LMFDB - Embedded newform 2166.4.a.j.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -12.0902 q^{5} +6.00000 q^{6} -0.673762 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -12.0902 q^{5} +6.00000 q^{6} -0.673762 q^{7} -8.00000 q^{8} +9.00000 q^{9} +24.1803 q^{10} +15.9443 q^{11} -12.0000 q^{12} -3.90983 q^{13} +1.34752 q^{14} +36.2705 q^{15} +16.0000 q^{16} -44.0132 q^{17} -18.0000 q^{18} -48.3607 q^{20} +2.02129 q^{21} -31.8885 q^{22} -9.11146 q^{23} +24.0000 q^{24} +21.1722 q^{25} +7.81966 q^{26} -27.0000 q^{27} -2.69505 q^{28} +161.151 q^{29} -72.5410 q^{30} -60.5035 q^{31} -32.0000 q^{32} -47.8328 q^{33} +88.0263 q^{34} +8.14590 q^{35} +36.0000 q^{36} -104.885 q^{37} +11.7295 q^{39} +96.7214 q^{40} -249.353 q^{41} -4.04257 q^{42} -6.90481 q^{43} +63.7771 q^{44} -108.812 q^{45} +18.2229 q^{46} +145.257 q^{47} -48.0000 q^{48} -342.546 q^{49} -42.3444 q^{50} +132.039 q^{51} -15.6393 q^{52} +754.187 q^{53} +54.0000 q^{54} -192.769 q^{55} +5.39010 q^{56} -322.302 q^{58} +704.046 q^{59} +145.082 q^{60} -32.5085 q^{61} +121.007 q^{62} -6.06386 q^{63} +64.0000 q^{64} +47.2705 q^{65} +95.6656 q^{66} -77.7264 q^{67} -176.053 q^{68} +27.3344 q^{69} -16.2918 q^{70} -44.4183 q^{71} -72.0000 q^{72} +280.735 q^{73} +209.771 q^{74} -63.5166 q^{75} -10.7426 q^{77} -23.4590 q^{78} -779.508 q^{79} -193.443 q^{80} +81.0000 q^{81} +498.705 q^{82} -274.023 q^{83} +8.08514 q^{84} +532.127 q^{85} +13.8096 q^{86} -483.453 q^{87} -127.554 q^{88} +1005.91 q^{89} +217.623 q^{90} +2.63430 q^{91} -36.4458 q^{92} +181.510 q^{93} -290.515 q^{94} +96.0000 q^{96} +832.600 q^{97} +685.092 q^{98} +143.498 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} - 13 q^{5} + 12 q^{6} - 17 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 - 13 * q^5 + 12 * q^6 - 17 * q^7 - 16 * q^8 + 18 * q^9	\( 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} - 13 q^{5} + 12 q^{6} - 17 q^{7} - 16 q^{8} + 18 q^{9} + 26 q^{10} + 14 q^{11} - 24 q^{12} - 19 q^{13} + 34 q^{14} + 39 q^{15} + 32 q^{16} - 12 q^{17} - 36 q^{18} - 52 q^{20} + 51 q^{21} - 28 q^{22} - 54 q^{23} + 48 q^{24} - 103 q^{25} + 38 q^{26} - 54 q^{27} - 68 q^{28} + 130 q^{29} - 78 q^{30} + 239 q^{31} - 64 q^{32} - 42 q^{33} + 24 q^{34} + 23 q^{35} + 72 q^{36} + 148 q^{37} + 57 q^{39} + 104 q^{40} - 331 q^{41} - 102 q^{42} - 224 q^{43} + 56 q^{44} - 117 q^{45} + 108 q^{46} + 333 q^{47} - 96 q^{48} - 419 q^{49} + 206 q^{50} + 36 q^{51} - 76 q^{52} + 766 q^{53} + 108 q^{54} - 191 q^{55} + 136 q^{56} - 260 q^{58} + 460 q^{59} + 156 q^{60} + 494 q^{61} - 478 q^{62} - 153 q^{63} + 128 q^{64} + 61 q^{65} + 84 q^{66} + 133 q^{67} - 48 q^{68} + 162 q^{69} - 46 q^{70} + 459 q^{71} - 144 q^{72} + 396 q^{73} - 296 q^{74} + 309 q^{75} + 21 q^{77} - 114 q^{78} - 1000 q^{79} - 208 q^{80} + 162 q^{81} + 662 q^{82} - 74 q^{83} + 204 q^{84} + 503 q^{85} + 448 q^{86} - 390 q^{87} - 112 q^{88} + 1180 q^{89} + 234 q^{90} + 249 q^{91} - 216 q^{92} - 717 q^{93} - 666 q^{94} + 192 q^{96} + 1938 q^{97} + 838 q^{98} + 126 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 - 13 * q^5 + 12 * q^6 - 17 * q^7 - 16 * q^8 + 18 * q^9 + 26 * q^10 + 14 * q^11 - 24 * q^12 - 19 * q^13 + 34 * q^14 + 39 * q^15 + 32 * q^16 - 12 * q^17 - 36 * q^18 - 52 * q^20 + 51 * q^21 - 28 * q^22 - 54 * q^23 + 48 * q^24 - 103 * q^25 + 38 * q^26 - 54 * q^27 - 68 * q^28 + 130 * q^29 - 78 * q^30 + 239 * q^31 - 64 * q^32 - 42 * q^33 + 24 * q^34 + 23 * q^35 + 72 * q^36 + 148 * q^37 + 57 * q^39 + 104 * q^40 - 331 * q^41 - 102 * q^42 - 224 * q^43 + 56 * q^44 - 117 * q^45 + 108 * q^46 + 333 * q^47 - 96 * q^48 - 419 * q^49 + 206 * q^50 + 36 * q^51 - 76 * q^52 + 766 * q^53 + 108 * q^54 - 191 * q^55 + 136 * q^56 - 260 * q^58 + 460 * q^59 + 156 * q^60 + 494 * q^61 - 478 * q^62 - 153 * q^63 + 128 * q^64 + 61 * q^65 + 84 * q^66 + 133 * q^67 - 48 * q^68 + 162 * q^69 - 46 * q^70 + 459 * q^71 - 144 * q^72 + 396 * q^73 - 296 * q^74 + 309 * q^75 + 21 * q^77 - 114 * q^78 - 1000 * q^79 - 208 * q^80 + 162 * q^81 + 662 * q^82 - 74 * q^83 + 204 * q^84 + 503 * q^85 + 448 * q^86 - 390 * q^87 - 112 * q^88 + 1180 * q^89 + 234 * q^90 + 249 * q^91 - 216 * q^92 - 717 * q^93 - 666 * q^94 + 192 * q^96 + 1938 * q^97 + 838 * q^98 + 126 * 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
$5$	−12.0902	−1.08138	−0.540689	−	0.841223i	$-0.681836\pi$
$5$	−12.0902	−1.08138	−0.540689	+	0.841223i	$0.681836\pi$
$6$	6.00000	0.408248
$7$	−0.673762	−0.0363797	−0.0181899	−	0.999835i	$-0.505790\pi$
$7$	−0.673762	−0.0363797	−0.0181899	+	0.999835i	$0.505790\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	24.1803	0.764649
$11$	15.9443	0.437034	0.218517	−	0.975833i	$-0.429878\pi$
$11$	15.9443	0.437034	0.218517	+	0.975833i	$0.429878\pi$
$12$	−12.0000	−0.288675
$13$	−3.90983	−0.0834147	−0.0417074	−	0.999130i	$-0.513280\pi$
$13$	−3.90983	−0.0834147	−0.0417074	+	0.999130i	$0.513280\pi$
$14$	1.34752	0.0257244
$15$	36.2705	0.624334
$16$	16.0000	0.250000
$17$	−44.0132	−0.627927	−0.313963	−	0.949435i	$-0.601657\pi$
$17$	−44.0132	−0.627927	−0.313963	+	0.949435i	$0.601657\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	−48.3607	−0.540689
$21$	2.02129	0.0210038
$22$	−31.8885	−0.309030
$23$	−9.11146	−0.0826030	−0.0413015	−	0.999147i	$-0.513150\pi$
$23$	−9.11146	−0.0826030	−0.0413015	+	0.999147i	$0.513150\pi$
$24$	24.0000	0.204124
$25$	21.1722	0.169378
$26$	7.81966	0.0589831
$27$	−27.0000	−0.192450
$28$	−2.69505	−0.0181899
$29$	161.151	1.03190	0.515948	−	0.856620i	$-0.327440\pi$
$29$	161.151	1.03190	0.515948	+	0.856620i	$0.327440\pi$
$30$	−72.5410	−0.441471
$31$	−60.5035	−0.350540	−0.175270	−	0.984520i	$-0.556080\pi$
$31$	−60.5035	−0.350540	−0.175270	+	0.984520i	$0.556080\pi$
$32$	−32.0000	−0.176777
$33$	−47.8328	−0.252322
$34$	88.0263	0.444011
$35$	8.14590	0.0393402
$36$	36.0000	0.166667
$37$	−104.885	−0.466029	−0.233014	−	0.972473i	$-0.574859\pi$
$37$	−104.885	−0.466029	−0.233014	+	0.972473i	$0.574859\pi$
$38$	0	0
$39$	11.7295	0.0481595
$40$	96.7214	0.382325
$41$	−249.353	−0.949813	−0.474906	−	0.880036i	$-0.657518\pi$
