LMFDB - Embedded newform 2075.2.a.f.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2075,2,Mod(1,2075)] 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("2075.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2075, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2075 = 5^{2} \cdot 83 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2075.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-2,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$16.5689584194$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7783241.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 9x^{3} + 5x^{2} - 6x - 1 $ x^6 - 2x^5 - 5x^4 + 9x^3 + 5x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 415)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.92537$ of defining polynomial
Character	$\chi$	$=$	2075.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.92537 q^{2} -0.431741 q^{3} +1.70704 q^{4} -0.831260 q^{6} +3.01948 q^{7} -0.564055 q^{8} -2.81360 q^{9} +0.386511 q^{11} -0.736999 q^{12} -6.99372 q^{13} +5.81360 q^{14} -4.50009 q^{16} -6.58773 q^{17} -5.41721 q^{18} -5.57007 q^{19} -1.30363 q^{21} +0.744176 q^{22} -2.83126 q^{23} +0.243526 q^{24} -13.4655 q^{26} +2.50997 q^{27} +5.15437 q^{28} +6.46422 q^{29} +3.65295 q^{31} -7.53623 q^{32} -0.166873 q^{33} -12.6838 q^{34} -4.80293 q^{36} +2.48887 q^{37} -10.7244 q^{38} +3.01948 q^{39} +8.52322 q^{41} -2.50997 q^{42} -6.95309 q^{43} +0.659790 q^{44} -5.45122 q^{46} -1.08451 q^{47} +1.94287 q^{48} +2.11723 q^{49} +2.84419 q^{51} -11.9386 q^{52} -5.27625 q^{53} +4.83261 q^{54} -1.70315 q^{56} +2.40483 q^{57} +12.4460 q^{58} +4.70284 q^{59} +5.05383 q^{61} +7.03328 q^{62} -8.49559 q^{63} -5.50982 q^{64} -0.321291 q^{66} -4.19631 q^{67} -11.2455 q^{68} +1.22237 q^{69} -10.3287 q^{71} +1.58702 q^{72} +13.8414 q^{73} +4.79198 q^{74} -9.50834 q^{76} +1.16706 q^{77} +5.81360 q^{78} +7.75829 q^{79} +7.35714 q^{81} +16.4103 q^{82} -1.00000 q^{83} -2.22535 q^{84} -13.3873 q^{86} -2.79087 q^{87} -0.218013 q^{88} +0.0634123 q^{89} -21.1174 q^{91} -4.83308 q^{92} -1.57713 q^{93} -2.08808 q^{94} +3.25370 q^{96} -6.29407 q^{97} +4.07645 q^{98} -1.08749 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} - 3 q^{3} + 2 q^{4} + 4 q^{6} - 3 q^{8} + 9 q^{9} + 4 q^{11} - 13 q^{12} - 7 q^{13} + 9 q^{14} - 6 q^{16} - 21 q^{17} - 11 q^{18} + 2 q^{19} - 9 q^{21} - 5 q^{22} - 8 q^{23} + 11 q^{24} - 15 q^{26}+ \cdots - 22 q^{99}+O(q^{100}) $ 6 * q - 2 * q^2 - 3 * q^3 + 2 * q^4 + 4 * q^6 - 3 * q^8 + 9 * q^9 + 4 * q^11 - 13 * q^12 - 7 * q^13 + 9 * q^14 - 6 * q^16 - 21 * q^17 - 11 * q^18 + 2 * q^19 - 9 * q^21 - 5 * q^22 - 8 * q^23 + 11 * q^24 - 15 * q^26 - 12 * q^27 + 5 * q^28 + 10 * q^29 - 13 * q^31 + 7 * q^32 - 6 * q^33 - 5 * q^34 - 10 * q^36 - q^37 - 3 * q^38 + 10 * q^41 + 12 * q^42 + 3 * q^43 + 6 * q^44 - 15 * q^46 - 14 * q^47 + 28 * q^48 - 12 * q^49 - 6 * q^51 - 5 * q^52 - 9 * q^53 + 21 * q^54 - 2 * q^56 - 19 * q^57 + 27 * q^58 + 20 * q^59 - 13 * q^61 + 12 * q^62 + 11 * q^63 - 19 * q^64 - 20 * q^66 - 16 * q^67 - 11 * q^69 + 5 * q^71 + 2 * q^72 - 3 * q^73 + 11 * q^74 - 23 * q^76 - 30 * q^77 + 9 * q^78 + 2 * q^79 + 2 * q^81 + q^82 - 6 * q^83 - 3 * q^84 - 42 * q^86 + 5 * q^87 + 4 * q^88 + 45 * q^89 - 21 * q^91 + 15 * q^92 + 27 * q^93 + 2 * q^94 - 45 * q^96 - 22 * q^97 + 3 * q^98 - 22 * 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 $

$2$	1.92537	1.36144	0.680720	−	0.732543i	$-0.261667\pi$
$2$	1.92537	1.36144	0.680720	+	0.732543i	$0.261667\pi$
$3$	−0.431741	−0.249266	−0.124633	−	0.992203i	$-0.539775\pi$
$3$	−0.431741	−0.249266	−0.124633	+	0.992203i	$0.539775\pi$
$4$	1.70704	0.853520
$5$	0	0
$5$	0	0
$6$	−0.831260	−0.339360
$7$	3.01948	1.14125	0.570627	−	0.821209i	$-0.306700\pi$
$7$	3.01948	1.14125	0.570627	+	0.821209i	$0.306700\pi$
$8$	−0.564055	−0.199424
$9$	−2.81360	−0.937867
$10$	0	0
$11$	0.386511	0.116537	0.0582687	−	0.998301i	$-0.481442\pi$
$11$	0.386511	0.116537	0.0582687	+	0.998301i	$0.481442\pi$
$12$	−0.736999	−0.212753
$13$	−6.99372	−1.93971	−0.969855	−	0.243685i	$-0.921644\pi$
$13$	−6.99372	−1.93971	−0.969855	+	0.243685i	$0.921644\pi$
$14$	5.81360	1.55375
$15$	0	0
$16$	−4.50009	−1.12502
$17$	−6.58773	−1.59776	−0.798880	−	0.601490i	$-0.794574\pi$
$17$	−6.58773	−1.59776	−0.798880	+	0.601490i	$0.794574\pi$
$18$	−5.41721	−1.27685
$19$	−5.57007	−1.27786	−0.638931	−	0.769264i	$-0.720623\pi$
$19$	−5.57007	−1.27786	−0.638931	+	0.769264i	$0.720623\pi$
$20$	0	0
$21$	−1.30363	−0.284476
$22$	0.744176	0.158659
$23$	−2.83126	−0.590359	−0.295179	−	0.955442i	$-0.595379\pi$
$23$	−2.83126	−0.590359	−0.295179	+	0.955442i	$0.595379\pi$
$24$	0.243526	0.0497095
$25$	0	0
$26$	−13.4655	−2.64080
$27$	2.50997	0.483044
$28$	5.15437	0.974084
$29$	6.46422	1.20038	0.600188	−	0.799859i	$-0.295092\pi$
$29$	6.46422	1.20038	0.600188	+	0.799859i	$0.295092\pi$
$30$	0	0
$31$	3.65295	0.656090	0.328045	−	0.944662i	$-0.393610\pi$
$31$	3.65295	0.656090	0.328045	+	0.944662i	$0.393610\pi$
$32$	−7.53623	−1.33223
$33$	−0.166873	−0.0290488
$34$	−12.6838	−2.17526
$35$	0	0
$36$	−4.80293	−0.800488
$37$	2.48887	0.409167	0.204584	−	0.978849i	$-0.434416\pi$
$37$	2.48887	0.409167	0.204584	+	0.978849i	$0.434416\pi$
$38$	−10.7244	−1.73973
$39$	3.01948	0.483503
$40$	0	0
$41$	8.52322	1.33110	0.665551	−	0.746352i	$-0.268196\pi$
$41$	8.52322	1.33110	0.665551	+	0.746352i	$0.268196\pi$
$42$	−2.50997	−0.387297
$43$	−6.95309	−1.06034	−0.530168	−	0.847893i	$-0.677871\pi$
$43$	−6.95309	−1.06034	−0.530168	+	0.847893i	$0.677871\pi$
