LMFDB - Embedded newform 2175.2.a.bc.1.7

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			1.7
Root			$-1.98486$ 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)

$f(q)$	$=$	\(q+1.98486 q^{2} -1.00000 q^{3} +1.93969 q^{4} -1.98486 q^{6} -2.16633 q^{7} -0.119715 q^{8} +1.00000 q^{9} +3.44851 q^{11} -1.93969 q^{12} +3.74214 q^{13} -4.29987 q^{14} -4.11699 q^{16} -4.33550 q^{17} +1.98486 q^{18} +3.90548 q^{19} +2.16633 q^{21} +6.84482 q^{22} +3.78401 q^{23} +0.119715 q^{24} +7.42765 q^{26} -1.00000 q^{27} -4.20200 q^{28} +1.00000 q^{29} +10.3323 q^{31} -7.93224 q^{32} -3.44851 q^{33} -8.60539 q^{34} +1.93969 q^{36} +7.70559 q^{37} +7.75186 q^{38} -3.74214 q^{39} +7.88284 q^{41} +4.29987 q^{42} -0.975329 q^{43} +6.68902 q^{44} +7.51075 q^{46} -12.1066 q^{47} +4.11699 q^{48} -2.30702 q^{49} +4.33550 q^{51} +7.25858 q^{52} +13.1410 q^{53} -1.98486 q^{54} +0.259342 q^{56} -3.90548 q^{57} +1.98486 q^{58} +1.47781 q^{59} -4.24195 q^{61} +20.5082 q^{62} -2.16633 q^{63} -7.51043 q^{64} -6.84482 q^{66} +6.74456 q^{67} -8.40952 q^{68} -3.78401 q^{69} +10.3646 q^{71} -0.119715 q^{72} +12.3613 q^{73} +15.2946 q^{74} +7.57541 q^{76} -7.47060 q^{77} -7.42765 q^{78} -5.02051 q^{79} +1.00000 q^{81} +15.6464 q^{82} -10.4380 q^{83} +4.20200 q^{84} -1.93590 q^{86} -1.00000 q^{87} -0.412839 q^{88} +4.35405 q^{89} -8.10671 q^{91} +7.33979 q^{92} -10.3323 q^{93} -24.0300 q^{94} +7.93224 q^{96} -3.75171 q^{97} -4.57913 q^{98} +3.44851 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.98486	1.40351	0.701755	−	0.712418i	$-0.252400\pi$
$2$	1.98486	1.40351	0.701755	+	0.712418i	$0.252400\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.93969	0.969843
$5$	0	0
$5$	0	0
$6$	−1.98486	−0.810317
$7$	−2.16633	−0.818795	−0.409398	−	0.912356i	$-0.634261\pi$
$7$	−2.16633	−0.818795	−0.409398	+	0.912356i	$0.634261\pi$
$8$	−0.119715	−0.0423257
$9$	1.00000	0.333333
$10$	0	0
$11$	3.44851	1.03976	0.519882	−	0.854238i	$-0.325976\pi$
$11$	3.44851	1.03976	0.519882	+	0.854238i	$0.325976\pi$
$12$	−1.93969	−0.559939
$13$	3.74214	1.03788	0.518942	−	0.854809i	$-0.326326\pi$
$13$	3.74214	1.03788	0.518942	+	0.854809i	$0.326326\pi$
$14$	−4.29987	−1.14919
$15$	0	0
$16$	−4.11699	−1.02925
$17$	−4.33550	−1.05151	−0.525757	−	0.850635i	$-0.676218\pi$
$17$	−4.33550	−1.05151	−0.525757	+	0.850635i	$0.676218\pi$
$18$	1.98486	0.467837
$19$	3.90548	0.895979	0.447990	−	0.894039i	$-0.352140\pi$
$19$	3.90548	0.895979	0.447990	+	0.894039i	$0.352140\pi$
$20$	0	0
$21$	2.16633	0.472732
$22$	6.84482	1.45932
$23$	3.78401	0.789021	0.394510	−	0.918891i	$-0.370914\pi$
$23$	3.78401	0.789021	0.394510	+	0.918891i	$0.370914\pi$
$24$	0.119715	0.0244368
$25$	0	0
$26$	7.42765	1.45668
$27$	−1.00000	−0.192450
$28$	−4.20200	−0.794103
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	10.3323	1.85574	0.927869	−	0.372907i	$-0.121639\pi$
$31$	10.3323	1.85574	0.927869	+	0.372907i	$0.121639\pi$
$32$	−7.93224	−1.40223
$33$	−3.44851	−0.600308
$34$	−8.60539	−1.47581
$35$	0	0
$36$	1.93969	0.323281
$37$	7.70559	1.26679	0.633396	−	0.773828i	$-0.281661\pi$
$37$	7.70559	1.26679	0.633396	+	0.773828i	$0.281661\pi$
$38$	7.75186	1.25752
$39$	−3.74214	−0.599223
$40$	0	0
$41$	7.88284	1.23109	0.615546	−	0.788101i	$-0.288935\pi$
$41$	7.88284	1.23109	0.615546	+	0.788101i	$0.288935\pi$
$42$	4.29987	0.663484
$43$	−0.975329	−0.148736	−0.0743681	−	0.997231i	$-0.523694\pi$
$43$	−0.975329	−0.148736	−0.0743681	+	0.997231i	$0.523694\pi$
$44$	6.68902	1.00841
$45$	0	0
$46$	7.51075	1.10740
$47$	−12.1066	−1.76593	−0.882967	−	0.469436i	$-0.844457\pi$
$47$	−12.1066	−1.76593	−0.882967	+	0.469436i	$0.844457\pi$
$48$	4.11699	0.594236
$49$	−2.30702	−0.329575
$50$	0	0
$51$	4.33550	0.607092
$52$	7.25858	1.00658
$53$	13.1410	1.80505	0.902527	−	0.430633i	$-0.141710\pi$
$53$	13.1410	1.80505	0.902527	+	0.430633i	$0.141710\pi$
$54$	−1.98486	−0.270106
$55$	0	0
$56$	0.259342	0.0346561
$57$	−3.90548	−0.517294
$58$	1.98486	0.260625
$59$	1.47781	0.192395	0.0961976	−	0.995362i	$-0.469332\pi$
$59$	1.47781	0.192395	0.0961976	+	0.995362i	$0.469332\pi$
$60$	0	0
$61$	−4.24195	−0.543126	−0.271563	−	0.962421i	$-0.587541\pi$
$61$	−4.24195	−0.543126	−0.271563	+	0.962421i	$0.587541\pi$
$62$	20.5082	2.60455
$63$	−2.16633	−0.272932
$64$	−7.51043	−0.938804
$65$	0	0
$66$	−6.84482	−0.842539
$67$	6.74456	0.823979	0.411989	−	0.911189i	$-0.364834\pi$
$67$	6.74456	0.823979	0.411989	+	0.911189i	$0.364834\pi$
$68$	−8.40952	−1.01980
$69$	−3.78401	−0.455541
$70$	0	0
$71$	10.3646	1.23005	0.615026	−	0.788507i	$-0.289145\pi$
$71$	10.3646	1.23005	0.615026	+	0.788507i	$0.289145\pi$
$72$	−0.119715	−0.0141086
$73$	12.3613	1.44678	0.723390	−	0.690440i	$-0.242583\pi$
$73$	12.3613	1.44678	0.723390	+	0.690440i	$0.242583\pi$
$74$	15.2946	1.77796
$75$	0	0
$76$	7.57541	0.868959
$77$	−7.47060	−0.851354
$78$	−7.42765	−0.841015
$79$	−5.02051	−0.564852	−0.282426	−	0.959289i	$-0.591139\pi$
$79$	−5.02051	−0.564852	−0.282426	+	0.959289i	$0.591139\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	15.6464	1.72785
$83$	−10.4380	−1.14572	−0.572861	−	0.819652i	$-0.694167\pi$
$83$	−10.4380	−1.14572	−0.572861	+	0.819652i	$0.694167\pi$
$84$	4.20200	0.458475
