LMFDB - Embedded newform 2166.4.a.r.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.373564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 106x - 348 $ x^3 - 106*x - 348
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.80115$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -18.3722 q^{5} +6.00000 q^{6} -27.9745 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -18.3722 q^{5} +6.00000 q^{6} -27.9745 q^{7} -8.00000 q^{8} +9.00000 q^{9} +36.7444 q^{10} +65.1791 q^{11} -12.0000 q^{12} +89.0910 q^{13} +55.9490 q^{14} +55.1166 q^{15} +16.0000 q^{16} -27.5143 q^{17} -18.0000 q^{18} -73.4887 q^{20} +83.9235 q^{21} -130.358 q^{22} -79.9605 q^{23} +24.0000 q^{24} +212.537 q^{25} -178.182 q^{26} -27.0000 q^{27} -111.898 q^{28} +288.131 q^{29} -110.233 q^{30} -7.47726 q^{31} -32.0000 q^{32} -195.537 q^{33} +55.0285 q^{34} +513.952 q^{35} +36.0000 q^{36} -18.9090 q^{37} -267.273 q^{39} +146.977 q^{40} +129.177 q^{41} -167.847 q^{42} +90.9800 q^{43} +260.716 q^{44} -165.350 q^{45} +159.921 q^{46} +33.8549 q^{47} -48.0000 q^{48} +439.572 q^{49} -425.074 q^{50} +82.5428 q^{51} +356.364 q^{52} -227.188 q^{53} +54.0000 q^{54} -1197.48 q^{55} +223.796 q^{56} -576.262 q^{58} -726.808 q^{59} +220.466 q^{60} +658.958 q^{61} +14.9545 q^{62} -251.770 q^{63} +64.0000 q^{64} -1636.80 q^{65} +391.074 q^{66} -545.672 q^{67} -110.057 q^{68} +239.881 q^{69} -1027.90 q^{70} -136.091 q^{71} -72.0000 q^{72} +174.940 q^{73} +37.8179 q^{74} -637.612 q^{75} -1823.35 q^{77} +534.546 q^{78} -260.421 q^{79} -293.955 q^{80} +81.0000 q^{81} -258.353 q^{82} -562.814 q^{83} +335.694 q^{84} +505.497 q^{85} -181.960 q^{86} -864.393 q^{87} -521.433 q^{88} +627.905 q^{89} +330.699 q^{90} -2492.28 q^{91} -319.842 q^{92} +22.4318 q^{93} -67.7098 q^{94} +96.0000 q^{96} +1089.92 q^{97} -879.144 q^{98} +586.612 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} - 5 q^{5} + 18 q^{6} - 11 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 - 5 * q^5 + 18 * q^6 - 11 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} - 5 q^{5} + 18 q^{6} - 11 q^{7} - 24 q^{8} + 27 q^{9} + 10 q^{10} + 77 q^{11} - 36 q^{12} + 44 q^{13} + 22 q^{14} + 15 q^{15} + 48 q^{16} + 45 q^{17} - 54 q^{18} - 20 q^{20} + 33 q^{21} - 154 q^{22} + 20 q^{23} + 72 q^{24} + 282 q^{25} - 88 q^{26} - 81 q^{27} - 44 q^{28} + 272 q^{29} - 30 q^{30} + 64 q^{31} - 96 q^{32} - 231 q^{33} - 90 q^{34} + 439 q^{35} + 108 q^{36} - 280 q^{37} - 132 q^{39} + 40 q^{40} - 32 q^{41} - 66 q^{42} + 173 q^{43} + 308 q^{44} - 45 q^{45} - 40 q^{46} + 507 q^{47} - 144 q^{48} + 52 q^{49} - 564 q^{50} - 135 q^{51} + 176 q^{52} - 208 q^{53} + 162 q^{54} - 93 q^{55} + 88 q^{56} - 544 q^{58} + 344 q^{59} + 60 q^{60} + 847 q^{61} - 128 q^{62} - 99 q^{63} + 192 q^{64} - 2440 q^{65} + 462 q^{66} + 268 q^{67} + 180 q^{68} - 60 q^{69} - 878 q^{70} + 384 q^{71} - 216 q^{72} + 225 q^{73} + 560 q^{74} - 846 q^{75} - 2563 q^{77} + 264 q^{78} - 576 q^{79} - 80 q^{80} + 243 q^{81} + 64 q^{82} + 1492 q^{83} + 132 q^{84} + 2099 q^{85} - 346 q^{86} - 816 q^{87} - 616 q^{88} + 816 q^{89} + 90 q^{90} - 2464 q^{91} + 80 q^{92} - 192 q^{93} - 1014 q^{94} + 288 q^{96} + 4008 q^{97} - 104 q^{98} + 693 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 - 5 * q^5 + 18 * q^6 - 11 * q^7 - 24 * q^8 + 27 * q^9 + 10 * q^10 + 77 * q^11 - 36 * q^12 + 44 * q^13 + 22 * q^14 + 15 * q^15 + 48 * q^16 + 45 * q^17 - 54 * q^18 - 20 * q^20 + 33 * q^21 - 154 * q^22 + 20 * q^23 + 72 * q^24 + 282 * q^25 - 88 * q^26 - 81 * q^27 - 44 * q^28 + 272 * q^29 - 30 * q^30 + 64 * q^31 - 96 * q^32 - 231 * q^33 - 90 * q^34 + 439 * q^35 + 108 * q^36 - 280 * q^37 - 132 * q^39 + 40 * q^40 - 32 * q^41 - 66 * q^42 + 173 * q^43 + 308 * q^44 - 45 * q^45 - 40 * q^46 + 507 * q^47 - 144 * q^48 + 52 * q^49 - 564 * q^50 - 135 * q^51 + 176 * q^52 - 208 * q^53 + 162 * q^54 - 93 * q^55 + 88 * q^56 - 544 * q^58 + 344 * q^59 + 60 * q^60 + 847 * q^61 - 128 * q^62 - 99 * q^63 + 192 * q^64 - 2440 * q^65 + 462 * q^66 + 268 * q^67 + 180 * q^68 - 60 * q^69 - 878 * q^70 + 384 * q^71 - 216 * q^72 + 225 * q^73 + 560 * q^74 - 846 * q^75 - 2563 * q^77 + 264 * q^78 - 576 * q^79 - 80 * q^80 + 243 * q^81 + 64 * q^82 + 1492 * q^83 + 132 * q^84 + 2099 * q^85 - 346 * q^86 - 816 * q^87 - 616 * q^88 + 816 * q^89 + 90 * q^90 - 2464 * q^91 + 80 * q^92 - 192 * q^93 - 1014 * q^94 + 288 * q^96 + 4008 * q^97 - 104 * q^98 + 693 * 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
$5$	−18.3722	−1.64326	−0.821629	−	0.570022i	$-0.806935\pi$
$5$	−18.3722	−1.64326	−0.821629	+	0.570022i	$0.806935\pi$
$6$	6.00000	0.408248
$7$	−27.9745	−1.51048	−0.755240	−	0.655448i	$-0.772480\pi$
$7$	−27.9745	−1.51048	−0.755240	+	0.655448i	$0.772480\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	36.7444	1.16196
$11$	65.1791	1.78657	0.893283	−	0.449494i	$-0.148396\pi$
$11$	65.1791	1.78657	0.893283	+	0.449494i	$0.148396\pi$
$12$	−12.0000	−0.288675
$13$	89.0910	1.90072	0.950362	−	0.311147i	$-0.100713\pi$
$13$	89.0910	1.90072	0.950362	+	0.311147i	$0.100713\pi$
$14$	55.9490	1.06807
$15$	55.1166	0.948736
$16$	16.0000	0.250000
$17$	−27.5143	−0.392541	−0.196270	−	0.980550i	$-0.562883\pi$
$17$	−27.5143	−0.392541	−0.196270	+	0.980550i	$0.562883\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	−73.4887	−0.821629
$21$	83.9235	0.872076
$22$	−130.358	−1.26329
$23$	−79.9605	−0.724909	−0.362455	−	0.932001i	$-0.618061\pi$
$23$	−79.9605	−0.724909	−0.362455	+	0.932001i	$0.618061\pi$
$24$	24.0000	0.204124
$25$	212.537	1.70030
$26$	−178.182	−1.34401
$27$	−27.0000	−0.192450
$28$	−111.898	−0.755240
$29$	288.131	1.84499	0.922493	−	0.386014i	$-0.126148\pi$
$29$	288.131	1.84499	0.922493	+	0.386014i	$0.126148\pi$
$30$	−110.233	−0.670857
$31$	−7.47726	−0.0433211	−0.0216606	−	0.999765i	$-0.506895\pi$
$31$	−7.47726	−0.0433211	−0.0216606	+	0.999765i	$0.506895\pi$
$32$	−32.0000	−0.176777
$33$	−195.537	−1.03147
$34$	55.0285	0.277568
$35$	513.952	2.48211
$36$	36.0000	0.166667
$37$	−18.9090	−0.0840166	−0.0420083	−	0.999117i	$-0.513376\pi$
