LMFDB - Embedded newform 4026.2.a.l.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
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			$-1.56155$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.561553 q^{10} +1.00000 q^{11} -1.00000 q^{12} +4.56155 q^{13} -5.12311 q^{14} -0.561553 q^{15} +1.00000 q^{16} -3.12311 q^{17} +1.00000 q^{18} -4.00000 q^{19} +0.561553 q^{20} +5.12311 q^{21} +1.00000 q^{22} -1.00000 q^{24} -4.68466 q^{25} +4.56155 q^{26} -1.00000 q^{27} -5.12311 q^{28} +4.56155 q^{29} -0.561553 q^{30} +7.68466 q^{31} +1.00000 q^{32} -1.00000 q^{33} -3.12311 q^{34} -2.87689 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} -4.56155 q^{39} +0.561553 q^{40} +7.43845 q^{41} +5.12311 q^{42} +10.2462 q^{43} +1.00000 q^{44} +0.561553 q^{45} -6.24621 q^{47} -1.00000 q^{48} +19.2462 q^{49} -4.68466 q^{50} +3.12311 q^{51} +4.56155 q^{52} +3.12311 q^{53} -1.00000 q^{54} +0.561553 q^{55} -5.12311 q^{56} +4.00000 q^{57} +4.56155 q^{58} -2.56155 q^{59} -0.561553 q^{60} +1.00000 q^{61} +7.68466 q^{62} -5.12311 q^{63} +1.00000 q^{64} +2.56155 q^{65} -1.00000 q^{66} +4.00000 q^{67} -3.12311 q^{68} -2.87689 q^{70} -5.12311 q^{71} +1.00000 q^{72} -3.12311 q^{73} -6.00000 q^{74} +4.68466 q^{75} -4.00000 q^{76} -5.12311 q^{77} -4.56155 q^{78} +0.561553 q^{80} +1.00000 q^{81} +7.43845 q^{82} +10.2462 q^{83} +5.12311 q^{84} -1.75379 q^{85} +10.2462 q^{86} -4.56155 q^{87} +1.00000 q^{88} -6.80776 q^{89} +0.561553 q^{90} -23.3693 q^{91} -7.68466 q^{93} -6.24621 q^{94} -2.24621 q^{95} -1.00000 q^{96} +5.68466 q^{97} +19.2462 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 3 q^{10} + 2 q^{11} - 2 q^{12} + 5 q^{13} - 2 q^{14} + 3 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 8 q^{19} - 3 q^{20} + 2 q^{21} + 2 q^{22} - 2 q^{24} + 3 q^{25} + 5 q^{26} - 2 q^{27} - 2 q^{28} + 5 q^{29} + 3 q^{30} + 3 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{34} - 14 q^{35} + 2 q^{36} - 12 q^{37} - 8 q^{38} - 5 q^{39} - 3 q^{40} + 19 q^{41} + 2 q^{42} + 4 q^{43} + 2 q^{44} - 3 q^{45} + 4 q^{47} - 2 q^{48} + 22 q^{49} + 3 q^{50} - 2 q^{51} + 5 q^{52} - 2 q^{53} - 2 q^{54} - 3 q^{55} - 2 q^{56} + 8 q^{57} + 5 q^{58} - q^{59} + 3 q^{60} + 2 q^{61} + 3 q^{62} - 2 q^{63} + 2 q^{64} + q^{65} - 2 q^{66} + 8 q^{67} + 2 q^{68} - 14 q^{70} - 2 q^{71} + 2 q^{72} + 2 q^{73} - 12 q^{74} - 3 q^{75} - 8 q^{76} - 2 q^{77} - 5 q^{78} - 3 q^{80} + 2 q^{81} + 19 q^{82} + 4 q^{83} + 2 q^{84} - 20 q^{85} + 4 q^{86} - 5 q^{87} + 2 q^{88} + 7 q^{89} - 3 q^{90} - 22 q^{91} - 3 q^{93} + 4 q^{94} + 12 q^{95} - 2 q^{96} - q^{97} + 22 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 - 3 * q^10 + 2 * q^11 - 2 * q^12 + 5 * q^13 - 2 * q^14 + 3 * q^15 + 2 * q^16 + 2 * q^17 + 2 * q^18 - 8 * q^19 - 3 * q^20 + 2 * q^21 + 2 * q^22 - 2 * q^24 + 3 * q^25 + 5 * q^26 - 2 * q^27 - 2 * q^28 + 5 * q^29 + 3 * q^30 + 3 * q^31 + 2 * q^32 - 2 * q^33 + 2 * q^34 - 14 * q^35 + 2 * q^36 - 12 * q^37 - 8 * q^38 - 5 * q^39 - 3 * q^40 + 19 * q^41 + 2 * q^42 + 4 * q^43 + 2 * q^44 - 3 * q^45 + 4 * q^47 - 2 * q^48 + 22 * q^49 + 3 * q^50 - 2 * q^51 + 5 * q^52 - 2 * q^53 - 2 * q^54 - 3 * q^55 - 2 * q^56 + 8 * q^57 + 5 * q^58 - q^59 + 3 * q^60 + 2 * q^61 + 3 * q^62 - 2 * q^63 + 2 * q^64 + q^65 - 2 * q^66 + 8 * q^67 + 2 * q^68 - 14 * q^70 - 2 * q^71 + 2 * q^72 + 2 * q^73 - 12 * q^74 - 3 * q^75 - 8 * q^76 - 2 * q^77 - 5 * q^78 - 3 * q^80 + 2 * q^81 + 19 * q^82 + 4 * q^83 + 2 * q^84 - 20 * q^85 + 4 * q^86 - 5 * q^87 + 2 * q^88 + 7 * q^89 - 3 * q^90 - 22 * q^91 - 3 * q^93 + 4 * q^94 + 12 * q^95 - 2 * q^96 - q^97 + 22 * q^98 + 2 * 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
$5$	0.561553	0.251134	0.125567	−	0.992085i	$-0.459925\pi$
$5$	0.561553	0.251134	0.125567	+	0.992085i	$0.459925\pi$
$6$	−1.00000	−0.408248
$7$	−5.12311	−1.93635	−0.968176	−	0.250270i	$-0.919480\pi$
$7$	−5.12311	−1.93635	−0.968176	+	0.250270i	$0.919480\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0.561553	0.177579
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	4.56155	1.26515	0.632574	−	0.774500i	$-0.281999\pi$
$13$	4.56155	1.26515	0.632574	+	0.774500i	$0.281999\pi$
$14$	−5.12311	−1.36921
$15$	−0.561553	−0.144992
$16$	1.00000	0.250000
$17$	−3.12311	−0.757464	−0.378732	−	0.925506i	$-0.623640\pi$
$17$	−3.12311	−0.757464	−0.378732	+	0.925506i	$0.623640\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0.561553	0.125567
$21$	5.12311	1.11795
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	−4.68466	−0.936932
$26$	4.56155	0.894594
$27$	−1.00000	−0.192450
$28$	−5.12311	−0.968176
$29$	4.56155	0.847059	0.423530	−	0.905882i	$-0.360791\pi$
$29$	4.56155	0.847059	0.423530	+	0.905882i	$0.360791\pi$
$30$	−0.561553	−0.102525
$31$	7.68466	1.38021	0.690103	−	0.723711i	$-0.257565\pi$
$31$	7.68466	1.38021	0.690103	+	0.723711i	$0.257565\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−3.12311	−0.535608
$35$	−2.87689	−0.486284
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−4.00000	−0.648886
$39$	−4.56155	−0.730433
$40$	0.561553	0.0887893
$41$	7.43845	1.16169	0.580845	−	0.814014i	$-0.302722\pi$
$41$	7.43845	1.16169	0.580845	+	0.814014i	$0.302722\pi$
$42$	5.12311	0.790512
$43$	10.2462	1.56253	0.781266	−	0.624198i	$-0.214574\pi$
$43$	10.2462	1.56253	0.781266	+	0.624198i	$0.214574\pi$
$44$	1.00000	0.150756
$45$	0.561553	0.0837114
$46$	0	0
$47$	−6.24621	−0.911104	−0.455552	−	0.890209i	$-0.650558\pi$
$47$	−6.24621	−0.911104	−0.455552	+	0.890209i	$0.650558\pi$
$48$	−1.00000	−0.144338
$49$	19.2462	2.74946
$50$	−4.68466	−0.662511
$51$	3.12311	0.437322
$52$	4.56155	0.632574
$53$	3.12311	0.428992	0.214496	−	0.976725i	$-0.431189\pi$
