LMFDB - Embedded newform 2366.4.a.bg.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2366,4,Mod(1,2366)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2366, 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("2366.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2366 = 2 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2366.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:	$139.598519074$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 219 x^{10} + 1022 x^{9} + 17084 x^{8} - 65540 x^{7} - 566763 x^{6} + 1871300 x^{5} + \cdots + 543166 $ x^12 - 6x^11 - 219x^10 + 1022x^9 + 17084x^8 - 65540x^7 - 566763x^6 + 1871300x^5 + 6589295x^4 - 20167176x^3 + 8073768x^2 + 5594524*x + 543166
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 13 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-6.44172$ of defining polynomial
Character	$\chi$	$=$	2366.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -4.70967 q^{3} +4.00000 q^{4} -13.2105 q^{5} -9.41934 q^{6} -7.00000 q^{7} +8.00000 q^{8} -4.81899 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -4.70967 q^{3} +4.00000 q^{4} -13.2105 q^{5} -9.41934 q^{6} -7.00000 q^{7} +8.00000 q^{8} -4.81899 q^{9} -26.4211 q^{10} -37.3726 q^{11} -18.8387 q^{12} -14.0000 q^{14} +62.2173 q^{15} +16.0000 q^{16} -12.0696 q^{17} -9.63799 q^{18} -54.8394 q^{19} -52.8422 q^{20} +32.9677 q^{21} -74.7453 q^{22} -31.6661 q^{23} -37.6774 q^{24} +49.5185 q^{25} +149.857 q^{27} -28.0000 q^{28} -292.123 q^{29} +124.435 q^{30} -261.672 q^{31} +32.0000 q^{32} +176.013 q^{33} -24.1393 q^{34} +92.4738 q^{35} -19.2760 q^{36} -79.6807 q^{37} -109.679 q^{38} -105.684 q^{40} -31.1851 q^{41} +65.9354 q^{42} -343.448 q^{43} -149.491 q^{44} +63.6615 q^{45} -63.3322 q^{46} -220.680 q^{47} -75.3547 q^{48} +49.0000 q^{49} +99.0369 q^{50} +56.8441 q^{51} -386.886 q^{53} +299.714 q^{54} +493.713 q^{55} -56.0000 q^{56} +258.276 q^{57} -584.247 q^{58} -902.603 q^{59} +248.869 q^{60} +455.560 q^{61} -523.344 q^{62} +33.7330 q^{63} +64.0000 q^{64} +352.026 q^{66} +792.021 q^{67} -48.2786 q^{68} +149.137 q^{69} +184.948 q^{70} -929.282 q^{71} -38.5520 q^{72} -1085.58 q^{73} -159.361 q^{74} -233.216 q^{75} -219.358 q^{76} +261.608 q^{77} +683.181 q^{79} -211.369 q^{80} -575.664 q^{81} -62.3703 q^{82} +978.764 q^{83} +131.871 q^{84} +159.447 q^{85} -686.897 q^{86} +1375.80 q^{87} -298.981 q^{88} -304.607 q^{89} +127.323 q^{90} -126.664 q^{92} +1232.39 q^{93} -441.359 q^{94} +724.458 q^{95} -150.709 q^{96} -283.318 q^{97} +98.0000 q^{98} +180.098 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 24 q^{2} + 6 q^{3} + 48 q^{4} + 28 q^{5} + 12 q^{6} - 84 q^{7} + 96 q^{8} + 162 q^{9}+O(q^{10}) $ 12 * q + 24 * q^2 + 6 * q^3 + 48 * q^4 + 28 * q^5 + 12 * q^6 - 84 * q^7 + 96 * q^8 + 162 * q^9	\( 12 q + 24 q^{2} + 6 q^{3} + 48 q^{4} + 28 q^{5} + 12 q^{6} - 84 q^{7} + 96 q^{8} + 162 q^{9} + 56 q^{10} + 6 q^{11} + 24 q^{12} - 168 q^{14} - 138 q^{15} + 192 q^{16} - 56 q^{17} + 324 q^{18} + 158 q^{19} + 112 q^{20} - 42 q^{21} + 12 q^{22} + 414 q^{23} + 48 q^{24} + 478 q^{25} + 390 q^{27} - 336 q^{28} + 222 q^{29} - 276 q^{30} - 200 q^{31} + 384 q^{32} + 844 q^{33} - 112 q^{34} - 196 q^{35} + 648 q^{36} - 560 q^{37} + 316 q^{38} + 224 q^{40} - 66 q^{41} - 84 q^{42} + 484 q^{43} + 24 q^{44} + 542 q^{45} + 828 q^{46} + 618 q^{47} + 96 q^{48} + 588 q^{49} + 956 q^{50} + 992 q^{51} + 504 q^{53} + 780 q^{54} + 2584 q^{55} - 672 q^{56} - 1164 q^{57} + 444 q^{58} + 1460 q^{59} - 552 q^{60} - 2 q^{61} - 400 q^{62} - 1134 q^{63} + 768 q^{64} + 1688 q^{66} + 334 q^{67} - 224 q^{68} + 4660 q^{69} - 392 q^{70} - 196 q^{71} + 1296 q^{72} + 490 q^{73} - 1120 q^{74} - 338 q^{75} + 632 q^{76} - 42 q^{77} + 2942 q^{79} + 448 q^{80} + 2824 q^{81} - 132 q^{82} + 236 q^{83} - 168 q^{84} + 1352 q^{85} + 968 q^{86} + 1456 q^{87} + 48 q^{88} + 3566 q^{89} + 1084 q^{90} + 1656 q^{92} + 1884 q^{93} + 1236 q^{94} + 5754 q^{95} + 192 q^{96} + 3032 q^{97} + 1176 q^{98} + 3670 q^{99}+O(q^{100}) \) 12 * q + 24 * q^2 + 6 * q^3 + 48 * q^4 + 28 * q^5 + 12 * q^6 - 84 * q^7 + 96 * q^8 + 162 * q^9 + 56 * q^10 + 6 * q^11 + 24 * q^12 - 168 * q^14 - 138 * q^15 + 192 * q^16 - 56 * q^17 + 324 * q^18 + 158 * q^19 + 112 * q^20 - 42 * q^21 + 12 * q^22 + 414 * q^23 + 48 * q^24 + 478 * q^25 + 390 * q^27 - 336 * q^28 + 222 * q^29 - 276 * q^30 - 200 * q^31 + 384 * q^32 + 844 * q^33 - 112 * q^34 - 196 * q^35 + 648 * q^36 - 560 * q^37 + 316 * q^38 + 224 * q^40 - 66 * q^41 - 84 * q^42 + 484 * q^43 + 24 * q^44 + 542 * q^45 + 828 * q^46 + 618 * q^47 + 96 * q^48 + 588 * q^49 + 956 * q^50 + 992 * q^51 + 504 * q^53 + 780 * q^54 + 2584 * q^55 - 672 * q^56 - 1164 * q^57 + 444 * q^58 + 1460 * q^59 - 552 * q^60 - 2 * q^61 - 400 * q^62 - 1134 * q^63 + 768 * q^64 + 1688 * q^66 + 334 * q^67 - 224 * q^68 + 4660 * q^69 - 392 * q^70 - 196 * q^71 + 1296 * q^72 + 490 * q^73 - 1120 * q^74 - 338 * q^75 + 632 * q^76 - 42 * q^77 + 2942 * q^79 + 448 * q^80 + 2824 * q^81 - 132 * q^82 + 236 * q^83 - 168 * q^84 + 1352 * q^85 + 968 * q^86 + 1456 * q^87 + 48 * q^88 + 3566 * q^89 + 1084 * q^90 + 1656 * q^92 + 1884 * q^93 + 1236 * q^94 + 5754 * q^95 + 192 * q^96 + 3032 * q^97 + 1176 * q^98 + 3670 * 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
$3$	−4.70967	−0.906377	−0.453188	−	0.891415i	$-0.649714\pi$
$3$	−4.70967	−0.906377	−0.453188	+	0.891415i	$0.649714\pi$
$4$	4.00000	0.500000
$5$	−13.2105	−1.18159	−0.590793	−	0.806823i	$-0.701185\pi$
$5$	−13.2105	−1.18159	−0.590793	+	0.806823i	$0.701185\pi$
$6$	−9.41934	−0.640905
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	−4.81899	−0.178481
$10$	−26.4211	−0.835508
$11$	−37.3726	−1.02439	−0.512194	−	0.858870i	$-0.671167\pi$
$11$	−37.3726	−1.02439	−0.512194	+	0.858870i	$0.671167\pi$
$12$	−18.8387	−0.453188
$13$	0	0
$13$	0	0
$14$	−14.0000	−0.267261
$15$	62.2173	1.07096
$16$	16.0000	0.250000
$17$	−12.0696	−0.172195	−0.0860976	−	0.996287i	$-0.527440\pi$
$17$	−12.0696	−0.172195	−0.0860976	+	0.996287i	$0.527440\pi$
$18$	−9.63799	−0.126205
$19$	−54.8394	−0.662159	−0.331079	−	0.943603i	$-0.607413\pi$
$19$	−54.8394	−0.662159	−0.331079	+	0.943603i	$0.607413\pi$
