LMFDB - Embedded newform 4026.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4026,2,Mod(1,4026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4026, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4026.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4026 = 2 \cdot 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4026.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:	$32.1477718538$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1129.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 3 $ x^3 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.39766$ of defining polynomial
Character	$\chi$	$=$	4026.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.39766 q^{5} -1.00000 q^{6} -4.14644 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.39766 q^{5} -1.00000 q^{6} -4.14644 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.39766 q^{10} +1.00000 q^{11} -1.00000 q^{12} +3.54410 q^{13} -4.14644 q^{14} +3.39766 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +6.54410 q^{19} -3.39766 q^{20} +4.14644 q^{21} +1.00000 q^{22} +4.54410 q^{23} -1.00000 q^{24} +6.54410 q^{25} +3.54410 q^{26} -1.00000 q^{27} -4.14644 q^{28} -1.35112 q^{29} +3.39766 q^{30} -9.14644 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} +14.0882 q^{35} +1.00000 q^{36} -5.74878 q^{37} +6.54410 q^{38} -3.54410 q^{39} -3.39766 q^{40} +0.497562 q^{41} +4.14644 q^{42} -2.39766 q^{43} +1.00000 q^{44} -3.39766 q^{45} +4.54410 q^{46} -1.90010 q^{47} -1.00000 q^{48} +10.1930 q^{49} +6.54410 q^{50} -1.00000 q^{51} +3.54410 q^{52} -6.69055 q^{53} -1.00000 q^{54} -3.39766 q^{55} -4.14644 q^{56} -6.54410 q^{57} -1.35112 q^{58} +10.4442 q^{59} +3.39766 q^{60} -1.00000 q^{61} -9.14644 q^{62} -4.14644 q^{63} +1.00000 q^{64} -12.0417 q^{65} -1.00000 q^{66} +9.14644 q^{67} +1.00000 q^{68} -4.54410 q^{69} +14.0882 q^{70} -9.14644 q^{71} +1.00000 q^{72} +4.64888 q^{73} -5.74878 q^{74} -6.54410 q^{75} +6.54410 q^{76} -4.14644 q^{77} -3.54410 q^{78} +3.54410 q^{79} -3.39766 q^{80} +1.00000 q^{81} +0.497562 q^{82} -17.7371 q^{83} +4.14644 q^{84} -3.39766 q^{85} -2.39766 q^{86} +1.35112 q^{87} +1.00000 q^{88} -12.6954 q^{89} -3.39766 q^{90} -14.6954 q^{91} +4.54410 q^{92} +9.14644 q^{93} -1.90010 q^{94} -22.2347 q^{95} -1.00000 q^{96} +10.7371 q^{97} +10.1930 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 2 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{11} - 3 q^{12} - 7 q^{13} - 2 q^{14} + 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} + 3 q^{22} - 4 q^{23} - 3 q^{24} + 2 q^{25} - 7 q^{26} - 3 q^{27} - 2 q^{28} - 8 q^{29} + 3 q^{30} - 17 q^{31} + 3 q^{32} - 3 q^{33} + 3 q^{34} + 7 q^{35} + 3 q^{36} - 14 q^{37} + 2 q^{38} + 7 q^{39} - 3 q^{40} - 5 q^{41} + 2 q^{42} + 3 q^{44} - 3 q^{45} - 4 q^{46} - 5 q^{47} - 3 q^{48} + 9 q^{49} + 2 q^{50} - 3 q^{51} - 7 q^{52} + 8 q^{53} - 3 q^{54} - 3 q^{55} - 2 q^{56} - 2 q^{57} - 8 q^{58} + 13 q^{59} + 3 q^{60} - 3 q^{61} - 17 q^{62} - 2 q^{63} + 3 q^{64} - 12 q^{65} - 3 q^{66} + 17 q^{67} + 3 q^{68} + 4 q^{69} + 7 q^{70} - 17 q^{71} + 3 q^{72} + 10 q^{73} - 14 q^{74} - 2 q^{75} + 2 q^{76} - 2 q^{77} + 7 q^{78} - 7 q^{79} - 3 q^{80} + 3 q^{81} - 5 q^{82} - 14 q^{83} + 2 q^{84} - 3 q^{85} + 8 q^{87} + 3 q^{88} - 23 q^{89} - 3 q^{90} - 29 q^{91} - 4 q^{92} + 17 q^{93} - 5 q^{94} - 21 q^{95} - 3 q^{96} - 7 q^{97} + 9 q^{98} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 2 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^11 - 3 * q^12 - 7 * q^13 - 2 * q^14 + 3 * q^15 + 3 * q^16 + 3 * q^17 + 3 * q^18 + 2 * q^19 - 3 * q^20 + 2 * q^21 + 3 * q^22 - 4 * q^23 - 3 * q^24 + 2 * q^25 - 7 * q^26 - 3 * q^27 - 2 * q^28 - 8 * q^29 + 3 * q^30 - 17 * q^31 + 3 * q^32 - 3 * q^33 + 3 * q^34 + 7 * q^35 + 3 * q^36 - 14 * q^37 + 2 * q^38 + 7 * q^39 - 3 * q^40 - 5 * q^41 + 2 * q^42 + 3 * q^44 - 3 * q^45 - 4 * q^46 - 5 * q^47 - 3 * q^48 + 9 * q^49 + 2 * q^50 - 3 * q^51 - 7 * q^52 + 8 * q^53 - 3 * q^54 - 3 * q^55 - 2 * q^56 - 2 * q^57 - 8 * q^58 + 13 * q^59 + 3 * q^60 - 3 * q^61 - 17 * q^62 - 2 * q^63 + 3 * q^64 - 12 * q^65 - 3 * q^66 + 17 * q^67 + 3 * q^68 + 4 * q^69 + 7 * q^70 - 17 * q^71 + 3 * q^72 + 10 * q^73 - 14 * q^74 - 2 * q^75 + 2 * q^76 - 2 * q^77 + 7 * q^78 - 7 * q^79 - 3 * q^80 + 3 * q^81 - 5 * q^82 - 14 * q^83 + 2 * q^84 - 3 * q^85 + 8 * q^87 + 3 * q^88 - 23 * q^89 - 3 * q^90 - 29 * q^91 - 4 * q^92 + 17 * q^93 - 5 * q^94 - 21 * q^95 - 3 * q^96 - 7 * q^97 + 9 * q^98 + 3 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−3.39766	−1.51948	−0.759740	−	0.650227i	$-0.774674\pi$
$5$	−3.39766	−1.51948	−0.759740	+	0.650227i	$0.774674\pi$
$6$	−1.00000	−0.408248
$7$	−4.14644	−1.56721	−0.783604	−	0.621261i	$-0.786621\pi$
$7$	−4.14644	−1.56721	−0.783604	+	0.621261i	$0.786621\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.39766	−1.07443
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	3.54410	0.982958	0.491479	−	0.870890i	$-0.336457\pi$
$13$	3.54410	0.982958	0.491479	+	0.870890i	$0.336457\pi$
$14$	−4.14644	−1.10818
$15$	3.39766	0.877272
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	1.00000	0.235702
$19$	6.54410	1.50132	0.750660	−	0.660688i	$-0.229736\pi$
$19$	6.54410	1.50132	0.750660	+	0.660688i	$0.229736\pi$
$20$	−3.39766	−0.759740
$21$	4.14644	0.904828
$22$	1.00000	0.213201
$23$	4.54410	0.947511	0.473756	−	0.880656i	$-0.342898\pi$
$23$	4.54410	0.947511	0.473756	+	0.880656i	$0.342898\pi$
$24$	−1.00000	−0.204124
$25$	6.54410	1.30882
$26$	3.54410	0.695056
$27$	−1.00000	−0.192450
$28$	−4.14644	−0.783604
$29$	−1.35112	−0.250897	−0.125448	−	0.992100i	$-0.540037\pi$
$29$	−1.35112	−0.250897	−0.125448	+	0.992100i	$0.540037\pi$
$30$	3.39766	0.620325
$31$	−9.14644	−1.64275	−0.821375	−	0.570389i	$-0.806793\pi$
$31$	−9.14644	−1.64275	−0.821375	+	0.570389i	$0.806793\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	1.00000	0.171499
$35$	14.0882	2.38134
$36$	1.00000	0.166667
$37$	−5.74878	−0.945094	−0.472547	−	0.881306i	$-0.656665\pi$
$37$	−5.74878	−0.945094	−0.472547	+	0.881306i	$0.656665\pi$
$38$	6.54410	1.06159
