LMFDB - Embedded newform 9075.2.a.be.1.2

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:	$2$
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:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +1.61803 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +1.61803 q^{7} -2.23607 q^{8} +1.00000 q^{9} -1.61803 q^{12} +2.61803 q^{13} +1.00000 q^{14} +1.85410 q^{16} -4.23607 q^{17} +0.618034 q^{18} -0.236068 q^{19} +1.61803 q^{21} -3.85410 q^{23} -2.23607 q^{24} +1.61803 q^{26} +1.00000 q^{27} -2.61803 q^{28} -4.23607 q^{29} -7.85410 q^{31} +5.61803 q^{32} -2.61803 q^{34} -1.61803 q^{36} +8.94427 q^{37} -0.145898 q^{38} +2.61803 q^{39} +6.09017 q^{41} +1.00000 q^{42} -4.47214 q^{43} -2.38197 q^{46} +1.70820 q^{47} +1.85410 q^{48} -4.38197 q^{49} -4.23607 q^{51} -4.23607 q^{52} -11.0902 q^{53} +0.618034 q^{54} -3.61803 q^{56} -0.236068 q^{57} -2.61803 q^{58} +8.70820 q^{59} -0.763932 q^{61} -4.85410 q^{62} +1.61803 q^{63} -0.236068 q^{64} +0.763932 q^{67} +6.85410 q^{68} -3.85410 q^{69} -1.76393 q^{71} -2.23607 q^{72} -3.38197 q^{73} +5.52786 q^{74} +0.381966 q^{76} +1.61803 q^{78} -9.70820 q^{79} +1.00000 q^{81} +3.76393 q^{82} +5.85410 q^{83} -2.61803 q^{84} -2.76393 q^{86} -4.23607 q^{87} +1.32624 q^{89} +4.23607 q^{91} +6.23607 q^{92} -7.85410 q^{93} +1.05573 q^{94} +5.61803 q^{96} -8.18034 q^{97} -2.70820 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 + 2 * q^3 - q^4 - q^6 + q^7 + 2 * q^9	$ 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} + q^{7} + 2 q^{9} - q^{12} + 3 q^{13} + 2 q^{14} - 3 q^{16} - 4 q^{17} - q^{18} + 4 q^{19} + q^{21} - q^{23} + q^{26} + 2 q^{27} - 3 q^{28} - 4 q^{29} - 9 q^{31} + 9 q^{32} - 3 q^{34} - q^{36} - 7 q^{38} + 3 q^{39} + q^{41} + 2 q^{42} - 7 q^{46} - 10 q^{47} - 3 q^{48} - 11 q^{49} - 4 q^{51} - 4 q^{52} - 11 q^{53} - q^{54} - 5 q^{56} + 4 q^{57} - 3 q^{58} + 4 q^{59} - 6 q^{61} - 3 q^{62} + q^{63} + 4 q^{64} + 6 q^{67} + 7 q^{68} - q^{69} - 8 q^{71} - 9 q^{73} + 20 q^{74} + 3 q^{76} + q^{78} - 6 q^{79} + 2 q^{81} + 12 q^{82} + 5 q^{83} - 3 q^{84} - 10 q^{86} - 4 q^{87} - 13 q^{89} + 4 q^{91} + 8 q^{92} - 9 q^{93} + 20 q^{94} + 9 q^{96} + 6 q^{97} + 8 q^{98}+O(q^{100}) $ 2 * q - q^2 + 2 * q^3 - q^4 - q^6 + q^7 + 2 * q^9 - q^12 + 3 * q^13 + 2 * q^14 - 3 * q^16 - 4 * q^17 - q^18 + 4 * q^19 + q^21 - q^23 + q^26 + 2 * q^27 - 3 * q^28 - 4 * q^29 - 9 * q^31 + 9 * q^32 - 3 * q^34 - q^36 - 7 * q^38 + 3 * q^39 + q^41 + 2 * q^42 - 7 * q^46 - 10 * q^47 - 3 * q^48 - 11 * q^49 - 4 * q^51 - 4 * q^52 - 11 * q^53 - q^54 - 5 * q^56 + 4 * q^57 - 3 * q^58 + 4 * q^59 - 6 * q^61 - 3 * q^62 + q^63 + 4 * q^64 + 6 * q^67 + 7 * q^68 - q^69 - 8 * q^71 - 9 * q^73 + 20 * q^74 + 3 * q^76 + q^78 - 6 * q^79 + 2 * q^81 + 12 * q^82 + 5 * q^83 - 3 * q^84 - 10 * q^86 - 4 * q^87 - 13 * q^89 + 4 * q^91 + 8 * q^92 - 9 * q^93 + 20 * q^94 + 9 * q^96 + 6 * q^97 + 8 * 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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	0.618034	0.252311
$7$	1.61803	0.611559	0.305780	−	0.952102i	$-0.401083\pi$
$7$	1.61803	0.611559	0.305780	+	0.952102i	$0.401083\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−1.61803	−0.467086
$13$	2.61803	0.726112	0.363056	−	0.931767i	$-0.381733\pi$
$13$	2.61803	0.726112	0.363056	+	0.931767i	$0.381733\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.85410	0.463525
$17$	−4.23607	−1.02740	−0.513699	−	0.857971i	$-0.671725\pi$
$17$	−4.23607	−1.02740	−0.513699	+	0.857971i	$0.671725\pi$
$18$	0.618034	0.145672
$19$	−0.236068	−0.0541577	−0.0270789	−	0.999633i	$-0.508621\pi$
$19$	−0.236068	−0.0541577	−0.0270789	+	0.999633i	$0.508621\pi$
$20$	0	0
$21$	1.61803	0.353084
$22$	0	0
$23$	−3.85410	−0.803636	−0.401818	−	0.915720i	$-0.631622\pi$
$23$	−3.85410	−0.803636	−0.401818	+	0.915720i	$0.631622\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	1.61803	0.317323
$27$	1.00000	0.192450
$28$	−2.61803	−0.494762
$29$	−4.23607	−0.786618	−0.393309	−	0.919406i	$-0.628670\pi$
$29$	−4.23607	−0.786618	−0.393309	+	0.919406i	$0.628670\pi$
$30$	0	0
$31$	−7.85410	−1.41064	−0.705319	−	0.708890i	$-0.749196\pi$
$31$	−7.85410	−1.41064	−0.705319	+	0.708890i	$0.749196\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	−2.61803	−0.448989
$35$	0	0
$36$	−1.61803	−0.269672
$37$	8.94427	1.47043	0.735215	−	0.677834i	$-0.237081\pi$
$37$	8.94427	1.47043	0.735215	+	0.677834i	$0.237081\pi$
$38$	−0.145898	−0.0236678
$39$	2.61803	0.419221
$40$	0	0
$41$	6.09017	0.951125	0.475562	−	0.879682i	$-0.342245\pi$
$41$	6.09017	0.951125	0.475562	+	0.879682i	$0.342245\pi$
$42$	1.00000	0.154303
$43$	−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
$46$	−2.38197	−0.351202
$47$	1.70820	0.249167	0.124584	−	0.992209i	$-0.460241\pi$
$47$	1.70820	0.249167	0.124584	+	0.992209i	$0.460241\pi$
$48$	1.85410	0.267617
$49$	−4.38197	−0.625995
$50$	0	0
$51$	−4.23607	−0.593168
$52$	−4.23607	−0.587437
$53$	−11.0902	−1.52335	−0.761676	−	0.647958i	$-0.775623\pi$
$53$	−11.0902	−1.52335	−0.761676	+	0.647958i	$0.775623\pi$
$54$	0.618034	0.0841038
$55$	0	0
$56$	−3.61803	−0.483480
$57$	−0.236068	−0.0312680
$58$	−2.61803	−0.343765
$59$	8.70820	1.13371	0.566856	−	0.823817i	$-0.308160\pi$
$59$	8.70820	1.13371	0.566856	+	0.823817i	$0.308160\pi$
$60$	0	0
$61$	−0.763932	−0.0978115	−0.0489057	−	0.998803i	$-0.515573\pi$
