LMFDB - Embedded newform 2175.2.a.bc.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,2,Mod(1,2175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2175.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-2,-8,12,0,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 $ x^8 - 2x^7 - 12x^6 + 23x^5 + 36x^4 - 62x^3 - 15x^2 + 14*x + 2
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			1.3
Root			$1.64893$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.64893 q^{2} -1.00000 q^{3} +0.718971 q^{4} +1.64893 q^{6} -1.44112 q^{7} +2.11233 q^{8} +1.00000 q^{9} +0.191176 q^{11} -0.718971 q^{12} -5.14501 q^{13} +2.37631 q^{14} -4.92102 q^{16} +3.58379 q^{17} -1.64893 q^{18} +7.84780 q^{19} +1.44112 q^{21} -0.315236 q^{22} -7.39261 q^{23} -2.11233 q^{24} +8.48377 q^{26} -1.00000 q^{27} -1.03612 q^{28} +1.00000 q^{29} -7.18783 q^{31} +3.88977 q^{32} -0.191176 q^{33} -5.90942 q^{34} +0.718971 q^{36} +8.70773 q^{37} -12.9405 q^{38} +5.14501 q^{39} +2.33703 q^{41} -2.37631 q^{42} +3.67511 q^{43} +0.137450 q^{44} +12.1899 q^{46} -9.52376 q^{47} +4.92102 q^{48} -4.92317 q^{49} -3.58379 q^{51} -3.69911 q^{52} -10.9053 q^{53} +1.64893 q^{54} -3.04412 q^{56} -7.84780 q^{57} -1.64893 q^{58} +14.2051 q^{59} +8.45202 q^{61} +11.8522 q^{62} -1.44112 q^{63} +3.42809 q^{64} +0.315236 q^{66} -9.96623 q^{67} +2.57664 q^{68} +7.39261 q^{69} -4.72562 q^{71} +2.11233 q^{72} +9.06473 q^{73} -14.3584 q^{74} +5.64234 q^{76} -0.275508 q^{77} -8.48377 q^{78} -8.78711 q^{79} +1.00000 q^{81} -3.85361 q^{82} -15.8701 q^{83} +1.03612 q^{84} -6.06000 q^{86} -1.00000 q^{87} +0.403827 q^{88} -11.6085 q^{89} +7.41459 q^{91} -5.31507 q^{92} +7.18783 q^{93} +15.7040 q^{94} -3.88977 q^{96} +9.85482 q^{97} +8.11796 q^{98} +0.191176 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 8 q^{3} + 12 q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} + 6 q^{11} - 12 q^{12} + 6 q^{13} + 9 q^{14} + 32 q^{16} - 12 q^{17} - 2 q^{18} - 2 q^{21} + 3 q^{22} - 14 q^{23} + 3 q^{24} + 18 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) $ 8 * q - 2 * q^2 - 8 * q^3 + 12 * q^4 + 2 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9 + 6 * q^11 - 12 * q^12 + 6 * q^13 + 9 * q^14 + 32 * q^16 - 12 * q^17 - 2 * q^18 - 2 * q^21 + 3 * q^22 - 14 * q^23 + 3 * q^24 + 18 * q^26 - 8 * q^27 + 14 * q^28 + 8 * q^29 + 8 * q^31 + 2 * q^32 - 6 * q^33 - 13 * q^34 + 12 * q^36 + 4 * q^37 - 26 * q^38 - 6 * q^39 + 2 * q^41 - 9 * q^42 + 2 * q^43 - 15 * q^44 + 24 * q^46 - 12 * q^47 - 32 * q^48 + 38 * q^49 + 12 * q^51 + 49 * q^52 - 4 * q^53 + 2 * q^54 + 58 * q^56 - 2 * q^58 + 18 * q^59 + 12 * q^61 + 4 * q^62 + 2 * q^63 + 21 * q^64 - 3 * q^66 + 26 * q^67 - 81 * q^68 + 14 * q^69 + 24 * q^71 - 3 * q^72 - 14 * q^73 - 22 * q^74 + 26 * q^77 - 18 * q^78 + 10 * q^79 + 8 * q^81 + 48 * q^82 - 40 * q^83 - 14 * q^84 + 8 * q^86 - 8 * q^87 - 10 * q^88 + 34 * q^89 + 26 * q^91 + 18 * q^92 - 8 * q^93 - 43 * q^94 - 2 * q^96 + 30 * q^97 - 29 * q^98 + 6 * q^99

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 $

$2$	−1.64893	−1.16597	−0.582985	−	0.812483i	$-0.698115\pi$
$2$	−1.64893	−1.16597	−0.582985	+	0.812483i	$0.698115\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.718971	0.359485
$5$	0	0
$5$	0	0
$6$	1.64893	0.673173
$7$	−1.44112	−0.544693	−0.272346	−	0.962199i	$-0.587800\pi$
$7$	−1.44112	−0.544693	−0.272346	+	0.962199i	$0.587800\pi$
$8$	2.11233	0.746821
$9$	1.00000	0.333333
$10$	0	0
$11$	0.191176	0.0576418	0.0288209	−	0.999585i	$-0.490825\pi$
$11$	0.191176	0.0576418	0.0288209	+	0.999585i	$0.490825\pi$
$12$	−0.718971	−0.207549
$13$	−5.14501	−1.42697	−0.713485	−	0.700671i	$-0.752884\pi$
$13$	−5.14501	−1.42697	−0.713485	+	0.700671i	$0.752884\pi$
$14$	2.37631	0.635095
$15$	0	0
$16$	−4.92102	−1.23026
$17$	3.58379	0.869197	0.434598	−	0.900624i	$-0.356890\pi$
$17$	3.58379	0.869197	0.434598	+	0.900624i	$0.356890\pi$
$18$	−1.64893	−0.388657
$19$	7.84780	1.80041	0.900205	−	0.435467i	$-0.143417\pi$
$19$	7.84780	1.80041	0.900205	+	0.435467i	$0.143417\pi$
$20$	0	0
$21$	1.44112	0.314479
$22$	−0.315236	−0.0672086
$23$	−7.39261	−1.54147	−0.770733	−	0.637158i	$-0.780110\pi$
$23$	−7.39261	−1.54147	−0.770733	+	0.637158i	$0.780110\pi$
$24$	−2.11233	−0.431177
$25$	0	0
$26$	8.48377	1.66380
$27$	−1.00000	−0.192450
$28$	−1.03612	−0.195809
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−7.18783	−1.29097	−0.645486	−	0.763772i	$-0.723345\pi$
$31$	−7.18783	−1.29097	−0.645486	+	0.763772i	$0.723345\pi$
$32$	3.88977	0.687620
$33$	−0.191176	−0.0332795
$34$	−5.90942	−1.01346
$35$	0	0
$36$	0.718971	0.119828
$37$	8.70773	1.43154	0.715771	−	0.698335i	$-0.246075\pi$
$37$	8.70773	1.43154	0.715771	+	0.698335i	$0.246075\pi$
$38$	−12.9405	−2.09922
$39$	5.14501	0.823861
$40$	0	0
$41$	2.33703	0.364983	0.182492	−	0.983207i	$-0.441584\pi$
$41$	2.33703	0.364983	0.182492	+	0.983207i	$0.441584\pi$
$42$	−2.37631	−0.366672
$43$	3.67511	0.560449	0.280225	−	0.959934i	$-0.409591\pi$
$43$	3.67511	0.560449	0.280225	+	0.959934i	$0.409591\pi$
$44$	0.137450	0.0207214
$45$	0	0
$46$	12.1899	1.79730
$47$	−9.52376	−1.38918	−0.694591	−	0.719404i	$-0.744415\pi$
$47$	−9.52376	−1.38918	−0.694591	+	0.719404i	$0.744415\pi$
$48$	4.92102	0.710288
$49$	−4.92317	−0.703310
$50$	0	0
$51$	−3.58379	−0.501831
$52$	−3.69911	−0.512975
$53$	−10.9053	−1.49796	−0.748981	−	0.662591i	$-0.769457\pi$
$53$	−10.9053	−1.49796	−0.748981	+	0.662591i	$0.769457\pi$
$54$	1.64893	0.224391
$55$	0	0
$56$	−3.04412	−0.406788
