LMFDB - Embedded newform 117.4.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(1,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, 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("117.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.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:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.73549$ of defining polynomial
Character	$\chi$	$=$	117.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.73549 q^{2} +5.95388 q^{4} +3.90776 q^{5} +36.4129 q^{7} -7.64325 q^{8} +O(q^{10})$	\(q+3.73549 q^{2} +5.95388 q^{4} +3.90776 q^{5} +36.4129 q^{7} -7.64325 q^{8} +14.5974 q^{10} -19.1943 q^{11} +13.0000 q^{13} +136.020 q^{14} -76.1823 q^{16} +83.8839 q^{17} +46.8492 q^{19} +23.2664 q^{20} -71.7000 q^{22} -103.905 q^{23} -109.729 q^{25} +48.5614 q^{26} +216.798 q^{28} -108.341 q^{29} -147.532 q^{31} -223.432 q^{32} +313.347 q^{34} +142.293 q^{35} -160.012 q^{37} +175.005 q^{38} -29.8680 q^{40} -231.490 q^{41} -340.314 q^{43} -114.280 q^{44} -388.135 q^{46} -119.653 q^{47} +982.902 q^{49} -409.893 q^{50} +77.4005 q^{52} +732.879 q^{53} -75.0067 q^{55} -278.313 q^{56} -404.706 q^{58} +229.782 q^{59} +108.943 q^{61} -551.104 q^{62} -225.170 q^{64} +50.8009 q^{65} +10.3955 q^{67} +499.435 q^{68} +531.535 q^{70} +869.201 q^{71} -1099.07 q^{73} -597.724 q^{74} +278.934 q^{76} -698.920 q^{77} +140.410 q^{79} -297.703 q^{80} -864.729 q^{82} +159.474 q^{83} +327.799 q^{85} -1271.24 q^{86} +146.707 q^{88} -1067.93 q^{89} +473.368 q^{91} -618.636 q^{92} -446.964 q^{94} +183.075 q^{95} +858.881 q^{97} +3671.62 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 30 q^{7} + 6 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 + 10 * q^4 - 4 * q^5 + 30 * q^7 + 6 * q^8	$ 3 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 30 q^{7} + 6 q^{8} - 4 q^{10} + 16 q^{11} + 39 q^{13} + 176 q^{14} - 110 q^{16} + 146 q^{17} + 94 q^{19} + 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 26 q^{26} + 80 q^{28} + 2 q^{29} + 302 q^{31} - 154 q^{32} + 164 q^{34} - 80 q^{35} + 374 q^{37} - 312 q^{38} - 516 q^{40} - 480 q^{41} - 260 q^{43} - 712 q^{44} - 1104 q^{46} + 24 q^{47} + 447 q^{49} - 814 q^{50} + 130 q^{52} + 678 q^{53} - 1552 q^{55} - 96 q^{56} - 628 q^{58} + 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 750 q^{64} - 52 q^{65} + 74 q^{67} + 460 q^{68} + 1216 q^{70} + 948 q^{71} - 222 q^{73} - 1724 q^{74} + 2392 q^{76} - 112 q^{77} - 24 q^{79} - 1100 q^{80} + 564 q^{82} + 796 q^{83} - 248 q^{85} - 1800 q^{86} + 1608 q^{88} - 1436 q^{89} + 390 q^{91} + 1296 q^{92} - 1920 q^{94} + 4032 q^{95} + 3242 q^{97} + 5070 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 10 * q^4 - 4 * q^5 + 30 * q^7 + 6 * q^8 - 4 * q^10 + 16 * q^11 + 39 * q^13 + 176 * q^14 - 110 * q^16 + 146 * q^17 + 94 * q^19 + 244 * q^20 - 56 * q^22 + 48 * q^23 + 145 * q^25 - 26 * q^26 + 80 * q^28 + 2 * q^29 + 302 * q^31 - 154 * q^32 + 164 * q^34 - 80 * q^35 + 374 * q^37 - 312 * q^38 - 516 * q^40 - 480 * q^41 - 260 * q^43 - 712 * q^44 - 1104 * q^46 + 24 * q^47 + 447 * q^49 - 814 * q^50 + 130 * q^52 + 678 * q^53 - 1552 * q^55 - 96 * q^56 - 628 * q^58 + 1788 * q^59 + 230 * q^61 - 1952 * q^62 - 750 * q^64 - 52 * q^65 + 74 * q^67 + 460 * q^68 + 1216 * q^70 + 948 * q^71 - 222 * q^73 - 1724 * q^74 + 2392 * q^76 - 112 * q^77 - 24 * q^79 - 1100 * q^80 + 564 * q^82 + 796 * q^83 - 248 * q^85 - 1800 * q^86 + 1608 * q^88 - 1436 * q^89 + 390 * q^91 + 1296 * q^92 - 1920 * q^94 + 4032 * q^95 + 3242 * q^97 + 5070 * 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 $

$2$	3.73549	1.32069	0.660347	−	0.750960i	$-0.270409\pi$
$2$	3.73549	1.32069	0.660347	+	0.750960i	$0.270409\pi$
$3$	0	0
$3$	0	0
$4$	5.95388	0.744235
$5$	3.90776	0.349521	0.174761	−	0.984611i	$-0.444085\pi$
$5$	3.90776	0.349521	0.174761	+	0.984611i	$0.444085\pi$
$6$	0	0
$7$	36.4129	1.96611	0.983057	−	0.183301i	$-0.0586782\pi$
$7$	36.4129	1.96611	0.983057	+	0.183301i	$0.0586782\pi$
$8$	−7.64325	−0.337787
$9$	0	0
$10$	14.5974	0.461611
$11$	−19.1943	−0.526117	−0.263059	−	0.964780i	$-0.584731\pi$
$11$	−19.1943	−0.526117	−0.263059	+	0.964780i	$0.584731\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	136.020	2.59664
$15$	0	0
$16$	−76.1823	−1.19035
$17$	83.8839	1.19676	0.598378	−	0.801214i	$-0.295812\pi$
$17$	83.8839	1.19676	0.598378	+	0.801214i	$0.295812\pi$
$18$	0	0
$19$	46.8492	0.565681	0.282840	−	0.959167i	$-0.408723\pi$
$19$	46.8492	0.565681	0.282840	+	0.959167i	$0.408723\pi$
$20$	23.2664	0.260126
$21$	0	0
$22$	−71.7000	−0.694840
$23$	−103.905	−0.941983	−0.470991	−	0.882138i	$-0.656104\pi$
$23$	−103.905	−0.941983	−0.470991	+	0.882138i	$0.656104\pi$
$24$	0	0
$25$	−109.729	−0.877835
$26$	48.5614	0.366295
$27$	0	0
$28$	216.798	1.46325
$29$	−108.341	−0.693738	−0.346869	−	0.937914i	$-0.612755\pi$
$29$	−108.341	−0.693738	−0.346869	+	0.937914i	$0.612755\pi$
$30$	0	0
$31$	−147.532	−0.854759	−0.427379	−	0.904072i	$-0.640563\pi$
$31$	−147.532	−0.854759	−0.427379	+	0.904072i	$0.640563\pi$
$32$	−223.432	−1.23430
$33$	0	0
$34$	313.347	1.58055
$35$	142.293	0.687198
$36$	0	0
$37$	−160.012	−0.710969	−0.355484	−	0.934682i	$-0.615684\pi$
$37$	−160.012	−0.710969	−0.355484	+	0.934682i	$0.615684\pi$
$38$	175.005	0.747092
$39$	0	0
$40$	−29.8680	−0.118064
$41$	−231.490	−0.881772	−0.440886	−	0.897563i	$-0.645336\pi$
$41$	−231.490	−0.881772	−0.440886	+	0.897563i	$0.645336\pi$