$53$	3.12311	0.428992	0.214496	+	0.976725i	$0.431189\pi$
$54$	−1.00000	−0.136083
$55$	0.561553	0.0757198
$56$	−5.12311	−0.684604
$57$	4.00000	0.529813
$58$	4.56155	0.598961
$59$	−2.56155	−0.333486	−0.166743	−	0.986000i	$-0.553325\pi$
$59$	−2.56155	−0.333486	−0.166743	+	0.986000i	$0.553325\pi$
$60$	−0.561553	−0.0724962
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	7.68466	0.975953
$63$	−5.12311	−0.645451
$64$	1.00000	0.125000
$65$	2.56155	0.317722
$66$	−1.00000	−0.123091
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−3.12311	−0.378732
$69$	0	0
$70$	−2.87689	−0.343855
$71$	−5.12311	−0.608001	−0.304000	−	0.952672i	$-0.598322\pi$
$71$	−5.12311	−0.608001	−0.304000	+	0.952672i	$0.598322\pi$
$72$	1.00000	0.117851
$73$	−3.12311	−0.365532	−0.182766	−	0.983156i	$-0.558505\pi$
$73$	−3.12311	−0.365532	−0.182766	+	0.983156i	$0.558505\pi$
$74$	−6.00000	−0.697486
$75$	4.68466	0.540938
$76$	−4.00000	−0.458831
$77$	−5.12311	−0.583832
$78$	−4.56155	−0.516494
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0.561553	0.0627835
$81$	1.00000	0.111111
$82$	7.43845	0.821439
$83$	10.2462	1.12467	0.562334	−	0.826910i	$-0.309904\pi$
$83$	10.2462	1.12467	0.562334	+	0.826910i	$0.309904\pi$
$84$	5.12311	0.558977
$85$	−1.75379	−0.190225
$86$	10.2462	1.10488
$87$	−4.56155	−0.489050
$88$	1.00000	0.106600
$89$	−6.80776	−0.721622	−0.360811	−	0.932639i	$-0.617500\pi$
$89$	−6.80776	−0.721622	−0.360811	+	0.932639i	$0.617500\pi$
$90$	0.561553	0.0591929
$91$	−23.3693	−2.44977
$92$	0	0
$93$	−7.68466	−0.796862
$94$	−6.24621	−0.644247
$95$	−2.24621	−0.230456
$96$	−1.00000	−0.102062
$97$	5.68466	0.577190	0.288595	−	0.957451i	$-0.406812\pi$
$97$	5.68466	0.577190	0.288595	+	0.957451i	$0.406812\pi$
$98$	19.2462	1.94416
$99$	1.00000	0.100504
$100$	−4.68466	−0.468466
$101$	12.5616	1.24992	0.624961	−	0.780656i	$-0.285115\pi$
$101$	12.5616	1.24992	0.624961	+	0.780656i	$0.285115\pi$
$102$	3.12311	0.309234
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	4.56155	0.447297
$105$	2.87689	0.280756
$106$	3.12311	0.303343
$107$	1.43845	0.139060	0.0695300	−	0.997580i	$-0.477850\pi$
$107$	1.43845	0.139060	0.0695300	+	0.997580i	$0.477850\pi$
$108$	−1.00000	−0.0962250
$109$	17.6847	1.69388	0.846942	−	0.531686i	$-0.178441\pi$
$109$	17.6847	1.69388	0.846942	+	0.531686i	$0.178441\pi$
$110$	0.561553	0.0535420
$111$	6.00000	0.569495
$112$	−5.12311	−0.484088
$113$	20.2462	1.90460	0.952302	−	0.305158i	$-0.0987093\pi$
$113$	20.2462	1.90460	0.952302	+	0.305158i	$0.0987093\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	4.56155	0.423530
$117$	4.56155	0.421716
$118$	−2.56155	−0.235810
$119$	16.0000	1.46672
$120$	−0.561553	−0.0512625
$121$	1.00000	0.0909091
$122$	1.00000	0.0905357
$123$	−7.43845	−0.670702
$124$	7.68466	0.690103
$125$	−5.43845	−0.486430
$126$	−5.12311	−0.456403
$127$	−12.4924	−1.10852	−0.554262	−	0.832343i	$-0.686999\pi$
$127$	−12.4924	−1.10852	−0.554262	+	0.832343i	$0.686999\pi$
$128$	1.00000	0.0883883
$129$	−10.2462	−0.902129
$130$	2.56155	0.224663
$131$	−8.80776	−0.769538	−0.384769	−	0.923013i	$-0.625719\pi$
$131$	−8.80776	−0.769538	−0.384769	+	0.923013i	$0.625719\pi$
$132$	−1.00000	−0.0870388
$133$	20.4924	1.77692
$134$	4.00000	0.345547
$135$	−0.561553	−0.0483308
$136$	−3.12311	−0.267804
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	16.8078	1.42562	0.712808	−	0.701359i	$-0.247423\pi$
$139$	16.8078	1.42562	0.712808	+	0.701359i	$0.247423\pi$
$140$	−2.87689	−0.243142
$141$	6.24621	0.526026
$142$	−5.12311	−0.429921
$143$	4.56155	0.381456
$144$	1.00000	0.0833333
$145$	2.56155	0.212725
$146$	−3.12311	−0.258470
$147$	−19.2462	−1.58740
$148$	−6.00000	−0.493197
$149$	11.1231	0.911240	0.455620	−	0.890174i	$-0.349418\pi$
$149$	11.1231	0.911240	0.455620	+	0.890174i	$0.349418\pi$
$150$	4.68466	0.382501
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−4.00000	−0.324443
$153$	−3.12311	−0.252488
$154$	−5.12311	−0.412832
$155$	4.31534	0.346617
$156$	−4.56155	−0.365217
$157$	6.31534	0.504019	0.252010	−	0.967725i	$-0.418909\pi$
$157$	6.31534	0.504019	0.252010	+	0.967725i	$0.418909\pi$
$158$	0	0
$159$	−3.12311	−0.247678
$160$	0.561553	0.0443946
$161$	0	0
$162$	1.00000	0.0785674
$163$	−0.315342	−0.0246995	−0.0123497	−	0.999924i	$-0.503931\pi$
$163$	−0.315342	−0.0246995	−0.0123497	+	0.999924i	$0.503931\pi$
$164$	7.43845	0.580845
$165$	−0.561553	−0.0437168
$166$	10.2462	0.795260
$167$	−2.24621	−0.173817	−0.0869085	−	0.996216i	$-0.527699\pi$
$167$	−2.24621	−0.173817	−0.0869085	+	0.996216i	$0.527699\pi$
$168$	5.12311	0.395256
$169$	7.80776	0.600597
$170$	−1.75379	−0.134509
$171$	−4.00000	−0.305888
$172$	10.2462	0.781266
$173$	10.3153	0.784261	0.392130	−	0.919910i	$-0.371738\pi$
$173$	10.3153	0.784261	0.392130	+	0.919910i	$0.371738\pi$
$174$	−4.56155	−0.345810
$175$	24.0000	1.81423
$176$	1.00000	0.0753778
$177$	2.56155	0.192538
$178$	−6.80776	−0.510263
$179$	1.12311	0.0839449	0.0419724	−	0.999119i	$-0.486636\pi$
$179$	1.12311	0.0839449	0.0419724	+	0.999119i	$0.486636\pi$
$180$	0.561553	0.0418557
$181$	18.8078	1.39797	0.698985	−	0.715136i	$-0.253635\pi$
$181$	18.8078	1.39797	0.698985	+	0.715136i	$0.253635\pi$
$182$	−23.3693	−1.73225
$183$	−1.00000	−0.0739221
$184$	0	0
$185$	−3.36932	−0.247717
$186$	−7.68466	−0.563466
$187$	−3.12311	−0.228384
$188$	−6.24621	−0.455552
$189$	5.12311	0.372651
$190$	−2.24621	−0.162957
$191$	13.1231	0.949555	0.474777	−	0.880106i	$-0.342529\pi$
$191$	13.1231	0.949555	0.474777	+	0.880106i	$0.342529\pi$
$192$	−1.00000	−0.0721688
$193$	6.80776	0.490034	0.245017	−	0.969519i	$-0.421207\pi$
$193$	6.80776	0.490034	0.245017	+	0.969519i	$0.421207\pi$
$194$	5.68466	0.408135
$195$	−2.56155	−0.183437
