LMFDB - Embedded newform 945.2.a.l.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:	$0$
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			$1.61803$ 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)

$f(q)$	$=$	$q+2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} +7.47214 q^{8} +O(q^{10})$	\(q+2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} +7.47214 q^{8} -2.61803 q^{10} -0.236068 q^{11} -0.381966 q^{13} +2.61803 q^{14} +9.85410 q^{16} +5.85410 q^{17} -3.61803 q^{19} -4.85410 q^{20} -0.618034 q^{22} +8.85410 q^{23} +1.00000 q^{25} -1.00000 q^{26} +4.85410 q^{28} -0.854102 q^{29} -7.47214 q^{31} +10.8541 q^{32} +15.3262 q^{34} -1.00000 q^{35} -9.94427 q^{37} -9.47214 q^{38} -7.47214 q^{40} -7.56231 q^{41} +2.23607 q^{43} -1.14590 q^{44} +23.1803 q^{46} -3.47214 q^{47} +1.00000 q^{49} +2.61803 q^{50} -1.85410 q^{52} -3.09017 q^{53} +0.236068 q^{55} +7.47214 q^{56} -2.23607 q^{58} -11.9443 q^{59} -6.85410 q^{61} -19.5623 q^{62} +8.70820 q^{64} +0.381966 q^{65} -11.8541 q^{67} +28.4164 q^{68} -2.61803 q^{70} +14.0902 q^{71} +9.94427 q^{73} -26.0344 q^{74} -17.5623 q^{76} -0.236068 q^{77} +14.7984 q^{79} -9.85410 q^{80} -19.7984 q^{82} +7.76393 q^{83} -5.85410 q^{85} +5.85410 q^{86} -1.76393 q^{88} +4.70820 q^{89} -0.381966 q^{91} +42.9787 q^{92} -9.09017 q^{94} +3.61803 q^{95} -3.09017 q^{97} +2.61803 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8	$ 2 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8} - 3 q^{10} + 4 q^{11} - 3 q^{13} + 3 q^{14} + 13 q^{16} + 5 q^{17} - 5 q^{19} - 3 q^{20} + q^{22} + 11 q^{23} + 2 q^{25} - 2 q^{26} + 3 q^{28} + 5 q^{29} - 6 q^{31} + 15 q^{32} + 15 q^{34} - 2 q^{35} - 2 q^{37} - 10 q^{38} - 6 q^{40} + 5 q^{41} - 9 q^{44} + 24 q^{46} + 2 q^{47} + 2 q^{49} + 3 q^{50} + 3 q^{52} + 5 q^{53} - 4 q^{55} + 6 q^{56} - 6 q^{59} - 7 q^{61} - 19 q^{62} + 4 q^{64} + 3 q^{65} - 17 q^{67} + 30 q^{68} - 3 q^{70} + 17 q^{71} + 2 q^{73} - 23 q^{74} - 15 q^{76} + 4 q^{77} + 5 q^{79} - 13 q^{80} - 15 q^{82} + 20 q^{83} - 5 q^{85} + 5 q^{86} - 8 q^{88} - 4 q^{89} - 3 q^{91} + 39 q^{92} - 7 q^{94} + 5 q^{95} + 5 q^{97} + 3 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8 - 3 * q^10 + 4 * q^11 - 3 * q^13 + 3 * q^14 + 13 * q^16 + 5 * q^17 - 5 * q^19 - 3 * q^20 + q^22 + 11 * q^23 + 2 * q^25 - 2 * q^26 + 3 * q^28 + 5 * q^29 - 6 * q^31 + 15 * q^32 + 15 * q^34 - 2 * q^35 - 2 * q^37 - 10 * q^38 - 6 * q^40 + 5 * q^41 - 9 * q^44 + 24 * q^46 + 2 * q^47 + 2 * q^49 + 3 * q^50 + 3 * q^52 + 5 * q^53 - 4 * q^55 + 6 * q^56 - 6 * q^59 - 7 * q^61 - 19 * q^62 + 4 * q^64 + 3 * q^65 - 17 * q^67 + 30 * q^68 - 3 * q^70 + 17 * q^71 + 2 * q^73 - 23 * q^74 - 15 * q^76 + 4 * q^77 + 5 * q^79 - 13 * q^80 - 15 * q^82 + 20 * q^83 - 5 * q^85 + 5 * q^86 - 8 * q^88 - 4 * q^89 - 3 * q^91 + 39 * q^92 - 7 * q^94 + 5 * q^95 + 5 * q^97 + 3 * 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 $

$2$	2.61803	1.85123	0.925615	−	0.378467i	$-0.123549\pi$
$2$	2.61803	1.85123	0.925615	+	0.378467i	$0.123549\pi$
$3$	0	0
$3$	0	0
$4$	4.85410	2.42705
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	7.47214	2.64180
$9$	0	0
$10$	−2.61803	−0.827895
$11$	−0.236068	−0.0711772	−0.0355886	−	0.999367i	$-0.511331\pi$
$11$	−0.236068	−0.0711772	−0.0355886	+	0.999367i	$0.511331\pi$
$12$	0	0
$13$	−0.381966	−0.105938	−0.0529692	−	0.998596i	$-0.516869\pi$
$13$	−0.381966	−0.105938	−0.0529692	+	0.998596i	$0.516869\pi$
$14$	2.61803	0.699699
$15$	0	0
$16$	9.85410	2.46353
$17$	5.85410	1.41983	0.709914	−	0.704288i	$-0.248734\pi$
$17$	5.85410	1.41983	0.709914	+	0.704288i	$0.248734\pi$
$18$	0	0
$19$	−3.61803	−0.830034	−0.415017	−	0.909814i	$-0.636224\pi$
$19$	−3.61803	−0.830034	−0.415017	+	0.909814i	$0.636224\pi$
$20$	−4.85410	−1.08541
$21$	0	0
$22$	−0.618034	−0.131765
$23$	8.85410	1.84621	0.923104	−	0.384551i	$-0.125644\pi$
$23$	8.85410	1.84621	0.923104	+	0.384551i	$0.125644\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	4.85410	0.917339
$29$	−0.854102	−0.158603	−0.0793014	−	0.996851i	$-0.525269\pi$
$29$	−0.854102	−0.158603	−0.0793014	+	0.996851i	$0.525269\pi$
$30$	0	0
$31$	−7.47214	−1.34204	−0.671018	−	0.741441i	$-0.734143\pi$
$31$	−7.47214	−1.34204	−0.671018	+	0.741441i	$0.734143\pi$
$32$	10.8541	1.91875
$33$	0	0
$34$	15.3262	2.62843
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−9.94427	−1.63483	−0.817414	−	0.576050i	$-0.804593\pi$
$37$	−9.94427	−1.63483	−0.817414	+	0.576050i	$0.804593\pi$
$38$	−9.47214	−1.53658
$39$	0	0
$40$	−7.47214	−1.18145
$41$	−7.56231	−1.18103	−0.590517	−	0.807025i	$-0.701076\pi$
$41$	−7.56231	−1.18103	−0.590517	+	0.807025i	$0.701076\pi$
$42$	0	0
$43$	2.23607	0.340997	0.170499	−	0.985358i	$-0.445462\pi$
$43$	2.23607	0.340997	0.170499	+	0.985358i	$0.445462\pi$
$44$	−1.14590	−0.172751
$45$	0	0
$46$	23.1803	3.41775
$47$	−3.47214	−0.506463	−0.253232	−	0.967406i	$-0.581493\pi$
$47$	−3.47214	−0.506463	−0.253232	+	0.967406i	$0.581493\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.61803	0.370246
$51$	0	0
$52$	−1.85410	−0.257118
$53$	−3.09017	−0.424467	−0.212234	−	0.977219i	$-0.568074\pi$
$53$	−3.09017	−0.424467	−0.212234	+	0.977219i	$0.568074\pi$
$54$	0	0
$55$	0.236068	0.0318314
$56$	7.47214	0.998506