$57$	−7.84780	−1.03947
$58$	−1.64893	−0.216515
$59$	14.2051	1.84934	0.924672	−	0.380766i	$-0.124340\pi$
$59$	14.2051	1.84934	0.924672	+	0.380766i	$0.124340\pi$
$60$	0	0
$61$	8.45202	1.08217	0.541085	−	0.840968i	$-0.318014\pi$
$61$	8.45202	1.08217	0.541085	+	0.840968i	$0.318014\pi$
$62$	11.8522	1.50523
$63$	−1.44112	−0.181564
$64$	3.42809	0.428511
$65$	0	0
$66$	0.315236	0.0388029
$67$	−9.96623	−1.21757	−0.608785	−	0.793336i	$-0.708343\pi$
$67$	−9.96623	−1.21757	−0.608785	+	0.793336i	$0.708343\pi$
$68$	2.57664	0.312464
$69$	7.39261	0.889966
$70$	0	0
$71$	−4.72562	−0.560828	−0.280414	−	0.959879i	$-0.590472\pi$
$71$	−4.72562	−0.560828	−0.280414	+	0.959879i	$0.590472\pi$
$72$	2.11233	0.248940
$73$	9.06473	1.06095	0.530473	−	0.847702i	$-0.322014\pi$
$73$	9.06473	1.06095	0.530473	+	0.847702i	$0.322014\pi$
$74$	−14.3584	−1.66913
$75$	0	0
$76$	5.64234	0.647221
$77$	−0.275508	−0.0313971
$78$	−8.48377	−0.960598
$79$	−8.78711	−0.988627	−0.494314	−	0.869284i	$-0.664581\pi$
$79$	−8.78711	−0.988627	−0.494314	+	0.869284i	$0.664581\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.85361	−0.425560
$83$	−15.8701	−1.74197	−0.870986	−	0.491308i	$-0.836519\pi$
$83$	−15.8701	−1.74197	−0.870986	+	0.491308i	$0.836519\pi$
$84$	1.03612	0.113050
$85$	0	0
$86$	−6.06000	−0.653467
$87$	−1.00000	−0.107211
$88$	0.403827	0.0430481
$89$	−11.6085	−1.23050	−0.615250	−	0.788332i	$-0.710945\pi$
$89$	−11.6085	−1.23050	−0.615250	+	0.788332i	$0.710945\pi$
$90$	0	0
$91$	7.41459	0.777260
$92$	−5.31507	−0.554135
$93$	7.18783	0.745343
$94$	15.7040	1.61975
$95$	0	0
$96$	−3.88977	−0.396998
$97$	9.85482	1.00061	0.500303	−	0.865851i	$-0.333222\pi$
$97$	9.85482	1.00061	0.500303	+	0.865851i	$0.333222\pi$
$98$	8.11796	0.820038
$99$	0.191176	0.0192139
$100$	0	0
$101$	−5.33326	−0.530679	−0.265340	−	0.964155i	$-0.585484\pi$
$101$	−5.33326	−0.530679	−0.265340	+	0.964155i	$0.585484\pi$
$102$	5.90942	0.585120
$103$	13.1476	1.29547	0.647736	−	0.761865i	$-0.275716\pi$
$103$	13.1476	1.29547	0.647736	+	0.761865i	$0.275716\pi$
$104$	−10.8680	−1.06569
$105$	0	0
$106$	17.9821	1.74658
$107$	15.2800	1.47717	0.738586	−	0.674159i	$-0.235494\pi$
$107$	15.2800	1.47717	0.738586	+	0.674159i	$0.235494\pi$
$108$	−0.718971	−0.0691830
$109$	9.88042	0.946372	0.473186	−	0.880962i	$-0.343104\pi$
$109$	9.88042	0.946372	0.473186	+	0.880962i	$0.343104\pi$
$110$	0	0
$111$	−8.70773	−0.826501
$112$	7.09179	0.670111
$113$	−6.30656	−0.593271	−0.296636	−	0.954991i	$-0.595865\pi$
$113$	−6.30656	−0.593271	−0.296636	+	0.954991i	$0.595865\pi$
$114$	12.9405	1.21199
$115$	0	0
$116$	0.718971	0.0667548
$117$	−5.14501	−0.475657
$118$	−23.4232	−2.15628
$119$	−5.16468	−0.473445
$120$	0	0
$121$	−10.9635	−0.996677
$122$	−13.9368	−1.26178
$123$	−2.33703	−0.210723
$124$	−5.16784	−0.464086
$125$	0	0
$126$	2.37631	0.211698
$127$	19.4653	1.72727	0.863635	−	0.504118i	$-0.168182\pi$
$127$	19.4653	1.72727	0.863635	+	0.504118i	$0.168182\pi$
$128$	−13.4322	−1.18725
$129$	−3.67511	−0.323575
$130$	0	0
$131$	11.8940	1.03919	0.519594	−	0.854413i	$-0.326083\pi$
$131$	11.8940	1.03919	0.519594	+	0.854413i	$0.326083\pi$
$132$	−0.137450	−0.0119635
$133$	−11.3096	−0.980670
$134$	16.4336	1.41965
$135$	0	0
$136$	7.57014	0.649134
$137$	16.2830	1.39115	0.695574	−	0.718454i	$-0.255150\pi$
$137$	16.2830	1.39115	0.695574	+	0.718454i	$0.255150\pi$
$138$	−12.1899	−1.03767
$139$	8.78796	0.745385	0.372692	−	0.927955i	$-0.378435\pi$
$139$	8.78796	0.745385	0.372692	+	0.927955i	$0.378435\pi$
$140$	0	0
$141$	9.52376	0.802045
$142$	7.79222	0.653909
$143$	−0.983604	−0.0822531
$144$	−4.92102	−0.410085
$145$	0	0
$146$	−14.9471	−1.23703
$147$	4.92317	0.406056
$148$	6.26060	0.514618
$149$	3.09748	0.253755	0.126878	−	0.991918i	$-0.459504\pi$
$149$	3.09748	0.253755	0.126878	+	0.991918i	$0.459504\pi$
$150$	0	0
$151$	−7.39067	−0.601444	−0.300722	−	0.953712i	$-0.597228\pi$
$151$	−7.39067	−0.601444	−0.300722	+	0.953712i	$0.597228\pi$
$152$	16.5771	1.34458
$153$	3.58379	0.289732
$154$	0.454294	0.0366080
$155$	0	0
$156$	3.69911	0.296166
$157$	9.63295	0.768793	0.384396	−	0.923168i	$-0.374410\pi$
$157$	9.63295	0.768793	0.384396	+	0.923168i	$0.374410\pi$
$158$	14.4893	1.15271
$159$	10.9053	0.864849
$160$	0	0
$161$	10.6537	0.839626
$162$	−1.64893	−0.129552
$163$	19.6807	1.54151	0.770756	−	0.637131i	$-0.219879\pi$
$163$	19.6807	1.54151	0.770756	+	0.637131i	$0.219879\pi$
$164$	1.68026	0.131206
$165$	0	0
$166$	26.1687	2.03109
$167$	3.69350	0.285811	0.142906	−	0.989736i	$-0.454355\pi$
$167$	3.69350	0.285811	0.142906	+	0.989736i	$0.454355\pi$
$168$	3.04412	0.234859
$169$	13.4712	1.03624
$170$	0	0
$171$	7.84780	0.600136
$172$	2.64230	0.201473
$173$	8.10495	0.616208	0.308104	−	0.951353i	$-0.400306\pi$
$173$	8.10495	0.616208	0.308104	+	0.951353i	$0.400306\pi$
$174$	1.64893	0.125005
$175$	0	0
$176$	−0.940782	−0.0709141
$177$	−14.2051	−1.06772
$178$	19.1416	1.43473
$179$	−7.63453	−0.570631	−0.285316	−	0.958434i	$-0.592098\pi$
$179$	−7.63453	−0.570631	−0.285316	+	0.958434i	$0.592098\pi$
$180$	0	0
$181$	−7.18140	−0.533790	−0.266895	−	0.963726i	$-0.585998\pi$
$181$	−7.18140	−0.533790	−0.266895	+	0.963726i	$0.585998\pi$
$182$	−12.2261	−0.906262
$183$	−8.45202	−0.624792
$184$	−15.6156	−1.15120
$185$	0	0
$186$	−11.8522	−0.869047
$187$	0.685135	0.0501020
$188$	−6.84730	−0.499391
$189$	1.44112	0.104826
$190$	0	0
$191$	5.24617	0.379600	0.189800	−	0.981823i	$-0.439216\pi$
$191$	5.24617	0.379600	0.189800	+	0.981823i	$0.439216\pi$
$192$	−3.42809	−0.247401
