LMFDB - Embedded newform 2166.4.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,4,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, 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("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2166.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:	$127.798137072$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1524.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 1 $ x^3 - x^2 - 7*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.140435$ of defining polynomial
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +20.2020 q^{5} +6.00000 q^{6} -22.8872 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +20.2020 q^{5} +6.00000 q^{6} -22.8872 q^{7} -8.00000 q^{8} +9.00000 q^{9} -40.4040 q^{10} -57.5614 q^{11} -12.0000 q^{12} +27.3148 q^{13} +45.7744 q^{14} -60.6059 q^{15} +16.0000 q^{16} -73.4486 q^{17} -18.0000 q^{18} +80.8079 q^{20} +68.6616 q^{21} +115.123 q^{22} -188.584 q^{23} +24.0000 q^{24} +283.120 q^{25} -54.6296 q^{26} -27.0000 q^{27} -91.5488 q^{28} -7.15594 q^{29} +121.212 q^{30} -117.144 q^{31} -32.0000 q^{32} +172.684 q^{33} +146.897 q^{34} -462.367 q^{35} +36.0000 q^{36} -332.600 q^{37} -81.9443 q^{39} -161.616 q^{40} -42.7158 q^{41} -137.323 q^{42} +47.2469 q^{43} -230.245 q^{44} +181.818 q^{45} +377.168 q^{46} -507.995 q^{47} -48.0000 q^{48} +180.824 q^{49} -566.240 q^{50} +220.346 q^{51} +109.259 q^{52} +492.609 q^{53} +54.0000 q^{54} -1162.85 q^{55} +183.098 q^{56} +14.3119 q^{58} +460.041 q^{59} -242.424 q^{60} +450.745 q^{61} +234.287 q^{62} -205.985 q^{63} +64.0000 q^{64} +551.813 q^{65} -345.368 q^{66} +522.465 q^{67} -293.794 q^{68} +565.751 q^{69} +924.734 q^{70} +931.419 q^{71} -72.0000 q^{72} +350.123 q^{73} +665.199 q^{74} -849.360 q^{75} +1317.42 q^{77} +163.889 q^{78} +194.868 q^{79} +323.232 q^{80} +81.0000 q^{81} +85.4317 q^{82} -286.083 q^{83} +274.646 q^{84} -1483.81 q^{85} -94.4937 q^{86} +21.4678 q^{87} +460.491 q^{88} +839.690 q^{89} -363.636 q^{90} -625.159 q^{91} -754.335 q^{92} +351.431 q^{93} +1015.99 q^{94} +96.0000 q^{96} +605.516 q^{97} -361.648 q^{98} -518.052 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} + 2 q^{5} + 18 q^{6} - 17 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 + 2 * q^5 + 18 * q^6 - 17 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} + 2 q^{5} + 18 q^{6} - 17 q^{7} - 24 q^{8} + 27 q^{9} - 4 q^{10} - 52 q^{11} - 36 q^{12} + 75 q^{13} + 34 q^{14} - 6 q^{15} + 48 q^{16} - 48 q^{17} - 54 q^{18} + 8 q^{20} + 51 q^{21} + 104 q^{22} - 238 q^{23} + 72 q^{24} + 229 q^{25} - 150 q^{26} - 81 q^{27} - 68 q^{28} - 8 q^{29} + 12 q^{30} + 107 q^{31} - 96 q^{32} + 156 q^{33} + 96 q^{34} - 294 q^{35} + 108 q^{36} + 305 q^{37} - 225 q^{39} - 16 q^{40} + 16 q^{41} - 102 q^{42} - 331 q^{43} - 208 q^{44} + 18 q^{45} + 476 q^{46} - 766 q^{47} - 144 q^{48} + 1142 q^{49} - 458 q^{50} + 144 q^{51} + 300 q^{52} - 118 q^{53} + 162 q^{54} - 1400 q^{55} + 136 q^{56} + 16 q^{58} + 936 q^{59} - 24 q^{60} - 399 q^{61} - 214 q^{62} - 153 q^{63} + 192 q^{64} + 370 q^{65} - 312 q^{66} + 61 q^{67} - 192 q^{68} + 714 q^{69} + 588 q^{70} + 974 q^{71} - 216 q^{72} + 91 q^{73} - 610 q^{74} - 687 q^{75} - 36 q^{77} + 450 q^{78} - 321 q^{79} + 32 q^{80} + 243 q^{81} - 32 q^{82} - 2148 q^{83} + 204 q^{84} - 1680 q^{85} + 662 q^{86} + 24 q^{87} + 416 q^{88} + 1116 q^{89} - 36 q^{90} + 1367 q^{91} - 952 q^{92} - 321 q^{93} + 1532 q^{94} + 288 q^{96} + 1382 q^{97} - 2284 q^{98} - 468 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 + 2 * q^5 + 18 * q^6 - 17 * q^7 - 24 * q^8 + 27 * q^9 - 4 * q^10 - 52 * q^11 - 36 * q^12 + 75 * q^13 + 34 * q^14 - 6 * q^15 + 48 * q^16 - 48 * q^17 - 54 * q^18 + 8 * q^20 + 51 * q^21 + 104 * q^22 - 238 * q^23 + 72 * q^24 + 229 * q^25 - 150 * q^26 - 81 * q^27 - 68 * q^28 - 8 * q^29 + 12 * q^30 + 107 * q^31 - 96 * q^32 + 156 * q^33 + 96 * q^34 - 294 * q^35 + 108 * q^36 + 305 * q^37 - 225 * q^39 - 16 * q^40 + 16 * q^41 - 102 * q^42 - 331 * q^43 - 208 * q^44 + 18 * q^45 + 476 * q^46 - 766 * q^47 - 144 * q^48 + 1142 * q^49 - 458 * q^50 + 144 * q^51 + 300 * q^52 - 118 * q^53 + 162 * q^54 - 1400 * q^55 + 136 * q^56 + 16 * q^58 + 936 * q^59 - 24 * q^60 - 399 * q^61 - 214 * q^62 - 153 * q^63 + 192 * q^64 + 370 * q^65 - 312 * q^66 + 61 * q^67 - 192 * q^68 + 714 * q^69 + 588 * q^70 + 974 * q^71 - 216 * q^72 + 91 * q^73 - 610 * q^74 - 687 * q^75 - 36 * q^77 + 450 * q^78 - 321 * q^79 + 32 * q^80 + 243 * q^81 - 32 * q^82 - 2148 * q^83 + 204 * q^84 - 1680 * q^85 + 662 * q^86 + 24 * q^87 + 416 * q^88 + 1116 * q^89 - 36 * q^90 + 1367 * q^91 - 952 * q^92 - 321 * q^93 + 1532 * q^94 + 288 * q^96 + 1382 * q^97 - 2284 * q^98 - 468 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	20.2020	1.80692	0.903460	−	0.428672i	$-0.141018\pi$
$5$	20.2020	1.80692	0.903460	+	0.428672i	$0.141018\pi$
$6$	6.00000	0.408248
$7$	−22.8872	−1.23579	−0.617897	−	0.786259i	$-0.712015\pi$
$7$	−22.8872	−1.23579	−0.617897	+	0.786259i	$0.712015\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−40.4040	−1.27769
$11$	−57.5614	−1.57776	−0.788882	−	0.614545i	$-0.789340\pi$
$11$	−57.5614	−1.57776	−0.788882	+	0.614545i	$0.789340\pi$
$12$	−12.0000	−0.288675
$13$	27.3148	0.582750	0.291375	−	0.956609i	$-0.405887\pi$
$13$	27.3148	0.582750	0.291375	+	0.956609i	$0.405887\pi$
$14$	45.7744	0.873838
$15$	−60.6059	−1.04323
$16$	16.0000	0.250000
$17$	−73.4486	−1.04788	−0.523938	−	0.851756i	$-0.675538\pi$
$17$	−73.4486	−1.04788	−0.523938	+	0.851756i	$0.675538\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	80.8079	0.903460
$21$	68.6616	0.713485
$22$	115.123	1.11565
$23$	−188.584	−1.70967	−0.854836	−	0.518899i	$-0.826342\pi$
$23$	−188.584	−1.70967	−0.854836	+	0.518899i	$0.826342\pi$
$24$	24.0000	0.204124
$25$	283.120	2.26496
$26$	−54.6296	−0.412067
$27$	−27.0000	−0.192450
$28$	−91.5488	−0.617897
$29$	−7.15594	−0.0458215	−0.0229108	−	0.999738i	$-0.507293\pi$
$29$	−7.15594	−0.0458215	−0.0229108	+	0.999738i	$0.507293\pi$
$30$	121.212	0.737672
$31$	−117.144	−0.678698	−0.339349	−	0.940661i	$-0.610207\pi$
