LMFDB - Embedded newform 9075.2.a.dx.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$72.4642398343$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 18 x^{8} - 220 x^{7} + 61 x^{6} + 330 x^{5} - 145 x^{4} + \cdots - 5 $ x^12 - 4x^11 - 10x^10 + 52x^9 + 18x^8 - 220x^7 + 61x^6 + 330x^5 - 145x^4 - 148x^3 + 71x^2 + 10*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.90743$ 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)

$f(q)$	$=$	$q-1.90743 q^{2} -1.00000 q^{3} +1.63829 q^{4} +1.90743 q^{6} -4.45734 q^{7} +0.689943 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.90743 q^{2} -1.00000 q^{3} +1.63829 q^{4} +1.90743 q^{6} -4.45734 q^{7} +0.689943 q^{8} +1.00000 q^{9} -1.63829 q^{12} +0.534008 q^{13} +8.50206 q^{14} -4.59259 q^{16} -4.54422 q^{17} -1.90743 q^{18} -1.92453 q^{19} +4.45734 q^{21} -5.18745 q^{23} -0.689943 q^{24} -1.01858 q^{26} -1.00000 q^{27} -7.30240 q^{28} +4.75453 q^{29} +3.49728 q^{31} +7.38016 q^{32} +8.66777 q^{34} +1.63829 q^{36} -0.527144 q^{37} +3.67091 q^{38} -0.534008 q^{39} -4.69578 q^{41} -8.50206 q^{42} -3.02459 q^{43} +9.89470 q^{46} +11.9878 q^{47} +4.59259 q^{48} +12.8679 q^{49} +4.54422 q^{51} +0.874857 q^{52} +5.02443 q^{53} +1.90743 q^{54} -3.07531 q^{56} +1.92453 q^{57} -9.06894 q^{58} +13.6495 q^{59} -11.6915 q^{61} -6.67082 q^{62} -4.45734 q^{63} -4.89194 q^{64} -11.5622 q^{67} -7.44473 q^{68} +5.18745 q^{69} +3.36587 q^{71} +0.689943 q^{72} +0.650391 q^{73} +1.00549 q^{74} -3.15294 q^{76} +1.01858 q^{78} +17.7199 q^{79} +1.00000 q^{81} +8.95686 q^{82} -4.24738 q^{83} +7.30240 q^{84} +5.76920 q^{86} -4.75453 q^{87} -8.24094 q^{89} -2.38025 q^{91} -8.49854 q^{92} -3.49728 q^{93} -22.8658 q^{94} -7.38016 q^{96} +1.13839 q^{97} -24.5445 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} - 12 q^{3} + 12 q^{4} + 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 4 * q^2 - 12 * q^3 + 12 * q^4 + 4 * q^6 - 8 * q^7 - 12 * q^8 + 12 * q^9	$ 12 q - 4 q^{2} - 12 q^{3} + 12 q^{4} + 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9} - 12 q^{12} - 18 q^{13} + 6 q^{14} + 24 q^{16} - 18 q^{17} - 4 q^{18} + 16 q^{19} + 8 q^{21} + 12 q^{24} + 16 q^{26} - 12 q^{27} - 30 q^{28} - 28 q^{32} + 6 q^{34} + 12 q^{36} + 28 q^{38} + 18 q^{39} - 6 q^{42} - 32 q^{43} + 28 q^{46} + 4 q^{47} - 24 q^{48} + 12 q^{49} + 18 q^{51} - 48 q^{52} + 12 q^{53} + 4 q^{54} + 6 q^{56} - 16 q^{57} + 10 q^{58} + 20 q^{59} + 20 q^{61} - 20 q^{62} - 8 q^{63} - 6 q^{64} - 10 q^{67} - 26 q^{68} + 32 q^{71} - 12 q^{72} - 26 q^{73} - 68 q^{74} + 34 q^{76} - 16 q^{78} + 32 q^{79} + 12 q^{81} + 62 q^{82} - 26 q^{83} + 30 q^{84} - 36 q^{86} - 10 q^{89} - 8 q^{92} - 2 q^{94} + 28 q^{96} + 22 q^{97} - 60 q^{98}+O(q^{100}) $ 12 * q - 4 * q^2 - 12 * q^3 + 12 * q^4 + 4 * q^6 - 8 * q^7 - 12 * q^8 + 12 * q^9 - 12 * q^12 - 18 * q^13 + 6 * q^14 + 24 * q^16 - 18 * q^17 - 4 * q^18 + 16 * q^19 + 8 * q^21 + 12 * q^24 + 16 * q^26 - 12 * q^27 - 30 * q^28 - 28 * q^32 + 6 * q^34 + 12 * q^36 + 28 * q^38 + 18 * q^39 - 6 * q^42 - 32 * q^43 + 28 * q^46 + 4 * q^47 - 24 * q^48 + 12 * q^49 + 18 * q^51 - 48 * q^52 + 12 * q^53 + 4 * q^54 + 6 * q^56 - 16 * q^57 + 10 * q^58 + 20 * q^59 + 20 * q^61 - 20 * q^62 - 8 * q^63 - 6 * q^64 - 10 * q^67 - 26 * q^68 + 32 * q^71 - 12 * q^72 - 26 * q^73 - 68 * q^74 + 34 * q^76 - 16 * q^78 + 32 * q^79 + 12 * q^81 + 62 * q^82 - 26 * q^83 + 30 * q^84 - 36 * q^86 - 10 * q^89 - 8 * q^92 - 2 * q^94 + 28 * q^96 + 22 * q^97 - 60 * q^98

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 $

$2$	−1.90743	−1.34876	−0.674378	−	0.738386i	$-0.735588\pi$
$2$	−1.90743	−1.34876	−0.674378	+	0.738386i	$0.735588\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.63829	0.819143
$5$	0	0
$5$	0	0
$6$	1.90743	0.778705
$7$	−4.45734	−1.68472	−0.842358	−	0.538919i	$-0.818833\pi$
$7$	−4.45734	−1.68472	−0.842358	+	0.538919i	$0.818833\pi$
$8$	0.689943	0.243932
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−1.63829	−0.472933
$13$	0.534008	0.148107	0.0740535	−	0.997254i	$-0.476406\pi$
$13$	0.534008	0.148107	0.0740535	+	0.997254i	$0.476406\pi$
$14$	8.50206	2.27227
$15$	0	0
$16$	−4.59259	−1.14815
$17$	−4.54422	−1.10213	−0.551067	−	0.834461i	$-0.685779\pi$
$17$	−4.54422	−1.10213	−0.551067	+	0.834461i	$0.685779\pi$
$18$	−1.90743	−0.449585
$19$	−1.92453	−0.441518	−0.220759	−	0.975328i	$-0.570853\pi$
$19$	−1.92453	−0.441518	−0.220759	+	0.975328i	$0.570853\pi$
$20$	0	0
$21$	4.45734	0.972671
$22$	0	0
$23$	−5.18745	−1.08166	−0.540830	−	0.841132i	$-0.681890\pi$
$23$	−5.18745	−1.08166	−0.540830	+	0.841132i	$0.681890\pi$
$24$	−0.689943	−0.140834
$25$	0	0
$26$	−1.01858	−0.199760
$27$	−1.00000	−0.192450
$28$	−7.30240	−1.38002
$29$	4.75453	0.882895	0.441447	−	0.897287i	$-0.354465\pi$
$29$	4.75453	0.882895	0.441447	+	0.897287i	$0.354465\pi$
$30$	0	0
$31$	3.49728	0.628130	0.314065	−	0.949401i	$-0.398309\pi$
$31$	3.49728	0.628130	0.314065	+	0.949401i	$0.398309\pi$
$32$	7.38016	1.30464
$33$	0	0
$34$	8.66777	1.48651
$35$	0	0
$36$	1.63829	0.273048
$37$	−0.527144	−0.0866620	−0.0433310	−	0.999061i	$-0.513797\pi$
$37$	−0.527144	−0.0866620	−0.0433310	+	0.999061i	$0.513797\pi$
$38$	3.67091	0.595501
$39$	−0.534008	−0.0855096
$40$	0	0
$41$	−4.69578	−0.733357	−0.366678	−	0.930348i	$-0.619505\pi$
$41$	−4.69578	−0.733357	−0.366678	+	0.930348i	$0.619505\pi$
$42$	−8.50206	−1.31190
$43$	−3.02459	−0.461246	−0.230623	−	0.973043i	$-0.574076\pi$
$43$	−3.02459	−0.461246	−0.230623	+	0.973043i	$0.574076\pi$
$44$	0	0
$45$	0	0
$46$	9.89470	1.45889
$47$	11.9878	1.74860	0.874298	−	0.485390i	$-0.161323\pi$
$47$	11.9878	1.74860	0.874298	+	0.485390i	$0.161323\pi$
