LMFDB - Embedded newform 9075.2.a.dx.1.6

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			1.6
Root			$0.488299$ 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)

$f(q)$	$=$	$q-0.488299 q^{2} -1.00000 q^{3} -1.76156 q^{4} +0.488299 q^{6} -5.10895 q^{7} +1.83677 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.488299 q^{2} -1.00000 q^{3} -1.76156 q^{4} +0.488299 q^{6} -5.10895 q^{7} +1.83677 q^{8} +1.00000 q^{9} +1.76156 q^{12} -1.81774 q^{13} +2.49469 q^{14} +2.62624 q^{16} -0.639966 q^{17} -0.488299 q^{18} +4.39788 q^{19} +5.10895 q^{21} -4.37977 q^{23} -1.83677 q^{24} +0.887599 q^{26} -1.00000 q^{27} +8.99974 q^{28} -7.56018 q^{29} +2.20501 q^{31} -4.95592 q^{32} +0.312494 q^{34} -1.76156 q^{36} -5.97194 q^{37} -2.14748 q^{38} +1.81774 q^{39} +2.23182 q^{41} -2.49469 q^{42} -3.38068 q^{43} +2.13864 q^{46} +1.51581 q^{47} -2.62624 q^{48} +19.1013 q^{49} +0.639966 q^{51} +3.20206 q^{52} +9.17695 q^{53} +0.488299 q^{54} -9.38395 q^{56} -4.39788 q^{57} +3.69163 q^{58} +5.37174 q^{59} -7.26768 q^{61} -1.07670 q^{62} -5.10895 q^{63} -2.83250 q^{64} +7.25598 q^{67} +1.12734 q^{68} +4.37977 q^{69} +5.50864 q^{71} +1.83677 q^{72} +7.14322 q^{73} +2.91609 q^{74} -7.74714 q^{76} -0.887599 q^{78} -1.90516 q^{79} +1.00000 q^{81} -1.08979 q^{82} -0.161902 q^{83} -8.99974 q^{84} +1.65078 q^{86} +7.56018 q^{87} -1.19539 q^{89} +9.28673 q^{91} +7.71525 q^{92} -2.20501 q^{93} -0.740167 q^{94} +4.95592 q^{96} -14.2823 q^{97} -9.32716 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.488299	−0.345279	−0.172640	−	0.984985i	$-0.555230\pi$
$2$	−0.488299	−0.345279	−0.172640	+	0.984985i	$0.555230\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.76156	−0.880782
$5$	0	0
$5$	0	0
$6$	0.488299	0.199347
$7$	−5.10895	−1.93100	−0.965500	−	0.260403i	$-0.916145\pi$
$7$	−5.10895	−1.93100	−0.965500	+	0.260403i	$0.916145\pi$
$8$	1.83677	0.649395
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	1.76156	0.508520
$13$	−1.81774	−0.504150	−0.252075	−	0.967708i	$-0.581113\pi$
$13$	−1.81774	−0.504150	−0.252075	+	0.967708i	$0.581113\pi$
$14$	2.49469	0.666735
$15$	0	0
$16$	2.62624	0.656559
$17$	−0.639966	−0.155214	−0.0776072	−	0.996984i	$-0.524728\pi$
$17$	−0.639966	−0.155214	−0.0776072	+	0.996984i	$0.524728\pi$
$18$	−0.488299	−0.115093
$19$	4.39788	1.00894	0.504471	−	0.863429i	$-0.331688\pi$
$19$	4.39788	1.00894	0.504471	+	0.863429i	$0.331688\pi$
$20$	0	0
$21$	5.10895	1.11486
$22$	0	0
$23$	−4.37977	−0.913246	−0.456623	−	0.889660i	$-0.650941\pi$
$23$	−4.37977	−0.913246	−0.456623	+	0.889660i	$0.650941\pi$
$24$	−1.83677	−0.374929
$25$	0	0
$26$	0.887599	0.174073
$27$	−1.00000	−0.192450
$28$	8.99974	1.70079
$29$	−7.56018	−1.40389	−0.701945	−	0.712231i	$-0.747685\pi$
$29$	−7.56018	−1.40389	−0.701945	+	0.712231i	$0.747685\pi$
$30$	0	0
$31$	2.20501	0.396032	0.198016	−	0.980199i	$-0.436550\pi$
$31$	2.20501	0.396032	0.198016	+	0.980199i	$0.436550\pi$
$32$	−4.95592	−0.876092
$33$	0	0
$34$	0.312494	0.0535924
$35$	0	0
$36$	−1.76156	−0.293594
$37$	−5.97194	−0.981780	−0.490890	−	0.871221i	$-0.663328\pi$
$37$	−5.97194	−0.981780	−0.490890	+	0.871221i	$0.663328\pi$
$38$	−2.14748	−0.348367
$39$	1.81774	0.291071
$40$	0	0
$41$	2.23182	0.348551	0.174276	−	0.984697i	$-0.444242\pi$
$41$	2.23182	0.348551	0.174276	+	0.984697i	$0.444242\pi$
$42$	−2.49469	−0.384939
$43$	−3.38068	−0.515549	−0.257775	−	0.966205i	$-0.582989\pi$
$43$	−3.38068	−0.515549	−0.257775	+	0.966205i	$0.582989\pi$
$44$	0	0
$45$	0	0
$46$	2.13864	0.315325
$47$	1.51581	0.221103	0.110552	−	0.993870i	$-0.464738\pi$
$47$	1.51581	0.221103	0.110552	+	0.993870i	$0.464738\pi$
$48$	−2.62624	−0.379065
$49$	19.1013	2.72876
$50$	0	0
$51$	0.639966	0.0896131
$52$	3.20206	0.444046
$53$	9.17695	1.26055	0.630275	−	0.776372i	$-0.282942\pi$
$53$	9.17695	1.26055	0.630275	+	0.776372i	$0.282942\pi$
$54$	0.488299	0.0664491
$55$	0	0
$56$	−9.38395	−1.25398
$57$	−4.39788	−0.582513
$58$	3.69163	0.484734
$59$	5.37174	0.699341	0.349671	−	0.936873i	$-0.386294\pi$
$59$	5.37174	0.699341	0.349671	+	0.936873i	$0.386294\pi$
$60$	0	0
$61$	−7.26768	−0.930531	−0.465266	−	0.885171i	$-0.654041\pi$
$61$	−7.26768	−0.930531	−0.465266	+	0.885171i	$0.654041\pi$
$62$	−1.07670	−0.136742
$63$	−5.10895	−0.643667
$64$	−2.83250	−0.354063
$65$	0	0
$66$	0	0
$67$	7.25598	0.886459	0.443230	−	0.896408i	$-0.353833\pi$
$67$	7.25598	0.886459	0.443230	+	0.896408i	$0.353833\pi$
$68$	1.12734	0.136710
$69$	4.37977	0.527263
$70$	0	0
$71$	5.50864	0.653756	0.326878	−	0.945067i	$-0.394003\pi$
$71$	5.50864	0.653756	0.326878	+	0.945067i	$0.394003\pi$
$72$	1.83677	0.216465
$73$	7.14322	0.836051	0.418026	−	0.908435i	$-0.362722\pi$
$73$	7.14322	0.836051	0.418026	+	0.908435i	$0.362722\pi$
$74$	2.91609	0.338988
$75$	0	0
$76$	−7.74714	−0.888658
$77$	0	0
$78$	−0.887599	−0.100501
$79$	−1.90516	−0.214347	−0.107174	−	0.994240i	$-0.534180\pi$
$79$	−1.90516	−0.214347	−0.107174	+	0.994240i	$0.534180\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−1.08979	−0.120348
$83$	−0.161902	−0.0177710	−0.00888551	−	0.999961i	$-0.502828\pi$
$83$	−0.161902	−0.0177710	−0.00888551	+	0.999961i	$0.502828\pi$
$84$	−8.99974	−0.981952