$20$	−52.8422	−0.590793
$21$	32.9677	0.342578
$22$	−74.7453	−0.724352
$23$	−31.6661	−0.287080	−0.143540	−	0.989645i	$-0.545849\pi$
$23$	−31.6661	−0.287080	−0.143540	+	0.989645i	$0.545849\pi$
$24$	−37.6774	−0.320453
$25$	49.5185	0.396148
$26$	0	0
$27$	149.857	1.06815
$28$	−28.0000	−0.188982
$29$	−292.123	−1.87055	−0.935275	−	0.353922i	$-0.884848\pi$
$29$	−292.123	−1.87055	−0.935275	+	0.353922i	$0.884848\pi$
$30$	124.435	0.757285
$31$	−261.672	−1.51605	−0.758026	−	0.652224i	$-0.773836\pi$
$31$	−261.672	−1.51605	−0.758026	+	0.652224i	$0.773836\pi$
$32$	32.0000	0.176777
$33$	176.013	0.928482
$34$	−24.1393	−0.121760
$35$	92.4738	0.446598
$36$	−19.2760	−0.0892406
$37$	−79.6807	−0.354038	−0.177019	−	0.984207i	$-0.556645\pi$
$37$	−79.6807	−0.354038	−0.177019	+	0.984207i	$0.556645\pi$
$38$	−109.679	−0.468217
$39$	0	0
$40$	−105.684	−0.417754
$41$	−31.1851	−0.118788	−0.0593939	−	0.998235i	$-0.518917\pi$
$41$	−31.1851	−0.118788	−0.0593939	+	0.998235i	$0.518917\pi$
$42$	65.9354	0.242239
$43$	−343.448	−1.21803	−0.609016	−	0.793158i	$-0.708436\pi$
$43$	−343.448	−1.21803	−0.609016	+	0.793158i	$0.708436\pi$
$44$	−149.491	−0.512194
$45$	63.6615	0.210891
$46$	−63.3322	−0.202996
$47$	−220.680	−0.684882	−0.342441	−	0.939539i	$-0.611254\pi$
$47$	−220.680	−0.684882	−0.342441	+	0.939539i	$0.611254\pi$
$48$	−75.3547	−0.226594
$49$	49.0000	0.142857
$50$	99.0369	0.280119
$51$	56.8441	0.156074
$52$	0	0
$53$	−386.886	−1.00270	−0.501348	−	0.865246i	$-0.667162\pi$
$53$	−386.886	−1.00270	−0.501348	+	0.865246i	$0.667162\pi$
$54$	299.714	0.755295
$55$	493.713	1.21040
$56$	−56.0000	−0.133631
$57$	258.276	0.600165
$58$	−584.247	−1.32268
$59$	−902.603	−1.99168	−0.995838	−	0.0911366i	$-0.970950\pi$
$59$	−902.603	−1.99168	−0.995838	+	0.0911366i	$0.970950\pi$
$60$	248.869	0.535481
$61$	455.560	0.956205	0.478103	−	0.878304i	$-0.341325\pi$
$61$	455.560	0.956205	0.478103	+	0.878304i	$0.341325\pi$
$62$	−523.344	−1.07201
$63$	33.7330	0.0674596
$64$	64.0000	0.125000
$65$	0	0
$66$	352.026	0.656536
$67$	792.021	1.44419	0.722096	−	0.691793i	$-0.243179\pi$
$67$	792.021	1.44419	0.722096	+	0.691793i	$0.243179\pi$
$68$	−48.2786	−0.0860976
$69$	149.137	0.260202
$70$	184.948	0.315792
$71$	−929.282	−1.55332	−0.776658	−	0.629922i	$-0.783087\pi$
$71$	−929.282	−1.55332	−0.776658	+	0.629922i	$0.783087\pi$
$72$	−38.5520	−0.0631027
$73$	−1085.58	−1.74051	−0.870256	−	0.492599i	$-0.836047\pi$
$73$	−1085.58	−1.74051	−0.870256	+	0.492599i	$0.836047\pi$
$74$	−159.361	−0.250343
$75$	−233.216	−0.359059
$76$	−219.358	−0.331079
$77$	261.608	0.387182
$78$	0	0
$79$	683.181	0.972960	0.486480	−	0.873692i	$-0.338281\pi$
$79$	683.181	0.972960	0.486480	+	0.873692i	$0.338281\pi$
$80$	−211.369	−0.295397
$81$	−575.664	−0.789663
$82$	−62.3703	−0.0839957
$83$	978.764	1.29438	0.647189	−	0.762330i	$-0.275945\pi$
$83$	978.764	1.29438	0.647189	+	0.762330i	$0.275945\pi$
$84$	131.871	0.171289
$85$	159.447	0.203464
$86$	−686.897	−0.861279
$87$	1375.80	1.69542
$88$	−298.981	−0.362176
$89$	−304.607	−0.362790	−0.181395	−	0.983410i	$-0.558061\pi$
$89$	−304.607	−0.362790	−0.181395	+	0.983410i	$0.558061\pi$
$90$	127.323	0.149123
$91$	0	0
$92$	−126.664	−0.143540
$93$	1232.39	1.37411
$94$	−441.359	−0.484284
$95$	724.458	0.782398
$96$	−150.709	−0.160226
$97$	−283.318	−0.296562	−0.148281	−	0.988945i	$-0.547374\pi$
$97$	−283.318	−0.296562	−0.148281	+	0.988945i	$0.547374\pi$
$98$	98.0000	0.101015
$99$	180.098	0.182834
$100$	198.074	0.198074
$101$	−520.008	−0.512304	−0.256152	−	0.966637i	$-0.582455\pi$
$101$	−520.008	−0.512304	−0.256152	+	0.966637i	$0.582455\pi$
$102$	113.688	0.110361
$103$	778.599	0.744832	0.372416	−	0.928066i	$-0.378529\pi$
$103$	778.599	0.744832	0.372416	+	0.928066i	$0.378529\pi$
$104$	0	0
$105$	−435.521	−0.404786
$106$	−773.772	−0.709013
$107$	−168.488	−0.152227	−0.0761137	−	0.997099i	$-0.524251\pi$
$107$	−168.488	−0.152227	−0.0761137	+	0.997099i	$0.524251\pi$
$108$	599.428	0.534074
$109$	−433.499	−0.380932	−0.190466	−	0.981694i	$-0.561000\pi$
$109$	−433.499	−0.380932	−0.190466	+	0.981694i	$0.561000\pi$
$110$	987.426	0.855885
$111$	375.270	0.320892
$112$	−112.000	−0.0944911
$113$	46.0139	0.0383064	0.0191532	−	0.999817i	$-0.493903\pi$
$113$	46.0139	0.0383064	0.0191532	+	0.999817i	$0.493903\pi$
$114$	516.551	0.424381
$115$	418.326	0.339210
$116$	−1168.49	−0.935275
$117$	0	0
$118$	−1805.21	−1.40833
$119$	84.4875	0.0650837
$120$	497.739	0.378643
$121$	65.7136	0.0493716
$122$	911.121	0.676139
$123$	146.872	0.107667
$124$	−1046.69	−0.758026
$125$	997.152	0.713504
$126$	67.4659	0.0477011
$127$	−1012.18	−0.707213	−0.353607	−	0.935394i	$-0.615045\pi$
$127$	−1012.18	−0.707213	−0.353607	+	0.935394i	$0.615045\pi$
$128$	128.000	0.0883883
$129$	1617.53	1.10400
$130$	0	0
$131$	249.396	0.166334	0.0831672	−	0.996536i	$-0.473496\pi$
$131$	249.396	0.166334	0.0831672	+	0.996536i	$0.473496\pi$
$132$	704.051	0.464241
$133$	383.876	0.250273
$134$	1584.04	1.02120
$135$	−1979.69	−1.26211
$136$	−96.5572	−0.0608802
$137$	1612.91	1.00584	0.502920	−	0.864333i	$-0.332259\pi$
$137$	1612.91	1.00584	0.502920	+	0.864333i	$0.332259\pi$
$138$	298.274	0.183991
$139$	563.979	0.344144	0.172072	−	0.985084i	$-0.444954\pi$
$139$	563.979	0.344144	0.172072	+	0.985084i	$0.444954\pi$
$140$	369.895	0.223299
$141$	1039.33	0.620761
$142$	−1858.56	−1.09836
$143$	0	0
$144$	−77.1039	−0.0446203
$145$	3859.11	2.21022
$146$	−2171.16	−1.23073
$147$	−230.774	−0.129482
$148$	−318.723	−0.177019
$149$	−1528.46	−0.840379	−0.420189	−	0.907436i	$-0.638036\pi$
$149$	−1528.46	−0.840379	−0.420189	+	0.907436i	$0.638036\pi$
$150$	−466.431	−0.253893
$151$	1279.72	0.689683	0.344841	−	0.938661i	$-0.387933\pi$
$151$	1279.72	0.689683	0.344841	+	0.938661i	$0.387933\pi$
$152$	−438.715	−0.234109
$153$	58.1636	0.0307336
$154$	523.217	0.273779
$155$	3456.83	1.79135
$156$	0	0
$157$	869.072	0.441780	0.220890	−	0.975299i	$-0.429104\pi$
$157$	869.072	0.441780	0.220890	+	0.975299i	$0.429104\pi$
$158$	1366.36	0.687987
$159$	1822.11	0.908820
$160$	−422.737	−0.208877
$161$	221.663	0.108506
$162$	−1151.33	−0.558376
$163$	1078.15	0.518082	0.259041	−	0.965866i	$-0.416594\pi$
$163$	1078.15	0.518082	0.259041	+	0.965866i	$0.416594\pi$