$196$	19.2462	1.37473
$197$	−12.2462	−0.872506	−0.436253	−	0.899824i	$-0.643695\pi$
$197$	−12.2462	−0.872506	−0.436253	+	0.899824i	$0.643695\pi$
$198$	1.00000	0.0710669
$199$	−7.36932	−0.522397	−0.261199	−	0.965285i	$-0.584118\pi$
$199$	−7.36932	−0.522397	−0.261199	+	0.965285i	$0.584118\pi$
$200$	−4.68466	−0.331255
$201$	−4.00000	−0.282138
$202$	12.5616	0.883828
$203$	−23.3693	−1.64020
$204$	3.12311	0.218661
$205$	4.17708	0.291740
$206$	16.0000	1.11477
$207$	0	0
$208$	4.56155	0.316287
$209$	−4.00000	−0.276686
$210$	2.87689	0.198525
$211$	9.43845	0.649770	0.324885	−	0.945754i	$-0.394674\pi$
$211$	9.43845	0.649770	0.324885	+	0.945754i	$0.394674\pi$
$212$	3.12311	0.214496
$213$	5.12311	0.351029
$214$	1.43845	0.0983302
$215$	5.75379	0.392405
$216$	−1.00000	−0.0680414
$217$	−39.3693	−2.67256
$218$	17.6847	1.19776
$219$	3.12311	0.211040
$220$	0.561553	0.0378599
$221$	−14.2462	−0.958304
$222$	6.00000	0.402694
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−5.12311	−0.342302
$225$	−4.68466	−0.312311
$226$	20.2462	1.34676
$227$	−14.2462	−0.945554	−0.472777	−	0.881182i	$-0.656748\pi$
$227$	−14.2462	−0.945554	−0.472777	+	0.881182i	$0.656748\pi$
$228$	4.00000	0.264906
$229$	11.1231	0.735036	0.367518	−	0.930017i	$-0.380208\pi$
$229$	11.1231	0.735036	0.367518	+	0.930017i	$0.380208\pi$
$230$	0	0
$231$	5.12311	0.337076
$232$	4.56155	0.299481
$233$	2.63068	0.172342	0.0861709	−	0.996280i	$-0.472537\pi$
$233$	2.63068	0.172342	0.0861709	+	0.996280i	$0.472537\pi$
$234$	4.56155	0.298198
$235$	−3.50758	−0.228809
$236$	−2.56155	−0.166743
$237$	0	0
$238$	16.0000	1.03713
$239$	−26.2462	−1.69773	−0.848863	−	0.528613i	$-0.822712\pi$
$239$	−26.2462	−1.69773	−0.848863	+	0.528613i	$0.822712\pi$
$240$	−0.561553	−0.0362481
$241$	9.36932	0.603531	0.301765	−	0.953382i	$-0.402424\pi$
$241$	9.36932	0.603531	0.301765	+	0.953382i	$0.402424\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	1.00000	0.0640184
$245$	10.8078	0.690483
$246$	−7.43845	−0.474258
$247$	−18.2462	−1.16098
$248$	7.68466	0.487976
$249$	−10.2462	−0.649327
$250$	−5.43845	−0.343958
$251$	21.4384	1.35318	0.676591	−	0.736359i	$-0.263456\pi$
$251$	21.4384	1.35318	0.676591	+	0.736359i	$0.263456\pi$
$252$	−5.12311	−0.322725
$253$	0	0
$254$	−12.4924	−0.783844
$255$	1.75379	0.109827
$256$	1.00000	0.0625000
$257$	6.49242	0.404986	0.202493	−	0.979284i	$-0.435096\pi$
$257$	6.49242	0.404986	0.202493	+	0.979284i	$0.435096\pi$
$258$	−10.2462	−0.637901
$259$	30.7386	1.91001
$260$	2.56155	0.158861
$261$	4.56155	0.282353
$262$	−8.80776	−0.544145
$263$	−13.1231	−0.809205	−0.404603	−	0.914493i	$-0.632590\pi$
$263$	−13.1231	−0.809205	−0.404603	+	0.914493i	$0.632590\pi$
$264$	−1.00000	−0.0615457
$265$	1.75379	0.107734
$266$	20.4924	1.25647
$267$	6.80776	0.416628
$268$	4.00000	0.244339
$269$	29.0540	1.77145	0.885726	−	0.464208i	$-0.153661\pi$
$269$	29.0540	1.77145	0.885726	+	0.464208i	$0.153661\pi$
$270$	−0.561553	−0.0341750
$271$	−10.2462	−0.622413	−0.311207	−	0.950342i	$-0.600733\pi$
$271$	−10.2462	−0.622413	−0.311207	+	0.950342i	$0.600733\pi$
$272$	−3.12311	−0.189366
$273$	23.3693	1.41438
$274$	−6.00000	−0.362473
$275$	−4.68466	−0.282496
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	16.8078	1.00806
$279$	7.68466	0.460068
$280$	−2.87689	−0.171927
$281$	9.36932	0.558927	0.279463	−	0.960156i	$-0.409843\pi$
$281$	9.36932	0.558927	0.279463	+	0.960156i	$0.409843\pi$
$282$	6.24621	0.371956
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	−5.12311	−0.304000
$285$	2.24621	0.133054
$286$	4.56155	0.269730
$287$	−38.1080	−2.24944
$288$	1.00000	0.0589256
$289$	−7.24621	−0.426248
$290$	2.56155	0.150420
$291$	−5.68466	−0.333241
$292$	−3.12311	−0.182766
$293$	24.8769	1.45332	0.726662	−	0.686995i	$-0.241071\pi$
$293$	24.8769	1.45332	0.726662	+	0.686995i	$0.241071\pi$
$294$	−19.2462	−1.12246
$295$	−1.43845	−0.0837496
$296$	−6.00000	−0.348743
$297$	−1.00000	−0.0580259
$298$	11.1231	0.644344
$299$	0	0
$300$	4.68466	0.270469
$301$	−52.4924	−3.02561
$302$	−16.0000	−0.920697
$303$	−12.5616	−0.721642
$304$	−4.00000	−0.229416
$305$	0.561553	0.0321544
$306$	−3.12311	−0.178536
$307$	−32.1771	−1.83644	−0.918222	−	0.396067i	$-0.870375\pi$
$307$	−32.1771	−1.83644	−0.918222	+	0.396067i	$0.870375\pi$
$308$	−5.12311	−0.291916
$309$	−16.0000	−0.910208
$310$	4.31534	0.245095
$311$	2.24621	0.127371	0.0636855	−	0.997970i	$-0.479715\pi$
$311$	2.24621	0.127371	0.0636855	+	0.997970i	$0.479715\pi$
$312$	−4.56155	−0.258247
$313$	−0.876894	−0.0495650	−0.0247825	−	0.999693i	$-0.507889\pi$
$313$	−0.876894	−0.0495650	−0.0247825	+	0.999693i	$0.507889\pi$
$314$	6.31534	0.356395
$315$	−2.87689	−0.162095
$316$	0	0
$317$	−1.68466	−0.0946198	−0.0473099	−	0.998880i	$-0.515065\pi$
$317$	−1.68466	−0.0946198	−0.0473099	+	0.998880i	$0.515065\pi$
$318$	−3.12311	−0.175135
$319$	4.56155	0.255398
$320$	0.561553	0.0313918
$321$	−1.43845	−0.0802863
$322$	0	0
$323$	12.4924	0.695097
$324$	1.00000	0.0555556
$325$	−21.3693	−1.18536
$326$	−0.315342	−0.0174652
$327$	−17.6847	−0.977964
$328$	7.43845	0.410720
$329$	32.0000	1.76422
$330$	−0.561553	−0.0309125
$331$	22.2462	1.22276	0.611381	−	0.791336i	$-0.290614\pi$
$331$	22.2462	1.22276	0.611381	+	0.791336i	$0.290614\pi$
$332$	10.2462	0.562334
$333$	−6.00000	−0.328798
$334$	−2.24621	−0.122907
$335$	2.24621	0.122724
$336$	5.12311	0.279488
$337$	18.3153	0.997700	0.498850	−	0.866688i	$-0.333756\pi$
$337$	18.3153	0.997700	0.498850	+	0.866688i	$0.333756\pi$
$338$	7.80776	0.424686
$339$	−20.2462	−1.09962
$340$	−1.75379	−0.0951125
$341$	7.68466	0.416148
$342$	−4.00000	−0.216295
$343$	−62.7386	−3.38757
$344$	10.2462	0.552439
$345$	0	0
$346$	10.3153	0.554556