$39$	−3.54410	−0.567511
$40$	−3.39766	−0.537217
$41$	0.497562	0.0777061	0.0388530	−	0.999245i	$-0.487630\pi$
$41$	0.497562	0.0777061	0.0388530	+	0.999245i	$0.487630\pi$
$42$	4.14644	0.639810
$43$	−2.39766	−0.365640	−0.182820	−	0.983146i	$-0.558523\pi$
$43$	−2.39766	−0.365640	−0.182820	+	0.983146i	$0.558523\pi$
$44$	1.00000	0.150756
$45$	−3.39766	−0.506493
$46$	4.54410	0.669992
$47$	−1.90010	−0.277158	−0.138579	−	0.990351i	$-0.544253\pi$
$47$	−1.90010	−0.277158	−0.138579	+	0.990351i	$0.544253\pi$
$48$	−1.00000	−0.144338
$49$	10.1930	1.45614
$50$	6.54410	0.925476
$51$	−1.00000	−0.140028
$52$	3.54410	0.491479
$53$	−6.69055	−0.919017	−0.459509	−	0.888173i	$-0.651974\pi$
$53$	−6.69055	−0.919017	−0.459509	+	0.888173i	$0.651974\pi$
$54$	−1.00000	−0.136083
$55$	−3.39766	−0.458141
$56$	−4.14644	−0.554092
$57$	−6.54410	−0.866788
$58$	−1.35112	−0.177411
$59$	10.4442	1.35972	0.679860	−	0.733342i	$-0.262041\pi$
$59$	10.4442	1.35972	0.679860	+	0.733342i	$0.262041\pi$
$60$	3.39766	0.438636
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−9.14644	−1.16160
$63$	−4.14644	−0.522403
$64$	1.00000	0.125000
$65$	−12.0417	−1.49358
$66$	−1.00000	−0.123091
$67$	9.14644	1.11742	0.558708	−	0.829365i	$-0.311297\pi$
$67$	9.14644	1.11742	0.558708	+	0.829365i	$0.311297\pi$
$68$	1.00000	0.121268
$69$	−4.54410	−0.547046
$70$	14.0882	1.68386
$71$	−9.14644	−1.08548	−0.542742	−	0.839900i	$-0.682614\pi$
$71$	−9.14644	−1.08548	−0.542742	+	0.839900i	$0.682614\pi$
$72$	1.00000	0.117851
$73$	4.64888	0.544110	0.272055	−	0.962282i	$-0.412297\pi$
$73$	4.64888	0.544110	0.272055	+	0.962282i	$0.412297\pi$
$74$	−5.74878	−0.668282
$75$	−6.54410	−0.755648
$76$	6.54410	0.750660
$77$	−4.14644	−0.472531
$78$	−3.54410	−0.401291
$79$	3.54410	0.398743	0.199371	−	0.979924i	$-0.436110\pi$
$79$	3.54410	0.398743	0.199371	+	0.979924i	$0.436110\pi$
$80$	−3.39766	−0.379870
$81$	1.00000	0.111111
$82$	0.497562	0.0549465
$83$	−17.7371	−1.94690	−0.973449	−	0.228903i	$-0.926486\pi$
$83$	−17.7371	−1.94690	−0.973449	+	0.228903i	$0.926486\pi$
$84$	4.14644	0.452414
$85$	−3.39766	−0.368528
$86$	−2.39766	−0.258546
$87$	1.35112	0.144855
$88$	1.00000	0.106600
$89$	−12.6954	−1.34571	−0.672856	−	0.739773i	$-0.734933\pi$
$89$	−12.6954	−1.34571	−0.672856	+	0.739773i	$0.734933\pi$
$90$	−3.39766	−0.358145
$91$	−14.6954	−1.54050
$92$	4.54410	0.473756
$93$	9.14644	0.948442
$94$	−1.90010	−0.195980
$95$	−22.2347	−2.28123
$96$	−1.00000	−0.102062
$97$	10.7371	1.09019	0.545093	−	0.838376i	$-0.316494\pi$
$97$	10.7371	1.09019	0.545093	+	0.838376i	$0.316494\pi$
$98$	10.1930	1.02965
$99$	1.00000	0.100504
$100$	6.54410	0.654410
$101$	−10.9883	−1.09338	−0.546689	−	0.837336i	$-0.684112\pi$
$101$	−10.9883	−1.09338	−0.546689	+	0.837336i	$0.684112\pi$
$102$	−1.00000	−0.0990148
$103$	−12.1930	−1.20141	−0.600705	−	0.799471i	$-0.705113\pi$
$103$	−12.1930	−1.20141	−0.600705	+	0.799471i	$0.705113\pi$
$104$	3.54410	0.347528
$105$	−14.0882	−1.37487
$106$	−6.69055	−0.649843
$107$	5.04167	0.487396	0.243698	−	0.969851i	$-0.421639\pi$
$107$	5.04167	0.487396	0.243698	+	0.969851i	$0.421639\pi$
$108$	−1.00000	−0.0962250
$109$	−18.5392	−1.77574	−0.887868	−	0.460098i	$-0.847814\pi$
$109$	−18.5392	−1.77574	−0.887868	+	0.460098i	$0.847814\pi$
$110$	−3.39766	−0.323954
$111$	5.74878	0.545650
$112$	−4.14644	−0.391802
$113$	7.19298	0.676659	0.338330	−	0.941028i	$-0.390138\pi$
$113$	7.19298	0.676659	0.338330	+	0.941028i	$0.390138\pi$
$114$	−6.54410	−0.612911
$115$	−15.4393	−1.43972
$116$	−1.35112	−0.125448
$117$	3.54410	0.327653
$118$	10.4442	0.961467
$119$	−4.14644	−0.380104
$120$	3.39766	0.310163
$121$	1.00000	0.0909091
$122$	−1.00000	−0.0905357
$123$	−0.497562	−0.0448636
$124$	−9.14644	−0.821375
$125$	−5.24634	−0.469247
$126$	−4.14644	−0.369394
$127$	5.79532	0.514252	0.257126	−	0.966378i	$-0.417224\pi$
$127$	5.79532	0.514252	0.257126	+	0.966378i	$0.417224\pi$
$128$	1.00000	0.0883883
$129$	2.39766	0.211102
$130$	−12.0417	−1.05612
$131$	−8.90010	−0.777605	−0.388803	−	0.921321i	$-0.627111\pi$
$131$	−8.90010	−0.777605	−0.388803	+	0.921321i	$0.627111\pi$
$132$	−1.00000	−0.0870388
$133$	−27.1348	−2.35288
$134$	9.14644	0.790132
$135$	3.39766	0.292424
$136$	1.00000	0.0857493
$137$	−13.7953	−1.17861	−0.589307	−	0.807909i	$-0.700599\pi$
$137$	−13.7953	−1.17861	−0.589307	+	0.807909i	$0.700599\pi$
$138$	−4.54410	−0.386820
$139$	−20.6323	−1.75001	−0.875005	−	0.484114i	$-0.839142\pi$
$139$	−20.6323	−1.75001	−0.875005	+	0.484114i	$0.839142\pi$
$140$	14.0882	1.19067
$141$	1.90010	0.160017
$142$	−9.14644	−0.767552
$143$	3.54410	0.296373
$144$	1.00000	0.0833333
$145$	4.59065	0.381232
$146$	4.64888	0.384744
$147$	−10.1930	−0.840703
$148$	−5.74878	−0.472547
$149$	−21.3860	−1.75201	−0.876003	−	0.482305i	$-0.839800\pi$
$149$	−21.3860	−1.75201	−0.876003	+	0.482305i	$0.839800\pi$
$150$	−6.54410	−0.534324
$151$	15.5906	1.26875	0.634374	−	0.773026i	$-0.281258\pi$
$151$	15.5906	1.26875	0.634374	+	0.773026i	$0.281258\pi$
$152$	6.54410	0.530797
$153$	1.00000	0.0808452
$154$	−4.14644	−0.334130
$155$	31.0765	2.49613
$156$	−3.54410	−0.283755
$157$	3.44420	0.274877	0.137439	−	0.990510i	$-0.456113\pi$
$157$	3.44420	0.274877	0.137439	+	0.990510i	$0.456113\pi$
$158$	3.54410	0.281954
$159$	6.69055	0.530595
$160$	−3.39766	−0.268609
$161$	−18.8419	−1.48495
$162$	1.00000	0.0785674
$163$	12.1464	0.951383	0.475691	−	0.879612i	$-0.342198\pi$
$163$	12.1464	0.951383	0.475691	+	0.879612i	$0.342198\pi$
$164$	0.497562	0.0388530
$165$	3.39766	0.264508
$166$	−17.7371	−1.37667
$167$	−2.10478	−0.162873	−0.0814363	−	0.996679i	$-0.525951\pi$
$167$	−2.10478	−0.162873	−0.0814363	+	0.996679i	$0.525951\pi$
$168$	4.14644	0.319905
$169$	−0.439327	−0.0337944
$170$	−3.39766	−0.260589
$171$	6.54410	0.500440
$172$	−2.39766	−0.182820
$173$	−3.35112	−0.254781	−0.127390	−	0.991853i	$-0.540660\pi$
$173$	−3.35112	−0.254781	−0.127390	+	0.991853i	$0.540660\pi$
$174$	1.35112	0.102428
$175$	−27.1348	−2.05119
$176$	1.00000	0.0753778
$177$	−10.4442	−0.785034
$178$	−12.6954	−0.951562
$179$	20.6857	1.54612	0.773060	−	0.634333i	$-0.218725\pi$
