LMFDB - Embedded newform 1935.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1935,2,Mod(1,1935)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1935, 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("1935.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1935 = 3^{2} \cdot 5 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1935.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:	$15.4510527911$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1933097.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 13x^{2} + 5x - 4 $ x^5 - 2x^4 - 7x^3 + 13x^2 + 5x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.20940$ of defining polynomial
Character	$\chi$	$=$	1935.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.20940 q^{2} +2.88146 q^{4} -1.00000 q^{5} -0.988801 q^{7} -1.94750 q^{8} +O(q^{10})$	\(q-2.20940 q^{2} +2.88146 q^{4} -1.00000 q^{5} -0.988801 q^{7} -1.94750 q^{8} +2.20940 q^{10} +3.47130 q^{11} +5.78805 q^{13} +2.18466 q^{14} -1.46010 q^{16} +7.13216 q^{17} +6.26433 q^{19} -2.88146 q^{20} -7.66951 q^{22} -1.36924 q^{23} +1.00000 q^{25} -12.7881 q^{26} -2.84919 q^{28} -7.74052 q^{29} -9.06357 q^{31} +7.12096 q^{32} -15.7578 q^{34} +0.988801 q^{35} +6.07967 q^{37} -13.8404 q^{38} +1.94750 q^{40} -3.31147 q^{41} -1.00000 q^{43} +10.0024 q^{44} +3.02520 q^{46} +6.18173 q^{47} -6.02227 q^{49} -2.20940 q^{50} +16.6780 q^{52} +0.867837 q^{53} -3.47130 q^{55} +1.92569 q^{56} +17.1019 q^{58} -0.261901 q^{59} +13.5761 q^{61} +20.0251 q^{62} -12.8129 q^{64} -5.78805 q^{65} -0.229250 q^{67} +20.5510 q^{68} -2.18466 q^{70} -12.2419 q^{71} -2.02482 q^{73} -13.4324 q^{74} +18.0504 q^{76} -3.43243 q^{77} +9.66086 q^{79} +1.46010 q^{80} +7.31636 q^{82} -1.04956 q^{83} -7.13216 q^{85} +2.20940 q^{86} -6.76037 q^{88} +6.15933 q^{89} -5.72323 q^{91} -3.94541 q^{92} -13.6579 q^{94} -6.26433 q^{95} +1.10492 q^{97} +13.3056 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 8 q^{4} - 5 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) $ 5 * q - 2 * q^2 + 8 * q^4 - 5 * q^5 + 5 * q^7 - 3 * q^8	$ 5 q - 2 q^{2} + 8 q^{4} - 5 q^{5} + 5 q^{7} - 3 q^{8} + 2 q^{10} + 6 q^{11} + 5 q^{13} - q^{14} + 14 q^{16} + 17 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{22} - q^{23} + 5 q^{25} - 22 q^{26} + 26 q^{28} - 6 q^{29} + 6 q^{31} + 7 q^{32} - 5 q^{35} + 5 q^{37} + 16 q^{38} + 3 q^{40} - 2 q^{41} - 5 q^{43} + 15 q^{44} - 14 q^{46} + 18 q^{49} - 2 q^{50} - 38 q^{52} + 23 q^{53} - 6 q^{55} + 19 q^{56} + 12 q^{58} + q^{59} + 20 q^{61} + 3 q^{62} - 25 q^{64} - 5 q^{65} + 21 q^{67} + 48 q^{68} + q^{70} - 4 q^{71} + 5 q^{73} - 24 q^{74} + 32 q^{76} + 26 q^{77} + 41 q^{79} - 14 q^{80} + 38 q^{82} + 7 q^{83} - 17 q^{85} + 2 q^{86} + 12 q^{88} - 20 q^{89} - 42 q^{91} + 52 q^{92} - 42 q^{94} + 6 q^{95} + 37 q^{97} + 26 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 8 * q^4 - 5 * q^5 + 5 * q^7 - 3 * q^8 + 2 * q^10 + 6 * q^11 + 5 * q^13 - q^14 + 14 * q^16 + 17 * q^17 - 6 * q^19 - 8 * q^20 - 8 * q^22 - q^23 + 5 * q^25 - 22 * q^26 + 26 * q^28 - 6 * q^29 + 6 * q^31 + 7 * q^32 - 5 * q^35 + 5 * q^37 + 16 * q^38 + 3 * q^40 - 2 * q^41 - 5 * q^43 + 15 * q^44 - 14 * q^46 + 18 * q^49 - 2 * q^50 - 38 * q^52 + 23 * q^53 - 6 * q^55 + 19 * q^56 + 12 * q^58 + q^59 + 20 * q^61 + 3 * q^62 - 25 * q^64 - 5 * q^65 + 21 * q^67 + 48 * q^68 + q^70 - 4 * q^71 + 5 * q^73 - 24 * q^74 + 32 * q^76 + 26 * q^77 + 41 * q^79 - 14 * q^80 + 38 * q^82 + 7 * q^83 - 17 * q^85 + 2 * q^86 + 12 * q^88 - 20 * q^89 - 42 * q^91 + 52 * q^92 - 42 * q^94 + 6 * q^95 + 37 * q^97 + 26 * 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 $

$2$	−2.20940	−1.56228	−0.781142	−	0.624354i	$-0.785362\pi$
$2$	−2.20940	−1.56228	−0.781142	+	0.624354i	$0.785362\pi$
$3$	0	0
$3$	0	0
$4$	2.88146	1.44073
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.988801	−0.373732	−0.186866	−	0.982385i	$-0.559833\pi$
$7$	−0.988801	−0.373732	−0.186866	+	0.982385i	$0.559833\pi$
$8$	−1.94750	−0.688546
$9$	0	0
$10$	2.20940	0.698675
$11$	3.47130	1.04664	0.523319	−	0.852137i	$-0.324694\pi$
$11$	3.47130	1.04664	0.523319	+	0.852137i	$0.324694\pi$
$12$	0	0
$13$	5.78805	1.60532	0.802658	−	0.596440i	$-0.203419\pi$
$13$	5.78805	1.60532	0.802658	+	0.596440i	$0.203419\pi$
$14$	2.18466	0.583875
$15$	0	0
$16$	−1.46010	−0.365026
$17$	7.13216	1.72980	0.864902	−	0.501941i	$-0.167381\pi$
$17$	7.13216	1.72980	0.864902	+	0.501941i	$0.167381\pi$
$18$	0	0
$19$	6.26433	1.43713	0.718567	−	0.695457i	$-0.244798\pi$
$19$	6.26433	1.43713	0.718567	+	0.695457i	$0.244798\pi$
$20$	−2.88146	−0.644314
$21$	0	0
$22$	−7.66951	−1.63514
$23$	−1.36924	−0.285506	−0.142753	−	0.989758i	$-0.545596\pi$
$23$	−1.36924	−0.285506	−0.142753	+	0.989758i	$0.545596\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−12.7881	−2.50796
$27$	0	0
$28$	−2.84919	−0.538447
$29$	−7.74052	−1.43738	−0.718690	−	0.695331i	$-0.755258\pi$
$29$	−7.74052	−1.43738	−0.718690	+	0.695331i	$0.755258\pi$
$30$	0	0
$31$	−9.06357	−1.62787	−0.813933	−	0.580959i	$-0.802678\pi$
$31$	−9.06357	−1.62787	−0.813933	+	0.580959i	$0.802678\pi$
$32$	7.12096	1.25882
$33$	0	0
$34$	−15.7578	−2.70244
$35$	0.988801	0.167138
$36$	0	0
$37$	6.07967	0.999491	0.499745	−	0.866172i	$-0.333427\pi$
$37$	6.07967	0.999491	0.499745	+	0.866172i	$0.333427\pi$
$38$	−13.8404	−2.24521
$39$	0	0
$40$	1.94750	0.307927
