LMFDB - Embedded newform 2166.4.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2166.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$127.798137072$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{313}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 78 $ x^2 - x - 78
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$9.34590$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -16.3459 q^{5} -6.00000 q^{6} +31.0377 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -16.3459 q^{5} -6.00000 q^{6} +31.0377 q^{7} +8.00000 q^{8} +9.00000 q^{9} -32.6918 q^{10} +4.34590 q^{11} -12.0000 q^{12} -24.0000 q^{13} +62.0754 q^{14} +49.0377 q^{15} +16.0000 q^{16} -6.42132 q^{17} +18.0000 q^{18} -65.3836 q^{20} -93.1131 q^{21} +8.69181 q^{22} -157.459 q^{23} -24.0000 q^{24} +142.189 q^{25} -48.0000 q^{26} -27.0000 q^{27} +124.151 q^{28} +248.075 q^{29} +98.0754 q^{30} -308.302 q^{31} +32.0000 q^{32} -13.0377 q^{33} -12.8426 q^{34} -507.339 q^{35} +36.0000 q^{36} +84.0000 q^{37} +72.0000 q^{39} -130.767 q^{40} +152.075 q^{41} -186.226 q^{42} -481.566 q^{43} +17.3836 q^{44} -147.113 q^{45} -314.918 q^{46} +426.874 q^{47} -48.0000 q^{48} +620.339 q^{49} +284.377 q^{50} +19.2640 q^{51} -96.0000 q^{52} +184.377 q^{53} -54.0000 q^{54} -71.0377 q^{55} +248.302 q^{56} +496.151 q^{58} +584.754 q^{59} +196.151 q^{60} -291.566 q^{61} -616.603 q^{62} +279.339 q^{63} +64.0000 q^{64} +392.302 q^{65} -26.0754 q^{66} +420.453 q^{67} -25.6853 q^{68} +472.377 q^{69} -1014.68 q^{70} -1088.30 q^{71} +72.0000 q^{72} -628.887 q^{73} +168.000 q^{74} -426.566 q^{75} +134.887 q^{77} +144.000 q^{78} -197.056 q^{79} -261.534 q^{80} +81.0000 q^{81} +304.151 q^{82} +519.597 q^{83} -372.453 q^{84} +104.962 q^{85} -963.131 q^{86} -744.226 q^{87} +34.7672 q^{88} +992.980 q^{89} -294.226 q^{90} -744.905 q^{91} -629.836 q^{92} +924.905 q^{93} +853.748 q^{94} -96.0000 q^{96} -1049.06 q^{97} +1240.68 q^{98} +39.1131 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} - 15 q^{5} - 12 q^{6} + 9 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 - 15 * q^5 - 12 * q^6 + 9 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} - 15 q^{5} - 12 q^{6} + 9 q^{7} + 16 q^{8} + 18 q^{9} - 30 q^{10} - 9 q^{11} - 24 q^{12} - 48 q^{13} + 18 q^{14} + 45 q^{15} + 32 q^{16} + 111 q^{17} + 36 q^{18} - 60 q^{20} - 27 q^{21} - 18 q^{22} - 138 q^{23} - 48 q^{24} + 19 q^{25} - 96 q^{26} - 54 q^{27} + 36 q^{28} + 390 q^{29} + 90 q^{30} - 192 q^{31} + 64 q^{32} + 27 q^{33} + 222 q^{34} - 537 q^{35} + 72 q^{36} + 168 q^{37} + 144 q^{39} - 120 q^{40} + 198 q^{41} - 54 q^{42} - 167 q^{43} - 36 q^{44} - 135 q^{45} - 276 q^{46} + 93 q^{47} - 96 q^{48} + 763 q^{49} + 38 q^{50} - 333 q^{51} - 192 q^{52} - 162 q^{53} - 108 q^{54} - 89 q^{55} + 72 q^{56} + 780 q^{58} + 108 q^{59} + 180 q^{60} + 213 q^{61} - 384 q^{62} + 81 q^{63} + 128 q^{64} + 360 q^{65} + 54 q^{66} + 204 q^{67} + 444 q^{68} + 414 q^{69} - 1074 q^{70} - 1752 q^{71} + 144 q^{72} - 1417 q^{73} + 336 q^{74} - 57 q^{75} + 429 q^{77} + 288 q^{78} + 1092 q^{79} - 240 q^{80} + 162 q^{81} + 396 q^{82} - 270 q^{83} - 108 q^{84} + 263 q^{85} - 334 q^{86} - 1170 q^{87} - 72 q^{88} + 606 q^{89} - 270 q^{90} - 216 q^{91} - 552 q^{92} + 576 q^{93} + 186 q^{94} - 192 q^{96} - 612 q^{97} + 1526 q^{98} - 81 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 - 15 * q^5 - 12 * q^6 + 9 * q^7 + 16 * q^8 + 18 * q^9 - 30 * q^10 - 9 * q^11 - 24 * q^12 - 48 * q^13 + 18 * q^14 + 45 * q^15 + 32 * q^16 + 111 * q^17 + 36 * q^18 - 60 * q^20 - 27 * q^21 - 18 * q^22 - 138 * q^23 - 48 * q^24 + 19 * q^25 - 96 * q^26 - 54 * q^27 + 36 * q^28 + 390 * q^29 + 90 * q^30 - 192 * q^31 + 64 * q^32 + 27 * q^33 + 222 * q^34 - 537 * q^35 + 72 * q^36 + 168 * q^37 + 144 * q^39 - 120 * q^40 + 198 * q^41 - 54 * q^42 - 167 * q^43 - 36 * q^44 - 135 * q^45 - 276 * q^46 + 93 * q^47 - 96 * q^48 + 763 * q^49 + 38 * q^50 - 333 * q^51 - 192 * q^52 - 162 * q^53 - 108 * q^54 - 89 * q^55 + 72 * q^56 + 780 * q^58 + 108 * q^59 + 180 * q^60 + 213 * q^61 - 384 * q^62 + 81 * q^63 + 128 * q^64 + 360 * q^65 + 54 * q^66 + 204 * q^67 + 444 * q^68 + 414 * q^69 - 1074 * q^70 - 1752 * q^71 + 144 * q^72 - 1417 * q^73 + 336 * q^74 - 57 * q^75 + 429 * q^77 + 288 * q^78 + 1092 * q^79 - 240 * q^80 + 162 * q^81 + 396 * q^82 - 270 * q^83 - 108 * q^84 + 263 * q^85 - 334 * q^86 - 1170 * q^87 - 72 * q^88 + 606 * q^89 - 270 * q^90 - 216 * q^91 - 552 * q^92 + 576 * q^93 + 186 * q^94 - 192 * q^96 - 612 * q^97 + 1526 * q^98 - 81 * 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
$5$	−16.3459	−1.46202	−0.731011	−	0.682366i	$-0.760951\pi$
$5$	−16.3459	−1.46202	−0.731011	+	0.682366i	$0.760951\pi$
$6$	−6.00000	−0.408248
$7$	31.0377	1.67588	0.837939	−	0.545763i	$-0.183760\pi$
$7$	31.0377	1.67588	0.837939	+	0.545763i	$0.183760\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−32.6918	−1.03381
$11$	4.34590	0.119122	0.0595609	−	0.998225i	$-0.481030\pi$
$11$	4.34590	0.119122	0.0595609	+	0.998225i	$0.481030\pi$
$12$	−12.0000	−0.288675
$13$	−24.0000	−0.512031	−0.256015	−	0.966673i	$-0.582410\pi$
$13$	−24.0000	−0.512031	−0.256015	+	0.966673i	$0.582410\pi$
$14$	62.0754	1.18503
$15$	49.0377	0.844099
$16$	16.0000	0.250000
$17$	−6.42132	−0.0916117	−0.0458059	−	0.998950i	$-0.514586\pi$
$17$	−6.42132	−0.0916117	−0.0458059	+	0.998950i	$0.514586\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	−65.3836	−0.731011
$21$	−93.1131	−0.967569
$22$	8.69181	0.0842318
$23$	−157.459	−1.42750	−0.713750	−	0.700401i	$-0.753005\pi$
$23$	−157.459	−1.42750	−0.713750	+	0.700401i	$0.753005\pi$
$24$	−24.0000	−0.204124
$25$	142.189	1.13751
$26$	−48.0000	−0.362061
$27$	−27.0000	−0.192450
$28$	124.151	0.837939
$29$	248.075	1.58850	0.794249	−	0.607592i	$-0.207864\pi$
$29$	248.075	1.58850	0.794249	+	0.607592i	$0.207864\pi$
$30$	98.0754	0.596868
$31$	−308.302	−1.78621	−0.893107	−	0.449845i	$-0.851479\pi$
$31$	−308.302	−1.78621	−0.893107	+	0.449845i	$0.851479\pi$
$32$	32.0000	0.176777
$33$	−13.0377	−0.0687750
$34$	−12.8426	−0.0647793
$35$	−507.339	−2.45017
