LMFDB - Embedded newform 1122.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1122,2,Mod(1,1122)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1122, 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("1122.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1122 = 2 \cdot 3 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1122.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:	$8.95921510679$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1122.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.82843 q^{5} -1.00000 q^{6} -4.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.82843 q^{5} -1.00000 q^{6} -4.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.82843 q^{10} -1.00000 q^{11} +1.00000 q^{12} -6.82843 q^{13} +4.82843 q^{14} +2.82843 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +2.82843 q^{20} -4.82843 q^{21} +1.00000 q^{22} -8.82843 q^{23} -1.00000 q^{24} +3.00000 q^{25} +6.82843 q^{26} +1.00000 q^{27} -4.82843 q^{28} -6.00000 q^{29} -2.82843 q^{30} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} -13.6569 q^{35} +1.00000 q^{36} -3.65685 q^{37} -6.82843 q^{39} -2.82843 q^{40} -2.00000 q^{41} +4.82843 q^{42} +9.65685 q^{43} -1.00000 q^{44} +2.82843 q^{45} +8.82843 q^{46} +6.48528 q^{47} +1.00000 q^{48} +16.3137 q^{49} -3.00000 q^{50} +1.00000 q^{51} -6.82843 q^{52} -5.17157 q^{53} -1.00000 q^{54} -2.82843 q^{55} +4.82843 q^{56} +6.00000 q^{58} +4.00000 q^{59} +2.82843 q^{60} +14.1421 q^{61} +4.00000 q^{62} -4.82843 q^{63} +1.00000 q^{64} -19.3137 q^{65} +1.00000 q^{66} -7.31371 q^{67} +1.00000 q^{68} -8.82843 q^{69} +13.6569 q^{70} -8.82843 q^{71} -1.00000 q^{72} +9.31371 q^{73} +3.65685 q^{74} +3.00000 q^{75} +4.82843 q^{77} +6.82843 q^{78} -14.4853 q^{79} +2.82843 q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} -4.82843 q^{84} +2.82843 q^{85} -9.65685 q^{86} -6.00000 q^{87} +1.00000 q^{88} +10.0000 q^{89} -2.82843 q^{90} +32.9706 q^{91} -8.82843 q^{92} -4.00000 q^{93} -6.48528 q^{94} -1.00000 q^{96} +9.31371 q^{97} -16.3137 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{11} + 2 q^{12} - 8 q^{13} + 4 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 4 q^{21} + 2 q^{22} - 12 q^{23} - 2 q^{24} + 6 q^{25} + 8 q^{26} + 2 q^{27} - 4 q^{28} - 12 q^{29} - 8 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} - 16 q^{35} + 2 q^{36} + 4 q^{37} - 8 q^{39} - 4 q^{41} + 4 q^{42} + 8 q^{43} - 2 q^{44} + 12 q^{46} - 4 q^{47} + 2 q^{48} + 10 q^{49} - 6 q^{50} + 2 q^{51} - 8 q^{52} - 16 q^{53} - 2 q^{54} + 4 q^{56} + 12 q^{58} + 8 q^{59} + 8 q^{62} - 4 q^{63} + 2 q^{64} - 16 q^{65} + 2 q^{66} + 8 q^{67} + 2 q^{68} - 12 q^{69} + 16 q^{70} - 12 q^{71} - 2 q^{72} - 4 q^{73} - 4 q^{74} + 6 q^{75} + 4 q^{77} + 8 q^{78} - 12 q^{79} + 2 q^{81} + 4 q^{82} - 8 q^{83} - 4 q^{84} - 8 q^{86} - 12 q^{87} + 2 q^{88} + 20 q^{89} + 32 q^{91} - 12 q^{92} - 8 q^{93} + 4 q^{94} - 2 q^{96} - 4 q^{97} - 10 q^{98} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^11 + 2 * q^12 - 8 * q^13 + 4 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^18 - 4 * q^21 + 2 * q^22 - 12 * q^23 - 2 * q^24 + 6 * q^25 + 8 * q^26 + 2 * q^27 - 4 * q^28 - 12 * q^29 - 8 * q^31 - 2 * q^32 - 2 * q^33 - 2 * q^34 - 16 * q^35 + 2 * q^36 + 4 * q^37 - 8 * q^39 - 4 * q^41 + 4 * q^42 + 8 * q^43 - 2 * q^44 + 12 * q^46 - 4 * q^47 + 2 * q^48 + 10 * q^49 - 6 * q^50 + 2 * q^51 - 8 * q^52 - 16 * q^53 - 2 * q^54 + 4 * q^56 + 12 * q^58 + 8 * q^59 + 8 * q^62 - 4 * q^63 + 2 * q^64 - 16 * q^65 + 2 * q^66 + 8 * q^67 + 2 * q^68 - 12 * q^69 + 16 * q^70 - 12 * q^71 - 2 * q^72 - 4 * q^73 - 4 * q^74 + 6 * q^75 + 4 * q^77 + 8 * q^78 - 12 * q^79 + 2 * q^81 + 4 * q^82 - 8 * q^83 - 4 * q^84 - 8 * q^86 - 12 * q^87 + 2 * q^88 + 20 * q^89 + 32 * q^91 - 12 * q^92 - 8 * q^93 + 4 * q^94 - 2 * q^96 - 4 * q^97 - 10 * 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
