\pi$
$284$	24.9282	1.47922
$285$	0	0
$286$	26.7685	1.58286
$287$	0	0
$288$	−7.58871	−0.447169
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−5.07180	−0.297314
$292$	−18.9282	−1.10769
$293$	−4.53590	−0.264990	−0.132495	−	0.991184i	$-0.542299\pi$
$293$	−4.53590	−0.264990	−0.132495	+	0.991184i	$0.542299\pi$
$294$	0	0
$295$	0	0
$296$	−0.392305	−0.0228023
$297$	−3.46410	−0.201008
$298$	40.4302	2.34206
$299$	25.2528	1.46041
$300$	0	0
$301$	0	0
$302$	−24.2175	−1.39356
$303$	−12.6264	−0.725367
$304$	1.69161	0.0970208
$305$	0	0
$306$	−7.72741	−0.441746
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	13.8564	0.788263
$310$	0	0
$311$	15.4548	0.876362	0.438181	−	0.898887i	$-0.355623\pi$
$311$	15.4548	0.876362	0.438181	+	0.898887i	$0.355623\pi$
$312$	2.07055	0.117222
$313$	−10.9282	−0.617699	−0.308849	−	0.951111i	$-0.599944\pi$
$313$	−10.9282	−0.617699	−0.308849	+	0.951111i	$0.599944\pi$
$314$	−36.5665	−2.06357
$315$	0	0
$316$	−19.8564	−1.11701
$317$	−29.2180	−1.64105	−0.820524	−	0.571613i	$-0.806318\pi$
$317$	−29.2180	−1.64105	−0.820524	+	0.571613i	$0.806318\pi$
$318$	14.1962	0.796081
$319$	−30.9282	−1.73165
$320$	0	0
$321$	15.5563	0.868271
$322$	0	0
$323$	1.51575	0.0843386
$324$	1.73205	0.0962250
$325$	0	0
$326$	32.3923	1.79404
$327$	−11.8564	−0.655661
$328$	4.39230	0.242524
$329$	0	0
$330$	0	0
$331$	32.7846	1.80201	0.901003	−	0.433814i	$-0.142832\pi$
$331$	32.7846	1.80201	0.901003	+	0.433814i	$0.142832\pi$
$332$	−10.3923	−0.570352
$333$	0.757875	0.0415313
$334$	−9.79796	−0.536120
$335$	0	0
$336$	0	0
$337$	−27.5264	−1.49946	−0.749729	−	0.661745i	$-0.769816\pi$
$337$	−27.5264	−1.49946	−0.749729	+	0.661745i	$0.769816\pi$
$338$	5.79555	0.315237
$339$	−0.378937	−0.0205811
$340$	0	0
$341$	25.4558	1.37851
$342$	−0.732051	−0.0395848
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−30.9096	−1.66171
$347$	−11.4152	−0.612802	−0.306401	−	0.951902i	$-0.599125\pi$
$347$	−11.4152	−0.612802	−0.306401	+	0.951902i	$0.599125\pi$
$348$	15.4641	0.828963
$349$	7.82894	0.419074	0.209537	−	0.977801i	$-0.432804\pi$
$349$	7.82894	0.419074	0.209537	+	0.977801i	$0.432804\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	26.2880	1.40116
$353$	−4.53590	−0.241422	−0.120711	−	0.992688i	$-0.538517\pi$
$353$	−4.53590	−0.241422	−0.120711	+	0.992688i	$0.538517\pi$
$354$	20.3923	1.08384
$355$	0	0
$356$	−7.17260	−0.380147
$357$	0	0
$358$	−20.0764	−1.06107
$359$	8.53590	0.450507	0.225254	−	0.974300i	$-0.427679\pi$
$359$	8.53590	0.450507	0.225254	+	0.974300i	$0.427679\pi$
$360$	0	0
$361$	−18.8564	−0.992442
$362$	−6.73205	−0.353829
$363$	1.00000	0.0524864
$364$	0	0
$365$	0	0
$366$	17.6603	0.923116
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	28.1827	1.46913
$369$	−8.48528	−0.441726
$370$	0	0
$371$	0	0
$372$	−12.7279	−0.659912
$373$	26.7685	1.38602	0.693011	−	0.720927i	$-0.256284\pi$
$373$	26.7685	1.38602	0.693011	+	0.720927i	$0.256284\pi$
$374$	26.7685	1.38417
$375$	0	0
$376$	3.10583	0.160171
$377$	−35.7128	−1.83930
$378$	0	0
$379$	4.53590	0.232993	0.116497	−	0.993191i	$-0.462834\pi$
$379$	4.53590	0.232993	0.116497	+	0.993191i	$0.462834\pi$
$380$	0	0
$381$	2.82843	0.144905
$382$	−1.03528	−0.0529693
$383$	−3.85641	−0.197053	−0.0985266	−	0.995134i	$-0.531413\pi$
$383$	−3.85641	−0.197053	−0.0985266	+	0.995134i	$0.531413\pi$
$384$	4.10394	0.209428
$385$	0	0
$386$	−31.3205	−1.59417
$387$	0	0
$388$	−8.78461	−0.445971
