LMFDB - Embedded newform 2175.2.a.bc.1.5

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			1.5
Root			$-0.135002$ 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)

$f(q)$	$=$	\(q+0.135002 q^{2} -1.00000 q^{3} -1.98177 q^{4} -0.135002 q^{6} -1.12340 q^{7} -0.537546 q^{8} +1.00000 q^{9} -5.08097 q^{11} +1.98177 q^{12} +0.338903 q^{13} -0.151660 q^{14} +3.89098 q^{16} -1.77011 q^{17} +0.135002 q^{18} -5.13115 q^{19} +1.12340 q^{21} -0.685940 q^{22} -7.31086 q^{23} +0.537546 q^{24} +0.0457525 q^{26} -1.00000 q^{27} +2.22632 q^{28} +1.00000 q^{29} +8.74398 q^{31} +1.60038 q^{32} +5.08097 q^{33} -0.238968 q^{34} -1.98177 q^{36} +9.40342 q^{37} -0.692714 q^{38} -0.338903 q^{39} +8.95349 q^{41} +0.151660 q^{42} -12.3626 q^{43} +10.0693 q^{44} -0.986978 q^{46} +2.12044 q^{47} -3.89098 q^{48} -5.73798 q^{49} +1.77011 q^{51} -0.671630 q^{52} -10.9295 q^{53} -0.135002 q^{54} +0.603878 q^{56} +5.13115 q^{57} +0.135002 q^{58} -9.22244 q^{59} +1.48958 q^{61} +1.18045 q^{62} -1.12340 q^{63} -7.56591 q^{64} +0.685940 q^{66} -9.41936 q^{67} +3.50796 q^{68} +7.31086 q^{69} +11.1960 q^{71} -0.537546 q^{72} +2.51452 q^{73} +1.26948 q^{74} +10.1688 q^{76} +5.70795 q^{77} -0.0457525 q^{78} +11.2602 q^{79} +1.00000 q^{81} +1.20874 q^{82} +11.4189 q^{83} -2.22632 q^{84} -1.66898 q^{86} -1.00000 q^{87} +2.73126 q^{88} +16.0114 q^{89} -0.380723 q^{91} +14.4885 q^{92} -8.74398 q^{93} +0.286263 q^{94} -1.60038 q^{96} +9.76285 q^{97} -0.774637 q^{98} -5.08097 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.135002	0.0954606	0.0477303	−	0.998860i	$-0.484801\pi$
$2$	0.135002	0.0954606	0.0477303	+	0.998860i	$0.484801\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.98177	−0.990887
$5$	0	0
$5$	0	0
$6$	−0.135002	−0.0551142
$7$	−1.12340	−0.424604	−0.212302	−	0.977204i	$-0.568096\pi$
$7$	−1.12340	−0.424604	−0.212302	+	0.977204i	$0.568096\pi$
$8$	−0.537546	−0.190051
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.08097	−1.53197	−0.765986	−	0.642858i	$-0.777749\pi$
$11$	−5.08097	−1.53197	−0.765986	+	0.642858i	$0.777749\pi$
$12$	1.98177	0.572089
$13$	0.338903	0.0939949	0.0469975	−	0.998895i	$-0.485035\pi$
$13$	0.338903	0.0939949	0.0469975	+	0.998895i	$0.485035\pi$
$14$	−0.151660	−0.0405329
$15$	0	0
$16$	3.89098	0.972745
$17$	−1.77011	−0.429315	−0.214658	−	0.976689i	$-0.568864\pi$
$17$	−1.77011	−0.429315	−0.214658	+	0.976689i	$0.568864\pi$
$18$	0.135002	0.0318202
$19$	−5.13115	−1.17717	−0.588584	−	0.808436i	$-0.700314\pi$
$19$	−5.13115	−1.17717	−0.588584	+	0.808436i	$0.700314\pi$
$20$	0	0
$21$	1.12340	0.245145
$22$	−0.685940	−0.146243
$23$	−7.31086	−1.52442	−0.762210	−	0.647330i	$-0.775886\pi$
$23$	−7.31086	−1.52442	−0.762210	+	0.647330i	$0.775886\pi$
$24$	0.537546	0.109726
$25$	0	0
$26$	0.0457525	0.00897281
$27$	−1.00000	−0.192450
$28$	2.22632	0.420735
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	8.74398	1.57046	0.785232	−	0.619201i	$-0.212544\pi$
$31$	8.74398	1.57046	0.785232	+	0.619201i	$0.212544\pi$
$32$	1.60038	0.282910
$33$	5.08097	0.884484
$34$	−0.238968	−0.0409827
$35$	0	0
$36$	−1.98177	−0.330296
$37$	9.40342	1.54591	0.772956	−	0.634460i	$-0.218777\pi$
$37$	9.40342	1.54591	0.772956	+	0.634460i	$0.218777\pi$
$38$	−0.692714	−0.112373
$39$	−0.338903	−0.0542680
$40$	0	0
$41$	8.95349	1.39830	0.699150	−	0.714975i	$-0.253562\pi$
$41$	8.95349	1.39830	0.699150	+	0.714975i	$0.253562\pi$
$42$	0.151660	0.0234017
$43$	−12.3626	−1.88529	−0.942643	−	0.333802i	$-0.891668\pi$
$43$	−12.3626	−1.88529	−0.942643	+	0.333802i	$0.891668\pi$
$44$	10.0693	1.51801
$45$	0	0
$46$	−0.986978	−0.145522
$47$	2.12044	0.309298	0.154649	−	0.987970i	$-0.450575\pi$
$47$	2.12044	0.309298	0.154649	+	0.987970i	$0.450575\pi$
$48$	−3.89098	−0.561615
$49$	−5.73798	−0.819711
$50$	0	0
$51$	1.77011	0.247865
$52$	−0.671630	−0.0931384
$53$	−10.9295	−1.50129	−0.750643	−	0.660708i	$-0.770256\pi$
$53$	−10.9295	−1.50129	−0.750643	+	0.660708i	$0.770256\pi$
$54$	−0.135002	−0.0183714
$55$	0	0
$56$	0.603878	0.0806965
$57$	5.13115	0.679638
$58$	0.135002	0.0177266
$59$	−9.22244	−1.20066	−0.600330	−	0.799753i	$-0.704964\pi$
$59$	−9.22244	−1.20066	−0.600330	+	0.799753i	$0.704964\pi$
$60$	0	0
$61$	1.48958	0.190721	0.0953604	−	0.995443i	$-0.469600\pi$
$61$	1.48958	0.190721	0.0953604	+	0.995443i	$0.469600\pi$
$62$	1.18045	0.149917
$63$	−1.12340	−0.141535
$64$	−7.56591	−0.945738
$65$	0	0
$66$	0.685940	0.0844333
$67$	−9.41936	−1.15076	−0.575379	−	0.817887i	$-0.695145\pi$
$67$	−9.41936	−1.15076	−0.575379	+	0.817887i	$0.695145\pi$
$68$	3.50796	0.425403
$69$	7.31086	0.880124
$70$	0	0
$71$	11.1960	1.32872	0.664358	−	0.747414i	$-0.268705\pi$
$71$	11.1960	1.32872	0.664358	+	0.747414i	$0.268705\pi$
$72$	−0.537546	−0.0633504
$73$	2.51452	0.294302	0.147151	−	0.989114i	$-0.452990\pi$
$73$	2.51452	0.294302	0.147151	+	0.989114i	$0.452990\pi$
$74$	1.26948	0.147574
$75$	0	0
$76$	10.1688	1.16644
$77$	5.70795	0.650481
$78$	−0.0457525	−0.00518045
$79$	11.2602	1.26688	0.633438	−	0.773793i	$-0.281643\pi$
$79$	11.2602	1.26688	0.633438	+	0.773793i	$0.281643\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.20874	0.133482
$83$	11.4189	1.25339	0.626695	−	0.779265i	$-0.284407\pi$
