LMFDB - Embedded newform 1035.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1035,4,Mod(1,1035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1035, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1035.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1035 = 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1035.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:	$61.0669768559$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.72015$ of defining polynomial
Character	$\chi$	$=$	1035.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{2} +1.00000 q^{4} -5.00000 q^{5} -25.6008 q^{7} -21.0000 q^{8} +O(q^{10})$	\(q+3.00000 q^{2} +1.00000 q^{4} -5.00000 q^{5} -25.6008 q^{7} -21.0000 q^{8} -15.0000 q^{10} -12.6008 q^{11} +8.16046 q^{13} -76.8023 q^{14} -71.0000 q^{16} +76.0411 q^{17} -103.362 q^{19} -5.00000 q^{20} -37.8023 q^{22} +23.0000 q^{23} +25.0000 q^{25} +24.4814 q^{26} -25.6008 q^{28} +267.202 q^{29} -63.7574 q^{31} -45.0000 q^{32} +228.123 q^{34} +128.004 q^{35} +112.164 q^{37} -310.086 q^{38} +105.000 q^{40} +239.078 q^{41} +282.239 q^{43} -12.6008 q^{44} +69.0000 q^{46} -577.291 q^{47} +312.399 q^{49} +75.0000 q^{50} +8.16046 q^{52} +2.31326 q^{53} +63.0038 q^{55} +537.616 q^{56} +801.605 q^{58} -272.888 q^{59} +294.049 q^{61} -191.272 q^{62} +433.000 q^{64} -40.8023 q^{65} +426.732 q^{67} +76.0411 q^{68} +384.011 q^{70} +1020.85 q^{71} -286.650 q^{73} +336.493 q^{74} -103.362 q^{76} +322.589 q^{77} -551.866 q^{79} +355.000 q^{80} +717.235 q^{82} -21.7021 q^{83} -380.205 q^{85} +846.716 q^{86} +264.616 q^{88} +1049.63 q^{89} -208.914 q^{91} +23.0000 q^{92} -1731.87 q^{94} +516.810 q^{95} +1729.19 q^{97} +937.198 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{2} + 2 q^{4} - 10 q^{5} + q^{7} - 42 q^{8}+O(q^{10}) $ 2 * q + 6 * q^2 + 2 * q^4 - 10 * q^5 + q^7 - 42 * q^8	$ 2 q + 6 q^{2} + 2 q^{4} - 10 q^{5} + q^{7} - 42 q^{8} - 30 q^{10} + 27 q^{11} - 15 q^{13} + 3 q^{14} - 142 q^{16} + 79 q^{17} - 71 q^{19} - 10 q^{20} + 81 q^{22} + 46 q^{23} + 50 q^{25} - 45 q^{26} + q^{28} + 430 q^{29} - 305 q^{31} - 90 q^{32} + 237 q^{34} - 5 q^{35} - 68 q^{37} - 213 q^{38} + 210 q^{40} + 593 q^{41} + 648 q^{43} + 27 q^{44} + 138 q^{46} - 382 q^{47} + 677 q^{49} + 150 q^{50} - 15 q^{52} + 464 q^{53} - 135 q^{55} - 21 q^{56} + 1290 q^{58} + 18 q^{59} - 7 q^{61} - 915 q^{62} + 866 q^{64} + 75 q^{65} + 60 q^{67} + 79 q^{68} - 15 q^{70} + 1029 q^{71} + 74 q^{73} - 204 q^{74} - 71 q^{76} + 1376 q^{77} + 692 q^{79} + 710 q^{80} + 1779 q^{82} + 1460 q^{83} - 395 q^{85} + 1944 q^{86} - 567 q^{88} + 220 q^{89} - 825 q^{91} + 46 q^{92} - 1146 q^{94} + 355 q^{95} + 1339 q^{97} + 2031 q^{98}+O(q^{100}) $ 2 * q + 6 * q^2 + 2 * q^4 - 10 * q^5 + q^7 - 42 * q^8 - 30 * q^10 + 27 * q^11 - 15 * q^13 + 3 * q^14 - 142 * q^16 + 79 * q^17 - 71 * q^19 - 10 * q^20 + 81 * q^22 + 46 * q^23 + 50 * q^25 - 45 * q^26 + q^28 + 430 * q^29 - 305 * q^31 - 90 * q^32 + 237 * q^34 - 5 * q^35 - 68 * q^37 - 213 * q^38 + 210 * q^40 + 593 * q^41 + 648 * q^43 + 27 * q^44 + 138 * q^46 - 382 * q^47 + 677 * q^49 + 150 * q^50 - 15 * q^52 + 464 * q^53 - 135 * q^55 - 21 * q^56 + 1290 * q^58 + 18 * q^59 - 7 * q^61 - 915 * q^62 + 866 * q^64 + 75 * q^65 + 60 * q^67 + 79 * q^68 - 15 * q^70 + 1029 * q^71 + 74 * q^73 - 204 * q^74 - 71 * q^76 + 1376 * q^77 + 692 * q^79 + 710 * q^80 + 1779 * q^82 + 1460 * q^83 - 395 * q^85 + 1944 * q^86 - 567 * q^88 + 220 * q^89 - 825 * q^91 + 46 * q^92 - 1146 * q^94 + 355 * q^95 + 1339 * q^97 + 2031 * 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 $

$2$	3.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	3.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−25.6008	−1.38231	−0.691156	−	0.722706i	$-0.742898\pi$
$7$	−25.6008	−1.38231	−0.691156	+	0.722706i	$0.742898\pi$
$8$	−21.0000	−0.928078
$9$	0	0
$10$	−15.0000	−0.474342
$11$	−12.6008	−0.345389	−0.172694	−	0.984975i	$-0.555247\pi$
$11$	−12.6008	−0.345389	−0.172694	+	0.984975i	$0.555247\pi$
$12$	0	0
$13$	8.16046	0.174100	0.0870502	−	0.996204i	$-0.472256\pi$
$13$	8.16046	0.174100	0.0870502	+	0.996204i	$0.472256\pi$
$14$	−76.8023	−1.46616
$15$	0	0
$16$	−71.0000	−1.10938
$17$	76.0411	1.08486	0.542431	−	0.840100i	$-0.317504\pi$
$17$	76.0411	1.08486	0.542431	+	0.840100i	$0.317504\pi$
$18$	0	0
$19$	−103.362	−1.24805	−0.624023	−	0.781406i	$-0.714503\pi$
$19$	−103.362	−1.24805	−0.624023	+	0.781406i	$0.714503\pi$
$20$	−5.00000	−0.0559017
$21$	0	0
$22$	−37.8023	−0.366340
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	24.4814	0.184661
$27$	0	0
$28$	−25.6008	−0.172789
$29$	267.202	1.71097	0.855484	−	0.517829i	$-0.173260\pi$
$29$	267.202	1.71097	0.855484	+	0.517829i	$0.173260\pi$
$30$	0	0
$31$	−63.7574	−0.369392	−0.184696	−	0.982796i	$-0.559130\pi$
$31$	−63.7574	−0.369392	−0.184696	+	0.982796i	$0.559130\pi$
$32$	−45.0000	−0.248592
$33$	0	0
$34$	228.123	1.15067
$35$	128.004	0.618188
$36$	0	0
$37$	112.164	0.498370	0.249185	−	0.968456i	$-0.419837\pi$
$37$	112.164	0.498370	0.249185	+	0.968456i	$0.419837\pi$
$38$	−310.086	−1.32375
$39$	0	0
$40$	105.000	0.415049
$41$	239.078	0.910677	0.455339	−	0.890318i	$-0.349518\pi$
$41$	239.078	0.910677	0.455339	+	0.890318i	$0.349518\pi$
$42$	0	0
$43$	282.239	1.00095	0.500477	−	0.865750i	$-0.333158\pi$
$43$	282.239	1.00095	0.500477	+	0.865750i	$0.333158\pi$
