LMFDB - Embedded newform 2166.4.a.u.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:	$3$
Coefficient field:	3.3.14457.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 32x - 3 $ x^3 - x^2 - 32*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.22121$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.44242 q^{5} -6.00000 q^{6} -9.44242 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.44242 q^{5} -6.00000 q^{6} -9.44242 q^{7} +8.00000 q^{8} +9.00000 q^{9} -16.8848 q^{10} -47.4865 q^{11} -12.0000 q^{12} +67.6545 q^{13} -18.8848 q^{14} +25.3272 q^{15} +16.0000 q^{16} -77.6104 q^{17} +18.0000 q^{18} -33.7697 q^{20} +28.3272 q^{21} -94.9729 q^{22} +170.955 q^{23} -24.0000 q^{24} -53.7256 q^{25} +135.309 q^{26} -27.0000 q^{27} -37.7697 q^{28} +240.123 q^{29} +50.6545 q^{30} +279.104 q^{31} +32.0000 q^{32} +142.459 q^{33} -155.221 q^{34} +79.7168 q^{35} +36.0000 q^{36} -20.0617 q^{37} -202.963 q^{39} -67.5393 q^{40} +71.8213 q^{41} +56.6545 q^{42} +147.318 q^{43} -189.946 q^{44} -75.9817 q^{45} +341.909 q^{46} -207.343 q^{47} -48.0000 q^{48} -253.841 q^{49} -107.451 q^{50} +232.831 q^{51} +270.618 q^{52} -261.523 q^{53} -54.0000 q^{54} +400.901 q^{55} -75.5393 q^{56} +480.245 q^{58} -21.9817 q^{59} +101.309 q^{60} -653.103 q^{61} +558.209 q^{62} -84.9817 q^{63} +64.0000 q^{64} -571.167 q^{65} +284.919 q^{66} -413.795 q^{67} -310.442 q^{68} -512.864 q^{69} +159.434 q^{70} -157.945 q^{71} +72.0000 q^{72} -1102.15 q^{73} -40.1233 q^{74} +161.177 q^{75} +448.387 q^{77} -405.927 q^{78} -417.813 q^{79} -135.079 q^{80} +81.0000 q^{81} +143.643 q^{82} +1431.80 q^{83} +113.309 q^{84} +655.220 q^{85} +294.636 q^{86} -720.368 q^{87} -379.892 q^{88} +1289.39 q^{89} -151.963 q^{90} -638.822 q^{91} +683.819 q^{92} -837.313 q^{93} -414.686 q^{94} -96.0000 q^{96} +138.762 q^{97} -507.682 q^{98} -427.378 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 10 * q^5 - 18 * q^6 + 7 * q^7 + 24 * q^8 + 27 * q^9	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} + 24 q^{8} + 27 q^{9} + 20 q^{10} - 44 q^{11} - 36 q^{12} - 9 q^{13} + 14 q^{14} - 30 q^{15} + 48 q^{16} - 84 q^{17} + 54 q^{18} + 40 q^{20} - 21 q^{21} - 88 q^{22} - 2 q^{23} - 72 q^{24} - 83 q^{25} - 18 q^{26} - 81 q^{27} + 28 q^{28} + 92 q^{29} - 60 q^{30} - 109 q^{31} + 96 q^{32} + 132 q^{33} - 168 q^{34} + 282 q^{35} + 108 q^{36} + 245 q^{37} + 27 q^{39} + 80 q^{40} - 688 q^{41} - 42 q^{42} - 103 q^{43} - 176 q^{44} + 90 q^{45} - 4 q^{46} + 322 q^{47} - 144 q^{48} - 754 q^{49} - 166 q^{50} + 252 q^{51} - 36 q^{52} - 1322 q^{53} - 162 q^{54} - 248 q^{55} + 56 q^{56} + 184 q^{58} + 252 q^{59} - 120 q^{60} - 435 q^{61} - 218 q^{62} + 63 q^{63} + 192 q^{64} - 1582 q^{65} + 264 q^{66} - 719 q^{67} - 336 q^{68} + 6 q^{69} + 564 q^{70} - 62 q^{71} + 216 q^{72} - 581 q^{73} + 490 q^{74} + 249 q^{75} - 204 q^{77} + 54 q^{78} - 489 q^{79} + 160 q^{80} + 243 q^{81} - 1376 q^{82} + 2496 q^{83} - 84 q^{84} + 1632 q^{85} - 206 q^{86} - 276 q^{87} - 352 q^{88} + 1584 q^{89} + 180 q^{90} - 1573 q^{91} - 8 q^{92} + 327 q^{93} + 644 q^{94} - 288 q^{96} + 974 q^{97} - 1508 q^{98} - 396 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 10 * q^5 - 18 * q^6 + 7 * q^7 + 24 * q^8 + 27 * q^9 + 20 * q^10 - 44 * q^11 - 36 * q^12 - 9 * q^13 + 14 * q^14 - 30 * q^15 + 48 * q^16 - 84 * q^17 + 54 * q^18 + 40 * q^20 - 21 * q^21 - 88 * q^22 - 2 * q^23 - 72 * q^24 - 83 * q^25 - 18 * q^26 - 81 * q^27 + 28 * q^28 + 92 * q^29 - 60 * q^30 - 109 * q^31 + 96 * q^32 + 132 * q^33 - 168 * q^34 + 282 * q^35 + 108 * q^36 + 245 * q^37 + 27 * q^39 + 80 * q^40 - 688 * q^41 - 42 * q^42 - 103 * q^43 - 176 * q^44 + 90 * q^45 - 4 * q^46 + 322 * q^47 - 144 * q^48 - 754 * q^49 - 166 * q^50 + 252 * q^51 - 36 * q^52 - 1322 * q^53 - 162 * q^54 - 248 * q^55 + 56 * q^56 + 184 * q^58 + 252 * q^59 - 120 * q^60 - 435 * q^61 - 218 * q^62 + 63 * q^63 + 192 * q^64 - 1582 * q^65 + 264 * q^66 - 719 * q^67 - 336 * q^68 + 6 * q^69 + 564 * q^70 - 62 * q^71 + 216 * q^72 - 581 * q^73 + 490 * q^74 + 249 * q^75 - 204 * q^77 + 54 * q^78 - 489 * q^79 + 160 * q^80 + 243 * q^81 - 1376 * q^82 + 2496 * q^83 - 84 * q^84 + 1632 * q^85 - 206 * q^86 - 276 * q^87 - 352 * q^88 + 1584 * q^89 + 180 * q^90 - 1573 * q^91 - 8 * q^92 + 327 * q^93 + 644 * q^94 - 288 * q^96 + 974 * q^97 - 1508 * q^98 - 396 * 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
$5$	−8.44242	−0.755113	−0.377556	−	0.925987i	$-0.623236\pi$
$5$	−8.44242	−0.755113	−0.377556	+	0.925987i	$0.623236\pi$
$6$	−6.00000	−0.408248
$7$	−9.44242	−0.509843	−0.254921	−	0.966962i	$-0.582050\pi$
$7$	−9.44242	−0.509843	−0.254921	+	0.966962i	$0.582050\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−16.8848	−0.533945
$11$	−47.4865	−1.30161	−0.650805	−	0.759245i	$-0.725568\pi$
$11$	−47.4865	−1.30161	−0.650805	+	0.759245i	$0.725568\pi$
$12$	−12.0000	−0.288675
$13$	67.6545	1.44338	0.721692	−	0.692215i	$-0.243365\pi$
$13$	67.6545	1.44338	0.721692	+	0.692215i	$0.243365\pi$
$14$	−18.8848	−0.360513
$15$	25.3272	0.435964
$16$	16.0000	0.250000
$17$	−77.6104	−1.10725	−0.553626	−	0.832765i	$-0.686756\pi$
$17$	−77.6104	−1.10725	−0.553626	+	0.832765i	$0.686756\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	−33.7697	−0.377556
$21$	28.3272	0.294358
$22$	−94.9729	−0.920377
$23$	170.955	1.54985	0.774924	−	0.632054i	$-0.217788\pi$
$23$	170.955	1.54985	0.774924	+	0.632054i	$0.217788\pi$
$24$	−24.0000	−0.204124
$25$	−53.7256	−0.429805
$26$	135.309	1.02063
$27$	−27.0000	−0.192450
$28$	−37.7697	−0.254921
$29$	240.123	1.53757	0.768787	−	0.639504i	$-0.220860\pi$
$29$	240.123	1.53757	0.768787	+	0.639504i	$0.220860\pi$
$30$	50.6545	0.308273
$31$	279.104	1.61705	0.808526	−	0.588460i	$-0.200265\pi$
$31$	279.104	1.61705	0.808526	+	0.588460i	$0.200265\pi$
$32$	32.0000	0.176777
$33$	142.459	0.751485
$34$	−155.221	−0.782946
$35$	79.7168	0.384989
$36$	36.0000	0.166667
$37$	−20.0617	−0.0891383	−0.0445692	−	0.999006i	$-0.514192\pi$
$37$	−20.0617	−0.0891383	−0.0445692	+	0.999006i	$0.514192\pi$
$38$	0	0
$39$	−202.963	−0.833338
$40$	−67.5393	−0.266973