$193$	22.9175	1.64964	0.824818	−	0.565399i	$-0.191278\pi$
$193$	22.9175	1.64964	0.824818	+	0.565399i	$0.191278\pi$
$194$	−16.2499	−1.16668
$195$	0	0
$196$	−3.53961	−0.252830
$197$	−3.56991	−0.254346	−0.127173	−	0.991881i	$-0.540590\pi$
$197$	−3.56991	−0.254346	−0.127173	+	0.991881i	$0.540590\pi$
$198$	−0.315236	−0.0224029
$199$	0.652738	0.0462714	0.0231357	−	0.999732i	$-0.492635\pi$
$199$	0.652738	0.0462714	0.0231357	+	0.999732i	$0.492635\pi$
$200$	0	0
$201$	9.96623	0.702964
$202$	8.79418	0.618756
$203$	−1.44112	−0.101147
$204$	−2.57664	−0.180401
$205$	0	0
$206$	−21.6795	−1.51048
$207$	−7.39261	−0.513822
$208$	25.3187	1.75554
$209$	1.50031	0.103779
$210$	0	0
$211$	8.86583	0.610349	0.305175	−	0.952296i	$-0.401285\pi$
$211$	8.86583	0.610349	0.305175	+	0.952296i	$0.401285\pi$
$212$	−7.84061	−0.538496
$213$	4.72562	0.323794
$214$	−25.1956	−1.72234
$215$	0	0
$216$	−2.11233	−0.143726
$217$	10.3585	0.703183
$218$	−16.2921	−1.10344
$219$	−9.06473	−0.612538
$220$	0	0
$221$	−18.4386	−1.24032
$222$	14.3584	0.963675
$223$	−8.04069	−0.538444	−0.269222	−	0.963078i	$-0.586767\pi$
$223$	−8.04069	−0.538444	−0.269222	+	0.963078i	$0.586767\pi$
$224$	−5.60563	−0.374542
$225$	0	0
$226$	10.3991	0.691736
$227$	24.5188	1.62737	0.813684	−	0.581307i	$-0.197458\pi$
$227$	24.5188	1.62737	0.813684	+	0.581307i	$0.197458\pi$
$228$	−5.64234	−0.373673
$229$	24.6405	1.62829	0.814145	−	0.580661i	$-0.197206\pi$
$229$	24.6405	1.62829	0.814145	+	0.580661i	$0.197206\pi$
$230$	0	0
$231$	0.275508	0.0181271
$232$	2.11233	0.138681
$233$	4.19800	0.275020	0.137510	−	0.990500i	$-0.456090\pi$
$233$	4.19800	0.275020	0.137510	+	0.990500i	$0.456090\pi$
$234$	8.48377	0.554601
$235$	0	0
$236$	10.2130	0.664812
$237$	8.78711	0.570784
$238$	8.51619	0.552023
$239$	9.04541	0.585099	0.292549	−	0.956250i	$-0.405496\pi$
$239$	9.04541	0.585099	0.292549	+	0.956250i	$0.405496\pi$
$240$	0	0
$241$	4.83690	0.311572	0.155786	−	0.987791i	$-0.450209\pi$
$241$	4.83690	0.311572	0.155786	+	0.987791i	$0.450209\pi$
$242$	18.0780	1.16210
$243$	−1.00000	−0.0641500
$244$	6.07676	0.389025
$245$	0	0
$246$	3.85361	0.245697
$247$	−40.3770	−2.56913
$248$	−15.1830	−0.964124
$249$	15.8701	1.00573
$250$	0	0
$251$	−3.88015	−0.244913	−0.122456	−	0.992474i	$-0.539077\pi$
$251$	−3.88015	−0.244913	−0.122456	+	0.992474i	$0.539077\pi$
$252$	−1.03612	−0.0652697
$253$	−1.41329	−0.0888529
$254$	−32.0970	−2.01394
$255$	0	0
$256$	15.2926	0.955788
$257$	−16.6439	−1.03822	−0.519109	−	0.854708i	$-0.673736\pi$
$257$	−16.6439	−1.03822	−0.519109	+	0.854708i	$0.673736\pi$
$258$	6.06000	0.377279
$259$	−12.5489	−0.779751
$260$	0	0
$261$	1.00000	0.0618984
$262$	−19.6125	−1.21166
$263$	10.2587	0.632580	0.316290	−	0.948663i	$-0.397563\pi$
$263$	10.2587	0.632580	0.316290	+	0.948663i	$0.397563\pi$
$264$	−0.403827	−0.0248538
$265$	0	0
$266$	18.6488	1.14343
$267$	11.6085	0.710430
$268$	−7.16543	−0.437698
$269$	3.35149	0.204344	0.102172	−	0.994767i	$-0.467421\pi$
$269$	3.35149	0.204344	0.102172	+	0.994767i	$0.467421\pi$
$270$	0	0
$271$	12.5730	0.763754	0.381877	−	0.924213i	$-0.375278\pi$
$271$	12.5730	0.763754	0.381877	+	0.924213i	$0.375278\pi$
$272$	−17.6359	−1.06933
$273$	−7.41459	−0.448751
$274$	−26.8495	−1.62204
$275$	0	0
$276$	5.31507	0.319930
$277$	−6.65219	−0.399692	−0.199846	−	0.979827i	$-0.564044\pi$
$277$	−6.65219	−0.399692	−0.199846	+	0.979827i	$0.564044\pi$
$278$	−14.4907	−0.869096
$279$	−7.18783	−0.430324
$280$	0	0
$281$	29.6058	1.76613	0.883067	−	0.469246i	$-0.155474\pi$
$281$	29.6058	1.76613	0.883067	+	0.469246i	$0.155474\pi$
$282$	−15.7040	−0.935160
$283$	18.4403	1.09616	0.548082	−	0.836425i	$-0.315358\pi$
$283$	18.4403	1.09616	0.548082	+	0.836425i	$0.315358\pi$
$284$	−3.39759	−0.201610
$285$	0	0
$286$	1.62189	0.0959046
$287$	−3.36795	−0.198804
$288$	3.88977	0.229207
$289$	−4.15645	−0.244497
$290$	0	0
$291$	−9.85482	−0.577700
$292$	6.51728	0.381395
$293$	14.8320	0.866495	0.433248	−	0.901275i	$-0.357368\pi$
$293$	14.8320	0.866495	0.433248	+	0.901275i	$0.357368\pi$
$294$	−8.11796	−0.473449
$295$	0	0
$296$	18.3936	1.06911
$297$	−0.191176	−0.0110932
$298$	−5.10752	−0.295871
$299$	38.0351	2.19963
$300$	0	0
$301$	−5.29628	−0.305273
$302$	12.1867	0.701265
$303$	5.33326	0.306388
$304$	−38.6192	−2.21496
$305$	0	0
$306$	−5.90942	−0.337819
$307$	−17.8595	−1.01929	−0.509647	−	0.860383i	$-0.670224\pi$
$307$	−17.8595	−1.01929	−0.509647	+	0.860383i	$0.670224\pi$
$308$	−0.198082	−0.0112868
$309$	−13.1476	−0.747941
$310$	0	0
$311$	−22.8466	−1.29551	−0.647756	−	0.761848i	$-0.724292\pi$
$311$	−22.8466	−1.29551	−0.647756	+	0.761848i	$0.724292\pi$
$312$	10.8680	0.615277
$313$	27.4021	1.54886	0.774428	−	0.632662i	$-0.218038\pi$
$313$	27.4021	1.54886	0.774428	+	0.632662i	$0.218038\pi$
$314$	−15.8841	−0.896389
$315$	0	0
$316$	−6.31768	−0.355397
$317$	11.5466	0.648519	0.324260	−	0.945968i	$-0.394885\pi$
$317$	11.5466	0.648519	0.324260	+	0.945968i	$0.394885\pi$
$318$	−17.9821	−1.00839
$319$	0.191176	0.0107038
$320$	0	0
$321$	−15.2800	−0.852846
$322$	−17.5671	−0.978978
$323$	28.1249	1.56491
$324$	0.718971	0.0399428
$325$	0	0
$326$	−32.4521	−1.79736
$327$	−9.88042	−0.546388
$328$	4.93658	0.272577
$329$	13.7249	0.756678
$330$	0	0
$331$	−26.0804	−1.43351	−0.716753	−	0.697327i	$-0.754373\pi$
$331$	−26.0804	−1.43351	−0.716753	+	0.697327i	$0.754373\pi$
$332$	−11.4102	−0.626213
$333$	8.70773	0.477181
$334$	−6.09032	−0.333248
$335$	0	0
$336$	−7.09179	−0.386889
$337$	7.08114	0.385734	0.192867	−	0.981225i	$-0.438221\pi$