$31$	−117.144	−0.678698	−0.339349	+	0.940661i	$0.610207\pi$
$32$	−32.0000	−0.176777
$33$	172.684	0.910922
$34$	146.897	0.740960
$35$	−462.367	−2.23298
$36$	36.0000	0.166667
$37$	−332.600	−1.47781	−0.738906	−	0.673809i	$-0.764657\pi$
$37$	−332.600	−1.47781	−0.738906	+	0.673809i	$0.764657\pi$
$38$	0	0
$39$	−81.9443	−0.336451
$40$	−161.616	−0.638843
$41$	−42.7158	−0.162710	−0.0813548	−	0.996685i	$-0.525925\pi$
$41$	−42.7158	−0.162710	−0.0813548	+	0.996685i	$0.525925\pi$
$42$	−137.323	−0.504510
$43$	47.2469	0.167560	0.0837800	−	0.996484i	$-0.473301\pi$
$43$	47.2469	0.167560	0.0837800	+	0.996484i	$0.473301\pi$
$44$	−230.245	−0.788882
$45$	181.818	0.602307
$46$	377.168	1.20892
$47$	−507.995	−1.57657	−0.788284	−	0.615312i	$-0.789030\pi$
$47$	−507.995	−1.57657	−0.788284	+	0.615312i	$0.789030\pi$
$48$	−48.0000	−0.144338
$49$	180.824	0.527184
$50$	−566.240	−1.60157
$51$	220.346	0.604991
$52$	109.259	0.291375
$53$	492.609	1.27670	0.638350	−	0.769746i	$-0.279617\pi$
$53$	492.609	1.27670	0.638350	+	0.769746i	$0.279617\pi$
$54$	54.0000	0.136083
$55$	−1162.85	−2.85089
$56$	183.098	0.436919
$57$	0	0
$58$	14.3119	0.0324007
$59$	460.041	1.01512	0.507561	−	0.861616i	$-0.330547\pi$
$59$	460.041	1.01512	0.507561	+	0.861616i	$0.330547\pi$
$60$	−242.424	−0.521613
$61$	450.745	0.946097	0.473049	−	0.881036i	$-0.343153\pi$
$61$	450.745	0.946097	0.473049	+	0.881036i	$0.343153\pi$
$62$	234.287	0.479912
$63$	−205.985	−0.411931
$64$	64.0000	0.125000
$65$	551.813	1.05298
$66$	−345.368	−0.644119
$67$	522.465	0.952675	0.476337	−	0.879263i	$-0.341964\pi$
$67$	522.465	0.952675	0.476337	+	0.879263i	$0.341964\pi$
$68$	−293.794	−0.523938
$69$	565.751	0.987079
$70$	924.734	1.57895
$71$	931.419	1.55689	0.778444	−	0.627714i	$-0.216009\pi$
$71$	931.419	1.55689	0.778444	+	0.627714i	$0.216009\pi$
$72$	−72.0000	−0.117851
$73$	350.123	0.561354	0.280677	−	0.959802i	$-0.409441\pi$
$73$	350.123	0.561354	0.280677	+	0.959802i	$0.409441\pi$
$74$	665.199	1.04497
$75$	−849.360	−1.30768
$76$	0	0
$77$	1317.42	1.94979
$78$	163.889	0.237907
$79$	194.868	0.277524	0.138762	−	0.990326i	$-0.455688\pi$
$79$	194.868	0.277524	0.138762	+	0.990326i	$0.455688\pi$
$80$	323.232	0.451730
$81$	81.0000	0.111111
$82$	85.4317	0.115053
$83$	−286.083	−0.378333	−0.189167	−	0.981945i	$-0.560579\pi$
$83$	−286.083	−0.378333	−0.189167	+	0.981945i	$0.560579\pi$
$84$	274.646	0.356743
$85$	−1483.81	−1.89343
$86$	−94.4937	−0.118483
$87$	21.4678	0.0264551
$88$	460.491	0.557824
$89$	839.690	1.00008	0.500039	−	0.866003i	$-0.333319\pi$
$89$	839.690	1.00008	0.500039	+	0.866003i	$0.333319\pi$
$90$	−363.636	−0.425895
$91$	−625.159	−0.720159
$92$	−754.335	−0.854836
$93$	351.431	0.391846
$94$	1015.99	1.11480
$95$	0	0
$96$	96.0000	0.102062
$97$	605.516	0.633823	0.316911	−	0.948455i	$-0.397354\pi$
$97$	605.516	0.633823	0.316911	+	0.948455i	$0.397354\pi$
$98$	−361.648	−0.372776
$99$	−518.052	−0.525921
$100$	1132.48	1.13248
$101$	47.2760	0.0465756	0.0232878	−	0.999729i	$-0.492587\pi$
$101$	47.2760	0.0465756	0.0232878	+	0.999729i	$0.492587\pi$
$102$	−440.691	−0.427794
$103$	−430.750	−0.412069	−0.206034	−	0.978545i	$-0.566056\pi$
$103$	−430.750	−0.412069	−0.206034	+	0.978545i	$0.566056\pi$
$104$	−218.518	−0.206033
$105$	1387.10	1.28921
$106$	−985.219	−0.902763
$107$	1055.78	0.953893	0.476946	−	0.878932i	$-0.341744\pi$
$107$	1055.78	0.953893	0.476946	+	0.878932i	$0.341744\pi$
$108$	−108.000	−0.0962250
$109$	−282.571	−0.248307	−0.124153	−	0.992263i	$-0.539621\pi$
$109$	−282.571	−0.248307	−0.124153	+	0.992263i	$0.539621\pi$
$110$	2325.71	2.01589
$111$	997.799	0.853215
$112$	−366.195	−0.308948
$113$	−841.860	−0.700846	−0.350423	−	0.936592i	$-0.613962\pi$
$113$	−841.860	−0.700846	−0.350423	+	0.936592i	$0.613962\pi$
$114$	0	0
$115$	−3809.77	−3.08924
$116$	−28.6238	−0.0229108
$117$	245.833	0.194250
$118$	−920.081	−0.717799
$119$	1681.03	1.29496
$120$	484.848	0.368836
$121$	1982.31	1.48934
$122$	−901.489	−0.668992
$123$	128.148	0.0939404
$124$	−468.575	−0.339349
$125$	3194.34	2.28568
$126$	411.970	0.291279
$127$	−955.393	−0.667539	−0.333769	−	0.942655i	$-0.608321\pi$
$127$	−955.393	−0.667539	−0.333769	+	0.942655i	$0.608321\pi$
$128$	−128.000	−0.0883883
$129$	−141.741	−0.0967408
$130$	−1103.63	−0.744572
$131$	−1639.11	−1.09320	−0.546601	−	0.837393i	$-0.684079\pi$
$131$	−1639.11	−1.09320	−0.546601	+	0.837393i	$0.684079\pi$
$132$	690.736	0.455461
$133$	0	0
$134$	−1044.93	−0.673643
$135$	−545.454	−0.347742
$136$	587.588	0.370480
$137$	−847.711	−0.528649	−0.264324	−	0.964434i	$-0.585149\pi$
$137$	−847.711	−0.528649	−0.264324	+	0.964434i	$0.585149\pi$
$138$	−1131.50	−0.697970
$139$	1167.60	0.712481	0.356240	−	0.934394i	$-0.384058\pi$
$139$	1167.60	0.712481	0.356240	+	0.934394i	$0.384058\pi$
$140$	−1849.47	−1.11649
$141$	1523.98	0.910231
$142$	−1862.84	−1.10089
$143$	−1572.28	−0.919442
$144$	144.000	0.0833333
$145$	−144.564	−0.0827959
$146$	−700.247	−0.396937
$147$	−542.473	−0.304370
$148$	−1330.40	−0.738906
$149$	−1721.57	−0.946552	−0.473276	−	0.880914i	$-0.656929\pi$
$149$	−1721.57	−0.946552	−0.473276	+	0.880914i	$0.656929\pi$
$150$	1698.72	0.924666
$151$	2327.14	1.25417	0.627085	−	0.778951i	$-0.284248\pi$
$151$	2327.14	1.25417	0.627085	+	0.778951i	$0.284248\pi$
$152$	0	0
$153$	−661.037	−0.349292
$154$	−2634.84	−1.37871
$155$	−2366.53	−1.22635
$156$	−327.777	−0.168226
$157$	−954.679	−0.485297	−0.242649	−	0.970114i	$-0.578016\pi$
$157$	−954.679	−0.485297	−0.242649	+	0.970114i	$0.578016\pi$
$158$	−389.736	−0.196239
$159$	−1477.83	−0.737103
$160$	−646.463	−0.319421
$161$	4316.16	2.11280
$162$	−162.000	−0.0785674
$163$	1826.52	0.877696	0.438848	−	0.898561i	$-0.355387\pi$
$163$	1826.52	0.877696	0.438848	+	0.898561i	$0.355387\pi$
$164$	−170.863	−0.0813548
$165$	3488.56	1.64596
$166$	572.166	0.267522
$167$	1465.61	0.679116	0.339558	−	0.940585i	$-0.389722\pi$
$167$	1465.61	0.679116	0.339558	+	0.940585i	$0.389722\pi$
$168$	−549.293	−0.252255
$169$	−1450.90	−0.660402
$170$	2967.61	1.33886
$171$	0	0
$172$	188.987	0.0837800
$173$	4162.86	1.82946	0.914730	−	0.404065i	$-0.132403\pi$
$173$	4162.86	1.82946	0.914730	+	0.404065i	$0.132403\pi$
