LMFDB - Embedded newform 4026.2.a.o.1.2

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			1.2
Root			$-0.440808$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.44081 q^{5} -1.00000 q^{6} +3.36488 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.44081 q^{5} -1.00000 q^{6} +3.36488 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.44081 q^{10} +1.00000 q^{11} -1.00000 q^{12} -5.92407 q^{13} +3.36488 q^{14} +1.44081 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -2.92407 q^{19} -1.44081 q^{20} -3.36488 q^{21} +1.00000 q^{22} -4.92407 q^{23} -1.00000 q^{24} -2.92407 q^{25} -5.92407 q^{26} -1.00000 q^{27} +3.36488 q^{28} +2.24650 q^{29} +1.44081 q^{30} -1.63512 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} -4.84815 q^{35} +1.00000 q^{36} -0.194311 q^{37} -2.92407 q^{38} +5.92407 q^{39} -1.44081 q^{40} -10.6114 q^{41} -3.36488 q^{42} -0.440808 q^{43} +1.00000 q^{44} -1.44081 q^{45} -4.92407 q^{46} -11.0522 q^{47} -1.00000 q^{48} +4.32242 q^{49} -2.92407 q^{50} -1.00000 q^{51} -5.92407 q^{52} +10.2890 q^{53} -1.00000 q^{54} -1.44081 q^{55} +3.36488 q^{56} +2.92407 q^{57} +2.24650 q^{58} +10.1281 q^{59} +1.44081 q^{60} -1.00000 q^{61} -1.63512 q^{62} +3.36488 q^{63} +1.00000 q^{64} +8.53545 q^{65} -1.00000 q^{66} +1.63512 q^{67} +1.00000 q^{68} +4.92407 q^{69} -4.84815 q^{70} -1.63512 q^{71} +1.00000 q^{72} +8.24650 q^{73} -0.194311 q^{74} +2.92407 q^{75} -2.92407 q^{76} +3.36488 q^{77} +5.92407 q^{78} -5.92407 q^{79} -1.44081 q^{80} +1.00000 q^{81} -10.6114 q^{82} -2.39835 q^{83} -3.36488 q^{84} -1.44081 q^{85} -0.440808 q^{86} -2.24650 q^{87} +1.00000 q^{88} -17.9338 q^{89} -1.44081 q^{90} -19.9338 q^{91} -4.92407 q^{92} +1.63512 q^{93} -11.0522 q^{94} +4.21303 q^{95} -1.00000 q^{96} -4.60165 q^{97} +4.32242 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.44081	−0.644349	−0.322174	−	0.946680i	$-0.604414\pi$
$5$	−1.44081	−0.644349	−0.322174	+	0.946680i	$0.604414\pi$
$6$	−1.00000	−0.408248
$7$	3.36488	1.27181	0.635903	−	0.771769i	$-0.280628\pi$
$7$	3.36488	1.27181	0.635903	+	0.771769i	$0.280628\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.44081	−0.455623
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	−5.92407	−1.64304	−0.821521	−	0.570178i	$-0.806874\pi$
$13$	−5.92407	−1.64304	−0.821521	+	0.570178i	$0.806874\pi$
$14$	3.36488	0.899302
$15$	1.44081	0.372015
$16$	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
$19$	−2.92407	−0.670828	−0.335414	−	0.942071i	$-0.608876\pi$
$19$	−2.92407	−0.670828	−0.335414	+	0.942071i	$0.608876\pi$
$20$	−1.44081	−0.322174
$21$	−3.36488	−0.734277
$22$	1.00000	0.213201
$23$	−4.92407	−1.02674	−0.513370	−	0.858167i	$-0.671603\pi$
$23$	−4.92407	−1.02674	−0.513370	+	0.858167i	$0.671603\pi$
$24$	−1.00000	−0.204124
$25$	−2.92407	−0.584815
$26$	−5.92407	−1.16181
$27$	−1.00000	−0.192450
$28$	3.36488	0.635903
$29$	2.24650	0.417164	0.208582	−	0.978005i	$-0.433115\pi$
$29$	2.24650	0.417164	0.208582	+	0.978005i	$0.433115\pi$
$30$	1.44081	0.263054
$31$	−1.63512	−0.293676	−0.146838	−	0.989161i	$-0.546910\pi$
$31$	−1.63512	−0.293676	−0.146838	+	0.989161i	$0.546910\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	1.00000	0.171499
$35$	−4.84815	−0.819486
$36$	1.00000	0.166667
$37$	−0.194311	−0.0319446	−0.0159723	−	0.999872i	$-0.505084\pi$
$37$	−0.194311	−0.0319446	−0.0159723	+	0.999872i	$0.505084\pi$
$38$	−2.92407	−0.474347
$39$	5.92407	0.948611
$40$	−1.44081	−0.227812
$41$	−10.6114	−1.65722	−0.828609	−	0.559827i	$-0.810867\pi$
$41$	−10.6114	−1.65722	−0.828609	+	0.559827i	$0.810867\pi$
$42$	−3.36488	−0.519212
$43$	−0.440808	−0.0672225	−0.0336113	−	0.999435i	$-0.510701\pi$
$43$	−0.440808	−0.0672225	−0.0336113	+	0.999435i	$0.510701\pi$
$44$	1.00000	0.150756
$45$	−1.44081	−0.214783
$46$	−4.92407	−0.726015
$47$	−11.0522	−1.61213	−0.806063	−	0.591829i	$-0.798406\pi$
$47$	−11.0522	−1.61213	−0.806063	+	0.591829i	$0.798406\pi$
$48$	−1.00000	−0.144338
$49$	4.32242	0.617489
$50$	−2.92407	−0.413526
$51$	−1.00000	−0.140028
$52$	−5.92407	−0.821521
$53$	10.2890	1.41330	0.706648	−	0.707565i	$-0.250206\pi$
$53$	10.2890	1.41330	0.706648	+	0.707565i	$0.250206\pi$
$54$	−1.00000	−0.136083
$55$	−1.44081	−0.194278
$56$	3.36488	0.449651
$57$	2.92407	0.387303
$58$	2.24650	0.294979
$59$	10.1281	1.31857	0.659284	−	0.751894i	$-0.270860\pi$
$59$	10.1281	1.31857	0.659284	+	0.751894i	$0.270860\pi$
$60$	1.44081	0.186007
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−1.63512	−0.207660
$63$	3.36488	0.423935
$64$	1.00000	0.125000
$65$	8.53545	1.05869
$66$	−1.00000	−0.123091
$67$	1.63512	0.199762	0.0998808	−	0.994999i	$-0.468154\pi$
$67$	1.63512	0.199762	0.0998808	+	0.994999i	$0.468154\pi$
$68$	1.00000	0.121268
$69$	4.92407	0.592789
$70$	−4.84815	−0.579464
$71$	−1.63512	−0.194053	−0.0970265	−	0.995282i	$-0.530933\pi$
$71$	−1.63512	−0.194053	−0.0970265	+	0.995282i	$0.530933\pi$
$72$	1.00000	0.117851
$73$	8.24650	0.965179	0.482590	−	0.875847i	$-0.339696\pi$
$73$	8.24650	0.965179	0.482590	+	0.875847i	$0.339696\pi$
$74$	−0.194311	−0.0225882
$75$	2.92407	0.337643
$76$	−2.92407	−0.335414
$77$	3.36488	0.383464
$78$	5.92407	0.670769
$79$	−5.92407	−0.666510	−0.333255	−	0.942837i	$-0.608147\pi$
$79$	−5.92407	−0.666510	−0.333255	+	0.942837i	$0.608147\pi$
$80$	−1.44081	−0.161087
$81$	1.00000	0.111111
$82$	−10.6114	−1.17183
$83$	−2.39835	−0.263253	−0.131627	−	0.991299i	$-0.542020\pi$
$83$	−2.39835	−0.263253	−0.131627	+	0.991299i	$0.542020\pi$
$84$	−3.36488	−0.367139