$85$	0	0
$86$	−1.93590	−0.208753
$87$	−1.00000	−0.107211
$88$	−0.412839	−0.0440087
$89$	4.35405	0.461528	0.230764	−	0.973010i	$-0.425877\pi$
$89$	4.35405	0.461528	0.230764	+	0.973010i	$0.425877\pi$
$90$	0	0
$91$	−8.10671	−0.849814
$92$	7.33979	0.765226
$93$	−10.3323	−1.07141
$94$	−24.0300	−2.47851
$95$	0	0
$96$	7.93224	0.809580
$97$	−3.75171	−0.380929	−0.190464	−	0.981694i	$-0.560999\pi$
$97$	−3.75171	−0.380929	−0.190464	+	0.981694i	$0.560999\pi$
$98$	−4.57913	−0.462562
$99$	3.44851	0.346588
$100$	0	0
$101$	4.04201	0.402195	0.201097	−	0.979571i	$-0.435549\pi$
$101$	4.04201	0.402195	0.201097	+	0.979571i	$0.435549\pi$
$102$	8.60539	0.852060
$103$	−1.44387	−0.142269	−0.0711343	−	0.997467i	$-0.522662\pi$
$103$	−1.44387	−0.142269	−0.0711343	+	0.997467i	$0.522662\pi$
$104$	−0.447991	−0.0439292
$105$	0	0
$106$	26.0831	2.53341
$107$	16.1134	1.55774	0.778868	−	0.627188i	$-0.215794\pi$
$107$	16.1134	1.55774	0.778868	+	0.627188i	$0.215794\pi$
$108$	−1.93969	−0.186646
$109$	−6.97181	−0.667779	−0.333889	−	0.942612i	$-0.608361\pi$
$109$	−6.97181	−0.667779	−0.333889	+	0.942612i	$0.608361\pi$
$110$	0	0
$111$	−7.70559	−0.731382
$112$	8.91875	0.842743
$113$	−14.7342	−1.38608	−0.693038	−	0.720901i	$-0.743728\pi$
$113$	−14.7342	−1.38608	−0.693038	+	0.720901i	$0.743728\pi$
$114$	−7.75186	−0.726028
$115$	0	0
$116$	1.93969	0.180095
$117$	3.74214	0.345961
$118$	2.93326	0.270029
$119$	9.39213	0.860975
$120$	0	0
$121$	0.892199	0.0811090
$122$	−8.41970	−0.762284
$123$	−7.88284	−0.710772
$124$	20.0414	1.79977
$125$	0	0
$126$	−4.29987	−0.383063
$127$	−13.2013	−1.17143	−0.585714	−	0.810518i	$-0.699186\pi$
$127$	−13.2013	−1.17143	−0.585714	+	0.810518i	$0.699186\pi$
$128$	0.957286	0.0846129
$129$	0.975329	0.0858729
$130$	0	0
$131$	−7.85914	−0.686656	−0.343328	−	0.939216i	$-0.611554\pi$
$131$	−7.85914	−0.686656	−0.343328	+	0.939216i	$0.611554\pi$
$132$	−6.68902	−0.582204
$133$	−8.46056	−0.733624
$134$	13.3870	1.15646
$135$	0	0
$136$	0.519026	0.0445061
$137$	−2.00728	−0.171493	−0.0857466	−	0.996317i	$-0.527328\pi$
$137$	−2.00728	−0.171493	−0.0857466	+	0.996317i	$0.527328\pi$
$138$	−7.51075	−0.639357
$139$	16.1970	1.37381	0.686905	−	0.726747i	$-0.258969\pi$
$139$	16.1970	1.37381	0.686905	+	0.726747i	$0.258969\pi$
$140$	0	0
$141$	12.1066	1.01956
$142$	20.5723	1.72639
$143$	12.9048	1.07915
$144$	−4.11699	−0.343083
$145$	0	0
$146$	24.5355	2.03057
$147$	2.30702	0.190280
$148$	14.9464	1.22859
$149$	8.07183	0.661270	0.330635	−	0.943759i	$-0.392737\pi$
$149$	8.07183	0.661270	0.330635	+	0.943759i	$0.392737\pi$
$150$	0	0
$151$	0.404386	0.0329085	0.0164542	−	0.999865i	$-0.494762\pi$
$151$	0.404386	0.0329085	0.0164542	+	0.999865i	$0.494762\pi$
$152$	−0.467546	−0.0379230
$153$	−4.33550	−0.350505
$154$	−14.8281	−1.19488
$155$	0	0
$156$	−7.25858	−0.581152
$157$	11.2927	0.901258	0.450629	−	0.892711i	$-0.351200\pi$
$157$	11.2927	0.901258	0.450629	+	0.892711i	$0.351200\pi$
$158$	−9.96503	−0.792775
$159$	−13.1410	−1.04215
$160$	0	0
$161$	−8.19741	−0.646046
$162$	1.98486	0.155946
$163$	6.76053	0.529526	0.264763	−	0.964314i	$-0.414706\pi$
$163$	6.76053	0.529526	0.264763	+	0.964314i	$0.414706\pi$
$164$	15.2902	1.19397
$165$	0	0
$166$	−20.7181	−1.60803
$167$	−23.6942	−1.83351	−0.916755	−	0.399449i	$-0.869201\pi$
$167$	−23.6942	−1.83351	−0.916755	+	0.399449i	$0.869201\pi$
$168$	−0.259342	−0.0200087
$169$	1.00364	0.0772032
$170$	0	0
$171$	3.90548	0.298660
$172$	−1.89183	−0.144251
$173$	8.47285	0.644179	0.322089	−	0.946709i	$-0.395615\pi$
$173$	8.47285	0.644179	0.322089	+	0.946709i	$0.395615\pi$
$174$	−1.98486	−0.150472
$175$	0	0
$176$	−14.1975	−1.07017
$177$	−1.47781	−0.111079
$178$	8.64220	0.647760
$179$	−9.43589	−0.705271	−0.352636	−	0.935761i	$-0.614715\pi$
$179$	−9.43589	−0.705271	−0.352636	+	0.935761i	$0.614715\pi$
$180$	0	0
$181$	−15.9628	−1.18650	−0.593252	−	0.805017i	$-0.702156\pi$
$181$	−15.9628	−1.18650	−0.593252	+	0.805017i	$0.702156\pi$
$182$	−16.0907	−1.19272
$183$	4.24195	0.313574
$184$	−0.453003	−0.0333959
$185$	0	0
$186$	−20.5082	−1.50374
$187$	−14.9510	−1.09333
$188$	−23.4831	−1.71268
$189$	2.16633	0.157577
$190$	0	0
$191$	−12.2070	−0.883269	−0.441635	−	0.897195i	$-0.645601\pi$
$191$	−12.2070	−0.883269	−0.441635	+	0.897195i	$0.645601\pi$
$192$	7.51043	0.542019
$193$	−12.8827	−0.927316	−0.463658	−	0.886014i	$-0.653463\pi$
$193$	−12.8827	−0.927316	−0.463658	+	0.886014i	$0.653463\pi$
$194$	−7.44664	−0.534638
$195$	0	0
$196$	−4.47490	−0.319636
$197$	−5.64807	−0.402409	−0.201204	−	0.979549i	$-0.564486\pi$
$197$	−5.64807	−0.402409	−0.201204	+	0.979549i	$0.564486\pi$
$198$	6.84482	0.486440
$199$	−2.79194	−0.197915	−0.0989577	−	0.995092i	$-0.531551\pi$
$199$	−2.79194	−0.197915	−0.0989577	+	0.995092i	$0.531551\pi$
$200$	0	0
$201$	−6.74456	−0.475724
$202$	8.02284	0.564485
$203$	−2.16633	−0.152046
$204$	8.40952	0.588784
$205$	0	0
$206$	−2.86589	−0.199676
$207$	3.78401	0.263007
$208$	−15.4064	−1.06824
$209$	13.4681	0.931607
$210$	0	0
$211$	19.7072	1.35670	0.678349	−	0.734740i	$-0.262696\pi$
$211$	19.7072	1.35670	0.678349	+	0.734740i	$0.262696\pi$
$212$	25.4894	1.75062
$213$	−10.3646	−0.710171
$214$	31.9828	2.18630
$215$	0	0
$216$	0.119715	0.00814558
