LMFDB - Embedded newform 2151.4.a.d.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35922 q^{2} -2.43409 q^{4} -3.14815 q^{5} -17.8040 q^{7} +24.6163 q^{8} +O(q^{10})$	\(q-2.35922 q^{2} -2.43409 q^{4} -3.14815 q^{5} -17.8040 q^{7} +24.6163 q^{8} +7.42716 q^{10} -0.0693792 q^{11} +41.9255 q^{13} +42.0035 q^{14} -38.6024 q^{16} -23.5753 q^{17} +90.8284 q^{19} +7.66288 q^{20} +0.163681 q^{22} -139.082 q^{23} -115.089 q^{25} -98.9113 q^{26} +43.3366 q^{28} -86.6706 q^{29} +13.4422 q^{31} -105.859 q^{32} +55.6192 q^{34} +56.0496 q^{35} +199.623 q^{37} -214.284 q^{38} -77.4957 q^{40} -229.859 q^{41} +273.386 q^{43} +0.168875 q^{44} +328.126 q^{46} -419.359 q^{47} -26.0175 q^{49} +271.520 q^{50} -102.050 q^{52} +250.803 q^{53} +0.218416 q^{55} -438.269 q^{56} +204.475 q^{58} +514.343 q^{59} +754.684 q^{61} -31.7131 q^{62} +558.563 q^{64} -131.987 q^{65} -224.227 q^{67} +57.3844 q^{68} -132.233 q^{70} +343.312 q^{71} +702.326 q^{73} -470.953 q^{74} -221.085 q^{76} +1.23523 q^{77} +113.109 q^{79} +121.526 q^{80} +542.288 q^{82} +1075.52 q^{83} +74.2184 q^{85} -644.977 q^{86} -1.70786 q^{88} +898.508 q^{89} -746.441 q^{91} +338.540 q^{92} +989.359 q^{94} -285.941 q^{95} -424.675 q^{97} +61.3808 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

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 $

$2$	−2.35922	−0.834109	−0.417055	−	0.908881i	$-0.636938\pi$
$2$	−2.35922	−0.834109	−0.417055	+	0.908881i	$0.636938\pi$
$3$	0	0
$3$	0	0
$4$	−2.43409	−0.304262
$5$	−3.14815	−0.281579	−0.140789	−	0.990040i	$-0.544964\pi$
$5$	−3.14815	−0.281579	−0.140789	+	0.990040i	$0.544964\pi$
$6$	0	0
$7$	−17.8040	−0.961326	−0.480663	−	0.876905i	$-0.659604\pi$
$7$	−17.8040	−0.961326	−0.480663	+	0.876905i	$0.659604\pi$
$8$	24.6163	1.08790
$9$	0	0
$10$	7.42716	0.234867
$11$	−0.0693792	−0.00190169	−0.000950846	−	1.00000i	$-0.500303\pi$
$11$	−0.0693792	−0.00190169	−0.000950846		1.00000i	$0.500303\pi$
$12$	0	0
$13$	41.9255	0.894464	0.447232	−	0.894418i	$-0.352410\pi$
$13$	41.9255	0.894464	0.447232	+	0.894418i	$0.352410\pi$
$14$	42.0035	0.801851
$15$	0	0
$16$	−38.6024	−0.603163
$17$	−23.5753	−0.336344	−0.168172	−	0.985758i	$-0.553786\pi$
$17$	−23.5753	−0.336344	−0.168172	+	0.985758i	$0.553786\pi$
$18$	0	0
$19$	90.8284	1.09671	0.548354	−	0.836246i	$-0.315255\pi$
$19$	90.8284	1.09671	0.548354	+	0.836246i	$0.315255\pi$
$20$	7.66288	0.0856736
$21$	0	0
$22$	0.163681	0.00158622
$23$	−139.082	−1.26090	−0.630450	−	0.776230i	$-0.717130\pi$
$23$	−139.082	−1.26090	−0.630450	+	0.776230i	$0.717130\pi$
$24$	0	0
$25$	−115.089	−0.920713
$26$	−98.9113	−0.746081
$27$	0	0
$28$	43.3366	0.292495
$29$	−86.6706	−0.554977	−0.277488	−	0.960729i	$-0.589502\pi$
$29$	−86.6706	−0.554977	−0.277488	+	0.960729i	$0.589502\pi$
$30$	0	0
$31$	13.4422	0.0778805	0.0389402	−	0.999242i	$-0.487602\pi$
$31$	13.4422	0.0778805	0.0389402	+	0.999242i	$0.487602\pi$
$32$	−105.859	−0.584793
$33$	0	0
$34$	55.6192	0.280547
$35$	56.0496	0.270689
$36$	0	0
$37$	199.623	0.886966	0.443483	−	0.896283i	$-0.353743\pi$
$37$	199.623	0.886966	0.443483	+	0.896283i	$0.353743\pi$
$38$	−214.284	−0.914775
$39$	0	0
$40$	−77.4957	−0.306329
$41$	−229.859	−0.875561	−0.437780	−	0.899082i	$-0.644235\pi$
$41$	−229.859	−0.875561	−0.437780	+	0.899082i	$0.644235\pi$
$42$	0	0
$43$	273.386	0.969558	0.484779	−	0.874637i	$-0.338900\pi$
$43$	273.386	0.969558	0.484779	+	0.874637i	$0.338900\pi$
$44$	0.168875	0.000578612 0
$45$	0	0
$46$	328.126	1.05173
$47$	−419.359	−1.30148	−0.650742	−	0.759299i	$-0.725542\pi$
$47$	−419.359	−1.30148	−0.650742	+	0.759299i	$0.725542\pi$
$48$	0	0
$49$	−26.0175	−0.0758526
$50$	271.520	0.767976
$51$	0	0
$52$	−102.050	−0.272151
$53$	250.803	0.650008	0.325004	−	0.945713i	$-0.394634\pi$
$53$	250.803	0.650008	0.325004	+	0.945713i	$0.394634\pi$
$54$	0	0
$55$	0.218416	0.000535476 0
$56$	−438.269	−1.04582
$57$	0	0
$58$	204.475	0.462911
$59$	514.343	1.13494	0.567472	−	0.823393i	$-0.307921\pi$
$59$	514.343	1.13494	0.567472	+	0.823393i	$0.307921\pi$
$60$	0	0
$61$	754.684	1.58406	0.792028	−	0.610485i	$-0.209025\pi$
$61$	754.684	1.58406	0.792028	+	0.610485i	$0.209025\pi$
$62$	−31.7131	−0.0649608
$63$	0	0
$64$	558.563	1.09094
$65$	−131.987	−0.251862
$66$	0	0
$67$	−224.227	−0.408861	−0.204430	−	0.978881i	$-0.565534\pi$
$67$	−224.227	−0.408861	−0.204430	+	0.978881i	$0.565534\pi$
$68$	57.3844	0.102336
$69$	0	0
$70$	−132.233	−0.225784
$71$	343.312	0.573854	0.286927	−	0.957952i	$-0.407366\pi$
$71$	343.312	0.573854	0.286927	+	0.957952i	$0.407366\pi$
$72$	0	0
$73$	702.326	1.12604	0.563021	−	0.826443i	$-0.309639\pi$
$73$	702.326	1.12604	0.563021	+	0.826443i	$0.309639\pi$
$74$	−470.953	−0.739827
$75$	0	0
$76$	−221.085	−0.333686
$77$	1.23523	0.00182815
$78$	0	0
$79$	113.109	0.161085	0.0805426	−	0.996751i	$-0.474335\pi$
$79$	113.109	0.161085	0.0805426	+	0.996751i	$0.474335\pi$