$179$	20.6857	1.54612	0.773060	+	0.634333i	$0.218725\pi$
$180$	−3.39766	−0.253247
$181$	−15.0931	−1.12186	−0.560930	−	0.827863i	$-0.689556\pi$
$181$	−15.0931	−1.12186	−0.560930	+	0.827863i	$0.689556\pi$
$182$	−14.6954	−1.08930
$183$	1.00000	0.0739221
$184$	4.54410	0.334996
$185$	19.5324	1.43605
$186$	9.14644	0.670650
$187$	1.00000	0.0731272
$188$	−1.90010	−0.138579
$189$	4.14644	0.301609
$190$	−22.2347	−1.61307
$191$	8.24634	0.596684	0.298342	−	0.954459i	$-0.403566\pi$
$191$	8.24634	0.596684	0.298342	+	0.954459i	$0.403566\pi$
$192$	−1.00000	−0.0721688
$193$	23.4393	1.68720	0.843600	−	0.536972i	$-0.180432\pi$
$193$	23.4393	1.68720	0.843600	+	0.536972i	$0.180432\pi$
$194$	10.7371	0.770878
$195$	12.0417	0.862322
$196$	10.1930	0.728070
$197$	−9.64400	−0.687107	−0.343553	−	0.939133i	$-0.611631\pi$
$197$	−9.64400	−0.687107	−0.343553	+	0.939133i	$0.611631\pi$
$198$	1.00000	0.0710669
$199$	−6.70224	−0.475109	−0.237555	−	0.971374i	$-0.576346\pi$
$199$	−6.70224	−0.475109	−0.237555	+	0.971374i	$0.576346\pi$
$200$	6.54410	0.462738
$201$	−9.14644	−0.645140
$202$	−10.9883	−0.773135
$203$	5.60234	0.393207
$204$	−1.00000	−0.0700140
$205$	−1.69055	−0.118073
$206$	−12.1930	−0.849525
$207$	4.54410	0.315837
$208$	3.54410	0.245739
$209$	6.54410	0.452665
$210$	−14.0882	−0.972179
$211$	−16.8787	−1.16197	−0.580987	−	0.813913i	$-0.697333\pi$
$211$	−16.8787	−1.16197	−0.580987	+	0.813913i	$0.697333\pi$
$212$	−6.69055	−0.459509
$213$	9.14644	0.626704
$214$	5.04167	0.344641
$215$	8.14644	0.555583
$216$	−1.00000	−0.0680414
$217$	37.9252	2.57453
$218$	−18.5392	−1.25563
$219$	−4.64888	−0.314142
$220$	−3.39766	−0.229070
$221$	3.54410	0.238402
$222$	5.74878	0.385833
$223$	20.3977	1.36593	0.682964	−	0.730452i	$-0.260690\pi$
$223$	20.3977	1.36593	0.682964	+	0.730452i	$0.260690\pi$
$224$	−4.14644	−0.277046
$225$	6.54410	0.436274
$226$	7.19298	0.478470
$227$	−0.765350	−0.0507980	−0.0253990	−	0.999677i	$-0.508086\pi$
$227$	−0.765350	−0.0507980	−0.0253990	+	0.999677i	$0.508086\pi$
$228$	−6.54410	−0.433394
$229$	−5.95833	−0.393738	−0.196869	−	0.980430i	$-0.563077\pi$
$229$	−5.95833	−0.393738	−0.196869	+	0.980430i	$0.563077\pi$
$230$	−15.4393	−1.01804
$231$	4.14644	0.272816
$232$	−1.35112	−0.0887053
$233$	−21.7904	−1.42754	−0.713770	−	0.700380i	$-0.753014\pi$
$233$	−21.7904	−1.42754	−0.713770	+	0.700380i	$0.753014\pi$
$234$	3.54410	0.231685
$235$	6.45590	0.421136
$236$	10.4442	0.679860
$237$	−3.54410	−0.230214
$238$	−4.14644	−0.268774
$239$	17.1881	1.11181	0.555903	−	0.831247i	$-0.312373\pi$
$239$	17.1881	1.11181	0.555903	+	0.831247i	$0.312373\pi$
$240$	3.39766	0.219318
$241$	−3.33943	−0.215111	−0.107556	−	0.994199i	$-0.534302\pi$
$241$	−3.33943	−0.215111	−0.107556	+	0.994199i	$0.534302\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	−1.00000	−0.0640184
$245$	−34.6323	−2.21258
$246$	−0.497562	−0.0317234
$247$	23.1930	1.47573
$248$	−9.14644	−0.580800
$249$	17.7371	1.12404
$250$	−5.24634	−0.331808
$251$	−30.3229	−1.91396	−0.956981	−	0.290151i	$-0.906295\pi$
$251$	−30.3229	−1.91396	−0.956981	+	0.290151i	$0.906295\pi$
$252$	−4.14644	−0.261201
$253$	4.54410	0.285685
$254$	5.79532	0.363631
$255$	3.39766	0.212770
$256$	1.00000	0.0625000
$257$	15.2395	0.950616	0.475308	−	0.879820i	$-0.342337\pi$
$257$	15.2395	0.950616	0.475308	+	0.879820i	$0.342337\pi$
$258$	2.39766	0.149272
$259$	23.8370	1.48116
$260$	−12.0417	−0.746792
$261$	−1.35112	−0.0836322
$262$	−8.90010	−0.549850
$263$	−19.9466	−1.22996	−0.614981	−	0.788542i	$-0.710836\pi$
$263$	−19.9466	−1.22996	−0.614981	+	0.788542i	$0.710836\pi$
$264$	−1.00000	−0.0615457
$265$	22.7322	1.39643
$266$	−27.1348	−1.66374
$267$	12.6954	0.776947
$268$	9.14644	0.558708
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	3.39766	0.206775
$271$	−18.5441	−1.12647	−0.563237	−	0.826295i	$-0.690444\pi$
$271$	−18.5441	−1.12647	−0.563237	+	0.826295i	$0.690444\pi$
$272$	1.00000	0.0606339
$273$	14.6954	0.889407
$274$	−13.7953	−0.833406
$275$	6.54410	0.394624
$276$	−4.54410	−0.273523
$277$	−6.54898	−0.393490	−0.196745	−	0.980455i	$-0.563037\pi$
$277$	−6.54898	−0.393490	−0.196745	+	0.980455i	$0.563037\pi$
$278$	−20.6323	−1.23744
$279$	−9.14644	−0.547583
$280$	14.0882	0.841931
$281$	−15.7371	−0.938796	−0.469398	−	0.882987i	$-0.655529\pi$
$281$	−15.7371	−0.938796	−0.469398	+	0.882987i	$0.655529\pi$
$282$	1.90010	0.113149
$283$	13.6023	0.808575	0.404288	−	0.914632i	$-0.367520\pi$
$283$	13.6023	0.808575	0.404288	+	0.914632i	$0.367520\pi$
$284$	−9.14644	−0.542742
$285$	22.2347	1.31707
$286$	3.54410	0.209567
$287$	−2.06311	−0.121782
$288$	1.00000	0.0589256
$289$	−16.0000	−0.941176
$290$	4.59065	0.269572
$291$	−10.7371	−0.629419
$292$	4.64888	0.272055
$293$	4.80702	0.280829	0.140414	−	0.990093i	$-0.455157\pi$
$293$	4.80702	0.280829	0.140414	+	0.990093i	$0.455157\pi$
$294$	−10.1930	−0.594467
$295$	−35.4859	−2.06607
$296$	−5.74878	−0.334141
$297$	−1.00000	−0.0580259
$298$	−21.3860	−1.23886
$299$	16.1048	0.931363
$300$	−6.54410	−0.377824
$301$	9.94177	0.573034
$302$	15.5906	0.897141
$303$	10.9883	0.631262
$304$	6.54410	0.375330
$305$	3.39766	0.194550
$306$	1.00000	0.0571662
$307$	14.5441	0.830076	0.415038	−	0.909804i	$-0.363768\pi$
$307$	14.5441	0.830076	0.415038	+	0.909804i	$0.363768\pi$
$308$	−4.14644	−0.236265
$309$	12.1930	0.693635
$310$	31.0765	1.76503
$311$	−11.9486	−0.677542	−0.338771	−	0.940869i	$-0.610011\pi$
$311$	−11.9486	−0.677542	−0.338771	+	0.940869i	$0.610011\pi$
$312$	−3.54410	−0.200645
$313$	−13.0368	−0.736883	−0.368441	−	0.929651i	$-0.620108\pi$
$313$	−13.0368	−0.736883	−0.368441	+	0.929651i	$0.620108\pi$
$314$	3.44420	0.194368
$315$	14.0882	0.793781
$316$	3.54410	0.199371
$317$	30.3743	1.70599	0.852995	−	0.521920i	$-0.174784\pi$
$317$	30.3743	1.70599	0.852995	+	0.521920i	$0.174784\pi$
$318$	6.69055	0.375187
$319$	−1.35112	−0.0756482
$320$	−3.39766	−0.189935
$321$	−5.04167	−0.281398
$322$	−18.8419	−1.05002
$323$	6.54410	0.364124
$324$	1.00000	0.0555556
$325$	23.1930	1.28652
$326$	12.1464	0.672729
$327$	18.5392	1.02522
$328$	0.497562	0.0274733
$329$	7.87865	0.434364
$330$	3.39766	0.187035