$83$	11.4189	1.25339	0.626695	+	0.779265i	$0.284407\pi$
$84$	−2.22632	−0.242911
$85$	0	0
$86$	−1.66898	−0.179970
$87$	−1.00000	−0.107211
$88$	2.73126	0.291153
$89$	16.0114	1.69721	0.848604	−	0.529029i	$-0.177444\pi$
$89$	16.0114	1.69721	0.848604	+	0.529029i	$0.177444\pi$
$90$	0	0
$91$	−0.380723	−0.0399106
$92$	14.4885	1.51053
$93$	−8.74398	−0.906708
$94$	0.286263	0.0295257
$95$	0	0
$96$	−1.60038	−0.163338
$97$	9.76285	0.991267	0.495633	−	0.868532i	$-0.334936\pi$
$97$	9.76285	0.991267	0.495633	+	0.868532i	$0.334936\pi$
$98$	−0.774637	−0.0782501
$99$	−5.08097	−0.510657
$100$	0	0
$101$	1.42037	0.141332	0.0706660	−	0.997500i	$-0.477488\pi$
$101$	1.42037	0.141332	0.0706660	+	0.997500i	$0.477488\pi$
$102$	0.238968	0.0236614
$103$	16.3022	1.60630	0.803150	−	0.595777i	$-0.203156\pi$
$103$	16.3022	1.60630	0.803150	+	0.595777i	$0.203156\pi$
$104$	−0.182176	−0.0178638
$105$	0	0
$106$	−1.47550	−0.143314
$107$	−7.74551	−0.748787	−0.374393	−	0.927270i	$-0.622149\pi$
$107$	−7.74551	−0.748787	−0.374393	+	0.927270i	$0.622149\pi$
$108$	1.98177	0.190696
$109$	15.5762	1.49193	0.745967	−	0.665983i	$-0.231988\pi$
$109$	15.5762	1.49193	0.745967	+	0.665983i	$0.231988\pi$
$110$	0	0
$111$	−9.40342	−0.892533
$112$	−4.37111	−0.413031
$113$	8.56372	0.805607	0.402804	−	0.915286i	$-0.368036\pi$
$113$	8.56372	0.805607	0.402804	+	0.915286i	$0.368036\pi$
$114$	0.692714	0.0648786
$115$	0	0
$116$	−1.98177	−0.184003
$117$	0.338903	0.0313316
$118$	−1.24504	−0.114616
$119$	1.98854	0.182289
$120$	0	0
$121$	14.8163	1.34694
$122$	0.201095	0.0182063
$123$	−8.95349	−0.807309
$124$	−17.3286	−1.55615
$125$	0	0
$126$	−0.151660	−0.0135110
$127$	−2.09778	−0.186147	−0.0930737	−	0.995659i	$-0.529669\pi$
$127$	−2.09778	−0.186147	−0.0930737	+	0.995659i	$0.529669\pi$
$128$	−4.22217	−0.373191
$129$	12.3626	1.08847
$130$	0	0
$131$	2.63717	0.230411	0.115206	−	0.993342i	$-0.463247\pi$
$131$	2.63717	0.230411	0.115206	+	0.993342i	$0.463247\pi$
$132$	−10.0693	−0.876424
$133$	5.76432	0.499830
$134$	−1.27163	−0.109852
$135$	0	0
$136$	0.951517	0.0815919
$137$	−16.3118	−1.39361	−0.696806	−	0.717259i	$-0.745396\pi$
$137$	−16.3118	−1.39361	−0.696806	+	0.717259i	$0.745396\pi$
$138$	0.986978	0.0840172
$139$	6.11311	0.518508	0.259254	−	0.965809i	$-0.416523\pi$
$139$	6.11311	0.518508	0.259254	+	0.965809i	$0.416523\pi$
$140$	0	0
$141$	−2.12044	−0.178573
$142$	1.51147	0.126840
$143$	−1.72196	−0.143997
$144$	3.89098	0.324248
$145$	0	0
$146$	0.339464	0.0280943
$147$	5.73798	0.473261
$148$	−18.6355	−1.53182
$149$	22.0616	1.80736	0.903680	−	0.428208i	$-0.140855\pi$
$149$	22.0616	1.80736	0.903680	+	0.428208i	$0.140855\pi$
$150$	0	0
$151$	12.3925	1.00848	0.504242	−	0.863563i	$-0.331772\pi$
$151$	12.3925	1.00848	0.504242	+	0.863563i	$0.331772\pi$
$152$	2.75823	0.223722
$153$	−1.77011	−0.143105
$154$	0.770582	0.0620953
$155$	0	0
$156$	0.671630	0.0537735
$157$	−7.01983	−0.560243	−0.280122	−	0.959965i	$-0.590375\pi$
$157$	−7.01983	−0.560243	−0.280122	+	0.959965i	$0.590375\pi$
$158$	1.52015	0.120937
$159$	10.9295	0.866768
$160$	0	0
$161$	8.21300	0.647275
$162$	0.135002	0.0106067
$163$	0.770765	0.0603710	0.0301855	−	0.999544i	$-0.490390\pi$
$163$	0.770765	0.0603710	0.0301855	+	0.999544i	$0.490390\pi$
$164$	−17.7438	−1.38556
$165$	0	0
$166$	1.54157	0.119649
$167$	−0.785442	−0.0607793	−0.0303897	−	0.999538i	$-0.509675\pi$
$167$	−0.785442	−0.0607793	−0.0303897	+	0.999538i	$0.509675\pi$
$168$	−0.603878	−0.0465902
$169$	−12.8851	−0.991165
$170$	0	0
$171$	−5.13115	−0.392389
$172$	24.5000	1.86811
$173$	−17.5316	−1.33290	−0.666452	−	0.745548i	$-0.732188\pi$
$173$	−17.5316	−1.33290	−0.666452	+	0.745548i	$0.732188\pi$
$174$	−0.135002	−0.0102344
$175$	0	0
$176$	−19.7700	−1.49022
$177$	9.22244	0.693201
$178$	2.16157	0.162016
$179$	−6.59833	−0.493182	−0.246591	−	0.969120i	$-0.579310\pi$
$179$	−6.59833	−0.493182	−0.246591	+	0.969120i	$0.579310\pi$
$180$	0	0
$181$	13.3511	0.992377	0.496188	−	0.868215i	$-0.334733\pi$
$181$	13.3511	0.992377	0.496188	+	0.868215i	$0.334733\pi$
$182$	−0.0513982	−0.00380989
$183$	−1.48958	−0.110113
$184$	3.92992	0.289718
$185$	0	0
$186$	−1.18045	−0.0865549
$187$	8.99390	0.657699
$188$	−4.20223	−0.306479
$189$	1.12340	0.0817151
$190$	0	0
$191$	−3.16949	−0.229336	−0.114668	−	0.993404i	$-0.536580\pi$
$191$	−3.16949	−0.229336	−0.114668	+	0.993404i	$0.536580\pi$
$192$	7.56591	0.546022
$193$	1.67935	0.120882	0.0604411	−	0.998172i	$-0.480749\pi$
$193$	1.67935	0.120882	0.0604411	+	0.998172i	$0.480749\pi$
$194$	1.31800	0.0946269
$195$	0	0
$196$	11.3714	0.812242
$197$	19.7035	1.40382	0.701909	−	0.712266i	$-0.252331\pi$
$197$	19.7035	1.40382	0.701909	+	0.712266i	$0.252331\pi$
$198$	−0.685940	−0.0487476
$199$	12.4776	0.884516	0.442258	−	0.896888i	$-0.354177\pi$
$199$	12.4776	0.884516	0.442258	+	0.896888i	$0.354177\pi$
$200$	0	0
$201$	9.41936	0.664390
$202$	0.191752	0.0134916
$203$	−1.12340	−0.0788470
$204$	−3.50796	−0.245607
$205$	0	0
$206$	2.20082	0.153338
$207$	−7.31086	−0.508140
$208$	1.31867	0.0914331
$209$	26.0712	1.80339
$210$	0	0
$211$	−25.8358	−1.77861	−0.889306	−	0.457313i	$-0.848812\pi$
$211$	−25.8358	−1.77861	−0.889306	+	0.457313i	$0.848812\pi$
$212$	21.6599	1.48761
$213$	−11.1960	−0.767135
$214$	−1.04566	−0.0714796