$44$	0.659790	0.0994671
$45$	0	0
$46$	−5.45122	−0.803738
$47$	−1.08451	−0.158192	−0.0790958	−	0.996867i	$-0.525203\pi$
$47$	−1.08451	−0.158192	−0.0790958	+	0.996867i	$0.525203\pi$
$48$	1.94287	0.280430
$49$	2.11723	0.302462
$50$	0	0
$51$	2.84419	0.398267
$52$	−11.9386	−1.65558
$53$	−5.27625	−0.724749	−0.362374	−	0.932033i	$-0.618034\pi$
$53$	−5.27625	−0.724749	−0.362374	+	0.932033i	$0.618034\pi$
$54$	4.83261	0.657635
$55$	0	0
$56$	−1.70315	−0.227593
$57$	2.40483	0.318527
$58$	12.4460	1.63424
$59$	4.70284	0.612257	0.306129	−	0.951990i	$-0.400966\pi$
$59$	4.70284	0.612257	0.306129	+	0.951990i	$0.400966\pi$
$60$	0	0
$61$	5.05383	0.647076	0.323538	−	0.946215i	$-0.395128\pi$
$61$	5.05383	0.647076	0.323538	+	0.946215i	$0.395128\pi$
$62$	7.03328	0.893227
$63$	−8.49559	−1.07034
$64$	−5.50982	−0.688727
$65$	0	0
$66$	−0.321291	−0.0395482
$67$	−4.19631	−0.512661	−0.256330	−	0.966589i	$-0.582514\pi$
$67$	−4.19631	−0.512661	−0.256330	+	0.966589i	$0.582514\pi$
$68$	−11.2455	−1.36372
$69$	1.22237	0.147156
$70$	0	0
$71$	−10.3287	−1.22579	−0.612897	−	0.790163i	$-0.709996\pi$
$71$	−10.3287	−1.22579	−0.612897	+	0.790163i	$0.709996\pi$
$72$	1.58702	0.187033
$73$	13.8414	1.62001	0.810004	−	0.586424i	$-0.199465\pi$
$73$	13.8414	1.62001	0.810004	+	0.586424i	$0.199465\pi$
$74$	4.79198	0.557057
$75$	0	0
$76$	−9.50834	−1.09068
$77$	1.16706	0.132999
$78$	5.81360	0.658261
$79$	7.75829	0.872876	0.436438	−	0.899734i	$-0.356240\pi$
$79$	7.75829	0.872876	0.436438	+	0.899734i	$0.356240\pi$
$80$	0	0
$81$	7.35714	0.817460
$82$	16.4103	1.81222
$83$	−1.00000	−0.109764
$83$	−1.00000	−0.109764
$84$	−2.22535	−0.242806
$85$	0	0
$86$	−13.3873	−1.44358
$87$	−2.79087	−0.299213
$88$	−0.218013	−0.0232403
$89$	0.0634123	0.00672169	0.00336084	−	0.999994i	$-0.498930\pi$
$89$	0.0634123	0.00672169	0.00336084	+	0.999994i	$0.498930\pi$
$90$	0	0
$91$	−21.1174	−2.21370
$92$	−4.83308	−0.503883
$93$	−1.57713	−0.163541
$94$	−2.08808	−0.215369
$95$	0	0
$96$	3.25370	0.332079
$97$	−6.29407	−0.639066	−0.319533	−	0.947575i	$-0.603526\pi$
$97$	−6.29407	−0.639066	−0.319533	+	0.947575i	$0.603526\pi$
$98$	4.07645	0.411783
$99$	−1.08749	−0.109297
$100$	0	0
$101$	−5.38701	−0.536028	−0.268014	−	0.963415i	$-0.586367\pi$
$101$	−5.38701	−0.536028	−0.268014	+	0.963415i	$0.586367\pi$
$102$	5.47612	0.542217
$103$	9.41240	0.927431	0.463716	−	0.885984i	$-0.346516\pi$
$103$	9.41240	0.927431	0.463716	+	0.885984i	$0.346516\pi$
$104$	3.94484	0.386824
$105$	0	0
$106$	−10.1587	−0.986702
$107$	−13.1270	−1.26903	−0.634517	−	0.772909i	$-0.718801\pi$
$107$	−13.1270	−1.26903	−0.634517	+	0.772909i	$0.718801\pi$
$108$	4.28462	0.412288
$109$	−16.5975	−1.58975	−0.794876	−	0.606773i	$-0.792464\pi$
$109$	−16.5975	−1.58975	−0.794876	+	0.606773i	$0.792464\pi$
$110$	0	0
$111$	−1.07455	−0.101991
$112$	−13.5879	−1.28394
$113$	4.56004	0.428972	0.214486	−	0.976727i	$-0.431192\pi$
$113$	4.56004	0.428972	0.214486	+	0.976727i	$0.431192\pi$
$114$	4.63018	0.433656
$115$	0	0
$116$	11.0347	1.02455
$117$	19.6775	1.81919
$118$	9.05469	0.833552
$119$	−19.8915	−1.82345
$120$	0	0
$121$	−10.8506	−0.986419
$122$	9.73047	0.880956
$123$	−3.67982	−0.331798
$124$	6.23574	0.559986
$125$	0	0
$126$	−16.3571	−1.45721
$127$	5.94008	0.527097	0.263549	−	0.964646i	$-0.415107\pi$
$127$	5.94008	0.527097	0.263549	+	0.964646i	$0.415107\pi$
$128$	4.46403	0.394568
$129$	3.00193	0.264306
$130$	0	0
$131$	−13.6795	−1.19518	−0.597590	−	0.801802i	$-0.703875\pi$
$131$	−13.6795	−1.19518	−0.597590	+	0.801802i	$0.703875\pi$
$132$	−0.284858	−0.0247937
$133$	−16.8187	−1.45837
$134$	−8.07944	−0.697957
$135$	0	0
$136$	3.71584	0.318631
$137$	−21.1218	−1.80456	−0.902278	−	0.431155i	$-0.858106\pi$
$137$	−21.1218	−1.80456	−0.902278	+	0.431155i	$0.858106\pi$
$138$	2.35351	0.200344
$139$	9.91452	0.840939	0.420469	−	0.907307i	$-0.361865\pi$
$139$	9.91452	0.840939	0.420469	+	0.907307i	$0.361865\pi$
$140$	0	0
$141$	0.468226	0.0394318
$142$	−19.8866	−1.66885
$143$	−2.70315	−0.226049
$144$	12.6615	1.05512
$145$	0	0
$146$	26.6497	2.20554
$147$	−0.914095	−0.0753933
$148$	4.24860	0.349232
$149$	17.7934	1.45769	0.728847	−	0.684677i	$-0.240057\pi$
$149$	17.7934	1.45769	0.728847	+	0.684677i	$0.240057\pi$
$150$	0	0
$151$	−16.2437	−1.32189	−0.660945	−	0.750434i	$-0.729845\pi$
$151$	−16.2437	−1.32189	−0.660945	+	0.750434i	$0.729845\pi$
$152$	3.14183	0.254836
$153$	18.5352	1.49849
$154$	2.24702	0.181070
$155$	0	0
$156$	5.15437	0.412680
$157$	10.1188	0.807572	0.403786	−	0.914854i	$-0.367694\pi$
$157$	10.1188	0.807572	0.403786	+	0.914854i	$0.367694\pi$
$158$	14.9376	1.18837
$159$	2.27797	0.180655
$160$	0	0
$161$	−8.54892	−0.673749
$162$	14.1652	1.11292
$163$	−16.9814	−1.33008	−0.665041	−	0.746806i	$-0.731586\pi$
$163$	−16.9814	−1.33008	−0.665041	+	0.746806i	$0.731586\pi$
$164$	14.5495	1.13612
$165$	0	0
$166$	−1.92537	−0.149438
$167$	4.22535	0.326967	0.163484	−	0.986546i	$-0.447727\pi$
$167$	4.22535	0.326967	0.163484	+	0.986546i	$0.447727\pi$
$168$	0.735320	0.0567311
$169$	35.9121	2.76247
$170$	0	0
$171$	15.6720	1.19846
$172$	−11.8692	−0.905018
$173$	−5.62999	−0.428040	−0.214020	−	0.976829i	$-0.568656\pi$
$173$	−5.62999	−0.428040	−0.214020	+	0.976829i	$0.568656\pi$
$174$	−5.37345	−0.407360
$175$	0	0
$176$	−1.73934	−0.131107
$177$	−2.03041	−0.152615
$178$	0.122092	0.00915118
$179$	−19.6729	−1.47042	−0.735210	−	0.677839i	$-0.762916\pi$
$179$	−19.6729	−1.47042	−0.735210	+	0.677839i	$0.762916\pi$
$180$	0	0
$181$	12.0848	0.898256	0.449128	−	0.893467i	$-0.351735\pi$