$37$	−18.9090	−0.0840166	−0.0420083	+	0.999117i	$0.513376\pi$
$38$	0	0
$39$	−267.273	−1.09738
$40$	146.977	0.580980
$41$	129.177	0.492048	0.246024	−	0.969264i	$-0.420876\pi$
$41$	129.177	0.492048	0.246024	+	0.969264i	$0.420876\pi$
$42$	−167.847	−0.616651
$43$	90.9800	0.322659	0.161329	−	0.986901i	$-0.448422\pi$
$43$	90.9800	0.322659	0.161329	+	0.986901i	$0.448422\pi$
$44$	260.716	0.893283
$45$	−165.350	−0.547753
$46$	159.921	0.512588
$47$	33.8549	0.105069	0.0525345	−	0.998619i	$-0.483270\pi$
$47$	33.8549	0.105069	0.0525345	+	0.998619i	$0.483270\pi$
$48$	−48.0000	−0.144338
$49$	439.572	1.28155
$50$	−425.074	−1.20229
$51$	82.5428	0.226633
$52$	356.364	0.950362
$53$	−227.188	−0.588804	−0.294402	−	0.955682i	$-0.595120\pi$
$53$	−227.188	−0.588804	−0.294402	+	0.955682i	$0.595120\pi$
$54$	54.0000	0.136083
$55$	−1197.48	−2.93579
$56$	223.796	0.534035
$57$	0	0
$58$	−576.262	−1.30460
$59$	−726.808	−1.60377	−0.801885	−	0.597479i	$-0.796169\pi$
$59$	−726.808	−1.60377	−0.801885	+	0.597479i	$0.796169\pi$
$60$	220.466	0.474368
$61$	658.958	1.38313	0.691565	−	0.722314i	$-0.256922\pi$
$61$	658.958	1.38313	0.691565	+	0.722314i	$0.256922\pi$
$62$	14.9545	0.0306327
$63$	−251.770	−0.503493
$64$	64.0000	0.125000
$65$	−1636.80	−3.12338
$66$	391.074	0.729363
$67$	−545.672	−0.994991	−0.497496	−	0.867466i	$-0.665747\pi$
$67$	−545.672	−0.994991	−0.497496	+	0.867466i	$0.665747\pi$
$68$	−110.057	−0.196270
$69$	239.881	0.418526
$70$	−1027.90	−1.75512
$71$	−136.091	−0.227479	−0.113740	−	0.993511i	$-0.536283\pi$
$71$	−136.091	−0.227479	−0.113740	+	0.993511i	$0.536283\pi$
$72$	−72.0000	−0.117851
$73$	174.940	0.280482	0.140241	−	0.990117i	$-0.455212\pi$
$73$	174.940	0.280482	0.140241	+	0.990117i	$0.455212\pi$
$74$	37.8179	0.0594087
$75$	−637.612	−0.981667
$76$	0	0
$77$	−1823.35	−2.69857
$78$	534.546	0.775967
$79$	−260.421	−0.370881	−0.185441	−	0.982655i	$-0.559371\pi$
$79$	−260.421	−0.370881	−0.185441	+	0.982655i	$0.559371\pi$
$80$	−293.955	−0.410815
$81$	81.0000	0.111111
$82$	−258.353	−0.347931
$83$	−562.814	−0.744300	−0.372150	−	0.928173i	$-0.621379\pi$
$83$	−562.814	−0.744300	−0.372150	+	0.928173i	$0.621379\pi$
$84$	335.694	0.436038
$85$	505.497	0.645045
$86$	−181.960	−0.228154
$87$	−864.393	−1.06520
$88$	−521.433	−0.631647
$89$	627.905	0.747840	0.373920	−	0.927461i	$-0.378013\pi$
$89$	627.905	0.747840	0.373920	+	0.927461i	$0.378013\pi$
$90$	330.699	0.387320
$91$	−2492.28	−2.87101
$92$	−319.842	−0.362455
$93$	22.4318	0.0250115
$94$	−67.7098	−0.0742951
$95$	0	0
$96$	96.0000	0.102062
$97$	1089.92	1.14087	0.570436	−	0.821342i	$-0.306774\pi$
$97$	1089.92	1.14087	0.570436	+	0.821342i	$0.306774\pi$
$98$	−879.144	−0.906193
$99$	586.612	0.595522
$100$	850.149	0.850149
$101$	−904.524	−0.891123	−0.445562	−	0.895251i	$-0.646996\pi$
$101$	−904.524	−0.891123	−0.445562	+	0.895251i	$0.646996\pi$
$102$	−165.086	−0.160254
$103$	600.162	0.574133	0.287066	−	0.957911i	$-0.407320\pi$
$103$	600.162	0.574133	0.287066	+	0.957911i	$0.407320\pi$
$104$	−712.728	−0.672007
$105$	−1541.86	−1.43305
$106$	454.375	0.416347
$107$	1023.13	0.924388	0.462194	−	0.886779i	$-0.347062\pi$
$107$	1023.13	0.924388	0.462194	+	0.886779i	$0.347062\pi$
$108$	−108.000	−0.0962250
$109$	−1580.94	−1.38924	−0.694620	−	0.719377i	$-0.744427\pi$
$109$	−1580.94	−1.38924	−0.694620	+	0.719377i	$0.744427\pi$
$110$	2394.96	2.07592
$111$	56.7269	0.0485070
$112$	−447.592	−0.377620
$113$	−1552.87	−1.29276	−0.646380	−	0.763016i	$-0.723718\pi$
$113$	−1552.87	−1.29276	−0.646380	+	0.763016i	$0.723718\pi$
$114$	0	0
$115$	1469.05	1.19121
$116$	1152.52	0.922493
$117$	801.819	0.633575
$118$	1453.62	1.13404
$119$	769.697	0.592925
$120$	−440.932	−0.335429
$121$	2917.31	2.19182
$122$	−1317.92	−0.978021
$123$	−387.530	−0.284084
$124$	−29.9090	−0.0216606
$125$	−1608.25	−1.15077
$126$	503.541	0.356024
$127$	−790.821	−0.552551	−0.276275	−	0.961078i	$-0.589100\pi$
$127$	−790.821	−0.552551	−0.276275	+	0.961078i	$0.589100\pi$
$128$	−128.000	−0.0883883
$129$	−272.940	−0.186287
$130$	3273.59	2.20856
$131$	1549.85	1.03367	0.516835	−	0.856085i	$-0.327110\pi$
$131$	1549.85	1.03367	0.516835	+	0.856085i	$0.327110\pi$
$132$	−782.149	−0.515737
$133$	0	0
$134$	1091.34	0.703565
$135$	496.049	0.316245
$136$	220.114	0.138784
$137$	−890.286	−0.555199	−0.277599	−	0.960697i	$-0.589539\pi$
$137$	−890.286	−0.555199	−0.277599	+	0.960697i	$0.589539\pi$
$138$	−479.763	−0.295943
$139$	−2272.69	−1.38681	−0.693407	−	0.720546i	$-0.743891\pi$
$139$	−2272.69	−1.38681	−0.693407	+	0.720546i	$0.743891\pi$
$140$	2055.81	1.24105
$141$	−101.565	−0.0606617
$142$	272.182	0.160852
$143$	5806.87	3.39577
$144$	144.000	0.0833333
$145$	−5293.60	−3.03179
$146$	−349.880	−0.198331
$147$	−1318.72	−0.739903
$148$	−75.6358	−0.0420083
$149$	1145.68	0.629918	0.314959	−	0.949105i	$-0.398009\pi$
$149$	1145.68	0.629918	0.314959	+	0.949105i	$0.398009\pi$
$150$	1275.22	0.694144
$151$	2001.84	1.07886	0.539430	−	0.842031i	$-0.318640\pi$
$151$	2001.84	1.07886	0.539430	+	0.842031i	$0.318640\pi$
$152$	0	0
$153$	−247.628	−0.130847
$154$	3646.70	1.90818
$155$	137.374	0.0711878
$156$	−1069.09	−0.548692
$157$	1897.03	0.964330	0.482165	−	0.876080i	$-0.339851\pi$
$157$	1897.03	0.964330	0.482165	+	0.876080i	$0.339851\pi$
$158$	520.841	0.262253
$159$	681.563	0.339946
$160$	587.910	0.290490
$161$	2236.85	1.09496
$162$	−162.000	−0.0785674
$163$	826.207	0.397015	0.198508	−	0.980099i	$-0.436391\pi$
$163$	826.207	0.397015	0.198508	+	0.980099i	$0.436391\pi$
$164$	516.706	0.246024
$165$	3592.45	1.69498
$166$	1125.63	0.526300
$167$	−716.397	−0.331955	−0.165977	−	0.986130i	$-0.553078\pi$
$167$	−716.397	−0.331955	−0.165977	+	0.986130i	$0.553078\pi$
$168$	−671.388	−0.308325
$169$	5740.21	2.61275
$170$	−1010.99	−0.456116
$171$	0	0
$172$	363.920	0.161329
$173$	−2149.19	−0.944508	−0.472254	−	0.881462i	$-0.656560\pi$
$173$	−2149.19	−0.944508	−0.472254	+	0.881462i	$0.656560\pi$
$174$	1728.79	0.753212
$175$	−5945.62	−2.56827
$176$	1042.87	0.446642
$177$	2180.43	0.925937
$178$	−1255.81	−0.528803
$179$	−3096.34	−1.29291	−0.646457	−	0.762951i	$-0.723750\pi$