$44$	−12.6008	−0.0431736
$45$	0	0
$46$	69.0000	0.221163
$47$	−577.291	−1.79163	−0.895815	−	0.444427i	$-0.853407\pi$
$47$	−577.291	−1.79163	−0.895815	+	0.444427i	$0.853407\pi$
$48$	0	0
$49$	312.399	0.910785
$50$	75.0000	0.212132
$51$	0	0
$52$	8.16046	0.0217625
$53$	2.31326	0.00599529	0.00299764	−	0.999996i	$-0.499046\pi$
$53$	2.31326	0.00599529	0.00299764	+	0.999996i	$0.499046\pi$
$54$	0	0
$55$	63.0038	0.154462
$56$	537.616	1.28289
$57$	0	0
$58$	801.605	1.81476
$59$	−272.888	−0.602153	−0.301077	−	0.953600i	$-0.597346\pi$
$59$	−272.888	−0.602153	−0.301077	+	0.953600i	$0.597346\pi$
$60$	0	0
$61$	294.049	0.617198	0.308599	−	0.951192i	$-0.400140\pi$
$61$	294.049	0.617198	0.308599	+	0.951192i	$0.400140\pi$
$62$	−191.272	−0.391800
$63$	0	0
$64$	433.000	0.845703
$65$	−40.8023	−0.0778600
$66$	0	0
$67$	426.732	0.778113	0.389056	−	0.921214i	$-0.372801\pi$
$67$	426.732	0.778113	0.389056	+	0.921214i	$0.372801\pi$
$68$	76.0411	0.135608
$69$	0	0
$70$	384.011	0.655688
$71$	1020.85	1.70638	0.853191	−	0.521598i	$-0.174664\pi$
$71$	1020.85	1.70638	0.853191	+	0.521598i	$0.174664\pi$
$72$	0	0
$73$	−286.650	−0.459586	−0.229793	−	0.973240i	$-0.573805\pi$
$73$	−286.650	−0.459586	−0.229793	+	0.973240i	$0.573805\pi$
$74$	336.493	0.528601
$75$	0	0
$76$	−103.362	−0.156006
$77$	322.589	0.477435
$78$	0	0
$79$	−551.866	−0.785947	−0.392974	−	0.919550i	$-0.628554\pi$
$79$	−551.866	−0.785947	−0.392974	+	0.919550i	$0.628554\pi$
$80$	355.000	0.496128
$81$	0	0
$82$	717.235	0.965919
$83$	−21.7021	−0.0287001	−0.0143501	−	0.999897i	$-0.504568\pi$
$83$	−21.7021	−0.0287001	−0.0143501	+	0.999897i	$0.504568\pi$
$84$	0	0
$85$	−380.205	−0.485165
$86$	846.716	1.06167
$87$	0	0
$88$	264.616	0.320547
$89$	1049.63	1.25012	0.625058	−	0.780578i	$-0.285075\pi$
$89$	1049.63	1.25012	0.625058	+	0.780578i	$0.285075\pi$
$90$	0	0
$91$	−208.914	−0.240661
$92$	23.0000	0.0260643
$93$	0	0
$94$	−1731.87	−1.90031
$95$	516.810	0.558143
$96$	0	0
$97$	1729.19	1.81003	0.905014	−	0.425381i	$-0.139860\pi$
$97$	1729.19	1.81003	0.905014	+	0.425381i	$0.139860\pi$
$98$	937.198	0.966033
$99$	0	0
$100$	25.0000	0.0250000
$101$	−855.874	−0.843195	−0.421597	−	0.906783i	$-0.638530\pi$
$101$	−855.874	−0.843195	−0.421597	+	0.906783i	$0.638530\pi$
$102$	0	0
$103$	−632.534	−0.605101	−0.302551	−	0.953133i	$-0.597838\pi$
$103$	−632.534	−0.605101	−0.302551	+	0.953133i	$0.597838\pi$
$104$	−171.370	−0.161579
$105$	0	0
$106$	6.93977	0.00635896
$107$	−1309.62	−1.18323	−0.591616	−	0.806220i	$-0.701510\pi$
$107$	−1309.62	−1.18323	−0.591616	+	0.806220i	$0.701510\pi$
$108$	0	0
$109$	1726.21	1.51689	0.758443	−	0.651739i	$-0.225960\pi$
$109$	1726.21	1.51689	0.758443	+	0.651739i	$0.225960\pi$
$110$	189.011	0.163832
$111$	0	0
$112$	1817.65	1.53350
$113$	1492.24	1.24228	0.621142	−	0.783698i	$-0.286669\pi$
$113$	1492.24	1.24228	0.621142	+	0.783698i	$0.286669\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	267.202	0.213871
$117$	0	0
$118$	−818.665	−0.638680
$119$	−1946.71	−1.49962
$120$	0	0
$121$	−1172.22	−0.880707
$122$	882.146	0.654637
$123$	0	0
$124$	−63.7574	−0.0461741
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1368.29	−0.956034	−0.478017	−	0.878351i	$-0.658644\pi$
$127$	−1368.29	−0.956034	−0.478017	+	0.878351i	$0.658644\pi$
$128$	1659.00	1.14560
$129$	0	0
$130$	−122.407	−0.0825830
$131$	−1984.24	−1.32339	−0.661694	−	0.749774i	$-0.730162\pi$
$131$	−1984.24	−1.32339	−0.661694	+	0.749774i	$0.730162\pi$
$132$	0	0
$133$	2646.15	1.72519
$134$	1280.19	0.825313
$135$	0	0
$136$	−1596.86	−1.00684
$137$	−273.331	−0.170455	−0.0852273	−	0.996362i	$-0.527162\pi$
$137$	−273.331	−0.170455	−0.0852273	+	0.996362i	$0.527162\pi$
$138$	0	0
$139$	−557.889	−0.340428	−0.170214	−	0.985407i	$-0.554446\pi$
$139$	−557.889	−0.340428	−0.170214	+	0.985407i	$0.554446\pi$
$140$	128.004	0.0772736
$141$	0	0
$142$	3062.56	1.80989
$143$	−102.828	−0.0601323
$144$	0	0
$145$	−1336.01	−0.765168
$146$	−859.949	−0.487465
$147$	0	0
$148$	112.164	0.0622963
$149$	1354.16	0.744545	0.372272	−	0.928124i	$-0.378579\pi$
$149$	1354.16	0.744545	0.372272	+	0.928124i	$0.378579\pi$
$150$	0	0
$151$	162.652	0.0876586	0.0438293	−	0.999039i	$-0.486044\pi$
$151$	162.652	0.0876586	0.0438293	+	0.999039i	$0.486044\pi$
$152$	2170.60	1.15828
$153$	0	0
$154$	967.768	0.506396
$155$	318.787	0.165197
$156$	0	0
$157$	−2364.83	−1.20213	−0.601063	−	0.799201i	$-0.705256\pi$
$157$	−2364.83	−1.20213	−0.601063	+	0.799201i	$0.705256\pi$
$158$	−1655.60	−0.833623
$159$	0	0
$160$	225.000	0.111174
$161$	−588.818	−0.288232
$162$	0	0
$163$	471.162	0.226406	0.113203	−	0.993572i	$-0.463889\pi$
$163$	471.162	0.226406	0.113203	+	0.993572i	$0.463889\pi$
$164$	239.078	0.113835
$165$	0	0
$166$	−65.1062	−0.0304411
$167$	3735.73	1.73102	0.865508	−	0.500895i	$-0.166996\pi$
$167$	3735.73	1.73102	0.865508	+	0.500895i	$0.166996\pi$
$168$	0	0
$169$	−2130.41	−0.969689
$170$	−1140.62	−0.514596
$171$	0	0
$172$	282.239	0.125119
$173$	2709.33	1.19067	0.595336	−	0.803477i	$-0.297019\pi$
$173$	2709.33	1.19067	0.595336	+	0.803477i	$0.297019\pi$
$174$	0	0
$175$	−640.019	−0.276462
$176$	894.654	0.383165
$177$	0	0
$178$	3148.88	1.32595
$179$	−3300.43	−1.37813	−0.689066	−	0.724699i	$-0.741979\pi$
$179$	−3300.43	−1.37813	−0.689066	+	0.724699i	$0.741979\pi$
$180$	0	0
$181$	−583.868	−0.239771	−0.119886	−	0.992788i	$-0.538253\pi$