$57$	0	0
$58$	−2.23607	−0.293610
$59$	−11.9443	−1.55501	−0.777506	−	0.628876i	$-0.783515\pi$
$59$	−11.9443	−1.55501	−0.777506	+	0.628876i	$0.783515\pi$
$60$	0	0
$61$	−6.85410	−0.877578	−0.438789	−	0.898590i	$-0.644592\pi$
$61$	−6.85410	−0.877578	−0.438789	+	0.898590i	$0.644592\pi$
$62$	−19.5623	−2.48442
$63$	0	0
$64$	8.70820	1.08853
$65$	0.381966	0.0473771
$66$	0	0
$67$	−11.8541	−1.44821	−0.724105	−	0.689690i	$-0.757747\pi$
$67$	−11.8541	−1.44821	−0.724105	+	0.689690i	$0.757747\pi$
$68$	28.4164	3.44600
$69$	0	0
$70$	−2.61803	−0.312915
$71$	14.0902	1.67220	0.836098	−	0.548580i	$-0.184832\pi$
$71$	14.0902	1.67220	0.836098	+	0.548580i	$0.184832\pi$
$72$	0	0
$73$	9.94427	1.16389	0.581944	−	0.813229i	$-0.302292\pi$
$73$	9.94427	1.16389	0.581944	+	0.813229i	$0.302292\pi$
$74$	−26.0344	−3.02644
$75$	0	0
$76$	−17.5623	−2.01453
$77$	−0.236068	−0.0269024
$78$	0	0
$79$	14.7984	1.66495	0.832474	−	0.554065i	$-0.186924\pi$
$79$	14.7984	1.66495	0.832474	+	0.554065i	$0.186924\pi$
$80$	−9.85410	−1.10172
$81$	0	0
$82$	−19.7984	−2.18636
$83$	7.76393	0.852202	0.426101	−	0.904676i	$-0.359887\pi$
$83$	7.76393	0.852202	0.426101	+	0.904676i	$0.359887\pi$
$84$	0	0
$85$	−5.85410	−0.634967
$86$	5.85410	0.631264
$87$	0	0
$88$	−1.76393	−0.188036
$89$	4.70820	0.499069	0.249534	−	0.968366i	$-0.419722\pi$
$89$	4.70820	0.499069	0.249534	+	0.968366i	$0.419722\pi$
$90$	0	0
$91$	−0.381966	−0.0400409
$92$	42.9787	4.48084
$93$	0	0
$94$	−9.09017	−0.937579
$95$	3.61803	0.371202
$96$	0	0
$97$	−3.09017	−0.313759	−0.156880	−	0.987618i	$-0.550143\pi$
$97$	−3.09017	−0.313759	−0.156880	+	0.987618i	$0.550143\pi$
$98$	2.61803	0.264461
$99$	0	0
$100$	4.85410	0.485410
$101$	11.4164	1.13598	0.567988	−	0.823037i	$-0.307722\pi$
$101$	11.4164	1.13598	0.567988	+	0.823037i	$0.307722\pi$
$102$	0	0
$103$	−13.1459	−1.29530	−0.647652	−	0.761936i	$-0.724249\pi$
$103$	−13.1459	−1.29530	−0.647652	+	0.761936i	$0.724249\pi$
$104$	−2.85410	−0.279868
$105$	0	0
$106$	−8.09017	−0.785787
$107$	−8.52786	−0.824420	−0.412210	−	0.911089i	$-0.635243\pi$
$107$	−8.52786	−0.824420	−0.412210	+	0.911089i	$0.635243\pi$
$108$	0	0
$109$	−7.56231	−0.724338	−0.362169	−	0.932113i	$-0.617964\pi$
$109$	−7.56231	−0.724338	−0.362169	+	0.932113i	$0.617964\pi$
$110$	0.618034	0.0589272
$111$	0	0
$112$	9.85410	0.931125
$113$	−2.09017	−0.196627	−0.0983133	−	0.995156i	$-0.531345\pi$
$113$	−2.09017	−0.196627	−0.0983133	+	0.995156i	$0.531345\pi$
$114$	0	0
$115$	−8.85410	−0.825649
$116$	−4.14590	−0.384937
$117$	0	0
$118$	−31.2705	−2.87868
$119$	5.85410	0.536645
$120$	0	0
$121$	−10.9443	−0.994934
$122$	−17.9443	−1.62460
$123$	0	0
$124$	−36.2705	−3.25719
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.09017	−0.806622	−0.403311	−	0.915063i	$-0.632141\pi$
$127$	−9.09017	−0.806622	−0.403311	+	0.915063i	$0.632141\pi$
$128$	1.09017	0.0963583
$129$	0	0
$130$	1.00000	0.0877058
$131$	9.56231	0.835463	0.417731	−	0.908571i	$-0.362825\pi$
$131$	9.56231	0.835463	0.417731	+	0.908571i	$0.362825\pi$
$132$	0	0
$133$	−3.61803	−0.313723
$134$	−31.0344	−2.68097
$135$	0	0
$136$	43.7426	3.75090
$137$	−2.47214	−0.211209	−0.105604	−	0.994408i	$-0.533678\pi$
$137$	−2.47214	−0.211209	−0.105604	+	0.994408i	$0.533678\pi$
$138$	0	0
$139$	9.00000	0.763370	0.381685	−	0.924292i	$-0.375344\pi$
$139$	9.00000	0.763370	0.381685	+	0.924292i	$0.375344\pi$
$140$	−4.85410	−0.410246
$141$	0	0
$142$	36.8885	3.09562
$143$	0.0901699	0.00754039
$144$	0	0
$145$	0.854102	0.0709293
$146$	26.0344	2.15463
$147$	0	0
$148$	−48.2705	−3.96781
$149$	6.32624	0.518266	0.259133	−	0.965842i	$-0.416563\pi$
$149$	6.32624	0.518266	0.259133	+	0.965842i	$0.416563\pi$
$150$	0	0
$151$	23.0000	1.87171	0.935857	−	0.352381i	$-0.114628\pi$
$151$	23.0000	1.87171	0.935857	+	0.352381i	$0.114628\pi$
$152$	−27.0344	−2.19278
$153$	0	0
$154$	−0.618034	−0.0498026
$155$	7.47214	0.600176
$156$	0	0
$157$	22.6525	1.80786	0.903932	−	0.427676i	$-0.140668\pi$
$157$	22.6525	1.80786	0.903932	+	0.427676i	$0.140668\pi$
$158$	38.7426	3.08220
$159$	0	0
$160$	−10.8541	−0.858092
$161$	8.85410	0.697801
$162$	0	0
$163$	−22.9443	−1.79713	−0.898567	−	0.438836i	$-0.855391\pi$
$163$	−22.9443	−1.79713	−0.898567	+	0.438836i	$0.855391\pi$
$164$	−36.7082	−2.86643
$165$	0	0
$166$	20.3262	1.57762
$167$	−12.4164	−0.960810	−0.480405	−	0.877047i	$-0.659510\pi$
$167$	−12.4164	−0.960810	−0.480405	+	0.877047i	$0.659510\pi$
$168$	0	0
$169$	−12.8541	−0.988777
$170$	−15.3262	−1.17547
$171$	0	0
$172$	10.8541	0.827618
$173$	−4.52786	−0.344247	−0.172124	−	0.985075i	$-0.555063\pi$
$173$	−4.52786	−0.344247	−0.172124	+	0.985075i	$0.555063\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−2.32624	−0.175347
$177$	0	0
$178$	12.3262	0.923891
$179$	−3.94427	−0.294809	−0.147404	−	0.989076i	$-0.547092\pi$
$179$	−3.94427	−0.294809	−0.147404	+	0.989076i	$0.547092\pi$
$180$	0	0
$181$	−4.23607	−0.314864	−0.157432	−	0.987530i	$-0.550322\pi$
$181$	−4.23607	−0.314864	−0.157432	+	0.987530i	$0.550322\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	66.1591	4.87731
$185$	9.94427	0.731117
$186$	0	0
$187$	−1.38197	−0.101059
$188$	−16.8541	−1.22921
$189$	0	0
$190$	9.47214	0.687181
$191$	15.1459	1.09592	0.547959	−	0.836505i	$-0.315405\pi$