$215$	0	0
$216$	0.537546	0.0365754
$217$	−9.82295	−0.666826
$218$	2.10282	0.142421
$219$	−2.51452	−0.169916
$220$	0	0
$221$	−0.599897	−0.0403535
$222$	−1.26948	−0.0852017
$223$	11.5403	0.772795	0.386397	−	0.922332i	$-0.373719\pi$
$223$	11.5403	0.772795	0.386397	+	0.922332i	$0.373719\pi$
$224$	−1.79786	−0.120125
$225$	0	0
$226$	1.15612	0.0769037
$227$	5.11378	0.339413	0.169707	−	0.985495i	$-0.445718\pi$
$227$	5.11378	0.339413	0.169707	+	0.985495i	$0.445718\pi$
$228$	−10.1688	−0.673444
$229$	−11.2580	−0.743949	−0.371974	−	0.928243i	$-0.621319\pi$
$229$	−11.2580	−0.743949	−0.371974	+	0.928243i	$0.621319\pi$
$230$	0	0
$231$	−5.70795	−0.375555
$232$	−0.537546	−0.0352916
$233$	−16.6226	−1.08898	−0.544491	−	0.838767i	$-0.683277\pi$
$233$	−16.6226	−1.08898	−0.544491	+	0.838767i	$0.683277\pi$
$234$	0.0457525	0.00299094
$235$	0	0
$236$	18.2768	1.18972
$237$	−11.2602	−0.731432
$238$	0.268456	0.0174014
$239$	7.23599	0.468057	0.234029	−	0.972230i	$-0.424809\pi$
$239$	7.23599	0.468057	0.234029	+	0.972230i	$0.424809\pi$
$240$	0	0
$241$	−1.49733	−0.0964517	−0.0482258	−	0.998836i	$-0.515357\pi$
$241$	−1.49733	−0.0964517	−0.0482258	+	0.998836i	$0.515357\pi$
$242$	2.00022	0.128579
$243$	−1.00000	−0.0641500
$244$	−2.95201	−0.188983
$245$	0	0
$246$	−1.20874	−0.0770661
$247$	−1.73897	−0.110648
$248$	−4.70029	−0.298469
$249$	−11.4189	−0.723645
$250$	0	0
$251$	−12.7786	−0.806581	−0.403290	−	0.915072i	$-0.632134\pi$
$251$	−12.7786	−0.806581	−0.403290	+	0.915072i	$0.632134\pi$
$252$	2.22632	0.140245
$253$	37.1463	2.33537
$254$	−0.283203	−0.0177697
$255$	0	0
$256$	14.5618	0.910113
$257$	−28.0469	−1.74952	−0.874760	−	0.484557i	$-0.838981\pi$
$257$	−28.0469	−1.74952	−0.874760	+	0.484557i	$0.838981\pi$
$258$	1.66898	0.103906
$259$	−10.5638	−0.656401
$260$	0	0
$261$	1.00000	0.0618984
$262$	0.356023	0.0219952
$263$	−15.8863	−0.979593	−0.489797	−	0.871837i	$-0.662929\pi$
$263$	−15.8863	−0.979593	−0.489797	+	0.871837i	$0.662929\pi$
$264$	−2.73126	−0.168097
$265$	0	0
$266$	0.778193	0.0477141
$267$	−16.0114	−0.979883
$268$	18.6670	1.14027
$269$	−20.8193	−1.26937	−0.634687	−	0.772769i	$-0.718871\pi$
$269$	−20.8193	−1.26937	−0.634687	+	0.772769i	$0.718871\pi$
$270$	0	0
$271$	−23.9227	−1.45320	−0.726599	−	0.687061i	$-0.758900\pi$
$271$	−23.9227	−1.45320	−0.726599	+	0.687061i	$0.758900\pi$
$272$	−6.88747	−0.417614
$273$	0.380723	0.0230424
$274$	−2.20212	−0.133035
$275$	0	0
$276$	−14.4885	−0.872104
$277$	15.2028	0.913448	0.456724	−	0.889608i	$-0.349023\pi$
$277$	15.2028	0.913448	0.456724	+	0.889608i	$0.349023\pi$
$278$	0.825280	0.0494970
$279$	8.74398	0.523488
$280$	0	0
$281$	−13.9400	−0.831592	−0.415796	−	0.909458i	$-0.636497\pi$
$281$	−13.9400	−0.831592	−0.415796	+	0.909458i	$0.636497\pi$
$282$	−0.286263	−0.0170467
$283$	28.4943	1.69381	0.846905	−	0.531744i	$-0.178463\pi$
$283$	28.4943	1.69381	0.846905	+	0.531744i	$0.178463\pi$
$284$	−22.1879	−1.31661
$285$	0	0
$286$	−0.232467	−0.0137461
$287$	−10.0583	−0.593724
$288$	1.60038	0.0943033
$289$	−13.8667	−0.815688
$290$	0	0
$291$	−9.76285	−0.572308
$292$	−4.98321	−0.291621
$293$	−1.06730	−0.0623523	−0.0311762	−	0.999514i	$-0.509925\pi$
$293$	−1.06730	−0.0623523	−0.0311762	+	0.999514i	$0.509925\pi$
$294$	0.774637	0.0451777
$295$	0	0
$296$	−5.05477	−0.293803
$297$	5.08097	0.294828
$298$	2.97836	0.172532
$299$	−2.47768	−0.143288
$300$	0	0
$301$	13.8882	0.800500
$302$	1.67300	0.0962704
$303$	−1.42037	−0.0815981
$304$	−19.9652	−1.14508
$305$	0	0
$306$	−0.238968	−0.0136609
$307$	23.9769	1.36843	0.684217	−	0.729279i	$-0.260144\pi$
$307$	23.9769	1.36843	0.684217	+	0.729279i	$0.260144\pi$
$308$	−11.3119	−0.644554
$309$	−16.3022	−0.927397
$310$	0	0
$311$	11.5897	0.657190	0.328595	−	0.944471i	$-0.393425\pi$
$311$	11.5897	0.657190	0.328595	+	0.944471i	$0.393425\pi$
$312$	0.182176	0.0103137
$313$	−3.82356	−0.216120	−0.108060	−	0.994144i	$-0.534464\pi$
$313$	−3.82356	−0.216120	−0.108060	+	0.994144i	$0.534464\pi$
$314$	−0.947688	−0.0534811
$315$	0	0
$316$	−22.3153	−1.25533
$317$	−25.2499	−1.41817	−0.709087	−	0.705121i	$-0.750893\pi$
$317$	−25.2499	−1.41817	−0.709087	+	0.705121i	$0.750893\pi$
$318$	1.47550	0.0827422
$319$	−5.08097	−0.284480
$320$	0	0
$321$	7.74551	0.432312
$322$	1.10877	0.0617892
$323$	9.08272	0.505376
$324$	−1.98177	−0.110099
$325$	0	0
$326$	0.104055	0.00576305
$327$	−15.5762	−0.861369
$328$	−4.81291	−0.265749
$329$	−2.38209	−0.131329
$330$	0	0
$331$	4.25823	0.234054	0.117027	−	0.993129i	$-0.462664\pi$
$331$	4.25823	0.234054	0.117027	+	0.993129i	$0.462664\pi$
$332$	−22.6297	−1.24197
$333$	9.40342	0.515304
$334$	−0.106036	−0.00580203
$335$	0	0
$336$	4.37111	0.238464
$337$	34.5971	1.88463	0.942313	−	0.334733i	$-0.108646\pi$
$337$	34.5971	1.88463	0.942313	+	0.334733i	$0.108646\pi$
$338$	−1.73952	−0.0946172
$339$	−8.56372	−0.465117
$340$	0	0
$341$	−44.4279	−2.40591
$342$	−0.692714	−0.0374577
$343$	14.3098	0.772657
$344$	6.64549	0.358301
$345$	0	0
$346$	−2.36680	−0.127240
$347$	−11.3706	−0.610405	−0.305202	−	0.952288i	$-0.598724\pi$
$347$	−11.3706	−0.610405	−0.305202	+	0.952288i	$0.598724\pi$
$348$	1.98177	0.106234
$349$	22.2680	1.19198	0.595990	−	0.802992i	$-0.296760\pi$
$349$	22.2680	1.19198	0.595990	+	0.802992i	$0.296760\pi$