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	−1.00000	−0.408248
$7$	−4.82843	−1.82497	−0.912487	−	0.409106i	$-0.865841\pi$
$7$	−4.82843	−1.82497	−0.912487	+	0.409106i	$0.865841\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.82843	−0.894427
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	−6.82843	−1.89386	−0.946932	−	0.321433i	$-0.895836\pi$
$13$	−6.82843	−1.89386	−0.946932	+	0.321433i	$0.895836\pi$
$14$	4.82843	1.29045
$15$	2.82843	0.730297
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	2.82843	0.632456
$21$	−4.82843	−1.05365
$22$	1.00000	0.213201
$23$	−8.82843	−1.84085	−0.920427	−	0.390914i	$-0.872159\pi$
$23$	−8.82843	−1.84085	−0.920427	+	0.390914i	$0.872159\pi$
$24$	−1.00000	−0.204124
$25$	3.00000	0.600000
$26$	6.82843	1.33916
$27$	1.00000	0.192450
$28$	−4.82843	−0.912487
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−2.82843	−0.516398
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−1.00000	−0.171499
$35$	−13.6569	−2.30843
$36$	1.00000	0.166667
$37$	−3.65685	−0.601183	−0.300592	−	0.953753i	$-0.597184\pi$
$37$	−3.65685	−0.601183	−0.300592	+	0.953753i	$0.597184\pi$
$38$	0	0
$39$	−6.82843	−1.09342
$40$	−2.82843	−0.447214
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	4.82843	0.745042
$43$	9.65685	1.47266	0.736328	−	0.676625i	$-0.236558\pi$
$43$	9.65685	1.47266	0.736328	+	0.676625i	$0.236558\pi$
$44$	−1.00000	−0.150756
$45$	2.82843	0.421637
$46$	8.82843	1.30168
$47$	6.48528	0.945976	0.472988	−	0.881069i	$-0.343175\pi$
$47$	6.48528	0.945976	0.472988	+	0.881069i	$0.343175\pi$
$48$	1.00000	0.144338
$49$	16.3137	2.33053
$50$	−3.00000	−0.424264
$51$	1.00000	0.140028
$52$	−6.82843	−0.946932
$53$	−5.17157	−0.710370	−0.355185	−	0.934796i	$-0.615582\pi$
$53$	−5.17157	−0.710370	−0.355185	+	0.934796i	$0.615582\pi$
$54$	−1.00000	−0.136083
$55$	−2.82843	−0.381385
$56$	4.82843	0.645226
$57$	0	0
$58$	6.00000	0.787839
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	2.82843	0.365148
$61$	14.1421	1.81071	0.905357	−	0.424650i	$-0.139603\pi$
$61$	14.1421	1.81071	0.905357	+	0.424650i	$0.139603\pi$
$62$	4.00000	0.508001
$63$	−4.82843	−0.608325
$64$	1.00000	0.125000
$65$	−19.3137	−2.39557
$66$	1.00000	0.123091
$67$	−7.31371	−0.893512	−0.446756	−	0.894656i	$-0.647421\pi$
$67$	−7.31371	−0.893512	−0.446756	+	0.894656i	$0.647421\pi$
$68$	1.00000	0.121268
$69$	−8.82843	−1.06282
$70$	13.6569	1.63231
$71$	−8.82843	−1.04774	−0.523871	−	0.851798i	$-0.675513\pi$
$71$	−8.82843	−1.04774	−0.523871	+	0.851798i	$0.675513\pi$
$72$	−1.00000	−0.117851
$73$	9.31371	1.09009	0.545044	−	0.838408i	$-0.316513\pi$
$73$	9.31371	1.09009	0.545044	+	0.838408i	$0.316513\pi$
$74$	3.65685	0.425101
$75$	3.00000	0.346410
$76$	0	0
$77$	4.82843	0.550250
$78$	6.82843	0.773167
$79$	−14.4853	−1.62972	−0.814861	−	0.579657i	$-0.803187\pi$
$79$	−14.4853	−1.62972	−0.814861	+	0.579657i	$0.803187\pi$
$80$	2.82843	0.316228
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−4.82843	−0.526825
$85$	2.82843	0.306786
$86$	−9.65685	−1.04133
$87$	−6.00000	−0.643268
$88$	1.00000	0.106600
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	−2.82843	−0.298142
$91$	32.9706	3.45625
$92$	−8.82843	−0.920427
$93$	−4.00000	−0.414781
$94$	−6.48528	−0.668906
$95$	0	0
$96$	−1.00000	−0.102062
$97$	9.31371	0.945664	0.472832	−	0.881153i	$-0.343232\pi$
$97$	9.31371	0.945664	0.472832	+	0.881153i	$0.343232\pi$
$98$	−16.3137	−1.64793
$99$	−1.00000	−0.100504
$100$	3.00000	0.300000