$85$	0	0
$86$	1.65078	0.178009
$87$	7.56018	0.810536
$88$	0	0
$89$	−1.19539	−0.126711	−0.0633554	−	0.997991i	$-0.520180\pi$
$89$	−1.19539	−0.126711	−0.0633554	+	0.997991i	$0.520180\pi$
$90$	0	0
$91$	9.28673	0.973514
$92$	7.71525	0.804371
$93$	−2.20501	−0.228649
$94$	−0.740167	−0.0763424
$95$	0	0
$96$	4.95592	0.505812
$97$	−14.2823	−1.45015	−0.725073	−	0.688672i	$-0.758194\pi$
$97$	−14.2823	−1.45015	−0.725073	+	0.688672i	$0.758194\pi$
$98$	−9.32716	−0.942185
$99$	0	0
$100$	0	0
$101$	4.29089	0.426960	0.213480	−	0.976947i	$-0.431520\pi$
$101$	4.29089	0.426960	0.213480	+	0.976947i	$0.431520\pi$
$102$	−0.312494	−0.0309416
$103$	−9.49802	−0.935868	−0.467934	−	0.883763i	$-0.655002\pi$
$103$	−9.49802	−0.935868	−0.467934	+	0.883763i	$0.655002\pi$
$104$	−3.33876	−0.327393
$105$	0	0
$106$	−4.48109	−0.435242
$107$	8.27470	0.799945	0.399973	−	0.916527i	$-0.369020\pi$
$107$	8.27470	0.799945	0.399973	+	0.916527i	$0.369020\pi$
$108$	1.76156	0.169507
$109$	12.4980	1.19709	0.598546	−	0.801089i	$-0.295745\pi$
$109$	12.4980	1.19709	0.598546	+	0.801089i	$0.295745\pi$
$110$	0	0
$111$	5.97194	0.566831
$112$	−13.4173	−1.26782
$113$	16.2096	1.52487	0.762435	−	0.647065i	$-0.224004\pi$
$113$	16.2096	1.52487	0.762435	+	0.647065i	$0.224004\pi$
$114$	2.14748	0.201130
$115$	0	0
$116$	13.3177	1.23652
$117$	−1.81774	−0.168050
$118$	−2.62302	−0.241468
$119$	3.26955	0.299719
$120$	0	0
$121$	0	0
$122$	3.54880	0.321293
$123$	−2.23182	−0.201236
$124$	−3.88427	−0.348817
$125$	0	0
$126$	2.49469	0.222245
$127$	9.88086	0.876785	0.438392	−	0.898784i	$-0.355548\pi$
$127$	9.88086	0.876785	0.438392	+	0.898784i	$0.355548\pi$
$128$	11.2950	0.998342
$129$	3.38068	0.297653
$130$	0	0
$131$	12.7829	1.11685	0.558425	−	0.829555i	$-0.311406\pi$
$131$	12.7829	1.11685	0.558425	+	0.829555i	$0.311406\pi$
$132$	0	0
$133$	−22.4685	−1.94827
$134$	−3.54309	−0.306076
$135$	0	0
$136$	−1.17547	−0.100796
$137$	19.0589	1.62831	0.814157	−	0.580645i	$-0.197200\pi$
$137$	19.0589	1.62831	0.814157	+	0.580645i	$0.197200\pi$
$138$	−2.13864	−0.182053
$139$	11.7146	0.993622	0.496811	−	0.867859i	$-0.334504\pi$
$139$	11.7146	0.993622	0.496811	+	0.867859i	$0.334504\pi$
$140$	0	0
$141$	−1.51581	−0.127654
$142$	−2.68986	−0.225728
$143$	0	0
$144$	2.62624	0.218853
$145$	0	0
$146$	−3.48803	−0.288671
$147$	−19.1013	−1.57545
$148$	10.5199	0.864735
$149$	−7.45724	−0.610921	−0.305461	−	0.952205i	$-0.598810\pi$
$149$	−7.45724	−0.610921	−0.305461	+	0.952205i	$0.598810\pi$
$150$	0	0
$151$	4.95905	0.403562	0.201781	−	0.979431i	$-0.435327\pi$
$151$	4.95905	0.403562	0.201781	+	0.979431i	$0.435327\pi$
$152$	8.07787	0.655202
$153$	−0.639966	−0.0517382
$154$	0	0
$155$	0	0
$156$	−3.20206	−0.256370
$157$	20.5916	1.64339	0.821693	−	0.569930i	$-0.193030\pi$
$157$	20.5916	1.64339	0.821693	+	0.569930i	$0.193030\pi$
$158$	0.930287	0.0740096
$159$	−9.17695	−0.727779
$160$	0	0
$161$	22.3760	1.76348
$162$	−0.488299	−0.0383644
$163$	−7.22774	−0.566120	−0.283060	−	0.959102i	$-0.591350\pi$
$163$	−7.22774	−0.566120	−0.283060	+	0.959102i	$0.591350\pi$
$164$	−3.93149	−0.306998
$165$	0	0
$166$	0.0790564	0.00613597
$167$	18.3130	1.41710	0.708550	−	0.705660i	$-0.249349\pi$
$167$	18.3130	1.41710	0.708550	+	0.705660i	$0.249349\pi$
$168$	9.38395	0.723987
$169$	−9.69583	−0.745833
$170$	0	0
$171$	4.39788	0.336314
$172$	5.95529	0.454087
$173$	−12.0157	−0.913536	−0.456768	−	0.889586i	$-0.650993\pi$
$173$	−12.0157	−0.913536	−0.456768	+	0.889586i	$0.650993\pi$
$174$	−3.69163	−0.279862
$175$	0	0
$176$	0	0
$177$	−5.37174	−0.403765
$178$	0.583706	0.0437506
$179$	−6.71004	−0.501532	−0.250766	−	0.968048i	$-0.580682\pi$
$179$	−6.71004	−0.501532	−0.250766	+	0.968048i	$0.580682\pi$
$180$	0	0
$181$	−11.3348	−0.842508	−0.421254	−	0.906943i	$-0.638410\pi$
$181$	−11.3348	−0.842508	−0.421254	+	0.906943i	$0.638410\pi$
$182$	−4.53470	−0.336134
$183$	7.26768	0.537242
$184$	−8.04463	−0.593058
$185$	0	0
$186$	1.07670	0.0789478
$187$	0	0
$188$	−2.67019	−0.194744
$189$	5.10895	0.371621
$190$	0	0
$191$	11.0864	0.802186	0.401093	−	0.916037i	$-0.368630\pi$
$191$	11.0864	0.802186	0.401093	+	0.916037i	$0.368630\pi$
$192$	2.83250	0.204418
$193$	−1.19205	−0.0858056	−0.0429028	−	0.999079i	$-0.513661\pi$
$193$	−1.19205	−0.0858056	−0.0429028	+	0.999079i	$0.513661\pi$
$194$	6.97402	0.500705
$195$	0	0
$196$	−33.6482	−2.40344
$197$	−18.3551	−1.30775	−0.653873	−	0.756604i	$-0.726857\pi$
$197$	−18.3551	−1.30775	−0.653873	+	0.756604i	$0.726857\pi$
$198$	0	0
$199$	22.1761	1.57202	0.786011	−	0.618213i	$-0.212143\pi$
$199$	22.1761	1.57202	0.786011	+	0.618213i	$0.212143\pi$
$200$	0	0
$201$	−7.25598	−0.511798
$202$	−2.09524	−0.147420
$203$	38.6246	2.71091
$204$	−1.12734	−0.0789296
$205$	0	0
$206$	4.63787	0.323136
$207$	−4.37977	−0.304415
$208$	−4.77381	−0.331004
$209$	0	0
$210$	0	0
$211$	−8.30729	−0.571897	−0.285949	−	0.958245i	$-0.592309\pi$
$211$	−8.30729	−0.571897	−0.285949	+	0.958245i	$0.592309\pi$
$212$	−16.1658	−1.11027
$213$	−5.50864	−0.377446
$214$	−4.04052	−0.276205
$215$	0	0
$216$	−1.83677	−0.124976
$217$	−11.2653	−0.764737
$218$	−6.10276	−0.413331
$219$	−7.14322	−0.482694
$220$	0	0
$221$	1.16329	0.0782514
$222$	−2.91609	−0.195715