$42$	0	0
$43$	−340.314	−1.20692	−0.603458	−	0.797395i	$-0.706211\pi$
$43$	−340.314	−1.20692	−0.603458	+	0.797395i	$0.706211\pi$
$44$	−114.280	−0.391555
$45$	0	0
$46$	−388.135	−1.24407
$47$	−119.653	−0.371346	−0.185673	−	0.982612i	$-0.559446\pi$
$47$	−119.653	−0.371346	−0.185673	+	0.982612i	$0.559446\pi$
$48$	0	0
$49$	982.902	2.86560
$50$	−409.893	−1.15935
$51$	0	0
$52$	77.4005	0.206414
$53$	732.879	1.89941	0.949705	−	0.313146i	$-0.101383\pi$
$53$	732.879	1.89941	0.949705	+	0.313146i	$0.101383\pi$
$54$	0	0
$55$	−75.0067	−0.183889
$56$	−278.313	−0.664128
$57$	0	0
$58$	−404.706	−0.916216
$59$	229.782	0.507035	0.253518	−	0.967331i	$-0.418412\pi$
$59$	229.782	0.507035	0.253518	+	0.967331i	$0.418412\pi$
$60$	0	0
$61$	108.943	0.228668	0.114334	−	0.993442i	$-0.463527\pi$
$61$	108.943	0.228668	0.114334	+	0.993442i	$0.463527\pi$
$62$	−551.104	−1.12888
$63$	0	0
$64$	−225.170	−0.439786
$65$	50.8009	0.0969397
$66$	0	0
$67$	10.3955	0.0189555	0.00947774	−	0.999955i	$-0.496983\pi$
$67$	10.3955	0.0189555	0.00947774	+	0.999955i	$0.496983\pi$
$68$	499.435	0.890667
$69$	0	0
$70$	531.535	0.907579
$71$	869.201	1.45289	0.726445	−	0.687224i	$-0.241171\pi$
$71$	869.201	1.45289	0.726445	+	0.687224i	$0.241171\pi$
$72$	0	0
$73$	−1099.07	−1.76214	−0.881072	−	0.472982i	$-0.843178\pi$
$73$	−1099.07	−1.76214	−0.881072	+	0.472982i	$0.843178\pi$
$74$	−597.724	−0.938973
$75$	0	0
$76$	278.934	0.421000
$77$	−698.920	−1.03441
$78$	0	0
$79$	140.410	0.199967	0.0999835	−	0.994989i	$-0.468121\pi$
$79$	140.410	0.199967	0.0999835	+	0.994989i	$0.468121\pi$
$80$	−297.703	−0.416052
$81$	0	0
$82$	−864.729	−1.16455
$83$	159.474	0.210898	0.105449	−	0.994425i	$-0.466372\pi$
$83$	159.474	0.210898	0.105449	+	0.994425i	$0.466372\pi$
$84$	0	0
$85$	327.799	0.418291
$86$	−1271.24	−1.59397
$87$	0	0
$88$	146.707	0.177716
$89$	−1067.93	−1.27192	−0.635959	−	0.771723i	$-0.719395\pi$
$89$	−1067.93	−1.27192	−0.635959	+	0.771723i	$0.719395\pi$
$90$	0	0
$91$	473.368	0.545302
$92$	−618.636	−0.701057
$93$	0	0
$94$	−446.964	−0.490434
$95$	183.075	0.197717
$96$	0	0
$97$	858.881	0.899032	0.449516	−	0.893272i	$-0.351596\pi$
$97$	858.881	0.899032	0.449516	+	0.893272i	$0.351596\pi$
$98$	3671.62	3.78459
$99$	0	0
$100$	−653.316	−0.653316
$101$	1574.16	1.55084	0.775421	−	0.631444i	$-0.217538\pi$
$101$	1574.16	1.55084	0.775421	+	0.631444i	$0.217538\pi$
$102$	0	0
$103$	−129.724	−0.124098	−0.0620489	−	0.998073i	$-0.519763\pi$
$103$	−129.724	−0.124098	−0.0620489	+	0.998073i	$0.519763\pi$
$104$	−99.3623	−0.0936853
$105$	0	0
$106$	2737.66	2.50854
$107$	1957.43	1.76853	0.884263	−	0.466990i	$-0.154661\pi$
$107$	1957.43	1.76853	0.884263	+	0.466990i	$0.154661\pi$
$108$	0	0
$109$	1228.77	1.07977	0.539886	−	0.841738i	$-0.318467\pi$
$109$	1228.77	1.07977	0.539886	+	0.841738i	$0.318467\pi$
$110$	−280.187	−0.242861
$111$	0	0
$112$	−2774.02	−2.34036
$113$	−1629.50	−1.35655	−0.678275	−	0.734808i	$-0.737272\pi$
$113$	−1629.50	−1.35655	−0.678275	+	0.734808i	$0.737272\pi$
$114$	0	0
$115$	−406.035	−0.329243
$116$	−645.049	−0.516304
$117$	0	0
$118$	858.349	0.669639
$119$	3054.46	2.35296
$120$	0	0
$121$	−962.580	−0.723201
$122$	406.956	0.302000
$123$	0	0
$124$	−878.388	−0.636142
$125$	−917.267	−0.656343
$126$	0	0
$127$	276.112	0.192921	0.0964607	−	0.995337i	$-0.469248\pi$
$127$	276.112	0.192921	0.0964607	+	0.995337i	$0.469248\pi$
$128$	946.337	0.653478
$129$	0	0
$130$	189.766	0.128028
$131$	96.2240	0.0641765	0.0320883	−	0.999485i	$-0.489784\pi$
$131$	96.2240	0.0641765	0.0320883	+	0.999485i	$0.489784\pi$
$132$	0	0
$133$	1705.92	1.11219
$134$	38.8324	0.0250344
$135$	0	0
$136$	−641.146	−0.404249
$137$	−2618.38	−1.63287	−0.816435	−	0.577438i	$-0.804053\pi$
$137$	−2618.38	−1.63287	−0.816435	+	0.577438i	$0.804053\pi$
$138$	0	0
$139$	1963.34	1.19805	0.599023	−	0.800732i	$-0.295556\pi$
$139$	1963.34	1.19805	0.599023	+	0.800732i	$0.295556\pi$
$140$	847.197	0.511437
$141$	0	0
$142$	3246.89	1.91882
$143$	−249.526	−0.145919
$144$	0	0
$145$	−423.370	−0.242476
$146$	−4105.57	−2.32725
$147$	0	0
$148$	−952.694	−0.529128
$149$	301.111	0.165557	0.0827784	−	0.996568i	$-0.473621\pi$
$149$	301.111	0.165557	0.0827784	+	0.996568i	$0.473621\pi$
$150$	0	0
$151$	342.973	0.184839	0.0924197	−	0.995720i	$-0.470540\pi$
$151$	342.973	0.184839	0.0924197	+	0.995720i	$0.470540\pi$
$152$	−358.080	−0.191080
$153$	0	0
$154$	−2610.81	−1.36614
$155$	−576.520	−0.298756
$156$	0	0
$157$	−1286.97	−0.654211	−0.327106	−	0.944988i	$-0.606073\pi$
$157$	−1286.97	−0.654211	−0.327106	+	0.944988i	$0.606073\pi$
$158$	524.501	0.264095
$159$	0	0
$160$	−873.121	−0.431414
$161$	−3783.47	−1.85205
$162$	0	0
$163$	532.561	0.255910	0.127955	−	0.991780i	$-0.459159\pi$
$163$	532.561	0.255910	0.127955	+	0.991780i	$0.459159\pi$
$164$	−1378.26	−0.656246
$165$	0	0
$166$	595.714	0.278532
$167$	−41.9542	−0.0194402	−0.00972011	−	0.999953i	$-0.503094\pi$
$167$	−41.9542	−0.0194402	−0.00972011	+	0.999953i	$0.503094\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	1224.49	0.552435
$171$	0	0
$172$	−2026.19	−0.898229
$173$	1066.50	0.468694	0.234347	−	0.972153i	$-0.424705\pi$
$173$	1066.50	0.468694	0.234347	+	0.972153i	$0.424705\pi$
$174$	0	0
$175$	−3995.57	−1.72592
$176$	1462.26	0.626263
$177$	0	0
$178$	−3989.25	−1.67982
$179$	−3174.61	−1.32559	−0.662797	−	0.748799i	$-0.730631\pi$
$179$	−3174.61	−1.32559	−0.662797	+	0.748799i	$0.730631\pi$
$180$	0	0
$181$	−2725.43	−1.11923	−0.559613	−	0.828754i	$-0.689050\pi$