$174$	−42.9356	−0.0187066
$175$	−6479.83	−2.79902
$176$	−920.982	−0.394441
$177$	−1380.12	−0.586081
$178$	−1679.38	−0.707162
$179$	−2637.74	−1.10142	−0.550710	−	0.834697i	$-0.685643\pi$
$179$	−2637.74	−1.10142	−0.550710	+	0.834697i	$0.685643\pi$
$180$	727.271	0.301153
$181$	−779.832	−0.320246	−0.160123	−	0.987097i	$-0.551189\pi$
$181$	−779.832	−0.320246	−0.160123	+	0.987097i	$0.551189\pi$
$182$	1250.32	0.509229
$183$	−1352.23	−0.546230
$184$	1508.67	0.604460
$185$	−6719.17	−2.67029
$186$	−702.862	−0.277077
$187$	4227.80	1.65330
$188$	−2031.98	−0.788284
$189$	617.955	0.237828
$190$	0	0
$191$	3961.70	1.50083	0.750415	−	0.660967i	$-0.229854\pi$
$191$	3961.70	1.50083	0.750415	+	0.660967i	$0.229854\pi$
$192$	−192.000	−0.0721688
$193$	−364.405	−0.135909	−0.0679545	−	0.997688i	$-0.521647\pi$
$193$	−364.405	−0.135909	−0.0679545	+	0.997688i	$0.521647\pi$
$194$	−1211.03	−0.448180
$195$	−1655.44	−0.607940
$196$	723.297	0.263592
$197$	2309.32	0.835190	0.417595	−	0.908633i	$-0.362873\pi$
$197$	2309.32	0.835190	0.417595	+	0.908633i	$0.362873\pi$
$198$	1036.10	0.371882
$199$	3676.55	1.30967	0.654833	−	0.755774i	$-0.272739\pi$
$199$	3676.55	1.30967	0.654833	+	0.755774i	$0.272739\pi$
$200$	−2264.96	−0.800785
$201$	−1567.39	−0.550027
$202$	−94.5519	−0.0329339
$203$	163.780	0.0566259
$204$	881.383	0.302496
$205$	−862.945	−0.294003
$206$	861.501	0.291377
$207$	−1697.25	−0.569890
$208$	437.036	0.145688
$209$	0	0
$210$	−2774.20	−0.911610
$211$	1342.60	0.438051	0.219025	−	0.975719i	$-0.429712\pi$
$211$	1342.60	0.438051	0.219025	+	0.975719i	$0.429712\pi$
$212$	1970.44	0.638350
$213$	−2794.26	−0.898870
$214$	−2111.57	−0.674504
$215$	954.480	0.302767
$216$	216.000	0.0680414
$217$	2681.09	0.838730
$218$	565.143	0.175579
$219$	−1050.37	−0.324098
$220$	−4651.41	−1.42545
$221$	−2006.23	−0.610650
$222$	−1995.60	−0.603314
$223$	−2239.58	−0.672528	−0.336264	−	0.941768i	$-0.609163\pi$
$223$	−2239.58	−0.672528	−0.336264	+	0.941768i	$0.609163\pi$
$224$	732.391	0.218459
$225$	2548.08	0.754987
$226$	1683.72	0.495573
$227$	−6233.41	−1.82258	−0.911291	−	0.411763i	$-0.864913\pi$
$227$	−6233.41	−1.82258	−0.911291	+	0.411763i	$0.864913\pi$
$228$	0	0
$229$	−2291.74	−0.661320	−0.330660	−	0.943750i	$-0.607271\pi$
$229$	−2291.74	−0.661320	−0.330660	+	0.943750i	$0.607271\pi$
$230$	7619.53	2.18442
$231$	−3952.26	−1.12571
$232$	57.2475	0.0162004
$233$	1599.07	0.449609	0.224804	−	0.974404i	$-0.427826\pi$
$233$	1599.07	0.449609	0.224804	+	0.974404i	$0.427826\pi$
$234$	−491.666	−0.137356
$235$	−10262.5	−2.84873
$236$	1840.16	0.507561
$237$	−584.604	−0.160228
$238$	−3362.06	−0.915673
$239$	−2443.37	−0.661291	−0.330646	−	0.943755i	$-0.607266\pi$
$239$	−2443.37	−0.661291	−0.330646	+	0.943755i	$0.607266\pi$
$240$	−969.695	−0.260806
$241$	−996.491	−0.266347	−0.133173	−	0.991093i	$-0.542517\pi$
$241$	−996.491	−0.266347	−0.133173	+	0.991093i	$0.542517\pi$
$242$	−3964.62	−1.05312
$243$	−243.000	−0.0641500
$244$	1802.98	0.473049
$245$	3653.01	0.952580
$246$	−256.295	−0.0664259
$247$	0	0
$248$	937.150	0.239956
$249$	858.248	0.218431
$250$	−6388.68	−1.61622
$251$	759.890	0.191091	0.0955455	−	0.995425i	$-0.469540\pi$
$251$	759.890	0.191091	0.0955455	+	0.995425i	$0.469540\pi$
$252$	−823.939	−0.205966
$253$	10855.1	2.69746
$254$	1910.79	0.472021
$255$	4451.42	1.09317
$256$	256.000	0.0625000
$257$	−4773.20	−1.15854	−0.579268	−	0.815137i	$-0.696662\pi$
$257$	−4773.20	−1.15854	−0.579268	+	0.815137i	$0.696662\pi$
$258$	283.481	0.0684061
$259$	7612.27	1.82627
$260$	2207.25	0.526492
$261$	−64.4035	−0.0152738
$262$	3278.22	0.773011
$263$	4356.43	1.02140	0.510702	−	0.859758i	$-0.329386\pi$
$263$	4356.43	1.02140	0.510702	+	0.859758i	$0.329386\pi$
$264$	−1381.47	−0.322060
$265$	9951.69	2.30690
$266$	0	0
$267$	−2519.07	−0.577396
$268$	2089.86	0.476337
$269$	−140.715	−0.0318941	−0.0159471	−	0.999873i	$-0.505076\pi$
$269$	−140.715	−0.0318941	−0.0159471	+	0.999873i	$0.505076\pi$
$270$	1090.91	0.245891
$271$	1523.06	0.341399	0.170700	−	0.985323i	$-0.445397\pi$
$271$	1523.06	0.341399	0.170700	+	0.985323i	$0.445397\pi$
$272$	−1175.18	−0.261969
$273$	1875.48	0.415784
$274$	1695.42	0.373811
$275$	−16296.8	−3.57357
$276$	2263.01	0.493540
$277$	2217.54	0.481008	0.240504	−	0.970648i	$-0.422687\pi$
$277$	2217.54	0.481008	0.240504	+	0.970648i	$0.422687\pi$
$278$	−2335.21	−0.503800
$279$	−1054.29	−0.226233
$280$	3698.94	0.789477
$281$	170.520	0.0362005	0.0181003	−	0.999836i	$-0.494238\pi$
$281$	170.520	0.0362005	0.0181003	+	0.999836i	$0.494238\pi$
$282$	−3047.97	−0.643631
$283$	2176.30	0.457130	0.228565	−	0.973529i	$-0.426597\pi$
$283$	2176.30	0.457130	0.228565	+	0.973529i	$0.426597\pi$
$284$	3725.68	0.778444
$285$	0	0
$286$	3144.55	0.650144
$287$	977.646	0.201075
$288$	−288.000	−0.0589256
$289$	481.691	0.0980441
$290$	289.128	0.0585455
$291$	−1816.55	−0.365938
$292$	1400.49	0.280677
$293$	−9879.00	−1.96975	−0.984876	−	0.173261i	$-0.944569\pi$
$293$	−9879.00	−1.96975	−0.984876	+	0.173261i	$0.944569\pi$
$294$	1084.95	0.215222
$295$	9293.73	1.83424
$296$	2660.80	0.522485
$297$	1554.16	0.303641
$298$	3443.13	0.669313
$299$	−5151.12	−0.996312
$300$	−3397.44	−0.653838
$301$	−1081.35	−0.207069
$302$	−4654.27	−0.886832
$303$	−141.828	−0.0268904
$304$	0	0
$305$	9105.94	1.70952
$306$	1322.07	0.246987
$307$	−1343.71	−0.249803	−0.124902	−	0.992169i	$-0.539862\pi$
$307$	−1343.71	−0.249803	−0.124902	+	0.992169i	$0.539862\pi$
$308$	5269.67	0.974895
$309$	1292.25	0.237908
$310$	4733.07	0.867162
$311$	88.3559	0.0161100	0.00805499	−	0.999968i	$-0.497436\pi$
$311$	88.3559	0.0161100	0.00805499	+	0.999968i	$0.497436\pi$
$312$	655.555	0.118953
$313$	−1892.46	−0.341751	−0.170876	−	0.985293i	$-0.554660\pi$
$313$	−1892.46	−0.341751	−0.170876	+	0.985293i	$0.554660\pi$
$314$	1909.36	0.343157
$315$	−4161.30	−0.744326
$316$	779.472	0.138762
$317$	8378.86	1.48455	0.742277	−	0.670093i	$-0.233746\pi$
$317$	8378.86	1.48455	0.742277	+	0.670093i	$0.233746\pi$
$318$	2955.66	0.521211
$319$	411.906	0.0722956
$320$	1292.93	0.225865
$321$	−3167.35	−0.550730
$322$	−8632.31	−1.49398
$323$	0	0
$324$	324.000	0.0555556
$325$	7733.36	1.31991
