LMFDB - Embedded newform 945.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(1,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, 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("945.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.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:	$7.54586299101$
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	$\chi$	$=$	945.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +0.618034 q^{10} +3.47214 q^{11} -6.09017 q^{13} -0.618034 q^{14} +1.85410 q^{16} -6.61803 q^{17} +0.381966 q^{19} -1.61803 q^{20} +2.14590 q^{22} -4.38197 q^{23} +1.00000 q^{25} -3.76393 q^{26} +1.61803 q^{28} -2.85410 q^{29} +3.00000 q^{31} +5.61803 q^{32} -4.09017 q^{34} -1.00000 q^{35} -3.00000 q^{37} +0.236068 q^{38} -2.23607 q^{40} +0.618034 q^{41} -7.76393 q^{43} -5.61803 q^{44} -2.70820 q^{46} -10.7082 q^{47} +1.00000 q^{49} +0.618034 q^{50} +9.85410 q^{52} -6.61803 q^{53} +3.47214 q^{55} +2.23607 q^{56} -1.76393 q^{58} -10.7082 q^{59} +1.14590 q^{61} +1.85410 q^{62} -0.236068 q^{64} -6.09017 q^{65} +12.7984 q^{67} +10.7082 q^{68} -0.618034 q^{70} -8.85410 q^{71} -2.52786 q^{73} -1.85410 q^{74} -0.618034 q^{76} -3.47214 q^{77} +10.7984 q^{79} +1.85410 q^{80} +0.381966 q^{82} +14.4164 q^{83} -6.61803 q^{85} -4.79837 q^{86} -7.76393 q^{88} -13.4721 q^{89} +6.09017 q^{91} +7.09017 q^{92} -6.61803 q^{94} +0.381966 q^{95} +16.0344 q^{97} +0.618034 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7	$ 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7} - q^{10} - 2 q^{11} - q^{13} + q^{14} - 3 q^{16} - 11 q^{17} + 3 q^{19} - q^{20} + 11 q^{22} - 11 q^{23} + 2 q^{25} - 12 q^{26} + q^{28} + q^{29} + 6 q^{31} + 9 q^{32} + 3 q^{34} - 2 q^{35} - 6 q^{37} - 4 q^{38} - q^{41} - 20 q^{43} - 9 q^{44} + 8 q^{46} - 8 q^{47} + 2 q^{49} - q^{50} + 13 q^{52} - 11 q^{53} - 2 q^{55} - 8 q^{58} - 8 q^{59} + 9 q^{61} - 3 q^{62} + 4 q^{64} - q^{65} + q^{67} + 8 q^{68} + q^{70} - 11 q^{71} - 14 q^{73} + 3 q^{74} + q^{76} + 2 q^{77} - 3 q^{79} - 3 q^{80} + 3 q^{82} + 2 q^{83} - 11 q^{85} + 15 q^{86} - 20 q^{88} - 18 q^{89} + q^{91} + 3 q^{92} - 11 q^{94} + 3 q^{95} + 3 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7 - q^10 - 2 * q^11 - q^13 + q^14 - 3 * q^16 - 11 * q^17 + 3 * q^19 - q^20 + 11 * q^22 - 11 * q^23 + 2 * q^25 - 12 * q^26 + q^28 + q^29 + 6 * q^31 + 9 * q^32 + 3 * q^34 - 2 * q^35 - 6 * q^37 - 4 * q^38 - q^41 - 20 * q^43 - 9 * q^44 + 8 * q^46 - 8 * q^47 + 2 * q^49 - q^50 + 13 * q^52 - 11 * q^53 - 2 * q^55 - 8 * q^58 - 8 * q^59 + 9 * q^61 - 3 * q^62 + 4 * q^64 - q^65 + q^67 + 8 * q^68 + q^70 - 11 * q^71 - 14 * q^73 + 3 * q^74 + q^76 + 2 * q^77 - 3 * q^79 - 3 * q^80 + 3 * q^82 + 2 * q^83 - 11 * q^85 + 15 * q^86 - 20 * q^88 - 18 * q^89 + q^91 + 3 * q^92 - 11 * q^94 + 3 * q^95 + 3 * q^97 - 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	$0.429881\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	0.618034	0.195440
$11$	3.47214	1.04689	0.523444	−	0.852060i	$-0.324647\pi$
$11$	3.47214	1.04689	0.523444	+	0.852060i	$0.324647\pi$
$12$	0	0
$13$	−6.09017	−1.68911	−0.844555	−	0.535469i	$-0.820135\pi$
$13$	−6.09017	−1.68911	−0.844555	+	0.535469i	$0.820135\pi$
$14$	−0.618034	−0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	−6.61803	−1.60511	−0.802555	−	0.596579i	$-0.796526\pi$
$17$	−6.61803	−1.60511	−0.802555	+	0.596579i	$0.796526\pi$
$18$	0	0
$19$	0.381966	0.0876290	0.0438145	−	0.999040i	$-0.486049\pi$
$19$	0.381966	0.0876290	0.0438145	+	0.999040i	$0.486049\pi$
$20$	−1.61803	−0.361803
$21$	0	0
$22$	2.14590	0.457507
$23$	−4.38197	−0.913703	−0.456852	−	0.889543i	$-0.651023\pi$
$23$	−4.38197	−0.913703	−0.456852	+	0.889543i	$0.651023\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.76393	−0.738168
$27$	0	0
$28$	1.61803	0.305780
$29$	−2.85410	−0.529993	−0.264997	−	0.964249i	$-0.585371\pi$
$29$	−2.85410	−0.529993	−0.264997	+	0.964249i	$0.585371\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	−4.09017	−0.701458
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0.236068	0.0382953
$39$	0	0
$40$	−2.23607	−0.353553
$41$	0.618034	0.0965207	0.0482603	−	0.998835i	$-0.484632\pi$
$41$	0.618034	0.0965207	0.0482603	+	0.998835i	$0.484632\pi$
$42$	0	0
$43$	−7.76393	−1.18399	−0.591994	−	0.805942i	$-0.701659\pi$
$43$	−7.76393	−1.18399	−0.591994	+	0.805942i	$0.701659\pi$
$44$	−5.61803	−0.846950
$45$	0	0
$46$	−2.70820	−0.399303
$47$	−10.7082	−1.56195	−0.780976	−	0.624561i	$-0.785278\pi$
$47$	−10.7082	−1.56195	−0.780976	+	0.624561i	$0.785278\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.618034	0.0874032
$51$	0	0
$52$	9.85410	1.36652
$53$	−6.61803	−0.909057	−0.454528	−	0.890732i	$-0.650192\pi$
$53$	−6.61803	−0.909057	−0.454528	+	0.890732i	$0.650192\pi$
$54$	0	0
$55$	3.47214	0.468183
$56$	2.23607	0.298807
$57$	0	0
$58$	−1.76393	−0.231616
$59$	−10.7082	−1.39409	−0.697045	−	0.717028i	$-0.745502\pi$
$59$	−10.7082	−1.39409	−0.697045	+	0.717028i	$0.745502\pi$
$60$	0	0
$61$	1.14590	0.146717	0.0733586	−	0.997306i	$-0.476628\pi$
$61$	1.14590	0.146717	0.0733586	+	0.997306i	$0.476628\pi$
$62$	1.85410	0.235471
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	−6.09017	−0.755393