$85$	−1.44081	−0.156278
$86$	−0.440808	−0.0475335
$87$	−2.24650	−0.240850
$88$	1.00000	0.106600
$89$	−17.9338	−1.90098	−0.950490	−	0.310757i	$-0.899418\pi$
$89$	−17.9338	−1.90098	−0.950490	+	0.310757i	$0.899418\pi$
$90$	−1.44081	−0.151874
$91$	−19.9338	−2.08963
$92$	−4.92407	−0.513370
$93$	1.63512	0.169554
$94$	−11.0522	−1.13995
$95$	4.21303	0.432247
$96$	−1.00000	−0.102062
$97$	−4.60165	−0.467227	−0.233613	−	0.972330i	$-0.575055\pi$
$97$	−4.60165	−0.467227	−0.233613	+	0.972330i	$0.575055\pi$
$98$	4.32242	0.436631
$99$	1.00000	0.100504
$100$	−2.92407	−0.292407
$101$	−1.20404	−0.119806	−0.0599032	−	0.998204i	$-0.519079\pi$
$101$	−1.20404	−0.119806	−0.0599032	+	0.998204i	$0.519079\pi$
$102$	−1.00000	−0.0990148
$103$	−6.32242	−0.622967	−0.311483	−	0.950252i	$-0.600826\pi$
$103$	−6.32242	−0.622967	−0.311483	+	0.950252i	$0.600826\pi$
$104$	−5.92407	−0.580903
$105$	4.84815	0.473131
$106$	10.2890	0.999352
$107$	−15.5355	−1.50187	−0.750934	−	0.660377i	$-0.770397\pi$
$107$	−15.5355	−1.50187	−0.750934	+	0.660377i	$0.770397\pi$
$108$	−1.00000	−0.0962250
$109$	13.1468	1.25924	0.629619	−	0.776904i	$-0.283211\pi$
$109$	13.1468	1.25924	0.629619	+	0.776904i	$0.283211\pi$
$110$	−1.44081	−0.137376
$111$	0.194311	0.0184432
$112$	3.36488	0.317951
$113$	1.32242	0.124403	0.0622016	−	0.998064i	$-0.480188\pi$
$113$	1.32242	0.124403	0.0622016	+	0.998064i	$0.480188\pi$
$114$	2.92407	0.273865
$115$	7.09464	0.661579
$116$	2.24650	0.208582
$117$	−5.92407	−0.547681
$118$	10.1281	0.932368
$119$	3.36488	0.308458
$120$	1.44081	0.131527
$121$	1.00000	0.0909091
$122$	−1.00000	−0.0905357
$123$	10.6114	0.956796
$124$	−1.63512	−0.146838
$125$	11.4171	1.02117
$126$	3.36488	0.299767
$127$	1.88162	0.166966	0.0834832	−	0.996509i	$-0.473396\pi$
$127$	1.88162	0.166966	0.0834832	+	0.996509i	$0.473396\pi$
$128$	1.00000	0.0883883
$129$	0.440808	0.0388110
$130$	8.53545	0.748609
$131$	−18.0522	−1.57723	−0.788613	−	0.614889i	$-0.789201\pi$
$131$	−18.0522	−1.57723	−0.788613	+	0.614889i	$0.789201\pi$
$132$	−1.00000	−0.0870388
$133$	−9.83916	−0.853163
$134$	1.63512	0.141253
$135$	1.44081	0.124005
$136$	1.00000	0.0857493
$137$	−9.88162	−0.844243	−0.422122	−	0.906539i	$-0.638715\pi$
$137$	−9.88162	−0.844243	−0.422122	+	0.906539i	$0.638715\pi$
$138$	4.92407	0.419165
$139$	7.77222	0.659231	0.329616	−	0.944115i	$-0.393081\pi$
$139$	7.77222	0.659231	0.329616	+	0.944115i	$0.393081\pi$
$140$	−4.84815	−0.409743
$141$	11.0522	0.930762
$142$	−1.63512	−0.137216
$143$	−5.92407	−0.495396
$144$	1.00000	0.0833333
$145$	−3.23677	−0.268799
$146$	8.24650	0.682485
$147$	−4.32242	−0.356507
$148$	−0.194311	−0.0159723
$149$	−9.64485	−0.790137	−0.395068	−	0.918652i	$-0.629279\pi$
$149$	−9.64485	−0.790137	−0.395068	+	0.918652i	$0.629279\pi$
$150$	2.92407	0.238750
$151$	7.76323	0.631763	0.315881	−	0.948799i	$-0.397700\pi$
$151$	7.76323	0.631763	0.315881	+	0.948799i	$0.397700\pi$
$152$	−2.92407	−0.237174
$153$	1.00000	0.0808452
$154$	3.36488	0.271150
$155$	2.35589	0.189230
$156$	5.92407	0.474305
$157$	3.12811	0.249650	0.124825	−	0.992179i	$-0.460163\pi$
$157$	3.12811	0.249650	0.124825	+	0.992179i	$0.460163\pi$
$158$	−5.92407	−0.471294
$159$	−10.2890	−0.815967
$160$	−1.44081	−0.113906
$161$	−16.5689	−1.30581
$162$	1.00000	0.0785674
$163$	4.63512	0.363051	0.181525	−	0.983386i	$-0.441897\pi$
$163$	4.63512	0.363051	0.181525	+	0.983386i	$0.441897\pi$
$164$	−10.6114	−0.828609
$165$	1.44081	0.112167
$166$	−2.39835	−0.186148
$167$	−15.1706	−1.17393	−0.586967	−	0.809611i	$-0.699678\pi$
$167$	−15.1706	−1.17393	−0.586967	+	0.809611i	$0.699678\pi$
$168$	−3.36488	−0.259606
$169$	22.0946	1.69959
$170$	−1.44081	−0.110505
$171$	−2.92407	−0.223609
$172$	−0.440808	−0.0336113
$173$	0.246496	0.0187408	0.00937038	−	0.999956i	$-0.497017\pi$
$173$	0.246496	0.0187408	0.00937038	+	0.999956i	$0.497017\pi$
$174$	−2.24650	−0.170306
$175$	−9.83916	−0.743770
$176$	1.00000	0.0753778
$177$	−10.1281	−0.761275
$178$	−17.9338	−1.34420
$179$	−18.5117	−1.38363	−0.691815	−	0.722075i	$-0.743189\pi$
$179$	−18.5117	−1.38363	−0.691815	+	0.722075i	$0.743189\pi$
$180$	−1.44081	−0.107391
$181$	−18.3746	−1.36577	−0.682887	−	0.730524i	$-0.739276\pi$
$181$	−18.3746	−1.36577	−0.682887	+	0.730524i	$0.739276\pi$
$182$	−19.9338	−1.47759
$183$	1.00000	0.0739221
$184$	−4.92407	−0.363007
$185$	0.279965	0.0205835
$186$	1.63512	0.119893
$187$	1.00000	0.0731272
$188$	−11.0522	−0.806063
$189$	−3.36488	−0.244759
$190$	4.21303	0.305645
$191$	−8.41707	−0.609037	−0.304519	−	0.952506i	$-0.598496\pi$
$191$	−8.41707	−0.609037	−0.304519	+	0.952506i	$0.598496\pi$
$192$	−1.00000	−0.0721688
$193$	0.905357	0.0651691	0.0325845	−	0.999469i	$-0.489626\pi$
$193$	0.905357	0.0651691	0.0325845	+	0.999469i	$0.489626\pi$
$194$	−4.60165	−0.330379
$195$	−8.53545	−0.611236
$196$	4.32242	0.308745
$197$	8.97626	0.639532	0.319766	−	0.947497i	$-0.396396\pi$
$197$	8.97626	0.639532	0.319766	+	0.947497i	$0.396396\pi$
$198$	1.00000	0.0710669
$199$	0.492993	0.0349473	0.0174737	−	0.999847i	$-0.494438\pi$
$199$	0.492993	0.0349473	0.0174737	+	0.999847i	$0.494438\pi$
$200$	−2.92407	−0.206763
$201$	−1.63512	−0.115332
$202$	−1.20404	−0.0847159
$203$	7.55919	0.530551
$204$	−1.00000	−0.0700140
$205$	15.2890	1.06783
$206$	−6.32242	−0.440504
$207$	−4.92407	−0.342247
$208$	−5.92407	−0.410761
$209$	−2.92407	−0.202262
$210$	4.84815	0.334554
$211$	28.1893	1.94063	0.970315	−	0.241844i	$-0.0777521\pi$
$211$	28.1893	1.94063	0.970315	+	0.241844i	$0.0777521\pi$
$212$	10.2890	0.706648