$217$	−22.3832	−1.51947
$218$	−13.8381	−0.937235
$219$	−12.3613	−0.835299
$220$	0	0
$221$	−16.2241	−1.09135
$222$	−15.2946	−1.02650
$223$	−23.7198	−1.58840	−0.794199	−	0.607658i	$-0.792109\pi$
$223$	−23.7198	−1.58840	−0.794199	+	0.607658i	$0.792109\pi$
$224$	17.1838	1.14814
$225$	0	0
$226$	−29.2454	−1.94537
$227$	−21.4371	−1.42283	−0.711414	−	0.702773i	$-0.751945\pi$
$227$	−21.4371	−1.42283	−0.711414	+	0.702773i	$0.751945\pi$
$228$	−7.57541	−0.501694
$229$	8.66186	0.572392	0.286196	−	0.958171i	$-0.407609\pi$
$229$	8.66186	0.572392	0.286196	+	0.958171i	$0.407609\pi$
$230$	0	0
$231$	7.47060	0.491529
$232$	−0.119715	−0.00785969
$233$	−15.7073	−1.02902	−0.514509	−	0.857485i	$-0.672026\pi$
$233$	−15.7073	−1.02902	−0.514509	+	0.857485i	$0.672026\pi$
$234$	7.42765	0.485561
$235$	0	0
$236$	2.86650	0.186593
$237$	5.02051	0.326117
$238$	18.6421	1.20839
$239$	−5.29189	−0.342304	−0.171152	−	0.985245i	$-0.554749\pi$
$239$	−5.29189	−0.342304	−0.171152	+	0.985245i	$0.554749\pi$
$240$	0	0
$241$	14.3138	0.922031	0.461015	−	0.887392i	$-0.347485\pi$
$241$	14.3138	0.922031	0.461015	+	0.887392i	$0.347485\pi$
$242$	1.77089	0.113837
$243$	−1.00000	−0.0641500
$244$	−8.22805	−0.526747
$245$	0	0
$246$	−15.6464	−0.997576
$247$	14.6149	0.929923
$248$	−1.23693	−0.0785454
$249$	10.4380	0.661483
$250$	0	0
$251$	−0.111554	−0.00704121	−0.00352061	−	0.999994i	$-0.501121\pi$
$251$	−0.111554	−0.00704121	−0.00352061	+	0.999994i	$0.501121\pi$
$252$	−4.20200	−0.264701
$253$	13.0492	0.820395
$254$	−26.2028	−1.64411
$255$	0	0
$256$	16.9209	1.05756
$257$	−8.53838	−0.532609	−0.266305	−	0.963889i	$-0.585803\pi$
$257$	−8.53838	−0.532609	−0.266305	+	0.963889i	$0.585803\pi$
$258$	1.93590	0.120524
$259$	−16.6928	−1.03724
$260$	0	0
$261$	1.00000	0.0618984
$262$	−15.5993	−0.963729
$263$	9.48901	0.585117	0.292559	−	0.956248i	$-0.405493\pi$
$263$	9.48901	0.585117	0.292559	+	0.956248i	$0.405493\pi$
$264$	0.412839	0.0254085
$265$	0	0
$266$	−16.7931	−1.02965
$267$	−4.35405	−0.266463
$268$	13.0823	0.799130
$269$	1.98710	0.121156	0.0605778	−	0.998163i	$-0.480706\pi$
$269$	1.98710	0.121156	0.0605778	+	0.998163i	$0.480706\pi$
$270$	0	0
$271$	25.0189	1.51979	0.759896	−	0.650045i	$-0.225250\pi$
$271$	25.0189	1.51979	0.759896	+	0.650045i	$0.225250\pi$
$272$	17.8492	1.08227
$273$	8.10671	0.490641
$274$	−3.98417	−0.240693
$275$	0	0
$276$	−7.33979	−0.441804
$277$	25.7950	1.54987	0.774935	−	0.632041i	$-0.217783\pi$
$277$	25.7950	1.54987	0.774935	+	0.632041i	$0.217783\pi$
$278$	32.1488	1.92816
$279$	10.3323	0.618579
$280$	0	0
$281$	9.78709	0.583849	0.291924	−	0.956441i	$-0.405704\pi$
$281$	9.78709	0.583849	0.291924	+	0.956441i	$0.405704\pi$
$282$	24.0300	1.43097
$283$	−26.6275	−1.58284	−0.791421	−	0.611272i	$-0.790658\pi$
$283$	−26.6275	−1.58284	−0.791421	+	0.611272i	$0.790658\pi$
$284$	20.1041	1.19296
$285$	0	0
$286$	25.6143	1.51460
$287$	−17.0768	−1.00801
$288$	−7.93224	−0.467412
$289$	1.79660	0.105682
$290$	0	0
$291$	3.75171	0.219929
$292$	23.9770	1.40315
$293$	27.9226	1.63126	0.815628	−	0.578576i	$-0.196392\pi$
$293$	27.9226	1.63126	0.815628	+	0.578576i	$0.196392\pi$
$294$	4.57913	0.267060
$295$	0	0
$296$	−0.922476	−0.0536178
$297$	−3.44851	−0.200103
$298$	16.0215	0.928100
$299$	14.1603	0.818912
$300$	0	0
$301$	2.11288	0.121785
$302$	0.802652	0.0461874
$303$	−4.04201	−0.232207
$304$	−16.0788	−0.922185
$305$	0	0
$306$	−8.60539	−0.491937
$307$	−33.6418	−1.92004	−0.960021	−	0.279930i	$-0.909689\pi$
$307$	−33.6418	−1.92004	−0.960021	+	0.279930i	$0.909689\pi$
$308$	−14.4906	−0.825679
$309$	1.44387	0.0821389
$310$	0	0
$311$	−11.2518	−0.638030	−0.319015	−	0.947750i	$-0.603352\pi$
$311$	−11.2518	−0.638030	−0.319015	+	0.947750i	$0.603352\pi$
$312$	0.447991	0.0253625
$313$	−5.97942	−0.337977	−0.168988	−	0.985618i	$-0.554050\pi$
$313$	−5.97942	−0.337977	−0.168988	+	0.985618i	$0.554050\pi$
$314$	22.4145	1.26493
$315$	0	0
$316$	−9.73822	−0.547817
$317$	7.06727	0.396938	0.198469	−	0.980107i	$-0.436403\pi$
$317$	7.06727	0.396938	0.198469	+	0.980107i	$0.436403\pi$
$318$	−26.0831	−1.46267
$319$	3.44851	0.193079
$320$	0	0
$321$	−16.1134	−0.899360
$322$	−16.2707	−0.906733
$323$	−16.9322	−0.942135
$324$	1.93969	0.107760
$325$	0	0
$326$	13.4187	0.743195
$327$	6.97181	0.385542
$328$	−0.943695	−0.0521069
$329$	26.2269	1.44594
$330$	0	0
$331$	1.01858	0.0559864	0.0279932	−	0.999608i	$-0.491088\pi$
$331$	1.01858	0.0559864	0.0279932	+	0.999608i	$0.491088\pi$
$332$	−20.2465	−1.11117
$333$	7.70559	0.422264
$334$	−47.0297	−2.57335
$335$	0	0
$336$	−8.91875	−0.486558
$337$	0.986769	0.0537527	0.0268764	−	0.999639i	$-0.491444\pi$
$337$	0.986769	0.0537527	0.0268764	+	0.999639i	$0.491444\pi$
$338$	1.99209	0.108356
$339$	14.7342	0.800251
$340$	0	0
$341$	35.6310	1.92953
$342$	7.75186	0.419172
$343$	20.1621	1.08865
$344$	0.116762	0.00629537
$345$	0	0
$346$	16.8175	0.904112
$347$	−11.7873	−0.632777	−0.316388	−	0.948630i	$-0.602470\pi$
$347$	−11.7873	−0.632777	−0.316388	+	0.948630i	$0.602470\pi$
$348$	−1.93969	−0.103978
$349$	3.27618	0.175370	0.0876850	−	0.996148i	$-0.472053\pi$
$349$	3.27618	0.175370	0.0876850	+	0.996148i	$0.472053\pi$
$350$	0	0
$351$	−3.74214	−0.199741
$352$	−27.3544	−1.45799