$350$	0	0
$351$	−0.338903	−0.0180893
$352$	−8.13149	−0.433410
$353$	−15.5367	−0.826936	−0.413468	−	0.910519i	$-0.635683\pi$
$353$	−15.5367	−0.826936	−0.413468	+	0.910519i	$0.635683\pi$
$354$	1.24504	0.0661734
$355$	0	0
$356$	−31.7310	−1.68174
$357$	−1.98854	−0.105245
$358$	−0.890785	−0.0470795
$359$	12.5835	0.664131	0.332066	−	0.943256i	$-0.392254\pi$
$359$	12.5835	0.664131	0.332066	+	0.943256i	$0.392254\pi$
$360$	0	0
$361$	7.32873	0.385722
$362$	1.80242	0.0947329
$363$	−14.8163	−0.777654
$364$	0.754507	0.0395469
$365$	0	0
$366$	−0.201095	−0.0105114
$367$	−2.61584	−0.136546	−0.0682729	−	0.997667i	$-0.521749\pi$
$367$	−2.61584	−0.136546	−0.0682729	+	0.997667i	$0.521749\pi$
$368$	−28.4464	−1.48287
$369$	8.95349	0.466100
$370$	0	0
$371$	12.2782	0.637453
$372$	17.3286	0.898445
$373$	4.57070	0.236662	0.118331	−	0.992974i	$-0.462246\pi$
$373$	4.57070	0.236662	0.118331	+	0.992974i	$0.462246\pi$
$374$	1.21419	0.0627843
$375$	0	0
$376$	−1.13983	−0.0587824
$377$	0.338903	0.0174544
$378$	0.151660	0.00780057
$379$	−13.8895	−0.713453	−0.356727	−	0.934209i	$-0.616107\pi$
$379$	−13.8895	−0.713453	−0.356727	+	0.934209i	$0.616107\pi$
$380$	0	0
$381$	2.09778	0.107472
$382$	−0.427886	−0.0218925
$383$	27.1213	1.38583	0.692916	−	0.721018i	$-0.256326\pi$
$383$	27.1213	1.38583	0.692916	+	0.721018i	$0.256326\pi$
$384$	4.22217	0.215462
$385$	0	0
$386$	0.226715	0.0115395
$387$	−12.3626	−0.628429
$388$	−19.3478	−0.982234
$389$	11.3172	0.573807	0.286903	−	0.957960i	$-0.407374\pi$
$389$	11.3172	0.573807	0.286903	+	0.957960i	$0.407374\pi$
$390$	0	0
$391$	12.9410	0.654457
$392$	3.08443	0.155787
$393$	−2.63717	−0.133028
$394$	2.66001	0.134009
$395$	0	0
$396$	10.0693	0.506004
$397$	23.6436	1.18664	0.593319	−	0.804968i	$-0.297817\pi$
$397$	23.6436	1.18664	0.593319	+	0.804968i	$0.297817\pi$
$398$	1.68450	0.0844364
$399$	−5.76432	−0.288577
$400$	0	0
$401$	18.9114	0.944388	0.472194	−	0.881495i	$-0.343462\pi$
$401$	18.9114	0.944388	0.472194	+	0.881495i	$0.343462\pi$
$402$	1.27163	0.0634231
$403$	2.96336	0.147616
$404$	−2.81485	−0.140044
$405$	0	0
$406$	−0.151660	−0.00752678
$407$	−47.7785	−2.36829
$408$	−0.951517	−0.0471071
$409$	−23.0388	−1.13920	−0.569598	−	0.821924i	$-0.692901\pi$
$409$	−23.0388	−1.13920	−0.569598	+	0.821924i	$0.692901\pi$
$410$	0	0
$411$	16.3118	0.804603
$412$	−32.3072	−1.59166
$413$	10.3605	0.509805
$414$	−0.986978	−0.0485073
$415$	0	0
$416$	0.542375	0.0265921
$417$	−6.11311	−0.299361
$418$	3.51966	0.172152
$419$	−0.115766	−0.00565553	−0.00282777	−	0.999996i	$-0.500900\pi$
$419$	−0.115766	−0.00565553	−0.00282777	+	0.999996i	$0.500900\pi$
$420$	0	0
$421$	11.1734	0.544560	0.272280	−	0.962218i	$-0.412222\pi$
$421$	11.1734	0.544560	0.272280	+	0.962218i	$0.412222\pi$
$422$	−3.48788	−0.169787
$423$	2.12044	0.103099
$424$	5.87513	0.285321
$425$	0	0
$426$	−1.51147	−0.0732311
$427$	−1.67339	−0.0809808
$428$	15.3499	0.741963
$429$	1.72196	0.0831370
$430$	0	0
$431$	2.85694	0.137614	0.0688069	−	0.997630i	$-0.478081\pi$
$431$	2.85694	0.137614	0.0688069	+	0.997630i	$0.478081\pi$
$432$	−3.89098	−0.187205
$433$	−8.31709	−0.399694	−0.199847	−	0.979827i	$-0.564044\pi$
$433$	−8.31709	−0.399694	−0.199847	+	0.979827i	$0.564044\pi$
$434$	−1.32611	−0.0636556
$435$	0	0
$436$	−30.8686	−1.47834
$437$	37.5131	1.79450
$438$	−0.339464	−0.0162202
$439$	18.9364	0.903786	0.451893	−	0.892072i	$-0.350749\pi$
$439$	18.9364	0.903786	0.451893	+	0.892072i	$0.350749\pi$
$440$	0	0
$441$	−5.73798	−0.273237
$442$	−0.0809871	−0.00385217
$443$	−8.99879	−0.427545	−0.213773	−	0.976883i	$-0.568575\pi$
$443$	−8.99879	−0.427545	−0.213773	+	0.976883i	$0.568575\pi$
$444$	18.6355	0.884399
$445$	0	0
$446$	1.55796	0.0737714
$447$	−22.0616	−1.04348
$448$	8.49951	0.401564
$449$	9.15082	0.431854	0.215927	−	0.976410i	$-0.430723\pi$
$449$	9.15082	0.431854	0.215927	+	0.976410i	$0.430723\pi$
$450$	0	0
$451$	−45.4924	−2.14215
$452$	−16.9714	−0.798266
$453$	−12.3925	−0.582248
$454$	0.690368	0.0324006
$455$	0	0
$456$	−2.75823	−0.129166
$457$	−33.4356	−1.56405	−0.782026	−	0.623246i	$-0.785814\pi$
$457$	−33.4356	−1.56405	−0.782026	+	0.623246i	$0.785814\pi$
$458$	−1.51985	−0.0710178
$459$	1.77011	0.0826218
$460$	0	0
$461$	−1.30316	−0.0606943	−0.0303471	−	0.999539i	$-0.509661\pi$
$461$	−1.30316	−0.0606943	−0.0303471	+	0.999539i	$0.509661\pi$
$462$	−0.770582	−0.0358507
$463$	3.69332	0.171643	0.0858215	−	0.996311i	$-0.472649\pi$
$463$	3.69332	0.171643	0.0858215	+	0.996311i	$0.472649\pi$
$464$	3.89098	0.180634
$465$	0	0
$466$	−2.24408	−0.103955
$467$	12.6950	0.587454	0.293727	−	0.955889i	$-0.405104\pi$
$467$	12.6950	0.587454	0.293727	+	0.955889i	$0.405104\pi$
$468$	−0.671630	−0.0310461
$469$	10.5817	0.488617
$470$	0	0
$471$	7.01983	0.323457
$472$	4.95749	0.228187
$473$	62.8143	2.88820
$474$	−1.52015	−0.0698229
$475$	0	0
$476$	−3.94084	−0.180628
$477$	−10.9295	−0.500429
$478$	0.976870	0.0446810
$479$	−39.7992	−1.81847	−0.909236	−	0.416280i	$-0.863334\pi$
$479$	−39.7992	−1.81847	−0.909236	+	0.416280i	$0.863334\pi$
$480$	0	0
$481$	3.18685	0.145308
$482$	−0.202142	−0.00920733
$483$	−8.21300	−0.373704
$484$	−29.3625	−1.33466
$485$	0	0
$486$	−0.135002	−0.00612380