$331$	7.23953	0.397920	0.198960	−	0.980008i	$-0.436244\pi$
$331$	7.23953	0.397920	0.198960	+	0.980008i	$0.436244\pi$
$332$	−17.7371	−0.973449
$333$	−5.74878	−0.315031
$334$	−2.10478	−0.115168
$335$	−31.0765	−1.69789
$336$	4.14644	0.226207
$337$	29.5809	1.61137	0.805687	−	0.592342i	$-0.201796\pi$
$337$	29.5809	1.61137	0.805687	+	0.592342i	$0.201796\pi$
$338$	−0.439327	−0.0238963
$339$	−7.19298	−0.390669
$340$	−3.39766	−0.184264
$341$	−9.14644	−0.495308
$342$	6.54410	0.353865
$343$	−13.2395	−0.714867
$344$	−2.39766	−0.129273
$345$	15.4393	0.831225
$346$	−3.35112	−0.180157
$347$	−2.48587	−0.133448	−0.0667242	−	0.997771i	$-0.521255\pi$
$347$	−2.48587	−0.133448	−0.0667242	+	0.997771i	$0.521255\pi$
$348$	1.35112	0.0724276
$349$	18.8884	1.01107	0.505537	−	0.862805i	$-0.331295\pi$
$349$	18.8884	1.01107	0.505537	+	0.862805i	$0.331295\pi$
$350$	−27.1348	−1.45041
$351$	−3.54410	−0.189170
$352$	1.00000	0.0533002
$353$	−31.1348	−1.65714	−0.828568	−	0.559889i	$-0.810844\pi$
$353$	−31.1348	−1.65714	−0.828568	+	0.559889i	$0.810844\pi$
$354$	−10.4442	−0.555103
$355$	31.0765	1.64937
$356$	−12.6954	−0.672856
$357$	4.14644	0.219453
$358$	20.6857	1.09327
$359$	15.1764	0.800981	0.400490	−	0.916301i	$-0.368840\pi$
$359$	15.1764	0.800981	0.400490	+	0.916301i	$0.368840\pi$
$360$	−3.39766	−0.179072
$361$	23.8253	1.25396
$362$	−15.0931	−0.793275
$363$	−1.00000	−0.0524864
$364$	−14.6954	−0.770249
$365$	−15.7953	−0.826765
$366$	1.00000	0.0522708
$367$	−26.6740	−1.39237	−0.696185	−	0.717862i	$-0.745121\pi$
$367$	−26.6740	−1.39237	−0.696185	+	0.717862i	$0.745121\pi$
$368$	4.54410	0.236878
$369$	0.497562	0.0259020
$370$	19.5324	1.01544
$371$	27.7420	1.44029
$372$	9.14644	0.474221
$373$	−1.43933	−0.0745255	−0.0372628	−	0.999306i	$-0.511864\pi$
$373$	−1.43933	−0.0745255	−0.0372628	+	0.999306i	$0.511864\pi$
$374$	1.00000	0.0517088
$375$	5.24634	0.270920
$376$	−1.90010	−0.0979902
$377$	−4.78851	−0.246621
$378$	4.14644	0.213270
$379$	−6.80020	−0.349303	−0.174651	−	0.984630i	$-0.555880\pi$
$379$	−6.80020	−0.349303	−0.174651	+	0.984630i	$0.555880\pi$
$380$	−22.2347	−1.14061
$381$	−5.79532	−0.296903
$382$	8.24634	0.421920
$383$	24.5906	1.25652	0.628262	−	0.778002i	$-0.283767\pi$
$383$	24.5906	1.25652	0.628262	+	0.778002i	$0.283767\pi$
$384$	−1.00000	−0.0510310
$385$	14.0882	0.718002
$386$	23.4393	1.19303
$387$	−2.39766	−0.121880
$388$	10.7371	0.545093
$389$	28.1464	1.42708	0.713541	−	0.700614i	$-0.247090\pi$
$389$	28.1464	1.42708	0.713541	+	0.700614i	$0.247090\pi$
$390$	12.0417	0.609753
$391$	4.54410	0.229805
$392$	10.1930	0.514823
$393$	8.90010	0.448951
$394$	−9.64400	−0.485858
$395$	−12.0417	−0.605882
$396$	1.00000	0.0502519
$397$	4.03485	0.202503	0.101252	−	0.994861i	$-0.467715\pi$
$397$	4.03485	0.202503	0.101252	+	0.994861i	$0.467715\pi$
$398$	−6.70224	−0.335953
$399$	27.1348	1.35844
$400$	6.54410	0.327205
$401$	−4.29288	−0.214376	−0.107188	−	0.994239i	$-0.534185\pi$
$401$	−4.29288	−0.214376	−0.107188	+	0.994239i	$0.534185\pi$
$402$	−9.14644	−0.456183
$403$	−32.4159	−1.61475
$404$	−10.9883	−0.546689
$405$	−3.39766	−0.168831
$406$	5.60234	0.278039
$407$	−5.74878	−0.284956
$408$	−1.00000	−0.0495074
$409$	−21.2298	−1.04974	−0.524872	−	0.851181i	$-0.675887\pi$
$409$	−21.2298	−1.04974	−0.524872	+	0.851181i	$0.675887\pi$
$410$	−1.69055	−0.0834901
$411$	13.7953	0.680473
$412$	−12.1930	−0.600705
$413$	−43.3063	−2.13096
$414$	4.54410	0.223331
$415$	60.2646	2.95827
$416$	3.54410	0.173764
$417$	20.6323	1.01037
$418$	6.54410	0.320083
$419$	5.83699	0.285156	0.142578	−	0.989784i	$-0.454461\pi$
$419$	5.83699	0.285156	0.142578	+	0.989784i	$0.454461\pi$
$420$	−14.0882	−0.687434
$421$	21.5392	1.04976	0.524879	−	0.851177i	$-0.324111\pi$
$421$	21.5392	1.04976	0.524879	+	0.851177i	$0.324111\pi$
$422$	−16.8787	−0.821640
$423$	−1.90010	−0.0923860
$424$	−6.69055	−0.324922
$425$	6.54410	0.317436
$426$	9.14644	0.443147
$427$	4.14644	0.200660
$428$	5.04167	0.243698
$429$	−3.54410	−0.171111
$430$	8.14644	0.392856
$431$	−14.4976	−0.698323	−0.349161	−	0.937063i	$-0.613534\pi$
$431$	−14.4976	−0.698323	−0.349161	+	0.937063i	$0.613534\pi$
$432$	−1.00000	−0.0481125
$433$	−25.8884	−1.24412	−0.622059	−	0.782971i	$-0.713704\pi$
$433$	−25.8884	−1.24412	−0.622059	+	0.782971i	$0.713704\pi$
$434$	37.9252	1.82047
$435$	−4.59065	−0.220105
$436$	−18.5392	−0.887868
$437$	29.7371	1.42252
$438$	−4.64888	−0.222132
$439$	−12.7488	−0.608466	−0.304233	−	0.952598i	$-0.598400\pi$
$439$	−12.7488	−0.608466	−0.304233	+	0.952598i	$0.598400\pi$
$440$	−3.39766	−0.161977
$441$	10.1930	0.485380
$442$	3.54410	0.168576
$443$	−40.0017	−1.90054	−0.950269	−	0.311429i	$-0.899192\pi$
$443$	−40.0017	−1.90054	−0.950269	+	0.311429i	$0.899192\pi$
$444$	5.74878	0.272825
$445$	43.1348	2.04478
$446$	20.3977	0.965857
$447$	21.3860	1.01152
$448$	−4.14644	−0.195901
$449$	−27.8884	−1.31614	−0.658068	−	0.752959i	$-0.728626\pi$
$449$	−27.8884	−1.31614	−0.658068	+	0.752959i	$0.728626\pi$
$450$	6.54410	0.308492
$451$	0.497562	0.0234293
$452$	7.19298	0.338330
$453$	−15.5906	−0.732512
$454$	−0.765350	−0.0359196
$455$	49.9301	2.34076
$456$	−6.54410	−0.306456
$457$	−10.6440	−0.497906	−0.248953	−	0.968516i	$-0.580086\pi$
$457$	−10.6440	−0.497906	−0.248953	+	0.968516i	$0.580086\pi$
$458$	−5.95833	−0.278415
$459$	−1.00000	−0.0466760
$460$	−15.4393	−0.719862
$461$	2.20955	0.102909	0.0514546	−	0.998675i	$-0.483614\pi$
$461$	2.20955	0.102909	0.0514546	+	0.998675i	$0.483614\pi$
$462$	4.14644	0.192910
$463$	0.163011	0.00757577	0.00378789	−	0.999993i	$-0.498794\pi$
$463$	0.163011	0.00757577	0.00378789	+	0.999993i	$0.498794\pi$
$464$	−1.35112	−0.0627241
$465$	−31.0765	−1.44114
$466$	−21.7904	−1.00942
$467$	16.3443	0.756324	0.378162	−	0.925739i	$-0.376556\pi$
$467$	16.3443	0.756324	0.378162	+	0.925739i	$0.376556\pi$
$468$	3.54410	0.163826
$469$	−37.9252	−1.75122
$470$	6.45590	0.297788
$471$	−3.44420	−0.158701
$472$	10.4442	0.480733
$473$	−2.39766	−0.110245
$474$	−3.54410	−0.162786
$475$	42.8253	1.96496
$476$	−4.14644	−0.190052
$477$	−6.69055	−0.306339
$478$	17.1881	0.786166
$479$	−11.4859	−0.524803	−0.262401	−	0.964959i	$-0.584514\pi$
