LMFDB - Embedded newform 2107.4.a.h.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2107,4,Mod(1,2107)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2107, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2107.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2107 = 7^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2107.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$124.317024382$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	2107.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.740371 q^{2} -9.08049 q^{3} -7.45185 q^{4} -1.06386 q^{5} -6.72294 q^{6} -11.4401 q^{8} +55.4554 q^{9} +O(q^{10})$	\(q+0.740371 q^{2} -9.08049 q^{3} -7.45185 q^{4} -1.06386 q^{5} -6.72294 q^{6} -11.4401 q^{8} +55.4554 q^{9} -0.787653 q^{10} -37.4463 q^{11} +67.6665 q^{12} +27.1250 q^{13} +9.66039 q^{15} +51.1449 q^{16} -57.9750 q^{17} +41.0576 q^{18} -98.6761 q^{19} +7.92774 q^{20} -27.7241 q^{22} -59.0993 q^{23} +103.882 q^{24} -123.868 q^{25} +20.0826 q^{26} -258.389 q^{27} -230.243 q^{29} +7.15228 q^{30} +203.771 q^{31} +129.387 q^{32} +340.031 q^{33} -42.9230 q^{34} -413.245 q^{36} +331.812 q^{37} -73.0570 q^{38} -246.309 q^{39} +12.1707 q^{40} +169.999 q^{41} -43.0000 q^{43} +279.044 q^{44} -58.9969 q^{45} -43.7554 q^{46} +540.191 q^{47} -464.421 q^{48} -91.7085 q^{50} +526.442 q^{51} -202.132 q^{52} -237.811 q^{53} -191.304 q^{54} +39.8377 q^{55} +896.028 q^{57} -170.465 q^{58} -211.862 q^{59} -71.9878 q^{60} -285.670 q^{61} +150.866 q^{62} -313.365 q^{64} -28.8573 q^{65} +251.749 q^{66} +862.150 q^{67} +432.021 q^{68} +536.651 q^{69} -105.954 q^{71} -634.416 q^{72} +935.014 q^{73} +245.664 q^{74} +1124.78 q^{75} +735.320 q^{76} -182.360 q^{78} +579.105 q^{79} -54.4111 q^{80} +849.004 q^{81} +125.862 q^{82} +608.033 q^{83} +61.6774 q^{85} -31.8360 q^{86} +2090.72 q^{87} +428.389 q^{88} +659.318 q^{89} -43.6796 q^{90} +440.399 q^{92} -1850.34 q^{93} +399.942 q^{94} +104.978 q^{95} -1174.90 q^{96} -572.237 q^{97} -2076.60 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9}+O(q^{10}) $ 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9	$ 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9} - 60 q^{11} - 96 q^{15} + 372 q^{16} - 1008 q^{18} - 234 q^{22} - 214 q^{23} + 520 q^{25} - 870 q^{29} - 12 q^{30} - 1548 q^{32} + 1142 q^{36} - 1246 q^{37} - 416 q^{39} - 1290 q^{43} - 446 q^{44} - 660 q^{46} - 278 q^{50} - 3702 q^{51} - 2960 q^{53} + 620 q^{57} - 3634 q^{58} - 898 q^{60} + 2578 q^{64} - 4848 q^{65} + 928 q^{67} - 1708 q^{71} - 7900 q^{72} + 1714 q^{74} - 138 q^{78} - 3562 q^{79} + 2210 q^{81} - 948 q^{85} + 344 q^{86} + 2502 q^{88} - 3848 q^{92} - 11986 q^{93} - 2894 q^{95} - 2804 q^{99}+O(q^{100}) $ 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9 - 60 * q^11 - 96 * q^15 + 372 * q^16 - 1008 * q^18 - 234 * q^22 - 214 * q^23 + 520 * q^25 - 870 * q^29 - 12 * q^30 - 1548 * q^32 + 1142 * q^36 - 1246 * q^37 - 416 * q^39 - 1290 * q^43 - 446 * q^44 - 660 * q^46 - 278 * q^50 - 3702 * q^51 - 2960 * q^53 + 620 * q^57 - 3634 * q^58 - 898 * q^60 + 2578 * q^64 - 4848 * q^65 + 928 * q^67 - 1708 * q^71 - 7900 * q^72 + 1714 * q^74 - 138 * q^78 - 3562 * q^79 + 2210 * q^81 - 948 * q^85 + 344 * q^86 + 2502 * q^88 - 3848 * q^92 - 11986 * q^93 - 2894 * q^95 - 2804 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.740371	0.261761	0.130880	−	0.991398i	$-0.458220\pi$
$2$	0.740371	0.261761	0.130880	+	0.991398i	$0.458220\pi$
$3$	−9.08049	−1.74754	−0.873771	−	0.486338i	$-0.838333\pi$
$3$	−9.08049	−1.74754	−0.873771	+	0.486338i	$0.838333\pi$
$4$	−7.45185	−0.931481
$5$	−1.06386	−0.0951547	−0.0475774	−	0.998868i	$-0.515150\pi$
$5$	−1.06386	−0.0951547	−0.0475774	+	0.998868i	$0.515150\pi$
$6$	−6.72294	−0.457438
$7$	0	0
$7$	0	0
$8$	−11.4401	−0.505586
$9$	55.4554	2.05390
$10$	−0.787653	−0.0249078
$11$	−37.4463	−1.02641	−0.513203	−	0.858267i	$-0.671541\pi$
$11$	−37.4463	−1.02641	−0.513203	+	0.858267i	$0.671541\pi$
$12$	67.6665	1.62780
$13$	27.1250	0.578702	0.289351	−	0.957223i	$-0.406560\pi$
$13$	27.1250	0.578702	0.289351	+	0.957223i	$0.406560\pi$
$14$	0	0
$15$	9.66039	0.166287
$16$	51.1449	0.799139
$17$	−57.9750	−0.827118	−0.413559	−	0.910477i	$-0.635714\pi$
$17$	−57.9750	−0.827118	−0.413559	+	0.910477i	$0.635714\pi$
$18$	41.0576	0.537631
$19$	−98.6761	−1.19147	−0.595733	−	0.803183i	$-0.703138\pi$
$19$	−98.6761	−1.19147	−0.595733	+	0.803183i	$0.703138\pi$
$20$	7.92774	0.0886348
$21$	0	0
$22$	−27.7241	−0.268673
$23$	−59.0993	−0.535785	−0.267892	−	0.963449i	$-0.586327\pi$
$23$	−59.0993	−0.535785	−0.267892	+	0.963449i	$0.586327\pi$
$24$	103.882	0.883533
$25$	−123.868	−0.990946
$26$	20.0826	0.151482
$27$	−258.389	−1.84174
$28$	0	0
$29$	−230.243	−1.47431	−0.737157	−	0.675722i	$-0.763832\pi$
$29$	−230.243	−1.47431	−0.737157	+	0.675722i	$0.763832\pi$
$30$	7.15228	0.0435274
$31$	203.771	1.18059	0.590296	−	0.807187i	$-0.299011\pi$
$31$	203.771	1.18059	0.590296	+	0.807187i	$0.299011\pi$
$32$	129.387	0.714769
$33$	340.031	1.79369
$34$	−42.9230	−0.216507
$35$	0	0
$36$	−413.245	−1.91317
$37$	331.812	1.47431	0.737155	−	0.675723i	$-0.236169\pi$
$37$	331.812	1.47431	0.737155	+	0.675723i	$0.236169\pi$
$38$	−73.0570	−0.311879
$39$	−246.309	−1.01131
$40$	12.1707	0.0481089
$41$	169.999	0.647545	0.323773	−	0.946135i	$-0.395049\pi$
$41$	169.999	0.647545	0.323773	+	0.946135i	$0.395049\pi$
$42$	0	0
$43$	−43.0000	−0.152499
$43$	−43.0000	−0.152499
$44$	279.044	0.956079
$45$	−58.9969	−0.195439
$46$	−43.7554	−0.140247
$47$	540.191	1.67649	0.838245	−	0.545294i	$-0.183582\pi$
$47$	540.191	1.67649	0.838245	+	0.545294i	$0.183582\pi$
$48$	−464.421	−1.39653