$61$	−0.763932	−0.0978115	−0.0489057	+	0.998803i	$0.515573\pi$
$62$	−4.85410	−0.616472
$63$	1.61803	0.203853
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0	0
$67$	0.763932	0.0933292	0.0466646	−	0.998911i	$-0.485141\pi$
$67$	0.763932	0.0933292	0.0466646	+	0.998911i	$0.485141\pi$
$68$	6.85410	0.831182
$69$	−3.85410	−0.463979
$70$	0	0
$71$	−1.76393	−0.209340	−0.104670	−	0.994507i	$-0.533379\pi$
$71$	−1.76393	−0.209340	−0.104670	+	0.994507i	$0.533379\pi$
$72$	−2.23607	−0.263523
$73$	−3.38197	−0.395829	−0.197915	−	0.980219i	$-0.563417\pi$
$73$	−3.38197	−0.395829	−0.197915	+	0.980219i	$0.563417\pi$
$74$	5.52786	0.642601
$75$	0	0
$76$	0.381966	0.0438145
$77$	0	0
$78$	1.61803	0.183206
$79$	−9.70820	−1.09226	−0.546129	−	0.837701i	$-0.683899\pi$
$79$	−9.70820	−1.09226	−0.546129	+	0.837701i	$0.683899\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.76393	0.415657
$83$	5.85410	0.642571	0.321286	−	0.946982i	$-0.395885\pi$
$83$	5.85410	0.642571	0.321286	+	0.946982i	$0.395885\pi$
$84$	−2.61803	−0.285651
$85$	0	0
$86$	−2.76393	−0.298042
$87$	−4.23607	−0.454154
$88$	0	0
$89$	1.32624	0.140581	0.0702905	−	0.997527i	$-0.477607\pi$
$89$	1.32624	0.140581	0.0702905	+	0.997527i	$0.477607\pi$
$90$	0	0
$91$	4.23607	0.444061
$92$	6.23607	0.650155
$93$	−7.85410	−0.814432
$94$	1.05573	0.108890
$95$	0	0
$96$	5.61803	0.573388
$97$	−8.18034	−0.830588	−0.415294	−	0.909687i	$-0.636321\pi$
$97$	−8.18034	−0.830588	−0.415294	+	0.909687i	$0.636321\pi$
$98$	−2.70820	−0.273570
$99$	0	0
$100$	0	0
$101$	0.909830	0.0905315	0.0452657	−	0.998975i	$-0.485587\pi$
$101$	0.909830	0.0905315	0.0452657	+	0.998975i	$0.485587\pi$
$102$	−2.61803	−0.259224
$103$	−12.4721	−1.22892	−0.614458	−	0.788950i	$-0.710625\pi$
$103$	−12.4721	−1.22892	−0.614458	+	0.788950i	$0.710625\pi$
$104$	−5.85410	−0.574042
$105$	0	0
$106$	−6.85410	−0.665729
$107$	−7.79837	−0.753897	−0.376949	−	0.926234i	$-0.623027\pi$
$107$	−7.79837	−0.753897	−0.376949	+	0.926234i	$0.623027\pi$
$108$	−1.61803	−0.155695
$109$	−17.1803	−1.64558	−0.822789	−	0.568347i	$-0.807583\pi$
$109$	−17.1803	−1.64558	−0.822789	+	0.568347i	$0.807583\pi$
$110$	0	0
$111$	8.94427	0.848953
$112$	3.00000	0.283473
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	−0.145898	−0.0136646
$115$	0	0
$116$	6.85410	0.636387
$117$	2.61803	0.242037
$118$	5.38197	0.495450
$119$	−6.85410	−0.628314
$120$	0	0
$121$	0	0
$122$	−0.472136	−0.0427452
$123$	6.09017	0.549132
$124$	12.7082	1.14123
$125$	0	0
$126$	1.00000	0.0890871
$127$	−9.00000	−0.798621	−0.399310	−	0.916816i	$-0.630750\pi$
$127$	−9.00000	−0.798621	−0.399310	+	0.916816i	$0.630750\pi$
$128$	−11.3820	−1.00603
$129$	−4.47214	−0.393750
$130$	0	0
$131$	21.5066	1.87904	0.939519	−	0.342496i	$-0.111272\pi$
$131$	21.5066	1.87904	0.939519	+	0.342496i	$0.111272\pi$
$132$	0	0
$133$	−0.381966	−0.0331207
$134$	0.472136	0.0407863
$135$	0	0
$136$	9.47214	0.812229
$137$	9.52786	0.814020	0.407010	−	0.913424i	$-0.366571\pi$
$137$	9.52786	0.814020	0.407010	+	0.913424i	$0.366571\pi$
$138$	−2.38197	−0.202766
$139$	−15.8541	−1.34473	−0.672364	−	0.740221i	$-0.734721\pi$
$139$	−15.8541	−1.34473	−0.672364	+	0.740221i	$0.734721\pi$
$140$	0	0
$141$	1.70820	0.143857
$142$	−1.09017	−0.0914850
$143$	0	0
$144$	1.85410	0.154508
$145$	0	0
$146$	−2.09017	−0.172984
$147$	−4.38197	−0.361418
$148$	−14.4721	−1.18960
$149$	17.8541	1.46267	0.731333	−	0.682021i	$-0.238899\pi$
$149$	17.8541	1.46267	0.731333	+	0.682021i	$0.238899\pi$
$150$	0	0
$151$	14.2705	1.16132	0.580659	−	0.814147i	$-0.302795\pi$
$151$	14.2705	1.16132	0.580659	+	0.814147i	$0.302795\pi$
$152$	0.527864	0.0428154
$153$	−4.23607	−0.342466
$154$	0	0
$155$	0	0
$156$	−4.23607	−0.339157
$157$	−5.29180	−0.422331	−0.211166	−	0.977450i	$-0.567726\pi$
$157$	−5.29180	−0.422331	−0.211166	+	0.977450i	$0.567726\pi$
$158$	−6.00000	−0.477334
$159$	−11.0902	−0.879508
$160$	0	0
$161$	−6.23607	−0.491471
$162$	0.618034	0.0485573
$163$	−2.90983	−0.227915	−0.113958	−	0.993486i	$-0.536353\pi$
$163$	−2.90983	−0.227915	−0.113958	+	0.993486i	$0.536353\pi$
$164$	−9.85410	−0.769476
$165$	0	0
$166$	3.61803	0.280814
$167$	−20.0344	−1.55031	−0.775156	−	0.631770i	$-0.782329\pi$
$167$	−20.0344	−1.55031	−0.775156	+	0.631770i	$0.782329\pi$
$168$	−3.61803	−0.279137
$169$	−6.14590	−0.472761
$170$	0	0
$171$	−0.236068	−0.0180526
$172$	7.23607	0.551745
$173$	0.0557281	0.00423693	0.00211846	−	0.999998i	$-0.499326\pi$
$173$	0.0557281	0.00423693	0.00211846	+	0.999998i	$0.499326\pi$
$174$	−2.61803	−0.198473
$175$	0	0
$176$	0	0
$177$	8.70820	0.654549
$178$	0.819660	0.0614361
$179$	17.9443	1.34122	0.670609	−	0.741811i	$-0.266033\pi$
$179$	17.9443	1.34122	0.670609	+	0.741811i	$0.266033\pi$
$180$	0	0
$181$	5.41641	0.402598	0.201299	−	0.979530i	$-0.435484\pi$
$181$	5.41641	0.402598	0.201299	+	0.979530i	$0.435484\pi$
$182$	2.61803	0.194062
$183$	−0.763932	−0.0564715
$184$	8.61803	0.635330
$185$	0	0
$186$	−4.85410	−0.355920
$187$	0	0
$188$	−2.76393	−0.201580
$189$	1.61803	0.117695
$190$	0	0
$191$	−22.7984	−1.64963	−0.824816	−	0.565401i	$-0.808721\pi$
$191$	−22.7984	−1.64963	−0.824816	+	0.565401i	$0.808721\pi$
$192$	−0.236068	−0.0170367
$193$	−13.8885	−0.999719	−0.499860	−	0.866106i	$-0.666615\pi$
$193$	−13.8885	−0.999719	−0.499860	+	0.866106i	$0.666615\pi$
$194$	−5.05573	−0.362980
