LMFDB - Embedded newform 2175.2.a.bc.1.8

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			1.8
Root			$-2.72810$ 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)

$f(q)$	$=$	\(q+2.72810 q^{2} -1.00000 q^{3} +5.44251 q^{4} -2.72810 q^{6} +4.61695 q^{7} +9.39150 q^{8} +1.00000 q^{9} -1.87758 q^{11} -5.44251 q^{12} +5.48400 q^{13} +12.5955 q^{14} +14.7359 q^{16} -5.74056 q^{17} +2.72810 q^{18} -5.40455 q^{19} -4.61695 q^{21} -5.12221 q^{22} -0.137016 q^{23} -9.39150 q^{24} +14.9609 q^{26} -1.00000 q^{27} +25.1278 q^{28} +1.00000 q^{29} -8.18167 q^{31} +21.4180 q^{32} +1.87758 q^{33} -15.6608 q^{34} +5.44251 q^{36} -4.45785 q^{37} -14.7441 q^{38} -5.48400 q^{39} -0.215243 q^{41} -12.5955 q^{42} +7.17115 q^{43} -10.2187 q^{44} -0.373793 q^{46} -9.91293 q^{47} -14.7359 q^{48} +14.3162 q^{49} +5.74056 q^{51} +29.8467 q^{52} -1.14754 q^{53} -2.72810 q^{54} +43.3601 q^{56} +5.40455 q^{57} +2.72810 q^{58} -0.244913 q^{59} +8.75284 q^{61} -22.3204 q^{62} +4.61695 q^{63} +28.9585 q^{64} +5.12221 q^{66} -6.53295 q^{67} -31.2431 q^{68} +0.137016 q^{69} -0.277412 q^{71} +9.39150 q^{72} -11.4933 q^{73} -12.1614 q^{74} -29.4143 q^{76} -8.66866 q^{77} -14.9609 q^{78} +1.66223 q^{79} +1.00000 q^{81} -0.587203 q^{82} +9.58514 q^{83} -25.1278 q^{84} +19.5636 q^{86} -1.00000 q^{87} -17.6333 q^{88} +16.3332 q^{89} +25.3193 q^{91} -0.745712 q^{92} +8.18167 q^{93} -27.0434 q^{94} -21.4180 q^{96} +8.04128 q^{97} +39.0559 q^{98} -1.87758 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.72810	1.92906	0.964528	−	0.263981i	$-0.0850357\pi$
$2$	2.72810	1.92906	0.964528	+	0.263981i	$0.0850357\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.44251	2.72126
$5$	0	0
$5$	0	0
$6$	−2.72810	−1.11374
$7$	4.61695	1.74504	0.872521	−	0.488577i	$-0.162484\pi$
$7$	4.61695	1.74504	0.872521	+	0.488577i	$0.162484\pi$
$8$	9.39150	3.32040
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.87758	−0.566110	−0.283055	−	0.959104i	$-0.591348\pi$
$11$	−1.87758	−0.566110	−0.283055	+	0.959104i	$0.591348\pi$
$12$	−5.44251	−1.57112
$13$	5.48400	1.52099	0.760494	−	0.649345i	$-0.224957\pi$
$13$	5.48400	1.52099	0.760494	+	0.649345i	$0.224957\pi$
$14$	12.5955	3.36628
$15$	0	0
$16$	14.7359	3.68398
$17$	−5.74056	−1.39229	−0.696145	−	0.717901i	$-0.745103\pi$
$17$	−5.74056	−1.39229	−0.696145	+	0.717901i	$0.745103\pi$
$18$	2.72810	0.643019
$19$	−5.40455	−1.23989	−0.619944	−	0.784646i	$-0.712845\pi$
$19$	−5.40455	−1.23989	−0.619944	+	0.784646i	$0.712845\pi$
$20$	0	0
$21$	−4.61695	−1.00750
$22$	−5.12221	−1.09206
$23$	−0.137016	−0.0285698	−0.0142849	−	0.999898i	$-0.504547\pi$
$23$	−0.137016	−0.0285698	−0.0142849	+	0.999898i	$0.504547\pi$
$24$	−9.39150	−1.91703
$25$	0	0
$26$	14.9609	2.93407
$27$	−1.00000	−0.192450
$28$	25.1278	4.74870
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−8.18167	−1.46947	−0.734736	−	0.678353i	$-0.762694\pi$
$31$	−8.18167	−1.46947	−0.734736	+	0.678353i	$0.762694\pi$
$32$	21.4180	3.78620
$33$	1.87758	0.326844
$34$	−15.6608	−2.68581
$35$	0	0
$36$	5.44251	0.907085
$37$	−4.45785	−0.732866	−0.366433	−	0.930444i	$-0.619421\pi$
$37$	−4.45785	−0.732866	−0.366433	+	0.930444i	$0.619421\pi$
$38$	−14.7441	−2.39181
$39$	−5.48400	−0.878142
$40$	0	0
$41$	−0.215243	−0.0336153	−0.0168076	−	0.999859i	$-0.505350\pi$
$41$	−0.215243	−0.0336153	−0.0168076	+	0.999859i	$0.505350\pi$
$42$	−12.5955	−1.94352
$43$	7.17115	1.09359	0.546795	−	0.837266i	$-0.315848\pi$
$43$	7.17115	1.09359	0.546795	+	0.837266i	$0.315848\pi$
$44$	−10.2187	−1.54053
$45$	0	0
$46$	−0.373793	−0.0551128
$47$	−9.91293	−1.44595	−0.722975	−	0.690874i	$-0.757226\pi$
$47$	−9.91293	−1.44595	−0.722975	+	0.690874i	$0.757226\pi$
$48$	−14.7359	−2.12695
$49$	14.3162	2.04517
$50$	0	0
$51$	5.74056	0.803839
$52$	29.8467	4.13899
$53$	−1.14754	−0.157627	−0.0788134	−	0.996889i	$-0.525113\pi$
$53$	−1.14754	−0.157627	−0.0788134	+	0.996889i	$0.525113\pi$
$54$	−2.72810	−0.371247
$55$	0	0
$56$	43.3601	5.79423
$57$	5.40455	0.715850
$58$	2.72810	0.358217
$59$	−0.244913	−0.0318849	−0.0159425	−	0.999873i	$-0.505075\pi$
$59$	−0.244913	−0.0318849	−0.0159425	+	0.999873i	$0.505075\pi$
$60$	0	0
$61$	8.75284	1.12069	0.560343	−	0.828260i	$-0.310669\pi$
$61$	8.75284	1.12069	0.560343	+	0.828260i	$0.310669\pi$
$62$	−22.3204	−2.83469
$63$	4.61695	0.581680
$64$	28.9585	3.61981
$65$	0	0
$66$	5.12221	0.630500
$67$	−6.53295	−0.798127	−0.399064	−	0.916923i	$-0.630665\pi$
$67$	−6.53295	−0.798127	−0.399064	+	0.916923i	$0.630665\pi$
$68$	−31.2431	−3.78878
$69$	0.137016	0.0164948
$70$	0	0
$71$	−0.277412	−0.0329227	−0.0164614	−	0.999865i	$-0.505240\pi$
$71$	−0.277412	−0.0329227	−0.0164614	+	0.999865i	$0.505240\pi$
$72$	9.39150	1.10680
$73$	−11.4933	−1.34519	−0.672594	−	0.740011i	$-0.734820\pi$
$73$	−11.4933	−1.34519	−0.672594	+	0.740011i	$0.734820\pi$
$74$	−12.1614	−1.41374
$75$	0	0
$76$	−29.4143	−3.37405
$77$	−8.66866	−0.987886
$78$	−14.9609	−1.69399
$79$	1.66223	0.187015	0.0935076	−	0.995619i	$-0.470192\pi$
$79$	1.66223	0.187015	0.0935076	+	0.995619i	$0.470192\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.587203	−0.0648457
$83$	9.58514	1.05211	0.526053	−	0.850452i	$-0.323671\pi$