$80$	121.526	0.169838
$81$	0	0
$82$	542.288	0.730313
$83$	1075.52	1.42233	0.711167	−	0.703023i	$-0.248167\pi$
$83$	1075.52	1.42233	0.711167	+	0.703023i	$0.248167\pi$
$84$	0	0
$85$	74.2184	0.0947072
$86$	−644.977	−0.808717
$87$	0	0
$88$	−1.70786	−0.00206884
$89$	898.508	1.07013	0.535066	−	0.844810i	$-0.320287\pi$
$89$	898.508	1.07013	0.535066	+	0.844810i	$0.320287\pi$
$90$	0	0
$91$	−746.441	−0.859871
$92$	338.540	0.383644
$93$	0	0
$94$	989.359	1.08558
$95$	−285.941	−0.308810
$96$	0	0
$97$	−424.675	−0.444528	−0.222264	−	0.974986i	$-0.571345\pi$
$97$	−424.675	−0.444528	−0.222264	+	0.974986i	$0.571345\pi$
$98$	61.3808	0.0632694
$99$	0	0
$100$	280.138	0.280138
$101$	−1195.40	−1.17769	−0.588844	−	0.808247i	$-0.700417\pi$
$101$	−1195.40	−1.17769	−0.588844	+	0.808247i	$0.700417\pi$
$102$	0	0
$103$	478.463	0.457712	0.228856	−	0.973460i	$-0.426502\pi$
$103$	478.463	0.457712	0.228856	+	0.973460i	$0.426502\pi$
$104$	1032.05	0.973085
$105$	0	0
$106$	−591.698	−0.542177
$107$	−656.735	−0.593355	−0.296678	−	0.954978i	$-0.595879\pi$
$107$	−656.735	−0.593355	−0.296678	+	0.954978i	$0.595879\pi$
$108$	0	0
$109$	−4.09105	−0.00359497	−0.00179748	−	0.999998i	$-0.500572\pi$
$109$	−4.09105	−0.00359497	−0.00179748	+	0.999998i	$0.500572\pi$
$110$	−0.515290	−0.000446645 0
$111$	0	0
$112$	687.278	0.579836
$113$	995.073	0.828395	0.414197	−	0.910187i	$-0.364062\pi$
$113$	995.073	0.828395	0.414197	+	0.910187i	$0.364062\pi$
$114$	0	0
$115$	437.852	0.355043
$116$	210.964	0.168858
$117$	0	0
$118$	−1213.45	−0.946668
$119$	419.734	0.323336
$120$	0	0
$121$	−1331.00	−0.999996
$122$	−1780.46	−1.32128
$123$	0	0
$124$	−32.7196	−0.0236960
$125$	755.836	0.540832
$126$	0	0
$127$	22.6155	0.0158016	0.00790080	−	0.999969i	$-0.497485\pi$
$127$	22.6155	0.0158016	0.00790080	+	0.999969i	$0.497485\pi$
$128$	−470.902	−0.325174
$129$	0	0
$130$	311.387	0.210080
$131$	−1804.77	−1.20369	−0.601845	−	0.798613i	$-0.705567\pi$
$131$	−1804.77	−1.20369	−0.601845	+	0.798613i	$0.705567\pi$
$132$	0	0
$133$	−1617.11	−1.05429
$134$	529.000	0.341035
$135$	0	0
$136$	−580.336	−0.365907
$137$	−80.1712	−0.0499962	−0.0249981	−	0.999687i	$-0.507958\pi$
$137$	−80.1712	−0.0499962	−0.0249981	+	0.999687i	$0.507958\pi$
$138$	0	0
$139$	2983.42	1.82050	0.910252	−	0.414054i	$-0.135887\pi$
$139$	2983.42	1.82050	0.910252	+	0.414054i	$0.135887\pi$
$140$	−136.430	−0.0823602
$141$	0	0
$142$	−809.948	−0.478657
$143$	−2.90876	−0.00170100
$144$	0	0
$145$	272.852	0.156270
$146$	−1656.94	−0.939242
$147$	0	0
$148$	−485.900	−0.269870
$149$	−3412.40	−1.87621	−0.938103	−	0.346355i	$-0.887419\pi$
$149$	−3412.40	−1.87621	−0.938103	+	0.346355i	$0.887419\pi$
$150$	0	0
$151$	1747.13	0.941586	0.470793	−	0.882244i	$-0.343968\pi$
$151$	1747.13	0.941586	0.470793	+	0.882244i	$0.343968\pi$
$152$	2235.86	1.19311
$153$	0	0
$154$	−2.91417	−0.00152487
$155$	−42.3181	−0.0219295
$156$	0	0
$157$	−624.020	−0.317212	−0.158606	−	0.987342i	$-0.550700\pi$
$157$	−624.020	−0.317212	−0.158606	+	0.987342i	$0.550700\pi$
$158$	−266.848	−0.134363
$159$	0	0
$160$	333.259	0.164665
$161$	2476.23	1.21214
$162$	0	0
$163$	−991.466	−0.476427	−0.238214	−	0.971213i	$-0.576562\pi$
$163$	−991.466	−0.476427	−0.238214	+	0.971213i	$0.576562\pi$
$164$	559.499	0.266399
$165$	0	0
$166$	−2537.39	−1.18638
$167$	−1227.79	−0.568916	−0.284458	−	0.958689i	$-0.591814\pi$
$167$	−1227.79	−0.568916	−0.284458	+	0.958689i	$0.591814\pi$
$168$	0	0
$169$	−439.255	−0.199934
$170$	−175.097	−0.0789961
$171$	0	0
$172$	−665.447	−0.294999
$173$	3612.86	1.58775	0.793875	−	0.608081i	$-0.208061\pi$
$173$	3612.86	1.58775	0.793875	+	0.608081i	$0.208061\pi$
$174$	0	0
$175$	2049.05	0.885106
$176$	2.67821	0.00114703
$177$	0	0
$178$	−2119.78	−0.892607
$179$	−797.734	−0.333103	−0.166552	−	0.986033i	$-0.553263\pi$
$179$	−797.734	−0.333103	−0.166552	+	0.986033i	$0.553263\pi$
$180$	0	0
$181$	−3.96129	−0.00162674	−0.000813371	−	1.00000i	$-0.500259\pi$
$181$	−3.96129	−0.00162674	−0.000813371		1.00000i	$0.500259\pi$
$182$	1761.02	0.717227
$183$	0	0
$184$	−3423.70	−1.37173
$185$	−628.441	−0.249751
$186$	0	0
$187$	1.63563	0.000639622 0
$188$	1020.76	0.395992
$189$	0	0
$190$	674.597	0.257581
$191$	2070.43	0.784352	0.392176	−	0.919890i	$-0.371722\pi$
$191$	2070.43	0.784352	0.392176	+	0.919890i	$0.371722\pi$
$192$	0	0
$193$	396.476	0.147870	0.0739352	−	0.997263i	$-0.476444\pi$
$193$	396.476	0.147870	0.0739352	+	0.997263i	$0.476444\pi$
$194$	1001.90	0.370785
$195$	0	0
$196$	63.3289	0.0230790
$197$	3173.16	1.14761	0.573803	−	0.818994i	$-0.305468\pi$
$197$	3173.16	1.14761	0.573803	+	0.818994i	$0.305468\pi$
$198$	0	0
$199$	−4327.70	−1.54162	−0.770811	−	0.637064i	$-0.780149\pi$
$199$	−4327.70	−1.54162	−0.770811	+	0.637064i	$0.780149\pi$
$200$	−2833.07	−1.00164
$201$	0	0
$202$	2820.20	0.982320
$203$	1543.08	0.533513
$204$	0	0
$205$	723.630	0.246539
$206$	−1128.80	−0.381782