$181$	−583.868	−0.239771	−0.119886	+	0.992788i	$0.538253\pi$
$182$	−626.742	−0.255259
$183$	0	0
$184$	−483.000	−0.193518
$185$	−560.821	−0.222878
$186$	0	0
$187$	−958.176	−0.374699
$188$	−577.291	−0.223954
$189$	0	0
$190$	1550.43	0.592000
$191$	3058.98	1.15885	0.579424	−	0.815026i	$-0.303278\pi$
$191$	3058.98	1.15885	0.579424	+	0.815026i	$0.303278\pi$
$192$	0	0
$193$	−1336.93	−0.498625	−0.249313	−	0.968423i	$-0.580205\pi$
$193$	−1336.93	−0.498625	−0.249313	+	0.968423i	$0.580205\pi$
$194$	5187.57	1.91983
$195$	0	0
$196$	312.399	0.113848
$197$	2449.21	0.885783	0.442892	−	0.896575i	$-0.353953\pi$
$197$	2449.21	0.885783	0.442892	+	0.896575i	$0.353953\pi$
$198$	0	0
$199$	1138.66	0.405614	0.202807	−	0.979219i	$-0.434994\pi$
$199$	1138.66	0.405614	0.202807	+	0.979219i	$0.434994\pi$
$200$	−525.000	−0.185616
$201$	0	0
$202$	−2567.62	−0.894343
$203$	−6840.56	−2.36509
$204$	0	0
$205$	−1195.39	−0.407267
$206$	−1897.60	−0.641807
$207$	0	0
$208$	−579.393	−0.193143
$209$	1302.44	0.431061
$210$	0	0
$211$	5596.81	1.82607	0.913034	−	0.407884i	$-0.133733\pi$
$211$	5596.81	1.82607	0.913034	+	0.407884i	$0.133733\pi$
$212$	2.31326	0.000749411 0
$213$	0	0
$214$	−3928.86	−1.25501
$215$	−1411.19	−0.447640
$216$	0	0
$217$	1632.24	0.510615
$218$	5178.62	1.60890
$219$	0	0
$220$	63.0038	0.0193078
$221$	620.530	0.188875
$222$	0	0
$223$	5580.60	1.67581	0.837903	−	0.545820i	$-0.183782\pi$
$223$	5580.60	1.67581	0.837903	+	0.545820i	$0.183782\pi$
$224$	1152.03	0.343632
$225$	0	0
$226$	4476.72	1.31764
$227$	566.323	0.165587	0.0827934	−	0.996567i	$-0.473616\pi$
$227$	566.323	0.165587	0.0827934	+	0.996567i	$0.473616\pi$
$228$	0	0
$229$	−693.702	−0.200180	−0.100090	−	0.994978i	$-0.531913\pi$
$229$	−693.702	−0.200180	−0.100090	+	0.994978i	$0.531913\pi$
$230$	−345.000	−0.0989071
$231$	0	0
$232$	−5611.23	−1.58791
$233$	2208.68	0.621011	0.310506	−	0.950572i	$-0.399502\pi$
$233$	2208.68	0.621011	0.310506	+	0.950572i	$0.399502\pi$
$234$	0	0
$235$	2886.46	0.801241
$236$	−272.888	−0.0752691
$237$	0	0
$238$	−5840.13	−1.59059
$239$	3876.79	1.04924	0.524621	−	0.851336i	$-0.324207\pi$
$239$	3876.79	1.04924	0.524621	+	0.851336i	$0.324207\pi$
$240$	0	0
$241$	−2024.57	−0.541136	−0.270568	−	0.962701i	$-0.587211\pi$
$241$	−2024.57	−0.541136	−0.270568	+	0.962701i	$0.587211\pi$
$242$	−3516.66	−0.934131
$243$	0	0
$244$	294.049	0.0771498
$245$	−1562.00	−0.407315
$246$	0	0
$247$	−843.481	−0.217285
$248$	1338.91	0.342825
$249$	0	0
$250$	−375.000	−0.0948683
$251$	733.444	0.184441	0.0922203	−	0.995739i	$-0.470604\pi$
$251$	733.444	0.184441	0.0922203	+	0.995739i	$0.470604\pi$
$252$	0	0
$253$	−289.818	−0.0720185
$254$	−4104.88	−1.01403
$255$	0	0
$256$	1513.00	0.369385
$257$	−2367.83	−0.574713	−0.287356	−	0.957824i	$-0.592776\pi$
$257$	−2367.83	−0.574713	−0.287356	+	0.957824i	$0.592776\pi$
$258$	0	0
$259$	−2871.49	−0.688903
$260$	−40.8023	−0.00973250
$261$	0	0
$262$	−5952.72	−1.40366
$263$	−6632.76	−1.55511	−0.777554	−	0.628816i	$-0.783540\pi$
$263$	−6632.76	−1.55511	−0.777554	+	0.628816i	$0.783540\pi$
$264$	0	0
$265$	−11.5663	−0.00268117
$266$	7938.44	1.82984
$267$	0	0
$268$	426.732	0.0972641
$269$	6092.97	1.38102	0.690511	−	0.723322i	$-0.257386\pi$
$269$	6092.97	1.38102	0.690511	+	0.723322i	$0.257386\pi$
$270$	0	0
$271$	−1981.30	−0.444115	−0.222058	−	0.975034i	$-0.571277\pi$
$271$	−1981.30	−0.444115	−0.222058	+	0.975034i	$0.571277\pi$
$272$	−5398.92	−1.20352
$273$	0	0
$274$	−819.994	−0.180794
$275$	−315.019	−0.0690777
$276$	0	0
$277$	−4097.42	−0.888773	−0.444386	−	0.895835i	$-0.646578\pi$
$277$	−4097.42	−0.888773	−0.444386	+	0.895835i	$0.646578\pi$
$278$	−1673.67	−0.361079
$279$	0	0
$280$	−2688.08	−0.573727
$281$	3866.40	0.820819	0.410409	−	0.911901i	$-0.365386\pi$
$281$	3866.40	0.820819	0.410409	+	0.911901i	$0.365386\pi$
$282$	0	0
$283$	3982.00	0.836415	0.418208	−	0.908351i	$-0.362658\pi$
$283$	3982.00	0.836415	0.418208	+	0.908351i	$0.362658\pi$
$284$	1020.85	0.213298
$285$	0	0
$286$	−308.484	−0.0637799
$287$	−6120.59	−1.25884
$288$	0	0
$289$	869.245	0.176927
$290$	−4008.02	−0.811583
$291$	0	0
$292$	−286.650	−0.0574483
$293$	7491.70	1.49375	0.746877	−	0.664962i	$-0.231552\pi$
$293$	7491.70	1.49375	0.746877	+	0.664962i	$0.231552\pi$
$294$	0	0
$295$	1364.44	0.269291
$296$	−2355.45	−0.462526
$297$	0	0
$298$	4062.48	0.789709
$299$	187.691	0.0363024
$300$	0	0
$301$	−7225.53	−1.38363
$302$	487.957	0.0929760
$303$	0	0
$304$	7338.70	1.38455
$305$	−1470.24	−0.276019
$306$	0	0
$307$	−5813.01	−1.08067	−0.540335	−	0.841450i	$-0.681703\pi$
$307$	−5813.01	−1.08067	−0.540335	+	0.841450i	$0.681703\pi$
$308$	322.589	0.0596793
$309$	0	0
$310$	956.361	0.175218
$311$	−4769.43	−0.869613	−0.434807	−	0.900524i	$-0.643183\pi$
$311$	−4769.43	−0.869613	−0.434807	+	0.900524i	$0.643183\pi$
$312$	0	0
$313$	7053.43	1.27375	0.636875	−	0.770967i	$-0.280227\pi$
$313$	7053.43	1.27375	0.636875	+	0.770967i	$0.280227\pi$
$314$	−7094.49	−1.27505
$315$	0	0
$316$	−551.866	−0.0982434
$317$	−8845.36	−1.56721	−0.783604	−	0.621261i	$-0.786621\pi$
$317$	−8845.36	−1.56721	−0.783604	+	0.621261i	$0.786621\pi$
$318$	0	0
$319$	−3366.94	−0.590949
$320$	−2165.00	−0.378210
$321$	0	0
$322$	−1766.45	−0.305716
$323$	−7859.76	−1.35396
$324$	0	0
$325$	204.011	0.0348201
$326$	1413.48	0.240140
$327$	0	0
$328$	−5020.64	−0.845179
$329$	14779.1	2.47659