$181$	12.0848	0.898256	0.449128	+	0.893467i	$0.351735\pi$
$182$	−40.6587	−3.01382
$183$	−2.18194	−0.161294
$184$	1.59699	0.117731
$185$	0	0
$186$	−3.03655	−0.222651
$187$	−2.54623	−0.186199
$188$	−1.85130	−0.135020
$189$	7.57879	0.551276
$190$	0	0
$191$	25.1044	1.81649	0.908246	−	0.418437i	$-0.137422\pi$
$191$	25.1044	1.81649	0.908246	+	0.418437i	$0.137422\pi$
$192$	2.37881	0.171676
$193$	−5.20785	−0.374869	−0.187435	−	0.982277i	$-0.560017\pi$
$193$	−5.20785	−0.374869	−0.187435	+	0.982277i	$0.560017\pi$
$194$	−12.1184	−0.870050
$195$	0	0
$196$	3.61420	0.258157
$197$	−14.6607	−1.04453	−0.522267	−	0.852782i	$-0.674913\pi$
$197$	−14.6607	−1.04453	−0.522267	+	0.852782i	$0.674913\pi$
$198$	−2.09381	−0.148801
$199$	−16.2836	−1.15432	−0.577159	−	0.816632i	$-0.695839\pi$
$199$	−16.2836	−1.15432	−0.577159	+	0.816632i	$0.695839\pi$
$200$	0	0
$201$	1.81172	0.127789
$202$	−10.3720	−0.729770
$203$	19.5186	1.36993
$204$	4.85515	0.339929
$205$	0	0
$206$	18.1223	1.26264
$207$	7.96603	0.553678
$208$	31.4724	2.18222
$209$	−2.15289	−0.148919
$210$	0	0
$211$	8.44791	0.581578	0.290789	−	0.956787i	$-0.406082\pi$
$211$	8.44791	0.581578	0.290789	+	0.956787i	$0.406082\pi$
$212$	−9.00677	−0.618588
$213$	4.45933	0.305548
$214$	−25.2743	−1.72771
$215$	0	0
$216$	−1.41576	−0.0963303
$217$	11.0300	0.748766
$218$	−31.9563	−2.16435
$219$	−5.97588	−0.403813
$220$	0	0
$221$	46.0728	3.09919
$222$	−2.06889	−0.138855
$223$	−12.9884	−0.869765	−0.434882	−	0.900487i	$-0.643210\pi$
$223$	−12.9884	−0.869765	−0.434882	+	0.900487i	$0.643210\pi$
$224$	−22.7554	−1.52041
$225$	0	0
$226$	8.77975	0.584020
$227$	22.1599	1.47080	0.735401	−	0.677632i	$-0.236994\pi$
$227$	22.1599	1.47080	0.735401	+	0.677632i	$0.236994\pi$
$228$	4.10514	0.271870
$229$	−9.23725	−0.610414	−0.305207	−	0.952286i	$-0.598726\pi$
$229$	−9.23725	−0.610414	−0.305207	+	0.952286i	$0.598726\pi$
$230$	0	0
$231$	−0.503868	−0.0331521
$232$	−3.64618	−0.239383
$233$	−10.6482	−0.697584	−0.348792	−	0.937200i	$-0.613408\pi$
$233$	−10.6482	−0.697584	−0.348792	+	0.937200i	$0.613408\pi$
$234$	37.8865	2.47672
$235$	0	0
$236$	8.02793	0.522574
$237$	−3.34957	−0.217578
$238$	−38.2985	−2.48252
$239$	−20.4925	−1.32555	−0.662775	−	0.748819i	$-0.730621\pi$
$239$	−20.4925	−1.32555	−0.662775	+	0.748819i	$0.730621\pi$
$240$	0	0
$241$	−1.19273	−0.0768303	−0.0384151	−	0.999262i	$-0.512231\pi$
$241$	−1.19273	−0.0768303	−0.0384151	+	0.999262i	$0.512231\pi$
$242$	−20.8914	−1.34295
$243$	−10.7063	−0.686809
$244$	8.62708	0.552293
$245$	0	0
$246$	−7.08501	−0.451724
$247$	38.9555	2.47868
$248$	−2.06047	−0.130840
$249$	0.431741	0.0273605
$250$	0	0
$251$	12.4779	0.787598	0.393799	−	0.919197i	$-0.371161\pi$
$251$	12.4779	0.787598	0.393799	+	0.919197i	$0.371161\pi$
$252$	−14.5023	−0.913561
$253$	−1.09431	−0.0687989
$254$	11.4368	0.717611
$255$	0	0
$256$	19.6145	1.22591
$257$	1.18701	0.0740434	0.0370217	−	0.999314i	$-0.488213\pi$
$257$	1.18701	0.0740434	0.0370217	+	0.999314i	$0.488213\pi$
$258$	5.77983	0.359836
$259$	7.51507	0.466964
$260$	0	0
$261$	−18.1877	−1.12579
$262$	−26.3380	−1.62717
$263$	28.3723	1.74951	0.874757	−	0.484562i	$-0.161021\pi$
$263$	28.3723	1.74951	0.874757	+	0.484562i	$0.161021\pi$
$264$	0.0941253	0.00579301
$265$	0	0
$266$	−32.3822	−1.98548
$267$	−0.0273777	−0.00167549
$268$	−7.16327	−0.437566
$269$	0.759625	0.0463152	0.0231576	−	0.999732i	$-0.492628\pi$
$269$	0.759625	0.0463152	0.0231576	+	0.999732i	$0.492628\pi$
$270$	0	0
$271$	10.0695	0.611678	0.305839	−	0.952083i	$-0.401063\pi$
$271$	10.0695	0.611678	0.305839	+	0.952083i	$0.401063\pi$
$272$	29.6454	1.79752
$273$	9.11723	0.551800
$274$	−40.6672	−2.45679
$275$	0	0
$276$	2.08664	0.125601
$277$	−3.70252	−0.222463	−0.111232	−	0.993795i	$-0.535480\pi$
$277$	−3.70252	−0.222463	−0.111232	+	0.993795i	$0.535480\pi$
$278$	19.0891	1.14489
$279$	−10.2780	−0.615325
$280$	0	0
$281$	−4.53568	−0.270576	−0.135288	−	0.990806i	$-0.543196\pi$
$281$	−4.53568	−0.270576	−0.135288	+	0.990806i	$0.543196\pi$
$282$	0.901507	0.0536840
$283$	−27.3069	−1.62323	−0.811614	−	0.584194i	$-0.801411\pi$
$283$	−27.3069	−1.62323	−0.811614	+	0.584194i	$0.801411\pi$
$284$	−17.6315	−1.04624
$285$	0	0
$286$	−5.20456	−0.307752
$287$	25.7356	1.51913
$288$	21.2039	1.24945
$289$	26.3982	1.55284
$290$	0	0
$291$	2.71741	0.159297
$292$	23.6278	1.38271
$293$	20.2487	1.18294	0.591471	−	0.806326i	$-0.298547\pi$
$293$	20.2487	1.18294	0.591471	+	0.806326i	$0.298547\pi$
$294$	−1.75997	−0.102643
$295$	0	0
$296$	−1.40386	−0.0815976
$297$	0.970131	0.0562927
$298$	34.2589	1.98456
$299$	19.8010	1.14512
$300$	0	0
$301$	−20.9947	−1.21011
$302$	−31.2750	−1.79968
$303$	2.32579	0.133613
$304$	25.0659	1.43763
$305$	0	0
$306$	35.6872	2.04010
$307$	−27.4709	−1.56785	−0.783925	−	0.620856i	$-0.786785\pi$
$307$	−27.4709	−1.56785	−0.783925	+	0.620856i	$0.786785\pi$
$308$	1.99222	0.113517
$309$	−4.06372	−0.231177
$310$	0	0
$311$	4.51650	0.256107	0.128053	−	0.991767i	$-0.459127\pi$
$311$	4.51650	0.256107	0.128053	+	0.991767i	$0.459127\pi$
$312$	−1.70315	−0.0964219
$313$	−3.91415	−0.221241	−0.110620	−	0.993863i	$-0.535284\pi$
$313$	−3.91415	−0.221241	−0.110620	+	0.993863i	$0.535284\pi$
$314$	19.4825	1.09946
$315$	0	0
$316$	13.2437	0.745017
$317$	15.5806	0.875095	0.437548	−	0.899195i	$-0.355847\pi$
$317$	15.5806	0.875095	0.437548	+	0.899195i	$0.355847\pi$
$318$	4.38593	0.245951
$319$	2.49849	0.139889
$320$	0	0
$321$	5.66746	0.316327
$322$	−16.4598	−0.917269
$323$	36.6942	2.04172
$324$	12.5589	0.697719
$325$	0	0
$326$	−32.6954	−1.81083
$327$	7.16581	0.396270