$223$	−16.5497	−1.10825	−0.554124	−	0.832434i	$-0.686947\pi$
$223$	−16.5497	−1.10825	−0.554124	+	0.832434i	$0.686947\pi$
$224$	25.3195	1.69173
$225$	0	0
$226$	−7.91513	−0.526506
$227$	−8.48300	−0.563036	−0.281518	−	0.959556i	$-0.590838\pi$
$227$	−8.48300	−0.563036	−0.281518	+	0.959556i	$0.590838\pi$
$228$	7.74714	0.513067
$229$	−3.39892	−0.224607	−0.112303	−	0.993674i	$-0.535823\pi$
$229$	−3.39892	−0.224607	−0.112303	+	0.993674i	$0.535823\pi$
$230$	0	0
$231$	0	0
$232$	−13.8863	−0.911680
$233$	−15.7123	−1.02935	−0.514675	−	0.857386i	$-0.672087\pi$
$233$	−15.7123	−1.02935	−0.514675	+	0.857386i	$0.672087\pi$
$234$	0.887599	0.0580242
$235$	0	0
$236$	−9.46267	−0.615967
$237$	1.90516	0.123753
$238$	−1.59652	−0.103487
$239$	−30.6972	−1.98564	−0.992819	−	0.119624i	$-0.961831\pi$
$239$	−30.6972	−1.98564	−0.992819	+	0.119624i	$0.961831\pi$
$240$	0	0
$241$	−5.59097	−0.360146	−0.180073	−	0.983653i	$-0.557633\pi$
$241$	−5.59097	−0.360146	−0.180073	+	0.983653i	$0.557633\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	12.8025	0.819595
$245$	0	0
$246$	1.08979	0.0694827
$247$	−7.99419	−0.508658
$248$	4.05009	0.257181
$249$	0.161902	0.0102601
$250$	0	0
$251$	18.7237	1.18183	0.590915	−	0.806734i	$-0.298767\pi$
$251$	18.7237	1.18183	0.590915	+	0.806734i	$0.298767\pi$
$252$	8.99974	0.566930
$253$	0	0
$254$	−4.82481	−0.302736
$255$	0	0
$256$	0.149694	0.00935585
$257$	4.98215	0.310778	0.155389	−	0.987853i	$-0.450337\pi$
$257$	4.98215	0.310778	0.155389	+	0.987853i	$0.450337\pi$
$258$	−1.65078	−0.102773
$259$	30.5103	1.89582
$260$	0	0
$261$	−7.56018	−0.467963
$262$	−6.24190	−0.385626
$263$	−20.0788	−1.23811	−0.619055	−	0.785347i	$-0.712484\pi$
$263$	−20.0788	−1.23811	−0.619055	+	0.785347i	$0.712484\pi$
$264$	0	0
$265$	0	0
$266$	10.9713	0.672696
$267$	1.19539	0.0731565
$268$	−12.7819	−0.780778
$269$	−9.53027	−0.581071	−0.290535	−	0.956864i	$-0.593833\pi$
$269$	−9.53027	−0.581071	−0.290535	+	0.956864i	$0.593833\pi$
$270$	0	0
$271$	1.18095	0.0717375	0.0358688	−	0.999357i	$-0.488580\pi$
$271$	1.18095	0.0717375	0.0358688	+	0.999357i	$0.488580\pi$
$272$	−1.68070	−0.101908
$273$	−9.28673	−0.562058
$274$	−9.30645	−0.562223
$275$	0	0
$276$	−7.71525	−0.464404
$277$	−8.22177	−0.493998	−0.246999	−	0.969016i	$-0.579444\pi$
$277$	−8.22177	−0.493998	−0.246999	+	0.969016i	$0.579444\pi$
$278$	−5.72024	−0.343077
$279$	2.20501	0.132011
$280$	0	0
$281$	−10.4453	−0.623116	−0.311558	−	0.950227i	$-0.600851\pi$
$281$	−10.4453	−0.623116	−0.311558	+	0.950227i	$0.600851\pi$
$282$	0.740167	0.0440763
$283$	−24.8463	−1.47696	−0.738480	−	0.674275i	$-0.764456\pi$
$283$	−24.8463	−1.47696	−0.738480	+	0.674275i	$0.764456\pi$
$284$	−9.70383	−0.575816
$285$	0	0
$286$	0	0
$287$	−11.4022	−0.673052
$288$	−4.95592	−0.292031
$289$	−16.5904	−0.975908
$290$	0	0
$291$	14.2823	0.837242
$292$	−12.5832	−0.736379
$293$	−24.8264	−1.45037	−0.725186	−	0.688553i	$-0.758246\pi$
$293$	−24.8264	−1.45037	−0.725186	+	0.688553i	$0.758246\pi$
$294$	9.32716	0.543971
$295$	0	0
$296$	−10.9691	−0.637563
$297$	0	0
$298$	3.64136	0.210939
$299$	7.96128	0.460413
$300$	0	0
$301$	17.2717	0.995526
$302$	−2.42150	−0.139342
$303$	−4.29089	−0.246505
$304$	11.5499	0.662430
$305$	0	0
$306$	0.312494	0.0178641
$307$	−12.4687	−0.711624	−0.355812	−	0.934557i	$-0.615796\pi$
$307$	−12.4687	−0.711624	−0.355812	+	0.934557i	$0.615796\pi$
$308$	0	0
$309$	9.49802	0.540324
$310$	0	0
$311$	−3.15914	−0.179138	−0.0895692	−	0.995981i	$-0.528549\pi$
$311$	−3.15914	−0.179138	−0.0895692	+	0.995981i	$0.528549\pi$
$312$	3.33876	0.189020
$313$	14.2948	0.807988	0.403994	−	0.914762i	$-0.367622\pi$
$313$	14.2948	0.807988	0.403994	+	0.914762i	$0.367622\pi$
$314$	−10.0548	−0.567427
$315$	0	0
$316$	3.35606	0.188793
$317$	26.8771	1.50957	0.754784	−	0.655974i	$-0.227742\pi$
$317$	26.8771	1.50957	0.754784	+	0.655974i	$0.227742\pi$
$318$	4.48109	0.251287
$319$	0	0
$320$	0	0
$321$	−8.27470	−0.461849
$322$	−10.9262	−0.608893
$323$	−2.81449	−0.156602
$324$	−1.76156	−0.0978647
$325$	0	0
$326$	3.52930	0.195470
$327$	−12.4980	−0.691141
$328$	4.09933	0.226347
$329$	−7.74418	−0.426950
$330$	0	0
$331$	19.7763	1.08700	0.543502	−	0.839408i	$-0.317098\pi$
$331$	19.7763	1.08700	0.543502	+	0.839408i	$0.317098\pi$
$332$	0.285200	0.0156524
$333$	−5.97194	−0.327260
$334$	−8.94221	−0.489296
$335$	0	0
$336$	13.4173	0.731974
$337$	14.2580	0.776685	0.388343	−	0.921515i	$-0.373048\pi$
$337$	14.2580	0.776685	0.388343	+	0.921515i	$0.373048\pi$
$338$	4.73446	0.257521
$339$	−16.2096	−0.880384
$340$	0	0
$341$	0	0
$342$	−2.14748	−0.116122
$343$	−61.8251	−3.33824
$344$	−6.20953	−0.334795
$345$	0	0
$346$	5.86725	0.315425
$347$	15.7954	0.847940	0.423970	−	0.905676i	$-0.360636\pi$
$347$	15.7954	0.847940	0.423970	+	0.905676i	$0.360636\pi$
$348$	−13.3177	−0.713906
$349$	−1.71442	−0.0917705	−0.0458853	−	0.998947i	$-0.514611\pi$
$349$	−1.71442	−0.0917705	−0.0458853	+	0.998947i	$0.514611\pi$
$350$	0	0
$351$	1.81774	0.0970237
$352$	0	0
$353$	−13.9610	−0.743072	−0.371536	−	0.928419i	$-0.621169\pi$
$353$	−13.9610	−0.743072	−0.371536	+	0.928419i	$0.621169\pi$
$354$	2.62302	0.139412
$355$	0	0
$356$	2.10575	0.111605