$207$	0	0
$208$	−1618.43	−0.539508
$209$	−6.30160	−0.00208560
$210$	0	0
$211$	−4586.84	−1.49655	−0.748273	−	0.663391i	$-0.769117\pi$
$211$	−4586.84	−1.49655	−0.748273	+	0.663391i	$0.769117\pi$
$212$	−610.477	−0.197772
$213$	0	0
$214$	1549.38	0.494923
$215$	−860.659	−0.273007
$216$	0	0
$217$	−239.325	−0.0748685
$218$	9.65168	0.00299860
$219$	0	0
$220$	−0.531644	−0.000162925 0
$221$	−988.404	−0.300847
$222$	0	0
$223$	−2227.66	−0.668946	−0.334473	−	0.942405i	$-0.608558\pi$
$223$	−2227.66	−0.668946	−0.334473	+	0.942405i	$0.608558\pi$
$224$	1884.71	0.562176
$225$	0	0
$226$	−2347.59	−0.690972
$227$	−5973.51	−1.74659	−0.873294	−	0.487193i	$-0.838021\pi$
$227$	−5973.51	−1.74659	−0.873294	+	0.487193i	$0.838021\pi$
$228$	0	0
$229$	1817.66	0.524517	0.262258	−	0.964998i	$-0.415533\pi$
$229$	1817.66	0.524517	0.262258	+	0.964998i	$0.415533\pi$
$230$	−1032.99	−0.296144
$231$	0	0
$232$	−2133.51	−0.603757
$233$	−2614.21	−0.735032	−0.367516	−	0.930017i	$-0.619792\pi$
$233$	−2614.21	−0.735032	−0.367516	+	0.930017i	$0.619792\pi$
$234$	0	0
$235$	1320.20	0.366470
$236$	−1251.96	−0.345320
$237$	0	0
$238$	−990.244	−0.269697
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	4459.46	1.19195	0.595973	−	0.803004i	$-0.296766\pi$
$241$	4459.46	1.19195	0.595973	+	0.803004i	$0.296766\pi$
$242$	3140.11	0.834106
$243$	0	0
$244$	−1836.97	−0.481967
$245$	81.9067	0.0213585
$246$	0	0
$247$	3808.02	0.980966
$248$	330.898	0.0847259
$249$	0	0
$250$	−1783.18	−0.451113
$251$	−1278.31	−0.321458	−0.160729	−	0.986999i	$-0.551385\pi$
$251$	−1278.31	−0.321458	−0.160729	+	0.986999i	$0.551385\pi$
$252$	0	0
$253$	9.64943	0.00239784
$254$	−53.3549	−0.0131803
$255$	0	0
$256$	−3357.55	−0.819714
$257$	−6005.95	−1.45775	−0.728873	−	0.684649i	$-0.759956\pi$
$257$	−6005.95	−1.45775	−0.728873	+	0.684649i	$0.759956\pi$
$258$	0	0
$259$	−3554.08	−0.852663
$260$	321.270	0.0766319
$261$	0	0
$262$	4257.84	1.00401
$263$	−1992.09	−0.467063	−0.233531	−	0.972349i	$-0.575028\pi$
$263$	−1992.09	−0.467063	−0.233531	+	0.972349i	$0.575028\pi$
$264$	0	0
$265$	−789.563	−0.183028
$266$	3815.11	0.879397
$267$	0	0
$268$	545.789	0.124401
$269$	4047.31	0.917356	0.458678	−	0.888602i	$-0.348323\pi$
$269$	4047.31	0.917356	0.458678	+	0.888602i	$0.348323\pi$
$270$	0	0
$271$	6077.67	1.36233	0.681166	−	0.732129i	$-0.261473\pi$
$271$	6077.67	1.36233	0.681166	+	0.732129i	$0.261473\pi$
$272$	910.063	0.202870
$273$	0	0
$274$	189.141	0.0417023
$275$	7.98480	0.00175091
$276$	0	0
$277$	−5610.59	−1.21700	−0.608498	−	0.793555i	$-0.708228\pi$
$277$	−5610.59	−1.21700	−0.608498	+	0.793555i	$0.708228\pi$
$278$	−7038.53	−1.51850
$279$	0	0
$280$	1379.73	0.294482
$281$	281.687	0.0598008	0.0299004	−	0.999553i	$-0.490481\pi$
$281$	281.687	0.0598008	0.0299004	+	0.999553i	$0.490481\pi$
$282$	0	0
$283$	−1188.19	−0.249578	−0.124789	−	0.992183i	$-0.539825\pi$
$283$	−1188.19	−0.249578	−0.124789	+	0.992183i	$0.539825\pi$
$284$	−835.654	−0.174602
$285$	0	0
$286$	6.86239	0.00141882
$287$	4092.41	0.841699
$288$	0	0
$289$	−4357.21	−0.886873
$290$	−643.716	−0.130346
$291$	0	0
$292$	−1709.53	−0.342611
$293$	895.592	0.178570	0.0892851	−	0.996006i	$-0.471542\pi$
$293$	895.592	0.178570	0.0892851	+	0.996006i	$0.471542\pi$
$294$	0	0
$295$	−1619.23	−0.319576
$296$	4913.97	0.964927
$297$	0	0
$298$	8050.60	1.56496
$299$	−5831.10	−1.12783
$300$	0	0
$301$	−4867.37	−0.932061
$302$	−4121.86	−0.785386
$303$	0	0
$304$	−3506.20	−0.661494
$305$	−2375.85	−0.446036
$306$	0	0
$307$	1750.55	0.325437	0.162718	−	0.986673i	$-0.447974\pi$
$307$	1750.55	0.325437	0.162718	+	0.986673i	$0.447974\pi$
$308$	−3.00666	−0.000556235 0
$309$	0	0
$310$	99.8375	0.0182916
$311$	−3530.10	−0.643645	−0.321822	−	0.946800i	$-0.604295\pi$
$311$	−3530.10	−0.643645	−0.321822	+	0.946800i	$0.604295\pi$
$312$	0	0
$313$	6507.33	1.17513	0.587565	−	0.809177i	$-0.300087\pi$
$313$	6507.33	1.17513	0.587565	+	0.809177i	$0.300087\pi$
$314$	1472.20	0.264589
$315$	0	0
$316$	−275.317	−0.0490121
$317$	5674.24	1.00535	0.502677	−	0.864475i	$-0.332349\pi$
$317$	5674.24	1.00535	0.502677	+	0.864475i	$0.332349\pi$
$318$	0	0
$319$	6.01313	0.00105539
$320$	−1758.44	−0.307187
$321$	0	0
$322$	−5841.95	−1.01105
$323$	−2141.30	−0.368871
$324$	0	0
$325$	−4825.17	−0.823545
$326$	2339.08	0.397392
$327$	0	0
$328$	−5658.28	−0.952520
$329$	7466.27	1.25115
$330$	0	0
$331$	2217.45	0.368224	0.184112	−	0.982905i	$-0.441059\pi$
$331$	2217.45	0.368224	0.184112	+	0.982905i	$0.441059\pi$
$332$	−2617.92	−0.432762
$333$	0	0
$334$	2896.61	0.474538
$335$	705.899	0.115126
$336$	0	0
$337$	−8076.56	−1.30551	−0.652757	−	0.757568i	$-0.726388\pi$
$337$	−8076.56	−1.30551	−0.652757	+	0.757568i	$0.726388\pi$
$338$	1036.30	0.166767
$339$	0	0
$340$	−180.654	−0.0288158
$341$	−0.932610	−0.000148105 0
$342$	0	0
$343$	6569.99	1.03424
$344$	6729.75	1.05478
$345$	0	0
$346$	−8523.52	−1.32436