$326$	−3653.05	−0.620625
$327$	847.714	0.143360
$328$	341.727	0.0575265
$329$	11626.6	1.94831
$330$	−6977.12	−1.16387
$331$	4833.50	0.802639	0.401319	−	0.915938i	$-0.368552\pi$
$331$	4833.50	0.802639	0.401319	+	0.915938i	$0.368552\pi$
$332$	−1144.33	−0.189167
$333$	−2993.40	−0.492604
$334$	−2931.22	−0.480208
$335$	10554.8	1.72141
$336$	1098.59	0.178371
$337$	−2625.77	−0.424436	−0.212218	−	0.977222i	$-0.568069\pi$
$337$	−2625.77	−0.424436	−0.212218	+	0.977222i	$0.568069\pi$
$338$	2901.81	0.466975
$339$	2525.58	0.404633
$340$	−5935.23	−0.946714
$341$	6742.95	1.07082
$342$	0	0
$343$	3711.75	0.584302
$344$	−377.975	−0.0592414
$345$	11429.3	1.78357
$346$	−8325.73	−1.29362
$347$	8041.49	1.24406	0.622031	−	0.782992i	$-0.286308\pi$
$347$	8041.49	1.24406	0.622031	+	0.782992i	$0.286308\pi$
$348$	85.8713	0.0132275
$349$	8603.95	1.31965	0.659826	−	0.751418i	$-0.270630\pi$
$349$	8603.95	1.31965	0.659826	+	0.751418i	$0.270630\pi$
$350$	12959.7	1.97921
$351$	−737.499	−0.112150
$352$	1841.96	0.278912
$353$	−10359.4	−1.56198	−0.780988	−	0.624547i	$-0.785284\pi$
$353$	−10359.4	−1.56198	−0.780988	+	0.624547i	$0.785284\pi$
$354$	2760.24	0.414422
$355$	18816.5	2.81317
$356$	3358.76	0.500039
$357$	−5043.10	−0.747644
$358$	5275.49	0.778821
$359$	−3544.68	−0.521117	−0.260558	−	0.965458i	$-0.583907\pi$
$359$	−3544.68	−0.521117	−0.260558	+	0.965458i	$0.583907\pi$
$360$	−1454.54	−0.212948
$361$	0	0
$362$	1559.66	0.226448
$363$	−5946.93	−0.859870
$364$	−2500.64	−0.360079
$365$	7073.18	1.01432
$366$	2704.47	0.386243
$367$	330.253	0.0469729	0.0234864	−	0.999724i	$-0.492523\pi$
$367$	330.253	0.0469729	0.0234864	+	0.999724i	$0.492523\pi$
$368$	−3017.34	−0.427418
$369$	−384.443	−0.0542365
$370$	13438.3	1.88818
$371$	−11274.5	−1.57774
$372$	1405.72	0.195923
$373$	−12016.2	−1.66802	−0.834012	−	0.551747i	$-0.813961\pi$
$373$	−12016.2	−1.66802	−0.834012	+	0.551747i	$0.813961\pi$
$374$	−8455.60	−1.16906
$375$	−9583.02	−1.31964
$376$	4063.96	0.557401
$377$	−195.463	−0.0267025
$378$	−1235.91	−0.168170
$379$	5463.94	0.740538	0.370269	−	0.928925i	$-0.379266\pi$
$379$	5463.94	0.740538	0.370269	+	0.928925i	$0.379266\pi$
$380$	0	0
$381$	2866.18	0.385404
$382$	−7923.40	−1.06125
$383$	7380.58	0.984673	0.492337	−	0.870405i	$-0.336143\pi$
$383$	7380.58	0.984673	0.492337	+	0.870405i	$0.336143\pi$
$384$	384.000	0.0510310
$385$	26614.5	3.52311
$386$	728.809	0.0961021
$387$	425.222	0.0558533
$388$	2422.06	0.316911
$389$	2035.83	0.265349	0.132675	−	0.991160i	$-0.457643\pi$
$389$	2035.83	0.265349	0.132675	+	0.991160i	$0.457643\pi$
$390$	3310.88	0.429879
$391$	13851.2	1.79152
$392$	−1446.59	−0.186388
$393$	4917.32	0.631161
$394$	−4618.64	−0.590568
$395$	3936.72	0.501463
$396$	−2072.21	−0.262961
$397$	13314.3	1.68318	0.841592	−	0.540114i	$-0.181619\pi$
$397$	13314.3	1.68318	0.841592	+	0.540114i	$0.181619\pi$
$398$	−7353.09	−0.926074
$399$	0	0
$400$	4529.92	0.566240
$401$	13204.3	1.64437	0.822186	−	0.569219i	$-0.192754\pi$
$401$	13204.3	1.64437	0.822186	+	0.569219i	$0.192754\pi$
$402$	3134.79	0.388928
$403$	−3199.75	−0.395511
$404$	189.104	0.0232878
$405$	1636.36	0.200769
$406$	−327.559	−0.0400406
$407$	19144.9	2.33164
$408$	−1762.77	−0.213897
$409$	2761.86	0.333900	0.166950	−	0.985965i	$-0.446608\pi$
$409$	2761.86	0.333900	0.166950	+	0.985965i	$0.446608\pi$
$410$	1725.89	0.207892
$411$	2543.13	0.305215
$412$	−1723.00	−0.206034
$413$	−10529.0	−1.25448
$414$	3394.51	0.402973
$415$	−5779.44	−0.683618
$416$	−874.073	−0.103017
$417$	−3502.81	−0.411351
$418$	0	0
$419$	−15182.0	−1.77015	−0.885073	−	0.465453i	$-0.845891\pi$
$419$	−15182.0	−1.77015	−0.885073	+	0.465453i	$0.845891\pi$
$420$	5548.40	0.644606
$421$	15980.4	1.84996	0.924982	−	0.380011i	$-0.124080\pi$
$421$	15980.4	1.84996	0.924982	+	0.380011i	$0.124080\pi$
$422$	−2685.21	−0.309749
$423$	−4571.95	−0.525522
$424$	−3940.88	−0.451382
$425$	−20794.8	−2.37340
$426$	5588.51	0.635597
$427$	−10316.3	−1.16918
$428$	4223.14	0.476946
$429$	4716.83	0.530840
$430$	−1908.96	−0.214089
$431$	4762.00	0.532198	0.266099	−	0.963946i	$-0.414265\pi$
$431$	4762.00	0.532198	0.266099	+	0.963946i	$0.414265\pi$
$432$	−432.000	−0.0481125
$433$	9974.90	1.10707	0.553537	−	0.832825i	$-0.313278\pi$
$433$	9974.90	1.10707	0.553537	+	0.832825i	$0.313278\pi$
$434$	−5362.18	−0.593072
$435$	433.693	0.0478022
$436$	−1130.29	−0.124153
$437$	0	0
$438$	2100.74	0.229172
$439$	−8956.89	−0.973779	−0.486890	−	0.873464i	$-0.661869\pi$
$439$	−8956.89	−0.973779	−0.486890	+	0.873464i	$0.661869\pi$
$440$	9302.83	1.00794
$441$	1627.42	0.175728
$442$	4012.46	0.431795
$443$	11368.1	1.21922	0.609612	−	0.792700i	$-0.291325\pi$
$443$	11368.1	1.21922	0.609612	+	0.792700i	$0.291325\pi$
$444$	3991.19	0.426607
$445$	16963.4	1.80706
$446$	4479.17	0.475549
$447$	5164.70	0.546492
$448$	−1464.78	−0.154474
$449$	7630.20	0.801985	0.400993	−	0.916081i	$-0.368665\pi$
$449$	7630.20	0.801985	0.400993	+	0.916081i	$0.368665\pi$
$450$	−5096.16	−0.533856
$451$	2458.78	0.256717
$452$	−3367.44	−0.350423
$453$	−6981.41	−0.724095
$454$	12466.8	1.28876
$455$	−12629.4	−1.30127
$456$	0	0
$457$	−1500.40	−0.153579	−0.0767896	−	0.997047i	$-0.524467\pi$
$457$	−1500.40	−0.153579	−0.0767896	+	0.997047i	$0.524467\pi$
$458$	4583.47	0.467624
$459$	1983.11	0.201664
$460$	−15239.1	−1.54462
$461$	−14495.8	−1.46450	−0.732252	−	0.681034i	$-0.761531\pi$
$461$	−14495.8	−1.46450	−0.732252	+	0.681034i	$0.761531\pi$
$462$	7904.51	0.795998
$463$	−12265.2	−1.23113	−0.615566	−	0.788086i	$-0.711072\pi$
$463$	−12265.2	−1.23113	−0.615566	+	0.788086i	$0.711072\pi$
$464$	−114.495	−0.0114554
$465$	7099.60	0.708035
$466$	−3198.15	−0.317921
$467$	−1890.82	−0.187360	−0.0936798	−	0.995602i	$-0.529863\pi$
$467$	−1890.82	−0.187360	−0.0936798	+	0.995602i	$0.529863\pi$
$468$	983.332	0.0971251
$469$	−11957.8	−1.17731
$470$	20525.0	2.01436
$471$	2864.04	0.280187
$472$	−3680.32	−0.358900
$473$	−2719.59	−0.264370
$474$	1169.21	0.113299
$475$	0	0
$476$	6724.13	0.647479
$477$	4433.49	0.425567
$478$	4886.74	0.467603
$479$	−11662.9	−1.11251	−0.556254	−	0.831013i	$-0.687762\pi$
$479$	−11662.9	−1.11251	−0.556254	+	0.831013i	$0.687762\pi$