$328$	−4.80756	−0.265453
$329$	−3.27464	−0.180537
$330$	0	0
$331$	12.4747	0.685672	0.342836	−	0.939395i	$-0.388612\pi$
$331$	12.4747	0.685672	0.342836	+	0.939395i	$0.388612\pi$
$332$	−1.70704	−0.0936860
$333$	−7.00267	−0.383744
$334$	8.13535	0.445147
$335$	0	0
$336$	5.86646	0.320042
$337$	−15.2315	−0.829712	−0.414856	−	0.909887i	$-0.636168\pi$
$337$	−15.2315	−0.829712	−0.414856	+	0.909887i	$0.636168\pi$
$338$	69.1440	3.76094
$339$	−1.96876	−0.106928
$340$	0	0
$341$	1.41191	0.0764591
$342$	30.1743	1.63164
$343$	−14.7434	−0.796069
$344$	3.92193	0.211456
$345$	0	0
$346$	−10.8398	−0.582751
$347$	10.9993	0.590472	0.295236	−	0.955424i	$-0.404602\pi$
$347$	10.9993	0.590472	0.295236	+	0.955424i	$0.404602\pi$
$348$	−4.76413	−0.255384
$349$	−17.1177	−0.916288	−0.458144	−	0.888878i	$-0.651486\pi$
$349$	−17.1177	−0.916288	−0.458144	+	0.888878i	$0.651486\pi$
$350$	0	0
$351$	−17.5540	−0.936964
$352$	−2.91283	−0.155255
$353$	−13.7837	−0.733631	−0.366816	−	0.930294i	$-0.619552\pi$
$353$	−13.7837	−0.733631	−0.366816	+	0.930294i	$0.619552\pi$
$354$	−3.90928	−0.207776
$355$	0	0
$356$	0.108247	0.00573710
$357$	8.58797	0.454524
$358$	−37.8775	−2.00189
$359$	−5.87777	−0.310217	−0.155109	−	0.987897i	$-0.549573\pi$
$359$	−5.87777	−0.310217	−0.155109	+	0.987897i	$0.549573\pi$
$360$	0	0
$361$	12.0257	0.632933
$362$	23.2677	1.22292
$363$	4.68465	0.245880
$364$	−36.0482	−1.88944
$365$	0	0
$366$	−4.20104	−0.219592
$367$	−7.31349	−0.381761	−0.190880	−	0.981613i	$-0.561134\pi$
$367$	−7.31349	−0.381761	−0.190880	+	0.981613i	$0.561134\pi$
$368$	12.7409	0.664167
$369$	−23.9809	−1.24840
$370$	0	0
$371$	−15.9315	−0.827122
$372$	−2.69222	−0.139585
$373$	−1.89691	−0.0982185	−0.0491092	−	0.998793i	$-0.515638\pi$
$373$	−1.89691	−0.0982185	−0.0491092	+	0.998793i	$0.515638\pi$
$374$	−4.90243	−0.253499
$375$	0	0
$376$	0.611722	0.0315471
$377$	−45.2090	−2.32838
$378$	14.5920	0.750529
$379$	−34.6799	−1.78139	−0.890693	−	0.454605i	$-0.849781\pi$
$379$	−34.6799	−1.78139	−0.890693	+	0.454605i	$0.849781\pi$
$380$	0	0
$381$	−2.56458	−0.131387
$382$	48.3352	2.47305
$383$	1.13468	0.0579796	0.0289898	−	0.999580i	$-0.490771\pi$
$383$	1.13468	0.0579796	0.0289898	+	0.999580i	$0.490771\pi$
$384$	−1.92730	−0.0983523
$385$	0	0
$386$	−10.0270	−0.510362
$387$	19.5632	0.994454
$388$	−10.7442	−0.545455
$389$	13.4060	0.679709	0.339855	−	0.940478i	$-0.389622\pi$
$389$	13.4060	0.679709	0.339855	+	0.940478i	$0.389622\pi$
$390$	0	0
$391$	18.6516	0.943251
$392$	−1.19423	−0.0603180
$393$	5.90598	0.297917
$394$	−28.2273	−1.42207
$395$	0	0
$396$	−1.85638	−0.0932868
$397$	24.1747	1.21329	0.606647	−	0.794972i	$-0.292514\pi$
$397$	24.1747	1.21329	0.606647	+	0.794972i	$0.292514\pi$
$398$	−31.3520	−1.57153
$399$	7.26132	0.363521
$400$	0	0
$401$	16.0221	0.800105	0.400053	−	0.916492i	$-0.368992\pi$
$401$	16.0221	0.800105	0.400053	+	0.916492i	$0.368992\pi$
$402$	3.48822	0.173977
$403$	−25.5477	−1.27262
$404$	−9.19585	−0.457511
$405$	0	0
$406$	37.5804	1.86508
$407$	0.961974	0.0476833
$408$	−1.60428	−0.0794238
$409$	15.4202	0.762480	0.381240	−	0.924476i	$-0.375497\pi$
$409$	15.4202	0.762480	0.381240	+	0.924476i	$0.375497\pi$
$410$	0	0
$411$	9.11914	0.449814
$412$	16.0673	0.791581
$413$	14.2001	0.698741
$414$	15.3375	0.753799
$415$	0	0
$416$	52.7063	2.58414
$417$	−4.28051	−0.209617
$418$	−4.14511	−0.202744
$419$	6.59479	0.322177	0.161088	−	0.986940i	$-0.448500\pi$
$419$	6.59479	0.322177	0.161088	+	0.986940i	$0.448500\pi$
$420$	0	0
$421$	−36.1505	−1.76187	−0.880934	−	0.473240i	$-0.843084\pi$
$421$	−36.1505	−1.76187	−0.880934	+	0.473240i	$0.843084\pi$
$422$	16.2653	0.791784
$423$	3.05137	0.148363
$424$	2.97609	0.144532
$425$	0	0
$426$	8.58585	0.415986
$427$	15.2599	0.738478
$428$	−22.4083	−1.08315
$429$	1.16706	0.0563462
$430$	0	0
$431$	25.4782	1.22724	0.613622	−	0.789600i	$-0.289712\pi$
$431$	25.4782	1.22724	0.613622	+	0.789600i	$0.289712\pi$
$432$	−11.2951	−0.543436
$433$	−14.7128	−0.707053	−0.353527	−	0.935424i	$-0.615018\pi$
$433$	−14.7128	−0.707053	−0.353527	+	0.935424i	$0.615018\pi$
$434$	21.2368	1.01940
$435$	0	0
$436$	−28.3326	−1.35688
$437$	15.7703	0.754397
$438$	−11.5058	−0.549767
$439$	6.66548	0.318126	0.159063	−	0.987268i	$-0.449153\pi$
$439$	6.66548	0.318126	0.159063	+	0.987268i	$0.449153\pi$
$440$	0	0
$441$	−5.95704	−0.283669
$442$	88.7070	4.21936
$443$	−0.549488	−0.0261070	−0.0130535	−	0.999915i	$-0.504155\pi$
$443$	−0.549488	−0.0261070	−0.0130535	+	0.999915i	$0.504155\pi$
$444$	−1.83429	−0.0870517
$445$	0	0
$446$	−25.0074	−1.18413
$447$	−7.68215	−0.363353
$448$	−16.6368	−0.786013
$449$	−4.52148	−0.213382	−0.106691	−	0.994292i	$-0.534026\pi$
$449$	−4.52148	−0.213382	−0.106691	+	0.994292i	$0.534026\pi$
$450$	0	0
$451$	3.29432	0.155123
$452$	7.78417	0.366137
$453$	7.01306	0.329502
$454$	42.6659	2.00241
$455$	0	0
$456$	−1.35646	−0.0635219
$457$	−32.9160	−1.53974	−0.769872	−	0.638199i	$-0.779680\pi$
$457$	−32.9160	−1.53974	−0.769872	+	0.638199i	$0.779680\pi$
$458$	−17.7851	−0.831043
$459$	−16.5350	−0.771788
$460$	0	0
$461$	−0.628370	−0.0292661	−0.0146331	−	0.999893i	$-0.504658\pi$
$461$	−0.628370	−0.0292661	−0.0146331	+	0.999893i	$0.504658\pi$
$462$	−0.970131	−0.0451346
$463$	−21.2255	−0.986432	−0.493216	−	0.869907i	$-0.664179\pi$
$463$	−21.2255	−0.986432	−0.493216	+	0.869907i	$0.664179\pi$
$464$	−29.0896	−1.35045
$465$	0	0
$466$	−20.5016	−0.949719
$467$	−35.7171	−1.65279	−0.826396	−	0.563089i	$-0.809613\pi$
$467$	−35.7171	−1.65279	−0.826396	+	0.563089i	$0.809613\pi$
$468$	33.5903	1.55271