$347$	−19.6847	−1.05673	−0.528364	−	0.849018i	$-0.677194\pi$
$347$	−19.6847	−1.05673	−0.528364	+	0.849018i	$0.677194\pi$
$348$	−4.56155	−0.244525
$349$	−2.63068	−0.140817	−0.0704086	−	0.997518i	$-0.522430\pi$
$349$	−2.63068	−0.140817	−0.0704086	+	0.997518i	$0.522430\pi$
$350$	24.0000	1.28285
$351$	−4.56155	−0.243478
$352$	1.00000	0.0533002
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	2.56155	0.136145
$355$	−2.87689	−0.152690
$356$	−6.80776	−0.360811
$357$	−16.0000	−0.846810
$358$	1.12311	0.0593580
$359$	−16.8078	−0.887080	−0.443540	−	0.896255i	$-0.646278\pi$
$359$	−16.8078	−0.887080	−0.443540	+	0.896255i	$0.646278\pi$
$360$	0.561553	0.0295964
$361$	−3.00000	−0.157895
$362$	18.8078	0.988514
$363$	−1.00000	−0.0524864
$364$	−23.3693	−1.22489
$365$	−1.75379	−0.0917975
$366$	−1.00000	−0.0522708
$367$	−15.3693	−0.802272	−0.401136	−	0.916019i	$-0.631384\pi$
$367$	−15.3693	−0.802272	−0.401136	+	0.916019i	$0.631384\pi$
$368$	0	0
$369$	7.43845	0.387230
$370$	−3.36932	−0.175162
$371$	−16.0000	−0.830679
$372$	−7.68466	−0.398431
$373$	−29.8617	−1.54618	−0.773091	−	0.634295i	$-0.781290\pi$
$373$	−29.8617	−1.54618	−0.773091	+	0.634295i	$0.781290\pi$
$374$	−3.12311	−0.161492
$375$	5.43845	0.280840
$376$	−6.24621	−0.322124
$377$	20.8078	1.07165
$378$	5.12311	0.263504
$379$	11.1922	0.574907	0.287453	−	0.957795i	$-0.407191\pi$
$379$	11.1922	0.574907	0.287453	+	0.957795i	$0.407191\pi$
$380$	−2.24621	−0.115228
$381$	12.4924	0.640006
$382$	13.1231	0.671436
$383$	17.6155	0.900111	0.450056	−	0.893000i	$-0.351404\pi$
$383$	17.6155	0.900111	0.450056	+	0.893000i	$0.351404\pi$
$384$	−1.00000	−0.0510310
$385$	−2.87689	−0.146620
$386$	6.80776	0.346506
$387$	10.2462	0.520844
$388$	5.68466	0.288595
$389$	25.8617	1.31124	0.655621	−	0.755090i	$-0.272407\pi$
$389$	25.8617	1.31124	0.655621	+	0.755090i	$0.272407\pi$
$390$	−2.56155	−0.129709
$391$	0	0
$392$	19.2462	0.972080
$393$	8.80776	0.444293
$394$	−12.2462	−0.616955
$395$	0	0
$396$	1.00000	0.0502519
$397$	1.19224	0.0598366	0.0299183	−	0.999552i	$-0.490475\pi$
$397$	1.19224	0.0598366	0.0299183	+	0.999552i	$0.490475\pi$
$398$	−7.36932	−0.369390
$399$	−20.4924	−1.02590
$400$	−4.68466	−0.234233
$401$	29.0540	1.45089	0.725443	−	0.688282i	$-0.241635\pi$
$401$	29.0540	1.45089	0.725443	+	0.688282i	$0.241635\pi$
$402$	−4.00000	−0.199502
$403$	35.0540	1.74616
$404$	12.5616	0.624961
$405$	0.561553	0.0279038
$406$	−23.3693	−1.15980
$407$	−6.00000	−0.297409
$408$	3.12311	0.154617
$409$	−26.1771	−1.29437	−0.647187	−	0.762332i	$-0.724055\pi$
$409$	−26.1771	−1.29437	−0.647187	+	0.762332i	$0.724055\pi$
$410$	4.17708	0.206291
$411$	6.00000	0.295958
$412$	16.0000	0.788263
$413$	13.1231	0.645746
$414$	0	0
$415$	5.75379	0.282442
$416$	4.56155	0.223649
$417$	−16.8078	−0.823080
$418$	−4.00000	−0.195646
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	2.87689	0.140378
$421$	−1.68466	−0.0821052	−0.0410526	−	0.999157i	$-0.513071\pi$
$421$	−1.68466	−0.0821052	−0.0410526	+	0.999157i	$0.513071\pi$
$422$	9.43845	0.459456
$423$	−6.24621	−0.303701
$424$	3.12311	0.151671
$425$	14.6307	0.709692
$426$	5.12311	0.248215
$427$	−5.12311	−0.247924
$428$	1.43845	0.0695300
$429$	−4.56155	−0.220234
$430$	5.75379	0.277472
$431$	−26.8769	−1.29461	−0.647307	−	0.762229i	$-0.724105\pi$
$431$	−26.8769	−1.29461	−0.647307	+	0.762229i	$0.724105\pi$
$432$	−1.00000	−0.0481125
$433$	−27.1231	−1.30345	−0.651727	−	0.758454i	$-0.725955\pi$
$433$	−27.1231	−1.30345	−0.651727	+	0.758454i	$0.725955\pi$
$434$	−39.3693	−1.88979
$435$	−2.56155	−0.122817
$436$	17.6847	0.846942
$437$	0	0
$438$	3.12311	0.149228
$439$	−29.3002	−1.39842	−0.699211	−	0.714916i	$-0.746465\pi$
$439$	−29.3002	−1.39842	−0.699211	+	0.714916i	$0.746465\pi$
$440$	0.561553	0.0267710
$441$	19.2462	0.916486
$442$	−14.2462	−0.677623
$443$	−9.12311	−0.433452	−0.216726	−	0.976232i	$-0.569538\pi$
$443$	−9.12311	−0.433452	−0.216726	+	0.976232i	$0.569538\pi$
$444$	6.00000	0.284747
$445$	−3.82292	−0.181224
$446$	4.00000	0.189405
$447$	−11.1231	−0.526105
$448$	−5.12311	−0.242044
$449$	−14.6307	−0.690465	−0.345232	−	0.938517i	$-0.612200\pi$
$449$	−14.6307	−0.690465	−0.345232	+	0.938517i	$0.612200\pi$
$450$	−4.68466	−0.220837
$451$	7.43845	0.350263
$452$	20.2462	0.952302
$453$	16.0000	0.751746
$454$	−14.2462	−0.668608
$455$	−13.1231	−0.615221
$456$	4.00000	0.187317
$457$	−24.7386	−1.15722	−0.578612	−	0.815603i	$-0.696406\pi$
$457$	−24.7386	−1.15722	−0.578612	+	0.815603i	$0.696406\pi$
$458$	11.1231	0.519749
$459$	3.12311	0.145774
$460$	0	0
$461$	1.50758	0.0702149	0.0351074	−	0.999384i	$-0.488823\pi$
$461$	1.50758	0.0702149	0.0351074	+	0.999384i	$0.488823\pi$
$462$	5.12311	0.238348
$463$	−0.630683	−0.0293103	−0.0146552	−	0.999893i	$-0.504665\pi$
$463$	−0.630683	−0.0293103	−0.0146552	+	0.999893i	$0.504665\pi$
$464$	4.56155	0.211765
$465$	−4.31534	−0.200119
$466$	2.63068	0.121864
$467$	31.6847	1.46619	0.733096	−	0.680126i	$-0.238075\pi$
$467$	31.6847	1.46619	0.733096	+	0.680126i	$0.238075\pi$
$468$	4.56155	0.210858
$469$	−20.4924	−0.946252
$470$	−3.50758	−0.161792
$471$	−6.31534	−0.290996
$472$	−2.56155	−0.117905
$473$	10.2462	0.471121
$474$	0	0
$475$	18.7386	0.859787
$476$	16.0000	0.733359
$477$	3.12311	0.142997
$478$	−26.2462	−1.20047
$479$	2.24621	0.102632	0.0513160	−	0.998682i	$-0.483658\pi$
$479$	2.24621	0.102632	0.0513160	+	0.998682i	$0.483658\pi$
$480$	−0.561553	−0.0256313
$481$	−27.3693	−1.24793
$482$	9.36932	0.426761
$483$	0	0
$484$	1.00000	0.0454545
$485$	3.19224	0.144952
$486$	−1.00000	−0.0453609
$487$	30.7386	1.39290	0.696450	−	0.717605i	$-0.254762\pi$
$487$	30.7386	1.39290	0.696450	+	0.717605i	$0.254762\pi$
$488$	1.00000	0.0452679