$41$	−249.353	−0.949813	−0.474906	+	0.880036i	$0.657518\pi$
$42$	−4.04257	−0.0148520
$43$	−6.90481	−0.0244877	−0.0122439	−	0.999925i	$-0.503897\pi$
$43$	−6.90481	−0.0244877	−0.0122439	+	0.999925i	$0.503897\pi$
$44$	63.7771	0.218517
$45$	−108.812	−0.360459
$46$	18.2229	0.0584092
$47$	145.257	0.450808	0.225404	−	0.974265i	$-0.427630\pi$
$47$	145.257	0.450808	0.225404	+	0.974265i	$0.427630\pi$
$48$	−48.0000	−0.144338
$49$	−342.546	−0.998677
$50$	−42.3444	−0.119768
$51$	132.039	0.362534
$52$	−15.6393	−0.0417074
$53$	754.187	1.95463	0.977317	−	0.211782i	$-0.0679267\pi$
$53$	754.187	1.95463	0.977317	+	0.211782i	$0.0679267\pi$
$54$	54.0000	0.136083
$55$	−192.769	−0.472599
$56$	5.39010	0.0128622
$57$	0	0
$58$	−322.302	−0.729660
$59$	704.046	1.55354	0.776771	−	0.629783i	$-0.216856\pi$
$59$	704.046	1.55354	0.776771	+	0.629783i	$0.216856\pi$
$60$	145.082	0.312167
$61$	−32.5085	−0.0682342	−0.0341171	−	0.999418i	$-0.510862\pi$
$61$	−32.5085	−0.0682342	−0.0341171	+	0.999418i	$0.510862\pi$
$62$	121.007	0.247869
$63$	−6.06386	−0.0121266
$64$	64.0000	0.125000
$65$	47.2705	0.0902028
$66$	95.6656	0.178419
$67$	−77.7264	−0.141728	−0.0708641	−	0.997486i	$-0.522576\pi$
$67$	−77.7264	−0.141728	−0.0708641	+	0.997486i	$0.522576\pi$
$68$	−176.053	−0.313963
$69$	27.3344	0.0476909
$70$	−16.2918	−0.0278177
$71$	−44.4183	−0.0742463	−0.0371231	−	0.999311i	$-0.511819\pi$
$71$	−44.4183	−0.0742463	−0.0371231	+	0.999311i	$0.511819\pi$
$72$	−72.0000	−0.117851
$73$	280.735	0.450103	0.225051	−	0.974347i	$-0.427745\pi$
$73$	280.735	0.450103	0.225051	+	0.974347i	$0.427745\pi$
$74$	209.771	0.329532
$75$	−63.5166	−0.0977902
$76$	0	0
$77$	−10.7426	−0.0158992
$78$	−23.4590	−0.0340539
$79$	−779.508	−1.11015	−0.555073	−	0.831802i	$-0.687310\pi$
$79$	−779.508	−1.11015	−0.555073	+	0.831802i	$0.687310\pi$
$80$	−193.443	−0.270344
$81$	81.0000	0.111111
$82$	498.705	0.671619
$83$	−274.023	−0.362385	−0.181192	−	0.983448i	$-0.557996\pi$
$83$	−274.023	−0.362385	−0.181192	+	0.983448i	$0.557996\pi$
$84$	8.08514	0.0105019
$85$	532.127	0.679026
$86$	13.8096	0.0173154
$87$	−483.453	−0.595765
$88$	−127.554	−0.154515
$89$	1005.91	1.19805	0.599023	−	0.800732i	$-0.295556\pi$
$89$	1005.91	1.19805	0.599023	+	0.800732i	$0.295556\pi$
$90$	217.623	0.254883
$91$	2.63430	0.00303461
$92$	−36.4458	−0.0413015
$93$	181.510	0.202384
$94$	−290.515	−0.318769
$95$	0	0
$96$	96.0000	0.102062
$97$	832.600	0.871523	0.435761	−	0.900062i	$-0.356479\pi$
$97$	832.600	0.871523	0.435761	+	0.900062i	$0.356479\pi$
$98$	685.092	0.706171
$99$	143.498	0.145678
$100$	84.6888	0.0846888
$101$	910.371	0.896885	0.448442	−	0.893812i	$-0.351979\pi$
$101$	910.371	0.896885	0.448442	+	0.893812i	$0.351979\pi$
$102$	−264.079	−0.256350
$103$	1272.16	1.21699	0.608493	−	0.793559i	$-0.291774\pi$
$103$	1272.16	1.21699	0.608493	+	0.793559i	$0.291774\pi$
$104$	31.2786	0.0294916
$105$	−24.4377	−0.0227131
$106$	−1508.37	−1.38213
$107$	−783.565	−0.707945	−0.353972	−	0.935256i	$-0.615169\pi$
$107$	−783.565	−0.707945	−0.353972	+	0.935256i	$0.615169\pi$
$108$	−108.000	−0.0962250
$109$	2013.17	1.76905	0.884524	−	0.466494i	$-0.154483\pi$
$109$	2013.17	1.76905	0.884524	+	0.466494i	$0.154483\pi$
$110$	385.538	0.334178
$111$	314.656	0.269062
$112$	−10.7802	−0.00909493
$113$	10.6506	0.00886655	0.00443327	−	0.999990i	$-0.498589\pi$
$113$	10.6506	0.00886655	0.00443327	+	0.999990i	$0.498589\pi$
$114$	0	0
$115$	110.159	0.0893251
$116$	644.604	0.515948
$117$	−35.1885	−0.0278049
$118$	−1408.09	−1.09852
$119$	29.6544	0.0228438
$120$	−290.164	−0.220735
$121$	−1076.78	−0.809001
$122$	65.0170	0.0482489
$123$	748.058	0.548375
$124$	−242.014	−0.175270
$125$	1255.30	0.898216
$126$	12.1277	0.00857479
$127$	932.541	0.651572	0.325786	−	0.945444i	$-0.394371\pi$
$127$	932.541	0.651572	0.325786	+	0.945444i	$0.394371\pi$
$128$	−128.000	−0.0883883
$129$	20.7144	0.0141380
$130$	−94.5410	−0.0637830
$131$	208.795	0.139256	0.0696278	−	0.997573i	$-0.477819\pi$
$131$	208.795	0.139256	0.0696278	+	0.997573i	$0.477819\pi$
$132$	−191.331	−0.126161
$133$	0	0
$134$	155.453	0.100217
$135$	326.435	0.208111
$136$	352.105	0.222006
$137$	−368.275	−0.229663	−0.114832	−	0.993385i	$-0.536633\pi$
$137$	−368.275	−0.229663	−0.114832	+	0.993385i	$0.536633\pi$
$138$	−54.6687	−0.0337226
$139$	−1722.46	−1.05106	−0.525528	−	0.850776i	$-0.676132\pi$
$139$	−1722.46	−1.05106	−0.525528	+	0.850776i	$0.676132\pi$
$140$	32.5836	0.0196701
$141$	−435.772	−0.260274
$142$	88.8367	0.0525000
$143$	−62.3394	−0.0364551
$144$	144.000	0.0833333
$145$	−1948.34	−1.11587
$146$	−561.469	−0.318271
$147$	1027.64	0.576586
$148$	−419.542	−0.233014
$149$	−2688.05	−1.47795	−0.738973	−	0.673736i	$-0.764689\pi$
$149$	−2688.05	−1.47795	−0.738973	+	0.673736i	$0.764689\pi$
$150$	127.033	0.0691481
$151$	−2474.74	−1.33372	−0.666860	−	0.745183i	$-0.732362\pi$
$151$	−2474.74	−1.33372	−0.666860	+	0.745183i	$0.732362\pi$
$152$	0	0
$153$	−396.118	−0.209309
$154$	21.4853	0.0112424
$155$	731.497	0.379066
$156$	46.9180	0.0240798
$157$	−1209.81	−0.614991	−0.307495	−	0.951550i	$-0.599491\pi$
$157$	−1209.81	−0.614991	−0.307495	+	0.951550i	$0.599491\pi$
$158$	1559.02	0.784992
$159$	−2262.56	−1.12851
$160$	386.885	0.191162
$161$	6.13895	0.00300508
$162$	−162.000	−0.0785674
$163$	2745.03	1.31906	0.659531	−	0.751678i	$-0.270755\pi$
$163$	2745.03	1.31906	0.659531	+	0.751678i	$0.270755\pi$
$164$	−997.410	−0.474906
$165$	578.307	0.272855
$166$	548.046	0.256245
$167$	3241.89	1.50218	0.751092	−	0.660197i	$-0.229527\pi$
$167$	3241.89	1.50218	0.751092	+	0.660197i	$0.229527\pi$
$168$	−16.1703	−0.00742598
$169$	−2181.71	−0.993042
$170$	−1064.25	−0.480144
$171$	0	0
$172$	−27.6192	−0.0122439
$173$	2461.28	1.08166	0.540832	−	0.841130i	$-0.318109\pi$
$173$	2461.28	1.08166	0.540832	+	0.841130i	$0.318109\pi$
$174$	966.906	0.421270
$175$	−14.2650	−0.00616191
$176$	255.108	0.109259
$177$	−2112.14	−0.896938
$178$	−2011.82	−0.847147
$179$	2964.26	1.23776	0.618880	−	0.785485i	$-0.287587\pi$
$179$	2964.26	1.23776	0.618880	+	0.785485i	$0.287587\pi$
$180$	−435.246	−0.180230
$181$	−4403.67	−1.80841	−0.904204	−	0.427100i	$-0.859535\pi$