$347$	8569.10	1.32569	0.662843	−	0.748758i	$-0.269349\pi$
$347$	8569.10	1.32569	0.662843	+	0.748758i	$0.269349\pi$
$348$	0	0
$349$	930.975	0.142791	0.0713953	−	0.997448i	$-0.477255\pi$
$349$	930.975	0.142791	0.0713953	+	0.997448i	$0.477255\pi$
$350$	−4834.15	−0.738275
$351$	0	0
$352$	7.34440	0.00111210
$353$	5568.96	0.839677	0.419839	−	0.907599i	$-0.362087\pi$
$353$	5568.96	0.839677	0.419839	+	0.907599i	$0.362087\pi$
$354$	0	0
$355$	−1080.80	−0.161585
$356$	−2187.05	−0.325600
$357$	0	0
$358$	1882.03	0.277844
$359$	2166.09	0.318445	0.159222	−	0.987243i	$-0.449101\pi$
$359$	2166.09	0.318445	0.159222	+	0.987243i	$0.449101\pi$
$360$	0	0
$361$	1390.80	0.202769
$362$	9.34554	0.00135688
$363$	0	0
$364$	1816.91	0.261626
$365$	−2211.02	−0.317069
$366$	0	0
$367$	−4.05010	−0.000576059 0	−0.000288029	−	1.00000i	$-0.500092\pi$
$367$	−4.05010	−0.000576059 0	−0.000288029		1.00000i	$0.500092\pi$
$368$	5368.92	0.760529
$369$	0	0
$370$	1482.63	0.208319
$371$	−4465.29	−0.624869
$372$	0	0
$373$	−7915.86	−1.09884	−0.549421	−	0.835546i	$-0.685152\pi$
$373$	−7915.86	−1.09884	−0.549421	+	0.835546i	$0.685152\pi$
$374$	−3.85881	−0.000533515 0
$375$	0	0
$376$	−10323.1	−1.41588
$377$	−3633.70	−0.496407
$378$	0	0
$379$	−9416.77	−1.27627	−0.638136	−	0.769924i	$-0.720294\pi$
$379$	−9416.77	−1.27627	−0.638136	+	0.769924i	$0.720294\pi$
$380$	696.007	0.0939589
$381$	0	0
$382$	−4884.60	−0.654235
$383$	862.464	0.115065	0.0575325	−	0.998344i	$-0.481677\pi$
$383$	862.464	0.115065	0.0575325	+	0.998344i	$0.481677\pi$
$384$	0	0
$385$	−3.88868	−0.000514767 0
$386$	−935.374	−0.123340
$387$	0	0
$388$	1033.70	0.135253
$389$	−7995.95	−1.04219	−0.521094	−	0.853499i	$-0.674476\pi$
$389$	−7995.95	−1.04219	−0.521094	+	0.853499i	$0.674476\pi$
$390$	0	0
$391$	3278.91	0.424096
$392$	−640.453	−0.0825199
$393$	0	0
$394$	−7486.17	−0.957228
$395$	−356.083	−0.0453582
$396$	0	0
$397$	−8896.04	−1.12463	−0.562317	−	0.826922i	$-0.690090\pi$
$397$	−8896.04	−1.12463	−0.562317	+	0.826922i	$0.690090\pi$
$398$	10210.0	1.28588
$399$	0	0
$400$	4442.72	0.555341
$401$	−7688.78	−0.957505	−0.478752	−	0.877950i	$-0.658911\pi$
$401$	−7688.78	−0.957505	−0.478752	+	0.877950i	$0.658911\pi$
$402$	0	0
$403$	563.571	0.0696613
$404$	2909.71	0.358325
$405$	0	0
$406$	−3640.47	−0.445008
$407$	−13.8497	−0.00168674
$408$	0	0
$409$	−9588.34	−1.15920	−0.579600	−	0.814901i	$-0.696791\pi$
$409$	−9588.34	−1.15920	−0.579600	+	0.814901i	$0.696791\pi$
$410$	−1707.20	−0.205641
$411$	0	0
$412$	−1164.62	−0.139264
$413$	−9157.36	−1.09105
$414$	0	0
$415$	−3385.89	−0.400499
$416$	−4438.18	−0.523076
$417$	0	0
$418$	14.8668	0.00173962
$419$	5958.92	0.694779	0.347390	−	0.937721i	$-0.387068\pi$
$419$	5958.92	0.694779	0.347390	+	0.937721i	$0.387068\pi$
$420$	0	0
$421$	−14410.7	−1.66825	−0.834126	−	0.551574i	$-0.814028\pi$
$421$	−14410.7	−1.66825	−0.834126	+	0.551574i	$0.814028\pi$
$422$	10821.4	1.24828
$423$	0	0
$424$	6173.83	0.707141
$425$	2713.26	0.309676
$426$	0	0
$427$	−13436.4	−1.52279
$428$	1598.55	0.180535
$429$	0	0
$430$	2030.48	0.227718
$431$	9770.01	1.09189	0.545945	−	0.837821i	$-0.316171\pi$
$431$	9770.01	1.09189	0.545945	+	0.837821i	$0.316171\pi$
$432$	0	0
$433$	5789.43	0.642546	0.321273	−	0.946987i	$-0.395889\pi$
$433$	5789.43	0.642546	0.321273	+	0.946987i	$0.395889\pi$
$434$	564.621	0.0624485
$435$	0	0
$436$	9.95800	0.00109381
$437$	−12632.6	−1.38284
$438$	0	0
$439$	−13897.0	−1.51086	−0.755431	−	0.655229i	$-0.772572\pi$
$439$	−13897.0	−1.51086	−0.755431	+	0.655229i	$0.772572\pi$
$440$	5.37659	0.000582543 0
$441$	0	0
$442$	2331.86	0.250940
$443$	−10394.0	−1.11475	−0.557375	−	0.830261i	$-0.688192\pi$
$443$	−10394.0	−1.11475	−0.557375	+	0.830261i	$0.688192\pi$
$444$	0	0
$445$	−2828.63	−0.301326
$446$	5255.53	0.557974
$447$	0	0
$448$	−9944.67	−1.04875
$449$	−454.270	−0.0477469	−0.0238734	−	0.999715i	$-0.507600\pi$
$449$	−454.270	−0.0477469	−0.0238734	+	0.999715i	$0.507600\pi$
$450$	0	0
$451$	15.9475	0.00166505
$452$	−2422.10	−0.252049
$453$	0	0
$454$	14092.8	1.45685
$455$	2349.91	0.242121
$456$	0	0
$457$	7024.53	0.719023	0.359511	−	0.933141i	$-0.382943\pi$
$457$	7024.53	0.719023	0.359511	+	0.933141i	$0.382943\pi$
$458$	−4288.25	−0.437504
$459$	0	0
$460$	−1065.77	−0.108026
$461$	−16985.0	−1.71599	−0.857994	−	0.513660i	$-0.828289\pi$
$461$	−16985.0	−1.71599	−0.857994	+	0.513660i	$0.828289\pi$
$462$	0	0
$463$	18933.0	1.90041	0.950206	−	0.311623i	$-0.100873\pi$
$463$	18933.0	1.90041	0.950206	+	0.311623i	$0.100873\pi$
$464$	3345.70	0.334741
$465$	0	0
$466$	6167.49	0.613097
$467$	−13691.0	−1.35663	−0.678314	−	0.734773i	$-0.737289\pi$
$467$	−13691.0	−1.35663	−0.678314	+	0.734773i	$0.737289\pi$
$468$	0	0
$469$	3992.14	0.393049
$470$	−3114.65	−0.305676
$471$	0	0
$472$	12661.2	1.23470
$473$	−18.9673	−0.00184380
$474$	0	0
$475$	−10453.4	−1.00975
$476$	−1021.67	−0.0983787