$36$	36.0000	0.166667
$37$	84.0000	0.373230	0.186615	−	0.982433i	$-0.440248\pi$
$37$	84.0000	0.373230	0.186615	+	0.982433i	$0.440248\pi$
$38$	0	0
$39$	72.0000	0.295621
$40$	−130.767	−0.516903
$41$	152.075	0.579273	0.289636	−	0.957137i	$-0.406466\pi$
$41$	152.075	0.579273	0.289636	+	0.957137i	$0.406466\pi$
$42$	−186.226	−0.684175
$43$	−481.566	−1.70786	−0.853931	−	0.520386i	$-0.825788\pi$
$43$	−481.566	−1.70786	−0.853931	+	0.520386i	$0.825788\pi$
$44$	17.3836	0.0595609
$45$	−147.113	−0.487341
$46$	−314.918	−1.00939
$47$	426.874	1.32481	0.662404	−	0.749147i	$-0.269536\pi$
$47$	426.874	1.32481	0.662404	+	0.749147i	$0.269536\pi$
$48$	−48.0000	−0.144338
$49$	620.339	1.80857
$50$	284.377	0.804340
$51$	19.2640	0.0528920
$52$	−96.0000	−0.256015
$53$	184.377	0.477852	0.238926	−	0.971038i	$-0.423205\pi$
$53$	184.377	0.477852	0.238926	+	0.971038i	$0.423205\pi$
$54$	−54.0000	−0.136083
$55$	−71.0377	−0.174159
$56$	248.302	0.592513
$57$	0	0
$58$	496.151	1.12324
$59$	584.754	1.29031	0.645157	−	0.764050i	$-0.276792\pi$
$59$	584.754	1.29031	0.645157	+	0.764050i	$0.276792\pi$
$60$	196.151	0.422049
$61$	−291.566	−0.611986	−0.305993	−	0.952034i	$-0.598988\pi$
$61$	−291.566	−0.611986	−0.305993	+	0.952034i	$0.598988\pi$
$62$	−616.603	−1.26304
$63$	279.339	0.558626
$64$	64.0000	0.125000
$65$	392.302	0.748601
$66$	−26.0754	−0.0486312
$67$	420.453	0.766663	0.383332	−	0.923611i	$-0.374777\pi$
$67$	420.453	0.766663	0.383332	+	0.923611i	$0.374777\pi$
$68$	−25.6853	−0.0458059
$69$	472.377	0.824167
$70$	−1014.68	−1.73253
$71$	−1088.30	−1.81912	−0.909561	−	0.415571i	$-0.863582\pi$
$71$	−1088.30	−1.81912	−0.909561	+	0.415571i	$0.863582\pi$
$72$	72.0000	0.117851
$73$	−628.887	−1.00830	−0.504148	−	0.863617i	$-0.668193\pi$
$73$	−628.887	−1.00830	−0.504148	+	0.863617i	$0.668193\pi$
$74$	168.000	0.263914
$75$	−426.566	−0.656741
$76$	0	0
$77$	134.887	0.199634
$78$	144.000	0.209036
$79$	−197.056	−0.280639	−0.140320	−	0.990106i	$-0.544813\pi$
$79$	−197.056	−0.280639	−0.140320	+	0.990106i	$0.544813\pi$
$80$	−261.534	−0.365506
$81$	81.0000	0.111111
$82$	304.151	0.409608
$83$	519.597	0.687147	0.343573	−	0.939126i	$-0.388363\pi$
$83$	519.597	0.687147	0.343573	+	0.939126i	$0.388363\pi$
$84$	−372.453	−0.483785
$85$	104.962	0.133938
$86$	−963.131	−1.20764
$87$	−744.226	−0.917120
$88$	34.7672	0.0421159
$89$	992.980	1.18265	0.591324	−	0.806434i	$-0.298605\pi$
$89$	992.980	1.18265	0.591324	+	0.806434i	$0.298605\pi$
$90$	−294.226	−0.344602
$91$	−744.905	−0.858102
$92$	−629.836	−0.713750
$93$	924.905	1.03127
$94$	853.748	0.936780
$95$	0	0
$96$	−96.0000	−0.102062
$97$	−1049.06	−1.09810	−0.549049	−	0.835790i	$-0.685010\pi$
$97$	−1049.06	−1.09810	−0.549049	+	0.835790i	$0.685010\pi$
$98$	1240.68	1.27885
$99$	39.1131	0.0397072
$100$	568.754	0.568754
$101$	−280.767	−0.276608	−0.138304	−	0.990390i	$-0.544165\pi$
$101$	−280.767	−0.276608	−0.138304	+	0.990390i	$0.544165\pi$
$102$	38.5279	0.0374003
$103$	1546.11	1.47906	0.739529	−	0.673125i	$-0.235048\pi$
$103$	1546.11	1.47906	0.739529	+	0.673125i	$0.235048\pi$
$104$	−192.000	−0.181030
$105$	1522.02	1.41461
$106$	368.754	0.337892
$107$	−1584.45	−1.43154	−0.715770	−	0.698336i	$-0.753924\pi$
$107$	−1584.45	−1.43154	−0.715770	+	0.698336i	$0.753924\pi$
$108$	−108.000	−0.0962250
$109$	−603.246	−0.530096	−0.265048	−	0.964235i	$-0.585388\pi$
$109$	−603.246	−0.530096	−0.265048	+	0.964235i	$0.585388\pi$
$110$	−142.075	−0.123149
$111$	−252.000	−0.215485
$112$	496.603	0.418970
$113$	−770.036	−0.641052	−0.320526	−	0.947240i	$-0.603860\pi$
$113$	−770.036	−0.641052	−0.320526	+	0.947240i	$0.603860\pi$
$114$	0	0
$115$	2573.81	2.08703
$116$	992.302	0.794249
$117$	−216.000	−0.170677
$118$	1169.51	0.912390
$119$	−199.303	−0.153530
$120$	392.302	0.298434
$121$	−1312.11	−0.985810
$122$	−583.131	−0.432740
$123$	−456.226	−0.334443
$124$	−1233.21	−0.893107
$125$	−280.962	−0.201040
$126$	558.679	0.395008
$127$	−2734.11	−1.91034	−0.955170	−	0.296058i	$-0.904328\pi$
$127$	−2734.11	−1.91034	−0.955170	+	0.296058i	$0.904328\pi$
$128$	128.000	0.0883883
$129$	1444.70	0.986035
$130$	784.603	0.529341
$131$	−575.743	−0.383991	−0.191996	−	0.981396i	$-0.561496\pi$
$131$	−575.743	−0.383991	−0.191996	+	0.981396i	$0.561496\pi$
$132$	−52.1508	−0.0343875
$133$	0	0
$134$	840.905	0.542113
$135$	441.339	0.281366
$136$	−51.3706	−0.0323896
$137$	2126.46	1.32610	0.663049	−	0.748576i	$-0.269262\pi$
$137$	2126.46	1.32610	0.663049	+	0.748576i	$0.269262\pi$
$138$	944.754	0.582774
$139$	−1747.22	−1.06617	−0.533085	−	0.846062i	$-0.678967\pi$
$139$	−1747.22	−1.06617	−0.533085	+	0.846062i	$0.678967\pi$
$140$	−2029.36	−1.22509
$141$	−1280.62	−0.764878
$142$	−2176.60	−1.28631
$143$	−104.302	−0.0609940
$144$	144.000	0.0833333
$145$	−4055.02	−2.32242
$146$	−1257.77	−0.712973
$147$	−1861.02	−1.04418
$148$	336.000	0.186615
$149$	−1171.20	−0.643950	−0.321975	−	0.946748i	$-0.604347\pi$
$149$	−1171.20	−0.643950	−0.321975	+	0.946748i	$0.604347\pi$
$150$	−853.131	−0.464386
$151$	985.810	0.531285	0.265643	−	0.964072i	$-0.414416\pi$
$151$	985.810	0.531285	0.265643	+	0.964072i	$0.414416\pi$
$152$	0	0
$153$	−57.7919	−0.0305372
$154$	269.774	0.141162
$155$	5039.47	2.61148
$156$	288.000	0.147811
$157$	−653.547	−0.332221	−0.166111	−	0.986107i	$-0.553121\pi$
$157$	−653.547	−0.332221	−0.166111	+	0.986107i	$0.553121\pi$
$158$	−394.112	−0.198442
$159$	−553.131	−0.275888
$160$	−523.069	−0.258451
$161$	−4887.17	−2.39232
$162$	162.000	0.0785674
$163$	−3450.41	−1.65802	−0.829010	−	0.559234i	$-0.811095\pi$
$163$	−3450.41	−1.65802	−0.829010	+	0.559234i	$0.811095\pi$
$164$	608.302	0.289636
$165$	213.113	0.100551
$166$	1039.19	0.485886
$167$	−1589.96	−0.736736	−0.368368	−	0.929680i	$-0.620083\pi$
$167$	−1589.96	−0.736736	−0.368368	+	0.929680i	$0.620083\pi$
$168$	−744.905	−0.342087
$169$	−1621.00	−0.737824
$170$	209.925	0.0947087
$171$	0	0
$172$	−1926.26	−0.853931
$173$	2523.47	1.10899	0.554497	−	0.832186i	$-0.312911\pi$
$173$	2523.47	1.10899	0.554497	+	0.832186i	$0.312911\pi$
$174$	−1488.45	−0.648502
$175$	4413.21	1.90633