$213$	1.63512	0.112037
$214$	−15.5355	−1.06198
$215$	0.635119	0.0433148
$216$	−1.00000	−0.0680414
$217$	−5.50198	−0.373499
$218$	13.1468	0.890416
$219$	−8.24650	−0.557247
$220$	−1.44081	−0.0971392
$221$	−5.92407	−0.398496
$222$	0.194311	0.0130413
$223$	18.4408	1.23489	0.617444	−	0.786615i	$-0.288168\pi$
$223$	18.4408	1.23489	0.617444	+	0.786615i	$0.288168\pi$
$224$	3.36488	0.224826
$225$	−2.92407	−0.194938
$226$	1.32242	0.0879663
$227$	−27.2130	−1.80619	−0.903096	−	0.429439i	$-0.858711\pi$
$227$	−27.2130	−1.80619	−0.903096	+	0.429439i	$0.858711\pi$
$228$	2.92407	0.193651
$229$	−26.5355	−1.75351	−0.876756	−	0.480936i	$-0.840297\pi$
$229$	−26.5355	−1.75351	−0.876756	+	0.480936i	$0.840297\pi$
$230$	7.09464	0.467807
$231$	−3.36488	−0.221393
$232$	2.24650	0.147490
$233$	4.34114	0.284397	0.142199	−	0.989838i	$-0.454583\pi$
$233$	4.34114	0.284397	0.142199	+	0.989838i	$0.454583\pi$
$234$	−5.92407	−0.387269
$235$	15.9241	1.03877
$236$	10.1281	0.659284
$237$	5.92407	0.384810
$238$	3.36488	0.218113
$239$	−10.9003	−0.705084	−0.352542	−	0.935796i	$-0.614683\pi$
$239$	−10.9003	−0.705084	−0.352542	+	0.935796i	$0.614683\pi$
$240$	1.44081	0.0930037
$241$	10.0425	0.646892	0.323446	−	0.946247i	$-0.395159\pi$
$241$	10.0425	0.646892	0.323446	+	0.946247i	$0.395159\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	−1.00000	−0.0640184
$245$	−6.22778	−0.397878
$246$	10.6114	0.676557
$247$	17.3224	1.10220
$248$	−1.63512	−0.103830
$249$	2.39835	0.151989
$250$	11.4171	0.722079
$251$	15.0612	0.950653	0.475326	−	0.879810i	$-0.342330\pi$
$251$	15.0612	0.950653	0.475326	+	0.879810i	$0.342330\pi$
$252$	3.36488	0.211968
$253$	−4.92407	−0.309574
$254$	1.88162	0.118063
$255$	1.44081	0.0902269
$256$	1.00000	0.0625000
$257$	11.0097	0.686768	0.343384	−	0.939195i	$-0.388427\pi$
$257$	11.0097	0.686768	0.343384	+	0.939195i	$0.388427\pi$
$258$	0.440808	0.0274435
$259$	−0.653835	−0.0406273
$260$	8.53545	0.529346
$261$	2.24650	0.139055
$262$	−18.0522	−1.11527
$263$	−30.7395	−1.89548	−0.947739	−	0.319047i	$-0.896637\pi$
$263$	−30.7395	−1.89548	−0.947739	+	0.319047i	$0.896637\pi$
$264$	−1.00000	−0.0615457
$265$	−14.8244	−0.910656
$266$	−9.83916	−0.603277
$267$	17.9338	1.09753
$268$	1.63512	0.0998808
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	1.44081	0.0876848
$271$	−9.07593	−0.551323	−0.275662	−	0.961255i	$-0.588897\pi$
$271$	−9.07593	−0.551323	−0.275662	+	0.961255i	$0.588897\pi$
$272$	1.00000	0.0606339
$273$	19.9338	1.20645
$274$	−9.88162	−0.596970
$275$	−2.92407	−0.176328
$276$	4.92407	0.296394
$277$	−19.2987	−1.15955	−0.579773	−	0.814778i	$-0.696859\pi$
$277$	−19.2987	−1.15955	−0.579773	+	0.814778i	$0.696859\pi$
$278$	7.77222	0.466147
$279$	−1.63512	−0.0978920
$280$	−4.84815	−0.289732
$281$	−0.398350	−0.0237636	−0.0118818	−	0.999929i	$-0.503782\pi$
$281$	−0.398350	−0.0237636	−0.0118818	+	0.999929i	$0.503782\pi$
$282$	11.0522	0.658148
$283$	15.5592	0.924898	0.462449	−	0.886646i	$-0.346971\pi$
$283$	15.5592	0.924898	0.462449	+	0.886646i	$0.346971\pi$
$284$	−1.63512	−0.0970265
$285$	−4.21303	−0.249558
$286$	−5.92407	−0.350298
$287$	−35.7060	−2.10766
$288$	1.00000	0.0589256
$289$	−16.0000	−0.941176
$290$	−3.23677	−0.190070
$291$	4.60165	0.269754
$292$	8.24650	0.482590
$293$	10.6776	0.623791	0.311895	−	0.950116i	$-0.399036\pi$
$293$	10.6776	0.623791	0.311895	+	0.950116i	$0.399036\pi$
$294$	−4.32242	−0.252089
$295$	−14.5927	−0.849618
$296$	−0.194311	−0.0112941
$297$	−1.00000	−0.0580259
$298$	−9.64485	−0.558711
$299$	29.1706	1.68698
$300$	2.92407	0.168821
$301$	−1.48327	−0.0854940
$302$	7.76323	0.446724
$303$	1.20404	0.0691702
$304$	−2.92407	−0.167707
$305$	1.44081	0.0825004
$306$	1.00000	0.0571662
$307$	5.07593	0.289698	0.144849	−	0.989454i	$-0.453730\pi$
$307$	5.07593	0.289698	0.144849	+	0.989454i	$0.453730\pi$
$308$	3.36488	0.191732
$309$	6.32242	0.359670
$310$	2.35589	0.133806
$311$	11.9101	0.675357	0.337679	−	0.941261i	$-0.390358\pi$
$311$	11.9101	0.675357	0.337679	+	0.941261i	$0.390358\pi$
$312$	5.92407	0.335385
$313$	29.7582	1.68203	0.841017	−	0.541009i	$-0.181958\pi$
$313$	29.7582	1.68203	0.841017	+	0.541009i	$0.181958\pi$
$314$	3.12811	0.176530
$315$	−4.84815	−0.273162
$316$	−5.92407	−0.333255
$317$	8.84888	0.497003	0.248501	−	0.968632i	$-0.420062\pi$
$317$	8.84888	0.497003	0.248501	+	0.968632i	$0.420062\pi$
$318$	−10.2890	−0.576976
$319$	2.24650	0.125780
$320$	−1.44081	−0.0805436
$321$	15.5355	0.867104
$322$	−16.5689	−0.923350
$323$	−2.92407	−0.162700
$324$	1.00000	0.0555556
$325$	17.3224	0.960875
$326$	4.63512	0.256716
$327$	−13.1468	−0.727021
$328$	−10.6114	−0.585915
$329$	−37.1893	−2.05031
$330$	1.44081	0.0793139
$331$	3.00973	0.165430	0.0827148	−	0.996573i	$-0.473641\pi$
$331$	3.00973	0.165430	0.0827148	+	0.996573i	$0.473641\pi$
$332$	−2.39835	−0.131627
$333$	−0.194311	−0.0106482
$334$	−15.1706	−0.830097
$335$	−2.35589	−0.128716
$336$	−3.36488	−0.183569
$337$	−22.6823	−1.23558	−0.617791	−	0.786342i	$-0.711972\pi$
$337$	−22.6823	−1.23558	−0.617791	+	0.786342i	$0.711972\pi$
$338$	22.0946	1.20179
$339$	−1.32242	−0.0718242
$340$	−1.44081	−0.0781388
$341$	−1.63512	−0.0885467
$342$	−2.92407	−0.158116
$343$	−9.00973	−0.486480
$344$	−0.440808	−0.0237668
$345$	−7.09464	−0.381963
$346$	0.246496	0.0132517
$347$	18.4073	0.988158	0.494079	−	0.869417i	$-0.335505\pi$
$347$	18.4073	0.988158	0.494079	+	0.869417i	$0.335505\pi$
$348$	−2.24650	−0.120425
$349$	18.2562	0.977233	0.488617	−	0.872499i	$-0.337502\pi$