$480$	1939.39	0.184418
$481$	−9084.88	−0.861195
$482$	1992.98	0.188336
$483$	−12948.5	−1.21983
$484$	7929.24	0.744669
$485$	12232.6	1.14527
$486$	486.000	0.0453609
$487$	13286.6	1.23629	0.618147	−	0.786063i	$-0.287884\pi$
$487$	13286.6	1.23629	0.618147	+	0.786063i	$0.287884\pi$
$488$	−3605.96	−0.334496
$489$	−5479.57	−0.506738
$490$	−7306.02	−0.673576
$491$	4037.71	0.371119	0.185559	−	0.982633i	$-0.440590\pi$
$491$	4037.71	0.371119	0.185559	+	0.982633i	$0.440590\pi$
$492$	512.590	0.0469702
$493$	525.594	0.0480153
$494$	0	0
$495$	−10465.7	−0.950298
$496$	−1874.30	−0.169674
$497$	−21317.6	−1.92399
$498$	−1716.50	−0.154454
$499$	3544.83	0.318013	0.159006	−	0.987278i	$-0.449171\pi$
$499$	3544.83	0.318013	0.159006	+	0.987278i	$0.449171\pi$
$500$	12777.4	1.14284
$501$	−4396.83	−0.392088
$502$	−1519.78	−0.135122
$503$	−12958.4	−1.14868	−0.574339	−	0.818617i	$-0.694741\pi$
$503$	−12958.4	−1.14868	−0.574339	+	0.818617i	$0.694741\pi$
$504$	1647.88	0.145640
$505$	955.068	0.0841584
$506$	−21710.3	−1.90739
$507$	4352.71	0.381283
$508$	−3821.57	−0.333769
$509$	−8236.28	−0.717224	−0.358612	−	0.933487i	$-0.616750\pi$
$509$	−8236.28	−0.717224	−0.358612	+	0.933487i	$0.616750\pi$
$510$	−8902.84	−0.772989
$511$	−8013.34	−0.693717
$512$	−512.000	−0.0441942
$513$	0	0
$514$	9546.39	0.819209
$515$	−8702.01	−0.744575
$516$	−566.962	−0.0483704
$517$	29240.9	2.48745
$518$	−15224.5	−1.29137
$519$	−12488.6	−1.05624
$520$	−4414.50	−0.372286
$521$	7427.58	0.624584	0.312292	−	0.949986i	$-0.398903\pi$
$521$	7427.58	0.624584	0.312292	+	0.949986i	$0.398903\pi$
$522$	128.807	0.0108002
$523$	−4604.88	−0.385005	−0.192502	−	0.981297i	$-0.561660\pi$
$523$	−4604.88	−0.385005	−0.192502	+	0.981297i	$0.561660\pi$
$524$	−6556.43	−0.546601
$525$	19439.5	1.61602
$526$	−8712.87	−0.722241
$527$	8604.04	0.711191
$528$	2762.94	0.227731
$529$	23396.8	1.92298
$530$	−19903.4	−1.63122
$531$	4140.36	0.338374
$532$	0	0
$533$	−1166.77	−0.0948191
$534$	5038.14	0.408280
$535$	21328.9	1.72361
$536$	−4179.72	−0.336821
$537$	7913.23	0.635905
$538$	281.429	0.0225525
$539$	−10408.5	−0.831772
$540$	−2181.81	−0.173871
$541$	9030.87	0.717685	0.358842	−	0.933398i	$-0.383172\pi$
$541$	9030.87	0.717685	0.358842	+	0.933398i	$0.383172\pi$
$542$	−3046.12	−0.241406
$543$	2339.50	0.184894
$544$	2350.35	0.185240
$545$	−5708.50	−0.448670
$546$	−3750.95	−0.294004
$547$	12875.6	1.00644	0.503218	−	0.864159i	$-0.332149\pi$
$547$	12875.6	1.00644	0.503218	+	0.864159i	$0.332149\pi$
$548$	−3390.85	−0.264324
$549$	4056.70	0.315366
$550$	32593.6	2.52690
$551$	0	0
$552$	−4526.01	−0.348985
$553$	−4459.98	−0.342962
$554$	−4435.09	−0.340124
$555$	20157.5	1.54169
$556$	4670.41	0.356240
$557$	−4891.20	−0.372077	−0.186038	−	0.982542i	$-0.559565\pi$
$557$	−4891.20	−0.372077	−0.186038	+	0.982542i	$0.559565\pi$
$558$	2108.59	0.159971
$559$	1290.54	0.0976456
$560$	−7397.87	−0.558245
$561$	−12683.4	−0.954534
$562$	−341.039	−0.0255976
$563$	2367.05	0.177192	0.0885961	−	0.996068i	$-0.471762\pi$
$563$	2367.05	0.177192	0.0885961	+	0.996068i	$0.471762\pi$
$564$	6095.94	0.455116
$565$	−17007.2	−1.26637
$566$	−4352.61	−0.323240
$567$	−1853.86	−0.137310
$568$	−7451.35	−0.550443
$569$	−675.013	−0.0497329	−0.0248665	−	0.999691i	$-0.507916\pi$
$569$	−675.013	−0.0497329	−0.0248665	+	0.999691i	$0.507916\pi$
$570$	0	0
$571$	17245.3	1.26391	0.631955	−	0.775005i	$-0.282253\pi$
$571$	17245.3	1.26391	0.631955	+	0.775005i	$0.282253\pi$
$572$	−6289.10	−0.459721
$573$	−11885.1	−0.866504
$574$	−1955.29	−0.142182
$575$	−53391.9	−3.87234
$576$	576.000	0.0416667
$577$	24732.5	1.78445	0.892226	−	0.451589i	$-0.149143\pi$
$577$	24732.5	1.78445	0.892226	+	0.451589i	$0.149143\pi$
$578$	−963.382	−0.0693277
$579$	1093.21	0.0784671
$580$	−578.257	−0.0413979
$581$	6547.64	0.467542
$582$	3633.09	0.258757
$583$	−28355.3	−2.01433
$584$	−2800.99	−0.198469
$585$	4966.31	0.350995
$586$	19758.0	1.39282
$587$	13718.5	0.964607	0.482304	−	0.876004i	$-0.339800\pi$
$587$	13718.5	0.964607	0.482304	+	0.876004i	$0.339800\pi$
$588$	−2169.89	−0.152185
$589$	0	0
$590$	−18587.5	−1.29701
$591$	−6927.97	−0.482197
$592$	−5321.59	−0.369453
$593$	7164.95	0.496170	0.248085	−	0.968738i	$-0.420199\pi$
$593$	7164.95	0.496170	0.248085	+	0.968738i	$0.420199\pi$
$594$	−3108.31	−0.214706
$595$	33960.2	2.33989
$596$	−6886.26	−0.473276
$597$	−11029.6	−0.756136
$598$	10302.2	0.704499
$599$	−1507.84	−0.102852	−0.0514262	−	0.998677i	$-0.516377\pi$
$599$	−1507.84	−0.102852	−0.0514262	+	0.998677i	$0.516377\pi$
$600$	6794.88	0.462333
$601$	8574.64	0.581975	0.290987	−	0.956727i	$-0.406016\pi$
$601$	8574.64	0.581975	0.290987	+	0.956727i	$0.406016\pi$
$602$	2162.70	0.146420
$603$	4702.18	0.317558
$604$	9308.54	0.627085
$605$	40046.6	2.69112
$606$	283.656	0.0190144
$607$	504.688	0.0337474	0.0168737	−	0.999858i	$-0.494629\pi$
$607$	504.688	0.0337474	0.0168737	+	0.999858i	$0.494629\pi$
$608$	0	0
$609$	−491.339	−0.0326930
$610$	−18211.9	−1.20882
$611$	−13875.8	−0.918745
$612$	−2644.15	−0.174646
$613$	14944.4	0.984666	0.492333	−	0.870407i	$-0.336144\pi$
$613$	14944.4	0.984666	0.492333	+	0.870407i	$0.336144\pi$
$614$	2687.42	0.176637
$615$	2588.83	0.169743
$616$	−10539.3	−0.689355
$617$	−3482.10	−0.227203	−0.113601	−	0.993526i	$-0.536239\pi$
$617$	−3482.10	−0.227203	−0.113601	+	0.993526i	$0.536239\pi$
$618$	−2584.50	−0.168226
$619$	3577.83	0.232319	0.116159	−	0.993231i	$-0.462942\pi$
$619$	3577.83	0.232319	0.116159	+	0.993231i	$0.462942\pi$
$620$	−9466.14	−0.613176
$621$	5091.76	0.329026
$622$	−176.712	−0.0113915
$623$	−19218.2	−1.23589
$624$	−1311.11	−0.0841128
$625$	29142.0	1.86509
$626$	3784.92	0.241655
$627$	0	0
$628$	−3818.72	−0.242649
$629$	24429.0	1.54856
$630$	8322.61	0.526318
$631$	3649.43	0.230240	0.115120	−	0.993352i	$-0.463275\pi$
$631$	3649.43	0.230240	0.115120	+	0.993352i	$0.463275\pi$
$632$	−1558.94	−0.0981194
$633$	−4027.81	−0.252909
$634$	−16757.7	−1.04974
$635$	−19300.8	−1.20619
$636$	−5911.31	−0.368552
$637$	4939.17	0.307217
$638$	−823.811	−0.0511207
$639$	8382.77	0.518963
$640$	−2585.85	−0.159711
$641$	−14669.2	−0.903898	−0.451949	−	0.892044i	$-0.649271\pi$