$487$	−11.0760	−0.501901	−0.250950	−	0.968000i	$-0.580743\pi$
$487$	−11.0760	−0.501901	−0.250950	+	0.968000i	$0.580743\pi$
$488$	−0.800716	−0.0362467
$489$	−0.770765	−0.0348552
$490$	0	0
$491$	−6.51563	−0.294046	−0.147023	−	0.989133i	$-0.546969\pi$
$491$	−6.51563	−0.294046	−0.147023	+	0.989133i	$0.546969\pi$
$492$	17.7438	0.799952
$493$	−1.77011	−0.0797219
$494$	−0.234763	−0.0105625
$495$	0	0
$496$	34.0226	1.52766
$497$	−12.5775	−0.564178
$498$	−1.54157	−0.0690796
$499$	5.66263	0.253494	0.126747	−	0.991935i	$-0.459546\pi$
$499$	5.66263	0.253494	0.126747	+	0.991935i	$0.459546\pi$
$500$	0	0
$501$	0.785442	0.0350909
$502$	−1.72514	−0.0769966
$503$	−10.1876	−0.454242	−0.227121	−	0.973867i	$-0.572931\pi$
$503$	−10.1876	−0.454242	−0.227121	+	0.973867i	$0.572931\pi$
$504$	0.603878	0.0268988
$505$	0	0
$506$	5.01481	0.222935
$507$	12.8851	0.572249
$508$	4.15732	0.184451
$509$	−16.6696	−0.738865	−0.369433	−	0.929258i	$-0.620448\pi$
$509$	−16.6696	−0.738865	−0.369433	+	0.929258i	$0.620448\pi$
$510$	0	0
$511$	−2.82480	−0.124962
$512$	10.4102	0.460071
$513$	5.13115	0.226546
$514$	−3.78638	−0.167010
$515$	0	0
$516$	−24.5000	−1.07855
$517$	−10.7739	−0.473835
$518$	−1.42613	−0.0626604
$519$	17.5316	0.769552
$520$	0	0
$521$	−8.05302	−0.352809	−0.176405	−	0.984318i	$-0.556447\pi$
$521$	−8.05302	−0.352809	−0.176405	+	0.984318i	$0.556447\pi$
$522$	0.135002	0.00590886
$523$	36.9867	1.61732	0.808658	−	0.588278i	$-0.200194\pi$
$523$	36.9867	1.61732	0.808658	+	0.588278i	$0.200194\pi$
$524$	−5.22629	−0.228311
$525$	0	0
$526$	−2.14468	−0.0935125
$527$	−15.4778	−0.674225
$528$	19.7700	0.860377
$529$	30.4487	1.32386
$530$	0	0
$531$	−9.22244	−0.400220
$532$	−11.4236	−0.495275
$533$	3.03437	0.131433
$534$	−2.16157	−0.0935402
$535$	0	0
$536$	5.06334	0.218703
$537$	6.59833	0.284739
$538$	−2.81064	−0.121175
$539$	29.1545	1.25577
$540$	0	0
$541$	6.43246	0.276553	0.138277	−	0.990394i	$-0.455844\pi$
$541$	6.43246	0.276553	0.138277	+	0.990394i	$0.455844\pi$
$542$	−3.22960	−0.138723
$543$	−13.3511	−0.572949
$544$	−2.83285	−0.121458
$545$	0	0
$546$	0.0513982	0.00219964
$547$	−30.3891	−1.29934	−0.649671	−	0.760215i	$-0.725093\pi$
$547$	−30.3891	−1.29934	−0.649671	+	0.760215i	$0.725093\pi$
$548$	32.3263	1.38091
$549$	1.48958	0.0635736
$550$	0	0
$551$	−5.13115	−0.218594
$552$	−3.92992	−0.167269
$553$	−12.6497	−0.537921
$554$	2.05240	0.0871983
$555$	0	0
$556$	−12.1148	−0.513783
$557$	8.18245	0.346702	0.173351	−	0.984860i	$-0.444541\pi$
$557$	8.18245	0.346702	0.173351	+	0.984860i	$0.444541\pi$
$558$	1.18045	0.0499725
$559$	−4.18974	−0.177207
$560$	0	0
$561$	−8.99390	−0.379723
$562$	−1.88193	−0.0793843
$563$	−19.1211	−0.805859	−0.402930	−	0.915231i	$-0.632008\pi$
$563$	−19.1211	−0.805859	−0.402930	+	0.915231i	$0.632008\pi$
$564$	4.20223	0.176946
$565$	0	0
$566$	3.84678	0.161692
$567$	−1.12340	−0.0471782
$568$	−6.01835	−0.252524
$569$	−2.07403	−0.0869478	−0.0434739	−	0.999055i	$-0.513843\pi$
$569$	−2.07403	−0.0869478	−0.0434739	+	0.999055i	$0.513843\pi$
$570$	0	0
$571$	−11.7399	−0.491299	−0.245649	−	0.969359i	$-0.579001\pi$
$571$	−11.7399	−0.491299	−0.245649	+	0.969359i	$0.579001\pi$
$572$	3.41254	0.142685
$573$	3.16949	0.132407
$574$	−1.35789	−0.0566772
$575$	0	0
$576$	−7.56591	−0.315246
$577$	27.7355	1.15464	0.577321	−	0.816517i	$-0.304098\pi$
$577$	27.7355	1.15464	0.577321	+	0.816517i	$0.304098\pi$
$578$	−1.87203	−0.0778661
$579$	−1.67935	−0.0697913
$580$	0	0
$581$	−12.8280	−0.532195
$582$	−1.31800	−0.0546329
$583$	55.5327	2.29993
$584$	−1.35167	−0.0559325
$585$	0	0
$586$	−0.144087	−0.00595219
$587$	3.47883	0.143587	0.0717934	−	0.997420i	$-0.477128\pi$
$587$	3.47883	0.143587	0.0717934	+	0.997420i	$0.477128\pi$
$588$	−11.3714	−0.468948
$589$	−44.8667	−1.84870
$590$	0	0
$591$	−19.7035	−0.810495
$592$	36.5885	1.50378
$593$	11.3701	0.466916	0.233458	−	0.972367i	$-0.424996\pi$
$593$	11.3701	0.466916	0.233458	+	0.972367i	$0.424996\pi$
$594$	0.685940	0.0281444
$595$	0	0
$596$	−43.7212	−1.79089
$597$	−12.4776	−0.510676
$598$	−0.334490	−0.0136783
$599$	−2.74154	−0.112016	−0.0560082	−	0.998430i	$-0.517837\pi$
$599$	−2.74154	−0.112016	−0.0560082	+	0.998430i	$0.517837\pi$
$600$	0	0
$601$	−35.0177	−1.42840	−0.714202	−	0.699940i	$-0.753210\pi$
$601$	−35.0177	−1.42840	−0.714202	+	0.699940i	$0.753210\pi$
$602$	1.87492	0.0764162
$603$	−9.41936	−0.383586
$604$	−24.5590	−0.999293
$605$	0	0
$606$	−0.191752	−0.00778940
$607$	10.4419	0.423823	0.211911	−	0.977289i	$-0.432031\pi$
$607$	10.4419	0.423823	0.211911	+	0.977289i	$0.432031\pi$
$608$	−8.21180	−0.333032
$609$	1.12340	0.0455223
$610$	0	0
$611$	0.718624	0.0290724
$612$	3.50796	0.141801
$613$	36.3298	1.46735	0.733673	−	0.679502i	$-0.237804\pi$
$613$	36.3298	1.46735	0.733673	+	0.679502i	$0.237804\pi$
$614$	3.23692	0.130631
$615$	0	0
$616$	−3.06829	−0.123625
$617$	−4.40120	−0.177186	−0.0885928	−	0.996068i	$-0.528237\pi$
$617$	−4.40120	−0.177186	−0.0885928	+	0.996068i	$0.528237\pi$
$618$	−2.20082	−0.0885299
$619$	12.6011	0.506480	0.253240	−	0.967403i	$-0.418504\pi$
$619$	12.6011	0.506480	0.253240	+	0.967403i	$0.418504\pi$
$620$	0	0
$621$	7.31086	0.293375
$622$	1.56462	0.0627357
$623$	−17.9872	−0.720641
$624$	−1.31867	−0.0527889
$625$	0	0
$626$	−0.516186	−0.0206310
