LMFDB - Embedded newform 9075.2.a.bi.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9075,2,Mod(1,9075)] 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("9075.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9075, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,6,0,0,0,0,2,0,0,-6,0,-20,0,-2,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.4642398343$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 363)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} +2.23607 q^{6} +4.47214 q^{7} -2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{12} -10.0000 q^{14} -1.00000 q^{16} -4.47214 q^{17} -2.23607 q^{18} -4.47214 q^{19} -4.47214 q^{21} +4.00000 q^{23} +2.23607 q^{24} -1.00000 q^{27} +13.4164 q^{28} +4.47214 q^{29} +6.70820 q^{32} +10.0000 q^{34} +3.00000 q^{36} -2.00000 q^{37} +10.0000 q^{38} -4.47214 q^{41} +10.0000 q^{42} -4.47214 q^{43} -8.94427 q^{46} -8.00000 q^{47} +1.00000 q^{48} +13.0000 q^{49} +4.47214 q^{51} -6.00000 q^{53} +2.23607 q^{54} -10.0000 q^{56} +4.47214 q^{57} -10.0000 q^{58} +8.94427 q^{61} +4.47214 q^{63} -13.0000 q^{64} +12.0000 q^{67} -13.4164 q^{68} -4.00000 q^{69} -8.00000 q^{71} -2.23607 q^{72} +8.94427 q^{73} +4.47214 q^{74} -13.4164 q^{76} +13.4164 q^{79} +1.00000 q^{81} +10.0000 q^{82} +8.94427 q^{83} -13.4164 q^{84} +10.0000 q^{86} -4.47214 q^{87} -14.0000 q^{89} +12.0000 q^{92} +17.8885 q^{94} -6.70820 q^{96} -2.00000 q^{97} -29.0689 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} - 6 q^{12} - 20 q^{14} - 2 q^{16} + 8 q^{23} - 2 q^{27} + 20 q^{34} + 6 q^{36} - 4 q^{37} + 20 q^{38} + 20 q^{42} - 16 q^{47} + 2 q^{48} + 26 q^{49} - 12 q^{53} - 20 q^{56}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9 - 6 * q^12 - 20 * q^14 - 2 * q^16 + 8 * q^23 - 2 * q^27 + 20 * q^34 + 6 * q^36 - 4 * q^37 + 20 * q^38 + 20 * q^42 - 16 * q^47 + 2 * q^48 + 26 * q^49 - 12 * q^53 - 20 * q^56 - 20 * q^58 - 26 * q^64 + 24 * q^67 - 8 * q^69 - 16 * q^71 + 2 * q^81 + 20 * q^82 + 20 * q^86 - 28 * q^89 + 24 * q^92 - 4 * 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 $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.00000	1.50000
$5$	0	0
$5$	0	0
$6$	2.23607	0.912871
$7$	4.47214	1.69031	0.845154	−	0.534522i	$-0.179509\pi$
$7$	4.47214	1.69031	0.845154	+	0.534522i	$0.179509\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−3.00000	−0.866025
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−10.0000	−2.67261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	−2.23607	−0.527046
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	−4.47214	−0.975900
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	13.4164	2.53546
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	6.70820	1.18585
$33$	0	0
$34$	10.0000	1.71499
$35$	0	0
$36$	3.00000	0.500000
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	10.0000	1.62221
$39$	0	0
$40$	0	0
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	10.0000	1.54303
$43$	−4.47214	−0.681994	−0.340997	−	0.940064i	$-0.610765\pi$
$43$	−4.47214	−0.681994	−0.340997	+	0.940064i	$0.610765\pi$
$44$	0	0
$45$	0	0
$46$	−8.94427	−1.31876
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	13.0000	1.85714
$50$	0	0
$51$	4.47214	0.626224
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	2.23607	0.304290
$55$	0	0
$56$	−10.0000	−1.33631
$57$	4.47214	0.592349
$58$	−10.0000	−1.31306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	8.94427	1.14520	0.572598	−	0.819836i	$-0.305935\pi$
$61$	8.94427	1.14520	0.572598	+	0.819836i	$0.305935\pi$
$62$	0	0
$63$	4.47214	0.563436
$64$	−13.0000	−1.62500
$65$	0	0
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−13.4164	−1.62698
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−2.23607	−0.263523
$73$	8.94427	1.04685	0.523424	−	0.852072i	$-0.324654\pi$
$73$	8.94427	1.04685	0.523424	+	0.852072i	$0.324654\pi$
$74$	4.47214	0.519875
$75$	0	0
$76$	−13.4164	−1.53897