$49$	0	0
$50$	−91.7085	−0.259391
$51$	526.442	1.44542
$52$	−202.132	−0.539050
$53$	−237.811	−0.616338	−0.308169	−	0.951332i	$-0.599716\pi$
$53$	−237.811	−0.616338	−0.308169	+	0.951332i	$0.599716\pi$
$54$	−191.304	−0.482095
$55$	39.8377	0.0976675
$56$	0	0
$57$	896.028	2.08214
$58$	−170.465	−0.385917
$59$	−211.862	−0.467493	−0.233746	−	0.972298i	$-0.575099\pi$
$59$	−211.862	−0.467493	−0.233746	+	0.972298i	$0.575099\pi$
$60$	−71.9878	−0.154893
$61$	−285.670	−0.599612	−0.299806	−	0.954000i	$-0.596922\pi$
$61$	−285.670	−0.599612	−0.299806	+	0.954000i	$0.596922\pi$
$62$	150.866	0.309033
$63$	0	0
$64$	−313.365	−0.612040
$65$	−28.8573	−0.0550663
$66$	251.749	0.469518
$67$	862.150	1.57207	0.786033	−	0.618185i	$-0.212132\pi$
$67$	862.150	1.57207	0.786033	+	0.618185i	$0.212132\pi$
$68$	432.021	0.770445
$69$	536.651	0.936306
$70$	0	0
$71$	−105.954	−0.177105	−0.0885524	−	0.996072i	$-0.528224\pi$
$71$	−105.954	−0.177105	−0.0885524	+	0.996072i	$0.528224\pi$
$72$	−634.416	−1.03842
$73$	935.014	1.49911	0.749556	−	0.661941i	$-0.230267\pi$
$73$	935.014	1.49911	0.749556	+	0.661941i	$0.230267\pi$
$74$	245.664	0.385917
$75$	1124.78	1.73172
$76$	735.320	1.10983
$77$	0	0
$78$	−182.360	−0.264720
$79$	579.105	0.824739	0.412370	−	0.911017i	$-0.364701\pi$
$79$	579.105	0.824739	0.412370	+	0.911017i	$0.364701\pi$
$80$	−54.4111	−0.0760418
$81$	849.004	1.16462
$82$	125.862	0.169502
$83$	608.033	0.804100	0.402050	−	0.915618i	$-0.368298\pi$
$83$	608.033	0.804100	0.402050	+	0.915618i	$0.368298\pi$
$84$	0	0
$85$	61.6774	0.0787042
$86$	−31.8360	−0.0399181
$87$	2090.72	2.57642
$88$	428.389	0.518937
$89$	659.318	0.785254	0.392627	−	0.919698i	$-0.371566\pi$
$89$	659.318	0.785254	0.392627	+	0.919698i	$0.371566\pi$
$90$	−43.6796	−0.0511582
$91$	0	0
$92$	440.399	0.499073
$93$	−1850.34	−2.06313
$94$	399.942	0.438839
$95$	104.978	0.113374
$96$	−1174.90	−1.24909
$97$	−572.237	−0.598988	−0.299494	−	0.954098i	$-0.596818\pi$
$97$	−572.237	−0.598988	−0.299494	+	0.954098i	$0.596818\pi$
$98$	0	0
$99$	−2076.60	−2.10814
$100$	923.047	0.923047
$101$	267.215	0.263256	0.131628	−	0.991299i	$-0.457979\pi$
$101$	267.215	0.263256	0.131628	+	0.991299i	$0.457979\pi$
$102$	389.762	0.378355
$103$	1940.56	1.85640	0.928201	−	0.372079i	$-0.121355\pi$
$103$	1940.56	1.85640	0.928201	+	0.372079i	$0.121355\pi$
$104$	−310.313	−0.292584
$105$	0	0
$106$	−176.069	−0.161333
$107$	−1263.69	−1.14173	−0.570867	−	0.821042i	$-0.693393\pi$
$107$	−1263.69	−1.14173	−0.570867	+	0.821042i	$0.693393\pi$
$108$	1925.48	1.71555
$109$	−1079.90	−0.948953	−0.474477	−	0.880268i	$-0.657363\pi$
$109$	−1079.90	−0.948953	−0.474477	+	0.880268i	$0.657363\pi$
$110$	29.4947	0.0255655
$111$	−3013.01	−2.57642
$112$	0	0
$113$	18.9711	0.0157933	0.00789667	−	0.999969i	$-0.497486\pi$
$113$	18.9711	0.0157933	0.00789667	+	0.999969i	$0.497486\pi$
$114$	663.393	0.545022
$115$	62.8735	0.0509824
$116$	1715.74	1.37330
$117$	1504.23	1.18860
$118$	−156.856	−0.122371
$119$	0	0
$120$	−110.516	−0.0840723
$121$	71.2234	0.0535112
$122$	−211.502	−0.156955
$123$	−1543.67	−1.13161
$124$	−1518.47	−1.09970
$125$	264.761	0.189448
$126$	0	0
$127$	2552.01	1.78310	0.891552	−	0.452918i	$-0.149617\pi$
$127$	2552.01	1.78310	0.891552	+	0.452918i	$0.149617\pi$
$128$	−1267.10	−0.874977
$129$	390.461	0.266498
$130$	−21.3651	−0.0144142
$131$	1353.66	0.902822	0.451411	−	0.892316i	$-0.350921\pi$
$131$	1353.66	0.902822	0.451411	+	0.892316i	$0.350921\pi$
$132$	−2533.86	−1.67079
$133$	0	0
$134$	638.311	0.411505
$135$	274.890	0.175250
$136$	663.240	0.418179
$137$	610.674	0.380828	0.190414	−	0.981704i	$-0.439017\pi$
$137$	610.674	0.380828	0.190414	+	0.981704i	$0.439017\pi$
$138$	397.321	0.245088
$139$	1001.35	0.611032	0.305516	−	0.952187i	$-0.401171\pi$
$139$	1001.35	0.611032	0.305516	+	0.952187i	$0.401171\pi$
$140$	0	0
$141$	−4905.21	−2.92974
$142$	−78.4454	−0.0463591
$143$	−1015.73	−0.593984
$144$	2836.26	1.64135
$145$	244.947	0.140288
$146$	692.258	0.392409
$147$	0	0
$148$	−2472.61	−1.37329
$149$	−2325.76	−1.27875	−0.639375	−	0.768895i	$-0.720807\pi$
$149$	−2325.76	−1.27875	−0.639375	+	0.768895i	$0.720807\pi$
$150$	832.758	0.453296
$151$	−1262.63	−0.680471	−0.340236	−	0.940340i	$-0.610507\pi$
$151$	−1262.63	−0.680471	−0.340236	+	0.940340i	$0.610507\pi$
$152$	1128.87	0.602389
$153$	−3215.03	−1.69882
$154$	0	0
$155$	−216.784	−0.112339
$156$	1835.46	0.942013
$157$	−2330.55	−1.18470	−0.592351	−	0.805680i	$-0.701800\pi$
$157$	−2330.55	−1.18470	−0.592351	+	0.805680i	$0.701800\pi$
$158$	428.753	0.215884
$159$	2159.45	1.07708
$160$	−137.650	−0.0680137
$161$	0	0
$162$	628.579	0.304851
$163$	−117.944	−0.0566753	−0.0283377	−	0.999598i	$-0.509021\pi$
$163$	−117.944	−0.0566753	−0.0283377	+	0.999598i	$0.509021\pi$
$164$	−1266.81	−0.603176
$165$	−361.746	−0.170678
$166$	450.171	0.210482
$167$	−242.357	−0.112300	−0.0561501	−	0.998422i	$-0.517883\pi$
$167$	−242.357	−0.112300	−0.0561501	+	0.998422i	$0.517883\pi$
$168$	0	0
$169$	−1461.23	−0.665104
$170$	45.6642	0.0206017
$171$	−5472.12	−2.44716
$172$	320.430	0.142050
$173$	−2315.14	−1.01744	−0.508719	−	0.860932i	$-0.669881\pi$
$173$	−2315.14	−1.01744	−0.508719	+	0.860932i	$0.669881\pi$
$174$	1547.91	0.674407
$175$	0	0
$176$	−1915.18	−0.820241
$177$	1923.81	0.816963
$178$	488.140	0.205549
$179$	−2016.44	−0.841988	−0.420994	−	0.907063i	$-0.638319\pi$
$179$	−2016.44	−0.841988	−0.420994	+	0.907063i	$0.638319\pi$
$180$	439.636	0.182047
$181$	1118.68	0.459399	0.229699	−	0.973262i	$-0.426226\pi$
$181$	1118.68	0.459399	0.229699	+	0.973262i	$0.426226\pi$
$182$	0	0
$183$	2594.03	1.04785
$184$	676.102	0.270885
$185$	−353.002	−0.140288
$186$	−1369.94	−0.540048