$66$	0	0
$67$	12.7984	1.56357	0.781785	−	0.623548i	$-0.214309\pi$
$67$	12.7984	1.56357	0.781785	+	0.623548i	$0.214309\pi$
$68$	10.7082	1.29856
$69$	0	0
$70$	−0.618034	−0.0738692
$71$	−8.85410	−1.05079	−0.525394	−	0.850859i	$-0.676082\pi$
$71$	−8.85410	−1.05079	−0.525394	+	0.850859i	$0.676082\pi$
$72$	0	0
$73$	−2.52786	−0.295864	−0.147932	−	0.988998i	$-0.547262\pi$
$73$	−2.52786	−0.295864	−0.147932	+	0.988998i	$0.547262\pi$
$74$	−1.85410	−0.215535
$75$	0	0
$76$	−0.618034	−0.0708934
$77$	−3.47214	−0.395687
$78$	0	0
$79$	10.7984	1.21491	0.607456	−	0.794353i	$-0.292190\pi$
$79$	10.7984	1.21491	0.607456	+	0.794353i	$0.292190\pi$
$80$	1.85410	0.207295
$81$	0	0
$82$	0.381966	0.0421811
$83$	14.4164	1.58241	0.791203	−	0.611553i	$-0.209455\pi$
$83$	14.4164	1.58241	0.791203	+	0.611553i	$0.209455\pi$
$84$	0	0
$85$	−6.61803	−0.717827
$86$	−4.79837	−0.517422
$87$	0	0
$88$	−7.76393	−0.827638
$89$	−13.4721	−1.42804	−0.714022	−	0.700123i	$-0.753128\pi$
$89$	−13.4721	−1.42804	−0.714022	+	0.700123i	$0.753128\pi$
$90$	0	0
$91$	6.09017	0.638423
$92$	7.09017	0.739201
$93$	0	0
$94$	−6.61803	−0.682598
$95$	0.381966	0.0391889
$96$	0	0
$97$	16.0344	1.62805	0.814025	−	0.580829i	$-0.197272\pi$
$97$	16.0344	1.62805	0.814025	+	0.580829i	$0.197272\pi$
$98$	0.618034	0.0624309
$99$	0	0
$100$	−1.61803	−0.161803
$101$	2.47214	0.245987	0.122993	−	0.992407i	$-0.460751\pi$
$101$	2.47214	0.245987	0.122993	+	0.992407i	$0.460751\pi$
$102$	0	0
$103$	−1.32624	−0.130678	−0.0653391	−	0.997863i	$-0.520813\pi$
$103$	−1.32624	−0.130678	−0.0653391	+	0.997863i	$0.520813\pi$
$104$	13.6180	1.33536
$105$	0	0
$106$	−4.09017	−0.397272
$107$	−1.29180	−0.124883	−0.0624413	−	0.998049i	$-0.519889\pi$
$107$	−1.29180	−0.124883	−0.0624413	+	0.998049i	$0.519889\pi$
$108$	0	0
$109$	−0.145898	−0.0139745	−0.00698725	−	0.999976i	$-0.502224\pi$
$109$	−0.145898	−0.0139745	−0.00698725	+	0.999976i	$0.502224\pi$
$110$	2.14590	0.204603
$111$	0	0
$112$	−1.85410	−0.175196
$113$	18.5623	1.74619	0.873097	−	0.487546i	$-0.162108\pi$
$113$	18.5623	1.74619	0.873097	+	0.487546i	$0.162108\pi$
$114$	0	0
$115$	−4.38197	−0.408620
$116$	4.61803	0.428774
$117$	0	0
$118$	−6.61803	−0.609239
$119$	6.61803	0.606674
$120$	0	0
$121$	1.05573	0.0959753
$122$	0.708204	0.0641178
$123$	0	0
$124$	−4.85410	−0.435911
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.79837	−0.780729	−0.390365	−	0.920660i	$-0.627651\pi$
$127$	−8.79837	−0.780729	−0.390365	+	0.920660i	$0.627651\pi$
$128$	−11.3820	−1.00603
$129$	0	0
$130$	−3.76393	−0.330119
$131$	−1.67376	−0.146237	−0.0731186	−	0.997323i	$-0.523295\pi$
$131$	−1.67376	−0.146237	−0.0731186	+	0.997323i	$0.523295\pi$
$132$	0	0
$133$	−0.381966	−0.0331207
$134$	7.90983	0.683305
$135$	0	0
$136$	14.7984	1.26895
$137$	9.52786	0.814020	0.407010	−	0.913424i	$-0.366571\pi$
$137$	9.52786	0.814020	0.407010	+	0.913424i	$0.366571\pi$
$138$	0	0
$139$	−9.47214	−0.803416	−0.401708	−	0.915768i	$-0.631583\pi$
$139$	−9.47214	−0.803416	−0.401708	+	0.915768i	$0.631583\pi$
$140$	1.61803	0.136749
$141$	0	0
$142$	−5.47214	−0.459211
$143$	−21.1459	−1.76831
$144$	0	0
$145$	−2.85410	−0.237020
$146$	−1.56231	−0.129297
$147$	0	0
$148$	4.85410	0.399005
$149$	11.0902	0.908542	0.454271	−	0.890864i	$-0.349900\pi$
$149$	11.0902	0.908542	0.454271	+	0.890864i	$0.349900\pi$
$150$	0	0
$151$	20.8885	1.69989	0.849943	−	0.526875i	$-0.176636\pi$
$151$	20.8885	1.69989	0.849943	+	0.526875i	$0.176636\pi$
$152$	−0.854102	−0.0692768
$153$	0	0
$154$	−2.14590	−0.172921
$155$	3.00000	0.240966
$156$	0	0
$157$	−0.763932	−0.0609684	−0.0304842	−	0.999535i	$-0.509705\pi$
$157$	−0.763932	−0.0609684	−0.0304842	+	0.999535i	$0.509705\pi$
$158$	6.67376	0.530936
$159$	0	0
$160$	5.61803	0.444145
$161$	4.38197	0.345347
$162$	0	0
$163$	−16.4721	−1.29020	−0.645099	−	0.764099i	$-0.723184\pi$
$163$	−16.4721	−1.29020	−0.645099	+	0.764099i	$0.723184\pi$
$164$	−1.00000	−0.0780869
$165$	0	0
$166$	8.90983	0.691537
$167$	3.76393	0.291262	0.145631	−	0.989339i	$-0.453479\pi$
$167$	3.76393	0.291262	0.145631	+	0.989339i	$0.453479\pi$
$168$	0	0
$169$	24.0902	1.85309
$170$	−4.09017	−0.313702
$171$	0	0
$172$	12.5623	0.957867
$173$	−7.18034	−0.545911	−0.272956	−	0.962027i	$-0.588001\pi$
$173$	−7.18034	−0.545911	−0.272956	+	0.962027i	$0.588001\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	6.43769	0.485259
$177$	0	0
$178$	−8.32624	−0.624078
$179$	1.29180	0.0965534	0.0482767	−	0.998834i	$-0.484627\pi$
$179$	1.29180	0.0965534	0.0482767	+	0.998834i	$0.484627\pi$
$180$	0	0
$181$	23.1803	1.72298	0.861491	−	0.507774i	$-0.169531\pi$
$181$	23.1803	1.72298	0.861491	+	0.507774i	$0.169531\pi$
$182$	3.76393	0.279001
$183$	0	0
$184$	9.79837	0.722346
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−22.9787	−1.68037
$188$	17.3262	1.26365
$189$	0	0
$190$	0.236068	0.0171262
$191$	−16.3820	−1.18536	−0.592679	−	0.805439i	$-0.701930\pi$
$191$	−16.3820	−1.18536	−0.592679	+	0.805439i	$0.701930\pi$
$192$	0	0
$193$	3.56231	0.256420	0.128210	−	0.991747i	$-0.459077\pi$
$193$	3.56231	0.256420	0.128210	+	0.991747i	$0.459077\pi$
$194$	9.90983	0.711484
$195$	0	0
$196$	−1.61803	−0.115574