$41$	−3.31147	−0.517164	−0.258582	−	0.965989i	$-0.583255\pi$
$41$	−3.31147	−0.517164	−0.258582	+	0.965989i	$0.583255\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	10.0024	1.50792
$45$	0	0
$46$	3.02520	0.446042
$47$	6.18173	0.901698	0.450849	−	0.892600i	$-0.351121\pi$
$47$	6.18173	0.901698	0.450849	+	0.892600i	$0.351121\pi$
$48$	0	0
$49$	−6.02227	−0.860325
$50$	−2.20940	−0.312457
$51$	0	0
$52$	16.6780	2.31283
$53$	0.867837	0.119207	0.0596033	−	0.998222i	$-0.481016\pi$
$53$	0.867837	0.119207	0.0596033	+	0.998222i	$0.481016\pi$
$54$	0	0
$55$	−3.47130	−0.468070
$56$	1.92569	0.257332
$57$	0	0
$58$	17.1019	2.24559
$59$	−0.261901	−0.0340965	−0.0170483	−	0.999855i	$-0.505427\pi$
$59$	−0.261901	−0.0340965	−0.0170483	+	0.999855i	$0.505427\pi$
$60$	0	0
$61$	13.5761	1.73824	0.869120	−	0.494601i	$-0.164686\pi$
$61$	13.5761	1.73824	0.869120	+	0.494601i	$0.164686\pi$
$62$	20.0251	2.54319
$63$	0	0
$64$	−12.8129	−1.60161
$65$	−5.78805	−0.717919
$66$	0	0
$67$	−0.229250	−0.0280073	−0.0140037	−	0.999902i	$-0.504458\pi$
$67$	−0.229250	−0.0280073	−0.0140037	+	0.999902i	$0.504458\pi$
$68$	20.5510	2.49218
$69$	0	0
$70$	−2.18466	−0.261117
$71$	−12.2419	−1.45285	−0.726425	−	0.687246i	$-0.758819\pi$
$71$	−12.2419	−1.45285	−0.726425	+	0.687246i	$0.758819\pi$
$72$	0	0
$73$	−2.02482	−0.236988	−0.118494	−	0.992955i	$-0.537807\pi$
$73$	−2.02482	−0.236988	−0.118494	+	0.992955i	$0.537807\pi$
$74$	−13.4324	−1.56149
$75$	0	0
$76$	18.0504	2.07052
$77$	−3.43243	−0.391162
$78$	0	0
$79$	9.66086	1.08693	0.543466	−	0.839431i	$-0.317112\pi$
$79$	9.66086	1.08693	0.543466	+	0.839431i	$0.317112\pi$
$80$	1.46010	0.163245
$81$	0	0
$82$	7.31636	0.807957
$83$	−1.04956	−0.115205	−0.0576023	−	0.998340i	$-0.518346\pi$
$83$	−1.04956	−0.115205	−0.0576023	+	0.998340i	$0.518346\pi$
$84$	0	0
$85$	−7.13216	−0.773592
$86$	2.20940	0.238246
$87$	0	0
$88$	−6.76037	−0.720658
$89$	6.15933	0.652888	0.326444	−	0.945217i	$-0.394150\pi$
$89$	6.15933	0.652888	0.326444	+	0.945217i	$0.394150\pi$
$90$	0	0
$91$	−5.72323	−0.599957
$92$	−3.94541	−0.411338
$93$	0	0
$94$	−13.6579	−1.40871
$95$	−6.26433	−0.642706
$96$	0	0
$97$	1.10492	0.112187	0.0560936	−	0.998426i	$-0.482135\pi$
$97$	1.10492	0.112187	0.0560936	+	0.998426i	$0.482135\pi$
$98$	13.3056	1.34407
$99$	0	0
$100$	2.88146	0.288146
$101$	4.12096	0.410051	0.205026	−	0.978757i	$-0.434272\pi$
$101$	4.12096	0.410051	0.205026	+	0.978757i	$0.434272\pi$
$102$	0	0
$103$	−11.3965	−1.12293	−0.561465	−	0.827501i	$-0.689762\pi$
$103$	−11.3965	−1.12293	−0.561465	+	0.827501i	$0.689762\pi$
$104$	−11.2722	−1.10533
$105$	0	0
$106$	−1.91740	−0.186235
$107$	13.2069	1.27676	0.638381	−	0.769720i	$-0.279604\pi$
$107$	13.2069	1.27676	0.638381	+	0.769720i	$0.279604\pi$
$108$	0	0
$109$	14.6019	1.39860	0.699302	−	0.714826i	$-0.253494\pi$
$109$	14.6019	1.39860	0.699302	+	0.714826i	$0.253494\pi$
$110$	7.66951	0.731259
$111$	0	0
$112$	1.44375	0.136422
$113$	−2.35531	−0.221569	−0.110785	−	0.993844i	$-0.535336\pi$
$113$	−2.35531	−0.221569	−0.110785	+	0.993844i	$0.535336\pi$
$114$	0	0
$115$	1.36924	0.127682
$116$	−22.3040	−2.07088
$117$	0	0
$118$	0.578644	0.0532685
$119$	−7.05229	−0.646483
$120$	0	0
$121$	1.04995	0.0954497
$122$	−29.9951	−2.71563
$123$	0	0
$124$	−26.1163	−2.34532
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.30904	0.293630	0.146815	−	0.989164i	$-0.453098\pi$
$127$	3.30904	0.293630	0.146815	+	0.989164i	$0.453098\pi$
$128$	14.0669	1.24335
$129$	0	0
$130$	12.7881	1.12159
$131$	3.17968	0.277810	0.138905	−	0.990306i	$-0.455642\pi$
$131$	3.17968	0.277810	0.138905	+	0.990306i	$0.455642\pi$
$132$	0	0
$133$	−6.19417	−0.537103
$134$	0.506505	0.0437554
$135$	0	0
$136$	−13.8899	−1.19105
$137$	−3.40189	−0.290643	−0.145322	−	0.989384i	$-0.546422\pi$
$137$	−3.40189	−0.290643	−0.145322	+	0.989384i	$0.546422\pi$
$138$	0	0
$139$	8.33667	0.707107	0.353554	−	0.935414i	$-0.384973\pi$
$139$	8.33667	0.707107	0.353554	+	0.935414i	$0.384973\pi$
$140$	2.84919	0.240801
$141$	0	0
$142$	27.0474	2.26976
$143$	20.0921	1.68018
$144$	0	0
$145$	7.74052	0.642816
$146$	4.47365	0.370242
$147$	0	0
$148$	17.5183	1.44000
$149$	−14.9699	−1.22638	−0.613189	−	0.789936i	$-0.710114\pi$
$149$	−14.9699	−1.22638	−0.613189	+	0.789936i	$0.710114\pi$
$150$	0	0
$151$	−16.0999	−1.31019	−0.655095	−	0.755546i	$-0.727372\pi$
$151$	−16.0999	−1.31019	−0.655095	+	0.755546i	$0.727372\pi$
$152$	−12.1998	−0.989534
$153$	0	0
$154$	7.58362	0.611105
$155$	9.06357	0.728004
$156$	0	0
$157$	−2.87809	−0.229697	−0.114848	−	0.993383i	$-0.536638\pi$
$157$	−2.87809	−0.229697	−0.114848	+	0.993383i	$0.536638\pi$
$158$	−21.3447	−1.69810
$159$	0	0
$160$	−7.12096	−0.562962
$161$	1.35391	0.106703
$162$	0	0
$163$	−14.8678	−1.16454	−0.582268	−	0.812997i	$-0.697834\pi$
$163$	−14.8678	−1.16454	−0.582268	+	0.812997i	$0.697834\pi$
$164$	−9.54186	−0.745094
$165$	0	0
$166$	2.31891	0.179982
$167$	19.6336	1.51929	0.759645	−	0.650337i	$-0.225372\pi$
$167$	19.6336	1.51929	0.759645	+	0.650337i	$0.225372\pi$
$168$	0	0
$169$	20.5015	1.57704
$170$	15.7578	1.20857
$171$	0	0
$172$	−2.88146	−0.219709
$173$	12.6607	0.962578	0.481289	−	0.876562i	$-0.340169\pi$
$173$	12.6607	0.962578	0.481289	+	0.876562i	$0.340169\pi$
$174$	0	0
$175$	−0.988801	−0.0747464
$176$	−5.06847	−0.382050
$177$	0	0
$178$	−13.6084	−1.02000
$179$	5.10704	0.381718	0.190859	−	0.981617i	$-0.438873\pi$
