LMFDB - Embedded newform 2166.4.a.v.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	$-0.899287\pi$
$13$	−89.0910	−1.90072	−0.950362	+	0.311147i	$0.899287\pi$
$14$	−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	$-0.873852\pi$
$29$	−288.131	−1.84499	−0.922493	+	0.386014i	$0.873852\pi$
$30$	−110.233	−0.670857
$31$	7.47726	0.0433211	0.0216606	−	0.999765i	$-0.493105\pi$
$31$	7.47726	0.0433211	0.0216606	+	0.999765i	$0.493105\pi$
$32$	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	$-0.486624\pi$
$37$	18.9090	0.0840166	0.0420083	+	0.999117i	$0.486624\pi$
$38$	0	0
$39$	−267.273	−1.09738
$40$	−146.977	−0.580980
$41$	−129.177	−0.492048	−0.246024	−	0.969264i	$-0.579124\pi$
$41$	−129.177	−0.492048	−0.246024	+	0.969264i	$0.579124\pi$
$42$	−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	$-0.404880\pi$
$53$	227.188	0.588804	0.294402	+	0.955682i	$0.404880\pi$
$54$	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	$-0.203831\pi$
$59$	726.808	1.60377	0.801885	+	0.597479i	$0.203831\pi$
$60$	−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	$-0.334253\pi$
$67$	545.672	0.994991	0.497496	+	0.867466i	$0.334253\pi$
$68$	−110.057	−0.196270
$69$	−239.881	−0.418526
$70$	1027.90	1.75512
$71$	136.091	0.227479	0.113740	−	0.993511i	$-0.463717\pi$
$71$	136.091	0.227479	0.113740	+	0.993511i	$0.463717\pi$
$72$	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	$-0.440629\pi$
$79$	260.421	0.370881	0.185441	+	0.982655i	$0.440629\pi$
$80$	−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	$-0.621987\pi$
$89$	−627.905	−0.747840	−0.373920	+	0.927461i	$0.621987\pi$
$90$	−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	$-0.693226\pi$
$97$	−1089.92	−1.14087	−0.570436	+	0.821342i	$0.693226\pi$
$98$	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	$-0.592680\pi$
$103$	−600.162	−0.574133	−0.287066	+	0.957911i	$0.592680\pi$
$104$	−712.728	−0.672007
$105$	1541.86	1.43305
$106$	454.375	0.416347
$107$	−1023.13	−0.924388	−0.462194	−	0.886779i	$-0.652938\pi$
$107$	−1023.13	−0.924388	−0.462194	+	0.886779i	$0.652938\pi$
$108$	108.000	0.0962250
$109$	1580.94	1.38924	0.694620	−	0.719377i	$-0.255573\pi$
$109$	1580.94	1.38924	0.694620	+	0.719377i	$0.255573\pi$
$110$	−2394.96	−2.07592
$111$	56.7269	0.0485070
$112$	−447.592	−0.377620
$113$	1552.87	1.29276	0.646380	−	0.763016i	$-0.276282\pi$
$113$	1552.87	1.29276	0.646380	+	0.763016i	$0.276282\pi$
$114$	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	$-0.410900\pi$
$127$	790.821	0.552551	0.276275	+	0.961078i	$0.410900\pi$
$128$	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	$-0.681360\pi$
$151$	−2001.84	−1.07886	−0.539430	+	0.842031i	$0.681360\pi$
$152$	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	$-0.446922\pi$
$167$	716.397	0.331955	0.165977	+	0.986130i	$0.446922\pi$
$168$	−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	$-0.343440\pi$
$173$	2149.19	0.944508	0.472254	+	0.881462i	$0.343440\pi$
$174$	−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	$-0.276250\pi$
$179$	3096.34	1.29291	0.646457	+	0.762951i	$0.276250\pi$
$180$	−661.399	−0.273876
$181$	−635.770	−0.261085	−0.130542	−	0.991443i	$-0.541672\pi$
$181$	−635.770	−0.261085	−0.130542	+	0.991443i	$0.541672\pi$
$182$	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	$-0.302351\pi$
$193$	3119.86	1.16359	0.581793	+	0.813337i	$0.302351\pi$
$194$	−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	$-0.430791\pi$
$211$	1322.32	0.431433	0.215717	+	0.976456i	$0.430791\pi$
$212$	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	$-0.594664\pi$
$223$	−1951.65	−0.586065	−0.293032	+	0.956103i	$0.594664\pi$
$224$	−895.184	−0.267018
$225$	1912.84	0.566766
$226$	3105.74	0.914119
$227$	2592.81	0.758110	0.379055	−	0.925374i	$-0.376249\pi$
$227$	2592.81	0.758110	0.379055	+	0.925374i	$0.376249\pi$
$228$	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	$-0.778227\pi$
$241$	−5738.83	−1.53390	−0.766951	+	0.641706i	$0.778227\pi$