$176$	69.5344	0.0297804
$177$	−1754.26	−0.744963
$178$	1985.96	0.836259
$179$	1086.41	0.453644	0.226822	−	0.973936i	$-0.427166\pi$
$179$	1086.41	0.453644	0.226822	+	0.973936i	$0.427166\pi$
$180$	−588.453	−0.243670
$181$	3353.51	1.37715	0.688575	−	0.725165i	$-0.258236\pi$
$181$	3353.51	1.37715	0.688575	+	0.725165i	$0.258236\pi$
$182$	−1489.81	−0.606770
$183$	874.697	0.353330
$184$	−1259.67	−0.504697
$185$	−1373.06	−0.545671
$186$	1849.81	0.729219
$187$	−27.9064	−0.0109129
$188$	1707.50	0.662404
$189$	−838.018	−0.322523
$190$	0	0
$191$	−2644.86	−1.00197	−0.500983	−	0.865457i	$-0.667028\pi$
$191$	−2644.86	−1.00197	−0.500983	+	0.865457i	$0.667028\pi$
$192$	−192.000	−0.0721688
$193$	−1440.37	−0.537204	−0.268602	−	0.963251i	$-0.586562\pi$
$193$	−1440.37	−0.537204	−0.268602	+	0.963251i	$0.586562\pi$
$194$	−2098.11	−0.776472
$195$	−1176.91	−0.432205
$196$	2481.36	0.904285
$197$	−1197.37	−0.433041	−0.216521	−	0.976278i	$-0.569471\pi$
$197$	−1197.37	−0.433041	−0.216521	+	0.976278i	$0.569471\pi$
$198$	78.2263	0.0280773
$199$	−3012.66	−1.07318	−0.536588	−	0.843845i	$-0.680287\pi$
$199$	−3012.66	−1.07318	−0.536588	+	0.843845i	$0.680287\pi$
$200$	1137.51	0.402170
$201$	−1261.36	−0.442633
$202$	−561.534	−0.195591
$203$	7699.69	2.66213
$204$	77.0559	0.0264460
$205$	−2485.81	−0.846910
$206$	3092.22	1.04585
$207$	−1417.13	−0.475833
$208$	−384.000	−0.128008
$209$	0	0
$210$	3044.04	1.00028
$211$	−5040.91	−1.64469	−0.822346	−	0.568987i	$-0.807335\pi$
$211$	−5040.91	−1.64469	−0.822346	+	0.568987i	$0.807335\pi$
$212$	737.508	0.238926
$213$	3264.91	1.05027
$214$	−3168.91	−1.01225
$215$	7871.63	2.49693
$216$	−216.000	−0.0680414
$217$	−9568.98	−2.99348
$218$	−1206.49	−0.374835
$219$	1886.66	0.582140
$220$	−284.151	−0.0870793
$221$	154.112	0.0469080
$222$	−504.000	−0.152371
$223$	−29.0559	−0.00872522	−0.00436261	−	0.999990i	$-0.501389\pi$
$223$	−29.0559	−0.00872522	−0.00436261	+	0.999990i	$0.501389\pi$
$224$	993.207	0.296256
$225$	1279.70	0.379169
$226$	−1540.07	−0.453292
$227$	4940.68	1.44460	0.722300	−	0.691580i	$-0.243085\pi$
$227$	4940.68	1.44460	0.722300	+	0.691580i	$0.243085\pi$
$228$	0	0
$229$	−2852.36	−0.823096	−0.411548	−	0.911388i	$-0.635012\pi$
$229$	−2852.36	−0.823096	−0.411548	+	0.911388i	$0.635012\pi$
$230$	5147.62	1.47576
$231$	−404.661	−0.115259
$232$	1984.60	0.561619
$233$	−4659.44	−1.31009	−0.655043	−	0.755591i	$-0.727350\pi$
$233$	−4659.44	−1.31009	−0.655043	+	0.755591i	$0.727350\pi$
$234$	−432.000	−0.120687
$235$	−6977.64	−1.93690
$236$	2339.02	0.645157
$237$	591.168	0.162027
$238$	−398.606	−0.108562
$239$	3902.83	1.05629	0.528145	−	0.849154i	$-0.322888\pi$
$239$	3902.83	1.05629	0.528145	+	0.849154i	$0.322888\pi$
$240$	784.603	0.211025
$241$	−5584.53	−1.49266	−0.746330	−	0.665576i	$-0.768186\pi$
$241$	−5584.53	−1.49266	−0.746330	+	0.665576i	$0.768186\pi$
$242$	−2624.23	−0.697073
$243$	−243.000	−0.0641500
$244$	−1166.26	−0.305993
$245$	−10140.0	−2.64417
$246$	−912.453	−0.236487
$247$	0	0
$248$	−2466.41	−0.631522
$249$	−1558.79	−0.396724
$250$	−561.925	−0.142157
$251$	1943.56	0.488750	0.244375	−	0.969681i	$-0.421417\pi$
$251$	1943.56	0.488750	0.244375	+	0.969681i	$0.421417\pi$
$252$	1117.36	0.279313
$253$	−684.302	−0.170046
$254$	−5468.22	−1.35081
$255$	−314.887	−0.0773293
$256$	256.000	0.0625000
$257$	−2120.45	−0.514669	−0.257335	−	0.966322i	$-0.582844\pi$
$257$	−2120.45	−0.514669	−0.257335	+	0.966322i	$0.582844\pi$
$258$	2889.39	0.697232
$259$	2607.17	0.625488
$260$	1569.21	0.374300
$261$	2232.68	0.529499
$262$	−1151.49	−0.271523
$263$	−5039.09	−1.18146	−0.590729	−	0.806870i	$-0.701160\pi$
$263$	−5039.09	−1.18146	−0.590729	+	0.806870i	$0.701160\pi$
$264$	−104.302	−0.0243156
$265$	−3013.81	−0.698630
$266$	0	0
$267$	−2978.94	−0.682802
$268$	1681.81	0.383332
$269$	−5587.09	−1.26636	−0.633180	−	0.774004i	$-0.718251\pi$
$269$	−5587.09	−1.26636	−0.633180	+	0.774004i	$0.718251\pi$
$270$	882.679	0.198956
$271$	−2707.55	−0.606907	−0.303454	−	0.952846i	$-0.598140\pi$
$271$	−2707.55	−0.606907	−0.303454	+	0.952846i	$0.598140\pi$
$272$	−102.741	−0.0229029
$273$	2234.72	0.495425
$274$	4252.92	0.937693
$275$	617.938	0.135502
$276$	1889.51	0.412083
$277$	−2841.87	−0.616431	−0.308216	−	0.951317i	$-0.599732\pi$
$277$	−2841.87	−0.616431	−0.308216	+	0.951317i	$0.599732\pi$
$278$	−3494.45	−0.753896
$279$	−2774.72	−0.595404
$280$	−4058.72	−0.866266
$281$	−2365.13	−0.502106	−0.251053	−	0.967973i	$-0.580777\pi$
$281$	−2365.13	−0.502106	−0.251053	+	0.967973i	$0.580777\pi$
$282$	−2561.24	−0.540850
$283$	3582.43	0.752486	0.376243	−	0.926521i	$-0.377216\pi$
$283$	3582.43	0.752486	0.376243	+	0.926521i	$0.377216\pi$
$284$	−4353.21	−0.909561
$285$	0	0
$286$	−208.603	−0.0431293
$287$	4720.07	0.970791
$288$	288.000	0.0589256
$289$	−4871.77	−0.991607
$290$	−8110.03	−1.64220
$291$	3147.17	0.633987
$292$	−2515.55	−0.504148
$293$	−2840.45	−0.566351	−0.283175	−	0.959068i	$-0.591388\pi$
$293$	−2840.45	−0.566351	−0.283175	+	0.959068i	$0.591388\pi$
$294$	−3722.04	−0.738345
$295$	−9558.34	−1.88647
$296$	672.000	0.131957
$297$	−117.339	−0.0229250
$298$	−2342.40	−0.455342
$299$	3779.02	0.730924
$300$	−1706.26	−0.328370
$301$	−14946.7	−2.86217
$302$	1971.62	0.375675
$303$	842.302	0.159700
$304$	0	0
$305$	4765.90	0.894737
$306$	−115.584	−0.0215931
$307$	9295.16	1.72802	0.864011	−	0.503472i	$-0.167944\pi$
$307$	9295.16	1.72802	0.864011	+	0.503472i	$0.167944\pi$
$308$	539.547	0.0998168
$309$	−4638.34	−0.853934
$310$	10078.9	1.84660
$311$	1866.72	0.340360	0.170180	−	0.985413i	$-0.445565\pi$
$311$	1866.72	0.340360	0.170180	+	0.985413i	$0.445565\pi$
$312$	576.000	0.104518
$313$	−5516.34	−0.996173	−0.498086	−	0.867127i	$-0.665964\pi$
$313$	−5516.34	−0.996173	−0.498086	+	0.867127i	$0.665964\pi$
$314$	−1307.09	−0.234916
$315$	−4566.05	−0.816724
$316$	−788.223	−0.140320
$317$	−5247.69	−0.929778	−0.464889	−	0.885369i	$-0.653906\pi$
$317$	−5247.69	−0.929778	−0.464889	+	0.885369i	$0.653906\pi$
$318$	−1106.26	−0.195082
$319$	1078.11	0.189225
$320$	−1046.14	−0.182753
$321$	4753.36	0.826500
$322$	−9774.34	−1.69162
$323$	0	0