$101$	11.6569	1.15990	0.579950	−	0.814652i	$-0.303072\pi$
$101$	11.6569	1.15990	0.579950	+	0.814652i	$0.303072\pi$
$102$	−1.00000	−0.0990148
$103$	−9.65685	−0.951518	−0.475759	−	0.879576i	$-0.657827\pi$
$103$	−9.65685	−0.951518	−0.475759	+	0.879576i	$0.657827\pi$
$104$	6.82843	0.669582
$105$	−13.6569	−1.33277
$106$	5.17157	0.502308
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	1.00000	0.0962250
$109$	9.17157	0.878477	0.439239	−	0.898370i	$-0.355248\pi$
$109$	9.17157	0.878477	0.439239	+	0.898370i	$0.355248\pi$
$110$	2.82843	0.269680
$111$	−3.65685	−0.347093
$112$	−4.82843	−0.456243
$113$	−13.3137	−1.25245	−0.626224	−	0.779643i	$-0.715401\pi$
$113$	−13.3137	−1.25245	−0.626224	+	0.779643i	$0.715401\pi$
$114$	0	0
$115$	−24.9706	−2.32852
$116$	−6.00000	−0.557086
$117$	−6.82843	−0.631288
$118$	−4.00000	−0.368230
$119$	−4.82843	−0.442621
$120$	−2.82843	−0.258199
$121$	1.00000	0.0909091
$122$	−14.1421	−1.28037
$123$	−2.00000	−0.180334
$124$	−4.00000	−0.359211
$125$	−5.65685	−0.505964
$126$	4.82843	0.430150
$127$	−16.8284	−1.49328	−0.746641	−	0.665228i	$-0.768335\pi$
$127$	−16.8284	−1.49328	−0.746641	+	0.665228i	$0.768335\pi$
$128$	−1.00000	−0.0883883
$129$	9.65685	0.850239
$130$	19.3137	1.69392
$131$	−9.65685	−0.843723	−0.421862	−	0.906660i	$-0.638623\pi$
$131$	−9.65685	−0.843723	−0.421862	+	0.906660i	$0.638623\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	7.31371	0.631808
$135$	2.82843	0.243432
$136$	−1.00000	−0.0857493
$137$	−4.34315	−0.371060	−0.185530	−	0.982639i	$-0.559400\pi$
$137$	−4.34315	−0.371060	−0.185530	+	0.982639i	$0.559400\pi$
$138$	8.82843	0.751526
$139$	−6.34315	−0.538019	−0.269009	−	0.963138i	$-0.586696\pi$
$139$	−6.34315	−0.538019	−0.269009	+	0.963138i	$0.586696\pi$
$140$	−13.6569	−1.15421
$141$	6.48528	0.546159
$142$	8.82843	0.740865
$143$	6.82843	0.571022
$144$	1.00000	0.0833333
$145$	−16.9706	−1.40933
$146$	−9.31371	−0.770808
$147$	16.3137	1.34553
$148$	−3.65685	−0.300592
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	−3.00000	−0.244949
$151$	10.4853	0.853280	0.426640	−	0.904422i	$-0.359697\pi$
$151$	10.4853	0.853280	0.426640	+	0.904422i	$0.359697\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	−4.82843	−0.389086
$155$	−11.3137	−0.908739
$156$	−6.82843	−0.546712
$157$	−11.6569	−0.930318	−0.465159	−	0.885227i	$-0.654003\pi$
$157$	−11.6569	−0.930318	−0.465159	+	0.885227i	$0.654003\pi$
$158$	14.4853	1.15239
$159$	−5.17157	−0.410132
$160$	−2.82843	−0.223607
$161$	42.6274	3.35951
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−2.00000	−0.156174
$165$	−2.82843	−0.220193
$166$	4.00000	0.310460
$167$	−2.34315	−0.181318	−0.0906590	−	0.995882i	$-0.528897\pi$
$167$	−2.34315	−0.181318	−0.0906590	+	0.995882i	$0.528897\pi$
$168$	4.82843	0.372521
$169$	33.6274	2.58672
$170$	−2.82843	−0.216930
$171$	0	0
$172$	9.65685	0.736328
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	6.00000	0.454859
$175$	−14.4853	−1.09498
$176$	−1.00000	−0.0753778
$177$	4.00000	0.300658
$178$	−10.0000	−0.749532
$179$	−7.31371	−0.546652	−0.273326	−	0.961921i	$-0.588124\pi$
$179$	−7.31371	−0.546652	−0.273326	+	0.961921i	$0.588124\pi$
$180$	2.82843	0.210819
$181$	−13.3137	−0.989600	−0.494800	−	0.869007i	$-0.664759\pi$
$181$	−13.3137	−0.989600	−0.494800	+	0.869007i	$0.664759\pi$
$182$	−32.9706	−2.44394
$183$	14.1421	1.04542
$184$	8.82843	0.650840
$185$	−10.3431	−0.760443
$186$	4.00000	0.293294
$187$	−1.00000	−0.0731272
$188$	6.48528	0.472988
$189$	−4.82843	−0.351216
$190$	0	0
$191$	−16.1421	−1.16800	−0.584002	−	0.811752i	$-0.698514\pi$
$191$	−16.1421	−1.16800	−0.584002	+	0.811752i	$0.698514\pi$
$192$	1.00000	0.0721688