$187$	2170.95	0.848960
$188$	−4025.43	−1.56162
$189$	0	0
$190$	77.7225	0.0296768
$191$	−1184.23	−0.448626	−0.224313	−	0.974517i	$-0.572014\pi$
$191$	−1184.23	−0.448626	−0.224313	+	0.974517i	$0.572014\pi$
$192$	2845.50	1.06957
$193$	4458.35	1.66279	0.831397	−	0.555679i	$-0.187542\pi$
$193$	4458.35	1.66279	0.831397	+	0.555679i	$0.187542\pi$
$194$	−423.668	−0.156792
$195$	262.038	0.0962306
$196$	0	0
$197$	−4242.89	−1.53448	−0.767241	−	0.641359i	$-0.778371\pi$
$197$	−4242.89	−1.53448	−0.767241	+	0.641359i	$0.778371\pi$
$198$	−1537.45	−0.551829
$199$	4118.96	1.46726	0.733631	−	0.679548i	$-0.237824\pi$
$199$	4118.96	1.46726	0.733631	+	0.679548i	$0.237824\pi$
$200$	1417.07	0.501008
$201$	−7828.75	−2.74725
$202$	197.838	0.0689102
$203$	0	0
$204$	−3922.97	−1.34638
$205$	−180.855	−0.0616170
$206$	1436.74	0.485933
$207$	−3277.37	−1.10045
$208$	1387.31	0.462463
$209$	3695.05	1.22293
$210$	0	0
$211$	1360.13	0.443768	0.221884	−	0.975073i	$-0.428779\pi$
$211$	1360.13	0.443768	0.221884	+	0.975073i	$0.428779\pi$
$212$	1772.14	0.574107
$213$	962.115	0.309498
$214$	−935.600	−0.298861
$215$	45.7461	0.0145110
$216$	2956.00	0.931158
$217$	0	0
$218$	−799.529	−0.248399
$219$	−8490.39	−2.61976
$220$	−296.864	−0.0909754
$221$	−1572.57	−0.478655
$222$	−2230.75	−0.674406
$223$	−333.430	−0.100126	−0.0500631	−	0.998746i	$-0.515942\pi$
$223$	−333.430	−0.100126	−0.0500631	+	0.998746i	$0.515942\pi$
$224$	0	0
$225$	−6869.16	−2.03531
$226$	14.0456	0.00413408
$227$	−89.9847	−0.0263105	−0.0131553	−	0.999913i	$-0.504188\pi$
$227$	−89.9847	−0.0263105	−0.0131553	+	0.999913i	$0.504188\pi$
$228$	−6677.07	−1.93947
$229$	−1355.08	−0.391030	−0.195515	−	0.980701i	$-0.562638\pi$
$229$	−1355.08	−0.391030	−0.195515	+	0.980701i	$0.562638\pi$
$230$	46.5497	0.0133452
$231$	0	0
$232$	2634.01	0.745392
$233$	1301.72	0.366002	0.183001	−	0.983113i	$-0.441419\pi$
$233$	1301.72	0.366002	0.183001	+	0.983113i	$0.441419\pi$
$234$	1113.69	0.311129
$235$	−574.689	−0.159526
$236$	1578.76	0.435461
$237$	−5258.56	−1.44127
$238$	0	0
$239$	1136.32	0.307543	0.153771	−	0.988106i	$-0.450858\pi$
$239$	1136.32	0.307543	0.153771	+	0.988106i	$0.450858\pi$
$240$	494.080	0.132886
$241$	774.282	0.206954	0.103477	−	0.994632i	$-0.467003\pi$
$241$	774.282	0.206954	0.103477	+	0.994632i	$0.467003\pi$
$242$	52.7317	0.0140071
$243$	−732.878	−0.193474
$244$	2128.77	0.558527
$245$	0	0
$246$	−1142.89	−0.296212
$247$	−2676.59	−0.689504
$248$	−2331.16	−0.596891
$249$	−5521.24	−1.40520
$250$	196.022	0.0495900
$251$	7380.86	1.85608	0.928039	−	0.372483i	$-0.121494\pi$
$251$	7380.86	1.85608	0.928039	+	0.372483i	$0.121494\pi$
$252$	0	0
$253$	2213.05	0.549933
$254$	1889.44	0.466747
$255$	−560.061	−0.137539
$256$	1568.79	0.383005
$257$	−1911.49	−0.463951	−0.231976	−	0.972722i	$-0.574519\pi$
$257$	−1911.49	−0.463951	−0.231976	+	0.972722i	$0.574519\pi$
$258$	289.086	0.0697586
$259$	0	0
$260$	215.040	0.0512932
$261$	−12768.2	−3.02810
$262$	1002.21	0.236323
$263$	−5939.54	−1.39258	−0.696288	−	0.717762i	$-0.745167\pi$
$263$	−5939.54	−1.39258	−0.696288	+	0.717762i	$0.745167\pi$
$264$	−3889.99	−0.906864
$265$	252.999	0.0586475
$266$	0	0
$267$	−5986.93	−1.37226
$268$	−6424.61	−1.46435
$269$	−4720.05	−1.06984	−0.534920	−	0.844903i	$-0.679658\pi$
$269$	−4720.05	−1.06984	−0.534920	+	0.844903i	$0.679658\pi$
$270$	203.521	0.0458736
$271$	5228.97	1.17209	0.586046	−	0.810277i	$-0.300684\pi$
$271$	5228.97	1.17209	0.586046	+	0.810277i	$0.300684\pi$
$272$	−2965.12	−0.660982
$273$	0	0
$274$	452.126	0.0996859
$275$	4638.40	1.01711
$276$	−3999.04	−0.872152
$277$	3295.18	0.714759	0.357379	−	0.933959i	$-0.383670\pi$
$277$	3295.18	0.714759	0.357379	+	0.933959i	$0.383670\pi$
$278$	741.371	0.159944
$279$	11300.2	2.42482
$280$	0	0
$281$	2801.88	0.594827	0.297414	−	0.954749i	$-0.403876\pi$
$281$	2801.88	0.594827	0.297414	+	0.954749i	$0.403876\pi$
$282$	−3631.67	−0.766890
$283$	−6682.98	−1.40375	−0.701876	−	0.712299i	$-0.747654\pi$
$283$	−6682.98	−1.40375	−0.701876	+	0.712299i	$0.747654\pi$
$284$	789.554	0.164970
$285$	−953.250	−0.198125
$286$	−752.018	−0.155482
$287$	0	0
$288$	7175.21	1.46807
$289$	−1551.90	−0.315876
$290$	181.352	0.0367219
$291$	5196.19	1.04676
$292$	−6967.58	−1.39639
$293$	9904.59	1.97486	0.987428	−	0.158072i	$-0.0505279\pi$
$293$	9904.59	1.97486	0.987428	+	0.158072i	$0.0505279\pi$
$294$	0	0
$295$	225.392	0.0444841
$296$	−3795.96	−0.745391
$297$	9675.70	1.89037
$298$	−1721.93	−0.334727
$299$	−1603.07	−0.310060
$300$	−8381.73	−1.61306
$301$	0	0
$302$	−934.813	−0.178121
$303$	−2426.45	−0.460052
$304$	−5046.78	−0.952146
$305$	303.914	0.0570559
$306$	−2380.31	−0.444685
$307$	−2149.36	−0.399579	−0.199789	−	0.979839i	$-0.564026\pi$
$307$	−2149.36	−0.399579	−0.199789	+	0.979839i	$0.564026\pi$
$308$	0	0
$309$	−17621.3	−3.24414
$310$	−160.501	−0.0294059
$311$	−4566.58	−0.832627	−0.416314	−	0.909221i	$-0.636678\pi$
$311$	−4566.58	−0.832627	−0.416314	+	0.909221i	$0.636678\pi$
$312$	2817.80	0.511303
$313$	−5650.32	−1.02037	−0.510184	−	0.860066i	$-0.670423\pi$
$313$	−5650.32	−1.02037	−0.510184	+	0.860066i	$0.670423\pi$
$314$	−1725.47	−0.310108
$315$	0	0
$316$	−4315.40	−0.768229
$317$	−681.341	−0.120719	−0.0603595	−	0.998177i	$-0.519225\pi$
$317$	−681.341	−0.120719	−0.0603595	+	0.998177i	$0.519225\pi$
$318$	1598.79	0.281936
$319$	8621.75	1.51325
$320$	333.377	0.0582385
$321$	11474.9	1.99523
$322$	0	0
$323$	5720.75	0.985483
$324$	−6326.65	−1.08482
$325$	−3359.93	−0.573463
$326$	−87.3223	−0.0148354
$327$	9806.05	1.65834
$328$	−1944.80	−0.327390
$329$	0	0
$330$	−267.826	−0.0446768
$331$	9915.86	1.64660	0.823301	−	0.567605i	$-0.192130\pi$