$197$	20.8885	1.48825	0.744124	−	0.668042i	$-0.232867\pi$
$197$	20.8885	1.48825	0.744124	+	0.668042i	$0.232867\pi$
$198$	0	0
$199$	−2.03444	−0.144218	−0.0721089	−	0.997397i	$-0.522973\pi$
$199$	−2.03444	−0.144218	−0.0721089	+	0.997397i	$0.522973\pi$
$200$	−2.23607	−0.158114
$201$	0	0
$202$	1.52786	0.107500
$203$	2.85410	0.200319
$204$	0	0
$205$	0.618034	0.0431654
$206$	−0.819660	−0.0571084
$207$	0	0
$208$	−11.2918	−0.782945
$209$	1.32624	0.0917378
$210$	0	0
$211$	3.70820	0.255283	0.127642	−	0.991820i	$-0.459259\pi$
$211$	3.70820	0.255283	0.127642	+	0.991820i	$0.459259\pi$
$212$	10.7082	0.735442
$213$	0	0
$214$	−0.798374	−0.0545757
$215$	−7.76393	−0.529496
$216$	0	0
$217$	−3.00000	−0.203653
$218$	−0.0901699	−0.00610708
$219$	0	0
$220$	−5.61803	−0.378768
$221$	40.3050	2.71120
$222$	0	0
$223$	−19.4721	−1.30395	−0.651975	−	0.758240i	$-0.726059\pi$
$223$	−19.4721	−1.30395	−0.651975	+	0.758240i	$0.726059\pi$
$224$	−5.61803	−0.375371
$225$	0	0
$226$	11.4721	0.763115
$227$	−5.27051	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$227$	−5.27051	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$228$	0	0
$229$	−17.1803	−1.13531	−0.567654	−	0.823267i	$-0.692149\pi$
$229$	−17.1803	−1.13531	−0.567654	+	0.823267i	$0.692149\pi$
$230$	−2.70820	−0.178574
$231$	0	0
$232$	6.38197	0.418997
$233$	17.9098	1.17331	0.586656	−	0.809836i	$-0.300444\pi$
$233$	17.9098	1.17331	0.586656	+	0.809836i	$0.300444\pi$
$234$	0	0
$235$	−10.7082	−0.698526
$236$	17.3262	1.12784
$237$	0	0
$238$	4.09017	0.265126
$239$	−23.3607	−1.51108	−0.755538	−	0.655104i	$-0.772625\pi$
$239$	−23.3607	−1.51108	−0.755538	+	0.655104i	$0.772625\pi$
$240$	0	0
$241$	−18.7984	−1.21091	−0.605455	−	0.795880i	$-0.707009\pi$
$241$	−18.7984	−1.21091	−0.605455	+	0.795880i	$0.707009\pi$
$242$	0.652476	0.0419427
$243$	0	0
$244$	−1.85410	−0.118697
$245$	1.00000	0.0638877
$246$	0	0
$247$	−2.32624	−0.148015
$248$	−6.70820	−0.425971
$249$	0	0
$250$	0.618034	0.0390879
$251$	−10.6525	−0.672378	−0.336189	−	0.941794i	$-0.609138\pi$
$251$	−10.6525	−0.672378	−0.336189	+	0.941794i	$0.609138\pi$
$252$	0	0
$253$	−15.2148	−0.956545
$254$	−5.43769	−0.341191
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−10.4721	−0.653234	−0.326617	−	0.945157i	$-0.605909\pi$
$257$	−10.4721	−0.653234	−0.326617	+	0.945157i	$0.605909\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	9.85410	0.611125
$261$	0	0
$262$	−1.03444	−0.0639080
$263$	8.90983	0.549404	0.274702	−	0.961529i	$-0.411421\pi$
$263$	8.90983	0.549404	0.274702	+	0.961529i	$0.411421\pi$
$264$	0	0
$265$	−6.61803	−0.406543
$266$	−0.236068	−0.0144743
$267$	0	0
$268$	−20.7082	−1.26495
$269$	−28.1246	−1.71479	−0.857394	−	0.514661i	$-0.827918\pi$
$269$	−28.1246	−1.71479	−0.857394	+	0.514661i	$0.827918\pi$
$270$	0	0
$271$	18.1459	1.10228	0.551142	−	0.834411i	$-0.314192\pi$
$271$	18.1459	1.10228	0.551142	+	0.834411i	$0.314192\pi$
$272$	−12.2705	−0.744009
$273$	0	0
$274$	5.88854	0.355740
$275$	3.47214	0.209378
$276$	0	0
$277$	−8.79837	−0.528643	−0.264322	−	0.964435i	$-0.585148\pi$
$277$	−8.79837	−0.528643	−0.264322	+	0.964435i	$0.585148\pi$
$278$	−5.85410	−0.351106
$279$	0	0
$280$	2.23607	0.133631
$281$	22.3262	1.33187	0.665936	−	0.746009i	$-0.268032\pi$
$281$	22.3262	1.33187	0.665936	+	0.746009i	$0.268032\pi$
$282$	0	0
$283$	26.9787	1.60372	0.801859	−	0.597513i	$-0.203844\pi$
$283$	26.9787	1.60372	0.801859	+	0.597513i	$0.203844\pi$
$284$	14.3262	0.850106
$285$	0	0
$286$	−13.0689	−0.772779
$287$	−0.618034	−0.0364814
$288$	0	0
$289$	26.7984	1.57637
$290$	−1.76393	−0.103582
$291$	0	0
$292$	4.09017	0.239359
$293$	−27.1246	−1.58464	−0.792318	−	0.610108i	$-0.791126\pi$
$293$	−27.1246	−1.58464	−0.792318	+	0.610108i	$0.791126\pi$
$294$	0	0
$295$	−10.7082	−0.623456
$296$	6.70820	0.389906
$297$	0	0
$298$	6.85410	0.397047
$299$	26.6869	1.54334
$300$	0	0
$301$	7.76393	0.447506
$302$	12.9098	0.742877
$303$	0	0
$304$	0.708204	0.0406183
$305$	1.14590	0.0656139
$306$	0	0
$307$	−31.4164	−1.79303	−0.896515	−	0.443014i	$-0.853909\pi$
$307$	−31.4164	−1.79303	−0.896515	+	0.443014i	$0.853909\pi$
$308$	5.61803	0.320117
$309$	0	0
$310$	1.85410	0.105306
$311$	−35.0344	−1.98662	−0.993310	−	0.115474i	$-0.963161\pi$
$311$	−35.0344	−1.98662	−0.993310	+	0.115474i	$0.963161\pi$
$312$	0	0
$313$	−6.81966	−0.385470	−0.192735	−	0.981251i	$-0.561736\pi$
$313$	−6.81966	−0.385470	−0.192735	+	0.981251i	$0.561736\pi$
$314$	−0.472136	−0.0266442
$315$	0	0
$316$	−17.4721	−0.982884
$317$	−23.9443	−1.34484	−0.672422	−	0.740168i	$-0.734746\pi$
$317$	−23.9443	−1.34484	−0.672422	+	0.740168i	$0.734746\pi$
$318$	0	0
$319$	−9.90983	−0.554844
$320$	−0.236068	−0.0131966
$321$	0	0
$322$	2.70820	0.150922
$323$	−2.52786	−0.140654
$324$	0	0
$325$	−6.09017	−0.337822
$326$	−10.1803	−0.563837
$327$	0	0
$328$	−1.38197	−0.0763063
$329$	10.7082	0.590362
$330$	0	0
$331$	10.8541	0.596595	0.298298	−	0.954473i	$-0.403581\pi$
$331$	10.8541	0.596595	0.298298	+	0.954473i	$0.403581\pi$
$332$	−23.3262	−1.28019
$333$	0	0
$334$	2.32624	0.127286
$335$	12.7984	0.699250
$336$	0	0
$337$	28.7426	1.56571	0.782856	−	0.622203i	$-0.213762\pi$
$337$	28.7426	1.56571	0.782856	+	0.622203i	$0.213762\pi$
$338$	14.8885	0.809830