$469$	−12.6707	−0.585076
$470$	0	0
$471$	−4.36872	−0.201300
$472$	−2.65266	−0.122098
$473$	−2.68745	−0.123569
$474$	−6.44916	−0.296220
$475$	0	0
$476$	−33.9556	−1.55635
$477$	14.8453	0.679717
$478$	−39.4556	−1.80466
$479$	−24.3814	−1.11401	−0.557007	−	0.830508i	$-0.688050\pi$
$479$	−24.3814	−1.11401	−0.557007	+	0.830508i	$0.688050\pi$
$480$	0	0
$481$	−17.4064	−0.793665
$482$	−2.29644	−0.104600
$483$	3.69092	0.167943
$484$	−18.5224	−0.841929
$485$	0	0
$486$	−20.6135	−0.935049
$487$	23.1661	1.04976	0.524879	−	0.851177i	$-0.324111\pi$
$487$	23.1661	1.04976	0.524879	+	0.851177i	$0.324111\pi$
$488$	−2.85064	−0.129042
$489$	7.33155	0.331544
$490$	0	0
$491$	0.895629	0.0404192	0.0202096	−	0.999796i	$-0.493567\pi$
$491$	0.895629	0.0404192	0.0202096	+	0.999796i	$0.493567\pi$
$492$	−6.28160	−0.283197
$493$	−42.5846	−1.91791
$494$	75.0037	3.37458
$495$	0	0
$496$	−16.4386	−0.738117
$497$	−31.1873	−1.39894
$498$	0.831260	0.0372496
$499$	−11.8061	−0.528514	−0.264257	−	0.964452i	$-0.585127\pi$
$499$	−11.8061	−0.528514	−0.264257	+	0.964452i	$0.585127\pi$
$500$	0	0
$501$	−1.82426	−0.0815018
$502$	24.0246	1.07227
$503$	19.7931	0.882531	0.441265	−	0.897377i	$-0.354530\pi$
$503$	19.7931	0.882531	0.441265	+	0.897377i	$0.354530\pi$
$504$	4.79198	0.213452
$505$	0	0
$506$	−2.10696	−0.0936656
$507$	−15.5047	−0.688589
$508$	10.1400	0.449888
$509$	3.69547	0.163799	0.0818995	−	0.996641i	$-0.473901\pi$
$509$	3.69547	0.163799	0.0818995	+	0.996641i	$0.473901\pi$
$510$	0	0
$511$	41.7936	1.84884
$512$	28.8371	1.27443
$513$	−13.9807	−0.617264
$514$	2.28542	0.100806
$515$	0	0
$516$	5.12442	0.225590
$517$	−0.419174	−0.0184353
$518$	14.4693	0.635743
$519$	2.43070	0.106696
$520$	0	0
$521$	17.7211	0.776376	0.388188	−	0.921580i	$-0.373101\pi$
$521$	17.7211	0.776376	0.388188	+	0.921580i	$0.373101\pi$
$522$	−35.0181	−1.53270
$523$	31.6186	1.38258	0.691292	−	0.722575i	$-0.257042\pi$
$523$	31.6186	1.38258	0.691292	+	0.722575i	$0.257042\pi$
$524$	−23.3514	−1.02011
$525$	0	0
$526$	54.6272	2.38186
$527$	−24.0647	−1.04827
$528$	0.750942	0.0326806
$529$	−14.9840	−0.651477
$530$	0	0
$531$	−13.2319	−0.574216
$532$	−28.7102	−1.24475
$533$	−59.6090	−2.58195
$534$	−0.0527121	−0.00228108
$535$	0	0
$536$	2.36695	0.102237
$537$	8.49359	0.366525
$538$	1.46256	0.0630553
$539$	0.818333	0.0352481
$540$	0	0
$541$	4.35506	0.187239	0.0936193	−	0.995608i	$-0.470156\pi$
$541$	4.35506	0.187239	0.0936193	+	0.995608i	$0.470156\pi$
$542$	19.3875	0.832763
$543$	−5.21750	−0.223904
$544$	49.6466	2.12858
$545$	0	0
$546$	17.5540	0.751243
$547$	42.4490	1.81499	0.907493	−	0.420067i	$-0.137993\pi$
$547$	42.4490	1.81499	0.907493	+	0.420067i	$0.137993\pi$
$548$	−36.0557	−1.54022
$549$	−14.2194	−0.606871
$550$	0	0
$551$	−36.0062	−1.53392
$552$	−0.689484	−0.0293464
$553$	23.4260	0.996173
$554$	−7.12872	−0.302870
$555$	0	0
$556$	16.9245	0.717758
$557$	6.63302	0.281050	0.140525	−	0.990077i	$-0.455121\pi$
$557$	6.63302	0.281050	0.140525	+	0.990077i	$0.455121\pi$
$558$	−19.7888	−0.837728
$559$	48.6280	2.05674
$560$	0	0
$561$	1.09931	0.0464130
$562$	−8.73285	−0.368373
$563$	−35.7791	−1.50791	−0.753955	−	0.656927i	$-0.771856\pi$
$563$	−35.7791	−1.50791	−0.753955	+	0.656927i	$0.771856\pi$
$564$	0.799281	0.0336558
$565$	0	0
$566$	−52.5759	−2.20993
$567$	22.2147	0.932930
$568$	5.82597	0.244452
$569$	−35.5696	−1.49116	−0.745578	−	0.666419i	$-0.767826\pi$
$569$	−35.5696	−1.49116	−0.745578	+	0.666419i	$0.767826\pi$
$570$	0	0
$571$	−20.4274	−0.854858	−0.427429	−	0.904049i	$-0.640581\pi$
$571$	−20.4274	−0.854858	−0.427429	+	0.904049i	$0.640581\pi$
$572$	−4.61439	−0.192937
$573$	−10.8386	−0.452789
$574$	49.5506	2.06820
$575$	0	0
$576$	15.5024	0.645934
$577$	−30.7540	−1.28030	−0.640152	−	0.768248i	$-0.721129\pi$
$577$	−30.7540	−1.28030	−0.640152	+	0.768248i	$0.721129\pi$
$578$	50.8263	2.11410
$579$	2.24844	0.0934420
$580$	0	0
$581$	−3.01948	−0.125269
$582$	5.23200	0.216874
$583$	−2.03933	−0.0844604
$584$	−7.80729	−0.323068
$585$	0	0
$586$	38.9862	1.61051
$587$	47.9182	1.97780	0.988898	−	0.148595i	$-0.0474751\pi$
$587$	47.9182	1.97780	0.988898	+	0.148595i	$0.0474751\pi$
$588$	−1.56040	−0.0643497
$589$	−20.3472	−0.838393
$590$	0	0
$591$	6.32963	0.260366
$592$	−11.2001	−0.460323
$593$	17.1170	0.702910	0.351455	−	0.936205i	$-0.385687\pi$
$593$	17.1170	0.702910	0.351455	+	0.936205i	$0.385687\pi$
$594$	1.86786	0.0766391
$595$	0	0
$596$	30.3741	1.24417
$597$	7.03032	0.287732
$598$	38.1243	1.55902
$599$	1.61307	0.0659081	0.0329540	−	0.999457i	$-0.489509\pi$
$599$	1.61307	0.0659081	0.0329540	+	0.999457i	$0.489509\pi$
$600$	0	0
$601$	−44.6538	−1.82147	−0.910734	−	0.412994i	$-0.864483\pi$
$601$	−44.6538	−1.82147	−0.910734	+	0.412994i	$0.864483\pi$
$602$	−40.4225	−1.64750
$603$	11.8067	0.480807
$604$	−27.7286	−1.12826
$605$	0	0
$606$	4.47801	0.181907
$607$	44.2902	1.79768	0.898841	−	0.438274i	$-0.144410\pi$
$607$	44.2902	1.79768	0.898841	+	0.438274i	$0.144410\pi$
$608$	41.9773	1.70241
$609$	−8.42696	−0.341478
$610$	0	0
$611$	7.58474	0.306846
$612$	31.6404	1.27899
$613$	44.7545	1.80762	0.903808	−	0.427939i	$-0.140760\pi$
$613$	44.7545	1.80762	0.903808	+	0.427939i	$0.140760\pi$
$614$	−52.8917	−2.13453
$615$	0	0
$616$	−0.658286	−0.0265231
$617$	−2.19344	−0.0883047	−0.0441523	−	0.999025i	$-0.514059\pi$
$617$	−2.19344	−0.0883047	−0.0441523	+	0.999025i	$0.514059\pi$
$618$	−7.82415	−0.314734
$619$	−20.1843	−0.811277	−0.405639	−	0.914033i	$-0.632951\pi$
$619$	−20.1843	−0.811277	−0.405639	+	0.914033i	$0.632951\pi$
$620$	0	0