$331$	9915.86	1.64660	0.823301	+	0.567605i	$0.192130\pi$
$332$	−4530.97	−0.749005
$333$	18400.7	3.02809
$334$	−179.434	−0.0293958
$335$	−917.209	−0.149589
$336$	0	0
$337$	−1387.48	−0.224276	−0.112138	−	0.993693i	$-0.535770\pi$
$337$	−1387.48	−0.224276	−0.112138	+	0.993693i	$0.535770\pi$
$338$	−1081.85	−0.174098
$339$	−172.267	−0.0275995
$340$	−459.611	−0.0733115
$341$	−7630.47	−1.21177
$342$	−4051.40	−0.640569
$343$	0	0
$344$	491.925	0.0771012
$345$	−570.922	−0.0890939
$346$	−1714.06	−0.266326
$347$	10105.8	1.56342	0.781712	−	0.623640i	$-0.214347\pi$
$347$	10105.8	1.56342	0.781712	+	0.623640i	$0.214347\pi$
$348$	−15579.8	−2.39989
$349$	−2694.15	−0.413223	−0.206611	−	0.978423i	$-0.566244\pi$
$349$	−2694.15	−0.413223	−0.206611	+	0.978423i	$0.566244\pi$
$350$	0	0
$351$	−7008.81	−1.06582
$352$	−4845.06	−0.733644
$353$	−941.643	−0.141979	−0.0709895	−	0.997477i	$-0.522616\pi$
$353$	−941.643	−0.141979	−0.0709895	+	0.997477i	$0.522616\pi$
$354$	1424.33	0.213849
$355$	112.721	0.0168523
$356$	−4913.14	−0.731449
$357$	0	0
$358$	−1492.91	−0.220399
$359$	−2844.28	−0.418149	−0.209074	−	0.977900i	$-0.567045\pi$
$359$	−2844.28	−0.418149	−0.209074	+	0.977900i	$0.567045\pi$
$360$	674.931	0.0988110
$361$	2877.97	0.419591
$362$	828.242	0.120253
$363$	−646.743	−0.0935130
$364$	0	0
$365$	−994.726	−0.142647
$366$	1920.54	0.274285
$367$	−12874.8	−1.83123	−0.915613	−	0.402060i	$-0.868294\pi$
$367$	−12874.8	−1.83123	−0.915613	+	0.402060i	$0.868294\pi$
$368$	−3022.62	−0.428166
$369$	9427.35	1.33000
$370$	−261.352	−0.0367218
$371$	0	0
$372$	13788.5	1.92177
$373$	−1047.82	−0.145453	−0.0727266	−	0.997352i	$-0.523170\pi$
$373$	−1047.82	−0.145453	−0.0727266	+	0.997352i	$0.523170\pi$
$374$	1607.31	0.222224
$375$	−2404.16	−0.331068
$376$	−6179.85	−0.847610
$377$	−6245.35	−0.853189
$378$	0	0
$379$	7621.84	1.03300	0.516500	−	0.856287i	$-0.327234\pi$
$379$	7621.84	1.03300	0.516500	+	0.856287i	$0.327234\pi$
$380$	−782.279	−0.105605
$381$	−23173.5	−3.11605
$382$	−876.766	−0.117433
$383$	−11662.6	−1.55596	−0.777980	−	0.628289i	$-0.783755\pi$
$383$	−11662.6	−1.55596	−0.777980	+	0.628289i	$0.783755\pi$
$384$	11505.9	1.52906
$385$	0	0
$386$	3300.84	0.435254
$387$	−2384.58	−0.313217
$388$	4264.22	0.557946
$389$	−791.828	−0.103206	−0.0516032	−	0.998668i	$-0.516433\pi$
$389$	−791.828	−0.103206	−0.0516032	+	0.998668i	$0.516433\pi$
$390$	194.006	0.0251894
$391$	3426.28	0.443157
$392$	0	0
$393$	−12291.9	−1.57772
$394$	−3141.31	−0.401667
$395$	−616.088	−0.0784778
$396$	15474.5	1.96369
$397$	11003.5	1.39106	0.695531	−	0.718496i	$-0.255169\pi$
$397$	11003.5	1.39106	0.695531	+	0.718496i	$0.255169\pi$
$398$	3049.56	0.384072
$399$	0	0
$400$	−6335.22	−0.791903
$401$	−3600.39	−0.448367	−0.224183	−	0.974547i	$-0.571971\pi$
$401$	−3600.39	−0.448367	−0.224183	+	0.974547i	$0.571971\pi$
$402$	−5796.18	−0.719123
$403$	5527.30	0.683212
$404$	−1991.25	−0.245218
$405$	−903.224	−0.110819
$406$	0	0
$407$	−12425.1	−1.51324
$408$	−6022.55	−0.730786
$409$	−11151.9	−1.34823	−0.674117	−	0.738625i	$-0.735476\pi$
$409$	−11151.9	−1.34823	−0.674117	+	0.738625i	$0.735476\pi$
$410$	−133.900	−0.0161289
$411$	−5545.23	−0.665513
$412$	−14460.8	−1.72920
$413$	0	0
$414$	−2426.47	−0.288055
$415$	−646.864	−0.0765139
$416$	3509.63	0.413639
$417$	−9092.75	−1.06780
$418$	2735.71	0.320115
$419$	−13631.7	−1.58938	−0.794692	−	0.607012i	$-0.792368\pi$
$419$	−13631.7	−1.58938	−0.794692	+	0.607012i	$0.792368\pi$
$420$	0	0
$421$	−4948.91	−0.572910	−0.286455	−	0.958094i	$-0.592477\pi$
$421$	−4948.91	−0.572910	−0.286455	+	0.958094i	$0.592477\pi$
$422$	1007.00	0.116161
$423$	29956.5	3.44335
$424$	2720.59	0.311612
$425$	7181.26	0.819629
$426$	712.323	0.0810144
$427$	0	0
$428$	9416.83	1.06350
$429$	9223.34	1.03801
$430$	33.8691	0.00379840
$431$	4579.00	0.511747	0.255873	−	0.966710i	$-0.417637\pi$
$431$	4579.00	0.511747	0.255873	+	0.966710i	$0.417637\pi$
$432$	−13215.3	−1.47181
$433$	15031.9	1.66833	0.834167	−	0.551512i	$-0.185949\pi$
$433$	15031.9	1.66833	0.834167	+	0.551512i	$0.185949\pi$
$434$	0	0
$435$	−2224.24	−0.245159
$436$	8047.27	0.883932
$437$	5831.68	0.638369
$438$	−6286.04	−0.685750
$439$	−805.710	−0.0875955	−0.0437977	−	0.999040i	$-0.513946\pi$
$439$	−805.710	−0.0875955	−0.0437977	+	0.999040i	$0.513946\pi$
$440$	−455.747	−0.0493793
$441$	0	0
$442$	−1164.29	−0.125293
$443$	2764.74	0.296516	0.148258	−	0.988949i	$-0.452633\pi$
$443$	2764.74	0.296516	0.148258	+	0.988949i	$0.452633\pi$
$444$	22452.5	2.39989
$445$	−701.424	−0.0747206
$446$	−246.862	−0.0262091
$447$	21119.1	2.23467
$448$	0	0
$449$	5922.39	0.622483	0.311242	−	0.950331i	$-0.399255\pi$
$449$	5922.39	0.622483	0.311242	+	0.950331i	$0.399255\pi$
$450$	−5085.73	−0.532763
$451$	−6365.82	−0.664645
$452$	−141.370	−0.0147112
$453$	11465.3	1.18915
$454$	−66.6221	−0.00688707
$455$	0	0
$456$	−10250.7	−1.05270
$457$	−7955.92	−0.814359	−0.407179	−	0.913348i	$-0.633488\pi$
$457$	−7955.92	−0.814359	−0.407179	+	0.913348i	$0.633488\pi$
$458$	−1003.26	−0.102356
$459$	14980.1	1.52334
$460$	−468.524	−0.0474892
$461$	−4758.54	−0.480754	−0.240377	−	0.970680i	$-0.577271\pi$
$461$	−4758.54	−0.480754	−0.240377	+	0.970680i	$0.577271\pi$
$462$	0	0
$463$	−17308.9	−1.73739	−0.868695	−	0.495348i	$-0.835041\pi$
$463$	−17308.9	−1.73739	−0.868695	+	0.495348i	$0.835041\pi$
$464$	−11775.8	−1.17818
$465$	1968.51	0.196317
$466$	963.755	0.0958049
$467$	−1452.07	−0.143884	−0.0719418	−	0.997409i	$-0.522920\pi$
$467$	−1452.07	−0.143884	−0.0719418	+	0.997409i	$0.522920\pi$
$468$	−11209.3	−1.10716
$469$	0	0
$470$	−425.483	−0.0417576
$471$	21162.6	2.07032
$472$	2423.72	0.236358
$473$	1610.19	0.156526
