LMFDB - Embedded newform 539.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,4,Mod(1,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, 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("539.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	539.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:	$31.8020294931$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.522072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 5x + 1 $ x^4 - x^3 - 12x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.555307$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.24550 q^{2} +5.49244 q^{3} +2.53327 q^{4} +16.0955 q^{5} -17.8257 q^{6} +17.7423 q^{8} +3.16692 q^{9} -52.2379 q^{10} -11.0000 q^{11} +13.9139 q^{12} -35.3712 q^{13} +88.4036 q^{15} -77.8487 q^{16} -40.4757 q^{17} -10.2782 q^{18} -118.159 q^{19} +40.7743 q^{20} +35.7005 q^{22} -174.510 q^{23} +97.4484 q^{24} +134.065 q^{25} +114.797 q^{26} -130.902 q^{27} -262.725 q^{29} -286.914 q^{30} +36.1894 q^{31} +110.720 q^{32} -60.4169 q^{33} +131.364 q^{34} +8.02266 q^{36} +19.0464 q^{37} +383.487 q^{38} -194.274 q^{39} +285.571 q^{40} -156.996 q^{41} +287.182 q^{43} -27.8660 q^{44} +50.9731 q^{45} +566.371 q^{46} -397.244 q^{47} -427.580 q^{48} -435.108 q^{50} -222.311 q^{51} -89.6049 q^{52} +272.483 q^{53} +424.842 q^{54} -177.050 q^{55} -648.984 q^{57} +852.674 q^{58} +507.466 q^{59} +223.950 q^{60} -35.5608 q^{61} -117.453 q^{62} +263.448 q^{64} -569.317 q^{65} +196.083 q^{66} +979.229 q^{67} -102.536 q^{68} -958.484 q^{69} +750.404 q^{71} +56.1882 q^{72} -395.594 q^{73} -61.8152 q^{74} +736.344 q^{75} -299.330 q^{76} +630.517 q^{78} -736.516 q^{79} -1253.01 q^{80} -804.477 q^{81} +509.531 q^{82} -582.975 q^{83} -651.477 q^{85} -932.050 q^{86} -1443.00 q^{87} -195.165 q^{88} +806.201 q^{89} -165.433 q^{90} -442.081 q^{92} +198.768 q^{93} +1289.26 q^{94} -1901.84 q^{95} +608.123 q^{96} +957.232 q^{97} -34.8361 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 14 q^{3} + 26 q^{4} - 10 q^{5} - 14 q^{6} - 18 q^{8} + 76 q^{9} + 2 q^{10} - 44 q^{11} - 70 q^{12} - 58 q^{13} + 284 q^{15} + 2 q^{16} - 4 q^{17} - 62 q^{18} - 258 q^{19} - 182 q^{20} + 22 q^{22}+ \cdots - 836 q^{99}+O(q^{100}) $ 4 * q - 2 * q^2 - 14 * q^3 + 26 * q^4 - 10 * q^5 - 14 * q^6 - 18 * q^8 + 76 * q^9 + 2 * q^10 - 44 * q^11 - 70 * q^12 - 58 * q^13 + 284 * q^15 + 2 * q^16 - 4 * q^17 - 62 * q^18 - 258 * q^19 - 182 * q^20 + 22 * q^22 + 8 * q^23 + 498 * q^24 + 80 * q^25 + 482 * q^26 - 428 * q^27 - 396 * q^29 - 628 * q^30 + 56 * q^31 + 134 * q^32 + 154 * q^33 - 472 * q^34 - 418 * q^36 + 84 * q^37 + 942 * q^38 - 412 * q^39 + 1026 * q^40 - 52 * q^41 + 408 * q^43 - 286 * q^44 - 826 * q^45 + 368 * q^46 - 8 * q^47 - 982 * q^48 - 1642 * q^50 - 388 * q^51 - 2030 * q^52 + 624 * q^53 - 92 * q^54 + 110 * q^55 + 48 * q^57 + 864 * q^58 + 238 * q^59 + 1420 * q^60 + 162 * q^61 - 688 * q^62 - 902 * q^64 - 32 * q^65 + 154 * q^66 + 1340 * q^67 + 1384 * q^68 - 2416 * q^69 + 1788 * q^71 - 2622 * q^72 - 1456 * q^73 + 996 * q^74 + 806 * q^75 - 3042 * q^76 - 2632 * q^78 - 1324 * q^79 - 2342 * q^80 + 1444 * q^81 - 1984 * q^82 - 450 * q^83 - 1736 * q^85 - 4380 * q^86 - 588 * q^87 + 198 * q^88 + 3072 * q^89 + 218 * q^90 + 544 * q^92 - 1264 * q^93 + 1696 * q^94 + 24 * q^95 - 862 * q^96 + 652 * q^97 - 836 * 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 $

$2$	−3.24550	−1.14746	−0.573729	−	0.819045i	$-0.694504\pi$
$2$	−3.24550	−1.14746	−0.573729	+	0.819045i	$0.694504\pi$
$3$	5.49244	1.05702	0.528510	−	0.848927i	$-0.322751\pi$
$3$	5.49244	1.05702	0.528510	+	0.848927i	$0.322751\pi$
$4$	2.53327	0.316659
$5$	16.0955	1.43962	0.719812	−	0.694169i	$-0.244228\pi$
$5$	16.0955	1.43962	0.719812	+	0.694169i	$0.244228\pi$
$6$	−17.8257	−1.21289
$7$	0	0
$7$	0	0
$8$	17.7423	0.784105
$9$	3.16692	0.117293
$10$	−52.2379	−1.65191
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	13.9139	0.334715
$13$	−35.3712	−0.754631	−0.377316	−	0.926085i	$-0.623153\pi$
$13$	−35.3712	−0.754631	−0.377316	+	0.926085i	$0.623153\pi$
$14$	0	0
$15$	88.4036	1.52171
$16$	−77.8487	−1.21639
$17$	−40.4757	−0.577459	−0.288730	−	0.957411i	$-0.593233\pi$
$17$	−40.4757	−0.577459	−0.288730	+	0.957411i	$0.593233\pi$
$18$	−10.2782	−0.134589
$19$	−118.159	−1.42672	−0.713359	−	0.700799i	$-0.752827\pi$
$19$	−118.159	−1.42672	−0.713359	+	0.700799i	$0.752827\pi$
$20$	40.7743	0.455870
$21$	0	0
$22$	35.7005	0.345972
$23$	−174.510	−1.58208	−0.791039	−	0.611766i	$-0.790459\pi$
$23$	−174.510	−1.58208	−0.791039	+	0.611766i	$0.790459\pi$
$24$	97.4484	0.828815
$25$	134.065	1.07252
$26$	114.797	0.865907
$27$	−130.902	−0.933040
$28$	0	0
$29$	−262.725	−1.68230	−0.841152	−	0.540799i	$-0.818122\pi$
$29$	−262.725	−1.68230	−0.841152	+	0.540799i	$0.818122\pi$
$30$	−286.914	−1.74610
$31$	36.1894	0.209671	0.104836	−	0.994490i	$-0.466568\pi$
$31$	36.1894	0.209671	0.104836	+	0.994490i	$0.466568\pi$
$32$	110.720	0.611647
$33$	−60.4169	−0.318704
$34$	131.364	0.662610
$35$	0	0
$36$	8.02266	0.0371420
$37$	19.0464	0.0846274	0.0423137	−	0.999104i	$-0.486527\pi$
$37$	19.0464	0.0846274	0.0423137	+	0.999104i	$0.486527\pi$
$38$	383.487	1.63710
$39$	−194.274	−0.797661
$40$	285.571	1.12882
$41$	−156.996	−0.598017	−0.299008	−	0.954250i	$-0.596656\pi$
$41$	−156.996	−0.598017	−0.299008	+	0.954250i	$0.596656\pi$
$42$	0	0
$43$	287.182	1.01849	0.509243	−	0.860623i	$-0.329926\pi$
$43$	287.182	1.01849	0.509243	+	0.860623i	$0.329926\pi$
$44$	−27.8660	−0.0954763
$45$	50.9731	0.168858
$46$	566.371	1.81537
$47$	−397.244	−1.23285	−0.616425	−	0.787413i	$-0.711420\pi$
$47$	−397.244	−1.23285	−0.616425	+	0.787413i	$0.711420\pi$