$179$	−3096.34	−1.29291	−0.646457	+	0.762951i	$0.723750\pi$
$180$	−661.399	−0.273876
$181$	635.770	0.261085	0.130542	−	0.991443i	$-0.458328\pi$
$181$	635.770	0.261085	0.130542	+	0.991443i	$0.458328\pi$
$182$	4984.55	2.03011
$183$	−1976.87	−0.798550
$184$	639.684	0.256294
$185$	347.399	0.138061
$186$	−44.8635	−0.0176858
$187$	−1793.35	−0.701300
$188$	135.420	0.0525345
$189$	755.311	0.290692
$190$	0	0
$191$	1005.89	0.381065	0.190533	−	0.981681i	$-0.438979\pi$
$191$	1005.89	0.381065	0.190533	+	0.981681i	$0.438979\pi$
$192$	−192.000	−0.0721688
$193$	−3119.86	−1.16359	−0.581793	−	0.813337i	$-0.697649\pi$
$193$	−3119.86	−1.16359	−0.581793	+	0.813337i	$0.697649\pi$
$194$	−2179.84	−0.806718
$195$	4910.39	1.80328
$196$	1758.29	0.640775
$197$	4018.89	1.45347	0.726736	−	0.686917i	$-0.241036\pi$
$197$	4018.89	1.45347	0.726736	+	0.686917i	$0.241036\pi$
$198$	−1173.22	−0.421098
$199$	4225.98	1.50539	0.752693	−	0.658371i	$-0.228754\pi$
$199$	4225.98	1.50539	0.752693	+	0.658371i	$0.228754\pi$
$200$	−1700.30	−0.601146
$201$	1637.02	0.574459
$202$	1809.05	0.630119
$203$	−8060.32	−2.78681
$204$	330.171	0.113317
$205$	−2373.26	−0.808563
$206$	−1200.32	−0.405973
$207$	−719.644	−0.241636
$208$	1425.46	0.475181
$209$	0	0
$210$	3083.71	1.01332
$211$	−1322.32	−0.431433	−0.215717	−	0.976456i	$-0.569209\pi$
$211$	−1322.32	−0.431433	−0.215717	+	0.976456i	$0.569209\pi$
$212$	−908.751	−0.294402
$213$	408.273	0.131335
$214$	−2046.26	−0.653641
$215$	−1671.50	−0.530212
$216$	216.000	0.0680414
$217$	209.172	0.0654357
$218$	3161.89	0.982340
$219$	−524.820	−0.161936
$220$	−4789.93	−1.46790
$221$	−2451.27	−0.746111
$222$	−113.454	−0.0342996
$223$	1951.65	0.586065	0.293032	−	0.956103i	$-0.405336\pi$
$223$	1951.65	0.586065	0.293032	+	0.956103i	$0.405336\pi$
$224$	895.184	0.267018
$225$	1912.84	0.566766
$226$	3105.74	0.914119
$227$	−2592.81	−0.758110	−0.379055	−	0.925374i	$-0.623751\pi$
$227$	−2592.81	−0.758110	−0.379055	+	0.925374i	$0.623751\pi$
$228$	0	0
$229$	2517.84	0.726566	0.363283	−	0.931679i	$-0.381656\pi$
$229$	2517.84	0.726566	0.363283	+	0.931679i	$0.381656\pi$
$230$	−2938.10	−0.842315
$231$	5470.05	1.55802
$232$	−2305.05	−0.652301
$233$	−4616.68	−1.29806	−0.649032	−	0.760761i	$-0.724826\pi$
$233$	−4616.68	−1.29806	−0.649032	+	0.760761i	$0.724826\pi$
$234$	−1603.64	−0.448005
$235$	−621.989	−0.172656
$236$	−2907.23	−0.801885
$237$	781.262	0.214128
$238$	−1539.39	−0.419261
$239$	1825.28	0.494006	0.247003	−	0.969015i	$-0.420554\pi$
$239$	1825.28	0.494006	0.247003	+	0.969015i	$0.420554\pi$
$240$	881.865	0.237184
$241$	5738.83	1.53390	0.766951	−	0.641706i	$-0.221773\pi$
$241$	5738.83	1.53390	0.766951	+	0.641706i	$0.221773\pi$
$242$	−5834.62	−1.54985
$243$	−243.000	−0.0641500
$244$	2635.83	0.691565
$245$	−8075.89	−2.10592
$246$	775.059	0.200878
$247$	0	0
$248$	59.8181	0.0153163
$249$	1688.44	0.429722
$250$	3216.50	0.813718
$251$	−4617.24	−1.16110	−0.580552	−	0.814223i	$-0.697163\pi$
$251$	−4617.24	−1.16110	−0.580552	+	0.814223i	$0.697163\pi$
$252$	−1007.08	−0.251747
$253$	−5211.75	−1.29510
$254$	1581.64	0.390713
$255$	−1516.49	−0.372417
$256$	256.000	0.0625000
$257$	−5790.72	−1.40551	−0.702753	−	0.711433i	$-0.748046\pi$
$257$	−5790.72	−1.40551	−0.702753	+	0.711433i	$0.748046\pi$
$258$	545.880	0.131725
$259$	528.968	0.126905
$260$	−6547.19	−1.56169
$261$	2593.18	0.614995
$262$	−3099.69	−0.730915
$263$	825.091	0.193450	0.0967249	−	0.995311i	$-0.469163\pi$
$263$	825.091	0.193450	0.0967249	+	0.995311i	$0.469163\pi$
$264$	1564.30	0.364681
$265$	4173.93	0.967557
$266$	0	0
$267$	−1883.71	−0.431766
$268$	−2182.69	−0.497496
$269$	−2140.39	−0.485137	−0.242568	−	0.970134i	$-0.577990\pi$
$269$	−2140.39	−0.485137	−0.242568	+	0.970134i	$0.577990\pi$
$270$	−992.098	−0.223619
$271$	3265.58	0.731991	0.365996	−	0.930617i	$-0.380729\pi$
$271$	3265.58	0.731991	0.365996	+	0.930617i	$0.380729\pi$
$272$	−440.228	−0.0981351
$273$	7476.83	1.65758
$274$	1780.57	0.392585
$275$	13853.0	3.03770
$276$	959.525	0.209263
$277$	1560.60	0.338510	0.169255	−	0.985572i	$-0.445864\pi$
$277$	1560.60	0.338510	0.169255	+	0.985572i	$0.445864\pi$
$278$	4545.38	0.980626
$279$	−67.2953	−0.0144404
$280$	−4111.62	−0.877558
$281$	5449.39	1.15688	0.578440	−	0.815725i	$-0.303662\pi$
$281$	5449.39	1.15688	0.578440	+	0.815725i	$0.303662\pi$
$282$	203.129	0.0428943
$283$	−3911.92	−0.821694	−0.410847	−	0.911704i	$-0.634767\pi$
$283$	−3911.92	−0.821694	−0.410847	+	0.911704i	$0.634767\pi$
$284$	−544.364	−0.113740
$285$	0	0
$286$	−11613.7	−2.40117
$287$	−3613.65	−0.743229
$288$	−288.000	−0.0589256
$289$	−4155.97	−0.845912
$290$	10587.2	2.14380
$291$	−3269.76	−0.658683
$292$	699.760	0.140241
$293$	989.750	0.197344	0.0986720	−	0.995120i	$-0.468541\pi$
$293$	989.750	0.197344	0.0986720	+	0.995120i	$0.468541\pi$
$294$	2637.43	0.523191
$295$	13353.1	2.63541
$296$	151.272	0.0297043
$297$	−1759.84	−0.343825
$298$	−2291.36	−0.445419
$299$	−7123.76	−1.37785
$300$	−2550.45	−0.490834
$301$	−2545.12	−0.487370
$302$	−4003.69	−0.762869
$303$	2713.57	0.514490
$304$	0	0
$305$	−12106.5	−2.27284
$306$	495.257	0.0925227
$307$	6992.74	1.29999	0.649994	−	0.759939i	$-0.274771\pi$
$307$	6992.74	1.29999	0.649994	+	0.759939i	$0.274771\pi$
$308$	−7293.40	−1.34929
$309$	−1800.48	−0.331476
$310$	−274.747	−0.0503374
$311$	−2876.02	−0.524385	−0.262193	−	0.965016i	$-0.584446\pi$
$311$	−2876.02	−0.524385	−0.262193	+	0.965016i	$0.584446\pi$
$312$	2138.19	0.387984
$313$	6200.47	1.11972	0.559858	−	0.828588i	$-0.310856\pi$
$313$	6200.47	1.11972	0.559858	+	0.828588i	$0.310856\pi$
$314$	−3794.07	−0.681884
$315$	4625.57	0.827370
$316$	−1041.68	−0.185441
$317$	674.175	0.119449	0.0597247	−	0.998215i	$-0.480978\pi$
$317$	674.175	0.119449	0.0597247	+	0.998215i	$0.480978\pi$
$318$	−1363.13	−0.240378
$319$	18780.1	3.29619
$320$	−1175.82	−0.205407
$321$	−3069.39	−0.533696
$322$	−4473.71	−0.774254
$323$	0	0
$324$	324.000	0.0555556
$325$	18935.2	3.23180
$326$	−1652.41	−0.280732
$327$	4742.83	0.802078
$328$	−1033.41	−0.173965
$329$	−947.074	−0.158705