$77$	0	0
$78$	0	0
$79$	13.4164	1.50946	0.754732	−	0.656033i	$-0.227767\pi$
$79$	13.4164	1.50946	0.754732	+	0.656033i	$0.227767\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	8.94427	0.981761	0.490881	−	0.871227i	$-0.336675\pi$
$83$	8.94427	0.981761	0.490881	+	0.871227i	$0.336675\pi$
$84$	−13.4164	−1.46385
$85$	0	0
$86$	10.0000	1.07833
$87$	−4.47214	−0.479463
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	0	0
$92$	12.0000	1.25109
$93$	0	0
$94$	17.8885	1.84506
$95$	0	0
$96$	−6.70820	−0.684653
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−29.0689	−2.93640
$99$	0	0
$100$	0	0
$101$	4.47214	0.444994	0.222497	−	0.974933i	$-0.428579\pi$
$101$	4.47214	0.444994	0.222497	+	0.974933i	$0.428579\pi$
$102$	−10.0000	−0.990148
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	0	0
$106$	13.4164	1.30312
$107$	−8.94427	−0.864675	−0.432338	−	0.901712i	$-0.642311\pi$
$107$	−8.94427	−0.864675	−0.432338	+	0.901712i	$0.642311\pi$
$108$	−3.00000	−0.288675
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	−4.47214	−0.422577
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−10.0000	−0.936586
$115$	0	0
$116$	13.4164	1.24568
$117$	0	0
$118$	0	0
$119$	−20.0000	−1.83340
$120$	0	0
$121$	0	0
$122$	−20.0000	−1.81071
$123$	4.47214	0.403239
$124$	0	0
$125$	0	0
$126$	−10.0000	−0.890871
$127$	−13.4164	−1.19051	−0.595257	−	0.803535i	$-0.702950\pi$
$127$	−13.4164	−1.19051	−0.595257	+	0.803535i	$0.702950\pi$
$128$	15.6525	1.38350
$129$	4.47214	0.393750
$130$	0	0
$131$	−17.8885	−1.56293	−0.781465	−	0.623949i	$-0.785527\pi$
$131$	−17.8885	−1.56293	−0.781465	+	0.623949i	$0.785527\pi$
$132$	0	0
$133$	−20.0000	−1.73422
$134$	−26.8328	−2.31800
$135$	0	0
$136$	10.0000	0.857493
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	8.94427	0.761387
$139$	13.4164	1.13796	0.568982	−	0.822350i	$-0.307337\pi$
$139$	13.4164	1.13796	0.568982	+	0.822350i	$0.307337\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	17.8885	1.50117
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−20.0000	−1.65521
$147$	−13.0000	−1.07222
$148$	−6.00000	−0.493197
$149$	−22.3607	−1.83186	−0.915929	−	0.401340i	$-0.868545\pi$
$149$	−22.3607	−1.83186	−0.915929	+	0.401340i	$0.868545\pi$
$150$	0	0
$151$	13.4164	1.09181	0.545906	−	0.837846i	$-0.316186\pi$
$151$	13.4164	1.09181	0.545906	+	0.837846i	$0.316186\pi$
$152$	10.0000	0.811107
$153$	−4.47214	−0.361551
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−30.0000	−2.38667
$159$	6.00000	0.475831
$160$	0	0
$161$	17.8885	1.40981
$162$	−2.23607	−0.175682
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−13.4164	−1.04765
$165$	0	0
$166$	−20.0000	−1.55230
$167$	−8.94427	−0.692129	−0.346064	−	0.938211i	$-0.612482\pi$
$167$	−8.94427	−0.692129	−0.346064	+	0.938211i	$0.612482\pi$
$168$	10.0000	0.771517
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−4.47214	−0.341993
$172$	−13.4164	−1.02299
$173$	13.4164	1.02003	0.510015	−	0.860165i	$-0.329640\pi$
$173$	13.4164	1.02003	0.510015	+	0.860165i	$0.329640\pi$
$174$	10.0000	0.758098
$175$	0	0
$176$	0	0
$177$	0	0
$178$	31.3050	2.34641
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−8.94427	−0.661180
$184$	−8.94427	−0.659380
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−24.0000	−1.75038
$189$	−4.47214	−0.325300
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	13.0000	0.938194
$193$	17.8885	1.28765	0.643823	−	0.765175i	$-0.277347\pi$
$193$	17.8885	1.28765	0.643823	+	0.765175i	$0.277347\pi$
$194$	4.47214	0.321081
$195$	0	0
$196$	39.0000	2.78571
$197$	−22.3607	−1.59313	−0.796566	−	0.604551i	$-0.793352\pi$
$197$	−22.3607	−1.59313	−0.796566	+	0.604551i	$0.793352\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	−10.0000	−0.703598
$203$	20.0000	1.40372
$204$	13.4164	0.939336
$205$	0	0
$206$	35.7771	2.49271
$207$	4.00000	0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−4.47214	−0.307875	−0.153937	−	0.988081i	$-0.549195\pi$