$191$	15.1459	1.09592	0.547959	+	0.836505i	$0.315405\pi$
$192$	0	0
$193$	19.2705	1.38712	0.693561	−	0.720398i	$-0.256041\pi$
$193$	19.2705	1.38712	0.693561	+	0.720398i	$0.256041\pi$
$194$	−8.09017	−0.580840
$195$	0	0
$196$	4.85410	0.346722
$197$	−1.76393	−0.125675	−0.0628375	−	0.998024i	$-0.520015\pi$
$197$	−1.76393	−0.125675	−0.0628375	+	0.998024i	$0.520015\pi$
$198$	0	0
$199$	−3.56231	−0.252525	−0.126263	−	0.991997i	$-0.540298\pi$
$199$	−3.56231	−0.252525	−0.126263	+	0.991997i	$0.540298\pi$
$200$	7.47214	0.528360
$201$	0	0
$202$	29.8885	2.10295
$203$	−0.854102	−0.0599462
$204$	0	0
$205$	7.56231	0.528174
$206$	−34.4164	−2.39790
$207$	0	0
$208$	−3.76393	−0.260982
$209$	0.854102	0.0590795
$210$	0	0
$211$	−8.65248	−0.595661	−0.297831	−	0.954619i	$-0.596263\pi$
$211$	−8.65248	−0.595661	−0.297831	+	0.954619i	$0.596263\pi$
$212$	−15.0000	−1.03020
$213$	0	0
$214$	−22.3262	−1.52619
$215$	−2.23607	−0.152499
$216$	0	0
$217$	−7.47214	−0.507242
$218$	−19.7984	−1.34092
$219$	0	0
$220$	1.14590	0.0772564
$221$	−2.23607	−0.150414
$222$	0	0
$223$	15.4721	1.03609	0.518045	−	0.855353i	$-0.326660\pi$
$223$	15.4721	1.03609	0.518045	+	0.855353i	$0.326660\pi$
$224$	10.8541	0.725220
$225$	0	0
$226$	−5.47214	−0.364001
$227$	24.5066	1.62656	0.813279	−	0.581873i	$-0.197680\pi$
$227$	24.5066	1.62656	0.813279	+	0.581873i	$0.197680\pi$
$228$	0	0
$229$	−3.65248	−0.241362	−0.120681	−	0.992691i	$-0.538508\pi$
$229$	−3.65248	−0.241362	−0.120681	+	0.992691i	$0.538508\pi$
$230$	−23.1803	−1.52847
$231$	0	0
$232$	−6.38197	−0.418997
$233$	6.56231	0.429911	0.214955	−	0.976624i	$-0.431039\pi$
$233$	6.56231	0.429911	0.214955	+	0.976624i	$0.431039\pi$
$234$	0	0
$235$	3.47214	0.226497
$236$	−57.9787	−3.77409
$237$	0	0
$238$	15.3262	0.993452
$239$	−17.1803	−1.11130	−0.555652	−	0.831415i	$-0.687531\pi$
$239$	−17.1803	−1.11130	−0.555652	+	0.831415i	$0.687531\pi$
$240$	0	0
$241$	16.6180	1.07046	0.535231	−	0.844706i	$-0.320225\pi$
$241$	16.6180	1.07046	0.535231	+	0.844706i	$0.320225\pi$
$242$	−28.6525	−1.84185
$243$	0	0
$244$	−33.2705	−2.12993
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	1.38197	0.0879324
$248$	−55.8328	−3.54539
$249$	0	0
$250$	−2.61803	−0.165579
$251$	26.6525	1.68229	0.841145	−	0.540810i	$-0.181882\pi$
$251$	26.6525	1.68229	0.841145	+	0.540810i	$0.181882\pi$
$252$	0	0
$253$	−2.09017	−0.131408
$254$	−23.7984	−1.49324
$255$	0	0
$256$	−14.5623	−0.910144
$257$	−18.4721	−1.15226	−0.576130	−	0.817358i	$-0.695438\pi$
$257$	−18.4721	−1.15226	−0.576130	+	0.817358i	$0.695438\pi$
$258$	0	0
$259$	−9.94427	−0.617907
$260$	1.85410	0.114987
$261$	0	0
$262$	25.0344	1.54663
$263$	14.9098	0.919380	0.459690	−	0.888080i	$-0.347961\pi$
$263$	14.9098	0.919380	0.459690	+	0.888080i	$0.347961\pi$
$264$	0	0
$265$	3.09017	0.189828
$266$	−9.47214	−0.580774
$267$	0	0
$268$	−57.5410	−3.51488
$269$	1.47214	0.0897577	0.0448789	−	0.998992i	$-0.485710\pi$
$269$	1.47214	0.0897577	0.0448789	+	0.998992i	$0.485710\pi$
$270$	0	0
$271$	−15.3820	−0.934388	−0.467194	−	0.884155i	$-0.654735\pi$
$271$	−15.3820	−0.934388	−0.467194	+	0.884155i	$0.654735\pi$
$272$	57.6869	3.49778
$273$	0	0
$274$	−6.47214	−0.390996
$275$	−0.236068	−0.0142354
$276$	0	0
$277$	20.7984	1.24965	0.624827	−	0.780764i	$-0.285170\pi$
$277$	20.7984	1.24965	0.624827	+	0.780764i	$0.285170\pi$
$278$	23.5623	1.41317
$279$	0	0
$280$	−7.47214	−0.446546
$281$	−6.79837	−0.405557	−0.202778	−	0.979225i	$-0.564997\pi$
$281$	−6.79837	−0.405557	−0.202778	+	0.979225i	$0.564997\pi$
$282$	0	0
$283$	15.2705	0.907738	0.453869	−	0.891069i	$-0.350043\pi$
$283$	15.2705	0.907738	0.453869	+	0.891069i	$0.350043\pi$
$284$	68.3951	4.05850
$285$	0	0
$286$	0.236068	0.0139590
$287$	−7.56231	−0.446389
$288$	0	0
$289$	17.2705	1.01591
$290$	2.23607	0.131306
$291$	0	0
$292$	48.2705	2.82482
$293$	−1.81966	−0.106306	−0.0531528	−	0.998586i	$-0.516927\pi$
$293$	−1.81966	−0.106306	−0.0531528	+	0.998586i	$0.516927\pi$
$294$	0	0
$295$	11.9443	0.695422
$296$	−74.3050	−4.31889
$297$	0	0
$298$	16.5623	0.959429
$299$	−3.38197	−0.195584
$300$	0	0
$301$	2.23607	0.128885
$302$	60.2148	3.46497
$303$	0	0
$304$	−35.6525	−2.04481
$305$	6.85410	0.392465
$306$	0	0
$307$	8.94427	0.510477	0.255238	−	0.966878i	$-0.417846\pi$
$307$	8.94427	0.510477	0.255238	+	0.966878i	$0.417846\pi$
$308$	−1.14590	−0.0652936
$309$	0	0
$310$	19.5623	1.11106
$311$	30.7426	1.74326	0.871628	−	0.490168i	$-0.163065\pi$
$311$	30.7426	1.74326	0.871628	+	0.490168i	$0.163065\pi$
$312$	0	0
$313$	20.7082	1.17050	0.585248	−	0.810854i	$-0.300997\pi$
$313$	20.7082	1.17050	0.585248	+	0.810854i	$0.300997\pi$
$314$	59.3050	3.34677
$315$	0	0
$316$	71.8328	4.04091
$317$	−26.5967	−1.49382	−0.746911	−	0.664924i	$-0.768464\pi$
$317$	−26.5967	−1.49382	−0.746911	+	0.664924i	$0.768464\pi$
$318$	0	0
$319$	0.201626	0.0112889
$320$	−8.70820	−0.486803
$321$	0	0
$322$	23.1803	1.29179
$323$	−21.1803	−1.17851
$324$	0	0
$325$	−0.381966	−0.0211877
$326$	−60.0689	−3.32691
$327$	0	0
$328$	−56.5066	−3.12005
$329$	−3.47214	−0.191425
$330$	0	0
$331$	−15.0344	−0.826368	−0.413184	−	0.910648i	$-0.635583\pi$
$331$	−15.0344	−0.826368	−0.413184	+	0.910648i	$0.635583\pi$
$332$	37.6869	2.06834
$333$	0	0
$334$	−32.5066	−1.77868
$335$	11.8541	0.647659
$336$	0	0