$357$	−3.26955	−0.173043
$358$	3.27650	0.173169
$359$	34.3062	1.81061	0.905306	−	0.424759i	$-0.139641\pi$
$359$	34.3062	1.81061	0.905306	+	0.424759i	$0.139641\pi$
$360$	0	0
$361$	0.341312	0.0179638
$362$	5.53476	0.290901
$363$	0	0
$364$	−16.3592	−0.857453
$365$	0	0
$366$	−3.54880	−0.185499
$367$	−3.61245	−0.188569	−0.0942843	−	0.995545i	$-0.530056\pi$
$367$	−3.61245	−0.188569	−0.0942843	+	0.995545i	$0.530056\pi$
$368$	−11.5023	−0.599600
$369$	2.23182	0.116184
$370$	0	0
$371$	−46.8845	−2.43412
$372$	3.88427	0.201390
$373$	−18.7064	−0.968579	−0.484290	−	0.874908i	$-0.660922\pi$
$373$	−18.7064	−0.968579	−0.484290	+	0.874908i	$0.660922\pi$
$374$	0	0
$375$	0	0
$376$	2.78418	0.143583
$377$	13.7424	0.707771
$378$	−2.49469	−0.128313
$379$	−9.37468	−0.481545	−0.240772	−	0.970582i	$-0.577401\pi$
$379$	−9.37468	−0.481545	−0.240772	+	0.970582i	$0.577401\pi$
$380$	0	0
$381$	−9.88086	−0.506212
$382$	−5.41349	−0.276978
$383$	19.2155	0.981865	0.490932	−	0.871198i	$-0.336656\pi$
$383$	19.2155	0.981865	0.490932	+	0.871198i	$0.336656\pi$
$384$	−11.2950	−0.576393
$385$	0	0
$386$	0.582076	0.0296269
$387$	−3.38068	−0.171850
$388$	25.1591	1.27726
$389$	−9.29700	−0.471376	−0.235688	−	0.971829i	$-0.575734\pi$
$389$	−9.29700	−0.471376	−0.235688	+	0.971829i	$0.575734\pi$
$390$	0	0
$391$	2.80290	0.141749
$392$	35.0847	1.77205
$393$	−12.7829	−0.644814
$394$	8.96277	0.451538
$395$	0	0
$396$	0	0
$397$	−13.8693	−0.696078	−0.348039	−	0.937480i	$-0.613152\pi$
$397$	−13.8693	−0.696078	−0.348039	+	0.937480i	$0.613152\pi$
$398$	−10.8286	−0.542787
$399$	22.4685	1.12483
$400$	0	0
$401$	10.6336	0.531015	0.265507	−	0.964109i	$-0.414461\pi$
$401$	10.6336	0.531015	0.265507	+	0.964109i	$0.414461\pi$
$402$	3.54309	0.176713
$403$	−4.00813	−0.199659
$404$	−7.55868	−0.376058
$405$	0	0
$406$	−18.8603	−0.936022
$407$	0	0
$408$	1.17547	0.0581943
$409$	−3.71320	−0.183606	−0.0918030	−	0.995777i	$-0.529263\pi$
$409$	−3.71320	−0.183606	−0.0918030	+	0.995777i	$0.529263\pi$
$410$	0	0
$411$	−19.0589	−0.940107
$412$	16.7314	0.824296
$413$	−27.4439	−1.35043
$414$	2.13864	0.105108
$415$	0	0
$416$	9.00857	0.441682
$417$	−11.7146	−0.573668
$418$	0	0
$419$	−15.8996	−0.776745	−0.388372	−	0.921503i	$-0.626963\pi$
$419$	−15.8996	−0.776745	−0.388372	+	0.921503i	$0.626963\pi$
$420$	0	0
$421$	4.36664	0.212817	0.106408	−	0.994323i	$-0.466065\pi$
$421$	4.36664	0.212817	0.106408	+	0.994323i	$0.466065\pi$
$422$	4.05644	0.197464
$423$	1.51581	0.0737011
$424$	16.8559	0.818596
$425$	0	0
$426$	2.68986	0.130324
$427$	37.1302	1.79686
$428$	−14.5764	−0.704577
$429$	0	0
$430$	0	0
$431$	−20.1005	−0.968209	−0.484105	−	0.875010i	$-0.660855\pi$
$431$	−20.1005	−0.968209	−0.484105	+	0.875010i	$0.660855\pi$
$432$	−2.62624	−0.126355
$433$	30.7232	1.47646	0.738230	−	0.674549i	$-0.235662\pi$
$433$	30.7232	1.47646	0.738230	+	0.674549i	$0.235662\pi$
$434$	5.50082	0.264048
$435$	0	0
$436$	−22.0160	−1.05438
$437$	−19.2617	−0.921412
$438$	3.48803	0.166664
$439$	20.6104	0.983679	0.491840	−	0.870686i	$-0.336325\pi$
$439$	20.6104	0.983679	0.491840	+	0.870686i	$0.336325\pi$
$440$	0	0
$441$	19.1013	0.909587
$442$	−0.568033	−0.0270186
$443$	−13.7997	−0.655644	−0.327822	−	0.944740i	$-0.606315\pi$
$443$	−13.7997	−0.655644	−0.327822	+	0.944740i	$0.606315\pi$
$444$	−10.5199	−0.499255
$445$	0	0
$446$	8.08118	0.382655
$447$	7.45724	0.352716
$448$	14.4711	0.683696
$449$	27.2508	1.28605	0.643023	−	0.765847i	$-0.277680\pi$
$449$	27.2508	1.28605	0.643023	+	0.765847i	$0.277680\pi$
$450$	0	0
$451$	0	0
$452$	−28.5542	−1.34308
$453$	−4.95905	−0.232997
$454$	4.14224	0.194405
$455$	0	0
$456$	−8.07787	−0.378281
$457$	4.62255	0.216234	0.108117	−	0.994138i	$-0.465518\pi$
$457$	4.62255	0.216234	0.108117	+	0.994138i	$0.465518\pi$
$458$	1.65969	0.0775521
$459$	0.639966	0.0298710
$460$	0	0
$461$	25.3940	1.18272	0.591359	−	0.806409i	$-0.298592\pi$
$461$	25.3940	1.18272	0.591359	+	0.806409i	$0.298592\pi$
$462$	0	0
$463$	12.8949	0.599275	0.299638	−	0.954053i	$-0.403134\pi$
$463$	12.8949	0.599275	0.299638	+	0.954053i	$0.403134\pi$
$464$	−19.8548	−0.921737
$465$	0	0
$466$	7.67231	0.355413
$467$	3.81025	0.176317	0.0881587	−	0.996106i	$-0.471902\pi$
$467$	3.81025	0.176317	0.0881587	+	0.996106i	$0.471902\pi$
$468$	3.20206	0.148015
$469$	−37.0704	−1.71175
$470$	0	0
$471$	−20.5916	−0.948809
$472$	9.86664	0.454149
$473$	0	0
$474$	−0.930287	−0.0427295
$475$	0	0
$476$	−5.75952	−0.263987
$477$	9.17695	0.420184
$478$	14.9894	0.685600
$479$	−12.2918	−0.561625	−0.280812	−	0.959763i	$-0.590604\pi$
$479$	−12.2918	−0.561625	−0.280812	+	0.959763i	$0.590604\pi$
$480$	0	0
$481$	10.8554	0.494964
$482$	2.73006	0.124351
$483$	−22.3760	−1.01814
$484$	0	0
$485$	0	0
$486$	0.488299	0.0221497
$487$	13.4389	0.608975	0.304488	−	0.952516i	$-0.401515\pi$
$487$	13.4389	0.608975	0.304488	+	0.952516i	$0.401515\pi$
$488$	−13.3490	−0.604283
$489$	7.22774	0.326850
$490$	0	0
$491$	6.08182	0.274469	0.137234	−	0.990539i	$-0.456179\pi$
$491$	6.08182	0.274469	0.137234	+	0.990539i	$0.456179\pi$
$492$	3.93149	0.177245
$493$	4.83826	0.217904
$494$	3.90355	0.175629
$495$	0	0
$496$	5.79088	0.260018
$497$	−28.1434	−1.26240
$498$	−0.0790564	−0.00354260