$337$	7.08114	0.385734	0.192867	+	0.981225i	$0.438221\pi$
$338$	−22.2130	−1.20823
$339$	6.30656	0.342525
$340$	0	0
$341$	−1.37414	−0.0744139
$342$	−12.9405	−0.699741
$343$	17.1827	0.927781
$344$	7.76304	0.418555
$345$	0	0
$346$	−13.3645	−0.718480
$347$	11.1830	0.600333	0.300167	−	0.953887i	$-0.402958\pi$
$347$	11.1830	0.600333	0.300167	+	0.953887i	$0.402958\pi$
$348$	−0.718971	−0.0385409
$349$	12.9183	0.691499	0.345749	−	0.938327i	$-0.387625\pi$
$349$	12.9183	0.691499	0.345749	+	0.938327i	$0.387625\pi$
$350$	0	0
$351$	5.14501	0.274620
$352$	0.743631	0.0396357
$353$	15.7409	0.837805	0.418903	−	0.908031i	$-0.362415\pi$
$353$	15.7409	0.837805	0.418903	+	0.908031i	$0.362415\pi$
$354$	23.4232	1.24493
$355$	0	0
$356$	−8.34618	−0.442347
$357$	5.16468	0.273344
$358$	12.5888	0.665339
$359$	16.7472	0.883881	0.441941	−	0.897044i	$-0.354290\pi$
$359$	16.7472	0.883881	0.441941	+	0.897044i	$0.354290\pi$
$360$	0	0
$361$	42.5880	2.24147
$362$	11.8416	0.622382
$363$	10.9635	0.575432
$364$	5.33087	0.279414
$365$	0	0
$366$	13.9368	0.728488
$367$	12.2086	0.637282	0.318641	−	0.947875i	$-0.396774\pi$
$367$	12.2086	0.637282	0.318641	+	0.947875i	$0.396774\pi$
$368$	36.3792	1.89640
$369$	2.33703	0.121661
$370$	0	0
$371$	15.7159	0.815929
$372$	5.16784	0.267940
$373$	−11.7029	−0.605953	−0.302976	−	0.952998i	$-0.597980\pi$
$373$	−11.7029	−0.605953	−0.302976	+	0.952998i	$0.597980\pi$
$374$	−1.12974	−0.0584175
$375$	0	0
$376$	−20.1173	−1.03747
$377$	−5.14501	−0.264982
$378$	−2.37631	−0.122224
$379$	−30.8446	−1.58438	−0.792191	−	0.610273i	$-0.791060\pi$
$379$	−30.8446	−1.58438	−0.792191	+	0.610273i	$0.791060\pi$
$380$	0	0
$381$	−19.4653	−0.997239
$382$	−8.65057	−0.442602
$383$	−10.4576	−0.534356	−0.267178	−	0.963647i	$-0.586091\pi$
$383$	−10.4576	−0.534356	−0.267178	+	0.963647i	$0.586091\pi$
$384$	13.4322	0.685460
$385$	0	0
$386$	−37.7893	−1.92342
$387$	3.67511	0.186816
$388$	7.08533	0.359703
$389$	−34.4133	−1.74483	−0.872413	−	0.488770i	$-0.837446\pi$
$389$	−34.4133	−1.74483	−0.872413	+	0.488770i	$0.837446\pi$
$390$	0	0
$391$	−26.4936	−1.33984
$392$	−10.3993	−0.525246
$393$	−11.8940	−0.599975
$394$	5.88653	0.296559
$395$	0	0
$396$	0.137450	0.00690713
$397$	−15.7312	−0.789525	−0.394762	−	0.918783i	$-0.629173\pi$
$397$	−15.7312	−0.789525	−0.394762	+	0.918783i	$0.629173\pi$
$398$	−1.07632	−0.0539510
$399$	11.3096	0.566190
$400$	0	0
$401$	−1.02583	−0.0512274	−0.0256137	−	0.999672i	$-0.508154\pi$
$401$	−1.02583	−0.0512274	−0.0256137	+	0.999672i	$0.508154\pi$
$402$	−16.4336	−0.819635
$403$	36.9815	1.84218
$404$	−3.83446	−0.190771
$405$	0	0
$406$	2.37631	0.117934
$407$	1.66471	0.0825166
$408$	−7.57014	−0.374778
$409$	12.6252	0.624274	0.312137	−	0.950037i	$-0.398955\pi$
$409$	12.6252	0.624274	0.312137	+	0.950037i	$0.398955\pi$
$410$	0	0
$411$	−16.2830	−0.803180
$412$	9.45275	0.465703
$413$	−20.4712	−1.00732
$414$	12.1899	0.599101
$415$	0	0
$416$	−20.0129	−0.981213
$417$	−8.78796	−0.430348
$418$	−2.47391	−0.121003
$419$	4.86499	0.237670	0.118835	−	0.992914i	$-0.462084\pi$
$419$	4.86499	0.237670	0.118835	+	0.992914i	$0.462084\pi$
$420$	0	0
$421$	31.7766	1.54870	0.774349	−	0.632759i	$-0.218078\pi$
$421$	31.7766	1.54870	0.774349	+	0.632759i	$0.218078\pi$
$422$	−14.6191	−0.711649
$423$	−9.52376	−0.463061
$424$	−23.0356	−1.11871
$425$	0	0
$426$	−7.79222	−0.377534
$427$	−12.1804	−0.589451
$428$	10.9859	0.531022
$429$	0.983604	0.0474888
$430$	0	0
$431$	−4.92223	−0.237095	−0.118548	−	0.992948i	$-0.537824\pi$
$431$	−4.92223	−0.237095	−0.118548	+	0.992948i	$0.537824\pi$
$432$	4.92102	0.236763
$433$	−17.0581	−0.819762	−0.409881	−	0.912139i	$-0.634430\pi$
$433$	−17.0581	−0.819762	−0.409881	+	0.912139i	$0.634430\pi$
$434$	−17.0805	−0.819890
$435$	0	0
$436$	7.10373	0.340207
$437$	−58.0158	−2.77527
$438$	14.9471	0.714201
$439$	−0.213932	−0.0102104	−0.00510520	−	0.999987i	$-0.501625\pi$
$439$	−0.213932	−0.0102104	−0.00510520	+	0.999987i	$0.501625\pi$
$440$	0	0
$441$	−4.92317	−0.234437
$442$	30.4040	1.44617
$443$	−24.1909	−1.14935	−0.574673	−	0.818383i	$-0.694871\pi$
$443$	−24.1909	−1.14935	−0.574673	+	0.818383i	$0.694871\pi$
$444$	−6.26060	−0.297115
$445$	0	0
$446$	13.2585	0.627810
$447$	−3.09748	−0.146506
$448$	−4.94029	−0.233407
$449$	6.64235	0.313472	0.156736	−	0.987641i	$-0.449903\pi$
$449$	6.64235	0.313472	0.156736	+	0.987641i	$0.449903\pi$
$450$	0	0
$451$	0.446785	0.0210383
$452$	−4.53423	−0.213272
$453$	7.39067	0.347244
$454$	−40.4298	−1.89746
$455$	0	0
$456$	−16.5771	−0.776295
$457$	25.7984	1.20680	0.603400	−	0.797438i	$-0.293812\pi$
$457$	25.7984	1.20680	0.603400	+	0.797438i	$0.293812\pi$
$458$	−40.6305	−1.89854
$459$	−3.58379	−0.167277
$460$	0	0
$461$	−9.89153	−0.460695	−0.230347	−	0.973108i	$-0.573986\pi$
$461$	−9.89153	−0.460695	−0.230347	+	0.973108i	$0.573986\pi$
$462$	−0.454294	−0.0211356
$463$	−17.9312	−0.833335	−0.416667	−	0.909059i	$-0.636802\pi$
$463$	−17.9312	−0.833335	−0.416667	+	0.909059i	$0.636802\pi$
$464$	−4.92102	−0.228453
$465$	0	0
$466$	−6.92220	−0.320665
$467$	−29.7640	−1.37731	−0.688657	−	0.725087i	$-0.741799\pi$
$467$	−29.7640	−1.37731	−0.688657	+	0.725087i	$0.741799\pi$
$468$	−3.69911	−0.170992
$469$	14.3626	0.663201
$470$	0	0
$471$	−9.63295	−0.443863
$472$	30.0058	1.38113
$473$	0.702594	0.0323053
$474$	−14.4893	−0.665517
$475$	0	0
$476$	−3.71325	−0.170197
$477$	−10.9053	−0.499321
$478$	−14.9153	−0.682208
$479$	−24.3451	−1.11236	−0.556178	−	0.831063i	$-0.687733\pi$