$389$	24.9282	1.26391	0.631955	−	0.775005i	$-0.282253\pi$
$389$	24.9282	1.26391	0.631955	+	0.775005i	$0.282253\pi$
$390$	0	0
$391$	25.2528	1.27709
$392$	0	0
$393$	5.65685	0.285351
$394$	22.1962	1.11823
$395$	0	0
$396$	−6.00000	−0.301511
$397$	31.7128	1.59162	0.795810	−	0.605546i	$-0.207045\pi$
$397$	31.7128	1.59162	0.795810	+	0.605546i	$0.207045\pi$
$398$	−6.19615	−0.310585
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	13.4641	0.671528
$403$	29.3939	1.46421
$404$	−21.8695	−1.08805
$405$	0	0
$406$	0	0
$407$	−2.62536	−0.130134
$408$	2.07055	0.102508
$409$	31.0112	1.53340	0.766702	−	0.642004i	$-0.221897\pi$
$409$	31.0112	1.53340	0.766702	+	0.642004i	$0.221897\pi$
$410$	0	0
$411$	−4.52004	−0.222957
$412$	24.0000	1.18240
$413$	0	0
$414$	−12.1962	−0.599408
$415$	0	0
$416$	30.3548	1.48827
$417$	5.27792	0.258461
$418$	2.53590	0.124035
$419$	1.51575	0.0740492	0.0370246	−	0.999314i	$-0.488212\pi$
$419$	1.51575	0.0740492	0.0370246	+	0.999314i	$0.488212\pi$
$420$	0	0
$421$	−28.7846	−1.40288	−0.701438	−	0.712730i	$-0.747458\pi$
$421$	−28.7846	−1.40288	−0.701438	+	0.712730i	$0.747458\pi$
$422$	−25.2528	−1.22929
$423$	−6.00000	−0.291730
$424$	−3.80385	−0.184731
$425$	0	0
$426$	27.8038	1.34710
$427$	0	0
$428$	26.9444	1.30241
$429$	13.8564	0.668994
$430$	0	0
$431$	2.67949	0.129067	0.0645333	−	0.997916i	$-0.479444\pi$
$431$	2.67949	0.129067	0.0645333	+	0.997916i	$0.479444\pi$
$432$	−4.46410	−0.214779
$433$	−9.85641	−0.473669	−0.236834	−	0.971550i	$-0.576110\pi$
$433$	−9.85641	−0.473669	−0.236834	+	0.971550i	$0.576110\pi$
$434$	0	0
$435$	0	0
$436$	−20.5359	−0.983491
$437$	2.39230	0.114439
$438$	−21.1117	−1.00875
$439$	1.69161	0.0807364	0.0403682	−	0.999185i	$-0.487147\pi$
$439$	1.69161	0.0807364	0.0403682	+	0.999185i	$0.487147\pi$
$440$	0	0
$441$	0	0
$442$	30.9096	1.47022
$443$	−26.6670	−1.26699	−0.633493	−	0.773748i	$-0.718380\pi$
$443$	−26.6670	−1.26699	−0.633493	+	0.773748i	$0.718380\pi$
$444$	1.31268	0.0622969
$445$	0	0
$446$	−15.4548	−0.731807
$447$	20.9282	0.989870
$448$	0	0
$449$	8.14359	0.384320	0.192160	−	0.981364i	$-0.438451\pi$
$449$	8.14359	0.384320	0.192160	+	0.981364i	$0.438451\pi$
$450$	0	0
$451$	29.3939	1.38410
$452$	−0.656339	−0.0308716
$453$	−12.5359	−0.588988
$454$	−7.72741	−0.362665
$455$	0	0
$456$	0.196152	0.00918568
$457$	−12.0716	−0.564685	−0.282342	−	0.959314i	$-0.591111\pi$
$457$	−12.0716	−0.564685	−0.282342	+	0.959314i	$0.591111\pi$
$458$	44.9808	2.10181
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−12.8295	−0.597527	−0.298764	−	0.954327i	$-0.596574\pi$
$461$	−12.8295	−0.597527	−0.298764	+	0.954327i	$0.596574\pi$
$462$	0	0
$463$	−16.7675	−0.779251	−0.389626	−	0.920973i	$-0.627396\pi$
$463$	−16.7675	−0.779251	−0.389626	+	0.920973i	$0.627396\pi$
$464$	−39.8564	−1.85029
$465$	0	0
$466$	−0.732051	−0.0339116
$467$	−17.8564	−0.826296	−0.413148	−	0.910664i	$-0.635571\pi$
$467$	−17.8564	−0.826296	−0.413148	+	0.910664i	$0.635571\pi$
$468$	−6.92820	−0.320256
$469$	0	0
$470$	0	0
$471$	−18.9282	−0.872166
$472$	−5.46410	−0.251506
$473$	0	0
$474$	−22.1469	−1.01724
$475$	0	0
$476$	0	0
$477$	7.34847	0.336463
$478$	−29.8744	−1.36642
$479$	−13.9391	−0.636892	−0.318446	−	0.947941i	$-0.603161\pi$
$479$	−13.9391	−0.636892	−0.318446	+	0.947941i	$0.603161\pi$
$480$	0	0
$481$	−3.03150	−0.138224
$482$	10.7321	0.488832
$483$	0	0
$484$	1.73205	0.0787296
$485$	0	0
$486$	1.93185	0.0876306