$181$	−2725.43	−1.11923	−0.559613	+	0.828754i	$0.689050\pi$
$182$	1768.26	0.720177
$183$	0	0
$184$	794.169	0.318190
$185$	−625.290	−0.248498
$186$	0	0
$187$	−1610.09	−0.629634
$188$	−712.402	−0.276368
$189$	0	0
$190$	683.877	0.261124
$191$	−784.888	−0.297343	−0.148672	−	0.988887i	$-0.547500\pi$
$191$	−784.888	−0.297343	−0.148672	+	0.988887i	$0.547500\pi$
$192$	0	0
$193$	−1255.87	−0.468391	−0.234195	−	0.972190i	$-0.575246\pi$
$193$	−1255.87	−0.468391	−0.234195	+	0.972190i	$0.575246\pi$
$194$	3208.34	1.18735
$195$	0	0
$196$	5852.08	2.13268
$197$	2777.35	1.00446	0.502229	−	0.864734i	$-0.332513\pi$
$197$	2777.35	1.00446	0.502229	+	0.864734i	$0.332513\pi$
$198$	0	0
$199$	1490.43	0.530924	0.265462	−	0.964121i	$-0.414476\pi$
$199$	1490.43	0.530924	0.265462	+	0.964121i	$0.414476\pi$
$200$	838.689	0.296521
$201$	0	0
$202$	5880.27	2.04819
$203$	−3945.01	−1.36397
$204$	0	0
$205$	−904.608	−0.308198
$206$	−484.582	−0.163895
$207$	0	0
$208$	−990.370	−0.330143
$209$	−899.236	−0.297614
$210$	0	0
$211$	2305.63	0.752255	0.376127	−	0.926568i	$-0.377255\pi$
$211$	2305.63	0.752255	0.376127	+	0.926568i	$0.377255\pi$
$212$	4363.48	1.41361
$213$	0	0
$214$	7311.97	2.33568
$215$	−1329.87	−0.421842
$216$	0	0
$217$	−5372.07	−1.68055
$218$	4590.07	1.42605
$219$	0	0
$220$	−446.581	−0.136857
$221$	1090.49	0.331920
$222$	0	0
$223$	1241.98	0.372956	0.186478	−	0.982459i	$-0.440293\pi$
$223$	1241.98	0.372956	0.186478	+	0.982459i	$0.440293\pi$
$224$	−8135.83	−2.42678
$225$	0	0
$226$	−6086.97	−1.79159
$227$	1724.76	0.504300	0.252150	−	0.967688i	$-0.418862\pi$
$227$	1724.76	0.504300	0.252150	+	0.967688i	$0.418862\pi$
$228$	0	0
$229$	−3273.72	−0.944688	−0.472344	−	0.881414i	$-0.656592\pi$
$229$	−3273.72	−0.944688	−0.472344	+	0.881414i	$0.656592\pi$
$230$	−1516.74	−0.434829
$231$	0	0
$232$	828.076	0.234336
$233$	2129.52	0.598752	0.299376	−	0.954135i	$-0.403222\pi$
$233$	2129.52	0.598752	0.299376	+	0.954135i	$0.403222\pi$
$234$	0	0
$235$	−467.577	−0.129793
$236$	1368.10	0.377354
$237$	0	0
$238$	11409.9	3.10754
$239$	5082.38	1.37553	0.687765	−	0.725933i	$-0.258592\pi$
$239$	5082.38	1.37553	0.687765	+	0.725933i	$0.258592\pi$
$240$	0	0
$241$	4765.65	1.27379	0.636893	−	0.770953i	$-0.280219\pi$
$241$	4765.65	1.27379	0.636893	+	0.770953i	$0.280219\pi$
$242$	−3595.71	−0.955127
$243$	0	0
$244$	648.634	0.170183
$245$	3840.95	1.00159
$246$	0	0
$247$	609.039	0.156892
$248$	1127.62	0.288727
$249$	0	0
$250$	−3426.44	−0.866829
$251$	4339.96	1.09138	0.545689	−	0.837988i	$-0.316268\pi$
$251$	4339.96	1.09138	0.545689	+	0.837988i	$0.316268\pi$
$252$	0	0
$253$	1994.37	0.495594
$254$	1031.41	0.254790
$255$	0	0
$256$	5336.40	1.30283
$257$	−4359.49	−1.05812	−0.529062	−	0.848583i	$-0.677456\pi$
$257$	−4359.49	−1.05812	−0.529062	+	0.848583i	$0.677456\pi$
$258$	0	0
$259$	−5826.51	−1.39785
$260$	302.463	0.0721459
$261$	0	0
$262$	359.444	0.0847576
$263$	−608.077	−0.142569	−0.0712844	−	0.997456i	$-0.522710\pi$
$263$	−608.077	−0.142569	−0.0712844	+	0.997456i	$0.522710\pi$
$264$	0	0
$265$	2863.92	0.663884
$266$	6372.43	1.46887
$267$	0	0
$268$	61.8938	0.0141073
$269$	−3454.29	−0.782942	−0.391471	−	0.920190i	$-0.628034\pi$
$269$	−3454.29	−0.782942	−0.391471	+	0.920190i	$0.628034\pi$
$270$	0	0
$271$	3703.72	0.830204	0.415102	−	0.909775i	$-0.363746\pi$
$271$	3703.72	0.830204	0.415102	+	0.909775i	$0.363746\pi$
$272$	−6390.47	−1.42456
$273$	0	0
$274$	−9780.92	−2.15652
$275$	2106.18	0.461844
$276$	0	0
$277$	−3566.89	−0.773696	−0.386848	−	0.922144i	$-0.626436\pi$
$277$	−3566.89	−0.773696	−0.386848	+	0.922144i	$0.626436\pi$
$278$	7334.04	1.58225
$279$	0	0
$280$	−1087.58	−0.232127
$281$	117.474	0.0249392	0.0124696	−	0.999922i	$-0.496031\pi$
$281$	117.474	0.0249392	0.0124696	+	0.999922i	$0.496031\pi$
$282$	0	0
$283$	−1737.62	−0.364984	−0.182492	−	0.983207i	$-0.558416\pi$
$283$	−1737.62	−0.364984	−0.182492	+	0.983207i	$0.558416\pi$
$284$	5175.12	1.08129
$285$	0	0
$286$	−932.100	−0.192714
$287$	−8429.23	−1.73366
$288$	0	0
$289$	2123.51	0.432223
$290$	−1581.50	−0.320237
$291$	0	0
$292$	−6543.74	−1.31145
$293$	−1904.05	−0.379643	−0.189822	−	0.981819i	$-0.560791\pi$
$293$	−1904.05	−0.379643	−0.189822	+	0.981819i	$0.560791\pi$
$294$	0	0
$295$	897.934	0.177219
$296$	1223.01	0.240156
$297$	0	0
$298$	1124.80	0.218650
$299$	−1350.76	−0.261259
$300$	0	0
$301$	−12391.8	−2.37293
$302$	1281.17	0.244117
$303$	0	0
$304$	−3569.08	−0.673358
$305$	425.724	0.0799242
$306$	0	0
$307$	2862.39	0.532134	0.266067	−	0.963955i	$-0.414276\pi$
$307$	2862.39	0.532134	0.266067	+	0.963955i	$0.414276\pi$
$308$	−4161.29	−0.769842
$309$	0	0
$310$	−2153.58	−0.394566
$311$	−4201.55	−0.766071	−0.383036	−	0.923734i	$-0.625121\pi$
$311$	−4201.55	−0.766071	−0.383036	+	0.923734i	$0.625121\pi$
$312$	0	0
$313$	3427.74	0.619002	0.309501	−	0.950899i	$-0.399838\pi$
$313$	3427.74	0.619002	0.309501	+	0.950899i	$0.399838\pi$
$314$	−4807.45	−0.864014
$315$	0	0
$316$	835.987	0.148823
$317$	−1676.09	−0.296966	−0.148483	−	0.988915i	$-0.547439\pi$
$317$	−1676.09	−0.296966	−0.148483	+	0.988915i	$0.547439\pi$
$318$	0	0
$319$	2079.52	0.364987
$320$	−879.912	−0.153714
$321$	0	0
$322$	−14133.1	−2.44599
$323$	3929.89	0.676982
$324$	0	0
$325$	−1426.48	−0.243468
$326$	1989.37	0.337979
$327$	0	0
$328$	1769.34	0.297851
$329$	−4356.93	−0.730108
$330$	0	0
$331$	11156.6	1.85264	0.926319	−	0.376740i	$-0.122955\pi$
$331$	11156.6	1.85264	0.926319	+	0.376740i	$0.122955\pi$