$83$	9.58514	1.05211	0.526053	+	0.850452i	$0.323671\pi$
$84$	−25.1278	−2.74167
$85$	0	0
$86$	19.5636	2.10960
$87$	−1.00000	−0.107211
$88$	−17.6333	−1.87971
$89$	16.3332	1.73132	0.865659	−	0.500634i	$-0.166900\pi$
$89$	16.3332	1.73132	0.865659	+	0.500634i	$0.166900\pi$
$90$	0	0
$91$	25.3193	2.65419
$92$	−0.745712	−0.0777458
$93$	8.18167	0.848400
$94$	−27.0434	−2.78932
$95$	0	0
$96$	−21.4180	−2.18596
$97$	8.04128	0.816468	0.408234	−	0.912877i	$-0.366145\pi$
$97$	8.04128	0.816468	0.408234	+	0.912877i	$0.366145\pi$
$98$	39.0559	3.94525
$99$	−1.87758	−0.188703
$100$	0	0
$101$	−12.3239	−1.22627	−0.613135	−	0.789978i	$-0.710092\pi$
$101$	−12.3239	−1.22627	−0.613135	+	0.789978i	$0.710092\pi$
$102$	15.6608	1.55065
$103$	−0.660912	−0.0651216	−0.0325608	−	0.999470i	$-0.510366\pi$
$103$	−0.660912	−0.0651216	−0.0325608	+	0.999470i	$0.510366\pi$
$104$	51.5030	5.05028
$105$	0	0
$106$	−3.13060	−0.304071
$107$	−14.5868	−1.41016	−0.705079	−	0.709128i	$-0.749089\pi$
$107$	−14.5868	−1.41016	−0.705079	+	0.709128i	$0.749089\pi$
$108$	−5.44251	−0.523706
$109$	15.5050	1.48511	0.742557	−	0.669783i	$-0.233613\pi$
$109$	15.5050	1.48511	0.742557	+	0.669783i	$0.233613\pi$
$110$	0	0
$111$	4.45785	0.423120
$112$	68.0349	6.42869
$113$	−7.80660	−0.734383	−0.367191	−	0.930145i	$-0.619681\pi$
$113$	−7.80660	−0.734383	−0.367191	+	0.930145i	$0.619681\pi$
$114$	14.7441	1.38091
$115$	0	0
$116$	5.44251	0.505325
$117$	5.48400	0.506996
$118$	−0.668145	−0.0615078
$119$	−26.5038	−2.42960
$120$	0	0
$121$	−7.47471	−0.679519
$122$	23.8786	2.16187
$123$	0.215243	0.0194078
$124$	−44.5289	−3.99881
$125$	0	0
$126$	12.5955	1.12209
$127$	19.7106	1.74903	0.874515	−	0.484999i	$-0.161180\pi$
$127$	19.7106	1.74903	0.874515	+	0.484999i	$0.161180\pi$
$128$	36.1656	3.19662
$129$	−7.17115	−0.631385
$130$	0	0
$131$	−1.05738	−0.0923834	−0.0461917	−	0.998933i	$-0.514709\pi$
$131$	−1.05738	−0.0923834	−0.0461917	+	0.998933i	$0.514709\pi$
$132$	10.2187	0.889426
$133$	−24.9525	−2.16366
$134$	−17.8225	−1.53963
$135$	0	0
$136$	−53.9125	−4.62296
$137$	3.21467	0.274648	0.137324	−	0.990526i	$-0.456150\pi$
$137$	3.21467	0.274648	0.137324	+	0.990526i	$0.456150\pi$
$138$	0.373793	0.0318194
$139$	8.48410	0.719612	0.359806	−	0.933027i	$-0.382843\pi$
$139$	8.48410	0.719612	0.359806	+	0.933027i	$0.382843\pi$
$140$	0	0
$141$	9.91293	0.834820
$142$	−0.756807	−0.0635098
$143$	−10.2966	−0.861046
$144$	14.7359	1.22799
$145$	0	0
$146$	−31.3548	−2.59494
$147$	−14.3162	−1.18078
$148$	−24.2619	−1.99432
$149$	−5.93205	−0.485972	−0.242986	−	0.970030i	$-0.578127\pi$
$149$	−5.93205	−0.485972	−0.242986	+	0.970030i	$0.578127\pi$
$150$	0	0
$151$	8.06281	0.656142	0.328071	−	0.944653i	$-0.393601\pi$
$151$	8.06281	0.656142	0.328071	+	0.944653i	$0.393601\pi$
$152$	−50.7568	−4.11692
$153$	−5.74056	−0.464097
$154$	−23.6490	−1.90569
$155$	0	0
$156$	−29.8467	−2.38965
$157$	−9.87127	−0.787813	−0.393906	−	0.919151i	$-0.628877\pi$
$157$	−9.87127	−0.787813	−0.393906	+	0.919151i	$0.628877\pi$
$158$	4.53472	0.360763
$159$	1.14754	0.0910058
$160$	0	0
$161$	−0.632596	−0.0498555
$162$	2.72810	0.214340
$163$	1.25711	0.0984643	0.0492321	−	0.998787i	$-0.484323\pi$
$163$	1.25711	0.0984643	0.0492321	+	0.998787i	$0.484323\pi$
$164$	−1.17146	−0.0914757
$165$	0	0
$166$	26.1492	2.02957
$167$	−15.1462	−1.17205	−0.586024	−	0.810294i	$-0.699308\pi$
$167$	−15.1462	−1.17205	−0.586024	+	0.810294i	$0.699308\pi$
$168$	−43.3601	−3.34530
$169$	17.0742	1.31340
$170$	0	0
$171$	−5.40455	−0.413296
$172$	39.0291	2.97594
$173$	2.34929	0.178613	0.0893065	−	0.996004i	$-0.471535\pi$
$173$	2.34929	0.178613	0.0893065	+	0.996004i	$0.471535\pi$
$174$	−2.72810	−0.206816
$175$	0	0
$176$	−27.6678	−2.08554
$177$	0.244913	0.0184088
$178$	44.5586	3.33981
$179$	−17.7814	−1.32904	−0.664520	−	0.747270i	$-0.731364\pi$
$179$	−17.7814	−1.32904	−0.664520	+	0.747270i	$0.731364\pi$
$180$	0	0
$181$	−2.69871	−0.200594	−0.100297	−	0.994958i	$-0.531979\pi$
$181$	−2.69871	−0.200594	−0.100297	+	0.994958i	$0.531979\pi$
$182$	69.0735	5.12007
$183$	−8.75284	−0.647029
$184$	−1.28679	−0.0948632
$185$	0	0
$186$	22.3204	1.63661
$187$	10.7783	0.788190
$188$	−53.9512	−3.93480
$189$	−4.61695	−0.335833
$190$	0	0
$191$	10.7997	0.781442	0.390721	−	0.920509i	$-0.372226\pi$
$191$	10.7997	0.781442	0.390721	+	0.920509i	$0.372226\pi$
$192$	−28.9585	−2.08990
$193$	−4.18447	−0.301205	−0.150602	−	0.988594i	$-0.548121\pi$
$193$	−4.18447	−0.301205	−0.150602	+	0.988594i	$0.548121\pi$
$194$	21.9374	1.57501
$195$	0	0
$196$	77.9160	5.56543
$197$	5.76856	0.410993	0.205496	−	0.978658i	$-0.434119\pi$
$197$	5.76856	0.410993	0.205496	+	0.978658i	$0.434119\pi$
$198$	−5.12221	−0.364019
$199$	−5.45094	−0.386407	−0.193203	−	0.981159i	$-0.561888\pi$
$199$	−5.45094	−0.386407	−0.193203	+	0.981159i	$0.561888\pi$
$200$	0	0
$201$	6.53295	0.460799
$202$	−33.6207	−2.36554
$203$	4.61695	0.324046
$204$	31.2431	2.18745
$205$	0	0
$206$	−1.80303	−0.125623
$207$	−0.137016	−0.00952328
$208$	80.8117	5.60328
$209$	10.1474	0.701914
$210$	0	0
$211$	0.863332	0.0594342	0.0297171	−	0.999558i	$-0.490539\pi$
$211$	0.863332	0.0594342	0.0297171	+	0.999558i	$0.490539\pi$
$212$	−6.24550	−0.428943