$339$	0	0
$340$	10.7082	0.580734
$341$	10.4164	0.564080
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	17.3607	0.936025
$345$	0	0
$346$	−4.43769	−0.238572
$347$	2.18034	0.117047	0.0585234	−	0.998286i	$-0.481361\pi$
$347$	2.18034	0.117047	0.0585234	+	0.998286i	$0.481361\pi$
$348$	0	0
$349$	35.1803	1.88316	0.941580	−	0.336789i	$-0.109341\pi$
$349$	35.1803	1.88316	0.941580	+	0.336789i	$0.109341\pi$
$350$	−0.618034	−0.0330353
$351$	0	0
$352$	19.5066	1.03970
$353$	26.4508	1.40784	0.703918	−	0.710281i	$-0.251432\pi$
$353$	26.4508	1.40784	0.703918	+	0.710281i	$0.251432\pi$
$354$	0	0
$355$	−8.85410	−0.469927
$356$	21.7984	1.15531
$357$	0	0
$358$	0.798374	0.0421954
$359$	1.29180	0.0681784	0.0340892	−	0.999419i	$-0.489147\pi$
$359$	1.29180	0.0681784	0.0340892	+	0.999419i	$0.489147\pi$
$360$	0	0
$361$	−18.8541	−0.992321
$362$	14.3262	0.752970
$363$	0	0
$364$	−9.85410	−0.516495
$365$	−2.52786	−0.132314
$366$	0	0
$367$	14.9787	0.781882	0.390941	−	0.920416i	$-0.372150\pi$
$367$	14.9787	0.781882	0.390941	+	0.920416i	$0.372150\pi$
$368$	−8.12461	−0.423525
$369$	0	0
$370$	−1.85410	−0.0963902
$371$	6.61803	0.343591
$372$	0	0
$373$	9.14590	0.473557	0.236778	−	0.971564i	$-0.423908\pi$
$373$	9.14590	0.473557	0.236778	+	0.971564i	$0.423908\pi$
$374$	−14.2016	−0.734349
$375$	0	0
$376$	23.9443	1.23483
$377$	17.3820	0.895217
$378$	0	0
$379$	−11.8885	−0.610673	−0.305337	−	0.952244i	$-0.598769\pi$
$379$	−11.8885	−0.610673	−0.305337	+	0.952244i	$0.598769\pi$
$380$	−0.618034	−0.0317045
$381$	0	0
$382$	−10.1246	−0.518020
$383$	−33.4721	−1.71035	−0.855173	−	0.518342i	$-0.826549\pi$
$383$	−33.4721	−1.71035	−0.855173	+	0.518342i	$0.826549\pi$
$384$	0	0
$385$	−3.47214	−0.176956
$386$	2.20163	0.112060
$387$	0	0
$388$	−25.9443	−1.31712
$389$	10.9098	0.553150	0.276575	−	0.960992i	$-0.410801\pi$
$389$	10.9098	0.553150	0.276575	+	0.960992i	$0.410801\pi$
$390$	0	0
$391$	29.0000	1.46659
$392$	−2.23607	−0.112938
$393$	0	0
$394$	12.9098	0.650388
$395$	10.7984	0.543325
$396$	0	0
$397$	10.3607	0.519988	0.259994	−	0.965610i	$-0.416279\pi$
$397$	10.3607	0.519988	0.259994	+	0.965610i	$0.416279\pi$
$398$	−1.25735	−0.0630255
$399$	0	0
$400$	1.85410	0.0927051
$401$	−13.6180	−0.680052	−0.340026	−	0.940416i	$-0.610436\pi$
$401$	−13.6180	−0.680052	−0.340026	+	0.940416i	$0.610436\pi$
$402$	0	0
$403$	−18.2705	−0.910119
$404$	−4.00000	−0.199007
$405$	0	0
$406$	1.76393	0.0875425
$407$	−10.4164	−0.516322
$408$	0	0
$409$	0.472136	0.0233456	0.0116728	−	0.999932i	$-0.496284\pi$
$409$	0.472136	0.0233456	0.0116728	+	0.999932i	$0.496284\pi$
$410$	0.381966	0.0188640
$411$	0	0
$412$	2.14590	0.105721
$413$	10.7082	0.526916
$414$	0	0
$415$	14.4164	0.707674
$416$	−34.2148	−1.67752
$417$	0	0
$418$	0.819660	0.0400909
$419$	−19.4721	−0.951276	−0.475638	−	0.879641i	$-0.657783\pi$
$419$	−19.4721	−0.951276	−0.475638	+	0.879641i	$0.657783\pi$
$420$	0	0
$421$	−38.8541	−1.89363	−0.946817	−	0.321774i	$-0.895721\pi$
$421$	−38.8541	−1.89363	−0.946817	+	0.321774i	$0.895721\pi$
$422$	2.29180	0.111563
$423$	0	0
$424$	14.7984	0.718673
$425$	−6.61803	−0.321022
$426$	0	0
$427$	−1.14590	−0.0554539
$428$	2.09017	0.101032
$429$	0	0
$430$	−4.79837	−0.231398
$431$	−20.3262	−0.979080	−0.489540	−	0.871981i	$-0.662835\pi$
$431$	−20.3262	−0.979080	−0.489540	+	0.871981i	$0.662835\pi$
$432$	0	0
$433$	7.09017	0.340732	0.170366	−	0.985381i	$-0.445505\pi$
$433$	7.09017	0.340732	0.170366	+	0.985381i	$0.445505\pi$
$434$	−1.85410	−0.0889997
$435$	0	0
$436$	0.236068	0.0113056
$437$	−1.67376	−0.0800669
$438$	0	0
$439$	7.58359	0.361945	0.180973	−	0.983488i	$-0.442075\pi$
$439$	7.58359	0.361945	0.180973	+	0.983488i	$0.442075\pi$
$440$	−7.76393	−0.370131
$441$	0	0
$442$	24.9098	1.18484
$443$	−33.9787	−1.61438	−0.807189	−	0.590293i	$-0.799012\pi$
$443$	−33.9787	−1.61438	−0.807189	+	0.590293i	$0.799012\pi$
$444$	0	0
$445$	−13.4721	−0.638640
$446$	−12.0344	−0.569847
$447$	0	0
$448$	0.236068	0.0111532
$449$	30.9443	1.46035	0.730175	−	0.683260i	$-0.239438\pi$
$449$	30.9443	1.46035	0.730175	+	0.683260i	$0.239438\pi$
$450$	0	0
$451$	2.14590	0.101046
$452$	−30.0344	−1.41270
$453$	0	0
$454$	−3.25735	−0.152875
$455$	6.09017	0.285512
$456$	0	0
$457$	−32.0344	−1.49851	−0.749254	−	0.662283i	$-0.769588\pi$
$457$	−32.0344	−1.49851	−0.749254	+	0.662283i	$0.769588\pi$
$458$	−10.6180	−0.496148
$459$	0	0
$460$	7.09017	0.330581
$461$	−22.7426	−1.05923	−0.529615	−	0.848238i	$-0.677664\pi$
$461$	−22.7426	−1.05923	−0.529615	+	0.848238i	$0.677664\pi$
$462$	0	0
$463$	−11.7984	−0.548317	−0.274158	−	0.961685i	$-0.588399\pi$
$463$	−11.7984	−0.548317	−0.274158	+	0.961685i	$0.588399\pi$
$464$	−5.29180	−0.245665
$465$	0	0
$466$	11.0689	0.512756
$467$	−17.5279	−0.811093	−0.405546	−	0.914074i	$-0.632919\pi$
$467$	−17.5279	−0.811093	−0.405546	+	0.914074i	$0.632919\pi$
$468$	0	0
$469$	−12.7984	−0.590974
$470$	−6.61803	−0.305267
$471$	0	0
$472$	23.9443	1.10212
$473$	−26.9574	−1.23950
$474$	0	0
$475$	0.381966	0.0175258
$476$	−10.7082	−0.490810
$477$	0	0
$478$	−14.4377	−0.660365
$479$	4.79837	0.219243	0.109622	−	0.993973i	$-0.465036\pi$
$479$	4.79837	0.219243	0.109622	+	0.993973i	$0.465036\pi$
$480$	0	0
$481$	18.2705	0.833064