$195$	0	0
$196$	7.09017	0.506441
$197$	22.8541	1.62829	0.814144	−	0.580663i	$-0.197207\pi$
$197$	22.8541	1.62829	0.814144	+	0.580663i	$0.197207\pi$
$198$	0	0
$199$	−18.1803	−1.28877	−0.644385	−	0.764701i	$-0.722887\pi$
$199$	−18.1803	−1.28877	−0.644385	+	0.764701i	$0.722887\pi$
$200$	0	0
$201$	0.763932	0.0538836
$202$	0.562306	0.0395637
$203$	−6.85410	−0.481064
$204$	6.85410	0.479883
$205$	0	0
$206$	−7.70820	−0.537056
$207$	−3.85410	−0.267879
$208$	4.85410	0.336571
$209$	0	0
$210$	0	0
$211$	9.18034	0.632001	0.316000	−	0.948759i	$-0.397660\pi$
$211$	9.18034	0.632001	0.316000	+	0.948759i	$0.397660\pi$
$212$	17.9443	1.23242
$213$	−1.76393	−0.120863
$214$	−4.81966	−0.329465
$215$	0	0
$216$	−2.23607	−0.152145
$217$	−12.7082	−0.862689
$218$	−10.6180	−0.719144
$219$	−3.38197	−0.228532
$220$	0	0
$221$	−11.0902	−0.746006
$222$	5.52786	0.371006
$223$	15.6525	1.04817	0.524084	−	0.851667i	$-0.324408\pi$
$223$	15.6525	1.04817	0.524084	+	0.851667i	$0.324408\pi$
$224$	9.09017	0.607363
$225$	0	0
$226$	−2.47214	−0.164444
$227$	8.70820	0.577984	0.288992	−	0.957332i	$-0.406680\pi$
$227$	8.70820	0.577984	0.288992	+	0.957332i	$0.406680\pi$
$228$	0.381966	0.0252963
$229$	−29.9787	−1.98105	−0.990525	−	0.137336i	$-0.956146\pi$
$229$	−29.9787	−1.98105	−0.990525	+	0.137336i	$0.956146\pi$
$230$	0	0
$231$	0	0
$232$	9.47214	0.621876
$233$	−20.3262	−1.33162	−0.665808	−	0.746123i	$-0.731913\pi$
$233$	−20.3262	−1.33162	−0.665808	+	0.746123i	$0.731913\pi$
$234$	1.61803	0.105774
$235$	0	0
$236$	−14.0902	−0.917192
$237$	−9.70820	−0.630616
$238$	−4.23607	−0.274584
$239$	28.4721	1.84171	0.920855	−	0.389906i	$-0.127492\pi$
$239$	28.4721	1.84171	0.920855	+	0.389906i	$0.127492\pi$
$240$	0	0
$241$	−2.23607	−0.144038	−0.0720189	−	0.997403i	$-0.522944\pi$
$241$	−2.23607	−0.144038	−0.0720189	+	0.997403i	$0.522944\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	1.23607	0.0791311
$245$	0	0
$246$	3.76393	0.239980
$247$	−0.618034	−0.0393246
$248$	17.5623	1.11521
$249$	5.85410	0.370989
$250$	0	0
$251$	2.41641	0.152522	0.0762612	−	0.997088i	$-0.475702\pi$
$251$	2.41641	0.152522	0.0762612	+	0.997088i	$0.475702\pi$
$252$	−2.61803	−0.164921
$253$	0	0
$254$	−5.56231	−0.349010
$255$	0	0
$256$	−6.56231	−0.410144
$257$	20.0902	1.25319	0.626595	−	0.779345i	$-0.284448\pi$
$257$	20.0902	1.25319	0.626595	+	0.779345i	$0.284448\pi$
$258$	−2.76393	−0.172075
$259$	14.4721	0.899255
$260$	0	0
$261$	−4.23607	−0.262206
$262$	13.2918	0.821170
$263$	10.4164	0.642303	0.321152	−	0.947028i	$-0.395930\pi$
$263$	10.4164	0.642303	0.321152	+	0.947028i	$0.395930\pi$
$264$	0	0
$265$	0	0
$266$	−0.236068	−0.0144743
$267$	1.32624	0.0811644
$268$	−1.23607	−0.0755049
$269$	−30.7984	−1.87781	−0.938905	−	0.344176i	$-0.888158\pi$
$269$	−30.7984	−1.87781	−0.938905	+	0.344176i	$0.888158\pi$
$270$	0	0
$271$	−25.3820	−1.54184	−0.770922	−	0.636929i	$-0.780204\pi$
$271$	−25.3820	−1.54184	−0.770922	+	0.636929i	$0.780204\pi$
$272$	−7.85410	−0.476225
$273$	4.23607	0.256378
$274$	5.88854	0.355740
$275$	0	0
$276$	6.23607	0.375367
$277$	23.6525	1.42114	0.710570	−	0.703627i	$-0.248437\pi$
$277$	23.6525	1.42114	0.710570	+	0.703627i	$0.248437\pi$
$278$	−9.79837	−0.587667
$279$	−7.85410	−0.470213
$280$	0	0
$281$	−0.909830	−0.0542759	−0.0271380	−	0.999632i	$-0.508639\pi$
$281$	−0.909830	−0.0542759	−0.0271380	+	0.999632i	$0.508639\pi$
$282$	1.05573	0.0628677
$283$	5.67376	0.337270	0.168635	−	0.985679i	$-0.446064\pi$
$283$	5.67376	0.337270	0.168635	+	0.985679i	$0.446064\pi$
$284$	2.85410	0.169360
$285$	0	0
$286$	0	0
$287$	9.85410	0.581669
$288$	5.61803	0.331046
$289$	0.944272	0.0555454
$290$	0	0
$291$	−8.18034	−0.479540
$292$	5.47214	0.320233
$293$	−8.76393	−0.511994	−0.255997	−	0.966678i	$-0.582404\pi$
$293$	−8.76393	−0.511994	−0.255997	+	0.966678i	$0.582404\pi$
$294$	−2.70820	−0.157946
$295$	0	0
$296$	−20.0000	−1.16248
$297$	0	0
$298$	11.0344	0.639208
$299$	−10.0902	−0.583530
$300$	0	0
$301$	−7.23607	−0.417080
$302$	8.81966	0.507514
$303$	0.909830	0.0522684
$304$	−0.437694	−0.0251035
$305$	0	0
$306$	−2.61803	−0.149663
$307$	−2.94427	−0.168038	−0.0840192	−	0.996464i	$-0.526776\pi$
$307$	−2.94427	−0.168038	−0.0840192	+	0.996464i	$0.526776\pi$
$308$	0	0
$309$	−12.4721	−0.709515
$310$	0	0
$311$	−4.38197	−0.248478	−0.124239	−	0.992252i	$-0.539649\pi$
$311$	−4.38197	−0.248478	−0.124239	+	0.992252i	$0.539649\pi$
$312$	−5.85410	−0.331423
$313$	−34.5967	−1.95552	−0.977762	−	0.209718i	$-0.932745\pi$
$313$	−34.5967	−1.95552	−0.977762	+	0.209718i	$0.932745\pi$
$314$	−3.27051	−0.184566
$315$	0	0
$316$	15.7082	0.883656
$317$	−13.1803	−0.740282	−0.370141	−	0.928976i	$-0.620691\pi$
$317$	−13.1803	−0.740282	−0.370141	+	0.928976i	$0.620691\pi$
$318$	−6.85410	−0.384359
$319$	0	0
$320$	0	0
$321$	−7.79837	−0.435263
$322$	−3.85410	−0.214781
$323$	1.00000	0.0556415
$324$	−1.61803	−0.0898908
$325$	0	0
$326$	−1.79837	−0.0996027
$327$	−17.1803	−0.950075
$328$	−13.6180	−0.751930
$329$	2.76393	0.152381
$330$	0	0
$331$	−8.88854	−0.488559	−0.244279	−	0.969705i	$-0.578551\pi$
$331$	−8.88854	−0.488559	−0.244279	+	0.969705i	$0.578551\pi$
$332$	−9.47214	−0.519851
$333$	8.94427	0.490143
$334$	−12.3820	−0.677511
$335$	0	0
$336$	3.00000	0.163663
$337$	34.8885	1.90050	0.950250	−	0.311488i	$-0.100827\pi$
$337$	34.8885	1.90050	0.950250	+	0.311488i	$0.100827\pi$