$487$	−4.14110	−0.187651	−0.0938257	−	0.995589i	$-0.529910\pi$
$487$	−4.14110	−0.187651	−0.0938257	+	0.995589i	$0.529910\pi$
$488$	−4.73205	−0.214210
$489$	16.7675	0.758252
$490$	0	0
$491$	6.67949	0.301441	0.150721	−	0.988576i	$-0.451841\pi$
$491$	6.67949	0.301441	0.150721	+	0.988576i	$0.451841\pi$
$492$	−14.6969	−0.662589
$493$	−35.7128	−1.60842
$494$	2.92820	0.131746
$495$	0	0
$496$	32.8043	1.47296
$497$	0	0
$498$	−11.5911	−0.519410
$499$	16.2487	0.727392	0.363696	−	0.931518i	$-0.381515\pi$
$499$	16.2487	0.727392	0.363696	+	0.931518i	$0.381515\pi$
$500$	0	0
$501$	−5.07180	−0.226591
$502$	17.4641	0.779461
$503$	3.85641	0.171949	0.0859743	−	0.996297i	$-0.472600\pi$
$503$	3.85641	0.171949	0.0859743	+	0.996297i	$0.472600\pi$
$504$	0	0
$505$	0	0
$506$	42.2487	1.87818
$507$	3.00000	0.133235
$508$	4.89898	0.217357
$509$	42.2233	1.87152	0.935758	−	0.352642i	$-0.114717\pi$
$509$	42.2233	1.87152	0.935758	+	0.352642i	$0.114717\pi$
$510$	0	0
$511$	0	0
$512$	29.2552	1.29291
$513$	−0.378937	−0.0167305
$514$	−6.69213	−0.295177
$515$	0	0
$516$	0	0
$517$	20.7846	0.914106
$518$	0	0
$519$	−16.0000	−0.702322
$520$	0	0
$521$	28.2843	1.23916	0.619578	−	0.784935i	$-0.287304\pi$
$521$	28.2843	1.23916	0.619578	+	0.784935i	$0.287304\pi$
$522$	17.2480	0.754923
$523$	−11.7128	−0.512166	−0.256083	−	0.966655i	$-0.582432\pi$
$523$	−11.7128	−0.512166	−0.256083	+	0.966655i	$0.582432\pi$
$524$	9.79796	0.428026
$525$	0	0
$526$	−42.0526	−1.83358
$527$	29.3939	1.28042
$528$	15.4641	0.672989
$529$	16.8564	0.732887
$530$	0	0
$531$	10.5558	0.458084
$532$	0	0
$533$	33.9411	1.47015
$534$	−8.00000	−0.346194
$535$	0	0
$536$	−3.60770	−0.155829
$537$	−10.3923	−0.448461
$538$	16.0000	0.689809
$539$	0	0
$540$	0	0
$541$	−42.6410	−1.83328	−0.916640	−	0.399713i	$-0.869110\pi$
$541$	−42.6410	−1.83328	−0.916640	+	0.399713i	$0.869110\pi$
$542$	−25.1244	−1.07918
$543$	−3.48477	−0.149546
$544$	30.3548	1.30145
$545$	0	0
$546$	0	0
$547$	29.5969	1.26547	0.632737	−	0.774367i	$-0.281931\pi$
$547$	29.5969	1.26547	0.632737	+	0.774367i	$0.281931\pi$
$548$	−7.82894	−0.334436
$549$	9.14162	0.390155
$550$	0	0
$551$	−3.38323	−0.144130
$552$	3.26795	0.139093
$553$	0	0
$554$	−21.8564	−0.928590
$555$	0	0
$556$	9.14162	0.387691
$557$	13.7632	0.583165	0.291583	−	0.956546i	$-0.405818\pi$
$557$	13.7632	0.583165	0.291583	+	0.956546i	$0.405818\pi$
$558$	−14.1962	−0.600971
$559$	0	0
$560$	0	0
$561$	13.8564	0.585018
$562$	0.277401	0.0117015
$563$	−31.8564	−1.34259	−0.671294	−	0.741191i	$-0.734261\pi$
$563$	−31.8564	−1.34259	−0.671294	+	0.741191i	$0.734261\pi$
$564$	−10.3923	−0.437595
$565$	0	0
$566$	−57.6781	−2.42439
$567$	0	0
$568$	−7.45001	−0.312595
$569$	11.8564	0.497046	0.248523	−	0.968626i	$-0.420055\pi$
$569$	11.8564	0.497046	0.248523	+	0.968626i	$0.420055\pi$
$570$	0	0
$571$	39.7128	1.66193	0.830965	−	0.556325i	$-0.187789\pi$
$571$	39.7128	1.66193	0.830965	+	0.556325i	$0.187789\pi$
$572$	24.0000	1.00349
$573$	−0.535898	−0.0223875
$574$	0	0
$575$	0	0
$576$	−5.73205	−0.238835
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	−1.93185	−0.0803544
$579$	−16.2127	−0.673776
$580$	0	0
$581$	0	0
$582$	−9.79796	−0.406138
$583$	−25.4558	−1.05427
$584$	5.65685	0.234082
$585$	0	0
$586$	−8.76268	−0.361983
$587$	31.8564	1.31485	0.657427	−	0.753518i	$-0.271645\pi$
$587$	31.8564	1.31485	0.657427	+	0.753518i	$0.271645\pi$
$588$	0	0
$589$	2.78461	0.114738
$590$	0	0
$591$	11.4896	0.472618
$592$	−3.38323	−0.139050