$477$	0	0
$478$	563.853	0.0539540
$479$	−918.211	−0.0875869	−0.0437935	−	0.999041i	$-0.513944\pi$
$479$	−918.211	−0.0875869	−0.0437935	+	0.999041i	$0.513944\pi$
$480$	0	0
$481$	8369.27	0.793359
$482$	−10520.8	−0.994213
$483$	0	0
$484$	3239.77	0.304261
$485$	1336.94	0.125170
$486$	0	0
$487$	−8356.42	−0.777548	−0.388774	−	0.921333i	$-0.627101\pi$
$487$	−8356.42	−0.777548	−0.388774	+	0.921333i	$0.627101\pi$
$488$	18577.5	1.72329
$489$	0	0
$490$	−193.236	−0.0178153
$491$	415.591	0.0381983	0.0190991	−	0.999818i	$-0.493920\pi$
$491$	415.591	0.0381983	0.0190991	+	0.999818i	$0.493920\pi$
$492$	0	0
$493$	2043.28	0.186663
$494$	−8983.95	−0.818233
$495$	0	0
$496$	−518.903	−0.0469746
$497$	−6112.33	−0.551661
$498$	0	0
$499$	−2197.11	−0.197106	−0.0985531	−	0.995132i	$-0.531421\pi$
$499$	−2197.11	−0.197106	−0.0985531	+	0.995132i	$0.531421\pi$
$500$	−1839.77	−0.164554
$501$	0	0
$502$	3015.80	0.268131
$503$	−10193.1	−0.903559	−0.451779	−	0.892130i	$-0.649211\pi$
$503$	−10193.1	−0.903559	−0.451779	+	0.892130i	$0.649211\pi$
$504$	0	0
$505$	3763.28	0.331612
$506$	−22.7651	−0.00200006
$507$	0	0
$508$	−55.0483	−0.00480782
$509$	−4235.60	−0.368841	−0.184420	−	0.982847i	$-0.559041\pi$
$509$	−4235.60	−0.368841	−0.184420	+	0.982847i	$0.559041\pi$
$510$	0	0
$511$	−12504.2	−1.08249
$512$	11688.4	1.00890
$513$	0	0
$514$	14169.3	1.21592
$515$	−1506.27	−0.128882
$516$	0	0
$517$	29.0948	0.00247502
$518$	8384.85	0.711214
$519$	0	0
$520$	−3249.04	−0.274000
$521$	6296.32	0.529456	0.264728	−	0.964323i	$-0.414718\pi$
$521$	6296.32	0.529456	0.264728	+	0.964323i	$0.414718\pi$
$522$	0	0
$523$	4338.21	0.362709	0.181354	−	0.983418i	$-0.441952\pi$
$523$	4338.21	0.362709	0.181354	+	0.983418i	$0.441952\pi$
$524$	4392.97	0.366236
$525$	0	0
$526$	4699.77	0.389581
$527$	−316.904	−0.0261946
$528$	0	0
$529$	7176.94	0.589869
$530$	1862.75	0.152666
$531$	0	0
$532$	3936.19	0.320781
$533$	−9636.96	−0.783158
$534$	0	0
$535$	2067.50	0.167076
$536$	−5519.64	−0.444798
$537$	0	0
$538$	−9548.48	−0.765175
$539$	1.80507	0.000144248 0
$540$	0	0
$541$	−5502.27	−0.437267	−0.218633	−	0.975807i	$-0.570160\pi$
$541$	−5502.27	−0.437267	−0.218633	+	0.975807i	$0.570160\pi$
$542$	−14338.5	−1.13633
$543$	0	0
$544$	2495.65	0.196691
$545$	12.8792	0.00101227
$546$	0	0
$547$	−16300.2	−1.27412	−0.637061	−	0.770813i	$-0.719850\pi$
$547$	−16300.2	−1.27412	−0.637061	+	0.770813i	$0.719850\pi$
$548$	195.144	0.0152119
$549$	0	0
$550$	−18.8379	−0.00146045
$551$	−7872.15	−0.608647
$552$	0	0
$553$	−2013.79	−0.154855
$554$	13236.6	1.01511
$555$	0	0
$556$	−7261.92	−0.553910
$557$	−16238.9	−1.23530	−0.617652	−	0.786451i	$-0.711916\pi$
$557$	−16238.9	−1.23530	−0.617652	+	0.786451i	$0.711916\pi$
$558$	0	0
$559$	11461.8	0.867235
$560$	−2163.65	−0.163270
$561$	0	0
$562$	−664.560	−0.0498804
$563$	−2152.09	−0.161101	−0.0805504	−	0.996751i	$-0.525668\pi$
$563$	−2152.09	−0.161101	−0.0805504	+	0.996751i	$0.525668\pi$
$564$	0	0
$565$	−3132.63	−0.233258
$566$	2803.20	0.208175
$567$	0	0
$568$	8451.08	0.624294
$569$	−10581.0	−0.779574	−0.389787	−	0.920905i	$-0.627451\pi$
$569$	−10581.0	−0.779574	−0.389787	+	0.920905i	$0.627451\pi$
$570$	0	0
$571$	−8378.07	−0.614030	−0.307015	−	0.951705i	$-0.599330\pi$
$571$	−8378.07	−0.614030	−0.307015	+	0.951705i	$0.599330\pi$
$572$	7.08018	0.000517548 0
$573$	0	0
$574$	−9654.90	−0.702069
$575$	16006.9	1.16093
$576$	0	0
$577$	−14133.1	−1.01970	−0.509852	−	0.860262i	$-0.670300\pi$
$577$	−14133.1	−1.01970	−0.509852	+	0.860262i	$0.670300\pi$
$578$	10279.6	0.739749
$579$	0	0
$580$	−664.146	−0.0475468
$581$	−19148.6	−1.36733
$582$	0	0
$583$	−17.4005	−0.00123611
$584$	17288.7	1.22502
$585$	0	0
$586$	−2112.90	−0.148947
$587$	−11159.4	−0.784660	−0.392330	−	0.919824i	$-0.628331\pi$
$587$	−11159.4	−0.784660	−0.392330	+	0.919824i	$0.628331\pi$
$588$	0	0
$589$	1220.94	0.0854121
$590$	3820.11	0.266561
$591$	0	0
$592$	−7705.92	−0.534985
$593$	−28640.3	−1.98333	−0.991665	−	0.128840i	$-0.958875\pi$
$593$	−28640.3	−1.98333	−0.991665	+	0.128840i	$0.958875\pi$
$594$	0	0
$595$	−1321.38	−0.0910445
$596$	8306.10	0.570858
$597$	0	0
$598$	13756.8	0.940733
$599$	17337.3	1.18261	0.591304	−	0.806449i	$-0.298614\pi$
$599$	17337.3	1.18261	0.591304	+	0.806449i	$0.298614\pi$
$600$	0	0
$601$	7312.66	0.496322	0.248161	−	0.968719i	$-0.420174\pi$
$601$	7312.66	0.496322	0.248161	+	0.968719i	$0.420174\pi$
$602$	11483.2	0.777441
$603$	0	0
$604$	−4252.68	−0.286489
$605$	4190.17	0.281578
$606$	0	0
$607$	2900.41	0.193944	0.0969719	−	0.995287i	$-0.469084\pi$
$607$	2900.41	0.193944	0.0969719	+	0.995287i	$0.469084\pi$
$608$	−9614.98	−0.641347
$609$	0	0
$610$	5605.16	0.372043
$611$	−17581.8	−1.16413
$612$	0	0
$613$	−21518.0	−1.41779	−0.708894	−	0.705315i	$-0.750806\pi$
$613$	−21518.0	−1.41779	−0.708894	+	0.705315i	$0.750806\pi$
$614$	−4129.92	−0.271450
$615$	0	0
$616$	30.4067	0.00198883