$179$	5.10704	0.381718	0.190859	+	0.981617i	$0.438873\pi$
$180$	0	0
$181$	16.5533	1.23040	0.615199	−	0.788372i	$-0.289076\pi$
$181$	16.5533	1.23040	0.615199	+	0.788372i	$0.289076\pi$
$182$	12.6449	0.937304
$183$	0	0
$184$	2.66660	0.196584
$185$	−6.07967	−0.446986
$186$	0	0
$187$	24.7579	1.81048
$188$	17.8124	1.29910
$189$	0	0
$190$	13.8404	1.00409
$191$	−9.31662	−0.674127	−0.337063	−	0.941482i	$-0.609434\pi$
$191$	−9.31662	−0.674127	−0.337063	+	0.941482i	$0.609434\pi$
$192$	0	0
$193$	3.79315	0.273037	0.136518	−	0.990638i	$-0.456409\pi$
$193$	3.79315	0.273037	0.136518	+	0.990638i	$0.456409\pi$
$194$	−2.44120	−0.175268
$195$	0	0
$196$	−17.3529	−1.23950
$197$	18.7452	1.33554	0.667771	−	0.744367i	$-0.267249\pi$
$197$	18.7452	1.33554	0.667771	+	0.744367i	$0.267249\pi$
$198$	0	0
$199$	−18.7881	−1.33185	−0.665927	−	0.746016i	$-0.731964\pi$
$199$	−18.7881	−1.33185	−0.665927	+	0.746016i	$0.731964\pi$
$200$	−1.94750	−0.137709
$201$	0	0
$202$	−9.10487	−0.640616
$203$	7.65384	0.537194
$204$	0	0
$205$	3.31147	0.231283
$206$	25.1794	1.75433
$207$	0	0
$208$	−8.45115	−0.585982
$209$	21.7454	1.50416
$210$	0	0
$211$	5.92735	0.408056	0.204028	−	0.978965i	$-0.434597\pi$
$211$	5.92735	0.408056	0.204028	+	0.978965i	$0.434597\pi$
$212$	2.50064	0.171745
$213$	0	0
$214$	−29.1794	−1.99467
$215$	1.00000	0.0681994
$216$	0	0
$217$	8.96207	0.608385
$218$	−32.2614	−2.18502
$219$	0	0
$220$	−10.0024	−0.674363
$221$	41.2813	2.77688
$222$	0	0
$223$	−8.62370	−0.577485	−0.288743	−	0.957407i	$-0.593237\pi$
$223$	−8.62370	−0.577485	−0.288743	+	0.957407i	$0.593237\pi$
$224$	−7.04122	−0.470461
$225$	0	0
$226$	5.20384	0.346154
$227$	3.18032	0.211085	0.105543	−	0.994415i	$-0.466342\pi$
$227$	3.18032	0.211085	0.105543	+	0.994415i	$0.466342\pi$
$228$	0	0
$229$	−22.5238	−1.48841	−0.744206	−	0.667950i	$-0.767172\pi$
$229$	−22.5238	−1.48841	−0.744206	+	0.667950i	$0.767172\pi$
$230$	−3.02520	−0.199476
$231$	0	0
$232$	15.0747	0.989702
$233$	−7.12267	−0.466622	−0.233311	−	0.972402i	$-0.574956\pi$
$233$	−7.12267	−0.466622	−0.233311	+	0.972402i	$0.574956\pi$
$234$	0	0
$235$	−6.18173	−0.403251
$236$	−0.754656	−0.0491239
$237$	0	0
$238$	15.5814	1.00999
$239$	28.3353	1.83286	0.916430	−	0.400195i	$-0.131058\pi$
$239$	28.3353	1.83286	0.916430	+	0.400195i	$0.131058\pi$
$240$	0	0
$241$	8.06020	0.519203	0.259602	−	0.965716i	$-0.416409\pi$
$241$	8.06020	0.519203	0.259602	+	0.965716i	$0.416409\pi$
$242$	−2.31976	−0.149120
$243$	0	0
$244$	39.1190	2.50434
$245$	6.02227	0.384749
$246$	0	0
$247$	36.2582	2.30705
$248$	17.6513	1.12086
$249$	0	0
$250$	2.20940	0.139735
$251$	11.0272	0.696029	0.348014	−	0.937489i	$-0.386856\pi$
$251$	11.0272	0.696029	0.348014	+	0.937489i	$0.386856\pi$
$252$	0	0
$253$	−4.75305	−0.298822
$254$	−7.31100	−0.458733
$255$	0	0
$256$	−5.45362	−0.340852
$257$	27.5410	1.71796	0.858979	−	0.512010i	$-0.171099\pi$
$257$	27.5410	1.71796	0.858979	+	0.512010i	$0.171099\pi$
$258$	0	0
$259$	−6.01158	−0.373541
$260$	−16.6780	−1.03433
$261$	0	0
$262$	−7.02520	−0.434019
$263$	−7.08171	−0.436677	−0.218338	−	0.975873i	$-0.570064\pi$
$263$	−7.08171	−0.436677	−0.218338	+	0.975873i	$0.570064\pi$
$264$	0	0
$265$	−0.867837	−0.0533108
$266$	13.6854	0.839107
$267$	0	0
$268$	−0.660574	−0.0403510
$269$	−10.3644	−0.631930	−0.315965	−	0.948771i	$-0.602328\pi$
$269$	−10.3644	−0.631930	−0.315965	+	0.948771i	$0.602328\pi$
$270$	0	0
$271$	−14.6010	−0.886947	−0.443474	−	0.896287i	$-0.646254\pi$
$271$	−14.6010	−0.886947	−0.443474	+	0.896287i	$0.646254\pi$
$272$	−10.4137	−0.631424
$273$	0	0
$274$	7.51615	0.454067
$275$	3.47130	0.209327
$276$	0	0
$277$	−7.47965	−0.449409	−0.224704	−	0.974427i	$-0.572142\pi$
$277$	−7.47965	−0.449409	−0.224704	+	0.974427i	$0.572142\pi$
$278$	−18.4191	−1.10470
$279$	0	0
$280$	−1.92569	−0.115082
$281$	−4.90271	−0.292471	−0.146236	−	0.989250i	$-0.546716\pi$
$281$	−4.90271	−0.292471	−0.146236	+	0.989250i	$0.546716\pi$
$282$	0	0
$283$	31.4284	1.86823	0.934113	−	0.356977i	$-0.116193\pi$
$283$	31.4284	1.86823	0.934113	+	0.356977i	$0.116193\pi$
$284$	−35.2746	−2.09316
$285$	0	0
$286$	−44.3915	−2.62492
$287$	3.27438	0.193281
$288$	0	0
$289$	33.8677	1.99222
$290$	−17.1019	−1.00426
$291$	0	0
$292$	−5.83445	−0.341435
$293$	3.45380	0.201773	0.100887	−	0.994898i	$-0.467832\pi$
$293$	3.45380	0.201773	0.100887	+	0.994898i	$0.467832\pi$
$294$	0	0
$295$	0.261901	0.0152484
$296$	−11.8402	−0.688195
$297$	0	0
$298$	33.0744	1.91595
$299$	−7.92523	−0.458328
$300$	0	0
$301$	0.988801	0.0569936
$302$	35.5712	2.04689
$303$	0	0
$304$	−9.14657	−0.524592
$305$	−13.5761	−0.777365
$306$	0	0
$307$	33.1991	1.89477	0.947386	−	0.320092i	$-0.103714\pi$
$307$	33.1991	1.89477	0.947386	+	0.320092i	$0.103714\pi$
$308$	−9.89041	−0.563558
$309$	0	0
$310$	−20.0251	−1.13735
$311$	−4.33667	−0.245910	−0.122955	−	0.992412i	$-0.539237\pi$
$311$	−4.33667	−0.245910	−0.122955	+	0.992412i	$0.539237\pi$
$312$	0	0
$313$	18.3833	1.03909	0.519544	−	0.854444i	$-0.326102\pi$
$313$	18.3833	1.03909	0.519544	+	0.854444i	$0.326102\pi$
$314$	6.35886	0.358851
$315$	0	0
$316$	27.8374	1.56598
$317$	−6.07477	−0.341193	−0.170597	−	0.985341i	$-0.554569\pi$
$317$	−6.07477	−0.341193	−0.170597	+	0.985341i	$0.554569\pi$
$318$	0	0
$319$	−26.8697	−1.50441
$320$	12.8129	0.716261
$321$	0	0
$322$	−2.99133	−0.166700
$323$	44.6782	2.48596
$324$	0	0
$325$	5.78805	0.321063
$326$	32.8489	1.81933
$327$	0	0