$211$	−4.47214	−0.307875	−0.153937	+	0.988081i	$0.549195\pi$
$212$	−18.0000	−1.23625
$213$	8.00000	0.548151
$214$	20.0000	1.36717
$215$	0	0
$216$	2.23607	0.152145
$217$	0	0
$218$	0	0
$219$	−8.94427	−0.604398
$220$	0	0
$221$	0	0
$222$	−4.47214	−0.300150
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	30.0000	2.00446
$225$	0	0
$226$	13.4164	0.892446
$227$	−8.94427	−0.593652	−0.296826	−	0.954932i	$-0.595928\pi$
$227$	−8.94427	−0.593652	−0.296826	+	0.954932i	$0.595928\pi$
$228$	13.4164	0.888523
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	−10.0000	−0.656532
$233$	4.47214	0.292979	0.146490	−	0.989212i	$-0.453202\pi$
$233$	4.47214	0.292979	0.146490	+	0.989212i	$0.453202\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−13.4164	−0.871489
$238$	44.7214	2.89886
$239$	8.94427	0.578557	0.289278	−	0.957245i	$-0.406585\pi$
$239$	8.94427	0.578557	0.289278	+	0.957245i	$0.406585\pi$
$240$	0	0
$241$	8.94427	0.576151	0.288076	−	0.957608i	$-0.406985\pi$
$241$	8.94427	0.576151	0.288076	+	0.957608i	$0.406985\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	26.8328	1.71780
$245$	0	0
$246$	−10.0000	−0.637577
$247$	0	0
$248$	0	0
$249$	−8.94427	−0.566820
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	13.4164	0.845154
$253$	0	0
$254$	30.0000	1.88237
$255$	0	0
$256$	−9.00000	−0.562500
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	−10.0000	−0.622573
$259$	−8.94427	−0.555770
$260$	0	0
$261$	4.47214	0.276818
$262$	40.0000	2.47121
$263$	−8.94427	−0.551527	−0.275764	−	0.961225i	$-0.588931\pi$
$263$	−8.94427	−0.551527	−0.275764	+	0.961225i	$0.588931\pi$
$264$	0	0
$265$	0	0
$266$	44.7214	2.74204
$267$	14.0000	0.856786
$268$	36.0000	2.19905
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−13.4164	−0.814989	−0.407494	−	0.913208i	$-0.633597\pi$
$271$	−13.4164	−0.814989	−0.407494	+	0.913208i	$0.633597\pi$
$272$	4.47214	0.271163
$273$	0	0
$274$	49.1935	2.97189
$275$	0	0
$276$	−12.0000	−0.722315
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	−30.0000	−1.79928
$279$	0	0
$280$	0	0
$281$	31.3050	1.86750	0.933748	−	0.357930i	$-0.116517\pi$
$281$	31.3050	1.86750	0.933748	+	0.357930i	$0.116517\pi$
$282$	−17.8885	−1.06525
$283$	13.4164	0.797523	0.398761	−	0.917055i	$-0.369440\pi$
$283$	13.4164	0.797523	0.398761	+	0.917055i	$0.369440\pi$
$284$	−24.0000	−1.42414
$285$	0	0
$286$	0	0
$287$	−20.0000	−1.18056
$288$	6.70820	0.395285
$289$	3.00000	0.176471
$290$	0	0
$291$	2.00000	0.117242
$292$	26.8328	1.57027
$293$	22.3607	1.30632	0.653162	−	0.757218i	$-0.273442\pi$
$293$	22.3607	1.30632	0.653162	+	0.757218i	$0.273442\pi$
$294$	29.0689	1.69533
$295$	0	0
$296$	4.47214	0.259938
$297$	0	0
$298$	50.0000	2.89642
$299$	0	0
$300$	0	0
$301$	−20.0000	−1.15278
$302$	−30.0000	−1.72631
$303$	−4.47214	−0.256917
$304$	4.47214	0.256495
$305$	0	0
$306$	10.0000	0.571662
$307$	4.47214	0.255238	0.127619	−	0.991823i	$-0.459266\pi$
$307$	4.47214	0.255238	0.127619	+	0.991823i	$0.459266\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	4.47214	0.252377
$315$	0	0
$316$	40.2492	2.26420
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−13.4164	−0.752355
$319$	0	0
$320$	0	0
$321$	8.94427	0.499221
$322$	−40.0000	−2.22911
$323$	20.0000	1.11283
$324$	3.00000	0.166667
$325$	0	0
$326$	8.94427	0.495377
$327$	0	0
$328$	10.0000	0.552158
$329$	−35.7771	−1.97245
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	26.8328	1.47264
$333$	−2.00000	−0.109599
$334$	20.0000	1.09435
$335$	0	0
$336$	4.47214	0.243975
$337$	−8.94427	−0.487226	−0.243613	−	0.969873i	$-0.578333\pi$
$337$	−8.94427	−0.487226	−0.243613	+	0.969873i	$0.578333\pi$
$338$	29.0689	1.58114
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	10.0000	0.540738
$343$	26.8328	1.44884
$344$	10.0000	0.539164
$345$	0	0
$346$	−30.0000	−1.61281
$347$	−8.94427	−0.480154	−0.240077	−	0.970754i	$-0.577173\pi$
$347$	−8.94427	−0.480154	−0.240077	+	0.970754i	$0.577173\pi$