$338$	−3.79837	−0.206604
$339$	−4.00000	−0.217250
$340$	0	0
$341$	0	0
$342$	−0.145898	−0.00788926
$343$	−18.4164	−0.994393
$344$	10.0000	0.539164
$345$	0	0
$346$	0.0344419	0.00185161
$347$	−31.0344	−1.66602	−0.833008	−	0.553261i	$-0.813383\pi$
$347$	−31.0344	−1.66602	−0.833008	+	0.553261i	$0.813383\pi$
$348$	6.85410	0.367418
$349$	9.43769	0.505188	0.252594	−	0.967572i	$-0.418716\pi$
$349$	9.43769	0.505188	0.252594	+	0.967572i	$0.418716\pi$
$350$	0	0
$351$	2.61803	0.139740
$352$	0	0
$353$	−25.8328	−1.37494	−0.687471	−	0.726212i	$-0.741279\pi$
$353$	−25.8328	−1.37494	−0.687471	+	0.726212i	$0.741279\pi$
$354$	5.38197	0.286048
$355$	0	0
$356$	−2.14590	−0.113732
$357$	−6.85410	−0.362758
$358$	11.0902	0.586134
$359$	−23.2918	−1.22929	−0.614647	−	0.788802i	$-0.710702\pi$
$359$	−23.2918	−1.22929	−0.614647	+	0.788802i	$0.710702\pi$
$360$	0	0
$361$	−18.9443	−0.997067
$362$	3.34752	0.175942
$363$	0	0
$364$	−6.85410	−0.359253
$365$	0	0
$366$	−0.472136	−0.0246789
$367$	−17.4721	−0.912038	−0.456019	−	0.889970i	$-0.650725\pi$
$367$	−17.4721	−0.912038	−0.456019	+	0.889970i	$0.650725\pi$
$368$	−7.14590	−0.372506
$369$	6.09017	0.317042
$370$	0	0
$371$	−17.9443	−0.931620
$372$	12.7082	0.658890
$373$	−22.9098	−1.18623	−0.593113	−	0.805119i	$-0.702101\pi$
$373$	−22.9098	−1.18623	−0.593113	+	0.805119i	$0.702101\pi$
$374$	0	0
$375$	0	0
$376$	−3.81966	−0.196984
$377$	−11.0902	−0.571173
$378$	1.00000	0.0514344
$379$	−18.3820	−0.944218	−0.472109	−	0.881540i	$-0.656507\pi$
$379$	−18.3820	−0.944218	−0.472109	+	0.881540i	$0.656507\pi$
$380$	0	0
$381$	−9.00000	−0.461084
$382$	−14.0902	−0.720916
$383$	−9.56231	−0.488611	−0.244306	−	0.969698i	$-0.578560\pi$
$383$	−9.56231	−0.488611	−0.244306	+	0.969698i	$0.578560\pi$
$384$	−11.3820	−0.580834
$385$	0	0
$386$	−8.58359	−0.436893
$387$	−4.47214	−0.227331
$388$	13.2361	0.671960
$389$	−4.47214	−0.226746	−0.113373	−	0.993552i	$-0.536166\pi$
$389$	−4.47214	−0.226746	−0.113373	+	0.993552i	$0.536166\pi$
$390$	0	0
$391$	16.3262	0.825653
$392$	9.79837	0.494893
$393$	21.5066	1.08486
$394$	14.1246	0.711588
$395$	0	0
$396$	0	0
$397$	−2.70820	−0.135921	−0.0679604	−	0.997688i	$-0.521649\pi$
$397$	−2.70820	−0.135921	−0.0679604	+	0.997688i	$0.521649\pi$
$398$	−11.2361	−0.563213
$399$	−0.381966	−0.0191222
$400$	0	0
$401$	−36.5967	−1.82755	−0.913777	−	0.406216i	$-0.866848\pi$
$401$	−36.5967	−1.82755	−0.913777	+	0.406216i	$0.866848\pi$
$402$	0.472136	0.0235480
$403$	−20.5623	−1.02428
$404$	−1.47214	−0.0732415
$405$	0	0
$406$	−4.23607	−0.210233
$407$	0	0
$408$	9.47214	0.468941
$409$	21.5066	1.06343	0.531716	−	0.846923i	$-0.321547\pi$
$409$	21.5066	1.06343	0.531716	+	0.846923i	$0.321547\pi$
$410$	0	0
$411$	9.52786	0.469975
$412$	20.1803	0.994214
$413$	14.0902	0.693332
$414$	−2.38197	−0.117067
$415$	0	0
$416$	14.7082	0.721129
$417$	−15.8541	−0.776379
$418$	0	0
$419$	19.2705	0.941426	0.470713	−	0.882286i	$-0.343997\pi$
$419$	19.2705	0.941426	0.470713	+	0.882286i	$0.343997\pi$
$420$	0	0
$421$	6.70820	0.326938	0.163469	−	0.986548i	$-0.447732\pi$
$421$	6.70820	0.326938	0.163469	+	0.986548i	$0.447732\pi$
$422$	5.67376	0.276194
$423$	1.70820	0.0830557
$424$	24.7984	1.20432
$425$	0	0
$426$	−1.09017	−0.0528189
$427$	−1.23607	−0.0598175
$428$	12.6180	0.609916
$429$	0	0
$430$	0	0
$431$	20.4721	0.986108	0.493054	−	0.869999i	$-0.335880\pi$
$431$	20.4721	0.986108	0.493054	+	0.869999i	$0.335880\pi$
$432$	1.85410	0.0892055
$433$	20.0344	0.962794	0.481397	−	0.876503i	$-0.340130\pi$
$433$	20.0344	0.962794	0.481397	+	0.876503i	$0.340130\pi$
$434$	−7.85410	−0.377009
$435$	0	0
$436$	27.7984	1.33130
$437$	0.909830	0.0435231
$438$	−2.09017	−0.0998722
$439$	−19.4164	−0.926695	−0.463347	−	0.886177i	$-0.653352\pi$
$439$	−19.4164	−0.926695	−0.463347	+	0.886177i	$0.653352\pi$
$440$	0	0
$441$	−4.38197	−0.208665
$442$	−6.85410	−0.326016
$443$	31.4164	1.49264	0.746319	−	0.665588i	$-0.231819\pi$
$443$	31.4164	1.49264	0.746319	+	0.665588i	$0.231819\pi$
$444$	−14.4721	−0.686817
$445$	0	0
$446$	9.67376	0.458066
$447$	17.8541	0.844470
$448$	−0.381966	−0.0180462
$449$	−31.0000	−1.46298	−0.731490	−	0.681852i	$-0.761175\pi$
$449$	−31.0000	−1.46298	−0.731490	+	0.681852i	$0.761175\pi$
$450$	0	0
$451$	0	0
$452$	6.47214	0.304424
$453$	14.2705	0.670487
$454$	5.38197	0.252588
$455$	0	0
$456$	0.527864	0.0247195
$457$	−15.7639	−0.737406	−0.368703	−	0.929547i	$-0.620198\pi$
$457$	−15.7639	−0.737406	−0.368703	+	0.929547i	$0.620198\pi$
$458$	−18.5279	−0.865750
$459$	−4.23607	−0.197723
$460$	0	0
$461$	−25.9443	−1.20835	−0.604173	−	0.796853i	$-0.706496\pi$
$461$	−25.9443	−1.20835	−0.604173	+	0.796853i	$0.706496\pi$
$462$	0	0
$463$	−11.2918	−0.524774	−0.262387	−	0.964963i	$-0.584510\pi$
$463$	−11.2918	−0.524774	−0.262387	+	0.964963i	$0.584510\pi$
$464$	−7.85410	−0.364618
$465$	0	0
$466$	−12.5623	−0.581938
$467$	27.7639	1.28476	0.642381	−	0.766386i	$-0.277947\pi$
$467$	27.7639	1.28476	0.642381	+	0.766386i	$0.277947\pi$
$468$	−4.23607	−0.195812
$469$	1.23607	0.0570763
$470$	0	0
$471$	−5.29180	−0.243833
$472$	−19.4721	−0.896278
$473$	0	0
$474$	−6.00000	−0.275589
$475$	0	0
$476$	11.0902	0.508317
$477$	−11.0902	−0.507784
$478$	17.5967	0.804857
$479$	−11.4164	−0.521629	−0.260814	−	0.965389i	$-0.583991\pi$
$479$	−11.4164	−0.521629	−0.260814	+	0.965389i	$0.583991\pi$