$479$	−11.4859	−0.524803	−0.262401	+	0.964959i	$0.584514\pi$
$480$	3.39766	0.155081
$481$	−20.3743	−0.928987
$482$	−3.33943	−0.152107
$483$	18.8419	0.857335
$484$	1.00000	0.0454545
$485$	−36.4810	−1.65652
$486$	−1.00000	−0.0453609
$487$	28.6489	1.29820	0.649102	−	0.760701i	$-0.275145\pi$
$487$	28.6489	1.29820	0.649102	+	0.760701i	$0.275145\pi$
$488$	−1.00000	−0.0452679
$489$	−12.1464	−0.549281
$490$	−34.6323	−1.56453
$491$	7.98831	0.360507	0.180254	−	0.983620i	$-0.442308\pi$
$491$	7.98831	0.360507	0.180254	+	0.983620i	$0.442308\pi$
$492$	−0.497562	−0.0224318
$493$	−1.35112	−0.0608514
$494$	23.1930	1.04350
$495$	−3.39766	−0.152714
$496$	−9.14644	−0.410687
$497$	37.9252	1.70118
$498$	17.7371	0.794818
$499$	31.5441	1.41211	0.706054	−	0.708158i	$-0.250474\pi$
$499$	31.5441	1.41211	0.706054	+	0.708158i	$0.250474\pi$
$500$	−5.24634	−0.234624
$501$	2.10478	0.0940345
$502$	−30.3229	−1.35338
$503$	−31.2298	−1.39247	−0.696233	−	0.717815i	$-0.745142\pi$
$503$	−31.2298	−1.39247	−0.696233	+	0.717815i	$0.745142\pi$
$504$	−4.14644	−0.184697
$505$	37.3346	1.66137
$506$	4.54410	0.202010
$507$	0.439327	0.0195112
$508$	5.79532	0.257126
$509$	21.0134	0.931403	0.465701	−	0.884942i	$-0.345802\pi$
$509$	21.0134	0.931403	0.465701	+	0.884942i	$0.345802\pi$
$510$	3.39766	0.150451
$511$	−19.2763	−0.852734
$512$	1.00000	0.0441942
$513$	−6.54410	−0.288929
$514$	15.2395	0.672187
$515$	41.4276	1.82552
$516$	2.39766	0.105551
$517$	−1.90010	−0.0835663
$518$	23.8370	1.04734
$519$	3.35112	0.147098
$520$	−12.0417	−0.528062
$521$	−28.4393	−1.24595	−0.622975	−	0.782242i	$-0.714076\pi$
$521$	−28.4393	−1.24595	−0.622975	+	0.782242i	$0.714076\pi$
$522$	−1.35112	−0.0591369
$523$	32.1784	1.40706	0.703531	−	0.710665i	$-0.251606\pi$
$523$	32.1784	1.40706	0.703531	+	0.710665i	$0.251606\pi$
$524$	−8.90010	−0.388803
$525$	27.1348	1.18426
$526$	−19.9466	−0.869715
$527$	−9.14644	−0.398425
$528$	−1.00000	−0.0435194
$529$	−2.35112	−0.102223
$530$	22.7322	0.987424
$531$	10.4442	0.453240
$532$	−27.1348	−1.17644
$533$	1.76341	0.0763818
$534$	12.6954	0.549385
$535$	−17.1299	−0.740589
$536$	9.14644	0.395066
$537$	−20.6857	−0.892653
$538$	−21.0000	−0.905374
$539$	10.1930	0.439043
$540$	3.39766	0.146212
$541$	26.8903	1.15611	0.578053	−	0.815999i	$-0.303813\pi$
$541$	26.8903	1.15611	0.578053	+	0.815999i	$0.303813\pi$
$542$	−18.5441	−0.796537
$543$	15.0931	0.647706
$544$	1.00000	0.0428746
$545$	62.9900	2.69820
$546$	14.6954	0.628906
$547$	−12.6323	−0.540119	−0.270059	−	0.962844i	$-0.587043\pi$
$547$	−12.6323	−0.540119	−0.270059	+	0.962844i	$0.587043\pi$
$548$	−13.7953	−0.589307
$549$	−1.00000	−0.0426790
$550$	6.54410	0.279042
$551$	−8.84187	−0.376676
$552$	−4.54410	−0.193410
$553$	−14.6954	−0.624913
$554$	−6.54898	−0.278240
$555$	−19.5324	−0.829105
$556$	−20.6323	−0.875005
$557$	−20.4442	−0.866249	−0.433124	−	0.901334i	$-0.642589\pi$
$557$	−20.4442	−0.866249	−0.433124	+	0.901334i	$0.642589\pi$
$558$	−9.14644	−0.387200
$559$	−8.49756	−0.359409
$560$	14.0882	0.595335
$561$	−1.00000	−0.0422200
$562$	−15.7371	−0.663829
$563$	−38.0551	−1.60383	−0.801915	−	0.597438i	$-0.796185\pi$
$563$	−38.0551	−1.60383	−0.801915	+	0.597438i	$0.796185\pi$
$564$	1.90010	0.0800086
$565$	−24.4393	−1.02817
$566$	13.6023	0.571749
$567$	−4.14644	−0.174134
$568$	−9.14644	−0.383776
$569$	−18.1182	−0.759554	−0.379777	−	0.925078i	$-0.623999\pi$
$569$	−18.1182	−0.759554	−0.379777	+	0.925078i	$0.623999\pi$
$570$	22.2347	0.931307
$571$	−3.28119	−0.137314	−0.0686568	−	0.997640i	$-0.521871\pi$
$571$	−3.28119	−0.137314	−0.0686568	+	0.997640i	$0.521871\pi$
$572$	3.54410	0.148186
$573$	−8.24634	−0.344496
$574$	−2.06311	−0.0861126
$575$	29.7371	1.24012
$576$	1.00000	0.0416667
$577$	−43.6089	−1.81546	−0.907732	−	0.419551i	$-0.862187\pi$
$577$	−43.6089	−1.81546	−0.907732	+	0.419551i	$0.862187\pi$
$578$	−16.0000	−0.665512
$579$	−23.4393	−0.974105
$580$	4.59065	0.190616
$581$	73.5458	3.05119
$582$	−10.7371	−0.445067
$583$	−6.69055	−0.277094
$584$	4.64888	0.192372
$585$	−12.0417	−0.497862
$586$	4.80702	0.198576
$587$	−25.7088	−1.06112	−0.530558	−	0.847648i	$-0.678018\pi$
$587$	−25.7088	−1.06112	−0.530558	+	0.847648i	$0.678018\pi$
$588$	−10.1930	−0.420352
$589$	−59.8553	−2.46629
$590$	−35.4859	−1.46093
$591$	9.64400	0.396701
$592$	−5.74878	−0.236273
$593$	−0.314330	−0.0129080	−0.00645400	−	0.999979i	$-0.502054\pi$
$593$	−0.314330	−0.0129080	−0.00645400	+	0.999979i	$0.502054\pi$
$594$	−1.00000	−0.0410305
$595$	14.0882	0.577560
$596$	−21.3860	−0.876003
$597$	6.70224	0.274304
$598$	16.1048	0.658573
$599$	7.75366	0.316806	0.158403	−	0.987375i	$-0.449366\pi$
$599$	7.75366	0.316806	0.158403	+	0.987375i	$0.449366\pi$
$600$	−6.54410	−0.267162
$601$	−11.9951	−0.489291	−0.244646	−	0.969613i	$-0.578672\pi$
$601$	−11.9951	−0.489291	−0.244646	+	0.969613i	$0.578672\pi$
$602$	9.94177	0.405196
$603$	9.14644	0.372472
$604$	15.5906	0.634374
$605$	−3.39766	−0.138135
$606$	10.9883	0.446369
$607$	−12.9981	−0.527575	−0.263788	−	0.964581i	$-0.584972\pi$
$607$	−12.9981	−0.527575	−0.263788	+	0.964581i	$0.584972\pi$
$608$	6.54410	0.265398
$609$	−5.60234	−0.227018
$610$	3.39766	0.137567
$611$	−6.73415	−0.272435
$612$	1.00000	0.0404226
$613$	18.6274	0.752355	0.376178	−	0.926548i	$-0.377238\pi$
$613$	18.6274	0.752355	0.376178	+	0.926548i	$0.377238\pi$
$614$	14.5441	0.586952
$615$	1.69055	0.0681694
$616$	−4.14644	−0.167065
$617$	−44.7088	−1.79991	−0.899955	−	0.435983i	$-0.856401\pi$
$617$	−44.7088	−1.79991	−0.899955	+	0.435983i	$0.856401\pi$
$618$	12.1930	0.490474
$619$	−14.5926	−0.586526	−0.293263	−	0.956032i	$-0.594741\pi$
$619$	−14.5926	−0.586526	−0.293263	+	0.956032i	$0.594741\pi$
$620$	31.0765	1.24806
$621$	−4.54410	−0.182349
$622$	−11.9486	−0.479094
$623$	52.6408	2.10901
$624$	−3.54410	−0.141878
$625$	−14.8952	−0.595809
$626$	−13.0368	−0.521055
$627$	−6.54410	−0.261346
$628$	3.44420	0.137439
$629$	−5.74878	−0.229219
$630$	14.0882	0.561288
$631$	13.3462	0.531306	0.265653	−	0.964069i	$-0.414413\pi$
$631$	13.3462	0.531306	0.265653	+	0.964069i	$0.414413\pi$
$632$	3.54410	0.140977