$164$	−124.741	−0.0593939
$165$	−2325.23	−1.09708
$166$	1957.53	0.915263
$167$	1436.19	0.665484	0.332742	−	0.943018i	$-0.392026\pi$
$167$	1436.19	0.665484	0.332742	+	0.943018i	$0.392026\pi$
$168$	263.742	0.121120
$169$	0	0
$170$	318.893	0.143871
$171$	264.271	0.118183
$172$	−1373.79	−0.609016
$173$	2638.42	1.15951	0.579755	−	0.814791i	$-0.303148\pi$
$173$	2638.42	1.15951	0.579755	+	0.814791i	$0.303148\pi$
$174$	2751.61	1.19884
$175$	−346.629	−0.149730
$176$	−597.962	−0.256097
$177$	4250.96	1.80521
$178$	−609.214	−0.256531
$179$	−2228.01	−0.930332	−0.465166	−	0.885224i	$-0.654005\pi$
$179$	−2228.01	−0.930332	−0.465166	+	0.885224i	$0.654005\pi$
$180$	254.646	0.105446
$181$	−766.169	−0.314635	−0.157317	−	0.987548i	$-0.550285\pi$
$181$	−766.169	−0.314635	−0.157317	+	0.987548i	$0.550285\pi$
$182$	0	0
$183$	−2145.54	−0.866682
$184$	−253.329	−0.101498
$185$	1052.63	0.418327
$186$	2464.78	0.971646
$187$	451.075	0.176395
$188$	−882.718	−0.342441
$189$	−1049.00	−0.403722
$190$	1448.92	0.553239
$191$	−4078.96	−1.54525	−0.772627	−	0.634860i	$-0.781058\pi$
$191$	−4078.96	−1.54525	−0.772627	+	0.634860i	$0.781058\pi$
$192$	−301.419	−0.113297
$193$	−3735.51	−1.39320	−0.696600	−	0.717460i	$-0.745305\pi$
$193$	−3735.51	−1.39320	−0.696600	+	0.717460i	$0.745305\pi$
$194$	−566.635	−0.209701
$195$	0	0
$196$	196.000	0.0714286
$197$	−518.661	−0.187579	−0.0937895	−	0.995592i	$-0.529898\pi$
$197$	−518.661	−0.187579	−0.0937895	+	0.995592i	$0.529898\pi$
$198$	360.197	0.129283
$199$	1844.75	0.657141	0.328571	−	0.944479i	$-0.393433\pi$
$199$	1844.75	0.657141	0.328571	+	0.944479i	$0.393433\pi$
$200$	396.148	0.140059
$201$	−3730.16	−1.30898
$202$	−1040.02	−0.362254
$203$	2044.86	0.707001
$204$	227.376	0.0780369
$205$	411.973	0.140358
$206$	1557.20	0.526676
$207$	152.599	0.0512384
$208$	0	0
$209$	2049.49	0.678308
$210$	−871.043	−0.286227
$211$	−5030.78	−1.64139	−0.820695	−	0.571367i	$-0.806413\pi$
$211$	−5030.78	−1.64139	−0.820695	+	0.571367i	$0.806413\pi$
$212$	−1547.54	−0.501348
$213$	4376.61	1.40789
$214$	−336.976	−0.107641
$215$	4537.14	1.43921
$216$	1198.86	0.377647
$217$	1831.70	0.573014
$218$	−866.997	−0.269360
$219$	5112.72	1.57756
$220$	1974.85	0.605202
$221$	0	0
$222$	750.540	0.226905
$223$	−5918.72	−1.77734	−0.888669	−	0.458549i	$-0.848369\pi$
$223$	−5918.72	−1.77734	−0.888669	+	0.458549i	$0.848369\pi$
$224$	−224.000	−0.0668153
$225$	−238.629	−0.0707049
$226$	92.0278	0.0270867
$227$	1426.24	0.417017	0.208509	−	0.978021i	$-0.433139\pi$
$227$	1426.24	0.417017	0.208509	+	0.978021i	$0.433139\pi$
$228$	1033.10	0.300083
$229$	3687.51	1.06409	0.532047	−	0.846715i	$-0.321423\pi$
$229$	3687.51	1.06409	0.532047	+	0.846715i	$0.321423\pi$
$230$	836.652	0.239857
$231$	−1232.09	−0.350933
$232$	−2336.99	−0.661339
$233$	5427.87	1.52614	0.763072	−	0.646314i	$-0.223690\pi$
$233$	5427.87	1.52614	0.763072	+	0.646314i	$0.223690\pi$
$234$	0	0
$235$	2915.30	0.809247
$236$	−3610.41	−0.995838
$237$	−3217.56	−0.881868
$238$	168.975	0.0460211
$239$	−4398.31	−1.19039	−0.595194	−	0.803582i	$-0.702925\pi$
$239$	−4398.31	−1.19039	−0.595194	+	0.803582i	$0.702925\pi$
$240$	995.477	0.267741
$241$	5758.44	1.53914	0.769572	−	0.638561i	$-0.220470\pi$
$241$	5758.44	1.53914	0.769572	+	0.638561i	$0.220470\pi$
$242$	131.427	0.0349110
$243$	−1334.95	−0.352416
$244$	1822.24	0.478103
$245$	−647.317	−0.168798
$246$	293.744	0.0761317
$247$	0	0
$248$	−2093.37	−0.536006
$249$	−4609.66	−1.17319
$250$	1994.30	0.504523
$251$	−2090.01	−0.525578	−0.262789	−	0.964853i	$-0.584642\pi$
$251$	−2090.01	−0.525578	−0.262789	+	0.964853i	$0.584642\pi$
$252$	134.932	0.0337298
$253$	1183.44	0.294081
$254$	−2024.35	−0.500075
$255$	−750.941	−0.184415
$256$	256.000	0.0625000
$257$	−1792.43	−0.435054	−0.217527	−	0.976054i	$-0.569799\pi$
$257$	−1792.43	−0.435054	−0.217527	+	0.976054i	$0.569799\pi$
$258$	3235.06	0.780643
$259$	557.765	0.133814
$260$	0	0
$261$	1407.74	0.333858
$262$	498.791	0.117616
$263$	2996.92	0.702654	0.351327	−	0.936253i	$-0.385730\pi$
$263$	2996.92	0.702654	0.351327	+	0.936253i	$0.385730\pi$
$264$	1408.10	0.328268
$265$	5110.98	1.18477
$266$	767.751	0.176969
$267$	1434.60	0.328824
$268$	3168.09	0.722096
$269$	6649.27	1.50711	0.753556	−	0.657383i	$-0.228337\pi$
$269$	6649.27	1.50711	0.753556	+	0.657383i	$0.228337\pi$
$270$	−3959.39	−0.892446
$271$	5171.24	1.15915	0.579576	−	0.814918i	$-0.303218\pi$
$271$	5171.24	1.15915	0.579576	+	0.814918i	$0.303218\pi$
$272$	−193.114	−0.0430488
$273$	0	0
$274$	3225.82	0.711237
$275$	−1850.64	−0.405809
$276$	596.547	0.130101
$277$	1009.02	0.218867	0.109433	−	0.993994i	$-0.465096\pi$
$277$	1009.02	0.218867	0.109433	+	0.993994i	$0.465096\pi$
$278$	1127.96	0.243347
$279$	1260.99	0.270587
$280$	739.790	0.157896
$281$	−7461.08	−1.58395	−0.791977	−	0.610551i	$-0.790948\pi$
$281$	−7461.08	−1.58395	−0.791977	+	0.610551i	$0.790948\pi$
$282$	2078.66	0.438944
$283$	5569.70	1.16991	0.584955	−	0.811066i	$-0.301112\pi$
$283$	5569.70	1.16991	0.584955	+	0.811066i	$0.301112\pi$
$284$	−3717.13	−0.776658
$285$	−3411.96	−0.709148
$286$	0	0
$287$	218.296	0.0448976
$288$	−154.208	−0.0315513
$289$	−4767.32	−0.970349
$290$	7718.22	1.56286
$291$	1334.33	0.268797
$292$	−4342.32	−0.870256
$293$	−3386.58	−0.675243	−0.337622	−	0.941282i	$-0.609622\pi$
$293$	−3386.58	−0.675243	−0.337622	+	0.941282i	$0.609622\pi$
$294$	−461.548	−0.0915579
$295$	11923.9	2.35334
$296$	−637.445	−0.125171
$297$	−5600.55	−1.09420
$298$	−3056.92	−0.594238
$299$	0	0
$300$	−932.863	−0.179530
$301$	2404.14	0.460373
$302$	2559.44	0.487679
$303$	2449.07	0.464340
$304$	−877.430	−0.165540
$305$	−6018.20	−1.12984
$306$	116.327	0.0217320
$307$	5998.70	1.11519	0.557596	−	0.830112i	$-0.311724\pi$
$307$	5998.70	1.11519	0.557596	+	0.830112i	$0.311724\pi$
$308$	1046.43	0.193591
$309$	−3666.95	−0.675098
$310$	6913.65	1.26667
$311$	5385.01	0.981852	0.490926	−	0.871201i	$-0.336659\pi$
$311$	5385.01	0.981852	0.490926	+	0.871201i	$0.336659\pi$
$312$	0	0
$313$	351.372	0.0634528	0.0317264	−	0.999497i	$-0.489899\pi$
$313$	351.372	0.0634528	0.0317264	+	0.999497i	$0.489899\pi$
$314$	1738.14	0.312386
$315$	−445.631	−0.0797094
$316$	2732.72	0.486480