$41$	71.8213	0.273576	0.136788	−	0.990600i	$-0.456322\pi$
$41$	71.8213	0.273576	0.136788	+	0.990600i	$0.456322\pi$
$42$	56.6545	0.208142
$43$	147.318	0.522459	0.261230	−	0.965277i	$-0.415872\pi$
$43$	147.318	0.522459	0.261230	+	0.965277i	$0.415872\pi$
$44$	−189.946	−0.650805
$45$	−75.9817	−0.251704
$46$	341.909	1.09591
$47$	−207.343	−0.643491	−0.321746	−	0.946826i	$-0.604270\pi$
$47$	−207.343	−0.643491	−0.321746	+	0.946826i	$0.604270\pi$
$48$	−48.0000	−0.144338
$49$	−253.841	−0.740061
$50$	−107.451	−0.303918
$51$	232.831	0.639273
$52$	270.618	0.721692
$53$	−261.523	−0.677791	−0.338896	−	0.940824i	$-0.610053\pi$
$53$	−261.523	−0.677791	−0.338896	+	0.940824i	$0.610053\pi$
$54$	−54.0000	−0.136083
$55$	400.901	0.982862
$56$	−75.5393	−0.180257
$57$	0	0
$58$	480.245	1.08723
$59$	−21.9817	−0.0485047	−0.0242524	−	0.999706i	$-0.507721\pi$
$59$	−21.9817	−0.0485047	−0.0242524	+	0.999706i	$0.507721\pi$
$60$	101.309	0.217982
$61$	−653.103	−1.37084	−0.685420	−	0.728148i	$-0.740382\pi$
$61$	−653.103	−1.37084	−0.685420	+	0.728148i	$0.740382\pi$
$62$	558.209	1.14343
$63$	−84.9817	−0.169948
$64$	64.0000	0.125000
$65$	−571.167	−1.08992
$66$	284.919	0.531380
$67$	−413.795	−0.754525	−0.377262	−	0.926106i	$-0.623135\pi$
$67$	−413.795	−0.754525	−0.377262	+	0.926106i	$0.623135\pi$
$68$	−310.442	−0.553626
$69$	−512.864	−0.894805
$70$	159.434	0.272228
$71$	−157.945	−0.264009	−0.132005	−	0.991249i	$-0.542141\pi$
$71$	−157.945	−0.264009	−0.132005	+	0.991249i	$0.542141\pi$
$72$	72.0000	0.117851
$73$	−1102.15	−1.76707	−0.883537	−	0.468362i	$-0.844844\pi$
$73$	−1102.15	−1.76707	−0.883537	+	0.468362i	$0.844844\pi$
$74$	−40.1233	−0.0630303
$75$	161.177	0.248148
$76$	0	0
$77$	448.387	0.663616
$78$	−405.927	−0.589259
$79$	−417.813	−0.595033	−0.297517	−	0.954717i	$-0.596158\pi$
$79$	−417.813	−0.595033	−0.297517	+	0.954717i	$0.596158\pi$
$80$	−135.079	−0.188778
$81$	81.0000	0.111111
$82$	143.643	0.193447
$83$	1431.80	1.89350	0.946751	−	0.321967i	$-0.104344\pi$
$83$	1431.80	1.89350	0.946751	+	0.321967i	$0.104344\pi$
$84$	113.309	0.147179
$85$	655.220	0.836101
$86$	294.636	0.369435
$87$	−720.368	−0.887719
$88$	−379.892	−0.460189
$89$	1289.39	1.53568	0.767839	−	0.640642i	$-0.221332\pi$
$89$	1289.39	1.53568	0.767839	+	0.640642i	$0.221332\pi$
$90$	−151.963	−0.177982
$91$	−638.822	−0.735898
$92$	683.819	0.774924
$93$	−837.313	−0.933606
$94$	−414.686	−0.455017
$95$	0	0
$96$	−96.0000	−0.102062
$97$	138.762	0.145249	0.0726245	−	0.997359i	$-0.476863\pi$
$97$	138.762	0.145249	0.0726245	+	0.997359i	$0.476863\pi$
$98$	−507.682	−0.523302
$99$	−427.378	−0.433870
$100$	−214.902	−0.214902
$101$	−69.6306	−0.0685990	−0.0342995	−	0.999412i	$-0.510920\pi$
$101$	−69.6306	−0.0685990	−0.0342995	+	0.999412i	$0.510920\pi$
$102$	465.663	0.452034
$103$	−644.346	−0.616401	−0.308201	−	0.951321i	$-0.599727\pi$
$103$	−644.346	−0.616401	−0.308201	+	0.951321i	$0.599727\pi$
$104$	541.236	0.510313
$105$	−239.150	−0.222273
$106$	−523.046	−0.479271
$107$	−189.454	−0.171171	−0.0855853	−	0.996331i	$-0.527276\pi$
$107$	−189.454	−0.171171	−0.0855853	+	0.996331i	$0.527276\pi$
$108$	−108.000	−0.0962250
$109$	−239.152	−0.210153	−0.105076	−	0.994464i	$-0.533509\pi$
$109$	−239.152	−0.210153	−0.105076	+	0.994464i	$0.533509\pi$
$110$	801.801	0.694988
$111$	60.1850	0.0514640
$112$	−151.079	−0.127461
$113$	162.743	0.135483	0.0677413	−	0.997703i	$-0.478421\pi$
$113$	162.743	0.135483	0.0677413	+	0.997703i	$0.478421\pi$
$114$	0	0
$115$	−1443.27	−1.17031
$116$	960.491	0.768787
$117$	608.890	0.481128
$118$	−43.9635	−0.0342980
$119$	732.830	0.564525
$120$	202.618	0.154137
$121$	923.965	0.694188
$122$	−1306.21	−0.969331
$123$	−215.464	−0.157949
$124$	1116.42	0.808526
$125$	1508.88	1.07966
$126$	−169.963	−0.120171
$127$	−97.3783	−0.0680388	−0.0340194	−	0.999421i	$-0.510831\pi$
$127$	−97.3783	−0.0680388	−0.0340194	+	0.999421i	$0.510831\pi$
$128$	128.000	0.0883883
$129$	−441.953	−0.301642
$130$	−1142.33	−0.770688
$131$	379.398	0.253039	0.126520	−	0.991964i	$-0.459619\pi$
$131$	379.398	0.253039	0.126520	+	0.991964i	$0.459619\pi$
$132$	569.838	0.375742
$133$	0	0
$134$	−827.591	−0.533530
$135$	227.945	0.145321
$136$	−620.884	−0.391473
$137$	2413.68	1.50522	0.752609	−	0.658468i	$-0.228795\pi$
$137$	2413.68	1.50522	0.752609	+	0.658468i	$0.228795\pi$
$138$	−1025.73	−0.632723
$139$	−3109.04	−1.89716	−0.948582	−	0.316532i	$-0.897481\pi$
$139$	−3109.04	−1.89716	−0.948582	+	0.316532i	$0.897481\pi$
$140$	318.867	0.192494
$141$	622.029	0.371520
$142$	−315.891	−0.186683
$143$	−3212.67	−1.87872
$144$	144.000	0.0833333
$145$	−2027.22	−1.16104
$146$	−2204.29	−1.24951
$147$	761.522	0.427274
$148$	−80.2467	−0.0445692
$149$	−2919.10	−1.60498	−0.802491	−	0.596665i	$-0.796492\pi$
$149$	−2919.10	−1.60498	−0.802491	+	0.596665i	$0.796492\pi$
$150$	322.354	0.175467
$151$	−1915.72	−1.03244	−0.516221	−	0.856455i	$-0.672662\pi$
$151$	−1915.72	−1.03244	−0.516221	+	0.856455i	$0.672662\pi$
$152$	0	0
$153$	−698.494	−0.369084
$154$	896.774	0.469247
$155$	−2356.32	−1.22106
$156$	−811.854	−0.416669
$157$	−2536.69	−1.28949	−0.644744	−	0.764398i	$-0.723036\pi$
$157$	−2536.69	−1.28949	−0.644744	+	0.764398i	$0.723036\pi$
$158$	−835.626	−0.420752
$159$	784.569	0.391323
$160$	−270.157	−0.133486
$161$	−1614.23	−0.790179
$162$	162.000	0.0785674
$163$	−1731.52	−0.832044	−0.416022	−	0.909355i	$-0.636576\pi$
$163$	−1731.52	−0.832044	−0.416022	+	0.909355i	$0.636576\pi$
$164$	287.285	0.136788
$165$	−1202.70	−0.567456
$166$	2863.60	1.33891
$167$	−1430.12	−0.662673	−0.331336	−	0.943513i	$-0.607499\pi$
$167$	−1430.12	−0.662673	−0.331336	+	0.943513i	$0.607499\pi$
$168$	226.618	0.104071
$169$	2380.13	1.08335
$170$	1310.44	0.591212
$171$	0	0
$172$	589.271	0.261230
$173$	293.325	0.128908	0.0644539	−	0.997921i	$-0.479469\pi$
$173$	293.325	0.128908	0.0644539	+	0.997921i	$0.479469\pi$
$174$	−1440.74	−0.627712
$175$	507.300	0.219133
$176$	−759.784	−0.325402
$177$	65.9452	0.0280042
$178$	2578.79	1.08589
$179$	−3843.52	−1.60490	−0.802452	−	0.596716i	$-0.796472\pi$
$179$	−3843.52	−1.60490	−0.802452	+	0.596716i	$0.796472\pi$