$474$	−3893.29	−0.377267
$475$	12222.8	1.18068
$476$	0	0
$477$	−13187.9	−1.26590
$478$	841.302	0.0805027
$479$	−6867.49	−0.655081	−0.327541	−	0.944837i	$-0.606220\pi$
$479$	−6867.49	−0.655081	−0.327541	+	0.944837i	$0.606220\pi$
$480$	1249.93	0.118857
$481$	9000.40	0.853187
$482$	573.256	0.0541724
$483$	0	0
$484$	−530.746	−0.0498447
$485$	608.781	0.0569965
$486$	−542.602	−0.0506439
$487$	−14480.9	−1.34742	−0.673708	−	0.738998i	$-0.735299\pi$
$487$	−14480.9	−1.34742	−0.673708	+	0.738998i	$0.735299\pi$
$488$	3268.10	0.303155
$489$	1070.99	0.0990425
$490$	0	0
$491$	−19084.3	−1.75409	−0.877047	−	0.480404i	$-0.840490\pi$
$491$	−19084.3	−1.75409	−0.877047	+	0.480404i	$0.840490\pi$
$492$	11503.2	1.05408
$493$	13348.4	1.21943
$494$	−1981.67	−0.180485
$495$	2209.21	0.200599
$496$	10421.8	0.943457
$497$	0	0
$498$	−4087.77	−0.367826
$499$	11314.1	1.01501	0.507505	−	0.861649i	$-0.330568\pi$
$499$	11314.1	1.01501	0.507505	+	0.861649i	$0.330568\pi$
$500$	−1972.96	−0.176467
$501$	2200.72	0.196249
$502$	5464.57	0.485848
$503$	12241.9	1.08517	0.542585	−	0.840001i	$-0.317446\pi$
$503$	12241.9	1.08517	0.542585	+	0.840001i	$0.317446\pi$
$504$	0	0
$505$	−284.280	−0.0250501
$506$	1638.48	0.143951
$507$	13268.7	1.16230
$508$	−19017.2	−1.66093
$509$	11731.8	1.02162	0.510810	−	0.859694i	$-0.329346\pi$
$509$	11731.8	1.02162	0.510810	+	0.859694i	$0.329346\pi$
$510$	−414.653	−0.0360023
$511$	0	0
$512$	11298.3	0.975233
$513$	25496.8	2.19437
$514$	−1415.21	−0.121444
$515$	−2064.49	−0.176645
$516$	−2909.66	−0.248238
$517$	−20228.2	−1.72076
$518$	0	0
$519$	21022.6	1.77802
$520$	330.130	0.0278407
$521$	5383.87	0.452728	0.226364	−	0.974043i	$-0.427316\pi$
$521$	5383.87	0.452728	0.226364	+	0.974043i	$0.427316\pi$
$522$	−9453.23	−0.792637
$523$	−3062.11	−0.256016	−0.128008	−	0.991773i	$-0.540858\pi$
$523$	−3062.11	−0.256016	−0.128008	+	0.991773i	$0.540858\pi$
$524$	−10087.3	−0.840962
$525$	0	0
$526$	−4397.46	−0.364522
$527$	−11813.6	−0.976489
$528$	17390.8	1.43341
$529$	−8674.28	−0.712935
$530$	187.313	0.0153516
$531$	−11748.9	−0.960184
$532$	0	0
$533$	4611.22	0.374736
$534$	−4432.56	−0.359205
$535$	1344.39	0.108641
$536$	−9863.09	−0.794815
$537$	18310.3	1.47141
$538$	−3494.59	−0.280042
$539$	0	0
$540$	−2048.44	−0.163242
$541$	−2018.47	−0.160408	−0.0802040	−	0.996778i	$-0.525557\pi$
$541$	−2018.47	−0.160408	−0.0802040	+	0.996778i	$0.525557\pi$
$542$	3871.38	0.306808
$543$	−10158.2	−0.802819
$544$	−7501.22	−0.591199
$545$	1148.87	0.0902974
$546$	0	0
$547$	−5158.37	−0.403210	−0.201605	−	0.979467i	$-0.564616\pi$
$547$	−5158.37	−0.403210	−0.201605	+	0.979467i	$0.564616\pi$
$548$	−4550.65	−0.354734
$549$	−15842.0	−1.23154
$550$	3434.14	0.266240
$551$	22719.5	1.75659
$552$	−6139.34	−0.473383
$553$	0	0
$554$	2439.66	0.187096
$555$	3205.43	0.245159
$556$	−7461.91	−0.569164
$557$	3886.86	0.295676	0.147838	−	0.989012i	$-0.452769\pi$
$557$	3886.86	0.295676	0.147838	+	0.989012i	$0.452769\pi$
$558$	8366.35	0.634723
$559$	−1166.38	−0.0882513
$560$	0	0
$561$	−19713.3	−1.48359
$562$	2074.43	0.155702
$563$	−16728.2	−1.25224	−0.626120	−	0.779727i	$-0.715358\pi$
$563$	−16728.2	−1.25224	−0.626120	+	0.779727i	$0.715358\pi$
$564$	36552.9	2.72899
$565$	−20.1826	−0.00150281
$566$	−4947.89	−0.367447
$567$	0	0
$568$	1212.13	0.0895417
$569$	−25224.3	−1.85845	−0.929226	−	0.369511i	$-0.879525\pi$
$569$	−25224.3	−1.85845	−0.929226	+	0.369511i	$0.879525\pi$
$570$	−705.759	−0.0518614
$571$	17579.5	1.28840	0.644202	−	0.764855i	$-0.277190\pi$
$571$	17579.5	1.28840	0.644202	+	0.764855i	$0.277190\pi$
$572$	7569.08	0.553285
$573$	10753.4	0.783992
$574$	0	0
$575$	7320.52	0.530933
$576$	−17377.8	−1.25707
$577$	−10382.7	−0.749109	−0.374554	−	0.927205i	$-0.622204\pi$
$577$	−10382.7	−0.749109	−0.374554	+	0.927205i	$0.622204\pi$
$578$	−1148.98	−0.0826839
$579$	−40484.0	−2.90580
$580$	−1825.31	−0.130676
$581$	0	0
$582$	3847.11	0.274000
$583$	8905.15	0.632614
$584$	−10696.7	−0.757930
$585$	−1600.29	−0.113101
$586$	7333.08	0.516940
$587$	−21851.0	−1.53644	−0.768219	−	0.640188i	$-0.778857\pi$
$587$	−21851.0	−1.53644	−0.768219	+	0.640188i	$0.778857\pi$
$588$	0	0
$589$	−20107.3	−1.40664
$590$	166.874	0.0116442
$591$	38527.5	2.68157
$592$	16970.5	1.17818
$593$	−22484.5	−1.55705	−0.778524	−	0.627615i	$-0.784031\pi$
$593$	−22484.5	−1.55705	−0.778524	+	0.627615i	$0.784031\pi$
$594$	7163.61	0.494826
$595$	0	0
$596$	17331.2	1.19113
$597$	−37402.2	−2.56410
$598$	−1186.87	−0.0811615
$599$	11570.9	0.789273	0.394637	−	0.918837i	$-0.370870\pi$
$599$	11570.9	0.789273	0.394637	+	0.918837i	$0.370870\pi$
$600$	−12867.7	−0.875533
$601$	−12611.2	−0.855940	−0.427970	−	0.903793i	$-0.640771\pi$
$601$	−12611.2	−0.855940	−0.427970	+	0.903793i	$0.640771\pi$
$602$	0	0
$603$	47810.9	3.22887
$604$	9408.91	0.633846
$605$	−75.7718	−0.00509184
$606$	−1796.47	−0.120424
$607$	9679.84	0.647270	0.323635	−	0.946182i	$-0.395095\pi$
$607$	9679.84	0.647270	0.323635	+	0.946182i	$0.395095\pi$
$608$	−12767.4	−0.851623
$609$	0	0
$610$	225.009	0.0149350
$611$	14652.7	0.970189
$612$	23957.9	1.58242
$613$	−14614.0	−0.962896	−0.481448	−	0.876475i	$-0.659889\pi$
$613$	−14614.0	−0.962896	−0.481448	+	0.876475i	$0.659889\pi$
$614$	−1591.33	−0.104594
$615$	1642.26	0.107678
$616$	0	0
$617$	13929.3	0.908869	0.454434	−	0.890780i	$-0.349841\pi$
$617$	13929.3	0.908869	0.454434	+	0.890780i	$0.349841\pi$
$618$	−13046.3	−0.849189
$619$	15122.1	0.981919	0.490960	−	0.871182i	$-0.336646\pi$
$619$	15122.1	0.981919	0.490960	+	0.871182i	$0.336646\pi$
$620$	1615.44	0.104642
$621$	15270.6	0.986776
$622$	−3380.97	−0.217949
$623$	0	0
$624$	−12597.4	−0.808174
$625$	15201.9	0.972919