$317$	1730.21	0.306557	0.153278	−	0.988183i	$-0.451017\pi$
$317$	1730.21	0.306557	0.153278	+	0.988183i	$0.451017\pi$
$318$	3644.21	0.642633
$319$	10917.4	1.91617
$320$	−845.475	−0.147698
$321$	793.522	0.137975
$322$	443.325	0.0767253
$323$	661.892	0.114021
$324$	−2302.66	−0.394832
$325$	0	0
$326$	2156.30	0.366339
$327$	2041.64	0.345268
$328$	−249.481	−0.0419978
$329$	1544.76	0.258861
$330$	−4650.45	−0.775754
$331$	−9405.49	−1.56185	−0.780925	−	0.624624i	$-0.785252\pi$
$331$	−9405.49	−1.56185	−0.780925	+	0.624624i	$0.785252\pi$
$332$	3915.06	0.647189
$333$	383.981	0.0631892
$334$	2872.38	0.470568
$335$	−10463.0	−1.70644
$336$	527.483	0.0856445
$337$	4298.72	0.694856	0.347428	−	0.937707i	$-0.387055\pi$
$337$	4298.72	0.694856	0.347428	+	0.937707i	$0.387055\pi$
$338$	0	0
$339$	−216.710	−0.0347200
$340$	637.786	0.101732
$341$	9779.36	1.55303
$342$	528.541	0.0835680
$343$	−343.000	−0.0539949
$344$	−2747.59	−0.430639
$345$	−1970.18	−0.307452
$346$	5276.84	0.819897
$347$	−2047.91	−0.316822	−0.158411	−	0.987373i	$-0.550637\pi$
$347$	−2047.91	−0.316822	−0.158411	+	0.987373i	$0.550637\pi$
$348$	5503.22	0.847711
$349$	−7633.21	−1.17076	−0.585382	−	0.810758i	$-0.699055\pi$
$349$	−7633.21	−1.17076	−0.585382	+	0.810758i	$0.699055\pi$
$350$	−693.259	−0.105875
$351$	0	0
$352$	−1195.92	−0.181088
$353$	−10488.0	−1.58136	−0.790682	−	0.612228i	$-0.790274\pi$
$353$	−10488.0	−1.58136	−0.790682	+	0.612228i	$0.790274\pi$
$354$	8501.93	1.27648
$355$	12276.3	1.83538
$356$	−1218.43	−0.181395
$357$	−397.909	−0.0589903
$358$	−4456.02	−0.657844
$359$	5411.35	0.795543	0.397772	−	0.917484i	$-0.369784\pi$
$359$	5411.35	0.795543	0.397772	+	0.917484i	$0.369784\pi$
$360$	509.292	0.0745613
$361$	−3851.64	−0.561546
$362$	−1532.34	−0.222480
$363$	−309.489	−0.0447492
$364$	0	0
$365$	14341.1	2.05657
$366$	−4291.08	−0.612837
$367$	−4078.72	−0.580130	−0.290065	−	0.957007i	$-0.593677\pi$
$367$	−4078.72	−0.580130	−0.290065	+	0.957007i	$0.593677\pi$
$368$	−506.657	−0.0717699
$369$	150.281	0.0212014
$370$	2105.25	0.295802
$371$	2708.20	0.378983
$372$	4929.55	0.687057
$373$	6723.68	0.933349	0.466675	−	0.884429i	$-0.345452\pi$
$373$	6723.68	0.933349	0.466675	+	0.884429i	$0.345452\pi$
$374$	902.149	0.124730
$375$	−4696.26	−0.646703
$376$	−1765.44	−0.242142
$377$	0	0
$378$	−2098.00	−0.285475
$379$	−10780.1	−1.46105	−0.730525	−	0.682886i	$-0.760724\pi$
$379$	−10780.1	−1.46105	−0.730525	+	0.682886i	$0.760724\pi$
$380$	2897.83	0.391199
$381$	4767.02	0.641002
$382$	−8157.93	−1.09266
$383$	8995.14	1.20008	0.600039	−	0.799970i	$-0.295152\pi$
$383$	8995.14	1.20008	0.600039	+	0.799970i	$0.295152\pi$
$384$	−602.838	−0.0801131
$385$	−3455.99	−0.457490
$386$	−7471.01	−0.985141
$387$	1655.08	0.217396
$388$	−1133.27	−0.148281
$389$	1638.91	0.213614	0.106807	−	0.994280i	$-0.465937\pi$
$389$	1638.91	0.213614	0.106807	+	0.994280i	$0.465937\pi$
$390$	0	0
$391$	382.198	0.0494338
$392$	392.000	0.0505076
$393$	−1174.57	−0.150762
$394$	−1037.32	−0.132638
$395$	−9025.19	−1.14964
$396$	720.394	0.0914171
$397$	100.973	0.0127649	0.00638246	−	0.999980i	$-0.497968\pi$
$397$	100.973	0.0127649	0.00638246	+	0.999980i	$0.497968\pi$
$398$	3689.51	0.464669
$399$	−1807.93	−0.226841
$400$	792.295	0.0990369
$401$	5879.60	0.732203	0.366102	−	0.930575i	$-0.380692\pi$
$401$	5879.60	0.732203	0.366102	+	0.930575i	$0.380692\pi$
$402$	−7460.32	−0.925589
$403$	0	0
$404$	−2080.03	−0.256152
$405$	7604.84	0.933056
$406$	4089.73	0.499925
$407$	2977.88	0.362673
$408$	454.753	0.0551804
$409$	−3374.30	−0.407943	−0.203971	−	0.978977i	$-0.565385\pi$
$409$	−3374.30	−0.407943	−0.203971	+	0.978977i	$0.565385\pi$
$410$	823.945	0.0992482
$411$	−7596.28	−0.911671
$412$	3114.40	0.372416
$413$	6318.22	0.752783
$414$	305.197	0.0362310
$415$	−12930.0	−1.52942
$416$	0	0
$417$	−2656.16	−0.311924
$418$	4098.98	0.479636
$419$	12158.8	1.41765	0.708826	−	0.705384i	$-0.249225\pi$
$419$	12158.8	1.41765	0.708826	+	0.705384i	$0.249225\pi$
$420$	−1742.09	−0.202393
$421$	7252.94	0.839636	0.419818	−	0.907608i	$-0.362094\pi$
$421$	7252.94	0.839636	0.419818	+	0.907608i	$0.362094\pi$
$422$	−10061.6	−1.16064
$423$	1063.45	0.122239
$424$	−3095.09	−0.354507
$425$	−597.670	−0.0682148
$426$	8753.22	0.995528
$427$	−3188.92	−0.361412
$428$	−673.951	−0.0761137
$429$	0	0
$430$	9074.28	1.01768
$431$	−8624.56	−0.963875	−0.481938	−	0.876205i	$-0.660067\pi$
$431$	−8624.56	−0.963875	−0.481938	+	0.876205i	$0.660067\pi$
$432$	2397.71	0.267037
$433$	3953.00	0.438727	0.219364	−	0.975643i	$-0.429602\pi$
$433$	3953.00	0.438727	0.219364	+	0.975643i	$0.429602\pi$
$434$	3663.40	0.405182
$435$	−18175.1	−2.00329
$436$	−1733.99	−0.190466
$437$	1736.55	0.190092
$438$	10225.4	1.11550
$439$	827.763	0.0899931	0.0449965	−	0.998987i	$-0.485672\pi$
$439$	827.763	0.0899931	0.0449965	+	0.998987i	$0.485672\pi$
$440$	3949.70	0.427942
$441$	−236.131	−0.0254973
$442$	0	0
$443$	4180.64	0.448371	0.224186	−	0.974546i	$-0.428028\pi$
$443$	4180.64	0.448371	0.224186	+	0.974546i	$0.428028\pi$
$444$	1501.08	0.160446
$445$	4024.03	0.428668
$446$	−11837.4	−1.25677
$447$	7198.55	0.761700
$448$	−448.000	−0.0472456
$449$	−15288.5	−1.60693	−0.803463	−	0.595355i	$-0.797011\pi$
$449$	−15288.5	−1.60693	−0.803463	+	0.595355i	$0.797011\pi$
$450$	−477.258	−0.0499959
$451$	1165.47	0.121685
$452$	184.056	0.0191532
$453$	−6027.06	−0.625113
$454$	2852.48	0.294876
$455$	0	0
$456$	2066.20	0.212191
$457$	−18564.3	−1.90022	−0.950111	−	0.311912i	$-0.899031\pi$
$457$	−18564.3	−1.90022	−0.950111	+	0.311912i	$0.899031\pi$
$458$	7375.02	0.752429
$459$	−1808.72	−0.183930
$460$	1673.30	0.169605
$461$	−18976.1	−1.91715	−0.958574	−	0.284844i	$-0.908058\pi$
$461$	−18976.1	−1.91715	−0.958574	+	0.284844i	$0.908058\pi$
$462$	−2464.18	−0.248147
$463$	−4340.33	−0.435663	−0.217832	−	0.975986i	$-0.569898\pi$
$463$	−4340.33	−0.435663	−0.217832	+	0.975986i	$0.569898\pi$
$464$	−4673.97	−0.467637
$465$	−16280.5	−1.62364
$466$	10855.7	1.07915
$467$	8647.58	0.856878	0.428439	−	0.903571i	$-0.359064\pi$
$467$	8647.58	0.856878	0.428439	+	0.903571i	$0.359064\pi$
$468$	0	0
$469$	−5544.15	−0.545853
$470$	5830.60	0.572224
$471$	−4093.04	−0.400419
$472$	−7220.83	−0.704164
$473$	12835.6	1.24774