$337$	−13.9098	−0.757717	−0.378858	−	0.925455i	$-0.623683\pi$
$337$	−13.9098	−0.757717	−0.378858	+	0.925455i	$0.623683\pi$
$338$	−33.6525	−1.83045
$339$	0	0
$340$	−28.4164	−1.54110
$341$	1.76393	0.0955223
$342$	0	0
$343$	1.00000	0.0539949
$344$	16.7082	0.900846
$345$	0	0
$346$	−11.8541	−0.637280
$347$	27.1246	1.45613	0.728063	−	0.685511i	$-0.240421\pi$
$347$	27.1246	1.45613	0.728063	+	0.685511i	$0.240421\pi$
$348$	0	0
$349$	29.0689	1.55602	0.778011	−	0.628251i	$-0.216229\pi$
$349$	29.0689	1.55602	0.778011	+	0.628251i	$0.216229\pi$
$350$	2.61803	0.139940
$351$	0	0
$352$	−2.56231	−0.136571
$353$	−12.6738	−0.674556	−0.337278	−	0.941405i	$-0.609506\pi$
$353$	−12.6738	−0.674556	−0.337278	+	0.941405i	$0.609506\pi$
$354$	0	0
$355$	−14.0902	−0.747829
$356$	22.8541	1.21126
$357$	0	0
$358$	−10.3262	−0.545759
$359$	22.8885	1.20801	0.604006	−	0.796980i	$-0.293570\pi$
$359$	22.8885	1.20801	0.604006	+	0.796980i	$0.293570\pi$
$360$	0	0
$361$	−5.90983	−0.311044
$362$	−11.0902	−0.582886
$363$	0	0
$364$	−1.85410	−0.0971813
$365$	−9.94427	−0.520507
$366$	0	0
$367$	−13.0902	−0.683301	−0.341651	−	0.939827i	$-0.610986\pi$
$367$	−13.0902	−0.683301	−0.341651	+	0.939827i	$0.610986\pi$
$368$	87.2492	4.54818
$369$	0	0
$370$	26.0344	1.35347
$371$	−3.09017	−0.160434
$372$	0	0
$373$	−8.56231	−0.443339	−0.221670	−	0.975122i	$-0.571151\pi$
$373$	−8.56231	−0.443339	−0.221670	+	0.975122i	$0.571151\pi$
$374$	−3.61803	−0.187084
$375$	0	0
$376$	−25.9443	−1.33797
$377$	0.326238	0.0168021
$378$	0	0
$379$	26.3607	1.35406	0.677029	−	0.735957i	$-0.263267\pi$
$379$	26.3607	1.35406	0.677029	+	0.735957i	$0.263267\pi$
$380$	17.5623	0.900927
$381$	0	0
$382$	39.6525	2.02880
$383$	3.29180	0.168203	0.0841015	−	0.996457i	$-0.473198\pi$
$383$	3.29180	0.168203	0.0841015	+	0.996457i	$0.473198\pi$
$384$	0	0
$385$	0.236068	0.0120311
$386$	50.4508	2.56788
$387$	0	0
$388$	−15.0000	−0.761510
$389$	−26.7984	−1.35873	−0.679366	−	0.733800i	$-0.737745\pi$
$389$	−26.7984	−1.35873	−0.679366	+	0.733800i	$0.737745\pi$
$390$	0	0
$391$	51.8328	2.62130
$392$	7.47214	0.377400
$393$	0	0
$394$	−4.61803	−0.232653
$395$	−14.7984	−0.744587
$396$	0	0
$397$	−13.4164	−0.673350	−0.336675	−	0.941621i	$-0.609302\pi$
$397$	−13.4164	−0.673350	−0.336675	+	0.941621i	$0.609302\pi$
$398$	−9.32624	−0.467482
$399$	0	0
$400$	9.85410	0.492705
$401$	−1.72949	−0.0863666	−0.0431833	−	0.999067i	$-0.513750\pi$
$401$	−1.72949	−0.0863666	−0.0431833	+	0.999067i	$0.513750\pi$
$402$	0	0
$403$	2.85410	0.142173
$404$	55.4164	2.75707
$405$	0	0
$406$	−2.23607	−0.110974
$407$	2.34752	0.116362
$408$	0	0
$409$	−22.3607	−1.10566	−0.552832	−	0.833293i	$-0.686453\pi$
$409$	−22.3607	−1.10566	−0.552832	+	0.833293i	$0.686453\pi$
$410$	19.7984	0.977772
$411$	0	0
$412$	−63.8115	−3.14377
$413$	−11.9443	−0.587739
$414$	0	0
$415$	−7.76393	−0.381116
$416$	−4.14590	−0.203269
$417$	0	0
$418$	2.23607	0.109370
$419$	8.23607	0.402358	0.201179	−	0.979554i	$-0.435523\pi$
$419$	8.23607	0.402358	0.201179	+	0.979554i	$0.435523\pi$
$420$	0	0
$421$	−2.27051	−0.110658	−0.0553289	−	0.998468i	$-0.517621\pi$
$421$	−2.27051	−0.110658	−0.0553289	+	0.998468i	$0.517621\pi$
$422$	−22.6525	−1.10271
$423$	0	0
$424$	−23.0902	−1.12136
$425$	5.85410	0.283966
$426$	0	0
$427$	−6.85410	−0.331693
$428$	−41.3951	−2.00091
$429$	0	0
$430$	−5.85410	−0.282310
$431$	25.3820	1.22261	0.611303	−	0.791397i	$-0.290646\pi$
$431$	25.3820	1.22261	0.611303	+	0.791397i	$0.290646\pi$
$432$	0	0
$433$	4.90983	0.235951	0.117976	−	0.993016i	$-0.462360\pi$
$433$	4.90983	0.235951	0.117976	+	0.993016i	$0.462360\pi$
$434$	−19.5623	−0.939021
$435$	0	0
$436$	−36.7082	−1.75800
$437$	−32.0344	−1.53242
$438$	0	0
$439$	4.52786	0.216103	0.108052	−	0.994145i	$-0.465539\pi$
$439$	4.52786	0.216103	0.108052	+	0.994145i	$0.465539\pi$
$440$	1.76393	0.0840922
$441$	0	0
$442$	−5.85410	−0.278451
$443$	29.1459	1.38476	0.692382	−	0.721531i	$-0.256561\pi$
$443$	29.1459	1.38476	0.692382	+	0.721531i	$0.256561\pi$
$444$	0	0
$445$	−4.70820	−0.223190
$446$	40.5066	1.91804
$447$	0	0
$448$	8.70820	0.411424
$449$	14.5836	0.688242	0.344121	−	0.938925i	$-0.388177\pi$
$449$	14.5836	0.688242	0.344121	+	0.938925i	$0.388177\pi$
$450$	0	0
$451$	1.78522	0.0840626
$452$	−10.1459	−0.477223
$453$	0	0
$454$	64.1591	3.01113
$455$	0.381966	0.0179068
$456$	0	0
$457$	−32.3262	−1.51216	−0.756079	−	0.654481i	$-0.772887\pi$
$457$	−32.3262	−1.51216	−0.756079	+	0.654481i	$0.772887\pi$
$458$	−9.56231	−0.446817
$459$	0	0
$460$	−42.9787	−2.00389
$461$	3.61803	0.168509	0.0842543	−	0.996444i	$-0.473149\pi$
$461$	3.61803	0.168509	0.0842543	+	0.996444i	$0.473149\pi$
$462$	0	0
$463$	−4.20163	−0.195266	−0.0976331	−	0.995222i	$-0.531127\pi$
$463$	−4.20163	−0.195266	−0.0976331	+	0.995222i	$0.531127\pi$
$464$	−8.41641	−0.390722
$465$	0	0
$466$	17.1803	0.795864
$467$	−20.9443	−0.969185	−0.484593	−	0.874740i	$-0.661032\pi$
$467$	−20.9443	−0.969185	−0.484593	+	0.874740i	$0.661032\pi$
$468$	0	0
$469$	−11.8541	−0.547372
$470$	9.09017	0.419298
$471$	0	0
$472$	−89.2492	−4.10803
$473$	−0.527864	−0.0242712
$474$	0	0
$475$	−3.61803	−0.166007
$476$	28.4164	1.30246
$477$	0	0
$478$	−44.9787	−2.05728
$479$	7.09017	0.323958	0.161979	−	0.986794i	$-0.448212\pi$