$48$	−427.580	−1.28575
$49$	0	0
$50$	−435.108	−1.23067
$51$	−222.311	−0.610386
$52$	−89.6049	−0.238961
$53$	272.483	0.706196	0.353098	−	0.935586i	$-0.385128\pi$
$53$	272.483	0.706196	0.353098	+	0.935586i	$0.385128\pi$
$54$	424.842	1.07062
$55$	−177.050	−0.434063
$56$	0	0
$57$	−648.984	−1.50807
$58$	852.674	1.93037
$59$	507.466	1.11977	0.559885	−	0.828570i	$-0.310845\pi$
$59$	507.466	1.11977	0.559885	+	0.828570i	$0.310845\pi$
$60$	223.950	0.481865
$61$	−35.5608	−0.0746409	−0.0373205	−	0.999303i	$-0.511882\pi$
$61$	−35.5608	−0.0746409	−0.0373205	+	0.999303i	$0.511882\pi$
$62$	−117.453	−0.240589
$63$	0	0
$64$	263.448	0.514547
$65$	−569.317	−1.08639
$66$	196.083	0.365699
$67$	979.229	1.78555	0.892775	−	0.450502i	$-0.148755\pi$
$67$	979.229	1.78555	0.892775	+	0.450502i	$0.148755\pi$
$68$	−102.536	−0.182858
$69$	−958.484	−1.67229
$70$	0	0
$71$	750.404	1.25432	0.627159	−	0.778891i	$-0.284218\pi$
$71$	750.404	1.25432	0.627159	+	0.778891i	$0.284218\pi$
$72$	56.1882	0.0919701
$73$	−395.594	−0.634257	−0.317129	−	0.948383i	$-0.602719\pi$
$73$	−395.594	−0.634257	−0.317129	+	0.948383i	$0.602719\pi$
$74$	−61.8152	−0.0971063
$75$	736.344	1.13368
$76$	−299.330	−0.451783
$77$	0	0
$78$	630.517	0.915282
$79$	−736.516	−1.04892	−0.524459	−	0.851436i	$-0.675732\pi$
$79$	−736.516	−1.04892	−0.524459	+	0.851436i	$0.675732\pi$
$80$	−1253.01	−1.75114
$81$	−804.477	−1.10354
$82$	509.531	0.686199
$83$	−582.975	−0.770962	−0.385481	−	0.922716i	$-0.625964\pi$
$83$	−582.975	−0.770962	−0.385481	+	0.922716i	$0.625964\pi$
$84$	0	0
$85$	−651.477	−0.831325
$86$	−932.050	−1.16867
$87$	−1443.00	−1.77823
$88$	−195.165	−0.236416
$89$	806.201	0.960192	0.480096	−	0.877216i	$-0.340602\pi$
$89$	806.201	0.960192	0.480096	+	0.877216i	$0.340602\pi$
$90$	−165.433	−0.193758
$91$	0	0
$92$	−442.081	−0.500979
$93$	198.768	0.221627
$94$	1289.26	1.41464
$95$	−1901.84	−2.05394
$96$	608.123	0.646524
$97$	957.232	1.00198	0.500991	−	0.865453i	$-0.332969\pi$
$97$	957.232	1.00198	0.500991	+	0.865453i	$0.332969\pi$
$98$	0	0
$99$	−34.8361	−0.0353652
$100$	339.623	0.339623
$101$	996.143	0.981386	0.490693	−	0.871333i	$-0.336744\pi$
$101$	996.143	0.981386	0.490693	+	0.871333i	$0.336744\pi$
$102$	721.509	0.700393
$103$	−1338.55	−1.28050	−0.640248	−	0.768169i	$-0.721168\pi$
$103$	−1338.55	−1.28050	−0.640248	+	0.768169i	$0.721168\pi$
$104$	−627.565	−0.591710
$105$	0	0
$106$	−884.343	−0.810330
$107$	1449.25	1.30939	0.654693	−	0.755895i	$-0.272798\pi$
$107$	1449.25	1.30939	0.654693	+	0.755895i	$0.272798\pi$
$108$	−331.610	−0.295456
$109$	−654.535	−0.575166	−0.287583	−	0.957756i	$-0.592852\pi$
$109$	−654.535	−0.575166	−0.287583	+	0.957756i	$0.592852\pi$
$110$	574.617	0.498069
$111$	104.611	0.0894529
$112$	0	0
$113$	−1160.63	−0.966223	−0.483111	−	0.875559i	$-0.660493\pi$
$113$	−1160.63	−0.966223	−0.483111	+	0.875559i	$0.660493\pi$
$114$	2106.28	1.73045
$115$	−2808.82	−2.27760
$116$	−665.554	−0.532717
$117$	−112.018	−0.0885131
$118$	−1646.98	−1.28489
$119$	0	0
$120$	1568.48	1.19318
$121$	121.000	0.0909091
$122$	115.413	0.0856473
$123$	−862.293	−0.632116
$124$	91.6776	0.0663943
$125$	145.906	0.104402
$126$	0	0
$127$	−1055.41	−0.737420	−0.368710	−	0.929545i	$-0.620200\pi$
$127$	−1055.41	−0.737420	−0.368710	+	0.929545i	$0.620200\pi$
$128$	−1740.78	−1.20207
$129$	1577.33	1.07656
$130$	1847.72	1.24658
$131$	−2657.40	−1.77235	−0.886175	−	0.463351i	$-0.846647\pi$
$131$	−2657.40	−1.77235	−0.886175	+	0.463351i	$0.846647\pi$
$132$	−153.052	−0.100920
$133$	0	0
$134$	−3178.09	−2.04884
$135$	−2106.93	−1.34323
$136$	−718.131	−0.452788
$137$	−147.314	−0.0918676	−0.0459338	−	0.998944i	$-0.514626\pi$
$137$	−147.314	−0.0918676	−0.0459338	+	0.998944i	$0.514626\pi$
$138$	3110.76	1.91888
$139$	−902.634	−0.550794	−0.275397	−	0.961331i	$-0.588809\pi$
$139$	−902.634	−0.550794	−0.275397	+	0.961331i	$0.588809\pi$
$140$	0	0
$141$	−2181.84	−1.30315
$142$	−2435.44	−1.43928
$143$	389.083	0.227530
$144$	−246.540	−0.142674
$145$	−4228.69	−2.42189
$146$	1283.90	0.727783
$147$	0	0
$148$	48.2498	0.0267980
$149$	1212.63	0.666727	0.333363	−	0.942798i	$-0.391816\pi$
$149$	1212.63	0.666727	0.333363	+	0.942798i	$0.391816\pi$
$150$	−2389.81	−1.30085
$151$	2565.33	1.38254	0.691270	−	0.722597i	$-0.257052\pi$
$151$	2565.33	1.38254	0.691270	+	0.722597i	$0.257052\pi$
$152$	−2096.42	−1.11870
$153$	−128.183	−0.0677320
$154$	0	0
$155$	582.486	0.301848
$156$	−492.150	−0.252587
$157$	−702.237	−0.356972	−0.178486	−	0.983942i	$-0.557120\pi$
$157$	−702.237	−0.356972	−0.178486	+	0.983942i	$0.557120\pi$
$158$	2390.36	1.20359
$159$	1496.60	0.746464
$160$	1782.09	0.880542
$161$	0	0
$162$	2610.93	1.26626
$163$	−1146.27	−0.550814	−0.275407	−	0.961328i	$-0.588813\pi$
$163$	−1146.27	−0.550814	−0.275407	+	0.961328i	$0.588813\pi$
$164$	−397.714	−0.189368
$165$	−972.439	−0.458814
$166$	1892.05	0.884646
$167$	3255.04	1.50828	0.754140	−	0.656714i	$-0.228054\pi$
$167$	3255.04	1.50828	0.754140	+	0.656714i	$0.228054\pi$
$168$	0	0
$169$	−945.879	−0.430532
$170$	2114.37	0.953910
$171$	−374.201	−0.167344
$172$	727.511	0.322513
$173$	−4024.24	−1.76854	−0.884269	−	0.466977i	$-0.845343\pi$
$173$	−4024.24	−1.76854	−0.884269	+	0.466977i	$0.845343\pi$
$174$	4683.26	2.04044
$175$	0	0
$176$	856.336	0.366754
$177$	2787.23	1.18362
$178$	−2616.52	−1.10178
$179$	706.090	0.294836	0.147418	−	0.989074i	$-0.452904\pi$
$179$	706.090	0.294836	0.147418	+	0.989074i	$0.452904\pi$
$180$	129.129	0.0534705
$181$	1268.90	0.521087	0.260544	−	0.965462i	$-0.416098\pi$
$181$	1268.90	0.521087	0.260544	+	0.965462i	$0.416098\pi$
$182$	0	0
$183$	−195.316	−0.0788970
$184$	−3096.20	−1.24051