$474$	−6435.11	−0.623575
$475$	−2715.56	−0.262313
$476$	337.950	0.0325418
$477$	1864.40	0.178962
$478$	−8796.62	−0.841732
$479$	−14100.9	−1.34506	−0.672532	−	0.740068i	$-0.734793\pi$
$479$	−14100.9	−1.34506	−0.672532	+	0.740068i	$0.734793\pi$
$480$	1990.95	0.189321
$481$	0	0
$482$	11516.9	1.08834
$483$	−1043.96	−0.0983473
$484$	262.854	0.0246858
$485$	3742.78	0.350414
$486$	−2669.90	−0.249196
$487$	−588.324	−0.0547423	−0.0273711	−	0.999625i	$-0.508714\pi$
$487$	−588.324	−0.0547423	−0.0273711	+	0.999625i	$0.508714\pi$
$488$	3644.48	0.338070
$489$	−5077.74	−0.469577
$490$	−1294.63	−0.119358
$491$	−7236.48	−0.665128	−0.332564	−	0.943081i	$-0.607914\pi$
$491$	−7236.48	−0.665128	−0.332564	+	0.943081i	$0.607914\pi$
$492$	587.487	0.0538333
$493$	3525.83	0.322100
$494$	0	0
$495$	−2379.20	−0.216034
$496$	−4186.75	−0.379013
$497$	6504.97	0.587098
$498$	−9219.31	−0.829573
$499$	−10578.0	−0.948968	−0.474484	−	0.880264i	$-0.657365\pi$
$499$	−10578.0	−0.948968	−0.474484	+	0.880264i	$0.657365\pi$
$500$	3988.61	0.356752
$501$	−6763.99	−0.603179
$502$	−4180.01	−0.371640
$503$	−1546.28	−0.137068	−0.0685338	−	0.997649i	$-0.521832\pi$
$503$	−1546.28	−0.137068	−0.0685338	+	0.997649i	$0.521832\pi$
$504$	269.864	0.0238506
$505$	6869.58	0.605332
$506$	2366.89	0.207947
$507$	0	0
$508$	−4048.70	−0.353607
$509$	−12173.3	−1.06006	−0.530031	−	0.847978i	$-0.677820\pi$
$509$	−12173.3	−1.06006	−0.530031	+	0.847978i	$0.677820\pi$
$510$	−1501.88	−0.130401
$511$	7599.06	0.657852
$512$	512.000	0.0441942
$513$	−8218.07	−0.707284
$514$	−3584.87	−0.307630
$515$	−10285.7	−0.880084
$516$	6470.12	0.551998
$517$	8247.38	0.701585
$518$	1115.53	0.0946208
$519$	−12426.1	−1.05095
$520$	0	0
$521$	−3325.74	−0.279661	−0.139831	−	0.990175i	$-0.544656\pi$
$521$	−3325.74	−0.279661	−0.139831	+	0.990175i	$0.544656\pi$
$522$	2815.48	0.236073
$523$	7630.21	0.637946	0.318973	−	0.947764i	$-0.396662\pi$
$523$	7630.21	0.637946	0.318973	+	0.947764i	$0.396662\pi$
$524$	997.583	0.0831672
$525$	1632.51	0.135712
$526$	5993.84	0.496852
$527$	3158.29	0.261057
$528$	2816.21	0.232120
$529$	−11164.3	−0.917585
$530$	10222.0	0.837761
$531$	4349.64	0.355477
$532$	1535.50	0.125136
$533$	0	0
$534$	2869.20	0.232514
$535$	2225.82	0.179870
$536$	6336.17	0.510599
$537$	10493.2	0.843231
$538$	13298.5	1.06569
$539$	−1831.26	−0.146341
$540$	−7918.77	−0.631055
$541$	20063.0	1.59441	0.797206	−	0.603707i	$-0.206310\pi$
$541$	20063.0	1.59441	0.797206	+	0.603707i	$0.206310\pi$
$542$	10342.5	0.819644
$543$	3608.40	0.285178
$544$	−386.229	−0.0304401
$545$	5726.75	0.450105
$546$	0	0
$547$	12774.2	0.998510	0.499255	−	0.866455i	$-0.333607\pi$
$547$	12774.2	0.998510	0.499255	+	0.866455i	$0.333607\pi$
$548$	6451.64	0.502920
$549$	−2195.34	−0.170665
$550$	−3701.27	−0.286950
$551$	16019.9	1.23860
$552$	1193.09	0.0919954
$553$	−4782.27	−0.367744
$554$	2018.04	0.154762
$555$	−4957.52	−0.379162
$556$	2255.92	0.172072
$557$	17676.6	1.34467	0.672337	−	0.740245i	$-0.265291\pi$
$557$	17676.6	1.34467	0.672337	+	0.740245i	$0.265291\pi$
$558$	2521.99	0.191334
$559$	0	0
$560$	1479.58	0.111649
$561$	−2124.41	−0.159880
$562$	−14922.2	−1.12002
$563$	−7606.82	−0.569430	−0.284715	−	0.958612i	$-0.591899\pi$
$563$	−7606.82	−0.569430	−0.284715	+	0.958612i	$0.591899\pi$
$564$	4157.31	0.310380
$565$	−607.869	−0.0452624
$566$	11139.4	0.827251
$567$	4029.65	0.298465
$568$	−7434.25	−0.549180
$569$	−5770.31	−0.425139	−0.212570	−	0.977146i	$-0.568183\pi$
$569$	−5770.31	−0.425139	−0.212570	+	0.977146i	$0.568183\pi$
$570$	−6823.92	−0.501443
$571$	25468.2	1.86657	0.933284	−	0.359140i	$-0.116930\pi$
$571$	25468.2	1.86657	0.933284	+	0.359140i	$0.116930\pi$
$572$	0	0
$573$	19210.6	1.40058
$574$	436.592	0.0317474
$575$	−1568.06	−0.113726
$576$	−308.416	−0.0223102
$577$	−1084.20	−0.0782247	−0.0391123	−	0.999235i	$-0.512453\pi$
$577$	−1084.20	−0.0782247	−0.0391123	+	0.999235i	$0.512453\pi$
$578$	−9534.65	−0.686140
$579$	17593.0	1.26276
$580$	15436.4	1.10511
$581$	−6851.35	−0.489229
$582$	2668.67	0.190068
$583$	14459.0	1.02715
$584$	−8684.63	−0.615364
$585$	0	0
$586$	−6773.17	−0.477469
$587$	15820.9	1.11243	0.556216	−	0.831038i	$-0.312253\pi$
$587$	15820.9	1.11243	0.556216	+	0.831038i	$0.312253\pi$
$588$	−923.096	−0.0647412
$589$	14349.9	1.00387
$590$	23847.8	1.66406
$591$	2442.72	0.170017
$592$	−1274.89	−0.0885096
$593$	−15171.1	−1.05060	−0.525298	−	0.850919i	$-0.676046\pi$
$593$	−15171.1	−1.05060	−0.525298	+	0.850919i	$0.676046\pi$
$594$	−11201.1	−0.773715
$595$	−1116.13	−0.0769020
$596$	−6113.85	−0.420189
$597$	−8688.18	−0.595617
$598$	0	0
$599$	−13382.1	−0.912819	−0.456410	−	0.889770i	$-0.650865\pi$
$599$	−13382.1	−0.912819	−0.456410	+	0.889770i	$0.650865\pi$
$600$	−1865.73	−0.126947
$601$	14397.0	0.977150	0.488575	−	0.872522i	$-0.337517\pi$
$601$	14397.0	0.977150	0.488575	+	0.872522i	$0.337517\pi$
$602$	4808.28	0.325533
$603$	−3816.75	−0.257761
$604$	5118.88	0.344841
$605$	−868.112	−0.0583368
$606$	4898.13	0.328338
$607$	−15147.4	−1.01288	−0.506438	−	0.862277i	$-0.669038\pi$
$607$	−15147.4	−1.01288	−0.506438	+	0.862277i	$0.669038\pi$
$608$	−1754.86	−0.117054
$609$	−9630.63	−0.640809
$610$	−12036.4	−0.798917
$611$	0	0
$612$	232.654	0.0153668
$613$	−10062.5	−0.663002	−0.331501	−	0.943455i	$-0.607555\pi$
$613$	−10062.5	−0.663002	−0.331501	+	0.943455i	$0.607555\pi$
$614$	11997.4	0.788560
$615$	−1940.26	−0.127217
$616$	2092.87	0.136890
$617$	18496.6	1.20688	0.603441	−	0.797408i	$-0.293796\pi$
$617$	18496.6	1.20688	0.603441	+	0.797408i	$0.293796\pi$
$618$	−7333.90	−0.477367
$619$	416.201	0.0270251	0.0135125	−	0.999909i	$-0.495699\pi$
$619$	416.201	0.0270251	0.0135125	+	0.999909i	$0.495699\pi$
$620$	13827.3	0.895674
$621$	−4745.38	−0.306644
$622$	10770.0	0.694274
$623$	2132.25	0.137122
$624$	0	0
$625$	−19362.7	−1.23921
$626$	702.744	0.0448679
$627$	−9652.44	−0.614802
$628$	3476.29	0.220890
$629$	961.718	0.0609637
$630$	−891.261	−0.0563630
$631$	−4816.53	−0.303871	−0.151936	−	0.988390i	$-0.548551\pi$
$631$	−4816.53	−0.303871	−0.151936	+	0.988390i	$0.548551\pi$
$632$	5465.45	0.343993
$633$	23693.3	1.48772
$634$	3460.43	0.216768
$635$	13371.4	0.835634
$636$	7288.43	0.454410
$637$	0	0