$593$	9.32051	0.382747	0.191374	−	0.981517i	$-0.438706\pi$
$593$	9.32051	0.382747	0.191374	+	0.981517i	$0.438706\pi$
$594$	−6.69213	−0.274581
$595$	0	0
$596$	36.2487	1.48481
$597$	−3.20736	−0.131269
$598$	48.7846	1.99495
$599$	−5.32051	−0.217390	−0.108695	−	0.994075i	$-0.534667\pi$
$599$	−5.32051	−0.217390	−0.108695	+	0.994075i	$0.534667\pi$
$600$	0	0
$601$	−33.6365	−1.37206	−0.686031	−	0.727572i	$-0.740649\pi$
$601$	−33.6365	−1.37206	−0.686031	+	0.727572i	$0.740649\pi$
$602$	0	0
$603$	6.96953	0.283821
$604$	−21.7128	−0.883482
$605$	0	0
$606$	−24.3923	−0.990870
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	2.87564	0.116623
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	−6.92820	−0.280056
$613$	−9.79796	−0.395736	−0.197868	−	0.980229i	$-0.563402\pi$
$613$	−9.79796	−0.395736	−0.197868	+	0.980229i	$0.563402\pi$
$614$	7.72741	0.311853
$615$	0	0
$616$	0	0
$617$	12.4505	0.501239	0.250620	−	0.968086i	$-0.419366\pi$
$617$	12.4505	0.501239	0.250620	+	0.968086i	$0.419366\pi$
$618$	26.7685	1.07679
$619$	−3.76217	−0.151214	−0.0756071	−	0.997138i	$-0.524089\pi$
$619$	−3.76217	−0.151214	−0.0756071	+	0.997138i	$0.524089\pi$
$620$	0	0
$621$	−6.31319	−0.253340
$622$	29.8564	1.19713
$623$	0	0
$624$	17.8564	0.714828
$625$	0	0
$626$	−21.1117	−0.843792
$627$	1.31268	0.0524233
$628$	−32.7846	−1.30825
$629$	−3.03150	−0.120874
$630$	0	0
$631$	9.32051	0.371044	0.185522	−	0.982640i	$-0.440602\pi$
$631$	9.32051	0.371044	0.185522	+	0.982640i	$0.440602\pi$
$632$	5.93426	0.236052
$633$	−13.0718	−0.519557
$634$	−56.4449	−2.24171
$635$	0	0
$636$	12.7279	0.504695
$637$	0	0
$638$	−59.7487	−2.36547
$639$	14.3923	0.569351
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	30.0526	1.18608
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	0	0
$646$	2.92820	0.115209
$647$	29.0718	1.14293	0.571465	−	0.820626i	$-0.306375\pi$
$647$	29.0718	1.14293	0.571465	+	0.820626i	$0.306375\pi$
$648$	−0.517638	−0.0203347
$649$	−36.5665	−1.43536
$650$	0	0
$651$	0	0
$652$	29.0421	1.13738
$653$	−18.6622	−0.730307	−0.365154	−	0.930947i	$-0.618984\pi$
$653$	−18.6622	−0.730307	−0.365154	+	0.930947i	$0.618984\pi$
$654$	−22.9048	−0.895649
$655$	0	0
$656$	37.8792	1.47893
$657$	−10.9282	−0.426350
$658$	0	0
$659$	−10.3923	−0.404827	−0.202413	−	0.979300i	$-0.564878\pi$
$659$	−10.3923	−0.404827	−0.202413	+	0.979300i	$0.564878\pi$
$660$	0	0
$661$	−23.2838	−0.905633	−0.452817	−	0.891604i	$-0.649581\pi$
$661$	−23.2838	−0.905633	−0.452817	+	0.891604i	$0.649581\pi$
$662$	63.3350	2.46158
$663$	16.0000	0.621389
$664$	3.10583	0.120530
$665$	0	0
$666$	1.46410	0.0567328
$667$	−56.3655	−2.18248
$668$	−8.78461	−0.339887
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−31.6675	−1.22251
$672$	0	0
$673$	12.0716	0.465325	0.232663	−	0.972557i	$-0.425256\pi$
$673$	12.0716	0.465325	0.232663	+	0.972557i	$0.425256\pi$
$674$	−53.1769	−2.04830
$675$	0	0
$676$	5.19615	0.199852
$677$	−39.1769	−1.50569	−0.752846	−	0.658197i	$-0.771319\pi$
$677$	−39.1769	−1.50569	−0.752846	+	0.658197i	$0.771319\pi$
$678$	−0.732051	−0.0281142
$679$	0	0
$680$	0	0
$681$	−4.00000	−0.153280
$682$	49.1769	1.88308
$683$	−22.7290	−0.869699	−0.434850	−	0.900503i	$-0.643198\pi$
$683$	−22.7290	−0.869699	−0.434850	+	0.900503i	$0.643198\pi$
$684$	−0.656339	−0.0250957
$685$	0	0
$686$	0	0
$687$	23.2838	0.888331
$688$	0	0
$689$	−29.3939	−1.11982
$690$	0	0
$691$	−6.03579	−0.229612	−0.114806	−	0.993388i	$-0.536625\pi$