$641$	−14669.2	−0.903898	−0.451949	+	0.892044i	$0.649271\pi$
$642$	6334.71	0.389425
$643$	7008.11	0.429818	0.214909	−	0.976634i	$-0.431055\pi$
$643$	7008.11	0.429818	0.214909	+	0.976634i	$0.431055\pi$
$644$	17264.6	1.05640
$645$	−2863.44	−0.174803
$646$	0	0
$647$	9659.76	0.586962	0.293481	−	0.955965i	$-0.405186\pi$
$647$	9659.76	0.586962	0.293481	+	0.955965i	$0.405186\pi$
$648$	−648.000	−0.0392837
$649$	−26480.6	−1.60162
$650$	−15466.7	−0.933315
$651$	−8043.28	−0.484241
$652$	7306.10	0.438848
$653$	6799.09	0.407456	0.203728	−	0.979027i	$-0.434694\pi$
$653$	6799.09	0.407456	0.203728	+	0.979027i	$0.434694\pi$
$654$	−1695.43	−0.101371
$655$	−33113.2	−1.97533
$656$	−683.454	−0.0406774
$657$	3151.11	0.187118
$658$	−23253.2	−1.37766
$659$	−19348.2	−1.14370	−0.571851	−	0.820357i	$-0.693775\pi$
$659$	−19348.2	−1.14370	−0.571851	+	0.820357i	$0.693775\pi$
$660$	13954.2	0.822982
$661$	5128.00	0.301749	0.150875	−	0.988553i	$-0.451791\pi$
$661$	5128.00	0.301749	0.150875	+	0.988553i	$0.451791\pi$
$662$	−9667.01	−0.567551
$663$	6018.69	0.352559
$664$	2288.66	0.133761
$665$	0	0
$666$	5986.79	0.348324
$667$	1349.49	0.0783398
$668$	5862.45	0.339558
$669$	6718.75	0.388284
$670$	−21109.6	−1.21722
$671$	−25945.5	−1.49272
$672$	−2197.17	−0.126128
$673$	−23921.1	−1.37012	−0.685061	−	0.728486i	$-0.740224\pi$
$673$	−23921.1	−1.37012	−0.685061	+	0.728486i	$0.740224\pi$
$674$	5251.55	0.300122
$675$	−7644.24	−0.435892
$676$	−5803.61	−0.330201
$677$	−766.891	−0.0435362	−0.0217681	−	0.999763i	$-0.506930\pi$
$677$	−766.891	−0.0435362	−0.0217681	+	0.999763i	$0.506930\pi$
$678$	−5051.16	−0.286119
$679$	−13858.6	−0.783274
$680$	11870.5	0.669428
$681$	18700.2	1.05227
$682$	−13485.9	−0.757187
$683$	−6573.45	−0.368267	−0.184133	−	0.982901i	$-0.558948\pi$
$683$	−6573.45	−0.368267	−0.184133	+	0.982901i	$0.558948\pi$
$684$	0	0
$685$	−17125.5	−0.955226
$686$	−7423.50	−0.413164
$687$	6875.21	0.381813
$688$	755.950	0.0418900
$689$	13455.5	0.743998
$690$	−22858.6	−1.26118
$691$	−12816.9	−0.705613	−0.352806	−	0.935696i	$-0.614773\pi$
$691$	−12816.9	−0.705613	−0.352806	+	0.935696i	$0.614773\pi$
$692$	16651.5	0.914730
$693$	11856.8	0.649930
$694$	−16083.0	−0.879685
$695$	23587.9	1.28740
$696$	−171.743	−0.00935328
$697$	3137.42	0.170500
$698$	−17207.9	−0.933135
$699$	−4797.22	−0.259582
$700$	−25919.3	−1.39951
$701$	−31041.6	−1.67250	−0.836251	−	0.548347i	$-0.815257\pi$
$701$	−31041.6	−1.67250	−0.836251	+	0.548347i	$0.815257\pi$
$702$	1475.00	0.0793023
$703$	0	0
$704$	−3683.93	−0.197220
$705$	30787.5	1.64472
$706$	20718.9	1.10448
$707$	−1082.01	−0.0575578
$708$	−5520.49	−0.293040
$709$	−13641.0	−0.722564	−0.361282	−	0.932457i	$-0.617661\pi$
$709$	−13641.0	−0.722564	−0.361282	+	0.932457i	$0.617661\pi$
$710$	−37633.0	−1.98921
$711$	1753.81	0.0925079
$712$	−6717.52	−0.353581
$713$	22091.4	1.16035
$714$	10086.2	0.528664
$715$	−31763.1	−1.66136
$716$	−10551.0	−0.550710
$717$	7330.11	0.381797
$718$	7089.35	0.368485
$719$	27533.8	1.42815	0.714073	−	0.700071i	$-0.246848\pi$
$719$	27533.8	1.42815	0.714073	+	0.700071i	$0.246848\pi$
$720$	2909.09	0.150577
$721$	9858.67	0.509232
$722$	0	0
$723$	2989.47	0.153775
$724$	−3119.33	−0.160123
$725$	−2025.99	−0.103784
$726$	11893.9	0.608020
$727$	20867.3	1.06455	0.532273	−	0.846573i	$-0.321338\pi$
$727$	20867.3	1.06455	0.532273	+	0.846573i	$0.321338\pi$
$728$	5001.27	0.254615
$729$	729.000	0.0370370
$730$	−14146.4	−0.717234
$731$	−3470.21	−0.175582
$732$	−5408.94	−0.273115
$733$	−18422.0	−0.928282	−0.464141	−	0.885761i	$-0.653637\pi$
$733$	−18422.0	−0.928282	−0.464141	+	0.885761i	$0.653637\pi$
$734$	−660.505	−0.0332148
$735$	−10959.0	−0.549972
$736$	6034.68	0.302230
$737$	−30073.8	−1.50310
$738$	768.885	0.0383510
$739$	−15508.0	−0.771949	−0.385975	−	0.922509i	$-0.626135\pi$
$739$	−15508.0	−0.771949	−0.385975	+	0.922509i	$0.626135\pi$
$740$	−26876.7	−1.33514
$741$	0	0
$742$	22548.9	1.11563
$743$	−11877.0	−0.586441	−0.293221	−	0.956045i	$-0.594727\pi$
$743$	−11877.0	−0.586441	−0.293221	+	0.956045i	$0.594727\pi$
$744$	−2811.45	−0.138539
$745$	−34779.0	−1.71034
$746$	24032.3	1.17947
$747$	−2574.75	−0.126111
$748$	16911.2	0.826650
$749$	−24164.0	−1.17881
$750$	19166.0	0.933126
$751$	26834.2	1.30385	0.651927	−	0.758281i	$-0.273961\pi$
$751$	26834.2	1.30385	0.651927	+	0.758281i	$0.273961\pi$
$752$	−8127.92	−0.394142
$753$	−2279.67	−0.110326
$754$	390.926	0.0188815
$755$	47012.8	2.26618
$756$	2471.82	0.118914
$757$	−39104.9	−1.87753	−0.938767	−	0.344552i	$-0.888031\pi$
$757$	−39104.9	−1.87753	−0.938767	+	0.344552i	$0.888031\pi$
$758$	−10927.9	−0.523639
$759$	−32565.4	−1.55738
$760$	0	0
$761$	13627.0	0.649117	0.324559	−	0.945866i	$-0.394784\pi$
$761$	13627.0	0.649117	0.324559	+	0.945866i	$0.394784\pi$
$762$	−5732.36	−0.272522
$763$	6467.27	0.306856
$764$	15846.8	0.750415
$765$	−13354.3	−0.631143
$766$	−14761.2	−0.696269
$767$	12565.9	0.591563
$768$	−768.000	−0.0360844
$769$	23166.4	1.08635	0.543173	−	0.839621i	$-0.317223\pi$
$769$	23166.4	1.08635	0.543173	+	0.839621i	$0.317223\pi$
$770$	−53228.9	−2.49122
$771$	14319.6	0.668881
$772$	−1457.62	−0.0679545
$773$	32201.6	1.49833	0.749167	−	0.662381i	$-0.230454\pi$
$773$	32201.6	1.49833	0.749167	+	0.662381i	$0.230454\pi$
$774$	−850.443	−0.0394943
$775$	−33165.7	−1.53722
$776$	−4844.12	−0.224090
$777$	−22836.8	−1.05440
$778$	−4071.67	−0.187630
$779$	0	0
$780$	−6621.75	−0.303970
$781$	−53613.7	−2.45640
$782$	−27702.4	−1.26680
$783$	193.210	0.00881836
$784$	2893.19	0.131796
$785$	−19286.4	−0.876893
$786$	−9834.65	−0.446298
$787$	−36081.7	−1.63427	−0.817137	−	0.576443i	$-0.804440\pi$
$787$	−36081.7	−1.63427	−0.817137	+	0.576443i	$0.804440\pi$
$788$	9237.29	0.417595
$789$	−13069.3	−0.589708
$790$	−7873.44	−0.354588
$791$	19267.8	0.866100
$792$	4144.42	0.185941
$793$	12312.0	0.551339
$794$	−26628.5	−1.19019
$795$	−29855.1	−1.33189
$796$	14706.2	0.654833
$797$	−39094.9	−1.73753	−0.868765	−	0.495224i	$-0.835086\pi$
$797$	−39094.9	−1.73753	−0.868765	+	0.495224i	$0.835086\pi$
$798$	0	0
$799$	37311.5	1.65205
$800$	−9059.84	−0.400392
$801$	7557.21	0.333359
$802$	−26408.7	−1.16275
$803$	−20153.6	−0.885684