$480$	0	0
$481$	23.4164	1.06770
$482$	−1.38197	−0.0629468
$483$	−6.23607	−0.283751
$484$	0	0
$485$	0	0
$486$	0.618034	0.0280346
$487$	−9.29180	−0.421051	−0.210526	−	0.977588i	$-0.567518\pi$
$487$	−9.29180	−0.421051	−0.210526	+	0.977588i	$0.567518\pi$
$488$	1.70820	0.0773268
$489$	−2.90983	−0.131587
$490$	0	0
$491$	22.5066	1.01571	0.507854	−	0.861443i	$-0.330439\pi$
$491$	22.5066	1.01571	0.507854	+	0.861443i	$0.330439\pi$
$492$	−9.85410	−0.444257
$493$	17.9443	0.808169
$494$	−0.381966	−0.0171855
$495$	0	0
$496$	−14.5623	−0.653867
$497$	−2.85410	−0.128024
$498$	3.61803	0.162128
$499$	−11.5279	−0.516058	−0.258029	−	0.966137i	$-0.583073\pi$
$499$	−11.5279	−0.516058	−0.258029	+	0.966137i	$0.583073\pi$
$500$	0	0
$501$	−20.0344	−0.895073
$502$	1.49342	0.0666547
$503$	11.8885	0.530084	0.265042	−	0.964237i	$-0.414614\pi$
$503$	11.8885	0.530084	0.265042	+	0.964237i	$0.414614\pi$
$504$	−3.61803	−0.161160
$505$	0	0
$506$	0	0
$507$	−6.14590	−0.272949
$508$	14.5623	0.646098
$509$	−11.9443	−0.529421	−0.264710	−	0.964328i	$-0.585276\pi$
$509$	−11.9443	−0.529421	−0.264710	+	0.964328i	$0.585276\pi$
$510$	0	0
$511$	−5.47214	−0.242073
$512$	18.7082	0.826794
$513$	−0.236068	−0.0104227
$514$	12.4164	0.547664
$515$	0	0
$516$	7.23607	0.318550
$517$	0	0
$518$	8.94427	0.392989
$519$	0.0557281	0.00244619
$520$	0	0
$521$	−43.5967	−1.91001	−0.955004	−	0.296593i	$-0.904150\pi$
$521$	−43.5967	−1.91001	−0.955004	+	0.296593i	$0.904150\pi$
$522$	−2.61803	−0.114588
$523$	31.1459	1.36192	0.680958	−	0.732323i	$-0.261564\pi$
$523$	31.1459	1.36192	0.680958	+	0.732323i	$0.261564\pi$
$524$	−34.7984	−1.52017
$525$	0	0
$526$	6.43769	0.280697
$527$	33.2705	1.44929
$528$	0	0
$529$	−8.14590	−0.354169
$530$	0	0
$531$	8.70820	0.377904
$532$	0.618034	0.0267952
$533$	15.9443	0.690623
$534$	0.819660	0.0354702
$535$	0	0
$536$	−1.70820	−0.0737832
$537$	17.9443	0.774352
$538$	−19.0344	−0.820633
$539$	0	0
$540$	0	0
$541$	34.6312	1.48891	0.744456	−	0.667672i	$-0.232709\pi$
$541$	34.6312	1.48891	0.744456	+	0.667672i	$0.232709\pi$
$542$	−15.6869	−0.673811
$543$	5.41641	0.232440
$544$	−23.7984	−1.02035
$545$	0	0
$546$	2.61803	0.112042
$547$	6.59675	0.282057	0.141028	−	0.990006i	$-0.454959\pi$
$547$	6.59675	0.282057	0.141028	+	0.990006i	$0.454959\pi$
$548$	−15.4164	−0.658556
$549$	−0.763932	−0.0326038
$550$	0	0
$551$	1.00000	0.0426014
$552$	8.61803	0.366808
$553$	−15.7082	−0.667981
$554$	14.6180	0.621061
$555$	0	0
$556$	25.6525	1.08791
$557$	−24.0344	−1.01837	−0.509186	−	0.860657i	$-0.670053\pi$
$557$	−24.0344	−1.01837	−0.509186	+	0.860657i	$0.670053\pi$
$558$	−4.85410	−0.205491
$559$	−11.7082	−0.495204
$560$	0	0
$561$	0	0
$562$	−0.562306	−0.0237194
$563$	−6.20163	−0.261367	−0.130684	−	0.991424i	$-0.541717\pi$
$563$	−6.20163	−0.261367	−0.130684	+	0.991424i	$0.541717\pi$
$564$	−2.76393	−0.116383
$565$	0	0
$566$	3.50658	0.147392
$567$	1.61803	0.0679510
$568$	3.94427	0.165498
$569$	32.7082	1.37120	0.685600	−	0.727979i	$-0.259540\pi$
$569$	32.7082	1.37120	0.685600	+	0.727979i	$0.259540\pi$
$570$	0	0
$571$	−13.1246	−0.549248	−0.274624	−	0.961552i	$-0.588553\pi$
$571$	−13.1246	−0.549248	−0.274624	+	0.961552i	$0.588553\pi$
$572$	0	0
$573$	−22.7984	−0.952416
$574$	6.09017	0.254199
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	10.9787	0.457050	0.228525	−	0.973538i	$-0.426610\pi$
$577$	10.9787	0.457050	0.228525	+	0.973538i	$0.426610\pi$
$578$	0.583592	0.0242742
$579$	−13.8885	−0.577188
$580$	0	0
$581$	9.47214	0.392970
$582$	−5.05573	−0.209567
$583$	0	0
$584$	7.56231	0.312930
$585$	0	0
$586$	−5.41641	−0.223750
$587$	41.3820	1.70802	0.854008	−	0.520259i	$-0.174165\pi$
$587$	41.3820	1.70802	0.854008	+	0.520259i	$0.174165\pi$
$588$	7.09017	0.292394
$589$	1.85410	0.0763969
$590$	0	0
$591$	22.8541	0.940092
$592$	16.5836	0.681581
$593$	6.79837	0.279176	0.139588	−	0.990210i	$-0.455422\pi$
$593$	6.79837	0.279176	0.139588	+	0.990210i	$0.455422\pi$
$594$	0	0
$595$	0	0
$596$	−28.8885	−1.18332
$597$	−18.1803	−0.744072
$598$	−6.23607	−0.255012
$599$	−32.3262	−1.32081	−0.660407	−	0.750908i	$-0.729616\pi$
$599$	−32.3262	−1.32081	−0.660407	+	0.750908i	$0.729616\pi$
$600$	0	0
$601$	4.74265	0.193457	0.0967283	−	0.995311i	$-0.469162\pi$
$601$	4.74265	0.193457	0.0967283	+	0.995311i	$0.469162\pi$
$602$	−4.47214	−0.182271
$603$	0.763932	0.0311097
$604$	−23.0902	−0.939526
$605$	0	0
$606$	0.562306	0.0228421
$607$	19.8328	0.804989	0.402495	−	0.915422i	$-0.368143\pi$
$607$	19.8328	0.804989	0.402495	+	0.915422i	$0.368143\pi$
$608$	−1.32624	−0.0537861
$609$	−6.85410	−0.277742
$610$	0	0
$611$	4.47214	0.180923
$612$	6.85410	0.277061
$613$	24.0000	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$614$	−1.81966	−0.0734355
$615$	0	0
$616$	0	0
$617$	−11.5066	−0.463237	−0.231619	−	0.972807i	$-0.574402\pi$
$617$	−11.5066	−0.463237	−0.231619	+	0.972807i	$0.574402\pi$
$618$	−7.70820	−0.310069
$619$	−27.3607	−1.09972	−0.549859	−	0.835257i	$-0.685319\pi$
$619$	−27.3607	−1.09972	−0.549859	+	0.835257i	$0.685319\pi$
$620$	0	0
$621$	−3.85410	−0.154660
$622$	−2.70820	−0.108589
$623$	2.14590	0.0859736
$624$	4.85410	0.194320
$625$	0	0
$626$	−21.3820	−0.854595
$627$	0	0
$628$	8.56231	0.341673
$629$	−37.8885	−1.51072
$630$	0	0
$631$	37.1591	1.47928	0.739639	−	0.673004i	$-0.234996\pi$
$631$	37.1591	1.47928	0.739639	+	0.673004i	$0.234996\pi$