$193$	−11.6569	−0.839079	−0.419539	−	0.907737i	$-0.637808\pi$
$193$	−11.6569	−0.839079	−0.419539	+	0.907737i	$0.637808\pi$
$194$	−9.31371	−0.668685
$195$	−19.3137	−1.38308
$196$	16.3137	1.16526
$197$	13.3137	0.948562	0.474281	−	0.880373i	$-0.342708\pi$
$197$	13.3137	0.948562	0.474281	+	0.880373i	$0.342708\pi$
$198$	1.00000	0.0710669
$199$	24.9706	1.77012	0.885058	−	0.465481i	$-0.154118\pi$
$199$	24.9706	1.77012	0.885058	+	0.465481i	$0.154118\pi$
$200$	−3.00000	−0.212132
$201$	−7.31371	−0.515869
$202$	−11.6569	−0.820173
$203$	28.9706	2.03333
$204$	1.00000	0.0700140
$205$	−5.65685	−0.395092
$206$	9.65685	0.672825
$207$	−8.82843	−0.613618
$208$	−6.82843	−0.473466
$209$	0	0
$210$	13.6569	0.942412
$211$	−11.3137	−0.778868	−0.389434	−	0.921054i	$-0.627329\pi$
$211$	−11.3137	−0.778868	−0.389434	+	0.921054i	$0.627329\pi$
$212$	−5.17157	−0.355185
$213$	−8.82843	−0.604914
$214$	4.00000	0.273434
$215$	27.3137	1.86278
$216$	−1.00000	−0.0680414
$217$	19.3137	1.31110
$218$	−9.17157	−0.621177
$219$	9.31371	0.629362
$220$	−2.82843	−0.190693
$221$	−6.82843	−0.459330
$222$	3.65685	0.245432
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	4.82843	0.322613
$225$	3.00000	0.200000
$226$	13.3137	0.885615
$227$	−15.3137	−1.01641	−0.508203	−	0.861237i	$-0.669690\pi$
$227$	−15.3137	−1.01641	−0.508203	+	0.861237i	$0.669690\pi$
$228$	0	0
$229$	−14.9706	−0.989283	−0.494641	−	0.869097i	$-0.664701\pi$
$229$	−14.9706	−0.989283	−0.494641	+	0.869097i	$0.664701\pi$
$230$	24.9706	1.64651
$231$	4.82843	0.317687
$232$	6.00000	0.393919
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	6.82843	0.446388
$235$	18.3431	1.19657
$236$	4.00000	0.260378
$237$	−14.4853	−0.940920
$238$	4.82843	0.312980
$239$	−6.34315	−0.410304	−0.205152	−	0.978730i	$-0.565769\pi$
$239$	−6.34315	−0.410304	−0.205152	+	0.978730i	$0.565769\pi$
$240$	2.82843	0.182574
$241$	7.65685	0.493221	0.246611	−	0.969115i	$-0.420683\pi$
$241$	7.65685	0.493221	0.246611	+	0.969115i	$0.420683\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	14.1421	0.905357
$245$	46.1421	2.94791
$246$	2.00000	0.127515
$247$	0	0
$248$	4.00000	0.254000
$249$	−4.00000	−0.253490
$250$	5.65685	0.357771
$251$	−26.6274	−1.68071	−0.840354	−	0.542038i	$-0.817653\pi$
$251$	−26.6274	−1.68071	−0.840354	+	0.542038i	$0.817653\pi$
$252$	−4.82843	−0.304162
$253$	8.82843	0.555038
$254$	16.8284	1.05591
$255$	2.82843	0.177123
$256$	1.00000	0.0625000
$257$	−20.6274	−1.28670	−0.643351	−	0.765571i	$-0.722457\pi$
$257$	−20.6274	−1.28670	−0.643351	+	0.765571i	$0.722457\pi$
$258$	−9.65685	−0.601209
$259$	17.6569	1.09714
$260$	−19.3137	−1.19779
$261$	−6.00000	−0.371391
$262$	9.65685	0.596602
$263$	19.3137	1.19093	0.595467	−	0.803380i	$-0.296967\pi$
$263$	19.3137	1.19093	0.595467	+	0.803380i	$0.296967\pi$
$264$	1.00000	0.0615457
$265$	−14.6274	−0.898555
$266$	0	0
$267$	10.0000	0.611990
$268$	−7.31371	−0.446756
$269$	−0.485281	−0.0295881	−0.0147941	−	0.999891i	$-0.504709\pi$
$269$	−0.485281	−0.0295881	−0.0147941	+	0.999891i	$0.504709\pi$
$270$	−2.82843	−0.172133
$271$	16.8284	1.02225	0.511127	−	0.859505i	$-0.329228\pi$
$271$	16.8284	1.02225	0.511127	+	0.859505i	$0.329228\pi$
$272$	1.00000	0.0606339
$273$	32.9706	1.99547
$274$	4.34315	0.262379
$275$	−3.00000	−0.180907
$276$	−8.82843	−0.531409
$277$	−5.17157	−0.310730	−0.155365	−	0.987857i	$-0.549655\pi$
$277$	−5.17157	−0.310730	−0.155365	+	0.987857i	$0.549655\pi$
$278$	6.34315	0.380437
$279$	−4.00000	−0.239474
$280$	13.6569	0.816153
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	−6.48528	−0.386193
$283$	11.3137	0.672530	0.336265	−	0.941767i	$-0.390836\pi$
$283$	11.3137	0.672530	0.336265	+	0.941767i	$0.390836\pi$
$284$	−8.82843	−0.523871