$349$	18.2562	0.977233	0.488617	+	0.872499i	$0.337502\pi$
$350$	−9.83916	−0.525925
$351$	5.92407	0.316204
$352$	1.00000	0.0533002
$353$	−13.8392	−0.736584	−0.368292	−	0.929710i	$-0.620057\pi$
$353$	−13.8392	−0.736584	−0.368292	+	0.929710i	$0.620057\pi$
$354$	−10.1281	−0.538303
$355$	2.35589	0.125038
$356$	−17.9338	−0.950490
$357$	−3.36488	−0.178088
$358$	−18.5117	−0.978374
$359$	−22.6963	−1.19786	−0.598932	−	0.800800i	$-0.704408\pi$
$359$	−22.6963	−1.19786	−0.598932	+	0.800800i	$0.704408\pi$
$360$	−1.44081	−0.0759372
$361$	−10.4498	−0.549989
$362$	−18.3746	−0.965748
$363$	−1.00000	−0.0524864
$364$	−19.9338	−1.04482
$365$	−11.8816	−0.621912
$366$	1.00000	0.0522708
$367$	22.3077	1.16445	0.582225	−	0.813027i	$-0.302182\pi$
$367$	22.3077	1.16445	0.582225	+	0.813027i	$0.302182\pi$
$368$	−4.92407	−0.256685
$369$	−10.6114	−0.552406
$370$	0.279965	0.0145547
$371$	34.6211	1.79744
$372$	1.63512	0.0847770
$373$	21.0946	1.09224	0.546120	−	0.837707i	$-0.316104\pi$
$373$	21.0946	1.09224	0.546120	+	0.837707i	$0.316104\pi$
$374$	1.00000	0.0517088
$375$	−11.4171	−0.589575
$376$	−11.0522	−0.569973
$377$	−13.3084	−0.685418
$378$	−3.36488	−0.173071
$379$	−25.1044	−1.28952	−0.644762	−	0.764383i	$-0.723044\pi$
$379$	−25.1044	−1.28952	−0.644762	+	0.764383i	$0.723044\pi$
$380$	4.21303	0.216124
$381$	−1.88162	−0.0963981
$382$	−8.41707	−0.430655
$383$	16.7632	0.856561	0.428281	−	0.903646i	$-0.359120\pi$
$383$	16.7632	0.856561	0.428281	+	0.903646i	$0.359120\pi$
$384$	−1.00000	−0.0510310
$385$	−4.84815	−0.247084
$386$	0.905357	0.0460815
$387$	−0.440808	−0.0224075
$388$	−4.60165	−0.233613
$389$	20.6351	1.04624	0.523121	−	0.852258i	$-0.324768\pi$
$389$	20.6351	1.04624	0.523121	+	0.852258i	$0.324768\pi$
$390$	−8.53545	−0.432209
$391$	−4.92407	−0.249021
$392$	4.32242	0.218315
$393$	18.0522	0.910612
$394$	8.97626	0.452217
$395$	8.53545	0.429465
$396$	1.00000	0.0502519
$397$	−4.10866	−0.206208	−0.103104	−	0.994671i	$-0.532877\pi$
$397$	−4.10866	−0.206208	−0.103104	+	0.994671i	$0.532877\pi$
$398$	0.492993	0.0247115
$399$	9.83916	0.492574
$400$	−2.92407	−0.146204
$401$	10.7298	0.535819	0.267909	−	0.963444i	$-0.413667\pi$
$401$	10.7298	0.535819	0.267909	+	0.963444i	$0.413667\pi$
$402$	−1.63512	−0.0815523
$403$	9.68657	0.482522
$404$	−1.20404	−0.0599032
$405$	−1.44081	−0.0715943
$406$	7.55919	0.375156
$407$	−0.194311	−0.00963166
$408$	−1.00000	−0.0495074
$409$	27.4358	1.35661	0.678306	−	0.734780i	$-0.262714\pi$
$409$	27.4358	1.35661	0.678306	+	0.734780i	$0.262714\pi$
$410$	15.2890	0.755068
$411$	9.88162	0.487424
$412$	−6.32242	−0.311483
$413$	34.0799	1.67696
$414$	−4.92407	−0.242005
$415$	3.45556	0.169627
$416$	−5.92407	−0.290452
$417$	−7.77222	−0.380607
$418$	−2.92407	−0.143021
$419$	−18.6538	−0.911299	−0.455650	−	0.890159i	$-0.650593\pi$
$419$	−18.6538	−0.911299	−0.455650	+	0.890159i	$0.650593\pi$
$420$	4.84815	0.236565
$421$	−10.1468	−0.494526	−0.247263	−	0.968948i	$-0.579531\pi$
$421$	−10.1468	−0.494526	−0.247263	+	0.968948i	$0.579531\pi$
$422$	28.1893	1.37223
$423$	−11.0522	−0.537376
$424$	10.2890	0.499676
$425$	−2.92407	−0.141838
$426$	1.63512	0.0792218
$427$	−3.36488	−0.162838
$428$	−15.5355	−0.750934
$429$	5.92407	0.286017
$430$	0.635119	0.0306282
$431$	−3.38862	−0.163224	−0.0816121	−	0.996664i	$-0.526007\pi$
$431$	−3.38862	−0.163224	−0.0816121	+	0.996664i	$0.526007\pi$
$432$	−1.00000	−0.0481125
$433$	−25.2562	−1.21374	−0.606868	−	0.794802i	$-0.707574\pi$
$433$	−25.2562	−1.21374	−0.606868	+	0.794802i	$0.707574\pi$
$434$	−5.50198	−0.264104
$435$	3.23677	0.155191
$436$	13.1468	0.629619
$437$	14.3983	0.688766
$438$	−8.24650	−0.394033
$439$	−7.19431	−0.343366	−0.171683	−	0.985152i	$-0.554920\pi$
$439$	−7.19431	−0.343366	−0.171683	+	0.985152i	$0.554920\pi$
$440$	−1.44081	−0.0686878
$441$	4.32242	0.205830
$442$	−5.92407	−0.281779
$443$	32.1461	1.52731	0.763653	−	0.645626i	$-0.223404\pi$
$443$	32.1461	1.52731	0.763653	+	0.645626i	$0.223404\pi$
$444$	0.194311	0.00922161
$445$	25.8392	1.22489
$446$	18.4408	0.873198
$447$	9.64485	0.456186
$448$	3.36488	0.158976
$449$	−27.2562	−1.28630	−0.643150	−	0.765740i	$-0.722373\pi$
$449$	−27.2562	−1.28630	−0.643150	+	0.765740i	$0.722373\pi$
$450$	−2.92407	−0.137842
$451$	−10.6114	−0.499670
$452$	1.32242	0.0622016
$453$	−7.76323	−0.364748
$454$	−27.2130	−1.27717
$455$	28.7208	1.34645
$456$	2.92407	0.136932
$457$	7.97626	0.373114	0.186557	−	0.982444i	$-0.440267\pi$
$457$	7.97626	0.373114	0.186557	+	0.982444i	$0.440267\pi$
$458$	−26.5355	−1.23992
$459$	−1.00000	−0.0466760
$460$	7.09464	0.330789
$461$	28.3411	1.31998	0.659989	−	0.751275i	$-0.270561\pi$
$461$	28.3411	1.31998	0.659989	+	0.751275i	$0.270561\pi$
$462$	−3.36488	−0.156548
$463$	24.6538	1.14576	0.572880	−	0.819639i	$-0.305826\pi$
$463$	24.6538	1.14576	0.572880	+	0.819639i	$0.305826\pi$
$464$	2.24650	0.104291
$465$	−2.35589	−0.109252
$466$	4.34114	0.201099
$467$	25.1803	1.16521	0.582603	−	0.812757i	$-0.302034\pi$
$467$	25.1803	1.16521	0.582603	+	0.812757i	$0.302034\pi$
$468$	−5.92407	−0.273840
$469$	5.50198	0.254058
$470$	15.9241	0.734523
$471$	−3.12811	−0.144136
$472$	10.1281	0.466184
$473$	−0.440808	−0.0202684
$474$	5.92407	0.272102
$475$	8.55020	0.392310
$476$	3.36488	0.154229
$477$	10.2890	0.471099
$478$	−10.9003	−0.498570
$479$	9.40734	0.429832	0.214916	−	0.976633i	$-0.431052\pi$
$479$	9.40734	0.429832	0.214916	+	0.976633i	$0.431052\pi$
$480$	1.44081	0.0657636
$481$	1.15112	0.0524863
$482$	10.0425	0.457421
$483$	16.5689	0.753912
$484$	1.00000	0.0454545
$485$	6.63009	0.301057