$324$	324.000	0.0555556
$325$	−3412.53	−0.582439
$326$	−6900.83	−1.17240
$327$	1809.74	0.306051
$328$	1216.60	0.204804
$329$	13249.2	2.22022
$330$	426.226	0.0710999
$331$	−3649.89	−0.606091	−0.303045	−	0.952976i	$-0.598003\pi$
$331$	−3649.89	−0.606091	−0.303045	+	0.952976i	$0.598003\pi$
$332$	2078.39	0.343573
$333$	756.000	0.124410
$334$	−3179.92	−0.520951
$335$	−6872.68	−1.12088
$336$	−1489.81	−0.241892
$337$	5598.04	0.904880	0.452440	−	0.891795i	$-0.350554\pi$
$337$	5598.04	0.904880	0.452440	+	0.891795i	$0.350554\pi$
$338$	−3242.00	−0.521721
$339$	2310.11	0.370112
$340$	419.849	0.0669692
$341$	−1339.85	−0.212777
$342$	0	0
$343$	8607.98	1.35506
$344$	−3852.53	−0.603820
$345$	−7721.43	−1.20495
$346$	5046.94	0.784177
$347$	−12868.1	−1.99077	−0.995386	−	0.0959483i	$-0.969412\pi$
$347$	−12868.1	−1.99077	−0.995386	+	0.0959483i	$0.969412\pi$
$348$	−2976.91	−0.458560
$349$	−1940.05	−0.297561	−0.148780	−	0.988870i	$-0.547535\pi$
$349$	−1940.05	−0.297561	−0.148780	+	0.988870i	$0.547535\pi$
$350$	8826.41	1.34798
$351$	648.000	0.0985404
$352$	139.069	0.0210579
$353$	596.544	0.0899457	0.0449728	−	0.998988i	$-0.485680\pi$
$353$	596.544	0.0899457	0.0449728	+	0.998988i	$0.485680\pi$
$354$	−3508.53	−0.526768
$355$	17789.3	2.65960
$356$	3971.92	0.591324
$357$	597.909	0.0886407
$358$	2172.83	0.320775
$359$	−2921.84	−0.429552	−0.214776	−	0.976663i	$-0.568902\pi$
$359$	−2921.84	−0.429552	−0.214776	+	0.976663i	$0.568902\pi$
$360$	−1176.91	−0.172301
$361$	0	0
$362$	6707.02	0.973793
$363$	3936.34	0.569158
$364$	−2979.62	−0.429051
$365$	10279.7	1.47415
$366$	1749.39	0.249842
$367$	−4450.79	−0.633050	−0.316525	−	0.948584i	$-0.602516\pi$
$367$	−4450.79	−0.633050	−0.316525	+	0.948584i	$0.602516\pi$
$368$	−2519.34	−0.356875
$369$	1368.68	0.193091
$370$	−2746.11	−0.385847
$371$	5722.64	0.800822
$372$	3699.62	0.515635
$373$	12624.1	1.75242	0.876211	−	0.481928i	$-0.160063\pi$
$373$	12624.1	1.75242	0.876211	+	0.481928i	$0.160063\pi$
$374$	−55.8129	−0.00771662
$375$	842.887	0.116071
$376$	3414.99	0.468390
$377$	−5953.81	−0.813360
$378$	−1676.04	−0.228058
$379$	11622.9	1.57528	0.787639	−	0.616137i	$-0.211303\pi$
$379$	11622.9	1.57528	0.787639	+	0.616137i	$0.211303\pi$
$380$	0	0
$381$	8202.34	1.10294
$382$	−5289.72	−0.708497
$383$	9928.82	1.32465	0.662323	−	0.749219i	$-0.269571\pi$
$383$	9928.82	1.32465	0.662323	+	0.749219i	$0.269571\pi$
$384$	−384.000	−0.0510310
$385$	−2204.85	−0.291869
$386$	−2880.75	−0.379861
$387$	−4334.09	−0.569287
$388$	−4196.22	−0.549049
$389$	7151.20	0.932082	0.466041	−	0.884763i	$-0.345680\pi$
$389$	7151.20	0.932082	0.466041	+	0.884763i	$0.345680\pi$
$390$	−2353.81	−0.305615
$391$	1011.09	0.130776
$392$	4962.72	0.639426
$393$	1727.23	0.221697
$394$	−2394.74	−0.306206
$395$	3221.06	0.410301
$396$	156.453	0.0198536
$397$	12822.1	1.62097	0.810483	−	0.585762i	$-0.199205\pi$
$397$	12822.1	1.62097	0.810483	+	0.585762i	$0.199205\pi$
$398$	−6025.32	−0.758849
$399$	0	0
$400$	2275.02	0.284377
$401$	97.5838	0.0121524	0.00607619	−	0.999982i	$-0.498066\pi$
$401$	97.5838	0.0121524	0.00607619	+	0.999982i	$0.498066\pi$
$402$	−2522.72	−0.312989
$403$	7399.24	0.914597
$404$	−1123.07	−0.138304
$405$	−1324.02	−0.162447
$406$	15399.4	1.88241
$407$	365.056	0.0444598
$408$	154.112	0.0187002
$409$	−2995.55	−0.362153	−0.181076	−	0.983469i	$-0.557958\pi$
$409$	−2995.55	−0.362153	−0.181076	+	0.983469i	$0.557958\pi$
$410$	−4971.62	−0.598856
$411$	−6379.37	−0.765623
$412$	6184.45	0.739529
$413$	18149.4	2.16241
$414$	−2834.26	−0.336465
$415$	−8493.28	−1.00462
$416$	−768.000	−0.0905151
$417$	5241.67	0.615554
$418$	0	0
$419$	1153.01	0.134434	0.0672172	−	0.997738i	$-0.478588\pi$
$419$	1153.01	0.134434	0.0672172	+	0.997738i	$0.478588\pi$
$420$	6088.07	0.707304
$421$	−3850.49	−0.445751	−0.222876	−	0.974847i	$-0.571544\pi$
$421$	−3850.49	−0.445751	−0.222876	+	0.974847i	$0.571544\pi$
$422$	−10081.8	−1.16297
$423$	3841.86	0.441602
$424$	1475.02	0.168946
$425$	−913.038	−0.104209
$426$	6529.81	0.742653
$427$	−9049.53	−1.02561
$428$	−6337.81	−0.715770
$429$	312.905	0.0352149
$430$	15743.3	1.76560
$431$	−5049.96	−0.564380	−0.282190	−	0.959359i	$-0.591061\pi$
$431$	−5049.96	−0.564380	−0.282190	+	0.959359i	$0.591061\pi$
$432$	−432.000	−0.0481125
$433$	9340.74	1.03669	0.518346	−	0.855171i	$-0.326548\pi$
$433$	9340.74	1.03669	0.518346	+	0.855171i	$0.326548\pi$
$434$	−19138.0	−2.11671
$435$	12165.1	1.34085
$436$	−2412.98	−0.265048
$437$	0	0
$438$	3773.32	0.411635
$439$	6445.04	0.700695	0.350348	−	0.936620i	$-0.386063\pi$
$439$	6445.04	0.700695	0.350348	+	0.936620i	$0.386063\pi$
$440$	−568.302	−0.0615744
$441$	5583.05	0.602857
$442$	308.223	0.0331690
$443$	−8463.66	−0.907721	−0.453861	−	0.891073i	$-0.649954\pi$
$443$	−8463.66	−0.907721	−0.453861	+	0.891073i	$0.649954\pi$
$444$	−1008.00	−0.107742
$445$	−16231.2	−1.72906
$446$	−58.1117	−0.00616966
$447$	3513.60	0.371785
$448$	1986.41	0.209485
$449$	1737.96	0.182672	0.0913358	−	0.995820i	$-0.470886\pi$
$449$	1737.96	0.182672	0.0913358	+	0.995820i	$0.470886\pi$
$450$	2559.39	0.268113
$451$	660.905	0.0690040
$452$	−3080.15	−0.320526
$453$	−2957.43	−0.306738
$454$	9881.35	1.02149
$455$	12176.1	1.25456
$456$	0	0
$457$	17723.3	1.81414	0.907069	−	0.420983i	$-0.138315\pi$
$457$	17723.3	1.81414	0.907069	+	0.420983i	$0.138315\pi$
$458$	−5704.71	−0.582017
$459$	173.376	0.0176307
$460$	10295.2	1.04352
$461$	4851.58	0.490153	0.245077	−	0.969504i	$-0.421187\pi$
$461$	4851.58	0.490153	0.245077	+	0.969504i	$0.421187\pi$
$462$	−809.321	−0.0815001
$463$	−10677.1	−1.07173	−0.535863	−	0.844305i	$-0.680014\pi$
$463$	−10677.1	−1.07173	−0.535863	+	0.844305i	$0.680014\pi$
$464$	3969.21	0.397125
$465$	−15118.4	−1.50774
$466$	−9318.88	−0.926371
$467$	−8464.27	−0.838714	−0.419357	−	0.907821i	$-0.637744\pi$
$467$	−8464.27	−0.838714	−0.419357	+	0.907821i	$0.637744\pi$
$468$	−864.000	−0.0853385
$469$	13049.9	1.28483
$470$	−13955.3	−1.36959
$471$	1960.64	0.191808
$472$	4678.03	0.456195
$473$	−2092.84	−0.203443
$474$	1182.34	0.114571
$475$	0	0
$476$	−797.212	−0.0767651
$477$	1659.39	0.159284