$180$	−303.927	−0.125852
$181$	1769.28	0.726571	0.363286	−	0.931678i	$-0.381655\pi$
$181$	1769.28	0.726571	0.363286	+	0.931678i	$0.381655\pi$
$182$	−1277.64	−0.520359
$183$	1959.31	0.791455
$184$	1367.64	0.547954
$185$	169.369	0.0673095
$186$	−1674.63	−0.660159
$187$	3685.45	1.44121
$188$	−829.372	−0.321746
$189$	254.945	0.0981192
$190$	0	0
$191$	−504.006	−0.190935	−0.0954676	−	0.995433i	$-0.530435\pi$
$191$	−504.006	−0.190935	−0.0954676	+	0.995433i	$0.530435\pi$
$192$	−192.000	−0.0721688
$193$	−2167.59	−0.808427	−0.404213	−	0.914665i	$-0.632455\pi$
$193$	−2167.59	−0.808427	−0.404213	+	0.914665i	$0.632455\pi$
$194$	277.524	0.102707
$195$	1713.50	0.629264
$196$	−1015.36	−0.370030
$197$	2111.10	0.763501	0.381750	−	0.924265i	$-0.375321\pi$
$197$	2111.10	0.763501	0.381750	+	0.924265i	$0.375321\pi$
$198$	−854.756	−0.306792
$199$	2138.59	0.761814	0.380907	−	0.924613i	$-0.375612\pi$
$199$	2138.59	0.761814	0.380907	+	0.924613i	$0.375612\pi$
$200$	−429.805	−0.151959
$201$	1241.39	0.435625
$202$	−139.261	−0.0485068
$203$	−2267.34	−0.783921
$204$	931.325	0.319636
$205$	−606.345	−0.206580
$206$	−1288.69	−0.435861
$207$	1538.59	0.516616
$208$	1082.47	0.360846
$209$	0	0
$210$	−478.301	−0.157171
$211$	−975.953	−0.318424	−0.159212	−	0.987244i	$-0.550895\pi$
$211$	−975.953	−0.318424	−0.159212	+	0.987244i	$0.550895\pi$
$212$	−1046.09	−0.338896
$213$	473.836	0.152426
$214$	−378.909	−0.121036
$215$	−1243.72	−0.394516
$216$	−216.000	−0.0680414
$217$	−2635.42	−0.824442
$218$	−478.305	−0.148600
$219$	3306.44	1.02022
$220$	1603.60	0.491431
$221$	−5250.70	−1.59819
$222$	120.370	0.0363906
$223$	−3575.45	−1.07368	−0.536839	−	0.843685i	$-0.680382\pi$
$223$	−3575.45	−1.07368	−0.536839	+	0.843685i	$0.680382\pi$
$224$	−302.157	−0.0901283
$225$	−483.530	−0.143268
$226$	325.485	0.0958007
$227$	−86.5237	−0.0252986	−0.0126493	−	0.999920i	$-0.504027\pi$
$227$	−86.5237	−0.0252986	−0.0126493	+	0.999920i	$0.504027\pi$
$228$	0	0
$229$	−3928.05	−1.13351	−0.566753	−	0.823888i	$-0.691801\pi$
$229$	−3928.05	−1.13351	−0.566753	+	0.823888i	$0.691801\pi$
$230$	−2886.54	−0.827534
$231$	−1345.16	−0.383139
$232$	1920.98	0.543615
$233$	1020.03	0.286799	0.143399	−	0.989665i	$-0.454197\pi$
$233$	1020.03	0.286799	0.143399	+	0.989665i	$0.454197\pi$
$234$	1217.78	0.340209
$235$	1750.48	0.485908
$236$	−87.9270	−0.0242524
$237$	1253.44	0.343543
$238$	1465.66	0.399179
$239$	264.681	0.0716351	0.0358175	−	0.999358i	$-0.488596\pi$
$239$	264.681	0.0716351	0.0358175	+	0.999358i	$0.488596\pi$
$240$	405.236	0.108991
$241$	−172.387	−0.0460764	−0.0230382	−	0.999735i	$-0.507334\pi$
$241$	−172.387	−0.0460764	−0.0230382	+	0.999735i	$0.507334\pi$
$242$	1847.93	0.490865
$243$	−243.000	−0.0641500
$244$	−2612.41	−0.685420
$245$	2143.03	0.558829
$246$	−430.928	−0.111687
$247$	0	0
$248$	2232.84	0.571715
$249$	−4295.41	−1.09321
$250$	3017.75	0.763438
$251$	−4884.87	−1.22841	−0.614203	−	0.789148i	$-0.710522\pi$
$251$	−4884.87	−1.22841	−0.614203	+	0.789148i	$0.710522\pi$
$252$	−339.927	−0.0849738
$253$	−8118.03	−2.01730
$254$	−194.757	−0.0481107
$255$	−1965.66	−0.482723
$256$	256.000	0.0625000
$257$	4022.64	0.976363	0.488181	−	0.872742i	$-0.337660\pi$
$257$	4022.64	0.976363	0.488181	+	0.872742i	$0.337660\pi$
$258$	−883.907	−0.213293
$259$	189.431	0.0454465
$260$	−2284.67	−0.544958
$261$	2161.10	0.512525
$262$	758.795	0.178926
$263$	−7142.12	−1.67453	−0.837266	−	0.546796i	$-0.815847\pi$
$263$	−7142.12	−1.67453	−0.837266	+	0.546796i	$0.815847\pi$
$264$	1139.68	0.265690
$265$	2207.89	0.511809
$266$	0	0
$267$	−3868.18	−0.886625
$268$	−1655.18	−0.377262
$269$	5081.26	1.15171	0.575855	−	0.817552i	$-0.304669\pi$
$269$	5081.26	1.15171	0.575855	+	0.817552i	$0.304669\pi$
$270$	455.890	0.102758
$271$	−2973.80	−0.666590	−0.333295	−	0.942823i	$-0.608160\pi$
$271$	−2973.80	−0.666590	−0.333295	+	0.942823i	$0.608160\pi$
$272$	−1241.77	−0.276813
$273$	1916.47	0.424871
$274$	4827.37	1.06435
$275$	2551.24	0.559438
$276$	−2051.46	−0.447403
$277$	7778.48	1.68723	0.843617	−	0.536946i	$-0.180422\pi$
$277$	7778.48	1.68723	0.843617	+	0.536946i	$0.180422\pi$
$278$	−6218.09	−1.34150
$279$	2511.94	0.539018
$280$	637.734	0.136114
$281$	2220.68	0.471440	0.235720	−	0.971821i	$-0.424255\pi$
$281$	2220.68	0.471440	0.235720	+	0.971821i	$0.424255\pi$
$282$	1244.06	0.262704
$283$	−7616.34	−1.59980	−0.799902	−	0.600130i	$-0.795115\pi$
$283$	−7616.34	−1.59980	−0.799902	+	0.600130i	$0.795115\pi$
$284$	−631.781	−0.132005
$285$	0	0
$286$	−6425.35	−1.32846
$287$	−678.166	−0.139480
$288$	288.000	0.0589256
$289$	1110.38	0.226009
$290$	−4054.43	−0.820981
$291$	−416.286	−0.0838596
$292$	−4408.58	−0.883537
$293$	−9036.61	−1.80179	−0.900894	−	0.434038i	$-0.857088\pi$
$293$	−9036.61	−1.80179	−0.900894	+	0.434038i	$0.857088\pi$
$294$	1523.04	0.302128
$295$	185.579	0.0366265
$296$	−160.493	−0.0315151
$297$	1282.13	0.250495
$298$	−5838.21	−1.13489
$299$	11565.9	2.23703
$300$	644.707	0.124074
$301$	−1391.04	−0.266372
$302$	−3831.43	−0.730047
$303$	208.892	0.0396057
$304$	0	0
$305$	5513.77	1.03514
$306$	−1396.99	−0.260982
$307$	7797.17	1.44954	0.724769	−	0.688992i	$-0.241947\pi$
$307$	7797.17	1.44954	0.724769	+	0.688992i	$0.241947\pi$
$308$	1793.55	0.331808
$309$	1933.04	0.355879
$310$	−4712.63	−0.863418
$311$	2737.49	0.499128	0.249564	−	0.968358i	$-0.419713\pi$
$311$	2737.49	0.499128	0.249564	+	0.968358i	$0.419713\pi$
$312$	−1623.71	−0.294629
$313$	−7646.15	−1.38079	−0.690393	−	0.723435i	$-0.742562\pi$
$313$	−7646.15	−1.38079	−0.690393	+	0.723435i	$0.742562\pi$
$314$	−5073.37	−0.911806
$315$	717.451	0.128330
$316$	−1671.25	−0.297517
$317$	−6420.39	−1.13755	−0.568777	−	0.822491i	$-0.692583\pi$
$317$	−6420.39	−1.13755	−0.568777	+	0.822491i	$0.692583\pi$
$318$	1569.14	0.276707
$319$	−11402.6	−2.00132
$320$	−540.315	−0.0943891
$321$	568.363	0.0988253
$322$	−3228.45	−0.558741
$323$	0	0
$324$	324.000	0.0555556
$325$	−3634.78	−0.620373
$326$	−3463.04	−0.588344
$327$	717.457	0.121332
$328$	574.570	0.0967236
$329$	1957.82	0.328079
$330$	−2405.40	−0.401252
$331$	−4594.77	−0.762996	−0.381498	−	0.924370i	$-0.624592\pi$