$48$	4.59259	0.662883
$49$	12.8679	1.83827
$50$	0	0
$51$	4.54422	0.636318
$52$	0.874857	0.121321
$53$	5.02443	0.690158	0.345079	−	0.938574i	$-0.387852\pi$
$53$	5.02443	0.690158	0.345079	+	0.938574i	$0.387852\pi$
$54$	1.90743	0.259568
$55$	0	0
$56$	−3.07531	−0.410955
$57$	1.92453	0.254911
$58$	−9.06894	−1.19081
$59$	13.6495	1.77702	0.888508	−	0.458861i	$-0.151742\pi$
$59$	13.6495	1.77702	0.888508	+	0.458861i	$0.151742\pi$
$60$	0	0
$61$	−11.6915	−1.49695	−0.748473	−	0.663165i	$-0.769213\pi$
$61$	−11.6915	−1.49695	−0.748473	+	0.663165i	$0.769213\pi$
$62$	−6.67082	−0.847195
$63$	−4.45734	−0.561572
$64$	−4.89194	−0.611493
$65$	0	0
$66$	0	0
$67$	−11.5622	−1.41255	−0.706276	−	0.707936i	$-0.749626\pi$
$67$	−11.5622	−1.41255	−0.706276	+	0.707936i	$0.749626\pi$
$68$	−7.44473	−0.902806
$69$	5.18745	0.624496
$70$	0	0
$71$	3.36587	0.399455	0.199728	−	0.979851i	$-0.435994\pi$
$71$	3.36587	0.399455	0.199728	+	0.979851i	$0.435994\pi$
$72$	0.689943	0.0813105
$73$	0.650391	0.0761226	0.0380613	−	0.999275i	$-0.487882\pi$
$73$	0.650391	0.0761226	0.0380613	+	0.999275i	$0.487882\pi$
$74$	1.00549	0.116886
$75$	0	0
$76$	−3.15294	−0.361667
$77$	0	0
$78$	1.01858	0.115332
$79$	17.7199	1.99365	0.996823	−	0.0796525i	$-0.0253811\pi$
$79$	17.7199	1.99365	0.996823	+	0.0796525i	$0.0253811\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	8.95686	0.989120
$83$	−4.24738	−0.466210	−0.233105	−	0.972452i	$-0.574889\pi$
$83$	−4.24738	−0.466210	−0.233105	+	0.972452i	$0.574889\pi$
$84$	7.30240	0.796757
$85$	0	0
$86$	5.76920	0.622109
$87$	−4.75453	−0.509739
$88$	0	0
$89$	−8.24094	−0.873538	−0.436769	−	0.899574i	$-0.643877\pi$
$89$	−8.24094	−0.873538	−0.436769	+	0.899574i	$0.643877\pi$
$90$	0	0
$91$	−2.38025	−0.249518
$92$	−8.49854	−0.886034
$93$	−3.49728	−0.362651
$94$	−22.8658	−2.35843
$95$	0	0
$96$	−7.38016	−0.753234
$97$	1.13839	0.115586	0.0577931	−	0.998329i	$-0.481594\pi$
$97$	1.13839	0.115586	0.0577931	+	0.998329i	$0.481594\pi$
$98$	−24.5445	−2.47937
$99$	0	0
$100$	0	0
$101$	1.02375	0.101867	0.0509333	−	0.998702i	$-0.483780\pi$
$101$	1.02375	0.101867	0.0509333	+	0.998702i	$0.483780\pi$
$102$	−8.66777	−0.858237
$103$	4.79071	0.472042	0.236021	−	0.971748i	$-0.424157\pi$
$103$	4.79071	0.472042	0.236021	+	0.971748i	$0.424157\pi$
$104$	0.368435	0.0361280
$105$	0	0
$106$	−9.58374	−0.930855
$107$	−0.735971	−0.0711490	−0.0355745	−	0.999367i	$-0.511326\pi$
$107$	−0.735971	−0.0711490	−0.0355745	+	0.999367i	$0.511326\pi$
$108$	−1.63829	−0.157644
$109$	0.0433994	0.00415691	0.00207845	−	0.999998i	$-0.499338\pi$
$109$	0.0433994	0.00415691	0.00207845	+	0.999998i	$0.499338\pi$
$110$	0	0
$111$	0.527144	0.0500343
$112$	20.4707	1.93430
$113$	−11.2006	−1.05366	−0.526831	−	0.849970i	$-0.676620\pi$
$113$	−11.2006	−1.05366	−0.526831	+	0.849970i	$0.676620\pi$
$114$	−3.67091	−0.343813
$115$	0	0
$116$	7.78929	0.723217
$117$	0.534008	0.0493690
$118$	−26.0355	−2.39676
$119$	20.2551	1.85678
$120$	0	0
$121$	0	0
$122$	22.3007	2.01901
$123$	4.69578	0.423404
$124$	5.72955	0.514529
$125$	0	0
$126$	8.50206	0.757423
$127$	8.17488	0.725403	0.362702	−	0.931905i	$-0.381854\pi$
$127$	8.17488	0.725403	0.362702	+	0.931905i	$0.381854\pi$
$128$	−5.42927	−0.479884
$129$	3.02459	0.266301
$130$	0	0
$131$	−17.7054	−1.54693	−0.773465	−	0.633839i	$-0.781478\pi$
$131$	−17.7054	−1.54693	−0.773465	+	0.633839i	$0.781478\pi$
$132$	0	0
$133$	8.57830	0.743833
$134$	22.0542	1.90519
$135$	0	0
$136$	−3.13525	−0.268845
$137$	5.48584	0.468687	0.234343	−	0.972154i	$-0.424706\pi$
$137$	5.48584	0.468687	0.234343	+	0.972154i	$0.424706\pi$
$138$	−9.89470	−0.842293
$139$	−3.25952	−0.276469	−0.138235	−	0.990400i	$-0.544143\pi$
$139$	−3.25952	−0.276469	−0.138235	+	0.990400i	$0.544143\pi$
$140$	0	0
$141$	−11.9878	−1.00955
$142$	−6.42016	−0.538768
$143$	0	0
$144$	−4.59259	−0.382716
$145$	0	0
$146$	−1.24058	−0.102671
$147$	−12.8679	−1.06132
$148$	−0.863613	−0.0709886
$149$	−2.80635	−0.229905	−0.114953	−	0.993371i	$-0.536672\pi$
$149$	−2.80635	−0.229905	−0.114953	+	0.993371i	$0.536672\pi$
$150$	0	0
$151$	14.1638	1.15263	0.576317	−	0.817226i	$-0.304489\pi$
$151$	14.1638	1.15263	0.576317	+	0.817226i	$0.304489\pi$
$152$	−1.32782	−0.107700
$153$	−4.54422	−0.367378
$154$	0	0
$155$	0	0
$156$	−0.874857	−0.0700446
$157$	3.11084	0.248272	0.124136	−	0.992265i	$-0.460384\pi$
$157$	3.11084	0.248272	0.124136	+	0.992265i	$0.460384\pi$
$158$	−33.7995	−2.68894
$159$	−5.02443	−0.398463
$160$	0	0
$161$	23.1222	1.82229
$162$	−1.90743	−0.149862
$163$	19.3745	1.51752	0.758762	−	0.651368i	$-0.225805\pi$
$163$	19.3745	1.51752	0.758762	+	0.651368i	$0.225805\pi$
$164$	−7.69303	−0.600724
$165$	0	0
$166$	8.10157	0.628804
$167$	−4.14992	−0.321130	−0.160565	−	0.987025i	$-0.551332\pi$
$167$	−4.14992	−0.321130	−0.160565	+	0.987025i	$0.551332\pi$
$168$	3.07531	0.237265
$169$	−12.7148	−0.978064
$170$	0	0
$171$	−1.92453	−0.147173
$172$	−4.95515	−0.377827
$173$	10.6654	0.810878	0.405439	−	0.914122i	$-0.367119\pi$
$173$	10.6654	0.810878	0.405439	+	0.914122i	$0.367119\pi$
$174$	9.06894	0.687514
$175$	0	0
$176$	0	0
$177$	−13.6495	−1.02596
$178$	15.7190	1.17819
$179$	22.6337	1.69172	0.845862	−	0.533402i	$-0.179087\pi$
$179$	22.6337	1.69172	0.845862	+	0.533402i	$0.179087\pi$
$180$	0	0
$181$	14.5822	1.08389	0.541944	−	0.840415i	$-0.317689\pi$
$181$	14.5822	1.08389	0.541944	+	0.840415i	$0.317689\pi$
$182$	4.54016	0.336539
$183$	11.6915	0.864262
$184$	−3.57905	−0.263851
$185$	0	0
$186$	6.67082	0.489128
$187$	0	0
$188$	19.6394	1.43235
$189$	4.45734	0.324224
$190$	0	0
$191$	3.96434	0.286850	0.143425	−	0.989661i	$-0.454188\pi$