$348$	−13.4164	−0.719195
$349$	−26.8328	−1.43633	−0.718164	−	0.695874i	$-0.755017\pi$
$349$	−26.8328	−1.43633	−0.718164	+	0.695874i	$0.755017\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	−42.0000	−2.22600
$357$	20.0000	1.05851
$358$	8.94427	0.472719
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	22.3607	1.17525
$363$	0	0
$364$	0	0
$365$	0	0
$366$	20.0000	1.04542
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−4.00000	−0.208514
$369$	−4.47214	−0.232810
$370$	0	0
$371$	−26.8328	−1.39309
$372$	0	0
$373$	26.8328	1.38935	0.694675	−	0.719323i	$-0.255548\pi$
$373$	26.8328	1.38935	0.694675	+	0.719323i	$0.255548\pi$
$374$	0	0
$375$	0	0
$376$	17.8885	0.922531
$377$	0	0
$378$	10.0000	0.514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	13.4164	0.687343
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	−15.6525	−0.798762
$385$	0	0
$386$	−40.0000	−2.03595
$387$	−4.47214	−0.227331
$388$	−6.00000	−0.304604
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	−17.8885	−0.904663
$392$	−29.0689	−1.46820
$393$	17.8885	0.902358
$394$	50.0000	2.51896
$395$	0	0
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	20.0000	1.00125
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	26.8328	1.33830
$403$	0	0
$404$	13.4164	0.667491
$405$	0	0
$406$	−44.7214	−2.21948
$407$	0	0
$408$	−10.0000	−0.495074
$409$	−26.8328	−1.32680	−0.663399	−	0.748266i	$-0.730887\pi$
$409$	−26.8328	−1.32680	−0.663399	+	0.748266i	$0.730887\pi$
$410$	0	0
$411$	22.0000	1.08518
$412$	−48.0000	−2.36479
$413$	0	0
$414$	−8.94427	−0.439587
$415$	0	0
$416$	0	0
$417$	−13.4164	−0.657004
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	10.0000	0.486792
$423$	−8.00000	−0.388973
$424$	13.4164	0.651558
$425$	0	0
$426$	−17.8885	−0.866703
$427$	40.0000	1.93574
$428$	−26.8328	−1.29701
$429$	0	0
$430$	0	0
$431$	−8.94427	−0.430830	−0.215415	−	0.976523i	$-0.569110\pi$
$431$	−8.94427	−0.430830	−0.215415	+	0.976523i	$0.569110\pi$
$432$	1.00000	0.0481125
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17.8885	−0.855725
$438$	20.0000	0.955637
$439$	−13.4164	−0.640330	−0.320165	−	0.947362i	$-0.603738\pi$
$439$	−13.4164	−0.640330	−0.320165	+	0.947362i	$0.603738\pi$
$440$	0	0
$441$	13.0000	0.619048
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	−35.7771	−1.69409
$447$	22.3607	1.05762
$448$	−58.1378	−2.74675
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	−18.0000	−0.846649
$453$	−13.4164	−0.630358
$454$	20.0000	0.938647
$455$	0	0
$456$	−10.0000	−0.468293
$457$	−26.8328	−1.25519	−0.627593	−	0.778542i	$-0.715960\pi$
$457$	−26.8328	−1.25519	−0.627593	+	0.778542i	$0.715960\pi$
$458$	−22.3607	−1.04485
$459$	4.47214	0.208741
$460$	0	0
$461$	13.4164	0.624864	0.312432	−	0.949940i	$-0.398856\pi$
$461$	13.4164	0.624864	0.312432	+	0.949940i	$0.398856\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−4.47214	−0.207614
$465$	0	0
$466$	−10.0000	−0.463241
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	53.6656	2.47805
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	0	0
$474$	30.0000	1.37795
$475$	0	0
$476$	−60.0000	−2.75010
$477$	−6.00000	−0.274721
$478$	−20.0000	−0.914779
$479$	−8.94427	−0.408674	−0.204337	−	0.978901i	$-0.565504\pi$
$479$	−8.94427	−0.408674	−0.204337	+	0.978901i	$0.565504\pi$
$480$	0	0
$481$	0	0
$482$	−20.0000	−0.910975
$483$	−17.8885	−0.813957
$484$	0	0
$485$	0	0
$486$	2.23607	0.101430
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−20.0000	−0.905357
$489$	4.00000	0.180886
$490$	0	0
$491$	−26.8328	−1.21095	−0.605474	−	0.795865i	$-0.707016\pi$
$491$	−26.8328	−1.21095	−0.605474	+	0.795865i	$0.707016\pi$
$492$	13.4164	0.604858
$493$	−20.0000	−0.900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−35.7771	−1.60482
$498$	20.0000	0.896221
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	8.94427	0.399601
$502$	26.8328	1.19761
$503$	26.8328	1.19642	0.598208	−	0.801341i	$-0.295880\pi$