$627$	−26.0712	−1.04119
$628$	13.9117	0.555138
$629$	−16.6451	−0.663684
$630$	0	0
$631$	14.7899	0.588778	0.294389	−	0.955686i	$-0.404884\pi$
$631$	14.7899	0.588778	0.294389	+	0.955686i	$0.404884\pi$
$632$	−6.05290	−0.240772
$633$	25.8358	1.02688
$634$	−3.40877	−0.135380
$635$	0	0
$636$	−21.6599	−0.858870
$637$	−1.94462	−0.0770487
$638$	−0.685940	−0.0271566
$639$	11.1960	0.442905
$640$	0	0
$641$	−23.7059	−0.936328	−0.468164	−	0.883642i	$-0.655084\pi$
$641$	−23.7059	−0.936328	−0.468164	+	0.883642i	$0.655084\pi$
$642$	1.04566	0.0412688
$643$	10.8899	0.429454	0.214727	−	0.976674i	$-0.431114\pi$
$643$	10.8899	0.429454	0.214727	+	0.976674i	$0.431114\pi$
$644$	−16.2763	−0.641376
$645$	0	0
$646$	1.22618	0.0482435
$647$	2.61519	0.102814	0.0514068	−	0.998678i	$-0.483629\pi$
$647$	2.61519	0.102814	0.0514068	+	0.998678i	$0.483629\pi$
$648$	−0.537546	−0.0211168
$649$	46.8590	1.83938
$650$	0	0
$651$	9.82295	0.384992
$652$	−1.52748	−0.0598208
$653$	21.9682	0.859680	0.429840	−	0.902905i	$-0.358570\pi$
$653$	21.9682	0.859680	0.429840	+	0.902905i	$0.358570\pi$
$654$	−2.10282	−0.0822267
$655$	0	0
$656$	34.8378	1.36019
$657$	2.51452	0.0981008
$658$	−0.321586	−0.0125367
$659$	4.26772	0.166247	0.0831233	−	0.996539i	$-0.473510\pi$
$659$	4.26772	0.166247	0.0831233	+	0.996539i	$0.473510\pi$
$660$	0	0
$661$	−6.66320	−0.259168	−0.129584	−	0.991568i	$-0.541364\pi$
$661$	−6.66320	−0.259168	−0.129584	+	0.991568i	$0.541364\pi$
$662$	0.574869	0.0223429
$663$	0.599897	0.0232981
$664$	−6.13820	−0.238208
$665$	0	0
$666$	1.26948	0.0491912
$667$	−7.31086	−0.283078
$668$	1.55657	0.0602254
$669$	−11.5403	−0.446173
$670$	0	0
$671$	−7.56850	−0.292179
$672$	1.79786	0.0693541
$673$	−3.63552	−0.140139	−0.0700695	−	0.997542i	$-0.522322\pi$
$673$	−3.63552	−0.140139	−0.0700695	+	0.997542i	$0.522322\pi$
$674$	4.67067	0.179907
$675$	0	0
$676$	25.5355	0.982133
$677$	−15.5368	−0.597126	−0.298563	−	0.954390i	$-0.596507\pi$
$677$	−15.5368	−0.597126	−0.298563	+	0.954390i	$0.596507\pi$
$678$	−1.15612	−0.0444004
$679$	−10.9676	−0.420896
$680$	0	0
$681$	−5.11378	−0.195960
$682$	−5.99784	−0.229669
$683$	36.8380	1.40957	0.704784	−	0.709422i	$-0.251044\pi$
$683$	36.8380	1.40957	0.704784	+	0.709422i	$0.251044\pi$
$684$	10.1688	0.388813
$685$	0	0
$686$	1.93185	0.0737583
$687$	11.2580	0.429519
$688$	−48.1028	−1.83390
$689$	−3.70406	−0.141113
$690$	0	0
$691$	41.0574	1.56190	0.780948	−	0.624596i	$-0.214736\pi$
$691$	41.0574	1.56190	0.780948	+	0.624596i	$0.214736\pi$
$692$	34.7437	1.32076
$693$	5.70795	0.216827
$694$	−1.53505	−0.0582696
$695$	0	0
$696$	0.537546	0.0203756
$697$	−15.8487	−0.600312
$698$	3.00622	0.113787
$699$	16.6226	0.628724
$700$	0	0
$701$	−1.12022	−0.0423102	−0.0211551	−	0.999776i	$-0.506734\pi$
$701$	−1.12022	−0.0423102	−0.0211551	+	0.999776i	$0.506734\pi$
$702$	−0.0457525	−0.00172682
$703$	−48.2504	−1.81980
$704$	38.4422	1.44884
$705$	0	0
$706$	−2.09748	−0.0789398
$707$	−1.59564	−0.0600102
$708$	−18.2768	−0.686884
$709$	31.7312	1.19169	0.595845	−	0.803099i	$-0.296817\pi$
$709$	31.7312	1.19169	0.595845	+	0.803099i	$0.296817\pi$
$710$	0	0
$711$	11.2602	0.422292
$712$	−8.60688	−0.322556
$713$	−63.9260	−2.39405
$714$	−0.268456	−0.0100467
$715$	0	0
$716$	13.0764	0.488688
$717$	−7.23599	−0.270233
$718$	1.69879	0.0633984
$719$	48.2204	1.79832	0.899159	−	0.437621i	$-0.144179\pi$
$719$	48.2204	1.79832	0.899159	+	0.437621i	$0.144179\pi$
$720$	0	0
$721$	−18.3138	−0.682041
$722$	0.989390	0.0368213
$723$	1.49733	0.0556864
$724$	−26.4588	−0.983334
$725$	0	0
$726$	−2.00022	−0.0742353
$727$	9.45675	0.350731	0.175366	−	0.984503i	$-0.443889\pi$
$727$	9.45675	0.350731	0.175366	+	0.984503i	$0.443889\pi$
$728$	0.204656	0.00758506
$729$	1.00000	0.0370370
$730$	0	0
$731$	21.8833	0.809383
$732$	2.95201	0.109109
$733$	7.53155	0.278184	0.139092	−	0.990279i	$-0.455582\pi$
$733$	7.53155	0.278184	0.139092	+	0.990279i	$0.455582\pi$
$734$	−0.353143	−0.0130347
$735$	0	0
$736$	−11.7002	−0.431274
$737$	47.8595	1.76293
$738$	1.20874	0.0444942
$739$	−52.2221	−1.92102	−0.960510	−	0.278246i	$-0.910247\pi$
$739$	−52.2221	−1.92102	−0.960510	+	0.278246i	$0.910247\pi$
$740$	0	0
$741$	1.73897	0.0638825
$742$	1.65758	0.0608516
$743$	22.6403	0.830593	0.415296	−	0.909686i	$-0.363678\pi$
$743$	22.6403	0.830593	0.415296	+	0.909686i	$0.363678\pi$
$744$	4.70029	0.172321
$745$	0	0
$746$	0.617051	0.0225919
$747$	11.4189	0.417797
$748$	−17.8239	−0.651705
$749$	8.70128	0.317938
$750$	0	0
$751$	50.9170	1.85799	0.928994	−	0.370095i	$-0.120675\pi$
$751$	50.9170	1.85799	0.928994	+	0.370095i	$0.120675\pi$
$752$	8.25058	0.300868
$753$	12.7786	0.465680
$754$	0.0457525	0.00166621
$755$	0	0
$756$	−2.22632	−0.0809704
$757$	−44.3795	−1.61300	−0.806501	−	0.591233i	$-0.798641\pi$
$757$	−44.3795	−1.61300	−0.806501	+	0.591233i	$0.798641\pi$
$758$	−1.87510	−0.0681066
$759$	−37.1463	−1.34832
$760$	0	0
$761$	48.3911	1.75418	0.877089	−	0.480329i	$-0.159483\pi$
$761$	48.3911	1.75418	0.877089	+	0.480329i	$0.159483\pi$
$762$	0.283203	0.0102594
$763$	−17.4983	−0.633481
$764$	6.28121	0.227246
$765$	0	0
$766$	3.66142	0.132292
$767$	−3.12552	−0.112856
$768$	−14.5618	−0.525454
$769$	33.8522	1.22074	0.610371	−	0.792116i	$-0.291020\pi$