$191$	3.96434	0.286850	0.143425	+	0.989661i	$0.454188\pi$
$192$	4.89194	0.353046
$193$	0.0300531	0.00216327	0.00108164	−	0.999999i	$-0.499656\pi$
$193$	0.0300531	0.00216327	0.00108164	+	0.999999i	$0.499656\pi$
$194$	−2.17140	−0.155898
$195$	0	0
$196$	21.0812	1.50580
$197$	−5.50562	−0.392259	−0.196130	−	0.980578i	$-0.562837\pi$
$197$	−5.50562	−0.392259	−0.196130	+	0.980578i	$0.562837\pi$
$198$	0	0
$199$	−3.12623	−0.221612	−0.110806	−	0.993842i	$-0.535343\pi$
$199$	−3.12623	−0.221612	−0.110806	+	0.993842i	$0.535343\pi$
$200$	0	0
$201$	11.5622	0.815538
$202$	−1.95273	−0.137393
$203$	−21.1926	−1.48743
$204$	7.44473	0.521235
$205$	0	0
$206$	−9.13794	−0.636670
$207$	−5.18745	−0.360553
$208$	−2.45248	−0.170049
$209$	0	0
$210$	0	0
$211$	20.3904	1.40373	0.701867	−	0.712308i	$-0.252350\pi$
$211$	20.3904	1.40373	0.701867	+	0.712308i	$0.252350\pi$
$212$	8.23145	0.565338
$213$	−3.36587	−0.230626
$214$	1.40381	0.0959626
$215$	0	0
$216$	−0.689943	−0.0469447
$217$	−15.5886	−1.05822
$218$	−0.0827813	−0.00560666
$219$	−0.650391	−0.0439494
$220$	0	0
$221$	−2.42665	−0.163234
$222$	−1.00549	−0.0674841
$223$	10.3019	0.689865	0.344933	−	0.938627i	$-0.387902\pi$
$223$	10.3019	0.689865	0.344933	+	0.938627i	$0.387902\pi$
$224$	−32.8958	−2.19795
$225$	0	0
$226$	21.3643	1.42113
$227$	−6.27244	−0.416317	−0.208158	−	0.978095i	$-0.566747\pi$
$227$	−6.27244	−0.416317	−0.208158	+	0.978095i	$0.566747\pi$
$228$	3.15294	0.208808
$229$	1.38111	0.0912660	0.0456330	−	0.998958i	$-0.485470\pi$
$229$	1.38111	0.0912660	0.0456330	+	0.998958i	$0.485470\pi$
$230$	0	0
$231$	0	0
$232$	3.28036	0.215366
$233$	8.30444	0.544042	0.272021	−	0.962291i	$-0.412308\pi$
$233$	8.30444	0.544042	0.272021	+	0.962291i	$0.412308\pi$
$234$	−1.01858	−0.0665868
$235$	0	0
$236$	22.3618	1.45563
$237$	−17.7199	−1.15103
$238$	−38.6352	−2.50435
$239$	−10.2950	−0.665931	−0.332966	−	0.942939i	$-0.608049\pi$
$239$	−10.2950	−0.665931	−0.332966	+	0.942939i	$0.608049\pi$
$240$	0	0
$241$	−12.2340	−0.788060	−0.394030	−	0.919097i	$-0.628919\pi$
$241$	−12.2340	−0.788060	−0.394030	+	0.919097i	$0.628919\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−19.1541	−1.22621
$245$	0	0
$246$	−8.95686	−0.571068
$247$	−1.02772	−0.0653920
$248$	2.41292	0.153221
$249$	4.24738	0.269167
$250$	0	0
$251$	−4.65658	−0.293921	−0.146960	−	0.989142i	$-0.546949\pi$
$251$	−4.65658	−0.293921	−0.146960	+	0.989142i	$0.546949\pi$
$252$	−7.30240	−0.460008
$253$	0	0
$254$	−15.5930	−0.978392
$255$	0	0
$256$	20.1398	1.25874
$257$	18.3364	1.14380	0.571898	−	0.820325i	$-0.306207\pi$
$257$	18.3364	1.14380	0.571898	+	0.820325i	$0.306207\pi$
$258$	−5.76920	−0.359175
$259$	2.34966	0.146001
$260$	0	0
$261$	4.75453	0.294298
$262$	33.7719	2.08643
$263$	9.80198	0.604416	0.302208	−	0.953242i	$-0.402276\pi$
$263$	9.80198	0.604416	0.302208	+	0.953242i	$0.402276\pi$
$264$	0	0
$265$	0	0
$266$	−16.3625	−1.00325
$267$	8.24094	0.504337
$268$	−18.9423	−1.15708
$269$	11.9309	0.727441	0.363721	−	0.931508i	$-0.381506\pi$
$269$	11.9309	0.727441	0.363721	+	0.931508i	$0.381506\pi$
$270$	0	0
$271$	30.4048	1.84696	0.923481	−	0.383644i	$-0.125331\pi$
$271$	30.4048	1.84696	0.923481	+	0.383644i	$0.125331\pi$
$272$	20.8697	1.26541
$273$	2.38025	0.144059
$274$	−10.4638	−0.632144
$275$	0	0
$276$	8.49854	0.511552
$277$	−22.1444	−1.33053	−0.665264	−	0.746608i	$-0.731681\pi$
$277$	−22.1444	−1.33053	−0.665264	+	0.746608i	$0.731681\pi$
$278$	6.21731	0.372890
$279$	3.49728	0.209377
$280$	0	0
$281$	−17.0323	−1.01606	−0.508030	−	0.861340i	$-0.669626\pi$
$281$	−17.0323	−1.01606	−0.508030	+	0.861340i	$0.669626\pi$
$282$	22.8658	1.36164
$283$	7.92150	0.470884	0.235442	−	0.971888i	$-0.424346\pi$
$283$	7.92150	0.470884	0.235442	+	0.971888i	$0.424346\pi$
$284$	5.51426	0.327211
$285$	0	0
$286$	0	0
$287$	20.9307	1.23550
$288$	7.38016	0.434880
$289$	3.64990	0.214700
$290$	0	0
$291$	−1.13839	−0.0667337
$292$	1.06553	0.0623553
$293$	−22.3396	−1.30509	−0.652545	−	0.757750i	$-0.726299\pi$
$293$	−22.3396	−1.30509	−0.652545	+	0.757750i	$0.726299\pi$
$294$	24.5445	1.43147
$295$	0	0
$296$	−0.363699	−0.0211396
$297$	0	0
$298$	5.35291	0.310086
$299$	−2.77014	−0.160201
$300$	0	0
$301$	13.4816	0.777069
$302$	−27.0164	−1.55462
$303$	−1.02375	−0.0588128
$304$	8.83860	0.506928
$305$	0	0
$306$	8.66777	0.495503
$307$	9.33327	0.532678	0.266339	−	0.963879i	$-0.414186\pi$
$307$	9.33327	0.532678	0.266339	+	0.963879i	$0.414186\pi$
$308$	0	0
$309$	−4.79071	−0.272534
$310$	0	0
$311$	−11.2480	−0.637815	−0.318907	−	0.947786i	$-0.603316\pi$
$311$	−11.2480	−0.637815	−0.318907	+	0.947786i	$0.603316\pi$
$312$	−0.368435	−0.0208585
$313$	10.8640	0.614069	0.307035	−	0.951698i	$-0.400663\pi$
$313$	10.8640	0.614069	0.307035	+	0.951698i	$0.400663\pi$
$314$	−5.93371	−0.334859
$315$	0	0
$316$	29.0303	1.63308
$317$	−12.6895	−0.712714	−0.356357	−	0.934350i	$-0.615981\pi$
$317$	−12.6895	−0.712714	−0.356357	+	0.934350i	$0.615981\pi$
$318$	9.58374	0.537429
$319$	0	0
$320$	0	0
$321$	0.735971	0.0410779
$322$	−44.1040	−2.45782
$323$	8.74550	0.486613
$324$	1.63829	0.0910159
$325$	0	0
$326$	−36.9554	−2.04677
$327$	−0.0433994	−0.00239999
$328$	−3.23982	−0.178889
$329$	−53.4335	−2.94588
$330$	0	0
$331$	−23.0627	−1.26764	−0.633821	−	0.773480i	$-0.718514\pi$
$331$	−23.0627	−1.26764	−0.633821	+	0.773480i	$0.718514\pi$
$332$	−6.95842	−0.381893
$333$	−0.527144	−0.0288873
$334$	7.91568	0.433127
$335$	0	0
$336$	−20.4707	−1.11677
$337$	−32.3630	−1.76293	−0.881463	−	0.472254i	$-0.843441\pi$
$337$	−32.3630	−1.76293	−0.881463	+	0.472254i	$0.843441\pi$
$338$	24.2527	1.31917
$339$	11.2006	0.608332
$340$	0	0