$489$	0.315342	0.0142602
$490$	10.8078	0.488245
$491$	30.7386	1.38722	0.693608	−	0.720353i	$-0.256020\pi$
$491$	30.7386	1.38722	0.693608	+	0.720353i	$0.256020\pi$
$492$	−7.43845	−0.335351
$493$	−14.2462	−0.641617
$494$	−18.2462	−0.820936
$495$	0.561553	0.0252399
$496$	7.68466	0.345051
$497$	26.2462	1.17730
$498$	−10.2462	−0.459144
$499$	13.6155	0.609515	0.304757	−	0.952430i	$-0.401425\pi$
$499$	13.6155	0.609515	0.304757	+	0.952430i	$0.401425\pi$
$500$	−5.43845	−0.243215
$501$	2.24621	0.100353
$502$	21.4384	0.956845
$503$	13.1231	0.585130	0.292565	−	0.956246i	$-0.405491\pi$
$503$	13.1231	0.585130	0.292565	+	0.956246i	$0.405491\pi$
$504$	−5.12311	−0.228201
$505$	7.05398	0.313898
$506$	0	0
$507$	−7.80776	−0.346755
$508$	−12.4924	−0.554262
$509$	−20.8769	−0.925352	−0.462676	−	0.886527i	$-0.653111\pi$
$509$	−20.8769	−0.925352	−0.462676	+	0.886527i	$0.653111\pi$
$510$	1.75379	0.0776591
$511$	16.0000	0.707798
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	6.49242	0.286368
$515$	8.98485	0.395920
$516$	−10.2462	−0.451064
$517$	−6.24621	−0.274708
$518$	30.7386	1.35058
$519$	−10.3153	−0.452793
$520$	2.56155	0.112332
$521$	34.9848	1.53271	0.766357	−	0.642415i	$-0.222067\pi$
$521$	34.9848	1.53271	0.766357	+	0.642415i	$0.222067\pi$
$522$	4.56155	0.199654
$523$	−20.3153	−0.888328	−0.444164	−	0.895946i	$-0.646499\pi$
$523$	−20.3153	−0.888328	−0.444164	+	0.895946i	$0.646499\pi$
$524$	−8.80776	−0.384769
$525$	−24.0000	−1.04745
$526$	−13.1231	−0.572195
$527$	−24.0000	−1.04546
$528$	−1.00000	−0.0435194
$529$	−23.0000	−1.00000
$530$	1.75379	0.0761797
$531$	−2.56155	−0.111162
$532$	20.4924	0.888459
$533$	33.9309	1.46971
$534$	6.80776	0.294601
$535$	0.807764	0.0349227
$536$	4.00000	0.172774
$537$	−1.12311	−0.0484656
$538$	29.0540	1.25261
$539$	19.2462	0.828993
$540$	−0.561553	−0.0241654
$541$	21.3693	0.918739	0.459369	−	0.888245i	$-0.348075\pi$
$541$	21.3693	0.918739	0.459369	+	0.888245i	$0.348075\pi$
$542$	−10.2462	−0.440112
$543$	−18.8078	−0.807118
$544$	−3.12311	−0.133902
$545$	9.93087	0.425392
$546$	23.3693	1.00011
$547$	−4.31534	−0.184511	−0.0922553	−	0.995735i	$-0.529408\pi$
$547$	−4.31534	−0.184511	−0.0922553	+	0.995735i	$0.529408\pi$
$548$	−6.00000	−0.256307
$549$	1.00000	0.0426790
$550$	−4.68466	−0.199755
$551$	−18.2462	−0.777315
$552$	0	0
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	3.36932	0.143020
$556$	16.8078	0.712808
$557$	−26.1771	−1.10916	−0.554579	−	0.832131i	$-0.687121\pi$
$557$	−26.1771	−1.10916	−0.554579	+	0.832131i	$0.687121\pi$
$558$	7.68466	0.325318
$559$	46.7386	1.97683
$560$	−2.87689	−0.121571
$561$	3.12311	0.131858
$562$	9.36932	0.395221
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	6.24621	0.263013
$565$	11.3693	0.478311
$566$	−12.0000	−0.504398
$567$	−5.12311	−0.215150
$568$	−5.12311	−0.214961
$569$	−18.1771	−0.762023	−0.381011	−	0.924570i	$-0.624424\pi$
$569$	−18.1771	−0.762023	−0.381011	+	0.924570i	$0.624424\pi$
$570$	2.24621	0.0940834
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	4.56155	0.190728
$573$	−13.1231	−0.548226
$574$	−38.1080	−1.59060
$575$	0	0
$576$	1.00000	0.0416667
$577$	−13.3693	−0.556572	−0.278286	−	0.960498i	$-0.589766\pi$
$577$	−13.3693	−0.556572	−0.278286	+	0.960498i	$0.589766\pi$
$578$	−7.24621	−0.301403
$579$	−6.80776	−0.282921
$580$	2.56155	0.106363
$581$	−52.4924	−2.17775
$582$	−5.68466	−0.235637
$583$	3.12311	0.129346
$584$	−3.12311	−0.129235
$585$	2.56155	0.105907
$586$	24.8769	1.02766
$587$	−19.1922	−0.792148	−0.396074	−	0.918219i	$-0.629628\pi$
$587$	−19.1922	−0.792148	−0.396074	+	0.918219i	$0.629628\pi$
$588$	−19.2462	−0.793700
$589$	−30.7386	−1.26656
$590$	−1.43845	−0.0592199
$591$	12.2462	0.503742
$592$	−6.00000	−0.246598
$593$	46.4924	1.90921	0.954607	−	0.297867i	$-0.0962751\pi$
$593$	46.4924	1.90921	0.954607	+	0.297867i	$0.0962751\pi$
$594$	−1.00000	−0.0410305
$595$	8.98485	0.368343
$596$	11.1231	0.455620
$597$	7.36932	0.301606
$598$	0	0
$599$	30.7386	1.25595	0.627973	−	0.778235i	$-0.283885\pi$
$599$	30.7386	1.25595	0.627973	+	0.778235i	$0.283885\pi$
$600$	4.68466	0.191250
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	−52.4924	−2.13943
$603$	4.00000	0.162893
$604$	−16.0000	−0.651031
$605$	0.561553	0.0228304
$606$	−12.5616	−0.510278
$607$	−35.0540	−1.42280	−0.711398	−	0.702789i	$-0.751938\pi$
$607$	−35.0540	−1.42280	−0.711398	+	0.702789i	$0.751938\pi$
$608$	−4.00000	−0.162221
$609$	23.3693	0.946973
$610$	0.561553	0.0227366
$611$	−28.4924	−1.15268
$612$	−3.12311	−0.126244
$613$	15.4384	0.623553	0.311777	−	0.950155i	$-0.399076\pi$
$613$	15.4384	0.623553	0.311777	+	0.950155i	$0.399076\pi$
$614$	−32.1771	−1.29856
$615$	−4.17708	−0.168436
$616$	−5.12311	−0.206416
$617$	22.3153	0.898382	0.449191	−	0.893436i	$-0.351712\pi$
$617$	22.3153	0.898382	0.449191	+	0.893436i	$0.351712\pi$
$618$	−16.0000	−0.643614
$619$	15.0540	0.605070	0.302535	−	0.953138i	$-0.402167\pi$
$619$	15.0540	0.605070	0.302535	+	0.953138i	$0.402167\pi$
$620$	4.31534	0.173308
$621$	0	0
$622$	2.24621	0.0900649
$623$	34.8769	1.39731
$624$	−4.56155	−0.182608
$625$	20.3693	0.814773
$626$	−0.876894	−0.0350477
$627$	4.00000	0.159745
$628$	6.31534	0.252010
$629$	18.7386	0.747158
$630$	−2.87689	−0.114618
$631$	6.24621	0.248658	0.124329	−	0.992241i	$-0.460322\pi$
$631$	6.24621	0.248658	0.124329	+	0.992241i	$0.460322\pi$
$632$	0	0
$633$	−9.43845	−0.375145
$634$	−1.68466	−0.0669063
$635$	−7.01515	−0.278388
$636$	−3.12311	−0.123839
$637$	87.7926	3.47847
$638$	4.56155	0.180594
$639$	−5.12311	−0.202667
$640$	0.561553	0.0221973
$641$	−12.7386	−0.503146	−0.251573	−	0.967838i	$-0.580948\pi$
$641$	−12.7386	−0.503146	−0.251573	+	0.967838i	$0.580948\pi$