$242$	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	$-0.251954\pi$
$257$	5790.72	1.40551	0.702753	+	0.711433i	$0.251954\pi$
$258$	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	$-0.422010\pi$
$269$	2140.39	0.485137	0.242568	+	0.970134i	$0.422010\pi$
$270$	−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	$-0.696338\pi$
$281$	−5449.39	−1.15688	−0.578440	+	0.815725i	$0.696338\pi$
$282$	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	$-0.531459\pi$
$293$	−989.750	−0.197344	−0.0986720	+	0.995120i	$0.531459\pi$
$294$	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	$-0.725229\pi$
$307$	−6992.74	−1.29999	−0.649994	+	0.759939i	$0.725229\pi$
$308$	−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	$-0.519022\pi$
$317$	−674.175	−0.119449	−0.0597247	+	0.998215i	$0.519022\pi$
$318$	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	$-0.709158\pi$
$331$	−7356.68	−1.22163	−0.610815	+	0.791773i	$0.709158\pi$
$332$	−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	$-0.549501\pi$
$337$	−1916.40	−0.309771	−0.154885	+	0.987932i	$0.549501\pi$
$338$	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	$-0.494035\pi$
$373$	269.962	0.0374748	0.0187374	+	0.999824i	$0.494035\pi$
$374$	−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	$-0.434366\pi$
$379$	3021.23	0.409473	0.204736	+	0.978817i	$0.434366\pi$
$380$	0	0
$381$	2372.46	0.319015
$382$	2011.77	0.269454
$383$	−1804.33	−0.240723	−0.120361	−	0.992730i	$-0.538405\pi$
$383$	−1804.33	−0.240723	−0.120361	+	0.992730i	$0.538405\pi$
$384$	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	$-0.518578\pi$
$401$	−936.825	−0.116665	−0.0583327	+	0.998297i	$0.518578\pi$
$402$	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	$-0.455237\pi$
$409$	2318.72	0.280326	0.140163	+	0.990128i	$0.455237\pi$
$410$	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	$-0.654634\pi$
$421$	−8066.56	−0.933824	−0.466912	+	0.884304i	$0.654634\pi$
$422$	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	$-0.714636\pi$
$431$	−11173.1	−1.24870	−0.624349	+	0.781145i	$0.714636\pi$
$432$	432.000	0.0481125
$433$	−2150.59	−0.238686	−0.119343	−	0.992853i	$-0.538079\pi$
$433$	−2150.59	−0.238686	−0.119343	+	0.992853i	$0.538079\pi$
$434$	−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	$-0.139823\pi$
$439$	16649.7	1.81013	0.905063	+	0.425277i	$0.139823\pi$
$440$	−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	$-0.107270\pi$
$449$	17958.0	1.88750	0.943752	+	0.330655i	$0.107270\pi$
$450$	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	$-0.725551\pi$
$487$	−13987.7	−1.30153	−0.650763	+	0.759281i	$0.725551\pi$
$488$	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	$-0.229592\pi$
$509$	17247.4	1.50192	0.750958	+	0.660350i	$0.229592\pi$
$510$	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	$-0.178650\pi$
$521$	20135.5	1.69319	0.846593	+	0.532240i	$0.178650\pi$
$522$	−5186.36	−0.434867
$523$	−9426.66	−0.788143	−0.394072	−	0.919080i	$-0.628934\pi$
$523$	−9426.66	−0.788143	−0.394072	+	0.919080i	$0.628934\pi$
$524$	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	$-0.706616\pi$
$547$	−15466.4	−1.20895	−0.604473	+	0.796625i	$0.706616\pi$
$548$	−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	$-0.358706\pi$
$563$	11473.8	0.858908	0.429454	+	0.903089i	$0.358706\pi$
$564$	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	$-0.571454\pi$
$569$	−6042.56	−0.445197	−0.222599	+	0.974910i	$0.571454\pi$
$570$	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	$-0.374553\pi$
$599$	11258.5	0.767963	0.383981	+	0.923341i	$0.374553\pi$
$600$	5100.89	0.347072
$601$	−4693.83	−0.318578	−0.159289	−	0.987232i	$-0.550920\pi$
$601$	−4693.83	−0.318578	−0.159289	+	0.987232i	$0.550920\pi$
$602$	−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	$-0.433284\pi$
$607$	6223.14	0.416127	0.208064	+	0.978115i	$0.433284\pi$
$608$	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	$-0.749644\pi$