$691$	−6.03579	−0.229612	−0.114806	+	0.993388i	$0.536625\pi$
$692$	−27.7128	−1.05348
$693$	0	0
$694$	−22.0526	−0.837104
$695$	0	0
$696$	−4.62158	−0.175180
$697$	33.9411	1.28561
$698$	15.1244	0.572465
$699$	−0.378937	−0.0143327
$700$	0	0
$701$	−24.9282	−0.941525	−0.470763	−	0.882260i	$-0.656021\pi$
$701$	−24.9282	−0.941525	−0.470763	+	0.882260i	$0.656021\pi$
$702$	−7.72741	−0.291652
$703$	−0.287187	−0.0108315
$704$	19.8564	0.748366
$705$	0	0
$706$	−8.76268	−0.329788
$707$	0	0
$708$	18.2832	0.687126
$709$	2.92820	0.109971	0.0549855	−	0.998487i	$-0.482489\pi$
$709$	2.92820	0.109971	0.0549855	+	0.998487i	$0.482489\pi$
$710$	0	0
$711$	−11.4641	−0.429937
$712$	2.14359	0.0803346
$713$	46.3923	1.73741
$714$	0	0
$715$	0	0
$716$	−18.0000	−0.672692
$717$	−15.4641	−0.577517
$718$	16.4901	0.615405
$719$	14.6969	0.548103	0.274052	−	0.961715i	$-0.411636\pi$
$719$	14.6969	0.548103	0.274052	+	0.961715i	$0.411636\pi$
$720$	0	0
$721$	0	0
$722$	−36.4278	−1.35570
$723$	5.55532	0.206605
$724$	−6.03579	−0.224318
$725$	0	0
$726$	1.93185	0.0716977
$727$	−31.7128	−1.17616	−0.588082	−	0.808802i	$-0.700117\pi$
$727$	−31.7128	−1.17616	−0.588082	+	0.808802i	$0.700117\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	15.8338	0.585232
$733$	9.85641	0.364055	0.182027	−	0.983293i	$-0.441734\pi$
$733$	9.85641	0.364055	0.182027	+	0.983293i	$0.441734\pi$
$734$	−7.72741	−0.285224
$735$	0	0
$736$	47.9090	1.76595
$737$	−24.1432	−0.889325
$738$	−16.3923	−0.603409
$739$	1.85641	0.0682890	0.0341445	−	0.999417i	$-0.489129\pi$
$739$	1.85641	0.0682890	0.0341445	+	0.999417i	$0.489129\pi$
$740$	0	0
$741$	1.51575	0.0556825
$742$	0	0
$743$	37.0197	1.35812	0.679061	−	0.734081i	$-0.262387\pi$
$743$	37.0197	1.35812	0.679061	+	0.734081i	$0.262387\pi$
$744$	3.80385	0.139456
$745$	0	0
$746$	51.7128	1.89334
$747$	−6.00000	−0.219529
$748$	24.0000	0.877527
$749$	0	0
$750$	0	0
$751$	4.78461	0.174593	0.0872964	−	0.996182i	$-0.472177\pi$
$751$	4.78461	0.174593	0.0872964	+	0.996182i	$0.472177\pi$
$752$	26.7846	0.976734
$753$	9.04008	0.329439
$754$	−68.9919	−2.51254
$755$	0	0
$756$	0	0
$757$	−34.2929	−1.24640	−0.623198	−	0.782064i	$-0.714167\pi$
$757$	−34.2929	−1.24640	−0.623198	+	0.782064i	$0.714167\pi$
$758$	8.76268	0.318275
$759$	21.8695	0.793814
$760$	0	0
$761$	40.7076	1.47565	0.737824	−	0.674993i	$-0.235853\pi$
$761$	40.7076	1.47565	0.737824	+	0.674993i	$0.235853\pi$
$762$	5.46410	0.197944
$763$	0	0
$764$	−0.928203	−0.0335812
$765$	0	0
$766$	−7.45001	−0.269180
$767$	−42.2233	−1.52460
$768$	19.3923	0.699760
$769$	19.4944	0.702985	0.351493	−	0.936191i	$-0.385674\pi$
$769$	19.4944	0.702985	0.351493	+	0.936191i	$0.385674\pi$
$770$	0	0
$771$	−3.46410	−0.124757
$772$	−28.0812	−1.01066
$773$	−10.1436	−0.364840	−0.182420	−	0.983221i	$-0.558393\pi$
$773$	−10.1436	−0.364840	−0.182420	+	0.983221i	$0.558393\pi$
$774$	0	0
$775$	0	0
$776$	2.62536	0.0942448
$777$	0	0
$778$	48.1576	1.72653
$779$	3.21539	0.115203
$780$	0	0
$781$	−49.8564	−1.78400
$782$	48.7846	1.74453
$783$	8.92820	0.319068
$784$	0	0
$785$	0	0
$786$	10.9282	0.389796
$787$	18.1436	0.646749	0.323375	−	0.946271i	$-0.395183\pi$
$787$	18.1436	0.646749	0.323375	+	0.946271i	$0.395183\pi$
$788$	19.9005	0.708927
$789$	−21.7680	−0.774962
$790$	0	0
$791$	0	0
$792$	1.79315	0.0637168
$793$	−36.5665	−1.29851
$794$	61.2645	2.17419
$795$	0	0
$796$	−5.55532	−0.196903
$797$	37.8564	1.34094	0.670471	−	0.741935i	$-0.266092\pi$