$482$	−11.6180	−0.529187
$483$	0	0
$484$	−1.70820	−0.0776456
$485$	16.0344	0.728086
$486$	0	0
$487$	−0.472136	−0.0213945	−0.0106973	−	0.999943i	$-0.503405\pi$
$487$	−0.472136	−0.0213945	−0.0106973	+	0.999943i	$0.503405\pi$
$488$	−2.56231	−0.115990
$489$	0	0
$490$	0.618034	0.0279199
$491$	8.79837	0.397065	0.198533	−	0.980094i	$-0.436382\pi$
$491$	8.79837	0.397065	0.198533	+	0.980094i	$0.436382\pi$
$492$	0	0
$493$	18.8885	0.850697
$494$	−1.43769	−0.0646849
$495$	0	0
$496$	5.56231	0.249755
$497$	8.85410	0.397161
$498$	0	0
$499$	−41.3607	−1.85156	−0.925779	−	0.378065i	$-0.876590\pi$
$499$	−41.3607	−1.85156	−0.925779	+	0.378065i	$0.876590\pi$
$500$	−1.61803	−0.0723607
$501$	0	0
$502$	−6.58359	−0.293840
$503$	21.7984	0.971941	0.485971	−	0.873975i	$-0.338466\pi$
$503$	21.7984	0.971941	0.485971	+	0.873975i	$0.338466\pi$
$504$	0	0
$505$	2.47214	0.110009
$506$	−9.40325	−0.418026
$507$	0	0
$508$	14.2361	0.631623
$509$	−21.4721	−0.951736	−0.475868	−	0.879517i	$-0.657866\pi$
$509$	−21.4721	−0.951736	−0.475868	+	0.879517i	$0.657866\pi$
$510$	0	0
$511$	2.52786	0.111826
$512$	18.7082	0.826794
$513$	0	0
$514$	−6.47214	−0.285474
$515$	−1.32624	−0.0584410
$516$	0	0
$517$	−37.1803	−1.63519
$518$	1.85410	0.0814646
$519$	0	0
$520$	13.6180	0.597190
$521$	−18.4164	−0.806837	−0.403419	−	0.915015i	$-0.632178\pi$
$521$	−18.4164	−0.806837	−0.403419	+	0.915015i	$0.632178\pi$
$522$	0	0
$523$	22.7426	0.994466	0.497233	−	0.867617i	$-0.334349\pi$
$523$	22.7426	0.994466	0.497233	+	0.867617i	$0.334349\pi$
$524$	2.70820	0.118308
$525$	0	0
$526$	5.50658	0.240098
$527$	−19.8541	−0.864858
$528$	0	0
$529$	−3.79837	−0.165147
$530$	−4.09017	−0.177666
$531$	0	0
$532$	0.618034	0.0267952
$533$	−3.76393	−0.163034
$534$	0	0
$535$	−1.29180	−0.0558492
$536$	−28.6180	−1.23611
$537$	0	0
$538$	−17.3820	−0.749390
$539$	3.47214	0.149555
$540$	0	0
$541$	−9.74265	−0.418869	−0.209435	−	0.977823i	$-0.567162\pi$
$541$	−9.74265	−0.418869	−0.209435	+	0.977823i	$0.567162\pi$
$542$	11.2148	0.481716
$543$	0	0
$544$	−37.1803	−1.59409
$545$	−0.145898	−0.00624959
$546$	0	0
$547$	29.1803	1.24766	0.623831	−	0.781560i	$-0.285576\pi$
$547$	29.1803	1.24766	0.623831	+	0.781560i	$0.285576\pi$
$548$	−15.4164	−0.658556
$549$	0	0
$550$	2.14590	0.0915014
$551$	−1.09017	−0.0464428
$552$	0	0
$553$	−10.7984	−0.459194
$554$	−5.43769	−0.231025
$555$	0	0
$556$	15.3262	0.649977
$557$	31.7984	1.34734	0.673670	−	0.739032i	$-0.264717\pi$
$557$	31.7984	1.34734	0.673670	+	0.739032i	$0.264717\pi$
$558$	0	0
$559$	47.2837	1.99989
$560$	−1.85410	−0.0783501
$561$	0	0
$562$	13.7984	0.582049
$563$	−17.8541	−0.752461	−0.376230	−	0.926526i	$-0.622780\pi$
$563$	−17.8541	−0.752461	−0.376230	+	0.926526i	$0.622780\pi$
$564$	0	0
$565$	18.5623	0.780922
$566$	16.6738	0.700850
$567$	0	0
$568$	19.7984	0.830721
$569$	−21.9443	−0.919952	−0.459976	−	0.887931i	$-0.652142\pi$
$569$	−21.9443	−0.919952	−0.459976	+	0.887931i	$0.652142\pi$
$570$	0	0
$571$	36.3262	1.52021	0.760103	−	0.649803i	$-0.225149\pi$
$571$	36.3262	1.52021	0.760103	+	0.649803i	$0.225149\pi$
$572$	34.2148	1.43059
$573$	0	0
$574$	−0.381966	−0.0159430
$575$	−4.38197	−0.182741
$576$	0	0
$577$	−12.1246	−0.504754	−0.252377	−	0.967629i	$-0.581212\pi$
$577$	−12.1246	−0.504754	−0.252377	+	0.967629i	$0.581212\pi$
$578$	16.5623	0.688901
$579$	0	0
$580$	4.61803	0.191753
$581$	−14.4164	−0.598093
$582$	0	0
$583$	−22.9787	−0.951681
$584$	5.65248	0.233901
$585$	0	0
$586$	−16.7639	−0.692512
$587$	−23.5066	−0.970220	−0.485110	−	0.874453i	$-0.661220\pi$
$587$	−23.5066	−0.970220	−0.485110	+	0.874453i	$0.661220\pi$
$588$	0	0
$589$	1.14590	0.0472159
$590$	−6.61803	−0.272460
$591$	0	0
$592$	−5.56231	−0.228609
$593$	−23.0689	−0.947326	−0.473663	−	0.880706i	$-0.657068\pi$
$593$	−23.0689	−0.947326	−0.473663	+	0.880706i	$0.657068\pi$
$594$	0	0
$595$	6.61803	0.271313
$596$	−17.9443	−0.735026
$597$	0	0
$598$	16.4934	0.674466
$599$	−1.63932	−0.0669808	−0.0334904	−	0.999439i	$-0.510662\pi$
$599$	−1.63932	−0.0669808	−0.0334904	+	0.999439i	$0.510662\pi$
$600$	0	0
$601$	−12.6738	−0.516974	−0.258487	−	0.966015i	$-0.583224\pi$
$601$	−12.6738	−0.516974	−0.258487	+	0.966015i	$0.583224\pi$
$602$	4.79837	0.195567
$603$	0	0
$604$	−33.7984	−1.37524
$605$	1.05573	0.0429215
$606$	0	0
$607$	−20.4164	−0.828676	−0.414338	−	0.910123i	$-0.635987\pi$
$607$	−20.4164	−0.828676	−0.414338	+	0.910123i	$0.635987\pi$
$608$	2.14590	0.0870277
$609$	0	0
$610$	0.708204	0.0286743
$611$	65.2148	2.63831
$612$	0	0
$613$	19.0000	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$613$	19.0000	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$614$	−19.4164	−0.783582
$615$	0	0
$616$	7.76393	0.312818
$617$	−28.0344	−1.12862	−0.564312	−	0.825562i	$-0.690859\pi$
$617$	−28.0344	−1.12862	−0.564312	+	0.825562i	$0.690859\pi$
$618$	0	0
$619$	31.0000	1.24600	0.622998	−	0.782224i	$-0.285915\pi$
$619$	31.0000	1.24600	0.622998	+	0.782224i	$0.285915\pi$
$620$	−4.85410	−0.194945
$621$	0	0
$622$	−21.6525	−0.868185
$623$	13.4721	0.539750
$624$	0	0
$625$	1.00000	0.0400000
$626$	−4.21478	−0.168457
$627$	0	0
$628$	1.23607	0.0493245
$629$	19.8541	0.791635
$630$	0	0
$631$	−49.3050	−1.96280	−0.981400	−	0.191976i	$-0.938510\pi$