$641$	−22925.4	−1.41263	−0.706316	+	0.707896i	$0.749644\pi$
$642$	−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	$-0.660650\pi$
$659$	−16360.3	−0.967082	−0.483541	+	0.875322i	$0.660650\pi$
$660$	−14369.8	−0.847490
$661$	−14548.3	−0.856074	−0.428037	−	0.903761i	$-0.640795\pi$
$661$	−14548.3	−0.856074	−0.428037	+	0.903761i	$0.640795\pi$
$662$	−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	$-0.773717\pi$
$673$	−26460.4	−1.51556	−0.757781	+	0.652509i	$0.773717\pi$
$674$	−3832.79	−0.219041
$675$	5738.51	0.327222
$676$	22960.9	1.30638
$677$	−16458.2	−0.934327	−0.467163	−	0.884171i	$-0.654724\pi$
$677$	−16458.2	−0.934327	−0.467163	+	0.884171i	$0.654724\pi$
$678$	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	$-0.490064\pi$
$683$	1114.16	0.0624189	0.0312095	+	0.999513i	$0.490064\pi$
$684$	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	$-0.649831\pi$
$743$	−18369.9	−0.907034	−0.453517	+	0.891248i	$0.649831\pi$
$744$	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	$-0.575825\pi$
$751$	−9712.59	−0.471927	−0.235964	+	0.971762i	$0.575825\pi$
$752$	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	$-0.547224\pi$
$773$	−6353.52	−0.295628	−0.147814	+	0.989015i	$0.547224\pi$
$774$	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	$-0.309804\pi$
$787$	24842.0	1.12519	0.562593	+	0.826734i	$0.309804\pi$
$788$	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	$-0.237151\pi$
$797$	33078.4	1.47013	0.735067	+	0.677995i	$0.237151\pi$
$798$	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	$-0.716705\pi$
$811$	−29073.5	−1.25883	−0.629414	+	0.777070i	$0.716705\pi$
$812$	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	$-0.721750\pi$
$827$	−30520.1	−1.28330	−0.641650	+	0.766998i	$0.721750\pi$
$828$	−2878.58	−0.120818
$829$	29677.4	1.24335	0.621675	−	0.783275i	$-0.286452\pi$
$829$	29677.4	1.24335	0.621675	+	0.783275i	$0.286452\pi$
$830$	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	$-0.302250\pi$
$839$	28290.1	1.16410	0.582051	+	0.813152i	$0.302250\pi$
$840$	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	$-0.381479\pi$
$857$	18254.3	0.727602	0.363801	+	0.931477i	$0.381479\pi$
$858$	−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	$-0.506646\pi$
$863$	−1058.61	−0.0417562	−0.0208781	+	0.999782i	$0.506646\pi$
$864$	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	$-0.422533\pi$
$877$	12516.9	0.481946	0.240973	+	0.970532i	$0.422533\pi$
$878$	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	$-0.783973\pi$
$887$	−41126.7	−1.55682	−0.778410	+	0.627757i	$0.783973\pi$
$888$	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	$-0.504990\pi$
$907$	−856.416	−0.0313526	−0.0156763	+	0.999877i	$0.504990\pi$
$908$	10371.3	0.379055
$909$	−8140.71	−0.297041
$910$	−91577.1	−3.33599
$911$	1594.94	0.0580053	0.0290027	−	0.999579i	$-0.490767\pi$
$911$	1594.94	0.0580053	0.0290027	+	0.999579i	$0.490767\pi$
$912$	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	$-0.598783\pi$
$941$	−17630.1	−0.610759	−0.305380	+	0.952231i	$0.598783\pi$
$942$	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	$-0.362258\pi$
$953$	24674.4	0.838700	0.419350	+	0.907825i	$0.362258\pi$
$954$	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	$-0.449359\pi$
$971$	9586.90	0.316847	0.158423	+	0.987371i	$0.449359\pi$
$972$	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	$-0.457040\pi$
$977$	8218.01	0.269107	0.134553	+	0.990906i	$0.457040\pi$
$978$	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	$-0.375286\pi$
$983$	23537.3	0.763705	0.381852	+	0.924223i	$0.375286\pi$
$984$	−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	$-0.294685\pi$
$991$	37511.7	1.20242	0.601211	+	0.799091i	$0.294685\pi$
$992$	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.v.1.1	yes	3
19.18	odd	2		2166.4.a.r.1.1	✓	3

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
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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