$332$	949.490	0.156958
$333$	0	0
$334$	−156.720	−0.0256746
$335$	40.6233	0.00662534
$336$	0	0
$337$	1636.44	0.264517	0.132259	−	0.991215i	$-0.457777\pi$
$337$	1636.44	0.264517	0.132259	+	0.991215i	$0.457777\pi$
$338$	631.298	0.101592
$339$	0	0
$340$	1951.67	0.311307
$341$	2831.77	0.449703
$342$	0	0
$343$	23300.7	3.66799
$344$	2601.10	0.407681
$345$	0	0
$346$	3983.88	0.619002
$347$	2977.87	0.460693	0.230347	−	0.973109i	$-0.426014\pi$
$347$	2977.87	0.460693	0.230347	+	0.973109i	$0.426014\pi$
$348$	0	0
$349$	9847.29	1.51035	0.755177	−	0.655521i	$-0.227551\pi$
$349$	9847.29	1.51035	0.755177	+	0.655521i	$0.227551\pi$
$350$	−14925.4	−2.27942
$351$	0	0
$352$	4288.62	0.649387
$353$	−4687.34	−0.706747	−0.353374	−	0.935482i	$-0.614966\pi$
$353$	−4687.34	−0.706747	−0.353374	+	0.935482i	$0.614966\pi$
$354$	0	0
$355$	3396.63	0.507816
$356$	−6358.35	−0.946606
$357$	0	0
$358$	−11858.7	−1.75070
$359$	2069.88	0.304301	0.152151	−	0.988357i	$-0.451380\pi$
$359$	2069.88	0.304301	0.152151	+	0.988357i	$0.451380\pi$
$360$	0	0
$361$	−4664.16	−0.680005
$362$	−10180.8	−1.47816
$363$	0	0
$364$	2818.38	0.405833
$365$	−4294.91	−0.615906
$366$	0	0
$367$	−7299.16	−1.03818	−0.519092	−	0.854719i	$-0.673730\pi$
$367$	−7299.16	−1.03818	−0.519092	+	0.854719i	$0.673730\pi$
$368$	7915.70	1.12129
$369$	0	0
$370$	−2335.76	−0.328191
$371$	26686.3	3.73446
$372$	0	0
$373$	−8964.32	−1.24438	−0.622192	−	0.782865i	$-0.713758\pi$
$373$	−8964.32	−1.24438	−0.622192	+	0.782865i	$0.713758\pi$
$374$	−6014.48	−0.831554
$375$	0	0
$376$	914.541	0.125436
$377$	−1408.43	−0.192408
$378$	0	0
$379$	−4399.26	−0.596239	−0.298120	−	0.954529i	$-0.596359\pi$
$379$	−4399.26	−0.596239	−0.298120	+	0.954529i	$0.596359\pi$
$380$	1090.01	0.147148
$381$	0	0
$382$	−2931.94	−0.392700
$383$	−3529.74	−0.470917	−0.235459	−	0.971884i	$-0.575659\pi$
$383$	−3529.74	−0.470917	−0.235459	+	0.971884i	$0.575659\pi$
$384$	0	0
$385$	−2731.21	−0.361547
$386$	−4691.28	−0.618601
$387$	0	0
$388$	5113.68	0.669092
$389$	3034.77	0.395549	0.197775	−	0.980248i	$-0.436629\pi$
$389$	3034.77	0.395549	0.197775	+	0.980248i	$0.436629\pi$
$390$	0	0
$391$	−8715.93	−1.12732
$392$	−7512.57	−0.967964
$393$	0	0
$394$	10374.8	1.32658
$395$	548.690	0.0698927
$396$	0	0
$397$	−3997.36	−0.505344	−0.252672	−	0.967552i	$-0.581309\pi$
$397$	−3997.36	−0.505344	−0.252672	+	0.967552i	$0.581309\pi$
$398$	5567.49	0.701188
$399$	0	0
$400$	8359.44	1.04493
$401$	9092.88	1.13236	0.566181	−	0.824281i	$-0.308420\pi$
$401$	9092.88	1.13236	0.566181	+	0.824281i	$0.308420\pi$
$402$	0	0
$403$	−1917.92	−0.237067
$404$	9372.38	1.15419
$405$	0	0
$406$	−14736.5	−1.80138
$407$	3071.32	0.374053
$408$	0	0
$409$	7143.54	0.863631	0.431816	−	0.901962i	$-0.357873\pi$
$409$	7143.54	0.863631	0.431816	+	0.901962i	$0.357873\pi$
$410$	−3379.15	−0.407036
$411$	0	0
$412$	−772.361	−0.0923579
$413$	8367.04	0.996889
$414$	0	0
$415$	623.187	0.0737134
$416$	−2904.62	−0.342333
$417$	0	0
$418$	−3359.09	−0.393058
$419$	−8213.84	−0.957691	−0.478845	−	0.877899i	$-0.658945\pi$
$419$	−8213.84	−0.957691	−0.478845	+	0.877899i	$0.658945\pi$
$420$	0	0
$421$	−7997.40	−0.925818	−0.462909	−	0.886406i	$-0.653194\pi$
$421$	−7997.40	−0.925818	−0.462909	+	0.886406i	$0.653194\pi$
$422$	8612.64	0.993499
$423$	0	0
$424$	−5601.58	−0.641597
$425$	−9204.53	−1.05055
$426$	0	0
$427$	3966.94	0.449587
$428$	11654.3	1.31620
$429$	0	0
$430$	−4967.70	−0.557125
$431$	13694.8	1.53053	0.765263	−	0.643718i	$-0.222609\pi$
$431$	13694.8	1.53053	0.765263	+	0.643718i	$0.222609\pi$
$432$	0	0
$433$	6716.57	0.745445	0.372722	−	0.927943i	$-0.378424\pi$
$433$	6716.57	0.745445	0.372722	+	0.927943i	$0.378424\pi$
$434$	−20067.3	−2.21950
$435$	0	0
$436$	7315.97	0.803604
$437$	−4867.84	−0.532862
$438$	0	0
$439$	−5933.32	−0.645061	−0.322531	−	0.946559i	$-0.604534\pi$
$439$	−5933.32	−0.645061	−0.322531	+	0.946559i	$0.604534\pi$
$440$	573.295	0.0621154
$441$	0	0
$442$	4073.52	0.438365
$443$	6923.40	0.742530	0.371265	−	0.928527i	$-0.378924\pi$
$443$	6923.40	0.742530	0.371265	+	0.928527i	$0.378924\pi$
$444$	0	0
$445$	−4173.23	−0.444562
$446$	4639.41	0.492561
$447$	0	0
$448$	−8199.11	−0.864669
$449$	−8886.78	−0.934061	−0.467030	−	0.884241i	$-0.654676\pi$
$449$	−8886.78	−0.934061	−0.467030	+	0.884241i	$0.654676\pi$
$450$	0	0
$451$	4443.28	0.463916
$452$	−9701.84	−1.00959
$453$	0	0
$454$	6442.81	0.666026
$455$	1849.81	0.190594
$456$	0	0
$457$	10965.0	1.12237	0.561184	−	0.827691i	$-0.310346\pi$
$457$	10965.0	1.12237	0.561184	+	0.827691i	$0.310346\pi$
$458$	−12228.9	−1.24764
$459$	0	0
$460$	−2417.48	−0.245034
$461$	−10069.2	−1.01729	−0.508644	−	0.860977i	$-0.669853\pi$
$461$	−10069.2	−1.01729	−0.508644	+	0.860977i	$0.669853\pi$
$462$	0	0
$463$	5599.72	0.562076	0.281038	−	0.959697i	$-0.409321\pi$
$463$	5599.72	0.562076	0.281038	+	0.959697i	$0.409321\pi$
$464$	8253.66	0.825790
$465$	0	0
$466$	7954.78	0.790769
$467$	13247.8	1.31271	0.656355	−	0.754452i	$-0.272097\pi$
$467$	13247.8	1.31271	0.656355	+	0.754452i	$0.272097\pi$
$468$	0	0
$469$	378.532	0.0372686
$470$	−1746.63	−0.171417
$471$	0	0
$472$	−1756.28	−0.171270
$473$	6532.08	0.634979
$474$	0	0
$475$	−5140.73	−0.496575
$476$	18185.9	1.75115
$477$	0	0
$478$	18985.2	1.81666
$479$	−16725.4	−1.59541	−0.797707	−	0.603045i	$-0.793954\pi$
$479$	−16725.4	−1.59541	−0.797707	+	0.603045i	$0.793954\pi$
$480$	0	0
$481$	−2080.16	−0.197187
$482$	17802.0	1.68228
$483$	0	0