$797$	37.8564	1.34094	0.670471	+	0.741935i	$0.266092\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	−4.14110	−0.146319
$802$	3.86370	0.136432
$803$	37.8564	1.33592
$804$	12.0716	0.425732
$805$	0	0
$806$	56.7846	2.00015
$807$	8.28221	0.291548
$808$	6.53590	0.229932
$809$	−33.7128	−1.18528	−0.592640	−	0.805468i	$-0.701914\pi$
$809$	−33.7128	−1.18528	−0.592640	+	0.805468i	$0.701914\pi$
$810$	0	0
$811$	−8.66115	−0.304134	−0.152067	−	0.988370i	$-0.548593\pi$
$811$	−8.66115	−0.304134	−0.152067	+	0.988370i	$0.548593\pi$
$812$	0	0
$813$	−13.0053	−0.456117
$814$	−5.07180	−0.177766
$815$	0	0
$816$	17.8564	0.625099
$817$	0	0
$818$	59.9090	2.09467
$819$	0	0
$820$	0	0
$821$	−22.7846	−0.795188	−0.397594	−	0.917561i	$-0.630155\pi$
$821$	−22.7846	−0.795188	−0.397594	+	0.917561i	$0.630155\pi$
$822$	−8.73205	−0.304565
$823$	−47.6771	−1.66192	−0.830960	−	0.556332i	$-0.812208\pi$
$823$	−47.6771	−1.66192	−0.830960	+	0.556332i	$0.812208\pi$
$824$	−7.17260	−0.249869
$825$	0	0
$826$	0	0
$827$	50.6071	1.75978	0.879890	−	0.475177i	$-0.157616\pi$
$827$	50.6071	1.75978	0.879890	+	0.475177i	$0.157616\pi$
$828$	−10.9348	−0.380010
$829$	35.9101	1.24721	0.623605	−	0.781739i	$-0.285667\pi$
$829$	35.9101	1.24721	0.623605	+	0.781739i	$0.285667\pi$
$830$	0	0
$831$	−11.3137	−0.392468
$832$	22.9282	0.794892
$833$	0	0
$834$	10.1962	0.353064
$835$	0	0
$836$	2.27362	0.0786349
$837$	−7.34847	−0.254000
$838$	2.92820	0.101153
$839$	−9.04008	−0.312098	−0.156049	−	0.987749i	$-0.549876\pi$
$839$	−9.04008	−0.312098	−0.156049	+	0.987749i	$0.549876\pi$
$840$	0	0
$841$	50.7128	1.74872
$842$	−55.6076	−1.91636
$843$	0.143594	0.00494562
$844$	−22.6410	−0.779336
$845$	0	0
$846$	−11.5911	−0.398511
$847$	0	0
$848$	−32.8043	−1.12650
$849$	−29.8564	−1.02467
$850$	0	0
$851$	−4.78461	−0.164014
$852$	24.9282	0.854026
$853$	−26.9282	−0.922004	−0.461002	−	0.887399i	$-0.652510\pi$
$853$	−26.9282	−0.922004	−0.461002	+	0.887399i	$0.652510\pi$
$854$	0	0
$855$	0	0
$856$	−8.05256	−0.275231
$857$	−17.8564	−0.609963	−0.304982	−	0.952358i	$-0.598650\pi$
$857$	−17.8564	−0.609963	−0.304982	+	0.952358i	$0.598650\pi$
$858$	26.7685	0.913862
$859$	47.5013	1.62072	0.810361	−	0.585931i	$-0.199271\pi$
$859$	47.5013	1.62072	0.810361	+	0.585931i	$0.199271\pi$
$860$	0	0
$861$	0	0
$862$	5.17638	0.176308
$863$	52.6776	1.79317	0.896584	−	0.442874i	$-0.146041\pi$
$863$	52.6776	1.79317	0.896584	+	0.442874i	$0.146041\pi$
$864$	−7.58871	−0.258173
$865$	0	0
$866$	−19.0411	−0.647043
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	39.7128	1.34716
$870$	0	0
$871$	−27.8781	−0.944614
$872$	6.13733	0.207836
$873$	−5.07180	−0.171654
$874$	4.62158	0.156327
$875$	0	0
$876$	−18.9282	−0.639525
$877$	−33.9411	−1.14611	−0.573055	−	0.819517i	$-0.694242\pi$
$877$	−33.9411	−1.14611	−0.573055	+	0.819517i	$0.694242\pi$
$878$	3.26795	0.110288
$879$	−4.53590	−0.152992
$880$	0	0
$881$	10.0010	0.336943	0.168472	−	0.985707i	$-0.446117\pi$
$881$	10.0010	0.336943	0.168472	+	0.985707i	$0.446117\pi$
$882$	0	0
$883$	−29.3939	−0.989183	−0.494591	−	0.869126i	$-0.664682\pi$
$883$	−29.3939	−0.989183	−0.494591	+	0.869126i	$0.664682\pi$
$884$	27.7128	0.932083
$885$	0	0
$886$	−51.5167	−1.73074
$887$	41.7128	1.40058	0.700290	−	0.713859i	$-0.253054\pi$
$887$	41.7128	1.40058	0.700290	+	0.713859i	$0.253054\pi$
$888$	−0.392305	−0.0131649
$889$	0	0
$890$	0	0
$891$	−3.46410	−0.116052