$341$	0	0
$342$	3.67091	0.198500
$343$	−26.1550	−1.41224
$344$	−2.08680	−0.112513
$345$	0	0
$346$	−20.3436	−1.09368
$347$	−29.0843	−1.56133	−0.780663	−	0.624952i	$-0.785119\pi$
$347$	−29.0843	−1.56133	−0.780663	+	0.624952i	$0.785119\pi$
$348$	−7.78929	−0.417550
$349$	14.4037	0.771012	0.385506	−	0.922705i	$-0.374027\pi$
$349$	14.4037	0.771012	0.385506	+	0.922705i	$0.374027\pi$
$350$	0	0
$351$	−0.534008	−0.0285032
$352$	0	0
$353$	28.9227	1.53940	0.769700	−	0.638406i	$-0.220406\pi$
$353$	28.9227	1.53940	0.769700	+	0.638406i	$0.220406\pi$
$354$	26.0355	1.38377
$355$	0	0
$356$	−13.5010	−0.715553
$357$	−20.2551	−1.07201
$358$	−43.1722	−2.28172
$359$	−23.5159	−1.24112	−0.620562	−	0.784157i	$-0.713095\pi$
$359$	−23.5159	−1.24112	−0.620562	+	0.784157i	$0.713095\pi$
$360$	0	0
$361$	−15.2962	−0.805061
$362$	−27.8145	−1.46190
$363$	0	0
$364$	−3.89953	−0.204391
$365$	0	0
$366$	−22.3007	−1.16568
$367$	25.3037	1.32084	0.660421	−	0.750895i	$-0.270378\pi$
$367$	25.3037	1.32084	0.660421	+	0.750895i	$0.270378\pi$
$368$	23.8239	1.24190
$369$	−4.69578	−0.244452
$370$	0	0
$371$	−22.3956	−1.16272
$372$	−5.72955	−0.297063
$373$	16.2682	0.842334	0.421167	−	0.906983i	$-0.361621\pi$
$373$	16.2682	0.842334	0.421167	+	0.906983i	$0.361621\pi$
$374$	0	0
$375$	0	0
$376$	8.27087	0.426538
$377$	2.53896	0.130763
$378$	−8.50206	−0.437299
$379$	25.0263	1.28552	0.642758	−	0.766069i	$-0.277790\pi$
$379$	25.0263	1.28552	0.642758	+	0.766069i	$0.277790\pi$
$380$	0	0
$381$	−8.17488	−0.418812
$382$	−7.56170	−0.386890
$383$	13.0991	0.669331	0.334666	−	0.942337i	$-0.391377\pi$
$383$	13.0991	0.669331	0.334666	+	0.942337i	$0.391377\pi$
$384$	5.42927	0.277061
$385$	0	0
$386$	−0.0573242	−0.00291772
$387$	−3.02459	−0.153749
$388$	1.86501	0.0946816
$389$	16.7136	0.847416	0.423708	−	0.905799i	$-0.360728\pi$
$389$	16.7136	0.847416	0.423708	+	0.905799i	$0.360728\pi$
$390$	0	0
$391$	23.5729	1.19213
$392$	8.87809	0.448411
$393$	17.7054	0.893121
$394$	10.5016	0.529062
$395$	0	0
$396$	0	0
$397$	4.22375	0.211984	0.105992	−	0.994367i	$-0.466198\pi$
$397$	4.22375	0.211984	0.105992	+	0.994367i	$0.466198\pi$
$398$	5.96306	0.298901
$399$	−8.57830	−0.429452
$400$	0	0
$401$	−20.9464	−1.04601	−0.523005	−	0.852329i	$-0.675189\pi$
$401$	−20.9464	−1.04601	−0.523005	+	0.852329i	$0.675189\pi$
$402$	−22.0542	−1.09996
$403$	1.86757	0.0930305
$404$	1.67719	0.0834434
$405$	0	0
$406$	40.4233	2.00618
$407$	0	0
$408$	3.13525	0.155218
$409$	6.86017	0.339213	0.169607	−	0.985512i	$-0.445750\pi$
$409$	6.86017	0.339213	0.169607	+	0.985512i	$0.445750\pi$
$410$	0	0
$411$	−5.48584	−0.270596
$412$	7.84855	0.386670
$413$	−60.8405	−2.99377
$414$	9.89470	0.486298
$415$	0	0
$416$	3.94106	0.193226
$417$	3.25952	0.159620
$418$	0	0
$419$	−19.0865	−0.932435	−0.466217	−	0.884670i	$-0.654384\pi$
$419$	−19.0865	−0.932435	−0.466217	+	0.884670i	$0.654384\pi$
$420$	0	0
$421$	−12.2531	−0.597180	−0.298590	−	0.954381i	$-0.596516\pi$
$421$	−12.2531	−0.597180	−0.298590	+	0.954381i	$0.596516\pi$
$422$	−38.8933	−1.89329
$423$	11.9878	0.582865
$424$	3.46657	0.168351
$425$	0	0
$426$	6.42016	0.311058
$427$	52.1131	2.52193
$428$	−1.20573	−0.0582812
$429$	0	0
$430$	0	0
$431$	7.02245	0.338260	0.169130	−	0.985594i	$-0.445904\pi$
$431$	7.02245	0.338260	0.169130	+	0.985594i	$0.445904\pi$
$432$	4.59259	0.220961
$433$	−27.4900	−1.32108	−0.660542	−	0.750789i	$-0.729674\pi$
$433$	−27.4900	−1.32108	−0.660542	+	0.750789i	$0.729674\pi$
$434$	29.7341	1.42728
$435$	0	0
$436$	0.0711006	0.00340510
$437$	9.98343	0.477572
$438$	1.24058	0.0592770
$439$	17.6992	0.844736	0.422368	−	0.906425i	$-0.361199\pi$
$439$	17.6992	0.844736	0.422368	+	0.906425i	$0.361199\pi$
$440$	0	0
$441$	12.8679	0.612755
$442$	4.62865	0.220163
$443$	16.5900	0.788216	0.394108	−	0.919064i	$-0.371054\pi$
$443$	16.5900	0.788216	0.394108	+	0.919064i	$0.371054\pi$
$444$	0.863613	0.0409853
$445$	0	0
$446$	−19.6501	−0.930460
$447$	2.80635	0.132736
$448$	21.8050	1.03019
$449$	−26.2868	−1.24055	−0.620276	−	0.784383i	$-0.712980\pi$
$449$	−26.2868	−1.24055	−0.620276	+	0.784383i	$0.712980\pi$
$450$	0	0
$451$	0	0
$452$	−18.3498	−0.863100
$453$	−14.1638	−0.665473
$454$	11.9642	0.561510
$455$	0	0
$456$	1.32782	0.0621808
$457$	27.4515	1.28413	0.642063	−	0.766652i	$-0.278079\pi$
$457$	27.4515	1.28413	0.642063	+	0.766652i	$0.278079\pi$
$458$	−2.63436	−0.123096
$459$	4.54422	0.212106
$460$	0	0
$461$	−16.7991	−0.782410	−0.391205	−	0.920304i	$-0.627942\pi$
$461$	−16.7991	−0.782410	−0.391205	+	0.920304i	$0.627942\pi$
$462$	0	0
$463$	32.2574	1.49913	0.749565	−	0.661930i	$-0.230263\pi$
$463$	32.2574	1.49913	0.749565	+	0.661930i	$0.230263\pi$
$464$	−21.8356	−1.01369
$465$	0	0
$466$	−15.8401	−0.733780
$467$	−2.29620	−0.106256	−0.0531278	−	0.998588i	$-0.516919\pi$
$467$	−2.29620	−0.106256	−0.0531278	+	0.998588i	$0.516919\pi$
$468$	0.874857	0.0404403
$469$	51.5368	2.37975
$470$	0	0
$471$	−3.11084	−0.143340
$472$	9.41739	0.433470
$473$	0	0
$474$	33.7995	1.55246
$475$	0	0
$476$	33.1837	1.52097
$477$	5.02443	0.230053
$478$	19.6371	0.898179
$479$	−0.341117	−0.0155861	−0.00779303	−	0.999970i	$-0.502481\pi$
$479$	−0.341117	−0.0155861	−0.00779303	+	0.999970i	$0.502481\pi$
$480$	0	0
$481$	−0.281499	−0.0128352
$482$	23.3355	1.06290
$483$	−23.1222	−1.05210
$484$	0	0
$485$	0	0
$486$	1.90743	0.0865227
$487$	−15.8553	−0.718471	−0.359235	−	0.933247i	$-0.616962\pi$
$487$	−15.8553	−0.718471	−0.359235	+	0.933247i	$0.616962\pi$
$488$	−8.06648	−0.365152
$489$	−19.3745	−0.876143
$490$	0	0
$491$	−30.0103	−1.35435	−0.677173	−	0.735824i	$-0.736795\pi$
$491$	−30.0103	−1.35435	−0.677173	+	0.735824i	$0.736795\pi$