$353$	21.4744	1.14296	0.571482	−	0.820614i	$-0.306369\pi$
$353$	21.4744	1.14296	0.571482	+	0.820614i	$0.306369\pi$
$354$	−2.93326	−0.155901
$355$	0	0
$356$	8.44549	0.447610
$357$	−9.39213	−0.497084
$358$	−18.7290	−0.989856
$359$	2.82558	0.149129	0.0745643	−	0.997216i	$-0.476243\pi$
$359$	2.82558	0.149129	0.0745643	+	0.997216i	$0.476243\pi$
$360$	0	0
$361$	−3.74720	−0.197221
$362$	−31.6839	−1.66527
$363$	−0.892199	−0.0468283
$364$	−15.7245	−0.824186
$365$	0	0
$366$	8.41970	0.440105
$367$	−30.5656	−1.59551	−0.797757	−	0.602979i	$-0.793980\pi$
$367$	−30.5656	−1.59551	−0.797757	+	0.602979i	$0.793980\pi$
$368$	−15.5787	−0.812098
$369$	7.88284	0.410364
$370$	0	0
$371$	−28.4677	−1.47797
$372$	−20.0414	−1.03910
$373$	18.9296	0.980137	0.490069	−	0.871684i	$-0.336972\pi$
$373$	18.9296	0.980137	0.490069	+	0.871684i	$0.336972\pi$
$374$	−29.6757	−1.53450
$375$	0	0
$376$	1.44935	0.0747444
$377$	3.74214	0.192730
$378$	4.29987	0.221161
$379$	34.5194	1.77314	0.886572	−	0.462591i	$-0.153080\pi$
$379$	34.5194	1.77314	0.886572	+	0.462591i	$0.153080\pi$
$380$	0	0
$381$	13.2013	0.676324
$382$	−24.2293	−1.23968
$383$	−24.1113	−1.23203	−0.616014	−	0.787735i	$-0.711254\pi$
$383$	−24.1113	−1.23203	−0.616014	+	0.787735i	$0.711254\pi$
$384$	−0.957286	−0.0488513
$385$	0	0
$386$	−25.5704	−1.30150
$387$	−0.975329	−0.0495788
$388$	−7.27715	−0.369441
$389$	1.62643	0.0824633	0.0412317	−	0.999150i	$-0.486872\pi$
$389$	1.62643	0.0824633	0.0412317	+	0.999150i	$0.486872\pi$
$390$	0	0
$391$	−16.4056	−0.829667
$392$	0.276186	0.0139495
$393$	7.85914	0.396441
$394$	−11.2107	−0.564785
$395$	0	0
$396$	6.68902	0.336136
$397$	30.4581	1.52865	0.764325	−	0.644831i	$-0.223072\pi$
$397$	30.4581	1.52865	0.764325	+	0.644831i	$0.223072\pi$
$398$	−5.54162	−0.277777
$399$	8.46056	0.423558
$400$	0	0
$401$	21.5746	1.07739	0.538693	−	0.842502i	$-0.318918\pi$
$401$	21.5746	1.07739	0.538693	+	0.842502i	$0.318918\pi$
$402$	−13.3870	−0.667684
$403$	38.6650	1.92604
$404$	7.84023	0.390066
$405$	0	0
$406$	−4.29987	−0.213399
$407$	26.5728	1.31716
$408$	−0.519026	−0.0256956
$409$	−12.4960	−0.617886	−0.308943	−	0.951080i	$-0.599975\pi$
$409$	−12.4960	−0.617886	−0.308943	+	0.951080i	$0.599975\pi$
$410$	0	0
$411$	2.00728	0.0990117
$412$	−2.80065	−0.137978
$413$	−3.20143	−0.157532
$414$	7.51075	0.369133
$415$	0	0
$416$	−29.6836	−1.45536
$417$	−16.1970	−0.793170
$418$	26.7323	1.30752
$419$	−11.1267	−0.543574	−0.271787	−	0.962357i	$-0.587615\pi$
$419$	−11.1267	−0.543574	−0.271787	+	0.962357i	$0.587615\pi$
$420$	0	0
$421$	−5.92063	−0.288554	−0.144277	−	0.989537i	$-0.546086\pi$
$421$	−5.92063	−0.288554	−0.144277	+	0.989537i	$0.546086\pi$
$422$	39.1161	1.90414
$423$	−12.1066	−0.588644
$424$	−1.57318	−0.0764002
$425$	0	0
$426$	−20.5723	−0.996733
$427$	9.18946	0.444709
$428$	31.2548	1.51076
$429$	−12.9048	−0.623050
$430$	0	0
$431$	−26.7078	−1.28647	−0.643234	−	0.765670i	$-0.722408\pi$
$431$	−26.7078	−1.28647	−0.643234	+	0.765670i	$0.722408\pi$
$432$	4.11699	0.198079
$433$	−1.00946	−0.0485113	−0.0242557	−	0.999706i	$-0.507722\pi$
$433$	−1.00946	−0.0485113	−0.0242557	+	0.999706i	$0.507722\pi$
$434$	−44.4276	−2.13259
$435$	0	0
$436$	−13.5231	−0.647640
$437$	14.7784	0.706946
$438$	−24.5355	−1.17235
$439$	−27.9498	−1.33397	−0.666985	−	0.745071i	$-0.732415\pi$
$439$	−27.9498	−1.33397	−0.666985	+	0.745071i	$0.732415\pi$
$440$	0	0
$441$	−2.30702	−0.109858
$442$	−32.2026	−1.53172
$443$	13.6698	0.649472	0.324736	−	0.945805i	$-0.394725\pi$
$443$	13.6698	0.649472	0.324736	+	0.945805i	$0.394725\pi$
$444$	−14.9464	−0.709326
$445$	0	0
$446$	−47.0806	−2.22933
$447$	−8.07183	−0.381785
$448$	16.2701	0.768688
$449$	−16.8518	−0.795287	−0.397643	−	0.917540i	$-0.630172\pi$
$449$	−16.8518	−0.795287	−0.397643	+	0.917540i	$0.630172\pi$
$450$	0	0
$451$	27.1840	1.28005
$452$	−28.5797	−1.34428
$453$	−0.404386	−0.0189997
$454$	−42.5497	−1.99696
$455$	0	0
$456$	0.467546	0.0218948
$457$	31.0628	1.45306	0.726529	−	0.687135i	$-0.241132\pi$
$457$	31.0628	1.45306	0.726529	+	0.687135i	$0.241132\pi$
$458$	17.1926	0.803358
$459$	4.33550	0.202364
$460$	0	0
$461$	12.8686	0.599352	0.299676	−	0.954041i	$-0.403121\pi$
$461$	12.8686	0.599352	0.299676	+	0.954041i	$0.403121\pi$
$462$	14.8281	0.689867
$463$	15.7824	0.733469	0.366734	−	0.930326i	$-0.380476\pi$
$463$	15.7824	0.733469	0.366734	+	0.930326i	$0.380476\pi$
$464$	−4.11699	−0.191126
$465$	0	0
$466$	−31.1768	−1.44424
$467$	2.64850	0.122558	0.0612791	−	0.998121i	$-0.480482\pi$
$467$	2.64850	0.122558	0.0612791	+	0.998121i	$0.480482\pi$
$468$	7.25858	0.335528
$469$	−14.6109	−0.674670
$470$	0	0
$471$	−11.2927	−0.520342
$472$	−0.176917	−0.00814326
$473$	−3.36343	−0.154651
$474$	9.96503	0.457709
$475$	0	0
$476$	18.2178	0.835010
$477$	13.1410	0.601685
$478$	−10.5037	−0.480427
$479$	−42.8226	−1.95661	−0.978307	−	0.207161i	$-0.933578\pi$
$479$	−42.8226	−1.95661	−0.978307	+	0.207161i	$0.933578\pi$
$480$	0	0
$481$	28.8354	1.31478
$482$	28.4109	1.29408
$483$	8.19741	0.372995
$484$	1.73059	0.0786630
$485$	0	0
$486$	−1.98486	−0.0900353
$487$	17.5946	0.797289	0.398644	−	0.917106i	$-0.369481\pi$
$487$	17.5946	0.797289	0.398644	+	0.917106i	$0.369481\pi$