$479$	−24.3451	−1.11236	−0.556178	+	0.831063i	$0.687733\pi$
$480$	0	0
$481$	−44.8014	−2.04277
$482$	−7.97571	−0.363284
$483$	−10.6537	−0.484758
$484$	−7.88240	−0.358291
$485$	0	0
$486$	1.64893	0.0747970
$487$	−1.76708	−0.0800740	−0.0400370	−	0.999198i	$-0.512748\pi$
$487$	−1.76708	−0.0800740	−0.0400370	+	0.999198i	$0.512748\pi$
$488$	17.8534	0.808188
$489$	−19.6807	−0.889992
$490$	0	0
$491$	−2.47556	−0.111720	−0.0558602	−	0.998439i	$-0.517790\pi$
$491$	−2.47556	−0.111720	−0.0558602	+	0.998439i	$0.517790\pi$
$492$	−1.68026	−0.0757519
$493$	3.58379	0.161406
$494$	66.5789	2.99553
$495$	0	0
$496$	35.3715	1.58823
$497$	6.81020	0.305479
$498$	−26.1687	−1.17265
$499$	−6.82406	−0.305487	−0.152743	−	0.988266i	$-0.548811\pi$
$499$	−6.82406	−0.305487	−0.152743	+	0.988266i	$0.548811\pi$
$500$	0	0
$501$	−3.69350	−0.165013
$502$	6.39810	0.285561
$503$	−22.1554	−0.987861	−0.493930	−	0.869501i	$-0.664440\pi$
$503$	−22.1554	−0.987861	−0.493930	+	0.869501i	$0.664440\pi$
$504$	−3.04412	−0.135596
$505$	0	0
$506$	2.33042	0.103600
$507$	−13.4712	−0.598275
$508$	13.9950	0.620928
$509$	17.4284	0.772499	0.386250	−	0.922394i	$-0.373770\pi$
$509$	17.4284	0.772499	0.386250	+	0.922394i	$0.373770\pi$
$510$	0	0
$511$	−13.0634	−0.577890
$512$	1.64799	0.0728316
$513$	−7.84780	−0.346489
$514$	27.4446	1.21053
$515$	0	0
$516$	−2.64230	−0.116321
$517$	−1.82072	−0.0800750
$518$	20.6923	0.909166
$519$	−8.10495	−0.355768
$520$	0	0
$521$	−8.50733	−0.372713	−0.186356	−	0.982482i	$-0.559668\pi$
$521$	−8.50733	−0.372713	−0.186356	+	0.982482i	$0.559668\pi$
$522$	−1.64893	−0.0721717
$523$	−29.8852	−1.30679	−0.653395	−	0.757017i	$-0.726656\pi$
$523$	−29.8852	−1.30679	−0.653395	+	0.757017i	$0.726656\pi$
$524$	8.55147	0.373573
$525$	0	0
$526$	−16.9159	−0.737569
$527$	−25.7597	−1.12211
$528$	0.940782	0.0409423
$529$	31.6507	1.37612
$530$	0	0
$531$	14.2051	0.616448
$532$	−8.13130	−0.352537
$533$	−12.0241	−0.520820
$534$	−19.1416	−0.828339
$535$	0	0
$536$	−21.0520	−0.909306
$537$	7.63453	0.329454
$538$	−5.52638	−0.238259
$539$	−0.941192	−0.0405400
$540$	0	0
$541$	−18.8846	−0.811913	−0.405957	−	0.913892i	$-0.633062\pi$
$541$	−18.8846	−0.811913	−0.405957	+	0.913892i	$0.633062\pi$
$542$	−20.7320	−0.890514
$543$	7.18140	0.308184
$544$	13.9401	0.597677
$545$	0	0
$546$	12.2261	0.523230
$547$	−13.1168	−0.560834	−0.280417	−	0.959878i	$-0.590473\pi$
$547$	−13.1168	−0.560834	−0.280417	+	0.959878i	$0.590473\pi$
$548$	11.7070	0.500098
$549$	8.45202	0.360724
$550$	0	0
$551$	7.84780	0.334328
$552$	15.6156	0.664645
$553$	12.6633	0.538498
$554$	10.9690	0.466028
$555$	0	0
$556$	6.31828	0.267955
$557$	16.7574	0.710034	0.355017	−	0.934860i	$-0.384475\pi$
$557$	16.7574	0.710034	0.355017	+	0.934860i	$0.384475\pi$
$558$	11.8522	0.501745
$559$	−18.9085	−0.799744
$560$	0	0
$561$	−0.685135	−0.0289264
$562$	−48.8179	−2.05926
$563$	−2.10791	−0.0888379	−0.0444190	−	0.999013i	$-0.514144\pi$
$563$	−2.10791	−0.0888379	−0.0444190	+	0.999013i	$0.514144\pi$
$564$	6.84730	0.288324
$565$	0	0
$566$	−30.4068	−1.27809
$567$	−1.44112	−0.0605214
$568$	−9.98206	−0.418838
$569$	32.9090	1.37962	0.689809	−	0.723991i	$-0.257694\pi$
$569$	32.9090	1.37962	0.689809	+	0.723991i	$0.257694\pi$
$570$	0	0
$571$	−21.5951	−0.903725	−0.451863	−	0.892088i	$-0.649240\pi$
$571$	−21.5951	−0.903725	−0.451863	+	0.892088i	$0.649240\pi$
$572$	−0.707182	−0.0295688
$573$	−5.24617	−0.219162
$574$	5.55351	0.231799
$575$	0	0
$576$	3.42809	0.142837
$577$	35.1630	1.46385	0.731927	−	0.681384i	$-0.238621\pi$
$577$	35.1630	1.46385	0.731927	+	0.681384i	$0.238621\pi$
$578$	6.85370	0.285076
$579$	−22.9175	−0.952417
$580$	0	0
$581$	22.8708	0.948839
$582$	16.2499	0.673580
$583$	−2.08484	−0.0863452
$584$	19.1477	0.792337
$585$	0	0
$586$	−24.4570	−1.01031
$587$	−43.1149	−1.77954	−0.889771	−	0.456407i	$-0.849136\pi$
$587$	−43.1149	−1.77954	−0.889771	+	0.456407i	$0.849136\pi$
$588$	3.53961	0.145971
$589$	−56.4086	−2.32428
$590$	0	0
$591$	3.56991	0.146847
$592$	−42.8509	−1.76116
$593$	−4.03925	−0.165872	−0.0829361	−	0.996555i	$-0.526430\pi$
$593$	−4.03925	−0.165872	−0.0829361	+	0.996555i	$0.526430\pi$
$594$	0.315236	0.0129343
$595$	0	0
$596$	2.22700	0.0912213
$597$	−0.652738	−0.0267148
$598$	−62.7172	−2.56470
$599$	−21.7899	−0.890312	−0.445156	−	0.895453i	$-0.646852\pi$
$599$	−21.7899	−0.890312	−0.445156	+	0.895453i	$0.646852\pi$
$600$	0	0
$601$	22.4489	0.915710	0.457855	−	0.889027i	$-0.348618\pi$
$601$	22.4489	0.915710	0.457855	+	0.889027i	$0.348618\pi$
$602$	8.73320	0.355939
$603$	−9.96623	−0.405856
$604$	−5.31367	−0.216210
$605$	0	0
$606$	−8.79418	−0.357239
$607$	−4.28567	−0.173950	−0.0869750	−	0.996210i	$-0.527720\pi$
$607$	−4.28567	−0.173950	−0.0869750	+	0.996210i	$0.527720\pi$
$608$	30.5261	1.23800
$609$	1.44112	0.0583972
$610$	0	0
$611$	48.9999	1.98232
$612$	2.57664	0.104155
$613$	−2.41278	−0.0974514	−0.0487257	−	0.998812i	$-0.515516\pi$
$613$	−2.41278	−0.0974514	−0.0487257	+	0.998812i	$0.515516\pi$
$614$	29.4490	1.18847
$615$	0	0
$616$	−0.581963	−0.0234480
$617$	25.3384	1.02008	0.510042	−	0.860150i	$-0.329630\pi$
$617$	25.3384	1.02008	0.510042	+	0.860150i	$0.329630\pi$
$618$	21.6795	0.872077
$619$	48.4512	1.94742	0.973708	−	0.227798i	$-0.0731527\pi$
$619$	48.4512	1.94742	0.973708	+	0.227798i	$0.0731527\pi$
$620$	0	0
$621$	7.39261	0.296655
$622$	37.6724	1.51053
$623$	16.7293	0.670245
$624$	−25.3187	−1.01356
$625$	0	0
$626$	−45.1841	−1.80592
$627$	−1.50031	−0.0599167
$628$	6.92581	0.276370
$629$	31.2067	1.24429