$484$	−5731.09	−0.538231
$485$	3356.30	0.314231
$486$	0	0
$487$	−5305.86	−0.493699	−0.246850	−	0.969054i	$-0.579395\pi$
$487$	−5305.86	−0.493699	−0.246850	+	0.969054i	$0.579395\pi$
$488$	−832.679	−0.0772410
$489$	0	0
$490$	14347.8	1.32279
$491$	−16200.2	−1.48901	−0.744506	−	0.667616i	$-0.767315\pi$
$491$	−16200.2	−1.48901	−0.744506	+	0.667616i	$0.767315\pi$
$492$	0	0
$493$	−9088.05	−0.830234
$494$	2275.06	0.207206
$495$	0	0
$496$	11239.3	1.01746
$497$	31650.2	2.85655
$498$	0	0
$499$	4392.70	0.394076	0.197038	−	0.980396i	$-0.436868\pi$
$499$	4392.70	0.394076	0.197038	+	0.980396i	$0.436868\pi$
$500$	−5461.30	−0.488473
$501$	0	0
$502$	16211.9	1.44138
$503$	14955.2	1.32568	0.662841	−	0.748760i	$-0.269350\pi$
$503$	14955.2	1.32568	0.662841	+	0.748760i	$0.269350\pi$
$504$	0	0
$505$	6151.46	0.542052
$506$	7449.96	0.654528
$507$	0	0
$508$	1643.94	0.143579
$509$	−13403.4	−1.16719	−0.583593	−	0.812047i	$-0.698353\pi$
$509$	−13403.4	−1.16719	−0.583593	+	0.812047i	$0.698353\pi$
$510$	0	0
$511$	−40020.4	−3.46458
$512$	12363.4	1.06716
$513$	0	0
$514$	−16284.8	−1.39746
$515$	−506.930	−0.0433748
$516$	0	0
$517$	2296.66	0.195371
$518$	−21764.9	−1.84613
$519$	0	0
$520$	−388.284	−0.0327450
$521$	−19643.0	−1.65178	−0.825888	−	0.563834i	$-0.809326\pi$
$521$	−19643.0	−1.65178	−0.825888	+	0.563834i	$0.809326\pi$
$522$	0	0
$523$	14657.4	1.22548	0.612738	−	0.790286i	$-0.290068\pi$
$523$	14657.4	1.22548	0.612738	+	0.790286i	$0.290068\pi$
$524$	572.906	0.0477624
$525$	0	0
$526$	−2271.46	−0.188290
$527$	−12375.6	−1.02294
$528$	0	0
$529$	−1370.83	−0.112668
$530$	10698.1	0.876788
$531$	0	0
$532$	10156.8	0.827733
$533$	−3009.37	−0.244560
$534$	0	0
$535$	7649.19	0.618137
$536$	−79.4558	−0.00640292
$537$	0	0
$538$	−12903.4	−1.03403
$539$	−18866.1	−1.50764
$540$	0	0
$541$	13921.3	1.10633	0.553164	−	0.833072i	$-0.313420\pi$
$541$	13921.3	1.10633	0.553164	+	0.833072i	$0.313420\pi$
$542$	13835.2	1.09645
$543$	0	0
$544$	−18742.4	−1.47716
$545$	4801.75	0.377403
$546$	0	0
$547$	−2324.11	−0.181667	−0.0908335	−	0.995866i	$-0.528953\pi$
$547$	−2324.11	−0.181667	−0.0908335	+	0.995866i	$0.528953\pi$
$548$	−15589.5	−1.21524
$549$	0	0
$550$	7867.60	0.609955
$551$	−5075.68	−0.392434
$552$	0	0
$553$	5112.75	0.393158
$554$	−13324.1	−1.02182
$555$	0	0
$556$	11689.5	0.891628
$557$	16962.8	1.29037	0.645185	−	0.764027i	$-0.276780\pi$
$557$	16962.8	1.29037	0.645185	+	0.764027i	$0.276780\pi$
$558$	0	0
$559$	−4424.08	−0.334738
$560$	−10840.2	−0.818006
$561$	0	0
$562$	438.823	0.0329370
$563$	389.000	0.0291197	0.0145599	−	0.999894i	$-0.495365\pi$
$563$	389.000	0.0291197	0.0145599	+	0.999894i	$0.495365\pi$
$564$	0	0
$565$	−6367.69	−0.474143
$566$	−6490.85	−0.482033
$567$	0	0
$568$	−6643.53	−0.490768
$569$	−2217.56	−0.163383	−0.0816914	−	0.996658i	$-0.526032\pi$
$569$	−2217.56	−0.163383	−0.0816914	+	0.996658i	$0.526032\pi$
$570$	0	0
$571$	−17087.3	−1.25233	−0.626167	−	0.779689i	$-0.715377\pi$
$571$	−17087.3	−1.25233	−0.626167	+	0.779689i	$0.715377\pi$
$572$	−1485.65	−0.108598
$573$	0	0
$574$	−31487.3	−2.28964
$575$	11401.4	0.826906
$576$	0	0
$577$	−3977.26	−0.286959	−0.143480	−	0.989653i	$-0.545829\pi$
$577$	−3977.26	−0.286959	−0.143480	+	0.989653i	$0.545829\pi$
$578$	7932.35	0.570835
$579$	0	0
$580$	−2520.70	−0.180459
$581$	5806.92	0.414650
$582$	0	0
$583$	−14067.1	−0.999312
$584$	8400.48	0.595230
$585$	0	0
$586$	−7112.54	−0.501393
$587$	16880.3	1.18693	0.593463	−	0.804861i	$-0.297760\pi$
$587$	16880.3	1.18693	0.593463	+	0.804861i	$0.297760\pi$
$588$	0	0
$589$	−6911.75	−0.483521
$590$	3354.22	0.234053
$591$	0	0
$592$	12190.1	0.846301
$593$	−2423.25	−0.167810	−0.0839048	−	0.996474i	$-0.526739\pi$
$593$	−2423.25	−0.167810	−0.0839048	+	0.996474i	$0.526739\pi$
$594$	0	0
$595$	11936.1	0.822408
$596$	1792.78	0.123213
$597$	0	0
$598$	−5045.75	−0.345044
$599$	3900.55	0.266064	0.133032	−	0.991112i	$-0.457529\pi$
$599$	3900.55	0.266064	0.133032	+	0.991112i	$0.457529\pi$
$600$	0	0
$601$	28653.4	1.94476	0.972378	−	0.233413i	$-0.0749893\pi$
$601$	28653.4	1.94476	0.972378	+	0.233413i	$0.0749893\pi$
$602$	−46289.5	−3.13392
$603$	0	0
$604$	2042.02	0.137564
$605$	−3761.54	−0.252774
$606$	0	0
$607$	214.736	0.0143589	0.00717946	−	0.999974i	$-0.497715\pi$
$607$	214.736	0.0143589	0.00717946	+	0.999974i	$0.497715\pi$
$608$	−10467.6	−0.698220
$609$	0	0
$610$	1590.29	0.105555
$611$	−1555.49	−0.102993
$612$	0	0
$613$	26438.5	1.74199	0.870996	−	0.491290i	$-0.163475\pi$
$613$	26438.5	1.74199	0.870996	+	0.491290i	$0.163475\pi$
$614$	10692.4	0.702787
$615$	0	0
$616$	5342.02	0.349409
$617$	−6700.96	−0.437229	−0.218615	−	0.975811i	$-0.570154\pi$
$617$	−6700.96	−0.437229	−0.218615	+	0.975811i	$0.570154\pi$
$618$	0	0
$619$	−27319.1	−1.77391	−0.886953	−	0.461860i	$-0.847182\pi$
$619$	−27319.1	−1.77391	−0.886953	+	0.461860i	$0.847182\pi$
$620$	−3432.53	−0.222345
$621$	0	0
$622$	−15694.9	−1.01175
$623$	−38886.6	−2.50074
$624$	0	0
$625$	10131.7	0.648429
$626$	12804.3	0.817513
$627$	0	0
$628$	−7662.45	−0.486887
$629$	−13422.4	−0.850855
$630$	0	0
$631$	−7126.87	−0.449629	−0.224815	−	0.974402i	$-0.572178\pi$
$631$	−7126.87	−0.449629	−0.224815	+	0.974402i	$0.572178\pi$
$632$	−1073.19	−0.0675463
$633$	0	0
$634$	−6261.00	−0.392202
$635$	1078.98	0.0674301
$636$	0	0
$637$	12777.7	0.794775
$638$	7768.04	0.482037
$639$	0	0
$640$	3698.06	0.228404
$641$	−23615.0	−1.45513	−0.727565	−	0.686039i	$-0.759348\pi$