$769$	33.8522	1.22074	0.610371	+	0.792116i	$0.291020\pi$
$770$	0	0
$771$	28.0469	1.01009
$772$	−3.32809	−0.119781
$773$	14.6079	0.525408	0.262704	−	0.964876i	$-0.415386\pi$
$773$	14.6079	0.525408	0.262704	+	0.964876i	$0.415386\pi$
$774$	−1.66898	−0.0599902
$775$	0	0
$776$	−5.24798	−0.188392
$777$	10.5638	0.378973
$778$	1.52785	0.0547759
$779$	−45.9417	−1.64603
$780$	0	0
$781$	−56.8864	−2.03555
$782$	1.74706	0.0624748
$783$	−1.00000	−0.0357371
$784$	−22.3264	−0.797370
$785$	0	0
$786$	−0.356023	−0.0126989
$787$	21.2126	0.756147	0.378073	−	0.925776i	$-0.376587\pi$
$787$	21.2126	0.756147	0.378073	+	0.925776i	$0.376587\pi$
$788$	−39.0480	−1.39103
$789$	15.8863	0.565568
$790$	0	0
$791$	−9.62046	−0.342064
$792$	2.73126	0.0970510
$793$	0.504823	0.0179268
$794$	3.19192	0.113277
$795$	0	0
$796$	−24.7279	−0.876456
$797$	39.9415	1.41480	0.707399	−	0.706814i	$-0.249868\pi$
$797$	39.9415	1.41480	0.707399	+	0.706814i	$0.249868\pi$
$798$	−0.778193	−0.0275477
$799$	−3.75341	−0.132786
$800$	0	0
$801$	16.0114	0.565736
$802$	2.55306	0.0901518
$803$	−12.7762	−0.450863
$804$	−18.6670	−0.658336
$805$	0	0
$806$	0.400059	0.0140915
$807$	20.8193	0.732874
$808$	−0.763514	−0.0268603
$809$	−5.03220	−0.176923	−0.0884613	−	0.996080i	$-0.528195\pi$
$809$	−5.03220	−0.176923	−0.0884613	+	0.996080i	$0.528195\pi$
$810$	0	0
$811$	−2.13286	−0.0748947	−0.0374474	−	0.999299i	$-0.511923\pi$
$811$	−2.13286	−0.0748947	−0.0374474	+	0.999299i	$0.511923\pi$
$812$	2.22632	0.0781285
$813$	23.9227	0.839005
$814$	−6.45018	−0.226079
$815$	0	0
$816$	6.88747	0.241110
$817$	63.4346	2.21930
$818$	−3.11028	−0.108748
$819$	−0.380723	−0.0133035
$820$	0	0
$821$	8.28909	0.289291	0.144646	−	0.989484i	$-0.453796\pi$
$821$	8.28909	0.289291	0.144646	+	0.989484i	$0.453796\pi$
$822$	2.20212	0.0768078
$823$	8.05558	0.280800	0.140400	−	0.990095i	$-0.455161\pi$
$823$	8.05558	0.280800	0.140400	+	0.990095i	$0.455161\pi$
$824$	−8.76316	−0.305279
$825$	0	0
$826$	1.39868	0.0486663
$827$	40.6099	1.41214	0.706072	−	0.708140i	$-0.250465\pi$
$827$	40.6099	1.41214	0.706072	+	0.708140i	$0.250465\pi$
$828$	14.4885	0.503509
$829$	20.2557	0.703511	0.351755	−	0.936092i	$-0.385585\pi$
$829$	20.2557	0.703511	0.351755	+	0.936092i	$0.385585\pi$
$830$	0	0
$831$	−15.2028	−0.527379
$832$	−2.56411	−0.0888946
$833$	10.1569	0.351915
$834$	−0.825280	−0.0285771
$835$	0	0
$836$	−51.6673	−1.78695
$837$	−8.74398	−0.302236
$838$	−0.0156286	−0.000539881 0
$839$	−43.5654	−1.50405	−0.752023	−	0.659137i	$-0.770922\pi$
$839$	−43.5654	−1.50405	−0.752023	+	0.659137i	$0.770922\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	1.50843	0.0519840
$843$	13.9400	0.480120
$844$	51.2008	1.76240
$845$	0	0
$846$	0.286263	0.00984191
$847$	−16.6446	−0.571914
$848$	−42.5266	−1.46037
$849$	−28.4943	−0.977922
$850$	0	0
$851$	−68.7471	−2.35662
$852$	22.1879	0.760144
$853$	−29.7784	−1.01959	−0.509796	−	0.860295i	$-0.670279\pi$
$853$	−29.7784	−1.01959	−0.509796	+	0.860295i	$0.670279\pi$
$854$	−0.225910	−0.00773048
$855$	0	0
$856$	4.16357	0.142308
$857$	−53.1696	−1.81624	−0.908120	−	0.418710i	$-0.862482\pi$
$857$	−53.1696	−1.81624	−0.908120	+	0.418710i	$0.862482\pi$
$858$	0.232467	0.00793630
$859$	38.8864	1.32679	0.663393	−	0.748271i	$-0.269116\pi$
$859$	38.8864	1.32679	0.663393	+	0.748271i	$0.269116\pi$
$860$	0	0
$861$	10.0583	0.342787
$862$	0.385691	0.0131367
$863$	46.2241	1.57349	0.786743	−	0.617281i	$-0.211766\pi$
$863$	46.2241	1.57349	0.786743	+	0.617281i	$0.211766\pi$
$864$	−1.60038	−0.0544461
$865$	0	0
$866$	−1.12282	−0.0381550
$867$	13.8667	0.470938
$868$	19.4669	0.660749
$869$	−57.2130	−1.94082
$870$	0	0
$871$	−3.19225	−0.108165
$872$	−8.37295	−0.283544
$873$	9.76285	0.330422
$874$	5.06434	0.171304
$875$	0	0
$876$	4.98321	0.168367
$877$	26.0030	0.878059	0.439029	−	0.898473i	$-0.355322\pi$
$877$	26.0030	0.878059	0.439029	+	0.898473i	$0.355322\pi$
$878$	2.55645	0.0862759
$879$	1.06730	0.0359991
$880$	0	0
$881$	50.7177	1.70872	0.854362	−	0.519679i	$-0.173948\pi$
$881$	50.7177	1.70872	0.854362	+	0.519679i	$0.173948\pi$
$882$	−0.774637	−0.0260834
$883$	−8.51437	−0.286531	−0.143266	−	0.989684i	$-0.545760\pi$
$883$	−8.51437	−0.286531	−0.143266	+	0.989684i	$0.545760\pi$
$884$	1.18886	0.0399857
$885$	0	0
$886$	−1.21485	−0.0408137
$887$	−57.1432	−1.91868	−0.959341	−	0.282250i	$-0.908919\pi$
$887$	−57.1432	−1.91868	−0.959341	+	0.282250i	$0.908919\pi$
$888$	5.05477	0.169627
$889$	2.35663	0.0790390
$890$	0	0
$891$	−5.08097	−0.170219
$892$	−22.8702	−0.765752
$893$	−10.8803	−0.364095
$894$	−2.97836	−0.0996112
$895$	0	0
$896$	4.74317	0.158458
$897$	2.47768	0.0827272
$898$	1.23538	0.0412250
$899$	8.74398	0.291628
$900$	0	0
$901$	19.3465	0.644526
$902$	−6.14155	−0.204491
$903$	−13.8882	−0.462169
$904$	−4.60339	−0.153107
$905$	0	0
$906$	−1.67300	−0.0555817
$907$	−40.8755	−1.35725	−0.678624	−	0.734486i	$-0.737423\pi$
$907$	−40.8755	−1.35725	−0.678624	+	0.734486i	$0.737423\pi$
$908$	−10.1344	−0.336320
$909$	1.42037	0.0471107
$910$	0	0
$911$	−22.7602	−0.754078	−0.377039	−	0.926197i	$-0.623058\pi$
$911$	−22.7602	−0.754078	−0.377039	+	0.926197i	$0.623058\pi$
$912$	19.9652	0.661114
$913$	−58.0193	−1.92016
$914$	−4.51386	−0.149305
$915$	0	0