$486$	−1.00000	−0.0453609
$487$	32.2465	1.46123	0.730614	−	0.682791i	$-0.239234\pi$
$487$	32.2465	1.46123	0.730614	+	0.682791i	$0.239234\pi$
$488$	−1.00000	−0.0452679
$489$	−4.63512	−0.209607
$490$	−6.22778	−0.281342
$491$	−1.79596	−0.0810506	−0.0405253	−	0.999179i	$-0.512903\pi$
$491$	−1.79596	−0.0810506	−0.0405253	+	0.999179i	$0.512903\pi$
$492$	10.6114	0.478398
$493$	2.24650	0.101177
$494$	17.3224	0.779373
$495$	−1.44081	−0.0647595
$496$	−1.63512	−0.0734190
$497$	−5.50198	−0.246798
$498$	2.39835	0.107473
$499$	22.0759	0.988254	0.494127	−	0.869390i	$-0.335488\pi$
$499$	22.0759	0.988254	0.494127	+	0.869390i	$0.335488\pi$
$500$	11.4171	0.510587
$501$	15.1706	0.677771
$502$	15.0612	0.672213
$503$	17.4358	0.777423	0.388712	−	0.921360i	$-0.372920\pi$
$503$	17.4358	0.777423	0.388712	+	0.921360i	$0.372920\pi$
$504$	3.36488	0.149884
$505$	1.73479	0.0771971
$506$	−4.92407	−0.218902
$507$	−22.0946	−0.981258
$508$	1.88162	0.0834832
$509$	−41.3501	−1.83281	−0.916406	−	0.400250i	$-0.868923\pi$
$509$	−41.3501	−1.83281	−0.916406	+	0.400250i	$0.868923\pi$
$510$	1.44081	0.0638000
$511$	27.7485	1.22752
$512$	1.00000	0.0441942
$513$	2.92407	0.129101
$514$	11.0097	0.485618
$515$	9.10940	0.401408
$516$	0.440808	0.0194055
$517$	−11.0522	−0.486075
$518$	−0.653835	−0.0287279
$519$	−0.246496	−0.0108200
$520$	8.53545	0.374304
$521$	−5.90536	−0.258718	−0.129359	−	0.991598i	$-0.541292\pi$
$521$	−5.90536	−0.258718	−0.129359	+	0.991598i	$0.541292\pi$
$522$	2.24650	0.0983265
$523$	−40.3458	−1.76420	−0.882100	−	0.471062i	$-0.843871\pi$
$523$	−40.3458	−1.76420	−0.882100	+	0.471062i	$0.843871\pi$
$524$	−18.0522	−0.788613
$525$	9.83916	0.429416
$526$	−30.7395	−1.34031
$527$	−1.63512	−0.0712269
$528$	−1.00000	−0.0435194
$529$	1.24650	0.0541955
$530$	−14.8244	−0.643931
$531$	10.1281	0.439523
$532$	−9.83916	−0.426582
$533$	62.8626	2.72288
$534$	17.9338	0.776071
$535$	22.3836	0.967727
$536$	1.63512	0.0706264
$537$	18.5117	0.798839
$538$	−21.0000	−0.905374
$539$	4.32242	0.186180
$540$	1.44081	0.0620025
$541$	−8.39332	−0.360857	−0.180429	−	0.983588i	$-0.557748\pi$
$541$	−8.39332	−0.360857	−0.180429	+	0.983588i	$0.557748\pi$
$542$	−9.07593	−0.389844
$543$	18.3746	0.788530
$544$	1.00000	0.0428746
$545$	−18.9421	−0.811388
$546$	19.9338	0.853088
$547$	15.7722	0.674371	0.337186	−	0.941438i	$-0.390525\pi$
$547$	15.7722	0.674371	0.337186	+	0.941438i	$0.390525\pi$
$548$	−9.88162	−0.422122
$549$	−1.00000	−0.0426790
$550$	−2.92407	−0.124683
$551$	−6.56892	−0.279845
$552$	4.92407	0.209582
$553$	−19.9338	−0.847671
$554$	−19.2987	−0.819922
$555$	−0.279965	−0.0118839
$556$	7.77222	0.329616
$557$	−20.1281	−0.852855	−0.426428	−	0.904522i	$-0.640228\pi$
$557$	−20.1281	−0.852855	−0.426428	+	0.904522i	$0.640228\pi$
$558$	−1.63512	−0.0692201
$559$	2.61138	0.110449
$560$	−4.84815	−0.204872
$561$	−1.00000	−0.0422200
$562$	−0.398350	−0.0168034
$563$	44.8856	1.89170	0.945851	−	0.324602i	$-0.105230\pi$
$563$	44.8856	1.89170	0.945851	+	0.324602i	$0.105230\pi$
$564$	11.0522	0.465381
$565$	−1.90536	−0.0801590
$566$	15.5592	0.654002
$567$	3.36488	0.141312
$568$	−1.63512	−0.0686081
$569$	31.1796	1.30712	0.653558	−	0.756877i	$-0.273276\pi$
$569$	31.1796	1.30712	0.653558	+	0.756877i	$0.273276\pi$
$570$	−4.21303	−0.176464
$571$	21.5257	0.900823	0.450412	−	0.892821i	$-0.351277\pi$
$571$	21.5257	0.900823	0.450412	+	0.892821i	$0.351277\pi$
$572$	−5.92407	−0.247698
$573$	8.41707	0.351628
$574$	−35.7060	−1.49034
$575$	14.3983	0.600453
$576$	1.00000	0.0416667
$577$	4.36414	0.181682	0.0908408	−	0.995865i	$-0.471045\pi$
$577$	4.36414	0.181682	0.0908408	+	0.995865i	$0.471045\pi$
$578$	−16.0000	−0.665512
$579$	−0.905357	−0.0376254
$580$	−3.23677	−0.134400
$581$	−8.07016	−0.334807
$582$	4.60165	0.190745
$583$	10.2890	0.426125
$584$	8.24650	0.341242
$585$	8.53545	0.352897
$586$	10.6776	0.441087
$587$	31.4163	1.29669	0.648345	−	0.761346i	$-0.275461\pi$
$587$	31.4163	1.29669	0.648345	+	0.761346i	$0.275461\pi$
$588$	−4.32242	−0.178254
$589$	4.78121	0.197006
$590$	−14.5927	−0.600770
$591$	−8.97626	−0.369234
$592$	−0.194311	−0.00798615
$593$	−39.5117	−1.62255	−0.811276	−	0.584664i	$-0.801226\pi$
$593$	−39.5117	−1.62255	−0.811276	+	0.584664i	$0.801226\pi$
$594$	−1.00000	−0.0410305
$595$	−4.84815	−0.198755
$596$	−9.64485	−0.395068
$597$	−0.492993	−0.0201768
$598$	29.1706	1.19287
$599$	24.4171	0.997654	0.498827	−	0.866702i	$-0.333764\pi$
$599$	24.4171	0.997654	0.498827	+	0.866702i	$0.333764\pi$
$600$	2.92407	0.119375
$601$	10.2228	0.416995	0.208497	−	0.978023i	$-0.433143\pi$
$601$	10.2228	0.416995	0.208497	+	0.978023i	$0.433143\pi$
$602$	−1.48327	−0.0604534
$603$	1.63512	0.0665872
$604$	7.76323	0.315881
$605$	−1.44081	−0.0585772
$606$	1.20404	0.0489107
$607$	−47.6495	−1.93404	−0.967018	−	0.254709i	$-0.918020\pi$
$607$	−47.6495	−1.93404	−0.967018	+	0.254709i	$0.918020\pi$
$608$	−2.92407	−0.118587
$609$	−7.55919	−0.306314
$610$	1.44081	0.0583366
$611$	65.4740	2.64879
$612$	1.00000	0.0404226
$613$	−31.9950	−1.29226	−0.646132	−	0.763225i	$-0.723615\pi$
$613$	−31.9950	−1.29226	−0.646132	+	0.763225i	$0.723615\pi$
$614$	5.07593	0.204848
$615$	−15.2890	−0.616510
$616$	3.36488	0.135575
$617$	12.4163	0.499863	0.249931	−	0.968264i	$-0.419592\pi$
$617$	12.4163	0.499863	0.249931	+	0.968264i	$0.419592\pi$
$618$	6.32242	0.254325
$619$	27.8863	1.12085	0.560423	−	0.828207i	$-0.310639\pi$
$619$	27.8863	1.12085	0.560423	+	0.828207i	$0.310639\pi$
$620$	2.35589	0.0946149
$621$	4.92407	0.197596