$617$	20621.1	1.34550	0.672750	−	0.739870i	$-0.265113\pi$
$617$	20621.1	1.34550	0.672750	+	0.739870i	$0.265113\pi$
$618$	0	0
$619$	23948.1	1.55502	0.777508	−	0.628873i	$-0.216483\pi$
$619$	23948.1	1.55502	0.777508	+	0.628873i	$0.216483\pi$
$620$	103.006	0.00667230
$621$	0	0
$622$	8328.27	0.536870
$623$	−15997.0	−1.02874
$624$	0	0
$625$	12006.7	0.768427
$626$	−15352.2	−0.980187
$627$	0	0
$628$	1518.92	0.0965153
$629$	−4706.15	−0.298325
$630$	0	0
$631$	16867.8	1.06418	0.532088	−	0.846689i	$-0.321408\pi$
$631$	16867.8	1.06418	0.532088	+	0.846689i	$0.321408\pi$
$632$	2784.32	0.175244
$633$	0	0
$634$	−13386.8	−0.838574
$635$	−71.1969	−0.00444939
$636$	0	0
$637$	−1090.79	−0.0678475
$638$	−14.1863	−0.000880315 0
$639$	0	0
$640$	1482.47	0.0915621
$641$	−17379.4	−1.07089	−0.535447	−	0.844569i	$-0.679857\pi$
$641$	−17379.4	−1.07089	−0.535447	+	0.844569i	$0.679857\pi$
$642$	0	0
$643$	6455.40	0.395919	0.197960	−	0.980210i	$-0.436569\pi$
$643$	6455.40	0.395919	0.197960	+	0.980210i	$0.436569\pi$
$644$	−6027.36	−0.368806
$645$	0	0
$646$	5051.80	0.307679
$647$	−6400.31	−0.388906	−0.194453	−	0.980912i	$-0.562293\pi$
$647$	−6400.31	−0.388906	−0.194453	+	0.980912i	$0.562293\pi$
$648$	0	0
$649$	−35.6847	−0.00215832
$650$	11383.6	0.686927
$651$	0	0
$652$	2413.32	0.144959
$653$	−20646.4	−1.23730	−0.618650	−	0.785667i	$-0.712320\pi$
$653$	−20646.4	−1.23730	−0.618650	+	0.785667i	$0.712320\pi$
$654$	0	0
$655$	5681.67	0.338933
$656$	8873.13	0.528106
$657$	0	0
$658$	−17614.5	−1.04360
$659$	−23881.4	−1.41167	−0.705833	−	0.708378i	$-0.749427\pi$
$659$	−23881.4	−1.41167	−0.705833	+	0.708378i	$0.749427\pi$
$660$	0	0
$661$	−24661.3	−1.45116	−0.725578	−	0.688139i	$-0.758428\pi$
$661$	−24661.3	−1.45116	−0.725578	+	0.688139i	$0.758428\pi$
$662$	−5231.46	−0.307139
$663$	0	0
$664$	26475.3	1.54735
$665$	5090.89	0.296867
$666$	0	0
$667$	12054.4	0.699770
$668$	2988.55	0.173099
$669$	0	0
$670$	−1665.37	−0.0960281
$671$	−52.3594	−0.00301239
$672$	0	0
$673$	−2554.75	−0.146327	−0.0731637	−	0.997320i	$-0.523310\pi$
$673$	−2554.75	−0.146327	−0.0731637	+	0.997320i	$0.523310\pi$
$674$	19054.4	1.08894
$675$	0	0
$676$	1069.19	0.0608322
$677$	11300.3	0.641513	0.320757	−	0.947162i	$-0.396063\pi$
$677$	11300.3	0.641513	0.320757	+	0.947162i	$0.396063\pi$
$678$	0	0
$679$	7560.92	0.427337
$680$	1826.98	0.103032
$681$	0	0
$682$	2.20023	0.000123535 0
$683$	−17887.6	−1.00212	−0.501061	−	0.865412i	$-0.667057\pi$
$683$	−17887.6	−1.00212	−0.501061	+	0.865412i	$0.667057\pi$
$684$	0	0
$685$	252.390	0.0140779
$686$	−15500.0	−0.862673
$687$	0	0
$688$	−10553.4	−0.584802
$689$	10515.0	0.581408
$690$	0	0
$691$	19863.8	1.09357	0.546784	−	0.837274i	$-0.315852\pi$
$691$	19863.8	1.09357	0.546784	+	0.837274i	$0.315852\pi$
$692$	−8794.04	−0.483091
$693$	0	0
$694$	−20216.4	−1.10577
$695$	−9392.23	−0.512615
$696$	0	0
$697$	5418.99	0.294489
$698$	−2196.37	−0.119103
$699$	0	0
$700$	−4987.57	−0.269304
$701$	14583.4	0.785747	0.392874	−	0.919592i	$-0.371481\pi$
$701$	14583.4	0.785747	0.392874	+	0.919592i	$0.371481\pi$
$702$	0	0
$703$	18131.4	0.972743
$704$	−38.7527	−0.00207464
$705$	0	0
$706$	−13138.4	−0.700383
$707$	21282.9	1.13214
$708$	0	0
$709$	−29646.3	−1.57037	−0.785184	−	0.619262i	$-0.787432\pi$
$709$	−29646.3	−1.57037	−0.785184	+	0.619262i	$0.787432\pi$
$710$	2549.83	0.134780
$711$	0	0
$712$	22117.9	1.16419
$713$	−1869.58	−0.0981995
$714$	0	0
$715$	9.15718	0.000478964 0
$716$	1941.76	0.101351
$717$	0	0
$718$	−5110.27	−0.265618
$719$	4648.26	0.241100	0.120550	−	0.992707i	$-0.461534\pi$
$719$	4648.26	0.241100	0.120550	+	0.992707i	$0.461534\pi$
$720$	0	0
$721$	−8518.55	−0.440010
$722$	−3281.19	−0.169132
$723$	0	0
$724$	9.64215	0.000494955 0
$725$	9974.84	0.510974
$726$	0	0
$727$	10738.9	0.547845	0.273923	−	0.961752i	$-0.411679\pi$
$727$	10738.9	0.547845	0.273923	+	0.961752i	$0.411679\pi$
$728$	−18374.6	−0.935451
$729$	0	0
$730$	5216.29	0.264470
$731$	−6445.15	−0.326105
$732$	0	0
$733$	7624.22	0.384184	0.192092	−	0.981377i	$-0.438473\pi$
$733$	7624.22	0.384184	0.192092	+	0.981377i	$0.438473\pi$
$734$	9.55507	0.000480496 0
$735$	0	0
$736$	14723.1	0.737365
$737$	15.5567	0.000777528 0
$738$	0	0
$739$	10290.0	0.512208	0.256104	−	0.966649i	$-0.417561\pi$
$739$	10290.0	0.512208	0.256104	+	0.966649i	$0.417561\pi$
$740$	1529.68	0.0759896
$741$	0	0
$742$	10534.6	0.521209
$743$	553.505	0.0273299	0.0136650	−	0.999907i	$-0.495650\pi$
$743$	553.505	0.0273299	0.0136650	+	0.999907i	$0.495650\pi$
$744$	0	0
$745$	10742.7	0.528300
$746$	18675.2	0.916554
$747$	0	0
$748$	−3.98128	−0.000194612 0
$749$	11692.5	0.570408
$750$	0	0
$751$	11234.0	0.545851	0.272926	−	0.962035i	$-0.412009\pi$
$751$	11234.0	0.545851	0.272926	+	0.962035i	$0.412009\pi$
$752$	16188.3	0.785008
$753$	0	0
$754$	8572.70	0.414057
$755$	−5500.22	−0.265131
$756$	0	0
$757$	28525.2	1.36957	0.684785	−	0.728745i	$-0.259896\pi$