$185$	306.562	0.121832
$186$	−645.102	−0.254307
$187$	445.233	0.174110
$188$	−1006.33	−0.390393
$189$	0	0
$190$	6172.41	2.35681
$191$	4864.58	1.84287	0.921436	−	0.388529i	$-0.127017\pi$
$191$	4864.58	1.84287	0.921436	+	0.388529i	$0.127017\pi$
$192$	1446.97	0.543887
$193$	−2675.49	−0.997855	−0.498928	−	0.866644i	$-0.666273\pi$
$193$	−2675.49	−0.997855	−0.498928	+	0.866644i	$0.666273\pi$
$194$	−3106.70	−1.14973
$195$	−3126.94	−1.14833
$196$	0	0
$197$	1627.73	0.588684	0.294342	−	0.955700i	$-0.404899\pi$
$197$	1627.73	0.588684	0.294342	+	0.955700i	$0.404899\pi$
$198$	113.060	0.0405801
$199$	2254.07	0.802947	0.401474	−	0.915871i	$-0.368498\pi$
$199$	2254.07	0.802947	0.401474	+	0.915871i	$0.368498\pi$
$200$	2378.62	0.840968
$201$	5378.36	1.88736
$202$	−3232.98	−1.12610
$203$	0	0
$204$	−563.173	−0.193284
$205$	−2526.93	−0.860920
$206$	4344.26	1.46931
$207$	−552.657	−0.185567
$208$	2753.60	0.917923
$209$	1299.75	0.430172
$210$	0	0
$211$	3112.07	1.01537	0.507687	−	0.861542i	$-0.330501\pi$
$211$	3112.07	1.01537	0.507687	+	0.861542i	$0.330501\pi$
$212$	690.273	0.223623
$213$	4121.55	1.32584
$214$	−4703.54	−1.50247
$215$	4622.34	1.46624
$216$	−2322.49	−0.731601
$217$	0	0
$218$	2124.30	0.659979
$219$	−2172.78	−0.670423
$220$	−448.517	−0.137450
$221$	1431.67	0.435769
$222$	−339.516	−0.102643
$223$	3558.38	1.06855	0.534275	−	0.845311i	$-0.320585\pi$
$223$	3558.38	1.06855	0.534275	+	0.845311i	$0.320585\pi$
$224$	0	0
$225$	424.572	0.125799
$226$	3766.84	1.10870
$227$	2330.51	0.681416	0.340708	−	0.940169i	$-0.389333\pi$
$227$	2330.51	0.681416	0.340708	+	0.940169i	$0.389333\pi$
$228$	−1644.05	−0.477544
$229$	−676.106	−0.195102	−0.0975509	−	0.995231i	$-0.531101\pi$
$229$	−676.106	−0.195102	−0.0975509	+	0.995231i	$0.531101\pi$
$230$	9116.02	2.61345
$231$	0	0
$232$	−4661.34	−1.31910
$233$	1620.20	0.455548	0.227774	−	0.973714i	$-0.426855\pi$
$233$	1620.20	0.455548	0.227774	+	0.973714i	$0.426855\pi$
$234$	363.553	0.101565
$235$	−6393.84	−1.77484
$236$	1285.55	0.354586
$237$	−4045.27	−1.10873
$238$	0	0
$239$	−2001.39	−0.541669	−0.270835	−	0.962626i	$-0.587300\pi$
$239$	−2001.39	−0.541669	−0.270835	+	0.962626i	$0.587300\pi$
$240$	−6882.10	−1.85099
$241$	1586.39	0.424018	0.212009	−	0.977268i	$-0.431999\pi$
$241$	1586.39	0.424018	0.212009	+	0.977268i	$0.431999\pi$
$242$	−392.706	−0.104314
$243$	−884.196	−0.233420
$244$	−90.0853	−0.0236357
$245$	0	0
$246$	2798.57	0.725327
$247$	4179.44	1.07665
$248$	642.081	0.164404
$249$	−3201.96	−0.814923
$250$	−473.537	−0.119796
$251$	−2612.23	−0.656902	−0.328451	−	0.944521i	$-0.606527\pi$
$251$	−2612.23	−0.656902	−0.328451	+	0.944521i	$0.606527\pi$
$252$	0	0
$253$	1919.61	0.477014
$254$	3425.33	0.846158
$255$	−3578.20	−0.878727
$256$	3542.12	0.864775
$257$	−4198.23	−1.01898	−0.509491	−	0.860476i	$-0.670166\pi$
$257$	−4198.23	−1.01898	−0.509491	+	0.860476i	$0.670166\pi$
$258$	−5119.23	−1.23531
$259$	0	0
$260$	−1442.24	−0.344014
$261$	−832.028	−0.197323
$262$	8624.58	2.03370
$263$	−5170.31	−1.21222	−0.606112	−	0.795380i	$-0.707272\pi$
$263$	−5170.31	−1.21222	−0.606112	+	0.795380i	$0.707272\pi$
$264$	−1071.93	−0.249897
$265$	4385.74	1.01666
$266$	0	0
$267$	4428.01	1.01494
$268$	2480.66	0.565411
$269$	−1648.20	−0.373578	−0.186789	−	0.982400i	$-0.559808\pi$
$269$	−1648.20	−0.373578	−0.186789	+	0.982400i	$0.559808\pi$
$270$	6838.04	1.54130
$271$	−2562.79	−0.574459	−0.287230	−	0.957862i	$-0.592734\pi$
$271$	−2562.79	−0.574459	−0.287230	+	0.957862i	$0.592734\pi$
$272$	3150.98	0.702413
$273$	0	0
$274$	478.107	0.105414
$275$	−1474.72	−0.323377
$276$	−2428.10	−0.529545
$277$	2762.94	0.599310	0.299655	−	0.954048i	$-0.403128\pi$
$277$	2762.94	0.599310	0.299655	+	0.954048i	$0.403128\pi$
$278$	2929.50	0.632013
$279$	114.609	0.0245930
$280$	0	0
$281$	6453.68	1.37009	0.685044	−	0.728502i	$-0.259783\pi$
$281$	6453.68	1.37009	0.685044	+	0.728502i	$0.259783\pi$
$282$	7081.16	1.49531
$283$	−4540.78	−0.953787	−0.476893	−	0.878961i	$-0.658237\pi$
$283$	−4540.78	−0.953787	−0.476893	+	0.878961i	$0.658237\pi$
$284$	1900.98	0.397191
$285$	−10445.7	−2.17106
$286$	−1262.77	−0.261081
$287$	0	0
$288$	350.641	0.0717420
$289$	−3274.72	−0.666541
$290$	13724.2	2.77901
$291$	5257.54	1.05912
$292$	−1002.15	−0.200843
$293$	−1916.34	−0.382094	−0.191047	−	0.981581i	$-0.561188\pi$
$293$	−1916.34	−0.382094	−0.191047	+	0.981581i	$0.561188\pi$
$294$	0	0
$295$	8167.92	1.61205
$296$	337.927	0.0663567
$297$	1439.92	0.281322
$298$	−3935.58	−0.765040
$299$	6172.61	1.19388
$300$	1865.36	0.358989
$301$	0	0
$302$	−8325.78	−1.58641
$303$	5471.26	1.03735
$304$	9198.56	1.73544
$305$	−572.369	−0.107455
$306$	416.019	0.0777196
$307$	−6876.30	−1.27834	−0.639171	−	0.769064i	$-0.720722\pi$
$307$	−6876.30	−1.27834	−0.639171	+	0.769064i	$0.720722\pi$
$308$	0	0
$309$	−7351.89	−1.35351
$310$	−1890.46	−0.346357
$311$	−8469.32	−1.54422	−0.772108	−	0.635492i	$-0.780797\pi$
$311$	−8469.32	−1.54422	−0.772108	+	0.635492i	$0.780797\pi$
$312$	−3446.86	−0.625450
$313$	2882.28	0.520498	0.260249	−	0.965542i	$-0.416195\pi$
$313$	2882.28	0.520498	0.260249	+	0.965542i	$0.416195\pi$
$314$	2279.11	0.409610
$315$	0	0
$316$	−1865.80	−0.332150
$317$	9795.31	1.73552	0.867759	−	0.496985i	$-0.165560\pi$
$317$	9795.31	1.73552	0.867759	+	0.496985i	$0.165560\pi$
$318$	−4857.20	−0.856535
$319$	2889.98	0.507234
$320$	4240.33	0.740755
$321$	7959.92	1.38405
$322$	0	0
$323$	4782.59	0.823871
$324$	−2037.96	−0.349445
$325$	−4742.04	−0.809357
$326$	3720.22	0.632036
$327$	−3595.00	−0.607963
$328$	−2785.47	−0.468908
$329$	0	0
$330$	3156.05	0.526470
$331$	1813.70	0.301178	0.150589	−	0.988596i	$-0.451883\pi$