$181$	−4403.67	−1.80841	−0.904204	+	0.427100i	$0.859535\pi$
$182$	−5.26859	−0.00214579
$183$	97.5255	0.0393950
$184$	72.8916	0.0292046
$185$	1268.08	0.503953
$186$	−363.021	−0.143107
$187$	−701.758	−0.274426
$188$	581.029	0.225404
$189$	18.1916	0.00700128
$190$	0	0
$191$	−2167.97	−0.821302	−0.410651	−	0.911793i	$-0.634698\pi$
$191$	−2167.97	−0.821302	−0.410651	+	0.911793i	$0.634698\pi$
$192$	−192.000	−0.0721688
$193$	3122.55	1.16459	0.582296	−	0.812977i	$-0.302155\pi$
$193$	3122.55	1.16459	0.582296	+	0.812977i	$0.302155\pi$
$194$	−1665.20	−0.616260
$195$	−141.812	−0.0520786
$196$	−1370.18	−0.499338
$197$	202.429	0.0732106	0.0366053	−	0.999330i	$-0.488346\pi$
$197$	202.429	0.0732106	0.0366053	+	0.999330i	$0.488346\pi$
$198$	−286.997	−0.103010
$199$	−1649.25	−0.587497	−0.293749	−	0.955883i	$-0.594903\pi$
$199$	−1649.25	−0.587497	−0.293749	+	0.955883i	$0.594903\pi$
$200$	−169.378	−0.0598841
$201$	233.179	0.0818268
$202$	−1820.74	−0.634193
$203$	−108.577	−0.0375401
$204$	528.158	0.181267
$205$	3014.71	1.02711
$206$	−2544.32	−0.860540
$207$	−82.0031	−0.0275343
$208$	−62.5573	−0.0208537
$209$	0	0
$210$	48.8754	0.0160606
$211$	−1629.35	−0.531608	−0.265804	−	0.964027i	$-0.585637\pi$
$211$	−1629.35	−0.531608	−0.265804	+	0.964027i	$0.585637\pi$
$212$	3016.75	0.977317
$213$	133.255	0.0428661
$214$	1567.13	0.500593
$215$	83.4803	0.0264805
$216$	216.000	0.0680414
$217$	40.7649	0.0127526
$218$	−4026.33	−1.25091
$219$	−842.204	−0.259867
$220$	−771.076	−0.236300
$221$	172.084	0.0523784
$222$	−629.313	−0.190255
$223$	481.492	0.144588	0.0722939	−	0.997383i	$-0.476968\pi$
$223$	481.492	0.144588	0.0722939	+	0.997383i	$0.476968\pi$
$224$	21.5604	0.00643109
$225$	190.550	0.0564592
$226$	−21.3011	−0.00626960
$227$	−2786.59	−0.814768	−0.407384	−	0.913257i	$-0.633559\pi$
$227$	−2786.59	−0.814768	−0.407384	+	0.913257i	$0.633559\pi$
$228$	0	0
$229$	1437.72	0.414878	0.207439	−	0.978248i	$-0.433487\pi$
$229$	1437.72	0.414878	0.207439	+	0.978248i	$0.433487\pi$
$230$	−220.318	−0.0631624
$231$	32.2279	0.00917941
$232$	−1289.21	−0.364830
$233$	−2782.74	−0.782419	−0.391209	−	0.920302i	$-0.627943\pi$
$233$	−2782.74	−0.782419	−0.391209	+	0.920302i	$0.627943\pi$
$234$	70.3769	0.0196610
$235$	−1756.19	−0.487493
$236$	2816.19	0.776771
$237$	2338.53	0.640943
$238$	−59.3088	−0.0161530
$239$	4530.56	1.22618	0.613091	−	0.790013i	$-0.289926\pi$
$239$	4530.56	1.22618	0.613091	+	0.790013i	$0.289926\pi$
$240$	580.328	0.156083
$241$	2457.27	0.656792	0.328396	−	0.944540i	$-0.393492\pi$
$241$	2457.27	0.656792	0.328396	+	0.944540i	$0.393492\pi$
$242$	2153.56	0.572050
$243$	−243.000	−0.0641500
$244$	−130.034	−0.0341171
$245$	4141.44	1.07995
$246$	−1496.12	−0.387759
$247$	0	0
$248$	484.028	0.123935
$249$	822.070	0.209223
$250$	−2510.59	−0.635135
$251$	−3211.61	−0.807631	−0.403815	−	0.914841i	$-0.632316\pi$
$251$	−3211.61	−0.807631	−0.403815	+	0.914841i	$0.632316\pi$
$252$	−24.2554	−0.00606329
$253$	−145.276	−0.0361004
$254$	−1865.08	−0.460731
$255$	−1596.38	−0.392036
$256$	256.000	0.0625000
$257$	−1670.69	−0.405504	−0.202752	−	0.979230i	$-0.564989\pi$
$257$	−1670.69	−0.405504	−0.202752	+	0.979230i	$0.564989\pi$
$258$	−41.4288	−0.00999708
$259$	70.6678	0.0169540
$260$	189.082	0.0451014
$261$	1450.36	0.343965
$262$	−417.590	−0.0984686
$263$	−7731.93	−1.81282	−0.906409	−	0.422402i	$-0.861187\pi$
$263$	−7731.93	−1.81282	−0.906409	+	0.422402i	$0.861187\pi$
$264$	382.663	0.0892093
$265$	−9118.25	−2.11370
$266$	0	0
$267$	−3017.73	−0.691692
$268$	−310.906	−0.0708641
$269$	−1024.48	−0.232207	−0.116104	−	0.993237i	$-0.537040\pi$
$269$	−1024.48	−0.232207	−0.116104	+	0.993237i	$0.537040\pi$
$270$	−652.869	−0.147157
$271$	1067.86	0.239366	0.119683	−	0.992812i	$-0.461812\pi$
$271$	1067.86	0.239366	0.119683	+	0.992812i	$0.461812\pi$
$272$	−704.210	−0.156982
$273$	−7.90289	−0.00175203
$274$	736.550	0.162396
$275$	337.575	0.0740239
$276$	109.337	0.0238454
$277$	311.065	0.0674733	0.0337367	−	0.999431i	$-0.489259\pi$
$277$	311.065	0.0674733	0.0337367	+	0.999431i	$0.489259\pi$
$278$	3444.91	0.743209
$279$	−544.531	−0.116847
$280$	−65.1672	−0.0139089
$281$	7346.75	1.55968	0.779840	−	0.625979i	$-0.215300\pi$
$281$	7346.75	1.55968	0.779840	+	0.625979i	$0.215300\pi$
$282$	871.544	0.184041
$283$	4046.95	0.850056	0.425028	−	0.905180i	$-0.360264\pi$
$283$	4046.95	0.850056	0.425028	+	0.905180i	$0.360264\pi$
$284$	−177.673	−0.0371231
$285$	0	0
$286$	124.679	0.0257777
$287$	168.004	0.0345539
$288$	−288.000	−0.0589256
$289$	−2975.84	−0.605708
$290$	3896.68	0.789039
$291$	−2497.80	−0.503174
$292$	1122.94	0.225051
$293$	3641.56	0.726083	0.363041	−	0.931773i	$-0.381738\pi$
$293$	3641.56	0.726083	0.363041	+	0.931773i	$0.381738\pi$
$294$	−2055.28	−0.407708
$295$	−8512.04	−1.67997
$296$	839.084	0.164766
$297$	−430.495	−0.0841073
$298$	5376.10	1.04506
$299$	35.6242	0.00689031
$300$	−254.067	−0.0488951
$301$	4.65220	0.000890858 0
$302$	4949.49	0.943082
$303$	−2731.11	−0.517817
$304$	0	0
$305$	393.033	0.0737869
$306$	792.237	0.148004
$307$	−4685.42	−0.871045	−0.435523	−	0.900178i	$-0.643436\pi$
$307$	−4685.42	−0.871045	−0.435523	+	0.900178i	$0.643436\pi$
$308$	−42.9706	−0.00794960
$309$	−3816.48	−0.702628
$310$	−1462.99	−0.268040
$311$	−2413.55	−0.440065	−0.220032	−	0.975493i	$-0.570616\pi$
$311$	−2413.55	−0.440065	−0.220032	+	0.975493i	$0.570616\pi$
$312$	−93.8359	−0.0170270
$313$	−5062.23	−0.914167	−0.457084	−	0.889424i	$-0.651106\pi$
$313$	−5062.23	−0.914167	−0.457084	+	0.889424i	$0.651106\pi$
$314$	2419.63	0.434864
$315$	73.3131	0.0131134
$316$	−3118.03	−0.555073
$317$	−5704.03	−1.01063	−0.505316	−	0.862935i	$-0.668624\pi$
$317$	−5704.03	−1.01063	−0.505316	+	0.862935i	$0.668624\pi$
$318$	4525.12	0.797976
$319$	2569.43	0.450974
$320$	−773.771	−0.135172
$321$	2350.69	0.408732
$322$	−12.2779	−0.00212491
$323$	0	0
$324$	324.000	0.0555556
$325$	−82.7797	−0.0141286
$326$	−5490.05	−0.932717
$327$	−6039.50	−1.02136
$328$	1994.82	0.335810
$329$	−97.8689	−0.0164003
$330$	−1156.61	−0.192938
$331$	−8367.55	−1.38949	−0.694747	−	0.719254i	$-0.744484\pi$