$757$	28525.2	1.36957	0.684785	+	0.728745i	$0.259896\pi$
$758$	22216.2	1.06455
$759$	0	0
$760$	−7038.81	−0.335953
$761$	−16365.7	−0.779574	−0.389787	−	0.920905i	$-0.627451\pi$
$761$	−16365.7	−0.779574	−0.389787	+	0.920905i	$0.627451\pi$
$762$	0	0
$763$	72.8371	0.00345594
$764$	−5039.62	−0.238648
$765$	0	0
$766$	−2034.74	−0.0959767
$767$	21564.1	1.01517
$768$	0	0
$769$	19340.0	0.906915	0.453457	−	0.891278i	$-0.350190\pi$
$769$	19340.0	0.906915	0.453457	+	0.891278i	$0.350190\pi$
$770$	9.17423	0.000429372 0
$771$	0	0
$772$	−965.060	−0.0449913
$773$	27751.1	1.29125	0.645626	−	0.763654i	$-0.276597\pi$
$773$	27751.1	1.29125	0.645626	+	0.763654i	$0.276597\pi$
$774$	0	0
$775$	−1547.05	−0.0717056
$776$	−10453.9	−0.483601
$777$	0	0
$778$	18864.2	0.869298
$779$	−20877.7	−0.960235
$780$	0	0
$781$	−23.8187	−0.00109129
$782$	−7735.65	−0.353742
$783$	0	0
$784$	1004.34	0.0457515
$785$	1964.51	0.0893200
$786$	0	0
$787$	−24428.0	−1.10643	−0.553217	−	0.833037i	$-0.686600\pi$
$787$	−24428.0	−1.10643	−0.553217	+	0.833037i	$0.686600\pi$
$788$	−7723.76	−0.349172
$789$	0	0
$790$	840.077	0.0378337
$791$	−17716.3	−0.796357
$792$	0	0
$793$	31640.5	1.41688
$794$	20987.7	0.938068
$795$	0	0
$796$	10534.0	0.469056
$797$	16028.5	0.712371	0.356186	−	0.934415i	$-0.384077\pi$
$797$	16028.5	0.712371	0.356186	+	0.934415i	$0.384077\pi$
$798$	0	0
$799$	9886.50	0.437746
$800$	12183.2	0.538427
$801$	0	0
$802$	18139.5	0.798663
$803$	−48.7268	−0.00214138
$804$	0	0
$805$	−7795.52	−0.341312
$806$	−1329.59	−0.0581051
$807$	0	0
$808$	−29426.2	−1.28120
$809$	−17069.3	−0.741812	−0.370906	−	0.928670i	$-0.620953\pi$
$809$	−17069.3	−0.741812	−0.370906	+	0.928670i	$0.620953\pi$
$810$	0	0
$811$	9953.05	0.430948	0.215474	−	0.976510i	$-0.430870\pi$
$811$	9953.05	0.430948	0.215474	+	0.976510i	$0.430870\pi$
$812$	−3756.01	−0.162328
$813$	0	0
$814$	32.6743	0.00140692
$815$	3121.28	0.134152
$816$	0	0
$817$	24831.2	1.06332
$818$	22621.0	0.966900
$819$	0	0
$820$	−1761.38	−0.0750124
$821$	11075.2	0.470801	0.235400	−	0.971898i	$-0.424360\pi$
$821$	11075.2	0.470801	0.235400	+	0.971898i	$0.424360\pi$
$822$	0	0
$823$	22390.1	0.948322	0.474161	−	0.880438i	$-0.342751\pi$
$823$	22390.1	0.948322	0.474161	+	0.880438i	$0.342751\pi$
$824$	11778.0	0.497943
$825$	0	0
$826$	21604.2	0.910056
$827$	34281.0	1.44144	0.720718	−	0.693228i	$-0.243812\pi$
$827$	34281.0	1.44144	0.720718	+	0.693228i	$0.243812\pi$
$828$	0	0
$829$	9391.91	0.393480	0.196740	−	0.980456i	$-0.436965\pi$
$829$	9391.91	0.393480	0.196740	+	0.980456i	$0.436965\pi$
$830$	7988.06	0.334060
$831$	0	0
$832$	23418.0	0.975810
$833$	613.368	0.0255126
$834$	0	0
$835$	3865.25	0.160195
$836$	15.3387	0.000634569 0
$837$	0	0
$838$	−14058.4	−0.579522
$839$	30771.9	1.26623	0.633113	−	0.774060i	$-0.281777\pi$
$839$	30771.9	1.26623	0.633113	+	0.774060i	$0.281777\pi$
$840$	0	0
$841$	−16877.2	−0.692001
$842$	33998.0	1.39150
$843$	0	0
$844$	11164.8	0.455341
$845$	1382.84	0.0562971
$846$	0	0
$847$	23697.0	0.961322
$848$	−9681.60	−0.392061
$849$	0	0
$850$	−6401.17	−0.258304
$851$	−27764.0	−1.11838
$852$	0	0
$853$	−4209.79	−0.168981	−0.0844903	−	0.996424i	$-0.526926\pi$
$853$	−4209.79	−0.168981	−0.0844903	+	0.996424i	$0.526926\pi$
$854$	31699.4	1.27018
$855$	0	0
$856$	−16166.4	−0.645509
$857$	−7515.31	−0.299555	−0.149777	−	0.988720i	$-0.547856\pi$
$857$	−7515.31	−0.299555	−0.149777	+	0.988720i	$0.547856\pi$
$858$	0	0
$859$	1188.77	0.0472181	0.0236090	−	0.999721i	$-0.492484\pi$
$859$	1188.77	0.0472181	0.0236090	+	0.999721i	$0.492484\pi$
$860$	2094.92	0.0830655
$861$	0	0
$862$	−23049.6	−0.910756
$863$	24795.0	0.978022	0.489011	−	0.872278i	$-0.337358\pi$
$863$	24795.0	0.978022	0.489011	+	0.872278i	$0.337358\pi$
$864$	0	0
$865$	−11373.8	−0.447076
$866$	−13658.5	−0.535953
$867$	0	0
$868$	582.540	0.0227796
$869$	−7.84740	−0.000306335 0
$870$	0	0
$871$	−9400.82	−0.365711
$872$	−100.707	−0.00391096
$873$	0	0
$874$	29803.1	1.15344
$875$	−13456.9	−0.519916
$876$	0	0
$877$	−43660.6	−1.68109	−0.840544	−	0.541744i	$-0.817764\pi$
$877$	−43660.6	−1.68109	−0.840544	+	0.541744i	$0.817764\pi$
$878$	32786.1	1.26022
$879$	0	0
$880$	−8.43138	−0.000322979 0
$881$	−12449.8	−0.476099	−0.238050	−	0.971253i	$-0.576508\pi$
$881$	−12449.8	−0.476099	−0.238050	+	0.971253i	$0.576508\pi$
$882$	0	0
$883$	−33168.9	−1.26413	−0.632063	−	0.774917i	$-0.717792\pi$
$883$	−33168.9	−1.26413	−0.632063	+	0.774917i	$0.717792\pi$
$884$	2405.87	0.0915363
$885$	0	0
$886$	24521.7	0.929824
$887$	−13214.6	−0.500229	−0.250114	−	0.968216i	$-0.580468\pi$
$887$	−13214.6	−0.500229	−0.250114	+	0.968216i	$0.580468\pi$
$888$	0	0
$889$	−402.647	−0.0151905
$890$	6673.36	0.251339
$891$	0	0
$892$	5422.32	0.203535
$893$	−38089.7	−1.42735
$894$	0	0
$895$	2511.38	0.0937947
$896$	8383.95	0.312598
$897$	0	0
$898$	1071.72	0.0398261
$899$	−1165.04	−0.0432218