$804$	−6269.58	−0.275014
$805$	87194.9	3.81766
$806$	6399.51	0.279669
$807$	422.144	0.0184141
$808$	−378.208	−0.0164670
$809$	7261.33	0.315568	0.157784	−	0.987474i	$-0.449565\pi$
$809$	7261.33	0.315568	0.157784	+	0.987474i	$0.449565\pi$
$810$	−3272.72	−0.141965
$811$	34133.3	1.47791	0.738953	−	0.673757i	$-0.235321\pi$
$811$	34133.3	1.47791	0.738953	+	0.673757i	$0.235321\pi$
$812$	655.118	0.0283130
$813$	−4569.18	−0.197107
$814$	−38289.8	−1.64872
$815$	36899.4	1.58593
$816$	3525.53	0.151248
$817$	0	0
$818$	−5523.72	−0.236103
$819$	−5626.43	−0.240053
$820$	−3451.78	−0.147002
$821$	13880.3	0.590043	0.295022	−	0.955491i	$-0.404673\pi$
$821$	13880.3	0.590043	0.295022	+	0.955491i	$0.404673\pi$
$822$	−5086.27	−0.215820
$823$	13016.1	0.551291	0.275646	−	0.961259i	$-0.411108\pi$
$823$	13016.1	0.551291	0.275646	+	0.961259i	$0.411108\pi$
$824$	3446.00	0.145688
$825$	48890.3	2.06320
$826$	21058.1	0.887051
$827$	−19864.1	−0.835238	−0.417619	−	0.908622i	$-0.637135\pi$
$827$	−19864.1	−0.835238	−0.417619	+	0.908622i	$0.637135\pi$
$828$	−6789.02	−0.284945
$829$	11325.6	0.474494	0.237247	−	0.971449i	$-0.423755\pi$
$829$	11325.6	0.474494	0.237247	+	0.971449i	$0.423755\pi$
$830$	11558.9	0.483391
$831$	−6652.63	−0.277710
$832$	1748.15	0.0728438
$833$	−13281.3	−0.552424
$834$	7005.62	0.290869
$835$	29608.3	1.22711
$836$	0	0
$837$	3162.88	0.130615
$838$	30364.1	1.25168
$839$	4507.99	0.185498	0.0927491	−	0.995690i	$-0.470435\pi$
$839$	4507.99	0.185498	0.0927491	+	0.995690i	$0.470435\pi$
$840$	−11096.8	−0.455805
$841$	−24337.8	−0.997900
$842$	−31960.7	−1.30812
$843$	−511.559	−0.0209004
$844$	5370.42	0.219025
$845$	−29311.1	−1.19329
$846$	9143.91	0.371600
$847$	−45369.5	−1.84051
$848$	7881.75	0.319175
$849$	−6528.91	−0.263924
$850$	41589.5	1.67825
$851$	62722.9	2.52657
$852$	−11177.0	−0.449435
$853$	−31512.5	−1.26491	−0.632454	−	0.774598i	$-0.717952\pi$
$853$	−31512.5	−1.26491	−0.632454	+	0.774598i	$0.717952\pi$
$854$	20632.6	0.826736
$855$	0	0
$856$	−8446.28	−0.337252
$857$	−6340.51	−0.252728	−0.126364	−	0.991984i	$-0.540331\pi$
$857$	−6340.51	−0.252728	−0.126364	+	0.991984i	$0.540331\pi$
$858$	−9433.65	−0.375361
$859$	−20206.0	−0.802582	−0.401291	−	0.915951i	$-0.631438\pi$
$859$	−20206.0	−0.802582	−0.401291	+	0.915951i	$0.631438\pi$
$860$	3817.92	0.151384
$861$	−2932.94	−0.116091
$862$	−9524.00	−0.376321
$863$	−34553.8	−1.36295	−0.681474	−	0.731842i	$-0.738661\pi$
$863$	−34553.8	−1.36295	−0.681474	+	0.731842i	$0.738661\pi$
$864$	864.000	0.0340207
$865$	84098.1	3.30569
$866$	−19949.8	−0.782820
$867$	−1445.07	−0.0566058
$868$	10724.4	0.419365
$869$	−11216.9	−0.437867
$870$	−867.385	−0.0338013
$871$	14271.0	0.555172
$872$	2260.57	0.0877896
$873$	5449.64	0.211274
$874$	0	0
$875$	−73109.5	−2.82463
$876$	−4201.48	−0.162049
$877$	30588.2	1.17775	0.588877	−	0.808223i	$-0.299570\pi$
$877$	30588.2	1.17775	0.588877	+	0.808223i	$0.299570\pi$
$878$	17913.8	0.688566
$879$	29637.0	1.13724
$880$	−18605.7	−0.712723
$881$	48445.7	1.85264	0.926321	−	0.376735i	$-0.122953\pi$
$881$	48445.7	1.85264	0.926321	+	0.376735i	$0.122953\pi$
$882$	−3254.84	−0.124259
$883$	10270.4	0.391422	0.195711	−	0.980662i	$-0.437299\pi$
$883$	10270.4	0.391422	0.195711	+	0.980662i	$0.437299\pi$
$884$	−8024.92	−0.305325
$885$	−27881.2	−1.05900
$886$	−22736.2	−0.862121
$887$	27630.3	1.04592	0.522962	−	0.852356i	$-0.324827\pi$
$887$	27630.3	1.04592	0.522962	+	0.852356i	$0.324827\pi$
$888$	−7982.39	−0.301657
$889$	21866.3	0.824940
$890$	−33926.8	−1.27779
$891$	−4662.47	−0.175307
$892$	−8958.34	−0.336264
$893$	0	0
$894$	−10329.4	−0.386428
$895$	−53287.6	−1.99018
$896$	2929.56	0.109230
$897$	15453.4	0.575221
$898$	−15260.4	−0.567089
$899$	838.273	0.0310990
$900$	10192.3	0.377493
$901$	−36181.5	−1.33782
$902$	−4917.56	−0.181527
$903$	3244.05	0.119552
$904$	6734.88	0.247786
$905$	−15754.2	−0.578658
$906$	13962.8	0.512013
$907$	40746.5	1.49169	0.745847	−	0.666118i	$-0.232045\pi$
$907$	40746.5	1.49169	0.745847	+	0.666118i	$0.232045\pi$
$908$	−24933.6	−0.911291
$909$	425.484	0.0155252
$910$	25258.9	0.920137
$911$	13335.5	0.484989	0.242494	−	0.970153i	$-0.422034\pi$
$911$	13335.5	0.484989	0.242494	+	0.970153i	$0.422034\pi$
$912$	0	0
$913$	16467.3	0.596921
$914$	3000.80	0.108597
$915$	−27317.8	−0.986993
$916$	−9166.94	−0.330660
$917$	37514.6	1.35097
$918$	−3966.22	−0.142598
$919$	16604.5	0.596009	0.298005	−	0.954564i	$-0.403679\pi$
$919$	16604.5	0.596009	0.298005	+	0.954564i	$0.403679\pi$
$920$	30478.1	1.09221
$921$	4031.13	0.144224
$922$	28991.6	1.03556
$923$	25441.5	0.907277
$924$	−15809.0	−0.562856
$925$	−94165.6	−3.34719
$926$	24530.5	0.870541
$927$	−3876.75	−0.137356
$928$	228.990	0.00810018
$929$	−16679.4	−0.589056	−0.294528	−	0.955643i	$-0.595162\pi$
$929$	−16679.4	−0.589056	−0.294528	+	0.955643i	$0.595162\pi$
$930$	−14199.2	−0.500656
$931$	0	0
$932$	6396.29	0.224804
$933$	−265.068	−0.00930110
$934$	3781.65	0.132483
$935$	85409.9	2.98738
$936$	−1966.66	−0.0686778
$937$	6867.24	0.239427	0.119713	−	0.992808i	$-0.461802\pi$
$937$	6867.24	0.239427	0.119713	+	0.992808i	$0.461802\pi$
$938$	23915.5	0.832483
$939$	5677.38	0.197310
$940$	−41050.0	−1.42437
$941$	48163.9	1.66854	0.834272	−	0.551353i	$-0.185888\pi$
$941$	48163.9	1.66854	0.834272	+	0.551353i	$0.185888\pi$
$942$	−5728.07	−0.198122
$943$	8055.52	0.278180
$944$	7360.65	0.253780
$945$	12483.9	0.429737
$946$	5439.19	0.186938
$947$	−45311.2	−1.55482	−0.777412	−	0.628992i	$-0.783468\pi$
$947$	−45311.2	−1.55482	−0.777412	+	0.628992i	$0.783468\pi$
$948$	−2338.42	−0.0801142
$949$	9563.54	0.327129
$950$	0	0
$951$	−25136.6	−0.857108
$952$	−13448.3	−0.457837
$953$	6964.61	0.236732	0.118366	−	0.992970i	$-0.462234\pi$
$953$	6964.61	0.236732	0.118366	+	0.992970i	$0.462234\pi$
$954$	−8866.97	−0.300921
$955$	80034.1	2.71188
$956$	−9773.49	−0.330646
$957$	−1235.72	−0.0417399
$958$	23325.8	0.786661
$959$	19401.7	0.653300
$960$	−3878.78	−0.130403
$961$	−16068.4	−0.539369
$962$	18169.8	0.608957
$963$	9502.06	0.317964
$964$	−3985.96	−0.133173
$965$	−7361.70	−0.245577
$966$	25896.9	0.862547
$967$	46299.5	1.53970	0.769851	−	0.638224i	$-0.220331\pi$