$642$	−1.43845	−0.0567710
$643$	27.3693	1.07934	0.539670	−	0.841876i	$-0.318549\pi$
$643$	27.3693	1.07934	0.539670	+	0.841876i	$0.318549\pi$
$644$	0	0
$645$	−5.75379	−0.226555
$646$	12.4924	0.491508
$647$	19.8617	0.780846	0.390423	−	0.920636i	$-0.372329\pi$
$647$	19.8617	0.780846	0.390423	+	0.920636i	$0.372329\pi$
$648$	1.00000	0.0392837
$649$	−2.56155	−0.100550
$650$	−21.3693	−0.838174
$651$	39.3693	1.54301
$652$	−0.315342	−0.0123497
$653$	−9.36932	−0.366650	−0.183325	−	0.983052i	$-0.558686\pi$
$653$	−9.36932	−0.366650	−0.183325	+	0.983052i	$0.558686\pi$
$654$	−17.6847	−0.691525
$655$	−4.94602	−0.193257
$656$	7.43845	0.290423
$657$	−3.12311	−0.121844
$658$	32.0000	1.24749
$659$	23.1922	0.903441	0.451721	−	0.892159i	$-0.350810\pi$
$659$	23.1922	0.903441	0.451721	+	0.892159i	$0.350810\pi$
$660$	−0.561553	−0.0218584
$661$	−22.8078	−0.887119	−0.443560	−	0.896245i	$-0.646285\pi$
$661$	−22.8078	−0.887119	−0.443560	+	0.896245i	$0.646285\pi$
$662$	22.2462	0.864624
$663$	14.2462	0.553277
$664$	10.2462	0.397630
$665$	11.5076	0.446245
$666$	−6.00000	−0.232495
$667$	0	0
$668$	−2.24621	−0.0869085
$669$	−4.00000	−0.154649
$670$	2.24621	0.0867787
$671$	1.00000	0.0386046
$672$	5.12311	0.197628
$673$	−23.9309	−0.922467	−0.461234	−	0.887279i	$-0.652593\pi$
$673$	−23.9309	−0.922467	−0.461234	+	0.887279i	$0.652593\pi$
$674$	18.3153	0.705481
$675$	4.68466	0.180313
$676$	7.80776	0.300299
$677$	26.4924	1.01819	0.509093	−	0.860711i	$-0.329981\pi$
$677$	26.4924	1.01819	0.509093	+	0.860711i	$0.329981\pi$
$678$	−20.2462	−0.777551
$679$	−29.1231	−1.11764
$680$	−1.75379	−0.0672547
$681$	14.2462	0.545916
$682$	7.68466	0.294261
$683$	25.1231	0.961309	0.480654	−	0.876910i	$-0.340399\pi$
$683$	25.1231	0.961309	0.480654	+	0.876910i	$0.340399\pi$
$684$	−4.00000	−0.152944
$685$	−3.36932	−0.128735
$686$	−62.7386	−2.39537
$687$	−11.1231	−0.424373
$688$	10.2462	0.390633
$689$	14.2462	0.542737
$690$	0	0
$691$	26.5616	1.01045	0.505225	−	0.862988i	$-0.331410\pi$
$691$	26.5616	1.01045	0.505225	+	0.862988i	$0.331410\pi$
$692$	10.3153	0.392130
$693$	−5.12311	−0.194611
$694$	−19.6847	−0.747219
$695$	9.43845	0.358021
$696$	−4.56155	−0.172905
$697$	−23.2311	−0.879939
$698$	−2.63068	−0.0995728
$699$	−2.63068	−0.0995016
$700$	24.0000	0.907115
$701$	0.0691303	0.00261102	0.00130551	−	0.999999i	$-0.499584\pi$
$701$	0.0691303	0.00261102	0.00130551	+	0.999999i	$0.499584\pi$
$702$	−4.56155	−0.172165
$703$	24.0000	0.905177
$704$	1.00000	0.0376889
$705$	3.50758	0.132103
$706$	−30.0000	−1.12906
$707$	−64.3542	−2.42029
$708$	2.56155	0.0962690
$709$	−7.43845	−0.279357	−0.139678	−	0.990197i	$-0.544607\pi$
$709$	−7.43845	−0.279357	−0.139678	+	0.990197i	$0.544607\pi$
$710$	−2.87689	−0.107968
$711$	0	0
$712$	−6.80776	−0.255132
$713$	0	0
$714$	−16.0000	−0.598785
$715$	2.56155	0.0957966
$716$	1.12311	0.0419724
$717$	26.2462	0.980183
$718$	−16.8078	−0.627260
$719$	38.2462	1.42634	0.713171	−	0.700990i	$-0.247258\pi$
$719$	38.2462	1.42634	0.713171	+	0.700990i	$0.247258\pi$
$720$	0.561553	0.0209278
$721$	−81.9697	−3.05271
$722$	−3.00000	−0.111648
$723$	−9.36932	−0.348449
$724$	18.8078	0.698985
$725$	−21.3693	−0.793637
$726$	−1.00000	−0.0371135
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	−23.3693	−0.866125
$729$	1.00000	0.0370370
$730$	−1.75379	−0.0649106
$731$	−32.0000	−1.18356
$732$	−1.00000	−0.0369611
$733$	−16.7386	−0.618256	−0.309128	−	0.951021i	$-0.600037\pi$
$733$	−16.7386	−0.618256	−0.309128	+	0.951021i	$0.600037\pi$
$734$	−15.3693	−0.567292
$735$	−10.8078	−0.398650
$736$	0	0
$737$	4.00000	0.147342
$738$	7.43845	0.273813
$739$	−34.4233	−1.26628	−0.633141	−	0.774037i	$-0.718235\pi$
$739$	−34.4233	−1.26628	−0.633141	+	0.774037i	$0.718235\pi$
$740$	−3.36932	−0.123859
$741$	18.2462	0.670291
$742$	−16.0000	−0.587378
$743$	−22.5616	−0.827703	−0.413852	−	0.910344i	$-0.635817\pi$
$743$	−22.5616	−0.827703	−0.413852	+	0.910344i	$0.635817\pi$
$744$	−7.68466	−0.281733
$745$	6.24621	0.228843
$746$	−29.8617	−1.09332
$747$	10.2462	0.374889
$748$	−3.12311	−0.114192
$749$	−7.36932	−0.269269
$750$	5.43845	0.198584
$751$	25.6155	0.934724	0.467362	−	0.884066i	$-0.345204\pi$
$751$	25.6155	0.934724	0.467362	+	0.884066i	$0.345204\pi$
$752$	−6.24621	−0.227776
$753$	−21.4384	−0.781260
$754$	20.8078	0.757774
$755$	−8.98485	−0.326992
$756$	5.12311	0.186326
$757$	−33.3693	−1.21283	−0.606414	−	0.795149i	$-0.707393\pi$
$757$	−33.3693	−1.21283	−0.606414	+	0.795149i	$0.707393\pi$
$758$	11.1922	0.406520
$759$	0	0
$760$	−2.24621	−0.0814786
$761$	−14.6307	−0.530362	−0.265181	−	0.964199i	$-0.585432\pi$
$761$	−14.6307	−0.530362	−0.265181	+	0.964199i	$0.585432\pi$
$762$	12.4924	0.452553
$763$	−90.6004	−3.27995
$764$	13.1231	0.474777
$765$	−1.75379	−0.0634084
$766$	17.6155	0.636475
$767$	−11.6847	−0.421909
$768$	−1.00000	−0.0360844
$769$	−23.3002	−0.840226	−0.420113	−	0.907472i	$-0.638010\pi$
$769$	−23.3002	−0.840226	−0.420113	+	0.907472i	$0.638010\pi$
$770$	−2.87689	−0.103676
$771$	−6.49242	−0.233819
$772$	6.80776	0.245017
$773$	−0.0691303	−0.00248644	−0.00124322	−	0.999999i	$-0.500396\pi$
$773$	−0.0691303	−0.00248644	−0.00124322	+	0.999999i	$0.500396\pi$
$774$	10.2462	0.368292
$775$	−36.0000	−1.29316
$776$	5.68466	0.204067
$777$	−30.7386	−1.10274
$778$	25.8617	0.927188
$779$	−29.7538	−1.06604
$780$	−2.56155	−0.0917183
$781$	−5.12311	−0.183319
$782$	0	0
$783$	−4.56155	−0.163017
$784$	19.2462	0.687365
$785$	3.54640	0.126576
$786$	8.80776	0.314163
$787$	16.8078	0.599132	0.299566	−	0.954076i	$-0.403158\pi$
$787$	16.8078	0.599132	0.299566	+	0.954076i	$0.403158\pi$
$788$	−12.2462	−0.436253
$789$	13.1231	0.467195
$790$	0	0
$791$	−103.723	−3.68798