$331$	1813.70	0.301178	0.150589	+	0.988596i	$0.451883\pi$
$332$	−1476.84	−0.244132
$333$	60.3184	0.00992621
$334$	−10564.2	−1.73069
$335$	15761.2	2.57052
$336$	0	0
$337$	−11964.2	−1.93393	−0.966964	−	0.254913i	$-0.917953\pi$
$337$	−11964.2	−1.93393	−0.966964	+	0.254913i	$0.917953\pi$
$338$	3069.85	0.494017
$339$	−6374.71	−1.02132
$340$	−1650.37	−0.263247
$341$	−398.083	−0.0632182
$342$	1214.47	0.192020
$343$	0	0
$344$	5095.26	0.798599
$345$	−15427.3	−2.40747
$346$	13060.7	2.02932
$347$	−2283.89	−0.353330	−0.176665	−	0.984271i	$-0.556531\pi$
$347$	−2283.89	−0.353330	−0.176665	+	0.984271i	$0.556531\pi$
$348$	−3655.52	−0.563093
$349$	2472.48	0.379223	0.189612	−	0.981859i	$-0.439277\pi$
$349$	2472.48	0.379223	0.189612	+	0.981859i	$0.439277\pi$
$350$	0	0
$351$	4630.15	0.704101
$352$	−1217.92	−0.184418
$353$	−10190.0	−1.53642	−0.768211	−	0.640197i	$-0.778853\pi$
$353$	−10190.0	−1.53642	−0.768211	+	0.640197i	$0.778853\pi$
$354$	−9045.95	−1.35815
$355$	12078.1	1.80575
$356$	2042.33	0.304054
$357$	0	0
$358$	−2291.62	−0.338312
$359$	−43.4294	−0.00638473	−0.00319236	−	0.999995i	$-0.501016\pi$
$359$	−43.4294	−0.00638473	−0.00319236	+	0.999995i	$0.501016\pi$
$360$	904.378	0.132402
$361$	7102.66	1.03552
$362$	−4118.23	−0.597926
$363$	664.585	0.0960928
$364$	0	0
$365$	−6367.28	−0.913093
$366$	633.897	0.0905310
$367$	−8775.78	−1.24821	−0.624104	−	0.781342i	$-0.714536\pi$
$367$	−8775.78	−1.24821	−0.624104	+	0.781342i	$0.714536\pi$
$368$	13585.4	1.92442
$369$	−497.194	−0.0701433
$370$	−994.946	−0.139797
$371$	0	0
$372$	503.534	0.0701801
$373$	−2751.89	−0.382004	−0.191002	−	0.981590i	$-0.561174\pi$
$373$	−2751.89	−0.382004	−0.191002	+	0.981590i	$0.561174\pi$
$374$	−1445.00	−0.199784
$375$	801.379	0.110355
$376$	−7048.01	−0.966684
$377$	9292.90	1.26952
$378$	0	0
$379$	2605.59	0.353141	0.176570	−	0.984288i	$-0.443500\pi$
$379$	2605.59	0.353141	0.176570	+	0.984288i	$0.443500\pi$
$380$	−4817.87	−0.650399
$381$	−5796.77	−0.779468
$382$	−15788.0	−2.11462
$383$	14360.5	1.91590	0.957949	−	0.286940i	$-0.0926380\pi$
$383$	14360.5	1.91590	0.957949	+	0.286940i	$0.0926380\pi$
$384$	−9561.14	−1.27061
$385$	0	0
$386$	8683.31	1.14500
$387$	909.482	0.119461
$388$	2424.93	0.317287
$389$	−10607.9	−1.38262	−0.691311	−	0.722557i	$-0.742967\pi$
$389$	−10607.9	−1.38262	−0.691311	+	0.722557i	$0.742967\pi$
$390$	10148.5	1.31766
$391$	7063.40	0.913585
$392$	0	0
$393$	−14595.6	−1.87341
$394$	−5282.79	−0.675490
$395$	−11854.6	−1.51005
$396$	−88.2493	−0.0111987
$397$	−7362.35	−0.930745	−0.465373	−	0.885115i	$-0.654080\pi$
$397$	−7362.35	−0.930745	−0.465373	+	0.885115i	$0.654080\pi$
$398$	−7315.57	−0.921348
$399$	0	0
$400$	−10436.8	−1.30460
$401$	−264.514	−0.0329406	−0.0164703	−	0.999864i	$-0.505243\pi$
$401$	−264.514	−0.0329406	−0.0164703	+	0.999864i	$0.505243\pi$
$402$	−17455.5	−2.16567
$403$	−1280.06	−0.158224
$404$	2523.50	0.310765
$405$	−12948.5	−1.58868
$406$	0	0
$407$	−209.511	−0.0255161
$408$	−3944.29	−0.478607
$409$	1244.03	0.150400	0.0751999	−	0.997168i	$-0.476041\pi$
$409$	1244.03	0.150400	0.0751999	+	0.997168i	$0.476041\pi$
$410$	8201.16	0.987869
$411$	−809.112	−0.0971060
$412$	−3390.91	−0.405481
$413$	0	0
$414$	1793.65	0.212930
$415$	−9383.27	−1.10990
$416$	−3916.30	−0.461568
$417$	−4957.66	−0.582201
$418$	−4218.35	−0.493604
$419$	−3974.76	−0.463436	−0.231718	−	0.972783i	$-0.574435\pi$
$419$	−3974.76	−0.463436	−0.231718	+	0.972783i	$0.574435\pi$
$420$	0	0
$421$	−14910.2	−1.72608	−0.863041	−	0.505134i	$-0.831443\pi$
$421$	−14910.2	−1.72608	−0.863041	+	0.505134i	$0.831443\pi$
$422$	−10100.2	−1.16510
$423$	−1258.04	−0.144605
$424$	4834.46	0.553731
$425$	−5426.38	−0.619336
$426$	−13376.5	−1.52135
$427$	0	0
$428$	3671.35	0.414629
$429$	2137.02	0.240504
$430$	−15001.8	−1.68244
$431$	−10622.4	−1.18715	−0.593574	−	0.804779i	$-0.702284\pi$
$431$	−10622.4	−1.18715	−0.593574	+	0.804779i	$0.702284\pi$
$432$	10190.5	1.13494
$433$	−17440.5	−1.93565	−0.967823	−	0.251630i	$-0.919033\pi$
$433$	−17440.5	−1.93565	−0.967823	+	0.251630i	$0.919033\pi$
$434$	0	0
$435$	−23225.8	−2.55999
$436$	−1658.12	−0.182132
$437$	20620.0	2.25718
$438$	7051.75	0.769282
$439$	−13627.1	−1.48151	−0.740756	−	0.671774i	$-0.765533\pi$
$439$	−13627.1	−1.48151	−0.740756	+	0.671774i	$0.765533\pi$
$440$	−3141.28	−0.340351
$441$	0	0
$442$	−4646.50	−0.500026
$443$	2135.29	0.229008	0.114504	−	0.993423i	$-0.463472\pi$
$443$	2135.29	0.229008	0.114504	+	0.993423i	$0.463472\pi$
$444$	265.009	0.0283261
$445$	12976.2	1.38232
$446$	−11548.7	−1.22612
$447$	6660.28	0.704744
$448$	0	0
$449$	17780.8	1.86889	0.934443	−	0.356113i	$-0.115898\pi$
$449$	17780.8	1.86889	0.934443	+	0.356113i	$0.115898\pi$
$450$	−1377.95	−0.144349
$451$	1726.96	0.180309
$452$	−2940.20	−0.305963
$453$	14089.9	1.46137
$454$	−7563.67	−0.781896
$455$	0	0
$456$	−11514.4	−1.18249
$457$	10357.0	1.06013	0.530064	−	0.847957i	$-0.322168\pi$
$457$	10357.0	1.06013	0.530064	+	0.847957i	$0.322168\pi$
$458$	2194.30	0.223871
$459$	5298.35	0.538792
$460$	−7115.51	−0.721222
$461$	19679.5	1.98821	0.994106	−	0.108410i	$-0.0345760\pi$
$461$	19679.5	1.98821	0.994106	+	0.108410i	$0.0345760\pi$
$462$	0	0
$463$	−7171.43	−0.719838	−0.359919	−	0.932984i	$-0.617196\pi$
$463$	−7171.43	−0.719838	−0.359919	+	0.932984i	$0.617196\pi$
$464$	20452.8	2.04633
$465$	3199.27	0.319059
$466$	−5258.36	−0.522722
$467$	12192.8	1.20817	0.604085	−	0.796920i	$-0.293539\pi$
$467$	12192.8	1.20817	0.604085	+	0.796920i	$0.293539\pi$
$468$	−283.771	−0.0280285
$469$	0	0
$470$	20751.2	2.03656
$471$	−3856.99	−0.377327
$472$	9003.60	0.878017
$473$	−3159.00	−0.307085
$474$	13128.9	1.27222
$475$	−15841.1	−1.53018