$331$	−8367.55	−1.38949	−0.694747	+	0.719254i	$0.744484\pi$
$332$	−1096.09	−0.181192
$333$	−943.969	−0.155343
$334$	−6483.78	−1.06221
$335$	939.725	0.153262
$336$	32.3406	0.00525096
$337$	−263.413	−0.0425787	−0.0212894	−	0.999773i	$-0.506777\pi$
$337$	−263.413	−0.0425787	−0.0212894	+	0.999773i	$0.506777\pi$
$338$	4363.43	0.702187
$339$	−31.9517	−0.00511910
$340$	2128.51	0.339513
$341$	−964.684	−0.153198
$342$	0	0
$343$	461.895	0.0727113
$344$	55.2384	0.00865772
$345$	−330.477	−0.0515719
$346$	−4922.57	−0.764853
$347$	4021.50	0.622148	0.311074	−	0.950386i	$-0.399311\pi$
$347$	4021.50	0.622148	0.311074	+	0.950386i	$0.399311\pi$
$348$	−1933.81	−0.297883
$349$	453.400	0.0695414	0.0347707	−	0.999395i	$-0.488930\pi$
$349$	453.400	0.0695414	0.0347707	+	0.999395i	$0.488930\pi$
$350$	28.5301	0.00435713
$351$	105.565	0.0160532
$352$	−510.217	−0.0772575
$353$	−4547.61	−0.685680	−0.342840	−	0.939394i	$-0.611389\pi$
$353$	−4547.61	−0.685680	−0.342840	+	0.939394i	$0.611389\pi$
$354$	4224.28	0.634231
$355$	537.025	0.0802883
$356$	4023.63	0.599023
$357$	−88.9632	−0.0131889
$358$	−5928.52	−0.875229
$359$	8396.86	1.23445	0.617227	−	0.786785i	$-0.288256\pi$
$359$	8396.86	1.23445	0.617227	+	0.786785i	$0.288256\pi$
$360$	870.492	0.127442
$361$	0	0
$362$	8807.33	1.27874
$363$	3230.34	0.467077
$364$	10.5372	0.00151730
$365$	−3394.13	−0.486731
$366$	−195.051	−0.0278565
$367$	−5422.59	−0.771273	−0.385636	−	0.922651i	$-0.626018\pi$
$367$	−5422.59	−0.771273	−0.385636	+	0.922651i	$0.626018\pi$
$368$	−145.783	−0.0206508
$369$	−2244.17	−0.316604
$370$	−2536.17	−0.356349
$371$	−508.143	−0.0711091
$372$	726.042	0.101192
$373$	−7439.39	−1.03270	−0.516350	−	0.856378i	$-0.672710\pi$
$373$	−7439.39	−1.03270	−0.516350	+	0.856378i	$0.672710\pi$
$374$	1403.52	0.194048
$375$	−3765.89	−0.518586
$376$	−1162.06	−0.159385
$377$	−630.073	−0.0860753
$378$	−36.3832	−0.00495065
$379$	−12741.8	−1.72692	−0.863461	−	0.504415i	$-0.831708\pi$
$379$	−12741.8	−1.72692	−0.863461	+	0.504415i	$0.831708\pi$
$380$	0	0
$381$	−2797.62	−0.376185
$382$	4335.93	0.580748
$383$	−10550.9	−1.40764	−0.703822	−	0.710376i	$-0.748525\pi$
$383$	−10550.9	−1.40764	−0.703822	+	0.710376i	$0.748525\pi$
$384$	384.000	0.0510310
$385$	129.880	0.0171930
$386$	−6245.10	−0.823490
$387$	−62.1432	−0.00816258
$388$	3330.40	0.435761
$389$	6581.54	0.857834	0.428917	−	0.903344i	$-0.358895\pi$
$389$	6581.54	0.857834	0.428917	+	0.903344i	$0.358895\pi$
$390$	283.623	0.0368252
$391$	401.024	0.0518687
$392$	2740.37	0.353085
$393$	−626.384	−0.0803993
$394$	−404.859	−0.0517677
$395$	9424.39	1.20049
$396$	573.994	0.0728391
$397$	2148.76	0.271646	0.135823	−	0.990733i	$-0.456632\pi$
$397$	2148.76	0.271646	0.135823	+	0.990733i	$0.456632\pi$
$398$	3298.49	0.415423
$399$	0	0
$400$	338.755	0.0423444
$401$	−2532.20	−0.315342	−0.157671	−	0.987492i	$-0.550398\pi$
$401$	−2532.20	−0.315342	−0.157671	+	0.987492i	$0.550398\pi$
$402$	−466.358	−0.0578603
$403$	236.558	0.0292402
$404$	3641.49	0.448442
$405$	−979.304	−0.120153
$406$	217.155	0.0265449
$407$	−1672.32	−0.203671
$408$	−1056.32	−0.128175
$409$	−8046.03	−0.972740	−0.486370	−	0.873753i	$-0.661679\pi$
$409$	−8046.03	−0.972740	−0.486370	+	0.873753i	$0.661679\pi$
$410$	−6029.43	−0.726274
$411$	1104.83	0.132596
$412$	5088.64	0.608493
$413$	−474.360	−0.0565175
$414$	164.006	0.0194697
$415$	3312.99	0.391875
$416$	125.115	0.0147458
$417$	5167.37	0.606828
$418$	0	0
$419$	−10432.9	−1.21642	−0.608210	−	0.793776i	$-0.708112\pi$
$419$	−10432.9	−1.21642	−0.608210	+	0.793776i	$0.708112\pi$
$420$	−97.7508	−0.0113565
$421$	−4591.58	−0.531544	−0.265772	−	0.964036i	$-0.585627\pi$
$421$	−4591.58	−0.531544	−0.265772	+	0.964036i	$0.585627\pi$
$422$	3258.71	0.375904
$423$	1307.32	0.150269
$424$	−6033.50	−0.691067
$425$	−931.856	−0.106357
$426$	−266.510	−0.0303109
$427$	21.9030	0.00248234
$428$	−3134.26	−0.353972
$429$	187.018	0.0210474
$430$	−166.961	−0.0187245
$431$	−14117.1	−1.57772	−0.788861	−	0.614571i	$-0.789329\pi$
$431$	−14117.1	−1.57772	−0.788861	+	0.614571i	$0.789329\pi$
$432$	−432.000	−0.0481125
$433$	13929.0	1.54592	0.772962	−	0.634452i	$-0.218774\pi$
$433$	13929.0	1.54592	0.772962	+	0.634452i	$0.218774\pi$
$434$	−81.5299	−0.00901742
$435$	5845.03	0.644247
$436$	8052.66	0.884524
$437$	0	0
$438$	1684.41	0.183754
$439$	−5222.03	−0.567731	−0.283866	−	0.958864i	$-0.591617\pi$
$439$	−5222.03	−0.567731	−0.283866	+	0.958864i	$0.591617\pi$
$440$	1542.15	0.167089
$441$	−3082.91	−0.332892
$442$	−344.168	−0.0370371
$443$	−10163.4	−1.09002	−0.545010	−	0.838430i	$-0.683474\pi$
$443$	−10163.4	−1.09002	−0.545010	+	0.838430i	$0.683474\pi$
$444$	1258.63	0.134531
$445$	−12161.6	−1.29554
$446$	−962.984	−0.102239
$447$	8064.16	0.853292
$448$	−43.1208	−0.00454747
$449$	−17851.6	−1.87632	−0.938161	−	0.346199i	$-0.887472\pi$
$449$	−17851.6	−1.87632	−0.938161	+	0.346199i	$0.887472\pi$
$450$	−381.100	−0.0399227
$451$	−3975.74	−0.415101
$452$	42.6022	0.00443327
$453$	7424.23	0.770024
$454$	5573.18	0.576128
$455$	−31.8491	−0.00328156
$456$	0	0
$457$	−5037.95	−0.515679	−0.257839	−	0.966188i	$-0.583011\pi$
$457$	−5037.95	−0.515679	−0.257839	+	0.966188i	$0.583011\pi$
$458$	−2875.44	−0.293363
$459$	1188.36	0.120845
$460$	440.636	0.0446625
$461$	−11181.4	−1.12965	−0.564825	−	0.825211i	$-0.691056\pi$
$461$	−11181.4	−1.12965	−0.564825	+	0.825211i	$0.691056\pi$
$462$	−64.4559	−0.00649082
$463$	18455.1	1.85244	0.926222	−	0.376978i	$-0.123037\pi$
$463$	18455.1	1.85244	0.926222	+	0.376978i	$0.123037\pi$
$464$	2578.41	0.257974
$465$	−2194.49	−0.218854
$466$	5565.48	0.553253
$467$	6412.78	0.635435	0.317718	−	0.948185i	$-0.397084\pi$
$467$	6412.78	0.635435	0.317718	+	0.948185i	$0.397084\pi$
$468$	−140.754	−0.0139025
$469$	52.3691	0.00515603
$470$	3512.37	0.344710
$471$	3629.44	0.355065
$472$	−5632.37	−0.549260
$473$	−110.092	−0.0107020
$474$	−4677.05	−0.453215
$475$	0	0
$476$	118.618	0.0114219
$477$	6787.69	0.651545
$478$	−9061.11	−0.867041
$479$	−8519.67	−0.812680	−0.406340	−	0.913722i	$-0.633195\pi$
$479$	−8519.67	−0.812680	−0.406340	+	0.913722i	$0.633195\pi$
$480$	−1160.66	−0.110368