$632$	21.7082	0.863506
$633$	9.18034	0.364886
$634$	−8.14590	−0.323515
$635$	0	0
$636$	17.9443	0.711537
$637$	−11.4721	−0.454543
$638$	0	0
$639$	−1.76393	−0.0697801
$640$	0	0
$641$	−27.5623	−1.08865	−0.544323	−	0.838876i	$-0.683213\pi$
$641$	−27.5623	−1.08865	−0.544323	+	0.838876i	$0.683213\pi$
$642$	−4.81966	−0.190217
$643$	−21.1246	−0.833073	−0.416537	−	0.909119i	$-0.636756\pi$
$643$	−21.1246	−0.833073	−0.416537	+	0.909119i	$0.636756\pi$
$644$	10.0902	0.397608
$645$	0	0
$646$	0.618034	0.0243162
$647$	19.0689	0.749675	0.374838	−	0.927090i	$-0.377698\pi$
$647$	19.0689	0.749675	0.374838	+	0.927090i	$0.377698\pi$
$648$	−2.23607	−0.0878410
$649$	0	0
$650$	0	0
$651$	−12.7082	−0.498074
$652$	4.70820	0.184387
$653$	31.0000	1.21312	0.606562	−	0.795036i	$-0.292548\pi$
$653$	31.0000	1.21312	0.606562	+	0.795036i	$0.292548\pi$
$654$	−10.6180	−0.415198
$655$	0	0
$656$	11.2918	0.440871
$657$	−3.38197	−0.131943
$658$	1.70820	0.0665927
$659$	0.201626	0.00785424	0.00392712	−	0.999992i	$-0.498750\pi$
$659$	0.201626	0.00785424	0.00392712	+	0.999992i	$0.498750\pi$
$660$	0	0
$661$	−2.32624	−0.0904802	−0.0452401	−	0.998976i	$-0.514405\pi$
$661$	−2.32624	−0.0904802	−0.0452401	+	0.998976i	$0.514405\pi$
$662$	−5.49342	−0.213508
$663$	−11.0902	−0.430707
$664$	−13.0902	−0.507997
$665$	0	0
$666$	5.52786	0.214200
$667$	16.3262	0.632154
$668$	32.4164	1.25423
$669$	15.6525	0.605160
$670$	0	0
$671$	0	0
$672$	9.09017	0.350661
$673$	−4.09017	−0.157664	−0.0788322	−	0.996888i	$-0.525119\pi$
$673$	−4.09017	−0.157664	−0.0788322	+	0.996888i	$0.525119\pi$
$674$	21.5623	0.830549
$675$	0	0
$676$	9.94427	0.382472
$677$	17.9787	0.690978	0.345489	−	0.938423i	$-0.387713\pi$
$677$	17.9787	0.690978	0.345489	+	0.938423i	$0.387713\pi$
$678$	−2.47214	−0.0949418
$679$	−13.2361	−0.507954
$680$	0	0
$681$	8.70820	0.333699
$682$	0	0
$683$	−16.4164	−0.628156	−0.314078	−	0.949397i	$-0.601695\pi$
$683$	−16.4164	−0.628156	−0.314078	+	0.949397i	$0.601695\pi$
$684$	0.381966	0.0146048
$685$	0	0
$686$	−11.3820	−0.434565
$687$	−29.9787	−1.14376
$688$	−8.29180	−0.316122
$689$	−29.0344	−1.10612
$690$	0	0
$691$	−25.9787	−0.988277	−0.494138	−	0.869383i	$-0.664516\pi$
$691$	−25.9787	−0.988277	−0.494138	+	0.869383i	$0.664516\pi$
$692$	−0.0901699	−0.00342775
$693$	0	0
$694$	−19.1803	−0.728076
$695$	0	0
$696$	9.47214	0.359040
$697$	−25.7984	−0.977183
$698$	5.83282	0.220775
$699$	−20.3262	−0.768809
$700$	0	0
$701$	−17.6525	−0.666725	−0.333362	−	0.942799i	$-0.608183\pi$
$701$	−17.6525	−0.666725	−0.333362	+	0.942799i	$0.608183\pi$
$702$	1.61803	0.0610688
$703$	−2.11146	−0.0796351
$704$	0	0
$705$	0	0
$706$	−15.9656	−0.600872
$707$	1.47214	0.0553654
$708$	−14.0902	−0.529541
$709$	−36.3820	−1.36635	−0.683177	−	0.730253i	$-0.739402\pi$
$709$	−36.3820	−1.36635	−0.683177	+	0.730253i	$0.739402\pi$
$710$	0	0
$711$	−9.70820	−0.364086
$712$	−2.96556	−0.111139
$713$	30.2705	1.13364
$714$	−4.23607	−0.158531
$715$	0	0
$716$	−29.0344	−1.08507
$717$	28.4721	1.06331
$718$	−14.3951	−0.537221
$719$	−15.3050	−0.570778	−0.285389	−	0.958412i	$-0.592123\pi$
$719$	−15.3050	−0.570778	−0.285389	+	0.958412i	$0.592123\pi$
$720$	0	0
$721$	−20.1803	−0.751555
$722$	−11.7082	−0.435734
$723$	−2.23607	−0.0831603
$724$	−8.76393	−0.325709
$725$	0	0
$726$	0	0
$727$	−36.9787	−1.37146	−0.685732	−	0.727854i	$-0.740518\pi$
$727$	−36.9787	−1.37146	−0.685732	+	0.727854i	$0.740518\pi$
$728$	−9.47214	−0.351061
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.9443	0.700679
$732$	1.23607	0.0456864
$733$	7.52786	0.278048	0.139024	−	0.990289i	$-0.455603\pi$
$733$	7.52786	0.278048	0.139024	+	0.990289i	$0.455603\pi$
$734$	−10.7984	−0.398575
$735$	0	0
$736$	−21.6525	−0.798121
$737$	0	0
$738$	3.76393	0.138552
$739$	33.9443	1.24866	0.624330	−	0.781161i	$-0.285372\pi$
$739$	33.9443	1.24866	0.624330	+	0.781161i	$0.285372\pi$
$740$	0	0
$741$	−0.618034	−0.0227040
$742$	−11.0902	−0.407133
$743$	27.3262	1.00250	0.501251	−	0.865302i	$-0.332873\pi$
$743$	27.3262	1.00250	0.501251	+	0.865302i	$0.332873\pi$
$744$	17.5623	0.643865
$745$	0	0
$746$	−14.1591	−0.518400
$747$	5.85410	0.214190
$748$	0	0
$749$	−12.6180	−0.461053
$750$	0	0
$751$	18.5066	0.675315	0.337657	−	0.941269i	$-0.390366\pi$
$751$	18.5066	0.675315	0.337657	+	0.941269i	$0.390366\pi$
$752$	3.16718	0.115495
$753$	2.41641	0.0880588
$754$	−6.85410	−0.249612
$755$	0	0
$756$	−2.61803	−0.0952170
$757$	29.1591	1.05980	0.529902	−	0.848059i	$-0.322229\pi$
$757$	29.1591	1.05980	0.529902	+	0.848059i	$0.322229\pi$
$758$	−11.3607	−0.412638
$759$	0	0
$760$	0	0
$761$	18.3475	0.665097	0.332549	−	0.943086i	$-0.392091\pi$
$761$	18.3475	0.665097	0.332549	+	0.943086i	$0.392091\pi$
$762$	−5.56231	−0.201501
$763$	−27.7984	−1.00637
$764$	36.8885	1.33458
$765$	0	0
$766$	−5.90983	−0.213531
$767$	22.7984	0.823202
$768$	−6.56231	−0.236797
$769$	7.76393	0.279975	0.139987	−	0.990153i	$-0.455294\pi$
$769$	7.76393	0.279975	0.139987	+	0.990153i	$0.455294\pi$
$770$	0	0
$771$	20.0902	0.723530
$772$	22.4721	0.808790
$773$	29.2148	1.05078	0.525391	−	0.850861i	$-0.323919\pi$
$773$	29.2148	1.05078	0.525391	+	0.850861i	$0.323919\pi$
$774$	−2.76393	−0.0993475
$775$	0	0
$776$	18.2918	0.656637
$777$	14.4721	0.519185
$778$	−2.76393	−0.0990918
$779$	−1.43769	−0.0515107
$780$	0	0
$781$	0	0
$782$	10.0902	0.360824