$499$	18.7942	0.841344	0.420672	−	0.907213i	$-0.361794\pi$
$499$	18.7942	0.841344	0.420672	+	0.907213i	$0.361794\pi$
$500$	0	0
$501$	−18.3130	−0.818164
$502$	−9.14276	−0.408061
$503$	−6.06605	−0.270472	−0.135236	−	0.990813i	$-0.543179\pi$
$503$	−6.06605	−0.270472	−0.135236	+	0.990813i	$0.543179\pi$
$504$	−9.38395	−0.417994
$505$	0	0
$506$	0	0
$507$	9.69583	0.430607
$508$	−17.4058	−0.772256
$509$	−19.7444	−0.875154	−0.437577	−	0.899181i	$-0.644163\pi$
$509$	−19.7444	−0.875154	−0.437577	+	0.899181i	$0.644163\pi$
$510$	0	0
$511$	−36.4943	−1.61441
$512$	−22.6630	−1.00157
$513$	−4.39788	−0.194171
$514$	−2.43278	−0.107305
$515$	0	0
$516$	−5.95529	−0.262167
$517$	0	0
$518$	−14.8981	−0.654587
$519$	12.0157	0.527430
$520$	0	0
$521$	−42.1434	−1.84634	−0.923168	−	0.384396i	$-0.874410\pi$
$521$	−42.1434	−1.84634	−0.923168	+	0.384396i	$0.874410\pi$
$522$	3.69163	0.161578
$523$	6.37047	0.278561	0.139281	−	0.990253i	$-0.455521\pi$
$523$	6.37047	0.278561	0.139281	+	0.990253i	$0.455521\pi$
$524$	−22.5180	−0.983702
$525$	0	0
$526$	9.80445	0.427494
$527$	−1.41113	−0.0614698
$528$	0	0
$529$	−3.81758	−0.165982
$530$	0	0
$531$	5.37174	0.233114
$532$	39.5797	1.71600
$533$	−4.05686	−0.175722
$534$	−0.583706	−0.0252594
$535$	0	0
$536$	13.3276	0.575663
$537$	6.71004	0.289560
$538$	4.65362	0.200632
$539$	0	0
$540$	0	0
$541$	12.3075	0.529140	0.264570	−	0.964367i	$-0.414770\pi$
$541$	12.3075	0.529140	0.264570	+	0.964367i	$0.414770\pi$
$542$	−0.576656	−0.0247695
$543$	11.3348	0.486422
$544$	3.17162	0.135982
$545$	0	0
$546$	4.53470	0.194067
$547$	−5.48939	−0.234709	−0.117355	−	0.993090i	$-0.537441\pi$
$547$	−5.48939	−0.234709	−0.117355	+	0.993090i	$0.537441\pi$
$548$	−33.5735	−1.43419
$549$	−7.26768	−0.310177
$550$	0	0
$551$	−33.2487	−1.41644
$552$	8.04463	0.342402
$553$	9.73335	0.413904
$554$	4.01468	0.170567
$555$	0	0
$556$	−20.6361	−0.875165
$557$	−25.4636	−1.07893	−0.539464	−	0.842009i	$-0.681373\pi$
$557$	−25.4636	−1.07893	−0.539464	+	0.842009i	$0.681373\pi$
$558$	−1.07670	−0.0455805
$559$	6.14520	0.259914
$560$	0	0
$561$	0	0
$562$	5.10044	0.215149
$563$	−38.3527	−1.61638	−0.808188	−	0.588925i	$-0.799551\pi$
$563$	−38.3527	−1.61638	−0.808188	+	0.588925i	$0.799551\pi$
$564$	2.67019	0.112435
$565$	0	0
$566$	12.1324	0.509964
$567$	−5.10895	−0.214556
$568$	10.1181	0.424546
$569$	−9.15245	−0.383691	−0.191845	−	0.981425i	$-0.561447\pi$
$569$	−9.15245	−0.383691	−0.191845	+	0.981425i	$0.561447\pi$
$570$	0	0
$571$	18.3508	0.767957	0.383978	−	0.923342i	$-0.374554\pi$
$571$	18.3508	0.767957	0.383978	+	0.923342i	$0.374554\pi$
$572$	0	0
$573$	−11.0864	−0.463142
$574$	5.56769	0.232391
$575$	0	0
$576$	−2.83250	−0.118021
$577$	−19.9317	−0.829770	−0.414885	−	0.909874i	$-0.636178\pi$
$577$	−19.9317	−0.829770	−0.414885	+	0.909874i	$0.636178\pi$
$578$	8.10109	0.336961
$579$	1.19205	0.0495399
$580$	0	0
$581$	0.827147	0.0343158
$582$	−6.97402	−0.289082
$583$	0	0
$584$	13.1204	0.542928
$585$	0	0
$586$	12.1227	0.500784
$587$	−39.5465	−1.63226	−0.816129	−	0.577870i	$-0.803884\pi$
$587$	−39.5465	−1.63226	−0.816129	+	0.577870i	$0.803884\pi$
$588$	33.6482	1.38763
$589$	9.69736	0.399573
$590$	0	0
$591$	18.3551	0.755028
$592$	−15.6837	−0.644597
$593$	12.3314	0.506391	0.253195	−	0.967415i	$-0.418518\pi$
$593$	12.3314	0.506391	0.253195	+	0.967415i	$0.418518\pi$
$594$	0	0
$595$	0	0
$596$	13.1364	0.538088
$597$	−22.1761	−0.907607
$598$	−3.88748	−0.158971
$599$	34.7495	1.41983	0.709913	−	0.704289i	$-0.248734\pi$
$599$	34.7495	1.41983	0.709913	+	0.704289i	$0.248734\pi$
$600$	0	0
$601$	−38.6097	−1.57492	−0.787462	−	0.616364i	$-0.788605\pi$
$601$	−38.6097	−1.57492	−0.787462	+	0.616364i	$0.788605\pi$
$602$	−8.43376	−0.343735
$603$	7.25598	0.295486
$604$	−8.73569	−0.355450
$605$	0	0
$606$	2.09524	0.0851132
$607$	−6.26341	−0.254224	−0.127112	−	0.991888i	$-0.540571\pi$
$607$	−6.26341	−0.254224	−0.127112	+	0.991888i	$0.540571\pi$
$608$	−21.7955	−0.883926
$609$	−38.6246	−1.56515
$610$	0	0
$611$	−2.75534	−0.111469
$612$	1.12734	0.0455700
$613$	−18.9911	−0.767042	−0.383521	−	0.923532i	$-0.625289\pi$
$613$	−18.9911	−0.767042	−0.383521	+	0.923532i	$0.625289\pi$
$614$	6.08844	0.245709
$615$	0	0
$616$	0	0
$617$	−7.58574	−0.305390	−0.152695	−	0.988273i	$-0.548795\pi$
$617$	−7.58574	−0.305390	−0.152695	+	0.988273i	$0.548795\pi$
$618$	−4.63787	−0.186563
$619$	27.9795	1.12459	0.562296	−	0.826936i	$-0.309918\pi$
$619$	27.9795	1.12459	0.562296	+	0.826936i	$0.309918\pi$
$620$	0	0
$621$	4.37977	0.175754
$622$	1.54260	0.0618528
$623$	6.10717	0.244679
$624$	4.77381	0.191105
$625$	0	0
$626$	−6.98012	−0.278982
$627$	0	0
$628$	−36.2734	−1.44746
$629$	3.82183	0.152387
$630$	0	0
$631$	−25.4321	−1.01244	−0.506218	−	0.862406i	$-0.668957\pi$
$631$	−25.4321	−1.01244	−0.506218	+	0.862406i	$0.668957\pi$
$632$	−3.49933	−0.139196
$633$	8.30729	0.330185
$634$	−13.1240	−0.521223
$635$	0	0
$636$	16.1658	0.641015
$637$	−34.7212	−1.37570
$638$	0	0
$639$	5.50864	0.217919
$640$	0	0
$641$	−8.90150	−0.351588	−0.175794	−	0.984427i	$-0.556249\pi$
$641$	−8.90150	−0.351588	−0.175794	+	0.984427i	$0.556249\pi$
$642$	4.04052	0.159467
$643$	19.4814	0.768272	0.384136	−	0.923276i	$-0.374499\pi$