$492$	7.69303	0.346828
$493$	−21.6056	−0.973068
$494$	1.96030	0.0881978
$495$	0	0
$496$	−16.0616	−0.721186
$497$	−15.0028	−0.672969
$498$	−8.10157	−0.363040
$499$	3.65516	0.163627	0.0818137	−	0.996648i	$-0.473929\pi$
$499$	3.65516	0.163627	0.0818137	+	0.996648i	$0.473929\pi$
$500$	0	0
$501$	4.14992	0.185405
$502$	8.88210	0.396427
$503$	−31.1782	−1.39017	−0.695083	−	0.718930i	$-0.744632\pi$
$503$	−31.1782	−1.39017	−0.695083	+	0.718930i	$0.744632\pi$
$504$	−3.07531	−0.136985
$505$	0	0
$506$	0	0
$507$	12.7148	0.564686
$508$	13.3928	0.594209
$509$	−14.7299	−0.652893	−0.326446	−	0.945216i	$-0.605851\pi$
$509$	−14.7299	−0.652893	−0.326446	+	0.945216i	$0.605851\pi$
$510$	0	0
$511$	−2.89901	−0.128245
$512$	−27.5568	−1.21785
$513$	1.92453	0.0849703
$514$	−34.9755	−1.54270
$515$	0	0
$516$	4.95515	0.218138
$517$	0	0
$518$	−4.48181	−0.196919
$519$	−10.6654	−0.468160
$520$	0	0
$521$	−1.29570	−0.0567658	−0.0283829	−	0.999597i	$-0.509036\pi$
$521$	−1.29570	−0.0567658	−0.0283829	+	0.999597i	$0.509036\pi$
$522$	−9.06894	−0.396937
$523$	30.7487	1.34455	0.672274	−	0.740303i	$-0.265318\pi$
$523$	30.7487	1.34455	0.672274	+	0.740303i	$0.265318\pi$
$524$	−29.0066	−1.26716
$525$	0	0
$526$	−18.6966	−0.815210
$527$	−15.8924	−0.692284
$528$	0	0
$529$	3.90968	0.169986
$530$	0	0
$531$	13.6495	0.592339
$532$	14.0537	0.609306
$533$	−2.50758	−0.108615
$534$	−15.7190	−0.680228
$535$	0	0
$536$	−7.97729	−0.344566
$537$	−22.6337	−0.976717
$538$	−22.7574	−0.981141
$539$	0	0
$540$	0	0
$541$	−15.7550	−0.677362	−0.338681	−	0.940901i	$-0.609981\pi$
$541$	−15.7550	−0.677362	−0.338681	+	0.940901i	$0.609981\pi$
$542$	−57.9951	−2.49110
$543$	−14.5822	−0.625783
$544$	−33.5370	−1.43789
$545$	0	0
$546$	−4.54016	−0.194301
$547$	−15.3597	−0.656734	−0.328367	−	0.944550i	$-0.606498\pi$
$547$	−15.3597	−0.656734	−0.328367	+	0.944550i	$0.606498\pi$
$548$	8.98738	0.383922
$549$	−11.6915	−0.498982
$550$	0	0
$551$	−9.15026	−0.389814
$552$	3.57905	0.152334
$553$	−78.9836	−3.35873
$554$	42.2389	1.79456
$555$	0	0
$556$	−5.34003	−0.226468
$557$	35.9189	1.52193	0.760967	−	0.648791i	$-0.224725\pi$
$557$	35.9189	1.52193	0.760967	+	0.648791i	$0.224725\pi$
$558$	−6.67082	−0.282398
$559$	−1.61516	−0.0683138
$560$	0	0
$561$	0	0
$562$	32.4878	1.37042
$563$	30.9605	1.30483	0.652414	−	0.757863i	$-0.273756\pi$
$563$	30.9605	1.30483	0.652414	+	0.757863i	$0.273756\pi$
$564$	−19.6394	−0.826968
$565$	0	0
$566$	−15.1097	−0.635108
$567$	−4.45734	−0.187191
$568$	2.32226	0.0974398
$569$	−2.33375	−0.0978360	−0.0489180	−	0.998803i	$-0.515577\pi$
$569$	−2.33375	−0.0978360	−0.0489180	+	0.998803i	$0.515577\pi$
$570$	0	0
$571$	26.6093	1.11357	0.556783	−	0.830658i	$-0.312036\pi$
$571$	26.6093	1.11357	0.556783	+	0.830658i	$0.312036\pi$
$572$	0	0
$573$	−3.96434	−0.165613
$574$	−39.9237	−1.66639
$575$	0	0
$576$	−4.89194	−0.203831
$577$	3.69867	0.153978	0.0769889	−	0.997032i	$-0.475469\pi$
$577$	3.69867	0.153978	0.0769889	+	0.997032i	$0.475469\pi$
$578$	−6.96192	−0.289578
$579$	−0.0300531	−0.00124896
$580$	0	0
$581$	18.9320	0.785431
$582$	2.17140	0.0900075
$583$	0	0
$584$	0.448733	0.0185687
$585$	0	0
$586$	42.6111	1.76025
$587$	−31.3576	−1.29427	−0.647133	−	0.762377i	$-0.724032\pi$
$587$	−31.3576	−1.29427	−0.647133	+	0.762377i	$0.724032\pi$
$588$	−21.0812	−0.869376
$589$	−6.73064	−0.277331
$590$	0	0
$591$	5.50562	0.226471
$592$	2.42096	0.0995007
$593$	32.0728	1.31707	0.658535	−	0.752550i	$-0.271176\pi$
$593$	32.0728	1.31707	0.658535	+	0.752550i	$0.271176\pi$
$594$	0	0
$595$	0	0
$596$	−4.59761	−0.188325
$597$	3.12623	0.127948
$598$	5.28385	0.216073
$599$	−20.3261	−0.830504	−0.415252	−	0.909706i	$-0.636307\pi$
$599$	−20.3261	−0.830504	−0.415252	+	0.909706i	$0.636307\pi$
$600$	0	0
$601$	11.6257	0.474222	0.237111	−	0.971483i	$-0.423799\pi$
$601$	11.6257	0.474222	0.237111	+	0.971483i	$0.423799\pi$
$602$	−25.7153	−1.04808
$603$	−11.5622	−0.470851
$604$	23.2044	0.944172
$605$	0	0
$606$	1.95273	0.0793241
$607$	25.6773	1.04221	0.521104	−	0.853493i	$-0.325520\pi$
$607$	25.6773	1.04221	0.521104	+	0.853493i	$0.325520\pi$
$608$	−14.2034	−0.576022
$609$	21.1926	0.858766
$610$	0	0
$611$	6.40156	0.258979
$612$	−7.44473	−0.300935
$613$	−6.58440	−0.265941	−0.132971	−	0.991120i	$-0.542452\pi$
$613$	−6.58440	−0.265941	−0.132971	+	0.991120i	$0.542452\pi$
$614$	−17.8026	−0.718453
$615$	0	0
$616$	0	0
$617$	−25.4721	−1.02547	−0.512734	−	0.858548i	$-0.671367\pi$
$617$	−25.4721	−1.02547	−0.512734	+	0.858548i	$0.671367\pi$
$618$	9.13794	0.367582
$619$	−1.08299	−0.0435292	−0.0217646	−	0.999763i	$-0.506928\pi$
$619$	−1.08299	−0.0435292	−0.0217646	+	0.999763i	$0.506928\pi$
$620$	0	0
$621$	5.18745	0.208165
$622$	21.4547	0.860256
$623$	36.7327	1.47166
$624$	2.45248	0.0981777
$625$	0	0
$626$	−20.7223	−0.828230
$627$	0	0
$628$	5.09645	0.203371
$629$	2.39546	0.0955131
$630$	0	0
$631$	31.0566	1.23635	0.618173	−	0.786042i	$-0.287873\pi$
$631$	31.0566	1.23635	0.618173	+	0.786042i	$0.287873\pi$
$632$	12.2257	0.486313
$633$	−20.3904	−0.810446
$634$	24.2044	0.961278
$635$	0	0
$636$	−8.23145	−0.326398
$637$	6.87153	0.272260
$638$	0	0
$639$	3.36587	0.133152
$640$	0	0
$641$	−22.4187	−0.885485	−0.442743	−	0.896649i	$-0.645994\pi$
$641$	−22.4187	−0.885485	−0.442743	+	0.896649i	$0.645994\pi$
$642$	−1.40381	−0.0554041
$643$	−13.8174	−0.544905	−0.272453	−	0.962169i	$-0.587835\pi$
$643$	−13.8174	−0.544905	−0.272453	+	0.962169i	$0.587835\pi$
$644$	37.8809	1.49271
$645$	0	0
$646$	−16.6814	−0.656322
$647$	−24.9483	−0.980819	−0.490409	−	0.871492i	$-0.663153\pi$
$647$	−24.9483	−0.980819	−0.490409	+	0.871492i	$0.663153\pi$