$622$	11.9101	0.477550
$623$	−60.3451	−2.41768
$624$	5.92407	0.237153
$625$	−1.82943	−0.0731772
$626$	29.7582	1.18938
$627$	2.92407	0.116776
$628$	3.12811	0.124825
$629$	−0.194311	−0.00774770
$630$	−4.84815	−0.193155
$631$	−12.4693	−0.496393	−0.248197	−	0.968710i	$-0.579838\pi$
$631$	−12.4693	−0.496393	−0.248197	+	0.968710i	$0.579838\pi$
$632$	−5.92407	−0.235647
$633$	−28.1893	−1.12042
$634$	8.84888	0.351434
$635$	−2.71105	−0.107585
$636$	−10.2890	−0.407984
$637$	−25.6064	−1.01456
$638$	2.24650	0.0889396
$639$	−1.63512	−0.0646843
$640$	−1.44081	−0.0569529
$641$	49.0904	1.93895	0.969476	−	0.245185i	$-0.0788488\pi$
$641$	49.0904	1.93895	0.969476	+	0.245185i	$0.0788488\pi$
$642$	15.5355	0.613135
$643$	6.55919	0.258669	0.129335	−	0.991601i	$-0.458716\pi$
$643$	6.55919	0.258669	0.129335	+	0.991601i	$0.458716\pi$
$644$	−16.5689	−0.652907
$645$	−0.635119	−0.0250078
$646$	−2.92407	−0.115046
$647$	−19.8154	−0.779024	−0.389512	−	0.921021i	$-0.627356\pi$
$647$	−19.8154	−0.779024	−0.389512	+	0.921021i	$0.627356\pi$
$648$	1.00000	0.0392837
$649$	10.1281	0.397563
$650$	17.3224	0.679441
$651$	5.50198	0.215640
$652$	4.63512	0.181525
$653$	15.7298	0.615553	0.307777	−	0.951459i	$-0.400415\pi$
$653$	15.7298	0.615553	0.307777	+	0.951459i	$0.400415\pi$
$654$	−13.1468	−0.514082
$655$	26.0097	1.01628
$656$	−10.6114	−0.414305
$657$	8.24650	0.321726
$658$	−37.1893	−1.44979
$659$	10.7967	0.420580	0.210290	−	0.977639i	$-0.432559\pi$
$659$	10.7967	0.420580	0.210290	+	0.977639i	$0.432559\pi$
$660$	1.44081	0.0560834
$661$	−7.24576	−0.281827	−0.140914	−	0.990022i	$-0.545004\pi$
$661$	−7.24576	−0.281827	−0.140914	+	0.990022i	$0.545004\pi$
$662$	3.00973	0.116976
$663$	5.92407	0.230072
$664$	−2.39835	−0.0930740
$665$	14.1763	0.549735
$666$	−0.194311	−0.00752942
$667$	−11.0619	−0.428319
$668$	−15.1706	−0.586967
$669$	−18.4408	−0.712963
$670$	−2.35589	−0.0910161
$671$	−1.00000	−0.0386046
$672$	−3.36488	−0.129803
$673$	−14.6636	−0.565239	−0.282619	−	0.959232i	$-0.591203\pi$
$673$	−14.6636	−0.565239	−0.282619	+	0.959232i	$0.591203\pi$
$674$	−22.6823	−0.873689
$675$	2.92407	0.112548
$676$	22.0946	0.849794
$677$	−7.67758	−0.295073	−0.147537	−	0.989057i	$-0.547134\pi$
$677$	−7.67758	−0.295073	−0.147537	+	0.989057i	$0.547134\pi$
$678$	−1.32242	−0.0507874
$679$	−15.4840	−0.594222
$680$	−1.44081	−0.0552525
$681$	27.2130	1.04281
$682$	−1.63512	−0.0626119
$683$	8.32168	0.318420	0.159210	−	0.987245i	$-0.449105\pi$
$683$	8.32168	0.318420	0.159210	+	0.987245i	$0.449105\pi$
$684$	−2.92407	−0.111805
$685$	14.2375	0.543987
$686$	−9.00973	−0.343993
$687$	26.5355	1.01239
$688$	−0.440808	−0.0168056
$689$	−60.9525	−2.32211
$690$	−7.09464	−0.270088
$691$	9.35087	0.355724	0.177862	−	0.984055i	$-0.443082\pi$
$691$	9.35087	0.355724	0.177862	+	0.984055i	$0.443082\pi$
$692$	0.246496	0.00937038
$693$	3.36488	0.127821
$694$	18.4073	0.698733
$695$	−11.1983	−0.424775
$696$	−2.24650	−0.0851532
$697$	−10.6114	−0.401935
$698$	18.2562	0.691008
$699$	−4.34114	−0.164197
$700$	−9.83916	−0.371885
$701$	6.38360	0.241105	0.120553	−	0.992707i	$-0.461533\pi$
$701$	6.38360	0.241105	0.120553	+	0.992707i	$0.461533\pi$
$702$	5.92407	0.223590
$703$	0.568181	0.0214293
$704$	1.00000	0.0376889
$705$	−15.9241	−0.599735
$706$	−13.8392	−0.520844
$707$	−4.05145	−0.152370
$708$	−10.1281	−0.380638
$709$	30.0849	1.12986	0.564931	−	0.825138i	$-0.308903\pi$
$709$	30.0849	1.12986	0.564931	+	0.825138i	$0.308903\pi$
$710$	2.35589	0.0884151
$711$	−5.92407	−0.222170
$712$	−17.9338	−0.672098
$713$	8.05145	0.301529
$714$	−3.36488	−0.125928
$715$	8.53545	0.319208
$716$	−18.5117	−0.691815
$717$	10.9003	0.407080
$718$	−22.6963	−0.847018
$719$	20.5020	0.764595	0.382297	−	0.924039i	$-0.375133\pi$
$719$	20.5020	0.764595	0.382297	+	0.924039i	$0.375133\pi$
$720$	−1.44081	−0.0536957
$721$	−21.2742	−0.792293
$722$	−10.4498	−0.388901
$723$	−10.0425	−0.373483
$724$	−18.3746	−0.682887
$725$	−6.56892	−0.243964
$726$	−1.00000	−0.0371135
$727$	11.1990	0.415348	0.207674	−	0.978198i	$-0.433411\pi$
$727$	11.1990	0.415348	0.207674	+	0.978198i	$0.433411\pi$
$728$	−19.9338	−0.738796
$729$	1.00000	0.0370370
$730$	−11.8816	−0.439758
$731$	−0.440808	−0.0163039
$732$	1.00000	0.0369611
$733$	18.7730	0.693395	0.346698	−	0.937977i	$-0.387303\pi$
$733$	18.7730	0.693395	0.346698	+	0.937977i	$0.387303\pi$
$734$	22.3077	0.823391
$735$	6.22778	0.229715
$736$	−4.92407	−0.181504
$737$	1.63512	0.0602304
$738$	−10.6114	−0.390610
$739$	−8.11336	−0.298455	−0.149227	−	0.988803i	$-0.547679\pi$
$739$	−8.11336	−0.298455	−0.149227	+	0.988803i	$0.547679\pi$
$740$	0.279965	0.0102917
$741$	−17.3224	−0.636355
$742$	34.6211	1.27098
$743$	−7.37461	−0.270548	−0.135274	−	0.990808i	$-0.543191\pi$
$743$	−7.37461	−0.270548	−0.135274	+	0.990808i	$0.543191\pi$
$744$	1.63512	0.0599464
$745$	13.8964	0.509124
$746$	21.0946	0.772330
$747$	−2.39835	−0.0877510
$748$	1.00000	0.0365636
$749$	−52.2749	−1.91008
$750$	−11.4171	−0.416892
$751$	−21.2235	−0.774456	−0.387228	−	0.921984i	$-0.626567\pi$
$751$	−21.2235	−0.774456	−0.387228	+	0.921984i	$0.626567\pi$
$752$	−11.0522	−0.403032
$753$	−15.0612	−0.548860
$754$	−13.3084	−0.484664
$755$	−11.1853	−0.407076
$756$	−3.36488	−0.122380
$757$	28.6920	1.04283	0.521414	−	0.853304i	$-0.325405\pi$
$757$	28.6920	1.04283	0.521414	+	0.853304i	$0.325405\pi$
$758$	−25.1044	−0.911832
$759$	4.92407	0.178733
$760$	4.21303	0.152823
$761$	−37.4700	−1.35829	−0.679143	−	0.734006i	$-0.737648\pi$
$761$	−37.4700	−1.35829	−0.679143	+	0.734006i	$0.737648\pi$