$643$	19.4814	0.768272	0.384136	+	0.923276i	$0.374499\pi$
$644$	−39.4168	−1.55324
$645$	0	0
$646$	1.37431	0.0540716
$647$	−32.8093	−1.28987	−0.644934	−	0.764238i	$-0.723115\pi$
$647$	−32.8093	−1.28987	−0.644934	+	0.764238i	$0.723115\pi$
$648$	1.83677	0.0721550
$649$	0	0
$650$	0	0
$651$	11.2653	0.441521
$652$	12.7321	0.498629
$653$	−14.9010	−0.583122	−0.291561	−	0.956552i	$-0.594175\pi$
$653$	−14.9010	−0.583122	−0.291561	+	0.956552i	$0.594175\pi$
$654$	6.10276	0.238637
$655$	0	0
$656$	5.86128	0.228844
$657$	7.14322	0.278684
$658$	3.78147	0.147417
$659$	25.7059	1.00136	0.500680	−	0.865633i	$-0.333083\pi$
$659$	25.7059	1.00136	0.500680	+	0.865633i	$0.333083\pi$
$660$	0	0
$661$	−12.4887	−0.485753	−0.242876	−	0.970057i	$-0.578091\pi$
$661$	−12.4887	−0.485753	−0.242876	+	0.970057i	$0.578091\pi$
$662$	−9.65675	−0.375320
$663$	−1.16329	−0.0451784
$664$	−0.297376	−0.0115404
$665$	0	0
$666$	2.91609	0.112996
$667$	33.1119	1.28210
$668$	−32.2595	−1.24816
$669$	16.5497	0.639847
$670$	0	0
$671$	0	0
$672$	−25.3195	−0.976723
$673$	40.3440	1.55515	0.777573	−	0.628793i	$-0.216451\pi$
$673$	40.3440	1.55515	0.777573	+	0.628793i	$0.216451\pi$
$674$	−6.96219	−0.268173
$675$	0	0
$676$	17.0798	0.656916
$677$	−5.64803	−0.217071	−0.108536	−	0.994093i	$-0.534616\pi$
$677$	−5.64803	−0.217071	−0.108536	+	0.994093i	$0.534616\pi$
$678$	7.91513	0.303979
$679$	72.9674	2.80023
$680$	0	0
$681$	8.48300	0.325069
$682$	0	0
$683$	−26.7349	−1.02298	−0.511492	−	0.859288i	$-0.670907\pi$
$683$	−26.7349	−1.02298	−0.511492	+	0.859288i	$0.670907\pi$
$684$	−7.74714	−0.296219
$685$	0	0
$686$	30.1891	1.15263
$687$	3.39892	0.129677
$688$	−8.87848	−0.338489
$689$	−16.6813	−0.635506
$690$	0	0
$691$	39.2696	1.49389	0.746943	−	0.664888i	$-0.231521\pi$
$691$	39.2696	1.49389	0.746943	+	0.664888i	$0.231521\pi$
$692$	21.1664	0.804626
$693$	0	0
$694$	−7.71286	−0.292776
$695$	0	0
$696$	13.8863	0.526359
$697$	−1.42829	−0.0541002
$698$	0.837147	0.0316865
$699$	15.7123	0.594295
$700$	0	0
$701$	7.23493	0.273259	0.136630	−	0.990622i	$-0.456373\pi$
$701$	7.23493	0.273259	0.136630	+	0.990622i	$0.456373\pi$
$702$	−0.887599	−0.0335003
$703$	−26.2638	−0.990559
$704$	0	0
$705$	0	0
$706$	6.81716	0.256567
$707$	−21.9219	−0.824459
$708$	9.46267	0.355629
$709$	−9.96887	−0.374389	−0.187194	−	0.982323i	$-0.559939\pi$
$709$	−9.96887	−0.374389	−0.187194	+	0.982323i	$0.559939\pi$
$710$	0	0
$711$	−1.90516	−0.0714490
$712$	−2.19565	−0.0822854
$713$	−9.65745	−0.361674
$714$	1.59652	0.0597482
$715$	0	0
$716$	11.8202	0.441740
$717$	30.6972	1.14641
$718$	−16.7517	−0.625167
$719$	33.1006	1.23445	0.617223	−	0.786788i	$-0.288258\pi$
$719$	33.1006	1.23445	0.617223	+	0.786788i	$0.288258\pi$
$720$	0	0
$721$	48.5249	1.80716
$722$	−0.166662	−0.00620253
$723$	5.59097	0.207930
$724$	19.9669	0.742066
$725$	0	0
$726$	0	0
$727$	18.7123	0.694000	0.347000	−	0.937865i	$-0.387200\pi$
$727$	18.7123	0.694000	0.347000	+	0.937865i	$0.387200\pi$
$728$	17.0576	0.632195
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.16352	0.0800207
$732$	−12.8025	−0.473193
$733$	10.9499	0.404442	0.202221	−	0.979340i	$-0.435184\pi$
$733$	10.9499	0.404442	0.202221	+	0.979340i	$0.435184\pi$
$734$	1.76396	0.0651089
$735$	0	0
$736$	21.7058	0.800087
$737$	0	0
$738$	−1.08979	−0.0401158
$739$	−38.0842	−1.40095	−0.700474	−	0.713677i	$-0.747028\pi$
$739$	−38.0842	−1.40095	−0.700474	+	0.713677i	$0.747028\pi$
$740$	0	0
$741$	7.99419	0.293674
$742$	22.8937	0.840453
$743$	16.5258	0.606275	0.303137	−	0.952947i	$-0.401966\pi$
$743$	16.5258	0.606275	0.303137	+	0.952947i	$0.401966\pi$
$744$	−4.05009	−0.148484
$745$	0	0
$746$	9.13430	0.334430
$747$	−0.161902	−0.00592367
$748$	0	0
$749$	−42.2750	−1.54469
$750$	0	0
$751$	−17.9317	−0.654336	−0.327168	−	0.944966i	$-0.606094\pi$
$751$	−17.9317	−0.654336	−0.327168	+	0.944966i	$0.606094\pi$
$752$	3.98087	0.145167
$753$	−18.7237	−0.682329
$754$	−6.71041	−0.244379
$755$	0	0
$756$	−8.99974	−0.327317
$757$	14.0679	0.511307	0.255653	−	0.966768i	$-0.417709\pi$
$757$	14.0679	0.511307	0.255653	+	0.966768i	$0.417709\pi$
$758$	4.57764	0.166267
$759$	0	0
$760$	0	0
$761$	−10.1182	−0.366783	−0.183392	−	0.983040i	$-0.558708\pi$
$761$	−10.1182	−0.366783	−0.183392	+	0.983040i	$0.558708\pi$
$762$	4.82481	0.174785
$763$	−63.8516	−2.31158
$764$	−19.5295	−0.706551
$765$	0	0
$766$	−9.38289	−0.339018
$767$	−9.76442	−0.352573
$768$	−0.149694	−0.00540160
$769$	16.0423	0.578499	0.289250	−	0.957254i	$-0.406594\pi$
$769$	16.0423	0.578499	0.289250	+	0.957254i	$0.406594\pi$
$770$	0	0
$771$	−4.98215	−0.179428
$772$	2.09987	0.0755760
$773$	40.4357	1.45437	0.727185	−	0.686442i	$-0.240828\pi$
$773$	40.4357	1.45437	0.727185	+	0.686442i	$0.240828\pi$
$774$	1.65078	0.0593362
$775$	0	0
$776$	−26.2332	−0.941718
$777$	−30.5103	−1.09455
$778$	4.53971	0.162757
$779$	9.81525	0.351668
$780$	0	0
$781$	0	0
$782$	−1.36866	−0.0489430
$783$	7.56018	0.270179
$784$	50.1646	1.79159
$785$	0	0
$786$	6.24190	0.222641
$787$	−16.4314	−0.585716	−0.292858	−	0.956156i	$-0.594606\pi$
$787$	−16.4314	−0.585716	−0.292858	+	0.956156i	$0.594606\pi$
$788$	32.3337	1.15184
$789$	20.0788	0.714824
$790$	0	0