$792$	1.00000	0.0355335
$793$	4.56155	0.161985
$794$	1.19224	0.0423109
$795$	−1.75379	−0.0622005
$796$	−7.36932	−0.261199
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	−20.4924	−0.725424
$799$	19.5076	0.690128
$800$	−4.68466	−0.165628
$801$	−6.80776	−0.240541
$802$	29.0540	1.02593
$803$	−3.12311	−0.110212
$804$	−4.00000	−0.141069
$805$	0	0
$806$	35.0540	1.23472
$807$	−29.0540	−1.02275
$808$	12.5616	0.441914
$809$	−44.4233	−1.56184	−0.780920	−	0.624631i	$-0.785249\pi$
$809$	−44.4233	−1.56184	−0.780920	+	0.624631i	$0.785249\pi$
$810$	0.561553	0.0197310
$811$	1.26137	0.0442926	0.0221463	−	0.999755i	$-0.492950\pi$
$811$	1.26137	0.0442926	0.0221463	+	0.999755i	$0.492950\pi$
$812$	−23.3693	−0.820102
$813$	10.2462	0.359350
$814$	−6.00000	−0.210300
$815$	−0.177081	−0.00620287
$816$	3.12311	0.109331
$817$	−40.9848	−1.43388
$818$	−26.1771	−0.915260
$819$	−23.3693	−0.816590
$820$	4.17708	0.145870
$821$	−24.7386	−0.863384	−0.431692	−	0.902021i	$-0.642083\pi$
$821$	−24.7386	−0.863384	−0.431692	+	0.902021i	$0.642083\pi$
$822$	6.00000	0.209274
$823$	−11.1922	−0.390137	−0.195068	−	0.980790i	$-0.562493\pi$
$823$	−11.1922	−0.390137	−0.195068	+	0.980790i	$0.562493\pi$
$824$	16.0000	0.557386
$825$	4.68466	0.163099
$826$	13.1231	0.456611
$827$	26.2462	0.912670	0.456335	−	0.889808i	$-0.349162\pi$
$827$	26.2462	0.912670	0.456335	+	0.889808i	$0.349162\pi$
$828$	0	0
$829$	−9.36932	−0.325410	−0.162705	−	0.986675i	$-0.552022\pi$
$829$	−9.36932	−0.325410	−0.162705	+	0.986675i	$0.552022\pi$
$830$	5.75379	0.199717
$831$	2.00000	0.0693792
$832$	4.56155	0.158143
$833$	−60.1080	−2.08262
$834$	−16.8078	−0.582005
$835$	−1.26137	−0.0436514
$836$	−4.00000	−0.138343
$837$	−7.68466	−0.265621
$838$	12.0000	0.414533
$839$	−11.1922	−0.386399	−0.193199	−	0.981160i	$-0.561886\pi$
$839$	−11.1922	−0.386399	−0.193199	+	0.981160i	$0.561886\pi$
$840$	2.87689	0.0992623
$841$	−8.19224	−0.282491
$842$	−1.68466	−0.0580572
$843$	−9.36932	−0.322696
$844$	9.43845	0.324885
$845$	4.38447	0.150830
$846$	−6.24621	−0.214749
$847$	−5.12311	−0.176032
$848$	3.12311	0.107248
$849$	12.0000	0.411839
$850$	14.6307	0.501828
$851$	0	0
$852$	5.12311	0.175515
$853$	5.19224	0.177779	0.0888894	−	0.996042i	$-0.471668\pi$
$853$	5.19224	0.177779	0.0888894	+	0.996042i	$0.471668\pi$
$854$	−5.12311	−0.175309
$855$	−2.24621	−0.0768188
$856$	1.43845	0.0491651
$857$	17.0540	0.582553	0.291276	−	0.956639i	$-0.405920\pi$
$857$	17.0540	0.582553	0.291276	+	0.956639i	$0.405920\pi$
$858$	−4.56155	−0.155729
$859$	0.946025	0.0322779	0.0161390	−	0.999870i	$-0.494863\pi$
$859$	0.946025	0.0322779	0.0161390	+	0.999870i	$0.494863\pi$
$860$	5.75379	0.196203
$861$	38.1080	1.29872
$862$	−26.8769	−0.915431
$863$	−55.6847	−1.89553	−0.947764	−	0.318973i	$-0.896662\pi$
$863$	−55.6847	−1.89553	−0.947764	+	0.318973i	$0.896662\pi$
$864$	−1.00000	−0.0340207
$865$	5.79261	0.196955
$866$	−27.1231	−0.921681
$867$	7.24621	0.246094
$868$	−39.3693	−1.33628
$869$	0	0
$870$	−2.56155	−0.0868448
$871$	18.2462	0.618249
$872$	17.6847	0.598878
$873$	5.68466	0.192397
$874$	0	0
$875$	27.8617	0.941899
$876$	3.12311	0.105520
$877$	−33.3693	−1.12680	−0.563401	−	0.826184i	$-0.690507\pi$
$877$	−33.3693	−1.12680	−0.563401	+	0.826184i	$0.690507\pi$
$878$	−29.3002	−0.988833
$879$	−24.8769	−0.839077
$880$	0.561553	0.0189299
$881$	33.3693	1.12424	0.562120	−	0.827055i	$-0.309986\pi$
$881$	33.3693	1.12424	0.562120	+	0.827055i	$0.309986\pi$
$882$	19.2462	0.648054
$883$	−46.2462	−1.55631	−0.778154	−	0.628073i	$-0.783844\pi$
$883$	−46.2462	−1.55631	−0.778154	+	0.628073i	$0.783844\pi$
$884$	−14.2462	−0.479152
$885$	1.43845	0.0483529
$886$	−9.12311	−0.306497
$887$	−40.9848	−1.37614	−0.688068	−	0.725646i	$-0.741541\pi$
$887$	−40.9848	−1.37614	−0.688068	+	0.725646i	$0.741541\pi$
$888$	6.00000	0.201347
$889$	64.0000	2.14649
$890$	−3.82292	−0.128145
$891$	1.00000	0.0335013
$892$	4.00000	0.133930
$893$	24.9848	0.836086
$894$	−11.1231	−0.372012
$895$	0.630683	0.0210814
$896$	−5.12311	−0.171151
$897$	0	0
$898$	−14.6307	−0.488232
$899$	35.0540	1.16912
$900$	−4.68466	−0.156155
$901$	−9.75379	−0.324946
$902$	7.43845	0.247673
$903$	52.4924	1.74684
$904$	20.2462	0.673379
$905$	10.5616	0.351078
$906$	16.0000	0.531564
$907$	26.7386	0.887842	0.443921	−	0.896066i	$-0.353587\pi$
$907$	26.7386	0.887842	0.443921	+	0.896066i	$0.353587\pi$
$908$	−14.2462	−0.472777
$909$	12.5616	0.416640
$910$	−13.1231	−0.435027
$911$	7.68466	0.254604	0.127302	−	0.991864i	$-0.459368\pi$
$911$	7.68466	0.254604	0.127302	+	0.991864i	$0.459368\pi$
$912$	4.00000	0.132453
$913$	10.2462	0.339100
$914$	−24.7386	−0.818281
$915$	−0.561553	−0.0185644
$916$	11.1231	0.367518
$917$	45.1231	1.49010
$918$	3.12311	0.103078
$919$	−37.9309	−1.25122	−0.625612	−	0.780134i	$-0.715151\pi$
$919$	−37.9309	−1.25122	−0.625612	+	0.780134i	$0.715151\pi$
$920$	0	0
$921$	32.1771	1.06027
$922$	1.50758	0.0496494
$923$	−23.3693	−0.769210
$924$	5.12311	0.168538
$925$	28.1080	0.924184
$926$	−0.630683	−0.0207255
$927$	16.0000	0.525509
$928$	4.56155	0.149740
$929$	−52.1080	−1.70961	−0.854803	−	0.518952i	$-0.826322\pi$
$929$	−52.1080	−1.70961	−0.854803	+	0.518952i	$0.826322\pi$
$930$	−4.31534	−0.141506
$931$	−76.9848	−2.52308
$932$	2.63068	0.0861709
$933$	−2.24621	−0.0735377
$934$	31.6847	1.03675
$935$	−1.75379	−0.0573550
$936$	4.56155	0.149099
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	−20.4924	−0.669101
$939$	0.876894	0.0286164
$940$	−3.50758	−0.114405
$941$	10.3153	0.336271	0.168135	−	0.985764i	$-0.446225\pi$
$941$	10.3153	0.336271	0.168135	+	0.985764i	$0.446225\pi$
$942$	−6.31534	−0.205765
$943$	0	0
$944$	−2.56155	−0.0833714
$945$	2.87689	0.0935854