$328$	6.44909	0.356091
$329$	−6.11250	−0.336993
$330$	0	0
$331$	1.83557	0.100892	0.0504460	−	0.998727i	$-0.483936\pi$
$331$	1.83557	0.100892	0.0504460	+	0.998727i	$0.483936\pi$
$332$	−3.02428	−0.165979
$333$	0	0
$334$	−43.3785	−2.37356
$335$	0.229250	0.0125252
$336$	0	0
$337$	−13.2447	−0.721483	−0.360741	−	0.932666i	$-0.617476\pi$
$337$	−13.2447	−0.721483	−0.360741	+	0.932666i	$0.617476\pi$
$338$	−45.2960	−2.46378
$339$	0	0
$340$	−20.5510	−1.11454
$341$	−31.4624	−1.70378
$342$	0	0
$343$	12.8764	0.695262
$344$	1.94750	0.105002
$345$	0	0
$346$	−27.9727	−1.50382
$347$	−12.1293	−0.651135	−0.325568	−	0.945519i	$-0.605555\pi$
$347$	−12.1293	−0.651135	−0.325568	+	0.945519i	$0.605555\pi$
$348$	0	0
$349$	26.6285	1.42539	0.712697	−	0.701472i	$-0.247474\pi$
$349$	26.6285	1.42539	0.712697	+	0.701472i	$0.247474\pi$
$350$	2.18466	0.116775
$351$	0	0
$352$	24.7190	1.31753
$353$	−5.23129	−0.278434	−0.139217	−	0.990262i	$-0.544458\pi$
$353$	−5.23129	−0.278434	−0.139217	+	0.990262i	$0.544458\pi$
$354$	0	0
$355$	12.2419	0.649734
$356$	17.7479	0.940635
$357$	0	0
$358$	−11.2835	−0.596352
$359$	−4.75892	−0.251166	−0.125583	−	0.992083i	$-0.540080\pi$
$359$	−4.75892	−0.251166	−0.125583	+	0.992083i	$0.540080\pi$
$360$	0	0
$361$	20.2418	1.06536
$362$	−36.5729	−1.92223
$363$	0	0
$364$	−16.4913	−0.864377
$365$	2.02482	0.105984
$366$	0	0
$367$	−18.5236	−0.966921	−0.483461	−	0.875366i	$-0.660620\pi$
$367$	−18.5236	−0.966921	−0.483461	+	0.875366i	$0.660620\pi$
$368$	1.99924	0.104217
$369$	0	0
$370$	13.4324	0.698319
$371$	−0.858119	−0.0445513
$372$	0	0
$373$	−27.6040	−1.42928	−0.714640	−	0.699493i	$-0.753409\pi$
$373$	−27.6040	−1.42928	−0.714640	+	0.699493i	$0.753409\pi$
$374$	−54.7002	−2.82848
$375$	0	0
$376$	−12.0389	−0.620860
$377$	−44.8025	−2.30745
$378$	0	0
$379$	−1.35652	−0.0696796	−0.0348398	−	0.999393i	$-0.511092\pi$
$379$	−1.35652	−0.0696796	−0.0348398	+	0.999393i	$0.511092\pi$
$380$	−18.0504	−0.925966
$381$	0	0
$382$	20.5842	1.05318
$383$	−18.9896	−0.970322	−0.485161	−	0.874425i	$-0.661239\pi$
$383$	−18.9896	−0.970322	−0.485161	+	0.874425i	$0.661239\pi$
$384$	0	0
$385$	3.43243	0.174933
$386$	−8.38059	−0.426561
$387$	0	0
$388$	3.18377	0.161631
$389$	−24.9697	−1.26601	−0.633007	−	0.774146i	$-0.718180\pi$
$389$	−24.9697	−1.26601	−0.633007	+	0.774146i	$0.718180\pi$
$390$	0	0
$391$	−9.76565	−0.493870
$392$	11.7284	0.592373
$393$	0	0
$394$	−41.4157	−2.08650
$395$	−9.66086	−0.486091
$396$	0	0
$397$	−20.8259	−1.04522	−0.522612	−	0.852571i	$-0.675042\pi$
$397$	−20.8259	−1.04522	−0.522612	+	0.852571i	$0.675042\pi$
$398$	41.5105	2.08074
$399$	0	0
$400$	−1.46010	−0.0730052
$401$	4.19770	0.209623	0.104811	−	0.994492i	$-0.466576\pi$
$401$	4.19770	0.209623	0.104811	+	0.994492i	$0.466576\pi$
$402$	0	0
$403$	−52.4604	−2.61324
$404$	11.8744	0.590773
$405$	0	0
$406$	−16.9104	−0.839250
$407$	21.1044	1.04610
$408$	0	0
$409$	−4.18173	−0.206773	−0.103387	−	0.994641i	$-0.532968\pi$
$409$	−4.18173	−0.206773	−0.103387	+	0.994641i	$0.532968\pi$
$410$	−7.31636	−0.361329
$411$	0	0
$412$	−32.8385	−1.61784
$413$	0.258968	0.0127430
$414$	0	0
$415$	1.04956	0.0515211
$416$	41.2165	2.02080
$417$	0	0
$418$	−48.0443	−2.34992
$419$	0.937506	0.0458002	0.0229001	−	0.999738i	$-0.492710\pi$
$419$	0.937506	0.0458002	0.0229001	+	0.999738i	$0.492710\pi$
$420$	0	0
$421$	15.1945	0.740534	0.370267	−	0.928925i	$-0.379266\pi$
$421$	15.1945	0.740534	0.370267	+	0.928925i	$0.379266\pi$
$422$	−13.0959	−0.637499
$423$	0	0
$424$	−1.69011	−0.0820792
$425$	7.13216	0.345961
$426$	0	0
$427$	−13.4241	−0.649636
$428$	38.0553	1.83947
$429$	0	0
$430$	−2.20940	−0.106547
$431$	−0.650480	−0.0313325	−0.0156663	−	0.999877i	$-0.504987\pi$
$431$	−0.650480	−0.0313325	−0.0156663	+	0.999877i	$0.504987\pi$
$432$	0	0
$433$	20.1847	0.970013	0.485006	−	0.874511i	$-0.338817\pi$
$433$	20.1847	0.970013	0.485006	+	0.874511i	$0.338817\pi$
$434$	−19.8008	−0.950470
$435$	0	0
$436$	42.0747	2.01501
$437$	−8.57737	−0.410311
$438$	0	0
$439$	−29.4009	−1.40323	−0.701614	−	0.712557i	$-0.747537\pi$
$439$	−29.4009	−1.40323	−0.701614	+	0.712557i	$0.747537\pi$
$440$	6.76037	0.322288
$441$	0	0
$442$	−91.2070	−4.33827
$443$	−34.5625	−1.64211	−0.821056	−	0.570847i	$-0.806615\pi$
$443$	−34.5625	−1.64211	−0.821056	+	0.570847i	$0.806615\pi$
$444$	0	0
$445$	−6.15933	−0.291980
$446$	19.0532	0.902196
$447$	0	0
$448$	12.6694	0.598572
$449$	−7.23402	−0.341395	−0.170697	−	0.985324i	$-0.554602\pi$
$449$	−7.23402	−0.341395	−0.170697	+	0.985324i	$0.554602\pi$
$450$	0	0
$451$	−11.4951	−0.541283
$452$	−6.78675	−0.319222
$453$	0	0
$454$	−7.02661	−0.329775
$455$	5.72323	0.268309
$456$	0	0
$457$	−9.93523	−0.464751	−0.232375	−	0.972626i	$-0.574650\pi$
$457$	−9.93523	−0.464751	−0.232375	+	0.972626i	$0.574650\pi$
$458$	49.7641	2.32532
$459$	0	0
$460$	3.94541	0.183956
$461$	4.34399	0.202320	0.101160	−	0.994870i	$-0.467745\pi$
$461$	4.34399	0.202320	0.101160	+	0.994870i	$0.467745\pi$
$462$	0	0
$463$	3.21037	0.149199	0.0745993	−	0.997214i	$-0.476232\pi$
$463$	3.21037	0.149199	0.0745993	+	0.997214i	$0.476232\pi$
$464$	11.3020	0.524681
$465$	0	0
$466$	15.7369	0.728996
$467$	−17.2961	−0.800369	−0.400184	−	0.916435i	$-0.631054\pi$
$467$	−17.2961	−0.800369	−0.400184	+	0.916435i	$0.631054\pi$
$468$	0	0
$469$	0.226682	0.0104672
$470$	13.6579	0.629993
$471$	0	0
$472$	0.510052	0.0234770
$473$	−3.47130	−0.159611