$892$	−13.8564	−0.463947
$893$	2.27362	0.0760839
$894$	40.4302	1.35219
$895$	0	0
$896$	0	0
$897$	25.2528	0.843166
$898$	15.7322	0.524991
$899$	−65.6086	−2.18817
$900$	0	0
$901$	−29.3939	−0.979252
$902$	56.7846	1.89072
$903$	0	0
$904$	0.196152	0.00652393
$905$	0	0
$906$	−24.2175	−0.804572
$907$	−52.0213	−1.72734	−0.863669	−	0.504059i	$-0.831839\pi$
$907$	−52.0213	−1.72734	−0.863669	+	0.504059i	$0.831839\pi$
$908$	−6.92820	−0.229920
$909$	−12.6264	−0.418791
$910$	0	0
$911$	30.3923	1.00694	0.503471	−	0.864012i	$-0.332056\pi$
$911$	30.3923	1.00694	0.503471	+	0.864012i	$0.332056\pi$
$912$	1.69161	0.0560150
$913$	20.7846	0.687870
$914$	−23.3205	−0.771374
$915$	0	0
$916$	40.3286	1.33250
$917$	0	0
$918$	−7.72741	−0.255042
$919$	−22.9282	−0.756332	−0.378166	−	0.925738i	$-0.623445\pi$
$919$	−22.9282	−0.756332	−0.378166	+	0.925738i	$0.623445\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	−24.7846	−0.816238
$923$	−57.5692	−1.89491
$924$	0	0
$925$	0	0
$926$	−32.3923	−1.06448
$927$	13.8564	0.455104
$928$	−67.7535	−2.22412
$929$	−40.7076	−1.33557	−0.667786	−	0.744353i	$-0.732758\pi$
$929$	−40.7076	−1.33557	−0.667786	+	0.744353i	$0.732758\pi$
$930$	0	0
$931$	0	0
$932$	−0.656339	−0.0214991
$933$	15.4548	0.505968
$934$	−34.4959	−1.12874
$935$	0	0
$936$	2.07055	0.0676781
$937$	−24.7846	−0.809678	−0.404839	−	0.914388i	$-0.632672\pi$
$937$	−24.7846	−0.809678	−0.404839	+	0.914388i	$0.632672\pi$
$938$	0	0
$939$	−10.9282	−0.356628
$940$	0	0
$941$	−39.3949	−1.28424	−0.642119	−	0.766605i	$-0.721944\pi$
$941$	−39.3949	−1.28424	−0.642119	+	0.766605i	$0.721944\pi$
$942$	−36.5665	−1.19140
$943$	53.5692	1.74445
$944$	−47.1223	−1.53370
$945$	0	0
$946$	0	0
$947$	18.3848	0.597425	0.298712	−	0.954343i	$-0.403443\pi$
$947$	18.3848	0.597425	0.298712	+	0.954343i	$0.403443\pi$
$948$	−19.8564	−0.644906
$949$	43.7128	1.41898
$950$	0	0
$951$	−29.2180	−0.947459
$952$	0	0
$953$	19.6231	0.635655	0.317828	−	0.948148i	$-0.397047\pi$
$953$	19.6231	0.635655	0.317828	+	0.948148i	$0.397047\pi$
$954$	14.1962	0.459617
$955$	0	0
$956$	−26.7846	−0.866276
$957$	−30.9282	−0.999767
$958$	−26.9282	−0.870011
$959$	0	0
$960$	0	0
$961$	23.0000	0.741935
$962$	−5.85641	−0.188818
$963$	15.5563	0.501296
$964$	9.62209	0.309907
$965$	0	0
$966$	0	0
$967$	29.5969	0.951774	0.475887	−	0.879507i	$-0.342127\pi$
$967$	29.5969	0.951774	0.475887	+	0.879507i	$0.342127\pi$
$968$	−0.517638	−0.0166375
$969$	1.51575	0.0486929
$970$	0	0
$971$	−56.9203	−1.82666	−0.913329	−	0.407222i	$-0.866498\pi$
$971$	−56.9203	−1.82666	−0.913329	+	0.407222i	$0.866498\pi$
$972$	1.73205	0.0555556
$973$	0	0
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−40.8091	−1.30627
$977$	50.5328	1.61669	0.808343	−	0.588712i	$-0.200365\pi$
$977$	50.5328	1.61669	0.808343	+	0.588712i	$0.200365\pi$
$978$	32.3923	1.03579
$979$	14.3452	0.458475
$980$	0	0
$981$	−11.8564	−0.378546
$982$	12.9038	0.411776
$983$	−16.7846	−0.535346	−0.267673	−	0.963510i	$-0.586255\pi$
$983$	−16.7846	−0.535346	−0.267673	+	0.963510i	$0.586255\pi$
$984$	4.39230	0.140022
$985$	0	0
$986$	−68.9919	−2.19715
$987$	0	0
$988$	2.62536	0.0835237
$989$	0	0
$990$	0	0
$991$	−28.7846	−0.914373	−0.457187	−	0.889371i	$-0.651143\pi$
$991$	−28.7846	−0.914373	−0.457187	+	0.889371i	$0.651143\pi$
$992$	55.7654	1.77055
$993$	32.7846	1.04039
$994$	0	0
$995$	0	0
$996$	−10.3923	−0.329293
$997$	−14.1436	−0.447932	−0.223966	−	0.974597i	$-0.571900\pi$