$641$	−23615.0	−1.45513	−0.727565	+	0.686039i	$0.759348\pi$
$642$	0	0
$643$	−8144.41	−0.499509	−0.249755	−	0.968309i	$-0.580350\pi$
$643$	−8144.41	−0.499509	−0.249755	+	0.968309i	$0.580350\pi$
$644$	−22526.3	−1.37836
$645$	0	0
$646$	14680.1	0.894086
$647$	−9682.00	−0.588313	−0.294157	−	0.955757i	$-0.595039\pi$
$647$	−9682.00	−0.588313	−0.294157	+	0.955757i	$0.595039\pi$
$648$	0	0
$649$	−4410.50	−0.266760
$650$	−5328.61	−0.321546
$651$	0	0
$652$	3170.80	0.190457
$653$	18193.6	1.09030	0.545152	−	0.838337i	$-0.316472\pi$
$653$	18193.6	1.09030	0.545152	+	0.838337i	$0.316472\pi$
$654$	0	0
$655$	376.020	0.0224310
$656$	17635.5	1.04962
$657$	0	0
$658$	−16275.3	−0.964250
$659$	−9300.88	−0.549789	−0.274895	−	0.961474i	$-0.588643\pi$
$659$	−9300.88	−0.549789	−0.274895	+	0.961474i	$0.588643\pi$
$660$	0	0
$661$	−5437.29	−0.319949	−0.159974	−	0.987121i	$-0.551141\pi$
$661$	−5437.29	−0.319949	−0.159974	+	0.987121i	$0.551141\pi$
$662$	41675.4	2.44677
$663$	0	0
$664$	−1218.90	−0.0712388
$665$	6666.32	0.388735
$666$	0	0
$667$	11257.1	0.653489
$668$	−249.791	−0.0144681
$669$	0	0
$670$	151.748	0.00875005
$671$	−2091.08	−0.120306
$672$	0	0
$673$	−8682.75	−0.497319	−0.248659	−	0.968591i	$-0.579990\pi$
$673$	−8682.75	−0.497319	−0.248659	+	0.968591i	$0.579990\pi$
$674$	6112.89	0.349347
$675$	0	0
$676$	1006.21	0.0572489
$677$	−13300.1	−0.755041	−0.377521	−	0.926001i	$-0.623223\pi$
$677$	−13300.1	−0.755041	−0.377521	+	0.926001i	$0.623223\pi$
$678$	0	0
$679$	31274.4	1.76760
$680$	−2505.45	−0.141293
$681$	0	0
$682$	10578.0	0.593921
$683$	504.175	0.0282455	0.0141228	−	0.999900i	$-0.495504\pi$
$683$	504.175	0.0282455	0.0141228	+	0.999900i	$0.495504\pi$
$684$	0	0
$685$	−10232.0	−0.570722
$686$	87039.6	4.84429
$687$	0	0
$688$	25925.9	1.43665
$689$	9527.43	0.526802
$690$	0	0
$691$	13443.8	0.740124	0.370062	−	0.929007i	$-0.379336\pi$
$691$	13443.8	0.740124	0.370062	+	0.929007i	$0.379336\pi$
$692$	6349.79	0.348819
$693$	0	0
$694$	11123.8	0.608435
$695$	7672.27	0.418742
$696$	0	0
$697$	−19418.3	−1.05527
$698$	36784.4	1.99472
$699$	0	0
$700$	−23789.1	−1.28449
$701$	−28735.6	−1.54826	−0.774128	−	0.633030i	$-0.781811\pi$
$701$	−28735.6	−1.54826	−0.774128	+	0.633030i	$0.781811\pi$
$702$	0	0
$703$	−7496.44	−0.402181
$704$	4321.98	0.231379
$705$	0	0
$706$	−17509.5	−0.933398
$707$	57319.9	3.04913
$708$	0	0
$709$	17610.2	0.932812	0.466406	−	0.884571i	$-0.345549\pi$
$709$	17610.2	0.932812	0.466406	+	0.884571i	$0.345549\pi$
$710$	12688.1	0.670670
$711$	0	0
$712$	8162.48	0.429638
$713$	15329.3	0.805168
$714$	0	0
$715$	−975.087	−0.0510016
$716$	−18901.2	−0.986553
$717$	0	0
$718$	7732.02	0.401889
$719$	9226.04	0.478544	0.239272	−	0.970953i	$-0.423091\pi$
$719$	9226.04	0.478544	0.239272	+	0.970953i	$0.423091\pi$
$720$	0	0
$721$	−4723.63	−0.243990
$722$	−17422.9	−0.898079
$723$	0	0
$724$	−16226.9	−0.832967
$725$	11888.2	0.608987
$726$	0	0
$727$	33246.0	1.69604	0.848022	−	0.529961i	$-0.177793\pi$
$727$	33246.0	1.69604	0.848022	+	0.529961i	$0.177793\pi$
$728$	−3618.07	−0.184196
$729$	0	0
$730$	−16043.6	−0.813424
$731$	−28546.9	−1.44438
$732$	0	0
$733$	−4423.26	−0.222888	−0.111444	−	0.993771i	$-0.535548\pi$
$733$	−4423.26	−0.222888	−0.111444	+	0.993771i	$0.535548\pi$
$734$	−27266.0	−1.37112
$735$	0	0
$736$	23215.6	1.16269
$737$	−199.535	−0.00997281
$738$	0	0
$739$	−4529.56	−0.225470	−0.112735	−	0.993625i	$-0.535961\pi$
$739$	−4529.56	−0.225470	−0.112735	+	0.993625i	$0.535961\pi$
$740$	−3722.90	−0.184941
$741$	0	0
$742$	99686.4	4.93208
$743$	−10851.5	−0.535803	−0.267901	−	0.963446i	$-0.586330\pi$
$743$	−10851.5	−0.535803	−0.267901	+	0.963446i	$0.586330\pi$
$744$	0	0
$745$	1176.67	0.0578656
$746$	−33486.1	−1.64345
$747$	0	0
$748$	−9586.29	−0.468595
$749$	71275.9	3.47712
$750$	0	0
$751$	−33022.6	−1.60454	−0.802272	−	0.596958i	$-0.796376\pi$
$751$	−33022.6	−1.60454	−0.802272	+	0.596958i	$0.796376\pi$
$752$	9115.48	0.442031
$753$	0	0
$754$	−5261.18	−0.254113
$755$	1340.26	0.0646053
$756$	0	0
$757$	−3443.77	−0.165345	−0.0826724	−	0.996577i	$-0.526346\pi$
$757$	−3443.77	−0.165345	−0.0826724	+	0.996577i	$0.526346\pi$
$758$	−16433.4	−0.787450
$759$	0	0
$760$	−1399.29	−0.0667864
$761$	−19562.6	−0.931858	−0.465929	−	0.884822i	$-0.654280\pi$
$761$	−19562.6	−0.931858	−0.465929	+	0.884822i	$0.654280\pi$
$762$	0	0
$763$	44743.2	2.12295
$764$	−4673.13	−0.221293
$765$	0	0
$766$	−13185.3	−0.621938
$767$	2987.17	0.140626
$768$	0	0
$769$	−17061.1	−0.800049	−0.400025	−	0.916504i	$-0.630998\pi$
$769$	−17061.1	−0.800049	−0.400025	+	0.916504i	$0.630998\pi$
$770$	−10202.4	−0.477493
$771$	0	0
$772$	−7477.29	−0.348593
$773$	−10798.4	−0.502448	−0.251224	−	0.967929i	$-0.580833\pi$
$773$	−10798.4	−0.502448	−0.251224	+	0.967929i	$0.580833\pi$
$774$	0	0
$775$	16188.6	0.750337
$776$	−6564.64	−0.303682
$777$	0	0
$778$	11336.3	0.522400
$779$	−10845.1	−0.498802
$780$	0	0
$781$	−16683.7	−0.764391
$782$	−32558.2	−1.48885
$783$	0	0
$784$	−74879.8	−3.41107
$785$	−5029.16	−0.228661
$786$	0	0
$787$	−35607.0	−1.61277	−0.806386	−	0.591390i	$-0.798580\pi$
$787$	−35607.0	−1.61277	−0.806386	+	0.591390i	$0.798580\pi$
$788$	16536.0	0.747553
$789$	0	0
$790$	2049.63	0.0923069
$791$	−59334.8	−2.66713
$792$	0	0
$793$	1416.26	0.0634210
$794$	−14932.1	−0.667405
$795$	0	0
$796$	8873.85	0.395132
$797$	−22155.3	−0.984668	−0.492334	−	0.870406i	$-0.663856\pi$
$797$	−22155.3	−0.984668	−0.492334	+	0.870406i	$0.663856\pi$
$798$	0	0
$799$	−10037.0	−0.444410