$474$	0	0
$475$	6.26433	0.287427
$476$	−20.3209	−0.931407
$477$	0	0
$478$	−62.6041	−2.86345
$479$	−5.90118	−0.269632	−0.134816	−	0.990871i	$-0.543044\pi$
$479$	−5.90118	−0.269632	−0.134816	+	0.990871i	$0.543044\pi$
$480$	0	0
$481$	35.1894	1.60450
$482$	−17.8082	−0.811143
$483$	0	0
$484$	3.02538	0.137517
$485$	−1.10492	−0.0501716
$486$	0	0
$487$	39.3612	1.78362	0.891812	−	0.452406i	$-0.149434\pi$
$487$	39.3612	1.78362	0.891812	+	0.452406i	$0.149434\pi$
$488$	−26.4395	−1.19686
$489$	0	0
$490$	−13.3056	−0.601087
$491$	4.35860	0.196701	0.0983505	−	0.995152i	$-0.468643\pi$
$491$	4.35860	0.196701	0.0983505	+	0.995152i	$0.468643\pi$
$492$	0	0
$493$	−55.2067	−2.48638
$494$	−80.1090	−3.60427
$495$	0	0
$496$	13.2338	0.594213
$497$	12.1048	0.542976
$498$	0	0
$499$	25.6899	1.15004	0.575018	−	0.818141i	$-0.304995\pi$
$499$	25.6899	1.15004	0.575018	+	0.818141i	$0.304995\pi$
$500$	−2.88146	−0.128863
$501$	0	0
$502$	−24.3635	−1.08739
$503$	−3.67703	−0.163951	−0.0819754	−	0.996634i	$-0.526123\pi$
$503$	−3.67703	−0.163951	−0.0819754	+	0.996634i	$0.526123\pi$
$504$	0	0
$505$	−4.12096	−0.183380
$506$	10.5014	0.466844
$507$	0	0
$508$	9.53487	0.423041
$509$	34.9139	1.54753	0.773767	−	0.633471i	$-0.218370\pi$
$509$	34.9139	1.54753	0.773767	+	0.633471i	$0.218370\pi$
$510$	0	0
$511$	2.00215	0.0885698
$512$	−16.0845	−0.710840
$513$	0	0
$514$	−60.8491	−2.68394
$515$	11.3965	0.502189
$516$	0	0
$517$	21.4587	0.943750
$518$	13.2820	0.583578
$519$	0	0
$520$	11.2722	0.494320
$521$	15.4257	0.675812	0.337906	−	0.941180i	$-0.390281\pi$
$521$	15.4257	0.675812	0.337906	+	0.941180i	$0.390281\pi$
$522$	0	0
$523$	32.2843	1.41169	0.705847	−	0.708364i	$-0.250567\pi$
$523$	32.2843	1.41169	0.705847	+	0.708364i	$0.250567\pi$
$524$	9.16214	0.400250
$525$	0	0
$526$	15.6463	0.682213
$527$	−64.6429	−2.81589
$528$	0	0
$529$	−21.1252	−0.918486
$530$	1.91740	0.0832866
$531$	0	0
$532$	−17.8483	−0.773821
$533$	−19.1669	−0.830211
$534$	0	0
$535$	−13.2069	−0.570985
$536$	0.446464	0.0192843
$537$	0	0
$538$	22.8992	0.987254
$539$	−20.9051	−0.900448
$540$	0	0
$541$	29.9794	1.28892	0.644459	−	0.764639i	$-0.277083\pi$
$541$	29.9794	1.28892	0.644459	+	0.764639i	$0.277083\pi$
$542$	32.2595	1.38566
$543$	0	0
$544$	50.7879	2.17751
$545$	−14.6019	−0.625475
$546$	0	0
$547$	−35.0520	−1.49871	−0.749357	−	0.662166i	$-0.769637\pi$
$547$	−35.0520	−1.49871	−0.749357	+	0.662166i	$0.769637\pi$
$548$	−9.80242	−0.418739
$549$	0	0
$550$	−7.66951	−0.327029
$551$	−48.4892	−2.06571
$552$	0	0
$553$	−9.55267	−0.406221
$554$	16.5256	0.702104
$555$	0	0
$556$	24.0218	1.01875
$557$	−26.4107	−1.11906	−0.559529	−	0.828811i	$-0.689018\pi$
$557$	−26.4107	−1.11906	−0.559529	+	0.828811i	$0.689018\pi$
$558$	0	0
$559$	−5.78805	−0.244808
$560$	−1.44375	−0.0610097
$561$	0	0
$562$	10.8321	0.456923
$563$	29.9152	1.26077	0.630387	−	0.776281i	$-0.282896\pi$
$563$	29.9152	1.26077	0.630387	+	0.776281i	$0.282896\pi$
$564$	0	0
$565$	2.35531	0.0990888
$566$	−69.4381	−2.91870
$567$	0	0
$568$	23.8412	1.00035
$569$	31.5162	1.32123	0.660614	−	0.750726i	$-0.270296\pi$
$569$	31.5162	1.32123	0.660614	+	0.750726i	$0.270296\pi$
$570$	0	0
$571$	17.6363	0.738056	0.369028	−	0.929418i	$-0.379691\pi$
$571$	17.6363	0.738056	0.369028	+	0.929418i	$0.379691\pi$
$572$	57.8945	2.42069
$573$	0	0
$574$	−7.23443	−0.301959
$575$	−1.36924	−0.0571013
$576$	0	0
$577$	28.5791	1.18976	0.594881	−	0.803814i	$-0.297199\pi$
$577$	28.5791	1.18976	0.594881	+	0.803814i	$0.297199\pi$
$578$	−74.8275	−3.11241
$579$	0	0
$580$	22.3040	0.926124
$581$	1.03781	0.0430557
$582$	0	0
$583$	3.01253	0.124766
$584$	3.94335	0.163177
$585$	0	0
$586$	−7.63084	−0.315227
$587$	32.9257	1.35899	0.679494	−	0.733681i	$-0.262199\pi$
$587$	32.9257	1.35899	0.679494	+	0.733681i	$0.262199\pi$
$588$	0	0
$589$	−56.7772	−2.33946
$590$	−0.578644	−0.0238224
$591$	0	0
$592$	−8.87695	−0.364840
$593$	28.5886	1.17399	0.586996	−	0.809590i	$-0.300310\pi$
$593$	28.5886	1.17399	0.586996	+	0.809590i	$0.300310\pi$
$594$	0	0
$595$	7.05229	0.289116
$596$	−43.1351	−1.76688
$597$	0	0
$598$	17.5100	0.716038
$599$	−25.8590	−1.05657	−0.528286	−	0.849067i	$-0.677165\pi$
$599$	−25.8590	−1.05657	−0.528286	+	0.849067i	$0.677165\pi$
$600$	0	0
$601$	−19.3020	−0.787344	−0.393672	−	0.919251i	$-0.628795\pi$
$601$	−19.3020	−0.787344	−0.393672	+	0.919251i	$0.628795\pi$
$602$	−2.18466	−0.0890401
$603$	0	0
$604$	−46.3912	−1.88763
$605$	−1.04995	−0.0426864
$606$	0	0
$607$	41.0596	1.66656	0.833278	−	0.552854i	$-0.186461\pi$
$607$	41.0596	1.66656	0.833278	+	0.552854i	$0.186461\pi$
$608$	44.6080	1.80909
$609$	0	0
$610$	29.9951	1.21446
$611$	35.7801	1.44751
$612$	0	0
$613$	1.39095	0.0561798	0.0280899	−	0.999605i	$-0.491058\pi$
$613$	1.39095	0.0561798	0.0280899	+	0.999605i	$0.491058\pi$
$614$	−73.3502	−2.96017
$615$	0	0
$616$	6.68466	0.269333
$617$	22.6207	0.910676	0.455338	−	0.890319i	$-0.349518\pi$
$617$	22.6207	0.910676	0.455338	+	0.890319i	$0.349518\pi$
$618$	0	0
$619$	−1.20272	−0.0483413	−0.0241706	−	0.999708i	$-0.507695\pi$
$619$	−1.20272	−0.0483413	−0.0241706	+	0.999708i	$0.507695\pi$
$620$	26.1163	1.04886
$621$	0	0
$622$	9.58145	0.384181
$623$	−6.09035	−0.244005
$624$	0	0
$625$	1.00000	0.0400000
$626$	−40.6162	−1.62335
$627$	0	0
$628$	−8.29311	−0.330931
$629$	43.3612	1.72892
$630$	0	0
$631$	−4.37325	−0.174096	−0.0870481	−	0.996204i	$-0.527743\pi$