$479$	7.09017	0.323958	0.161979	+	0.986794i	$0.448212\pi$
$480$	0	0
$481$	3.79837	0.173191
$482$	43.5066	1.98167
$483$	0	0
$484$	−53.1246	−2.41476
$485$	3.09017	0.140317
$486$	0	0
$487$	23.8885	1.08249	0.541247	−	0.840864i	$-0.317953\pi$
$487$	23.8885	1.08249	0.541247	+	0.840864i	$0.317953\pi$
$488$	−51.2148	−2.31838
$489$	0	0
$490$	−2.61803	−0.118271
$491$	−32.3262	−1.45886	−0.729431	−	0.684054i	$-0.760215\pi$
$491$	−32.3262	−1.45886	−0.729431	+	0.684054i	$0.760215\pi$
$492$	0	0
$493$	−5.00000	−0.225189
$494$	3.61803	0.162783
$495$	0	0
$496$	−73.6312	−3.30614
$497$	14.0902	0.632031
$498$	0	0
$499$	8.52786	0.381760	0.190880	−	0.981613i	$-0.438866\pi$
$499$	8.52786	0.381760	0.190880	+	0.981613i	$0.438866\pi$
$500$	−4.85410	−0.217082
$501$	0	0
$502$	69.7771	3.11430
$503$	−10.2705	−0.457939	−0.228970	−	0.973434i	$-0.573536\pi$
$503$	−10.2705	−0.457939	−0.228970	+	0.973434i	$0.573536\pi$
$504$	0	0
$505$	−11.4164	−0.508023
$506$	−5.47214	−0.243266
$507$	0	0
$508$	−44.1246	−1.95771
$509$	−33.7639	−1.49656	−0.748280	−	0.663383i	$-0.769120\pi$
$509$	−33.7639	−1.49656	−0.748280	+	0.663383i	$0.769120\pi$
$510$	0	0
$511$	9.94427	0.439909
$512$	−40.3050	−1.78124
$513$	0	0
$514$	−48.3607	−2.13310
$515$	13.1459	0.579277
$516$	0	0
$517$	0.819660	0.0360486
$518$	−26.0344	−1.14389
$519$	0	0
$520$	2.85410	0.125161
$521$	−8.59675	−0.376630	−0.188315	−	0.982109i	$-0.560303\pi$
$521$	−8.59675	−0.376630	−0.188315	+	0.982109i	$0.560303\pi$
$522$	0	0
$523$	13.0344	0.569956	0.284978	−	0.958534i	$-0.408014\pi$
$523$	13.0344	0.569956	0.284978	+	0.958534i	$0.408014\pi$
$524$	46.4164	2.02771
$525$	0	0
$526$	39.0344	1.70198
$527$	−43.7426	−1.90546
$528$	0	0
$529$	55.3951	2.40848
$530$	8.09017	0.351415
$531$	0	0
$532$	−17.5623	−0.761423
$533$	2.88854	0.125117
$534$	0	0
$535$	8.52786	0.368692
$536$	−88.5755	−3.82588
$537$	0	0
$538$	3.85410	0.166162
$539$	−0.236068	−0.0101682
$540$	0	0
$541$	1.09017	0.0468701	0.0234350	−	0.999725i	$-0.492540\pi$
$541$	1.09017	0.0468701	0.0234350	+	0.999725i	$0.492540\pi$
$542$	−40.2705	−1.72977
$543$	0	0
$544$	63.5410	2.72430
$545$	7.56231	0.323934
$546$	0	0
$547$	6.59675	0.282057	0.141028	−	0.990006i	$-0.454959\pi$
$547$	6.59675	0.282057	0.141028	+	0.990006i	$0.454959\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	−0.618034	−0.0263531
$551$	3.09017	0.131646
$552$	0	0
$553$	14.7984	0.629291
$554$	54.4508	2.31339
$555$	0	0
$556$	43.6869	1.85274
$557$	−17.2148	−0.729414	−0.364707	−	0.931122i	$-0.618831\pi$
$557$	−17.2148	−0.729414	−0.364707	+	0.931122i	$0.618831\pi$
$558$	0	0
$559$	−0.854102	−0.0361247
$560$	−9.85410	−0.416412
$561$	0	0
$562$	−17.7984	−0.750779
$563$	−25.3820	−1.06972	−0.534861	−	0.844940i	$-0.679636\pi$
$563$	−25.3820	−1.06972	−0.534861	+	0.844940i	$0.679636\pi$
$564$	0	0
$565$	2.09017	0.0879341
$566$	39.9787	1.68043
$567$	0	0
$568$	105.284	4.41760
$569$	8.81966	0.369739	0.184870	−	0.982763i	$-0.440814\pi$
$569$	8.81966	0.369739	0.184870	+	0.982763i	$0.440814\pi$
$570$	0	0
$571$	−14.1459	−0.591987	−0.295994	−	0.955190i	$-0.595651\pi$
$571$	−14.1459	−0.591987	−0.295994	+	0.955190i	$0.595651\pi$
$572$	0.437694	0.0183009
$573$	0	0
$574$	−19.7984	−0.826368
$575$	8.85410	0.369242
$576$	0	0
$577$	−29.1803	−1.21479	−0.607397	−	0.794399i	$-0.707786\pi$
$577$	−29.1803	−1.21479	−0.607397	+	0.794399i	$0.707786\pi$
$578$	45.2148	1.88069
$579$	0	0
$580$	4.14590	0.172149
$581$	7.76393	0.322102
$582$	0	0
$583$	0.729490	0.0302124
$584$	74.3050	3.07476
$585$	0	0
$586$	−4.76393	−0.196796
$587$	−10.2705	−0.423909	−0.211955	−	0.977280i	$-0.567983\pi$
$587$	−10.2705	−0.423909	−0.211955	+	0.977280i	$0.567983\pi$
$588$	0	0
$589$	27.0344	1.11393
$590$	31.2705	1.28739
$591$	0	0
$592$	−97.9919	−4.02744
$593$	−37.3607	−1.53422	−0.767110	−	0.641516i	$-0.778306\pi$
$593$	−37.3607	−1.53422	−0.767110	+	0.641516i	$0.778306\pi$
$594$	0	0
$595$	−5.85410	−0.239995
$596$	30.7082	1.25786
$597$	0	0
$598$	−8.85410	−0.362071
$599$	15.5279	0.634451	0.317226	−	0.948350i	$-0.397249\pi$
$599$	15.5279	0.634451	0.317226	+	0.948350i	$0.397249\pi$
$600$	0	0
$601$	−5.61803	−0.229164	−0.114582	−	0.993414i	$-0.536553\pi$
$601$	−5.61803	−0.229164	−0.114582	+	0.993414i	$0.536553\pi$
$602$	5.85410	0.238595
$603$	0	0
$604$	111.644	4.54274
$605$	10.9443	0.444948
$606$	0	0
$607$	−10.4164	−0.422789	−0.211394	−	0.977401i	$-0.567800\pi$
$607$	−10.4164	−0.422789	−0.211394	+	0.977401i	$0.567800\pi$
$608$	−39.2705	−1.59263
$609$	0	0
$610$	17.9443	0.726542
$611$	1.32624	0.0536538
$612$	0	0
$613$	8.05573	0.325368	0.162684	−	0.986678i	$-0.447985\pi$
$613$	8.05573	0.325368	0.162684	+	0.986678i	$0.447985\pi$
$614$	23.4164	0.945009
$615$	0	0
$616$	−1.76393	−0.0710708
$617$	−49.0902	−1.97630	−0.988148	−	0.153505i	$-0.950944\pi$
$617$	−49.0902	−1.97630	−0.988148	+	0.153505i	$0.950944\pi$
$618$	0	0
$619$	25.8328	1.03831	0.519154	−	0.854681i	$-0.326247\pi$
$619$	25.8328	1.03831	0.519154	+	0.854681i	$0.326247\pi$
$620$	36.2705	1.45666
$621$	0	0
$622$	80.4853	3.22717
$623$	4.70820	0.188630
$624$	0	0
$625$	1.00000	0.0400000
$626$	54.2148	2.16686
$627$	0	0
$628$	109.957	4.38778
$629$	−58.2148	−2.32118
$630$	0	0
$631$	−22.8328	−0.908960	−0.454480	−	0.890757i	$-0.650175\pi$