$213$	0.277412	0.0190080
$214$	−39.7942	−2.72027
$215$	0	0
$216$	−9.39150	−0.639011
$217$	−37.7743	−2.56429
$218$	42.2993	2.86487
$219$	11.4933	0.776645
$220$	0	0
$221$	−31.4812	−2.11766
$222$	12.1614	0.816223
$223$	−0.615197	−0.0411967	−0.0205983	−	0.999788i	$-0.506557\pi$
$223$	−0.615197	−0.0411967	−0.0205983	+	0.999788i	$0.506557\pi$
$224$	98.8856	6.60707
$225$	0	0
$226$	−21.2972	−1.41667
$227$	−23.7230	−1.57455	−0.787275	−	0.616602i	$-0.788509\pi$
$227$	−23.7230	−1.57455	−0.787275	+	0.616602i	$0.788509\pi$
$228$	29.4143	1.94801
$229$	−20.9808	−1.38645	−0.693226	−	0.720721i	$-0.743811\pi$
$229$	−20.9808	−1.38645	−0.693226	+	0.720721i	$0.743811\pi$
$230$	0	0
$231$	8.66866	0.570356
$232$	9.39150	0.616582
$233$	−7.95482	−0.521138	−0.260569	−	0.965455i	$-0.583910\pi$
$233$	−7.95482	−0.521138	−0.260569	+	0.965455i	$0.583910\pi$
$234$	14.9609	0.978023
$235$	0	0
$236$	−1.33294	−0.0867670
$237$	−1.66223	−0.107973
$238$	−72.3051	−4.68684
$239$	17.4888	1.13125	0.565627	−	0.824661i	$-0.308634\pi$
$239$	17.4888	1.13125	0.565627	+	0.824661i	$0.308634\pi$
$240$	0	0
$241$	−14.7743	−0.951699	−0.475849	−	0.879527i	$-0.657859\pi$
$241$	−14.7743	−0.951699	−0.475849	+	0.879527i	$0.657859\pi$
$242$	−20.3917	−1.31083
$243$	−1.00000	−0.0641500
$244$	47.6375	3.04968
$245$	0	0
$246$	0.587203	0.0374387
$247$	−29.6385	−1.88585
$248$	−76.8382	−4.87923
$249$	−9.58514	−0.607433
$250$	0	0
$251$	−3.30996	−0.208923	−0.104461	−	0.994529i	$-0.533312\pi$
$251$	−3.30996	−0.208923	−0.104461	+	0.994529i	$0.533312\pi$
$252$	25.1278	1.58290
$253$	0.257258	0.0161737
$254$	53.7723	3.37398
$255$	0	0
$256$	40.7463	2.54664
$257$	−30.8668	−1.92542	−0.962709	−	0.270539i	$-0.912798\pi$
$257$	−30.8668	−1.92542	−0.962709	+	0.270539i	$0.912798\pi$
$258$	−19.5636	−1.21798
$259$	−20.5816	−1.27888
$260$	0	0
$261$	1.00000	0.0618984
$262$	−2.88463	−0.178213
$263$	19.4833	1.20139	0.600697	−	0.799477i	$-0.294890\pi$
$263$	19.4833	1.20139	0.600697	+	0.799477i	$0.294890\pi$
$264$	17.6333	1.08525
$265$	0	0
$266$	−68.0728	−4.17381
$267$	−16.3332	−0.999577
$268$	−35.5557	−2.17191
$269$	20.8385	1.27055	0.635273	−	0.772287i	$-0.280887\pi$
$269$	20.8385	1.27055	0.635273	+	0.772287i	$0.280887\pi$
$270$	0	0
$271$	24.5849	1.49343	0.746713	−	0.665146i	$-0.231631\pi$
$271$	24.5849	1.49343	0.746713	+	0.665146i	$0.231631\pi$
$272$	−84.5924	−5.12916
$273$	−25.3193	−1.53239
$274$	8.76994	0.529811
$275$	0	0
$276$	0.745712	0.0448866
$277$	1.89904	0.114102	0.0570510	−	0.998371i	$-0.481830\pi$
$277$	1.89904	0.114102	0.0570510	+	0.998371i	$0.481830\pi$
$278$	23.1455	1.38817
$279$	−8.18167	−0.489824
$280$	0	0
$281$	32.4916	1.93828	0.969142	−	0.246503i	$-0.0792814\pi$
$281$	32.4916	1.93828	0.969142	+	0.246503i	$0.0792814\pi$
$282$	27.0434	1.61041
$283$	−11.9151	−0.708279	−0.354140	−	0.935193i	$-0.615226\pi$
$283$	−11.9151	−0.708279	−0.354140	+	0.935193i	$0.615226\pi$
$284$	−1.50982	−0.0895912
$285$	0	0
$286$	−28.0902	−1.66101
$287$	−0.993764	−0.0586600
$288$	21.4180	1.26207
$289$	15.9540	0.938472
$290$	0	0
$291$	−8.04128	−0.471388
$292$	−62.5524	−3.66060
$293$	20.7359	1.21140	0.605702	−	0.795692i	$-0.292892\pi$
$293$	20.7359	1.21140	0.605702	+	0.795692i	$0.292892\pi$
$294$	−39.0559	−2.27779
$295$	0	0
$296$	−41.8659	−2.43341
$297$	1.87758	0.108948
$298$	−16.1832	−0.937467
$299$	−0.751396	−0.0434544
$300$	0	0
$301$	33.1088	1.90836
$302$	21.9961	1.26574
$303$	12.3239	0.707987
$304$	−79.6409	−4.56772
$305$	0	0
$306$	−15.6608	−0.895268
$307$	−16.9755	−0.968842	−0.484421	−	0.874835i	$-0.660970\pi$
$307$	−16.9755	−0.968842	−0.484421	+	0.874835i	$0.660970\pi$
$308$	−47.1793	−2.68829
$309$	0.660912	0.0375980
$310$	0	0
$311$	−20.9019	−1.18524	−0.592619	−	0.805483i	$-0.701906\pi$
$311$	−20.9019	−1.18524	−0.592619	+	0.805483i	$0.701906\pi$
$312$	−51.5030	−2.91578
$313$	12.7754	0.722110	0.361055	−	0.932545i	$-0.382417\pi$
$313$	12.7754	0.722110	0.361055	+	0.932545i	$0.382417\pi$
$314$	−26.9298	−1.51973
$315$	0	0
$316$	9.04669	0.508916
$317$	15.0552	0.845582	0.422791	−	0.906227i	$-0.361050\pi$
$317$	15.0552	0.845582	0.422791	+	0.906227i	$0.361050\pi$
$318$	3.13060	0.175555
$319$	−1.87758	−0.105124
$320$	0	0
$321$	14.5868	0.814155
$322$	−1.72578	−0.0961741
$323$	31.0251	1.72628
$324$	5.44251	0.302362
$325$	0	0
$326$	3.42951	0.189943
$327$	−15.5050	−0.857431
$328$	−2.02145	−0.111616
$329$	−45.7675	−2.52324
$330$	0	0
$331$	−17.7325	−0.974668	−0.487334	−	0.873216i	$-0.662031\pi$
$331$	−17.7325	−0.974668	−0.487334	+	0.873216i	$0.662031\pi$
$332$	52.1672	2.86305
$333$	−4.45785	−0.244289
$334$	−41.3203	−2.26095
$335$	0	0
$336$	−68.0349	−3.71161
$337$	−17.5905	−0.958217	−0.479108	−	0.877756i	$-0.659040\pi$
$337$	−17.5905	−0.958217	−0.479108	+	0.877756i	$0.659040\pi$
$338$	46.5801	2.53363
$339$	7.80660	0.423996
$340$	0	0
$341$	15.3617	0.831883
$342$	−14.7441	−0.797271
$343$	33.7784	1.82386
$344$	67.3479	3.63115
$345$	0	0
$346$	6.40909	0.344555
$347$	−21.0789	−1.13157	−0.565787	−	0.824552i	$-0.691427\pi$
$347$	−21.0789	−1.13157	−0.565787	+	0.824552i	$0.691427\pi$
$348$	−5.44251	−0.291749
$349$	−0.723154	−0.0387096	−0.0193548	−	0.999813i	$-0.506161\pi$