$631$	−4.37325	−0.174096	−0.0870481	+	0.996204i	$0.527743\pi$
$632$	−18.8145	−0.748402
$633$	0	0
$634$	13.4216	0.533040
$635$	−3.30904	−0.131315
$636$	0	0
$637$	−34.8572	−1.38109
$638$	59.3660	2.35032
$639$	0	0
$640$	−14.0669	−0.556042
$641$	−16.6004	−0.655675	−0.327838	−	0.944734i	$-0.606320\pi$
$641$	−16.6004	−0.655675	−0.327838	+	0.944734i	$0.606320\pi$
$642$	0	0
$643$	19.3762	0.764123	0.382062	−	0.924137i	$-0.375214\pi$
$643$	19.3762	0.764123	0.382062	+	0.924137i	$0.375214\pi$
$644$	3.90123	0.153730
$645$	0	0
$646$	−98.7121	−3.88378
$647$	−27.1872	−1.06884	−0.534420	−	0.845219i	$-0.679470\pi$
$647$	−27.1872	−1.06884	−0.534420	+	0.845219i	$0.679470\pi$
$648$	0	0
$649$	−0.909136	−0.0356867
$650$	−12.7881	−0.501592
$651$	0	0
$652$	−42.8410	−1.67778
$653$	−22.8764	−0.895223	−0.447612	−	0.894228i	$-0.647725\pi$
$653$	−22.8764	−0.895223	−0.447612	+	0.894228i	$0.647725\pi$
$654$	0	0
$655$	−3.17968	−0.124241
$656$	4.83509	0.188778
$657$	0	0
$658$	13.5050	0.526479
$659$	−18.2273	−0.710036	−0.355018	−	0.934859i	$-0.615525\pi$
$659$	−18.2273	−0.710036	−0.355018	+	0.934859i	$0.615525\pi$
$660$	0	0
$661$	−26.6129	−1.03512	−0.517562	−	0.855646i	$-0.673160\pi$
$661$	−26.6129	−1.03512	−0.517562	+	0.855646i	$0.673160\pi$
$662$	−4.05551	−0.157622
$663$	0	0
$664$	2.04403	0.0793237
$665$	6.19417	0.240200
$666$	0	0
$667$	10.5986	0.410381
$668$	56.5734	2.18889
$669$	0	0
$670$	−0.506505	−0.0195680
$671$	47.1267	1.81931
$672$	0	0
$673$	−32.7129	−1.26099	−0.630495	−	0.776193i	$-0.717148\pi$
$673$	−32.7129	−1.26099	−0.630495	+	0.776193i	$0.717148\pi$
$674$	29.2628	1.12716
$675$	0	0
$676$	59.0742	2.27209
$677$	35.5858	1.36767	0.683836	−	0.729636i	$-0.260310\pi$
$677$	35.5858	1.36767	0.683836	+	0.729636i	$0.260310\pi$
$678$	0	0
$679$	−1.09254	−0.0419279
$680$	13.8899	0.532653
$681$	0	0
$682$	69.5131	2.66180
$683$	−25.4303	−0.973064	−0.486532	−	0.873663i	$-0.661738\pi$
$683$	−25.4303	−0.973064	−0.486532	+	0.873663i	$0.661738\pi$
$684$	0	0
$685$	3.40189	0.129980
$686$	−28.4492	−1.08620
$687$	0	0
$688$	1.46010	0.0556660
$689$	5.02308	0.191364
$690$	0	0
$691$	16.0016	0.608731	0.304366	−	0.952555i	$-0.401556\pi$
$691$	16.0016	0.608731	0.304366	+	0.952555i	$0.401556\pi$
$692$	36.4814	1.38682
$693$	0	0
$694$	26.7985	1.01726
$695$	−8.33667	−0.316228
$696$	0	0
$697$	−23.6179	−0.894592
$698$	−58.8332	−2.22687
$699$	0	0
$700$	−2.84919	−0.107689
$701$	−18.6181	−0.703196	−0.351598	−	0.936151i	$-0.614362\pi$
$701$	−18.6181	−0.703196	−0.351598	+	0.936151i	$0.614362\pi$
$702$	0	0
$703$	38.0850	1.43640
$704$	−44.4774	−1.67630
$705$	0	0
$706$	11.5580	0.434992
$707$	−4.07481	−0.153249
$708$	0	0
$709$	35.9746	1.35105	0.675527	−	0.737335i	$-0.263916\pi$
$709$	35.9746	1.35105	0.675527	+	0.737335i	$0.263916\pi$
$710$	−27.0474	−1.01507
$711$	0	0
$712$	−11.9953	−0.449543
$713$	12.4102	0.464766
$714$	0	0
$715$	−20.0921	−0.751401
$716$	14.7157	0.549953
$717$	0	0
$718$	10.5144	0.392393
$719$	−28.6926	−1.07006	−0.535028	−	0.844835i	$-0.679699\pi$
$719$	−28.6926	−1.07006	−0.535028	+	0.844835i	$0.679699\pi$
$720$	0	0
$721$	11.2689	0.419674
$722$	−44.7222	−1.66439
$723$	0	0
$724$	47.6977	1.77267
$725$	−7.74052	−0.287476
$726$	0	0
$727$	−32.2142	−1.19476	−0.597378	−	0.801960i	$-0.703791\pi$
$727$	−32.2142	−1.19476	−0.597378	+	0.801960i	$0.703791\pi$
$728$	11.1460	0.413098
$729$	0	0
$730$	−4.47365	−0.165577
$731$	−7.13216	−0.263793
$732$	0	0
$733$	−8.01974	−0.296216	−0.148108	−	0.988971i	$-0.547318\pi$
$733$	−8.01974	−0.296216	−0.148108	+	0.988971i	$0.547318\pi$
$734$	40.9260	1.51061
$735$	0	0
$736$	−9.75032	−0.359401
$737$	−0.795795	−0.0293135
$738$	0	0
$739$	−53.8031	−1.97918	−0.989590	−	0.143915i	$-0.954031\pi$
$739$	−53.8031	−1.97918	−0.989590	+	0.143915i	$0.954031\pi$
$740$	−17.5183	−0.643986
$741$	0	0
$742$	1.89593	0.0696018
$743$	−12.0967	−0.443785	−0.221893	−	0.975071i	$-0.571223\pi$
$743$	−12.0967	−0.443785	−0.221893	+	0.975071i	$0.571223\pi$
$744$	0	0
$745$	14.9699	0.548453
$746$	60.9883	2.23294
$747$	0	0
$748$	71.3389	2.60841
$749$	−13.0590	−0.477167
$750$	0	0
$751$	−29.0995	−1.06186	−0.530928	−	0.847417i	$-0.678156\pi$
$751$	−29.0995	−1.06186	−0.530928	+	0.847417i	$0.678156\pi$
$752$	−9.02597	−0.329143
$753$	0	0
$754$	98.9868	3.60489
$755$	16.0999	0.585935
$756$	0	0
$757$	18.5816	0.675359	0.337680	−	0.941261i	$-0.390358\pi$
$757$	18.5816	0.675359	0.337680	+	0.941261i	$0.390358\pi$
$758$	2.99709	0.108859
$759$	0	0
$760$	12.1998	0.442533
$761$	−1.78507	−0.0647087	−0.0323543	−	0.999476i	$-0.510301\pi$
$761$	−1.78507	−0.0647087	−0.0323543	+	0.999476i	$0.510301\pi$
$762$	0	0
$763$	−14.4383	−0.522703
$764$	−26.8455	−0.971235
$765$	0	0
$766$	41.9556	1.51592
$767$	−1.51589	−0.0547357
$768$	0	0
$769$	16.4096	0.591747	0.295874	−	0.955227i	$-0.404389\pi$
$769$	16.4096	0.591747	0.295874	+	0.955227i	$0.404389\pi$
$770$	−7.58362	−0.273295
$771$	0	0
$772$	10.9298	0.393372
$773$	30.1231	1.08345	0.541726	−	0.840555i	$-0.317771\pi$
$773$	30.1231	1.08345	0.541726	+	0.840555i	$0.317771\pi$
$774$	0	0
$775$	−9.06357	−0.325573
$776$	−2.15183	−0.0772460
$777$	0	0
$778$	55.1681	1.97787
$779$	−20.7441	−0.743234
$780$	0	0
$781$	−42.4954	−1.52061
$782$	21.5763	0.771565
$783$	0	0
$784$	8.79315	0.314041
$785$	2.87809	0.102723
$786$	0	0
$787$	−16.8814	−0.601757	−0.300878	−	0.953663i	$-0.597280\pi$
$787$	−16.8814	−0.601757	−0.300878	+	0.953663i	$0.597280\pi$