$648$	0.689943	0.0271035
$649$	0	0
$650$	0	0
$651$	15.5886	0.610964
$652$	31.7409	1.24307
$653$	18.8619	0.738125	0.369063	−	0.929405i	$-0.379679\pi$
$653$	18.8619	0.738125	0.369063	+	0.929405i	$0.379679\pi$
$654$	0.0827813	0.00323700
$655$	0	0
$656$	21.5658	0.842002
$657$	0.650391	0.0253742
$658$	101.921	3.97328
$659$	2.97553	0.115910	0.0579550	−	0.998319i	$-0.481542\pi$
$659$	2.97553	0.115910	0.0579550	+	0.998319i	$0.481542\pi$
$660$	0	0
$661$	−39.0164	−1.51756	−0.758782	−	0.651345i	$-0.774205\pi$
$661$	−39.0164	−1.51756	−0.758782	+	0.651345i	$0.774205\pi$
$662$	43.9905	1.70974
$663$	2.42665	0.0942431
$664$	−2.93045	−0.113723
$665$	0	0
$666$	1.00549	0.0389620
$667$	−24.6639	−0.954991
$668$	−6.79876	−0.263052
$669$	−10.3019	−0.398294
$670$	0	0
$671$	0	0
$672$	32.8958	1.26898
$673$	15.2282	0.587003	0.293502	−	0.955959i	$-0.405179\pi$
$673$	15.2282	0.587003	0.293502	+	0.955959i	$0.405179\pi$
$674$	61.7302	2.37776
$675$	0	0
$676$	−20.8305	−0.801175
$677$	−0.829064	−0.0318635	−0.0159318	−	0.999873i	$-0.505071\pi$
$677$	−0.829064	−0.0318635	−0.0159318	+	0.999873i	$0.505071\pi$
$678$	−21.3643	−0.820492
$679$	−5.07420	−0.194730
$680$	0	0
$681$	6.27244	0.240361
$682$	0	0
$683$	1.03468	0.0395910	0.0197955	−	0.999804i	$-0.493698\pi$
$683$	1.03468	0.0395910	0.0197955	+	0.999804i	$0.493698\pi$
$684$	−3.15294	−0.120556
$685$	0	0
$686$	49.8889	1.90477
$687$	−1.38111	−0.0526925
$688$	13.8907	0.529579
$689$	2.68308	0.102217
$690$	0	0
$691$	19.2788	0.733399	0.366700	−	0.930339i	$-0.380488\pi$
$691$	19.2788	0.733399	0.366700	+	0.930339i	$0.380488\pi$
$692$	17.4730	0.664225
$693$	0	0
$694$	55.4762	2.10585
$695$	0	0
$696$	−3.28036	−0.124342
$697$	21.3386	0.808258
$698$	−27.4740	−1.03991
$699$	−8.30444	−0.314103
$700$	0	0
$701$	−3.29844	−0.124580	−0.0622902	−	0.998058i	$-0.519840\pi$
$701$	−3.29844	−0.124580	−0.0622902	+	0.998058i	$0.519840\pi$
$702$	1.01858	0.0384439
$703$	1.01451	0.0382629
$704$	0	0
$705$	0	0
$706$	−55.1680	−2.07627
$707$	−4.56319	−0.171616
$708$	−22.3618	−0.840409
$709$	−37.7593	−1.41808	−0.709040	−	0.705169i	$-0.750871\pi$
$709$	−37.7593	−1.41808	−0.709040	+	0.705169i	$0.750871\pi$
$710$	0	0
$711$	17.7199	0.664548
$712$	−5.68578	−0.213084
$713$	−18.1420	−0.679423
$714$	38.6352	1.44589
$715$	0	0
$716$	37.0805	1.38576
$717$	10.2950	0.384476
$718$	44.8550	1.67397
$719$	−29.0980	−1.08517	−0.542585	−	0.840001i	$-0.682554\pi$
$719$	−29.0980	−1.08517	−0.542585	+	0.840001i	$0.682554\pi$
$720$	0	0
$721$	−21.3538	−0.795257
$722$	29.1764	1.08583
$723$	12.2340	0.454987
$724$	23.8898	0.887859
$725$	0	0
$726$	0	0
$727$	38.1331	1.41428	0.707140	−	0.707074i	$-0.249985\pi$
$727$	38.1331	1.41428	0.707140	+	0.707074i	$0.249985\pi$
$728$	−1.64224	−0.0608654
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.7444	0.508355
$732$	19.1541	0.707954
$733$	−40.0048	−1.47761	−0.738805	−	0.673919i	$-0.764610\pi$
$733$	−40.0048	−1.47761	−0.738805	+	0.673919i	$0.764610\pi$
$734$	−48.2650	−1.78150
$735$	0	0
$736$	−38.2842	−1.41118
$737$	0	0
$738$	8.95686	0.329707
$739$	41.4965	1.52647	0.763236	−	0.646119i	$-0.223609\pi$
$739$	41.4965	1.52647	0.763236	+	0.646119i	$0.223609\pi$
$740$	0	0
$741$	1.02772	0.0377541
$742$	42.7180	1.56823
$743$	−2.69500	−0.0988701	−0.0494350	−	0.998777i	$-0.515742\pi$
$743$	−2.69500	−0.0988701	−0.0494350	+	0.998777i	$0.515742\pi$
$744$	−2.41292	−0.0884621
$745$	0	0
$746$	−31.0304	−1.13610
$747$	−4.24738	−0.155403
$748$	0	0
$749$	3.28047	0.119866
$750$	0	0
$751$	−45.6347	−1.66524	−0.832618	−	0.553848i	$-0.813159\pi$
$751$	−45.6347	−1.66524	−0.832618	+	0.553848i	$0.813159\pi$
$752$	−55.0549	−2.00765
$753$	4.65658	0.169695
$754$	−4.84288	−0.176367
$755$	0	0
$756$	7.30240	0.265586
$757$	8.99188	0.326815	0.163408	−	0.986559i	$-0.447751\pi$
$757$	8.99188	0.326815	0.163408	+	0.986559i	$0.447751\pi$
$758$	−47.7359	−1.73385
$759$	0	0
$760$	0	0
$761$	15.0581	0.545855	0.272927	−	0.962035i	$-0.412008\pi$
$761$	15.0581	0.545855	0.272927	+	0.962035i	$0.412008\pi$
$762$	15.5930	0.564875
$763$	−0.193446	−0.00700321
$764$	6.49473	0.234971
$765$	0	0
$766$	−24.9856	−0.902765
$767$	7.28895	0.263189
$768$	−20.1398	−0.726734
$769$	48.1314	1.73566	0.867831	−	0.496859i	$-0.165513\pi$
$769$	48.1314	1.73566	0.867831	+	0.496859i	$0.165513\pi$
$770$	0	0
$771$	−18.3364	−0.660371
$772$	0.0492356	0.00177203
$773$	28.5551	1.02706	0.513528	−	0.858073i	$-0.328338\pi$
$773$	28.5551	1.02706	0.513528	+	0.858073i	$0.328338\pi$
$774$	5.76920	0.207370
$775$	0	0
$776$	0.785425	0.0281951
$777$	−2.34966	−0.0842936
$778$	−31.8801	−1.14296
$779$	9.03718	0.323791
$780$	0	0
$781$	0	0
$782$	−44.9637	−1.60790
$783$	−4.75453	−0.169913
$784$	−59.0968	−2.11060
$785$	0	0
$786$	−33.7719	−1.20460
$787$	−42.7119	−1.52252	−0.761258	−	0.648450i	$-0.775418\pi$
$787$	−42.7119	−1.52252	−0.761258	+	0.648450i	$0.775418\pi$
$788$	−9.01978	−0.321316
$789$	−9.80198	−0.348960
$790$	0	0
$791$	49.9248	1.77512
$792$	0	0
$793$	−6.24336	−0.221708
$794$	−8.05650	−0.285914
$795$	0	0
$796$	−5.12166	−0.181532
$797$	−8.54872	−0.302811	−0.151406	−	0.988472i	$-0.548380\pi$
$797$	−8.54872	−0.302811	−0.151406	+	0.988472i	$0.548380\pi$
$798$	16.3625	0.579226
$799$	−54.4750	−1.92719
$800$	0	0
$801$	−8.24094	−0.291179
$802$	39.9537	1.41081
$803$	0	0
$804$	18.9423	0.668042
$805$	0	0
$806$	−3.56227	−0.125476
$807$	−11.9309	−0.419989
$808$	0.706327	0.0248485
$809$	−12.8182	−0.450664	−0.225332	−	0.974282i	$-0.572347\pi$
$809$	−12.8182	−0.450664	−0.225332	+	0.974282i	$0.572347\pi$
$810$	0	0
$811$	−34.0522	−1.19573	−0.597867	−	0.801596i	$-0.703985\pi$
$811$	−34.0522	−1.19573	−0.597867	+	0.801596i	$0.703985\pi$