$791$	−82.8140	−2.94453
$792$	0	0
$793$	13.2107	0.469127
$794$	6.77235	0.240342
$795$	0	0
$796$	−39.0646	−1.38461
$797$	11.8928	0.421266	0.210633	−	0.977565i	$-0.432448\pi$
$797$	11.8928	0.421266	0.210633	+	0.977565i	$0.432448\pi$
$798$	−10.9713	−0.388381
$799$	−0.970064	−0.0343184
$800$	0	0
$801$	−1.19539	−0.0422370
$802$	−5.19236	−0.183349
$803$	0	0
$804$	12.7819	0.450782
$805$	0	0
$806$	1.95717	0.0689382
$807$	9.53027	0.335481
$808$	7.88137	0.277266
$809$	−39.7032	−1.39589	−0.697944	−	0.716152i	$-0.745902\pi$
$809$	−39.7032	−1.39589	−0.697944	+	0.716152i	$0.745902\pi$
$810$	0	0
$811$	−29.1509	−1.02363	−0.511814	−	0.859096i	$-0.671026\pi$
$811$	−29.1509	−1.02363	−0.511814	+	0.859096i	$0.671026\pi$
$812$	−68.0396	−2.38772
$813$	−1.18095	−0.0414177
$814$	0	0
$815$	0	0
$816$	1.68070	0.0588363
$817$	−14.8678	−0.520159
$818$	1.81315	0.0633954
$819$	9.28673	0.324505
$820$	0	0
$821$	−47.6040	−1.66139	−0.830696	−	0.556727i	$-0.812057\pi$
$821$	−47.6040	−1.66139	−0.830696	+	0.556727i	$0.812057\pi$
$822$	9.30645	0.324600
$823$	19.2238	0.670099	0.335049	−	0.942201i	$-0.391247\pi$
$823$	19.2238	0.670099	0.335049	+	0.942201i	$0.391247\pi$
$824$	−17.4457	−0.607748
$825$	0	0
$826$	13.4008	0.466275
$827$	−30.6312	−1.06515	−0.532576	−	0.846382i	$-0.678776\pi$
$827$	−30.6312	−1.06515	−0.532576	+	0.846382i	$0.678776\pi$
$828$	7.71525	0.268124
$829$	−44.8613	−1.55810	−0.779048	−	0.626964i	$-0.784297\pi$
$829$	−44.8613	−1.55810	−0.779048	+	0.626964i	$0.784297\pi$
$830$	0	0
$831$	8.22177	0.285210
$832$	5.14875	0.178501
$833$	−12.2242	−0.423543
$834$	5.72024	0.198076
$835$	0	0
$836$	0	0
$837$	−2.20501	−0.0762163
$838$	7.76374	0.268194
$839$	4.34435	0.149984	0.0749919	−	0.997184i	$-0.476107\pi$
$839$	4.34435	0.149984	0.0749919	+	0.997184i	$0.476107\pi$
$840$	0	0
$841$	28.1563	0.970908
$842$	−2.13222	−0.0734813
$843$	10.4453	0.359756
$844$	14.6338	0.503717
$845$	0	0
$846$	−0.740167	−0.0254475
$847$	0	0
$848$	24.1008	0.827626
$849$	24.8463	0.852724
$850$	0	0
$851$	26.1557	0.896607
$852$	9.70383	0.332448
$853$	−40.3581	−1.38183	−0.690917	−	0.722934i	$-0.742793\pi$
$853$	−40.3581	−1.38183	−0.690917	+	0.722934i	$0.742793\pi$
$854$	−18.1306	−0.620417
$855$	0	0
$856$	15.1987	0.519481
$857$	36.6122	1.25065	0.625324	−	0.780365i	$-0.284967\pi$
$857$	36.6122	1.25065	0.625324	+	0.780365i	$0.284967\pi$
$858$	0	0
$859$	28.5710	0.974831	0.487415	−	0.873170i	$-0.337940\pi$
$859$	28.5710	0.974831	0.487415	+	0.873170i	$0.337940\pi$
$860$	0	0
$861$	11.4022	0.388587
$862$	9.81507	0.334303
$863$	29.3094	0.997704	0.498852	−	0.866687i	$-0.333755\pi$
$863$	29.3094	0.997704	0.498852	+	0.866687i	$0.333755\pi$
$864$	4.95592	0.168604
$865$	0	0
$866$	−15.0021	−0.509791
$867$	16.5904	0.563441
$868$	19.8445	0.673567
$869$	0	0
$870$	0	0
$871$	−13.1895	−0.446908
$872$	22.9559	0.777386
$873$	−14.2823	−0.483382
$874$	9.40547	0.318145
$875$	0	0
$876$	12.5832	0.425149
$877$	−3.67475	−0.124087	−0.0620437	−	0.998073i	$-0.519762\pi$
$877$	−3.67475	−0.124087	−0.0620437	+	0.998073i	$0.519762\pi$
$878$	−10.0640	−0.339644
$879$	24.8264	0.837373
$880$	0	0
$881$	−11.1723	−0.376403	−0.188202	−	0.982130i	$-0.560266\pi$
$881$	−11.1723	−0.376403	−0.188202	+	0.982130i	$0.560266\pi$
$882$	−9.32716	−0.314062
$883$	42.2570	1.42206	0.711030	−	0.703161i	$-0.248229\pi$
$883$	42.2570	1.42206	0.711030	+	0.703161i	$0.248229\pi$
$884$	−2.04921	−0.0689224
$885$	0	0
$886$	6.73838	0.226380
$887$	−55.8459	−1.87512	−0.937562	−	0.347819i	$-0.886922\pi$
$887$	−55.8459	−1.87512	−0.937562	+	0.347819i	$0.886922\pi$
$888$	10.9691	0.368097
$889$	−50.4808	−1.69307
$890$	0	0
$891$	0	0
$892$	29.1533	0.976125
$893$	6.66633	0.223080
$894$	−3.64136	−0.121785
$895$	0	0
$896$	−57.7053	−1.92780
$897$	−7.96128	−0.265819
$898$	−13.3066	−0.444045
$899$	−16.6703	−0.555985
$900$	0	0
$901$	−5.87293	−0.195656
$902$	0	0
$903$	−17.2717	−0.574767
$904$	29.7733	0.990244
$905$	0	0
$906$	2.42150	0.0804490
$907$	12.0875	0.401358	0.200679	−	0.979657i	$-0.435685\pi$
$907$	12.0875	0.401358	0.200679	+	0.979657i	$0.435685\pi$
$908$	14.9433	0.495912
$909$	4.29089	0.142320
$910$	0	0
$911$	−40.2306	−1.33290	−0.666450	−	0.745550i	$-0.732187\pi$
$911$	−40.2306	−1.33290	−0.666450	+	0.745550i	$0.732187\pi$
$912$	−11.5499	−0.382454
$913$	0	0
$914$	−2.25719	−0.0746611
$915$	0	0
$916$	5.98741	0.197830
$917$	−65.3074	−2.15664
$918$	−0.312494	−0.0103139
$919$	−3.65013	−0.120407	−0.0602033	−	0.998186i	$-0.519175\pi$
$919$	−3.65013	−0.120407	−0.0602033	+	0.998186i	$0.519175\pi$
$920$	0	0
$921$	12.4687	0.410857
$922$	−12.3999	−0.408368
$923$	−10.0133	−0.329591
$924$	0	0
$925$	0	0
$926$	−6.29655	−0.206917
$927$	−9.49802	−0.311956
$928$	37.4677	1.22994
$929$	10.0510	0.329762	0.164881	−	0.986313i	$-0.447276\pi$
$929$	10.0510	0.329762	0.164881	+	0.986313i	$0.447276\pi$
$930$	0	0
$931$	84.0053	2.75316
$932$	27.6783	0.906632
$933$	3.15914	0.103426
$934$	−1.86054	−0.0608788
$935$	0	0
$936$	−3.33876	−0.109131
$937$	−28.2337	−0.922355	−0.461177	−	0.887308i	$-0.652573\pi$
$937$	−28.2337	−0.922355	−0.461177	+	0.887308i	$0.652573\pi$
$938$	18.1014	0.591033
$939$	−14.2948	−0.466492
$940$	0	0
$941$	−32.6668	−1.06491	−0.532453	−	0.846459i	$-0.678730\pi$