$800$	24517.1	1.08351
$801$	0	0
$802$	33966.4	1.49550
$803$	21095.9	0.927094
$804$	0	0
$805$	−14784.9	−0.647329
$806$	−7164.35	−0.313094
$807$	0	0
$808$	−12031.7	−0.523855
$809$	22524.6	0.978889	0.489445	−	0.872034i	$-0.337200\pi$
$809$	22524.6	0.978889	0.489445	+	0.872034i	$0.337200\pi$
$810$	0	0
$811$	4452.39	0.192780	0.0963900	−	0.995344i	$-0.469270\pi$
$811$	4452.39	0.192780	0.0963900	+	0.995344i	$0.469270\pi$
$812$	−23488.1	−1.01511
$813$	0	0
$814$	11472.9	0.494010
$815$	2081.12	0.0894460
$816$	0	0
$817$	−15943.4	−0.682729
$818$	26684.6	1.14059
$819$	0	0
$820$	−5385.93	−0.229372
$821$	7097.27	0.301701	0.150851	−	0.988557i	$-0.451799\pi$
$821$	7097.27	0.301701	0.150851	+	0.988557i	$0.451799\pi$
$822$	0	0
$823$	−12193.8	−0.516463	−0.258231	−	0.966083i	$-0.583140\pi$
$823$	−12193.8	−0.516463	−0.258231	+	0.966083i	$0.583140\pi$
$824$	991.512	0.0419187
$825$	0	0
$826$	31255.0	1.31659
$827$	−7427.97	−0.312329	−0.156164	−	0.987731i	$-0.549913\pi$
$827$	−7427.97	−0.312329	−0.156164	+	0.987731i	$0.549913\pi$
$828$	0	0
$829$	16966.2	0.710810	0.355405	−	0.934712i	$-0.384343\pi$
$829$	16966.2	0.710810	0.355405	+	0.934712i	$0.384343\pi$
$830$	2327.91	0.0973529
$831$	0	0
$832$	−2927.21	−0.121975
$833$	82449.7	3.42943
$834$	0	0
$835$	−163.947	−0.00679477
$836$	−5353.94	−0.221495
$837$	0	0
$838$	−30682.7	−1.26482
$839$	−12025.7	−0.494844	−0.247422	−	0.968908i	$-0.579583\pi$
$839$	−12025.7	−0.494844	−0.247422	+	0.968908i	$0.579583\pi$
$840$	0	0
$841$	−12651.3	−0.518728
$842$	−29874.2	−1.22272
$843$	0	0
$844$	13727.4	0.559855
$845$	660.412	0.0268862
$846$	0	0
$847$	−35050.4	−1.42189
$848$	−55832.5	−2.26096
$849$	0	0
$850$	−34383.4	−1.38746
$851$	16626.0	0.669720
$852$	0	0
$853$	−22187.2	−0.890593	−0.445297	−	0.895383i	$-0.646902\pi$
$853$	−22187.2	−0.890593	−0.445297	+	0.895383i	$0.646902\pi$
$854$	14818.5	0.593767
$855$	0	0
$856$	−14961.2	−0.597385
$857$	−5746.19	−0.229038	−0.114519	−	0.993421i	$-0.536533\pi$
$857$	−5746.19	−0.229038	−0.114519	+	0.993421i	$0.536533\pi$
$858$	0	0
$859$	8305.66	0.329902	0.164951	−	0.986302i	$-0.447253\pi$
$859$	8305.66	0.329902	0.164951	+	0.986302i	$0.447253\pi$
$860$	−7917.87	−0.313950
$861$	0	0
$862$	51156.9	2.02136
$863$	−38086.4	−1.50229	−0.751146	−	0.660137i	$-0.770498\pi$
$863$	−38086.4	−1.50229	−0.751146	+	0.660137i	$0.770498\pi$
$864$	0	0
$865$	4167.61	0.163819
$866$	25089.7	0.984505
$867$	0	0
$868$	−31984.7	−1.25073
$869$	−2695.07	−0.105206
$870$	0	0
$871$	135.142	0.00525731
$872$	−9391.82	−0.364733
$873$	0	0
$874$	−18183.8	−0.703748
$875$	−33400.4	−1.29044
$876$	0	0
$877$	−2098.53	−0.0808009	−0.0404005	−	0.999184i	$-0.512863\pi$
$877$	−2098.53	−0.0808009	−0.0404005	+	0.999184i	$0.512863\pi$
$878$	−22163.8	−0.851929
$879$	0	0
$880$	5714.18	0.218892
$881$	−14555.3	−0.556619	−0.278309	−	0.960491i	$-0.589774\pi$
$881$	−14555.3	−0.556619	−0.278309	+	0.960491i	$0.589774\pi$
$882$	0	0
$883$	2122.88	0.0809066	0.0404533	−	0.999181i	$-0.487120\pi$
$883$	2122.88	0.0809066	0.0404533	+	0.999181i	$0.487120\pi$
$884$	6492.65	0.247027
$885$	0	0
$886$	25862.3	0.980656
$887$	12487.3	0.472696	0.236348	−	0.971668i	$-0.424049\pi$
$887$	12487.3	0.472696	0.236348	+	0.971668i	$0.424049\pi$
$888$	0	0
$889$	10054.1	0.379305
$890$	−15589.1	−0.587131
$891$	0	0
$892$	7394.61	0.277567
$893$	−5605.66	−0.210063
$894$	0	0
$895$	−12405.6	−0.463323
$896$	34458.9	1.28481
$897$	0	0
$898$	−33196.5	−1.23361
$899$	15983.7	0.592978
$900$	0	0
$901$	61476.8	2.27313
$902$	16597.8	0.612691
$903$	0	0
$904$	12454.7	0.458226
$905$	−10650.3	−0.391193
$906$	0	0
$907$	29679.8	1.08655	0.543275	−	0.839555i	$-0.317184\pi$
$907$	29679.8	1.08655	0.543275	+	0.839555i	$0.317184\pi$
$908$	10269.0	0.375318
$909$	0	0
$910$	6909.95	0.251717
$911$	−24800.0	−0.901934	−0.450967	−	0.892541i	$-0.648921\pi$
$911$	−24800.0	−0.901934	−0.450967	+	0.892541i	$0.648921\pi$
$912$	0	0
$913$	−3060.99	−0.110957
$914$	40959.7	1.48230
$915$	0	0
$916$	−19491.3	−0.703070
$917$	3503.80	0.126178
$918$	0	0
$919$	−6597.90	−0.236828	−0.118414	−	0.992964i	$-0.537781\pi$
$919$	−6597.90	−0.236828	−0.118414	+	0.992964i	$0.537781\pi$
$920$	3103.43	0.111214
$921$	0	0
$922$	−37613.4	−1.34353
$923$	11299.6	0.402959
$924$	0	0
$925$	17558.0	0.624113
$926$	20917.7	0.742331
$927$	0	0
$928$	24206.8	0.856281
$929$	−15056.0	−0.531724	−0.265862	−	0.964011i	$-0.585656\pi$
$929$	−15056.0	−0.531724	−0.265862	+	0.964011i	$0.585656\pi$
$930$	0	0
$931$	46048.1	1.62102
$932$	12678.9	0.445612
$933$	0	0
$934$	49487.1	1.73369
$935$	−6291.85	−0.220070
$936$	0	0
$937$	−35777.0	−1.24737	−0.623683	−	0.781677i	$-0.714365\pi$
$937$	−35777.0	−1.24737	−0.623683	+	0.781677i	$0.714365\pi$
$938$	1414.00	0.0492205
$939$	0	0
$940$	−2783.90	−0.0965966
$941$	22973.6	0.795873	0.397937	−	0.917413i	$-0.369726\pi$
$941$	22973.6	0.795873	0.397937	+	0.917413i	$0.369726\pi$
$942$	0	0
$943$	24052.9	0.830615
$944$	−17505.3	−0.603549
$945$	0	0
$946$	24400.5	0.838614
$947$	51038.3	1.75134	0.875671	−	0.482908i	$-0.160420\pi$
$947$	51038.3	1.75134	0.875671	+	0.482908i	$0.160420\pi$
$948$	0	0
$949$	−14287.9	−0.488731
$950$	−19203.1	−0.655823
$951$	0	0
$952$	−23346.0	−0.794799
$953$	22586.5	0.767733	0.383866	−	0.923389i	$-0.374592\pi$
$953$	22586.5	0.767733	0.383866	+	0.923389i	$0.374592\pi$
$954$	0	0
$955$	−3067.16	−0.103928
$956$	30259.9	1.02372
$957$	0	0
$958$	−62477.6	−2.10706
$959$	−95342.8	−3.21041
$960$	0	0