$476$	0	0
$477$	862.930	0.0828319
$478$	6495.50	0.621542
$479$	−8475.11	−0.808429	−0.404215	−	0.914664i	$-0.632455\pi$
$479$	−8475.11	−0.808429	−0.404215	+	0.914664i	$0.632455\pi$
$480$	9788.04	0.930751
$481$	−673.695	−0.0638624
$482$	−5148.63	−0.486543
$483$	0	0
$484$	306.526	0.0287872
$485$	15407.1	1.44248
$486$	2869.66	0.267840
$487$	2735.29	0.254513	0.127257	−	0.991870i	$-0.459383\pi$
$487$	2735.29	0.254513	0.127257	+	0.991870i	$0.459383\pi$
$488$	−630.929	−0.0585263
$489$	−6295.81	−0.582222
$490$	0	0
$491$	−2233.82	−0.205317	−0.102659	−	0.994717i	$-0.532735\pi$
$491$	−2233.82	−0.205317	−0.102659	+	0.994717i	$0.532735\pi$
$492$	−2184.42	−0.200165
$493$	10634.0	0.971462
$494$	−13564.4	−1.23541
$495$	−560.704	−0.0509126
$496$	−2817.30	−0.255041
$497$	0	0
$498$	10392.0	0.935089
$499$	−18100.3	−1.62381	−0.811904	−	0.583791i	$-0.801569\pi$
$499$	−18100.3	−1.62381	−0.811904	+	0.583791i	$0.801569\pi$
$500$	369.619	0.0330597
$501$	17878.1	1.59428
$502$	8477.99	0.753768
$503$	−6149.06	−0.545076	−0.272538	−	0.962145i	$-0.587863\pi$
$503$	−6149.06	−0.545076	−0.272538	+	0.962145i	$0.587863\pi$
$504$	0	0
$505$	16033.4	1.41283
$506$	−6230.08	−0.547354
$507$	−5195.18	−0.455081
$508$	−2673.64	−0.233511
$509$	−14193.9	−1.23602	−0.618008	−	0.786172i	$-0.712060\pi$
$509$	−14193.9	−1.23602	−0.618008	+	0.786172i	$0.712060\pi$
$510$	11613.0	1.00830
$511$	0	0
$512$	2430.30	0.209775
$513$	15467.3	1.33118
$514$	13625.4	1.16924
$515$	−21544.6	−1.84343
$516$	3995.81	0.340903
$517$	4369.68	0.371718
$518$	0	0
$519$	−22102.9	−1.86938
$520$	−10101.0	−0.851840
$521$	−10371.8	−0.872163	−0.436082	−	0.899907i	$-0.643634\pi$
$521$	−10371.8	−0.872163	−0.436082	+	0.899907i	$0.643634\pi$
$522$	2700.35	0.226420
$523$	11369.4	0.950569	0.475285	−	0.879832i	$-0.342345\pi$
$523$	11369.4	0.950569	0.475285	+	0.879832i	$0.342345\pi$
$524$	−6731.91	−0.561231
$525$	0	0
$526$	16780.2	1.39098
$527$	−1464.79	−0.121076
$528$	4703.37	0.387667
$529$	18286.6	1.50297
$530$	−14233.9	−1.16657
$531$	1607.10	0.131341
$532$	0	0
$533$	5553.14	0.451282
$534$	−14371.1	−1.16460
$535$	23326.4	1.88503
$536$	17373.7	1.40006
$537$	3878.16	0.311648
$538$	5349.24	0.428665
$539$	0	0
$540$	−5337.43	−0.425345
$541$	5386.88	0.428096	0.214048	−	0.976823i	$-0.431335\pi$
$541$	5386.88	0.428096	0.214048	+	0.976823i	$0.431335\pi$
$542$	8317.54	0.659167
$543$	6969.38	0.550800
$544$	−4481.47	−0.353201
$545$	−10535.1	−0.828024
$546$	0	0
$547$	13890.8	1.08579	0.542894	−	0.839801i	$-0.317328\pi$
$547$	13890.8	1.08579	0.542894	+	0.839801i	$0.317328\pi$
$548$	−373.186	−0.0290907
$549$	−112.618	−0.00875487
$550$	4786.19	0.371061
$551$	31043.5	2.40017
$552$	−17005.7	−1.31125
$553$	0	0
$554$	−8967.12	−0.687683
$555$	1683.77	0.128779
$556$	−2286.62	−0.174414
$557$	17498.3	1.33111	0.665553	−	0.746351i	$-0.268196\pi$
$557$	17498.3	1.33111	0.665553	+	0.746351i	$0.268196\pi$
$558$	−371.962	−0.0282194
$559$	−10158.0	−0.768581
$560$	0	0
$561$	2445.42	0.184038
$562$	−20945.4	−1.57212
$563$	−147.373	−0.0110320	−0.00551600	−	0.999985i	$-0.501756\pi$
$563$	−147.373	−0.0110320	−0.00551600	+	0.999985i	$0.501756\pi$
$564$	−5527.19	−0.412654
$565$	−18681.0	−1.39100
$566$	14737.1	1.09443
$567$	0	0
$568$	13313.9	0.983517
$569$	−14155.3	−1.04292	−0.521459	−	0.853276i	$-0.674612\pi$
$569$	−14155.3	−1.04292	−0.521459	+	0.853276i	$0.674612\pi$
$570$	33901.6	2.49120
$571$	−248.361	−0.0182025	−0.00910123	−	0.999959i	$-0.502897\pi$
$571$	−248.361	−0.0182025	−0.00910123	+	0.999959i	$0.502897\pi$
$572$	985.654	0.0720494
$573$	26718.4	1.94795
$574$	0	0
$575$	−23395.6	−1.69681
$576$	834.318	0.0603529
$577$	−19364.6	−1.39715	−0.698576	−	0.715536i	$-0.746183\pi$
$577$	−19364.6	−1.39715	−0.698576	+	0.715536i	$0.746183\pi$
$578$	10628.1	0.764828
$579$	−14695.0	−1.05475
$580$	−10712.4	−0.766913
$581$	0	0
$582$	−17063.4	−1.21529
$583$	−2997.31	−0.212926
$584$	−7018.73	−0.497324
$585$	−1802.98	−0.127426
$586$	6219.47	0.438437
$587$	9135.18	0.642332	0.321166	−	0.947023i	$-0.395925\pi$
$587$	9135.18	0.642332	0.321166	+	0.947023i	$0.395925\pi$
$588$	0	0
$589$	−4276.12	−0.299141
$590$	−26509.0	−1.84976
$591$	8940.20	0.622252
$592$	−1482.74	−0.102940
$593$	12287.6	0.850916	0.425458	−	0.904978i	$-0.360113\pi$
$593$	12287.6	0.850916	0.425458	+	0.904978i	$0.360113\pi$
$594$	−4673.26	−0.322805
$595$	0	0
$596$	3071.91	0.211125
$597$	12380.3	0.848732
$598$	−20033.2	−1.36993
$599$	12351.9	0.842549	0.421275	−	0.906933i	$-0.361583\pi$
$599$	12351.9	0.842549	0.421275	+	0.906933i	$0.361583\pi$
$600$	13064.4	0.888921
$601$	−21624.2	−1.46767	−0.733836	−	0.679327i	$-0.762272\pi$
$601$	−21624.2	−1.46767	−0.733836	+	0.679327i	$0.762272\pi$
$602$	0	0
$603$	3101.14	0.209433
$604$	6498.68	0.437794
$605$	1947.56	0.130875
$606$	−17757.0	−1.19031
$607$	2086.03	0.139488	0.0697442	−	0.997565i	$-0.477782\pi$
$607$	2086.03	0.139488	0.0697442	+	0.997565i	$0.477782\pi$
$608$	−13082.6	−0.872648
$609$	0	0
$610$	1857.62	0.123300
$611$	14051.0	0.930347
$612$	−324.723	−0.0214480
$613$	−8338.46	−0.549408	−0.274704	−	0.961529i	$-0.588580\pi$
$613$	−8338.46	−0.549408	−0.274704	+	0.961529i	$0.588580\pi$
$614$	22317.0	1.46684
$615$	−13879.0	−0.910011
$616$	0	0
$617$	−4771.20	−0.311315	−0.155657	−	0.987811i	$-0.549750\pi$
$617$	−4771.20	−0.311315	−0.155657	+	0.987811i	$0.549750\pi$
$618$	23860.6	1.55310
$619$	16609.4	1.07850	0.539248	−	0.842147i	$-0.318708\pi$
$619$	16609.4	1.07850	0.539248	+	0.842147i	$0.318708\pi$
$620$	1475.60	0.0955828
$621$	22843.6	1.47614
$622$	27487.2	1.77192
$623$	0	0
$624$	15124.0	0.970264
$625$	−14409.7	−0.922221
$626$	−9354.43	−0.597250
$627$	7138.82	0.454700