$488$	0.507826	0.0229882
$489$	−6.76053	−0.305722
$490$	0	0
$491$	−31.7466	−1.43270	−0.716351	−	0.697740i	$-0.754189\pi$
$491$	−31.7466	−1.43270	−0.716351	+	0.697740i	$0.754189\pi$
$492$	−15.2902	−0.689337
$493$	−4.33550	−0.195261
$494$	29.0086	1.30516
$495$	0	0
$496$	−42.5380	−1.91001
$497$	−22.4531	−1.00716
$498$	20.7181	0.928399
$499$	10.7922	0.483126	0.241563	−	0.970385i	$-0.422340\pi$
$499$	10.7922	0.483126	0.241563	+	0.970385i	$0.422340\pi$
$500$	0	0
$501$	23.6942	1.05858
$502$	−0.221419	−0.00988242
$503$	−11.7995	−0.526114	−0.263057	−	0.964780i	$-0.584731\pi$
$503$	−11.7995	−0.526114	−0.263057	+	0.964780i	$0.584731\pi$
$504$	0.259342	0.0115520
$505$	0	0
$506$	25.9009	1.15143
$507$	−1.00364	−0.0445733
$508$	−25.6064	−1.13610
$509$	13.6611	0.605516	0.302758	−	0.953068i	$-0.402093\pi$
$509$	13.6611	0.605516	0.302758	+	0.953068i	$0.402093\pi$
$510$	0	0
$511$	−26.7786	−1.18462
$512$	31.6712	1.39968
$513$	−3.90548	−0.172431
$514$	−16.9475	−0.747523
$515$	0	0
$516$	1.89183	0.0832833
$517$	−41.7498	−1.83615
$518$	−33.1330	−1.45578
$519$	−8.47285	−0.371917
$520$	0	0
$521$	5.55126	0.243205	0.121603	−	0.992579i	$-0.461197\pi$
$521$	5.55126	0.243205	0.121603	+	0.992579i	$0.461197\pi$
$522$	1.98486	0.0868751
$523$	28.8905	1.26329	0.631647	−	0.775256i	$-0.282379\pi$
$523$	28.8905	1.26329	0.631647	+	0.775256i	$0.282379\pi$
$524$	−15.2443	−0.665949
$525$	0	0
$526$	18.8344	0.821218
$527$	−44.7958	−1.95133
$528$	14.1975	0.617866
$529$	−8.68126	−0.377446
$530$	0	0
$531$	1.47781	0.0641317
$532$	−16.4108	−0.711500
$533$	29.4987	1.27773
$534$	−8.64220	−0.373984
$535$	0	0
$536$	−0.807426	−0.0348755
$537$	9.43589	0.407189
$538$	3.94412	0.170043
$539$	−7.95578	−0.342680
$540$	0	0
$541$	−26.7544	−1.15026	−0.575132	−	0.818061i	$-0.695049\pi$
$541$	−26.7544	−1.15026	−0.575132	+	0.818061i	$0.695049\pi$
$542$	49.6592	2.13304
$543$	15.9628	0.685028
$544$	34.3902	1.47447
$545$	0	0
$546$	16.0907	0.688619
$547$	43.9711	1.88007	0.940035	−	0.341077i	$-0.110792\pi$
$547$	43.9711	1.88007	0.940035	+	0.341077i	$0.110792\pi$
$548$	−3.89349	−0.166322
$549$	−4.24195	−0.181042
$550$	0	0
$551$	3.90548	0.166379
$552$	0.453003	0.0192811
$553$	10.8761	0.462498
$554$	51.1995	2.17526
$555$	0	0
$556$	31.4171	1.33238
$557$	−29.0132	−1.22933	−0.614665	−	0.788788i	$-0.710709\pi$
$557$	−29.0132	−1.22933	−0.614665	+	0.788788i	$0.710709\pi$
$558$	20.5082	0.868183
$559$	−3.64982	−0.154371
$560$	0	0
$561$	14.9510	0.631232
$562$	19.4260	0.819438
$563$	−32.7636	−1.38082	−0.690410	−	0.723419i	$-0.742570\pi$
$563$	−32.7636	−1.38082	−0.690410	+	0.723419i	$0.742570\pi$
$564$	23.4831	0.988815
$565$	0	0
$566$	−52.8520	−2.22154
$567$	−2.16633	−0.0909772
$568$	−1.24080	−0.0520628
$569$	−46.5934	−1.95330	−0.976648	−	0.214848i	$-0.931074\pi$
$569$	−46.5934	−1.95330	−0.976648	+	0.214848i	$0.931074\pi$
$570$	0	0
$571$	−13.2917	−0.556238	−0.278119	−	0.960547i	$-0.589711\pi$
$571$	−13.2917	−0.556238	−0.278119	+	0.960547i	$0.589711\pi$
$572$	25.0313	1.04661
$573$	12.2070	0.509956
$574$	−33.8952	−1.41476
$575$	0	0
$576$	−7.51043	−0.312935
$577$	5.53380	0.230375	0.115187	−	0.993344i	$-0.463253\pi$
$577$	5.53380	0.230375	0.115187	+	0.993344i	$0.463253\pi$
$578$	3.56600	0.148326
$579$	12.8827	0.535386
$580$	0	0
$581$	22.6122	0.938112
$582$	7.44664	0.308673
$583$	45.3168	1.87683
$584$	−1.47983	−0.0612360
$585$	0	0
$586$	55.4226	2.28949
$587$	−33.5077	−1.38301	−0.691505	−	0.722372i	$-0.743052\pi$
$587$	−33.5077	−1.38301	−0.691505	+	0.722372i	$0.743052\pi$
$588$	4.47490	0.184542
$589$	40.3527	1.66270
$590$	0	0
$591$	5.64807	0.232331
$592$	−31.7238	−1.30384
$593$	16.3242	0.670355	0.335178	−	0.942155i	$-0.391204\pi$
$593$	16.3242	0.670355	0.335178	+	0.942155i	$0.391204\pi$
$594$	−6.84482	−0.280846
$595$	0	0
$596$	15.6568	0.641328
$597$	2.79194	0.114267
$598$	28.1063	1.14935
$599$	−19.8302	−0.810242	−0.405121	−	0.914263i	$-0.632771\pi$
$599$	−19.8302	−0.810242	−0.405121	+	0.914263i	$0.632771\pi$
$600$	0	0
$601$	−11.2018	−0.456933	−0.228466	−	0.973552i	$-0.573371\pi$
$601$	−11.2018	−0.456933	−0.228466	+	0.973552i	$0.573371\pi$
$602$	4.19379	0.170926
$603$	6.74456	0.274660
$604$	0.784382	0.0319161
$605$	0	0
$606$	−8.02284	−0.325905
$607$	26.7066	1.08399	0.541994	−	0.840382i	$-0.317669\pi$
$607$	26.7066	1.08399	0.541994	+	0.840382i	$0.317669\pi$
$608$	−30.9792	−1.25637
$609$	2.16633	0.0877840
$610$	0	0
$611$	−45.3048	−1.83283
$612$	−8.40952	−0.339935
$613$	22.0053	0.888784	0.444392	−	0.895832i	$-0.353420\pi$
$613$	22.0053	0.888784	0.444392	+	0.895832i	$0.353420\pi$
$614$	−66.7745	−2.69480
$615$	0	0
$616$	0.894344	0.0360341
$617$	35.1749	1.41609	0.708044	−	0.706168i	$-0.249578\pi$
$617$	35.1749	1.41609	0.708044	+	0.706168i	$0.249578\pi$
$618$	2.86589	0.115283
$619$	−15.6201	−0.627826	−0.313913	−	0.949452i	$-0.601640\pi$
$619$	−15.6201	−0.627826	−0.313913	+	0.949452i	$0.601640\pi$
$620$	0	0
$621$	−3.78401	−0.151847
$622$	−22.3333	−0.895483
$623$	−9.43230	−0.377897
$624$	15.4064	0.616748
$625$	0	0
$626$	−11.8683	−0.474354
$627$	−13.4681	−0.537864
$628$	21.9044	0.874079
$629$	−33.4076	−1.33205
$630$	0	0
$631$	37.6084	1.49717	0.748583	−	0.663041i	$-0.230735\pi$