$503$	26.8328	1.19642	0.598208	+	0.801341i	$0.295880\pi$
$504$	−10.0000	−0.445435
$505$	0	0
$506$	0	0
$507$	13.0000	0.577350
$508$	−40.2492	−1.78577
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	−11.1803	−0.494106
$513$	4.47214	0.197450
$514$	−49.1935	−2.16983
$515$	0	0
$516$	13.4164	0.590624
$517$	0	0
$518$	20.0000	0.878750
$519$	−13.4164	−0.588915
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	−10.0000	−0.437688
$523$	−4.47214	−0.195553	−0.0977764	−	0.995208i	$-0.531173\pi$
$523$	−4.47214	−0.195553	−0.0977764	+	0.995208i	$0.531173\pi$
$524$	−53.6656	−2.34439
$525$	0	0
$526$	20.0000	0.872041
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	−60.0000	−2.60133
$533$	0	0
$534$	−31.3050	−1.35470
$535$	0	0
$536$	−26.8328	−1.15900
$537$	4.00000	0.172613
$538$	22.3607	0.964037
$539$	0	0
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	30.0000	1.28861
$543$	10.0000	0.429141
$544$	−30.0000	−1.28624
$545$	0	0
$546$	0	0
$547$	−40.2492	−1.72093	−0.860466	−	0.509507i	$-0.829828\pi$
$547$	−40.2492	−1.72093	−0.860466	+	0.509507i	$0.829828\pi$
$548$	−66.0000	−2.81938
$549$	8.94427	0.381732
$550$	0	0
$551$	−20.0000	−0.852029
$552$	8.94427	0.380693
$553$	60.0000	2.55146
$554$	0	0
$555$	0	0
$556$	40.2492	1.70695
$557$	40.2492	1.70541	0.852707	−	0.522389i	$-0.174959\pi$
$557$	40.2492	1.70541	0.852707	+	0.522389i	$0.174959\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−70.0000	−2.95277
$563$	−17.8885	−0.753912	−0.376956	−	0.926231i	$-0.623029\pi$
$563$	−17.8885	−0.753912	−0.376956	+	0.926231i	$0.623029\pi$
$564$	24.0000	1.01058
$565$	0	0
$566$	−30.0000	−1.26099
$567$	4.47214	0.187812
$568$	17.8885	0.750587
$569$	4.47214	0.187482	0.0937408	−	0.995597i	$-0.470117\pi$
$569$	4.47214	0.187482	0.0937408	+	0.995597i	$0.470117\pi$
$570$	0	0
$571$	−13.4164	−0.561459	−0.280730	−	0.959787i	$-0.590576\pi$
$571$	−13.4164	−0.561459	−0.280730	+	0.959787i	$0.590576\pi$
$572$	0	0
$573$	0	0
$574$	44.7214	1.86663
$575$	0	0
$576$	−13.0000	−0.541667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−6.70820	−0.279024
$579$	−17.8885	−0.743423
$580$	0	0
$581$	40.0000	1.65948
$582$	−4.47214	−0.185376
$583$	0	0
$584$	−20.0000	−0.827606
$585$	0	0
$586$	−50.0000	−2.06548
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	−39.0000	−1.60833
$589$	0	0
$590$	0	0
$591$	22.3607	0.919795
$592$	2.00000	0.0821995
$593$	22.3607	0.918243	0.459122	−	0.888373i	$-0.348164\pi$
$593$	22.3607	0.918243	0.459122	+	0.888373i	$0.348164\pi$
$594$	0	0
$595$	0	0
$596$	−67.0820	−2.74779
$597$	0	0
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	44.7214	1.82271
$603$	12.0000	0.488678
$604$	40.2492	1.63772
$605$	0	0
$606$	10.0000	0.406222
$607$	4.47214	0.181518	0.0907592	−	0.995873i	$-0.471071\pi$
$607$	4.47214	0.181518	0.0907592	+	0.995873i	$0.471071\pi$
$608$	−30.0000	−1.21666
$609$	−20.0000	−0.810441
$610$	0	0
$611$	0	0
$612$	−13.4164	−0.542326
$613$	−44.7214	−1.80628	−0.903139	−	0.429348i	$-0.858744\pi$
$613$	−44.7214	−1.80628	−0.903139	+	0.429348i	$0.858744\pi$
$614$	−10.0000	−0.403567
$615$	0	0
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	−35.7771	−1.43917
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	26.8328	1.07590
$623$	−62.6099	−2.50841
$624$	0	0
$625$	0	0
$626$	31.3050	1.25120
$627$	0	0
$628$	−6.00000	−0.239426
$629$	8.94427	0.356631
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	−30.0000	−1.19334
$633$	4.47214	0.177751
$634$	40.2492	1.59850
$635$	0	0
$636$	18.0000	0.713746
$637$	0	0
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−20.0000	−0.789337
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	53.6656	2.11472
$645$	0	0
$646$	−44.7214	−1.75954
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−2.23607	−0.0878410
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−12.0000	−0.469956
$653$	−46.0000	−1.80012	−0.900060	−	0.435767i	$-0.856477\pi$