$285$	0	0
$286$	−6.82843	−0.403773
$287$	9.65685	0.570026
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	16.9706	0.996546
$291$	9.31371	0.545979
$292$	9.31371	0.545044
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	−16.3137	−0.951435
$295$	11.3137	0.658710
$296$	3.65685	0.212550
$297$	−1.00000	−0.0580259
$298$	−2.00000	−0.115857
$299$	60.2843	3.48633
$300$	3.00000	0.173205
$301$	−46.6274	−2.68756
$302$	−10.4853	−0.603360
$303$	11.6569	0.669669
$304$	0	0
$305$	40.0000	2.29039
$306$	−1.00000	−0.0571662
$307$	−27.3137	−1.55888	−0.779438	−	0.626480i	$-0.784495\pi$
$307$	−27.3137	−1.55888	−0.779438	+	0.626480i	$0.784495\pi$
$308$	4.82843	0.275125
$309$	−9.65685	−0.549359
$310$	11.3137	0.642575
$311$	13.7990	0.782469	0.391234	−	0.920291i	$-0.372048\pi$
$311$	13.7990	0.782469	0.391234	+	0.920291i	$0.372048\pi$
$312$	6.82843	0.386584
$313$	6.68629	0.377932	0.188966	−	0.981984i	$-0.439486\pi$
$313$	6.68629	0.377932	0.188966	+	0.981984i	$0.439486\pi$
$314$	11.6569	0.657834
$315$	−13.6569	−0.769477
$316$	−14.4853	−0.814861
$317$	−26.1421	−1.46829	−0.734144	−	0.678993i	$-0.762416\pi$
$317$	−26.1421	−1.46829	−0.734144	+	0.678993i	$0.762416\pi$
$318$	5.17157	0.290007
$319$	6.00000	0.335936
$320$	2.82843	0.158114
$321$	−4.00000	−0.223258
$322$	−42.6274	−2.37553
$323$	0	0
$324$	1.00000	0.0555556
$325$	−20.4853	−1.13632
$326$	−4.00000	−0.221540
$327$	9.17157	0.507189
$328$	2.00000	0.110432
$329$	−31.3137	−1.72638
$330$	2.82843	0.155700
$331$	25.6569	1.41023	0.705114	−	0.709094i	$-0.250896\pi$
$331$	25.6569	1.41023	0.705114	+	0.709094i	$0.250896\pi$
$332$	−4.00000	−0.219529
$333$	−3.65685	−0.200394
$334$	2.34315	0.128211
$335$	−20.6863	−1.13021
$336$	−4.82843	−0.263412
$337$	−21.3137	−1.16103	−0.580516	−	0.814249i	$-0.697149\pi$
$337$	−21.3137	−1.16103	−0.580516	+	0.814249i	$0.697149\pi$
$338$	−33.6274	−1.82909
$339$	−13.3137	−0.723101
$340$	2.82843	0.153393
$341$	4.00000	0.216612
$342$	0	0
$343$	−44.9706	−2.42818
$344$	−9.65685	−0.520663
$345$	−24.9706	−1.34437
$346$	−2.00000	−0.107521
$347$	−17.6569	−0.947870	−0.473935	−	0.880560i	$-0.657167\pi$
$347$	−17.6569	−0.947870	−0.473935	+	0.880560i	$0.657167\pi$
$348$	−6.00000	−0.321634
$349$	9.17157	0.490943	0.245472	−	0.969404i	$-0.421057\pi$
$349$	9.17157	0.490943	0.245472	+	0.969404i	$0.421057\pi$
$350$	14.4853	0.774271
$351$	−6.82843	−0.364474
$352$	1.00000	0.0533002
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−4.00000	−0.212598
$355$	−24.9706	−1.32530
$356$	10.0000	0.529999
$357$	−4.82843	−0.255547
$358$	7.31371	0.386542
$359$	6.34315	0.334778	0.167389	−	0.985891i	$-0.446466\pi$
$359$	6.34315	0.334778	0.167389	+	0.985891i	$0.446466\pi$
$360$	−2.82843	−0.149071
$361$	−19.0000	−1.00000
$362$	13.3137	0.699753
$363$	1.00000	0.0524864
$364$	32.9706	1.72813
$365$	26.3431	1.37886
$366$	−14.1421	−0.739221
$367$	18.6274	0.972343	0.486172	−	0.873863i	$-0.338393\pi$
$367$	18.6274	0.972343	0.486172	+	0.873863i	$0.338393\pi$
$368$	−8.82843	−0.460214
$369$	−2.00000	−0.104116
$370$	10.3431	0.537715
$371$	24.9706	1.29641
$372$	−4.00000	−0.207390
$373$	23.7990	1.23226	0.616132	−	0.787643i	$-0.288699\pi$
$373$	23.7990	1.23226	0.616132	+	0.787643i	$0.288699\pi$
$374$	1.00000	0.0517088
$375$	−5.65685	−0.292119
$376$	−6.48528	−0.334453
$377$	40.9706	2.11009
$378$	4.82843	0.248347
$379$	−7.31371	−0.375680	−0.187840	−	0.982200i	$-0.560149\pi$
$379$	−7.31371	−0.375680	−0.187840	+	0.982200i	$0.560149\pi$
$380$	0	0
$381$	−16.8284	−0.862146
$382$	16.1421	0.825904
$383$	−14.4853	−0.740163	−0.370082	−	0.928999i	$-0.620670\pi$
$383$	−14.4853	−0.740163	−0.370082	+	0.928999i	$0.620670\pi$
$384$	−1.00000	−0.0510310