$631$	−49.3050	−1.96280	−0.981400	+	0.191976i	$0.938510\pi$
$632$	−24.1459	−0.960472
$633$	0	0
$634$	−14.7984	−0.587719
$635$	−8.79837	−0.349153
$636$	0	0
$637$	−6.09017	−0.241301
$638$	−6.12461	−0.242476
$639$	0	0
$640$	−11.3820	−0.449912
$641$	37.3951	1.47702	0.738509	−	0.674243i	$-0.235530\pi$
$641$	37.3951	1.47702	0.738509	+	0.674243i	$0.235530\pi$
$642$	0	0
$643$	16.3820	0.646042	0.323021	−	0.946392i	$-0.395302\pi$
$643$	16.3820	0.646042	0.323021	+	0.946392i	$0.395302\pi$
$644$	−7.09017	−0.279392
$645$	0	0
$646$	−1.56231	−0.0614681
$647$	−18.9443	−0.744776	−0.372388	−	0.928077i	$-0.621461\pi$
$647$	−18.9443	−0.744776	−0.372388	+	0.928077i	$0.621461\pi$
$648$	0	0
$649$	−37.1803	−1.45946
$650$	−3.76393	−0.147634
$651$	0	0
$652$	26.6525	1.04379
$653$	−6.45085	−0.252441	−0.126221	−	0.992002i	$-0.540285\pi$
$653$	−6.45085	−0.252441	−0.126221	+	0.992002i	$0.540285\pi$
$654$	0	0
$655$	−1.67376	−0.0653993
$656$	1.14590	0.0447398
$657$	0	0
$658$	6.61803	0.257998
$659$	−17.2361	−0.671422	−0.335711	−	0.941965i	$-0.608977\pi$
$659$	−17.2361	−0.671422	−0.335711	+	0.941965i	$0.608977\pi$
$660$	0	0
$661$	−31.3262	−1.21845	−0.609225	−	0.792998i	$-0.708519\pi$
$661$	−31.3262	−1.21845	−0.609225	+	0.792998i	$0.708519\pi$
$662$	6.70820	0.260722
$663$	0	0
$664$	−32.2361	−1.25100
$665$	−0.381966	−0.0148120
$666$	0	0
$667$	12.5066	0.484257
$668$	−6.09017	−0.235636
$669$	0	0
$670$	7.90983	0.305583
$671$	3.97871	0.153597
$672$	0	0
$673$	3.05573	0.117790	0.0588948	−	0.998264i	$-0.481242\pi$
$673$	3.05573	0.117790	0.0588948	+	0.998264i	$0.481242\pi$
$674$	17.7639	0.684241
$675$	0	0
$676$	−38.9787	−1.49918
$677$	20.5279	0.788950	0.394475	−	0.918907i	$-0.370926\pi$
$677$	20.5279	0.788950	0.394475	+	0.918907i	$0.370926\pi$
$678$	0	0
$679$	−16.0344	−0.615345
$680$	14.7984	0.567492
$681$	0	0
$682$	6.43769	0.246512
$683$	13.1459	0.503014	0.251507	−	0.967856i	$-0.419074\pi$
$683$	13.1459	0.503014	0.251507	+	0.967856i	$0.419074\pi$
$684$	0	0
$685$	9.52786	0.364041
$686$	−0.618034	−0.0235966
$687$	0	0
$688$	−14.3951	−0.548809
$689$	40.3050	1.53550
$690$	0	0
$691$	13.6525	0.519365	0.259682	−	0.965694i	$-0.416382\pi$
$691$	13.6525	0.519365	0.259682	+	0.965694i	$0.416382\pi$
$692$	11.6180	0.441651
$693$	0	0
$694$	1.34752	0.0511513
$695$	−9.47214	−0.359299
$696$	0	0
$697$	−4.09017	−0.154926
$698$	21.7426	0.822971
$699$	0	0
$700$	1.61803	0.0611559
$701$	−1.18034	−0.0445808	−0.0222904	−	0.999752i	$-0.507096\pi$
$701$	−1.18034	−0.0445808	−0.0222904	+	0.999752i	$0.507096\pi$
$702$	0	0
$703$	−1.14590	−0.0432184
$704$	−0.819660	−0.0308921
$705$	0	0
$706$	16.3475	0.615247
$707$	−2.47214	−0.0929742
$708$	0	0
$709$	12.7639	0.479360	0.239680	−	0.970852i	$-0.422958\pi$
$709$	12.7639	0.479360	0.239680	+	0.970852i	$0.422958\pi$
$710$	−5.47214	−0.205366
$711$	0	0
$712$	30.1246	1.12897
$713$	−13.1459	−0.492318
$714$	0	0
$715$	−21.1459	−0.790812
$716$	−2.09017	−0.0781133
$717$	0	0
$718$	0.798374	0.0297950
$719$	9.74265	0.363339	0.181670	−	0.983360i	$-0.441850\pi$
$719$	9.74265	0.363339	0.181670	+	0.983360i	$0.441850\pi$
$720$	0	0
$721$	1.32624	0.0493917
$722$	−11.6525	−0.433660
$723$	0	0
$724$	−37.5066	−1.39392
$725$	−2.85410	−0.105999
$726$	0	0
$727$	−1.25735	−0.0466327	−0.0233163	−	0.999728i	$-0.507422\pi$
$727$	−1.25735	−0.0466327	−0.0233163	+	0.999728i	$0.507422\pi$
$728$	−13.6180	−0.504718
$729$	0	0
$730$	−1.56231	−0.0578235
$731$	51.3820	1.90043
$732$	0	0
$733$	37.5623	1.38740	0.693698	−	0.720266i	$-0.255980\pi$
$733$	37.5623	1.38740	0.693698	+	0.720266i	$0.255980\pi$
$734$	9.25735	0.341695
$735$	0	0
$736$	−24.6180	−0.907433
$737$	44.4377	1.63688
$738$	0	0
$739$	19.8197	0.729078	0.364539	−	0.931188i	$-0.381227\pi$
$739$	19.8197	0.729078	0.364539	+	0.931188i	$0.381227\pi$
$740$	4.85410	0.178440
$741$	0	0
$742$	4.09017	0.150155
$743$	27.1591	0.996369	0.498185	−	0.867071i	$-0.334000\pi$
$743$	27.1591	0.996369	0.498185	+	0.867071i	$0.334000\pi$
$744$	0	0
$745$	11.0902	0.406312
$746$	5.65248	0.206952
$747$	0	0
$748$	37.1803	1.35945
$749$	1.29180	0.0472012
$750$	0	0
$751$	−2.94427	−0.107438	−0.0537190	−	0.998556i	$-0.517108\pi$
$751$	−2.94427	−0.107438	−0.0537190	+	0.998556i	$0.517108\pi$
$752$	−19.8541	−0.724005
$753$	0	0
$754$	10.7426	0.391224
$755$	20.8885	0.760212
$756$	0	0
$757$	28.6869	1.04264	0.521322	−	0.853360i	$-0.325439\pi$
$757$	28.6869	1.04264	0.521322	+	0.853360i	$0.325439\pi$
$758$	−7.34752	−0.266874
$759$	0	0
$760$	−0.854102	−0.0309815
$761$	−9.90983	−0.359231	−0.179616	−	0.983737i	$-0.557485\pi$
$761$	−9.90983	−0.359231	−0.179616	+	0.983737i	$0.557485\pi$
$762$	0	0
$763$	0.145898	0.00528186
$764$	26.5066	0.958974
$765$	0	0
$766$	−20.6869	−0.747449
$767$	65.2148	2.35477
$768$	0	0
$769$	−4.88854	−0.176285	−0.0881427	−	0.996108i	$-0.528093\pi$
$769$	−4.88854	−0.176285	−0.0881427	+	0.996108i	$0.528093\pi$
$770$	−2.14590	−0.0773328
$771$	0	0
$772$	−5.76393	−0.207448
$773$	37.7984	1.35951	0.679757	−	0.733438i	$-0.262085\pi$
$773$	37.7984	1.35951	0.679757	+	0.733438i	$0.262085\pi$
$774$	0	0
$775$	3.00000	0.107763
$776$	−35.8541	−1.28709
$777$	0	0
$778$	6.74265	0.241736
$779$	0.236068	0.00845801