$638$	21834.8	1.35494
$639$	4478.20	0.277238
$640$	−1690.95	−0.104439
$641$	−23114.5	−1.42429	−0.712144	−	0.702033i	$-0.752276\pi$
$641$	−23114.5	−1.42429	−0.712144	+	0.702033i	$0.752276\pi$
$642$	1587.04	0.0975633
$643$	−13010.3	−0.797941	−0.398970	−	0.916964i	$-0.630632\pi$
$643$	−13010.3	−0.797941	−0.398970	+	0.916964i	$0.630632\pi$
$644$	886.650	0.0542530
$645$	−21368.4	−1.30447
$646$	1323.78	0.0806248
$647$	−2568.67	−0.156082	−0.0780408	−	0.996950i	$-0.524866\pi$
$647$	−2568.67	−0.156082	−0.0780408	+	0.996950i	$0.524866\pi$
$648$	−4605.32	−0.279188
$649$	33732.7	2.04025
$650$	0	0
$651$	−8626.72	−0.519367
$652$	4312.60	0.259041
$653$	−19455.7	−1.16594	−0.582972	−	0.812492i	$-0.698110\pi$
$653$	−19455.7	−1.16594	−0.582972	+	0.812492i	$0.698110\pi$
$654$	4083.27	0.244141
$655$	−3294.65	−0.196538
$656$	−498.962	−0.0296970
$657$	5231.40	0.310649
$658$	3089.51	0.183042
$659$	−24103.7	−1.42480	−0.712402	−	0.701772i	$-0.752393\pi$
$659$	−24103.7	−1.42480	−0.712402	+	0.701772i	$0.752393\pi$
$660$	−9300.90	−0.548541
$661$	−29175.0	−1.71676	−0.858378	−	0.513017i	$-0.828528\pi$
$661$	−29175.0	−1.71676	−0.858378	+	0.513017i	$0.828528\pi$
$662$	−18811.0	−1.10440
$663$	0	0
$664$	7830.11	0.457631
$665$	−5071.21	−0.295719
$666$	767.961	0.0446815
$667$	9250.40	0.536997
$668$	5744.76	0.332742
$669$	27875.2	1.61094
$670$	−20926.1	−1.20663
$671$	−17025.5	−0.979526
$672$	1054.97	0.0605598
$673$	−3443.69	−0.197243	−0.0986214	−	0.995125i	$-0.531443\pi$
$673$	−3443.69	−0.197243	−0.0986214	+	0.995125i	$0.531443\pi$
$674$	8597.45	0.491337
$675$	7420.69	0.423144
$676$	0	0
$677$	−20359.6	−1.15581	−0.577904	−	0.816105i	$-0.696129\pi$
$677$	−20359.6	−1.15581	−0.577904	+	0.816105i	$0.696129\pi$
$678$	−433.421	−0.0245508
$679$	1983.22	0.112090
$680$	1275.57	0.0719353
$681$	−6717.12	−0.377975
$682$	19558.7	1.09816
$683$	21335.9	1.19531	0.597655	−	0.801754i	$-0.296099\pi$
$683$	21335.9	1.19531	0.597655	+	0.801754i	$0.296099\pi$
$684$	1057.08	0.0590915
$685$	−21307.4	−1.18849
$686$	−686.000	−0.0381802
$687$	−17367.0	−0.964471
$688$	−5495.17	−0.304508
$689$	0	0
$690$	−3940.36	−0.217401
$691$	23931.6	1.31751	0.658756	−	0.752357i	$-0.271083\pi$
$691$	23931.6	1.31751	0.658756	+	0.752357i	$0.271083\pi$
$692$	10553.7	0.579755
$693$	−1260.69	−0.0691048
$694$	−4095.81	−0.224027
$695$	−7450.47	−0.406636
$696$	11006.4	0.599422
$697$	376.394	0.0204547
$698$	−15266.4	−0.827855
$699$	−25563.5	−1.38326
$700$	−1386.52	−0.0748649
$701$	−8076.20	−0.435141	−0.217571	−	0.976045i	$-0.569813\pi$
$701$	−8076.20	−0.435141	−0.217571	+	0.976045i	$0.569813\pi$
$702$	0	0
$703$	4369.64	0.234430
$704$	−2391.85	−0.128049
$705$	−13730.1	−0.733483
$706$	−20976.0	−1.11819
$707$	3640.05	0.193633
$708$	17003.9	0.902605
$709$	−1088.36	−0.0576505	−0.0288252	−	0.999584i	$-0.509177\pi$
$709$	−1088.36	−0.0576505	−0.0288252	+	0.999584i	$0.509177\pi$
$710$	24552.6	1.29781
$711$	−3292.24	−0.173655
$712$	−2436.86	−0.128266
$713$	8286.12	0.435228
$714$	−795.817	−0.0417125
$715$	0	0
$716$	−8912.04	−0.465166
$717$	20714.6	1.07894
$718$	10822.7	0.562534
$719$	11773.5	0.610680	0.305340	−	0.952243i	$-0.401230\pi$
$719$	11773.5	0.610680	0.305340	+	0.952243i	$0.401230\pi$
$720$	1018.58	0.0527228
$721$	−5450.20	−0.281520
$722$	−7703.28	−0.397073
$723$	−27120.3	−1.39504
$724$	−3064.68	−0.157317
$725$	−14465.5	−0.741014
$726$	−618.978	−0.0316425
$727$	22522.4	1.14898	0.574492	−	0.818511i	$-0.305200\pi$
$727$	22522.4	1.14898	0.574492	+	0.818511i	$0.305200\pi$
$728$	0	0
$729$	21830.1	1.10908
$730$	28682.2	1.45421
$731$	4145.30	0.209739
$732$	−8582.16	−0.433341
$733$	−1960.56	−0.0987924	−0.0493962	−	0.998779i	$-0.515730\pi$
$733$	−1960.56	−0.0987924	−0.0493962	+	0.998779i	$0.515730\pi$
$734$	−8157.45	−0.410214
$735$	3048.65	0.152995
$736$	−1013.31	−0.0507490
$737$	−29599.9	−1.47941
$738$	300.562	0.0149917
$739$	−30744.9	−1.53041	−0.765203	−	0.643789i	$-0.777361\pi$
$739$	−30744.9	−1.53041	−0.765203	+	0.643789i	$0.777361\pi$
$740$	4210.50	0.209164
$741$	0	0
$742$	5416.40	0.267982
$743$	−19873.6	−0.981279	−0.490640	−	0.871363i	$-0.663237\pi$
$743$	−19873.6	−0.981279	−0.490640	+	0.871363i	$0.663237\pi$
$744$	9859.10	0.485823
$745$	20191.8	0.992981
$746$	13447.4	0.659977
$747$	−4716.66	−0.231022
$748$	1804.30	0.0881974
$749$	1179.41	0.0575366
$750$	−9392.52	−0.457288
$751$	−1282.02	−0.0622923	−0.0311462	−	0.999515i	$-0.509916\pi$
$751$	−1282.02	−0.0622923	−0.0311462	+	0.999515i	$0.509916\pi$
$752$	−3530.87	−0.171220
$753$	9843.24	0.476371
$754$	0	0
$755$	−16905.8	−0.814920
$756$	−4196.00	−0.201861
$757$	1954.16	0.0938243	0.0469121	−	0.998899i	$-0.485062\pi$
$757$	1954.16	0.0938243	0.0469121	+	0.998899i	$0.485062\pi$
$758$	−21560.2	−1.03312
$759$	−5573.64	−0.266548
$760$	5795.67	0.276620
$761$	−28636.7	−1.36410	−0.682049	−	0.731306i	$-0.738911\pi$
$761$	−28636.7	−1.36410	−0.682049	+	0.731306i	$0.738911\pi$
$762$	9534.03	0.453257
$763$	3034.49	0.143979
$764$	−16315.9	−0.772627
$765$	−768.372	−0.0363145
$766$	17990.3	0.848584
$767$	0	0
$768$	−1205.68	−0.0566485
$769$	−1596.84	−0.0748811	−0.0374406	−	0.999299i	$-0.511920\pi$
$769$	−1596.84	−0.0748811	−0.0374406	+	0.999299i	$0.511920\pi$
$770$	−6911.98	−0.323494
$771$	8441.77	0.394323
$772$	−14942.0	−0.696600
$773$	−36657.9	−1.70568	−0.852841	−	0.522171i	$-0.825122\pi$
$773$	−36657.9	−1.70568	−0.852841	+	0.522171i	$0.825122\pi$
$774$	3310.15	0.153722
$775$	−12957.6	−0.600581
$776$	−2266.54	−0.104851
$777$	−2626.89	−0.121286
$778$	3277.81	0.151048
$779$	1710.17	0.0786564
$780$	0	0
$781$	34729.7	1.59120
$782$	764.397	0.0349550
$783$	−43776.7	−1.99802
$784$	784.000	0.0357143
$785$	−11480.9	−0.522002
$786$	−2349.14	−0.106605
$787$	576.010	0.0260896	0.0130448	−	0.999915i	$-0.495848\pi$
$787$	576.010	0.0260896	0.0130448	+	0.999915i	$0.495848\pi$
$788$	−2074.64	−0.0937895
$789$	−14114.5	−0.636870
$790$	−18050.4	−0.812916
$791$	−322.097	−0.0144785
$792$	1440.79	0.0646416
$793$	0	0
$794$	201.945	0.00902616
$795$	−24071.0	−1.07385
$796$	7379.01	0.328571
$797$	−23835.9	−1.05936	−0.529682	−	0.848197i	$-0.677689\pi$
$797$	−23835.9	−1.05936	−0.529682	+	0.848197i	$0.677689\pi$
$798$	−3615.86	−0.160401
$799$	2663.53	0.117933