$631$	37.6084	1.49717	0.748583	+	0.663041i	$0.230735\pi$
$632$	0.601031	0.0239077
$633$	−19.7072	−0.783290
$634$	14.0276	0.557106
$635$	0	0
$636$	−25.4894	−1.01072
$637$	−8.63321	−0.342060
$638$	6.84482	0.270989
$639$	10.3646	0.410017
$640$	0	0
$641$	5.74582	0.226946	0.113473	−	0.993541i	$-0.463802\pi$
$641$	5.74582	0.226946	0.113473	+	0.993541i	$0.463802\pi$
$642$	−31.9828	−1.26226
$643$	−23.9831	−0.945802	−0.472901	−	0.881116i	$-0.656793\pi$
$643$	−23.9831	−0.945802	−0.472901	+	0.881116i	$0.656793\pi$
$644$	−15.9004	−0.626564
$645$	0	0
$646$	−33.6082	−1.32230
$647$	3.39096	0.133312	0.0666562	−	0.997776i	$-0.478767\pi$
$647$	3.39096	0.133312	0.0666562	+	0.997776i	$0.478767\pi$
$648$	−0.119715	−0.00470286
$649$	5.09625	0.200045
$650$	0	0
$651$	22.3832	0.877266
$652$	13.1133	0.513557
$653$	−5.53243	−0.216501	−0.108250	−	0.994124i	$-0.534525\pi$
$653$	−5.53243	−0.216501	−0.108250	+	0.994124i	$0.534525\pi$
$654$	13.8381	0.541113
$655$	0	0
$656$	−32.4536	−1.26710
$657$	12.3613	0.482260
$658$	52.0569	2.02939
$659$	19.5303	0.760794	0.380397	−	0.924823i	$-0.375787\pi$
$659$	19.5303	0.760794	0.380397	+	0.924823i	$0.375787\pi$
$660$	0	0
$661$	−11.2225	−0.436506	−0.218253	−	0.975892i	$-0.570036\pi$
$661$	−11.2225	−0.436506	−0.218253	+	0.975892i	$0.570036\pi$
$662$	2.02175	0.0785775
$663$	16.2241	0.630091
$664$	1.24959	0.0484935
$665$	0	0
$666$	15.2946	0.592652
$667$	3.78401	0.146517
$668$	−45.9593	−1.77822
$669$	23.7198	0.917061
$670$	0	0
$671$	−14.6284	−0.564723
$672$	−17.1838	−0.662881
$673$	−44.2403	−1.70534	−0.852670	−	0.522450i	$-0.825018\pi$
$673$	−44.2403	−1.70534	−0.852670	+	0.522450i	$0.825018\pi$
$674$	1.95860	0.0754425
$675$	0	0
$676$	1.94675	0.0748750
$677$	−30.7265	−1.18092	−0.590458	−	0.807068i	$-0.701053\pi$
$677$	−30.7265	−1.18092	−0.590458	+	0.807068i	$0.701053\pi$
$678$	29.2454	1.12316
$679$	8.12745	0.311903
$680$	0	0
$681$	21.4371	0.821471
$682$	70.7228	2.70811
$683$	33.1261	1.26753	0.633767	−	0.773524i	$-0.281508\pi$
$683$	33.1261	1.26753	0.633767	+	0.773524i	$0.281508\pi$
$684$	7.57541	0.289653
$685$	0	0
$686$	40.0190	1.52793
$687$	−8.66186	−0.330470
$688$	4.01542	0.153086
$689$	49.1755	1.87344
$690$	0	0
$691$	−16.3416	−0.621663	−0.310831	−	0.950465i	$-0.600607\pi$
$691$	−16.3416	−0.621663	−0.310831	+	0.950465i	$0.600607\pi$
$692$	16.4347	0.624752
$693$	−7.47060	−0.283785
$694$	−23.3962	−0.888109
$695$	0	0
$696$	0.119715	0.00453779
$697$	−34.1761	−1.29451
$698$	6.50278	0.246134
$699$	15.7073	0.594103
$700$	0	0
$701$	22.4186	0.846737	0.423369	−	0.905957i	$-0.360847\pi$
$701$	22.4186	0.846737	0.423369	+	0.905957i	$0.360847\pi$
$702$	−7.42765	−0.280338
$703$	30.0941	1.13502
$704$	−25.8998	−0.976134
$705$	0	0
$706$	42.6237	1.60416
$707$	−8.75632	−0.329315
$708$	−2.86650	−0.107730
$709$	39.4966	1.48332	0.741662	−	0.670773i	$-0.234038\pi$
$709$	39.4966	1.48332	0.741662	+	0.670773i	$0.234038\pi$
$710$	0	0
$711$	−5.02051	−0.188284
$712$	−0.521246	−0.0195345
$713$	39.0976	1.46422
$714$	−18.6421	−0.697663
$715$	0	0
$716$	−18.3027	−0.684003
$717$	5.29189	0.197629
$718$	5.60840	0.209304
$719$	−26.4473	−0.986317	−0.493158	−	0.869940i	$-0.664158\pi$
$719$	−26.4473	−0.986317	−0.493158	+	0.869940i	$0.664158\pi$
$720$	0	0
$721$	3.12790	0.116489
$722$	−7.43768	−0.276802
$723$	−14.3138	−0.532335
$724$	−30.9628	−1.15072
$725$	0	0
$726$	−1.77089	−0.0657240
$727$	26.9951	1.00119	0.500597	−	0.865681i	$-0.333114\pi$
$727$	26.9951	1.00119	0.500597	+	0.865681i	$0.333114\pi$
$728$	0.970496	0.0359690
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.22854	0.156398
$732$	8.22805	0.304118
$733$	−43.0982	−1.59187	−0.795934	−	0.605383i	$-0.793020\pi$
$733$	−43.0982	−1.59187	−0.795934	+	0.605383i	$0.793020\pi$
$734$	−60.6687	−2.23932
$735$	0	0
$736$	−30.0157	−1.10639
$737$	23.2586	0.856743
$738$	15.6464	0.575951
$739$	−5.59739	−0.205903	−0.102952	−	0.994686i	$-0.532829\pi$
$739$	−5.59739	−0.205903	−0.102952	+	0.994686i	$0.532829\pi$
$740$	0	0
$741$	−14.6149	−0.536891
$742$	−56.5045	−2.07435
$743$	29.3572	1.07701	0.538506	−	0.842621i	$-0.318989\pi$
$743$	29.3572	1.07701	0.538506	+	0.842621i	$0.318989\pi$
$744$	1.23693	0.0453482
$745$	0	0
$746$	37.5727	1.37563
$747$	−10.4380	−0.381908
$748$	−29.0003	−1.06036
$749$	−34.9068	−1.27547
$750$	0	0
$751$	33.2141	1.21200	0.606000	−	0.795465i	$-0.292773\pi$
$751$	33.2141	1.21200	0.606000	+	0.795465i	$0.292773\pi$
$752$	49.8429	1.81758
$753$	0.111554	0.00406525
$754$	7.42765	0.270499
$755$	0	0
$756$	4.20200	0.152825
$757$	51.1932	1.86065	0.930324	−	0.366738i	$-0.119525\pi$
$757$	51.1932	1.86065	0.930324	+	0.366738i	$0.119525\pi$
$758$	68.5164	2.48863
$759$	−13.0492	−0.473656
$760$	0	0
$761$	−22.2603	−0.806935	−0.403467	−	0.914994i	$-0.632195\pi$
$761$	−22.2603	−0.806935	−0.403467	+	0.914994i	$0.632195\pi$
$762$	26.2028	0.949229
$763$	15.1032	0.546774
$764$	−23.6778	−0.856632
$765$	0	0
$766$	−47.8576	−1.72917
$767$	5.53020	0.199684
$768$	−16.9209	−0.610582
$769$	41.5103	1.49690	0.748450	−	0.663192i	$-0.230799\pi$
$769$	41.5103	1.49690	0.748450	+	0.663192i	$0.230799\pi$
$770$	0	0
$771$	8.53838	0.307502
$772$	−24.9884	−0.899351
$773$	−12.9815	−0.466913	−0.233457	−	0.972367i	$-0.575004\pi$