$478$	7805.66	0.746910
$479$	−8449.99	−0.806033	−0.403017	−	0.915193i	$-0.632038\pi$
$479$	−8449.99	−0.806033	−0.403017	+	0.915193i	$0.632038\pi$
$480$	1569.21	0.149217
$481$	−2016.00	−0.191105
$482$	−11169.1	−1.05547
$483$	14661.5	1.38120
$484$	−5248.45	−0.492905
$485$	17147.8	1.60544
$486$	−486.000	−0.0453609
$487$	7855.10	0.730901	0.365450	−	0.930831i	$-0.380915\pi$
$487$	7855.10	0.730901	0.365450	+	0.930831i	$0.380915\pi$
$488$	−2332.53	−0.216370
$489$	10351.2	0.957258
$490$	−20280.0	−1.86971
$491$	−5341.38	−0.490943	−0.245472	−	0.969404i	$-0.578943\pi$
$491$	−5341.38	−0.490943	−0.245472	+	0.969404i	$0.578943\pi$
$492$	−1824.91	−0.167222
$493$	−1592.97	−0.145525
$494$	0	0
$495$	−639.339	−0.0580529
$496$	−4932.83	−0.446553
$497$	−33778.4	−3.04863
$498$	−3117.58	−0.280526
$499$	−3157.08	−0.283227	−0.141613	−	0.989922i	$-0.545229\pi$
$499$	−3157.08	−0.283227	−0.141613	+	0.989922i	$0.545229\pi$
$500$	−1123.85	−0.100520
$501$	4769.88	0.425355
$502$	3887.11	0.345598
$503$	−16111.5	−1.42818	−0.714092	−	0.700052i	$-0.753160\pi$
$503$	−16111.5	−1.42818	−0.714092	+	0.700052i	$0.753160\pi$
$504$	2234.72	0.197504
$505$	4589.39	0.404407
$506$	−1368.60	−0.120241
$507$	4863.00	0.425983
$508$	−10936.4	−0.955170
$509$	15652.4	1.36302	0.681512	−	0.731807i	$-0.261323\pi$
$509$	15652.4	1.36302	0.681512	+	0.731807i	$0.261323\pi$
$510$	−629.774	−0.0546801
$511$	−19519.2	−1.68978
$512$	512.000	0.0441942
$513$	0	0
$514$	−4240.90	−0.363926
$515$	−25272.6	−2.16241
$516$	5778.79	0.493017
$517$	1855.15	0.157813
$518$	5214.34	0.442287
$519$	−7570.42	−0.640278
$520$	3138.41	0.264670
$521$	−6090.64	−0.512161	−0.256080	−	0.966655i	$-0.582431\pi$
$521$	−6090.64	−0.512161	−0.256080	+	0.966655i	$0.582431\pi$
$522$	4465.36	0.374413
$523$	4300.68	0.359571	0.179786	−	0.983706i	$-0.442460\pi$
$523$	4300.68	0.359571	0.179786	+	0.983706i	$0.442460\pi$
$524$	−2302.97	−0.191996
$525$	−13239.6	−1.10062
$526$	−10078.2	−0.835417
$527$	1979.70	0.163638
$528$	−208.603	−0.0171937
$529$	12626.3	1.03775
$530$	−6027.62	−0.494006
$531$	5262.79	0.430105
$532$	0	0
$533$	−3649.81	−0.296606
$534$	−5957.88	−0.482814
$535$	25899.3	2.09294
$536$	3363.62	0.271056
$537$	−3259.24	−0.261912
$538$	−11174.2	−0.895452
$539$	2695.93	0.215440
$540$	1765.36	0.140683
$541$	−7444.62	−0.591625	−0.295812	−	0.955246i	$-0.595590\pi$
$541$	−7444.62	−0.591625	−0.295812	+	0.955246i	$0.595590\pi$
$542$	−5415.09	−0.429148
$543$	−10060.5	−0.795098
$544$	−205.482	−0.0161948
$545$	9860.60	0.775012
$546$	4469.43	0.350319
$547$	−10139.6	−0.792576	−0.396288	−	0.918126i	$-0.629702\pi$
$547$	−10139.6	−0.792576	−0.396288	+	0.918126i	$0.629702\pi$
$548$	8505.83	0.663049
$549$	−2624.09	−0.203995
$550$	1235.88	0.0958144
$551$	0	0
$552$	3779.02	0.291387
$553$	−6116.16	−0.470318
$554$	−5683.74	−0.435883
$555$	4119.17	0.315043
$556$	−6988.90	−0.533085
$557$	17922.0	1.36334	0.681669	−	0.731661i	$-0.261254\pi$
$557$	17922.0	1.36334	0.681669	+	0.731661i	$0.261254\pi$
$558$	−5549.43	−0.421015
$559$	11557.6	0.874478
$560$	−8117.43	−0.612543
$561$	83.7193	0.00630059
$562$	−4730.26	−0.355043
$563$	−1620.91	−0.121337	−0.0606687	−	0.998158i	$-0.519323\pi$
$563$	−1620.91	−0.121337	−0.0606687	+	0.998158i	$0.519323\pi$
$564$	−5122.49	−0.382439
$565$	12586.9	0.937233
$566$	7164.87	0.532088
$567$	2514.05	0.186209
$568$	−8706.41	−0.643157
$569$	−4613.42	−0.339903	−0.169951	−	0.985452i	$-0.554361\pi$
$569$	−4613.42	−0.339903	−0.169951	+	0.985452i	$0.554361\pi$
$570$	0	0
$571$	6112.45	0.447983	0.223991	−	0.974591i	$-0.428091\pi$
$571$	6112.45	0.447983	0.223991	+	0.974591i	$0.428091\pi$
$572$	−417.207	−0.0304970
$573$	7934.58	0.578485
$574$	9440.15	0.686453
$575$	−22388.9	−1.62379
$576$	576.000	0.0416667
$577$	−9098.80	−0.656478	−0.328239	−	0.944595i	$-0.606455\pi$
$577$	−9098.80	−0.656478	−0.328239	+	0.944595i	$0.606455\pi$
$578$	−9743.53	−0.701172
$579$	4321.12	0.310155
$580$	−16220.1	−1.16121
$581$	16127.1	1.15157
$582$	6294.34	0.448297
$583$	801.285	0.0569225
$584$	−5031.09	−0.356487
$585$	3530.72	0.249534
$586$	−5680.90	−0.400471
$587$	−402.016	−0.0282675	−0.0141337	−	0.999900i	$-0.504499\pi$
$587$	−402.016	−0.0282675	−0.0141337	+	0.999900i	$0.504499\pi$
$588$	−7444.07	−0.522089
$589$	0	0
$590$	−19116.7	−1.33393
$591$	3592.11	0.250016
$592$	1344.00	0.0933075
$593$	−11044.5	−0.764831	−0.382415	−	0.923991i	$-0.624908\pi$
$593$	−11044.5	−0.764831	−0.382415	+	0.923991i	$0.624908\pi$
$594$	−234.679	−0.0162104
$595$	3257.79	0.224464
$596$	−4684.81	−0.321975
$597$	9037.98	0.619598
$598$	7558.03	0.516841
$599$	12554.3	0.856355	0.428177	−	0.903695i	$-0.359156\pi$
$599$	12554.3	0.856355	0.428177	+	0.903695i	$0.359156\pi$
$600$	−3412.53	−0.232193
$601$	−2024.89	−0.137433	−0.0687164	−	0.997636i	$-0.521890\pi$
$601$	−2024.89	−0.137433	−0.0687164	+	0.997636i	$0.521890\pi$
$602$	−29893.4	−2.02386
$603$	3784.07	0.255554
$604$	3943.24	0.265643
$605$	21447.7	1.44128
$606$	1684.60	0.112925
$607$	23449.2	1.56800	0.783998	−	0.620763i	$-0.213177\pi$
$607$	23449.2	1.56800	0.783998	+	0.620763i	$0.213177\pi$
$608$	0	0
$609$	−23099.1	−1.53698
$610$	9531.81	0.632675
$611$	−10245.0	−0.678342
$612$	−231.168	−0.0152686
$613$	27561.9	1.81601	0.908005	−	0.418958i	$-0.137605\pi$
$613$	27561.9	1.81601	0.908005	+	0.418958i	$0.137605\pi$
$614$	18590.3	1.22190
$615$	7457.43	0.488964
$616$	1079.09	0.0705811
$617$	12506.6	0.816040	0.408020	−	0.912973i	$-0.366219\pi$
$617$	12506.6	0.816040	0.408020	+	0.912973i	$0.366219\pi$
$618$	−9276.67	−0.603823
$619$	5823.54	0.378139	0.189069	−	0.981964i	$-0.439453\pi$
$619$	5823.54	0.378139	0.189069	+	0.981964i	$0.439453\pi$
$620$	20157.9	1.30574
$621$	4251.39	0.274722
$622$	3733.43	0.240671
$623$	30819.8	1.98198
$624$	1152.00	0.0739053
$625$	−13181.0	−0.843583
$626$	−11032.7	−0.704400
$627$	0	0
$628$	−2614.19	−0.166111
$629$	−539.391	−0.0341923
$630$	−9132.11	−0.577511
$631$	−17701.4	−1.11677	−0.558385	−	0.829582i	$-0.688579\pi$
$631$	−17701.4	−1.11677	−0.558385	+	0.829582i	$0.688579\pi$
$632$	−1576.45	−0.0992210
$633$	15122.7	0.949564