$349$	−0.723154	−0.0387096	−0.0193548	+	0.999813i	$0.506161\pi$
$350$	0	0
$351$	−5.48400	−0.292714
$352$	−40.2139	−2.14341
$353$	5.93891	0.316096	0.158048	−	0.987431i	$-0.449480\pi$
$353$	5.93891	0.316096	0.158048	+	0.987431i	$0.449480\pi$
$354$	0.668145	0.0355115
$355$	0	0
$356$	88.8938	4.71136
$357$	26.5038	1.40273
$358$	−48.5092	−2.56379
$359$	−2.30973	−0.121903	−0.0609514	−	0.998141i	$-0.519413\pi$
$359$	−2.30973	−0.121903	−0.0609514	+	0.998141i	$0.519413\pi$
$360$	0	0
$361$	10.2091	0.537323
$362$	−7.36234	−0.386956
$363$	7.47471	0.392321
$364$	137.801	7.22272
$365$	0	0
$366$	−23.8786	−1.24815
$367$	22.9235	1.19660	0.598298	−	0.801274i	$-0.295844\pi$
$367$	22.9235	1.19660	0.598298	+	0.801274i	$0.295844\pi$
$368$	−2.01906	−0.105251
$369$	−0.215243	−0.0112051
$370$	0	0
$371$	−5.29813	−0.275065
$372$	44.5289	2.30871
$373$	−2.83182	−0.146626	−0.0733129	−	0.997309i	$-0.523357\pi$
$373$	−2.83182	−0.146626	−0.0733129	+	0.997309i	$0.523357\pi$
$374$	29.4043	1.52046
$375$	0	0
$376$	−93.0973	−4.80113
$377$	5.48400	0.282440
$378$	−12.5955	−0.647841
$379$	−29.2539	−1.50267	−0.751335	−	0.659921i	$-0.770590\pi$
$379$	−29.2539	−1.50267	−0.751335	+	0.659921i	$0.770590\pi$
$380$	0	0
$381$	−19.7106	−1.00980
$382$	29.4627	1.50744
$383$	−2.99063	−0.152814	−0.0764070	−	0.997077i	$-0.524345\pi$
$383$	−2.99063	−0.152814	−0.0764070	+	0.997077i	$0.524345\pi$
$384$	−36.1656	−1.84557
$385$	0	0
$386$	−11.4156	−0.581040
$387$	7.17115	0.364530
$388$	43.7648	2.22182
$389$	−8.57339	−0.434688	−0.217344	−	0.976095i	$-0.569739\pi$
$389$	−8.57339	−0.434688	−0.217344	+	0.976095i	$0.569739\pi$
$390$	0	0
$391$	0.786549	0.0397775
$392$	134.451	6.79078
$393$	1.05738	0.0533376
$394$	15.7372	0.792828
$395$	0	0
$396$	−10.2187	−0.513510
$397$	9.29605	0.466556	0.233278	−	0.972410i	$-0.425055\pi$
$397$	9.29605	0.466556	0.233278	+	0.972410i	$0.425055\pi$
$398$	−14.8707	−0.745400
$399$	24.9525	1.24919
$400$	0	0
$401$	19.5392	0.975739	0.487870	−	0.872916i	$-0.337774\pi$
$401$	19.5392	0.975739	0.487870	+	0.872916i	$0.337774\pi$
$402$	17.8225	0.888907
$403$	−44.8683	−2.23505
$404$	−67.0728	−3.33700
$405$	0	0
$406$	12.5955	0.625103
$407$	8.36995	0.414883
$408$	53.9125	2.66907
$409$	37.9582	1.87691	0.938456	−	0.345398i	$-0.112256\pi$
$409$	37.9582	1.87691	0.938456	+	0.345398i	$0.112256\pi$
$410$	0	0
$411$	−3.21467	−0.158568
$412$	−3.59702	−0.177213
$413$	−1.13075	−0.0556405
$414$	−0.373793	−0.0183709
$415$	0	0
$416$	117.456	5.75876
$417$	−8.48410	−0.415468
$418$	27.6832	1.35403
$419$	−10.0952	−0.493184	−0.246592	−	0.969119i	$-0.579311\pi$
$419$	−10.0952	−0.493184	−0.246592	+	0.969119i	$0.579311\pi$
$420$	0	0
$421$	13.8521	0.675111	0.337556	−	0.941306i	$-0.390400\pi$
$421$	13.8521	0.675111	0.337556	+	0.941306i	$0.390400\pi$
$422$	2.35525	0.114652
$423$	−9.91293	−0.481983
$424$	−10.7771	−0.523384
$425$	0	0
$426$	0.756807	0.0366674
$427$	40.4114	1.95565
$428$	−79.3888	−3.83740
$429$	10.2966	0.497125
$430$	0	0
$431$	18.2149	0.877380	0.438690	−	0.898638i	$-0.355443\pi$
$431$	18.2149	0.877380	0.438690	+	0.898638i	$0.355443\pi$
$432$	−14.7359	−0.708982
$433$	6.73476	0.323652	0.161826	−	0.986819i	$-0.448262\pi$
$433$	6.73476	0.323652	0.161826	+	0.986819i	$0.448262\pi$
$434$	−103.052	−4.94666
$435$	0	0
$436$	84.3864	4.04137
$437$	0.740510	0.0354234
$438$	31.3548	1.49819
$439$	30.9590	1.47759	0.738795	−	0.673930i	$-0.235395\pi$
$439$	30.9590	1.47759	0.738795	+	0.673930i	$0.235395\pi$
$440$	0	0
$441$	14.3162	0.681723
$442$	−85.8838	−4.08508
$443$	21.4432	1.01880	0.509398	−	0.860531i	$-0.329868\pi$
$443$	21.4432	1.01880	0.509398	+	0.860531i	$0.329868\pi$
$444$	24.2619	1.15142
$445$	0	0
$446$	−1.67832	−0.0794707
$447$	5.93205	0.280576
$448$	133.700	6.31672
$449$	25.6895	1.21236	0.606180	−	0.795327i	$-0.292701\pi$
$449$	25.6895	1.21236	0.606180	+	0.795327i	$0.292701\pi$
$450$	0	0
$451$	0.404134	0.0190299
$452$	−42.4875	−1.99844
$453$	−8.06281	−0.378824
$454$	−64.7186	−3.03739
$455$	0	0
$456$	50.7568	2.37691
$457$	30.3278	1.41868	0.709338	−	0.704868i	$-0.248994\pi$
$457$	30.3278	1.41868	0.709338	+	0.704868i	$0.248994\pi$
$458$	−57.2377	−2.67454
$459$	5.74056	0.267946
$460$	0	0
$461$	−31.0026	−1.44393	−0.721967	−	0.691928i	$-0.756762\pi$
$461$	−31.0026	−1.44393	−0.721967	+	0.691928i	$0.756762\pi$
$462$	23.6490	1.10025
$463$	−27.6990	−1.28728	−0.643640	−	0.765328i	$-0.722577\pi$
$463$	−27.6990	−1.28728	−0.643640	+	0.765328i	$0.722577\pi$
$464$	14.7359	0.684097
$465$	0	0
$466$	−21.7015	−1.00530
$467$	−4.03243	−0.186598	−0.0932992	−	0.995638i	$-0.529741\pi$
$467$	−4.03243	−0.186598	−0.0932992	+	0.995638i	$0.529741\pi$
$468$	29.8467	1.37966
$469$	−30.1623	−1.39277
$470$	0	0
$471$	9.87127	0.454844
$472$	−2.30010	−0.105871
$473$	−13.4644	−0.619093
$474$	−4.53472	−0.208286
$475$	0	0
$476$	−144.248	−6.61157
$477$	−1.14754	−0.0525422
$478$	47.7110	2.18225
$479$	1.32186	0.0603973	0.0301986	−	0.999544i	$-0.490386\pi$
$479$	1.32186	0.0603973	0.0301986	+	0.999544i	$0.490386\pi$
$480$	0	0
$481$	−24.4468	−1.11468
$482$	−40.3058	−1.83588
$483$	0.632596	0.0287841
$484$	−40.6812	−1.84915
$485$	0	0
$486$	−2.72810	−0.123749
$487$	−9.89390	−0.448335	−0.224168	−	0.974551i	$-0.571966\pi$