$783$	−4.23607	−0.151385
$784$	−8.12461	−0.290165
$785$	0	0
$786$	13.2918	0.474103
$787$	−36.6525	−1.30652	−0.653260	−	0.757134i	$-0.726599\pi$
$787$	−36.6525	−1.30652	−0.653260	+	0.757134i	$0.726599\pi$
$788$	−36.9787	−1.31731
$789$	10.4164	0.370834
$790$	0	0
$791$	−6.47214	−0.230123
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	−1.67376	−0.0593996
$795$	0	0
$796$	29.4164	1.04264
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	−0.236068	−0.00835672
$799$	−7.23607	−0.255994
$800$	0	0
$801$	1.32624	0.0468603
$802$	−22.6180	−0.798671
$803$	0	0
$804$	−1.23607	−0.0435928
$805$	0	0
$806$	−12.7082	−0.447627
$807$	−30.7984	−1.08415
$808$	−2.03444	−0.0715714
$809$	24.3951	0.857687	0.428843	−	0.903379i	$-0.358921\pi$
$809$	24.3951	0.857687	0.428843	+	0.903379i	$0.358921\pi$
$810$	0	0
$811$	−29.9787	−1.05270	−0.526348	−	0.850270i	$-0.676439\pi$
$811$	−29.9787	−1.05270	−0.526348	+	0.850270i	$0.676439\pi$
$812$	11.0902	0.389189
$813$	−25.3820	−0.890184
$814$	0	0
$815$	0	0
$816$	−7.85410	−0.274949
$817$	1.05573	0.0369353
$818$	13.2918	0.464737
$819$	4.23607	0.148020
$820$	0	0
$821$	−15.5066	−0.541183	−0.270592	−	0.962694i	$-0.587219\pi$
$821$	−15.5066	−0.541183	−0.270592	+	0.962694i	$0.587219\pi$
$822$	5.88854	0.205387
$823$	24.4164	0.851102	0.425551	−	0.904934i	$-0.360080\pi$
$823$	24.4164	0.851102	0.425551	+	0.904934i	$0.360080\pi$
$824$	27.8885	0.971543
$825$	0	0
$826$	8.70820	0.302997
$827$	−42.3820	−1.47377	−0.736883	−	0.676021i	$-0.763703\pi$
$827$	−42.3820	−1.47377	−0.736883	+	0.676021i	$0.763703\pi$
$828$	6.23607	0.216718
$829$	16.2705	0.565098	0.282549	−	0.959253i	$-0.408820\pi$
$829$	16.2705	0.565098	0.282549	+	0.959253i	$0.408820\pi$
$830$	0	0
$831$	23.6525	0.820495
$832$	−0.618034	−0.0214265
$833$	18.5623	0.643146
$834$	−9.79837	−0.339290
$835$	0	0
$836$	0	0
$837$	−7.85410	−0.271477
$838$	11.9098	0.411418
$839$	7.25735	0.250552	0.125276	−	0.992122i	$-0.460018\pi$
$839$	7.25735	0.250552	0.125276	+	0.992122i	$0.460018\pi$
$840$	0	0
$841$	−11.0557	−0.381232
$842$	4.14590	0.142877
$843$	−0.909830	−0.0313362
$844$	−14.8541	−0.511299
$845$	0	0
$846$	1.05573	0.0362967
$847$	0	0
$848$	−20.5623	−0.706112
$849$	5.67376	0.194723
$850$	0	0
$851$	−34.4721	−1.18169
$852$	2.85410	0.0977799
$853$	32.5967	1.11609	0.558046	−	0.829810i	$-0.311551\pi$
$853$	32.5967	1.11609	0.558046	+	0.829810i	$0.311551\pi$
$854$	−0.763932	−0.0261412
$855$	0	0
$856$	17.4377	0.596008
$857$	47.2705	1.61473	0.807365	−	0.590052i	$-0.200893\pi$
$857$	47.2705	1.61473	0.807365	+	0.590052i	$0.200893\pi$
$858$	0	0
$859$	22.8541	0.779772	0.389886	−	0.920863i	$-0.372514\pi$
$859$	22.8541	0.779772	0.389886	+	0.920863i	$0.372514\pi$
$860$	0	0
$861$	9.85410	0.335827
$862$	12.6525	0.430945
$863$	8.76393	0.298328	0.149164	−	0.988812i	$-0.452342\pi$
$863$	8.76393	0.298328	0.149164	+	0.988812i	$0.452342\pi$
$864$	5.61803	0.191129
$865$	0	0
$866$	12.3820	0.420756
$867$	0.944272	0.0320692
$868$	20.5623	0.697930
$869$	0	0
$870$	0	0
$871$	2.00000	0.0677674
$872$	38.4164	1.30094
$873$	−8.18034	−0.276863
$874$	0.562306	0.0190203
$875$	0	0
$876$	5.47214	0.184886
$877$	11.9230	0.402611	0.201305	−	0.979529i	$-0.435482\pi$
$877$	11.9230	0.402611	0.201305	+	0.979529i	$0.435482\pi$
$878$	−12.0000	−0.404980
$879$	−8.76393	−0.295600
$880$	0	0
$881$	47.1246	1.58767	0.793834	−	0.608134i	$-0.208082\pi$
$881$	47.1246	1.58767	0.793834	+	0.608134i	$0.208082\pi$
$882$	−2.70820	−0.0911900
$883$	10.3820	0.349381	0.174690	−	0.984623i	$-0.444108\pi$
$883$	10.3820	0.349381	0.174690	+	0.984623i	$0.444108\pi$
$884$	17.9443	0.603531
$885$	0	0
$886$	19.4164	0.652307
$887$	−39.3607	−1.32160	−0.660801	−	0.750561i	$-0.729783\pi$
$887$	−39.3607	−1.32160	−0.660801	+	0.750561i	$0.729783\pi$
$888$	−20.0000	−0.671156
$889$	−14.5623	−0.488404
$890$	0	0
$891$	0	0
$892$	−25.3262	−0.847985
$893$	−0.403252	−0.0134943
$894$	11.0344	0.369047
$895$	0	0
$896$	−18.4164	−0.615249
$897$	−10.0902	−0.336901
$898$	−19.1591	−0.639346
$899$	33.2705	1.10963
$900$	0	0
$901$	46.9787	1.56509
$902$	0	0
$903$	−7.23607	−0.240801
$904$	8.94427	0.297482
$905$	0	0
$906$	8.81966	0.293014
$907$	−3.43769	−0.114147	−0.0570734	−	0.998370i	$-0.518177\pi$
$907$	−3.43769	−0.114147	−0.0570734	+	0.998370i	$0.518177\pi$
$908$	−14.0902	−0.467599
$909$	0.909830	0.0301772
$910$	0	0
$911$	−27.8541	−0.922848	−0.461424	−	0.887180i	$-0.652661\pi$
$911$	−27.8541	−0.922848	−0.461424	+	0.887180i	$0.652661\pi$
$912$	−0.437694	−0.0144935
$913$	0	0
$914$	−9.74265	−0.322258
$915$	0	0
$916$	48.5066	1.60270
$917$	34.7984	1.14914
$918$	−2.61803	−0.0864080
$919$	10.8328	0.357342	0.178671	−	0.983909i	$-0.442820\pi$
$919$	10.8328	0.357342	0.178671	+	0.983909i	$0.442820\pi$
$920$	0	0
$921$	−2.94427	−0.0970171
$922$	−16.0344	−0.528066
$923$	−4.61803	−0.152004
$924$	0	0
$925$	0	0
$926$	−6.97871	−0.229335
$927$	−12.4721	−0.409639
$928$	−23.7984	−0.781220
$929$	59.3394	1.94686	0.973431	−	0.228980i	$-0.0735390\pi$
$929$	59.3394	1.94686	0.973431	+	0.228980i	$0.0735390\pi$
$930$	0	0
$931$	1.03444	0.0339025
$932$	32.8885	1.07730
$933$	−4.38197	−0.143459
$934$	17.1591	0.561461
$935$	0	0
$936$	−5.85410	−0.191347
$937$	9.14590	0.298783	0.149392	−	0.988778i	$-0.452268\pi$
$937$	9.14590	0.298783	0.149392	+	0.988778i	$0.452268\pi$
$938$	0.763932	0.0249433
$939$	−34.5967	−1.12902