$331$	−4594.77	−0.762996	−0.381498	+	0.924370i	$0.624592\pi$
$332$	5727.21	0.946751
$333$	−180.555	−0.0297128
$334$	−2860.25	−0.468580
$335$	3493.43	0.569751
$336$	453.236	0.0735894
$337$	6611.27	1.06866	0.534331	−	0.845276i	$-0.320564\pi$
$337$	6611.27	1.06866	0.534331	+	0.845276i	$0.320564\pi$
$338$	4760.26	0.766048
$339$	−488.228	−0.0782209
$340$	2620.88	0.418050
$341$	−13253.7	−2.10477
$342$	0	0
$343$	5635.62	0.887157
$344$	1178.54	0.184717
$345$	4329.81	0.675679
$346$	586.649	0.0911516
$347$	−2498.63	−0.386552	−0.193276	−	0.981144i	$-0.561911\pi$
$347$	−2498.63	−0.386552	−0.193276	+	0.981144i	$0.561911\pi$
$348$	−2881.47	−0.443860
$349$	10330.5	1.58447	0.792235	−	0.610216i	$-0.208917\pi$
$349$	10330.5	1.58447	0.792235	+	0.610216i	$0.208917\pi$
$350$	1014.60	0.154950
$351$	−1826.67	−0.277779
$352$	−1519.57	−0.230094
$353$	2608.83	0.393354	0.196677	−	0.980468i	$-0.436985\pi$
$353$	2608.83	0.393354	0.196677	+	0.980468i	$0.436985\pi$
$354$	131.890	0.0198020
$355$	1333.44	0.199357
$356$	5157.57	0.767839
$357$	−2198.49	−0.325928
$358$	−7687.03	−1.13484
$359$	4150.33	0.610156	0.305078	−	0.952327i	$-0.401317\pi$
$359$	4150.33	0.610156	0.305078	+	0.952327i	$0.401317\pi$
$360$	−607.854	−0.0889909
$361$	0	0
$362$	3538.56	0.513764
$363$	−2771.89	−0.400790
$364$	−2555.29	−0.367949
$365$	9304.77	1.33434
$366$	3918.62	0.559643
$367$	−7370.48	−1.04833	−0.524163	−	0.851618i	$-0.675622\pi$
$367$	−7370.48	−1.04833	−0.524163	+	0.851618i	$0.675622\pi$
$368$	2735.27	0.387462
$369$	646.391	0.0911918
$370$	338.738	0.0475950
$371$	2469.41	0.345567
$372$	−3349.25	−0.466803
$373$	13166.5	1.82772	0.913858	−	0.406035i	$-0.133089\pi$
$373$	13166.5	1.82772	0.913858	+	0.406035i	$0.133089\pi$
$374$	7370.89	1.01909
$375$	−4526.63	−0.623344
$376$	−1658.74	−0.227508
$377$	16245.4	2.21931
$378$	509.890	0.0693808
$379$	−14284.2	−1.93596	−0.967978	−	0.251034i	$-0.919230\pi$
$379$	−14284.2	−1.93596	−0.967978	+	0.251034i	$0.919230\pi$
$380$	0	0
$381$	292.135	0.0392822
$382$	−1008.01	−0.135012
$383$	−9458.62	−1.26191	−0.630957	−	0.775818i	$-0.717337\pi$
$383$	−9458.62	−1.26191	−0.630957	+	0.775818i	$0.717337\pi$
$384$	−384.000	−0.0510310
$385$	−3785.47	−0.501105
$386$	−4335.17	−0.571644
$387$	1325.86	0.174153
$388$	555.048	0.0726245
$389$	−7100.96	−0.925534	−0.462767	−	0.886480i	$-0.653143\pi$
$389$	−7100.96	−0.925534	−0.462767	+	0.886480i	$0.653143\pi$
$390$	3427.00	0.444957
$391$	−13267.9	−1.71607
$392$	−2030.73	−0.261651
$393$	−1138.19	−0.146092
$394$	4222.20	0.539877
$395$	3527.35	0.449317
$396$	−1709.51	−0.216935
$397$	1916.16	0.242240	0.121120	−	0.992638i	$-0.461351\pi$
$397$	1916.16	0.242240	0.121120	+	0.992638i	$0.461351\pi$
$398$	4277.19	0.538684
$399$	0	0
$400$	−859.610	−0.107451
$401$	−5568.69	−0.693484	−0.346742	−	0.937961i	$-0.612712\pi$
$401$	−5568.69	−0.693484	−0.346742	+	0.937961i	$0.612712\pi$
$402$	2482.77	0.308033
$403$	18882.7	2.33403
$404$	−278.522	−0.0342995
$405$	−683.836	−0.0839014
$406$	−4534.68	−0.554316
$407$	952.658	0.116023
$408$	1862.65	0.226017
$409$	72.7133	0.00879081	0.00439541	−	0.999990i	$-0.498601\pi$
$409$	72.7133	0.00879081	0.00439541	+	0.999990i	$0.498601\pi$
$410$	−1212.69	−0.146074
$411$	−7241.05	−0.869038
$412$	−2577.38	−0.308201
$413$	207.561	0.0247298
$414$	3077.18	0.365303
$415$	−12087.9	−1.42981
$416$	2164.94	0.255157
$417$	9327.13	1.09533
$418$	0	0
$419$	340.170	0.0396620	0.0198310	−	0.999803i	$-0.493687\pi$
$419$	340.170	0.0396620	0.0198310	+	0.999803i	$0.493687\pi$
$420$	−956.602	−0.111137
$421$	1476.99	0.170984	0.0854920	−	0.996339i	$-0.472754\pi$
$421$	1476.99	0.170984	0.0854920	+	0.996339i	$0.472754\pi$
$422$	−1951.91	−0.225160
$423$	−1866.09	−0.214497
$424$	−2092.18	−0.239635
$425$	4169.67	0.475903
$426$	947.672	0.107781
$427$	6166.87	0.698913
$428$	−757.817	−0.0855853
$429$	9638.02	1.08468
$430$	−2487.44	−0.278965
$431$	−12279.0	−1.37230	−0.686149	−	0.727461i	$-0.740700\pi$
$431$	−12279.0	−1.37230	−0.686149	+	0.727461i	$0.740700\pi$
$432$	−432.000	−0.0481125
$433$	6211.53	0.689393	0.344697	−	0.938714i	$-0.387982\pi$
$433$	6211.53	0.689393	0.344697	+	0.938714i	$0.387982\pi$
$434$	−5270.84	−0.582969
$435$	6081.65	0.670328
$436$	−956.609	−0.105076
$437$	0	0
$438$	6612.87	0.721405
$439$	−13469.0	−1.46433	−0.732164	−	0.681129i	$-0.761489\pi$
$439$	−13469.0	−1.46433	−0.732164	+	0.681129i	$0.761489\pi$
$440$	3207.20	0.347494
$441$	−2284.57	−0.246687
$442$	−10501.4	−1.13009
$443$	7411.09	0.794834	0.397417	−	0.917638i	$-0.369907\pi$
$443$	7411.09	0.794834	0.397417	+	0.917638i	$0.369907\pi$
$444$	240.740	0.0257320
$445$	−10885.6	−1.15961
$446$	−7150.91	−0.759205
$447$	8757.31	0.926636
$448$	−604.315	−0.0637303
$449$	−3345.93	−0.351680	−0.175840	−	0.984419i	$-0.556264\pi$
$449$	−3345.93	−0.351680	−0.175840	+	0.984419i	$0.556264\pi$
$450$	−967.061	−0.101306
$451$	−3410.54	−0.356089
$452$	650.970	0.0677413
$453$	5747.15	0.596081
$454$	−173.047	−0.0178888
$455$	5393.20	0.555686
$456$	0	0
$457$	−542.238	−0.0555029	−0.0277515	−	0.999615i	$-0.508835\pi$
$457$	−542.238	−0.0555029	−0.0277515	+	0.999615i	$0.508835\pi$
$458$	−7856.10	−0.801510
$459$	2095.48	0.213091
$460$	−5773.08	−0.585155
$461$	3823.43	0.386280	0.193140	−	0.981171i	$-0.438133\pi$
$461$	3823.43	0.386280	0.193140	+	0.981171i	$0.438133\pi$
$462$	−2690.32	−0.270920
$463$	−3750.37	−0.376445	−0.188223	−	0.982126i	$-0.560273\pi$
$463$	−3750.37	−0.376445	−0.188223	+	0.982126i	$0.560273\pi$
$464$	3841.96	0.384394
$465$	7068.95	0.704978
$466$	2040.05	0.202797
$467$	−2981.61	−0.295444	−0.147722	−	0.989029i	$-0.547194\pi$
$467$	−2981.61	−0.295444	−0.147722	+	0.989029i	$0.547194\pi$
$468$	2435.56	0.240564
$469$	3907.23	0.384689
$470$	3500.95	0.343589
$471$	7610.06	0.744486
$472$	−175.854	−0.0171490
$473$	−6995.60	−0.680038
$474$	2506.88	0.242921
$475$	0	0
$476$	2931.32	0.282262
$477$	−2353.71	−0.225930
$478$	529.362	0.0506537
$479$	15355.8	1.46477	0.732387	−	0.680889i	$-0.238406\pi$
$479$	15355.8	1.46477	0.732387	+	0.680889i	$0.238406\pi$
$480$	810.472	0.0770684
$481$	−1357.26	−0.128661