$621$	−7.10637	−0.285169
$622$	8.69591	0.348674
$623$	0.191472	0.00767116
$624$	−13.5879	−0.543952
$625$	0	0
$626$	−7.53617	−0.301206
$627$	0.929493	0.0371204
$628$	17.2733	0.689279
$629$	−16.3960	−0.653751
$630$	0	0
$631$	−6.65239	−0.264827	−0.132414	−	0.991195i	$-0.542273\pi$
$631$	−6.65239	−0.264827	−0.132414	+	0.991195i	$0.542273\pi$
$632$	−4.37610	−0.174072
$633$	−3.64731	−0.144967
$634$	29.9984	1.19139
$635$	0	0
$636$	3.88859	0.154193
$637$	−14.8073	−0.586687
$638$	4.81052	0.190450
$639$	29.0609	1.14963
$640$	0	0
$641$	35.1221	1.38724	0.693620	−	0.720341i	$-0.256015\pi$
$641$	35.1221	1.38724	0.693620	+	0.720341i	$0.256015\pi$
$642$	10.9119	0.430660
$643$	33.0043	1.30156	0.650781	−	0.759266i	$-0.274442\pi$
$643$	33.0043	1.30156	0.650781	+	0.759266i	$0.274442\pi$
$644$	−14.5934	−0.575059
$645$	0	0
$646$	70.6498	2.77968
$647$	15.2774	0.600618	0.300309	−	0.953842i	$-0.402910\pi$
$647$	15.2774	0.600618	0.300309	+	0.953842i	$0.402910\pi$
$648$	−4.14983	−0.163021
$649$	1.81770	0.0713509
$650$	0	0
$651$	−4.76210	−0.186642
$652$	−28.9879	−1.13525
$653$	−34.4208	−1.34699	−0.673495	−	0.739192i	$-0.735208\pi$
$653$	−34.4208	−1.34699	−0.673495	+	0.739192i	$0.735208\pi$
$654$	13.7968	0.539499
$655$	0	0
$656$	−38.3553	−1.49752
$657$	−38.9440	−1.51935
$658$	−6.30489	−0.245790
$659$	−24.7850	−0.965488	−0.482744	−	0.875762i	$-0.660360\pi$
$659$	−24.7850	−0.965488	−0.482744	+	0.875762i	$0.660360\pi$
$660$	0	0
$661$	−3.33949	−0.129891	−0.0649456	−	0.997889i	$-0.520687\pi$
$661$	−3.33949	−0.129891	−0.0649456	+	0.997889i	$0.520687\pi$
$662$	24.0184	0.933502
$663$	−19.8915	−0.772522
$664$	0.564055	0.0218896
$665$	0	0
$666$	−13.4827	−0.522445
$667$	−18.3019	−0.708652
$668$	7.21284	0.279073
$669$	5.60761	0.216803
$670$	0	0
$671$	1.95336	0.0754086
$672$	9.82446	0.378987
$673$	−14.0825	−0.542841	−0.271421	−	0.962461i	$-0.587493\pi$
$673$	−14.0825	−0.542841	−0.271421	+	0.962461i	$0.587493\pi$
$674$	−29.3262	−1.12960
$675$	0	0
$676$	61.3035	2.35783
$677$	−35.9538	−1.38182	−0.690909	−	0.722941i	$-0.742790\pi$
$677$	−35.9538	−1.38182	−0.690909	+	0.722941i	$0.742790\pi$
$678$	−3.79058	−0.145576
$679$	−19.0048	−0.729336
$680$	0	0
$681$	−9.56732	−0.366620
$682$	2.71844	0.104094
$683$	3.40122	0.130144	0.0650720	−	0.997881i	$-0.479272\pi$
$683$	3.40122	0.130144	0.0650720	+	0.997881i	$0.479272\pi$
$684$	26.7527	1.02291
$685$	0	0
$686$	−28.3865	−1.08380
$687$	3.98810	0.152155
$688$	31.2896	1.19290
$689$	36.9006	1.40580
$690$	0	0
$691$	−18.9887	−0.722365	−0.361182	−	0.932495i	$-0.617627\pi$
$691$	−18.9887	−0.722365	−0.361182	+	0.932495i	$0.617627\pi$
$692$	−9.61062	−0.365341
$693$	−3.28364	−0.124735
$694$	21.1777	0.803893
$695$	0	0
$696$	1.57420	0.0596701
$697$	−56.1487	−2.12678
$698$	−32.9578	−1.24747
$699$	4.59725	0.173884
$700$	0	0
$701$	−43.2740	−1.63444	−0.817219	−	0.576327i	$-0.804485\pi$
$701$	−43.2740	−1.63444	−0.817219	+	0.576327i	$0.804485\pi$
$702$	−33.7979	−1.27562
$703$	−13.8632	−0.522859
$704$	−2.12960	−0.0802625
$705$	0	0
$706$	−26.5386	−0.998795
$707$	−16.2660	−0.611744
$708$	−3.46599	−0.130260
$709$	−9.47103	−0.355692	−0.177846	−	0.984058i	$-0.556913\pi$
$709$	−9.47103	−0.355692	−0.177846	+	0.984058i	$0.556913\pi$
$710$	0	0
$711$	−21.8287	−0.818641
$712$	−0.0357680	−0.00134046
$713$	−10.3425	−0.387328
$714$	16.5350	0.618807
$715$	0	0
$716$	−33.5824	−1.25503
$717$	8.84745	0.330414
$718$	−11.3169	−0.422342
$719$	24.6960	0.921004	0.460502	−	0.887659i	$-0.347669\pi$
$719$	24.6960	0.921004	0.460502	+	0.887659i	$0.347669\pi$
$720$	0	0
$721$	28.4205	1.05844
$722$	23.1539	0.861701
$723$	0.514949	0.0191512
$724$	20.6292	0.766680
$725$	0	0
$726$	9.01968	0.334752
$727$	23.3386	0.865582	0.432791	−	0.901494i	$-0.357529\pi$
$727$	23.3386	0.865582	0.432791	+	0.901494i	$0.357529\pi$
$728$	11.9114	0.441464
$729$	−17.4491	−0.646263
$730$	0	0
$731$	45.8051	1.69416
$732$	−3.72467	−0.137668
$733$	−40.6924	−1.50301	−0.751504	−	0.659728i	$-0.770671\pi$
$733$	−40.6924	−1.50301	−0.751504	+	0.659728i	$0.770671\pi$
$734$	−14.0812	−0.519745
$735$	0	0
$736$	21.3370	0.786493
$737$	−1.62192	−0.0597442
$738$	−46.1721	−1.69962
$739$	−16.3218	−0.600408	−0.300204	−	0.953875i	$-0.597055\pi$
$739$	−16.3218	−0.600408	−0.300204	+	0.953875i	$0.597055\pi$
$740$	0	0
$741$	−16.8187	−0.617850
$742$	−30.6740	−1.12608
$743$	1.50962	0.0553826	0.0276913	−	0.999617i	$-0.491184\pi$
$743$	1.50962	0.0553826	0.0276913	+	0.999617i	$0.491184\pi$
$744$	0.889588	0.0326139
$745$	0	0
$746$	−3.65226	−0.133719
$747$	2.81360	0.102944
$748$	−4.34652	−0.158925
$749$	−39.6366	−1.44829
$750$	0	0
$751$	19.7197	0.719582	0.359791	−	0.933033i	$-0.382848\pi$
$751$	19.7197	0.719582	0.359791	+	0.933033i	$0.382848\pi$
$752$	4.88038	0.177969
$753$	−5.38722	−0.196321
$754$	−87.0439	−3.16995
$755$	0	0
$756$	12.9373	0.470525
$757$	−14.9494	−0.543346	−0.271673	−	0.962390i	$-0.587577\pi$
$757$	−14.9494	−0.543346	−0.271673	+	0.962390i	$0.587577\pi$
$758$	−66.7716	−2.42525
$759$	0.472460	0.0171492
$760$	0	0
$761$	9.48843	0.343955	0.171978	−	0.985101i	$-0.444984\pi$
$761$	9.48843	0.343955	0.171978	+	0.985101i	$0.444984\pi$
$762$	−4.93775	−0.178876
$763$	−50.1157	−1.81431
$764$	42.8543	1.55041
$765$	0	0
$766$	2.18468	0.0789357
$767$	−32.8903	−1.18760
$768$	−8.46839	−0.305577
$769$	44.3031	1.59761	0.798806	−	0.601589i	$-0.205465\pi$
$769$	44.3031	1.59761	0.798806	+	0.601589i	$0.205465\pi$
$770$	0	0
$771$	−0.512479	−0.0184565
$772$	−8.89000	−0.319958
$773$	53.9911	1.94193	0.970963	−	0.239231i	$-0.0768952\pi$