$385$	13.6569	0.696018
$386$	11.6569	0.593318
$387$	9.65685	0.490885
$388$	9.31371	0.472832
$389$	10.8284	0.549023	0.274512	−	0.961584i	$-0.411484\pi$
$389$	10.8284	0.549023	0.274512	+	0.961584i	$0.411484\pi$
$390$	19.3137	0.977988
$391$	−8.82843	−0.446473
$392$	−16.3137	−0.823967
$393$	−9.65685	−0.487124
$394$	−13.3137	−0.670735
$395$	−40.9706	−2.06145
$396$	−1.00000	−0.0502519
$397$	1.31371	0.0659331	0.0329666	−	0.999456i	$-0.489505\pi$
$397$	1.31371	0.0659331	0.0329666	+	0.999456i	$0.489505\pi$
$398$	−24.9706	−1.25166
$399$	0	0
$400$	3.00000	0.150000
$401$	−30.9706	−1.54660	−0.773298	−	0.634043i	$-0.781394\pi$
$401$	−30.9706	−1.54660	−0.773298	+	0.634043i	$0.781394\pi$
$402$	7.31371	0.364775
$403$	27.3137	1.36059
$404$	11.6569	0.579950
$405$	2.82843	0.140546
$406$	−28.9706	−1.43778
$407$	3.65685	0.181264
$408$	−1.00000	−0.0495074
$409$	−10.9706	−0.542459	−0.271230	−	0.962515i	$-0.587430\pi$
$409$	−10.9706	−0.542459	−0.271230	+	0.962515i	$0.587430\pi$
$410$	5.65685	0.279372
$411$	−4.34315	−0.214232
$412$	−9.65685	−0.475759
$413$	−19.3137	−0.950365
$414$	8.82843	0.433894
$415$	−11.3137	−0.555368
$416$	6.82843	0.334791
$417$	−6.34315	−0.310625
$418$	0	0
$419$	24.9706	1.21989	0.609946	−	0.792443i	$-0.291191\pi$
$419$	24.9706	1.21989	0.609946	+	0.792443i	$0.291191\pi$
$420$	−13.6569	−0.666386
$421$	−5.31371	−0.258974	−0.129487	−	0.991581i	$-0.541333\pi$
$421$	−5.31371	−0.258974	−0.129487	+	0.991581i	$0.541333\pi$
$422$	11.3137	0.550743
$423$	6.48528	0.315325
$424$	5.17157	0.251154
$425$	3.00000	0.145521
$426$	8.82843	0.427739
$427$	−68.2843	−3.30451
$428$	−4.00000	−0.193347
$429$	6.82843	0.329680
$430$	−27.3137	−1.31718
$431$	23.3137	1.12298	0.561491	−	0.827483i	$-0.310228\pi$
$431$	23.3137	1.12298	0.561491	+	0.827483i	$0.310228\pi$
$432$	1.00000	0.0481125
$433$	5.31371	0.255361	0.127680	−	0.991815i	$-0.459247\pi$
$433$	5.31371	0.255361	0.127680	+	0.991815i	$0.459247\pi$
$434$	−19.3137	−0.927088
$435$	−16.9706	−0.813676
$436$	9.17157	0.439239
$437$	0	0
$438$	−9.31371	−0.445026
$439$	19.4558	0.928577	0.464288	−	0.885684i	$-0.346310\pi$
$439$	19.4558	0.928577	0.464288	+	0.885684i	$0.346310\pi$
$440$	2.82843	0.134840
$441$	16.3137	0.776843
$442$	6.82843	0.324795
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	−3.65685	−0.173547
$445$	28.2843	1.34080
$446$	−24.0000	−1.13643
$447$	2.00000	0.0945968
$448$	−4.82843	−0.228122
$449$	7.65685	0.361349	0.180675	−	0.983543i	$-0.442172\pi$
$449$	7.65685	0.361349	0.180675	+	0.983543i	$0.442172\pi$
$450$	−3.00000	−0.141421
$451$	2.00000	0.0941763
$452$	−13.3137	−0.626224
$453$	10.4853	0.492641
$454$	15.3137	0.718708
$455$	93.2548	4.37185
$456$	0	0
$457$	−28.3431	−1.32584	−0.662918	−	0.748692i	$-0.730682\pi$
$457$	−28.3431	−1.32584	−0.662918	+	0.748692i	$0.730682\pi$
$458$	14.9706	0.699528
$459$	1.00000	0.0466760
$460$	−24.9706	−1.16426
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	−4.82843	−0.224639
$463$	12.9706	0.602793	0.301397	−	0.953499i	$-0.402547\pi$
$463$	12.9706	0.602793	0.301397	+	0.953499i	$0.402547\pi$
$464$	−6.00000	−0.278543
$465$	−11.3137	−0.524661
$466$	−14.0000	−0.648537
$467$	16.9706	0.785304	0.392652	−	0.919687i	$-0.371558\pi$
$467$	16.9706	0.785304	0.392652	+	0.919687i	$0.371558\pi$
$468$	−6.82843	−0.315644
$469$	35.3137	1.63064
$470$	−18.3431	−0.846106
$471$	−11.6569	−0.537119
$472$	−4.00000	−0.184115
$473$	−9.65685	−0.444023
$474$	14.4853	0.665331
$475$	0	0
$476$	−4.82843	−0.221311
$477$	−5.17157	−0.236790
$478$	6.34315	0.290129
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	−2.82843	−0.129099
$481$	24.9706	1.13856
$482$	−7.65685	−0.348760
$483$	42.6274	1.93961