$812$	−34.7195	−1.21842
$813$	−30.4048	−1.06634
$814$	0	0
$815$	0	0
$816$	−20.8697	−0.730586
$817$	5.82093	0.203649
$818$	−13.0853	−0.457516
$819$	−2.38025	−0.0831727
$820$	0	0
$821$	−17.3373	−0.605076	−0.302538	−	0.953137i	$-0.597834\pi$
$821$	−17.3373	−0.605076	−0.302538	+	0.953137i	$0.597834\pi$
$822$	10.4638	0.364969
$823$	−23.5673	−0.821505	−0.410752	−	0.911747i	$-0.634734\pi$
$823$	−23.5673	−0.821505	−0.410752	+	0.911747i	$0.634734\pi$
$824$	3.30531	0.115146
$825$	0	0
$826$	116.049	4.03786
$827$	−15.3311	−0.533116	−0.266558	−	0.963819i	$-0.585886\pi$
$827$	−15.3311	−0.533116	−0.266558	+	0.963819i	$0.585886\pi$
$828$	−8.49854	−0.295345
$829$	−22.4463	−0.779593	−0.389797	−	0.920901i	$-0.627455\pi$
$829$	−22.4463	−0.779593	−0.389797	+	0.920901i	$0.627455\pi$
$830$	0	0
$831$	22.1444	0.768181
$832$	−2.61233	−0.0905664
$833$	−58.4743	−2.02602
$834$	−6.21731	−0.215288
$835$	0	0
$836$	0	0
$837$	−3.49728	−0.120884
$838$	36.4061	1.25763
$839$	−46.4511	−1.60367	−0.801835	−	0.597546i	$-0.796143\pi$
$839$	−46.4511	−1.60367	−0.801835	+	0.597546i	$0.796143\pi$
$840$	0	0
$841$	−6.39441	−0.220497
$842$	23.3719	0.805451
$843$	17.0323	0.586622
$844$	33.4053	1.14986
$845$	0	0
$846$	−22.8658	−0.786143
$847$	0	0
$848$	−23.0751	−0.792403
$849$	−7.92150	−0.271865
$850$	0	0
$851$	2.73454	0.0937387
$852$	−5.51426	−0.188916
$853$	−35.6321	−1.22002	−0.610009	−	0.792394i	$-0.708834\pi$
$853$	−35.6321	−1.22002	−0.610009	+	0.792394i	$0.708834\pi$
$854$	−99.4020	−3.40147
$855$	0	0
$856$	−0.507778	−0.0173555
$857$	−12.7176	−0.434424	−0.217212	−	0.976124i	$-0.569696\pi$
$857$	−12.7176	−0.434424	−0.217212	+	0.976124i	$0.569696\pi$
$858$	0	0
$859$	−12.9964	−0.443431	−0.221715	−	0.975111i	$-0.571166\pi$
$859$	−12.9964	−0.443431	−0.221715	+	0.975111i	$0.571166\pi$
$860$	0	0
$861$	−20.9307	−0.713315
$862$	−13.3948	−0.456230
$863$	−53.1839	−1.81040	−0.905201	−	0.424985i	$-0.860280\pi$
$863$	−53.1839	−1.81040	−0.905201	+	0.424985i	$0.860280\pi$
$864$	−7.38016	−0.251078
$865$	0	0
$866$	52.4352	1.78182
$867$	−3.64990	−0.123957
$868$	−25.5385	−0.866835
$869$	0	0
$870$	0	0
$871$	−6.17432	−0.209209
$872$	0.0299431	0.00101400
$873$	1.13839	0.0385287
$874$	−19.0427	−0.644129
$875$	0	0
$876$	−1.06553	−0.0360008
$877$	9.40681	0.317645	0.158823	−	0.987307i	$-0.449230\pi$
$877$	9.40681	0.317645	0.158823	+	0.987307i	$0.449230\pi$
$878$	−33.7599	−1.13934
$879$	22.3396	0.753494
$880$	0	0
$881$	−26.5160	−0.893345	−0.446673	−	0.894697i	$-0.647391\pi$
$881$	−26.5160	−0.893345	−0.446673	+	0.894697i	$0.647391\pi$
$882$	−24.5445	−0.826457
$883$	11.5024	0.387086	0.193543	−	0.981092i	$-0.438002\pi$
$883$	11.5024	0.387086	0.193543	+	0.981092i	$0.438002\pi$
$884$	−3.97554	−0.133712
$885$	0	0
$886$	−31.6443	−1.06311
$887$	−32.2772	−1.08376	−0.541881	−	0.840455i	$-0.682288\pi$
$887$	−32.2772	−1.08376	−0.541881	+	0.840455i	$0.682288\pi$
$888$	0.363699	0.0122050
$889$	−36.4382	−1.22210
$890$	0	0
$891$	0	0
$892$	16.8774	0.565098
$893$	−23.0709	−0.772037
$894$	−5.35291	−0.179028
$895$	0	0
$896$	24.2001	0.808469
$897$	2.77014	0.0924923
$898$	50.1403	1.67320
$899$	16.6279	0.554573
$900$	0	0
$901$	−22.8321	−0.760647
$902$	0	0
$903$	−13.4816	−0.448641
$904$	−7.72776	−0.257022
$905$	0	0
$906$	27.0164	0.897561
$907$	−23.0090	−0.764001	−0.382000	−	0.924162i	$-0.624765\pi$
$907$	−23.0090	−0.764001	−0.382000	+	0.924162i	$0.624765\pi$
$908$	−10.2761	−0.341023
$909$	1.02375	0.0339556
$910$	0	0
$911$	−30.1780	−0.999841	−0.499921	−	0.866071i	$-0.666637\pi$
$911$	−30.1780	−0.999841	−0.499921	+	0.866071i	$0.666637\pi$
$912$	−8.83860	−0.292675
$913$	0	0
$914$	−52.3618	−1.73197
$915$	0	0
$916$	2.26265	0.0747599
$917$	78.9191	2.60614
$918$	−8.66777	−0.286079
$919$	28.3493	0.935158	0.467579	−	0.883951i	$-0.345126\pi$
$919$	28.3493	0.935158	0.467579	+	0.883951i	$0.345126\pi$
$920$	0	0
$921$	−9.33327	−0.307542
$922$	32.0430	1.05528
$923$	1.79740	0.0591622
$924$	0	0
$925$	0	0
$926$	−61.5288	−2.02196
$927$	4.79071	0.157347
$928$	35.0892	1.15186
$929$	−31.8685	−1.04557	−0.522786	−	0.852464i	$-0.675107\pi$
$929$	−31.8685	−1.04557	−0.522786	+	0.852464i	$0.675107\pi$
$930$	0	0
$931$	−24.7646	−0.811628
$932$	13.6051	0.445648
$933$	11.2480	0.368242
$934$	4.37985	0.143313
$935$	0	0
$936$	0.368435	0.0120427
$937$	−22.2288	−0.726182	−0.363091	−	0.931754i	$-0.618279\pi$
$937$	−22.2288	−0.726182	−0.363091	+	0.931754i	$0.618279\pi$
$938$	−98.3028	−3.20970
$939$	−10.8640	−0.354533
$940$	0	0
$941$	1.43990	0.0469393	0.0234697	−	0.999725i	$-0.492529\pi$
$941$	1.43990	0.0469393	0.0234697	+	0.999725i	$0.492529\pi$
$942$	5.93371	0.193331
$943$	24.3591	0.793242
$944$	−62.6867	−2.04028
$945$	0	0
$946$	0	0
$947$	−22.8848	−0.743655	−0.371827	−	0.928302i	$-0.621269\pi$
$947$	−22.8848	−0.743655	−0.371827	+	0.928302i	$0.621269\pi$
$948$	−29.0303	−0.942860
$949$	0.347314	0.0112743
$950$	0	0
$951$	12.6895	0.411486
$952$	13.9749	0.452928
$953$	−13.9872	−0.453089	−0.226544	−	0.974001i	$-0.572743\pi$
$953$	−13.9872	−0.453089	−0.226544	+	0.974001i	$0.572743\pi$
$954$	−9.58374	−0.310285
$955$	0	0
$956$	−16.8662	−0.545493
$957$	0	0
$958$	0.650657	0.0210218
$959$	−24.4522	−0.789604
$960$	0	0
$961$	−18.7690	−0.605452
$962$	0.536939	0.0173116
$963$	−0.735971	−0.0237163
$964$	−20.0428	−0.645534
$965$	0	0
$966$	44.1040	1.41902
$967$	−34.3901	−1.10591	−0.552955	−	0.833211i	$-0.686500\pi$
$967$	−34.3901	−1.10591	−0.552955	+	0.833211i	$0.686500\pi$
$968$	0	0
$969$	−8.74550	−0.280946
$970$	0	0
$971$	40.4046	1.29665	0.648323	−	0.761365i	$-0.275471\pi$
$971$	40.4046	1.29665	0.648323	+	0.761365i	$0.275471\pi$