$788$	54.0136	1.92416
$789$	0	0
$790$	21.3447	0.759411
$791$	2.32894	0.0828075
$792$	0	0
$793$	78.5791	2.79042
$794$	46.0129	1.63294
$795$	0	0
$796$	−54.1373	−1.91884
$797$	37.1911	1.31738	0.658688	−	0.752416i	$-0.271112\pi$
$797$	37.1911	1.31738	0.658688	+	0.752416i	$0.271112\pi$
$798$	0	0
$799$	44.0891	1.55976
$800$	7.12096	0.251764
$801$	0	0
$802$	−9.27440	−0.327491
$803$	−7.02877	−0.248040
$804$	0	0
$805$	−1.35391	−0.0477190
$806$	115.906	4.08262
$807$	0	0
$808$	−8.02559	−0.282339
$809$	−36.3933	−1.27952	−0.639760	−	0.768575i	$-0.720966\pi$
$809$	−36.3933	−1.27952	−0.639760	+	0.768575i	$0.720966\pi$
$810$	0	0
$811$	11.7890	0.413968	0.206984	−	0.978344i	$-0.433635\pi$
$811$	11.7890	0.413968	0.206984	+	0.978344i	$0.433635\pi$
$812$	22.0542	0.773952
$813$	0	0
$814$	−46.6280	−1.63431
$815$	14.8678	0.520796
$816$	0	0
$817$	−6.26433	−0.219161
$818$	9.23912	0.323038
$819$	0	0
$820$	9.54186	0.333216
$821$	39.6755	1.38469	0.692343	−	0.721569i	$-0.256579\pi$
$821$	39.6755	1.38469	0.692343	+	0.721569i	$0.256579\pi$
$822$	0	0
$823$	17.3869	0.606068	0.303034	−	0.952980i	$-0.402000\pi$
$823$	17.3869	0.606068	0.303034	+	0.952980i	$0.402000\pi$
$824$	22.1947	0.773189
$825$	0	0
$826$	−0.572164	−0.0199081
$827$	−20.8700	−0.725720	−0.362860	−	0.931844i	$-0.618200\pi$
$827$	−20.8700	−0.725720	−0.362860	+	0.931844i	$0.618200\pi$
$828$	0	0
$829$	11.5430	0.400904	0.200452	−	0.979703i	$-0.435759\pi$
$829$	11.5430	0.400904	0.200452	+	0.979703i	$0.435759\pi$
$830$	−2.31891	−0.0804906
$831$	0	0
$832$	−74.1615	−2.57109
$833$	−42.9518	−1.48819
$834$	0	0
$835$	−19.6336	−0.679448
$836$	62.6584	2.16709
$837$	0	0
$838$	−2.07133	−0.0715528
$839$	−29.6682	−1.02426	−0.512131	−	0.858908i	$-0.671144\pi$
$839$	−29.6682	−1.02426	−0.512131	+	0.858908i	$0.671144\pi$
$840$	0	0
$841$	30.9157	1.06606
$842$	−33.5707	−1.15692
$843$	0	0
$844$	17.0794	0.587898
$845$	−20.5015	−0.705272
$846$	0	0
$847$	−1.03819	−0.0356726
$848$	−1.26713	−0.0435135
$849$	0	0
$850$	−15.7578	−0.540489
$851$	−8.32453	−0.285361
$852$	0	0
$853$	5.54118	0.189726	0.0948632	−	0.995490i	$-0.469759\pi$
$853$	5.54118	0.189726	0.0948632	+	0.995490i	$0.469759\pi$
$854$	29.6592	1.01492
$855$	0	0
$856$	−25.7205	−0.879110
$857$	15.6903	0.535970	0.267985	−	0.963423i	$-0.413642\pi$
$857$	15.6903	0.535970	0.267985	+	0.963423i	$0.413642\pi$
$858$	0	0
$859$	−17.1962	−0.586727	−0.293364	−	0.956001i	$-0.594775\pi$
$859$	−17.1962	−0.586727	−0.293364	+	0.956001i	$0.594775\pi$
$860$	2.88146	0.0982570
$861$	0	0
$862$	1.43717	0.0489503
$863$	0.805293	0.0274125	0.0137063	−	0.999906i	$-0.495637\pi$
$863$	0.805293	0.0274125	0.0137063	+	0.999906i	$0.495637\pi$
$864$	0	0
$865$	−12.6607	−0.430478
$866$	−44.5960	−1.51544
$867$	0	0
$868$	25.8239	0.876519
$869$	33.5358	1.13762
$870$	0	0
$871$	−1.32691	−0.0449605
$872$	−28.4372	−0.963004
$873$	0	0
$874$	18.9509	0.641023
$875$	0.988801	0.0334276
$876$	0	0
$877$	−21.0222	−0.709870	−0.354935	−	0.934891i	$-0.615497\pi$
$877$	−21.0222	−0.709870	−0.354935	+	0.934891i	$0.615497\pi$
$878$	64.9584	2.19224
$879$	0	0
$880$	5.06847	0.170858
$881$	−37.1848	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$881$	−37.1848	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$882$	0	0
$883$	−45.7461	−1.53948	−0.769739	−	0.638359i	$-0.779614\pi$
$883$	−45.7461	−1.53948	−0.769739	+	0.638359i	$0.779614\pi$
$884$	118.950	4.00074
$885$	0	0
$886$	76.3624	2.56545
$887$	28.6685	0.962594	0.481297	−	0.876558i	$-0.340166\pi$
$887$	28.6685	0.962594	0.481297	+	0.876558i	$0.340166\pi$
$888$	0	0
$889$	−3.27198	−0.109739
$890$	13.6084	0.456156
$891$	0	0
$892$	−24.8488	−0.832001
$893$	38.7244	1.29586
$894$	0	0
$895$	−5.10704	−0.170709
$896$	−13.9093	−0.464678
$897$	0	0
$898$	15.9829	0.533355
$899$	70.1568	2.33986
$900$	0	0
$901$	6.18956	0.206204
$902$	25.3973	0.845638
$903$	0	0
$904$	4.58698	0.152561
$905$	−16.5533	−0.550251
$906$	0	0
$907$	16.6413	0.552566	0.276283	−	0.961076i	$-0.410897\pi$
$907$	16.6413	0.552566	0.276283	+	0.961076i	$0.410897\pi$
$908$	9.16397	0.304117
$909$	0	0
$910$	−12.6449	−0.419175
$911$	26.1115	0.865112	0.432556	−	0.901607i	$-0.357612\pi$
$911$	26.1115	0.865112	0.432556	+	0.901607i	$0.357612\pi$
$912$	0	0
$913$	−3.64336	−0.120578
$914$	21.9509	0.726072
$915$	0	0
$916$	−64.9013	−2.14440
$917$	−3.14408	−0.103827
$918$	0	0
$919$	35.3733	1.16686	0.583429	−	0.812164i	$-0.301711\pi$
$919$	35.3733	1.16686	0.583429	+	0.812164i	$0.301711\pi$
$920$	−2.66660	−0.0879152
$921$	0	0
$922$	−9.59763	−0.316081
$923$	−70.8569	−2.33228
$924$	0	0
$925$	6.07967	0.199898
$926$	−7.09301	−0.233091
$927$	0	0
$928$	−55.1200	−1.80940
$929$	58.3105	1.91311	0.956553	−	0.291559i	$-0.0941739\pi$
$929$	58.3105	1.91311	0.956553	+	0.291559i	$0.0941739\pi$
$930$	0	0
$931$	−37.7255	−1.23640
$932$	−20.5237	−0.672276
$933$	0	0
$934$	38.2141	1.25040
$935$	−24.7579	−0.809670
$936$	0	0
$937$	−44.7746	−1.46272	−0.731362	−	0.681990i	$-0.761115\pi$
$937$	−44.7746	−1.46272	−0.731362	+	0.681990i	$0.761115\pi$
$938$	−0.500833	−0.0163528
$939$	0	0
$940$	−17.8124	−0.580977
$941$	11.8845	0.387424	0.193712	−	0.981058i	$-0.437947\pi$
$941$	11.8845	0.387424	0.193712	+	0.981058i	$0.437947\pi$
$942$	0	0
$943$	4.53419	0.147654
$944$	0.382402	0.0124461
$945$	0	0
$946$	7.66951	0.249357
$947$	−19.2420	−0.625281	−0.312641	−	0.949871i	$-0.601214\pi$