$780$	0	0
$781$	−30.7426	−1.10006
$782$	17.9230	0.640925
$783$	0	0
$784$	1.85410	0.0662179
$785$	−0.763932	−0.0272659
$786$	0	0
$787$	−25.5967	−0.912426	−0.456213	−	0.889871i	$-0.650794\pi$
$787$	−25.5967	−0.912426	−0.456213	+	0.889871i	$0.650794\pi$
$788$	−33.7984	−1.20402
$789$	0	0
$790$	6.67376	0.237442
$791$	−18.5623	−0.660000
$792$	0	0
$793$	−6.97871	−0.247821
$794$	6.40325	0.227243
$795$	0	0
$796$	3.29180	0.116675
$797$	15.5623	0.551245	0.275623	−	0.961266i	$-0.411116\pi$
$797$	15.5623	0.551245	0.275623	+	0.961266i	$0.411116\pi$
$798$	0	0
$799$	70.8673	2.50710
$800$	5.61803	0.198627
$801$	0	0
$802$	−8.41641	−0.297194
$803$	−8.77709	−0.309737
$804$	0	0
$805$	4.38197	0.154444
$806$	−11.2918	−0.397737
$807$	0	0
$808$	−5.52786	−0.194470
$809$	15.1803	0.533712	0.266856	−	0.963736i	$-0.414015\pi$
$809$	15.1803	0.533712	0.266856	+	0.963736i	$0.414015\pi$
$810$	0	0
$811$	−8.20163	−0.287998	−0.143999	−	0.989578i	$-0.545996\pi$
$811$	−8.20163	−0.287998	−0.143999	+	0.989578i	$0.545996\pi$
$812$	−4.61803	−0.162061
$813$	0	0
$814$	−6.43769	−0.225641
$815$	−16.4721	−0.576994
$816$	0	0
$817$	−2.96556	−0.103752
$818$	0.291796	0.0102024
$819$	0	0
$820$	−1.00000	−0.0349215
$821$	22.1246	0.772154	0.386077	−	0.922466i	$-0.373830\pi$
$821$	22.1246	0.772154	0.386077	+	0.922466i	$0.373830\pi$
$822$	0	0
$823$	12.2361	0.426523	0.213261	−	0.976995i	$-0.431591\pi$
$823$	12.2361	0.426523	0.213261	+	0.976995i	$0.431591\pi$
$824$	2.96556	0.103310
$825$	0	0
$826$	6.61803	0.230271
$827$	−48.9574	−1.70242	−0.851208	−	0.524828i	$-0.824130\pi$
$827$	−48.9574	−1.70242	−0.851208	+	0.524828i	$0.824130\pi$
$828$	0	0
$829$	14.2361	0.494439	0.247220	−	0.968959i	$-0.420483\pi$
$829$	14.2361	0.494439	0.247220	+	0.968959i	$0.420483\pi$
$830$	8.90983	0.309265
$831$	0	0
$832$	1.43769	0.0498431
$833$	−6.61803	−0.229301
$834$	0	0
$835$	3.76393	0.130256
$836$	−2.14590	−0.0742174
$837$	0	0
$838$	−12.0344	−0.415723
$839$	−8.50658	−0.293680	−0.146840	−	0.989160i	$-0.546910\pi$
$839$	−8.50658	−0.293680	−0.146840	+	0.989160i	$0.546910\pi$
$840$	0	0
$841$	−20.8541	−0.719107
$842$	−24.0132	−0.827548
$843$	0	0
$844$	−6.00000	−0.206529
$845$	24.0902	0.828727
$846$	0	0
$847$	−1.05573	−0.0362752
$848$	−12.2705	−0.421371
$849$	0	0
$850$	−4.09017	−0.140292
$851$	13.1459	0.450636
$852$	0	0
$853$	1.00000	0.0342393	0.0171197	−	0.999853i	$-0.494550\pi$
$853$	1.00000	0.0342393	0.0171197	+	0.999853i	$0.494550\pi$
$854$	−0.708204	−0.0242342
$855$	0	0
$856$	2.88854	0.0987284
$857$	9.11146	0.311241	0.155621	−	0.987817i	$-0.450262\pi$
$857$	9.11146	0.311241	0.155621	+	0.987817i	$0.450262\pi$
$858$	0	0
$859$	14.0689	0.480024	0.240012	−	0.970770i	$-0.422849\pi$
$859$	14.0689	0.480024	0.240012	+	0.970770i	$0.422849\pi$
$860$	12.5623	0.428371
$861$	0	0
$862$	−12.5623	−0.427874
$863$	16.9443	0.576790	0.288395	−	0.957512i	$-0.406878\pi$
$863$	16.9443	0.576790	0.288395	+	0.957512i	$0.406878\pi$
$864$	0	0
$865$	−7.18034	−0.244139
$866$	4.38197	0.148905
$867$	0	0
$868$	4.85410	0.164759
$869$	37.4934	1.27188
$870$	0	0
$871$	−77.9443	−2.64104
$872$	0.326238	0.0110478
$873$	0	0
$874$	−1.03444	−0.0349905
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−53.6525	−1.81172	−0.905858	−	0.423582i	$-0.860772\pi$
$877$	−53.6525	−1.81172	−0.905858	+	0.423582i	$0.860772\pi$
$878$	4.68692	0.158176
$879$	0	0
$880$	6.43769	0.217015
$881$	27.5279	0.927437	0.463719	−	0.885983i	$-0.346515\pi$
$881$	27.5279	0.927437	0.463719	+	0.885983i	$0.346515\pi$
$882$	0	0
$883$	7.94427	0.267346	0.133673	−	0.991025i	$-0.457323\pi$
$883$	7.94427	0.267346	0.133673	+	0.991025i	$0.457323\pi$
$884$	−65.2148	−2.19341
$885$	0	0
$886$	−21.0000	−0.705509
$887$	−23.8197	−0.799786	−0.399893	−	0.916562i	$-0.630953\pi$
$887$	−23.8197	−0.799786	−0.399893	+	0.916562i	$0.630953\pi$
$888$	0	0
$889$	8.79837	0.295088
$890$	−8.32624	−0.279096
$891$	0	0
$892$	31.5066	1.05492
$893$	−4.09017	−0.136872
$894$	0	0
$895$	1.29180	0.0431800
$896$	11.3820	0.380245
$897$	0	0
$898$	19.1246	0.638197
$899$	−8.56231	−0.285569
$900$	0	0
$901$	43.7984	1.45914
$902$	1.32624	0.0441589
$903$	0	0
$904$	−41.5066	−1.38049
$905$	23.1803	0.770541
$906$	0	0
$907$	35.6869	1.18496	0.592482	−	0.805583i	$-0.298148\pi$
$907$	35.6869	1.18496	0.592482	+	0.805583i	$0.298148\pi$
$908$	8.52786	0.283007
$909$	0	0
$910$	3.76393	0.124773
$911$	−34.4721	−1.14211	−0.571056	−	0.820911i	$-0.693466\pi$
$911$	−34.4721	−1.14211	−0.571056	+	0.820911i	$0.693466\pi$
$912$	0	0
$913$	50.0557	1.65660
$914$	−19.7984	−0.654872
$915$	0	0
$916$	27.7984	0.918484
$917$	1.67376	0.0552725
$918$	0	0
$919$	7.09017	0.233883	0.116942	−	0.993139i	$-0.462691\pi$
$919$	7.09017	0.233883	0.116942	+	0.993139i	$0.462691\pi$
$920$	9.79837	0.323043
$921$	0	0
$922$	−14.0557	−0.462901
$923$	53.9230	1.77490
$924$	0	0
$925$	−3.00000	−0.0986394
$926$	−7.29180	−0.239623
$927$	0	0
$928$	−16.0344	−0.526356
$929$	42.9787	1.41009	0.705043	−	0.709165i	$-0.250928\pi$
$929$	42.9787	1.41009	0.705043	+	0.709165i	$0.250928\pi$
$930$	0	0
$931$	0.381966	0.0125184
$932$	−28.9787	−0.949229
$933$	0	0
$934$	−10.8328	−0.354461
$935$	−22.9787	−0.751484
$936$	0	0
$937$	−43.5279	−1.42199	−0.710997	−	0.703195i	$-0.751756\pi$