$482$	−344.774	−0.0325810
$483$	4842.68	0.456210
$484$	3695.86	0.347094
$485$	−1171.49	−0.109679
$486$	−486.000	−0.0453609
$487$	439.463	0.0408911	0.0204455	−	0.999791i	$-0.493492\pi$
$487$	439.463	0.0408911	0.0204455	+	0.999791i	$0.493492\pi$
$488$	−5224.83	−0.484665
$489$	5194.56	0.480381
$490$	4286.06	0.395152
$491$	19765.8	1.81674	0.908370	−	0.418168i	$-0.137328\pi$
$491$	19765.8	1.81674	0.908370	+	0.418168i	$0.137328\pi$
$492$	−861.855	−0.0789745
$493$	−18636.0	−1.70248
$494$	0	0
$495$	3608.10	0.327621
$496$	4465.67	0.404263
$497$	1491.38	0.134603
$498$	−8590.81	−0.773019
$499$	8159.72	0.732023	0.366011	−	0.930610i	$-0.380723\pi$
$499$	8159.72	0.732023	0.366011	+	0.930610i	$0.380723\pi$
$500$	6035.50	0.539832
$501$	4290.37	0.382594
$502$	−9769.73	−0.868614
$503$	−19674.3	−1.74400	−0.872002	−	0.489503i	$-0.837178\pi$
$503$	−19674.3	−1.74400	−0.872002	+	0.489503i	$0.837178\pi$
$504$	−679.854	−0.0600855
$505$	587.850	0.0518000
$506$	−16236.1	−1.42645
$507$	−7140.39	−0.625475
$508$	−389.513	−0.0340194
$509$	−6855.74	−0.597005	−0.298502	−	0.954409i	$-0.596487\pi$
$509$	−6855.74	−0.597005	−0.298502	+	0.954409i	$0.596487\pi$
$510$	−3931.32	−0.341337
$511$	10406.9	0.900929
$512$	512.000	0.0441942
$513$	0	0
$514$	8045.28	0.690393
$515$	5439.84	0.465452
$516$	−1767.81	−0.150821
$517$	9845.99	0.837574
$518$	378.861	0.0321355
$519$	−879.974	−0.0744250
$520$	−4569.34	−0.385344
$521$	−12666.6	−1.06513	−0.532567	−	0.846388i	$-0.678773\pi$
$521$	−12666.6	−1.06513	−0.532567	+	0.846388i	$0.678773\pi$
$522$	4322.21	0.362410
$523$	4717.23	0.394398	0.197199	−	0.980363i	$-0.436815\pi$
$523$	4717.23	0.394398	0.197199	+	0.980363i	$0.436815\pi$
$524$	1517.59	0.126520
$525$	−1521.90	−0.126516
$526$	−14284.2	−1.18407
$527$	−21661.4	−1.79049
$528$	2279.35	0.187871
$529$	17058.5	1.40203
$530$	4415.77	0.361904
$531$	−197.836	−0.0161682
$532$	0	0
$533$	4859.03	0.394874
$534$	−7736.36	−0.626938
$535$	1599.45	0.129253
$536$	−3310.36	−0.266765
$537$	11530.5	0.926592
$538$	10162.5	0.814382
$539$	12054.0	0.963270
$540$	911.781	0.0726607
$541$	11741.2	0.933072	0.466536	−	0.884502i	$-0.345502\pi$
$541$	11741.2	0.933072	0.466536	+	0.884502i	$0.345502\pi$
$542$	−5947.61	−0.471350
$543$	−5307.83	−0.419486
$544$	−2483.53	−0.195736
$545$	2019.02	0.158689
$546$	3832.93	0.300429
$547$	−22262.1	−1.74015	−0.870073	−	0.492922i	$-0.835929\pi$
$547$	−22262.1	−1.74015	−0.870073	+	0.492922i	$0.835929\pi$
$548$	9654.73	0.752609
$549$	−5877.93	−0.456947
$550$	5102.48	0.395583
$551$	0	0
$552$	−4102.91	−0.316362
$553$	3945.16	0.303373
$554$	15557.0	1.19305
$555$	−508.107	−0.0388611
$556$	−12436.2	−0.948582
$557$	−19627.7	−1.49310	−0.746548	−	0.665332i	$-0.768290\pi$
$557$	−19627.7	−1.49310	−0.746548	+	0.665332i	$0.768290\pi$
$558$	5023.88	0.381143
$559$	9966.71	0.754109
$560$	1275.47	0.0962471
$561$	−11056.3	−0.832084
$562$	4441.36	0.333358
$563$	9810.37	0.734384	0.367192	−	0.930145i	$-0.380319\pi$
$563$	9810.37	0.734384	0.367192	+	0.930145i	$0.380319\pi$
$564$	2488.12	0.185760
$565$	−1373.94	−0.102305
$566$	−15232.7	−1.13123
$567$	−764.836	−0.0566492
$568$	−1263.56	−0.0933414
$569$	2528.37	0.186282	0.0931412	−	0.995653i	$-0.470309\pi$
$569$	2528.37	0.186282	0.0931412	+	0.995653i	$0.470309\pi$
$570$	0	0
$571$	−21892.0	−1.60447	−0.802233	−	0.597011i	$-0.796355\pi$
$571$	−21892.0	−1.60447	−0.802233	+	0.597011i	$0.796355\pi$
$572$	−12850.7	−0.939361
$573$	1512.02	0.110236
$574$	−1356.33	−0.0986276
$575$	−9184.64	−0.666132
$576$	576.000	0.0416667
$577$	−4002.61	−0.288788	−0.144394	−	0.989520i	$-0.546123\pi$
$577$	−4002.61	−0.288788	−0.144394	+	0.989520i	$0.546123\pi$
$578$	2220.76	0.159812
$579$	6502.76	0.466745
$580$	−8108.86	−0.580521
$581$	−13519.7	−0.965388
$582$	−832.573	−0.0592977
$583$	12418.8	0.882220
$584$	−8817.16	−0.624755
$585$	−5140.51	−0.363306
$586$	−18073.2	−1.27406
$587$	24518.5	1.72400	0.861998	−	0.506912i	$-0.169213\pi$
$587$	24518.5	1.72400	0.861998	+	0.506912i	$0.169213\pi$
$588$	3046.09	0.213637
$589$	0	0
$590$	371.158	0.0258989
$591$	−6333.30	−0.440807
$592$	−320.987	−0.0222846
$593$	−17698.7	−1.22563	−0.612815	−	0.790226i	$-0.709963\pi$
$593$	−17698.7	−1.22563	−0.612815	+	0.790226i	$0.709963\pi$
$594$	2564.27	0.177127
$595$	−6186.86	−0.426280
$596$	−11676.4	−0.802491
$597$	−6415.78	−0.439833
$598$	23131.7	1.58182
$599$	3608.01	0.246109	0.123054	−	0.992400i	$-0.460731\pi$
$599$	3608.01	0.246109	0.123054	+	0.992400i	$0.460731\pi$
$600$	1289.41	0.0877336
$601$	23195.0	1.57429	0.787143	−	0.616771i	$-0.211559\pi$
$601$	23195.0	1.57429	0.787143	+	0.616771i	$0.211559\pi$
$602$	−2782.07	−0.188353
$603$	−3724.16	−0.251508
$604$	−7662.86	−0.516221
$605$	−7800.50	−0.524190
$606$	417.783	0.0280054
$607$	−3485.49	−0.233067	−0.116534	−	0.993187i	$-0.537178\pi$
$607$	−3485.49	−0.233067	−0.116534	+	0.993187i	$0.537178\pi$
$608$	0	0
$609$	6802.02	0.452597
$610$	11027.5	0.731954
$611$	−14027.7	−0.928804
$612$	−2793.98	−0.184542
$613$	−10953.2	−0.721692	−0.360846	−	0.932625i	$-0.617512\pi$
$613$	−10953.2	−0.721692	−0.360846	+	0.932625i	$0.617512\pi$
$614$	15594.3	1.02498
$615$	1819.03	0.119269
$616$	3587.10	0.234624
$617$	7009.52	0.457362	0.228681	−	0.973501i	$-0.426559\pi$
$617$	7009.52	0.457362	0.228681	+	0.973501i	$0.426559\pi$
$618$	3866.08	0.251645
$619$	24225.1	1.57301	0.786503	−	0.617587i	$-0.211890\pi$
$619$	24225.1	1.57301	0.786503	+	0.617587i	$0.211890\pi$
$620$	−9425.26	−0.610529
$621$	−4615.78	−0.298268
$622$	5474.98	0.352937
$623$	−12175.0	−0.782954
$624$	−3247.42	−0.208334
$625$	−6022.86	−0.385463
$626$	−15292.3	−0.976363
$627$	0	0
$628$	−10146.7	−0.644744
$629$	1556.99	0.0986986
$630$	1434.90	0.0907427
$631$	14904.5	0.940314	0.470157	−	0.882583i	$-0.344197\pi$
$631$	14904.5	0.940314	0.470157	+	0.882583i	$0.344197\pi$
$632$	−3342.50	−0.210376
$633$	2927.86	0.183842
$634$	−12840.8	−0.804373
$635$	822.108	0.0513769
$636$	3138.28	0.195662
$637$	−17173.5	−1.06819
$638$	−22805.2	−1.41515
$639$	−1421.51	−0.0880031
$640$	−1080.63	−0.0667432
$641$	−16749.1	−1.03206	−0.516028	−	0.856572i	$-0.672590\pi$