$773$	53.9911	1.94193	0.970963	+	0.239231i	$0.0768952\pi$
$774$	37.6664	1.35389
$775$	0	0
$776$	3.55020	0.127445
$777$	−3.24456	−0.116398
$778$	25.8114	0.925383
$779$	−47.4750	−1.70097
$780$	0	0
$781$	−3.99216	−0.142851
$782$	35.9112	1.28418
$783$	16.2250	0.579834
$784$	−9.52774	−0.340276
$785$	0	0
$786$	11.3712	0.405597
$787$	−8.30642	−0.296092	−0.148046	−	0.988980i	$-0.547298\pi$
$787$	−8.30642	−0.296092	−0.148046	+	0.988980i	$0.547298\pi$
$788$	−25.0264	−0.891530
$789$	−12.2495	−0.436094
$790$	0	0
$791$	13.7689	0.489567
$792$	0.613403	0.0217963
$793$	−35.3450	−1.25514
$794$	46.5452	1.65183
$795$	0	0
$796$	−27.7968	−0.985233
$797$	−10.8857	−0.385590	−0.192795	−	0.981239i	$-0.561755\pi$
$797$	−10.8857	−0.385590	−0.192795	+	0.981239i	$0.561755\pi$
$798$	13.9807	0.494912
$799$	7.14445	0.252752
$800$	0	0
$801$	−0.178417	−0.00630405
$802$	30.8484	1.08930
$803$	5.34984	0.188792
$804$	3.09268	0.109070
$805$	0	0
$806$	−49.1888	−1.73260
$807$	−0.327961	−0.0115448
$808$	3.03857	0.106897
$809$	−49.2828	−1.73269	−0.866346	−	0.499444i	$-0.833537\pi$
$809$	−49.2828	−1.73269	−0.866346	+	0.499444i	$0.833537\pi$
$810$	0	0
$811$	−7.57591	−0.266026	−0.133013	−	0.991114i	$-0.542465\pi$
$811$	−7.57591	−0.266026	−0.133013	+	0.991114i	$0.542465\pi$
$812$	33.3190	1.16927
$813$	−4.34741	−0.152470
$814$	1.85215	0.0649180
$815$	0	0
$816$	−12.7991	−0.448060
$817$	38.7292	1.35496
$818$	29.6896	1.03807
$819$	59.4158	2.07616
$820$	0	0
$821$	−13.1781	−0.459919	−0.229959	−	0.973200i	$-0.573859\pi$
$821$	−13.1781	−0.459919	−0.229959	+	0.973200i	$0.573859\pi$
$822$	17.5577	0.612395
$823$	−20.1232	−0.701450	−0.350725	−	0.936479i	$-0.614065\pi$
$823$	−20.1232	−0.701450	−0.350725	+	0.936479i	$0.614065\pi$
$824$	−5.30911	−0.184952
$825$	0	0
$826$	27.3404	0.951294
$827$	0.667971	0.0232276	0.0116138	−	0.999933i	$-0.496303\pi$
$827$	0.667971	0.0232276	0.0116138	+	0.999933i	$0.496303\pi$
$828$	13.5983	0.472575
$829$	−52.7661	−1.83264	−0.916321	−	0.400445i	$-0.868855\pi$
$829$	−52.7661	−1.83264	−0.916321	+	0.400445i	$0.868855\pi$
$830$	0	0
$831$	1.59853	0.0554524
$832$	38.5341	1.33593
$833$	−13.9478	−0.483261
$834$	−8.24155	−0.285381
$835$	0	0
$836$	−3.67508	−0.127105
$837$	9.16880	0.316920
$838$	12.6974	0.438624
$839$	46.4586	1.60393	0.801965	−	0.597372i	$-0.203788\pi$
$839$	46.4586	1.60393	0.801965	+	0.597372i	$0.203788\pi$
$840$	0	0
$841$	12.7862	0.440903
$842$	−69.6030	−2.39868
$843$	1.95824	0.0674453
$844$	14.4209	0.496389
$845$	0	0
$846$	5.87501	0.201987
$847$	−32.7631	−1.12576
$848$	23.7436	0.815359
$849$	11.7895	0.404615
$850$	0	0
$851$	−7.04663	−0.241555
$852$	7.61226	0.260792
$853$	34.0142	1.16463	0.582313	−	0.812965i	$-0.302148\pi$
$853$	34.0142	1.16463	0.582313	+	0.812965i	$0.302148\pi$
$854$	29.3809	1.00539
$855$	0	0
$856$	7.40434	0.253075
$857$	23.6364	0.807404	0.403702	−	0.914891i	$-0.367723\pi$
$857$	23.6364	0.807404	0.403702	+	0.914891i	$0.367723\pi$
$858$	2.24702	0.0767120
$859$	−27.1844	−0.927519	−0.463759	−	0.885961i	$-0.653500\pi$
$859$	−27.1844	−0.927519	−0.463759	+	0.885961i	$0.653500\pi$
$860$	0	0
$861$	−11.1111	−0.378666
$862$	49.0550	1.67082
$863$	55.4152	1.88636	0.943178	−	0.332289i	$-0.107821\pi$
$863$	55.4152	1.88636	0.943178	+	0.332289i	$0.107821\pi$
$864$	−18.9157	−0.643525
$865$	0	0
$866$	−28.3276	−0.962611
$867$	−11.3972	−0.387069
$868$	18.8287	0.639086
$869$	2.99866	0.101723
$870$	0	0
$871$	29.3478	0.994413
$872$	9.36190	0.317034
$873$	17.7090	0.599358
$874$	30.3637	1.02707
$875$	0	0
$876$	−10.2011	−0.344662
$877$	31.3826	1.05971	0.529857	−	0.848087i	$-0.322246\pi$
$877$	31.3826	1.05971	0.529857	+	0.848087i	$0.322246\pi$
$878$	12.8335	0.433110
$879$	−8.74220	−0.294867
$880$	0	0
$881$	−10.8571	−0.365787	−0.182893	−	0.983133i	$-0.558546\pi$
$881$	−10.8571	−0.365787	−0.182893	+	0.983133i	$0.558546\pi$
$882$	−11.4695	−0.386198
$883$	14.8728	0.500511	0.250255	−	0.968180i	$-0.419485\pi$
$883$	14.8728	0.500511	0.250255	+	0.968180i	$0.419485\pi$
$884$	78.6481	2.64522
$885$	0	0
$886$	−1.05797	−0.0355431
$887$	−34.0443	−1.14309	−0.571547	−	0.820569i	$-0.693657\pi$
$887$	−34.0443	−1.14309	−0.571547	+	0.820569i	$0.693657\pi$
$888$	0.606103	0.0203395
$889$	17.9359	0.601552
$890$	0	0
$891$	2.84362	0.0952647
$892$	−22.1717	−0.742362
$893$	6.04079	0.202147
$894$	−14.7910	−0.494684
$895$	0	0
$896$	13.4790	0.450303
$897$	−8.54892	−0.285440
$898$	−8.70550	−0.290506
$899$	23.6135	0.787555
$900$	0	0
$901$	34.7585	1.15797
$902$	6.34277	0.211191
$903$	9.06426	0.301640
$904$	−2.57211	−0.0855472
$905$	0	0
$906$	13.5027	0.448598
$907$	−2.39080	−0.0793851	−0.0396925	−	0.999212i	$-0.512638\pi$
$907$	−2.39080	−0.0793851	−0.0396925	+	0.999212i	$0.512638\pi$
$908$	37.8278	1.25536
$909$	15.1569	0.502723
$910$	0	0
$911$	6.37553	0.211231	0.105615	−	0.994407i	$-0.466319\pi$
$911$	6.37553	0.211231	0.105615	+	0.994407i	$0.466319\pi$
$912$	−10.8220	−0.358351
$913$	−0.386511	−0.0127916
$914$	−63.3753	−2.09627
$915$	0	0
$916$	−15.7684	−0.521001
$917$	−41.3048	−1.36400
$918$	−31.8360	−1.05074
$919$	59.7294	1.97029	0.985146	−	0.171719i	$-0.0549322\pi$
$919$	59.7294	1.97029	0.985146	+	0.171719i	$0.0549322\pi$
$920$	0	0
$921$	11.8603	0.390811
$922$	−1.20984	−0.0398441
$923$	72.2362	2.37768
$924$	−0.860123	−0.0282960
$925$	0	0
$926$	−40.8669	−1.34297
$927$	−26.4827	−0.869807
$928$	−48.7158	−1.59918
$929$	4.54835	0.149226	0.0746132	−	0.997213i	$-0.476228\pi$
$929$	4.54835	0.149226	0.0746132	+	0.997213i	$0.476228\pi$
$930$	0	0
$931$	−11.7931	−0.386504
$932$	−18.1768	−0.595402