$484$	1.00000	0.0454545
$485$	26.3431	1.19618
$486$	−1.00000	−0.0453609
$487$	40.9706	1.85655	0.928277	−	0.371890i	$-0.121290\pi$
$487$	40.9706	1.85655	0.928277	+	0.371890i	$0.121290\pi$
$488$	−14.1421	−0.640184
$489$	4.00000	0.180886
$490$	−46.1421	−2.08449
$491$	−36.9706	−1.66846	−0.834229	−	0.551418i	$-0.814087\pi$
$491$	−36.9706	−1.66846	−0.834229	+	0.551418i	$0.814087\pi$
$492$	−2.00000	−0.0901670
$493$	−6.00000	−0.270226
$494$	0	0
$495$	−2.82843	−0.127128
$496$	−4.00000	−0.179605
$497$	42.6274	1.91210
$498$	4.00000	0.179244
$499$	0.686292	0.0307226	0.0153613	−	0.999882i	$-0.495110\pi$
$499$	0.686292	0.0307226	0.0153613	+	0.999882i	$0.495110\pi$
$500$	−5.65685	−0.252982
$501$	−2.34315	−0.104684
$502$	26.6274	1.18844
$503$	−24.9706	−1.11338	−0.556691	−	0.830720i	$-0.687929\pi$
$503$	−24.9706	−1.11338	−0.556691	+	0.830720i	$0.687929\pi$
$504$	4.82843	0.215075
$505$	32.9706	1.46717
$506$	−8.82843	−0.392471
$507$	33.6274	1.49345
$508$	−16.8284	−0.746641
$509$	−19.7990	−0.877575	−0.438787	−	0.898591i	$-0.644592\pi$
$509$	−19.7990	−0.877575	−0.438787	+	0.898591i	$0.644592\pi$
$510$	−2.82843	−0.125245
$511$	−44.9706	−1.98938
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	20.6274	0.909836
$515$	−27.3137	−1.20359
$516$	9.65685	0.425119
$517$	−6.48528	−0.285222
$518$	−17.6569	−0.775798
$519$	2.00000	0.0877903
$520$	19.3137	0.846962
$521$	−30.9706	−1.35684	−0.678422	−	0.734672i	$-0.737336\pi$
$521$	−30.9706	−1.35684	−0.678422	+	0.734672i	$0.737336\pi$
$522$	6.00000	0.262613
$523$	16.2843	0.712061	0.356031	−	0.934474i	$-0.384130\pi$
$523$	16.2843	0.712061	0.356031	+	0.934474i	$0.384130\pi$
$524$	−9.65685	−0.421862
$525$	−14.4853	−0.632190
$526$	−19.3137	−0.842118
$527$	−4.00000	−0.174243
$528$	−1.00000	−0.0435194
$529$	54.9411	2.38874
$530$	14.6274	0.635374
$531$	4.00000	0.173585
$532$	0	0
$533$	13.6569	0.591544
$534$	−10.0000	−0.432742
$535$	−11.3137	−0.489134
$536$	7.31371	0.315904
$537$	−7.31371	−0.315610
$538$	0.485281	0.0209220
$539$	−16.3137	−0.702681
$540$	2.82843	0.121716
$541$	1.17157	0.0503699	0.0251849	−	0.999683i	$-0.491983\pi$
$541$	1.17157	0.0503699	0.0251849	+	0.999683i	$0.491983\pi$
$542$	−16.8284	−0.722843
$543$	−13.3137	−0.571346
$544$	−1.00000	−0.0428746
$545$	25.9411	1.11120
$546$	−32.9706	−1.41101
$547$	−33.6569	−1.43906	−0.719532	−	0.694460i	$-0.755643\pi$
$547$	−33.6569	−1.43906	−0.719532	+	0.694460i	$0.755643\pi$
$548$	−4.34315	−0.185530
$549$	14.1421	0.603572
$550$	3.00000	0.127920
$551$	0	0
$552$	8.82843	0.375763
$553$	69.9411	2.97420
$554$	5.17157	0.219719
$555$	−10.3431	−0.439042
$556$	−6.34315	−0.269009
$557$	29.3137	1.24206	0.621031	−	0.783786i	$-0.286714\pi$
$557$	29.3137	1.24206	0.621031	+	0.783786i	$0.286714\pi$
$558$	4.00000	0.169334
$559$	−65.9411	−2.78901
$560$	−13.6569	−0.577107
$561$	−1.00000	−0.0422200
$562$	−2.00000	−0.0843649
$563$	20.9706	0.883804	0.441902	−	0.897063i	$-0.354304\pi$
$563$	20.9706	0.883804	0.441902	+	0.897063i	$0.354304\pi$
$564$	6.48528	0.273080
$565$	−37.6569	−1.58424
$566$	−11.3137	−0.475551
$567$	−4.82843	−0.202775
$568$	8.82843	0.370433
$569$	−12.6274	−0.529369	−0.264684	−	0.964335i	$-0.585268\pi$
$569$	−12.6274	−0.529369	−0.264684	+	0.964335i	$0.585268\pi$
$570$	0	0
$571$	−3.02944	−0.126778	−0.0633890	−	0.997989i	$-0.520191\pi$
$571$	−3.02944	−0.126778	−0.0633890	+	0.997989i	$0.520191\pi$
$572$	6.82843	0.285511
$573$	−16.1421	−0.674347
$574$	−9.65685	−0.403069
$575$	−26.4853	−1.10451
$576$	1.00000	0.0416667
$577$	−5.31371	−0.221213	−0.110606	−	0.993864i	$-0.535279\pi$
$577$	−5.31371	−0.221213	−0.110606	+	0.993864i	$0.535279\pi$
$578$	−1.00000	−0.0415945
$579$	−11.6569	−0.484442
$580$	−16.9706	−0.704664