$634$	−10495.4	−0.657452
$635$	44691.5	2.79296
$636$	−2212.53	−0.137944
$637$	−14888.1	−0.926044
$638$	2156.22	0.133802
$639$	−9794.72	−0.606374
$640$	−2092.28	−0.129226
$641$	8093.73	0.498726	0.249363	−	0.968410i	$-0.419779\pi$
$641$	8093.73	0.498726	0.249363	+	0.968410i	$0.419779\pi$
$642$	9506.72	0.584424
$643$	3232.10	0.198229	0.0991147	−	0.995076i	$-0.468399\pi$
$643$	3232.10	0.198229	0.0991147	+	0.995076i	$0.468399\pi$
$644$	−19548.7	−1.19616
$645$	−23614.9	−1.44160
$646$	0	0
$647$	−27741.4	−1.68567	−0.842835	−	0.538172i	$-0.819115\pi$
$647$	−27741.4	−1.68567	−0.842835	+	0.538172i	$0.819115\pi$
$648$	648.000	0.0392837
$649$	2541.28	0.153704
$650$	−6825.05	−0.411847
$651$	28706.9	1.72828
$652$	−13801.7	−0.829010
$653$	−25073.6	−1.50261	−0.751307	−	0.659953i	$-0.770576\pi$
$653$	−25073.6	−1.50261	−0.751307	+	0.659953i	$0.770576\pi$
$654$	3619.47	0.216411
$655$	9411.03	0.561404
$656$	2433.21	0.144818
$657$	−5659.98	−0.336099
$658$	26498.4	1.56993
$659$	−7758.63	−0.458624	−0.229312	−	0.973353i	$-0.573648\pi$
$659$	−7758.63	−0.458624	−0.229312	+	0.973353i	$0.573648\pi$
$660$	852.453	0.0502753
$661$	−24560.3	−1.44521	−0.722605	−	0.691261i	$-0.757056\pi$
$661$	−24560.3	−1.44521	−0.722605	+	0.691261i	$0.757056\pi$
$662$	−7299.78	−0.428571
$663$	−462.335	−0.0270824
$664$	4156.77	0.242943
$665$	0	0
$666$	1512.00	0.0879712
$667$	−39061.7	−2.26758
$668$	−6359.84	−0.368368
$669$	87.1676	0.00503751
$670$	−13745.4	−0.792581
$671$	−1267.12	−0.0729008
$672$	−2979.62	−0.171044
$673$	−24146.9	−1.38305	−0.691526	−	0.722352i	$-0.743061\pi$
$673$	−24146.9	−1.38305	−0.691526	+	0.722352i	$0.743061\pi$
$674$	11196.1	0.639847
$675$	−3839.09	−0.218914
$676$	−6484.00	−0.368912
$677$	−2559.33	−0.145293	−0.0726463	−	0.997358i	$-0.523144\pi$
$677$	−2559.33	−0.145293	−0.0726463	+	0.997358i	$0.523144\pi$
$678$	4620.22	0.261709
$679$	−32560.3	−1.84028
$680$	839.698	0.0473544
$681$	−14822.0	−0.834040
$682$	−2679.70	−0.150456
$683$	−21797.7	−1.22118	−0.610591	−	0.791946i	$-0.709068\pi$
$683$	−21797.7	−1.22118	−0.610591	+	0.791946i	$0.709068\pi$
$684$	0	0
$685$	−34758.9	−1.93879
$686$	17216.0	0.958175
$687$	8557.07	0.475215
$688$	−7705.05	−0.426966
$689$	−4425.05	−0.244675
$690$	−15442.9	−0.852028
$691$	−8254.22	−0.454421	−0.227211	−	0.973846i	$-0.572961\pi$
$691$	−8254.22	−0.454421	−0.227211	+	0.973846i	$0.572961\pi$
$692$	10093.9	0.554497
$693$	1213.98	0.0665445
$694$	−25736.3	−1.40769
$695$	28560.0	1.55876
$696$	−5953.81	−0.324251
$697$	−976.525	−0.0530682
$698$	−3880.11	−0.210407
$699$	13978.3	0.756379
$700$	17652.8	0.953163
$701$	−31060.9	−1.67354	−0.836771	−	0.547553i	$-0.815559\pi$
$701$	−31060.9	−1.67354	−0.836771	+	0.547553i	$0.815559\pi$
$702$	1296.00	0.0696786
$703$	0	0
$704$	278.138	0.0148902
$705$	20932.9	1.11827
$706$	1193.09	0.0636012
$707$	−8714.37	−0.463561
$708$	−7017.05	−0.372481
$709$	−15942.7	−0.844488	−0.422244	−	0.906482i	$-0.638758\pi$
$709$	−15942.7	−0.844488	−0.422244	+	0.906482i	$0.638758\pi$
$710$	35578.5	1.88062
$711$	−1773.50	−0.0935465
$712$	7943.84	0.418129
$713$	48544.9	2.54982
$714$	1195.82	0.0626784
$715$	1704.91	0.0891746
$716$	4345.65	0.226822
$717$	−11708.5	−0.609849
$718$	−5843.69	−0.303739
$719$	34289.0	1.77853	0.889265	−	0.457393i	$-0.151217\pi$
$719$	34289.0	1.77853	0.889265	+	0.457393i	$0.151217\pi$
$720$	−2353.81	−0.121835
$721$	47987.8	2.47872
$722$	0	0
$723$	16753.6	0.861787
$724$	13414.0	0.688575
$725$	35273.5	1.80693
$726$	7872.68	0.402455
$727$	−11249.9	−0.573914	−0.286957	−	0.957943i	$-0.592644\pi$
$727$	−11249.9	−0.573914	−0.286957	+	0.957943i	$0.592644\pi$
$728$	−5959.24	−0.303385
$729$	729.000	0.0370370
$730$	20559.4	1.04238
$731$	3092.29	0.156460
$732$	3498.79	0.176665
$733$	163.883	0.00825804	0.00412902	−	0.999991i	$-0.498686\pi$
$733$	163.883	0.00825804	0.00412902	+	0.999991i	$0.498686\pi$
$734$	−8901.58	−0.447634
$735$	30420.0	1.52661
$736$	−5038.69	−0.252349
$737$	1827.25	0.0913263
$738$	2737.36	0.136536
$739$	21950.5	1.09264	0.546321	−	0.837576i	$-0.316028\pi$
$739$	21950.5	1.09264	0.546321	+	0.837576i	$0.316028\pi$
$740$	−5492.22	−0.272835
$741$	0	0
$742$	11445.3	0.566266
$743$	−1194.51	−0.0589802	−0.0294901	−	0.999565i	$-0.509388\pi$
$743$	−1194.51	−0.0589802	−0.0294901	+	0.999565i	$0.509388\pi$
$744$	7399.24	0.364609
$745$	19144.3	0.941469
$746$	25248.3	1.23915
$747$	4676.37	0.229049
$748$	−111.626	−0.00545647
$749$	−49177.8	−2.39909
$750$	1685.77	0.0820743
$751$	15628.2	0.759363	0.379681	−	0.925117i	$-0.376034\pi$
$751$	15628.2	0.759363	0.379681	+	0.925117i	$0.376034\pi$
$752$	6829.98	0.331202
$753$	−5830.67	−0.282180
$754$	−11907.6	−0.575133
$755$	−16114.0	−0.776751
$756$	−3352.07	−0.161262
$757$	11865.5	0.569695	0.284847	−	0.958573i	$-0.408057\pi$
$757$	11865.5	0.569695	0.284847	+	0.958573i	$0.408057\pi$
$758$	23245.9	1.11389
$759$	2052.91	0.0981762
$760$	0	0
$761$	15761.0	0.750768	0.375384	−	0.926869i	$-0.377511\pi$
$761$	15761.0	0.750768	0.375384	+	0.926869i	$0.377511\pi$
$762$	16404.7	0.779893
$763$	−18723.4	−0.888377
$764$	−10579.4	−0.500983
$765$	944.661	0.0446461
$766$	19857.6	0.936666
$767$	−14034.1	−0.660681
$768$	−768.000	−0.0360844
$769$	3067.23	0.143833	0.0719163	−	0.997411i	$-0.477089\pi$
$769$	3067.23	0.143833	0.0719163	+	0.997411i	$0.477089\pi$
$770$	−4409.70	−0.206382
$771$	6361.35	0.297144
$772$	−5761.50	−0.268602
$773$	16600.1	0.772401	0.386200	−	0.922415i	$-0.373787\pi$
$773$	16600.1	0.772401	0.386200	+	0.922415i	$0.373787\pi$
$774$	−8668.18	−0.402547
$775$	−43837.0	−2.03183
$776$	−8392.45	−0.388236
$777$	−7821.50	−0.361126
$778$	14302.4	0.659082
$779$	0	0
$780$	−4707.62	−0.216102
$781$	−4729.65	−0.216697
$782$	2022.19	0.0924723
$783$	−6698.04	−0.305707
$784$	9925.43	0.452142
$785$	10682.8	0.485715
$786$	3454.46	0.156764
$787$	−18271.7	−0.827594	−0.413797	−	0.910369i	$-0.635798\pi$
$787$	−18271.7	−0.827594	−0.413797	+	0.910369i	$0.635798\pi$
$788$	−4789.48	−0.216521
$789$	15117.3	0.682115
$790$	6442.11	0.290127
$791$	−23900.2	−1.07433
$792$	312.905	0.0140386
$793$	6997.58	0.313356
$794$	25644.2	1.14620