$972$	−1.63829	−0.0525481
$973$	14.5288	0.465772
$974$	30.2428	0.969042
$975$	0	0
$976$	53.6944	1.71871
$977$	−48.8722	−1.56356	−0.781780	−	0.623554i	$-0.785688\pi$
$977$	−48.8722	−1.56356	−0.781780	+	0.623554i	$0.785688\pi$
$978$	36.9554	1.18170
$979$	0	0
$980$	0	0
$981$	0.0433994	0.00138564
$982$	57.2426	1.82668
$983$	−10.0095	−0.319253	−0.159626	−	0.987177i	$-0.551029\pi$
$983$	−10.0095	−0.319253	−0.159626	+	0.987177i	$0.551029\pi$
$984$	3.23982	0.103282
$985$	0	0
$986$	41.2112	1.31243
$987$	53.4335	1.70081
$988$	−1.68369	−0.0535654
$989$	15.6899	0.498911
$990$	0	0
$991$	31.6029	1.00390	0.501949	−	0.864897i	$-0.332617\pi$
$991$	31.6029	1.00390	0.501949	+	0.864897i	$0.332617\pi$
$992$	25.8105	0.819484
$993$	23.0627	0.731874
$994$	28.6168	0.907671
$995$	0	0
$996$	6.95842	0.220486
$997$	−53.4009	−1.69122	−0.845611	−	0.533799i	$-0.820764\pi$
$997$	−53.4009	−1.69122	−0.845611	+	0.533799i	$0.820764\pi$
$998$	−6.97195	−0.220693
$999$	0.527144	0.0166781

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.dx.1.4		12
5.2	odd	4		1815.2.c.k.364.6		24
5.3	odd	4		1815.2.c.k.364.19		24
5.4	even	2		9075.2.a.ea.1.9		12
11.2	odd	10		825.2.n.o.301.2		24
11.6	odd	10		825.2.n.o.751.2		24
11.10	odd	2		9075.2.a.dz.1.9		12
55.2	even	20		165.2.s.a.4.10	yes	48
55.13	even	20		165.2.s.a.4.3	✓	48
55.17	even	20		165.2.s.a.124.3	yes	48
55.24	odd	10		825.2.n.p.301.5		24
55.28	even	20		165.2.s.a.124.10	yes	48
55.32	even	4		1815.2.c.j.364.19		24
55.39	odd	10		825.2.n.p.751.5		24
55.43	even	4		1815.2.c.j.364.6		24
55.54	odd	2		9075.2.a.dy.1.4		12
165.2	odd	20		495.2.ba.c.334.3		48
165.17	odd	20		495.2.ba.c.289.10		48
165.68	odd	20		495.2.ba.c.334.10		48
165.83	odd	20		495.2.ba.c.289.3		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.s.a.4.3	✓	48	55.13	even	20
165.2.s.a.4.10	yes	48	55.2	even	20
165.2.s.a.124.3	yes	48	55.17	even	20
165.2.s.a.124.10	yes	48	55.28	even	20
495.2.ba.c.289.3		48	165.83	odd	20
495.2.ba.c.289.10		48	165.17	odd	20
495.2.ba.c.334.3		48	165.2	odd	20
495.2.ba.c.334.10		48	165.68	odd	20
825.2.n.o.301.2		24	11.2	odd	10
825.2.n.o.751.2		24	11.6	odd	10
825.2.n.p.301.5		24	55.24	odd	10
825.2.n.p.751.5		24	55.39	odd	10
1815.2.c.j.364.6		24	55.43	even	4
1815.2.c.j.364.19		24	55.32	even	4
1815.2.c.k.364.6		24	5.2	odd	4
1815.2.c.k.364.19		24	5.3	odd	4
9075.2.a.dx.1.4		12	1.1	even	1	trivial
9075.2.a.dy.1.4		12	55.54	odd	2
9075.2.a.dz.1.9		12	11.10	odd	2
9075.2.a.ea.1.9		12	5.4	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.90743 q^{2} -1.00000 q^{3} +1.63829 q^{4} +1.90743 q^{6} -4.45734 q^{7} +0.689943 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.90743 q^{2} -1.00000 q^{3} +1.63829 q^{4} +1.90743 q^{6} -4.45734 q^{7} +0.689943 q^{8} +1.00000 q^{9} -1.63829 q^{12} +0.534008 q^{13} +8.50206 q^{14} -4.59259 q^{16} -4.54422 q^{17} -1.90743 q^{18} -1.92453 q^{19} +4.45734 q^{21} -5.18745 q^{23} -0.689943 q^{24} -1.01858 q^{26} -1.00000 q^{27} -7.30240 q^{28} +4.75453 q^{29} +3.49728 q^{31} +7.38016 q^{32} +8.66777 q^{34} +1.63829 q^{36} -0.527144 q^{37} +3.67091 q^{38} -0.534008 q^{39} -4.69578 q^{41} -8.50206 q^{42} -3.02459 q^{43} +9.89470 q^{46} +11.9878 q^{47} +4.59259 q^{48} +12.8679 q^{49} +4.54422 q^{51} +0.874857 q^{52} +5.02443 q^{53} +1.90743 q^{54} -3.07531 q^{56} +1.92453 q^{57} -9.06894 q^{58} +13.6495 q^{59} -11.6915 q^{61} -6.67082 q^{62} -4.45734 q^{63} -4.89194 q^{64} -11.5622 q^{67} -7.44473 q^{68} +5.18745 q^{69} +3.36587 q^{71} +0.689943 q^{72} +0.650391 q^{73} +1.00549 q^{74} -3.15294 q^{76} +1.01858 q^{78} +17.7199 q^{79} +1.00000 q^{81} +8.95686 q^{82} -4.24738 q^{83} +7.30240 q^{84} +5.76920 q^{86} -4.75453 q^{87} -8.24094 q^{89} -2.38025 q^{91} -8.49854 q^{92} -3.49728 q^{93} -22.8658 q^{94} -7.38016 q^{96} +1.13839 q^{97} -24.5445 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} - 12 q^{3} + 12 q^{4} + 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 4 * q^2 - 12 * q^3 + 12 * q^4 + 4 * q^6 - 8 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 4 q^{2} - 12 q^{3} + 12 q^{4} + 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9} - 12 q^{12} - 18 q^{13} + 6 q^{14} + 24 q^{16} - 18 q^{17} - 4 q^{18} + 16 q^{19} + 8 q^{21} + 12 q^{24} + 16 q^{26} - 12 q^{27} - 30 q^{28} - 28 q^{32} + 6 q^{34} + 12 q^{36} + 28 q^{38} + 18 q^{39} - 6 q^{42} - 32 q^{43} + 28 q^{46} + 4 q^{47} - 24 q^{48} + 12 q^{49} + 18 q^{51} - 48 q^{52} + 12 q^{53} + 4 q^{54} + 6 q^{56} - 16 q^{57} + 10 q^{58} + 20 q^{59} + 20 q^{61} - 20 q^{62} - 8 q^{63} - 6 q^{64} - 10 q^{67} - 26 q^{68} + 32 q^{71} - 12 q^{72} - 26 q^{73} - 68 q^{74} + 34 q^{76} - 16 q^{78} + 32 q^{79} + 12 q^{81} + 62 q^{82} - 26 q^{83} + 30 q^{84} - 36 q^{86} - 10 q^{89} - 8 q^{92} - 2 q^{94} + 28 q^{96} + 22 q^{97} - 60 q^{98}+O(q^{100}) \) 12 * q - 4 * q^2 - 12 * q^3 + 12 * q^4 + 4 * q^6 - 8 * q^7 - 12 * q^8 + 12 * q^9 - 12 * q^12 - 18 * q^13 + 6 * q^14 + 24 * q^16 - 18 * q^17 - 4 * q^18 + 16 * q^19 + 8 * q^21 + 12 * q^24 + 16 * q^26 - 12 * q^27 - 30 * q^28 - 28 * q^32 + 6 * q^34 + 12 * q^36 + 28 * q^38 + 18 * q^39 - 6 * q^42 - 32 * q^43 + 28 * q^46 + 4 * q^47 - 24 * q^48 + 12 * q^49 + 18 * q^51 - 48 * q^52 + 12 * q^53 + 4 * q^54 + 6 * q^56 - 16 * q^57 + 10 * q^58 + 20 * q^59 + 20 * q^61 - 20 * q^62 - 8 * q^63 - 6 * q^64 - 10 * q^67 - 26 * q^68 + 32 * q^71 - 12 * q^72 - 26 * q^73 - 68 * q^74 + 34 * q^76 - 16 * q^78 + 32 * q^79 + 12 * q^81 + 62 * q^82 - 26 * q^83 + 30 * q^84 - 36 * q^86 - 10 * q^89 - 8 * q^92 - 2 * q^94 + 28 * q^96 + 22 * q^97 - 60 * q^98

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