$487$	−9.89390	−0.448335	−0.224168	+	0.974551i	$0.571966\pi$
$488$	82.2024	3.72113
$489$	−1.25711	−0.0568484
$490$	0	0
$491$	17.1305	0.773089	0.386545	−	0.922271i	$-0.373669\pi$
$491$	17.1305	0.773089	0.386545	+	0.922271i	$0.373669\pi$
$492$	1.17146	0.0528135
$493$	−5.74056	−0.258542
$494$	−80.8568	−3.63792
$495$	0	0
$496$	−120.564	−5.41350
$497$	−1.28080	−0.0574516
$498$	−26.1492	−1.17177
$499$	−7.95477	−0.356104	−0.178052	−	0.984021i	$-0.556980\pi$
$499$	−7.95477	−0.356104	−0.178052	+	0.984021i	$0.556980\pi$
$500$	0	0
$501$	15.1462	0.676683
$502$	−9.02989	−0.403024
$503$	13.3471	0.595119	0.297559	−	0.954703i	$-0.403827\pi$
$503$	13.3471	0.595119	0.297559	+	0.954703i	$0.403827\pi$
$504$	43.3601	1.93141
$505$	0	0
$506$	0.701825	0.0311999
$507$	−17.0742	−0.758293
$508$	107.275	4.75956
$509$	−5.55324	−0.246143	−0.123071	−	0.992398i	$-0.539274\pi$
$509$	−5.55324	−0.246143	−0.123071	+	0.992398i	$0.539274\pi$
$510$	0	0
$511$	−53.0639	−2.34741
$512$	38.8286	1.71600
$513$	5.40455	0.238617
$514$	−84.2076	−3.71424
$515$	0	0
$516$	−39.0291	−1.71816
$517$	18.6123	0.818567
$518$	−56.1487	−2.46703
$519$	−2.34929	−0.103122
$520$	0	0
$521$	41.5095	1.81857	0.909283	−	0.416179i	$-0.136631\pi$
$521$	41.5095	1.81857	0.909283	+	0.416179i	$0.136631\pi$
$522$	2.72810	0.119406
$523$	−11.6261	−0.508375	−0.254188	−	0.967155i	$-0.581808\pi$
$523$	−11.6261	−0.508375	−0.254188	+	0.967155i	$0.581808\pi$
$524$	−5.75479	−0.251399
$525$	0	0
$526$	53.1525	2.31756
$527$	46.9674	2.04593
$528$	27.6678	1.20409
$529$	−22.9812	−0.999184
$530$	0	0
$531$	−0.244913	−0.0106283
$532$	−135.804	−5.88786
$533$	−1.18039	−0.0511284
$534$	−44.5586	−1.92824
$535$	0	0
$536$	−61.3543	−2.65010
$537$	17.7814	0.767322
$538$	56.8495	2.45096
$539$	−26.8797	−1.15779
$540$	0	0
$541$	−5.29830	−0.227792	−0.113896	−	0.993493i	$-0.536333\pi$
$541$	−5.29830	−0.227792	−0.113896	+	0.993493i	$0.536333\pi$
$542$	67.0700	2.88090
$543$	2.69871	0.115813
$544$	−122.951	−5.27149
$545$	0	0
$546$	−69.0735	−2.95607
$547$	−6.78093	−0.289932	−0.144966	−	0.989437i	$-0.546307\pi$
$547$	−6.78093	−0.289932	−0.144966	+	0.989437i	$0.546307\pi$
$548$	17.4959	0.747387
$549$	8.75284	0.373562
$550$	0	0
$551$	−5.40455	−0.230241
$552$	1.28679	0.0547693
$553$	7.67441	0.326349
$554$	5.18076	0.220109
$555$	0	0
$556$	46.1748	1.95825
$557$	0.186307	0.00789410	0.00394705	−	0.999992i	$-0.498744\pi$
$557$	0.186307	0.00789410	0.00394705	+	0.999992i	$0.498744\pi$
$558$	−22.3204	−0.944898
$559$	39.3266	1.66334
$560$	0	0
$561$	−10.7783	−0.455062
$562$	88.6401	3.73906
$563$	−27.1382	−1.14374	−0.571869	−	0.820345i	$-0.693781\pi$
$563$	−27.1382	−1.14374	−0.571869	+	0.820345i	$0.693781\pi$
$564$	53.9512	2.27176
$565$	0	0
$566$	−32.5056	−1.36631
$567$	4.61695	0.193893
$568$	−2.60532	−0.109317
$569$	22.8795	0.959157	0.479578	−	0.877499i	$-0.340790\pi$
$569$	22.8795	0.959157	0.479578	+	0.877499i	$0.340790\pi$
$570$	0	0
$571$	38.3621	1.60541	0.802703	−	0.596379i	$-0.203394\pi$
$571$	38.3621	1.60541	0.802703	+	0.596379i	$0.203394\pi$
$572$	−56.0395	−2.34313
$573$	−10.7997	−0.451165
$574$	−2.71108	−0.113158
$575$	0	0
$576$	28.9585	1.20660
$577$	−12.0497	−0.501637	−0.250818	−	0.968034i	$-0.580700\pi$
$577$	−12.0497	−0.501637	−0.250818	+	0.968034i	$0.580700\pi$
$578$	43.5241	1.81036
$579$	4.18447	0.173901
$580$	0	0
$581$	44.2541	1.83597
$582$	−21.9374	−0.909334
$583$	2.15459	0.0892341
$584$	−107.939	−4.46656
$585$	0	0
$586$	56.5695	2.33687
$587$	−23.6704	−0.976982	−0.488491	−	0.872569i	$-0.662453\pi$
$587$	−23.6704	−0.976982	−0.488491	+	0.872569i	$0.662453\pi$
$588$	−77.9160	−3.21320
$589$	44.2183	1.82198
$590$	0	0
$591$	−5.76856	−0.237287
$592$	−65.6905	−2.69986
$593$	25.9414	1.06529	0.532643	−	0.846340i	$-0.321199\pi$
$593$	25.9414	1.06529	0.532643	+	0.846340i	$0.321199\pi$
$594$	5.12221	0.210167
$595$	0	0
$596$	−32.2852	−1.32245
$597$	5.45094	0.223092
$598$	−2.04988	−0.0838259
$599$	−17.2886	−0.706393	−0.353196	−	0.935549i	$-0.614905\pi$
$599$	−17.2886	−0.706393	−0.353196	+	0.935549i	$0.614905\pi$
$600$	0	0
$601$	−30.8997	−1.26042	−0.630212	−	0.776423i	$-0.717032\pi$
$601$	−30.8997	−1.26042	−0.630212	+	0.776423i	$0.717032\pi$
$602$	90.3240	3.68133
$603$	−6.53295	−0.266042
$604$	43.8820	1.78553
$605$	0	0
$606$	33.6207	1.36575
$607$	4.07152	0.165258	0.0826290	−	0.996580i	$-0.473668\pi$
$607$	4.07152	0.165258	0.0826290	+	0.996580i	$0.473668\pi$
$608$	−115.754	−4.69446
$609$	−4.61695	−0.187088
$610$	0	0
$611$	−54.3625	−2.19927
$612$	−31.2431	−1.26293
$613$	32.3595	1.30699	0.653494	−	0.756931i	$-0.273302\pi$
$613$	32.3595	1.30699	0.653494	+	0.756931i	$0.273302\pi$
$614$	−46.3108	−1.86895
$615$	0	0
$616$	−81.4118	−3.28017
$617$	−33.4957	−1.34849	−0.674243	−	0.738509i	$-0.735530\pi$
$617$	−33.4957	−1.34849	−0.674243	+	0.738509i	$0.735530\pi$
$618$	1.80303	0.0725286
$619$	−16.6857	−0.670656	−0.335328	−	0.942101i	$-0.608847\pi$
$619$	−16.6857	−0.670656	−0.335328	+	0.942101i	$0.608847\pi$
$620$	0	0
$621$	0.137016	0.00549827
$622$	−57.0224	−2.28639
$623$	75.4096	3.02122
$624$	−80.8117	−3.23506
$625$	0	0
$626$	34.8526	1.39299
$627$	−10.1474	−0.405250