$916$	22.3108	0.737169
$917$	−2.96259	−0.0978335
$918$	0.238968	0.00788712
$919$	−9.93884	−0.327852	−0.163926	−	0.986473i	$-0.552416\pi$
$919$	−9.93884	−0.327852	−0.163926	+	0.986473i	$0.552416\pi$
$920$	0	0
$921$	−23.9769	−0.790065
$922$	−0.175929	−0.00579391
$923$	3.79435	0.124893
$924$	11.3119	0.372133
$925$	0	0
$926$	0.498604	0.0163851
$927$	16.3022	0.535433
$928$	1.60038	0.0525351
$929$	45.1407	1.48102	0.740509	−	0.672046i	$-0.234584\pi$
$929$	45.1407	1.48102	0.740509	+	0.672046i	$0.234584\pi$
$930$	0	0
$931$	29.4424	0.964937
$932$	32.9422	1.07906
$933$	−11.5897	−0.379429
$934$	1.71385	0.0560787
$935$	0	0
$936$	−0.182176	−0.00595462
$937$	−19.6118	−0.640689	−0.320345	−	0.947301i	$-0.603799\pi$
$937$	−19.6118	−0.640689	−0.320345	+	0.947301i	$0.603799\pi$
$938$	1.42854	0.0466436
$939$	3.82356	0.124777
$940$	0	0
$941$	41.9039	1.36603	0.683014	−	0.730405i	$-0.260669\pi$
$941$	41.9039	1.36603	0.683014	+	0.730405i	$0.260669\pi$
$942$	0.947688	0.0308774
$943$	−65.4577	−2.13160
$944$	−35.8843	−1.16794
$945$	0	0
$946$	8.48003	0.275710
$947$	−22.3981	−0.727841	−0.363920	−	0.931430i	$-0.618562\pi$
$947$	−22.3981	−0.727841	−0.363920	+	0.931430i	$0.618562\pi$
$948$	22.3153	0.724766
$949$	0.852180	0.0276629
$950$	0	0
$951$	25.2499	0.818783
$952$	−1.06893	−0.0346443
$953$	−15.8510	−0.513463	−0.256732	−	0.966483i	$-0.582646\pi$
$953$	−15.8510	−0.513463	−0.256732	+	0.966483i	$0.582646\pi$
$954$	−1.47550	−0.0477712
$955$	0	0
$956$	−14.3401	−0.463792
$957$	5.08097	0.164245
$958$	−5.37296	−0.173592
$959$	18.3246	0.591734
$960$	0	0
$961$	45.4571	1.46636
$962$	0.430230	0.0138712
$963$	−7.74551	−0.249596
$964$	2.96738	0.0955727
$965$	0	0
$966$	−1.10877	−0.0356740
$967$	5.33341	0.171511	0.0857555	−	0.996316i	$-0.472670\pi$
$967$	5.33341	0.171511	0.0857555	+	0.996316i	$0.472670\pi$
$968$	−7.96444	−0.255987
$969$	−9.08272	−0.291779
$970$	0	0
$971$	−33.9975	−1.09103	−0.545516	−	0.838101i	$-0.683666\pi$
$971$	−33.9975	−1.09103	−0.545516	+	0.838101i	$0.683666\pi$
$972$	1.98177	0.0635654
$973$	−6.86745	−0.220160
$974$	−1.49528	−0.0479117
$975$	0	0
$976$	5.79591	0.185523
$977$	32.2235	1.03092	0.515460	−	0.856914i	$-0.327621\pi$
$977$	32.2235	1.03092	0.515460	+	0.856914i	$0.327621\pi$
$978$	−0.104055	−0.00332730
$979$	−81.3536	−2.60007
$980$	0	0
$981$	15.5762	0.497311
$982$	−0.879621	−0.0280698
$983$	−13.2017	−0.421069	−0.210534	−	0.977586i	$-0.567520\pi$
$983$	−13.2017	−0.421069	−0.210534	+	0.977586i	$0.567520\pi$
$984$	4.81291	0.153430
$985$	0	0
$986$	−0.238968	−0.00761030
$987$	2.38209	0.0758229
$988$	3.44624	0.109639
$989$	90.3816	2.87397
$990$	0	0
$991$	12.5678	0.399230	0.199615	−	0.979874i	$-0.436031\pi$
$991$	12.5678	0.399230	0.199615	+	0.979874i	$0.436031\pi$
$992$	13.9937	0.444300
$993$	−4.25823	−0.135131
$994$	−1.69798	−0.0538568
$995$	0	0
$996$	22.6297	0.717051
$997$	2.28244	0.0722855	0.0361427	−	0.999347i	$-0.488493\pi$
$997$	2.28244	0.0722855	0.0361427	+	0.999347i	$0.488493\pi$
$998$	0.764464	0.0241987
$999$	−9.40342	−0.297511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.bc.1.5	✓	8
3.2	odd	2		6525.2.a.bz.1.4		8
5.2	odd	4		2175.2.c.p.349.9		16
5.3	odd	4		2175.2.c.p.349.8		16
5.4	even	2		2175.2.a.bd.1.4	yes	8
15.14	odd	2		6525.2.a.by.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.bc.1.5	✓	8	1.1	even	1	trivial
2175.2.a.bd.1.4	yes	8	5.4	even	2
2175.2.c.p.349.8		16	5.3	odd	4
2175.2.c.p.349.9		16	5.2	odd	4
6525.2.a.by.1.5		8	15.14	odd	2
6525.2.a.bz.1.4		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)

\(f(q)\)	\(=\)	\(q+0.135002 q^{2} -1.00000 q^{3} -1.98177 q^{4} -0.135002 q^{6} -1.12340 q^{7} -0.537546 q^{8} +1.00000 q^{9} -5.08097 q^{11} +1.98177 q^{12} +0.338903 q^{13} -0.151660 q^{14} +3.89098 q^{16} -1.77011 q^{17} +0.135002 q^{18} -5.13115 q^{19} +1.12340 q^{21} -0.685940 q^{22} -7.31086 q^{23} +0.537546 q^{24} +0.0457525 q^{26} -1.00000 q^{27} +2.22632 q^{28} +1.00000 q^{29} +8.74398 q^{31} +1.60038 q^{32} +5.08097 q^{33} -0.238968 q^{34} -1.98177 q^{36} +9.40342 q^{37} -0.692714 q^{38} -0.338903 q^{39} +8.95349 q^{41} +0.151660 q^{42} -12.3626 q^{43} +10.0693 q^{44} -0.986978 q^{46} +2.12044 q^{47} -3.89098 q^{48} -5.73798 q^{49} +1.77011 q^{51} -0.671630 q^{52} -10.9295 q^{53} -0.135002 q^{54} +0.603878 q^{56} +5.13115 q^{57} +0.135002 q^{58} -9.22244 q^{59} +1.48958 q^{61} +1.18045 q^{62} -1.12340 q^{63} -7.56591 q^{64} +0.685940 q^{66} -9.41936 q^{67} +3.50796 q^{68} +7.31086 q^{69} +11.1960 q^{71} -0.537546 q^{72} +2.51452 q^{73} +1.26948 q^{74} +10.1688 q^{76} +5.70795 q^{77} -0.0457525 q^{78} +11.2602 q^{79} +1.00000 q^{81} +1.20874 q^{82} +11.4189 q^{83} -2.22632 q^{84} -1.66898 q^{86} -1.00000 q^{87} +2.73126 q^{88} +16.0114 q^{89} -0.380723 q^{91} +14.4885 q^{92} -8.74398 q^{93} +0.286263 q^{94} -1.60038 q^{96} +9.76285 q^{97} -0.774637 q^{98} -5.08097 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 8 q^{3} + 12 q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} + 6 q^{11} - 12 q^{12} + 6 q^{13} + 9 q^{14} + 32 q^{16} - 12 q^{17} - 2 q^{18} - 2 q^{21} + 3 q^{22} - 14 q^{23} + 3 q^{24} + 18 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

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