$628$	−1778.96	−0.113038
$629$	−770.918	−0.0488688
$630$	0	0
$631$	17254.9	1.08860	0.544299	−	0.838891i	$-0.316796\pi$
$631$	17254.9	1.08860	0.544299	+	0.838891i	$0.316796\pi$
$632$	−13067.5	−0.822462
$633$	17092.9	1.07327
$634$	−31790.7	−1.99143
$635$	−16987.3	−1.06161
$636$	3791.28	0.236375
$637$	0	0
$638$	−9379.42	−0.582029
$639$	2376.47	0.147123
$640$	−28018.7	−1.73053
$641$	−7350.77	−0.452945	−0.226473	−	0.974018i	$-0.572719\pi$
$641$	−7350.77	−0.452945	−0.226473	+	0.974018i	$0.572719\pi$
$642$	−25833.9	−1.58814
$643$	−10117.8	−0.620542	−0.310271	−	0.950648i	$-0.600420\pi$
$643$	−10117.8	−0.620542	−0.310271	+	0.950648i	$0.600420\pi$
$644$	0	0
$645$	25387.9	1.54984
$646$	−15521.9	−0.945358
$647$	24590.9	1.49423	0.747116	−	0.664693i	$-0.231438\pi$
$647$	24590.9	1.49423	0.747116	+	0.664693i	$0.231438\pi$
$648$	−14273.2	−0.865287
$649$	−5582.13	−0.337624
$650$	15390.3	0.928703
$651$	0	0
$652$	−2903.81	−0.174420
$653$	2339.03	0.140173	0.0700867	−	0.997541i	$-0.477672\pi$
$653$	2339.03	0.140173	0.0700867	+	0.997541i	$0.477672\pi$
$654$	11667.6	0.697612
$655$	−42772.1	−2.55152
$656$	12222.0	0.727420
$657$	−1252.81	−0.0743940
$658$	0	0
$659$	−15735.7	−0.930162	−0.465081	−	0.885268i	$-0.653975\pi$
$659$	−15735.7	−0.930162	−0.465081	+	0.885268i	$0.653975\pi$
$660$	−2463.45	−0.145288
$661$	−4846.75	−0.285199	−0.142600	−	0.989780i	$-0.545546\pi$
$661$	−4846.75	−0.285199	−0.142600	+	0.989780i	$0.545546\pi$
$662$	−5886.36	−0.345589
$663$	7863.39	0.460617
$664$	−10343.3	−0.604515
$665$	0	0
$666$	−195.763	−0.0113899
$667$	45848.1	2.66154
$668$	8245.91	0.477611
$669$	19544.2	1.12948
$670$	−51152.9	−2.94957
$671$	391.169	0.0225051
$672$	0	0
$673$	15716.8	0.900205	0.450103	−	0.892977i	$-0.351387\pi$
$673$	15716.8	0.900205	0.450103	+	0.892977i	$0.351387\pi$
$674$	38830.0	2.21910
$675$	−17549.4	−1.00070
$676$	−2396.17	−0.136332
$677$	856.968	0.0486499	0.0243249	−	0.999704i	$-0.492256\pi$
$677$	856.968	0.0486499	0.0243249	+	0.999704i	$0.492256\pi$
$678$	20689.1	1.17192
$679$	0	0
$680$	−11558.7	−0.651846
$681$	12800.2	0.720271
$682$	1291.98	0.0725402
$683$	24804.0	1.38960	0.694801	−	0.719202i	$-0.255492\pi$
$683$	24804.0	1.38960	0.694801	+	0.719202i	$0.255492\pi$
$684$	−947.953	−0.0529911
$685$	−2371.09	−0.132255
$686$	0	0
$687$	−3713.47	−0.206227
$688$	−22356.8	−1.23887
$689$	−9638.04	−0.532917
$690$	50069.2	2.76247
$691$	−3475.97	−0.191364	−0.0956818	−	0.995412i	$-0.530503\pi$
$691$	−3475.97	−0.191364	−0.0956818	+	0.995412i	$0.530503\pi$
$692$	−10194.5	−0.560024
$693$	0	0
$694$	7412.36	0.405431
$695$	−14528.3	−0.792937
$696$	−25602.1	−1.39432
$697$	6354.54	0.345330
$698$	−8024.44	−0.435143
$699$	8898.85	0.481524
$700$	0	0
$701$	17461.0	0.940790	0.470395	−	0.882456i	$-0.344111\pi$
$701$	17461.0	0.940790	0.470395	+	0.882456i	$0.344111\pi$
$702$	−15027.2	−0.807926
$703$	−2250.52	−0.120739
$704$	−2897.93	−0.155142
$705$	−35117.8	−1.87605
$706$	33071.5	1.76298
$707$	0	0
$708$	7060.81	0.374804
$709$	−18405.7	−0.974950	−0.487475	−	0.873137i	$-0.662082\pi$
$709$	−18405.7	−0.974950	−0.487475	+	0.873137i	$0.662082\pi$
$710$	−39199.6	−2.07202
$711$	−2332.48	−0.123031
$712$	14303.8	0.752891
$713$	−6315.39	−0.331716
$714$	0	0
$715$	6262.49	0.327558
$716$	1788.72	0.0933626
$717$	−10992.5	−0.572555
$718$	140.950	0.00732621
$719$	892.380	0.0462867	0.0231434	−	0.999732i	$-0.492633\pi$
$719$	892.380	0.0462867	0.0231434	+	0.999732i	$0.492633\pi$
$720$	−3968.19	−0.205397
$721$	0	0
$722$	−23051.7	−1.18822
$723$	8713.15	0.448196
$724$	3214.48	0.165007
$725$	−35222.2	−1.80431
$726$	−2156.91	−0.110262
$727$	−27919.2	−1.42430	−0.712149	−	0.702028i	$-0.752278\pi$
$727$	−27919.2	−1.42430	−0.712149	+	0.702028i	$0.752278\pi$
$728$	0	0
$729$	16864.5	0.856805
$730$	20665.0	1.04774
$731$	−11623.9	−0.588134
$732$	−494.788	−0.0249835
$733$	4769.38	0.240329	0.120164	−	0.992754i	$-0.461658\pi$
$733$	4769.38	0.240329	0.120164	+	0.992754i	$0.461658\pi$
$734$	28481.8	1.43227
$735$	0	0
$736$	−19321.7	−0.967673
$737$	−10771.5	−0.538364
$738$	1613.64	0.0804865
$739$	−5170.63	−0.257381	−0.128691	−	0.991685i	$-0.541077\pi$
$739$	−5170.63	−0.257381	−0.128691	+	0.991685i	$0.541077\pi$
$740$	776.604	0.0385791
$741$	22955.3	1.13804
$742$	0	0
$743$	−29407.9	−1.45205	−0.726024	−	0.687669i	$-0.758634\pi$
$743$	−29407.9	−1.45205	−0.726024	+	0.687669i	$0.758634\pi$
$744$	3526.59	0.173779
$745$	19517.8	0.959836
$746$	8931.26	0.438333
$747$	−1846.23	−0.0904286
$748$	1127.90	0.0551337
$749$	0	0
$750$	−2600.88	−0.126627
$751$	−16956.5	−0.823905	−0.411952	−	0.911205i	$-0.635153\pi$
$751$	−16956.5	−0.823905	−0.411952	+	0.911205i	$0.635153\pi$
$752$	30924.9	1.49962
$753$	−14347.5	−0.694360
$754$	−30160.1	−1.45672
$755$	41290.3	1.99034
$756$	0	0
$757$	−21322.0	−1.02373	−0.511864	−	0.859067i	$-0.671045\pi$
$757$	−21322.0	−1.02373	−0.511864	+	0.859067i	$0.671045\pi$
$758$	−8456.45	−0.405214
$759$	10543.3	0.504214
$760$	−33742.9	−1.61050
$761$	23548.0	1.12170	0.560851	−	0.827917i	$-0.310474\pi$
$761$	23548.0	1.12170	0.560851	+	0.827917i	$0.310474\pi$
$762$	18813.4	0.894406
$763$	0	0
$764$	12323.3	0.583563
$765$	−2063.17	−0.0975087
$766$	−46607.1	−2.19841
$767$	−17949.7	−0.845014
$768$	19454.9	0.914085
$769$	17230.9	0.808015	0.404007	−	0.914756i	$-0.367617\pi$
$769$	17230.9	0.808015	0.404007	+	0.914756i	$0.367617\pi$
$770$	0	0
$771$	−23058.5	−1.07709
$772$	−6777.75	−0.315980
$773$	−12285.8	−0.571655	−0.285828	−	0.958281i	$-0.592268\pi$
$773$	−12285.8	−0.571655	−0.285828	+	0.958281i	$0.592268\pi$
$774$	−2951.72	−0.137077
$775$	4851.73	0.224876
$776$	16983.5	0.785658
$777$	0	0
$778$	34427.8	1.58650