$641$	−16749.1	−1.03206	−0.516028	+	0.856572i	$0.672590\pi$
$642$	1136.73	0.0698801
$643$	−7109.61	−0.436043	−0.218022	−	0.975944i	$-0.569960\pi$
$643$	−7109.61	−0.436043	−0.218022	+	0.975944i	$0.569960\pi$
$644$	−6456.90	−0.395089
$645$	3731.15	0.227774
$646$	0	0
$647$	6637.82	0.403338	0.201669	−	0.979454i	$-0.435363\pi$
$647$	6637.82	0.403338	0.201669	+	0.979454i	$0.435363\pi$
$648$	648.000	0.0392837
$649$	1043.84	0.0631342
$650$	−7269.56	−0.438670
$651$	7906.26	0.475992
$652$	−6926.08	−0.416022
$653$	−18762.0	−1.12437	−0.562184	−	0.827012i	$-0.690039\pi$
$653$	−18762.0	−1.12437	−0.562184	+	0.827012i	$0.690039\pi$
$654$	1434.91	0.0857944
$655$	−3203.03	−0.191073
$656$	1149.14	0.0683939
$657$	−9919.31	−0.589024
$658$	3915.64	0.231987
$659$	6156.57	0.363924	0.181962	−	0.983306i	$-0.441755\pi$
$659$	6156.57	0.363924	0.181962	+	0.983306i	$0.441755\pi$
$660$	−4810.81	−0.283728
$661$	26165.2	1.53965	0.769824	−	0.638257i	$-0.220344\pi$
$661$	26165.2	1.53965	0.769824	+	0.638257i	$0.220344\pi$
$662$	−9189.55	−0.539520
$663$	15752.1	0.922715
$664$	11454.4	0.669454
$665$	0	0
$666$	−361.110	−0.0210101
$667$	41050.1	2.38301
$668$	−5720.50	−0.331336
$669$	10726.4	0.619888
$670$	6986.87	0.402875
$671$	31013.6	1.78430
$672$	906.472	0.0520356
$673$	31028.8	1.77723	0.888613	−	0.458657i	$-0.151669\pi$
$673$	31028.8	1.77723	0.888613	+	0.458657i	$0.151669\pi$
$674$	13222.5	0.755658
$675$	1450.59	0.0827160
$676$	9520.52	0.541677
$677$	−4341.55	−0.246469	−0.123234	−	0.992378i	$-0.539327\pi$
$677$	−4341.55	−0.246469	−0.123234	+	0.992378i	$0.539327\pi$
$678$	−976.455	−0.0553105
$679$	−1310.25	−0.0740542
$680$	5241.76	0.295606
$681$	259.571	0.0146062
$682$	−26507.4	−1.48830
$683$	16457.2	0.921986	0.460993	−	0.887404i	$-0.347493\pi$
$683$	16457.2	0.921986	0.460993	+	0.887404i	$0.347493\pi$
$684$	0	0
$685$	−20377.3	−1.13661
$686$	11271.2	0.627315
$687$	11784.2	0.654430
$688$	2357.08	0.130615
$689$	−17693.2	−0.978313
$690$	8659.62	0.477777
$691$	2522.21	0.138856	0.0694280	−	0.997587i	$-0.477883\pi$
$691$	2522.21	0.138856	0.0694280	+	0.997587i	$0.477883\pi$
$692$	1173.30	0.0644539
$693$	4035.48	0.221205
$694$	−4997.26	−0.273333
$695$	26247.8	1.43257
$696$	−5762.95	−0.313856
$697$	−5574.08	−0.302917
$698$	20661.1	1.12039
$699$	−3060.08	−0.165583
$700$	2029.20	0.109566
$701$	4019.82	0.216586	0.108293	−	0.994119i	$-0.465462\pi$
$701$	4019.82	0.216586	0.108293	+	0.994119i	$0.465462\pi$
$702$	−3653.34	−0.196420
$703$	0	0
$704$	−3039.13	−0.162701
$705$	−5251.43	−0.280539
$706$	5217.65	0.278143
$707$	657.481	0.0349747
$708$	263.781	0.0140021
$709$	−10052.9	−0.532503	−0.266252	−	0.963904i	$-0.585785\pi$
$709$	−10052.9	−0.532503	−0.266252	+	0.963904i	$0.585785\pi$
$710$	2666.88	0.140966
$711$	−3760.32	−0.198344
$712$	10315.1	0.542944
$713$	47714.2	2.50619
$714$	−4396.98	−0.230466
$715$	27122.7	1.41865
$716$	−15374.1	−0.802452
$717$	−794.043	−0.0413585
$718$	8300.66	0.431445
$719$	10430.4	0.541011	0.270505	−	0.962718i	$-0.412809\pi$
$719$	10430.4	0.541011	0.270505	+	0.962718i	$0.412809\pi$
$720$	−1215.71	−0.0629261
$721$	6084.18	0.314268
$722$	0	0
$723$	517.161	0.0266022
$724$	7077.11	0.363286
$725$	−12900.7	−0.660857
$726$	−5543.79	−0.283401
$727$	−21693.5	−1.10669	−0.553346	−	0.832951i	$-0.686649\pi$
$727$	−21693.5	−1.10669	−0.553346	+	0.832951i	$0.686649\pi$
$728$	−5110.58	−0.260179
$729$	729.000	0.0370370
$730$	18609.5	0.943520
$731$	−11433.4	−0.578495
$732$	7837.24	0.395728
$733$	−3382.86	−0.170462	−0.0852310	−	0.996361i	$-0.527163\pi$
$733$	−3382.86	−0.170462	−0.0852310	+	0.996361i	$0.527163\pi$
$734$	−14741.0	−0.741279
$735$	−6429.09	−0.322640
$736$	5470.55	0.273977
$737$	19649.7	0.982097
$738$	1292.78	0.0644824
$739$	8017.89	0.399111	0.199555	−	0.979887i	$-0.436050\pi$
$739$	8017.89	0.399111	0.199555	+	0.979887i	$0.436050\pi$
$740$	677.476	0.0336547
$741$	0	0
$742$	4938.82	0.244353
$743$	9357.91	0.462057	0.231029	−	0.972947i	$-0.425791\pi$
$743$	9357.91	0.462057	0.231029	+	0.972947i	$0.425791\pi$
$744$	−6698.51	−0.330080
$745$	24644.3	1.21194
$746$	26333.1	1.29239
$747$	12886.2	0.631167
$748$	14741.8	0.720606
$749$	1788.91	0.0872700
$750$	−9053.26	−0.440771
$751$	−12967.3	−0.630070	−0.315035	−	0.949080i	$-0.602016\pi$
$751$	−12967.3	−0.630070	−0.315035	+	0.949080i	$0.602016\pi$
$752$	−3317.49	−0.160873
$753$	14654.6	0.709221
$754$	32490.8	1.56929
$755$	16173.3	0.779610
$756$	1019.78	0.0490596
$757$	27781.4	1.33386	0.666931	−	0.745119i	$-0.267608\pi$
$757$	27781.4	1.33386	0.666931	+	0.745119i	$0.267608\pi$
$758$	−28568.3	−1.36893
$759$	24354.1	1.16469
$760$	0	0
$761$	26066.1	1.24165	0.620825	−	0.783949i	$-0.286798\pi$
$761$	26066.1	1.24165	0.620825	+	0.783949i	$0.286798\pi$
$762$	584.270	0.0277767
$763$	2258.18	0.107145
$764$	−2016.03	−0.0954676
$765$	5896.98	0.278700
$766$	−18917.2	−0.892307
$767$	−1487.16	−0.0700109
$768$	−768.000	−0.0360844
$769$	−36835.4	−1.72733	−0.863665	−	0.504066i	$-0.831837\pi$
$769$	−36835.4	−1.72733	−0.863665	+	0.504066i	$0.831837\pi$
$770$	−7570.94	−0.354335
$771$	−12067.9	−0.563703
$772$	−8670.35	−0.404213
$773$	11208.4	0.521524	0.260762	−	0.965403i	$-0.416026\pi$
$773$	11208.4	0.521524	0.260762	+	0.965403i	$0.416026\pi$
$774$	2651.72	0.123145
$775$	−14995.1	−0.695017
$776$	1110.10	0.0513533
$777$	−568.292	−0.0262385
$778$	−14201.9	−0.654452
$779$	0	0
$780$	6854.01	0.314632
$781$	7500.26	0.343637
$782$	−26535.7	−1.21345
$783$	−6483.31	−0.295906
$784$	−4061.45	−0.185015
$785$	21415.8	0.973709
$786$	−2276.39	−0.103303
$787$	−25466.9	−1.15349	−0.576745	−	0.816924i	$-0.695677\pi$
$787$	−25466.9	−1.15349	−0.576745	+	0.816924i	$0.695677\pi$
$788$	8444.40	0.381750
$789$	21426.4	0.966791
$790$	7054.70	0.317715
$791$	−1536.68	−0.0690748
$792$	−3419.03	−0.153396
$793$	−44185.4	−1.97865
$794$	3832.32	0.171290
$795$	−6623.66	−0.295493
$796$	8554.38	0.380907
$797$	31745.4	1.41089	0.705446	−	0.708764i	$-0.250747\pi$
$797$	31745.4	1.41089	0.705446	+	0.708764i	$0.250747\pi$
$798$	0	0
$799$	16092.0	0.712507
$800$	−1719.22	−0.0759795
$801$	11604.5	0.511893
$802$	−11137.4	−0.490367