$330$	0	0
$331$	4264.07	0.708079	0.354040	−	0.935230i	$-0.384808\pi$
$331$	4264.07	0.708079	0.354040	+	0.935230i	$0.384808\pi$
$332$	−21.7021	−0.00358752
$333$	0	0
$334$	11207.2	1.83602
$335$	−2133.66	−0.347983
$336$	0	0
$337$	−1290.58	−0.208613	−0.104306	−	0.994545i	$-0.533262\pi$
$337$	−1290.58	−0.208613	−0.104306	+	0.994545i	$0.533262\pi$
$338$	−6391.22	−1.02851
$339$	0	0
$340$	−380.205	−0.0606457
$341$	803.392	0.127584
$342$	0	0
$343$	783.403	0.123323
$344$	−5927.01	−0.928963
$345$	0	0
$346$	8127.98	1.26290
$347$	1805.09	0.279257	0.139629	−	0.990204i	$-0.455409\pi$
$347$	1805.09	0.279257	0.139629	+	0.990204i	$0.455409\pi$
$348$	0	0
$349$	−3586.81	−0.550136	−0.275068	−	0.961425i	$-0.588700\pi$
$349$	−3586.81	−0.550136	−0.275068	+	0.961425i	$0.588700\pi$
$350$	−1920.06	−0.293233
$351$	0	0
$352$	567.034	0.0858609
$353$	2292.90	0.345719	0.172859	−	0.984947i	$-0.444699\pi$
$353$	2292.90	0.345719	0.172859	+	0.984947i	$0.444699\pi$
$354$	0	0
$355$	−5104.27	−0.763118
$356$	1049.63	0.156264
$357$	0	0
$358$	−9901.28	−1.46173
$359$	3077.18	0.452388	0.226194	−	0.974082i	$-0.427372\pi$
$359$	3077.18	0.452388	0.226194	+	0.974082i	$0.427372\pi$
$360$	0	0
$361$	3824.70	0.557618
$362$	−1751.60	−0.254316
$363$	0	0
$364$	−208.914	−0.0300826
$365$	1433.25	0.205533
$366$	0	0
$367$	−1378.56	−0.196076	−0.0980382	−	0.995183i	$-0.531257\pi$
$367$	−1378.56	−0.196076	−0.0980382	+	0.995183i	$0.531257\pi$
$368$	−1633.00	−0.231321
$369$	0	0
$370$	−1682.46	−0.236398
$371$	−59.2211	−0.00828736
$372$	0	0
$373$	11182.3	1.55227	0.776135	−	0.630567i	$-0.217178\pi$
$373$	11182.3	1.55227	0.776135	+	0.630567i	$0.217178\pi$
$374$	−2874.53	−0.397429
$375$	0	0
$376$	12123.1	1.66277
$377$	2180.49	0.297880
$378$	0	0
$379$	12021.9	1.62935	0.814677	−	0.579915i	$-0.196914\pi$
$379$	12021.9	1.62935	0.814677	+	0.579915i	$0.196914\pi$
$380$	516.810	0.0697679
$381$	0	0
$382$	9176.93	1.22914
$383$	13274.6	1.77102	0.885512	−	0.464617i	$-0.153808\pi$
$383$	13274.6	1.77102	0.885512	+	0.464617i	$0.153808\pi$
$384$	0	0
$385$	−1612.95	−0.213515
$386$	−4010.80	−0.528872
$387$	0	0
$388$	1729.19	0.226254
$389$	−11016.4	−1.43587	−0.717934	−	0.696112i	$-0.754912\pi$
$389$	−11016.4	−1.43587	−0.717934	+	0.696112i	$0.754912\pi$
$390$	0	0
$391$	1748.94	0.226210
$392$	−6560.38	−0.845279
$393$	0	0
$394$	7347.64	0.939515
$395$	2759.33	0.351486
$396$	0	0
$397$	−1893.18	−0.239335	−0.119668	−	0.992814i	$-0.538183\pi$
$397$	−1893.18	−0.239335	−0.119668	+	0.992814i	$0.538183\pi$
$398$	3415.97	0.430219
$399$	0	0
$400$	−1775.00	−0.221875
$401$	−11281.8	−1.40495	−0.702475	−	0.711709i	$-0.747922\pi$
$401$	−11281.8	−1.40495	−0.702475	+	0.711709i	$0.747922\pi$
$402$	0	0
$403$	−520.290	−0.0643113
$404$	−855.874	−0.105399
$405$	0	0
$406$	−20521.7	−2.50856
$407$	−1413.36	−0.172131
$408$	0	0
$409$	−14677.6	−1.77447	−0.887236	−	0.461316i	$-0.847377\pi$
$409$	−14677.6	−1.77447	−0.887236	+	0.461316i	$0.847377\pi$
$410$	−3586.17	−0.431972
$411$	0	0
$412$	−632.534	−0.0756376
$413$	6986.15	0.832363
$414$	0	0
$415$	108.510	0.0128351
$416$	−367.221	−0.0432800
$417$	0	0
$418$	3907.32	0.457209
$419$	−2196.50	−0.256100	−0.128050	−	0.991768i	$-0.540872\pi$
$419$	−2196.50	−0.256100	−0.128050	+	0.991768i	$0.540872\pi$
$420$	0	0
$421$	−6571.28	−0.760723	−0.380362	−	0.924838i	$-0.624200\pi$
$421$	−6571.28	−0.760723	−0.380362	+	0.924838i	$0.624200\pi$
$422$	16790.4	1.93684
$423$	0	0
$424$	−48.5784	−0.00556409
$425$	1901.03	0.216973
$426$	0	0
$427$	−7527.87	−0.853160
$428$	−1309.62	−0.147904
$429$	0	0
$430$	−4233.58	−0.474794
$431$	583.607	0.0652235	0.0326118	−	0.999468i	$-0.489618\pi$
$431$	583.607	0.0652235	0.0326118	+	0.999468i	$0.489618\pi$
$432$	0	0
$433$	−2677.60	−0.297176	−0.148588	−	0.988899i	$-0.547473\pi$
$433$	−2677.60	−0.297176	−0.148588	+	0.988899i	$0.547473\pi$
$434$	4896.71	0.541589
$435$	0	0
$436$	1726.21	0.189611
$437$	−2377.33	−0.260236
$438$	0	0
$439$	−5509.18	−0.598949	−0.299475	−	0.954104i	$-0.596811\pi$
$439$	−5509.18	−0.598949	−0.299475	+	0.954104i	$0.596811\pi$
$440$	−1323.08	−0.143353
$441$	0	0
$442$	1861.59	0.200332
$443$	13054.6	1.40009	0.700047	−	0.714097i	$-0.253162\pi$
$443$	13054.6	1.40009	0.700047	+	0.714097i	$0.253162\pi$
$444$	0	0
$445$	−5248.14	−0.559069
$446$	16741.8	1.77746
$447$	0	0
$448$	−11085.1	−1.16903
$449$	11819.1	1.24227	0.621135	−	0.783704i	$-0.286672\pi$
$449$	11819.1	1.24227	0.621135	+	0.783704i	$0.286672\pi$
$450$	0	0
$451$	−3012.57	−0.314537
$452$	1492.24	0.155285
$453$	0	0
$454$	1698.97	0.175631
$455$	1044.57	0.107627
$456$	0	0
$457$	−4144.87	−0.424264	−0.212132	−	0.977241i	$-0.568041\pi$
$457$	−4144.87	−0.424264	−0.212132	+	0.977241i	$0.568041\pi$
$458$	−2081.11	−0.212323
$459$	0	0
$460$	−115.000	−0.0116563
$461$	6061.38	0.612379	0.306189	−	0.951971i	$-0.400946\pi$
$461$	6061.38	0.612379	0.306189	+	0.951971i	$0.400946\pi$
$462$	0	0
$463$	−5729.13	−0.575066	−0.287533	−	0.957771i	$-0.592835\pi$
$463$	−5729.13	−0.575066	−0.287533	+	0.957771i	$0.592835\pi$
$464$	−18971.3	−1.89811
$465$	0	0
$466$	6626.05	0.658682
$467$	−8958.05	−0.887643	−0.443821	−	0.896115i	$-0.646378\pi$
$467$	−8958.05	−0.887643	−0.443821	+	0.896115i	$0.646378\pi$
$468$	0	0
$469$	−10924.7	−1.07559
$470$	8659.37	0.849845
$471$	0	0
$472$	5730.65	0.558845
$473$	−3556.42	−0.345718
$474$	0	0
$475$	−2584.05	−0.249609