$330$	−7184.89	−1.19853
$331$	7356.68	1.22163	0.610815	−	0.791773i	$-0.290842\pi$
$331$	7356.68	1.22163	0.610815	+	0.791773i	$0.290842\pi$
$332$	−2251.26	−0.372150
$333$	−170.181	−0.0280055
$334$	1432.79	0.234728
$335$	10025.2	1.63503
$336$	1342.78	0.218019
$337$	1916.40	0.309771	0.154885	−	0.987932i	$-0.450499\pi$
$337$	1916.40	0.309771	0.154885	+	0.987932i	$0.450499\pi$
$338$	−11480.4	−1.84749
$339$	4658.61	0.746375
$340$	2021.99	0.322523
$341$	−487.361	−0.0773961
$342$	0	0
$343$	−2701.55	−0.425276
$344$	−727.840	−0.114077
$345$	−4407.15	−0.687747
$346$	4298.38	0.667868
$347$	−27.7822	−0.00429807	−0.00214903	−	0.999998i	$-0.500684\pi$
$347$	−27.7822	−0.00429807	−0.00214903	+	0.999998i	$0.500684\pi$
$348$	−3457.57	−0.532602
$349$	−1867.83	−0.286484	−0.143242	−	0.989688i	$-0.545753\pi$
$349$	−1867.83	−0.286484	−0.143242	+	0.989688i	$0.545753\pi$
$350$	11891.2	1.81604
$351$	−2405.46	−0.365794
$352$	−2085.73	−0.315823
$353$	5429.71	0.818680	0.409340	−	0.912382i	$-0.365759\pi$
$353$	5429.71	0.818680	0.409340	+	0.912382i	$0.365759\pi$
$354$	−4360.85	−0.654736
$355$	2500.29	0.373808
$356$	2511.62	0.373920
$357$	−2309.09	−0.342325
$358$	6192.68	0.914228
$359$	4944.58	0.726922	0.363461	−	0.931609i	$-0.381595\pi$
$359$	4944.58	0.726922	0.363461	+	0.931609i	$0.381595\pi$
$360$	1322.80	0.193660
$361$	0	0
$362$	−1271.54	−0.184615
$363$	−8751.94	−1.26545
$364$	−9969.10	−1.43550
$365$	−3214.03	−0.460904
$366$	3953.75	0.564660
$367$	−9812.06	−1.39560	−0.697801	−	0.716292i	$-0.745838\pi$
$367$	−9812.06	−1.39560	−0.697801	+	0.716292i	$0.745838\pi$
$368$	−1279.37	−0.181227
$369$	1162.59	0.164016
$370$	−694.798	−0.0976238
$371$	6355.46	0.889377
$372$	89.7271	0.0125057
$373$	−269.962	−0.0374748	−0.0187374	−	0.999824i	$-0.505965\pi$
$373$	−269.962	−0.0374748	−0.0187374	+	0.999824i	$0.505965\pi$
$374$	3586.71	0.495894
$375$	4824.75	0.664398
$376$	−270.839	−0.0371475
$377$	25669.9	3.50681
$378$	−1510.62	−0.205550
$379$	−3021.23	−0.409473	−0.204736	−	0.978817i	$-0.565634\pi$
$379$	−3021.23	−0.409473	−0.204736	+	0.978817i	$0.565634\pi$
$380$	0	0
$381$	2372.46	0.319015
$382$	−2011.77	−0.269454
$383$	1804.33	0.240723	0.120361	−	0.992730i	$-0.461595\pi$
$383$	1804.33	0.240723	0.120361	+	0.992730i	$0.461595\pi$
$384$	384.000	0.0510310
$385$	33498.9	4.43445
$386$	6239.71	0.822780
$387$	818.820	0.107553
$388$	4359.68	0.570436
$389$	−4112.87	−0.536069	−0.268035	−	0.963409i	$-0.586374\pi$
$389$	−4112.87	−0.536069	−0.268035	+	0.963409i	$0.586374\pi$
$390$	−9820.78	−1.27511
$391$	2200.05	0.284556
$392$	−3516.57	−0.453096
$393$	−4649.54	−0.596790
$394$	−8037.78	−1.02776
$395$	4784.50	0.609454
$396$	2346.45	0.297761
$397$	−2793.39	−0.353139	−0.176569	−	0.984288i	$-0.556500\pi$
$397$	−2793.39	−0.353139	−0.176569	+	0.984288i	$0.556500\pi$
$398$	−8451.96	−1.06447
$399$	0	0
$400$	3400.60	0.425074
$401$	936.825	0.116665	0.0583327	−	0.998297i	$-0.481422\pi$
$401$	936.825	0.116665	0.0583327	+	0.998297i	$0.481422\pi$
$402$	−3274.03	−0.406204
$403$	−666.157	−0.0823415
$404$	−3618.09	−0.445562
$405$	−1488.15	−0.182584
$406$	16120.6	1.97058
$407$	−1232.47	−0.150101
$408$	−660.342	−0.0801270
$409$	−2318.72	−0.280326	−0.140163	−	0.990128i	$-0.544763\pi$
$409$	−2318.72	−0.280326	−0.140163	+	0.990128i	$0.544763\pi$
$410$	4746.51	0.571740
$411$	2670.86	0.320544
$412$	2400.65	0.287066
$413$	20332.1	2.42246
$414$	1439.29	0.170863
$415$	10340.1	1.22308
$416$	−2850.91	−0.336004
$417$	6818.08	0.800678
$418$	0	0
$419$	−5725.14	−0.667521	−0.333761	−	0.942658i	$-0.608318\pi$
$419$	−5725.14	−0.667521	−0.333761	+	0.942658i	$0.608318\pi$
$420$	−6167.43	−0.716523
$421$	8066.56	0.933824	0.466912	−	0.884304i	$-0.345366\pi$
$421$	8066.56	0.933824	0.466912	+	0.884304i	$0.345366\pi$
$422$	2644.65	0.305070
$423$	304.694	0.0350230
$424$	1817.50	0.208174
$425$	−5847.81	−0.667436
$426$	−816.547	−0.0928681
$427$	−18434.0	−2.08919
$428$	4092.51	0.462194
$429$	−17420.6	−1.96055
$430$	3343.00	0.374916
$431$	11173.1	1.24870	0.624349	−	0.781145i	$-0.285364\pi$
$431$	11173.1	1.24870	0.624349	+	0.781145i	$0.285364\pi$
$432$	−432.000	−0.0481125
$433$	2150.59	0.238686	0.119343	−	0.992853i	$-0.461921\pi$
$433$	2150.59	0.238686	0.119343	+	0.992853i	$0.461921\pi$
$434$	−418.345	−0.0462700
$435$	15880.8	1.75040
$436$	−6323.78	−0.694620
$437$	0	0
$438$	1049.64	0.114506
$439$	−16649.7	−1.81013	−0.905063	−	0.425277i	$-0.860177\pi$
$439$	−16649.7	−1.81013	−0.905063	+	0.425277i	$0.860177\pi$
$440$	9579.86	1.03796
$441$	3956.15	0.427183
$442$	4902.55	0.527580
$443$	2507.67	0.268946	0.134473	−	0.990917i	$-0.457066\pi$
$443$	2507.67	0.268946	0.134473	+	0.990917i	$0.457066\pi$
$444$	226.907	0.0242535
$445$	−11536.0	−1.22889
$446$	−3903.31	−0.414410
$447$	−3437.04	−0.363683
$448$	−1790.37	−0.188810
$449$	−17958.0	−1.88750	−0.943752	−	0.330655i	$-0.892730\pi$
$449$	−17958.0	−1.88750	−0.943752	+	0.330655i	$0.892730\pi$
$450$	−3825.67	−0.400764
$451$	8419.61	0.879077
$452$	−6211.48	−0.646380
$453$	−6005.53	−0.622880
$454$	5185.63	0.536065
$455$	45788.6	4.71780
$456$	0	0
$457$	−9750.15	−0.998015	−0.499007	−	0.866598i	$-0.666302\pi$
$457$	−9750.15	−0.998015	−0.499007	+	0.866598i	$0.666302\pi$
$458$	−5035.68	−0.513760
$459$	742.885	0.0755445
$460$	5876.19	0.595606
$461$	2514.25	0.254014	0.127007	−	0.991902i	$-0.459463\pi$
$461$	2514.25	0.254014	0.127007	+	0.991902i	$0.459463\pi$
$462$	−10940.1	−1.10169
$463$	9017.89	0.905177	0.452589	−	0.891719i	$-0.350501\pi$
$463$	9017.89	0.905177	0.452589	+	0.891719i	$0.350501\pi$
$464$	4610.10	0.461247
$465$	−412.121	−0.0411003
$466$	9233.36	0.917869
$467$	−16395.0	−1.62457	−0.812283	−	0.583263i	$-0.801776\pi$
$467$	−16395.0	−1.62457	−0.812283	+	0.583263i	$0.801776\pi$
$468$	3207.28	0.316787
$469$	15264.9	1.50291
$470$	1243.98	0.122086
$471$	−5691.10	−0.556756
$472$	5814.47	0.567018
$473$	5929.99	0.576451
$474$	−1562.52	−0.151412
$475$	0	0
$476$	3078.79	0.296462
$477$	−2044.69	−0.196268
$478$	−3650.56	−0.349315
$479$	−10658.5	−1.01670	−0.508350	−	0.861151i	$-0.669744\pi$