$803$	52337.0	2.30004
$804$	4965.55	0.217813
$805$	13628.0	0.596674
$806$	37765.3	1.65041
$807$	−15243.8	−0.664940
$808$	−557.045	−0.0242534
$809$	−14106.2	−0.613040	−0.306520	−	0.951864i	$-0.599165\pi$
$809$	−14106.2	−0.613040	−0.306520	+	0.951864i	$0.599165\pi$
$810$	−1367.67	−0.0593273
$811$	42675.4	1.84776	0.923882	−	0.382678i	$-0.124998\pi$
$811$	42675.4	1.84776	0.923882	+	0.382678i	$0.124998\pi$
$812$	−9069.35	−0.391961
$813$	8921.41	0.384856
$814$	1905.32	0.0820409
$815$	14618.2	0.628287
$816$	3725.30	0.159818
$817$	0	0
$818$	145.427	0.00621604
$819$	−5749.40	−0.245299
$820$	−2425.38	−0.103290
$821$	−19439.3	−0.826354	−0.413177	−	0.910651i	$-0.635581\pi$
$821$	−19439.3	−0.826354	−0.413177	+	0.910651i	$0.635581\pi$
$822$	−14482.1	−0.614503
$823$	−25023.4	−1.05986	−0.529928	−	0.848042i	$-0.677781\pi$
$823$	−25023.4	−1.05986	−0.529928	+	0.848042i	$0.677781\pi$
$824$	−5154.77	−0.217931
$825$	−7653.72	−0.322992
$826$	415.122	0.0174866
$827$	−33545.7	−1.41052	−0.705260	−	0.708949i	$-0.749170\pi$
$827$	−33545.7	−1.41052	−0.705260	+	0.708949i	$0.749170\pi$
$828$	6154.37	0.258308
$829$	5555.52	0.232752	0.116376	−	0.993205i	$-0.462872\pi$
$829$	5555.52	0.232752	0.116376	+	0.993205i	$0.462872\pi$
$830$	−24175.7	−1.01103
$831$	−23335.4	−0.974124
$832$	4329.89	0.180423
$833$	19700.7	0.819434
$834$	18654.3	0.774514
$835$	12073.7	0.500392
$836$	0	0
$837$	−7535.82	−0.311202
$838$	680.340	0.0280453
$839$	40982.4	1.68638	0.843188	−	0.537618i	$-0.180676\pi$
$839$	40982.4	1.68638	0.843188	+	0.537618i	$0.180676\pi$
$840$	−1913.20	−0.0785855
$841$	33269.9	1.36414
$842$	2953.99	0.120904
$843$	−6662.04	−0.272186
$844$	−3903.81	−0.159212
$845$	−20094.1	−0.818055
$846$	−3732.17	−0.151672
$847$	−8724.46	−0.353927
$848$	−4184.37	−0.169448
$849$	22849.0	0.923648
$850$	8339.34	0.336514
$851$	−3429.64	−0.138151
$852$	1895.34	0.0762129
$853$	−31609.3	−1.26879	−0.634397	−	0.773007i	$-0.718752\pi$
$853$	−31609.3	−1.26879	−0.634397	+	0.773007i	$0.718752\pi$
$854$	12333.7	0.494206
$855$	0	0
$856$	−1515.63	−0.0605179
$857$	20963.7	0.835596	0.417798	−	0.908540i	$-0.362802\pi$
$857$	20963.7	0.835596	0.417798	+	0.908540i	$0.362802\pi$
$858$	19276.0	0.766985
$859$	3896.64	0.154775	0.0773875	−	0.997001i	$-0.475342\pi$
$859$	3896.64	0.154775	0.0773875	+	0.997001i	$0.475342\pi$
$860$	−4974.87	−0.197258
$861$	2034.50	0.0805291
$862$	−24558.1	−0.970361
$863$	2630.83	0.103771	0.0518856	−	0.998653i	$-0.483477\pi$
$863$	2630.83	0.103771	0.0518856	+	0.998653i	$0.483477\pi$
$864$	−864.000	−0.0340207
$865$	−2476.37	−0.0973400
$866$	12423.1	0.487475
$867$	−3331.14	−0.130486
$868$	−10541.7	−0.412221
$869$	19840.5	0.774501
$870$	12163.3	0.473993
$871$	−27995.1	−1.08907
$872$	−1913.22	−0.0743002
$873$	1248.86	0.0484164
$874$	0	0
$875$	−14247.4	−0.550459
$876$	13225.7	0.510110
$877$	−36577.3	−1.40836	−0.704178	−	0.710023i	$-0.748684\pi$
$877$	−36577.3	−1.40836	−0.704178	+	0.710023i	$0.748684\pi$
$878$	−26938.0	−1.03544
$879$	27109.8	1.04026
$880$	6414.41	0.245716
$881$	6389.06	0.244328	0.122164	−	0.992510i	$-0.461017\pi$
$881$	6389.06	0.244328	0.122164	+	0.992510i	$0.461017\pi$
$882$	−4569.13	−0.174434
$883$	−17737.8	−0.676020	−0.338010	−	0.941142i	$-0.609754\pi$
$883$	−17737.8	−0.676020	−0.338010	+	0.941142i	$0.609754\pi$
$884$	−21002.8	−0.799095
$885$	−556.737	−0.0211463
$886$	14822.2	0.562033
$887$	29621.0	1.12128	0.560641	−	0.828059i	$-0.310555\pi$
$887$	29621.0	1.12128	0.560641	+	0.828059i	$0.310555\pi$
$888$	481.480	0.0181953
$889$	919.486	0.0346891
$890$	−21771.2	−0.819968
$891$	−3846.40	−0.144623
$892$	−14301.8	−0.536839
$893$	0	0
$894$	17514.6	0.655231
$895$	32448.6	1.21188
$896$	−1208.63	−0.0450641
$897$	−34697.6	−1.29155
$898$	−6691.87	−0.248675
$899$	67019.3	2.48634
$900$	−1934.12	−0.0716341
$901$	20296.9	0.750486
$902$	−6821.08	−0.251793
$903$	4173.11	0.153790
$904$	1301.94	0.0479003
$905$	−14937.0	−0.548643
$906$	11494.3	0.421493
$907$	14352.3	0.525423	0.262712	−	0.964874i	$-0.415383\pi$
$907$	14352.3	0.525423	0.262712	+	0.964874i	$0.415383\pi$
$908$	−346.095	−0.0126493
$909$	−626.675	−0.0228663
$910$	10786.4	0.392929
$911$	50.5532	0.00183853	0.000919266	−	1.00000i	$-0.499707\pi$
$911$	50.5532	0.00183853	0.000919266		1.00000i	$0.499707\pi$
$912$	0	0
$913$	−67991.2	−2.46460
$914$	−1084.48	−0.0392465
$915$	−16541.3	−0.597638
$916$	−15712.2	−0.566753
$917$	−3582.43	−0.129010
$918$	4190.96	0.150678
$919$	−38362.9	−1.37701	−0.688507	−	0.725230i	$-0.741734\pi$
$919$	−38362.9	−1.37701	−0.688507	+	0.725230i	$0.741734\pi$
$920$	−11546.2	−0.413767
$921$	−23391.5	−0.836891
$922$	7646.86	0.273141
$923$	−10685.7	−0.381066
$924$	−5380.64	−0.191569
$925$	1077.83	0.0383121
$926$	−7500.73	−0.266187
$927$	−5799.12	−0.205467
$928$	7683.93	0.271807
$929$	−38087.0	−1.34510	−0.672548	−	0.740053i	$-0.734800\pi$
$929$	−38087.0	−1.34510	−0.672548	+	0.740053i	$0.734800\pi$
$930$	14137.9	0.498494
$931$	0	0
$932$	4080.10	0.143399
$933$	−8212.47	−0.288172
$934$	−5963.22	−0.208910
$935$	−31114.1	−1.08828
$936$	4871.12	0.170104
$937$	44240.1	1.54243	0.771217	−	0.636573i	$-0.219649\pi$
$937$	44240.1	1.54243	0.771217	+	0.636573i	$0.219649\pi$
$938$	7814.46	0.272016
$939$	22938.5	0.797197
$940$	7001.90	0.242954
$941$	24209.1	0.838675	0.419338	−	0.907830i	$-0.362262\pi$
$941$	24209.1	0.838675	0.419338	+	0.907830i	$0.362262\pi$
$942$	15220.1	0.526431
$943$	12278.2	0.424001
$944$	−351.708	−0.0121262
$945$	−2152.35	−0.0740911
$946$	−13991.2	−0.480860
$947$	−20524.7	−0.704292	−0.352146	−	0.935945i	$-0.614548\pi$
$947$	−20524.7	−0.704292	−0.352146	+	0.935945i	$0.614548\pi$
$948$	5013.76	0.171771
$949$	−74565.1	−2.55056
$950$	0	0
$951$	19261.2	0.656768
$952$	5862.64	0.199590
$953$	−4911.36	−0.166941	−0.0834703	−	0.996510i	$-0.526600\pi$
$953$	−4911.36	−0.166941	−0.0834703	+	0.996510i	$0.526600\pi$
$954$	−4707.41	−0.159757
$955$	4255.03	0.144178
$956$	1058.72	0.0358175
$957$	34207.7	1.15546
$958$	30711.7	1.03575
$959$	−22791.0	−0.767424
$960$	1620.94	0.0544956
$961$	48108.3	1.61486
$962$	−2714.52	−0.0909769
$963$	−1705.09	−0.0570568