$800$	1584.59	0.0700297
$801$	1467.90	0.0647512
$802$	11759.2	0.517746
$803$	40571.0	1.78296
$804$	−14920.6	−0.654491
$805$	−2928.28	−0.128209
$806$	0	0
$807$	−31315.9	−1.36601
$808$	−4160.06	−0.181127
$809$	14154.8	0.615151	0.307576	−	0.951524i	$-0.400482\pi$
$809$	14154.8	0.615151	0.307576	+	0.951524i	$0.400482\pi$
$810$	15209.7	0.659770
$811$	−44377.2	−1.92145	−0.960723	−	0.277508i	$-0.910492\pi$
$811$	−44377.2	−1.92145	−0.960723	+	0.277508i	$0.910492\pi$
$812$	8179.45	0.353501
$813$	−24354.8	−1.05063
$814$	5955.75	0.256448
$815$	−14243.0	−0.612159
$816$	909.505	0.0390184
$817$	18834.5	0.806531
$818$	−6748.61	−0.288459
$819$	0	0
$820$	1647.89	0.0701791
$821$	−40665.0	−1.72864	−0.864322	−	0.502938i	$-0.832252\pi$
$821$	−40665.0	−1.72864	−0.864322	+	0.502938i	$0.832252\pi$
$822$	−15192.6	−0.644649
$823$	−23221.5	−0.983538	−0.491769	−	0.870726i	$-0.663650\pi$
$823$	−23221.5	−0.983538	−0.491769	+	0.870726i	$0.663650\pi$
$824$	6228.80	0.263338
$825$	8715.88	0.367816
$826$	12636.4	0.532298
$827$	6351.39	0.267061	0.133530	−	0.991045i	$-0.457369\pi$
$827$	6351.39	0.267061	0.133530	+	0.991045i	$0.457369\pi$
$828$	610.395	0.0256192
$829$	20930.3	0.876885	0.438443	−	0.898759i	$-0.355530\pi$
$829$	20930.3	0.876885	0.438443	+	0.898759i	$0.355530\pi$
$830$	−25860.0	−1.08146
$831$	−4752.15	−0.198376
$832$	0	0
$833$	−591.413	−0.0245993
$834$	−5312.31	−0.220564
$835$	−18972.9	−0.786327
$836$	8197.97	0.339154
$837$	−39213.3	−1.61937
$838$	24317.6	1.00243
$839$	13623.5	0.560591	0.280296	−	0.959914i	$-0.409568\pi$
$839$	13623.5	0.560591	0.280296	+	0.959914i	$0.409568\pi$
$840$	−3484.17	−0.143113
$841$	60947.0	2.49896
$842$	14505.9	0.593712
$843$	35139.3	1.43566
$844$	−20123.1	−0.820695
$845$	0	0
$846$	2126.91	0.0864357
$847$	−459.995	−0.0186607
$848$	−6190.18	−0.250674
$849$	−26231.5	−1.06038
$850$	−1195.34	−0.0482351
$851$	2523.17	0.101637
$852$	17506.4	0.703945
$853$	11973.9	0.480633	0.240316	−	0.970695i	$-0.422749\pi$
$853$	11973.9	0.480633	0.240316	+	0.970695i	$0.422749\pi$
$854$	−6377.85	−0.255557
$855$	−3491.16	−0.139643
$856$	−1347.90	−0.0538205
$857$	−10649.6	−0.424484	−0.212242	−	0.977217i	$-0.568076\pi$
$857$	−10649.6	−0.424484	−0.212242	+	0.977217i	$0.568076\pi$
$858$	0	0
$859$	12761.2	0.506877	0.253438	−	0.967352i	$-0.418438\pi$
$859$	12761.2	0.506877	0.253438	+	0.967352i	$0.418438\pi$
$860$	18148.6	0.719605
$861$	−1028.10	−0.0406941
$862$	−17249.1	−0.681563
$863$	13715.4	0.540992	0.270496	−	0.962721i	$-0.412812\pi$
$863$	13715.4	0.540992	0.270496	+	0.962721i	$0.412812\pi$
$864$	4795.42	0.188824
$865$	−34854.9	−1.37006
$866$	7905.99	0.310227
$867$	22452.5	0.879502
$868$	7326.81	0.286507
$869$	−25532.3	−0.996689
$870$	−36350.3	−1.41654
$871$	0	0
$872$	−3467.99	−0.134680
$873$	1365.31	0.0529308
$874$	3473.10	0.134416
$875$	−6980.06	−0.269679
$876$	20450.9	0.788780
$877$	1900.54	0.0731776	0.0365888	−	0.999330i	$-0.488351\pi$
$877$	1900.54	0.0731776	0.0365888	+	0.999330i	$0.488351\pi$
$878$	1655.53	0.0636347
$879$	15949.7	0.612025
$880$	7899.40	0.302601
$881$	51828.0	1.98199	0.990993	−	0.133915i	$-0.0427548\pi$
$881$	51828.0	1.98199	0.990993	+	0.133915i	$0.0427548\pi$
$882$	−472.261	−0.0180293
$883$	−12165.4	−0.463644	−0.231822	−	0.972758i	$-0.574469\pi$
$883$	−12165.4	−0.463644	−0.231822	+	0.972758i	$0.574469\pi$
$884$	0	0
$885$	−56157.6	−2.13301
$886$	8361.29	0.317046
$887$	−34794.9	−1.31713	−0.658567	−	0.752522i	$-0.728837\pi$
$887$	−34794.9	−1.31713	−0.658567	+	0.752522i	$0.728837\pi$
$888$	3002.16	0.113453
$889$	7085.23	0.267301
$890$	8048.05	0.303114
$891$	21514.1	0.808922
$892$	−23674.9	−0.888669
$893$	12101.9	0.453500
$894$	14397.1	0.538603
$895$	29433.2	1.09927
$896$	−896.000	−0.0334077
$897$	0	0
$898$	−30577.0	−1.13627
$899$	76440.4	2.83585
$900$	−954.517	−0.0353525
$901$	4669.58	0.172660
$902$	2330.94	0.0860442
$903$	−11322.7	−0.417271
$904$	368.111	0.0135434
$905$	10121.5	0.371768
$906$	−12054.1	−0.442021
$907$	−11976.7	−0.438457	−0.219229	−	0.975673i	$-0.570354\pi$
$907$	−11976.7	−0.438457	−0.219229	+	0.975673i	$0.570354\pi$
$908$	5704.96	0.208509
$909$	2505.91	0.0914366
$910$	0	0
$911$	−1791.40	−0.0651500	−0.0325750	−	0.999469i	$-0.510371\pi$
$911$	−1791.40	−0.0651500	−0.0325750	+	0.999469i	$0.510371\pi$
$912$	4132.41	0.150041
$913$	−36579.0	−1.32595
$914$	−37128.6	−1.34366
$915$	28343.7	1.02406
$916$	14750.0	0.532047
$917$	−1745.77	−0.0628685
$918$	−3617.44	−0.130058
$919$	26169.1	0.939325	0.469663	−	0.882846i	$-0.344376\pi$
$919$	26169.1	0.939325	0.469663	+	0.882846i	$0.344376\pi$
$920$	3346.61	0.119929
$921$	−28251.9	−1.01078
$922$	−37952.2	−1.35563
$923$	0	0
$924$	−4928.36	−0.175467
$925$	−3945.66	−0.140252
$926$	−8680.66	−0.308061
$927$	−3752.07	−0.132939
$928$	−9347.95	−0.330670
$929$	−13067.6	−0.461500	−0.230750	−	0.973013i	$-0.574118\pi$
$929$	−13067.6	−0.461500	−0.230750	+	0.973013i	$0.574118\pi$
$930$	−32561.0	−1.14808
$931$	−2687.13	−0.0945941
$932$	21711.5	0.763072
$933$	−25361.6	−0.889928
$934$	17295.2	0.605904
$935$	−5958.94	−0.208426
$936$	0	0
$937$	−33477.3	−1.16719	−0.583594	−	0.812046i	$-0.698354\pi$
$937$	−33477.3	−1.16719	−0.583594	+	0.812046i	$0.698354\pi$
$938$	−11088.3	−0.385976
$939$	−1654.85	−0.0575121
$940$	11661.2	0.404624
$941$	−50388.6	−1.74561	−0.872807	−	0.488066i	$-0.837703\pi$
$941$	−50388.6	−1.74561	−0.872807	+	0.488066i	$0.837703\pi$
$942$	−8186.08	−0.283139
$943$	987.511	0.0341016
$944$	−14441.7	−0.497919
$945$	13857.8	0.477033
$946$	25671.1	0.882284
$947$	34825.5	1.19501	0.597506	−	0.801864i	$-0.296158\pi$
$947$	34825.5	1.19501	0.597506	+	0.801864i	$0.296158\pi$
$948$	−12870.2	−0.440934
$949$	0	0
$950$	−5431.13	−0.185483
$951$	−8148.74	−0.277856
$952$	675.900	0.0230106
$953$	−49163.7	−1.67111	−0.835555	−	0.549406i	$-0.814854\pi$
$953$	−49163.7	−1.67111	−0.835555	+	0.549406i	$0.814854\pi$
$954$	3728.80	0.126546
$955$	53885.3	1.82585
$956$	−17593.2	−0.595194
$957$	−51417.4	−1.73677
$958$	−28201.8	−0.951104
$959$	−11290.4	−0.380172
$960$	3981.91	0.133870
$961$	38681.1	1.29842
$962$	0	0
$963$	811.942	0.0271697
$964$	23033.7	0.769572
$965$	49348.1	1.64619
$966$	−2087.92	−0.0695420
$967$	−55880.6	−1.85832	−0.929162	−	0.369673i	$-0.879470\pi$