$631$	−22.8328	−0.908960	−0.454480	+	0.890757i	$0.650175\pi$
$632$	110.575	4.39846
$633$	0	0
$634$	−69.6312	−2.76541
$635$	9.09017	0.360732
$636$	0	0
$637$	−0.381966	−0.0151340
$638$	0.527864	0.0208983
$639$	0	0
$640$	−1.09017	−0.0430928
$641$	31.3951	1.24003	0.620016	−	0.784589i	$-0.287126\pi$
$641$	31.3951	1.24003	0.620016	+	0.784589i	$0.287126\pi$
$642$	0	0
$643$	22.0902	0.871151	0.435576	−	0.900152i	$-0.356545\pi$
$643$	22.0902	0.871151	0.435576	+	0.900152i	$0.356545\pi$
$644$	42.9787	1.69360
$645$	0	0
$646$	−55.4508	−2.18168
$647$	42.9443	1.68831	0.844157	−	0.536096i	$-0.180102\pi$
$647$	42.9443	1.68831	0.844157	+	0.536096i	$0.180102\pi$
$648$	0	0
$649$	2.81966	0.110681
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	−111.374	−4.36174
$653$	26.5623	1.03946	0.519732	−	0.854330i	$-0.326032\pi$
$653$	26.5623	1.03946	0.519732	+	0.854330i	$0.326032\pi$
$654$	0	0
$655$	−9.56231	−0.373630
$656$	−74.5197	−2.90951
$657$	0	0
$658$	−9.09017	−0.354372
$659$	2.76393	0.107668	0.0538338	−	0.998550i	$-0.482856\pi$
$659$	2.76393	0.107668	0.0538338	+	0.998550i	$0.482856\pi$
$660$	0	0
$661$	30.9230	1.20276	0.601382	−	0.798961i	$-0.294617\pi$
$661$	30.9230	1.20276	0.601382	+	0.798961i	$0.294617\pi$
$662$	−39.3607	−1.52980
$663$	0	0
$664$	58.0132	2.25135
$665$	3.61803	0.140301
$666$	0	0
$667$	−7.56231	−0.292814
$668$	−60.2705	−2.33194
$669$	0	0
$670$	31.0344	1.19897
$671$	1.61803	0.0624635
$672$	0	0
$673$	−34.2492	−1.32021	−0.660105	−	0.751173i	$-0.729488\pi$
$673$	−34.2492	−1.32021	−0.660105	+	0.751173i	$0.729488\pi$
$674$	−36.4164	−1.40271
$675$	0	0
$676$	−62.3951	−2.39981
$677$	26.0132	0.999767	0.499883	−	0.866093i	$-0.333376\pi$
$677$	26.0132	0.999767	0.499883	+	0.866093i	$0.333376\pi$
$678$	0	0
$679$	−3.09017	−0.118590
$680$	−43.7426	−1.67745
$681$	0	0
$682$	4.61803	0.176834
$683$	10.3820	0.397255	0.198627	−	0.980075i	$-0.436352\pi$
$683$	10.3820	0.397255	0.198627	+	0.980075i	$0.436352\pi$
$684$	0	0
$685$	2.47214	0.0944555
$686$	2.61803	0.0999570
$687$	0	0
$688$	22.0344	0.840055
$689$	1.18034	0.0449674
$690$	0	0
$691$	45.0689	1.71450	0.857251	−	0.514899i	$-0.172171\pi$
$691$	45.0689	1.71450	0.857251	+	0.514899i	$0.172171\pi$
$692$	−21.9787	−0.835505
$693$	0	0
$694$	71.0132	2.69562
$695$	−9.00000	−0.341389
$696$	0	0
$697$	−44.2705	−1.67687
$698$	76.1033	2.88055
$699$	0	0
$700$	4.85410	0.183468
$701$	43.8328	1.65554	0.827771	−	0.561066i	$-0.189608\pi$
$701$	43.8328	1.65554	0.827771	+	0.561066i	$0.189608\pi$
$702$	0	0
$703$	35.9787	1.35696
$704$	−2.05573	−0.0774782
$705$	0	0
$706$	−33.1803	−1.24876
$707$	11.4164	0.429358
$708$	0	0
$709$	−43.5967	−1.63731	−0.818655	−	0.574285i	$-0.805280\pi$
$709$	−43.5967	−1.63731	−0.818655	+	0.574285i	$0.805280\pi$
$710$	−36.8885	−1.38440
$711$	0	0
$712$	35.1803	1.31844
$713$	−66.1591	−2.47768
$714$	0	0
$715$	−0.0901699	−0.00337216
$716$	−19.1459	−0.715516
$717$	0	0
$718$	59.9230	2.23631
$719$	−5.85410	−0.218321	−0.109161	−	0.994024i	$-0.534816\pi$
$719$	−5.85410	−0.218321	−0.109161	+	0.994024i	$0.534816\pi$
$720$	0	0
$721$	−13.1459	−0.489579
$722$	−15.4721	−0.575813
$723$	0	0
$724$	−20.5623	−0.764192
$725$	−0.854102	−0.0317206
$726$	0	0
$727$	−19.6869	−0.730147	−0.365074	−	0.930979i	$-0.618956\pi$
$727$	−19.6869	−0.730147	−0.365074	+	0.930979i	$0.618956\pi$
$728$	−2.85410	−0.105780
$729$	0	0
$730$	−26.0344	−0.963578
$731$	13.0902	0.484157
$732$	0	0
$733$	13.8541	0.511713	0.255856	−	0.966715i	$-0.417643\pi$
$733$	13.8541	0.511713	0.255856	+	0.966715i	$0.417643\pi$
$734$	−34.2705	−1.26495
$735$	0	0
$736$	96.1033	3.54242
$737$	2.79837	0.103079
$738$	0	0
$739$	11.2361	0.413325	0.206663	−	0.978412i	$-0.433740\pi$
$739$	11.2361	0.413325	0.206663	+	0.978412i	$0.433740\pi$
$740$	48.2705	1.77446
$741$	0	0
$742$	−8.09017	−0.296999
$743$	38.1033	1.39788	0.698938	−	0.715183i	$-0.253656\pi$
$743$	38.1033	1.39788	0.698938	+	0.715183i	$0.253656\pi$
$744$	0	0
$745$	−6.32624	−0.231775
$746$	−22.4164	−0.820723
$747$	0	0
$748$	−6.70820	−0.245276
$749$	−8.52786	−0.311601
$750$	0	0
$751$	−0.111456	−0.00406709	−0.00203355	−	0.999998i	$-0.500647\pi$
$751$	−0.111456	−0.00406709	−0.00203355	+	0.999998i	$0.500647\pi$
$752$	−34.2148	−1.24768
$753$	0	0
$754$	0.854102	0.0311046
$755$	−23.0000	−0.837056
$756$	0	0
$757$	7.45085	0.270806	0.135403	−	0.990791i	$-0.456767\pi$
$757$	7.45085	0.270806	0.135403	+	0.990791i	$0.456767\pi$
$758$	69.0132	2.50667
$759$	0	0
$760$	27.0344	0.980642
$761$	−50.7426	−1.83942	−0.919710	−	0.392599i	$-0.871576\pi$
$761$	−50.7426	−1.83942	−0.919710	+	0.392599i	$0.871576\pi$
$762$	0	0
$763$	−7.56231	−0.273774
$764$	73.5197	2.65985
$765$	0	0
$766$	8.61803	0.311382
$767$	4.56231	0.164735
$768$	0	0
$769$	17.5836	0.634081	0.317040	−	0.948412i	$-0.397311\pi$
$769$	17.5836	0.634081	0.317040	+	0.948412i	$0.397311\pi$
$770$	0.618034	0.0222724
$771$	0	0
$772$	93.5410	3.36661
$773$	0.201626	0.00725199	0.00362599	−	0.999993i	$-0.498846\pi$
$773$	0.201626	0.00725199	0.00362599	+	0.999993i	$0.498846\pi$
$774$	0	0
$775$	−7.47214	−0.268407
$776$	−23.0902	−0.828889
$777$	0	0
$778$	−70.1591	−2.51532
$779$	27.3607	0.980298
$780$	0	0
$781$	−3.32624	−0.119022
$782$	135.700	4.85262
$783$	0	0
$784$	9.85410	0.351932
$785$	−22.6525	−0.808502
$786$	0	0