$773$	−12.9815	−0.466913	−0.233457	+	0.972367i	$0.575004\pi$
$774$	−1.93590	−0.0695843
$775$	0	0
$776$	0.449137	0.0161231
$777$	16.6928	0.598852
$778$	3.22824	0.115738
$779$	30.7863	1.10303
$780$	0	0
$781$	35.7424	1.27896
$782$	−32.5629	−1.16445
$783$	−1.00000	−0.0357371
$784$	9.49799	0.339214
$785$	0	0
$786$	15.5993	0.556409
$787$	22.7162	0.809745	0.404872	−	0.914373i	$-0.367316\pi$
$787$	22.7162	0.809745	0.404872	+	0.914373i	$0.367316\pi$
$788$	−10.9555	−0.390273
$789$	−9.48901	−0.337818
$790$	0	0
$791$	31.9191	1.13491
$792$	−0.412839	−0.0146696
$793$	−15.8740	−0.563702
$794$	60.4553	2.14548
$795$	0	0
$796$	−5.41549	−0.191947
$797$	−17.8673	−0.632892	−0.316446	−	0.948611i	$-0.602490\pi$
$797$	−17.8673	−0.632892	−0.316446	+	0.948611i	$0.602490\pi$
$798$	16.7931	0.594468
$799$	52.4883	1.85690
$800$	0	0
$801$	4.35405	0.153843
$802$	42.8227	1.51212
$803$	42.6280	1.50431
$804$	−13.0823	−0.461378
$805$	0	0
$806$	76.7447	2.70322
$807$	−1.98710	−0.0699492
$808$	−0.483890	−0.0170232
$809$	8.20702	0.288543	0.144272	−	0.989538i	$-0.453916\pi$
$809$	8.20702	0.288543	0.144272	+	0.989538i	$0.453916\pi$
$810$	0	0
$811$	−33.4661	−1.17516	−0.587578	−	0.809168i	$-0.699918\pi$
$811$	−33.4661	−1.17516	−0.587578	+	0.809168i	$0.699918\pi$
$812$	−4.20200	−0.147461
$813$	−25.0189	−0.877452
$814$	52.7434	1.84865
$815$	0	0
$816$	−17.8492	−0.624848
$817$	−3.80913	−0.133265
$818$	−24.8028	−0.867210
$819$	−8.10671	−0.283271
$820$	0	0
$821$	−44.7864	−1.56306	−0.781528	−	0.623870i	$-0.785560\pi$
$821$	−44.7864	−1.56306	−0.781528	+	0.623870i	$0.785560\pi$
$822$	3.98417	0.138964
$823$	15.4704	0.539263	0.269631	−	0.962964i	$-0.413098\pi$
$823$	15.4704	0.539263	0.269631	+	0.962964i	$0.413098\pi$
$824$	0.172853	0.00602162
$825$	0	0
$826$	−6.35441	−0.221098
$827$	3.28177	0.114118	0.0570591	−	0.998371i	$-0.481828\pi$
$827$	3.28177	0.114118	0.0570591	+	0.998371i	$0.481828\pi$
$828$	7.33979	0.255075
$829$	13.8994	0.482745	0.241373	−	0.970433i	$-0.422402\pi$
$829$	13.8994	0.482745	0.241373	+	0.970433i	$0.422402\pi$
$830$	0	0
$831$	−25.7950	−0.894818
$832$	−28.1051	−0.974370
$833$	10.0021	0.346552
$834$	−32.1488	−1.11322
$835$	0	0
$836$	26.1239	0.903513
$837$	−10.3323	−0.357137
$838$	−22.0849	−0.762911
$839$	23.2825	0.803801	0.401901	−	0.915683i	$-0.368350\pi$
$839$	23.2825	0.803801	0.401901	+	0.915683i	$0.368350\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−11.7517	−0.404989
$843$	−9.78709	−0.337085
$844$	38.2257	1.31578
$845$	0	0
$846$	−24.0300	−0.826169
$847$	−1.93280	−0.0664116
$848$	−54.1013	−1.85785
$849$	26.6275	0.913854
$850$	0	0
$851$	29.1580	0.999525
$852$	−20.1041	−0.688754
$853$	−0.183345	−0.00627762	−0.00313881	−	0.999995i	$-0.500999\pi$
$853$	−0.183345	−0.00627762	−0.00313881	+	0.999995i	$0.500999\pi$
$854$	18.2398	0.624154
$855$	0	0
$856$	−1.92901	−0.0659323
$857$	29.5304	1.00874	0.504369	−	0.863488i	$-0.331725\pi$
$857$	29.5304	1.00874	0.504369	+	0.863488i	$0.331725\pi$
$858$	−25.6143	−0.874458
$859$	14.0724	0.480144	0.240072	−	0.970755i	$-0.422829\pi$
$859$	14.0724	0.480144	0.240072	+	0.970755i	$0.422829\pi$
$860$	0	0
$861$	17.0768	0.581976
$862$	−53.0113	−1.80557
$863$	−37.7709	−1.28574	−0.642868	−	0.765977i	$-0.722256\pi$
$863$	−37.7709	−1.28574	−0.642868	+	0.765977i	$0.722256\pi$
$864$	7.93224	0.269860
$865$	0	0
$866$	−2.00363	−0.0680862
$867$	−1.79660	−0.0610157
$868$	−43.4163	−1.47365
$869$	−17.3133	−0.587312
$870$	0	0
$871$	25.2391	0.855194
$872$	0.834632	0.0282642
$873$	−3.75171	−0.126976
$874$	29.3331	0.992207
$875$	0	0
$876$	−23.9770	−0.810108
$877$	28.7011	0.969167	0.484583	−	0.874745i	$-0.338971\pi$
$877$	28.7011	0.969167	0.484583	+	0.874745i	$0.338971\pi$
$878$	−55.4765	−1.87224
$879$	−27.9226	−0.941806
$880$	0	0
$881$	−15.0242	−0.506179	−0.253089	−	0.967443i	$-0.581447\pi$
$881$	−15.0242	−0.506179	−0.253089	+	0.967443i	$0.581447\pi$
$882$	−4.57913	−0.154187
$883$	56.7832	1.91091	0.955453	−	0.295143i	$-0.0953673\pi$
$883$	56.7832	1.91091	0.955453	+	0.295143i	$0.0953673\pi$
$884$	−31.4696	−1.05844
$885$	0	0
$886$	27.1327	0.911541
$887$	−18.5123	−0.621584	−0.310792	−	0.950478i	$-0.600594\pi$
$887$	−18.5123	−0.621584	−0.310792	+	0.950478i	$0.600594\pi$
$888$	0.922476	0.0309563
$889$	28.5984	0.959160
$890$	0	0
$891$	3.44851	0.115529
$892$	−46.0090	−1.54050
$893$	−47.2822	−1.58224
$894$	−16.0215	−0.535839
$895$	0	0
$896$	−2.07380	−0.0692806
$897$	−14.1603	−0.472799
$898$	−33.4486	−1.11619
$899$	10.3323	0.344602
$900$	0	0
$901$	−56.9728	−1.89804
$902$	53.9566	1.79656
$903$	−2.11288	−0.0703123
$904$	1.76391	0.0586666
$905$	0	0
$906$	−0.802652	−0.0266663
$907$	−37.5849	−1.24798	−0.623992	−	0.781430i	$-0.714490\pi$
$907$	−37.5849	−1.24798	−0.623992	+	0.781430i	$0.714490\pi$
$908$	−41.5812	−1.37992
$909$	4.04201	0.134065
$910$	0	0
$911$	−17.4415	−0.577862	−0.288931	−	0.957350i	$-0.593300\pi$
$911$	−17.4415	−0.577862	−0.288931	+	0.957350i	$0.593300\pi$
$912$	16.0788	0.532424
$913$	−35.9956	−1.19128
$914$	61.6555	2.03938
$915$	0	0
$916$	16.8013	0.555130
$917$	17.0255	0.562231
$918$	8.60539	0.284020
$919$	2.23086	0.0735894	0.0367947	−	0.999323i	$-0.488285\pi$
$919$	2.23086	0.0735894	0.0367947	+	0.999323i	$0.488285\pi$