$946$	10.2462	0.333133
$947$	31.2311	1.01487	0.507436	−	0.861689i	$-0.330593\pi$
$947$	31.2311	1.01487	0.507436	+	0.861689i	$0.330593\pi$
$948$	0	0
$949$	−14.2462	−0.462452
$950$	18.7386	0.607962
$951$	1.68466	0.0546288
$952$	16.0000	0.518563
$953$	28.2462	0.914985	0.457492	−	0.889214i	$-0.348748\pi$
$953$	28.2462	0.914985	0.457492	+	0.889214i	$0.348748\pi$
$954$	3.12311	0.101114
$955$	7.36932	0.238465
$956$	−26.2462	−0.848863
$957$	−4.56155	−0.147454
$958$	2.24621	0.0725718
$959$	30.7386	0.992602
$960$	−0.561553	−0.0181240
$961$	28.0540	0.904967
$962$	−27.3693	−0.882422
$963$	1.43845	0.0463533
$964$	9.36932	0.301765
$965$	3.82292	0.123064
$966$	0	0
$967$	−59.0540	−1.89905	−0.949524	−	0.313695i	$-0.898433\pi$
$967$	−59.0540	−1.89905	−0.949524	+	0.313695i	$0.898433\pi$
$968$	1.00000	0.0321412
$969$	−12.4924	−0.401314
$970$	3.19224	0.102497
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	−1.00000	−0.0320750
$973$	−86.1080	−2.76049
$974$	30.7386	0.984929
$975$	21.3693	0.684366
$976$	1.00000	0.0320092
$977$	−46.3542	−1.48300	−0.741501	−	0.670952i	$-0.765886\pi$
$977$	−46.3542	−1.48300	−0.741501	+	0.670952i	$0.765886\pi$
$978$	0.315342	0.0100835
$979$	−6.80776	−0.217577
$980$	10.8078	0.345241
$981$	17.6847	0.564628
$982$	30.7386	0.980909
$983$	−37.1231	−1.18404	−0.592022	−	0.805922i	$-0.701670\pi$
$983$	−37.1231	−1.18404	−0.592022	+	0.805922i	$0.701670\pi$
$984$	−7.43845	−0.237129
$985$	−6.87689	−0.219116
$986$	−14.2462	−0.453692
$987$	−32.0000	−1.01857
$988$	−18.2462	−0.580489
$989$	0	0
$990$	0.561553	0.0178473
$991$	54.1080	1.71880	0.859398	−	0.511307i	$-0.170839\pi$
$991$	54.1080	1.71880	0.859398	+	0.511307i	$0.170839\pi$
$992$	7.68466	0.243988
$993$	−22.2462	−0.705962
$994$	26.2462	0.832479
$995$	−4.13826	−0.131192
$996$	−10.2462	−0.324664
$997$	41.8617	1.32577	0.662887	−	0.748719i	$-0.269331\pi$
$997$	41.8617	1.32577	0.662887	+	0.748719i	$0.269331\pi$
$998$	13.6155	0.430992
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4026.2.a.l.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.l.1.2	✓	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4026.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.561553 q^{10} +1.00000 q^{11} -1.00000 q^{12} +4.56155 q^{13} -5.12311 q^{14} -0.561553 q^{15} +1.00000 q^{16} -3.12311 q^{17} +1.00000 q^{18} -4.00000 q^{19} +0.561553 q^{20} +5.12311 q^{21} +1.00000 q^{22} -1.00000 q^{24} -4.68466 q^{25} +4.56155 q^{26} -1.00000 q^{27} -5.12311 q^{28} +4.56155 q^{29} -0.561553 q^{30} +7.68466 q^{31} +1.00000 q^{32} -1.00000 q^{33} -3.12311 q^{34} -2.87689 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} -4.56155 q^{39} +0.561553 q^{40} +7.43845 q^{41} +5.12311 q^{42} +10.2462 q^{43} +1.00000 q^{44} +0.561553 q^{45} -6.24621 q^{47} -1.00000 q^{48} +19.2462 q^{49} -4.68466 q^{50} +3.12311 q^{51} +4.56155 q^{52} +3.12311 q^{53} -1.00000 q^{54} +0.561553 q^{55} -5.12311 q^{56} +4.00000 q^{57} +4.56155 q^{58} -2.56155 q^{59} -0.561553 q^{60} +1.00000 q^{61} +7.68466 q^{62} -5.12311 q^{63} +1.00000 q^{64} +2.56155 q^{65} -1.00000 q^{66} +4.00000 q^{67} -3.12311 q^{68} -2.87689 q^{70} -5.12311 q^{71} +1.00000 q^{72} -3.12311 q^{73} -6.00000 q^{74} +4.68466 q^{75} -4.00000 q^{76} -5.12311 q^{77} -4.56155 q^{78} +0.561553 q^{80} +1.00000 q^{81} +7.43845 q^{82} +10.2462 q^{83} +5.12311 q^{84} -1.75379 q^{85} +10.2462 q^{86} -4.56155 q^{87} +1.00000 q^{88} -6.80776 q^{89} +0.561553 q^{90} -23.3693 q^{91} -7.68466 q^{93} -6.24621 q^{94} -2.24621 q^{95} -1.00000 q^{96} +5.68466 q^{97} +19.2462 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 3 q^{10} + 2 q^{11} - 2 q^{12} + 5 q^{13} - 2 q^{14} + 3 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 8 q^{19} - 3 q^{20} + 2 q^{21} + 2 q^{22} - 2 q^{24} + 3 q^{25} + 5 q^{26} - 2 q^{27} - 2 q^{28} + 5 q^{29} + 3 q^{30} + 3 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{34} - 14 q^{35} + 2 q^{36} - 12 q^{37} - 8 q^{38} - 5 q^{39} - 3 q^{40} + 19 q^{41} + 2 q^{42} + 4 q^{43} + 2 q^{44} - 3 q^{45} + 4 q^{47} - 2 q^{48} + 22 q^{49} + 3 q^{50} - 2 q^{51} + 5 q^{52} - 2 q^{53} - 2 q^{54} - 3 q^{55} - 2 q^{56} + 8 q^{57} + 5 q^{58} - q^{59} + 3 q^{60} + 2 q^{61} + 3 q^{62} - 2 q^{63} + 2 q^{64} + q^{65} - 2 q^{66} + 8 q^{67} + 2 q^{68} - 14 q^{70} - 2 q^{71} + 2 q^{72} + 2 q^{73} - 12 q^{74} - 3 q^{75} - 8 q^{76} - 2 q^{77} - 5 q^{78} - 3 q^{80} + 2 q^{81} + 19 q^{82} + 4 q^{83} + 2 q^{84} - 20 q^{85} + 4 q^{86} - 5 q^{87} + 2 q^{88} + 7 q^{89} - 3 q^{90} - 22 q^{91} - 3 q^{93} + 4 q^{94} + 12 q^{95} - 2 q^{96} - q^{97} + 22 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 - 3 * q^10 + 2 * q^11 - 2 * q^12 + 5 * q^13 - 2 * q^14 + 3 * q^15 + 2 * q^16 + 2 * q^17 + 2 * q^18 - 8 * q^19 - 3 * q^20 + 2 * q^21 + 2 * q^22 - 2 * q^24 + 3 * q^25 + 5 * q^26 - 2 * q^27 - 2 * q^28 + 5 * q^29 + 3 * q^30 + 3 * q^31 + 2 * q^32 - 2 * q^33 + 2 * q^34 - 14 * q^35 + 2 * q^36 - 12 * q^37 - 8 * q^38 - 5 * q^39 - 3 * q^40 + 19 * q^41 + 2 * q^42 + 4 * q^43 + 2 * q^44 - 3 * q^45 + 4 * q^47 - 2 * q^48 + 22 * q^49 + 3 * q^50 - 2 * q^51 + 5 * q^52 - 2 * q^53 - 2 * q^54 - 3 * q^55 - 2 * q^56 + 8 * q^57 + 5 * q^58 - q^59 + 3 * q^60 + 2 * q^61 + 3 * q^62 - 2 * q^63 + 2 * q^64 + q^65 - 2 * q^66 + 8 * q^67 + 2 * q^68 - 14 * q^70 - 2 * q^71 + 2 * q^72 + 2 * q^73 - 12 * q^74 - 3 * q^75 - 8 * q^76 - 2 * q^77 - 5 * q^78 - 3 * q^80 + 2 * q^81 + 19 * q^82 + 4 * q^83 + 2 * q^84 - 20 * q^85 + 4 * q^86 - 5 * q^87 + 2 * q^88 + 7 * q^89 - 3 * q^90 - 22 * q^91 - 3 * q^93 + 4 * q^94 + 12 * q^95 - 2 * q^96 - q^97 + 22 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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