$479$	−10658.5	−1.01670	−0.508350	+	0.861151i	$0.669744\pi$
$480$	−1763.73	−0.167714
$481$	−1684.62	−0.159692
$482$	−11477.7	−1.08463
$483$	−6710.56	−0.632176
$484$	11669.2	1.09591
$485$	−20024.2	−1.87475
$486$	486.000	0.0453609
$487$	13987.7	1.30153	0.650763	−	0.759281i	$-0.274449\pi$
$487$	13987.7	1.30153	0.650763	+	0.759281i	$0.274449\pi$
$488$	−5271.66	−0.489010
$489$	−2478.62	−0.229217
$490$	16151.8	1.48911
$491$	11659.2	1.07163	0.535815	−	0.844335i	$-0.320004\pi$
$491$	11659.2	1.07163	0.535815	+	0.844335i	$0.320004\pi$
$492$	−1550.12	−0.142042
$493$	−7927.71	−0.724232
$494$	0	0
$495$	−10777.3	−0.978597
$496$	−119.636	−0.0108303
$497$	3807.08	0.343603
$498$	−3376.89	−0.303859
$499$	−2154.41	−0.193276	−0.0966379	−	0.995320i	$-0.530809\pi$
$499$	−2154.41	−0.193276	−0.0966379	+	0.995320i	$0.530809\pi$
$500$	−6433.00	−0.575385
$501$	2149.19	0.191654
$502$	9234.47	0.821025
$503$	22014.5	1.95144	0.975722	−	0.219013i	$-0.0702837\pi$
$503$	22014.5	1.95144	0.975722	+	0.219013i	$0.0702837\pi$
$504$	2014.16	0.178012
$505$	16618.1	1.46435
$506$	10423.5	0.915773
$507$	−17220.6	−1.50847
$508$	−3163.28	−0.276275
$509$	−17247.4	−1.50192	−0.750958	−	0.660350i	$-0.770408\pi$
$509$	−17247.4	−1.50192	−0.750958	+	0.660350i	$0.770408\pi$
$510$	3032.98	0.263339
$511$	−4893.86	−0.423663
$512$	−512.000	−0.0441942
$513$	0	0
$514$	11581.4	0.993843
$515$	−11026.3	−0.943448
$516$	−1091.76	−0.0931435
$517$	2206.63	0.187713
$518$	−1057.94	−0.0897356
$519$	6447.57	0.545312
$520$	13094.4	1.10428
$521$	−20135.5	−1.69319	−0.846593	−	0.532240i	$-0.821350\pi$
$521$	−20135.5	−1.69319	−0.846593	+	0.532240i	$0.821350\pi$
$522$	−5186.36	−0.434867
$523$	9426.66	0.788143	0.394072	−	0.919080i	$-0.371066\pi$
$523$	9426.66	0.788143	0.394072	+	0.919080i	$0.371066\pi$
$524$	6199.39	0.516835
$525$	17836.9	1.48279
$526$	−1650.18	−0.136790
$527$	205.731	0.0170053
$528$	−3128.60	−0.257869
$529$	−5773.33	−0.474507
$530$	−8347.87	−0.684166
$531$	−6541.28	−0.534590
$532$	0	0
$533$	11508.5	0.935248
$534$	3767.43	0.305304
$535$	−18797.1	−1.51901
$536$	4365.37	0.351783
$537$	9289.03	0.746464
$538$	4280.78	0.343043
$539$	28650.9	2.28958
$540$	1984.20	0.158123
$541$	13265.7	1.05422	0.527112	−	0.849796i	$-0.323275\pi$
$541$	13265.7	1.05422	0.527112	+	0.849796i	$0.323275\pi$
$542$	−6531.15	−0.517596
$543$	−1907.31	−0.150737
$544$	880.456	0.0693920
$545$	29045.4	2.28288
$546$	−14953.7	−1.17208
$547$	15466.4	1.20895	0.604473	−	0.796625i	$-0.293384\pi$
$547$	15466.4	1.20895	0.604473	+	0.796625i	$0.293384\pi$
$548$	−3561.14	−0.277599
$549$	5930.62	0.461043
$550$	−27706.0	−2.14798
$551$	0	0
$552$	−1919.05	−0.147971
$553$	7285.13	0.560209
$554$	−3121.20	−0.239363
$555$	−1042.20	−0.0797095
$556$	−9090.77	−0.693407
$557$	9057.64	0.689021	0.344510	−	0.938783i	$-0.388045\pi$
$557$	9057.64	0.689021	0.344510	+	0.938783i	$0.388045\pi$
$558$	134.591	0.0102109
$559$	8105.51	0.613285
$560$	8223.24	0.620527
$561$	5380.06	0.404896
$562$	−10898.8	−0.818037
$563$	−11473.8	−0.858908	−0.429454	−	0.903089i	$-0.641294\pi$
$563$	−11473.8	−0.858908	−0.429454	+	0.903089i	$0.641294\pi$
$564$	−406.259	−0.0303308
$565$	28529.6	2.12434
$566$	7823.83	0.581025
$567$	−2265.93	−0.167831
$568$	1088.73	0.0804261
$569$	6042.56	0.445197	0.222599	−	0.974910i	$-0.428546\pi$
$569$	6042.56	0.445197	0.222599	+	0.974910i	$0.428546\pi$
$570$	0	0
$571$	18696.1	1.37024	0.685121	−	0.728429i	$-0.259749\pi$
$571$	18696.1	1.37024	0.685121	+	0.728429i	$0.259749\pi$
$572$	23227.5	1.69788
$573$	−3017.66	−0.220008
$574$	7227.29	0.525543
$575$	−16994.6	−1.23256
$576$	576.000	0.0416667
$577$	−10918.2	−0.787746	−0.393873	−	0.919165i	$-0.628865\pi$
$577$	−10918.2	−0.787746	−0.393873	+	0.919165i	$0.628865\pi$
$578$	8311.93	0.598150
$579$	9359.57	0.671797
$580$	−21174.4	−1.51589
$581$	15744.4	1.12425
$582$	6539.52	0.465759
$583$	−14807.9	−1.05194
$584$	−1399.52	−0.0991654
$585$	−14731.2	−1.04113
$586$	−1979.50	−0.139543
$587$	−8385.30	−0.589605	−0.294803	−	0.955558i	$-0.595254\pi$
$587$	−8385.30	−0.589605	−0.294803	+	0.955558i	$0.595254\pi$
$588$	−5274.86	−0.369952
$589$	0	0
$590$	−26706.1	−1.86351
$591$	−12056.7	−0.839163
$592$	−302.543	−0.0210041
$593$	−17150.0	−1.18763	−0.593817	−	0.804600i	$-0.702380\pi$
$593$	−17150.0	−1.18763	−0.593817	+	0.804600i	$0.702380\pi$
$594$	3519.67	0.243121
$595$	−14141.0	−0.974328
$596$	4582.72	0.314959
$597$	−12677.9	−0.869135
$598$	14247.5	0.974288
$599$	−11258.5	−0.767963	−0.383981	−	0.923341i	$-0.625447\pi$
$599$	−11258.5	−0.767963	−0.383981	+	0.923341i	$0.625447\pi$
$600$	5100.89	0.347072
$601$	4693.83	0.318578	0.159289	−	0.987232i	$-0.449080\pi$
$601$	4693.83	0.318578	0.159289	+	0.987232i	$0.449080\pi$
$602$	5090.24	0.344622
$603$	−4911.05	−0.331664
$604$	8007.38	0.539430
$605$	−53597.4	−3.60173
$606$	−5427.14	−0.363800
$607$	−6223.14	−0.416127	−0.208064	−	0.978115i	$-0.566716\pi$
$607$	−6223.14	−0.416127	−0.208064	+	0.978115i	$0.566716\pi$
$608$	0	0
$609$	24181.0	1.60897
$610$	24213.0	1.60714
$611$	3016.17	0.199707
$612$	−990.513	−0.0654234
$613$	−20857.8	−1.37429	−0.687144	−	0.726521i	$-0.741136\pi$
$613$	−20857.8	−1.37429	−0.687144	+	0.726521i	$0.741136\pi$
$614$	−13985.5	−0.919231
$615$	7119.77	0.466824
$616$	14586.8	0.954090
$617$	5605.82	0.365772	0.182886	−	0.983134i	$-0.441456\pi$
$617$	5605.82	0.365772	0.182886	+	0.983134i	$0.441456\pi$
$618$	3600.97	0.234389
$619$	−20849.5	−1.35382	−0.676908	−	0.736068i	$-0.736681\pi$
$619$	−20849.5	−1.35382	−0.676908	+	0.736068i	$0.736681\pi$
$620$	549.494	0.0355939
$621$	2158.93	0.139509
$622$	5752.03	0.370797
$623$	−17565.3	−1.12960
$624$	−4276.37	−0.274346
$625$	2979.92	0.190715
$626$	−12400.9	−0.791759
$627$	0	0
$628$	7588.14	0.482165
$629$	520.266	0.0329799
$630$	−9251.14	−0.585039
$631$	−608.986	−0.0384205	−0.0192103	−	0.999815i	$-0.506115\pi$
$631$	−608.986	−0.0384205	−0.0192103	+	0.999815i	$0.506115\pi$
$632$	2083.37	0.131126
$633$	3966.97	0.249088
$634$	−1348.35	−0.0844634
$635$	14529.1	0.907984
$636$	2726.25	0.169973
$637$	39161.9	2.43587
$638$	−37560.2	−2.33076