$481$	410.084	0.0388737
$482$	−4914.55	−0.464422
$483$	−18.4169	−0.00173498
$484$	−4307.12	−0.404500
$485$	−10066.3	−0.942445
$486$	486.000	0.0453609
$487$	−12746.8	−1.18607	−0.593033	−	0.805178i	$-0.702070\pi$
$487$	−12746.8	−1.18607	−0.593033	+	0.805178i	$0.702070\pi$
$488$	260.068	0.0241244
$489$	−8235.08	−0.761560
$490$	−8282.88	−0.763637
$491$	−5187.09	−0.476762	−0.238381	−	0.971172i	$-0.576617\pi$
$491$	−5187.09	−0.476762	−0.238381	+	0.971172i	$0.576617\pi$
$492$	2992.23	0.274187
$493$	−7092.76	−0.647955
$494$	0	0
$495$	−1734.92	−0.157533
$496$	−968.056	−0.0876350
$497$	29.9274	0.00270106
$498$	−1644.14	−0.147943
$499$	−10704.5	−0.960315	−0.480158	−	0.877182i	$-0.659421\pi$
$499$	−10704.5	−0.960315	−0.480158	+	0.877182i	$0.659421\pi$
$500$	5021.18	0.449108
$501$	−9725.67	−0.867287
$502$	6423.23	0.571081
$503$	13720.9	1.21627	0.608137	−	0.793832i	$-0.291917\pi$
$503$	13720.9	1.21627	0.608137	+	0.793832i	$0.291917\pi$
$504$	48.5109	0.00428739
$505$	−11006.5	−0.969871
$506$	290.551	0.0255268
$507$	6545.14	0.573333
$508$	3730.16	0.325786
$509$	6427.78	0.559737	0.279869	−	0.960038i	$-0.409709\pi$
$509$	6427.78	0.559737	0.279869	+	0.960038i	$0.409709\pi$
$510$	3192.76	0.277211
$511$	−189.148	−0.0163746
$512$	−512.000	−0.0441942
$513$	0	0
$514$	3341.37	0.286735
$515$	−15380.6	−1.31602
$516$	82.8577	0.00706900
$517$	2316.02	0.197018
$518$	−141.336	−0.0119883
$519$	−7383.85	−0.624500
$520$	−378.164	−0.0318915
$521$	2296.27	0.193093	0.0965464	−	0.995328i	$-0.469220\pi$
$521$	2296.27	0.193093	0.0965464	+	0.995328i	$0.469220\pi$
$522$	−2900.72	−0.243220
$523$	672.621	0.0562364	0.0281182	−	0.999605i	$-0.491049\pi$
$523$	672.621	0.0562364	0.0281182	+	0.999605i	$0.491049\pi$
$524$	835.179	0.0696278
$525$	42.7951	0.00355758
$526$	15463.9	1.28186
$527$	2662.95	0.220114
$528$	−765.325	−0.0630805
$529$	−12084.0	−0.993177
$530$	18236.5	1.49461
$531$	6336.42	0.517848
$532$	0	0
$533$	974.926	0.0792284
$534$	6035.45	0.489100
$535$	9473.43	0.765556
$536$	621.811	0.0501085
$537$	−8892.78	−0.714622
$538$	2048.96	0.164195
$539$	−5461.65	−0.436456
$540$	1305.74	0.104056
$541$	−18524.3	−1.47213	−0.736065	−	0.676911i	$-0.763318\pi$
$541$	−18524.3	−1.47213	−0.736065	+	0.676911i	$0.763318\pi$
$542$	−2135.73	−0.169257
$543$	13211.0	1.04409
$544$	1408.42	0.111003
$545$	−24339.5	−1.91301
$546$	15.8058	0.00123887
$547$	−6149.10	−0.480652	−0.240326	−	0.970692i	$-0.577254\pi$
$547$	−6149.10	−0.480652	−0.240326	+	0.970692i	$0.577254\pi$
$548$	−1473.10	−0.114832
$549$	−292.576	−0.0227447
$550$	−675.151	−0.0523428
$551$	0	0
$552$	−218.675	−0.0168613
$553$	525.203	0.0403868
$554$	−622.131	−0.0477109
$555$	−3804.25	−0.290957
$556$	−6889.82	−0.525528
$557$	−1151.85	−0.0876218	−0.0438109	−	0.999040i	$-0.513950\pi$
$557$	−1151.85	−0.0876218	−0.0438109	+	0.999040i	$0.513950\pi$
$558$	1089.06	0.0826231
$559$	26.9966	0.00204264
$560$	130.334	0.00983506
$561$	2105.27	0.158440
$562$	−14693.5	−1.10286
$563$	−3182.72	−0.238252	−0.119126	−	0.992879i	$-0.538009\pi$
$563$	−3182.72	−0.238252	−0.119126	+	0.992879i	$0.538009\pi$
$564$	−1743.09	−0.130137
$565$	−128.767	−0.00958809
$566$	−8093.89	−0.601081
$567$	−54.5747	−0.00404219
$568$	355.347	0.0262500
$569$	20679.8	1.52363	0.761814	−	0.647796i	$-0.224309\pi$
$569$	20679.8	1.52363	0.761814	+	0.647796i	$0.224309\pi$
$570$	0	0
$571$	−2820.58	−0.206721	−0.103360	−	0.994644i	$-0.532960\pi$
$571$	−2820.58	−0.206721	−0.103360	+	0.994644i	$0.532960\pi$
$572$	−249.358	−0.0182276
$573$	6503.90	0.474179
$574$	−336.009	−0.0244333
$575$	−192.910	−0.0139911
$576$	576.000	0.0416667
$577$	11208.6	0.808700	0.404350	−	0.914604i	$-0.367498\pi$
$577$	11208.6	0.808700	0.404350	+	0.914604i	$0.367498\pi$
$578$	5951.68	0.428300
$579$	−9367.65	−0.672377
$580$	−7793.37	−0.557934
$581$	184.626	0.0131835
$582$	4995.60	0.355798
$583$	12025.0	0.854242
$584$	−2245.88	−0.159135
$585$	425.435	0.0300676
$586$	−7283.12	−0.513418
$587$	−209.871	−0.0147569	−0.00737846	−	0.999973i	$-0.502349\pi$
$587$	−209.871	−0.0147569	−0.00737846	+	0.999973i	$0.502349\pi$
$588$	4110.55	0.288293
$589$	0	0
$590$	17024.1	1.18792
$591$	−607.288	−0.0422682
$592$	−1678.17	−0.116507
$593$	9737.59	0.674325	0.337163	−	0.941446i	$-0.390533\pi$
$593$	9737.59	0.674325	0.337163	+	0.941446i	$0.390533\pi$
$594$	860.991	0.0594729
$595$	−358.527	−0.0247028
$596$	−10752.2	−0.738973
$597$	4947.74	0.339192
$598$	−71.2485	−0.00487219
$599$	−3772.44	−0.257325	−0.128662	−	0.991688i	$-0.541068\pi$
$599$	−3772.44	−0.257325	−0.128662	+	0.991688i	$0.541068\pi$
$600$	508.133	0.0345741
$601$	−12419.2	−0.842910	−0.421455	−	0.906849i	$-0.638480\pi$
$601$	−12419.2	−0.842910	−0.421455	+	0.906849i	$0.638480\pi$
$602$	−9.30439	−0.000629931 0
$603$	−699.537	−0.0472427
$604$	−9898.97	−0.666860
$605$	13018.5	0.874836
$606$	5462.23	0.366152
$607$	3934.15	0.263068	0.131534	−	0.991312i	$-0.458010\pi$
$607$	3934.15	0.263068	0.131534	+	0.991312i	$0.458010\pi$
$608$	0	0
$609$	325.732	0.0216738
$610$	−786.067	−0.0521753
$611$	−567.932	−0.0376040
$612$	−1584.47	−0.104654
$613$	−22886.6	−1.50796	−0.753980	−	0.656897i	$-0.771868\pi$
$613$	−22886.6	−1.50796	−0.753980	+	0.656897i	$0.771868\pi$
$614$	9370.84	0.615922
$615$	−9044.14	−0.593000
$616$	85.9412	0.00562121
$617$	5961.78	0.388999	0.194500	−	0.980903i	$-0.437692\pi$
$617$	5961.78	0.388999	0.194500	+	0.980903i	$0.437692\pi$
$618$	7632.96	0.496833
$619$	15162.2	0.984526	0.492263	−	0.870447i	$-0.336170\pi$
$619$	15162.2	0.984526	0.492263	+	0.870447i	$0.336170\pi$
$620$	2925.99	0.189533
$621$	246.009	0.0158970
$622$	4827.11	0.311173
$623$	−677.743	−0.0435846
$624$	187.672	0.0120399
$625$	−17823.3	−1.14069
$626$	10124.5	0.646414
$627$	0	0
$628$	−4839.25	−0.307495
$629$	4616.34	0.292632
$630$	−146.626	−0.00927258
$631$	19490.7	1.22966	0.614829	−	0.788661i	$-0.289225\pi$
$631$	19490.7	1.22966	0.614829	+	0.788661i	$0.289225\pi$
$632$	6236.07	0.392496
$633$	4888.06	0.306924
$634$	11408.1	0.714624
$635$	−11274.6	−0.704595
$636$	−9050.25	−0.564254
$637$	1339.30	0.0833044
$638$	−5138.87	−0.318887
$639$	−399.765	−0.0247488