$997$	−14.1436	−0.447932	−0.223966	+	0.974597i	$0.571900\pi$
$998$	31.3901	0.993636
$999$	0.757875	0.0239781

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bu.1.4		4
5.2	odd	4		735.2.d.f.589.7	yes	8
5.3	odd	4		735.2.d.f.589.2	yes	8
5.4	even	2		3675.2.a.bs.1.1		4
7.6	odd	2		3675.2.a.bs.1.4		4
15.2	even	4		2205.2.d.t.1324.2		8
15.8	even	4		2205.2.d.t.1324.8		8
35.2	odd	12		735.2.q.c.214.4		8
35.3	even	12		735.2.q.d.79.4		8
35.12	even	12		735.2.q.d.214.4		8
35.13	even	4		735.2.d.f.589.1	✓	8
35.17	even	12		735.2.q.c.79.1		8
35.18	odd	12		735.2.q.c.79.4		8
35.23	odd	12		735.2.q.d.214.1		8
35.27	even	4		735.2.d.f.589.8	yes	8
35.32	odd	12		735.2.q.d.79.1		8
35.33	even	12		735.2.q.c.214.1		8
35.34	odd	2	inner	3675.2.a.bu.1.1		4
105.62	odd	4		2205.2.d.t.1324.1		8
105.83	odd	4		2205.2.d.t.1324.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.2.d.f.589.1	✓	8	35.13	even	4
735.2.d.f.589.2	yes	8	5.3	odd	4
735.2.d.f.589.7	yes	8	5.2	odd	4
735.2.d.f.589.8	yes	8	35.27	even	4
735.2.q.c.79.1		8	35.17	even	12
735.2.q.c.79.4		8	35.18	odd	12
735.2.q.c.214.1		8	35.33	even	12
735.2.q.c.214.4		8	35.2	odd	12
735.2.q.d.79.1		8	35.32	odd	12
735.2.q.d.79.4		8	35.3	even	12
735.2.q.d.214.1		8	35.23	odd	12
735.2.q.d.214.4		8	35.12	even	12
2205.2.d.t.1324.1		8	105.62	odd	4
2205.2.d.t.1324.2		8	15.2	even	4
2205.2.d.t.1324.7		8	105.83	odd	4
2205.2.d.t.1324.8		8	15.8	even	4
3675.2.a.bs.1.1		4	5.4	even	2
3675.2.a.bs.1.4		4	7.6	odd	2
3675.2.a.bu.1.1		4	35.34	odd	2	inner
3675.2.a.bu.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q+1.93185 q^{2} +1.00000 q^{3} +1.73205 q^{4} +1.93185 q^{6} -0.517638 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.93185 q^{2} +1.00000 q^{3} +1.73205 q^{4} +1.93185 q^{6} -0.517638 q^{8} +1.00000 q^{9} -3.46410 q^{11} +1.73205 q^{12} -4.00000 q^{13} -4.46410 q^{16} -4.00000 q^{17} +1.93185 q^{18} -0.378937 q^{19} -6.69213 q^{22} -6.31319 q^{23} -0.517638 q^{24} -7.72741 q^{26} +1.00000 q^{27} +8.92820 q^{29} -7.34847 q^{31} -7.58871 q^{32} -3.46410 q^{33} -7.72741 q^{34} +1.73205 q^{36} +0.757875 q^{37} -0.732051 q^{38} -4.00000 q^{39} -8.48528 q^{41} -6.00000 q^{44} -12.1962 q^{46} -6.00000 q^{47} -4.46410 q^{48} -4.00000 q^{51} -6.92820 q^{52} +7.34847 q^{53} +1.93185 q^{54} -0.378937 q^{57} +17.2480 q^{58} +10.5558 q^{59} +9.14162 q^{61} -14.1962 q^{62} -5.73205 q^{64} -6.69213 q^{66} +6.96953 q^{67} -6.92820 q^{68} -6.31319 q^{69} +14.3923 q^{71} -0.517638 q^{72} -10.9282 q^{73} +1.46410 q^{74} -0.656339 q^{76} -7.72741 q^{78} -11.4641 q^{79} +1.00000 q^{81} -16.3923 q^{82} -6.00000 q^{83} +8.92820 q^{87} +1.79315 q^{88} -4.14110 q^{89} -10.9348 q^{92} -7.34847 q^{93} -11.5911 q^{94} -7.58871 q^{96} -5.07180 q^{97} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} - 16 q^{13} - 4 q^{16} - 16 q^{17} + 4 q^{27} + 8 q^{29} + 4 q^{38} - 16 q^{39} - 24 q^{44} - 28 q^{46} - 24 q^{47} - 4 q^{48} - 16 q^{51} - 36 q^{62} - 16 q^{64} + 16 q^{71} - 16 q^{73} - 8 q^{74} - 32 q^{79} + 4 q^{81} - 24 q^{82} - 24 q^{83} + 8 q^{87} - 48 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 - 16 * q^13 - 4 * q^16 - 16 * q^17 + 4 * q^27 + 8 * q^29 + 4 * q^38 - 16 * q^39 - 24 * q^44 - 28 * q^46 - 24 * q^47 - 4 * q^48 - 16 * q^51 - 36 * q^62 - 16 * q^64 + 16 * q^71 - 16 * q^73 - 8 * q^74 - 32 * q^79 + 4 * q^81 - 24 * q^82 - 24 * q^83 + 8 * q^87 - 48 * q^97

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