$639$	−1224.82	−0.0758265
$640$	2351.64	0.145245
$641$	22925.4	1.41263	0.706316	−	0.707896i	$-0.250356\pi$
$641$	22925.4	1.41263	0.706316	+	0.707896i	$0.250356\pi$
$642$	6138.77	0.377380
$643$	840.855	0.0515709	0.0257855	−	0.999667i	$-0.491791\pi$
$643$	840.855	0.0515709	0.0257855	+	0.999667i	$0.491791\pi$
$644$	8947.41	0.547480
$645$	5014.51	0.306118
$646$	0	0
$647$	−17094.5	−1.03872	−0.519361	−	0.854555i	$-0.673830\pi$
$647$	−17094.5	−1.03872	−0.519361	+	0.854555i	$0.673830\pi$
$648$	−648.000	−0.0392837
$649$	−47372.7	−2.86524
$650$	−37870.3	−2.28523
$651$	−627.517	−0.0377793
$652$	3304.83	0.198508
$653$	15727.5	0.942520	0.471260	−	0.881994i	$-0.343799\pi$
$653$	15727.5	0.942520	0.471260	+	0.881994i	$0.343799\pi$
$654$	−9485.67	−0.567154
$655$	−28474.1	−1.69859
$656$	2066.82	0.123012
$657$	1574.46	0.0934940
$658$	1894.15	0.112221
$659$	16360.3	0.967082	0.483541	−	0.875322i	$-0.339350\pi$
$659$	16360.3	0.967082	0.483541	+	0.875322i	$0.339350\pi$
$660$	14369.8	0.847490
$661$	14548.3	0.856074	0.428037	−	0.903761i	$-0.359205\pi$
$661$	14548.3	0.856074	0.428037	+	0.903761i	$0.359205\pi$
$662$	−14713.4	−0.863823
$663$	7353.82	0.430767
$664$	4502.52	0.263150
$665$	0	0
$666$	340.361	0.0198029
$667$	−23039.1	−1.33745
$668$	−2865.59	−0.165977
$669$	−5854.96	−0.338365
$670$	−20050.4	−1.15614
$671$	42950.3	2.47105
$672$	−2685.55	−0.154163
$673$	26460.4	1.51556	0.757781	−	0.652509i	$-0.226283\pi$
$673$	26460.4	1.51556	0.757781	+	0.652509i	$0.226283\pi$
$674$	−3832.79	−0.219041
$675$	−5738.51	−0.327222
$676$	22960.9	1.30638
$677$	16458.2	0.934327	0.467163	−	0.884171i	$-0.345276\pi$
$677$	16458.2	0.934327	0.467163	+	0.884171i	$0.345276\pi$
$678$	−9317.22	−0.527767
$679$	−30489.9	−1.72326
$680$	−4043.98	−0.228058
$681$	7778.44	0.437695
$682$	974.721	0.0547273
$683$	−1114.16	−0.0624189	−0.0312095	−	0.999513i	$-0.509936\pi$
$683$	−1114.16	−0.0624189	−0.0312095	+	0.999513i	$0.509936\pi$
$684$	0	0
$685$	16356.5	0.912335
$686$	5403.09	0.300716
$687$	−7553.53	−0.419483
$688$	1455.68	0.0806647
$689$	−20240.4	−1.11915
$690$	8814.29	0.486311
$691$	−18406.3	−1.01333	−0.506663	−	0.862144i	$-0.669121\pi$
$691$	−18406.3	−1.01333	−0.506663	+	0.862144i	$0.669121\pi$
$692$	−8596.76	−0.472254
$693$	−16410.2	−0.899524
$694$	55.5645	0.00303919
$695$	41754.3	2.27889
$696$	6915.15	0.376606
$697$	−3554.20	−0.193149
$698$	3735.67	0.202575
$699$	13850.0	0.749437
$700$	−23782.5	−1.28413
$701$	−23488.3	−1.26554	−0.632768	−	0.774341i	$-0.718081\pi$
$701$	−23488.3	−1.26554	−0.632768	+	0.774341i	$0.718081\pi$
$702$	4810.92	0.258656
$703$	0	0
$704$	4171.46	0.223321
$705$	1865.97	0.0996828
$706$	−10859.4	−0.578894
$707$	25303.6	1.34602
$708$	8721.70	0.462968
$709$	18132.1	0.960458	0.480229	−	0.877143i	$-0.340554\pi$
$709$	18132.1	0.960458	0.480229	+	0.877143i	$0.340554\pi$
$710$	−5000.58	−0.264322
$711$	−2343.79	−0.123627
$712$	−5023.24	−0.264401
$713$	597.885	0.0314039
$714$	4618.18	0.242060
$715$	−106685.	−5.58013
$716$	−12385.4	−0.646457
$717$	−5475.84	−0.285215
$718$	−9889.16	−0.514011
$719$	13429.0	0.696547	0.348274	−	0.937393i	$-0.386768\pi$
$719$	13429.0	0.696547	0.348274	+	0.937393i	$0.386768\pi$
$720$	−2645.59	−0.136938
$721$	−16789.2	−0.867216
$722$	0	0
$723$	−17216.5	−0.885599
$724$	2543.08	0.130542
$725$	61238.6	3.13703
$726$	17503.9	0.894807
$727$	−34284.0	−1.74900	−0.874500	−	0.485025i	$-0.838811\pi$
$727$	−34284.0	−1.74900	−0.874500	+	0.485025i	$0.838811\pi$
$728$	19938.2	1.01505
$729$	729.000	0.0370370
$730$	6428.06	0.325909
$731$	−2503.25	−0.126657
$732$	−7907.50	−0.399275
$733$	21537.1	1.08525	0.542626	−	0.839975i	$-0.317430\pi$
$733$	21537.1	1.08525	0.542626	+	0.839975i	$0.317430\pi$
$734$	19624.1	0.986839
$735$	24227.7	1.21585
$736$	2558.73	0.128147
$737$	−35566.4	−1.77762
$738$	−2325.18	−0.115977
$739$	34455.2	1.71510	0.857548	−	0.514404i	$-0.171987\pi$
$739$	34455.2	1.71510	0.857548	+	0.514404i	$0.171987\pi$
$740$	1389.60	0.0690305
$741$	0	0
$742$	−12710.9	−0.628885
$743$	18369.9	0.907034	0.453517	−	0.891248i	$-0.350169\pi$
$743$	18369.9	0.907034	0.453517	+	0.891248i	$0.350169\pi$
$744$	−179.454	−0.00884289
$745$	−21048.6	−1.03512
$746$	539.925	0.0264987
$747$	−5065.33	−0.248100
$748$	−7173.42	−0.350650
$749$	−28621.5	−1.39627
$750$	−9649.50	−0.469800
$751$	9712.59	0.471927	0.235964	−	0.971762i	$-0.424175\pi$
$751$	9712.59	0.471927	0.235964	+	0.971762i	$0.424175\pi$
$752$	541.679	0.0262673
$753$	13851.7	0.670364
$754$	−51339.8	−2.47969
$755$	−36778.3	−1.77284
$756$	3021.24	0.145346
$757$	39989.4	1.92000	0.960000	−	0.280002i	$-0.0903350\pi$
$757$	39989.4	1.92000	0.960000	+	0.280002i	$0.0903350\pi$
$758$	6042.46	0.289541
$759$	15635.2	0.747725
$760$	0	0
$761$	19958.4	0.950714	0.475357	−	0.879793i	$-0.342319\pi$
$761$	19958.4	0.950714	0.475357	+	0.879793i	$0.342319\pi$
$762$	−4744.92	−0.225578
$763$	44226.1	2.09842
$764$	4023.55	0.190533
$765$	4549.47	0.215015
$766$	−3608.65	−0.170217
$767$	−64752.1	−3.04832
$768$	−768.000	−0.0360844
$769$	34776.3	1.63077	0.815387	−	0.578917i	$-0.196524\pi$
$769$	34776.3	1.63077	0.815387	+	0.578917i	$0.196524\pi$
$770$	−66997.9	−3.13563
$771$	17372.2	0.811470
$772$	−12479.4	−0.581793
$773$	6353.52	0.295628	0.147814	−	0.989015i	$-0.452776\pi$
$773$	6353.52	0.295628	0.147814	+	0.989015i	$0.452776\pi$
$774$	−1637.64	−0.0760514
$775$	−1589.20	−0.0736588
$776$	−8719.36	−0.403359
$777$	−1586.90	−0.0732688
$778$	8225.74	0.379058
$779$	0	0
$780$	19641.6	0.901642
$781$	−8870.29	−0.406407
$782$	−4400.11	−0.201212
$783$	−7779.54	−0.355068
$784$	7033.15	0.320388
$785$	−34852.7	−1.58464
$786$	9299.08	0.421994
$787$	−24842.0	−1.12519	−0.562593	−	0.826734i	$-0.690196\pi$
$787$	−24842.0	−1.12519	−0.562593	+	0.826734i	$0.690196\pi$
$788$	16075.6	0.726736
$789$	−2475.27	−0.111688
$790$	−9568.99	−0.430949
$791$	43440.8	1.95269
$792$	−4692.89	−0.210549
$793$	58707.3	2.62895
$794$	5586.77	0.249707
$795$	−12521.8	−0.558620
$796$	16903.9	0.752693
$797$	−33078.4	−1.47013	−0.735067	−	0.677995i	$-0.762849\pi$
$797$	−33078.4	−1.47013	−0.735067	+	0.677995i	$0.762849\pi$