$640$	1547.54	0.0955812
$641$	7848.89	0.483639	0.241819	−	0.970321i	$-0.422256\pi$
$641$	7848.89	0.483639	0.241819	+	0.970321i	$0.422256\pi$
$642$	−4701.39	−0.289017
$643$	−15640.8	−0.959272	−0.479636	−	0.877468i	$-0.659231\pi$
$643$	−15640.8	−0.959272	−0.479636	+	0.877468i	$0.659231\pi$
$644$	24.5558	0.00150254
$645$	−250.441	−0.0152885
$646$	0	0
$647$	5621.52	0.341584	0.170792	−	0.985307i	$-0.445367\pi$
$647$	5621.52	0.341584	0.170792	+	0.985307i	$0.445367\pi$
$648$	−648.000	−0.0392837
$649$	11225.5	0.678952
$650$	165.559	0.00999043
$651$	−122.295	−0.00736269
$652$	10980.1	0.659531
$653$	−21586.7	−1.29365	−0.646824	−	0.762640i	$-0.723903\pi$
$653$	−21586.7	−1.29365	−0.646824	+	0.762640i	$0.723903\pi$
$654$	12079.0	0.722211
$655$	−2524.36	−0.150588
$656$	−3989.64	−0.237453
$657$	2526.61	0.150034
$658$	195.738	0.0115967
$659$	−6062.41	−0.358358	−0.179179	−	0.983816i	$-0.557344\pi$
$659$	−6062.41	−0.358358	−0.179179	+	0.983816i	$0.557344\pi$
$660$	2313.23	0.136428
$661$	5076.16	0.298698	0.149349	−	0.988785i	$-0.452282\pi$
$661$	5076.16	0.298698	0.149349	+	0.988785i	$0.452282\pi$
$662$	16735.1	0.982520
$663$	−516.252	−0.0302407
$664$	2192.19	0.128122
$665$	0	0
$666$	1887.94	0.109844
$667$	−1468.32	−0.0852377
$668$	12967.6	0.751092
$669$	−1444.48	−0.0834778
$670$	−1879.45	−0.108372
$671$	−518.324	−0.0298207
$672$	−64.6812	−0.00371299
$673$	−15406.8	−0.882450	−0.441225	−	0.897397i	$-0.645456\pi$
$673$	−15406.8	−0.882450	−0.441225	+	0.897397i	$0.645456\pi$
$674$	526.826	0.0301077
$675$	−571.650	−0.0325967
$676$	−8726.85	−0.496521
$677$	7518.29	0.426811	0.213406	−	0.976964i	$-0.431544\pi$
$677$	7518.29	0.426811	0.213406	+	0.976964i	$0.431544\pi$
$678$	63.9033	0.00361975
$679$	−560.974	−0.0317058
$680$	−4257.01	−0.240072
$681$	8359.77	0.470407
$682$	1929.37	0.108327
$683$	18762.7	1.05115	0.525574	−	0.850748i	$-0.323851\pi$
$683$	18762.7	1.05115	0.525574	+	0.850748i	$0.323851\pi$
$684$	0	0
$685$	4452.51	0.248353
$686$	−923.790	−0.0514147
$687$	−4313.16	−0.239530
$688$	−110.477	−0.00612194
$689$	−2948.74	−0.163045
$690$	660.954	0.0364668
$691$	30372.1	1.67208	0.836042	−	0.548666i	$-0.184864\pi$
$691$	30372.1	1.67208	0.836042	+	0.548666i	$0.184864\pi$
$692$	9845.14	0.540832
$693$	−96.6838	−0.00529973
$694$	−8042.99	−0.439925
$695$	20824.8	1.13659
$696$	3867.62	0.210635
$697$	10974.8	0.596413
$698$	−906.800	−0.0491732
$699$	8348.23	0.451730
$700$	−57.0601	−0.00308096
$701$	−23215.5	−1.25084	−0.625420	−	0.780288i	$-0.715072\pi$
$701$	−23215.5	−1.25084	−0.625420	+	0.780288i	$0.715072\pi$
$702$	−211.131	−0.0113513
$703$	0	0
$704$	1020.43	0.0546293
$705$	5268.56	0.281454
$706$	9095.22	0.484849
$707$	−613.374	−0.0326284
$708$	−8448.56	−0.448469
$709$	−2093.79	−0.110908	−0.0554540	−	0.998461i	$-0.517661\pi$
$709$	−2093.79	−0.110908	−0.0554540	+	0.998461i	$0.517661\pi$
$710$	−1074.05	−0.0567724
$711$	−7015.58	−0.370049
$712$	−8047.27	−0.423573
$713$	551.275	0.0289557
$714$	177.926	0.00932595
$715$	753.694	0.0394218
$716$	11857.0	0.618880
$717$	−13591.7	−0.707936
$718$	−16793.7	−0.872891
$719$	10026.4	0.520059	0.260030	−	0.965601i	$-0.416268\pi$
$719$	10026.4	0.520059	0.260030	+	0.965601i	$0.416268\pi$
$720$	−1740.98	−0.0901148
$721$	−857.133	−0.0442737
$722$	0	0
$723$	−7371.82	−0.379199
$724$	−17614.7	−0.904204
$725$	3411.92	0.174780
$726$	−6460.68	−0.330273
$727$	31486.5	1.60628	0.803142	−	0.595787i	$-0.203160\pi$
$727$	31486.5	1.60628	0.803142	+	0.595787i	$0.203160\pi$
$728$	−21.0744	−0.00107290
$729$	729.000	0.0370370
$730$	6788.26	0.344171
$731$	303.902	0.0153765
$732$	390.102	0.0196975
$733$	−14789.2	−0.745227	−0.372614	−	0.927987i	$-0.621538\pi$
$733$	−14789.2	−0.745227	−0.372614	+	0.927987i	$0.621538\pi$
$734$	10845.2	0.545372
$735$	−12424.3	−0.623507
$736$	291.567	0.0146023
$737$	−1239.29	−0.0619401
$738$	4488.35	0.223873
$739$	−34580.3	−1.72132	−0.860660	−	0.509179i	$-0.829949\pi$
$739$	−34580.3	−1.72132	−0.860660	+	0.509179i	$0.829949\pi$
$740$	5072.33	0.251976
$741$	0	0
$742$	1016.29	0.0502817
$743$	−14656.3	−0.723670	−0.361835	−	0.932242i	$-0.617850\pi$
$743$	−14656.3	−0.723670	−0.361835	+	0.932242i	$0.617850\pi$
$744$	−1452.08	−0.0715537
$745$	32499.0	1.59822
$746$	14878.8	0.730229
$747$	−2466.21	−0.120795
$748$	−2807.03	−0.137213
$749$	527.936	0.0257548
$750$	7531.77	0.366695
$751$	15707.7	0.763224	0.381612	−	0.924323i	$-0.375369\pi$
$751$	15707.7	0.763224	0.381612	+	0.924323i	$0.375369\pi$
$752$	2324.12	0.112702
$753$	9634.84	0.466286
$754$	1260.15	0.0608644
$755$	29920.1	1.44226
$756$	72.7663	0.00350064
$757$	−3495.32	−0.167820	−0.0839099	−	0.996473i	$-0.526741\pi$
$757$	−3495.32	−0.167820	−0.0839099	+	0.996473i	$0.526741\pi$
$758$	25483.7	1.22112
$759$	435.827	0.0208426
$760$	0	0
$761$	−9634.36	−0.458929	−0.229465	−	0.973317i	$-0.573698\pi$
$761$	−9634.36	−0.458929	−0.229465	+	0.973317i	$0.573698\pi$
$762$	5595.25	0.266003
$763$	−1356.39	−0.0643575
$764$	−8671.87	−0.410651
$765$	4789.14	0.226342
$766$	21101.9	0.995354
$767$	−2752.70	−0.129588
$768$	−768.000	−0.0360844
$769$	3390.81	0.159006	0.0795031	−	0.996835i	$-0.474667\pi$
$769$	3390.81	0.159006	0.0795031	+	0.996835i	$0.474667\pi$
$770$	−259.761	−0.0121573
$771$	5012.06	0.234118
$772$	12490.2	0.582296
$773$	−34473.8	−1.60406	−0.802028	−	0.597286i	$-0.796246\pi$
$773$	−34473.8	−1.60406	−0.802028	+	0.597286i	$0.796246\pi$
$774$	124.286	0.00577182
$775$	−1280.99	−0.0593737
$776$	−6660.80	−0.308130
$777$	−212.003	−0.00978840
$778$	−13163.1	−0.606580
$779$	0	0
$780$	−567.246	−0.0260393
$781$	−708.218	−0.0324482
$782$	−802.048	−0.0366767
$783$	−4351.07	−0.198588
$784$	−5480.74	−0.249669
$785$	14626.8	0.665037
$786$	1252.77	0.0568509
$787$	−30099.5	−1.36332	−0.681658	−	0.731671i	$-0.738741\pi$
$787$	−30099.5	−1.36332	−0.681658	+	0.731671i	$0.738741\pi$
$788$	809.717	0.0366053
$789$	23195.8	1.04663
$790$	−18848.8	−0.848873
$791$	−7.17594	−0.000322563 0
$792$	−1147.99	−0.0515050
$793$	127.103	0.00569174
$794$	−4297.53	−0.192083
$795$	27354.8	1.22034
$796$	−6596.98	−0.293749
$797$	−2712.41	−0.120550	−0.0602752	−	0.998182i	$-0.519198\pi$