$630$	0	0
$631$	18.0190	0.717325	0.358662	−	0.933467i	$-0.383233\pi$
$631$	18.0190	0.717325	0.358662	+	0.933467i	$0.383233\pi$
$632$	−18.5613	−0.738327
$633$	−8.86583	−0.352385
$634$	−19.0395	−0.756154
$635$	0	0
$636$	7.84061	0.310901
$637$	25.3298	1.00360
$638$	−0.315236	−0.0124803
$639$	−4.72562	−0.186943
$640$	0	0
$641$	50.3755	1.98971	0.994855	−	0.101306i	$-0.0323020\pi$
$641$	50.3755	1.98971	0.994855	+	0.101306i	$0.0323020\pi$
$642$	25.1956	0.994392
$643$	12.6433	0.498602	0.249301	−	0.968426i	$-0.419799\pi$
$643$	12.6433	0.498602	0.249301	+	0.968426i	$0.419799\pi$
$644$	7.65967	0.301833
$645$	0	0
$646$	−46.3760	−1.82464
$647$	9.38392	0.368920	0.184460	−	0.982840i	$-0.440946\pi$
$647$	9.38392	0.368920	0.184460	+	0.982840i	$0.440946\pi$
$648$	2.11233	0.0829801
$649$	2.71567	0.106599
$650$	0	0
$651$	−10.3585	−0.405983
$652$	14.1499	0.554151
$653$	−0.265477	−0.0103889	−0.00519446	−	0.999987i	$-0.501653\pi$
$653$	−0.265477	−0.0103889	−0.00519446	+	0.999987i	$0.501653\pi$
$654$	16.2921	0.637072
$655$	0	0
$656$	−11.5006	−0.449023
$657$	9.06473	0.353649
$658$	−22.6314	−0.882263
$659$	−43.9795	−1.71320	−0.856598	−	0.515984i	$-0.827426\pi$
$659$	−43.9795	−1.71320	−0.856598	+	0.515984i	$0.827426\pi$
$660$	0	0
$661$	−7.73684	−0.300928	−0.150464	−	0.988615i	$-0.548077\pi$
$661$	−7.73684	−0.300928	−0.150464	+	0.988615i	$0.548077\pi$
$662$	43.0047	1.67142
$663$	18.4386	0.716098
$664$	−33.5229	−1.30094
$665$	0	0
$666$	−14.3584	−0.556378
$667$	−7.39261	−0.286243
$668$	2.65552	0.102745
$669$	8.04069	0.310871
$670$	0	0
$671$	1.61583	0.0623782
$672$	5.60563	0.216242
$673$	19.1293	0.737380	0.368690	−	0.929552i	$-0.379806\pi$
$673$	19.1293	0.737380	0.368690	+	0.929552i	$0.379806\pi$
$674$	−11.6763	−0.449755
$675$	0	0
$676$	9.68537	0.372514
$677$	29.2009	1.12228	0.561141	−	0.827720i	$-0.310362\pi$
$677$	29.2009	1.12228	0.561141	+	0.827720i	$0.310362\pi$
$678$	−10.3991	−0.399374
$679$	−14.2020	−0.545022
$680$	0	0
$681$	−24.5188	−0.939562
$682$	2.26586	0.0867644
$683$	−31.7459	−1.21472	−0.607362	−	0.794425i	$-0.707772\pi$
$683$	−31.7459	−1.21472	−0.607362	+	0.794425i	$0.707772\pi$
$684$	5.64234	0.215740
$685$	0	0
$686$	−28.3331	−1.08176
$687$	−24.6405	−0.940094
$688$	−18.0853	−0.689496
$689$	56.1081	2.13755
$690$	0	0
$691$	−37.7635	−1.43659	−0.718296	−	0.695737i	$-0.755078\pi$
$691$	−37.7635	−1.43659	−0.718296	+	0.695737i	$0.755078\pi$
$692$	5.82722	0.221518
$693$	−0.275508	−0.0104657
$694$	−18.4399	−0.699971
$695$	0	0
$696$	−2.11233	−0.0800676
$697$	8.37544	0.317242
$698$	−21.3013	−0.806267
$699$	−4.19800	−0.158783
$700$	0	0
$701$	−7.91040	−0.298772	−0.149386	−	0.988779i	$-0.547730\pi$
$701$	−7.91040	−0.298772	−0.149386	+	0.988779i	$0.547730\pi$
$702$	−8.48377	−0.320199
$703$	68.3365	2.57736
$704$	0.655369	0.0247002
$705$	0	0
$706$	−25.9557	−0.976856
$707$	7.68588	0.289057
$708$	−10.2130	−0.383829
$709$	23.5795	0.885548	0.442774	−	0.896633i	$-0.353994\pi$
$709$	23.5795	0.885548	0.442774	+	0.896633i	$0.353994\pi$
$710$	0	0
$711$	−8.78711	−0.329542
$712$	−24.5210	−0.918963
$713$	53.1368	1.98999
$714$	−8.51619	−0.318710
$715$	0	0
$716$	−5.48900	−0.205134
$717$	−9.04541	−0.337807
$718$	−27.6149	−1.03058
$719$	−16.8365	−0.627895	−0.313948	−	0.949440i	$-0.601652\pi$
$719$	−16.8365	−0.627895	−0.313948	+	0.949440i	$0.601652\pi$
$720$	0	0
$721$	−18.9473	−0.705634
$722$	−70.2246	−2.61349
$723$	−4.83690	−0.179886
$724$	−5.16322	−0.191890
$725$	0	0
$726$	−18.0780	−0.670936
$727$	−19.7661	−0.733086	−0.366543	−	0.930401i	$-0.619459\pi$
$727$	−19.7661	−0.733086	−0.366543	+	0.930401i	$0.619459\pi$
$728$	15.6620	0.580474
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.1708	0.487141
$732$	−6.07676	−0.224603
$733$	10.5624	0.390130	0.195065	−	0.980790i	$-0.437508\pi$
$733$	10.5624	0.390130	0.195065	+	0.980790i	$0.437508\pi$
$734$	−20.1311	−0.743052
$735$	0	0
$736$	−28.7555	−1.05994
$737$	−1.90531	−0.0701829
$738$	−3.85361	−0.141853
$739$	27.8049	1.02282	0.511410	−	0.859337i	$-0.329123\pi$
$739$	27.8049	1.02282	0.511410	+	0.859337i	$0.329123\pi$
$740$	0	0
$741$	40.3770	1.48329
$742$	−25.9144	−0.951349
$743$	1.43047	0.0524789	0.0262395	−	0.999656i	$-0.491647\pi$
$743$	1.43047	0.0524789	0.0262395	+	0.999656i	$0.491647\pi$
$744$	15.1830	0.556637
$745$	0	0
$746$	19.2972	0.706522
$747$	−15.8701	−0.580657
$748$	0.492592	0.0180110
$749$	−22.0203	−0.804605
$750$	0	0
$751$	9.87670	0.360406	0.180203	−	0.983629i	$-0.442325\pi$
$751$	9.87670	0.360406	0.180203	+	0.983629i	$0.442325\pi$
$752$	46.8666	1.70905
$753$	3.88015	0.141401
$754$	8.48377	0.308961
$755$	0	0
$756$	1.03612	0.0376835
$757$	−34.4041	−1.25044	−0.625219	−	0.780449i	$-0.714990\pi$
$757$	−34.4041	−1.25044	−0.625219	+	0.780449i	$0.714990\pi$
$758$	50.8606	1.84734
$759$	1.41329	0.0512992
$760$	0	0
$761$	−8.06456	−0.292340	−0.146170	−	0.989259i	$-0.546695\pi$
$761$	−8.06456	−0.292340	−0.146170	+	0.989259i	$0.546695\pi$
$762$	32.0970	1.16275
$763$	−14.2389	−0.515482
$764$	3.77185	0.136461
$765$	0	0
$766$	17.2438	0.623043
$767$	−73.0853	−2.63896
$768$	−15.2926	−0.551824
$769$	−3.51646	−0.126807	−0.0634034	−	0.997988i	$-0.520195\pi$
$769$	−3.51646	−0.126807	−0.0634034	+	0.997988i	$0.520195\pi$
$770$	0	0
$771$	16.6439	0.599415
$772$	16.4770	0.593020
$773$	15.6759	0.563822	0.281911	−	0.959441i	$-0.409032\pi$
$773$	15.6759	0.563822	0.281911	+	0.959441i	$0.409032\pi$
$774$	−6.06000	−0.217822
$775$	0	0
$776$	20.8166	0.747273
$777$	12.5489	0.450189
$778$	56.7452	2.03441