$628$	−53.7245	−2.14384
$629$	25.5905	1.02036
$630$	0	0
$631$	9.91590	0.394746	0.197373	−	0.980328i	$-0.436759\pi$
$631$	9.91590	0.394746	0.197373	+	0.980328i	$0.436759\pi$
$632$	15.6108	0.620965
$633$	−0.863332	−0.0343143
$634$	41.0719	1.63118
$635$	0	0
$636$	6.24550	0.247650
$637$	78.5099	3.11068
$638$	−5.12221	−0.202790
$639$	−0.277412	−0.0109742
$640$	0	0
$641$	−22.9862	−0.907902	−0.453951	−	0.891027i	$-0.649986\pi$
$641$	−22.9862	−0.907902	−0.453951	+	0.891027i	$0.649986\pi$
$642$	39.7942	1.57055
$643$	−12.9117	−0.509190	−0.254595	−	0.967048i	$-0.581942\pi$
$643$	−12.9117	−0.509190	−0.254595	+	0.967048i	$0.581942\pi$
$644$	−3.44291	−0.135670
$645$	0	0
$646$	84.6395	3.33010
$647$	16.6096	0.652989	0.326494	−	0.945199i	$-0.394133\pi$
$647$	16.6096	0.652989	0.326494	+	0.945199i	$0.394133\pi$
$648$	9.39150	0.368933
$649$	0.459842	0.0180504
$650$	0	0
$651$	37.7743	1.48049
$652$	6.84182	0.267947
$653$	−21.5638	−0.843858	−0.421929	−	0.906629i	$-0.638647\pi$
$653$	−21.5638	−0.843858	−0.421929	+	0.906629i	$0.638647\pi$
$654$	−42.2993	−1.65403
$655$	0	0
$656$	−3.17180	−0.123838
$657$	−11.4933	−0.448396
$658$	−124.858	−4.86747
$659$	31.7863	1.23822	0.619110	−	0.785305i	$-0.287494\pi$
$659$	31.7863	1.23822	0.619110	+	0.785305i	$0.287494\pi$
$660$	0	0
$661$	30.0046	1.16704	0.583521	−	0.812098i	$-0.301674\pi$
$661$	30.0046	1.16704	0.583521	+	0.812098i	$0.301674\pi$
$662$	−48.3760	−1.88019
$663$	31.4812	1.22263
$664$	90.0189	3.49341
$665$	0	0
$666$	−12.1614	−0.471246
$667$	−0.137016	−0.00530529
$668$	−82.4334	−3.18944
$669$	0.615197	0.0237849
$670$	0	0
$671$	−16.4341	−0.634432
$672$	−98.8856	−3.81460
$673$	−35.0842	−1.35240	−0.676199	−	0.736719i	$-0.736374\pi$
$673$	−35.0842	−1.35240	−0.676199	+	0.736719i	$0.736374\pi$
$674$	−47.9886	−1.84845
$675$	0	0
$676$	92.9267	3.57410
$677$	42.1433	1.61970	0.809849	−	0.586638i	$-0.199549\pi$
$677$	42.1433	1.61970	0.809849	+	0.586638i	$0.199549\pi$
$678$	21.2972	0.817912
$679$	37.1261	1.42477
$680$	0	0
$681$	23.7230	0.909067
$682$	41.9082	1.60475
$683$	−31.8895	−1.22022	−0.610109	−	0.792318i	$-0.708874\pi$
$683$	−31.8895	−1.22022	−0.610109	+	0.792318i	$0.708874\pi$
$684$	−29.4143	−1.12468
$685$	0	0
$686$	92.1508	3.51833
$687$	20.9808	0.800468
$688$	105.673	4.02876
$689$	−6.29311	−0.239748
$690$	0	0
$691$	11.2963	0.429732	0.214866	−	0.976644i	$-0.431069\pi$
$691$	11.2963	0.429732	0.214866	+	0.976644i	$0.431069\pi$
$692$	12.7860	0.486052
$693$	−8.66866	−0.329295
$694$	−57.5052	−2.18287
$695$	0	0
$696$	−9.39150	−0.355984
$697$	1.23561	0.0468022
$698$	−1.97283	−0.0746729
$699$	7.95482	0.300879
$700$	0	0
$701$	−1.39354	−0.0526332	−0.0263166	−	0.999654i	$-0.508378\pi$
$701$	−1.39354	−0.0526332	−0.0263166	+	0.999654i	$0.508378\pi$
$702$	−14.9609	−0.564662
$703$	24.0927	0.908672
$704$	−54.3717	−2.04921
$705$	0	0
$706$	16.2019	0.609767
$707$	−56.8986	−2.13989
$708$	1.33294	0.0500949
$709$	5.24967	0.197156	0.0985778	−	0.995129i	$-0.468571\pi$
$709$	5.24967	0.197156	0.0985778	+	0.995129i	$0.468571\pi$
$710$	0	0
$711$	1.66223	0.0623384
$712$	153.394	5.74867
$713$	1.12102	0.0419826
$714$	72.3051	2.70595
$715$	0	0
$716$	−96.7752	−3.61666
$717$	−17.4888	−0.653130
$718$	−6.30116	−0.235157
$719$	46.4801	1.73341	0.866707	−	0.498817i	$-0.166232\pi$
$719$	46.4801	1.73341	0.866707	+	0.498817i	$0.166232\pi$
$720$	0	0
$721$	−3.05140	−0.113640
$722$	27.8515	1.03653
$723$	14.7743	0.549464
$724$	−14.6878	−0.545866
$725$	0	0
$726$	20.3917	0.756808
$727$	43.3101	1.60628	0.803141	−	0.595790i	$-0.203161\pi$
$727$	43.3101	1.60628	0.803141	+	0.595790i	$0.203161\pi$
$728$	237.786	8.81295
$729$	1.00000	0.0370370
$730$	0	0
$731$	−41.1664	−1.52259
$732$	−47.6375	−1.76073
$733$	18.6793	0.689936	0.344968	−	0.938614i	$-0.387890\pi$
$733$	18.6793	0.689936	0.344968	+	0.938614i	$0.387890\pi$
$734$	62.5375	2.30830
$735$	0	0
$736$	−2.93461	−0.108171
$737$	12.2661	0.451828
$738$	−0.587203	−0.0216152
$739$	41.3256	1.52019	0.760093	−	0.649815i	$-0.225153\pi$
$739$	41.3256	1.52019	0.760093	+	0.649815i	$0.225153\pi$
$740$	0	0
$741$	29.6385	1.08880
$742$	−14.4538	−0.530616
$743$	32.3390	1.18640	0.593202	−	0.805054i	$-0.297864\pi$
$743$	32.3390	1.18640	0.593202	+	0.805054i	$0.297864\pi$
$744$	76.8382	2.81703
$745$	0	0
$746$	−7.72547	−0.282849
$747$	9.58514	0.350702
$748$	58.6612	2.14487
$749$	−67.3464	−2.46078
$750$	0	0
$751$	−9.34454	−0.340987	−0.170494	−	0.985359i	$-0.554536\pi$
$751$	−9.34454	−0.340987	−0.170494	+	0.985359i	$0.554536\pi$
$752$	−146.076	−5.32685
$753$	3.30996	0.120622
$754$	14.9609	0.544843
$755$	0	0
$756$	−25.1278	−0.913888
$757$	5.01932	0.182430	0.0912151	−	0.995831i	$-0.470925\pi$
$757$	5.01932	0.182430	0.0912151	+	0.995831i	$0.470925\pi$
$758$	−79.8074	−2.89873
$759$	−0.257258	−0.00933788
$760$	0	0
$761$	−34.9103	−1.26550	−0.632748	−	0.774358i	$-0.718073\pi$
$761$	−34.9103	−1.26550	−0.632748	+	0.774358i	$0.718073\pi$
$762$	−53.7723	−1.94797
$763$	71.5859	2.59159
$764$	58.7777	2.12650
$765$	0	0
$766$	−8.15872	−0.294787
$767$	−1.34310	−0.0484965
$768$	−40.7463	−1.47031
$769$	37.4889	1.35189	0.675943	−	0.736954i	$-0.263737\pi$
$769$	37.4889	1.35189	0.675943	+	0.736954i	$0.263737\pi$