$476$	−1946.71	−0.187452
$477$	0	0
$478$	11630.4	1.11289
$479$	−6356.60	−0.606347	−0.303174	−	0.952935i	$-0.598046\pi$
$479$	−6356.60	−0.606347	−0.303174	+	0.952935i	$0.598046\pi$
$480$	0	0
$481$	915.312	0.0867664
$482$	−6073.70	−0.573961
$483$	0	0
$484$	−1172.22	−0.110088
$485$	−8645.96	−0.809469
$486$	0	0
$487$	13991.2	1.30185	0.650926	−	0.759141i	$-0.274381\pi$
$487$	13991.2	1.30185	0.650926	+	0.759141i	$0.274381\pi$
$488$	−6175.02	−0.572808
$489$	0	0
$490$	−4685.99	−0.432023
$491$	5301.96	0.487320	0.243660	−	0.969861i	$-0.421652\pi$
$491$	5301.96	0.487320	0.243660	+	0.969861i	$0.421652\pi$
$492$	0	0
$493$	20318.3	1.85617
$494$	−2530.44	−0.230466
$495$	0	0
$496$	4526.78	0.409795
$497$	−26134.7	−2.35875
$498$	0	0
$499$	−8013.45	−0.718900	−0.359450	−	0.933164i	$-0.617036\pi$
$499$	−8013.45	−0.718900	−0.359450	+	0.933164i	$0.617036\pi$
$500$	−125.000	−0.0111803
$501$	0	0
$502$	2200.33	0.195629
$503$	11881.9	1.05325	0.526627	−	0.850097i	$-0.323456\pi$
$503$	11881.9	1.05325	0.526627	+	0.850097i	$0.323456\pi$
$504$	0	0
$505$	4279.37	0.377088
$506$	−869.453	−0.0763871
$507$	0	0
$508$	−1368.29	−0.119504
$509$	12113.8	1.05488	0.527441	−	0.849592i	$-0.323152\pi$
$509$	12113.8	1.05488	0.527441	+	0.849592i	$0.323152\pi$
$510$	0	0
$511$	7338.45	0.635291
$512$	−8733.00	−0.753804
$513$	0	0
$514$	−7103.49	−0.609575
$515$	3162.67	0.270609
$516$	0	0
$517$	7274.31	0.618808
$518$	−8614.48	−0.730692
$519$	0	0
$520$	856.848	0.0722602
$521$	15327.6	1.28890	0.644449	−	0.764647i	$-0.277087\pi$
$521$	15327.6	1.28890	0.644449	+	0.764647i	$0.277087\pi$
$522$	0	0
$523$	14545.8	1.21614	0.608070	−	0.793883i	$-0.291944\pi$
$523$	14545.8	1.21614	0.608070	+	0.793883i	$0.291944\pi$
$524$	−1984.24	−0.165423
$525$	0	0
$526$	−19898.3	−1.64944
$527$	−4848.18	−0.400740
$528$	0	0
$529$	529.000	0.0434783
$530$	−34.6989	−0.00284381
$531$	0	0
$532$	2646.15	0.215648
$533$	1950.99	0.158549
$534$	0	0
$535$	6548.10	0.529157
$536$	−8961.36	−0.722149
$537$	0	0
$538$	18278.9	1.46479
$539$	−3936.47	−0.314575
$540$	0	0
$541$	−5468.93	−0.434617	−0.217308	−	0.976103i	$-0.569728\pi$
$541$	−5468.93	−0.434617	−0.217308	+	0.976103i	$0.569728\pi$
$542$	−5943.89	−0.471055
$543$	0	0
$544$	−3421.85	−0.269688
$545$	−8631.03	−0.678372
$546$	0	0
$547$	−19566.3	−1.52943	−0.764713	−	0.644371i	$-0.777119\pi$
$547$	−19566.3	−1.52943	−0.764713	+	0.644371i	$0.777119\pi$
$548$	−273.331	−0.0213068
$549$	0	0
$550$	−945.057	−0.0732680
$551$	−27618.5	−2.13537
$552$	0	0
$553$	14128.2	1.08642
$554$	−12292.3	−0.942686
$555$	0	0
$556$	−557.889	−0.0425536
$557$	9803.11	0.745729	0.372865	−	0.927886i	$-0.378376\pi$
$557$	9803.11	0.745729	0.372865	+	0.927886i	$0.378376\pi$
$558$	0	0
$559$	2303.20	0.174266
$560$	−9088.27	−0.685803
$561$	0	0
$562$	11599.2	0.870610
$563$	4909.73	0.367532	0.183766	−	0.982970i	$-0.441171\pi$
$563$	4909.73	0.367532	0.183766	+	0.982970i	$0.441171\pi$
$564$	0	0
$565$	−7461.19	−0.555566
$566$	11946.0	0.887153
$567$	0	0
$568$	−21438.0	−1.58366
$569$	−11498.2	−0.847155	−0.423578	−	0.905860i	$-0.639226\pi$
$569$	−11498.2	−0.847155	−0.423578	+	0.905860i	$0.639226\pi$
$570$	0	0
$571$	14766.6	1.08225	0.541124	−	0.840943i	$-0.317999\pi$
$571$	14766.6	1.08225	0.541124	+	0.840943i	$0.317999\pi$
$572$	−102.828	−0.00751653
$573$	0	0
$574$	−18361.8	−1.33520
$575$	575.000	0.0417029
$576$	0	0
$577$	13364.0	0.964212	0.482106	−	0.876113i	$-0.339872\pi$
$577$	13364.0	0.964212	0.482106	+	0.876113i	$0.339872\pi$
$578$	2607.73	0.187660
$579$	0	0
$580$	−1336.01	−0.0956460
$581$	555.590	0.0396725
$582$	0	0
$583$	−29.1488	−0.00207070
$584$	6019.64	0.426532
$585$	0	0
$586$	22475.1	1.58437
$587$	−12524.7	−0.880664	−0.440332	−	0.897835i	$-0.645139\pi$
$587$	−12524.7	−0.880664	−0.440332	+	0.897835i	$0.645139\pi$
$588$	0	0
$589$	6590.09	0.461019
$590$	4093.32	0.285626
$591$	0	0
$592$	−7963.66	−0.552879
$593$	17938.6	1.24224	0.621121	−	0.783714i	$-0.286677\pi$
$593$	17938.6	1.24224	0.621121	+	0.783714i	$0.286677\pi$
$594$	0	0
$595$	9733.55	0.670650
$596$	1354.16	0.0930681
$597$	0	0
$598$	563.072	0.0385045
$599$	26735.8	1.82370	0.911848	−	0.410528i	$-0.134656\pi$
$599$	26735.8	1.82370	0.911848	+	0.410528i	$0.134656\pi$
$600$	0	0
$601$	−21043.6	−1.42826	−0.714131	−	0.700012i	$-0.753178\pi$
$601$	−21043.6	−1.42826	−0.714131	+	0.700012i	$0.753178\pi$
$602$	−21676.6	−1.46756
$603$	0	0
$604$	162.652	0.0109573
$605$	5861.10	0.393864
$606$	0	0
$607$	5098.83	0.340947	0.170474	−	0.985362i	$-0.445470\pi$
$607$	5098.83	0.340947	0.170474	+	0.985362i	$0.445470\pi$
$608$	4651.29	0.310254
$609$	0	0
$610$	−4410.73	−0.292763
$611$	−4710.96	−0.311923
$612$	0	0
$613$	−12784.8	−0.842369	−0.421185	−	0.906975i	$-0.638386\pi$
$613$	−12784.8	−0.842369	−0.421185	+	0.906975i	$0.638386\pi$
$614$	−17439.0	−1.14622
$615$	0	0
$616$	−6774.37	−0.443096
$617$	8940.73	0.583372	0.291686	−	0.956514i	$-0.405784\pi$
$617$	8940.73	0.583372	0.291686	+	0.956514i	$0.405784\pi$
$618$	0	0
$619$	1333.26	0.0865725	0.0432863	−	0.999063i	$-0.486217\pi$
$619$	1333.26	0.0865725	0.0432863	+	0.999063i	$0.486217\pi$
$620$	318.787	0.0206497
$621$	0	0
$622$	−14308.3	−0.922364
$623$	−26871.3	−1.72805
$624$	0	0
$625$	625.000	0.0400000
$626$	21160.3	1.35102
$627$	0	0
$628$	−2364.83	−0.150266
$629$	8529.09	0.540663
$630$	0	0
$631$	−16667.9	−1.05157	−0.525784	−	0.850618i	$-0.676228\pi$