$795$	9041.43	0.403354
$796$	−12050.6	−0.536588
$797$	11702.7	0.520114	0.260057	−	0.965593i	$-0.416259\pi$
$797$	11702.7	0.520114	0.260057	+	0.965593i	$0.416259\pi$
$798$	0	0
$799$	−2741.09	−0.121368
$800$	4550.03	0.201085
$801$	8936.82	0.394216
$802$	195.168	0.00859302
$803$	−2733.08	−0.120110
$804$	−5045.43	−0.221317
$805$	79885.2	3.49762
$806$	14798.5	0.646717
$807$	16761.3	0.731134
$808$	−2246.14	−0.0977956
$809$	−37702.6	−1.63851	−0.819254	−	0.573431i	$-0.805612\pi$
$809$	−37702.6	−1.63851	−0.819254	+	0.573431i	$0.805612\pi$
$810$	−2648.04	−0.114867
$811$	−6225.35	−0.269546	−0.134773	−	0.990877i	$-0.543030\pi$
$811$	−6225.35	−0.269546	−0.134773	+	0.990877i	$0.543030\pi$
$812$	30798.8	1.33107
$813$	8122.64	0.350398
$814$	730.112	0.0314378
$815$	56400.1	2.42406
$816$	308.223	0.0132230
$817$	0	0
$818$	−5991.11	−0.256081
$819$	−6704.15	−0.286034
$820$	−9943.24	−0.423455
$821$	−7967.79	−0.338706	−0.169353	−	0.985555i	$-0.554168\pi$
$821$	−7967.79	−0.338706	−0.169353	+	0.985555i	$0.554168\pi$
$822$	−12758.7	−0.541378
$823$	27328.1	1.15747	0.578736	−	0.815515i	$-0.303546\pi$
$823$	27328.1	1.15747	0.578736	+	0.815515i	$0.303546\pi$
$824$	12368.9	0.522926
$825$	−1853.81	−0.0782321
$826$	36298.9	1.52905
$827$	−3241.58	−0.136301	−0.0681503	−	0.997675i	$-0.521710\pi$
$827$	−3241.58	−0.136301	−0.0681503	+	0.997675i	$0.521710\pi$
$828$	−5668.53	−0.237917
$829$	16527.9	0.692444	0.346222	−	0.938153i	$-0.387464\pi$
$829$	16527.9	0.692444	0.346222	+	0.938153i	$0.387464\pi$
$830$	−16986.6	−0.710376
$831$	8525.61	0.355897
$832$	−1536.00	−0.0640039
$833$	−3983.40	−0.165686
$834$	10483.3	0.435262
$835$	25989.3	1.07712
$836$	0	0
$837$	8324.15	0.343757
$838$	2306.01	0.0950595
$839$	2868.98	0.118055	0.0590276	−	0.998256i	$-0.481200\pi$
$839$	2868.98	0.118055	0.0590276	+	0.998256i	$0.481200\pi$
$840$	12176.1	0.500139
$841$	37152.4	1.52333
$842$	−7700.97	−0.315194
$843$	7095.39	0.289891
$844$	−20163.6	−0.822346
$845$	26496.7	1.07872
$846$	7683.73	0.312260
$847$	−40725.0	−1.65210
$848$	2950.03	0.119463
$849$	−10747.3	−0.434448
$850$	−1826.08	−0.0736870
$851$	−13226.6	−0.532786
$852$	13059.6	0.525135
$853$	48181.8	1.93401	0.967007	−	0.254749i	$-0.0819930\pi$
$853$	48181.8	1.93401	0.967007	+	0.254749i	$0.0819930\pi$
$854$	−18099.1	−0.725219
$855$	0	0
$856$	−12675.6	−0.506126
$857$	−11377.9	−0.453516	−0.226758	−	0.973951i	$-0.572813\pi$
$857$	−11377.9	−0.453516	−0.226758	+	0.973951i	$0.572813\pi$
$858$	625.810	0.0249007
$859$	−2197.20	−0.0872731	−0.0436365	−	0.999047i	$-0.513894\pi$
$859$	−2197.20	−0.0872731	−0.0436365	+	0.999047i	$0.513894\pi$
$860$	31486.5	1.24847
$861$	−14160.2	−0.560487
$862$	−10099.9	−0.399077
$863$	−37507.8	−1.47947	−0.739734	−	0.672899i	$-0.765049\pi$
$863$	−37507.8	−1.47947	−0.739734	+	0.672899i	$0.765049\pi$
$864$	−864.000	−0.0340207
$865$	−41248.4	−1.62137
$866$	18681.5	0.733052
$867$	14615.3	0.572505
$868$	−38275.9	−1.49674
$869$	−856.386	−0.0334303
$870$	24330.1	0.948124
$871$	−10090.9	−0.392555
$872$	−4825.97	−0.187417
$873$	−9441.50	−0.366033
$874$	0	0
$875$	−8720.43	−0.336919
$876$	7546.64	0.291070
$877$	15932.7	0.613464	0.306732	−	0.951796i	$-0.400764\pi$
$877$	15932.7	0.613464	0.306732	+	0.951796i	$0.400764\pi$
$878$	12890.1	0.495466
$879$	8521.35	0.326983
$880$	−1136.60	−0.0435396
$881$	43405.8	1.65991	0.829953	−	0.557833i	$-0.188367\pi$
$881$	43405.8	1.65991	0.829953	+	0.557833i	$0.188367\pi$
$882$	11166.1	0.426284
$883$	15528.8	0.591829	0.295915	−	0.955214i	$-0.404376\pi$
$883$	15528.8	0.591829	0.295915	+	0.955214i	$0.404376\pi$
$884$	616.447	0.0234540
$885$	28675.0	1.08915
$886$	−16927.3	−0.641856
$887$	36209.0	1.37067	0.685333	−	0.728230i	$-0.259657\pi$
$887$	36209.0	1.37067	0.685333	+	0.728230i	$0.259657\pi$
$888$	−2016.00	−0.0761853
$889$	−84860.6	−3.20150
$890$	−32462.3	−1.22263
$891$	352.018	0.0132357
$892$	−116.223	−0.00436261
$893$	0	0
$894$	7027.21	0.262892
$895$	−17758.4	−0.663238
$896$	3972.83	0.148128
$897$	−11337.1	−0.421999
$898$	3475.93	0.129168
$899$	−76482.1	−2.83740
$900$	5118.79	0.189585
$901$	−1183.94	−0.0437768
$902$	1321.81	0.0487932
$903$	44840.1	1.65247
$904$	−6160.29	−0.226646
$905$	−54816.1	−2.01342
$906$	−5914.86	−0.216896
$907$	8592.97	0.314581	0.157290	−	0.987552i	$-0.449724\pi$
$907$	8592.97	0.314581	0.157290	+	0.987552i	$0.449724\pi$
$908$	19762.7	0.722300
$909$	−2526.91	−0.0922026
$910$	24352.3	0.887111
$911$	−25963.2	−0.944235	−0.472117	−	0.881536i	$-0.656510\pi$
$911$	−25963.2	−0.944235	−0.472117	+	0.881536i	$0.656510\pi$
$912$	0	0
$913$	2258.12	0.0818541
$914$	35446.6	1.28279
$915$	−14297.7	−0.516577
$916$	−11409.4	−0.411548
$917$	−17869.7	−0.643523
$918$	346.751	0.0124668
$919$	2229.83	0.0800385	0.0400193	−	0.999199i	$-0.487258\pi$
$919$	2229.83	0.0800385	0.0400193	+	0.999199i	$0.487258\pi$
$920$	20590.5	0.737878
$921$	−27885.5	−0.997674
$922$	9703.16	0.346591
$923$	26119.2	0.931447
$924$	−1618.64	−0.0576293
$925$	11943.8	0.424552
$926$	−21354.3	−0.757825
$927$	13915.0	0.493019
$928$	7938.41	0.280809
$929$	14164.9	0.500253	0.250126	−	0.968213i	$-0.419528\pi$
$929$	14164.9	0.500253	0.250126	+	0.968213i	$0.419528\pi$
$930$	−30236.8	−1.06613
$931$	0	0
$932$	−18637.8	−0.655043
$933$	−5600.15	−0.196507
$934$	−16928.5	−0.593061
$935$	456.156	0.0159550
$936$	−1728.00	−0.0603434
$937$	−42462.9	−1.48047	−0.740236	−	0.672347i	$-0.765286\pi$
$937$	−42462.9	−1.48047	−0.740236	+	0.672347i	$0.765286\pi$
$938$	26099.8	0.908516
$939$	16549.0	0.575141
$940$	−27910.6	−0.968449
$941$	−47645.2	−1.65057	−0.825287	−	0.564714i	$-0.808987\pi$
$941$	−47645.2	−1.65057	−0.825287	+	0.564714i	$0.808987\pi$
$942$	3921.28	0.135629
$943$	−23945.6	−0.826912
$944$	9356.07	0.322578
$945$	13698.2	0.471536
$946$	−4185.68	−0.143856
$947$	40549.9	1.39144	0.695721	−	0.718312i	$-0.255085\pi$
$947$	40549.9	1.39144	0.695721	+	0.718312i	$0.255085\pi$
$948$	2364.67	0.0810136
$949$	15093.3	0.516279
$950$	0	0
$951$	15743.1	0.536808
$952$	−1594.42	−0.0542811
$953$	8636.89	0.293574	0.146787	−	0.989168i	$-0.453107\pi$
$953$	8636.89	0.293574	0.146787	+	0.989168i	$0.453107\pi$