$961$	−8025.32	−0.269387
$962$	−7770.41	−0.260424
$963$	0	0
$964$	28374.1	0.947996
$965$	−4907.64	−0.163712
$966$	0	0
$967$	−36678.7	−1.21976	−0.609880	−	0.792494i	$-0.708782\pi$
$967$	−36678.7	−1.21976	−0.609880	+	0.792494i	$0.708782\pi$
$968$	7357.24	0.244288
$969$	0	0
$970$	12537.4	0.415003
$971$	−32635.2	−1.07859	−0.539296	−	0.842116i	$-0.681310\pi$
$971$	−32635.2	−1.07859	−0.539296	+	0.842116i	$0.681310\pi$
$972$	0	0
$973$	71491.0	2.35549
$974$	−19820.0	−0.652026
$975$	0	0
$976$	−8299.54	−0.272194
$977$	−44432.6	−1.45499	−0.727496	−	0.686112i	$-0.759316\pi$
$977$	−44432.6	−1.45499	−0.727496	+	0.686112i	$0.759316\pi$
$978$	0	0
$979$	20498.2	0.669178
$980$	22868.6	0.745418
$981$	0	0
$982$	−60515.6	−1.96653
$983$	−484.485	−0.0157199	−0.00785996	−	0.999969i	$-0.502502\pi$
$983$	−484.485	−0.0157199	−0.00785996	+	0.999969i	$0.502502\pi$
$984$	0	0
$985$	10853.2	0.351079
$986$	−33948.3	−1.09649
$987$	0	0
$988$	3626.15	0.116764
$989$	35360.2	1.13689
$990$	0	0
$991$	−48017.1	−1.53917	−0.769583	−	0.638546i	$-0.779536\pi$
$991$	−48017.1	−1.53917	−0.769583	+	0.638546i	$0.779536\pi$
$992$	32963.4	1.05503
$993$	0	0
$994$	118229.	3.77263
$995$	5824.25	0.185569
$996$	0	0
$997$	−26561.9	−0.843755	−0.421877	−	0.906653i	$-0.638629\pi$
$997$	−26561.9	−0.843755	−0.421877	+	0.906653i	$0.638629\pi$
$998$	16408.9	0.520455
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.a.f.1.3		3
3.2	odd	2		39.4.a.c.1.1	✓	3
4.3	odd	2		1872.4.a.bk.1.2		3
12.11	even	2		624.4.a.t.1.2		3
13.12	even	2		1521.4.a.u.1.1		3
15.14	odd	2		975.4.a.l.1.3		3
21.20	even	2		1911.4.a.k.1.1		3
24.5	odd	2		2496.4.a.bl.1.2		3
24.11	even	2		2496.4.a.bp.1.2		3
39.5	even	4		507.4.b.g.337.5		6
39.8	even	4		507.4.b.g.337.2		6
39.38	odd	2		507.4.a.h.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.1	✓	3	3.2	odd	2
117.4.a.f.1.3		3	1.1	even	1	trivial
507.4.a.h.1.3		3	39.38	odd	2
507.4.b.g.337.2		6	39.8	even	4
507.4.b.g.337.5		6	39.5	even	4
624.4.a.t.1.2		3	12.11	even	2
975.4.a.l.1.3		3	15.14	odd	2
1521.4.a.u.1.1		3	13.12	even	2
1872.4.a.bk.1.2		3	4.3	odd	2
1911.4.a.k.1.1		3	21.20	even	2
2496.4.a.bl.1.2		3	24.5	odd	2
2496.4.a.bp.1.2		3	24.11	even	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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)

\(f(q)\)	\(=\)	\(q+3.73549 q^{2} +5.95388 q^{4} +3.90776 q^{5} +36.4129 q^{7} -7.64325 q^{8} +O(q^{10})\)	\(q+3.73549 q^{2} +5.95388 q^{4} +3.90776 q^{5} +36.4129 q^{7} -7.64325 q^{8} +14.5974 q^{10} -19.1943 q^{11} +13.0000 q^{13} +136.020 q^{14} -76.1823 q^{16} +83.8839 q^{17} +46.8492 q^{19} +23.2664 q^{20} -71.7000 q^{22} -103.905 q^{23} -109.729 q^{25} +48.5614 q^{26} +216.798 q^{28} -108.341 q^{29} -147.532 q^{31} -223.432 q^{32} +313.347 q^{34} +142.293 q^{35} -160.012 q^{37} +175.005 q^{38} -29.8680 q^{40} -231.490 q^{41} -340.314 q^{43} -114.280 q^{44} -388.135 q^{46} -119.653 q^{47} +982.902 q^{49} -409.893 q^{50} +77.4005 q^{52} +732.879 q^{53} -75.0067 q^{55} -278.313 q^{56} -404.706 q^{58} +229.782 q^{59} +108.943 q^{61} -551.104 q^{62} -225.170 q^{64} +50.8009 q^{65} +10.3955 q^{67} +499.435 q^{68} +531.535 q^{70} +869.201 q^{71} -1099.07 q^{73} -597.724 q^{74} +278.934 q^{76} -698.920 q^{77} +140.410 q^{79} -297.703 q^{80} -864.729 q^{82} +159.474 q^{83} +327.799 q^{85} -1271.24 q^{86} +146.707 q^{88} -1067.93 q^{89} +473.368 q^{91} -618.636 q^{92} -446.964 q^{94} +183.075 q^{95} +858.881 q^{97} +3671.62 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 30 q^{7} + 6 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 + 10 * q^4 - 4 * q^5 + 30 * q^7 + 6 * q^8	\( 3 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 30 q^{7} + 6 q^{8} - 4 q^{10} + 16 q^{11} + 39 q^{13} + 176 q^{14} - 110 q^{16} + 146 q^{17} + 94 q^{19} + 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 26 q^{26} + 80 q^{28} + 2 q^{29} + 302 q^{31} - 154 q^{32} + 164 q^{34} - 80 q^{35} + 374 q^{37} - 312 q^{38} - 516 q^{40} - 480 q^{41} - 260 q^{43} - 712 q^{44} - 1104 q^{46} + 24 q^{47} + 447 q^{49} - 814 q^{50} + 130 q^{52} + 678 q^{53} - 1552 q^{55} - 96 q^{56} - 628 q^{58} + 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 750 q^{64} - 52 q^{65} + 74 q^{67} + 460 q^{68} + 1216 q^{70} + 948 q^{71} - 222 q^{73} - 1724 q^{74} + 2392 q^{76} - 112 q^{77} - 24 q^{79} - 1100 q^{80} + 564 q^{82} + 796 q^{83} - 248 q^{85} - 1800 q^{86} + 1608 q^{88} - 1436 q^{89} + 390 q^{91} + 1296 q^{92} - 1920 q^{94} + 4032 q^{95} + 3242 q^{97} + 5070 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 10 * q^4 - 4 * q^5 + 30 * q^7 + 6 * q^8 - 4 * q^10 + 16 * q^11 + 39 * q^13 + 176 * q^14 - 110 * q^16 + 146 * q^17 + 94 * q^19 + 244 * q^20 - 56 * q^22 + 48 * q^23 + 145 * q^25 - 26 * q^26 + 80 * q^28 + 2 * q^29 + 302 * q^31 - 154 * q^32 + 164 * q^34 - 80 * q^35 + 374 * q^37 - 312 * q^38 - 516 * q^40 - 480 * q^41 - 260 * q^43 - 712 * q^44 - 1104 * q^46 + 24 * q^47 + 447 * q^49 - 814 * q^50 + 130 * q^52 + 678 * q^53 - 1552 * q^55 - 96 * q^56 - 628 * q^58 + 1788 * q^59 + 230 * q^61 - 1952 * q^62 - 750 * q^64 - 52 * q^65 + 74 * q^67 + 460 * q^68 + 1216 * q^70 + 948 * q^71 - 222 * q^73 - 1724 * q^74 + 2392 * q^76 - 112 * q^77 - 24 * q^79 - 1100 * q^80 + 564 * q^82 + 796 * q^83 - 248 * q^85 - 1800 * q^86 + 1608 * q^88 - 1436 * q^89 + 390 * q^91 + 1296 * q^92 - 1920 * q^94 + 4032 * q^95 + 3242 * q^97 + 5070 * q^98

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