$779$	18550.6	0.853202
$780$	−7921.39	−0.363630
$781$	−8254.45	−0.378191
$782$	−22924.3	−1.04830
$783$	34391.2	1.56966
$784$	0	0
$785$	−11302.8	−0.513906
$786$	47370.0	2.14966
$787$	−663.152	−0.0300366	−0.0150183	−	0.999887i	$-0.504781\pi$
$787$	−663.152	−0.0300366	−0.0150183	+	0.999887i	$0.504781\pi$
$788$	4123.48	0.186412
$789$	−28397.6	−1.28135
$790$	38474.1	1.73272
$791$	0	0
$792$	−618.071	−0.0277300
$793$	1257.83	0.0563264
$794$	23894.5	1.06799
$795$	24088.4	1.07463
$796$	5710.17	0.254261
$797$	−19216.3	−0.854050	−0.427025	−	0.904240i	$-0.640438\pi$
$797$	−19216.3	−0.854050	−0.427025	+	0.904240i	$0.640438\pi$
$798$	0	0
$799$	16078.7	0.711921
$800$	14843.7	0.656004
$801$	2553.17	0.112624
$802$	858.480	0.0377980
$803$	4351.53	0.191236
$804$	13624.9	0.597651
$805$	0	0
$806$	4154.44	0.181556
$807$	−9052.65	−0.394880
$808$	17673.8	0.769509
$809$	−42881.8	−1.86359	−0.931794	−	0.362989i	$-0.881756\pi$
$809$	−42881.8	−1.86359	−0.931794	+	0.362989i	$0.881756\pi$
$810$	42024.2	1.82294
$811$	1205.73	0.0522058	0.0261029	−	0.999659i	$-0.491690\pi$
$811$	1205.73	0.0522058	0.0261029	+	0.999659i	$0.491690\pi$
$812$	0	0
$813$	−14076.0	−0.607215
$814$	679.967	0.0292787
$815$	−18449.8	−0.792966
$816$	17306.6	0.742466
$817$	−33933.3	−1.45309
$818$	−4037.51	−0.172577
$819$	0	0
$820$	−6401.41	−0.272618
$821$	−28577.6	−1.21482	−0.607408	−	0.794390i	$-0.707791\pi$
$821$	−28577.6	−1.21482	−0.607408	+	0.794390i	$0.707791\pi$
$822$	2625.97	0.111425
$823$	42524.8	1.80112	0.900561	−	0.434730i	$-0.143156\pi$
$823$	42524.8	1.80112	0.900561	+	0.434730i	$0.143156\pi$
$824$	−23748.9	−1.00404
$825$	−8099.79	−0.341816
$826$	0	0
$827$	30768.7	1.29375	0.646876	−	0.762595i	$-0.276075\pi$
$827$	30768.7	1.29375	0.646876	+	0.762595i	$0.276075\pi$
$828$	−1400.03	−0.0587614
$829$	17583.3	0.736661	0.368330	−	0.929695i	$-0.379930\pi$
$829$	17583.3	0.736661	0.368330	+	0.929695i	$0.379930\pi$
$830$	30453.4	1.27356
$831$	15175.3	0.633484
$832$	−9318.48	−0.388293
$833$	0	0
$834$	16090.1	0.668051
$835$	52391.5	2.17136
$836$	3292.63	0.136218
$837$	−4737.25	−0.195631
$838$	12900.1	0.531773
$839$	−19552.8	−0.804573	−0.402287	−	0.915514i	$-0.631784\pi$
$839$	−19552.8	−0.804573	−0.402287	+	0.915514i	$0.631784\pi$
$840$	0	0
$841$	44635.5	1.83015
$842$	48391.2	1.98061
$843$	35446.5	1.44821
$844$	7883.72	0.321527
$845$	−15224.4	−0.619805
$846$	4082.96	0.165928
$847$	0	0
$848$	−21212.4	−0.859007
$849$	−24940.0	−1.00817
$850$	17611.3	0.710662
$851$	−3323.78	−0.133887
$852$	10441.0	0.419840
$853$	18524.1	0.743557	0.371779	−	0.928321i	$-0.378748\pi$
$853$	18524.1	0.743557	0.371779	+	0.928321i	$0.378748\pi$
$854$	0	0
$855$	−6022.95	−0.240913
$856$	25713.0	1.02670
$857$	−24439.0	−0.974118	−0.487059	−	0.873369i	$-0.661930\pi$
$857$	−24439.0	−0.974118	−0.487059	+	0.873369i	$0.661930\pi$
$858$	−6935.69	−0.275968
$859$	−11301.4	−0.448893	−0.224447	−	0.974486i	$-0.572057\pi$
$859$	−11301.4	−0.448893	−0.224447	+	0.974486i	$0.572057\pi$
$860$	11709.6	0.464297
$861$	0	0
$862$	34474.9	1.36220
$863$	−26377.3	−1.04043	−0.520217	−	0.854034i	$-0.674149\pi$
$863$	−26377.3	−1.04043	−0.520217	+	0.854034i	$0.674149\pi$
$864$	−14493.4	−0.570691
$865$	−64772.1	−2.54603
$866$	56603.0	2.22107
$867$	−17986.2	−0.704548
$868$	0	0
$869$	8101.68	0.316261
$870$	75379.5	2.93747
$871$	−34636.5	−1.34743
$872$	−11612.9	−0.450991
$873$	3031.47	0.117526
$874$	−66922.1	−2.59002
$875$	0	0
$876$	−5504.24	−0.212296
$877$	−28425.0	−1.09446	−0.547232	−	0.836981i	$-0.684319\pi$
$877$	−28425.0	−1.09446	−0.547232	+	0.836981i	$0.684319\pi$
$878$	44226.6	1.69997
$879$	−10525.4	−0.403882
$880$	13783.2	0.527989
$881$	−16897.1	−0.646171	−0.323086	−	0.946370i	$-0.604720\pi$
$881$	−16897.1	−0.646171	−0.323086	+	0.946370i	$0.604720\pi$
$882$	0	0
$883$	−25538.9	−0.973331	−0.486665	−	0.873588i	$-0.661787\pi$
$883$	−25538.9	−0.973331	−0.486665	+	0.873588i	$0.661787\pi$
$884$	3626.82	0.137990
$885$	44861.8	1.70397
$886$	−6930.07	−0.262777
$887$	−6478.48	−0.245238	−0.122619	−	0.992454i	$-0.539129\pi$
$887$	−6478.48	−0.245238	−0.122619	+	0.992454i	$0.539129\pi$
$888$	1856.04	0.0701404
$889$	0	0
$890$	−42114.3	−1.58615
$891$	8849.25	0.332728
$892$	9014.35	0.338366
$893$	46938.1	1.75893
$894$	−21615.9	−0.808664
$895$	11364.9	0.424453
$896$	0	0
$897$	33902.7	1.26196
$898$	−57707.7	−2.14447
$899$	−9507.86	−0.352731
$900$	1075.56	0.0398355
$901$	−11028.9	−0.407799
$902$	−5604.85	−0.206897
$903$	0	0
$904$	−20592.3	−0.757620
$905$	20423.6	0.750171
$906$	−45728.8	−1.67686
$907$	−9356.17	−0.342521	−0.171260	−	0.985226i	$-0.554784\pi$
$907$	−9356.17	−0.342521	−0.171260	+	0.985226i	$0.554784\pi$
$908$	5903.82	0.215777
$909$	3154.70	0.115110
$910$	0	0
$911$	−16574.0	−0.602768	−0.301384	−	0.953503i	$-0.597449\pi$
$911$	−16574.0	−0.602768	−0.301384	+	0.953503i	$0.597449\pi$
$912$	50522.6	1.83440
$913$	6412.73	0.232454
$914$	−33613.6	−1.21645
$915$	−3143.70	−0.113582
$916$	−1712.76	−0.0617808
$917$	0	0
$918$	−17195.8	−0.618241
$919$	8214.92	0.294870	0.147435	−	0.989072i	$-0.452898\pi$
$919$	8214.92	0.294870	0.147435	+	0.989072i	$0.452898\pi$
$920$	−49834.8	−1.78588
$921$	−37767.7	−1.35123
$922$	−63869.8	−2.28139
$923$	−26542.7	−0.946548
$924$	0	0
$925$	2553.46	0.0907646
$926$	23274.9	0.825983
$927$	−4239.07	−0.150193
$928$	−29088.9	−1.02898
$929$	−42653.9	−1.50638	−0.753192	−	0.657801i	$-0.771487\pi$
$929$	−42653.9	−1.50638	−0.753192	+	0.657801i	$0.771487\pi$
$930$	−10383.2	−0.366107
$931$	0	0
$932$	4104.41	0.144254
$933$	−46517.2	−1.63227
$934$	−39571.7	−1.38632
$935$	7166.25	0.250654
$936$	−1987.45	−0.0694035
$937$	18484.8	0.644473	0.322237	−	0.946659i	$-0.395565\pi$