$653$	−46.0000	−1.80012	−0.900060	+	0.435767i	$0.856477\pi$
$654$	0	0
$655$	0	0
$656$	4.47214	0.174608
$657$	8.94427	0.348949
$658$	80.0000	3.11872
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	44.7214	1.73814
$663$	0	0
$664$	−20.0000	−0.776151
$665$	0	0
$666$	4.47214	0.173292
$667$	17.8885	0.692647
$668$	−26.8328	−1.03819
$669$	−16.0000	−0.618596
$670$	0	0
$671$	0	0
$672$	−30.0000	−1.15728
$673$	17.8885	0.689553	0.344776	−	0.938685i	$-0.387955\pi$
$673$	17.8885	0.689553	0.344776	+	0.938685i	$0.387955\pi$
$674$	20.0000	0.770371
$675$	0	0
$676$	−39.0000	−1.50000
$677$	31.3050	1.20315	0.601574	−	0.798817i	$-0.294541\pi$
$677$	31.3050	1.20315	0.601574	+	0.798817i	$0.294541\pi$
$678$	−13.4164	−0.515254
$679$	−8.94427	−0.343250
$680$	0	0
$681$	8.94427	0.342745
$682$	0	0
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	−13.4164	−0.512989
$685$	0	0
$686$	−60.0000	−2.29081
$687$	−10.0000	−0.381524
$688$	4.47214	0.170499
$689$	0	0
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	40.2492	1.53005
$693$	0	0
$694$	20.0000	0.759190
$695$	0	0
$696$	10.0000	0.379049
$697$	20.0000	0.757554
$698$	60.0000	2.27103
$699$	−4.47214	−0.169152
$700$	0	0
$701$	−22.3607	−0.844551	−0.422276	−	0.906467i	$-0.638769\pi$
$701$	−22.3607	−0.844551	−0.422276	+	0.906467i	$0.638769\pi$
$702$	0	0
$703$	8.94427	0.337340
$704$	0	0
$705$	0	0
$706$	−31.3050	−1.17818
$707$	20.0000	0.752177
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	13.4164	0.503155
$712$	31.3050	1.17320
$713$	0	0
$714$	−44.7214	−1.67365
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−8.94427	−0.334030
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	−71.5542	−2.66482
$722$	−2.23607	−0.0832178
$723$	−8.94427	−0.332641
$724$	−30.0000	−1.11494
$725$	0	0
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	20.0000	0.739727
$732$	−26.8328	−0.991769
$733$	17.8885	0.660728	0.330364	−	0.943854i	$-0.392828\pi$
$733$	17.8885	0.660728	0.330364	+	0.943854i	$0.392828\pi$
$734$	17.8885	0.660278
$735$	0	0
$736$	26.8328	0.989071
$737$	0	0
$738$	10.0000	0.368105
$739$	4.47214	0.164510	0.0822551	−	0.996611i	$-0.473788\pi$
$739$	4.47214	0.164510	0.0822551	+	0.996611i	$0.473788\pi$
$740$	0	0
$741$	0	0
$742$	60.0000	2.20267
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	−60.0000	−2.19676
$747$	8.94427	0.327254
$748$	0	0
$749$	−40.0000	−1.46157
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	8.00000	0.291730
$753$	12.0000	0.437304
$754$	0	0
$755$	0	0
$756$	−13.4164	−0.487950
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	44.7214	1.62435
$759$	0	0
$760$	0	0
$761$	−31.3050	−1.13480	−0.567402	−	0.823441i	$-0.692051\pi$
$761$	−31.3050	−1.13480	−0.567402	+	0.823441i	$0.692051\pi$
$762$	−30.0000	−1.08679
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−80.4984	−2.90853
$767$	0	0
$768$	9.00000	0.324760
$769$	35.7771	1.29015	0.645077	−	0.764117i	$-0.276825\pi$
$769$	35.7771	1.29015	0.645077	+	0.764117i	$0.276825\pi$
$770$	0	0
$771$	−22.0000	−0.792311
$772$	53.6656	1.93147
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	10.0000	0.359443
$775$	0	0
$776$	4.47214	0.160540
$777$	8.94427	0.320874
$778$	−22.3607	−0.801669
$779$	20.0000	0.716574
$780$	0	0
$781$	0	0
$782$	40.0000	1.43040
$783$	−4.47214	−0.159821
$784$	−13.0000	−0.464286
$785$	0	0
$786$	−40.0000	−1.42675
$787$	−40.2492	−1.43473	−0.717365	−	0.696698i	$-0.754652\pi$
$787$	−40.2492	−1.43473	−0.717365	+	0.696698i	$0.754652\pi$
$788$	−67.0820	−2.38970
$789$	8.94427	0.318425
$790$	0	0
$791$	−26.8328	−0.954065
$792$	0	0
$793$	0	0
$794$	49.1935	1.74581
$795$	0	0
$796$	0	0
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	−44.7214	−1.58312
$799$	35.7771	1.26570
$800$	0	0
$801$	−14.0000	−0.494666
$802$	−67.0820	−2.36875
$803$	0	0
$804$	−36.0000	−1.26962
$805$	0	0
$806$	0	0
$807$	10.0000	0.352017
$808$	−10.0000	−0.351799
$809$	−13.4164	−0.471696	−0.235848	−	0.971790i	$-0.575787\pi$