$626$	−4183.33	−0.267092
$627$	−33552.9	−2.13712
$628$	17366.9	1.10353
$629$	−19236.8	−1.21943
$630$	0	0
$631$	5374.97	0.339103	0.169552	−	0.985521i	$-0.445768\pi$
$631$	5374.97	0.339103	0.169552	+	0.985521i	$0.445768\pi$
$632$	−6625.02	−0.416977
$633$	−12350.6	−0.775503
$634$	−504.446	−0.0315995
$635$	−2714.99	−0.169671
$636$	−16091.9	−1.00328
$637$	0	0
$638$	6383.30	0.396108
$639$	−5875.72	−0.363756
$640$	1348.02	0.0832582
$641$	−6600.76	−0.406731	−0.203365	−	0.979103i	$-0.565188\pi$
$641$	−6600.76	−0.406731	−0.203365	+	0.979103i	$0.565188\pi$
$642$	8495.71	0.522273
$643$	5777.56	0.354347	0.177173	−	0.984180i	$-0.443305\pi$
$643$	5777.56	0.354347	0.177173	+	0.984180i	$0.443305\pi$
$644$	0	0
$645$	−415.397	−0.0253585
$646$	4235.48	0.257961
$647$	20034.9	1.21739	0.608696	−	0.793404i	$-0.291693\pi$
$647$	20034.9	1.21739	0.608696	+	0.793404i	$0.291693\pi$
$648$	−9712.70	−0.588813
$649$	7933.44	0.479838
$650$	−2487.59	−0.150110
$651$	0	0
$652$	878.900	0.0527920
$653$	24346.7	1.45905	0.729526	−	0.683953i	$-0.239741\pi$
$653$	24346.7	1.45905	0.729526	+	0.683953i	$0.239741\pi$
$654$	7260.12	0.434087
$655$	−1440.11	−0.0859077
$656$	8694.57	0.517479
$657$	51851.6	3.07903
$658$	0	0
$659$	16691.1	0.986636	0.493318	−	0.869849i	$-0.335784\pi$
$659$	16691.1	0.986636	0.493318	+	0.869849i	$0.335784\pi$
$660$	2695.67	0.158983
$661$	10246.7	0.602953	0.301477	−	0.953474i	$-0.402520\pi$
$661$	10246.7	0.602953	0.301477	+	0.953474i	$0.402520\pi$
$662$	7341.42	0.431016
$663$	14279.8	0.836470
$664$	−6955.97	−0.406542
$665$	0	0
$666$	13623.4	0.792636
$667$	13607.2	0.789915
$668$	1806.01	0.104605
$669$	3027.71	0.174975
$670$	−679.075	−0.0391567
$671$	10697.3	0.615446
$672$	0	0
$673$	4243.70	0.243065	0.121532	−	0.992587i	$-0.461219\pi$
$673$	4243.70	0.243065	0.121532	+	0.992587i	$0.461219\pi$
$674$	−1027.25	−0.0587067
$675$	32006.2	1.82506
$676$	10888.9	0.619532
$677$	16511.7	0.937364	0.468682	−	0.883367i	$-0.344729\pi$
$677$	16511.7	0.937364	0.468682	+	0.883367i	$0.344729\pi$
$678$	−127.541	−0.00722447
$679$	0	0
$680$	−705.596	−0.0397917
$681$	817.105	0.0459788
$682$	−5649.38	−0.317193
$683$	25219.8	1.41290	0.706449	−	0.707764i	$-0.250296\pi$
$683$	25219.8	1.41290	0.706449	+	0.707764i	$0.250296\pi$
$684$	40777.4	2.27948
$685$	−649.673	−0.0362376
$686$	0	0
$687$	12304.8	0.683342
$688$	−2199.23	−0.121868
$689$	−6450.64	−0.356676
$690$	−422.694	−0.0233213
$691$	15027.8	0.827331	0.413665	−	0.910429i	$-0.364248\pi$
$691$	15027.8	0.827331	0.413665	+	0.910429i	$0.364248\pi$
$692$	17252.1	0.947725
$693$	0	0
$694$	7482.05	0.409243
$695$	−1065.30	−0.0581425
$696$	−23918.1	−1.30260
$697$	−9855.69	−0.535596
$698$	−1994.67	−0.108165
$699$	−11820.3	−0.639604
$700$	0	0
$701$	27285.6	1.47013	0.735066	−	0.677995i	$-0.237151\pi$
$701$	27285.6	1.47013	0.735066	+	0.677995i	$0.237151\pi$
$702$	−5189.12	−0.278990
$703$	−32741.9	−1.75659
$704$	11734.3	0.628202
$705$	5218.46	0.278778
$706$	−697.166	−0.0371645
$707$	0	0
$708$	−14335.9	−0.760986
$709$	−20818.3	−1.10275	−0.551374	−	0.834258i	$-0.685896\pi$
$709$	−20818.3	−1.10275	−0.551374	+	0.834258i	$0.685896\pi$
$710$	83.4550	0.00441128
$711$	32114.5	1.69394
$712$	−7542.67	−0.397013
$713$	−12042.7	−0.632543
$714$	0	0
$715$	1080.60	0.0565204
$716$	15026.2	0.784296
$717$	−10318.4	−0.537444
$718$	−2105.82	−0.109455
$719$	5041.43	0.261493	0.130747	−	0.991416i	$-0.458263\pi$
$719$	5041.43	0.261493	0.130747	+	0.991416i	$0.458263\pi$
$720$	−3017.39	−0.156183
$721$	0	0
$722$	2130.77	0.109832
$723$	−7030.86	−0.361661
$724$	−8336.27	−0.427921
$725$	28519.8	1.46096
$726$	−478.830	−0.0244780
$727$	−19402.4	−0.989815	−0.494908	−	0.868946i	$-0.664798\pi$
$727$	−19402.4	−0.989815	−0.494908	+	0.868946i	$0.664798\pi$
$728$	0	0
$729$	−16268.2	−0.826512
$730$	−736.467	−0.0373395
$731$	2492.93	0.126134
$732$	−19330.3	−0.976050
$733$	−9996.27	−0.503712	−0.251856	−	0.967765i	$-0.581041\pi$
$733$	−9996.27	−0.503712	−0.251856	+	0.967765i	$0.581041\pi$
$734$	−9532.15	−0.479343
$735$	0	0
$736$	−7646.68	−0.382962
$737$	−32284.3	−1.61358
$738$	6979.74	0.348141
$739$	−12223.2	−0.608441	−0.304220	−	0.952602i	$-0.598396\pi$
$739$	−12223.2	−0.608441	−0.304220	+	0.952602i	$0.598396\pi$
$740$	2630.52	0.130675
$741$	24304.8	1.20494
$742$	0	0
$743$	−1097.99	−0.0542147	−0.0271073	−	0.999633i	$-0.508630\pi$
$743$	−1097.99	−0.0542147	−0.0271073	+	0.999633i	$0.508630\pi$
$744$	21168.1	1.04309
$745$	2474.29	0.121679
$746$	−775.776	−0.0380739
$747$	33718.7	1.65154
$748$	−16177.6	−0.790790
$749$	0	0
$750$	−1779.97	−0.0866606
$751$	23406.1	1.13729	0.568643	−	0.822584i	$-0.307468\pi$
$751$	23406.1	1.13729	0.568643	+	0.822584i	$0.307468\pi$
$752$	27628.0	1.33975
$753$	−67021.8	−3.24357
$754$	−4623.88	−0.223331
$755$	1343.26	0.0647500
$756$	0	0
$757$	−1979.59	−0.0950453	−0.0475226	−	0.998870i	$-0.515133\pi$
$757$	−1979.59	−0.0950453	−0.0475226	+	0.998870i	$0.515133\pi$
$758$	5642.99	0.270399
$759$	−20095.6	−0.961031
$760$	−1200.96	−0.0573201
$761$	2392.96	0.113988	0.0569939	−	0.998375i	$-0.481848\pi$
$761$	2392.96	0.113988	0.0569939	+	0.998375i	$0.481848\pi$
$762$	−17157.0	−0.815660
$763$	0	0
$764$	8824.67	0.417887
$765$	3420.34	0.161651
$766$	−8634.68	−0.407289
$767$	−5746.76	−0.270539
$768$	−14245.4	−0.669318
$769$	−15772.2	−0.739610	−0.369805	−	0.929109i	$-0.620576\pi$
$769$	−15772.2	−0.739610	−0.369805	+	0.929109i	$0.620576\pi$
$770$	0	0
$771$	17357.3	0.810774
$772$	−33223.0	−1.54886
$773$	−4748.03	−0.220925	−0.110462	−	0.993880i	$-0.535233\pi$
$773$	−4748.03	−0.220925	−0.110462	+	0.993880i	$0.535233\pi$
$774$	−1765.48	−0.0819880
$775$	−25240.8	−1.16990
$776$	6546.45	0.302840
$777$	0	0