$762$	−1.88162	−0.0681637
$763$	44.2375	1.60151
$764$	−8.41707	−0.304519
$765$	−1.44081	−0.0520925
$766$	16.7632	0.605680
$767$	−59.9997	−2.16646
$768$	−1.00000	−0.0360844
$769$	−23.9770	−0.864633	−0.432316	−	0.901722i	$-0.642304\pi$
$769$	−23.9770	−0.864633	−0.432316	+	0.901722i	$0.642304\pi$
$770$	−4.84815	−0.174715
$771$	−11.0097	−0.396506
$772$	0.905357	0.0325845
$773$	−9.53545	−0.342966	−0.171483	−	0.985187i	$-0.554856\pi$
$773$	−9.53545	−0.342966	−0.171483	+	0.985187i	$0.554856\pi$
$774$	−0.440808	−0.0158445
$775$	4.78121	0.171746
$776$	−4.60165	−0.165190
$777$	0.653835	0.0234562
$778$	20.6351	0.739805
$779$	31.0284	1.11171
$780$	−8.53545	−0.305618
$781$	−1.63512	−0.0585092
$782$	−4.92407	−0.176084
$783$	−2.24650	−0.0802832
$784$	4.32242	0.154372
$785$	−4.50701	−0.160862
$786$	18.0522	0.643900
$787$	9.74452	0.347354	0.173677	−	0.984803i	$-0.444435\pi$
$787$	9.74452	0.347354	0.173677	+	0.984803i	$0.444435\pi$
$788$	8.97626	0.319766
$789$	30.7395	1.09435
$790$	8.53545	0.303678
$791$	4.44980	0.158217
$792$	1.00000	0.0355335
$793$	5.92407	0.210370
$794$	−4.10866	−0.145811
$795$	14.8244	0.525767
$796$	0.492993	0.0174737
$797$	−53.6398	−1.90002	−0.950010	−	0.312220i	$-0.898928\pi$
$797$	−53.6398	−1.90002	−0.950010	+	0.312220i	$0.898928\pi$
$798$	9.83916	0.348302
$799$	−11.0522	−0.390998
$800$	−2.92407	−0.103382
$801$	−17.9338	−0.633660
$802$	10.7298	0.378881
$803$	8.24650	0.291013
$804$	−1.63512	−0.0576662
$805$	23.8726	0.841400
$806$	9.68657	0.341195
$807$	21.0000	0.739235
$808$	−1.20404	−0.0423579
$809$	−40.8816	−1.43732	−0.718661	−	0.695361i	$-0.755244\pi$
$809$	−40.8816	−1.43732	−0.718661	+	0.695361i	$0.755244\pi$
$810$	−1.44081	−0.0506248
$811$	24.5876	0.863389	0.431694	−	0.902020i	$-0.357916\pi$
$811$	24.5876	0.863389	0.431694	+	0.902020i	$0.357916\pi$
$812$	7.55919	0.265276
$813$	9.07593	0.318307
$814$	−0.194311	−0.00681061
$815$	−6.67832	−0.233931
$816$	−1.00000	−0.0350070
$817$	1.28895	0.0450948
$818$	27.4358	0.959270
$819$	−19.9338	−0.696543
$820$	15.2890	0.533913
$821$	−36.6308	−1.27842	−0.639212	−	0.769030i	$-0.720740\pi$
$821$	−36.6308	−1.27842	−0.639212	+	0.769030i	$0.720740\pi$
$822$	9.88162	0.344661
$823$	47.3084	1.64907	0.824534	−	0.565813i	$-0.191438\pi$
$823$	47.3084	1.64907	0.824534	+	0.565813i	$0.191438\pi$
$824$	−6.32242	−0.220252
$825$	2.92407	0.101803
$826$	34.0799	1.18579
$827$	37.7917	1.31415	0.657073	−	0.753827i	$-0.271794\pi$
$827$	37.7917	1.31415	0.657073	+	0.753827i	$0.271794\pi$
$828$	−4.92407	−0.171123
$829$	38.0806	1.32259	0.661297	−	0.750124i	$-0.270006\pi$
$829$	38.0806	1.32259	0.661297	+	0.750124i	$0.270006\pi$
$830$	3.45556	0.119944
$831$	19.2987	0.669464
$832$	−5.92407	−0.205380
$833$	4.32242	0.149763
$834$	−7.77222	−0.269130
$835$	21.8579	0.756423
$836$	−2.92407	−0.101131
$837$	1.63512	0.0565180
$838$	−18.6538	−0.644386
$839$	23.0709	0.796496	0.398248	−	0.917278i	$-0.369618\pi$
$839$	23.0709	0.796496	0.398248	+	0.917278i	$0.369618\pi$
$840$	4.84815	0.167277
$841$	−23.9533	−0.825974
$842$	−10.1468	−0.349683
$843$	0.398350	0.0137199
$844$	28.1893	0.970315
$845$	−31.8341	−1.09513
$846$	−11.0522	−0.379982
$847$	3.36488	0.115619
$848$	10.2890	0.353324
$849$	−15.5592	−0.533990
$850$	−2.92407	−0.100295
$851$	0.956804	0.0327988
$852$	1.63512	0.0560183
$853$	5.89060	0.201690	0.100845	−	0.994902i	$-0.467845\pi$
$853$	5.89060	0.201690	0.100845	+	0.994902i	$0.467845\pi$
$854$	−3.36488	−0.115144
$855$	4.21303	0.144082
$856$	−15.5355	−0.530991
$857$	−20.1706	−0.689014	−0.344507	−	0.938784i	$-0.611954\pi$
$857$	−20.1706	−0.689014	−0.344507	+	0.938784i	$0.611954\pi$
$858$	5.92407	0.202245
$859$	8.11336	0.276824	0.138412	−	0.990375i	$-0.455800\pi$
$859$	8.11336	0.276824	0.138412	+	0.990375i	$0.455800\pi$
$860$	0.635119	0.0216574
$861$	35.7060	1.21686
$862$	−3.38862	−0.115417
$863$	−22.8528	−0.777920	−0.388960	−	0.921255i	$-0.627166\pi$
$863$	−22.8528	−0.777920	−0.388960	+	0.921255i	$0.627166\pi$
$864$	−1.00000	−0.0340207
$865$	−0.355154	−0.0120756
$866$	−25.2562	−0.858241
$867$	16.0000	0.543388
$868$	−5.50198	−0.186749
$869$	−5.92407	−0.200960
$870$	3.23677	0.109737
$871$	−9.68657	−0.328217
$872$	13.1468	0.445208
$873$	−4.60165	−0.155742
$874$	14.3983	0.487031
$875$	38.4171	1.29873
$876$	−8.24650	−0.278623
$877$	3.67758	0.124183	0.0620915	−	0.998070i	$-0.480223\pi$
$877$	3.67758	0.124183	0.0620915	+	0.998070i	$0.480223\pi$
$878$	−7.19431	−0.242796
$879$	−10.6776	−0.360146
$880$	−1.44081	−0.0485696
$881$	−39.3933	−1.32719	−0.663597	−	0.748090i	$-0.730971\pi$
$881$	−39.3933	−1.32719	−0.663597	+	0.748090i	$0.730971\pi$
$882$	4.32242	0.145544
$883$	−26.3181	−0.885676	−0.442838	−	0.896602i	$-0.646028\pi$
$883$	−26.3181	−0.885676	−0.442838	+	0.896602i	$0.646028\pi$
$884$	−5.92407	−0.199248
$885$	14.5927	0.490527
$886$	32.1461	1.07997
$887$	−22.9950	−0.772096	−0.386048	−	0.922479i	$-0.626160\pi$
$887$	−22.9950	−0.772096	−0.386048	+	0.922479i	$0.626160\pi$
$888$	0.194311	0.00652066
$889$	6.33141	0.212349
$890$	25.8392	0.866131
$891$	1.00000	0.0335013
$892$	18.4408	0.617444
$893$	32.3174	1.08146
$894$	9.64485	0.322572
$895$	26.6718	0.891540
$896$	3.36488	0.112413
$897$	−29.1706	−0.973977
$898$	−27.2562	−0.909552
$899$	−3.67329	−0.122511
$900$	−2.92407	−0.0974691
$901$	10.2890	0.342775
$902$	−10.6114	−0.353320
$903$	1.48327	0.0493600
$904$	1.32242	0.0439831
$905$	26.4743	0.880035
$906$	−7.76323	−0.257916
$907$	29.4700	0.978535	0.489267	−	0.872134i	$-0.337264\pi$