$779$	18.3406	0.657119
$780$	0	0
$781$	−0.903426	−0.0323271
$782$	43.6861	1.56221
$783$	−1.00000	−0.0357371
$784$	24.2270	0.865251
$785$	0	0
$786$	19.6125	0.699553
$787$	18.3318	0.653458	0.326729	−	0.945118i	$-0.394054\pi$
$787$	18.3318	0.653458	0.326729	+	0.945118i	$0.394054\pi$
$788$	−2.56666	−0.0914336
$789$	−10.2587	−0.365220
$790$	0	0
$791$	9.08852	0.323150
$792$	0.403827	0.0143494
$793$	−43.4858	−1.54423
$794$	25.9396	0.920562
$795$	0	0
$796$	0.469300	0.0166339
$797$	43.7461	1.54957	0.774784	−	0.632226i	$-0.217859\pi$
$797$	43.7461	1.54957	0.774784	+	0.632226i	$0.217859\pi$
$798$	−18.6488	−0.660160
$799$	−34.1311	−1.20747
$800$	0	0
$801$	−11.6085	−0.410167
$802$	1.69152	0.0597296
$803$	1.73296	0.0611549
$804$	7.16543	0.252705
$805$	0	0
$806$	−60.9798	−2.14792
$807$	−3.35149	−0.117978
$808$	−11.2656	−0.396322
$809$	22.7075	0.798354	0.399177	−	0.916874i	$-0.369296\pi$
$809$	22.7075	0.798354	0.399177	+	0.916874i	$0.369296\pi$
$810$	0	0
$811$	39.3558	1.38197	0.690985	−	0.722869i	$-0.257177\pi$
$811$	39.3558	1.38197	0.690985	+	0.722869i	$0.257177\pi$
$812$	−1.03612	−0.0363608
$813$	−12.5730	−0.440953
$814$	−2.74499	−0.0962119
$815$	0	0
$816$	17.6359	0.617380
$817$	28.8415	1.00904
$818$	−20.8180	−0.727885
$819$	7.41459	0.259087
$820$	0	0
$821$	44.0845	1.53856	0.769280	−	0.638912i	$-0.220615\pi$
$821$	44.0845	1.53856	0.769280	+	0.638912i	$0.220615\pi$
$822$	26.8495	0.936484
$823$	50.4379	1.75815	0.879077	−	0.476681i	$-0.158160\pi$
$823$	50.4379	1.75815	0.879077	+	0.476681i	$0.158160\pi$
$824$	27.7721	0.967486
$825$	0	0
$826$	33.7556	1.17451
$827$	−39.9104	−1.38782	−0.693910	−	0.720062i	$-0.744113\pi$
$827$	−39.9104	−1.38782	−0.693910	+	0.720062i	$0.744113\pi$
$828$	−5.31507	−0.184712
$829$	17.1048	0.594075	0.297038	−	0.954866i	$-0.404001\pi$
$829$	17.1048	0.594075	0.297038	+	0.954866i	$0.404001\pi$
$830$	0	0
$831$	6.65219	0.230762
$832$	−17.6376	−0.611473
$833$	−17.6436	−0.611315
$834$	14.4907	0.501773
$835$	0	0
$836$	1.07868	0.0373070
$837$	7.18783	0.248448
$838$	−8.02202	−0.277116
$839$	1.88147	0.0649557	0.0324778	−	0.999472i	$-0.489660\pi$
$839$	1.88147	0.0649557	0.0324778	+	0.999472i	$0.489660\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−52.3974	−1.80573
$843$	−29.6058	−1.01968
$844$	6.37428	0.219412
$845$	0	0
$846$	15.7040	0.539915
$847$	15.7997	0.542883
$848$	53.6654	1.84288
$849$	−18.4403	−0.632871
$850$	0	0
$851$	−64.3729	−2.20667
$852$	3.39759	0.116399
$853$	−42.5766	−1.45780	−0.728898	−	0.684622i	$-0.759967\pi$
$853$	−42.5766	−1.45780	−0.728898	+	0.684622i	$0.759967\pi$
$854$	20.0846	0.687282
$855$	0	0
$856$	32.2763	1.10318
$857$	−3.22361	−0.110116	−0.0550582	−	0.998483i	$-0.517534\pi$
$857$	−3.22361	−0.110116	−0.0550582	+	0.998483i	$0.517534\pi$
$858$	−1.62189	−0.0553705
$859$	6.60992	0.225527	0.112764	−	0.993622i	$-0.464030\pi$
$859$	6.60992	0.225527	0.112764	+	0.993622i	$0.464030\pi$
$860$	0	0
$861$	3.36795	0.114779
$862$	8.11641	0.276446
$863$	48.2625	1.64287	0.821437	−	0.570299i	$-0.193173\pi$
$863$	48.2625	1.64287	0.821437	+	0.570299i	$0.193173\pi$
$864$	−3.88977	−0.132333
$865$	0	0
$866$	28.1277	0.955818
$867$	4.15645	0.141160
$868$	7.44748	0.252784
$869$	−1.67989	−0.0569862
$870$	0	0
$871$	51.2764	1.73743
$872$	20.8707	0.706770
$873$	9.85482	0.333535
$874$	95.6640	3.23588
$875$	0	0
$876$	−6.51728	−0.220198
$877$	12.7497	0.430525	0.215263	−	0.976556i	$-0.430939\pi$
$877$	12.7497	0.430525	0.215263	+	0.976556i	$0.430939\pi$
$878$	0.352758	0.0119050
$879$	−14.8320	−0.500271
$880$	0	0
$881$	−30.7457	−1.03585	−0.517924	−	0.855427i	$-0.673295\pi$
$881$	−30.7457	−1.03585	−0.517924	+	0.855427i	$0.673295\pi$
$882$	8.11796	0.273346
$883$	−8.68913	−0.292413	−0.146206	−	0.989254i	$-0.546706\pi$
$883$	−8.68913	−0.292413	−0.146206	+	0.989254i	$0.546706\pi$
$884$	−13.2568	−0.445876
$885$	0	0
$886$	39.8892	1.34010
$887$	13.4294	0.450916	0.225458	−	0.974253i	$-0.427612\pi$
$887$	13.4294	0.450916	0.225458	+	0.974253i	$0.427612\pi$
$888$	−18.3936	−0.617248
$889$	−28.0519	−0.940831
$890$	0	0
$891$	0.191176	0.00640464
$892$	−5.78102	−0.193563
$893$	−74.7406	−2.50110
$894$	5.10752	0.170821
$895$	0	0
$896$	19.3575	0.646687
$897$	−38.0351	−1.26995
$898$	−10.9528	−0.365499
$899$	−7.18783	−0.239727
$900$	0	0
$901$	−39.0824	−1.30202
$902$	−0.736718	−0.0245300
$903$	5.29628	0.176249
$904$	−13.3215	−0.443067
$905$	0	0
$906$	−12.1867	−0.404876
$907$	16.1270	0.535488	0.267744	−	0.963490i	$-0.413722\pi$
$907$	16.1270	0.535488	0.267744	+	0.963490i	$0.413722\pi$
$908$	17.6283	0.585015
$909$	−5.33326	−0.176893
$910$	0	0
$911$	16.7310	0.554323	0.277162	−	0.960823i	$-0.410606\pi$
$911$	16.7310	0.554323	0.277162	+	0.960823i	$0.410606\pi$
$912$	38.6192	1.27881
$913$	−3.03399	−0.100410
$914$	−42.5398	−1.40709
$915$	0	0
$916$	17.7158	0.585347
$917$	−17.1408	−0.566038
$918$	5.90942	0.195040
$919$	−21.0522	−0.694448	−0.347224	−	0.937782i	$-0.612876\pi$
$919$	−21.0522	−0.694448	−0.347224	+	0.937782i	$0.612876\pi$
$920$	0	0
$921$	17.8595	0.588490
$922$	16.3105	0.537156
$923$	24.3134	0.800285
$924$	0.198082	0.00651643
$925$	0	0
$926$	29.5673	0.971643
$927$	13.1476	0.431824
$928$	3.88977	0.127688
$929$	−14.9033	−0.488961	−0.244481	−	0.969654i	$-0.578617\pi$
$929$	−14.9033	−0.488961	−0.244481	+	0.969654i	$0.578617\pi$
$930$	0	0
$931$	−38.6361	−1.26625
$932$	3.01824	0.0988656
$933$	22.8466	0.747964
$934$	49.0788	1.60591
$935$	0	0
$936$	−10.8680	−0.355230
$937$	7.56849	0.247252	0.123626	−	0.992329i	$-0.460548\pi$