$967$	46299.5	1.53970	0.769851	+	0.638224i	$0.220331\pi$
$968$	−15858.5	−0.526561
$969$	0	0
$970$	−24465.2	−0.809826
$971$	44402.3	1.46750	0.733748	−	0.679422i	$-0.237769\pi$
$971$	44402.3	1.46750	0.733748	+	0.679422i	$0.237769\pi$
$972$	−972.000	−0.0320750
$973$	−26723.2	−0.880478
$974$	−26573.3	−0.874192
$975$	−23200.1	−0.762049
$976$	7211.92	0.236524
$977$	36365.1	1.19081	0.595405	−	0.803426i	$-0.296992\pi$
$977$	36365.1	1.19081	0.595405	+	0.803426i	$0.296992\pi$
$978$	10959.1	0.358318
$979$	−48333.7	−1.57789
$980$	14612.0	0.476290
$981$	−2543.14	−0.0827689
$982$	−8075.42	−0.262421
$983$	3682.23	0.119476	0.0597381	−	0.998214i	$-0.480973\pi$
$983$	3682.23	0.119476	0.0597381	+	0.998214i	$0.480973\pi$
$984$	−1025.18	−0.0332130
$985$	46652.9	1.50912
$986$	−1051.19	−0.0339519
$987$	−34879.7	−1.12486
$988$	0	0
$989$	−8909.99	−0.286472
$990$	20931.4	0.671962
$991$	−4412.76	−0.141449	−0.0707246	−	0.997496i	$-0.522531\pi$
$991$	−4412.76	−0.141449	−0.0707246	+	0.997496i	$0.522531\pi$
$992$	3748.60	0.119978
$993$	−14500.5	−0.463404
$994$	42635.2	1.36047
$995$	74273.5	2.36646
$996$	3432.99	0.109215
$997$	−24825.9	−0.788610	−0.394305	−	0.918980i	$-0.629015\pi$
$997$	−24825.9	−0.788610	−0.394305	+	0.918980i	$0.629015\pi$
$998$	−7089.66	−0.224869
$999$	8980.19	0.284405

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.s.1.3		3
19.7	even	3		114.4.e.e.49.1	yes	6
19.11	even	3		114.4.e.e.7.1	✓	6
19.18	odd	2		2166.4.a.w.1.3		3
57.11	odd	6		342.4.g.g.235.3		6
57.26	odd	6		342.4.g.g.163.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.e.e.7.1	✓	6	19.11	even	3
114.4.e.e.49.1	yes	6	19.7	even	3
342.4.g.g.163.3		6	57.26	odd	6
342.4.g.g.235.3		6	57.11	odd	6
2166.4.a.s.1.3		3	1.1	even	1	trivial
2166.4.a.w.1.3		3	19.18	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:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +20.2020 q^{5} +6.00000 q^{6} -22.8872 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +20.2020 q^{5} +6.00000 q^{6} -22.8872 q^{7} -8.00000 q^{8} +9.00000 q^{9} -40.4040 q^{10} -57.5614 q^{11} -12.0000 q^{12} +27.3148 q^{13} +45.7744 q^{14} -60.6059 q^{15} +16.0000 q^{16} -73.4486 q^{17} -18.0000 q^{18} +80.8079 q^{20} +68.6616 q^{21} +115.123 q^{22} -188.584 q^{23} +24.0000 q^{24} +283.120 q^{25} -54.6296 q^{26} -27.0000 q^{27} -91.5488 q^{28} -7.15594 q^{29} +121.212 q^{30} -117.144 q^{31} -32.0000 q^{32} +172.684 q^{33} +146.897 q^{34} -462.367 q^{35} +36.0000 q^{36} -332.600 q^{37} -81.9443 q^{39} -161.616 q^{40} -42.7158 q^{41} -137.323 q^{42} +47.2469 q^{43} -230.245 q^{44} +181.818 q^{45} +377.168 q^{46} -507.995 q^{47} -48.0000 q^{48} +180.824 q^{49} -566.240 q^{50} +220.346 q^{51} +109.259 q^{52} +492.609 q^{53} +54.0000 q^{54} -1162.85 q^{55} +183.098 q^{56} +14.3119 q^{58} +460.041 q^{59} -242.424 q^{60} +450.745 q^{61} +234.287 q^{62} -205.985 q^{63} +64.0000 q^{64} +551.813 q^{65} -345.368 q^{66} +522.465 q^{67} -293.794 q^{68} +565.751 q^{69} +924.734 q^{70} +931.419 q^{71} -72.0000 q^{72} +350.123 q^{73} +665.199 q^{74} -849.360 q^{75} +1317.42 q^{77} +163.889 q^{78} +194.868 q^{79} +323.232 q^{80} +81.0000 q^{81} +85.4317 q^{82} -286.083 q^{83} +274.646 q^{84} -1483.81 q^{85} -94.4937 q^{86} +21.4678 q^{87} +460.491 q^{88} +839.690 q^{89} -363.636 q^{90} -625.159 q^{91} -754.335 q^{92} +351.431 q^{93} +1015.99 q^{94} +96.0000 q^{96} +605.516 q^{97} -361.648 q^{98} -518.052 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} + 2 q^{5} + 18 q^{6} - 17 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 + 2 * q^5 + 18 * q^6 - 17 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} + 2 q^{5} + 18 q^{6} - 17 q^{7} - 24 q^{8} + 27 q^{9} - 4 q^{10} - 52 q^{11} - 36 q^{12} + 75 q^{13} + 34 q^{14} - 6 q^{15} + 48 q^{16} - 48 q^{17} - 54 q^{18} + 8 q^{20} + 51 q^{21} + 104 q^{22} - 238 q^{23} + 72 q^{24} + 229 q^{25} - 150 q^{26} - 81 q^{27} - 68 q^{28} - 8 q^{29} + 12 q^{30} + 107 q^{31} - 96 q^{32} + 156 q^{33} + 96 q^{34} - 294 q^{35} + 108 q^{36} + 305 q^{37} - 225 q^{39} - 16 q^{40} + 16 q^{41} - 102 q^{42} - 331 q^{43} - 208 q^{44} + 18 q^{45} + 476 q^{46} - 766 q^{47} - 144 q^{48} + 1142 q^{49} - 458 q^{50} + 144 q^{51} + 300 q^{52} - 118 q^{53} + 162 q^{54} - 1400 q^{55} + 136 q^{56} + 16 q^{58} + 936 q^{59} - 24 q^{60} - 399 q^{61} - 214 q^{62} - 153 q^{63} + 192 q^{64} + 370 q^{65} - 312 q^{66} + 61 q^{67} - 192 q^{68} + 714 q^{69} + 588 q^{70} + 974 q^{71} - 216 q^{72} + 91 q^{73} - 610 q^{74} - 687 q^{75} - 36 q^{77} + 450 q^{78} - 321 q^{79} + 32 q^{80} + 243 q^{81} - 32 q^{82} - 2148 q^{83} + 204 q^{84} - 1680 q^{85} + 662 q^{86} + 24 q^{87} + 416 q^{88} + 1116 q^{89} - 36 q^{90} + 1367 q^{91} - 952 q^{92} - 321 q^{93} + 1532 q^{94} + 288 q^{96} + 1382 q^{97} - 2284 q^{98} - 468 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 + 2 * q^5 + 18 * q^6 - 17 * q^7 - 24 * q^8 + 27 * q^9 - 4 * q^10 - 52 * q^11 - 36 * q^12 + 75 * q^13 + 34 * q^14 - 6 * q^15 + 48 * q^16 - 48 * q^17 - 54 * q^18 + 8 * q^20 + 51 * q^21 + 104 * q^22 - 238 * q^23 + 72 * q^24 + 229 * q^25 - 150 * q^26 - 81 * q^27 - 68 * q^28 - 8 * q^29 + 12 * q^30 + 107 * q^31 - 96 * q^32 + 156 * q^33 + 96 * q^34 - 294 * q^35 + 108 * q^36 + 305 * q^37 - 225 * q^39 - 16 * q^40 + 16 * q^41 - 102 * q^42 - 331 * q^43 - 208 * q^44 + 18 * q^45 + 476 * q^46 - 766 * q^47 - 144 * q^48 + 1142 * q^49 - 458 * q^50 + 144 * q^51 + 300 * q^52 - 118 * q^53 + 162 * q^54 - 1400 * q^55 + 136 * q^56 + 16 * q^58 + 936 * q^59 - 24 * q^60 - 399 * q^61 - 214 * q^62 - 153 * q^63 + 192 * q^64 + 370 * q^65 - 312 * q^66 + 61 * q^67 - 192 * q^68 + 714 * q^69 + 588 * q^70 + 974 * q^71 - 216 * q^72 + 91 * q^73 - 610 * q^74 - 687 * q^75 - 36 * q^77 + 450 * q^78 - 321 * q^79 + 32 * q^80 + 243 * q^81 - 32 * q^82 - 2148 * q^83 + 204 * q^84 - 1680 * q^85 + 662 * q^86 + 24 * q^87 + 416 * q^88 + 1116 * q^89 - 36 * q^90 + 1367 * q^91 - 952 * q^92 - 321 * q^93 + 1532 * q^94 + 288 * q^96 + 1382 * q^97 - 2284 * q^98 - 468 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

Properties

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