$937$	−43.5279	−1.42199	−0.710997	+	0.703195i	$0.751756\pi$
$938$	−7.90983	−0.258265
$939$	0	0
$940$	17.3262	0.565120
$941$	−3.65248	−0.119067	−0.0595337	−	0.998226i	$-0.518961\pi$
$941$	−3.65248	−0.119067	−0.0595337	+	0.998226i	$0.518961\pi$
$942$	0	0
$943$	−2.70820	−0.0881913
$944$	−19.8541	−0.646196
$945$	0	0
$946$	−16.6606	−0.541683
$947$	20.6869	0.672234	0.336117	−	0.941820i	$-0.390886\pi$
$947$	20.6869	0.672234	0.336117	+	0.941820i	$0.390886\pi$
$948$	0	0
$949$	15.3951	0.499747
$950$	0.236068	0.00765906
$951$	0	0
$952$	−14.7984	−0.479618
$953$	46.4853	1.50581	0.752903	−	0.658131i	$-0.228653\pi$
$953$	46.4853	1.50581	0.752903	+	0.658131i	$0.228653\pi$
$954$	0	0
$955$	−16.3820	−0.530108
$956$	37.7984	1.22249
$957$	0	0
$958$	2.96556	0.0958128
$959$	−9.52786	−0.307671
$960$	0	0
$961$	−22.0000	−0.709677
$962$	11.2918	0.364062
$963$	0	0
$964$	30.4164	0.979647
$965$	3.56231	0.114675
$966$	0	0
$967$	7.40325	0.238073	0.119036	−	0.992890i	$-0.462020\pi$
$967$	7.40325	0.238073	0.119036	+	0.992890i	$0.462020\pi$
$968$	−2.36068	−0.0758751
$969$	0	0
$970$	9.90983	0.318185
$971$	−43.4164	−1.39330	−0.696649	−	0.717412i	$-0.745327\pi$
$971$	−43.4164	−1.39330	−0.696649	+	0.717412i	$0.745327\pi$
$972$	0	0
$973$	9.47214	0.303663
$974$	−0.291796	−0.00934975
$975$	0	0
$976$	2.12461	0.0680072
$977$	−37.7771	−1.20860	−0.604298	−	0.796758i	$-0.706546\pi$
$977$	−37.7771	−1.20860	−0.604298	+	0.796758i	$0.706546\pi$
$978$	0	0
$979$	−46.7771	−1.49500
$980$	−1.61803	−0.0516862
$981$	0	0
$982$	5.43769	0.173524
$983$	37.9787	1.21133	0.605666	−	0.795719i	$-0.292907\pi$
$983$	37.9787	1.21133	0.605666	+	0.795719i	$0.292907\pi$
$984$	0	0
$985$	20.8885	0.665564
$986$	11.6738	0.371768
$987$	0	0
$988$	3.76393	0.119747
$989$	34.0213	1.08181
$990$	0	0
$991$	−21.9443	−0.697083	−0.348541	−	0.937293i	$-0.613323\pi$
$991$	−21.9443	−0.697083	−0.348541	+	0.937293i	$0.613323\pi$
$992$	16.8541	0.535118
$993$	0	0
$994$	5.47214	0.173566
$995$	−2.03444	−0.0644961
$996$	0	0
$997$	−12.1246	−0.383990	−0.191995	−	0.981396i	$-0.561496\pi$
$997$	−12.1246	−0.383990	−0.191995	+	0.981396i	$0.561496\pi$
$998$	−25.5623	−0.809161
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.d.1.2	✓	2
3.2	odd	2		945.2.a.i.1.1	yes	2
5.4	even	2		4725.2.a.bc.1.1		2
7.6	odd	2		6615.2.a.m.1.2		2
15.14	odd	2		4725.2.a.x.1.2		2
21.20	even	2		6615.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.d.1.2	✓	2	1.1	even	1	trivial
945.2.a.i.1.1	yes	2	3.2	odd	2
4725.2.a.x.1.2		2	15.14	odd	2
4725.2.a.bc.1.1		2	5.4	even	2
6615.2.a.m.1.2		2	7.6	odd	2
6615.2.a.s.1.1		2	21.20	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:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +0.618034 q^{10} +3.47214 q^{11} -6.09017 q^{13} -0.618034 q^{14} +1.85410 q^{16} -6.61803 q^{17} +0.381966 q^{19} -1.61803 q^{20} +2.14590 q^{22} -4.38197 q^{23} +1.00000 q^{25} -3.76393 q^{26} +1.61803 q^{28} -2.85410 q^{29} +3.00000 q^{31} +5.61803 q^{32} -4.09017 q^{34} -1.00000 q^{35} -3.00000 q^{37} +0.236068 q^{38} -2.23607 q^{40} +0.618034 q^{41} -7.76393 q^{43} -5.61803 q^{44} -2.70820 q^{46} -10.7082 q^{47} +1.00000 q^{49} +0.618034 q^{50} +9.85410 q^{52} -6.61803 q^{53} +3.47214 q^{55} +2.23607 q^{56} -1.76393 q^{58} -10.7082 q^{59} +1.14590 q^{61} +1.85410 q^{62} -0.236068 q^{64} -6.09017 q^{65} +12.7984 q^{67} +10.7082 q^{68} -0.618034 q^{70} -8.85410 q^{71} -2.52786 q^{73} -1.85410 q^{74} -0.618034 q^{76} -3.47214 q^{77} +10.7984 q^{79} +1.85410 q^{80} +0.381966 q^{82} +14.4164 q^{83} -6.61803 q^{85} -4.79837 q^{86} -7.76393 q^{88} -13.4721 q^{89} +6.09017 q^{91} +7.09017 q^{92} -6.61803 q^{94} +0.381966 q^{95} +16.0344 q^{97} +0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7	\( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7} - q^{10} - 2 q^{11} - q^{13} + q^{14} - 3 q^{16} - 11 q^{17} + 3 q^{19} - q^{20} + 11 q^{22} - 11 q^{23} + 2 q^{25} - 12 q^{26} + q^{28} + q^{29} + 6 q^{31} + 9 q^{32} + 3 q^{34} - 2 q^{35} - 6 q^{37} - 4 q^{38} - q^{41} - 20 q^{43} - 9 q^{44} + 8 q^{46} - 8 q^{47} + 2 q^{49} - q^{50} + 13 q^{52} - 11 q^{53} - 2 q^{55} - 8 q^{58} - 8 q^{59} + 9 q^{61} - 3 q^{62} + 4 q^{64} - q^{65} + q^{67} + 8 q^{68} + q^{70} - 11 q^{71} - 14 q^{73} + 3 q^{74} + q^{76} + 2 q^{77} - 3 q^{79} - 3 q^{80} + 3 q^{82} + 2 q^{83} - 11 q^{85} + 15 q^{86} - 20 q^{88} - 18 q^{89} + q^{91} + 3 q^{92} - 11 q^{94} + 3 q^{95} + 3 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7 - q^10 - 2 * q^11 - q^13 + q^14 - 3 * q^16 - 11 * q^17 + 3 * q^19 - q^20 + 11 * q^22 - 11 * q^23 + 2 * q^25 - 12 * q^26 + q^28 + q^29 + 6 * q^31 + 9 * q^32 + 3 * q^34 - 2 * q^35 - 6 * q^37 - 4 * q^38 - q^41 - 20 * q^43 - 9 * q^44 + 8 * q^46 - 8 * q^47 + 2 * q^49 - q^50 + 13 * q^52 - 11 * q^53 - 2 * q^55 - 8 * q^58 - 8 * q^59 + 9 * q^61 - 3 * q^62 + 4 * q^64 - q^65 + q^67 + 8 * q^68 + q^70 - 11 * q^71 - 14 * q^73 + 3 * q^74 + q^76 + 2 * q^77 - 3 * q^79 - 3 * q^80 + 3 * q^82 + 2 * q^83 - 11 * q^85 + 15 * q^86 - 20 * q^88 - 18 * q^89 + q^91 + 3 * q^92 - 11 * q^94 + 3 * q^95 + 3 * q^97 - q^98

Introduction

L-functions

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