$937$	7.56849	0.247252	0.123626	+	0.992329i	$0.460548\pi$
$938$	−23.6829	−0.773272
$939$	−27.4021	−0.894232
$940$	0	0
$941$	27.8686	0.908490	0.454245	−	0.890877i	$-0.349909\pi$
$941$	27.8686	0.908490	0.454245	+	0.890877i	$0.349909\pi$
$942$	15.8841	0.517531
$943$	−17.2768	−0.562610
$944$	−69.9035	−2.27516
$945$	0	0
$946$	−1.15853	−0.0376670
$947$	−6.21467	−0.201950	−0.100975	−	0.994889i	$-0.532196\pi$
$947$	−6.21467	−0.201950	−0.100975	+	0.994889i	$0.532196\pi$
$948$	6.31768	0.205189
$949$	−46.6382	−1.51394
$950$	0	0
$951$	−11.5466	−0.374423
$952$	−10.9095	−0.353579
$953$	−41.5221	−1.34503	−0.672517	−	0.740082i	$-0.734787\pi$
$953$	−41.5221	−1.34503	−0.672517	+	0.740082i	$0.734787\pi$
$954$	17.9821	0.582193
$955$	0	0
$956$	6.50339	0.210335
$957$	−0.191176	−0.00617985
$958$	40.1434	1.29697
$959$	−23.4657	−0.757749
$960$	0	0
$961$	20.6649	0.666608
$962$	73.8744	2.38180
$963$	15.2800	0.492391
$964$	3.47759	0.112006
$965$	0	0
$966$	17.5671	0.565213
$967$	27.9230	0.897942	0.448971	−	0.893546i	$-0.351791\pi$
$967$	27.9230	0.897942	0.448971	+	0.893546i	$0.351791\pi$
$968$	−23.1584	−0.744339
$969$	−28.1249	−0.903501
$970$	0	0
$971$	−3.57298	−0.114663	−0.0573313	−	0.998355i	$-0.518259\pi$
$971$	−3.57298	−0.114663	−0.0573313	+	0.998355i	$0.518259\pi$
$972$	−0.718971	−0.0230610
$973$	−12.6645	−0.406006
$974$	2.91379	0.0933639
$975$	0	0
$976$	−41.5926	−1.33135
$977$	−16.8174	−0.538037	−0.269019	−	0.963135i	$-0.586699\pi$
$977$	−16.8174	−0.538037	−0.269019	+	0.963135i	$0.586699\pi$
$978$	32.4521	1.03770
$979$	−2.21927	−0.0709282
$980$	0	0
$981$	9.88042	0.315457
$982$	4.08202	0.130263
$983$	39.7389	1.26747	0.633737	−	0.773549i	$-0.281520\pi$
$983$	39.7389	1.26747	0.633737	+	0.773549i	$0.281520\pi$
$984$	−4.93658	−0.157372
$985$	0	0
$986$	−5.90942	−0.188194
$987$	−13.7249	−0.436868
$988$	−29.0299	−0.923565
$989$	−27.1687	−0.863914
$990$	0	0
$991$	0.836771	0.0265809	0.0132905	−	0.999912i	$-0.495769\pi$
$991$	0.836771	0.0265809	0.0132905	+	0.999912i	$0.495769\pi$
$992$	−27.9590	−0.887698
$993$	26.0804	0.827635
$994$	−11.2295	−0.356179
$995$	0	0
$996$	11.4102	0.361544
$997$	−48.1571	−1.52515	−0.762576	−	0.646899i	$-0.776066\pi$
$997$	−48.1571	−1.52515	−0.762576	+	0.646899i	$0.776066\pi$
$998$	11.2524	0.356188
$999$	−8.70773	−0.275500

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.bc.1.3	✓	8
3.2	odd	2		6525.2.a.bz.1.6		8
5.2	odd	4		2175.2.c.p.349.5		16
5.3	odd	4		2175.2.c.p.349.12		16
5.4	even	2		2175.2.a.bd.1.6	yes	8
15.14	odd	2		6525.2.a.by.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.bc.1.3	✓	8	1.1	even	1	trivial
2175.2.a.bd.1.6	yes	8	5.4	even	2
2175.2.c.p.349.5		16	5.2	odd	4
2175.2.c.p.349.12		16	5.3	odd	4
6525.2.a.by.1.3		8	15.14	odd	2
6525.2.a.bz.1.6		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q-1.64893 q^{2} -1.00000 q^{3} +0.718971 q^{4} +1.64893 q^{6} -1.44112 q^{7} +2.11233 q^{8} +1.00000 q^{9} +0.191176 q^{11} -0.718971 q^{12} -5.14501 q^{13} +2.37631 q^{14} -4.92102 q^{16} +3.58379 q^{17} -1.64893 q^{18} +7.84780 q^{19} +1.44112 q^{21} -0.315236 q^{22} -7.39261 q^{23} -2.11233 q^{24} +8.48377 q^{26} -1.00000 q^{27} -1.03612 q^{28} +1.00000 q^{29} -7.18783 q^{31} +3.88977 q^{32} -0.191176 q^{33} -5.90942 q^{34} +0.718971 q^{36} +8.70773 q^{37} -12.9405 q^{38} +5.14501 q^{39} +2.33703 q^{41} -2.37631 q^{42} +3.67511 q^{43} +0.137450 q^{44} +12.1899 q^{46} -9.52376 q^{47} +4.92102 q^{48} -4.92317 q^{49} -3.58379 q^{51} -3.69911 q^{52} -10.9053 q^{53} +1.64893 q^{54} -3.04412 q^{56} -7.84780 q^{57} -1.64893 q^{58} +14.2051 q^{59} +8.45202 q^{61} +11.8522 q^{62} -1.44112 q^{63} +3.42809 q^{64} +0.315236 q^{66} -9.96623 q^{67} +2.57664 q^{68} +7.39261 q^{69} -4.72562 q^{71} +2.11233 q^{72} +9.06473 q^{73} -14.3584 q^{74} +5.64234 q^{76} -0.275508 q^{77} -8.48377 q^{78} -8.78711 q^{79} +1.00000 q^{81} -3.85361 q^{82} -15.8701 q^{83} +1.03612 q^{84} -6.06000 q^{86} -1.00000 q^{87} +0.403827 q^{88} -11.6085 q^{89} +7.41459 q^{91} -5.31507 q^{92} +7.18783 q^{93} +15.7040 q^{94} -3.88977 q^{96} +9.85482 q^{97} +8.11796 q^{98} +0.191176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 8 q^{3} + 12 q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} + 6 q^{11} - 12 q^{12} + 6 q^{13} + 9 q^{14} + 32 q^{16} - 12 q^{17} - 2 q^{18} - 2 q^{21} + 3 q^{22} - 14 q^{23} + 3 q^{24} + 18 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 - 8 * q^3 + 12 * q^4 + 2 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9 + 6 * q^11 - 12 * q^12 + 6 * q^13 + 9 * q^14 + 32 * q^16 - 12 * q^17 - 2 * q^18 - 2 * q^21 + 3 * q^22 - 14 * q^23 + 3 * q^24 + 18 * q^26 - 8 * q^27 + 14 * q^28 + 8 * q^29 + 8 * q^31 + 2 * q^32 - 6 * q^33 - 13 * q^34 + 12 * q^36 + 4 * q^37 - 26 * q^38 - 6 * q^39 + 2 * q^41 - 9 * q^42 + 2 * q^43 - 15 * q^44 + 24 * q^46 - 12 * q^47 - 32 * q^48 + 38 * q^49 + 12 * q^51 + 49 * q^52 - 4 * q^53 + 2 * q^54 + 58 * q^56 - 2 * q^58 + 18 * q^59 + 12 * q^61 + 4 * q^62 + 2 * q^63 + 21 * q^64 - 3 * q^66 + 26 * q^67 - 81 * q^68 + 14 * q^69 + 24 * q^71 - 3 * q^72 - 14 * q^73 - 22 * q^74 + 26 * q^77 - 18 * q^78 + 10 * q^79 + 8 * q^81 + 48 * q^82 - 40 * q^83 - 14 * q^84 + 8 * q^86 - 8 * q^87 - 10 * q^88 + 34 * q^89 + 26 * q^91 + 18 * q^92 - 8 * q^93 - 43 * q^94 - 2 * q^96 + 30 * q^97 - 29 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

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Database

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