$633$	16.8787	0.670866
$634$	30.3743	1.20632
$635$	−19.6905	−0.781395
$636$	6.69055	0.265297
$637$	36.1250	1.43132
$638$	−1.35112	−0.0534913
$639$	−9.14644	−0.361828
$640$	−3.39766	−0.134304
$641$	16.3957	0.647592	0.323796	−	0.946127i	$-0.395041\pi$
$641$	16.3957	0.647592	0.323796	+	0.946127i	$0.395041\pi$
$642$	−5.04167	−0.198979
$643$	4.60234	0.181499	0.0907493	−	0.995874i	$-0.471074\pi$
$643$	4.60234	0.181499	0.0907493	+	0.995874i	$0.471074\pi$
$644$	−18.8419	−0.742474
$645$	−8.14644	−0.320766
$646$	6.54410	0.257474
$647$	−18.4907	−0.726946	−0.363473	−	0.931605i	$-0.618409\pi$
$647$	−18.4907	−0.726946	−0.363473	+	0.931605i	$0.618409\pi$
$648$	1.00000	0.0392837
$649$	10.4442	0.409971
$650$	23.1930	0.909704
$651$	−37.9252	−1.48641
$652$	12.1464	0.475691
$653$	0.707115	0.0276716	0.0138358	−	0.999904i	$-0.495596\pi$
$653$	0.707115	0.0276716	0.0138358	+	0.999904i	$0.495596\pi$
$654$	18.5392	0.724941
$655$	30.2395	1.18156
$656$	0.497562	0.0194265
$657$	4.64888	0.181370
$658$	7.87865	0.307142
$659$	41.4742	1.61560	0.807802	−	0.589454i	$-0.200657\pi$
$659$	41.4742	1.61560	0.807802	+	0.589454i	$0.200657\pi$
$660$	3.39766	0.132254
$661$	36.8136	1.43188	0.715942	−	0.698160i	$-0.245998\pi$
$661$	36.8136	1.43188	0.715942	+	0.698160i	$0.245998\pi$
$662$	7.23953	0.281372
$663$	−3.54410	−0.137642
$664$	−17.7371	−0.688333
$665$	92.1947	3.57516
$666$	−5.74878	−0.222761
$667$	−6.13963	−0.237727
$668$	−2.10478	−0.0814363
$669$	−20.3977	−0.788619
$670$	−31.0765	−1.20059
$671$	−1.00000	−0.0386046
$672$	4.14644	0.159952
$673$	5.59746	0.215766	0.107883	−	0.994164i	$-0.465593\pi$
$673$	5.59746	0.215766	0.107883	+	0.994164i	$0.465593\pi$
$674$	29.5809	1.13941
$675$	−6.54410	−0.251883
$676$	−0.439327	−0.0168972
$677$	−1.80702	−0.0694492	−0.0347246	−	0.999397i	$-0.511055\pi$
$677$	−1.80702	−0.0694492	−0.0347246	+	0.999397i	$0.511055\pi$
$678$	−7.19298	−0.276245
$679$	−44.5207	−1.70855
$680$	−3.39766	−0.130294
$681$	0.765350	0.0293283
$682$	−9.14644	−0.350235
$683$	−26.2695	−1.00517	−0.502587	−	0.864526i	$-0.667618\pi$
$683$	−26.2695	−1.00517	−0.502587	+	0.864526i	$0.667618\pi$
$684$	6.54410	0.250220
$685$	46.8718	1.79088
$686$	−13.2395	−0.505487
$687$	5.95833	0.227325
$688$	−2.39766	−0.0914100
$689$	−23.7120	−0.903355
$690$	15.4393	0.587765
$691$	−12.5509	−0.477459	−0.238730	−	0.971086i	$-0.576731\pi$
$691$	−12.5509	−0.477459	−0.238730	+	0.971086i	$0.576731\pi$
$692$	−3.35112	−0.127390
$693$	−4.14644	−0.157510
$694$	−2.48587	−0.0943623
$695$	70.1016	2.65911
$696$	1.35112	0.0512140
$697$	0.497562	0.0188465
$698$	18.8884	0.714937
$699$	21.7904	0.824190
$700$	−27.1348	−1.02560
$701$	−33.1299	−1.25130	−0.625649	−	0.780105i	$-0.715166\pi$
$701$	−33.1299	−1.25130	−0.625649	+	0.780105i	$0.715166\pi$
$702$	−3.54410	−0.133764
$703$	−37.6206	−1.41889
$704$	1.00000	0.0376889
$705$	−6.45590	−0.243143
$706$	−31.1348	−1.17177
$707$	45.5624	1.71355
$708$	−10.4442	−0.392517
$709$	3.32115	0.124728	0.0623641	−	0.998053i	$-0.480136\pi$
$709$	3.32115	0.124728	0.0623641	+	0.998053i	$0.480136\pi$
$710$	31.0765	1.16628
$711$	3.54410	0.132914
$712$	−12.6954	−0.475781
$713$	−41.5624	−1.55652
$714$	4.14644	0.155177
$715$	−12.0417	−0.450333
$716$	20.6857	0.773060
$717$	−17.1881	−0.641902
$718$	15.1764	0.566379
$719$	−22.9252	−0.854966	−0.427483	−	0.904023i	$-0.640600\pi$
$719$	−22.9252	−0.854966	−0.427483	+	0.904023i	$0.640600\pi$
$720$	−3.39766	−0.126623
$721$	50.5575	1.88286
$722$	23.8253	0.886686
$723$	3.33943	0.124195
$724$	−15.0931	−0.560930
$725$	−8.84187	−0.328379
$726$	−1.00000	−0.0371135
$727$	−29.6391	−1.09925	−0.549627	−	0.835410i	$-0.685230\pi$
$727$	−29.6391	−1.09925	−0.549627	+	0.835410i	$0.685230\pi$
$728$	−14.6954	−0.544649
$729$	1.00000	0.0370370
$730$	−15.7953	−0.584611
$731$	−2.39766	−0.0886807
$732$	1.00000	0.0369611
$733$	30.8302	1.13874	0.569369	−	0.822082i	$-0.307187\pi$
$733$	30.8302	1.13874	0.569369	+	0.822082i	$0.307187\pi$
$734$	−26.6740	−0.984554
$735$	34.6323	1.27743
$736$	4.54410	0.167498
$737$	9.14644	0.336914
$738$	0.497562	0.0183155
$739$	46.4228	1.70769	0.853844	−	0.520528i	$-0.174265\pi$
$739$	46.4228	1.70769	0.853844	+	0.520528i	$0.174265\pi$
$740$	19.5324	0.718026
$741$	−23.1930	−0.852016
$742$	27.7420	1.01844
$743$	−4.09308	−0.150161	−0.0750804	−	0.997177i	$-0.523921\pi$
$743$	−4.09308	−0.150161	−0.0750804	+	0.997177i	$0.523921\pi$
$744$	9.14644	0.335325
$745$	72.6623	2.66214
$746$	−1.43933	−0.0526975
$747$	−17.7371	−0.648966
$748$	1.00000	0.0365636
$749$	−20.9050	−0.763851
$750$	5.24634	0.191569
$751$	−39.4674	−1.44018	−0.720092	−	0.693878i	$-0.755901\pi$
$751$	−39.4674	−1.44018	−0.720092	+	0.693878i	$0.755901\pi$
$752$	−1.90010	−0.0692895
$753$	30.3229	1.10503
$754$	−4.78851	−0.174387
$755$	−52.9717	−1.92784
$756$	4.14644	0.150805
$757$	−19.3414	−0.702974	−0.351487	−	0.936193i	$-0.614324\pi$
$757$	−19.3414	−0.702974	−0.351487	+	0.936193i	$0.614324\pi$
$758$	−6.80020	−0.246994
$759$	−4.54410	−0.164941
$760$	−22.2347	−0.806536
$761$	−52.1162	−1.88921	−0.944606	−	0.328206i	$-0.893556\pi$
$761$	−52.1162	−1.88921	−0.944606	+	0.328206i	$0.893556\pi$
$762$	−5.79532	−0.209942
$763$	76.8718	2.78295
$764$	8.24634	0.298342
$765$	−3.39766	−0.122843
$766$	24.5906	0.888496
$767$	37.0153	1.33655
$768$	−1.00000	−0.0360844
$769$	−45.8185	−1.65226	−0.826128	−	0.563482i	$-0.809461\pi$
$769$	−45.8185	−1.65226	−0.826128	+	0.563482i	$0.809461\pi$
$770$	14.0882	0.507704
$771$	−15.2395	−0.548838
$772$	23.4393	0.843600
$773$	11.0417	0.397141	0.198571	−	0.980087i	$-0.436370\pi$
$773$	11.0417	0.397141	0.198571	+	0.980087i	$0.436370\pi$
$774$	−2.39766	−0.0861822
$775$	−59.8553	−2.15006
$776$	10.7371	0.385439
$777$	−23.8370	−0.855147
$778$	28.1464	1.00910
$779$	3.25610	0.116662
$780$	12.0417	0.431161
$781$	−9.14644	−0.327285
$782$	4.54410	0.162497
$783$	1.35112	0.0482851
$784$	10.1930	0.364035
$785$	−11.7022	−0.417671
$786$	8.90010	0.317456
$787$	49.5741	1.76713	0.883563	−	0.468313i	$-0.155138\pi$
$787$	49.5741	1.76713	0.883563	+	0.468313i	$0.155138\pi$
$788$	−9.64400	−0.343553
$789$	19.9466	0.710119