$631$	−16667.9	−1.05157	−0.525784	+	0.850618i	$0.676228\pi$
$632$	11589.2	0.729420
$633$	0	0
$634$	−26536.1	−1.66228
$635$	6841.46	0.427551
$636$	0	0
$637$	2549.32	0.158568
$638$	−10100.8	−0.626796
$639$	0	0
$640$	−8295.00	−0.512326
$641$	−27431.9	−1.69032	−0.845158	−	0.534516i	$-0.820494\pi$
$641$	−27431.9	−1.69032	−0.845158	+	0.534516i	$0.820494\pi$
$642$	0	0
$643$	9618.00	0.589886	0.294943	−	0.955515i	$-0.404699\pi$
$643$	9618.00	0.589886	0.294943	+	0.955515i	$0.404699\pi$
$644$	−588.818	−0.0360290
$645$	0	0
$646$	−23579.3	−1.43609
$647$	12548.6	0.762496	0.381248	−	0.924473i	$-0.375494\pi$
$647$	12548.6	0.762496	0.381248	+	0.924473i	$0.375494\pi$
$648$	0	0
$649$	3438.60	0.207977
$650$	612.034	0.0369323
$651$	0	0
$652$	471.162	0.0283008
$653$	8829.79	0.529152	0.264576	−	0.964365i	$-0.414768\pi$
$653$	8829.79	0.529152	0.264576	+	0.964365i	$0.414768\pi$
$654$	0	0
$655$	9921.19	0.591837
$656$	−16974.6	−1.01028
$657$	0	0
$658$	44337.3	2.62682
$659$	−5935.56	−0.350860	−0.175430	−	0.984492i	$-0.556132\pi$
$659$	−5935.56	−0.350860	−0.175430	+	0.984492i	$0.556132\pi$
$660$	0	0
$661$	−1182.03	−0.0695549	−0.0347775	−	0.999395i	$-0.511072\pi$
$661$	−1182.03	−0.0695549	−0.0347775	+	0.999395i	$0.511072\pi$
$662$	12792.2	0.751032
$663$	0	0
$664$	455.743	0.0266360
$665$	−13230.7	−0.771527
$666$	0	0
$667$	6145.64	0.356762
$668$	3735.73	0.216377
$669$	0	0
$670$	−6400.97	−0.369091
$671$	−3705.24	−0.213173
$672$	0	0
$673$	−6053.25	−0.346710	−0.173355	−	0.984859i	$-0.555461\pi$
$673$	−6053.25	−0.346710	−0.173355	+	0.984859i	$0.555461\pi$
$674$	−3871.74	−0.221267
$675$	0	0
$676$	−2130.41	−0.121211
$677$	−16694.2	−0.947728	−0.473864	−	0.880598i	$-0.657141\pi$
$677$	−16694.2	−0.947728	−0.473864	+	0.880598i	$0.657141\pi$
$678$	0	0
$679$	−44268.6	−2.50202
$680$	7984.31	0.450271
$681$	0	0
$682$	2410.18	0.135323
$683$	4150.03	0.232499	0.116249	−	0.993220i	$-0.462913\pi$
$683$	4150.03	0.232499	0.116249	+	0.993220i	$0.462913\pi$
$684$	0	0
$685$	1366.66	0.0762296
$686$	2350.21	0.130804
$687$	0	0
$688$	−20039.0	−1.11043
$689$	18.8772	0.00104378
$690$	0	0
$691$	33382.4	1.83781	0.918904	−	0.394481i	$-0.129076\pi$
$691$	33382.4	1.83781	0.918904	+	0.394481i	$0.129076\pi$
$692$	2709.33	0.148834
$693$	0	0
$694$	5415.27	0.296197
$695$	2789.45	0.152244
$696$	0	0
$697$	18179.8	0.987960
$698$	−10760.4	−0.583508
$699$	0	0
$700$	−640.019	−0.0345578
$701$	29528.7	1.59099	0.795494	−	0.605961i	$-0.207211\pi$
$701$	29528.7	1.59099	0.795494	+	0.605961i	$0.207211\pi$
$702$	0	0
$703$	−11593.5	−0.621989
$704$	−5456.13	−0.292096
$705$	0	0
$706$	6878.69	0.366690
$707$	21911.0	1.16556
$708$	0	0
$709$	27016.3	1.43106	0.715529	−	0.698583i	$-0.246186\pi$
$709$	27016.3	1.43106	0.715529	+	0.698583i	$0.246186\pi$
$710$	−15312.8	−0.809408
$711$	0	0
$712$	−22042.2	−1.16020
$713$	−1466.42	−0.0770237
$714$	0	0
$715$	514.140	0.0268920
$716$	−3300.43	−0.172266
$717$	0	0
$718$	9231.54	0.479830
$719$	−24752.9	−1.28390	−0.641952	−	0.766745i	$-0.721875\pi$
$719$	−24752.9	−1.28390	−0.641952	+	0.766745i	$0.721875\pi$
$720$	0	0
$721$	16193.4	0.836438
$722$	11474.1	0.591443
$723$	0	0
$724$	−583.868	−0.0299714
$725$	6680.04	0.342194
$726$	0	0
$727$	−17574.5	−0.896565	−0.448283	−	0.893892i	$-0.647964\pi$
$727$	−17574.5	−0.896565	−0.448283	+	0.893892i	$0.647964\pi$
$728$	4387.19	0.223352
$729$	0	0
$730$	4299.74	0.218001
$731$	21461.7	1.08590
$732$	0	0
$733$	27739.6	1.39780	0.698898	−	0.715221i	$-0.253674\pi$
$733$	27739.6	1.39780	0.698898	+	0.715221i	$0.253674\pi$
$734$	−4135.67	−0.207970
$735$	0	0
$736$	−1035.00	−0.0518351
$737$	−5377.15	−0.268751
$738$	0	0
$739$	−3140.78	−0.156340	−0.0781701	−	0.996940i	$-0.524908\pi$
$739$	−3140.78	−0.156340	−0.0781701	+	0.996940i	$0.524908\pi$
$740$	−560.821	−0.0278597
$741$	0	0
$742$	−177.663	−0.00879007
$743$	35881.5	1.77169	0.885844	−	0.463984i	$-0.153580\pi$
$743$	35881.5	1.77169	0.885844	+	0.463984i	$0.153580\pi$
$744$	0	0
$745$	−6770.80	−0.332970
$746$	33546.9	1.64643
$747$	0	0
$748$	−958.176	−0.0468374
$749$	33527.3	1.63559
$750$	0	0
$751$	−7009.03	−0.340563	−0.170282	−	0.985395i	$-0.554468\pi$
$751$	−7009.03	−0.340563	−0.170282	+	0.985395i	$0.554468\pi$
$752$	40987.7	1.98759
$753$	0	0
$754$	6541.46	0.315950
$755$	−813.261	−0.0392021
$756$	0	0
$757$	29998.3	1.44030	0.720150	−	0.693818i	$-0.244073\pi$
$757$	29998.3	1.44030	0.720150	+	0.693818i	$0.244073\pi$
$758$	36065.8	1.72819
$759$	0	0
$760$	−10853.0	−0.518000
$761$	−5283.49	−0.251677	−0.125839	−	0.992051i	$-0.540162\pi$
$761$	−5283.49	−0.251677	−0.125839	+	0.992051i	$0.540162\pi$
$762$	0	0
$763$	−44192.2	−2.09681
$764$	3058.98	0.144856
$765$	0	0
$766$	39823.9	1.87845
$767$	−2226.89	−0.104835
$768$	0	0
$769$	−15216.2	−0.713538	−0.356769	−	0.934193i	$-0.616122\pi$
$769$	−15216.2	−0.713538	−0.356769	+	0.934193i	$0.616122\pi$
$770$	−4838.84	−0.226467
$771$	0	0
$772$	−1336.93	−0.0623281
$773$	−15651.3	−0.728250	−0.364125	−	0.931350i	$-0.618632\pi$
$773$	−15651.3	−0.728250	−0.364125	+	0.931350i	$0.618632\pi$
$774$	0	0
$775$	−1593.93	−0.0738785
$776$	−36313.0	−1.67985
$777$	0	0
$778$	−33049.1	−1.52297
$779$	−24711.6	−1.13657
$780$	0	0
$781$	−12863.6	−0.589365
$782$	5246.83	0.239931
$783$	0	0
$784$	−22180.3	−1.01040
$785$	11824.1	0.537607
$786$	0	0
$787$	9401.09	0.425810	0.212905	−	0.977073i	$-0.431707\pi$