$798$	0	0
$799$	−931.493	−0.0412439
$800$	−6801.19	−0.300573
$801$	5651.14	0.249280
$802$	−1873.65	−0.0824948
$803$	11402.4	0.501100
$804$	6548.06	0.287229
$805$	−41095.9	−1.79930
$806$	1332.31	0.0582242
$807$	6421.16	0.280094
$808$	7236.19	0.315060
$809$	18929.2	0.822641	0.411320	−	0.911491i	$-0.365068\pi$
$809$	18929.2	0.822641	0.411320	+	0.911491i	$0.365068\pi$
$810$	2976.29	0.129107
$811$	29073.5	1.25883	0.629414	−	0.777070i	$-0.283295\pi$
$811$	29073.5	1.25883	0.629414	+	0.777070i	$0.283295\pi$
$812$	−32241.3	−1.39341
$813$	−9796.73	−0.422615
$814$	2464.94	0.106138
$815$	−15179.2	−0.652399
$816$	1320.68	0.0566583
$817$	0	0
$818$	4637.45	0.198221
$819$	−22430.5	−0.957002
$820$	−9493.02	−0.404281
$821$	30121.9	1.28047	0.640233	−	0.768181i	$-0.278838\pi$
$821$	30121.9	1.28047	0.640233	+	0.768181i	$0.278838\pi$
$822$	−5341.71	−0.226659
$823$	−774.261	−0.0327935	−0.0163967	−	0.999866i	$-0.505219\pi$
$823$	−774.261	−0.0327935	−0.0163967	+	0.999866i	$0.505219\pi$
$824$	−4801.29	−0.202987
$825$	−41558.9	−1.75381
$826$	−40664.2	−1.71294
$827$	30520.1	1.28330	0.641650	−	0.766998i	$-0.278250\pi$
$827$	30520.1	1.28330	0.641650	+	0.766998i	$0.278250\pi$
$828$	−2878.58	−0.120818
$829$	−29677.4	−1.24335	−0.621675	−	0.783275i	$-0.713548\pi$
$829$	−29677.4	−1.24335	−0.621675	+	0.783275i	$0.713548\pi$
$830$	−20680.3	−0.864846
$831$	−4681.79	−0.195439
$832$	5701.83	0.237590
$833$	−12094.5	−0.503060
$834$	−13636.2	−0.566165
$835$	13161.8	0.545488
$836$	0	0
$837$	201.886	0.00833716
$838$	11450.3	0.472009
$839$	−28290.1	−1.16410	−0.582051	−	0.813152i	$-0.697750\pi$
$839$	−28290.1	−1.16410	−0.582051	+	0.813152i	$0.697750\pi$
$840$	12334.9	0.506658
$841$	58630.5	2.40397
$842$	−16133.1	−0.660313
$843$	−16348.2	−0.667925
$844$	−5289.29	−0.215717
$845$	−105460.	−4.29343
$846$	−609.388	−0.0247650
$847$	−81610.3	−3.31070
$848$	−3635.00	−0.147201
$849$	11735.8	0.474405
$850$	11695.6	0.471948
$851$	1511.97	0.0609044
$852$	1633.09	0.0656677
$853$	8208.45	0.329486	0.164743	−	0.986336i	$-0.447320\pi$
$853$	8208.45	0.329486	0.164743	+	0.986336i	$0.447320\pi$
$854$	36868.0	1.47728
$855$	0	0
$856$	−8185.03	−0.326821
$857$	−18254.3	−0.727602	−0.363801	−	0.931477i	$-0.618521\pi$
$857$	−18254.3	−0.727602	−0.363801	+	0.931477i	$0.618521\pi$
$858$	34841.2	1.38632
$859$	26966.6	1.07111	0.535557	−	0.844499i	$-0.320102\pi$
$859$	26966.6	1.07111	0.535557	+	0.844499i	$0.320102\pi$
$860$	−6686.01	−0.265106
$861$	10840.9	0.429104
$862$	−22346.2	−0.882963
$863$	1058.61	0.0417562	0.0208781	−	0.999782i	$-0.493354\pi$
$863$	1058.61	0.0417562	0.0208781	+	0.999782i	$0.493354\pi$
$864$	864.000	0.0340207
$865$	39485.3	1.55207
$866$	−4301.18	−0.168776
$867$	12467.9	0.488387
$868$	836.690	0.0327179
$869$	−16974.0	−0.662604
$870$	−31761.6	−1.23772
$871$	−48614.5	−1.89120
$872$	12647.6	0.491170
$873$	9809.28	0.380291
$874$	0	0
$875$	44990.0	1.73822
$876$	−2099.28	−0.0809682
$877$	−12516.9	−0.481946	−0.240973	−	0.970532i	$-0.577467\pi$
$877$	−12516.9	−0.481946	−0.240973	+	0.970532i	$0.577467\pi$
$878$	33299.3	1.27995
$879$	−2969.25	−0.113937
$880$	−19159.7	−0.733948
$881$	46916.2	1.79415	0.897075	−	0.441879i	$-0.145688\pi$
$881$	46916.2	1.79415	0.897075	+	0.441879i	$0.145688\pi$
$882$	−7912.29	−0.302064
$883$	−36806.4	−1.40276	−0.701379	−	0.712789i	$-0.747432\pi$
$883$	−36806.4	−1.40276	−0.701379	+	0.712789i	$0.747432\pi$
$884$	−9805.10	−0.373056
$885$	−40059.2	−1.52155
$886$	−5015.35	−0.190174
$887$	41126.7	1.55682	0.778410	−	0.627757i	$-0.216027\pi$
$887$	41126.7	1.55682	0.778410	+	0.627757i	$0.216027\pi$
$888$	−453.815	−0.0171498
$889$	22122.8	0.834617
$890$	23072.0	0.868960
$891$	5279.51	0.198507
$892$	7806.61	0.293032
$893$	0	0
$894$	6874.08	0.257163
$895$	56886.6	2.12459
$896$	3580.73	0.133509
$897$	21371.3	0.795503
$898$	35915.9	1.33467
$899$	−2154.43	−0.0799269
$900$	7651.34	0.283383
$901$	6250.90	0.231129
$902$	−16839.2	−0.621602
$903$	7635.36	0.281383
$904$	12423.0	0.457059
$905$	−11680.5	−0.429030
$906$	12011.1	0.440443
$907$	856.416	0.0313526	0.0156763	−	0.999877i	$-0.495010\pi$
$907$	856.416	0.0313526	0.0156763	+	0.999877i	$0.495010\pi$
$908$	−10371.3	−0.379055
$909$	−8140.71	−0.297041
$910$	−91577.1	−3.33599
$911$	−1594.94	−0.0580053	−0.0290027	−	0.999579i	$-0.509233\pi$
$911$	−1594.94	−0.0580053	−0.0290027	+	0.999579i	$0.509233\pi$
$912$	0	0
$913$	−36683.7	−1.32974
$914$	19500.3	0.705703
$915$	36319.5	1.31222
$916$	10071.4	0.363283
$917$	−43356.2	−1.56134
$918$	−1485.77	−0.0534180
$919$	15691.8	0.563246	0.281623	−	0.959525i	$-0.409127\pi$
$919$	15691.8	0.563246	0.281623	+	0.959525i	$0.409127\pi$
$920$	−11752.4	−0.421157
$921$	−20978.2	−0.750549
$922$	−5028.51	−0.179615
$923$	−12124.5	−0.432376
$924$	21880.2	0.779011
$925$	−4018.86	−0.142853
$926$	−18035.8	−0.640057
$927$	5401.45	0.191378
$928$	−9220.19	−0.326151
$929$	−27745.1	−0.979856	−0.489928	−	0.871763i	$-0.662977\pi$
$929$	−27745.1	−0.979856	−0.489928	+	0.871763i	$0.662977\pi$
$930$	824.241	0.0290623
$931$	0	0
$932$	−18466.7	−0.649032
$933$	8628.05	0.302754
$934$	32790.1	1.14874
$935$	32947.8	1.15242
$936$	−6414.56	−0.224002
$937$	5892.63	0.205447	0.102724	−	0.994710i	$-0.467244\pi$
$937$	5892.63	0.205447	0.102724	+	0.994710i	$0.467244\pi$
$938$	−30529.8	−1.06272
$939$	−18601.4	−0.646469
$940$	−2487.96	−0.0863278
$941$	17630.1	0.610759	0.305380	−	0.952231i	$-0.401217\pi$
$941$	17630.1	0.610759	0.305380	+	0.952231i	$0.401217\pi$
$942$	11382.2	0.393686
$943$	−10329.0	−0.356690
$944$	−11628.9	−0.400942
$945$	−13876.7	−0.477682
$946$	−11860.0	−0.407613
$947$	35788.8	1.22807	0.614034	−	0.789280i	$-0.289546\pi$
$947$	35788.8	1.22807	0.614034	+	0.789280i	$0.289546\pi$
$948$	3125.05	0.107064
$949$	15585.6	0.533119
$950$	0	0
$951$	−2022.52	−0.0689641
$952$	−6157.58	−0.209631
$953$	−24674.4	−0.838700	−0.419350	−	0.907825i	$-0.637742\pi$
$953$	−24674.4	−0.838700	−0.419350	+	0.907825i	$0.637742\pi$
$954$	4089.38	0.138782
$955$	−18480.3	−0.626188
$956$	7301.12	0.247003
$957$	−56340.4	−1.90306
$958$	21317.0	0.718915