$809$	−13.4164	−0.471696	−0.235848	+	0.971790i	$0.575787\pi$
$810$	0	0
$811$	−4.47214	−0.157038	−0.0785190	−	0.996913i	$-0.525019\pi$
$811$	−4.47214	−0.157038	−0.0785190	+	0.996913i	$0.525019\pi$
$812$	60.0000	2.10559
$813$	13.4164	0.470534
$814$	0	0
$815$	0	0
$816$	−4.47214	−0.156556
$817$	20.0000	0.699711
$818$	60.0000	2.09785
$819$	0	0
$820$	0	0
$821$	22.3607	0.780393	0.390197	−	0.920732i	$-0.372407\pi$
$821$	22.3607	0.780393	0.390197	+	0.920732i	$0.372407\pi$
$822$	−49.1935	−1.71582
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	35.7771	1.24635
$825$	0	0
$826$	0	0
$827$	−44.7214	−1.55511	−0.777557	−	0.628812i	$-0.783541\pi$
$827$	−44.7214	−1.55511	−0.777557	+	0.628812i	$0.783541\pi$
$828$	12.0000	0.417029
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−58.1378	−2.01435
$834$	30.0000	1.03882
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−8.94427	−0.308975
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	−22.3607	−0.770600
$843$	−31.3050	−1.07820
$844$	−13.4164	−0.461812
$845$	0	0
$846$	17.8885	0.615021
$847$	0	0
$848$	6.00000	0.206041
$849$	−13.4164	−0.460450
$850$	0	0
$851$	−8.00000	−0.274236
$852$	24.0000	0.822226
$853$	−8.94427	−0.306246	−0.153123	−	0.988207i	$-0.548933\pi$
$853$	−8.94427	−0.306246	−0.153123	+	0.988207i	$0.548933\pi$
$854$	−89.4427	−3.06067
$855$	0	0
$856$	20.0000	0.683586
$857$	−40.2492	−1.37489	−0.687444	−	0.726238i	$-0.741267\pi$
$857$	−40.2492	−1.37489	−0.687444	+	0.726238i	$0.741267\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	20.0000	0.681598
$862$	20.0000	0.681203
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	−6.70820	−0.228218
$865$	0	0
$866$	−13.4164	−0.455908
$867$	−3.00000	−0.101885
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	40.0000	1.35302
$875$	0	0
$876$	−26.8328	−0.906597
$877$	8.94427	0.302027	0.151013	−	0.988532i	$-0.451746\pi$
$877$	8.94427	0.302027	0.151013	+	0.988532i	$0.451746\pi$
$878$	30.0000	1.01245
$879$	−22.3607	−0.754207
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	−29.0689	−0.978800
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	0	0
$886$	−53.6656	−1.80293
$887$	53.6656	1.80192	0.900958	−	0.433907i	$-0.142865\pi$
$887$	53.6656	1.80192	0.900958	+	0.433907i	$0.142865\pi$
$888$	−4.47214	−0.150075
$889$	−60.0000	−2.01234
$890$	0	0
$891$	0	0
$892$	48.0000	1.60716
$893$	35.7771	1.19723
$894$	−50.0000	−1.67225
$895$	0	0
$896$	70.0000	2.33854
$897$	0	0
$898$	13.4164	0.447711
$899$	0	0
$900$	0	0
$901$	26.8328	0.893931
$902$	0	0
$903$	20.0000	0.665558
$904$	13.4164	0.446223
$905$	0	0
$906$	30.0000	0.996683
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−26.8328	−0.890478
$909$	4.47214	0.148331
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	−4.47214	−0.148087
$913$	0	0
$914$	60.0000	1.98462
$915$	0	0
$916$	30.0000	0.991228
$917$	−80.0000	−2.64183
$918$	−10.0000	−0.330049
$919$	22.3607	0.737611	0.368805	−	0.929507i	$-0.379767\pi$
$919$	22.3607	0.737611	0.368805	+	0.929507i	$0.379767\pi$
$920$	0	0
$921$	−4.47214	−0.147362
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	0	0
$926$	53.6656	1.76356
$927$	−16.0000	−0.525509
$928$	30.0000	0.984798
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−58.1378	−1.90539
$932$	13.4164	0.439469
$933$	12.0000	0.392862
$934$	−17.8885	−0.585331
$935$	0	0
$936$	0	0
$937$	−53.6656	−1.75318	−0.876590	−	0.481238i	$-0.840187\pi$
$937$	−53.6656	−1.75318	−0.876590	+	0.481238i	$0.840187\pi$
$938$	−120.000	−3.91814
$939$	14.0000	0.456873
$940$	0	0
$941$	22.3607	0.728937	0.364469	−	0.931216i	$-0.381251\pi$
$941$	22.3607	0.728937	0.364469	+	0.931216i	$0.381251\pi$
$942$	−4.47214	−0.145710
$943$	−17.8885	−0.582531
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	−40.2492	−1.30723
$949$	0	0
$950$	0	0
$951$	18.0000	0.583690
$952$	44.7214	1.44943
$953$	−22.3607	−0.724333	−0.362167	−	0.932113i	$-0.617963\pi$