$933$	−1.94996	−0.0638387
$934$	−68.7686	−2.25018
$935$	0	0
$936$	−11.0992	−0.362789
$937$	1.52795	0.0499161	0.0249580	−	0.999688i	$-0.492055\pi$
$937$	1.52795	0.0499161	0.0249580	+	0.999688i	$0.492055\pi$
$938$	−24.3957	−0.796547
$939$	1.68990	0.0551477
$940$	0	0
$941$	−27.1537	−0.885184	−0.442592	−	0.896723i	$-0.645941\pi$
$941$	−27.1537	−0.885184	−0.442592	+	0.896723i	$0.645941\pi$
$942$	−8.41139	−0.274058
$943$	−24.1314	−0.785828
$944$	−21.1632	−0.688804
$945$	0	0
$946$	−5.17432	−0.168232
$947$	−58.1874	−1.89084	−0.945419	−	0.325858i	$-0.894347\pi$
$947$	−58.1874	−1.89084	−0.945419	+	0.325858i	$0.894347\pi$
$948$	−5.71785	−0.185707
$949$	−96.8026	−3.14234
$950$	0	0
$951$	−6.72679	−0.218131
$952$	11.2199	0.363639
$953$	19.3719	0.627518	0.313759	−	0.949503i	$-0.398412\pi$
$953$	19.3719	0.627518	0.313759	+	0.949503i	$0.398412\pi$
$954$	28.5826	0.925395
$955$	0	0
$956$	−34.9815	−1.13138
$957$	−1.07870	−0.0348695
$958$	−46.9432	−1.51666
$959$	−63.7767	−2.05946
$960$	0	0
$961$	−17.6559	−0.569546
$962$	−33.5138	−1.08053
$963$	36.9341	1.19018
$964$	−2.03603	−0.0655762
$965$	0	0
$966$	7.10637	0.228644
$967$	19.4636	0.625908	0.312954	−	0.949768i	$-0.398681\pi$
$967$	19.4636	0.625908	0.312954	+	0.949768i	$0.398681\pi$
$968$	6.12034	0.196715
$969$	−15.8424	−0.508930
$970$	0	0
$971$	44.6819	1.43391	0.716956	−	0.697119i	$-0.245535\pi$
$971$	44.6819	1.43391	0.716956	+	0.697119i	$0.245535\pi$
$972$	−18.2761	−0.586205
$973$	29.9367	0.959725
$974$	44.6033	1.42918
$975$	0	0
$976$	−22.7427	−0.727976
$977$	17.7202	0.566918	0.283459	−	0.958984i	$-0.408518\pi$
$977$	17.7202	0.566918	0.283459	+	0.958984i	$0.408518\pi$
$978$	14.1159	0.451378
$979$	0.0245095	0.000783328 0
$980$	0	0
$981$	46.6987	1.49097
$982$	1.72441	0.0550283
$983$	−45.6673	−1.45656	−0.728280	−	0.685279i	$-0.759680\pi$
$983$	−45.6673	−1.45656	−0.728280	+	0.685279i	$0.759680\pi$
$984$	2.07562	0.0661684
$985$	0	0
$986$	−81.9910	−2.61113
$987$	1.41380	0.0450017
$988$	66.4987	2.11560
$989$	19.6860	0.625979
$990$	0	0
$991$	16.7135	0.530922	0.265461	−	0.964122i	$-0.414476\pi$
$991$	16.7135	0.530922	0.265461	+	0.964122i	$0.414476\pi$
$992$	−27.5295	−0.874062
$993$	−5.38584	−0.170915
$994$	−60.0471	−1.90458
$995$	0	0
$996$	0.736999	0.0233527
$997$	14.3840	0.455546	0.227773	−	0.973714i	$-0.426856\pi$
$997$	14.3840	0.455546	0.227773	+	0.973714i	$0.426856\pi$
$998$	−22.7311	−0.719540
$999$	6.24698	0.197646

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2075.2.a.f.1.6		6
5.4	even	2		415.2.a.c.1.1	✓	6
15.14	odd	2		3735.2.a.l.1.6		6
20.19	odd	2		6640.2.a.bb.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
415.2.a.c.1.1	✓	6	5.4	even	2
2075.2.a.f.1.6		6	1.1	even	1	trivial
3735.2.a.l.1.6		6	15.14	odd	2
6640.2.a.bb.1.4		6	20.19	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:	\( N \)	\(=\)	\( 2075 = 5^{2} \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2075.a (trivial)

\(f(q)\)	\(=\)	\(q+1.92537 q^{2} -0.431741 q^{3} +1.70704 q^{4} -0.831260 q^{6} +3.01948 q^{7} -0.564055 q^{8} -2.81360 q^{9} +0.386511 q^{11} -0.736999 q^{12} -6.99372 q^{13} +5.81360 q^{14} -4.50009 q^{16} -6.58773 q^{17} -5.41721 q^{18} -5.57007 q^{19} -1.30363 q^{21} +0.744176 q^{22} -2.83126 q^{23} +0.243526 q^{24} -13.4655 q^{26} +2.50997 q^{27} +5.15437 q^{28} +6.46422 q^{29} +3.65295 q^{31} -7.53623 q^{32} -0.166873 q^{33} -12.6838 q^{34} -4.80293 q^{36} +2.48887 q^{37} -10.7244 q^{38} +3.01948 q^{39} +8.52322 q^{41} -2.50997 q^{42} -6.95309 q^{43} +0.659790 q^{44} -5.45122 q^{46} -1.08451 q^{47} +1.94287 q^{48} +2.11723 q^{49} +2.84419 q^{51} -11.9386 q^{52} -5.27625 q^{53} +4.83261 q^{54} -1.70315 q^{56} +2.40483 q^{57} +12.4460 q^{58} +4.70284 q^{59} +5.05383 q^{61} +7.03328 q^{62} -8.49559 q^{63} -5.50982 q^{64} -0.321291 q^{66} -4.19631 q^{67} -11.2455 q^{68} +1.22237 q^{69} -10.3287 q^{71} +1.58702 q^{72} +13.8414 q^{73} +4.79198 q^{74} -9.50834 q^{76} +1.16706 q^{77} +5.81360 q^{78} +7.75829 q^{79} +7.35714 q^{81} +16.4103 q^{82} -1.00000 q^{83} -2.22535 q^{84} -13.3873 q^{86} -2.79087 q^{87} -0.218013 q^{88} +0.0634123 q^{89} -21.1174 q^{91} -4.83308 q^{92} -1.57713 q^{93} -2.08808 q^{94} +3.25370 q^{96} -6.29407 q^{97} +4.07645 q^{98} -1.08749 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - 3 q^{3} + 2 q^{4} + 4 q^{6} - 3 q^{8} + 9 q^{9} + 4 q^{11} - 13 q^{12} - 7 q^{13} + 9 q^{14} - 6 q^{16} - 21 q^{17} - 11 q^{18} + 2 q^{19} - 9 q^{21} - 5 q^{22} - 8 q^{23} + 11 q^{24} - 15 q^{26}+ \cdots - 22 q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - 3 * q^3 + 2 * q^4 + 4 * q^6 - 3 * q^8 + 9 * q^9 + 4 * q^11 - 13 * q^12 - 7 * q^13 + 9 * q^14 - 6 * q^16 - 21 * q^17 - 11 * q^18 + 2 * q^19 - 9 * q^21 - 5 * q^22 - 8 * q^23 + 11 * q^24 - 15 * q^26 - 12 * q^27 + 5 * q^28 + 10 * q^29 - 13 * q^31 + 7 * q^32 - 6 * q^33 - 5 * q^34 - 10 * q^36 - q^37 - 3 * q^38 + 10 * q^41 + 12 * q^42 + 3 * q^43 + 6 * q^44 - 15 * q^46 - 14 * q^47 + 28 * q^48 - 12 * q^49 - 6 * q^51 - 5 * q^52 - 9 * q^53 + 21 * q^54 - 2 * q^56 - 19 * q^57 + 27 * q^58 + 20 * q^59 - 13 * q^61 + 12 * q^62 + 11 * q^63 - 19 * q^64 - 20 * q^66 - 16 * q^67 - 11 * q^69 + 5 * q^71 + 2 * q^72 - 3 * q^73 + 11 * q^74 - 23 * q^76 - 30 * q^77 + 9 * q^78 + 2 * q^79 + 2 * q^81 + q^82 - 6 * q^83 - 3 * q^84 - 42 * q^86 + 5 * q^87 + 4 * q^88 + 45 * q^89 - 21 * q^91 + 15 * q^92 + 27 * q^93 + 2 * q^94 - 45 * q^96 - 22 * q^97 + 3 * q^98 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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