$959$	24905.3	0.838617
$960$	3527.46	0.118592
$961$	−29735.1	−0.998123
$962$	3369.24	0.112920
$963$	9208.16	0.308129
$964$	22955.3	0.766951
$965$	57318.6	1.91207
$966$	13421.1	0.447016
$967$	−2214.12	−0.0736312	−0.0368156	−	0.999322i	$-0.511721\pi$
$967$	−2214.12	−0.0736312	−0.0368156	+	0.999322i	$0.511721\pi$
$968$	−23338.5	−0.774925
$969$	0	0
$970$	40048.4	1.32565
$971$	−9586.90	−0.316847	−0.158423	−	0.987371i	$-0.550641\pi$
$971$	−9586.90	−0.316847	−0.158423	+	0.987371i	$0.550641\pi$
$972$	−972.000	−0.0320750
$973$	63577.4	2.09476
$974$	−27975.4	−0.920318
$975$	−56805.5	−1.86588
$976$	10543.3	0.345782
$977$	−8218.01	−0.269107	−0.134553	−	0.990906i	$-0.542960\pi$
$977$	−8218.01	−0.269107	−0.134553	+	0.990906i	$0.542960\pi$
$978$	4957.24	0.162081
$979$	40926.3	1.33607
$980$	−32303.6	−1.05296
$981$	−14228.5	−0.463080
$982$	−23318.3	−0.757757
$983$	−23537.3	−0.763705	−0.381852	−	0.924223i	$-0.624714\pi$
$983$	−23537.3	−0.763705	−0.381852	+	0.924223i	$0.624714\pi$
$984$	3100.24	0.100439
$985$	−73835.8	−2.38843
$986$	15855.4	0.512109
$987$	2841.22	0.0916282
$988$	0	0
$989$	−7274.80	−0.233898
$990$	21554.7	0.691972
$991$	−37511.7	−1.20242	−0.601211	−	0.799091i	$-0.705315\pi$
$991$	−37511.7	−1.20242	−0.601211	+	0.799091i	$0.705315\pi$
$992$	239.272	0.00765817
$993$	−22070.0	−0.705309
$994$	−7614.16	−0.242964
$995$	−77640.5	−2.47374
$996$	6753.77	0.214861
$997$	−35906.3	−1.14058	−0.570292	−	0.821442i	$-0.693170\pi$
$997$	−35906.3	−1.14058	−0.570292	+	0.821442i	$0.693170\pi$
$998$	4308.82	0.136667
$999$	510.542	0.0161690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.r.1.1	✓	3
19.18	odd	2		2166.4.a.v.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.4.a.r.1.1	✓	3	1.1	even	1	trivial
2166.4.a.v.1.1	yes	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -18.3722 q^{5} +6.00000 q^{6} -27.9745 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -18.3722 q^{5} +6.00000 q^{6} -27.9745 q^{7} -8.00000 q^{8} +9.00000 q^{9} +36.7444 q^{10} +65.1791 q^{11} -12.0000 q^{12} +89.0910 q^{13} +55.9490 q^{14} +55.1166 q^{15} +16.0000 q^{16} -27.5143 q^{17} -18.0000 q^{18} -73.4887 q^{20} +83.9235 q^{21} -130.358 q^{22} -79.9605 q^{23} +24.0000 q^{24} +212.537 q^{25} -178.182 q^{26} -27.0000 q^{27} -111.898 q^{28} +288.131 q^{29} -110.233 q^{30} -7.47726 q^{31} -32.0000 q^{32} -195.537 q^{33} +55.0285 q^{34} +513.952 q^{35} +36.0000 q^{36} -18.9090 q^{37} -267.273 q^{39} +146.977 q^{40} +129.177 q^{41} -167.847 q^{42} +90.9800 q^{43} +260.716 q^{44} -165.350 q^{45} +159.921 q^{46} +33.8549 q^{47} -48.0000 q^{48} +439.572 q^{49} -425.074 q^{50} +82.5428 q^{51} +356.364 q^{52} -227.188 q^{53} +54.0000 q^{54} -1197.48 q^{55} +223.796 q^{56} -576.262 q^{58} -726.808 q^{59} +220.466 q^{60} +658.958 q^{61} +14.9545 q^{62} -251.770 q^{63} +64.0000 q^{64} -1636.80 q^{65} +391.074 q^{66} -545.672 q^{67} -110.057 q^{68} +239.881 q^{69} -1027.90 q^{70} -136.091 q^{71} -72.0000 q^{72} +174.940 q^{73} +37.8179 q^{74} -637.612 q^{75} -1823.35 q^{77} +534.546 q^{78} -260.421 q^{79} -293.955 q^{80} +81.0000 q^{81} -258.353 q^{82} -562.814 q^{83} +335.694 q^{84} +505.497 q^{85} -181.960 q^{86} -864.393 q^{87} -521.433 q^{88} +627.905 q^{89} +330.699 q^{90} -2492.28 q^{91} -319.842 q^{92} +22.4318 q^{93} -67.7098 q^{94} +96.0000 q^{96} +1089.92 q^{97} -879.144 q^{98} +586.612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} - 5 q^{5} + 18 q^{6} - 11 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 - 5 * q^5 + 18 * q^6 - 11 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} - 5 q^{5} + 18 q^{6} - 11 q^{7} - 24 q^{8} + 27 q^{9} + 10 q^{10} + 77 q^{11} - 36 q^{12} + 44 q^{13} + 22 q^{14} + 15 q^{15} + 48 q^{16} + 45 q^{17} - 54 q^{18} - 20 q^{20} + 33 q^{21} - 154 q^{22} + 20 q^{23} + 72 q^{24} + 282 q^{25} - 88 q^{26} - 81 q^{27} - 44 q^{28} + 272 q^{29} - 30 q^{30} + 64 q^{31} - 96 q^{32} - 231 q^{33} - 90 q^{34} + 439 q^{35} + 108 q^{36} - 280 q^{37} - 132 q^{39} + 40 q^{40} - 32 q^{41} - 66 q^{42} + 173 q^{43} + 308 q^{44} - 45 q^{45} - 40 q^{46} + 507 q^{47} - 144 q^{48} + 52 q^{49} - 564 q^{50} - 135 q^{51} + 176 q^{52} - 208 q^{53} + 162 q^{54} - 93 q^{55} + 88 q^{56} - 544 q^{58} + 344 q^{59} + 60 q^{60} + 847 q^{61} - 128 q^{62} - 99 q^{63} + 192 q^{64} - 2440 q^{65} + 462 q^{66} + 268 q^{67} + 180 q^{68} - 60 q^{69} - 878 q^{70} + 384 q^{71} - 216 q^{72} + 225 q^{73} + 560 q^{74} - 846 q^{75} - 2563 q^{77} + 264 q^{78} - 576 q^{79} - 80 q^{80} + 243 q^{81} + 64 q^{82} + 1492 q^{83} + 132 q^{84} + 2099 q^{85} - 346 q^{86} - 816 q^{87} - 616 q^{88} + 816 q^{89} + 90 q^{90} - 2464 q^{91} + 80 q^{92} - 192 q^{93} - 1014 q^{94} + 288 q^{96} + 4008 q^{97} - 104 q^{98} + 693 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 - 5 * q^5 + 18 * q^6 - 11 * q^7 - 24 * q^8 + 27 * q^9 + 10 * q^10 + 77 * q^11 - 36 * q^12 + 44 * q^13 + 22 * q^14 + 15 * q^15 + 48 * q^16 + 45 * q^17 - 54 * q^18 - 20 * q^20 + 33 * q^21 - 154 * q^22 + 20 * q^23 + 72 * q^24 + 282 * q^25 - 88 * q^26 - 81 * q^27 - 44 * q^28 + 272 * q^29 - 30 * q^30 + 64 * q^31 - 96 * q^32 - 231 * q^33 - 90 * q^34 + 439 * q^35 + 108 * q^36 - 280 * q^37 - 132 * q^39 + 40 * q^40 - 32 * q^41 - 66 * q^42 + 173 * q^43 + 308 * q^44 - 45 * q^45 - 40 * q^46 + 507 * q^47 - 144 * q^48 + 52 * q^49 - 564 * q^50 - 135 * q^51 + 176 * q^52 - 208 * q^53 + 162 * q^54 - 93 * q^55 + 88 * q^56 - 544 * q^58 + 344 * q^59 + 60 * q^60 + 847 * q^61 - 128 * q^62 - 99 * q^63 + 192 * q^64 - 2440 * q^65 + 462 * q^66 + 268 * q^67 + 180 * q^68 - 60 * q^69 - 878 * q^70 + 384 * q^71 - 216 * q^72 + 225 * q^73 + 560 * q^74 - 846 * q^75 - 2563 * q^77 + 264 * q^78 - 576 * q^79 - 80 * q^80 + 243 * q^81 + 64 * q^82 + 1492 * q^83 + 132 * q^84 + 2099 * q^85 - 346 * q^86 - 816 * q^87 - 616 * q^88 + 816 * q^89 + 90 * q^90 - 2464 * q^91 + 80 * q^92 - 192 * q^93 - 1014 * q^94 + 288 * q^96 + 4008 * q^97 - 104 * q^98 + 693 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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