$900$	0	0
$901$	−5912.74	−0.218626
$902$	−37.6235	−0.00138883
$903$	0	0
$904$	24495.0	0.901208
$905$	12.4707	0.000458056 0
$906$	0	0
$907$	−15804.3	−0.578579	−0.289290	−	0.957242i	$-0.593419\pi$
$907$	−15804.3	−0.578579	−0.289290	+	0.957242i	$0.593419\pi$
$908$	14540.1	0.531420
$909$	0	0
$910$	−5543.94	−0.201956
$911$	−36311.2	−1.32058	−0.660288	−	0.751013i	$-0.729566\pi$
$911$	−36311.2	−1.32058	−0.660288	+	0.751013i	$0.729566\pi$
$912$	0	0
$913$	−74.6187	−0.00270484
$914$	−16572.4	−0.599744
$915$	0	0
$916$	−4424.35	−0.159590
$917$	32132.1	1.15714
$918$	0	0
$919$	−22741.4	−0.816290	−0.408145	−	0.912917i	$-0.633824\pi$
$919$	−22741.4	−0.816290	−0.408145	+	0.912917i	$0.633824\pi$
$920$	10778.3	0.386250
$921$	0	0
$922$	40071.3	1.43132
$923$	14393.5	0.513292
$924$	0	0
$925$	−22974.4	−0.816642
$926$	−44667.0	−1.58515
$927$	0	0
$928$	9174.84	0.324546
$929$	−20879.1	−0.737376	−0.368688	−	0.929553i	$-0.620193\pi$
$929$	−20879.1	−0.737376	−0.368688	+	0.929553i	$0.620193\pi$
$930$	0	0
$931$	−2363.12	−0.0831882
$932$	6363.23	0.223642
$933$	0	0
$934$	32300.1	1.13158
$935$	−5.14921	−0.000180104 0
$936$	0	0
$937$	−13891.3	−0.484322	−0.242161	−	0.970236i	$-0.577856\pi$
$937$	−13891.3	−0.484322	−0.242161	+	0.970236i	$0.577856\pi$
$938$	−9418.32	−0.327845
$939$	0	0
$940$	−3213.50	−0.111503
$941$	36347.4	1.25918	0.629592	−	0.776926i	$-0.283222\pi$
$941$	36347.4	1.25918	0.629592	+	0.776926i	$0.283222\pi$
$942$	0	0
$943$	31969.4	1.10399
$944$	−19854.9	−0.684557
$945$	0	0
$946$	44.7480	0.00153793
$947$	−14658.2	−0.502986	−0.251493	−	0.967859i	$-0.580922\pi$
$947$	−14658.2	−0.502986	−0.251493	+	0.967859i	$0.580922\pi$
$948$	0	0
$949$	29445.3	1.00720
$950$	24661.8	0.842245
$951$	0	0
$952$	10332.3	0.351756
$953$	42626.0	1.44889	0.724444	−	0.689334i	$-0.242097\pi$
$953$	42626.0	1.44889	0.724444	+	0.689334i	$0.242097\pi$
$954$	0	0
$955$	−6518.02	−0.220857
$956$	581.748	0.0196810
$957$	0	0
$958$	2166.26	0.0730571
$959$	1427.37	0.0480627
$960$	0	0
$961$	−29610.3	−0.993935
$962$	−19744.9	−0.661748
$963$	0	0
$964$	−10854.7	−0.362664
$965$	−1248.17	−0.0416372
$966$	0	0
$967$	−43046.7	−1.43153	−0.715764	−	0.698342i	$-0.753921\pi$
$967$	−43046.7	−1.43153	−0.715764	+	0.698342i	$0.753921\pi$
$968$	−32764.2	−1.08789
$969$	0	0
$970$	−3154.13	−0.104405
$971$	−7977.53	−0.263657	−0.131829	−	0.991273i	$-0.542085\pi$
$971$	−7977.53	−0.263657	−0.131829	+	0.991273i	$0.542085\pi$
$972$	0	0
$973$	−53116.8	−1.75010
$974$	19714.6	0.648560
$975$	0	0
$976$	−29132.7	−0.955444
$977$	−3315.27	−0.108562	−0.0542809	−	0.998526i	$-0.517287\pi$
$977$	−3315.27	−0.108562	−0.0542809	+	0.998526i	$0.517287\pi$
$978$	0	0
$979$	−62.3378	−0.00203506
$980$	−199.369	−0.00649857
$981$	0	0
$982$	−980.469	−0.0318615
$983$	3632.65	0.117867	0.0589336	−	0.998262i	$-0.481230\pi$
$983$	3632.65	0.117867	0.0589336	+	0.998262i	$0.481230\pi$
$984$	0	0
$985$	−9989.56	−0.323141
$986$	−4820.55	−0.155697
$987$	0	0
$988$	−9269.08	−0.298470
$989$	−38023.2	−1.22252
$990$	0	0
$991$	−38959.4	−1.24883	−0.624413	−	0.781094i	$-0.714662\pi$
$991$	−38959.4	−1.24883	−0.624413	+	0.781094i	$0.714662\pi$
$992$	−1422.98	−0.0455439
$993$	0	0
$994$	14420.3	0.460146
$995$	13624.2	0.434088
$996$	0	0
$997$	14345.4	0.455690	0.227845	−	0.973697i	$-0.426832\pi$
$997$	14345.4	0.455690	0.227845	+	0.973697i	$0.426832\pi$
$998$	5183.45	0.164408
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.d.1.11		32
3.2	odd	2		717.4.a.d.1.22	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.d.1.22	✓	32	3.2	odd	2
2151.4.a.d.1.11		32	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35922 q^{2} -2.43409 q^{4} -3.14815 q^{5} -17.8040 q^{7} +24.6163 q^{8} +O(q^{10})\)	\(q-2.35922 q^{2} -2.43409 q^{4} -3.14815 q^{5} -17.8040 q^{7} +24.6163 q^{8} +7.42716 q^{10} -0.0693792 q^{11} +41.9255 q^{13} +42.0035 q^{14} -38.6024 q^{16} -23.5753 q^{17} +90.8284 q^{19} +7.66288 q^{20} +0.163681 q^{22} -139.082 q^{23} -115.089 q^{25} -98.9113 q^{26} +43.3366 q^{28} -86.6706 q^{29} +13.4422 q^{31} -105.859 q^{32} +55.6192 q^{34} +56.0496 q^{35} +199.623 q^{37} -214.284 q^{38} -77.4957 q^{40} -229.859 q^{41} +273.386 q^{43} +0.168875 q^{44} +328.126 q^{46} -419.359 q^{47} -26.0175 q^{49} +271.520 q^{50} -102.050 q^{52} +250.803 q^{53} +0.218416 q^{55} -438.269 q^{56} +204.475 q^{58} +514.343 q^{59} +754.684 q^{61} -31.7131 q^{62} +558.563 q^{64} -131.987 q^{65} -224.227 q^{67} +57.3844 q^{68} -132.233 q^{70} +343.312 q^{71} +702.326 q^{73} -470.953 q^{74} -221.085 q^{76} +1.23523 q^{77} +113.109 q^{79} +121.526 q^{80} +542.288 q^{82} +1075.52 q^{83} +74.2184 q^{85} -644.977 q^{86} -1.70786 q^{88} +898.508 q^{89} -746.441 q^{91} +338.540 q^{92} +989.359 q^{94} -285.941 q^{95} -424.675 q^{97} +61.3808 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

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