$940$	0	0
$941$	−28.6525	−0.934044	−0.467022	−	0.884246i	$-0.654673\pi$
$941$	−28.6525	−0.934044	−0.467022	+	0.884246i	$0.654673\pi$
$942$	−3.27051	−0.106559
$943$	−23.4721	−0.764358
$944$	16.1459	0.525504
$945$	0	0
$946$	0	0
$947$	60.3394	1.96077	0.980383	−	0.197100i	$-0.0631523\pi$
$947$	60.3394	1.96077	0.980383	+	0.197100i	$0.0631523\pi$
$948$	15.7082	0.510179
$949$	−8.85410	−0.287416
$950$	0	0
$951$	−13.1803	−0.427402
$952$	15.3262	0.496726
$953$	37.2148	1.20551	0.602753	−	0.797928i	$-0.294071\pi$
$953$	37.2148	1.20551	0.602753	+	0.797928i	$0.294071\pi$
$954$	−6.85410	−0.221910
$955$	0	0
$956$	−46.0689	−1.48997
$957$	0	0
$958$	−7.05573	−0.227960
$959$	15.4164	0.497822
$960$	0	0
$961$	30.6869	0.989901
$962$	14.4721	0.466600
$963$	−7.79837	−0.251299
$964$	3.61803	0.116529
$965$	0	0
$966$	−3.85410	−0.124004
$967$	47.8115	1.53752	0.768758	−	0.639540i	$-0.220875\pi$
$967$	47.8115	1.53752	0.768758	+	0.639540i	$0.220875\pi$
$968$	0	0
$969$	1.00000	0.0321246
$970$	0	0
$971$	−0.0557281	−0.00178840	−0.000894200	−	1.00000i	$-0.500285\pi$
$971$	−0.0557281	−0.00178840	−0.000894200		1.00000i	$0.500285\pi$
$972$	−1.61803	−0.0518985
$973$	−25.6525	−0.822381
$974$	−5.74265	−0.184006
$975$	0	0
$976$	−1.41641	−0.0453381
$977$	1.21478	0.0388643	0.0194322	−	0.999811i	$-0.493814\pi$
$977$	1.21478	0.0388643	0.0194322	+	0.999811i	$0.493814\pi$
$978$	−1.79837	−0.0575057
$979$	0	0
$980$	0	0
$981$	−17.1803	−0.548526
$982$	13.9098	0.443881
$983$	5.05573	0.161253	0.0806263	−	0.996744i	$-0.474308\pi$
$983$	5.05573	0.161253	0.0806263	+	0.996744i	$0.474308\pi$
$984$	−13.6180	−0.434127
$985$	0	0
$986$	11.0902	0.353183
$987$	2.76393	0.0879769
$988$	1.00000	0.0318142
$989$	17.2361	0.548075
$990$	0	0
$991$	−25.4508	−0.808473	−0.404236	−	0.914655i	$-0.632463\pi$
$991$	−25.4508	−0.808473	−0.404236	+	0.914655i	$0.632463\pi$
$992$	−44.1246	−1.40096
$993$	−8.88854	−0.282069
$994$	−1.76393	−0.0559485
$995$	0	0
$996$	−9.47214	−0.300136
$997$	18.9443	0.599971	0.299986	−	0.953944i	$-0.403018\pi$
$997$	18.9443	0.599971	0.299986	+	0.953944i	$0.403018\pi$
$998$	−7.12461	−0.225526
$999$	8.94427	0.282984

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.be.1.2	yes	2
5.4	even	2		9075.2.a.bs.1.1	yes	2
11.10	odd	2		9075.2.a.bw.1.1	yes	2
55.54	odd	2		9075.2.a.ba.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9075.2.a.ba.1.2	✓	2	55.54	odd	2
9075.2.a.be.1.2	yes	2	1.1	even	1	trivial
9075.2.a.bs.1.1	yes	2	5.4	even	2
9075.2.a.bw.1.1	yes	2	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +1.61803 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +1.61803 q^{7} -2.23607 q^{8} +1.00000 q^{9} -1.61803 q^{12} +2.61803 q^{13} +1.00000 q^{14} +1.85410 q^{16} -4.23607 q^{17} +0.618034 q^{18} -0.236068 q^{19} +1.61803 q^{21} -3.85410 q^{23} -2.23607 q^{24} +1.61803 q^{26} +1.00000 q^{27} -2.61803 q^{28} -4.23607 q^{29} -7.85410 q^{31} +5.61803 q^{32} -2.61803 q^{34} -1.61803 q^{36} +8.94427 q^{37} -0.145898 q^{38} +2.61803 q^{39} +6.09017 q^{41} +1.00000 q^{42} -4.47214 q^{43} -2.38197 q^{46} +1.70820 q^{47} +1.85410 q^{48} -4.38197 q^{49} -4.23607 q^{51} -4.23607 q^{52} -11.0902 q^{53} +0.618034 q^{54} -3.61803 q^{56} -0.236068 q^{57} -2.61803 q^{58} +8.70820 q^{59} -0.763932 q^{61} -4.85410 q^{62} +1.61803 q^{63} -0.236068 q^{64} +0.763932 q^{67} +6.85410 q^{68} -3.85410 q^{69} -1.76393 q^{71} -2.23607 q^{72} -3.38197 q^{73} +5.52786 q^{74} +0.381966 q^{76} +1.61803 q^{78} -9.70820 q^{79} +1.00000 q^{81} +3.76393 q^{82} +5.85410 q^{83} -2.61803 q^{84} -2.76393 q^{86} -4.23607 q^{87} +1.32624 q^{89} +4.23607 q^{91} +6.23607 q^{92} -7.85410 q^{93} +1.05573 q^{94} +5.61803 q^{96} -8.18034 q^{97} -2.70820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 + 2 * q^3 - q^4 - q^6 + q^7 + 2 * q^9	\( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} + q^{7} + 2 q^{9} - q^{12} + 3 q^{13} + 2 q^{14} - 3 q^{16} - 4 q^{17} - q^{18} + 4 q^{19} + q^{21} - q^{23} + q^{26} + 2 q^{27} - 3 q^{28} - 4 q^{29} - 9 q^{31} + 9 q^{32} - 3 q^{34} - q^{36} - 7 q^{38} + 3 q^{39} + q^{41} + 2 q^{42} - 7 q^{46} - 10 q^{47} - 3 q^{48} - 11 q^{49} - 4 q^{51} - 4 q^{52} - 11 q^{53} - q^{54} - 5 q^{56} + 4 q^{57} - 3 q^{58} + 4 q^{59} - 6 q^{61} - 3 q^{62} + q^{63} + 4 q^{64} + 6 q^{67} + 7 q^{68} - q^{69} - 8 q^{71} - 9 q^{73} + 20 q^{74} + 3 q^{76} + q^{78} - 6 q^{79} + 2 q^{81} + 12 q^{82} + 5 q^{83} - 3 q^{84} - 10 q^{86} - 4 q^{87} - 13 q^{89} + 4 q^{91} + 8 q^{92} - 9 q^{93} + 20 q^{94} + 9 q^{96} + 6 q^{97} + 8 q^{98}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 - q^4 - q^6 + q^7 + 2 * q^9 - q^12 + 3 * q^13 + 2 * q^14 - 3 * q^16 - 4 * q^17 - q^18 + 4 * q^19 + q^21 - q^23 + q^26 + 2 * q^27 - 3 * q^28 - 4 * q^29 - 9 * q^31 + 9 * q^32 - 3 * q^34 - q^36 - 7 * q^38 + 3 * q^39 + q^41 + 2 * q^42 - 7 * q^46 - 10 * q^47 - 3 * q^48 - 11 * q^49 - 4 * q^51 - 4 * q^52 - 11 * q^53 - q^54 - 5 * q^56 + 4 * q^57 - 3 * q^58 + 4 * q^59 - 6 * q^61 - 3 * q^62 + q^63 + 4 * q^64 + 6 * q^67 + 7 * q^68 - q^69 - 8 * q^71 - 9 * q^73 + 20 * q^74 + 3 * q^76 + q^78 - 6 * q^79 + 2 * q^81 + 12 * q^82 + 5 * q^83 - 3 * q^84 - 10 * q^86 - 4 * q^87 - 13 * q^89 + 4 * q^91 + 8 * q^92 - 9 * q^93 + 20 * q^94 + 9 * q^96 + 6 * q^97 + 8 * q^98

Introduction

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