$787$	−5.81966	−0.207448	−0.103724	−	0.994606i	$-0.533076\pi$
$787$	−5.81966	−0.207448	−0.103724	+	0.994606i	$0.533076\pi$
$788$	−8.56231	−0.305020
$789$	0	0
$790$	−38.7426	−1.37840
$791$	−2.09017	−0.0743179
$792$	0	0
$793$	2.61803	0.0929691
$794$	−35.1246	−1.24653
$795$	0	0
$796$	−17.2918	−0.612891
$797$	0.0344419	0.00121999	0.000609997	−	1.00000i	$-0.499806\pi$
$797$	0.0344419	0.00121999	0.000609997		1.00000i	$0.499806\pi$
$798$	0	0
$799$	−20.3262	−0.719091
$800$	10.8541	0.383750
$801$	0	0
$802$	−4.52786	−0.159884
$803$	−2.34752	−0.0828423
$804$	0	0
$805$	−8.85410	−0.312066
$806$	7.47214	0.263195
$807$	0	0
$808$	85.3050	3.00102
$809$	−1.47214	−0.0517575	−0.0258788	−	0.999665i	$-0.508238\pi$
$809$	−1.47214	−0.0517575	−0.0258788	+	0.999665i	$0.508238\pi$
$810$	0	0
$811$	13.6869	0.480613	0.240306	−	0.970697i	$-0.422752\pi$
$811$	13.6869	0.480613	0.240306	+	0.970697i	$0.422752\pi$
$812$	−4.14590	−0.145492
$813$	0	0
$814$	6.14590	0.215414
$815$	22.9443	0.803703
$816$	0	0
$817$	−8.09017	−0.283039
$818$	−58.5410	−2.04684
$819$	0	0
$820$	36.7082	1.28191
$821$	−3.47214	−0.121178	−0.0605892	−	0.998163i	$-0.519298\pi$
$821$	−3.47214	−0.121178	−0.0605892	+	0.998163i	$0.519298\pi$
$822$	0	0
$823$	−45.7639	−1.59523	−0.797615	−	0.603167i	$-0.793905\pi$
$823$	−45.7639	−1.59523	−0.797615	+	0.603167i	$0.793905\pi$
$824$	−98.2279	−3.42193
$825$	0	0
$826$	−31.2705	−1.08804
$827$	−34.8885	−1.21319	−0.606597	−	0.795010i	$-0.707466\pi$
$827$	−34.8885	−1.21319	−0.606597	+	0.795010i	$0.707466\pi$
$828$	0	0
$829$	25.0689	0.870678	0.435339	−	0.900267i	$-0.356628\pi$
$829$	25.0689	0.870678	0.435339	+	0.900267i	$0.356628\pi$
$830$	−20.3262	−0.705534
$831$	0	0
$832$	−3.32624	−0.115317
$833$	5.85410	0.202833
$834$	0	0
$835$	12.4164	0.429688
$836$	4.14590	0.143389
$837$	0	0
$838$	21.5623	0.744857
$839$	26.1459	0.902657	0.451328	−	0.892358i	$-0.350950\pi$
$839$	26.1459	0.902657	0.451328	+	0.892358i	$0.350950\pi$
$840$	0	0
$841$	−28.2705	−0.974845
$842$	−5.94427	−0.204853
$843$	0	0
$844$	−42.0000	−1.44570
$845$	12.8541	0.442195
$846$	0	0
$847$	−10.9443	−0.376050
$848$	−30.4508	−1.04569
$849$	0	0
$850$	15.3262	0.525686
$851$	−88.0476	−3.01823
$852$	0	0
$853$	35.9443	1.23071	0.615354	−	0.788251i	$-0.289013\pi$
$853$	35.9443	1.23071	0.615354	+	0.788251i	$0.289013\pi$
$854$	−17.9443	−0.614040
$855$	0	0
$856$	−63.7214	−2.17795
$857$	−31.2918	−1.06891	−0.534454	−	0.845198i	$-0.679483\pi$
$857$	−31.2918	−1.06891	−0.534454	+	0.845198i	$0.679483\pi$
$858$	0	0
$859$	−2.65248	−0.0905013	−0.0452507	−	0.998976i	$-0.514409\pi$
$859$	−2.65248	−0.0905013	−0.0452507	+	0.998976i	$0.514409\pi$
$860$	−10.8541	−0.370122
$861$	0	0
$862$	66.4508	2.26332
$863$	−40.9443	−1.39376	−0.696880	−	0.717188i	$-0.745429\pi$
$863$	−40.9443	−1.39376	−0.696880	+	0.717188i	$0.745429\pi$
$864$	0	0
$865$	4.52786	0.153952
$866$	12.8541	0.436800
$867$	0	0
$868$	−36.2705	−1.23110
$869$	−3.49342	−0.118506
$870$	0	0
$871$	4.52786	0.153421
$872$	−56.5066	−1.91355
$873$	0	0
$874$	−83.8673	−2.83685
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−18.7082	−0.631731	−0.315866	−	0.948804i	$-0.602295\pi$
$877$	−18.7082	−0.631731	−0.315866	+	0.948804i	$0.602295\pi$
$878$	11.8541	0.400057
$879$	0	0
$880$	2.32624	0.0784175
$881$	−1.05573	−0.0355684	−0.0177842	−	0.999842i	$-0.505661\pi$
$881$	−1.05573	−0.0355684	−0.0177842	+	0.999842i	$0.505661\pi$
$882$	0	0
$883$	−4.88854	−0.164513	−0.0822563	−	0.996611i	$-0.526213\pi$
$883$	−4.88854	−0.164513	−0.0822563	+	0.996611i	$0.526213\pi$
$884$	−10.8541	−0.365063
$885$	0	0
$886$	76.3050	2.56351
$887$	13.1246	0.440681	0.220341	−	0.975423i	$-0.429283\pi$
$887$	13.1246	0.440681	0.220341	+	0.975423i	$0.429283\pi$
$888$	0	0
$889$	−9.09017	−0.304875
$890$	−12.3262	−0.413176
$891$	0	0
$892$	75.1033	2.51465
$893$	12.5623	0.420382
$894$	0	0
$895$	3.94427	0.131842
$896$	1.09017	0.0364200
$897$	0	0
$898$	38.1803	1.27409
$899$	6.38197	0.212850
$900$	0	0
$901$	−18.0902	−0.602671
$902$	4.67376	0.155619
$903$	0	0
$904$	−15.6180	−0.519448
$905$	4.23607	0.140812
$906$	0	0
$907$	−49.7984	−1.65353	−0.826764	−	0.562549i	$-0.809821\pi$
$907$	−49.7984	−1.65353	−0.826764	+	0.562549i	$0.809821\pi$
$908$	118.957	3.94774
$909$	0	0
$910$	1.00000	0.0331497
$911$	7.41641	0.245717	0.122858	−	0.992424i	$-0.460794\pi$
$911$	7.41641	0.245717	0.122858	+	0.992424i	$0.460794\pi$
$912$	0	0
$913$	−1.83282	−0.0606573
$914$	−84.6312	−2.79935
$915$	0	0
$916$	−17.7295	−0.585799
$917$	9.56231	0.315775
$918$	0	0
$919$	−3.74265	−0.123458	−0.0617292	−	0.998093i	$-0.519662\pi$
$919$	−3.74265	−0.123458	−0.0617292	+	0.998093i	$0.519662\pi$
$920$	−66.1591	−2.18120
$921$	0	0
$922$	9.47214	0.311948
$923$	−5.38197	−0.177150
$924$	0	0
$925$	−9.94427	−0.326966
$926$	−11.0000	−0.361482
$927$	0	0
$928$	−9.27051	−0.304319
$929$	−35.0902	−1.15127	−0.575636	−	0.817706i	$-0.695245\pi$
$929$	−35.0902	−1.15127	−0.575636	+	0.817706i	$0.695245\pi$
$930$	0	0
$931$	−3.61803	−0.118576
$932$	31.8541	1.04342
$933$	0	0
$934$	−54.8328	−1.79418
$935$	1.38197	0.0451951
$936$	0	0
$937$	49.1935	1.60708	0.803541	−	0.595250i	$-0.202947\pi$
$937$	49.1935	1.60708	0.803541	+	0.595250i	$0.202947\pi$
$938$	−31.0344	−1.01331
$939$	0	0
$940$	16.8541	0.549720