$920$	0	0
$921$	33.6418	1.10854
$922$	25.5425	0.841197
$923$	38.7858	1.27665
$924$	14.4906	0.476706
$925$	0	0
$926$	31.3258	1.02943
$927$	−1.44387	−0.0474229
$928$	−7.93224	−0.260388
$929$	−51.7922	−1.69925	−0.849624	−	0.527388i	$-0.823171\pi$
$929$	−51.7922	−1.69925	−0.849624	+	0.527388i	$0.823171\pi$
$930$	0	0
$931$	−9.01004	−0.295292
$932$	−30.4672	−0.997985
$933$	11.2518	0.368367
$934$	5.25692	0.172012
$935$	0	0
$936$	−0.447991	−0.0146431
$937$	35.9764	1.17530	0.587648	−	0.809116i	$-0.300054\pi$
$937$	35.9764	1.17530	0.587648	+	0.809116i	$0.300054\pi$
$938$	−29.0007	−0.946906
$939$	5.97942	0.195131
$940$	0	0
$941$	−46.3203	−1.51000	−0.754999	−	0.655726i	$-0.772362\pi$
$941$	−46.3203	−1.51000	−0.754999	+	0.655726i	$0.772362\pi$
$942$	−22.4145	−0.730305
$943$	29.8287	0.971358
$944$	−6.08415	−0.198022
$945$	0	0
$946$	−6.67595	−0.217054
$947$	−4.98490	−0.161987	−0.0809937	−	0.996715i	$-0.525809\pi$
$947$	−4.98490	−0.161987	−0.0809937	+	0.996715i	$0.525809\pi$
$948$	9.73822	0.316283
$949$	46.2577	1.50159
$950$	0	0
$951$	−7.06727	−0.229172
$952$	−1.12438	−0.0364414
$953$	−8.50497	−0.275503	−0.137751	−	0.990467i	$-0.543988\pi$
$953$	−8.50497	−0.275503	−0.137751	+	0.990467i	$0.543988\pi$
$954$	26.0831	0.844471
$955$	0	0
$956$	−10.2646	−0.331981
$957$	−3.44851	−0.111474
$958$	−84.9970	−2.74613
$959$	4.34842	0.140418
$960$	0	0
$961$	75.7566	2.44376
$962$	57.2344	1.84531
$963$	16.1134	0.519246
$964$	27.7642	0.894225
$965$	0	0
$966$	16.2707	0.523503
$967$	−14.5668	−0.468436	−0.234218	−	0.972184i	$-0.575253\pi$
$967$	−14.5668	−0.468436	−0.234218	+	0.972184i	$0.575253\pi$
$968$	−0.106810	−0.00343299
$969$	16.9322	0.543942
$970$	0	0
$971$	10.1423	0.325482	0.162741	−	0.986669i	$-0.447967\pi$
$971$	10.1423	0.325482	0.162741	+	0.986669i	$0.447967\pi$
$972$	−1.93969	−0.0622155
$973$	−35.0880	−1.12487
$974$	34.9230	1.11900
$975$	0	0
$976$	17.4641	0.559011
$977$	39.0644	1.24978	0.624891	−	0.780712i	$-0.285143\pi$
$977$	39.0644	1.24978	0.624891	+	0.780712i	$0.285143\pi$
$978$	−13.4187	−0.429084
$979$	15.0150	0.479880
$980$	0	0
$981$	−6.97181	−0.222593
$982$	−63.0126	−2.01081
$983$	−28.5284	−0.909915	−0.454957	−	0.890513i	$-0.650346\pi$
$983$	−28.5284	−0.909915	−0.454957	+	0.890513i	$0.650346\pi$
$984$	0.943695	0.0300839
$985$	0	0
$986$	−8.60539	−0.274051
$987$	−26.2269	−0.834812
$988$	28.3483	0.901879
$989$	−3.69066	−0.117356
$990$	0	0
$991$	−56.7608	−1.80307	−0.901534	−	0.432709i	$-0.857558\pi$
$991$	−56.7608	−1.80307	−0.901534	+	0.432709i	$0.857558\pi$
$992$	−81.9583	−2.60218
$993$	−1.01858	−0.0323238
$994$	−44.5664	−1.41356
$995$	0	0
$996$	20.2465	0.641535
$997$	10.0683	0.318866	0.159433	−	0.987209i	$-0.449033\pi$
$997$	10.0683	0.318866	0.159433	+	0.987209i	$0.449033\pi$
$998$	21.4211	0.678073
$999$	−7.70559	−0.243794

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.bc.1.7	✓	8
3.2	odd	2		6525.2.a.bz.1.2		8
5.2	odd	4		2175.2.c.p.349.13		16
5.3	odd	4		2175.2.c.p.349.4		16
5.4	even	2		2175.2.a.bd.1.2	yes	8
15.14	odd	2		6525.2.a.by.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.bc.1.7	✓	8	1.1	even	1	trivial
2175.2.a.bd.1.2	yes	8	5.4	even	2
2175.2.c.p.349.4		16	5.3	odd	4
2175.2.c.p.349.13		16	5.2	odd	4
6525.2.a.by.1.7		8	15.14	odd	2
6525.2.a.bz.1.2		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)

\(f(q)\)	\(=\)	\(q+1.98486 q^{2} -1.00000 q^{3} +1.93969 q^{4} -1.98486 q^{6} -2.16633 q^{7} -0.119715 q^{8} +1.00000 q^{9} +3.44851 q^{11} -1.93969 q^{12} +3.74214 q^{13} -4.29987 q^{14} -4.11699 q^{16} -4.33550 q^{17} +1.98486 q^{18} +3.90548 q^{19} +2.16633 q^{21} +6.84482 q^{22} +3.78401 q^{23} +0.119715 q^{24} +7.42765 q^{26} -1.00000 q^{27} -4.20200 q^{28} +1.00000 q^{29} +10.3323 q^{31} -7.93224 q^{32} -3.44851 q^{33} -8.60539 q^{34} +1.93969 q^{36} +7.70559 q^{37} +7.75186 q^{38} -3.74214 q^{39} +7.88284 q^{41} +4.29987 q^{42} -0.975329 q^{43} +6.68902 q^{44} +7.51075 q^{46} -12.1066 q^{47} +4.11699 q^{48} -2.30702 q^{49} +4.33550 q^{51} +7.25858 q^{52} +13.1410 q^{53} -1.98486 q^{54} +0.259342 q^{56} -3.90548 q^{57} +1.98486 q^{58} +1.47781 q^{59} -4.24195 q^{61} +20.5082 q^{62} -2.16633 q^{63} -7.51043 q^{64} -6.84482 q^{66} +6.74456 q^{67} -8.40952 q^{68} -3.78401 q^{69} +10.3646 q^{71} -0.119715 q^{72} +12.3613 q^{73} +15.2946 q^{74} +7.57541 q^{76} -7.47060 q^{77} -7.42765 q^{78} -5.02051 q^{79} +1.00000 q^{81} +15.6464 q^{82} -10.4380 q^{83} +4.20200 q^{84} -1.93590 q^{86} -1.00000 q^{87} -0.412839 q^{88} +4.35405 q^{89} -8.10671 q^{91} +7.33979 q^{92} -10.3323 q^{93} -24.0300 q^{94} +7.93224 q^{96} -3.75171 q^{97} -4.57913 q^{98} +3.44851 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 8 q^{3} + 12 q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} + 6 q^{11} - 12 q^{12} + 6 q^{13} + 9 q^{14} + 32 q^{16} - 12 q^{17} - 2 q^{18} - 2 q^{21} + 3 q^{22} - 14 q^{23} + 3 q^{24} + 18 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) \) 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

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Representations

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Database

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