$790$	−12.0417	−0.428423
$791$	−29.8253	−1.06047
$792$	1.00000	0.0355335
$793$	−3.54410	−0.125855
$794$	4.03485	0.143191
$795$	−22.7322	−0.806228
$796$	−6.70224	−0.237555
$797$	−14.7585	−0.522774	−0.261387	−	0.965234i	$-0.584180\pi$
$797$	−14.7585	−0.522774	−0.261387	+	0.965234i	$0.584180\pi$
$798$	27.1348	0.960560
$799$	−1.90010	−0.0672207
$800$	6.54410	0.231369
$801$	−12.6954	−0.448571
$802$	−4.29288	−0.151587
$803$	4.64888	0.164055
$804$	−9.14644	−0.322570
$805$	64.0183	2.25635
$806$	−32.4159	−1.14180
$807$	21.0000	0.739235
$808$	−10.9883	−0.386567
$809$	−44.7953	−1.57492	−0.787460	−	0.616366i	$-0.788604\pi$
$809$	−44.7953	−1.57492	−0.787460	+	0.616366i	$0.788604\pi$
$810$	−3.39766	−0.119382
$811$	−5.14157	−0.180545	−0.0902724	−	0.995917i	$-0.528774\pi$
$811$	−5.14157	−0.180545	−0.0902724	+	0.995917i	$0.528774\pi$
$812$	5.60234	0.196604
$813$	18.5441	0.650370
$814$	−5.74878	−0.201495
$815$	−41.2695	−1.44561
$816$	−1.00000	−0.0350070
$817$	−15.6905	−0.548943
$818$	−21.2298	−0.742282
$819$	−14.6954	−0.513500
$820$	−1.69055	−0.0590364
$821$	−33.9815	−1.18596	−0.592981	−	0.805216i	$-0.702049\pi$
$821$	−33.9815	−1.18596	−0.592981	+	0.805216i	$0.702049\pi$
$822$	13.7953	0.481167
$823$	38.7885	1.35208	0.676041	−	0.736864i	$-0.263694\pi$
$823$	38.7885	1.35208	0.676041	+	0.736864i	$0.263694\pi$
$824$	−12.1930	−0.424763
$825$	−6.54410	−0.227836
$826$	−43.3063	−1.50682
$827$	17.8467	0.620592	0.310296	−	0.950640i	$-0.399572\pi$
$827$	17.8467	0.620592	0.310296	+	0.950640i	$0.399572\pi$
$828$	4.54410	0.157919
$829$	1.15620	0.0401563	0.0200782	−	0.999798i	$-0.493608\pi$
$829$	1.15620	0.0401563	0.0200782	+	0.999798i	$0.493608\pi$
$830$	60.2646	2.09182
$831$	6.54898	0.227182
$832$	3.54410	0.122870
$833$	10.1930	0.353166
$834$	20.6323	0.714439
$835$	7.15132	0.247482
$836$	6.54410	0.226333
$837$	9.14644	0.316147
$838$	5.83699	0.201635
$839$	−18.0833	−0.624306	−0.312153	−	0.950032i	$-0.601050\pi$
$839$	−18.0833	−0.624306	−0.312153	+	0.950032i	$0.601050\pi$
$840$	−14.0882	−0.486089
$841$	−27.1745	−0.937051
$842$	21.5392	0.742291
$843$	15.7371	0.542014
$844$	−16.8787	−0.580987
$845$	1.49269	0.0513499
$846$	−1.90010	−0.0653268
$847$	−4.14644	−0.142473
$848$	−6.69055	−0.229754
$849$	−13.6023	−0.466831
$850$	6.54410	0.224461
$851$	−26.1231	−0.895487
$852$	9.14644	0.313352
$853$	−26.4276	−0.904865	−0.452432	−	0.891799i	$-0.649444\pi$
$853$	−26.4276	−0.904865	−0.452432	+	0.891799i	$0.649444\pi$
$854$	4.14644	0.141888
$855$	−22.2347	−0.760409
$856$	5.04167	0.172321
$857$	−7.10478	−0.242695	−0.121347	−	0.992610i	$-0.538721\pi$
$857$	−7.10478	−0.242695	−0.121347	+	0.992610i	$0.538721\pi$
$858$	−3.54410	−0.120994
$859$	−46.4228	−1.58392	−0.791962	−	0.610570i	$-0.790940\pi$
$859$	−46.4228	−1.58392	−0.791962	+	0.610570i	$0.790940\pi$
$860$	8.14644	0.277791
$861$	2.06311	0.0703106
$862$	−14.4976	−0.493789
$863$	42.4761	1.44590	0.722952	−	0.690898i	$-0.242785\pi$
$863$	42.4761	1.44590	0.722952	+	0.690898i	$0.242785\pi$
$864$	−1.00000	−0.0340207
$865$	11.3860	0.387135
$866$	−25.8884	−0.879724
$867$	16.0000	0.543388
$868$	37.9252	1.28727
$869$	3.54410	0.120225
$870$	−4.59065	−0.155637
$871$	32.4159	1.09837
$872$	−18.5392	−0.627817
$873$	10.7371	0.363395
$874$	29.7371	1.00587
$875$	21.7537	0.735408
$876$	−4.64888	−0.157071
$877$	−2.19298	−0.0740518	−0.0370259	−	0.999314i	$-0.511788\pi$
$877$	−2.19298	−0.0740518	−0.0370259	+	0.999314i	$0.511788\pi$
$878$	−12.7488	−0.430251
$879$	−4.80702	−0.162137
$880$	−3.39766	−0.114535
$881$	−4.10965	−0.138458	−0.0692289	−	0.997601i	$-0.522054\pi$
$881$	−4.10965	−0.138458	−0.0692289	+	0.997601i	$0.522054\pi$
$882$	10.1930	0.343216
$883$	−22.0280	−0.741302	−0.370651	−	0.928772i	$-0.620865\pi$
$883$	−22.0280	−0.741302	−0.370651	+	0.928772i	$0.620865\pi$
$884$	3.54410	0.119201
$885$	35.4859	1.19284
$886$	−40.0017	−1.34388
$887$	27.6274	0.927638	0.463819	−	0.885930i	$-0.346479\pi$
$887$	27.6274	0.927638	0.463819	+	0.885930i	$0.346479\pi$
$888$	5.74878	0.192916
$889$	−24.0300	−0.805939
$890$	43.1348	1.44588
$891$	1.00000	0.0335013
$892$	20.3977	0.682964
$893$	−12.4345	−0.416103
$894$	21.3860	0.715254
$895$	−70.2829	−2.34930
$896$	−4.14644	−0.138523
$897$	−16.1048	−0.537723
$898$	−27.8884	−0.930648
$899$	12.3579	0.412160
$900$	6.54410	0.218137
$901$	−6.69055	−0.222894
$902$	0.497562	0.0165670
$903$	−9.94177	−0.330841
$904$	7.19298	0.239235
$905$	51.2812	1.70464
$906$	−15.5906	−0.517965
$907$	44.1162	1.46486	0.732428	−	0.680845i	$-0.238387\pi$
$907$	44.1162	1.46486	0.732428	+	0.680845i	$0.238387\pi$
$908$	−0.765350	−0.0253990
$909$	−10.9883	−0.364459
$910$	49.9301	1.65517
$911$	17.6720	0.585501	0.292750	−	0.956189i	$-0.405430\pi$
$911$	17.6720	0.585501	0.292750	+	0.956189i	$0.405430\pi$
$912$	−6.54410	−0.216697
$913$	−17.7371	−0.587012
$914$	−10.6440	−0.352072
$915$	−3.39766	−0.112323
$916$	−5.95833	−0.196869
$917$	36.9038	1.21867
$918$	−1.00000	−0.0330049
$919$	−12.5422	−0.413728	−0.206864	−	0.978370i	$-0.566326\pi$
$919$	−12.5422	−0.413728	−0.206864	+	0.978370i	$0.566326\pi$
$920$	−15.4393	−0.509020
$921$	−14.5441	−0.479245
$922$	2.20955	0.0727678
$923$	−32.4159	−1.06698
$924$	4.14644	0.136408
$925$	−37.6206	−1.23696
$926$	0.163011	0.00535688
$927$	−12.1930	−0.400470
$928$	−1.35112	−0.0443527
$929$	38.5906	1.26612	0.633059	−	0.774104i	$-0.281799\pi$
$929$	38.5906	1.26612	0.633059	+	0.774104i	$0.281799\pi$
$930$	−31.0765	−1.01904
$931$	66.7040	2.18613
$932$	−21.7904	−0.713770
$933$	11.9486	0.391179
$934$	16.3443	0.534802
$935$	−3.39766	−0.111115
$936$	3.54410	0.115843
$937$	40.0465	1.30826	0.654132	−	0.756381i	$-0.273034\pi$
$937$	40.0465	1.30826	0.654132	+	0.756381i	$0.273034\pi$
$938$	−37.9252	−1.23830
$939$	13.0368	0.425440
$940$	6.45590	0.210568
$941$	37.5858	1.22526	0.612631	−	0.790369i	$-0.290111\pi$
$941$	37.5858	1.22526	0.612631	+	0.790369i	$0.290111\pi$
$942$	−3.44420	−0.112218
$943$	2.26097	0.0736274
$944$	10.4442	0.339930
$945$	−14.0882	−0.458289
$946$	−2.39766	−0.0779547
$947$	4.72540	0.153555	0.0767774	−	0.997048i	$-0.475537\pi$
$947$	4.72540	0.153555	0.0767774	+	0.997048i	$0.475537\pi$