$964$	−689.548	−0.0230382
$965$	18299.7	0.610453
$966$	9685.35	0.322589
$967$	−40394.7	−1.34334	−0.671668	−	0.740852i	$-0.734422\pi$
$967$	−40394.7	−1.34334	−0.671668	+	0.740852i	$0.734422\pi$
$968$	7391.72	0.245433
$969$	0	0
$970$	−2342.98	−0.0775550
$971$	−26791.9	−0.885471	−0.442735	−	0.896652i	$-0.645992\pi$
$971$	−26791.9	−0.885471	−0.442735	+	0.896652i	$0.645992\pi$
$972$	−972.000	−0.0320750
$973$	29356.9	0.967255
$974$	878.926	0.0289144
$975$	10904.3	0.358173
$976$	−10449.7	−0.342710
$977$	−8122.31	−0.265973	−0.132987	−	0.991118i	$-0.542457\pi$
$977$	−8122.31	−0.265973	−0.132987	+	0.991118i	$0.542457\pi$
$978$	10389.1	0.339680
$979$	−61228.7	−1.99885
$980$	8572.12	0.279415
$981$	−2152.37	−0.0700509
$982$	39531.6	1.28463
$983$	−32918.1	−1.06808	−0.534041	−	0.845459i	$-0.679327\pi$
$983$	−32918.1	−1.06808	−0.534041	+	0.845459i	$0.679327\pi$
$984$	−1723.71	−0.0558434
$985$	−17822.8	−0.576529
$986$	−37272.1	−1.20384
$987$	−5873.46	−0.189417
$988$	0	0
$989$	25184.7	0.809733
$990$	7216.21	0.231663
$991$	−31770.1	−1.01838	−0.509189	−	0.860655i	$-0.670054\pi$
$991$	−31770.1	−1.01838	−0.509189	+	0.860655i	$0.670054\pi$
$992$	8931.34	0.285857
$993$	13784.3	0.440516
$994$	2982.77	0.0951788
$995$	−18054.9	−0.575255
$996$	−17181.6	−0.546607
$997$	15506.5	0.492574	0.246287	−	0.969197i	$-0.420789\pi$
$997$	15506.5	0.492574	0.246287	+	0.969197i	$0.420789\pi$
$998$	16319.4	0.517618
$999$	541.665	0.0171547

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.u.1.1		3
19.7	even	3		114.4.e.d.49.3	yes	6
19.11	even	3		114.4.e.d.7.3	✓	6
19.18	odd	2		2166.4.a.t.1.1		3
57.11	odd	6		342.4.g.h.235.1		6
57.26	odd	6		342.4.g.h.163.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.e.d.7.3	✓	6	19.11	even	3
114.4.e.d.49.3	yes	6	19.7	even	3
342.4.g.h.163.1		6	57.26	odd	6
342.4.g.h.235.1		6	57.11	odd	6
2166.4.a.t.1.1		3	19.18	odd	2
2166.4.a.u.1.1		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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.44242 q^{5} -6.00000 q^{6} -9.44242 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -8.44242 q^{5} -6.00000 q^{6} -9.44242 q^{7} +8.00000 q^{8} +9.00000 q^{9} -16.8848 q^{10} -47.4865 q^{11} -12.0000 q^{12} +67.6545 q^{13} -18.8848 q^{14} +25.3272 q^{15} +16.0000 q^{16} -77.6104 q^{17} +18.0000 q^{18} -33.7697 q^{20} +28.3272 q^{21} -94.9729 q^{22} +170.955 q^{23} -24.0000 q^{24} -53.7256 q^{25} +135.309 q^{26} -27.0000 q^{27} -37.7697 q^{28} +240.123 q^{29} +50.6545 q^{30} +279.104 q^{31} +32.0000 q^{32} +142.459 q^{33} -155.221 q^{34} +79.7168 q^{35} +36.0000 q^{36} -20.0617 q^{37} -202.963 q^{39} -67.5393 q^{40} +71.8213 q^{41} +56.6545 q^{42} +147.318 q^{43} -189.946 q^{44} -75.9817 q^{45} +341.909 q^{46} -207.343 q^{47} -48.0000 q^{48} -253.841 q^{49} -107.451 q^{50} +232.831 q^{51} +270.618 q^{52} -261.523 q^{53} -54.0000 q^{54} +400.901 q^{55} -75.5393 q^{56} +480.245 q^{58} -21.9817 q^{59} +101.309 q^{60} -653.103 q^{61} +558.209 q^{62} -84.9817 q^{63} +64.0000 q^{64} -571.167 q^{65} +284.919 q^{66} -413.795 q^{67} -310.442 q^{68} -512.864 q^{69} +159.434 q^{70} -157.945 q^{71} +72.0000 q^{72} -1102.15 q^{73} -40.1233 q^{74} +161.177 q^{75} +448.387 q^{77} -405.927 q^{78} -417.813 q^{79} -135.079 q^{80} +81.0000 q^{81} +143.643 q^{82} +1431.80 q^{83} +113.309 q^{84} +655.220 q^{85} +294.636 q^{86} -720.368 q^{87} -379.892 q^{88} +1289.39 q^{89} -151.963 q^{90} -638.822 q^{91} +683.819 q^{92} -837.313 q^{93} -414.686 q^{94} -96.0000 q^{96} +138.762 q^{97} -507.682 q^{98} -427.378 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 10 * q^5 - 18 * q^6 + 7 * q^7 + 24 * q^8 + 27 * q^9	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} + 24 q^{8} + 27 q^{9} + 20 q^{10} - 44 q^{11} - 36 q^{12} - 9 q^{13} + 14 q^{14} - 30 q^{15} + 48 q^{16} - 84 q^{17} + 54 q^{18} + 40 q^{20} - 21 q^{21} - 88 q^{22} - 2 q^{23} - 72 q^{24} - 83 q^{25} - 18 q^{26} - 81 q^{27} + 28 q^{28} + 92 q^{29} - 60 q^{30} - 109 q^{31} + 96 q^{32} + 132 q^{33} - 168 q^{34} + 282 q^{35} + 108 q^{36} + 245 q^{37} + 27 q^{39} + 80 q^{40} - 688 q^{41} - 42 q^{42} - 103 q^{43} - 176 q^{44} + 90 q^{45} - 4 q^{46} + 322 q^{47} - 144 q^{48} - 754 q^{49} - 166 q^{50} + 252 q^{51} - 36 q^{52} - 1322 q^{53} - 162 q^{54} - 248 q^{55} + 56 q^{56} + 184 q^{58} + 252 q^{59} - 120 q^{60} - 435 q^{61} - 218 q^{62} + 63 q^{63} + 192 q^{64} - 1582 q^{65} + 264 q^{66} - 719 q^{67} - 336 q^{68} + 6 q^{69} + 564 q^{70} - 62 q^{71} + 216 q^{72} - 581 q^{73} + 490 q^{74} + 249 q^{75} - 204 q^{77} + 54 q^{78} - 489 q^{79} + 160 q^{80} + 243 q^{81} - 1376 q^{82} + 2496 q^{83} - 84 q^{84} + 1632 q^{85} - 206 q^{86} - 276 q^{87} - 352 q^{88} + 1584 q^{89} + 180 q^{90} - 1573 q^{91} - 8 q^{92} + 327 q^{93} + 644 q^{94} - 288 q^{96} + 974 q^{97} - 1508 q^{98} - 396 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 10 * q^5 - 18 * q^6 + 7 * q^7 + 24 * q^8 + 27 * q^9 + 20 * q^10 - 44 * q^11 - 36 * q^12 - 9 * q^13 + 14 * q^14 - 30 * q^15 + 48 * q^16 - 84 * q^17 + 54 * q^18 + 40 * q^20 - 21 * q^21 - 88 * q^22 - 2 * q^23 - 72 * q^24 - 83 * q^25 - 18 * q^26 - 81 * q^27 + 28 * q^28 + 92 * q^29 - 60 * q^30 - 109 * q^31 + 96 * q^32 + 132 * q^33 - 168 * q^34 + 282 * q^35 + 108 * q^36 + 245 * q^37 + 27 * q^39 + 80 * q^40 - 688 * q^41 - 42 * q^42 - 103 * q^43 - 176 * q^44 + 90 * q^45 - 4 * q^46 + 322 * q^47 - 144 * q^48 - 754 * q^49 - 166 * q^50 + 252 * q^51 - 36 * q^52 - 1322 * q^53 - 162 * q^54 - 248 * q^55 + 56 * q^56 + 184 * q^58 + 252 * q^59 - 120 * q^60 - 435 * q^61 - 218 * q^62 + 63 * q^63 + 192 * q^64 - 1582 * q^65 + 264 * q^66 - 719 * q^67 - 336 * q^68 + 6 * q^69 + 564 * q^70 - 62 * q^71 + 216 * q^72 - 581 * q^73 + 490 * q^74 + 249 * q^75 - 204 * q^77 + 54 * q^78 - 489 * q^79 + 160 * q^80 + 243 * q^81 - 1376 * q^82 + 2496 * q^83 - 84 * q^84 + 1632 * q^85 - 206 * q^86 - 276 * q^87 - 352 * q^88 + 1584 * q^89 + 180 * q^90 - 1573 * q^91 - 8 * q^92 + 327 * q^93 + 644 * q^94 - 288 * q^96 + 974 * q^97 - 1508 * q^98 - 396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more