$907$	29.4700	0.978535	0.489267	+	0.872134i	$0.337264\pi$
$908$	−27.2130	−0.903096
$909$	−1.20404	−0.0399354
$910$	28.7208	0.952084
$911$	3.34188	0.110721	0.0553607	−	0.998466i	$-0.482369\pi$
$911$	3.34188	0.110721	0.0553607	+	0.998466i	$0.482369\pi$
$912$	2.92407	0.0968257
$913$	−2.39835	−0.0793738
$914$	7.97626	0.263831
$915$	−1.44081	−0.0476316
$916$	−26.5355	−0.876756
$917$	−60.7435	−2.00593
$918$	−1.00000	−0.0330049
$919$	−37.7255	−1.24445	−0.622224	−	0.782839i	$-0.713771\pi$
$919$	−37.7255	−1.24445	−0.622224	+	0.782839i	$0.713771\pi$
$920$	7.09464	0.233903
$921$	−5.07593	−0.167257
$922$	28.3411	0.933366
$923$	9.68657	0.318837
$924$	−3.36488	−0.110696
$925$	0.568181	0.0186817
$926$	24.6538	0.810175
$927$	−6.32242	−0.207656
$928$	2.24650	0.0737449
$929$	30.7632	1.00931	0.504654	−	0.863321i	$-0.331620\pi$
$929$	30.7632	1.00931	0.504654	+	0.863321i	$0.331620\pi$
$930$	−2.35589	−0.0772528
$931$	−12.6391	−0.414229
$932$	4.34114	0.142199
$933$	−11.9101	−0.389918
$934$	25.1803	0.823924
$935$	−1.44081	−0.0471195
$936$	−5.92407	−0.193634
$937$	41.6873	1.36186	0.680932	−	0.732346i	$-0.261575\pi$
$937$	41.6873	1.36186	0.680932	+	0.732346i	$0.261575\pi$
$938$	5.50198	0.179646
$939$	−29.7582	−0.971122
$940$	15.9241	0.519386
$941$	7.54048	0.245813	0.122906	−	0.992418i	$-0.460779\pi$
$941$	7.54048	0.245813	0.122906	+	0.992418i	$0.460779\pi$
$942$	−3.12811	−0.101919
$943$	52.2512	1.70153
$944$	10.1281	0.329642
$945$	4.84815	0.157710
$946$	−0.440808	−0.0143319
$947$	−20.3976	−0.662833	−0.331417	−	0.943484i	$-0.607527\pi$
$947$	−20.3976	−0.662833	−0.331417	+	0.943484i	$0.607527\pi$
$948$	5.92407	0.192405
$949$	−48.8528	−1.58583
$950$	8.55020	0.277405
$951$	−8.84888	−0.286945
$952$	3.36488	0.109056
$953$	−5.05721	−0.163819	−0.0819096	−	0.996640i	$-0.526102\pi$
$953$	−5.05721	−0.163819	−0.0819096	+	0.996640i	$0.526102\pi$
$954$	10.2890	0.333117
$955$	12.1274	0.392433
$956$	−10.9003	−0.352542
$957$	−2.24650	−0.0726189
$958$	9.40734	0.303937
$959$	−33.2505	−1.07371
$960$	1.44081	0.0465019
$961$	−28.3264	−0.913754
$962$	1.15112	0.0371134
$963$	−15.5355	−0.500623
$964$	10.0425	0.323446
$965$	−1.30445	−0.0419916
$966$	16.5689	0.533096
$967$	22.1796	0.713247	0.356623	−	0.934248i	$-0.383928\pi$
$967$	22.1796	0.713247	0.356623	+	0.934248i	$0.383928\pi$
$968$	1.00000	0.0321412
$969$	2.92407	0.0939348
$970$	6.63009	0.212879
$971$	14.4693	0.464340	0.232170	−	0.972675i	$-0.425417\pi$
$971$	14.4693	0.464340	0.232170	+	0.972675i	$0.425417\pi$
$972$	−1.00000	−0.0320750
$973$	26.1526	0.838414
$974$	32.2465	1.03324
$975$	−17.3224	−0.554762
$976$	−1.00000	−0.0320092
$977$	−54.1885	−1.73365	−0.866823	−	0.498617i	$-0.833842\pi$
$977$	−54.1885	−1.73365	−0.866823	+	0.498617i	$0.833842\pi$
$978$	−4.63512	−0.148215
$979$	−17.9338	−0.573167
$980$	−6.22778	−0.198939
$981$	13.1468	0.419746
$982$	−1.79596	−0.0573114
$983$	55.0619	1.75620	0.878101	−	0.478475i	$-0.158810\pi$
$983$	55.0619	1.75620	0.878101	+	0.478475i	$0.158810\pi$
$984$	10.6114	0.338278
$985$	−12.9331	−0.412082
$986$	2.24650	0.0715430
$987$	37.1893	1.18375
$988$	17.3224	0.551100
$989$	2.17057	0.0690201
$990$	−1.44081	−0.0457919
$991$	−47.0799	−1.49554	−0.747771	−	0.663957i	$-0.768876\pi$
$991$	−47.0799	−1.49554	−0.747771	+	0.663957i	$0.768876\pi$
$992$	−1.63512	−0.0519151
$993$	−3.00973	−0.0955108
$994$	−5.50198	−0.174512
$995$	−0.710307	−0.0225183
$996$	2.39835	0.0759946
$997$	−49.6251	−1.57164	−0.785821	−	0.618454i	$-0.787759\pi$
$997$	−49.6251	−1.57164	−0.785821	+	0.618454i	$0.787759\pi$
$998$	22.0759	0.698801
$999$	0.194311	0.00614774

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4026.2.a.o.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.o.1.2	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.44081 q^{5} -1.00000 q^{6} +3.36488 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.44081 q^{5} -1.00000 q^{6} +3.36488 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.44081 q^{10} +1.00000 q^{11} -1.00000 q^{12} -5.92407 q^{13} +3.36488 q^{14} +1.44081 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -2.92407 q^{19} -1.44081 q^{20} -3.36488 q^{21} +1.00000 q^{22} -4.92407 q^{23} -1.00000 q^{24} -2.92407 q^{25} -5.92407 q^{26} -1.00000 q^{27} +3.36488 q^{28} +2.24650 q^{29} +1.44081 q^{30} -1.63512 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} -4.84815 q^{35} +1.00000 q^{36} -0.194311 q^{37} -2.92407 q^{38} +5.92407 q^{39} -1.44081 q^{40} -10.6114 q^{41} -3.36488 q^{42} -0.440808 q^{43} +1.00000 q^{44} -1.44081 q^{45} -4.92407 q^{46} -11.0522 q^{47} -1.00000 q^{48} +4.32242 q^{49} -2.92407 q^{50} -1.00000 q^{51} -5.92407 q^{52} +10.2890 q^{53} -1.00000 q^{54} -1.44081 q^{55} +3.36488 q^{56} +2.92407 q^{57} +2.24650 q^{58} +10.1281 q^{59} +1.44081 q^{60} -1.00000 q^{61} -1.63512 q^{62} +3.36488 q^{63} +1.00000 q^{64} +8.53545 q^{65} -1.00000 q^{66} +1.63512 q^{67} +1.00000 q^{68} +4.92407 q^{69} -4.84815 q^{70} -1.63512 q^{71} +1.00000 q^{72} +8.24650 q^{73} -0.194311 q^{74} +2.92407 q^{75} -2.92407 q^{76} +3.36488 q^{77} +5.92407 q^{78} -5.92407 q^{79} -1.44081 q^{80} +1.00000 q^{81} -10.6114 q^{82} -2.39835 q^{83} -3.36488 q^{84} -1.44081 q^{85} -0.440808 q^{86} -2.24650 q^{87} +1.00000 q^{88} -17.9338 q^{89} -1.44081 q^{90} -19.9338 q^{91} -4.92407 q^{92} +1.63512 q^{93} -11.0522 q^{94} +4.21303 q^{95} -1.00000 q^{96} -4.60165 q^{97} +4.32242 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 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