$967$	−55880.6	−1.85832	−0.929162	+	0.369673i	$0.879470\pi$
$968$	525.708	0.0174555
$969$	−3117.29	−0.103346
$970$	7485.56	0.247780
$971$	−31479.6	−1.04040	−0.520199	−	0.854045i	$-0.674142\pi$
$971$	−31479.6	−1.04040	−0.520199	+	0.854045i	$0.674142\pi$
$972$	−5339.80	−0.176208
$973$	−3947.85	−0.130074
$974$	−1176.65	−0.0387086
$975$	0	0
$976$	7288.97	0.239051
$977$	17306.4	0.566716	0.283358	−	0.959014i	$-0.408552\pi$
$977$	17306.4	0.566716	0.283358	+	0.959014i	$0.408552\pi$
$978$	−10155.5	−0.332041
$979$	11384.0	0.371638
$980$	−2589.27	−0.0843991
$981$	2089.03	0.0679893
$982$	−14473.0	−0.470316
$983$	1613.34	0.0523474	0.0261737	−	0.999657i	$-0.491668\pi$
$983$	1613.34	0.0523474	0.0261737	+	0.999657i	$0.491668\pi$
$984$	1174.97	0.0380659
$985$	6851.79	0.221641
$986$	7051.65	0.227759
$987$	−7275.30	−0.234625
$988$	0	0
$989$	10875.7	0.349672
$990$	−4758.40	−0.152759
$991$	−60621.9	−1.94321	−0.971603	−	0.236617i	$-0.923961\pi$
$991$	−60621.9	−1.94321	−0.971603	+	0.236617i	$0.923961\pi$
$992$	−8373.50	−0.268003
$993$	44296.8	1.41562
$994$	13009.9	0.415141
$995$	−24370.2	−0.776469
$996$	−18438.6	−0.586597
$997$	52900.9	1.68043	0.840215	−	0.542253i	$-0.182429\pi$
$997$	52900.9	1.68043	0.840215	+	0.542253i	$0.182429\pi$
$998$	−21155.9	−0.671022
$999$	−11940.7	−0.378165

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.4.a.bg.1.4		12
13.2	odd	12		182.4.m.b.43.11	✓	24
13.7	odd	12		182.4.m.b.127.11	yes	24
13.12	even	2		2366.4.a.bd.1.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.4.m.b.43.11	✓	24	13.2	odd	12
182.4.m.b.127.11	yes	24	13.7	odd	12
2366.4.a.bd.1.4		12	13.12	even	2
2366.4.a.bg.1.4		12	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:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2366.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -4.70967 q^{3} +4.00000 q^{4} -13.2105 q^{5} -9.41934 q^{6} -7.00000 q^{7} +8.00000 q^{8} -4.81899 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -4.70967 q^{3} +4.00000 q^{4} -13.2105 q^{5} -9.41934 q^{6} -7.00000 q^{7} +8.00000 q^{8} -4.81899 q^{9} -26.4211 q^{10} -37.3726 q^{11} -18.8387 q^{12} -14.0000 q^{14} +62.2173 q^{15} +16.0000 q^{16} -12.0696 q^{17} -9.63799 q^{18} -54.8394 q^{19} -52.8422 q^{20} +32.9677 q^{21} -74.7453 q^{22} -31.6661 q^{23} -37.6774 q^{24} +49.5185 q^{25} +149.857 q^{27} -28.0000 q^{28} -292.123 q^{29} +124.435 q^{30} -261.672 q^{31} +32.0000 q^{32} +176.013 q^{33} -24.1393 q^{34} +92.4738 q^{35} -19.2760 q^{36} -79.6807 q^{37} -109.679 q^{38} -105.684 q^{40} -31.1851 q^{41} +65.9354 q^{42} -343.448 q^{43} -149.491 q^{44} +63.6615 q^{45} -63.3322 q^{46} -220.680 q^{47} -75.3547 q^{48} +49.0000 q^{49} +99.0369 q^{50} +56.8441 q^{51} -386.886 q^{53} +299.714 q^{54} +493.713 q^{55} -56.0000 q^{56} +258.276 q^{57} -584.247 q^{58} -902.603 q^{59} +248.869 q^{60} +455.560 q^{61} -523.344 q^{62} +33.7330 q^{63} +64.0000 q^{64} +352.026 q^{66} +792.021 q^{67} -48.2786 q^{68} +149.137 q^{69} +184.948 q^{70} -929.282 q^{71} -38.5520 q^{72} -1085.58 q^{73} -159.361 q^{74} -233.216 q^{75} -219.358 q^{76} +261.608 q^{77} +683.181 q^{79} -211.369 q^{80} -575.664 q^{81} -62.3703 q^{82} +978.764 q^{83} +131.871 q^{84} +159.447 q^{85} -686.897 q^{86} +1375.80 q^{87} -298.981 q^{88} -304.607 q^{89} +127.323 q^{90} -126.664 q^{92} +1232.39 q^{93} -441.359 q^{94} +724.458 q^{95} -150.709 q^{96} -283.318 q^{97} +98.0000 q^{98} +180.098 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 24 q^{2} + 6 q^{3} + 48 q^{4} + 28 q^{5} + 12 q^{6} - 84 q^{7} + 96 q^{8} + 162 q^{9}+O(q^{10}) \) 12 * q + 24 * q^2 + 6 * q^3 + 48 * q^4 + 28 * q^5 + 12 * q^6 - 84 * q^7 + 96 * q^8 + 162 * q^9	\( 12 q + 24 q^{2} + 6 q^{3} + 48 q^{4} + 28 q^{5} + 12 q^{6} - 84 q^{7} + 96 q^{8} + 162 q^{9} + 56 q^{10} + 6 q^{11} + 24 q^{12} - 168 q^{14} - 138 q^{15} + 192 q^{16} - 56 q^{17} + 324 q^{18} + 158 q^{19} + 112 q^{20} - 42 q^{21} + 12 q^{22} + 414 q^{23} + 48 q^{24} + 478 q^{25} + 390 q^{27} - 336 q^{28} + 222 q^{29} - 276 q^{30} - 200 q^{31} + 384 q^{32} + 844 q^{33} - 112 q^{34} - 196 q^{35} + 648 q^{36} - 560 q^{37} + 316 q^{38} + 224 q^{40} - 66 q^{41} - 84 q^{42} + 484 q^{43} + 24 q^{44} + 542 q^{45} + 828 q^{46} + 618 q^{47} + 96 q^{48} + 588 q^{49} + 956 q^{50} + 992 q^{51} + 504 q^{53} + 780 q^{54} + 2584 q^{55} - 672 q^{56} - 1164 q^{57} + 444 q^{58} + 1460 q^{59} - 552 q^{60} - 2 q^{61} - 400 q^{62} - 1134 q^{63} + 768 q^{64} + 1688 q^{66} + 334 q^{67} - 224 q^{68} + 4660 q^{69} - 392 q^{70} - 196 q^{71} + 1296 q^{72} + 490 q^{73} - 1120 q^{74} - 338 q^{75} + 632 q^{76} - 42 q^{77} + 2942 q^{79} + 448 q^{80} + 2824 q^{81} - 132 q^{82} + 236 q^{83} - 168 q^{84} + 1352 q^{85} + 968 q^{86} + 1456 q^{87} + 48 q^{88} + 3566 q^{89} + 1084 q^{90} + 1656 q^{92} + 1884 q^{93} + 1236 q^{94} + 5754 q^{95} + 192 q^{96} + 3032 q^{97} + 1176 q^{98} + 3670 q^{99}+O(q^{100}) \) 12 * q + 24 * q^2 + 6 * q^3 + 48 * q^4 + 28 * q^5 + 12 * q^6 - 84 * q^7 + 96 * q^8 + 162 * q^9 + 56 * q^10 + 6 * q^11 + 24 * q^12 - 168 * q^14 - 138 * q^15 + 192 * q^16 - 56 * q^17 + 324 * q^18 + 158 * q^19 + 112 * q^20 - 42 * q^21 + 12 * q^22 + 414 * q^23 + 48 * q^24 + 478 * q^25 + 390 * q^27 - 336 * q^28 + 222 * q^29 - 276 * q^30 - 200 * q^31 + 384 * q^32 + 844 * q^33 - 112 * q^34 - 196 * q^35 + 648 * q^36 - 560 * q^37 + 316 * q^38 + 224 * q^40 - 66 * q^41 - 84 * q^42 + 484 * q^43 + 24 * q^44 + 542 * q^45 + 828 * q^46 + 618 * q^47 + 96 * q^48 + 588 * q^49 + 956 * q^50 + 992 * q^51 + 504 * q^53 + 780 * q^54 + 2584 * q^55 - 672 * q^56 - 1164 * q^57 + 444 * q^58 + 1460 * q^59 - 552 * q^60 - 2 * q^61 - 400 * q^62 - 1134 * q^63 + 768 * q^64 + 1688 * q^66 + 334 * q^67 - 224 * q^68 + 4660 * q^69 - 392 * q^70 - 196 * q^71 + 1296 * q^72 + 490 * q^73 - 1120 * q^74 - 338 * q^75 + 632 * q^76 - 42 * q^77 + 2942 * q^79 + 448 * q^80 + 2824 * q^81 - 132 * q^82 + 236 * q^83 - 168 * q^84 + 1352 * q^85 + 968 * q^86 + 1456 * q^87 + 48 * q^88 + 3566 * q^89 + 1084 * q^90 + 1656 * q^92 + 1884 * q^93 + 1236 * q^94 + 5754 * q^95 + 192 * q^96 + 3032 * q^97 + 1176 * q^98 + 3670 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more