$778$	−586.247	−0.0270154
$779$	−16774.8	−0.771528
$780$	−1952.67	−0.0896370
$781$	3967.59	0.181782
$782$	2536.72	0.116001
$783$	59492.3	2.71530
$784$	0	0
$785$	2479.38	0.112730
$786$	−9100.56	−0.412985
$787$	1358.03	0.0615104	0.0307552	−	0.999527i	$-0.490209\pi$
$787$	1358.03	0.0615104	0.0307552	+	0.999527i	$0.490209\pi$
$788$	31617.3	1.42934
$789$	53934.0	2.43359
$790$	−456.134	−0.0205424
$791$	0	0
$792$	23756.5	1.06585
$793$	−7748.81	−0.346997
$794$	8146.71	0.364126
$795$	−2297.35	−0.102489
$796$	−30693.9	−1.36673
$797$	−26923.0	−1.19656	−0.598282	−	0.801286i	$-0.704150\pi$
$797$	−26923.0	−1.19656	−0.598282	+	0.801286i	$0.704150\pi$
$798$	0	0
$799$	−31317.6	−1.38665
$800$	−16026.9	−0.708297
$801$	36562.7	1.61284
$802$	−2665.63	−0.117365
$803$	−35012.8	−1.53870
$804$	58338.7	2.55901
$805$	0	0
$806$	4092.25	0.178838
$807$	42860.4	1.86959
$808$	−3056.97	−0.133099
$809$	−28875.1	−1.25488	−0.627438	−	0.778666i	$-0.715897\pi$
$809$	−28875.1	−1.25488	−0.627438	+	0.778666i	$0.715897\pi$
$810$	−668.721	−0.0290080
$811$	−20113.9	−0.870891	−0.435446	−	0.900215i	$-0.643409\pi$
$811$	−20113.9	−0.870891	−0.435446	+	0.900215i	$0.643409\pi$
$812$	0	0
$813$	−47481.6	−2.04828
$814$	−9199.20	−0.396108
$815$	125.476	0.00539292
$816$	26924.8	1.15509
$817$	4243.07	0.181697
$818$	−8256.57	−0.352915
$819$	0	0
$820$	1347.71	0.0573951
$821$	25645.2	1.09016	0.545082	−	0.838383i	$-0.316498\pi$
$821$	25645.2	1.09016	0.545082	+	0.838383i	$0.316498\pi$
$822$	−4105.53	−0.174205
$823$	−15807.9	−0.669536	−0.334768	−	0.942301i	$-0.608658\pi$
$823$	−15807.9	−0.669536	−0.334768	+	0.942301i	$0.608658\pi$
$824$	−22200.3	−0.938571
$825$	−42119.0	−1.77745
$826$	0	0
$827$	41594.4	1.74895	0.874474	−	0.485072i	$-0.161207\pi$
$827$	41594.4	1.74895	0.874474	+	0.485072i	$0.161207\pi$
$828$	24422.5	1.02505
$829$	10832.4	0.453829	0.226914	−	0.973915i	$-0.427136\pi$
$829$	10832.4	0.453829	0.226914	+	0.973915i	$0.427136\pi$
$830$	−478.919	−0.0200284
$831$	−29921.9	−1.24907
$832$	−8500.02	−0.354189
$833$	0	0
$834$	−6732.01	−0.279509
$835$	257.834	0.0106859
$836$	−27535.0	−1.13914
$837$	−52652.2	−2.17434
$838$	−10092.5	−0.416039
$839$	30720.0	1.26409	0.632045	−	0.774932i	$-0.282216\pi$
$839$	30720.0	1.26409	0.632045	+	0.774932i	$0.282216\pi$
$840$	0	0
$841$	28622.9	1.17360
$842$	−3664.03	−0.149965
$843$	−25442.5	−1.03949
$844$	−10135.5	−0.413362
$845$	1554.55	0.0632877
$846$	22179.0	0.901334
$847$	0	0
$848$	−12162.8	−0.492540
$849$	60684.8	2.45312
$850$	5316.80	0.214547
$851$	−19609.8	−0.789913
$852$	−7169.54	−0.288292
$853$	−16384.1	−0.657657	−0.328829	−	0.944390i	$-0.606654\pi$
$853$	−16384.1	−0.657657	−0.328829	+	0.944390i	$0.606654\pi$
$854$	0	0
$855$	5821.58	0.232858
$856$	14456.8	0.577245
$857$	16967.1	0.676296	0.338148	−	0.941093i	$-0.390199\pi$
$857$	16967.1	0.676296	0.338148	+	0.941093i	$0.390199\pi$
$858$	6828.70	0.271711
$859$	−29139.2	−1.15741	−0.578706	−	0.815536i	$-0.696442\pi$
$859$	−29139.2	−1.15741	−0.578706	+	0.815536i	$0.696442\pi$
$860$	−340.893	−0.0135167
$861$	0	0
$862$	3390.16	0.133955
$863$	696.418	0.0274697	0.0137348	−	0.999906i	$-0.495628\pi$
$863$	696.418	0.0274697	0.0137348	+	0.999906i	$0.495628\pi$
$864$	−33432.2	−1.31642
$865$	2462.99	0.0968141
$866$	11129.2	0.436704
$867$	14092.0	0.552006
$868$	0	0
$869$	−21685.3	−0.846518
$870$	−1646.76	−0.0641730
$871$	23385.9	0.909758
$872$	12354.2	0.479778
$873$	−31733.6	−1.23026
$874$	4317.61	0.167100
$875$	0	0
$876$	63269.1	2.44026
$877$	−6366.86	−0.245147	−0.122573	−	0.992459i	$-0.539115\pi$
$877$	−6366.86	−0.245147	−0.122573	+	0.992459i	$0.539115\pi$
$878$	−596.524	−0.0229291
$879$	−89938.6	−3.45114
$880$	2037.49	0.0780498
$881$	31972.8	1.22269	0.611346	−	0.791364i	$-0.290629\pi$
$881$	31972.8	1.22269	0.611346	+	0.791364i	$0.290629\pi$
$882$	0	0
$883$	−25911.2	−0.987521	−0.493761	−	0.869598i	$-0.664378\pi$
$883$	−25911.2	−0.987521	−0.493761	+	0.869598i	$0.664378\pi$
$884$	11718.6	0.445858
$885$	−2046.67	−0.0777379
$886$	2046.93	0.0776163
$887$	25214.9	0.954491	0.477246	−	0.878770i	$-0.341635\pi$
$887$	25214.9	0.954491	0.477246	+	0.878770i	$0.341635\pi$
$888$	34469.2	1.30260
$889$	0	0
$890$	−519.314	−0.0195589
$891$	−31792.1	−1.19537
$892$	2484.67	0.0932656
$893$	−53304.0	−1.99748
$894$	15636.0	0.584949
$895$	2145.21	0.0801191
$896$	0	0
$897$	14556.7	0.541843
$898$	4384.77	0.162942
$899$	−46916.9	−1.74056
$900$	51187.9	1.89585
$901$	13787.1	0.509784
$902$	−4713.07	−0.173978
$903$	0	0
$904$	−217.031	−0.00798489
$905$	−1190.13	−0.0437139
$906$	8488.56	0.311273
$907$	−49098.5	−1.79745	−0.898725	−	0.438512i	$-0.855506\pi$
$907$	−49098.5	−1.79745	−0.898725	+	0.438512i	$0.855506\pi$
$908$	670.552	0.0245078
$909$	14818.5	0.540703
$910$	0	0
$911$	−13512.6	−0.491429	−0.245715	−	0.969342i	$-0.579023\pi$
$911$	−13512.6	−0.491429	−0.245715	+	0.969342i	$0.579023\pi$
$912$	45827.2	1.66392
$913$	−22768.6	−0.825334
$914$	−5890.33	−0.213167
$915$	−2759.69	−0.0997076
$916$	10097.8	0.364237
$917$	0	0
$918$	11090.8	0.398750
$919$	10258.9	0.368238	0.184119	−	0.982904i	$-0.441057\pi$
$919$	10258.9	0.368238	0.184119	+	0.982904i	$0.441057\pi$
$920$	−719.279	−0.0257760
$921$	19517.3	0.698281
$922$	−3523.09	−0.125842
$923$	−2874.01	−0.102491
$924$	0	0
$925$	−41100.9	−1.46096
$926$	−12815.0	−0.454780
$927$	107615.	3.81287
$928$	−29790.5	−1.05379
$929$	−25686.3	−0.907146	−0.453573	−	0.891219i	$-0.649851\pi$
$929$	−25686.3	−0.907146	−0.453573	+	0.891219i	$0.649851\pi$
$930$	1457.43	0.0513881
$931$	0	0
$932$	−9700.21	−0.340924
$933$	41466.8	1.45505
$934$	−1075.07	−0.0376631
$935$	−2309.59	−0.0807825
$936$	−17208.5	−0.600939