$953$	−22.3607	−0.724333	−0.362167	+	0.932113i	$0.617963\pi$
$954$	13.4164	0.434372
$955$	0	0
$956$	26.8328	0.867835
$957$	0	0
$958$	20.0000	0.646171
$959$	−98.3870	−3.17708
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−8.94427	−0.288225
$964$	26.8328	0.864227
$965$	0	0
$966$	40.0000	1.28698
$967$	−13.4164	−0.431443	−0.215721	−	0.976455i	$-0.569210\pi$
$967$	−13.4164	−0.431443	−0.215721	+	0.976455i	$0.569210\pi$
$968$	0	0
$969$	−20.0000	−0.642493
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	−3.00000	−0.0962250
$973$	60.0000	1.92351
$974$	17.8885	0.573186
$975$	0	0
$976$	−8.94427	−0.286299
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	−8.94427	−0.286006
$979$	0	0
$980$	0	0
$981$	0	0
$982$	60.0000	1.91468
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	44.7214	1.42422
$987$	35.7771	1.13880
$988$	0	0
$989$	−17.8885	−0.568823
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	80.0000	2.53745
$995$	0	0
$996$	−26.8328	−0.850230
$997$	26.8328	0.849804	0.424902	−	0.905239i	$-0.360309\pi$
$997$	26.8328	0.849804	0.424902	+	0.905239i	$0.360309\pi$
$998$	44.7214	1.41563
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.bi.1.1		2
5.4	even	2		363.2.a.g.1.2	yes	2
11.10	odd	2	inner	9075.2.a.bi.1.2		2
15.14	odd	2		1089.2.a.p.1.1		2
20.19	odd	2		5808.2.a.bx.1.2		2
55.4	even	10		363.2.e.l.148.1		4
55.9	even	10		363.2.e.a.202.1		4
55.14	even	10		363.2.e.l.130.1		4
55.19	odd	10		363.2.e.a.130.1		4
55.24	odd	10		363.2.e.l.202.1		4
55.29	odd	10		363.2.e.a.148.1		4
55.39	odd	10		363.2.e.l.124.1		4
55.49	even	10		363.2.e.a.124.1		4
55.54	odd	2		363.2.a.g.1.1	✓	2
165.164	even	2		1089.2.a.p.1.2		2
220.219	even	2		5808.2.a.bx.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.g.1.1	✓	2	55.54	odd	2
363.2.a.g.1.2	yes	2	5.4	even	2
363.2.e.a.124.1		4	55.49	even	10
363.2.e.a.130.1		4	55.19	odd	10
363.2.e.a.148.1		4	55.29	odd	10
363.2.e.a.202.1		4	55.9	even	10
363.2.e.l.124.1		4	55.39	odd	10
363.2.e.l.130.1		4	55.14	even	10
363.2.e.l.148.1		4	55.4	even	10
363.2.e.l.202.1		4	55.24	odd	10
1089.2.a.p.1.1		2	15.14	odd	2
1089.2.a.p.1.2		2	165.164	even	2
5808.2.a.bx.1.1		2	220.219	even	2
5808.2.a.bx.1.2		2	20.19	odd	2
9075.2.a.bi.1.1		2	1.1	even	1	trivial
9075.2.a.bi.1.2		2	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} +2.23607 q^{6} +4.47214 q^{7} -2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{12} -10.0000 q^{14} -1.00000 q^{16} -4.47214 q^{17} -2.23607 q^{18} -4.47214 q^{19} -4.47214 q^{21} +4.00000 q^{23} +2.23607 q^{24} -1.00000 q^{27} +13.4164 q^{28} +4.47214 q^{29} +6.70820 q^{32} +10.0000 q^{34} +3.00000 q^{36} -2.00000 q^{37} +10.0000 q^{38} -4.47214 q^{41} +10.0000 q^{42} -4.47214 q^{43} -8.94427 q^{46} -8.00000 q^{47} +1.00000 q^{48} +13.0000 q^{49} +4.47214 q^{51} -6.00000 q^{53} +2.23607 q^{54} -10.0000 q^{56} +4.47214 q^{57} -10.0000 q^{58} +8.94427 q^{61} +4.47214 q^{63} -13.0000 q^{64} +12.0000 q^{67} -13.4164 q^{68} -4.00000 q^{69} -8.00000 q^{71} -2.23607 q^{72} +8.94427 q^{73} +4.47214 q^{74} -13.4164 q^{76} +13.4164 q^{79} +1.00000 q^{81} +10.0000 q^{82} +8.94427 q^{83} -13.4164 q^{84} +10.0000 q^{86} -4.47214 q^{87} -14.0000 q^{89} +12.0000 q^{92} +17.8885 q^{94} -6.70820 q^{96} -2.00000 q^{97} -29.0689 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} - 6 q^{12} - 20 q^{14} - 2 q^{16} + 8 q^{23} - 2 q^{27} + 20 q^{34} + 6 q^{36} - 4 q^{37} + 20 q^{38} + 20 q^{42} - 16 q^{47} + 2 q^{48} + 26 q^{49} - 12 q^{53} - 20 q^{56}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9 - 6 * q^12 - 20 * q^14 - 2 * q^16 + 8 * q^23 - 2 * q^27 + 20 * q^34 + 6 * q^36 - 4 * q^37 + 20 * q^38 + 20 * q^42 - 16 * q^47 + 2 * q^48 + 26 * q^49 - 12 * q^53 - 20 * q^56 - 20 * q^58 - 26 * q^64 + 24 * q^67 - 8 * q^69 - 16 * q^71 + 2 * q^81 + 20 * q^82 + 20 * q^86 - 28 * q^89 + 24 * q^92 - 4 * q^97

Introduction

L-functions

Modular forms

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