$937$	18484.8	0.644473	0.322237	+	0.946659i	$0.395565\pi$
$938$	0	0
$939$	15830.7	0.550177
$940$	−16197.3	−0.562020
$941$	7183.03	0.248842	0.124421	−	0.992230i	$-0.460293\pi$
$941$	7183.03	0.248842	0.124421	+	0.992230i	$0.460293\pi$
$942$	12517.9	0.432966
$943$	27397.4	0.946109
$944$	−39505.6	−1.36207
$945$	0	0
$946$	10252.5	0.352367
$947$	41443.3	1.42210	0.711049	−	0.703143i	$-0.248221\pi$
$947$	41443.3	1.42210	0.711049	+	0.703143i	$0.248221\pi$
$948$	−10247.8	−0.351089
$949$	13992.6	0.478630
$950$	51412.1	1.75582
$951$	53800.2	1.83448
$952$	0	0
$953$	7981.30	0.271290	0.135645	−	0.990757i	$-0.456689\pi$
$953$	7981.30	0.271290	0.135645	+	0.990757i	$0.456689\pi$
$954$	−2800.64	−0.0950461
$955$	78297.8	2.65305
$956$	−5070.06	−0.171524
$957$	15873.0	0.536157
$958$	27506.0	0.927638
$959$	0	0
$960$	23289.8	0.782993
$961$	−28481.3	−0.956038
$962$	2186.48	0.0732795
$963$	4589.65	0.153582
$964$	4018.76	0.134269
$965$	−43063.3	−1.43654
$966$	0	0
$967$	−18745.7	−0.623394	−0.311697	−	0.950182i	$-0.600897\pi$
$967$	−18745.7	−0.623394	−0.311697	+	0.950182i	$0.600897\pi$
$968$	2146.81	0.0712822
$969$	26268.1	0.870849
$970$	−50003.8	−1.65518
$971$	−3096.87	−0.102351	−0.0511757	−	0.998690i	$-0.516297\pi$
$971$	−3096.87	−0.102351	−0.0511757	+	0.998690i	$0.516297\pi$
$972$	−2239.91	−0.0739147
$973$	0	0
$974$	−8877.40	−0.292043
$975$	−26045.4	−0.855507
$976$	2768.36	0.0907922
$977$	19960.2	0.653618	0.326809	−	0.945090i	$-0.394027\pi$
$977$	19960.2	0.653618	0.326809	+	0.945090i	$0.394027\pi$
$978$	20433.1	0.668075
$979$	−8868.21	−0.289509
$980$	0	0
$981$	−2072.86	−0.0674631
$982$	7249.85	0.235593
$983$	33434.5	1.08484	0.542419	−	0.840108i	$-0.317509\pi$
$983$	33434.5	1.08484	0.542419	+	0.840108i	$0.317509\pi$
$984$	−15299.0	−0.495645
$985$	26199.1	0.847485
$986$	−34512.6	−1.11471
$987$	0	0
$988$	10587.7	0.340930
$989$	−50116.1	−1.61132
$990$	1819.76	0.0584201
$991$	24855.2	0.796722	0.398361	−	0.917229i	$-0.369579\pi$
$991$	24855.2	0.796722	0.398361	+	0.917229i	$0.369579\pi$
$992$	4006.88	0.128245
$993$	9961.63	0.318351
$994$	0	0
$995$	36280.3	1.15594
$996$	−8111.43	−0.258053
$997$	7810.38	0.248102	0.124051	−	0.992276i	$-0.460411\pi$
$997$	7810.38	0.248102	0.124051	+	0.992276i	$0.460411\pi$
$998$	58744.5	1.86325
$999$	−2493.21	−0.0789607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.4.a.g.1.2		4
7.6	odd	2		77.4.a.d.1.2	✓	4
21.20	even	2		693.4.a.l.1.3		4
28.27	even	2		1232.4.a.s.1.4		4
35.34	odd	2		1925.4.a.p.1.3		4
77.76	even	2		847.4.a.d.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.2	✓	4	7.6	odd	2
539.4.a.g.1.2		4	1.1	even	1	trivial
693.4.a.l.1.3		4	21.20	even	2
847.4.a.d.1.3		4	77.76	even	2
1232.4.a.s.1.4		4	28.27	even	2
1925.4.a.p.1.3		4	35.34	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q-3.24550 q^{2} +5.49244 q^{3} +2.53327 q^{4} +16.0955 q^{5} -17.8257 q^{6} +17.7423 q^{8} +3.16692 q^{9} -52.2379 q^{10} -11.0000 q^{11} +13.9139 q^{12} -35.3712 q^{13} +88.4036 q^{15} -77.8487 q^{16} -40.4757 q^{17} -10.2782 q^{18} -118.159 q^{19} +40.7743 q^{20} +35.7005 q^{22} -174.510 q^{23} +97.4484 q^{24} +134.065 q^{25} +114.797 q^{26} -130.902 q^{27} -262.725 q^{29} -286.914 q^{30} +36.1894 q^{31} +110.720 q^{32} -60.4169 q^{33} +131.364 q^{34} +8.02266 q^{36} +19.0464 q^{37} +383.487 q^{38} -194.274 q^{39} +285.571 q^{40} -156.996 q^{41} +287.182 q^{43} -27.8660 q^{44} +50.9731 q^{45} +566.371 q^{46} -397.244 q^{47} -427.580 q^{48} -435.108 q^{50} -222.311 q^{51} -89.6049 q^{52} +272.483 q^{53} +424.842 q^{54} -177.050 q^{55} -648.984 q^{57} +852.674 q^{58} +507.466 q^{59} +223.950 q^{60} -35.5608 q^{61} -117.453 q^{62} +263.448 q^{64} -569.317 q^{65} +196.083 q^{66} +979.229 q^{67} -102.536 q^{68} -958.484 q^{69} +750.404 q^{71} +56.1882 q^{72} -395.594 q^{73} -61.8152 q^{74} +736.344 q^{75} -299.330 q^{76} +630.517 q^{78} -736.516 q^{79} -1253.01 q^{80} -804.477 q^{81} +509.531 q^{82} -582.975 q^{83} -651.477 q^{85} -932.050 q^{86} -1443.00 q^{87} -195.165 q^{88} +806.201 q^{89} -165.433 q^{90} -442.081 q^{92} +198.768 q^{93} +1289.26 q^{94} -1901.84 q^{95} +608.123 q^{96} +957.232 q^{97} -34.8361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 14 q^{3} + 26 q^{4} - 10 q^{5} - 14 q^{6} - 18 q^{8} + 76 q^{9} + 2 q^{10} - 44 q^{11} - 70 q^{12} - 58 q^{13} + 284 q^{15} + 2 q^{16} - 4 q^{17} - 62 q^{18} - 258 q^{19} - 182 q^{20} + 22 q^{22}+ \cdots - 836 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - 14 * q^3 + 26 * q^4 - 10 * q^5 - 14 * q^6 - 18 * q^8 + 76 * q^9 + 2 * q^10 - 44 * q^11 - 70 * q^12 - 58 * q^13 + 284 * q^15 + 2 * q^16 - 4 * q^17 - 62 * q^18 - 258 * q^19 - 182 * q^20 + 22 * q^22 + 8 * q^23 + 498 * q^24 + 80 * q^25 + 482 * q^26 - 428 * q^27 - 396 * q^29 - 628 * q^30 + 56 * q^31 + 134 * q^32 + 154 * q^33 - 472 * q^34 - 418 * q^36 + 84 * q^37 + 942 * q^38 - 412 * q^39 + 1026 * q^40 - 52 * q^41 + 408 * q^43 - 286 * q^44 - 826 * q^45 + 368 * q^46 - 8 * q^47 - 982 * q^48 - 1642 * q^50 - 388 * q^51 - 2030 * q^52 + 624 * q^53 - 92 * q^54 + 110 * q^55 + 48 * q^57 + 864 * q^58 + 238 * q^59 + 1420 * q^60 + 162 * q^61 - 688 * q^62 - 902 * q^64 - 32 * q^65 + 154 * q^66 + 1340 * q^67 + 1384 * q^68 - 2416 * q^69 + 1788 * q^71 - 2622 * q^72 - 1456 * q^73 + 996 * q^74 + 806 * q^75 - 3042 * q^76 - 2632 * q^78 - 1324 * q^79 - 2342 * q^80 + 1444 * q^81 - 1984 * q^82 - 450 * q^83 - 1736 * q^85 - 4380 * q^86 - 588 * q^87 + 198 * q^88 + 3072 * q^89 + 218 * q^90 + 544 * q^92 - 1264 * q^93 + 1696 * q^94 + 24 * q^95 - 862 * q^96 + 652 * q^97 - 836 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more