$954$	3318.79	0.112631
$955$	43232.6	1.46490
$956$	15611.3	0.528145
$957$	−3234.34	−0.109249
$958$	−16900.0	−0.569952
$959$	66000.4	2.22238
$960$	3138.41	0.105512
$961$	65258.9	2.19056
$962$	−4032.00	−0.135132
$963$	−14260.1	−0.477180
$964$	−22338.1	−0.746330
$965$	23544.2	0.785404
$966$	29323.0	0.976659
$967$	32994.5	1.09724	0.548620	−	0.836072i	$-0.315154\pi$
$967$	32994.5	1.09724	0.548620	+	0.836072i	$0.315154\pi$
$968$	−10496.9	−0.348536
$969$	0	0
$970$	34295.5	1.13522
$971$	5801.68	0.191745	0.0958727	−	0.995394i	$-0.469436\pi$
$971$	5801.68	0.191745	0.0958727	+	0.995394i	$0.469436\pi$
$972$	−972.000	−0.0320750
$973$	−54229.9	−1.78677
$974$	15710.2	0.516825
$975$	10237.6	0.336272
$976$	−4665.05	−0.152997
$977$	35449.2	1.16082	0.580410	−	0.814324i	$-0.302892\pi$
$977$	35449.2	1.16082	0.580410	+	0.814324i	$0.302892\pi$
$978$	20702.5	0.676884
$979$	4315.40	0.140879
$980$	−40560.0	−1.32208
$981$	−5429.21	−0.176699
$982$	−10682.8	−0.347149
$983$	−22860.0	−0.741732	−0.370866	−	0.928686i	$-0.620939\pi$
$983$	−22860.0	−0.741732	−0.370866	+	0.928686i	$0.620939\pi$
$984$	−3649.81	−0.118244
$985$	19572.1	0.633116
$986$	−3185.94	−0.102902
$987$	−39747.6	−1.28184
$988$	0	0
$989$	75826.9	2.43797
$990$	−1278.68	−0.0410496
$991$	22985.4	0.736787	0.368394	−	0.929670i	$-0.379908\pi$
$991$	22985.4	0.736787	0.368394	+	0.929670i	$0.379908\pi$
$992$	−9865.65	−0.315761
$993$	10949.7	0.349927
$994$	−67556.8	−2.15571
$995$	49244.7	1.56901
$996$	−6235.16	−0.198362
$997$	13201.6	0.419357	0.209678	−	0.977770i	$-0.432758\pi$
$997$	13201.6	0.419357	0.209678	+	0.977770i	$0.432758\pi$
$998$	−6314.15	−0.200272
$999$	−2268.00	−0.0718282

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.4.a.m.1.1	✓	2	19.18	odd	2
2166.4.a.o.1.1	yes	2	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -16.3459 q^{5} -6.00000 q^{6} +31.0377 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -16.3459 q^{5} -6.00000 q^{6} +31.0377 q^{7} +8.00000 q^{8} +9.00000 q^{9} -32.6918 q^{10} +4.34590 q^{11} -12.0000 q^{12} -24.0000 q^{13} +62.0754 q^{14} +49.0377 q^{15} +16.0000 q^{16} -6.42132 q^{17} +18.0000 q^{18} -65.3836 q^{20} -93.1131 q^{21} +8.69181 q^{22} -157.459 q^{23} -24.0000 q^{24} +142.189 q^{25} -48.0000 q^{26} -27.0000 q^{27} +124.151 q^{28} +248.075 q^{29} +98.0754 q^{30} -308.302 q^{31} +32.0000 q^{32} -13.0377 q^{33} -12.8426 q^{34} -507.339 q^{35} +36.0000 q^{36} +84.0000 q^{37} +72.0000 q^{39} -130.767 q^{40} +152.075 q^{41} -186.226 q^{42} -481.566 q^{43} +17.3836 q^{44} -147.113 q^{45} -314.918 q^{46} +426.874 q^{47} -48.0000 q^{48} +620.339 q^{49} +284.377 q^{50} +19.2640 q^{51} -96.0000 q^{52} +184.377 q^{53} -54.0000 q^{54} -71.0377 q^{55} +248.302 q^{56} +496.151 q^{58} +584.754 q^{59} +196.151 q^{60} -291.566 q^{61} -616.603 q^{62} +279.339 q^{63} +64.0000 q^{64} +392.302 q^{65} -26.0754 q^{66} +420.453 q^{67} -25.6853 q^{68} +472.377 q^{69} -1014.68 q^{70} -1088.30 q^{71} +72.0000 q^{72} -628.887 q^{73} +168.000 q^{74} -426.566 q^{75} +134.887 q^{77} +144.000 q^{78} -197.056 q^{79} -261.534 q^{80} +81.0000 q^{81} +304.151 q^{82} +519.597 q^{83} -372.453 q^{84} +104.962 q^{85} -963.131 q^{86} -744.226 q^{87} +34.7672 q^{88} +992.980 q^{89} -294.226 q^{90} -744.905 q^{91} -629.836 q^{92} +924.905 q^{93} +853.748 q^{94} -96.0000 q^{96} -1049.06 q^{97} +1240.68 q^{98} +39.1131 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} - 15 q^{5} - 12 q^{6} + 9 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 - 15 * q^5 - 12 * q^6 + 9 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} - 15 q^{5} - 12 q^{6} + 9 q^{7} + 16 q^{8} + 18 q^{9} - 30 q^{10} - 9 q^{11} - 24 q^{12} - 48 q^{13} + 18 q^{14} + 45 q^{15} + 32 q^{16} + 111 q^{17} + 36 q^{18} - 60 q^{20} - 27 q^{21} - 18 q^{22} - 138 q^{23} - 48 q^{24} + 19 q^{25} - 96 q^{26} - 54 q^{27} + 36 q^{28} + 390 q^{29} + 90 q^{30} - 192 q^{31} + 64 q^{32} + 27 q^{33} + 222 q^{34} - 537 q^{35} + 72 q^{36} + 168 q^{37} + 144 q^{39} - 120 q^{40} + 198 q^{41} - 54 q^{42} - 167 q^{43} - 36 q^{44} - 135 q^{45} - 276 q^{46} + 93 q^{47} - 96 q^{48} + 763 q^{49} + 38 q^{50} - 333 q^{51} - 192 q^{52} - 162 q^{53} - 108 q^{54} - 89 q^{55} + 72 q^{56} + 780 q^{58} + 108 q^{59} + 180 q^{60} + 213 q^{61} - 384 q^{62} + 81 q^{63} + 128 q^{64} + 360 q^{65} + 54 q^{66} + 204 q^{67} + 444 q^{68} + 414 q^{69} - 1074 q^{70} - 1752 q^{71} + 144 q^{72} - 1417 q^{73} + 336 q^{74} - 57 q^{75} + 429 q^{77} + 288 q^{78} + 1092 q^{79} - 240 q^{80} + 162 q^{81} + 396 q^{82} - 270 q^{83} - 108 q^{84} + 263 q^{85} - 334 q^{86} - 1170 q^{87} - 72 q^{88} + 606 q^{89} - 270 q^{90} - 216 q^{91} - 552 q^{92} + 576 q^{93} + 186 q^{94} - 192 q^{96} - 612 q^{97} + 1526 q^{98} - 81 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 - 15 * q^5 - 12 * q^6 + 9 * q^7 + 16 * q^8 + 18 * q^9 - 30 * q^10 - 9 * q^11 - 24 * q^12 - 48 * q^13 + 18 * q^14 + 45 * q^15 + 32 * q^16 + 111 * q^17 + 36 * q^18 - 60 * q^20 - 27 * q^21 - 18 * q^22 - 138 * q^23 - 48 * q^24 + 19 * q^25 - 96 * q^26 - 54 * q^27 + 36 * q^28 + 390 * q^29 + 90 * q^30 - 192 * q^31 + 64 * q^32 + 27 * q^33 + 222 * q^34 - 537 * q^35 + 72 * q^36 + 168 * q^37 + 144 * q^39 - 120 * q^40 + 198 * q^41 - 54 * q^42 - 167 * q^43 - 36 * q^44 - 135 * q^45 - 276 * q^46 + 93 * q^47 - 96 * q^48 + 763 * q^49 + 38 * q^50 - 333 * q^51 - 192 * q^52 - 162 * q^53 - 108 * q^54 - 89 * q^55 + 72 * q^56 + 780 * q^58 + 108 * q^59 + 180 * q^60 + 213 * q^61 - 384 * q^62 + 81 * q^63 + 128 * q^64 + 360 * q^65 + 54 * q^66 + 204 * q^67 + 444 * q^68 + 414 * q^69 - 1074 * q^70 - 1752 * q^71 + 144 * q^72 - 1417 * q^73 + 336 * q^74 - 57 * q^75 + 429 * q^77 + 288 * q^78 + 1092 * q^79 - 240 * q^80 + 162 * q^81 + 396 * q^82 - 270 * q^83 - 108 * q^84 + 263 * q^85 - 334 * q^86 - 1170 * q^87 - 72 * q^88 + 606 * q^89 - 270 * q^90 - 216 * q^91 - 552 * q^92 + 576 * q^93 + 186 * q^94 - 192 * q^96 - 612 * q^97 + 1526 * q^98 - 81 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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