$947$	−19.2420	−0.625281	−0.312641	+	0.949871i	$0.601214\pi$
$948$	0	0
$949$	−11.7198	−0.380440
$950$	−13.8404	−0.449042
$951$	0	0
$952$	13.7344	0.445133
$953$	−31.5462	−1.02188	−0.510940	−	0.859616i	$-0.670703\pi$
$953$	−31.5462	−1.02188	−0.510940	+	0.859616i	$0.670703\pi$
$954$	0	0
$955$	9.31662	0.301479
$956$	81.6471	2.64066
$957$	0	0
$958$	13.0381	0.421242
$959$	3.36380	0.108623
$960$	0	0
$961$	51.1483	1.64995
$962$	−77.7475	−2.50668
$963$	0	0
$964$	23.2252	0.748032
$965$	−3.79315	−0.122106
$966$	0	0
$967$	19.0405	0.612302	0.306151	−	0.951983i	$-0.400959\pi$
$967$	19.0405	0.612302	0.306151	+	0.951983i	$0.400959\pi$
$968$	−2.04477	−0.0657215
$969$	0	0
$970$	2.44120	0.0783823
$971$	−42.0227	−1.34857	−0.674286	−	0.738470i	$-0.735549\pi$
$971$	−42.0227	−1.34857	−0.674286	+	0.738470i	$0.735549\pi$
$972$	0	0
$973$	−8.24331	−0.264268
$974$	−86.9647	−2.78653
$975$	0	0
$976$	−19.8225	−0.634503
$977$	16.7234	0.535028	0.267514	−	0.963554i	$-0.413798\pi$
$977$	16.7234	0.535028	0.267514	+	0.963554i	$0.413798\pi$
$978$	0	0
$979$	21.3809	0.683337
$980$	17.3529	0.554319
$981$	0	0
$982$	−9.62991	−0.307303
$983$	14.4984	0.462427	0.231213	−	0.972903i	$-0.425730\pi$
$983$	14.4984	0.462427	0.231213	+	0.972903i	$0.425730\pi$
$984$	0	0
$985$	−18.7452	−0.597272
$986$	121.974	3.88444
$987$	0	0
$988$	104.477	3.32384
$989$	1.36924	0.0435393
$990$	0	0
$991$	−14.3810	−0.456828	−0.228414	−	0.973564i	$-0.573354\pi$
$991$	−14.3810	−0.456828	−0.228414	+	0.973564i	$0.573354\pi$
$992$	−64.5414	−2.04919
$993$	0	0
$994$	−26.7445	−0.848283
$995$	18.7881	0.595624
$996$	0	0
$997$	36.5886	1.15877	0.579386	−	0.815053i	$-0.303292\pi$
$997$	36.5886	1.15877	0.579386	+	0.815053i	$0.303292\pi$
$998$	−56.7593	−1.79668
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1935.2.a.u.1.2		5
3.2	odd	2		215.2.a.c.1.4	✓	5
5.4	even	2		9675.2.a.ch.1.4		5
12.11	even	2		3440.2.a.w.1.3		5
15.2	even	4		1075.2.b.h.474.8		10
15.8	even	4		1075.2.b.h.474.3		10
15.14	odd	2		1075.2.a.m.1.2		5
129.128	even	2		9245.2.a.l.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.2.a.c.1.4	✓	5	3.2	odd	2
1075.2.a.m.1.2		5	15.14	odd	2
1075.2.b.h.474.3		10	15.8	even	4
1075.2.b.h.474.8		10	15.2	even	4
1935.2.a.u.1.2		5	1.1	even	1	trivial
3440.2.a.w.1.3		5	12.11	even	2
9245.2.a.l.1.2		5	129.128	even	2
9675.2.a.ch.1.4		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1935 = 3^{2} \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1935.a (trivial)

\(f(q)\)	\(=\)	\(q-2.20940 q^{2} +2.88146 q^{4} -1.00000 q^{5} -0.988801 q^{7} -1.94750 q^{8} +O(q^{10})\)	\(q-2.20940 q^{2} +2.88146 q^{4} -1.00000 q^{5} -0.988801 q^{7} -1.94750 q^{8} +2.20940 q^{10} +3.47130 q^{11} +5.78805 q^{13} +2.18466 q^{14} -1.46010 q^{16} +7.13216 q^{17} +6.26433 q^{19} -2.88146 q^{20} -7.66951 q^{22} -1.36924 q^{23} +1.00000 q^{25} -12.7881 q^{26} -2.84919 q^{28} -7.74052 q^{29} -9.06357 q^{31} +7.12096 q^{32} -15.7578 q^{34} +0.988801 q^{35} +6.07967 q^{37} -13.8404 q^{38} +1.94750 q^{40} -3.31147 q^{41} -1.00000 q^{43} +10.0024 q^{44} +3.02520 q^{46} +6.18173 q^{47} -6.02227 q^{49} -2.20940 q^{50} +16.6780 q^{52} +0.867837 q^{53} -3.47130 q^{55} +1.92569 q^{56} +17.1019 q^{58} -0.261901 q^{59} +13.5761 q^{61} +20.0251 q^{62} -12.8129 q^{64} -5.78805 q^{65} -0.229250 q^{67} +20.5510 q^{68} -2.18466 q^{70} -12.2419 q^{71} -2.02482 q^{73} -13.4324 q^{74} +18.0504 q^{76} -3.43243 q^{77} +9.66086 q^{79} +1.46010 q^{80} +7.31636 q^{82} -1.04956 q^{83} -7.13216 q^{85} +2.20940 q^{86} -6.76037 q^{88} +6.15933 q^{89} -5.72323 q^{91} -3.94541 q^{92} -13.6579 q^{94} -6.26433 q^{95} +1.10492 q^{97} +13.3056 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 8 q^{4} - 5 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) 5 * q - 2 * q^2 + 8 * q^4 - 5 * q^5 + 5 * q^7 - 3 * q^8	\( 5 q - 2 q^{2} + 8 q^{4} - 5 q^{5} + 5 q^{7} - 3 q^{8} + 2 q^{10} + 6 q^{11} + 5 q^{13} - q^{14} + 14 q^{16} + 17 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{22} - q^{23} + 5 q^{25} - 22 q^{26} + 26 q^{28} - 6 q^{29} + 6 q^{31} + 7 q^{32} - 5 q^{35} + 5 q^{37} + 16 q^{38} + 3 q^{40} - 2 q^{41} - 5 q^{43} + 15 q^{44} - 14 q^{46} + 18 q^{49} - 2 q^{50} - 38 q^{52} + 23 q^{53} - 6 q^{55} + 19 q^{56} + 12 q^{58} + q^{59} + 20 q^{61} + 3 q^{62} - 25 q^{64} - 5 q^{65} + 21 q^{67} + 48 q^{68} + q^{70} - 4 q^{71} + 5 q^{73} - 24 q^{74} + 32 q^{76} + 26 q^{77} + 41 q^{79} - 14 q^{80} + 38 q^{82} + 7 q^{83} - 17 q^{85} + 2 q^{86} + 12 q^{88} - 20 q^{89} - 42 q^{91} + 52 q^{92} - 42 q^{94} + 6 q^{95} + 37 q^{97} + 26 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 8 * q^4 - 5 * q^5 + 5 * q^7 - 3 * q^8 + 2 * q^10 + 6 * q^11 + 5 * q^13 - q^14 + 14 * q^16 + 17 * q^17 - 6 * q^19 - 8 * q^20 - 8 * q^22 - q^23 + 5 * q^25 - 22 * q^26 + 26 * q^28 - 6 * q^29 + 6 * q^31 + 7 * q^32 - 5 * q^35 + 5 * q^37 + 16 * q^38 + 3 * q^40 - 2 * q^41 - 5 * q^43 + 15 * q^44 - 14 * q^46 + 18 * q^49 - 2 * q^50 - 38 * q^52 + 23 * q^53 - 6 * q^55 + 19 * q^56 + 12 * q^58 + q^59 + 20 * q^61 + 3 * q^62 - 25 * q^64 - 5 * q^65 + 21 * q^67 + 48 * q^68 + q^70 - 4 * q^71 + 5 * q^73 - 24 * q^74 + 32 * q^76 + 26 * q^77 + 41 * q^79 - 14 * q^80 + 38 * q^82 + 7 * q^83 - 17 * q^85 + 2 * q^86 + 12 * q^88 - 20 * q^89 - 42 * q^91 + 52 * q^92 - 42 * q^94 + 6 * q^95 + 37 * q^97 + 26 * q^98

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