$787$	9401.09	0.425810	0.212905	+	0.977073i	$0.431707\pi$
$788$	2449.21	0.110723
$789$	0	0
$790$	8278.00	0.372807
$791$	−38202.5	−1.71722
$792$	0	0
$793$	2399.57	0.107454
$794$	−5679.55	−0.253854
$795$	0	0
$796$	1138.66	0.0507018
$797$	−37388.5	−1.66169	−0.830845	−	0.556504i	$-0.812143\pi$
$797$	−37388.5	−1.66169	−0.830845	+	0.556504i	$0.812143\pi$
$798$	0	0
$799$	−43897.9	−1.94367
$800$	−1125.00	−0.0497184
$801$	0	0
$802$	−33845.3	−1.49017
$803$	3612.00	0.158736
$804$	0	0
$805$	2944.09	0.128901
$806$	−1560.87	−0.0682125
$807$	0	0
$808$	17973.4	0.782550
$809$	−24390.6	−1.05999	−0.529993	−	0.848002i	$-0.677805\pi$
$809$	−24390.6	−1.05999	−0.529993	+	0.848002i	$0.677805\pi$
$810$	0	0
$811$	23750.8	1.02836	0.514181	−	0.857682i	$-0.328096\pi$
$811$	23750.8	1.02836	0.514181	+	0.857682i	$0.328096\pi$
$812$	−6840.56	−0.295636
$813$	0	0
$814$	−4240.07	−0.182573
$815$	−2355.81	−0.101252
$816$	0	0
$817$	−29172.8	−1.24924
$818$	−44032.7	−1.88211
$819$	0	0
$820$	−1195.39	−0.0509084
$821$	−3041.59	−0.129296	−0.0646481	−	0.997908i	$-0.520592\pi$
$821$	−3041.59	−0.129296	−0.0646481	+	0.997908i	$0.520592\pi$
$822$	0	0
$823$	28411.6	1.20336	0.601681	−	0.798737i	$-0.294498\pi$
$823$	28411.6	1.20336	0.601681	+	0.798737i	$0.294498\pi$
$824$	13283.2	0.561581
$825$	0	0
$826$	20958.4	0.882854
$827$	24351.5	1.02392	0.511962	−	0.859008i	$-0.328919\pi$
$827$	24351.5	1.02392	0.511962	+	0.859008i	$0.328919\pi$
$828$	0	0
$829$	−47240.8	−1.97918	−0.989590	−	0.143914i	$-0.954031\pi$
$829$	−47240.8	−1.97918	−0.989590	+	0.143914i	$0.954031\pi$
$830$	325.531	0.0136137
$831$	0	0
$832$	3533.48	0.147237
$833$	23755.2	0.988077
$834$	0	0
$835$	−18678.7	−0.774134
$836$	1302.44	0.0538826
$837$	0	0
$838$	−6589.49	−0.271635
$839$	−2646.33	−0.108893	−0.0544466	−	0.998517i	$-0.517339\pi$
$839$	−2646.33	−0.108893	−0.0544466	+	0.998517i	$0.517339\pi$
$840$	0	0
$841$	47007.7	1.92741
$842$	−19713.8	−0.806869
$843$	0	0
$844$	5596.81	0.228258
$845$	10652.0	0.433658
$846$	0	0
$847$	30009.7	1.21741
$848$	−164.241	−0.00665102
$849$	0	0
$850$	5703.08	0.230134
$851$	2579.78	0.103917
$852$	0	0
$853$	−40826.4	−1.63877	−0.819385	−	0.573244i	$-0.805685\pi$
$853$	−40826.4	−1.63877	−0.819385	+	0.573244i	$0.805685\pi$
$854$	−22583.6	−0.904913
$855$	0	0
$856$	27502.0	1.09813
$857$	−1516.61	−0.0604510	−0.0302255	−	0.999543i	$-0.509623\pi$
$857$	−1516.61	−0.0604510	−0.0302255	+	0.999543i	$0.509623\pi$
$858$	0	0
$859$	6514.74	0.258766	0.129383	−	0.991595i	$-0.458700\pi$
$859$	6514.74	0.258766	0.129383	+	0.991595i	$0.458700\pi$
$860$	−1411.19	−0.0559550
$861$	0	0
$862$	1750.82	0.0691800
$863$	7120.18	0.280850	0.140425	−	0.990091i	$-0.455153\pi$
$863$	7120.18	0.280850	0.140425	+	0.990091i	$0.455153\pi$
$864$	0	0
$865$	−13546.6	−0.532485
$866$	−8032.79	−0.315202
$867$	0	0
$868$	1632.24	0.0638269
$869$	6953.94	0.271457
$870$	0	0
$871$	3482.33	0.135470
$872$	−36250.3	−1.40779
$873$	0	0
$874$	−7131.98	−0.276021
$875$	3200.10	0.123638
$876$	0	0
$877$	−34013.4	−1.30964	−0.654818	−	0.755787i	$-0.727255\pi$
$877$	−34013.4	−1.30964	−0.654818	+	0.755787i	$0.727255\pi$
$878$	−16527.5	−0.635282
$879$	0	0
$880$	−4473.27	−0.171357
$881$	−6704.11	−0.256376	−0.128188	−	0.991750i	$-0.540916\pi$
$881$	−6704.11	−0.256376	−0.128188	+	0.991750i	$0.540916\pi$
$882$	0	0
$883$	47808.5	1.82207	0.911033	−	0.412333i	$-0.135286\pi$
$883$	47808.5	1.82207	0.911033	+	0.412333i	$0.135286\pi$
$884$	620.530	0.0236094
$885$	0	0
$886$	39163.7	1.48502
$887$	38955.7	1.47464	0.737320	−	0.675544i	$-0.236091\pi$
$887$	38955.7	1.47464	0.737320	+	0.675544i	$0.236091\pi$
$888$	0	0
$889$	35029.3	1.32154
$890$	−15744.4	−0.592982
$891$	0	0
$892$	5580.60	0.209476
$893$	59670.0	2.23604
$894$	0	0
$895$	16502.1	0.616319
$896$	−42471.7	−1.58357
$897$	0	0
$898$	35457.4	1.31763
$899$	−17036.1	−0.632019
$900$	0	0
$901$	175.903	0.00650407
$902$	−9037.71	−0.333617
$903$	0	0
$904$	−31337.0	−1.15294
$905$	2919.34	0.107229
$906$	0	0
$907$	7725.66	0.282830	0.141415	−	0.989950i	$-0.454835\pi$
$907$	7725.66	0.282830	0.141415	+	0.989950i	$0.454835\pi$
$908$	566.323	0.0206983
$909$	0	0
$910$	3133.71	0.114155
$911$	13150.7	0.478269	0.239135	−	0.970986i	$-0.423136\pi$
$911$	13150.7	0.478269	0.239135	+	0.970986i	$0.423136\pi$
$912$	0	0
$913$	273.463	0.00991270
$914$	−12434.6	−0.450000
$915$	0	0
$916$	−693.702	−0.0250224
$917$	50798.0	1.82933
$918$	0	0
$919$	33083.7	1.18752	0.593759	−	0.804643i	$-0.297643\pi$
$919$	33083.7	1.18752	0.593759	+	0.804643i	$0.297643\pi$
$920$	2415.00	0.0865437
$921$	0	0
$922$	18184.1	0.649526
$923$	8330.65	0.297082
$924$	0	0
$925$	2804.11	0.0996740
$926$	−17187.4	−0.609949
$927$	0	0
$928$	−12024.1	−0.425333
$929$	1310.12	0.0462686	0.0231343	−	0.999732i	$-0.492635\pi$
$929$	1310.12	0.0462686	0.0231343	+	0.999732i	$0.492635\pi$
$930$	0	0
$931$	−32290.2	−1.13670
$932$	2208.68	0.0776264
$933$	0	0
$934$	−26874.1	−0.941487
$935$	4790.88	0.167571
$936$	0	0
$937$	40623.1	1.41633	0.708163	−	0.706049i	$-0.249524\pi$
$937$	40623.1	1.41633	0.708163	+	0.706049i	$0.249524\pi$
$938$	−32774.0	−1.14084
$939$	0	0
$940$	2886.46	0.100155
$941$	10434.0	0.361466	0.180733	−	0.983532i	$-0.442153\pi$
$941$	10434.0	0.361466	0.180733	+	0.983532i	$0.442153\pi$
$942$	0	0
$943$	5498.80	0.189889
$944$	19375.1	0.668013
$945$	0	0
$946$	−10669.3	−0.366689
$947$	51333.4	1.76147	0.880734	−	0.473612i	$-0.157050\pi$