$948$	−3.54410	−0.115107
$949$	16.4761	0.534837
$950$	42.8253	1.38944
$951$	−30.3743	−0.984953
$952$	−4.14644	−0.134387
$953$	−46.5275	−1.50717	−0.753587	−	0.657348i	$-0.771678\pi$
$953$	−46.5275	−1.50717	−0.753587	+	0.657348i	$0.771678\pi$
$954$	−6.69055	−0.216614
$955$	−28.0183	−0.906650
$956$	17.1881	0.555903
$957$	1.35112	0.0436755
$958$	−11.4859	−0.371092
$959$	57.2015	1.84713
$960$	3.39766	0.109659
$961$	52.6574	1.69863
$962$	−20.3743	−0.656893
$963$	5.04167	0.162465
$964$	−3.33943	−0.107556
$965$	−79.6389	−2.56367
$966$	18.8419	0.606227
$967$	−27.1182	−0.872062	−0.436031	−	0.899932i	$-0.643616\pi$
$967$	−27.1182	−0.872062	−0.436031	+	0.899932i	$0.643616\pi$
$968$	1.00000	0.0321412
$969$	−6.54410	−0.210227
$970$	−36.4810	−1.17133
$971$	−11.3462	−0.364118	−0.182059	−	0.983288i	$-0.558276\pi$
$971$	−11.3462	−0.364118	−0.182059	+	0.983288i	$0.558276\pi$
$972$	−1.00000	−0.0320750
$973$	85.5507	2.74263
$974$	28.6489	0.917969
$975$	−23.1930	−0.742770
$976$	−1.00000	−0.0320092
$977$	31.3411	1.00269	0.501346	−	0.865247i	$-0.332838\pi$
$977$	31.3411	1.00269	0.501346	+	0.865247i	$0.332838\pi$
$978$	−12.1464	−0.388400
$979$	−12.6954	−0.405747
$980$	−34.6323	−1.10629
$981$	−18.5392	−0.591912
$982$	7.98831	0.254917
$983$	50.1396	1.59921	0.799603	−	0.600529i	$-0.205043\pi$
$983$	50.1396	1.59921	0.799603	+	0.600529i	$0.205043\pi$
$984$	−0.497562	−0.0158617
$985$	32.7671	1.04405
$986$	−1.35112	−0.0430284
$987$	−7.87865	−0.250780
$988$	23.1930	0.737867
$989$	−10.8952	−0.346448
$990$	−3.39766	−0.107985
$991$	30.3063	0.962711	0.481356	−	0.876525i	$-0.340145\pi$
$991$	30.3063	0.962711	0.481356	+	0.876525i	$0.340145\pi$
$992$	−9.14644	−0.290400
$993$	−7.23953	−0.229739
$994$	37.9252	1.20291
$995$	22.7719	0.721919
$996$	17.7371	0.562021
$997$	44.1084	1.39693	0.698464	−	0.715645i	$-0.253867\pi$
$997$	44.1084	1.39693	0.698464	+	0.715645i	$0.253867\pi$
$998$	31.5441	0.998511
$999$	5.74878	0.181883

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4026.2.a.o.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.o.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:	\( N \)	\(=\)	\( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4026.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.39766 q^{5} -1.00000 q^{6} -4.14644 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.39766 q^{5} -1.00000 q^{6} -4.14644 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.39766 q^{10} +1.00000 q^{11} -1.00000 q^{12} +3.54410 q^{13} -4.14644 q^{14} +3.39766 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +6.54410 q^{19} -3.39766 q^{20} +4.14644 q^{21} +1.00000 q^{22} +4.54410 q^{23} -1.00000 q^{24} +6.54410 q^{25} +3.54410 q^{26} -1.00000 q^{27} -4.14644 q^{28} -1.35112 q^{29} +3.39766 q^{30} -9.14644 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} +14.0882 q^{35} +1.00000 q^{36} -5.74878 q^{37} +6.54410 q^{38} -3.54410 q^{39} -3.39766 q^{40} +0.497562 q^{41} +4.14644 q^{42} -2.39766 q^{43} +1.00000 q^{44} -3.39766 q^{45} +4.54410 q^{46} -1.90010 q^{47} -1.00000 q^{48} +10.1930 q^{49} +6.54410 q^{50} -1.00000 q^{51} +3.54410 q^{52} -6.69055 q^{53} -1.00000 q^{54} -3.39766 q^{55} -4.14644 q^{56} -6.54410 q^{57} -1.35112 q^{58} +10.4442 q^{59} +3.39766 q^{60} -1.00000 q^{61} -9.14644 q^{62} -4.14644 q^{63} +1.00000 q^{64} -12.0417 q^{65} -1.00000 q^{66} +9.14644 q^{67} +1.00000 q^{68} -4.54410 q^{69} +14.0882 q^{70} -9.14644 q^{71} +1.00000 q^{72} +4.64888 q^{73} -5.74878 q^{74} -6.54410 q^{75} +6.54410 q^{76} -4.14644 q^{77} -3.54410 q^{78} +3.54410 q^{79} -3.39766 q^{80} +1.00000 q^{81} +0.497562 q^{82} -17.7371 q^{83} +4.14644 q^{84} -3.39766 q^{85} -2.39766 q^{86} +1.35112 q^{87} +1.00000 q^{88} -12.6954 q^{89} -3.39766 q^{90} -14.6954 q^{91} +4.54410 q^{92} +9.14644 q^{93} -1.90010 q^{94} -22.2347 q^{95} -1.00000 q^{96} +10.7371 q^{97} +10.1930 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 2 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{11} - 3 q^{12} - 7 q^{13} - 2 q^{14} + 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} + 3 q^{22} - 4 q^{23} - 3 q^{24} + 2 q^{25} - 7 q^{26} - 3 q^{27} - 2 q^{28} - 8 q^{29} + 3 q^{30} - 17 q^{31} + 3 q^{32} - 3 q^{33} + 3 q^{34} + 7 q^{35} + 3 q^{36} - 14 q^{37} + 2 q^{38} + 7 q^{39} - 3 q^{40} - 5 q^{41} + 2 q^{42} + 3 q^{44} - 3 q^{45} - 4 q^{46} - 5 q^{47} - 3 q^{48} + 9 q^{49} + 2 q^{50} - 3 q^{51} - 7 q^{52} + 8 q^{53} - 3 q^{54} - 3 q^{55} - 2 q^{56} - 2 q^{57} - 8 q^{58} + 13 q^{59} + 3 q^{60} - 3 q^{61} - 17 q^{62} - 2 q^{63} + 3 q^{64} - 12 q^{65} - 3 q^{66} + 17 q^{67} + 3 q^{68} + 4 q^{69} + 7 q^{70} - 17 q^{71} + 3 q^{72} + 10 q^{73} - 14 q^{74} - 2 q^{75} + 2 q^{76} - 2 q^{77} + 7 q^{78} - 7 q^{79} - 3 q^{80} + 3 q^{81} - 5 q^{82} - 14 q^{83} + 2 q^{84} - 3 q^{85} + 8 q^{87} + 3 q^{88} - 23 q^{89} - 3 q^{90} - 29 q^{91} - 4 q^{92} + 17 q^{93} - 5 q^{94} - 21 q^{95} - 3 q^{96} - 7 q^{97} + 9 q^{98} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 2 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^11 - 3 * q^12 - 7 * q^13 - 2 * q^14 + 3 * q^15 + 3 * q^16 + 3 * q^17 + 3 * q^18 + 2 * q^19 - 3 * q^20 + 2 * q^21 + 3 * q^22 - 4 * q^23 - 3 * q^24 + 2 * q^25 - 7 * q^26 - 3 * q^27 - 2 * q^28 - 8 * q^29 + 3 * q^30 - 17 * q^31 + 3 * q^32 - 3 * q^33 + 3 * q^34 + 7 * q^35 + 3 * q^36 - 14 * q^37 + 2 * q^38 + 7 * q^39 - 3 * q^40 - 5 * q^41 + 2 * q^42 + 3 * q^44 - 3 * q^45 - 4 * q^46 - 5 * q^47 - 3 * q^48 + 9 * q^49 + 2 * q^50 - 3 * q^51 - 7 * q^52 + 8 * q^53 - 3 * q^54 - 3 * q^55 - 2 * q^56 - 2 * q^57 - 8 * q^58 + 13 * q^59 + 3 * q^60 - 3 * q^61 - 17 * q^62 - 2 * q^63 + 3 * q^64 - 12 * q^65 - 3 * q^66 + 17 * q^67 + 3 * q^68 + 4 * q^69 + 7 * q^70 - 17 * q^71 + 3 * q^72 + 10 * q^73 - 14 * q^74 - 2 * q^75 + 2 * q^76 - 2 * q^77 + 7 * q^78 - 7 * q^79 - 3 * q^80 + 3 * q^81 - 5 * q^82 - 14 * q^83 + 2 * q^84 - 3 * q^85 + 8 * q^87 + 3 * q^88 - 23 * q^89 - 3 * q^90 - 29 * q^91 - 4 * q^92 + 17 * q^93 - 5 * q^94 - 21 * q^95 - 3 * q^96 - 7 * q^97 + 9 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more