$797$	−2712.41	−0.120550	−0.0602752	+	0.998182i	$0.519198\pi$
$798$	0	0
$799$	−6393.23	−0.283074
$800$	−677.511	−0.0299420
$801$	9053.18	0.399349
$802$	5064.40	0.222980
$803$	4476.11	0.196710
$804$	932.717	0.0409134
$805$	−74.2210	−0.00324962
$806$	−473.117	−0.0206760
$807$	3073.45	0.134065
$808$	−7282.97	−0.317097
$809$	25242.2	1.09699	0.548497	−	0.836153i	$-0.315200\pi$
$809$	25242.2	1.09699	0.548497	+	0.836153i	$0.315200\pi$
$810$	1958.61	0.0849611
$811$	29503.8	1.27746	0.638728	−	0.769432i	$-0.279461\pi$
$811$	29503.8	1.27746	0.638728	+	0.769432i	$0.279461\pi$
$812$	−434.310	−0.0187700
$813$	−3203.59	−0.138198
$814$	3344.64	0.144017
$815$	−33187.8	−1.42640
$816$	2112.63	0.0906334
$817$	0	0
$818$	16092.1	0.687831
$819$	23.7087	0.00101154
$820$	12058.9	0.513553
$821$	401.572	0.0170706	0.00853530	−	0.999964i	$-0.497283\pi$
$821$	401.572	0.0170706	0.00853530	+	0.999964i	$0.497283\pi$
$822$	−2209.65	−0.0937596
$823$	21122.7	0.894642	0.447321	−	0.894374i	$-0.352378\pi$
$823$	21122.7	0.894642	0.447321	+	0.894374i	$0.352378\pi$
$824$	−10177.3	−0.430270
$825$	−1012.73	−0.0427377
$826$	948.720	0.0399639
$827$	43088.6	1.81177	0.905887	−	0.423519i	$-0.139205\pi$
$827$	43088.6	1.81177	0.905887	+	0.423519i	$0.139205\pi$
$828$	−328.012	−0.0137672
$829$	−33228.8	−1.39214	−0.696070	−	0.717974i	$-0.745070\pi$
$829$	−33228.8	−1.39214	−0.696070	+	0.717974i	$0.745070\pi$
$830$	−6625.97	−0.277097
$831$	−933.196	−0.0389557
$832$	−250.229	−0.0104268
$833$	15076.5	0.627096
$834$	−10334.7	−0.429092
$835$	−39195.0	−1.62443
$836$	0	0
$837$	1633.59	0.0674615
$838$	20865.8	0.860139
$839$	−35600.4	−1.46491	−0.732456	−	0.680814i	$-0.761626\pi$
$839$	−35600.4	−1.46491	−0.732456	+	0.680814i	$0.761626\pi$
$840$	195.502	0.00803029
$841$	1580.62	0.0648087
$842$	9183.16	0.375858
$843$	−22040.2	−0.900482
$844$	−6517.41	−0.265804
$845$	26377.3	1.07385
$846$	−2614.63	−0.106256
$847$	725.494	0.0294312
$848$	12067.0	0.488658
$849$	−12140.8	−0.490780
$850$	1863.71	0.0752056
$851$	955.659	0.0384954
$852$	533.020	0.0214331
$853$	−8614.63	−0.345791	−0.172895	−	0.984940i	$-0.555312\pi$
$853$	−8614.63	−0.345791	−0.172895	+	0.984940i	$0.555312\pi$
$854$	−43.8060	−0.00175528
$855$	0	0
$856$	6268.52	0.250296
$857$	26698.4	1.06418	0.532088	−	0.846689i	$-0.321407\pi$
$857$	26698.4	1.06418	0.532088	+	0.846689i	$0.321407\pi$
$858$	−374.036	−0.0148827
$859$	45356.8	1.80158	0.900788	−	0.434260i	$-0.142990\pi$
$859$	45356.8	1.80158	0.900788	+	0.434260i	$0.142990\pi$
$860$	333.921	0.0132402
$861$	−504.013	−0.0199497
$862$	28234.3	1.11562
$863$	20943.6	0.826104	0.413052	−	0.910707i	$-0.364463\pi$
$863$	20943.6	0.826104	0.413052	+	0.910707i	$0.364463\pi$
$864$	864.000	0.0340207
$865$	−29757.4	−1.16969
$866$	−27858.0	−1.09313
$867$	8927.53	0.349706
$868$	163.060	0.00637628
$869$	−12428.7	−0.485172
$870$	−11690.1	−0.455552
$871$	303.897	0.0118222
$872$	−16105.3	−0.625453
$873$	7493.40	0.290508
$874$	0	0
$875$	−845.771	−0.0326769
$876$	−3368.81	−0.129933
$877$	9561.82	0.368164	0.184082	−	0.982911i	$-0.441069\pi$
$877$	9561.82	0.368164	0.184082	+	0.982911i	$0.441069\pi$
$878$	10444.1	0.401447
$879$	−10924.7	−0.419204
$880$	−3084.30	−0.118150
$881$	41424.3	1.58413	0.792065	−	0.610436i	$-0.209006\pi$
$881$	41424.3	1.58413	0.792065	+	0.610436i	$0.209006\pi$
$882$	6165.83	0.235390
$883$	37078.3	1.41312	0.706559	−	0.707654i	$-0.250246\pi$
$883$	37078.3	1.41312	0.706559	+	0.707654i	$0.250246\pi$
$884$	688.336	0.0261892
$885$	25536.1	0.969929
$886$	20326.8	0.770760
$887$	−6509.98	−0.246430	−0.123215	−	0.992380i	$-0.539321\pi$
$887$	−6509.98	−0.246430	−0.123215	+	0.992380i	$0.539321\pi$
$888$	−2517.25	−0.0951277
$889$	−628.311	−0.0237040
$890$	24323.2	0.916085
$891$	1291.49	0.0485594
$892$	1925.97	0.0722939
$893$	0	0
$894$	−16128.3	−0.603369
$895$	−35838.4	−1.33849
$896$	86.2415	0.00321554
$897$	−106.873	−0.00397812
$898$	35703.2	1.32676
$899$	−9750.19	−0.361721
$900$	762.200	0.0282296
$901$	−33194.2	−1.22737
$902$	7951.49	0.293521
$903$	−13.9566	−0.000514337 0
$904$	−85.2045	−0.00313480
$905$	53241.1	1.95557
$906$	−14848.5	−0.544489
$907$	−52647.2	−1.92737	−0.963683	−	0.267048i	$-0.913952\pi$
$907$	−52647.2	−1.92737	−0.963683	+	0.267048i	$0.913952\pi$
$908$	−11146.4	−0.407384
$909$	8193.34	0.298962
$910$	63.6982	0.00232041
$911$	−16500.2	−0.600084	−0.300042	−	0.953926i	$-0.597001\pi$
$911$	−16500.2	−0.600084	−0.300042	+	0.953926i	$0.597001\pi$
$912$	0	0
$913$	−4369.10	−0.158375
$914$	10075.9	0.364640
$915$	−1179.10	−0.0426009
$916$	5750.88	0.207439
$917$	−140.678	−0.00506608
$918$	−2376.71	−0.0854500
$919$	8761.75	0.314498	0.157249	−	0.987559i	$-0.449738\pi$
$919$	8761.75	0.314498	0.157249	+	0.987559i	$0.449738\pi$
$920$	−881.272	−0.0315812
$921$	14056.3	0.502898
$922$	22362.8	0.798783
$923$	173.668	0.00619323
$924$	128.912	0.00458970
$925$	−2220.66	−0.0789348
$926$	−36910.2	−1.30988
$927$	11449.4	0.405662
$928$	−5156.83	−0.182415
$929$	−27264.8	−0.962893	−0.481446	−	0.876476i	$-0.659888\pi$
$929$	−27264.8	−0.962893	−0.481446	+	0.876476i	$0.659888\pi$
$930$	4388.98	0.154753
$931$	0	0
$932$	−11131.0	−0.391209
$933$	7240.66	0.254071
$934$	−12825.6	−0.449321
$935$	8484.37	0.296758
$936$	281.508	0.00983052
$937$	23381.2	0.815186	0.407593	−	0.913164i	$-0.366368\pi$
$937$	23381.2	0.815186	0.407593	+	0.913164i	$0.366368\pi$
$938$	−104.738	−0.00364587
$939$	15186.7	0.527795
$940$	−7024.74	−0.243747
$941$	39198.4	1.35795	0.678975	−	0.734161i	$-0.262424\pi$
$941$	39198.4	1.35795	0.678975	+	0.734161i	$0.262424\pi$
$942$	−7258.88	−0.251069
$943$	2271.96	0.0784574
$944$	11264.7	0.388386
$945$	−219.939	−0.00757103
$946$	220.184	0.00756745
$947$	18505.7	0.635010	0.317505	−	0.948257i	$-0.397155\pi$
$947$	18505.7	0.635010	0.317505	+	0.948257i	$0.397155\pi$
$948$	9354.10	0.320472
$949$	−1097.62	−0.0375452
$950$	0	0
$951$	17112.1	0.583488
$952$	−237.235	−0.00807651
$953$	14696.2	0.499535	0.249768	−	0.968306i	$-0.419646\pi$
$953$	14696.2	0.499535	0.249768	+	0.968306i	$0.419646\pi$
$954$	−13575.4	−0.460712
$955$	26211.1	0.888137
$956$	18122.2	0.613091