$770$	0	0
$771$	30.8668	1.11164
$772$	−22.7740	−0.819655
$773$	33.7332	1.21330	0.606649	−	0.794970i	$-0.292513\pi$
$773$	33.7332	1.21330	0.606649	+	0.794970i	$0.292513\pi$
$774$	19.5636	0.703199
$775$	0	0
$776$	75.5197	2.71100
$777$	20.5816	0.738362
$778$	−23.3890	−0.838538
$779$	1.16329	0.0416792
$780$	0	0
$781$	0.520862	0.0186379
$782$	2.14578	0.0767330
$783$	−1.00000	−0.0357371
$784$	210.962	7.53436
$785$	0	0
$786$	2.88463	0.102891
$787$	−11.5508	−0.411740	−0.205870	−	0.978579i	$-0.566002\pi$
$787$	−11.5508	−0.411740	−0.205870	+	0.978579i	$0.566002\pi$
$788$	31.3955	1.11842
$789$	−19.4833	−0.693626
$790$	0	0
$791$	−36.0426	−1.28153
$792$	−17.6333	−0.626571
$793$	48.0006	1.70455
$794$	25.3605	0.900012
$795$	0	0
$796$	−29.6668	−1.05151
$797$	−9.11795	−0.322974	−0.161487	−	0.986875i	$-0.551629\pi$
$797$	−9.11795	−0.322974	−0.161487	+	0.986875i	$0.551629\pi$
$798$	68.0728	2.40975
$799$	56.9058	2.01318
$800$	0	0
$801$	16.3332	0.577106
$802$	53.3047	1.88226
$803$	21.5795	0.761525
$804$	35.5557	1.25395
$805$	0	0
$806$	−122.405	−4.31153
$807$	−20.8385	−0.733551
$808$	−115.740	−4.07171
$809$	−11.9809	−0.421227	−0.210613	−	0.977569i	$-0.567546\pi$
$809$	−11.9809	−0.421227	−0.210613	+	0.977569i	$0.567546\pi$
$810$	0	0
$811$	−19.2172	−0.674807	−0.337403	−	0.941360i	$-0.609549\pi$
$811$	−19.2172	−0.674807	−0.337403	+	0.941360i	$0.609549\pi$
$812$	25.1278	0.881812
$813$	−24.5849	−0.862230
$814$	22.8340	0.800332
$815$	0	0
$816$	84.5924	2.96132
$817$	−38.7568	−1.35593
$818$	103.554	3.62067
$819$	25.3193	0.884728
$820$	0	0
$821$	−10.5611	−0.368585	−0.184292	−	0.982871i	$-0.558999\pi$
$821$	−10.5611	−0.368585	−0.184292	+	0.982871i	$0.558999\pi$
$822$	−8.76994	−0.305887
$823$	29.1655	1.01665	0.508323	−	0.861166i	$-0.330266\pi$
$823$	29.1655	1.01665	0.508323	+	0.861166i	$0.330266\pi$
$824$	−6.20696	−0.216230
$825$	0	0
$826$	−3.08479	−0.107334
$827$	−40.5140	−1.40881	−0.704405	−	0.709798i	$-0.748786\pi$
$827$	−40.5140	−1.40881	−0.704405	+	0.709798i	$0.748786\pi$
$828$	−0.745712	−0.0259153
$829$	51.3925	1.78494	0.892468	−	0.451111i	$-0.148972\pi$
$829$	51.3925	1.78494	0.892468	+	0.451111i	$0.148972\pi$
$830$	0	0
$831$	−1.89904	−0.0658769
$832$	158.808	5.50569
$833$	−82.1829	−2.84747
$834$	−23.1455	−0.801461
$835$	0	0
$836$	55.2276	1.91009
$837$	8.18167	0.282800
$838$	−27.5407	−0.951379
$839$	−30.2352	−1.04383	−0.521917	−	0.852996i	$-0.674783\pi$
$839$	−30.2352	−1.04383	−0.521917	+	0.852996i	$0.674783\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	37.7899	1.30233
$843$	−32.4916	−1.11907
$844$	4.69869	0.161736
$845$	0	0
$846$	−27.0434	−0.929773
$847$	−34.5103	−1.18579
$848$	−16.9100	−0.580693
$849$	11.9151	0.408925
$850$	0	0
$851$	0.610797	0.0209379
$852$	1.50982	0.0517255
$853$	−0.163336	−0.00559252	−0.00279626	−	0.999996i	$-0.500890\pi$
$853$	−0.163336	−0.00559252	−0.00279626	+	0.999996i	$0.500890\pi$
$854$	110.246	3.77255
$855$	0	0
$856$	−136.992	−4.68229
$857$	0.869730	0.0297094	0.0148547	−	0.999890i	$-0.495271\pi$
$857$	0.869730	0.0297094	0.0148547	+	0.999890i	$0.495271\pi$
$858$	28.0902	0.958983
$859$	−50.5362	−1.72427	−0.862136	−	0.506677i	$-0.830874\pi$
$859$	−50.5362	−1.72427	−0.862136	+	0.506677i	$0.830874\pi$
$860$	0	0
$861$	0.993764	0.0338674
$862$	49.6920	1.69251
$863$	−43.1139	−1.46761	−0.733807	−	0.679358i	$-0.762258\pi$
$863$	−43.1139	−1.46761	−0.733807	+	0.679358i	$0.762258\pi$
$864$	−21.4180	−0.728654
$865$	0	0
$866$	18.3731	0.624342
$867$	−15.9540	−0.541827
$868$	−205.587	−6.97809
$869$	−3.12096	−0.105871
$870$	0	0
$871$	−35.8267	−1.21394
$872$	145.616	4.93117
$873$	8.04128	0.272156
$874$	2.02018	0.0683337
$875$	0	0
$876$	62.5524	2.11345
$877$	−3.23901	−0.109374	−0.0546868	−	0.998504i	$-0.517416\pi$
$877$	−3.23901	−0.109374	−0.0546868	+	0.998504i	$0.517416\pi$
$878$	84.4590	2.85035
$879$	−20.7359	−0.699404
$880$	0	0
$881$	9.49218	0.319800	0.159900	−	0.987133i	$-0.448883\pi$
$881$	9.49218	0.319800	0.159900	+	0.987133i	$0.448883\pi$
$882$	39.0559	1.31508
$883$	−40.4268	−1.36047	−0.680235	−	0.732994i	$-0.738122\pi$
$883$	−40.4268	−1.36047	−0.680235	+	0.732994i	$0.738122\pi$
$884$	−171.337	−5.76268
$885$	0	0
$886$	58.4991	1.96532
$887$	17.5249	0.588430	0.294215	−	0.955739i	$-0.404942\pi$
$887$	17.5249	0.588430	0.294215	+	0.955739i	$0.404942\pi$
$888$	41.8659	1.40493
$889$	91.0026	3.05213
$890$	0	0
$891$	−1.87758	−0.0629011
$892$	−3.34822	−0.112107
$893$	53.5749	1.79282
$894$	16.1832	0.541247
$895$	0	0
$896$	166.975	5.57823
$897$	0.751396	0.0250884
$898$	70.0833	2.33871
$899$	−8.18167	−0.272874
$900$	0	0
$901$	6.58752	0.219462
$902$	1.10252	0.0367098
$903$	−33.1088	−1.10179
$904$	−73.3157	−2.43844
$905$	0	0
$906$	−21.9961	−0.730773
$907$	34.8878	1.15843	0.579216	−	0.815174i	$-0.303359\pi$
$907$	34.8878	1.15843	0.579216	+	0.815174i	$0.303359\pi$
$908$	−129.113	−4.28475
$909$	−12.3239	−0.408757
$910$	0	0
$911$	43.5212	1.44192	0.720960	−	0.692976i	$-0.243701\pi$
$911$	43.5212	1.44192	0.720960	+	0.692976i	$0.243701\pi$
$912$	79.6409	2.63717
$913$	−17.9968	−0.595608
$914$	82.7373	2.73671
$915$	0	0
$916$	−114.188	−3.77289
$917$	−4.88185	−0.161213
$918$	15.6608	0.516883