$947$	51333.4	1.76147	0.880734	+	0.473612i	$0.157050\pi$
$948$	0	0
$949$	−2339.19	−0.0800141
$950$	−7752.15	−0.264750
$951$	0	0
$952$	40880.9	1.39176
$953$	−40263.6	−1.36859	−0.684294	−	0.729206i	$-0.739890\pi$
$953$	−40263.6	−1.36859	−0.684294	+	0.729206i	$0.739890\pi$
$954$	0	0
$955$	−15294.9	−0.518252
$956$	3876.79	0.131155
$957$	0	0
$958$	−19069.8	−0.643128
$959$	6997.49	0.235621
$960$	0	0
$961$	−25726.0	−0.863549
$962$	2745.94	0.0920297
$963$	0	0
$964$	−2024.57	−0.0676420
$965$	6684.67	0.222992
$966$	0	0
$967$	−8353.12	−0.277785	−0.138893	−	0.990307i	$-0.544354\pi$
$967$	−8353.12	−0.277785	−0.138893	+	0.990307i	$0.544354\pi$
$968$	24616.6	0.817364
$969$	0	0
$970$	−25937.9	−0.858572
$971$	28020.1	0.926063	0.463032	−	0.886342i	$-0.346762\pi$
$971$	28020.1	0.926063	0.463032	+	0.886342i	$0.346762\pi$
$972$	0	0
$973$	14282.4	0.470578
$974$	41973.6	1.38082
$975$	0	0
$976$	−20877.5	−0.684704
$977$	50115.3	1.64107	0.820537	−	0.571593i	$-0.193674\pi$
$977$	50115.3	1.64107	0.820537	+	0.571593i	$0.193674\pi$
$978$	0	0
$979$	−13226.1	−0.431776
$980$	−1562.00	−0.0509144
$981$	0	0
$982$	15905.9	0.516881
$983$	38113.0	1.23664	0.618320	−	0.785926i	$-0.287814\pi$
$983$	38113.0	1.23664	0.618320	+	0.785926i	$0.287814\pi$
$984$	0	0
$985$	−12246.1	−0.396134
$986$	60954.9	1.96876
$987$	0	0
$988$	−843.481	−0.0271606
$989$	6491.49	0.208713
$990$	0	0
$991$	−5245.02	−0.168127	−0.0840634	−	0.996460i	$-0.526790\pi$
$991$	−5245.02	−0.168127	−0.0840634	+	0.996460i	$0.526790\pi$
$992$	2869.08	0.0918281
$993$	0	0
$994$	−78404.0	−2.50183
$995$	−5693.29	−0.181396
$996$	0	0
$997$	21896.1	0.695544	0.347772	−	0.937579i	$-0.386938\pi$
$997$	21896.1	0.695544	0.347772	+	0.937579i	$0.386938\pi$
$998$	−24040.3	−0.762509
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1035.4.a.g.1.1		2
3.2	odd	2		115.4.a.c.1.2	✓	2
12.11	even	2		1840.4.a.h.1.1		2
15.2	even	4		575.4.b.f.24.1		4
15.8	even	4		575.4.b.f.24.4		4
15.14	odd	2		575.4.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.c.1.2	✓	2	3.2	odd	2
575.4.a.h.1.1		2	15.14	odd	2
575.4.b.f.24.1		4	15.2	even	4
575.4.b.f.24.4		4	15.8	even	4
1035.4.a.g.1.1		2	1.1	even	1	trivial
1840.4.a.h.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1035.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{2} +1.00000 q^{4} -5.00000 q^{5} -25.6008 q^{7} -21.0000 q^{8} +O(q^{10})\)	\(q+3.00000 q^{2} +1.00000 q^{4} -5.00000 q^{5} -25.6008 q^{7} -21.0000 q^{8} -15.0000 q^{10} -12.6008 q^{11} +8.16046 q^{13} -76.8023 q^{14} -71.0000 q^{16} +76.0411 q^{17} -103.362 q^{19} -5.00000 q^{20} -37.8023 q^{22} +23.0000 q^{23} +25.0000 q^{25} +24.4814 q^{26} -25.6008 q^{28} +267.202 q^{29} -63.7574 q^{31} -45.0000 q^{32} +228.123 q^{34} +128.004 q^{35} +112.164 q^{37} -310.086 q^{38} +105.000 q^{40} +239.078 q^{41} +282.239 q^{43} -12.6008 q^{44} +69.0000 q^{46} -577.291 q^{47} +312.399 q^{49} +75.0000 q^{50} +8.16046 q^{52} +2.31326 q^{53} +63.0038 q^{55} +537.616 q^{56} +801.605 q^{58} -272.888 q^{59} +294.049 q^{61} -191.272 q^{62} +433.000 q^{64} -40.8023 q^{65} +426.732 q^{67} +76.0411 q^{68} +384.011 q^{70} +1020.85 q^{71} -286.650 q^{73} +336.493 q^{74} -103.362 q^{76} +322.589 q^{77} -551.866 q^{79} +355.000 q^{80} +717.235 q^{82} -21.7021 q^{83} -380.205 q^{85} +846.716 q^{86} +264.616 q^{88} +1049.63 q^{89} -208.914 q^{91} +23.0000 q^{92} -1731.87 q^{94} +516.810 q^{95} +1729.19 q^{97} +937.198 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{2} + 2 q^{4} - 10 q^{5} + q^{7} - 42 q^{8}+O(q^{10}) \) 2 * q + 6 * q^2 + 2 * q^4 - 10 * q^5 + q^7 - 42 * q^8	\( 2 q + 6 q^{2} + 2 q^{4} - 10 q^{5} + q^{7} - 42 q^{8} - 30 q^{10} + 27 q^{11} - 15 q^{13} + 3 q^{14} - 142 q^{16} + 79 q^{17} - 71 q^{19} - 10 q^{20} + 81 q^{22} + 46 q^{23} + 50 q^{25} - 45 q^{26} + q^{28} + 430 q^{29} - 305 q^{31} - 90 q^{32} + 237 q^{34} - 5 q^{35} - 68 q^{37} - 213 q^{38} + 210 q^{40} + 593 q^{41} + 648 q^{43} + 27 q^{44} + 138 q^{46} - 382 q^{47} + 677 q^{49} + 150 q^{50} - 15 q^{52} + 464 q^{53} - 135 q^{55} - 21 q^{56} + 1290 q^{58} + 18 q^{59} - 7 q^{61} - 915 q^{62} + 866 q^{64} + 75 q^{65} + 60 q^{67} + 79 q^{68} - 15 q^{70} + 1029 q^{71} + 74 q^{73} - 204 q^{74} - 71 q^{76} + 1376 q^{77} + 692 q^{79} + 710 q^{80} + 1779 q^{82} + 1460 q^{83} - 395 q^{85} + 1944 q^{86} - 567 q^{88} + 220 q^{89} - 825 q^{91} + 46 q^{92} - 1146 q^{94} + 355 q^{95} + 1339 q^{97} + 2031 q^{98}+O(q^{100}) \) 2 * q + 6 * q^2 + 2 * q^4 - 10 * q^5 + q^7 - 42 * q^8 - 30 * q^10 + 27 * q^11 - 15 * q^13 + 3 * q^14 - 142 * q^16 + 79 * q^17 - 71 * q^19 - 10 * q^20 + 81 * q^22 + 46 * q^23 + 50 * q^25 - 45 * q^26 + q^28 + 430 * q^29 - 305 * q^31 - 90 * q^32 + 237 * q^34 - 5 * q^35 - 68 * q^37 - 213 * q^38 + 210 * q^40 + 593 * q^41 + 648 * q^43 + 27 * q^44 + 138 * q^46 - 382 * q^47 + 677 * q^49 + 150 * q^50 - 15 * q^52 + 464 * q^53 - 135 * q^55 - 21 * q^56 + 1290 * q^58 + 18 * q^59 - 7 * q^61 - 915 * q^62 + 866 * q^64 + 75 * q^65 + 60 * q^67 + 79 * q^68 - 15 * q^70 + 1029 * q^71 + 74 * q^73 - 204 * q^74 - 71 * q^76 + 1376 * q^77 + 692 * q^79 + 710 * q^80 + 1779 * q^82 + 1460 * q^83 - 395 * q^85 + 1944 * q^86 - 567 * q^88 + 220 * q^89 - 825 * q^91 + 46 * q^92 - 1146 * q^94 + 355 * q^95 + 1339 * q^97 + 2031 * q^98

Introduction

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