$957$	−7708.30	−0.260370
$958$	17039.3	0.574651
$959$	248.130	0.00835509
$960$	2321.31	0.0780417
$961$	−26130.3	−0.877122
$962$	−820.168	−0.0274878
$963$	−7052.08	−0.235982
$964$	9829.09	0.328396
$965$	−37752.2	−1.25936
$966$	36.8337	0.00122682
$967$	−53986.4	−1.79533	−0.897666	−	0.440676i	$-0.854739\pi$
$967$	−53986.4	−1.79533	−0.897666	+	0.440676i	$0.854739\pi$
$968$	8614.24	0.286025
$969$	0	0
$970$	20132.5	0.666409
$971$	29949.9	0.989845	0.494922	−	0.868937i	$-0.335196\pi$
$971$	29949.9	0.989845	0.494922	+	0.868937i	$0.335196\pi$
$972$	−972.000	−0.0320750
$973$	1160.53	0.0382371
$974$	25493.7	0.838675
$975$	248.339	0.00815715
$976$	−520.136	−0.0170586
$977$	10119.4	0.331371	0.165685	−	0.986179i	$-0.447016\pi$
$977$	10119.4	0.331371	0.165685	+	0.986179i	$0.447016\pi$
$978$	16470.2	0.538505
$979$	16038.5	0.523587
$980$	16565.8	0.539973
$981$	18118.5	0.589683
$982$	10374.2	0.337122
$983$	−21988.0	−0.713438	−0.356719	−	0.934212i	$-0.616105\pi$
$983$	−21988.0	−0.713438	−0.356719	+	0.934212i	$0.616105\pi$
$984$	−5984.46	−0.193880
$985$	−2447.40	−0.0791683
$986$	14185.5	0.458173
$987$	293.607	0.00946870
$988$	0	0
$989$	62.9128	0.00202276
$990$	3469.84	0.111393
$991$	−2492.80	−0.0799056	−0.0399528	−	0.999202i	$-0.512721\pi$
$991$	−2492.80	−0.0799056	−0.0399528	+	0.999202i	$0.512721\pi$
$992$	1936.11	0.0619673
$993$	25102.7	0.802224
$994$	−59.8548	−0.00190994
$995$	19939.7	0.635306
$996$	3288.28	0.104612
$997$	45403.2	1.44226	0.721130	−	0.692799i	$-0.243623\pi$
$997$	45403.2	1.44226	0.721130	+	0.692799i	$0.243623\pi$
$998$	21408.9	0.679045
$999$	2831.91	0.0896873

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.j.1.1	✓	2
19.18	odd	2		2166.4.a.p.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.4.a.j.1.1	✓	2	1.1	even	1	trivial
2166.4.a.p.1.1	yes	2	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -12.0902 q^{5} +6.00000 q^{6} -0.673762 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -12.0902 q^{5} +6.00000 q^{6} -0.673762 q^{7} -8.00000 q^{8} +9.00000 q^{9} +24.1803 q^{10} +15.9443 q^{11} -12.0000 q^{12} -3.90983 q^{13} +1.34752 q^{14} +36.2705 q^{15} +16.0000 q^{16} -44.0132 q^{17} -18.0000 q^{18} -48.3607 q^{20} +2.02129 q^{21} -31.8885 q^{22} -9.11146 q^{23} +24.0000 q^{24} +21.1722 q^{25} +7.81966 q^{26} -27.0000 q^{27} -2.69505 q^{28} +161.151 q^{29} -72.5410 q^{30} -60.5035 q^{31} -32.0000 q^{32} -47.8328 q^{33} +88.0263 q^{34} +8.14590 q^{35} +36.0000 q^{36} -104.885 q^{37} +11.7295 q^{39} +96.7214 q^{40} -249.353 q^{41} -4.04257 q^{42} -6.90481 q^{43} +63.7771 q^{44} -108.812 q^{45} +18.2229 q^{46} +145.257 q^{47} -48.0000 q^{48} -342.546 q^{49} -42.3444 q^{50} +132.039 q^{51} -15.6393 q^{52} +754.187 q^{53} +54.0000 q^{54} -192.769 q^{55} +5.39010 q^{56} -322.302 q^{58} +704.046 q^{59} +145.082 q^{60} -32.5085 q^{61} +121.007 q^{62} -6.06386 q^{63} +64.0000 q^{64} +47.2705 q^{65} +95.6656 q^{66} -77.7264 q^{67} -176.053 q^{68} +27.3344 q^{69} -16.2918 q^{70} -44.4183 q^{71} -72.0000 q^{72} +280.735 q^{73} +209.771 q^{74} -63.5166 q^{75} -10.7426 q^{77} -23.4590 q^{78} -779.508 q^{79} -193.443 q^{80} +81.0000 q^{81} +498.705 q^{82} -274.023 q^{83} +8.08514 q^{84} +532.127 q^{85} +13.8096 q^{86} -483.453 q^{87} -127.554 q^{88} +1005.91 q^{89} +217.623 q^{90} +2.63430 q^{91} -36.4458 q^{92} +181.510 q^{93} -290.515 q^{94} +96.0000 q^{96} +832.600 q^{97} +685.092 q^{98} +143.498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} - 13 q^{5} + 12 q^{6} - 17 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 - 13 * q^5 + 12 * q^6 - 17 * q^7 - 16 * q^8 + 18 * q^9	\( 2 q - 4 q^{2} - 6 q^{3} + 8 q^{4} - 13 q^{5} + 12 q^{6} - 17 q^{7} - 16 q^{8} + 18 q^{9} + 26 q^{10} + 14 q^{11} - 24 q^{12} - 19 q^{13} + 34 q^{14} + 39 q^{15} + 32 q^{16} - 12 q^{17} - 36 q^{18} - 52 q^{20} + 51 q^{21} - 28 q^{22} - 54 q^{23} + 48 q^{24} - 103 q^{25} + 38 q^{26} - 54 q^{27} - 68 q^{28} + 130 q^{29} - 78 q^{30} + 239 q^{31} - 64 q^{32} - 42 q^{33} + 24 q^{34} + 23 q^{35} + 72 q^{36} + 148 q^{37} + 57 q^{39} + 104 q^{40} - 331 q^{41} - 102 q^{42} - 224 q^{43} + 56 q^{44} - 117 q^{45} + 108 q^{46} + 333 q^{47} - 96 q^{48} - 419 q^{49} + 206 q^{50} + 36 q^{51} - 76 q^{52} + 766 q^{53} + 108 q^{54} - 191 q^{55} + 136 q^{56} - 260 q^{58} + 460 q^{59} + 156 q^{60} + 494 q^{61} - 478 q^{62} - 153 q^{63} + 128 q^{64} + 61 q^{65} + 84 q^{66} + 133 q^{67} - 48 q^{68} + 162 q^{69} - 46 q^{70} + 459 q^{71} - 144 q^{72} + 396 q^{73} - 296 q^{74} + 309 q^{75} + 21 q^{77} - 114 q^{78} - 1000 q^{79} - 208 q^{80} + 162 q^{81} + 662 q^{82} - 74 q^{83} + 204 q^{84} + 503 q^{85} + 448 q^{86} - 390 q^{87} - 112 q^{88} + 1180 q^{89} + 234 q^{90} + 249 q^{91} - 216 q^{92} - 717 q^{93} - 666 q^{94} + 192 q^{96} + 1938 q^{97} + 838 q^{98} + 126 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 - 13 * q^5 + 12 * q^6 - 17 * q^7 - 16 * q^8 + 18 * q^9 + 26 * q^10 + 14 * q^11 - 24 * q^12 - 19 * q^13 + 34 * q^14 + 39 * q^15 + 32 * q^16 - 12 * q^17 - 36 * q^18 - 52 * q^20 + 51 * q^21 - 28 * q^22 - 54 * q^23 + 48 * q^24 - 103 * q^25 + 38 * q^26 - 54 * q^27 - 68 * q^28 + 130 * q^29 - 78 * q^30 + 239 * q^31 - 64 * q^32 - 42 * q^33 + 24 * q^34 + 23 * q^35 + 72 * q^36 + 148 * q^37 + 57 * q^39 + 104 * q^40 - 331 * q^41 - 102 * q^42 - 224 * q^43 + 56 * q^44 - 117 * q^45 + 108 * q^46 + 333 * q^47 - 96 * q^48 - 419 * q^49 + 206 * q^50 + 36 * q^51 - 76 * q^52 + 766 * q^53 + 108 * q^54 - 191 * q^55 + 136 * q^56 - 260 * q^58 + 460 * q^59 + 156 * q^60 + 494 * q^61 - 478 * q^62 - 153 * q^63 + 128 * q^64 + 61 * q^65 + 84 * q^66 + 133 * q^67 - 48 * q^68 + 162 * q^69 - 46 * q^70 + 459 * q^71 - 144 * q^72 + 396 * q^73 - 296 * q^74 + 309 * q^75 + 21 * q^77 - 114 * q^78 - 1000 * q^79 - 208 * q^80 + 162 * q^81 + 662 * q^82 - 74 * q^83 + 204 * q^84 + 503 * q^85 + 448 * q^86 - 390 * q^87 - 112 * q^88 + 1180 * q^89 + 234 * q^90 + 249 * q^91 - 216 * q^92 - 717 * q^93 - 666 * q^94 + 192 * q^96 + 1938 * q^97 + 838 * q^98 + 126 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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