$937$	41183.7	1.43587	0.717936	−	0.696109i	$-0.245087\pi$
$937$	41183.7	1.43587	0.717936	+	0.696109i	$0.245087\pi$
$938$	0	0
$939$	51307.7	1.78313
$940$	4282.50	0.148595
$941$	−35349.7	−1.22462	−0.612310	−	0.790618i	$-0.709759\pi$
$941$	−35349.7	−1.22462	−0.612310	+	0.790618i	$0.709759\pi$
$942$	15668.1	0.541928
$943$	−10046.8	−0.346945
$944$	−10835.6	−0.373591
$945$	0	0
$946$	1192.14	0.0409723
$947$	−31690.5	−1.08744	−0.543718	−	0.839268i	$-0.682984\pi$
$947$	−31690.5	−1.08744	−0.543718	+	0.839268i	$0.682984\pi$
$948$	39186.0	1.34251
$949$	25362.3	0.867539
$950$	9049.43	0.309055
$951$	6186.92	0.210962
$952$	0	0
$953$	47554.0	1.61640	0.808198	−	0.588910i	$-0.200443\pi$
$953$	47554.0	1.61640	0.808198	+	0.588910i	$0.200443\pi$
$954$	−9763.96	−0.331363
$955$	1259.85	0.0426889
$956$	−8467.72	−0.286470
$957$	−78289.8	−2.64446
$958$	−5084.50	−0.171475
$959$	0	0
$960$	−3027.22	−0.101774
$961$	11731.6	0.393798
$962$	6663.64	0.223331
$963$	−70078.4	−2.34501
$964$	−5769.83	−0.192774
$965$	−4743.07	−0.158223
$966$	0	0
$967$	44989.9	1.49615	0.748076	−	0.663613i	$-0.230978\pi$
$967$	44989.9	1.49615	0.748076	+	0.663613i	$0.230978\pi$
$968$	−814.803	−0.0270545
$969$	−51947.2	−1.72217
$970$	450.724	0.0149195
$971$	22780.8	0.752905	0.376453	−	0.926436i	$-0.377144\pi$
$971$	22780.8	0.752905	0.376453	+	0.926436i	$0.377144\pi$
$972$	5461.30	0.180217
$973$	0	0
$974$	−10721.2	−0.352701
$975$	30509.8	1.00215
$976$	−14610.6	−0.479173
$977$	9900.74	0.324210	0.162105	−	0.986774i	$-0.448172\pi$
$977$	9900.74	0.324210	0.162105	+	0.986774i	$0.448172\pi$
$978$	792.929	0.0259254
$979$	−24689.0	−0.805990
$980$	0	0
$981$	−59886.4	−1.94906
$982$	−14129.4	−0.459153
$983$	−22363.0	−0.725604	−0.362802	−	0.931866i	$-0.618180\pi$
$983$	−22363.0	−0.725604	−0.362802	+	0.931866i	$0.618180\pi$
$984$	17659.8	0.572128
$985$	4513.84	0.146013
$986$	9882.74	0.319199
$987$	0	0
$988$	19945.6	0.642260
$989$	2541.27	0.0817064
$990$	1635.64	0.0525091
$991$	−40839.1	−1.30908	−0.654539	−	0.756028i	$-0.727137\pi$
$991$	−40839.1	−1.30908	−0.654539	+	0.756028i	$0.727137\pi$
$992$	26365.3	0.843851
$993$	−90040.9	−2.87751
$994$	0	0
$995$	−4382.00	−0.139617
$996$	41143.5	1.30892
$997$	−50531.5	−1.60516	−0.802582	−	0.596541i	$-0.796541\pi$
$997$	−50531.5	−1.60516	−0.802582	+	0.596541i	$0.796541\pi$
$998$	8376.65	0.265690
$999$	−85736.5	−2.71530

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2107.4.a.h.1.17	✓	30
7.6	odd	2	inner	2107.4.a.h.1.18	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2107.4.a.h.1.17	✓	30	1.1	even	1	trivial
2107.4.a.h.1.18	yes	30	7.6	odd	2	inner

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2107 = 7^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2107.a (trivial)

\(f(q)\)	\(=\)	\(q+0.740371 q^{2} -9.08049 q^{3} -7.45185 q^{4} -1.06386 q^{5} -6.72294 q^{6} -11.4401 q^{8} +55.4554 q^{9} +O(q^{10})\)	\(q+0.740371 q^{2} -9.08049 q^{3} -7.45185 q^{4} -1.06386 q^{5} -6.72294 q^{6} -11.4401 q^{8} +55.4554 q^{9} -0.787653 q^{10} -37.4463 q^{11} +67.6665 q^{12} +27.1250 q^{13} +9.66039 q^{15} +51.1449 q^{16} -57.9750 q^{17} +41.0576 q^{18} -98.6761 q^{19} +7.92774 q^{20} -27.7241 q^{22} -59.0993 q^{23} +103.882 q^{24} -123.868 q^{25} +20.0826 q^{26} -258.389 q^{27} -230.243 q^{29} +7.15228 q^{30} +203.771 q^{31} +129.387 q^{32} +340.031 q^{33} -42.9230 q^{34} -413.245 q^{36} +331.812 q^{37} -73.0570 q^{38} -246.309 q^{39} +12.1707 q^{40} +169.999 q^{41} -43.0000 q^{43} +279.044 q^{44} -58.9969 q^{45} -43.7554 q^{46} +540.191 q^{47} -464.421 q^{48} -91.7085 q^{50} +526.442 q^{51} -202.132 q^{52} -237.811 q^{53} -191.304 q^{54} +39.8377 q^{55} +896.028 q^{57} -170.465 q^{58} -211.862 q^{59} -71.9878 q^{60} -285.670 q^{61} +150.866 q^{62} -313.365 q^{64} -28.8573 q^{65} +251.749 q^{66} +862.150 q^{67} +432.021 q^{68} +536.651 q^{69} -105.954 q^{71} -634.416 q^{72} +935.014 q^{73} +245.664 q^{74} +1124.78 q^{75} +735.320 q^{76} -182.360 q^{78} +579.105 q^{79} -54.4111 q^{80} +849.004 q^{81} +125.862 q^{82} +608.033 q^{83} +61.6774 q^{85} -31.8360 q^{86} +2090.72 q^{87} +428.389 q^{88} +659.318 q^{89} -43.6796 q^{90} +440.399 q^{92} -1850.34 q^{93} +399.942 q^{94} +104.978 q^{95} -1174.90 q^{96} -572.237 q^{97} -2076.60 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9}+O(q^{10}) \) 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9	\( 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9} - 60 q^{11} - 96 q^{15} + 372 q^{16} - 1008 q^{18} - 234 q^{22} - 214 q^{23} + 520 q^{25} - 870 q^{29} - 12 q^{30} - 1548 q^{32} + 1142 q^{36} - 1246 q^{37} - 416 q^{39} - 1290 q^{43} - 446 q^{44} - 660 q^{46} - 278 q^{50} - 3702 q^{51} - 2960 q^{53} + 620 q^{57} - 3634 q^{58} - 898 q^{60} + 2578 q^{64} - 4848 q^{65} + 928 q^{67} - 1708 q^{71} - 7900 q^{72} + 1714 q^{74} - 138 q^{78} - 3562 q^{79} + 2210 q^{81} - 948 q^{85} + 344 q^{86} + 2502 q^{88} - 3848 q^{92} - 11986 q^{93} - 2894 q^{95} - 2804 q^{99}+O(q^{100}) \) 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9 - 60 * q^11 - 96 * q^15 + 372 * q^16 - 1008 * q^18 - 234 * q^22 - 214 * q^23 + 520 * q^25 - 870 * q^29 - 12 * q^30 - 1548 * q^32 + 1142 * q^36 - 1246 * q^37 - 416 * q^39 - 1290 * q^43 - 446 * q^44 - 660 * q^46 - 278 * q^50 - 3702 * q^51 - 2960 * q^53 + 620 * q^57 - 3634 * q^58 - 898 * q^60 + 2578 * q^64 - 4848 * q^65 + 928 * q^67 - 1708 * q^71 - 7900 * q^72 + 1714 * q^74 - 138 * q^78 - 3562 * q^79 + 2210 * q^81 - 948 * q^85 + 344 * q^86 + 2502 * q^88 - 3848 * q^92 - 11986 * q^93 - 2894 * q^95 - 2804 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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