$919$	−38.2543	−1.26189	−0.630946	−	0.775827i	$-0.717333\pi$
$919$	−38.2543	−1.26189	−0.630946	+	0.775827i	$0.717333\pi$
$920$	0	0
$921$	16.9755	0.559361
$922$	−84.5780	−2.78543
$923$	−1.52133	−0.0500751
$924$	47.1793	1.55209
$925$	0	0
$926$	−75.5655	−2.48324
$927$	−0.660912	−0.0217072
$928$	21.4180	0.703079
$929$	40.3492	1.32381	0.661906	−	0.749586i	$-0.269748\pi$
$929$	40.3492	1.32381	0.661906	+	0.749586i	$0.269748\pi$
$930$	0	0
$931$	−77.3725	−2.53578
$932$	−43.2942	−1.41815
$933$	20.9019	0.684298
$934$	−11.0009	−0.359959
$935$	0	0
$936$	51.5030	1.68343
$937$	−4.95384	−0.161835	−0.0809174	−	0.996721i	$-0.525785\pi$
$937$	−4.95384	−0.161835	−0.0809174	+	0.996721i	$0.525785\pi$
$938$	−82.2857	−2.68672
$939$	−12.7754	−0.416910
$940$	0	0
$941$	−47.5220	−1.54917	−0.774587	−	0.632467i	$-0.782042\pi$
$941$	−47.5220	−1.54917	−0.774587	+	0.632467i	$0.782042\pi$
$942$	26.9298	0.877419
$943$	0.0294917	0.000960383 0
$944$	−3.60901	−0.117463
$945$	0	0
$946$	−36.7321	−1.19426
$947$	−37.1547	−1.20736	−0.603682	−	0.797225i	$-0.706300\pi$
$947$	−37.1547	−1.20736	−0.603682	+	0.797225i	$0.706300\pi$
$948$	−9.04669	−0.293823
$949$	−63.0292	−2.04601
$950$	0	0
$951$	−15.0552	−0.488197
$952$	−248.911	−8.06725
$953$	44.5351	1.44263	0.721316	−	0.692606i	$-0.243537\pi$
$953$	44.5351	1.44263	0.721316	+	0.692606i	$0.243537\pi$
$954$	−3.13060	−0.101357
$955$	0	0
$956$	95.1828	3.07843
$957$	1.87758	0.0606934
$958$	3.60616	0.116510
$959$	14.8420	0.479272
$960$	0	0
$961$	35.9398	1.15935
$962$	−66.6933	−2.15028
$963$	−14.5868	−0.470053
$964$	−80.4095	−2.58982
$965$	0	0
$966$	1.72578	0.0555262
$967$	48.7291	1.56702	0.783511	−	0.621379i	$-0.213427\pi$
$967$	48.7291	1.56702	0.783511	+	0.621379i	$0.213427\pi$
$968$	−70.1988	−2.25627
$969$	−31.0251	−0.996671
$970$	0	0
$971$	−51.7448	−1.66057	−0.830285	−	0.557339i	$-0.811822\pi$
$971$	−51.7448	−1.66057	−0.830285	+	0.557339i	$0.811822\pi$
$972$	−5.44251	−0.174569
$973$	39.1706	1.25575
$974$	−26.9915	−0.864864
$975$	0	0
$976$	128.981	4.12858
$977$	37.7726	1.20845	0.604227	−	0.796813i	$-0.293482\pi$
$977$	37.7726	1.20845	0.604227	+	0.796813i	$0.293482\pi$
$978$	−3.42951	−0.109664
$979$	−30.6669	−0.980117
$980$	0	0
$981$	15.5050	0.495038
$982$	46.7337	1.49133
$983$	−21.1318	−0.673999	−0.336999	−	0.941505i	$-0.609412\pi$
$983$	−21.1318	−0.673999	−0.336999	+	0.941505i	$0.609412\pi$
$984$	2.02145	0.0644416
$985$	0	0
$986$	−15.6608	−0.498742
$987$	45.7675	1.45679
$988$	−161.308	−5.13189
$989$	−0.982563	−0.0312437
$990$	0	0
$991$	11.0241	0.350191	0.175096	−	0.984551i	$-0.443977\pi$
$991$	11.0241	0.350191	0.175096	+	0.984551i	$0.443977\pi$
$992$	−175.235	−5.56371
$993$	17.7325	0.562725
$994$	−3.49414	−0.110827
$995$	0	0
$996$	−52.1672	−1.65298
$997$	28.2785	0.895589	0.447795	−	0.894136i	$-0.352210\pi$
$997$	28.2785	0.895589	0.447795	+	0.894136i	$0.352210\pi$
$998$	−21.7014	−0.686945
$999$	4.45785	0.141040

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.bc.1.8	✓	8
3.2	odd	2		6525.2.a.bz.1.1		8
5.2	odd	4		2175.2.c.p.349.16		16
5.3	odd	4		2175.2.c.p.349.1		16
5.4	even	2		2175.2.a.bd.1.1	yes	8
15.14	odd	2		6525.2.a.by.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.bc.1.8	✓	8	1.1	even	1	trivial
2175.2.a.bd.1.1	yes	8	5.4	even	2
2175.2.c.p.349.1		16	5.3	odd	4
2175.2.c.p.349.16		16	5.2	odd	4
6525.2.a.by.1.8		8	15.14	odd	2
6525.2.a.bz.1.1		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)

\(f(q)\)	\(=\)	\(q+2.72810 q^{2} -1.00000 q^{3} +5.44251 q^{4} -2.72810 q^{6} +4.61695 q^{7} +9.39150 q^{8} +1.00000 q^{9} -1.87758 q^{11} -5.44251 q^{12} +5.48400 q^{13} +12.5955 q^{14} +14.7359 q^{16} -5.74056 q^{17} +2.72810 q^{18} -5.40455 q^{19} -4.61695 q^{21} -5.12221 q^{22} -0.137016 q^{23} -9.39150 q^{24} +14.9609 q^{26} -1.00000 q^{27} +25.1278 q^{28} +1.00000 q^{29} -8.18167 q^{31} +21.4180 q^{32} +1.87758 q^{33} -15.6608 q^{34} +5.44251 q^{36} -4.45785 q^{37} -14.7441 q^{38} -5.48400 q^{39} -0.215243 q^{41} -12.5955 q^{42} +7.17115 q^{43} -10.2187 q^{44} -0.373793 q^{46} -9.91293 q^{47} -14.7359 q^{48} +14.3162 q^{49} +5.74056 q^{51} +29.8467 q^{52} -1.14754 q^{53} -2.72810 q^{54} +43.3601 q^{56} +5.40455 q^{57} +2.72810 q^{58} -0.244913 q^{59} +8.75284 q^{61} -22.3204 q^{62} +4.61695 q^{63} +28.9585 q^{64} +5.12221 q^{66} -6.53295 q^{67} -31.2431 q^{68} +0.137016 q^{69} -0.277412 q^{71} +9.39150 q^{72} -11.4933 q^{73} -12.1614 q^{74} -29.4143 q^{76} -8.66866 q^{77} -14.9609 q^{78} +1.66223 q^{79} +1.00000 q^{81} -0.587203 q^{82} +9.58514 q^{83} -25.1278 q^{84} +19.5636 q^{86} -1.00000 q^{87} -17.6333 q^{88} +16.3332 q^{89} +25.3193 q^{91} -0.745712 q^{92} +8.18167 q^{93} -27.0434 q^{94} -21.4180 q^{96} +8.04128 q^{97} +39.0559 q^{98} -1.87758 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 8 q^{3} + 12 q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} + 6 q^{11} - 12 q^{12} + 6 q^{13} + 9 q^{14} + 32 q^{16} - 12 q^{17} - 2 q^{18} - 2 q^{21} + 3 q^{22} - 14 q^{23} + 3 q^{24} + 18 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) \) 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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