$941$	−32.6668	−1.06491	−0.532453	+	0.846459i	$0.678730\pi$
$942$	10.0548	0.327604
$943$	−9.77485	−0.318313
$944$	14.1075	0.459159
$945$	0	0
$946$	0	0
$947$	57.1073	1.85574	0.927869	−	0.372906i	$-0.121639\pi$
$947$	57.1073	1.85574	0.927869	+	0.372906i	$0.121639\pi$
$948$	−3.35606	−0.109000
$949$	−12.9845	−0.421495
$950$	0	0
$951$	−26.8771	−0.871549
$952$	6.00540	0.194636
$953$	−4.51422	−0.146230	−0.0731150	−	0.997324i	$-0.523294\pi$
$953$	−4.51422	−0.146230	−0.0731150	+	0.997324i	$0.523294\pi$
$954$	−4.48109	−0.145081
$955$	0	0
$956$	54.0751	1.74891
$957$	0	0
$958$	6.00205	0.193917
$959$	−97.3710	−3.14427
$960$	0	0
$961$	−26.1379	−0.843159
$962$	−5.30069	−0.170901
$963$	8.27470	0.266648
$964$	9.84886	0.317210
$965$	0	0
$966$	10.9262	0.351544
$967$	1.33608	0.0429654	0.0214827	−	0.999769i	$-0.493161\pi$
$967$	1.33608	0.0429654	0.0214827	+	0.999769i	$0.493161\pi$
$968$	0	0
$969$	2.81449	0.0904144
$970$	0	0
$971$	2.71644	0.0871747	0.0435873	−	0.999050i	$-0.486121\pi$
$971$	2.71644	0.0871747	0.0435873	+	0.999050i	$0.486121\pi$
$972$	1.76156	0.0565022
$973$	−59.8494	−1.91868
$974$	−6.56221	−0.210267
$975$	0	0
$976$	−19.0867	−0.610949
$977$	16.6829	0.533733	0.266866	−	0.963734i	$-0.414012\pi$
$977$	16.6829	0.533733	0.266866	+	0.963734i	$0.414012\pi$
$978$	−3.52930	−0.112854
$979$	0	0
$980$	0	0
$981$	12.4980	0.399031
$982$	−2.96975	−0.0947684
$983$	−26.2517	−0.837298	−0.418649	−	0.908148i	$-0.637496\pi$
$983$	−26.2517	−0.837298	−0.418649	+	0.908148i	$0.637496\pi$
$984$	−4.09933	−0.130682
$985$	0	0
$986$	−2.36251	−0.0752378
$987$	7.74418	0.246500
$988$	14.0823	0.448017
$989$	14.8066	0.470823
$990$	0	0
$991$	−50.3117	−1.59821	−0.799103	−	0.601195i	$-0.794692\pi$
$991$	−50.3117	−1.59821	−0.799103	+	0.601195i	$0.794692\pi$
$992$	−10.9279	−0.346960
$993$	−19.7763	−0.627583
$994$	13.7424	0.435881
$995$	0	0
$996$	−0.285200	−0.00903691
$997$	−45.2449	−1.43292	−0.716460	−	0.697628i	$-0.754239\pi$
$997$	−45.2449	−1.43292	−0.716460	+	0.697628i	$0.754239\pi$
$998$	−9.17719	−0.290499
$999$	5.97194	0.188944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.dx.1.6		12
5.2	odd	4		1815.2.c.k.364.10		24
5.3	odd	4		1815.2.c.k.364.15		24
5.4	even	2		9075.2.a.ea.1.7		12
11.7	odd	10		825.2.n.o.676.4		24
11.8	odd	10		825.2.n.o.526.4		24
11.10	odd	2		9075.2.a.dz.1.7		12
55.7	even	20		165.2.s.a.49.5	✓	48
55.8	even	20		165.2.s.a.64.5	yes	48
55.18	even	20		165.2.s.a.49.8	yes	48
55.19	odd	10		825.2.n.p.526.3		24
55.29	odd	10		825.2.n.p.676.3		24
55.32	even	4		1815.2.c.j.364.15		24
55.43	even	4		1815.2.c.j.364.10		24
55.52	even	20		165.2.s.a.64.8	yes	48
55.54	odd	2		9075.2.a.dy.1.6		12
165.8	odd	20		495.2.ba.c.64.8		48
165.62	odd	20		495.2.ba.c.379.8		48
165.107	odd	20		495.2.ba.c.64.5		48
165.128	odd	20		495.2.ba.c.379.5		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.s.a.49.5	✓	48	55.7	even	20
165.2.s.a.49.8	yes	48	55.18	even	20
165.2.s.a.64.5	yes	48	55.8	even	20
165.2.s.a.64.8	yes	48	55.52	even	20
495.2.ba.c.64.5		48	165.107	odd	20
495.2.ba.c.64.8		48	165.8	odd	20
495.2.ba.c.379.5		48	165.128	odd	20
495.2.ba.c.379.8		48	165.62	odd	20
825.2.n.o.526.4		24	11.8	odd	10
825.2.n.o.676.4		24	11.7	odd	10
825.2.n.p.526.3		24	55.19	odd	10
825.2.n.p.676.3		24	55.29	odd	10
1815.2.c.j.364.10		24	55.43	even	4
1815.2.c.j.364.15		24	55.32	even	4
1815.2.c.k.364.10		24	5.2	odd	4
1815.2.c.k.364.15		24	5.3	odd	4
9075.2.a.dx.1.6		12	1.1	even	1	trivial
9075.2.a.dy.1.6		12	55.54	odd	2
9075.2.a.dz.1.7		12	11.10	odd	2
9075.2.a.ea.1.7		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)

\(f(q)\)	\(=\)	\(q-0.488299 q^{2} -1.00000 q^{3} -1.76156 q^{4} +0.488299 q^{6} -5.10895 q^{7} +1.83677 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.488299 q^{2} -1.00000 q^{3} -1.76156 q^{4} +0.488299 q^{6} -5.10895 q^{7} +1.83677 q^{8} +1.00000 q^{9} +1.76156 q^{12} -1.81774 q^{13} +2.49469 q^{14} +2.62624 q^{16} -0.639966 q^{17} -0.488299 q^{18} +4.39788 q^{19} +5.10895 q^{21} -4.37977 q^{23} -1.83677 q^{24} +0.887599 q^{26} -1.00000 q^{27} +8.99974 q^{28} -7.56018 q^{29} +2.20501 q^{31} -4.95592 q^{32} +0.312494 q^{34} -1.76156 q^{36} -5.97194 q^{37} -2.14748 q^{38} +1.81774 q^{39} +2.23182 q^{41} -2.49469 q^{42} -3.38068 q^{43} +2.13864 q^{46} +1.51581 q^{47} -2.62624 q^{48} +19.1013 q^{49} +0.639966 q^{51} +3.20206 q^{52} +9.17695 q^{53} +0.488299 q^{54} -9.38395 q^{56} -4.39788 q^{57} +3.69163 q^{58} +5.37174 q^{59} -7.26768 q^{61} -1.07670 q^{62} -5.10895 q^{63} -2.83250 q^{64} +7.25598 q^{67} +1.12734 q^{68} +4.37977 q^{69} +5.50864 q^{71} +1.83677 q^{72} +7.14322 q^{73} +2.91609 q^{74} -7.74714 q^{76} -0.887599 q^{78} -1.90516 q^{79} +1.00000 q^{81} -1.08979 q^{82} -0.161902 q^{83} -8.99974 q^{84} +1.65078 q^{86} +7.56018 q^{87} -1.19539 q^{89} +9.28673 q^{91} +7.71525 q^{92} -2.20501 q^{93} -0.740167 q^{94} +4.95592 q^{96} -14.2823 q^{97} -9.32716 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} - 12 q^{3} + 12 q^{4} + 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 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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