$941$	−22.4164	−0.730754	−0.365377	−	0.930860i	$-0.619060\pi$
$941$	−22.4164	−0.730754	−0.365377	+	0.930860i	$0.619060\pi$
$942$	0	0
$943$	−66.9574	−2.18043
$944$	−117.700	−3.83081
$945$	0	0
$946$	−1.38197	−0.0449316
$947$	21.3820	0.694821	0.347410	−	0.937713i	$-0.387061\pi$
$947$	21.3820	0.694821	0.347410	+	0.937713i	$0.387061\pi$
$948$	0	0
$949$	−3.79837	−0.123300
$950$	−9.47214	−0.307317
$951$	0	0
$952$	43.7426	1.41771
$953$	−3.58359	−0.116084	−0.0580420	−	0.998314i	$-0.518486\pi$
$953$	−3.58359	−0.116084	−0.0580420	+	0.998314i	$0.518486\pi$
$954$	0	0
$955$	−15.1459	−0.490110
$956$	−83.3951	−2.69719
$957$	0	0
$958$	18.5623	0.599721
$959$	−2.47214	−0.0798294
$960$	0	0
$961$	24.8328	0.801059
$962$	9.94427	0.320616
$963$	0	0
$964$	80.6656	2.59807
$965$	−19.2705	−0.620340
$966$	0	0
$967$	−10.8197	−0.347937	−0.173968	−	0.984751i	$-0.555659\pi$
$967$	−10.8197	−0.347937	−0.173968	+	0.984751i	$0.555659\pi$
$968$	−81.7771	−2.62842
$969$	0	0
$970$	8.09017	0.259760
$971$	−19.0557	−0.611527	−0.305764	−	0.952107i	$-0.598912\pi$
$971$	−19.0557	−0.611527	−0.305764	+	0.952107i	$0.598912\pi$
$972$	0	0
$973$	9.00000	0.288527
$974$	62.5410	2.00394
$975$	0	0
$976$	−67.5410	−2.16194
$977$	−29.4164	−0.941114	−0.470557	−	0.882370i	$-0.655947\pi$
$977$	−29.4164	−0.941114	−0.470557	+	0.882370i	$0.655947\pi$
$978$	0	0
$979$	−1.11146	−0.0355223
$980$	−4.85410	−0.155059
$981$	0	0
$982$	−84.6312	−2.70069
$983$	12.0213	0.383419	0.191710	−	0.981452i	$-0.438597\pi$
$983$	12.0213	0.383419	0.191710	+	0.981452i	$0.438597\pi$
$984$	0	0
$985$	1.76393	0.0562035
$986$	−13.0902	−0.416876
$987$	0	0
$988$	6.70820	0.213416
$989$	19.7984	0.629552
$990$	0	0
$991$	0.527864	0.0167682	0.00838408	−	0.999965i	$-0.497331\pi$
$991$	0.527864	0.0167682	0.00838408	+	0.999965i	$0.497331\pi$
$992$	−81.1033	−2.57503
$993$	0	0
$994$	36.8885	1.17003
$995$	3.56231	0.112933
$996$	0	0
$997$	−58.7082	−1.85931	−0.929654	−	0.368434i	$-0.879894\pi$
$997$	−58.7082	−1.85931	−0.929654	+	0.368434i	$0.879894\pi$
$998$	22.3262	0.706725
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.l.1.2	yes	2
3.2	odd	2		945.2.a.a.1.1	✓	2
5.4	even	2		4725.2.a.u.1.1		2
7.6	odd	2		6615.2.a.x.1.2		2
15.14	odd	2		4725.2.a.bh.1.2		2
21.20	even	2		6615.2.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.a.1.1	✓	2	3.2	odd	2
945.2.a.l.1.2	yes	2	1.1	even	1	trivial
4725.2.a.u.1.1		2	5.4	even	2
4725.2.a.bh.1.2		2	15.14	odd	2
6615.2.a.k.1.1		2	21.20	even	2
6615.2.a.x.1.2		2	7.6	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q+2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} +7.47214 q^{8} +O(q^{10})\)	\(q+2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} +7.47214 q^{8} -2.61803 q^{10} -0.236068 q^{11} -0.381966 q^{13} +2.61803 q^{14} +9.85410 q^{16} +5.85410 q^{17} -3.61803 q^{19} -4.85410 q^{20} -0.618034 q^{22} +8.85410 q^{23} +1.00000 q^{25} -1.00000 q^{26} +4.85410 q^{28} -0.854102 q^{29} -7.47214 q^{31} +10.8541 q^{32} +15.3262 q^{34} -1.00000 q^{35} -9.94427 q^{37} -9.47214 q^{38} -7.47214 q^{40} -7.56231 q^{41} +2.23607 q^{43} -1.14590 q^{44} +23.1803 q^{46} -3.47214 q^{47} +1.00000 q^{49} +2.61803 q^{50} -1.85410 q^{52} -3.09017 q^{53} +0.236068 q^{55} +7.47214 q^{56} -2.23607 q^{58} -11.9443 q^{59} -6.85410 q^{61} -19.5623 q^{62} +8.70820 q^{64} +0.381966 q^{65} -11.8541 q^{67} +28.4164 q^{68} -2.61803 q^{70} +14.0902 q^{71} +9.94427 q^{73} -26.0344 q^{74} -17.5623 q^{76} -0.236068 q^{77} +14.7984 q^{79} -9.85410 q^{80} -19.7984 q^{82} +7.76393 q^{83} -5.85410 q^{85} +5.85410 q^{86} -1.76393 q^{88} +4.70820 q^{89} -0.381966 q^{91} +42.9787 q^{92} -9.09017 q^{94} +3.61803 q^{95} -3.09017 q^{97} +2.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8	\( 2 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8} - 3 q^{10} + 4 q^{11} - 3 q^{13} + 3 q^{14} + 13 q^{16} + 5 q^{17} - 5 q^{19} - 3 q^{20} + q^{22} + 11 q^{23} + 2 q^{25} - 2 q^{26} + 3 q^{28} + 5 q^{29} - 6 q^{31} + 15 q^{32} + 15 q^{34} - 2 q^{35} - 2 q^{37} - 10 q^{38} - 6 q^{40} + 5 q^{41} - 9 q^{44} + 24 q^{46} + 2 q^{47} + 2 q^{49} + 3 q^{50} + 3 q^{52} + 5 q^{53} - 4 q^{55} + 6 q^{56} - 6 q^{59} - 7 q^{61} - 19 q^{62} + 4 q^{64} + 3 q^{65} - 17 q^{67} + 30 q^{68} - 3 q^{70} + 17 q^{71} + 2 q^{73} - 23 q^{74} - 15 q^{76} + 4 q^{77} + 5 q^{79} - 13 q^{80} - 15 q^{82} + 20 q^{83} - 5 q^{85} + 5 q^{86} - 8 q^{88} - 4 q^{89} - 3 q^{91} + 39 q^{92} - 7 q^{94} + 5 q^{95} + 5 q^{97} + 3 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8 - 3 * q^10 + 4 * q^11 - 3 * q^13 + 3 * q^14 + 13 * q^16 + 5 * q^17 - 5 * q^19 - 3 * q^20 + q^22 + 11 * q^23 + 2 * q^25 - 2 * q^26 + 3 * q^28 + 5 * q^29 - 6 * q^31 + 15 * q^32 + 15 * q^34 - 2 * q^35 - 2 * q^37 - 10 * q^38 - 6 * q^40 + 5 * q^41 - 9 * q^44 + 24 * q^46 + 2 * q^47 + 2 * q^49 + 3 * q^50 + 3 * q^52 + 5 * q^53 - 4 * q^55 + 6 * q^56 - 6 * q^59 - 7 * q^61 - 19 * q^62 + 4 * q^64 + 3 * q^65 - 17 * q^67 + 30 * q^68 - 3 * q^70 + 17 * q^71 + 2 * q^73 - 23 * q^74 - 15 * q^76 + 4 * q^77 + 5 * q^79 - 13 * q^80 - 15 * q^82 + 20 * q^83 - 5 * q^85 + 5 * q^86 - 8 * q^88 - 4 * q^89 - 3 * q^91 + 39 * q^92 - 7 * q^94 + 5 * q^95 + 5 * q^97 + 3 * q^98

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