$581$	19.3137	0.801268
$582$	−9.31371	−0.386066
$583$	5.17157	0.214185
$584$	−9.31371	−0.385404
$585$	−19.3137	−0.798524
$586$	−26.0000	−1.07405
$587$	−10.6274	−0.438640	−0.219320	−	0.975653i	$-0.570384\pi$
$587$	−10.6274	−0.438640	−0.219320	+	0.975653i	$0.570384\pi$
$588$	16.3137	0.672766
$589$	0	0
$590$	−11.3137	−0.465778
$591$	13.3137	0.547653
$592$	−3.65685	−0.150296
$593$	−28.6274	−1.17559	−0.587794	−	0.809011i	$-0.700003\pi$
$593$	−28.6274	−1.17559	−0.587794	+	0.809011i	$0.700003\pi$
$594$	1.00000	0.0410305
$595$	−13.6569	−0.559876
$596$	2.00000	0.0819232
$597$	24.9706	1.02198
$598$	−60.2843	−2.46521
$599$	−30.4853	−1.24559	−0.622797	−	0.782383i	$-0.714004\pi$
$599$	−30.4853	−1.24559	−0.622797	+	0.782383i	$0.714004\pi$
$600$	−3.00000	−0.122474
$601$	−5.31371	−0.216751	−0.108375	−	0.994110i	$-0.534565\pi$
$601$	−5.31371	−0.216751	−0.108375	+	0.994110i	$0.534565\pi$
$602$	46.6274	1.90039
$603$	−7.31371	−0.297837
$604$	10.4853	0.426640
$605$	2.82843	0.114992
$606$	−11.6569	−0.473527
$607$	8.14214	0.330479	0.165240	−	0.986253i	$-0.447160\pi$
$607$	8.14214	0.330479	0.165240	+	0.986253i	$0.447160\pi$
$608$	0	0
$609$	28.9706	1.17395
$610$	−40.0000	−1.61955
$611$	−44.2843	−1.79155
$612$	1.00000	0.0404226
$613$	−35.7990	−1.44591	−0.722954	−	0.690896i	$-0.757216\pi$
$613$	−35.7990	−1.44591	−0.722954	+	0.690896i	$0.757216\pi$
$614$	27.3137	1.10229
$615$	−5.65685	−0.228106
$616$	−4.82843	−0.194543
$617$	18.6863	0.752282	0.376141	−	0.926562i	$-0.377251\pi$
$617$	18.6863	0.752282	0.376141	+	0.926562i	$0.377251\pi$
$618$	9.65685	0.388456
$619$	6.34315	0.254953	0.127476	−	0.991842i	$-0.459312\pi$
$619$	6.34315	0.254953	0.127476	+	0.991842i	$0.459312\pi$
$620$	−11.3137	−0.454369
$621$	−8.82843	−0.354273
$622$	−13.7990	−0.553289
$623$	−48.2843	−1.93447
$624$	−6.82843	−0.273356
$625$	−31.0000	−1.24000
$626$	−6.68629	−0.267238
$627$	0	0
$628$	−11.6569	−0.465159
$629$	−3.65685	−0.145808
$630$	13.6569	0.544102
$631$	41.9411	1.66965	0.834825	−	0.550516i	$-0.185569\pi$
$631$	41.9411	1.66965	0.834825	+	0.550516i	$0.185569\pi$
$632$	14.4853	0.576194
$633$	−11.3137	−0.449680
$634$	26.1421	1.03824
$635$	−47.5980	−1.88887
$636$	−5.17157	−0.205066
$637$	−111.397	−4.41371
$638$	−6.00000	−0.237542
$639$	−8.82843	−0.349247
$640$	−2.82843	−0.111803
$641$	−3.65685	−0.144437	−0.0722185	−	0.997389i	$-0.523008\pi$
$641$	−3.65685	−0.144437	−0.0722185	+	0.997389i	$0.523008\pi$
$642$	4.00000	0.157867
$643$	−1.65685	−0.0653400	−0.0326700	−	0.999466i	$-0.510401\pi$
$643$	−1.65685	−0.0653400	−0.0326700	+	0.999466i	$0.510401\pi$
$644$	42.6274	1.67976
$645$	27.3137	1.07548
$646$	0	0
$647$	−15.8579	−0.623437	−0.311718	−	0.950175i	$-0.600905\pi$
$647$	−15.8579	−0.623437	−0.311718	+	0.950175i	$0.600905\pi$
$648$	−1.00000	−0.0392837
$649$	−4.00000	−0.157014
$650$	20.4853	0.803499
$651$	19.3137	0.756964
$652$	4.00000	0.156652
$653$	39.7990	1.55745	0.778727	−	0.627362i	$-0.215866\pi$
$653$	39.7990	1.55745	0.778727	+	0.627362i	$0.215866\pi$
$654$	−9.17157	−0.358637
$655$	−27.3137	−1.06723
$656$	−2.00000	−0.0780869
$657$	9.31371	0.363362
$658$	31.3137	1.22074
$659$	−25.6569	−0.999449	−0.499725	−	0.866184i	$-0.666565\pi$
$659$	−25.6569	−0.999449	−0.499725	+	0.866184i	$0.666565\pi$
$660$	−2.82843	−0.110096
$661$	14.2843	0.555594	0.277797	−	0.960640i	$-0.410396\pi$
$661$	14.2843	0.555594	0.277797	+	0.960640i	$0.410396\pi$
$662$	−25.6569	−0.997182
$663$	−6.82843	−0.265194
$664$	4.00000	0.155230
$665$	0	0
$666$	3.65685	0.141700
$667$	52.9706	2.05103
$668$	−2.34315	−0.0906590
$669$	24.0000	0.927894
$670$	20.6863	0.799181
$671$	−14.1421	−0.545951
$672$	4.82843	0.186261
$673$	−0.343146	−0.0132273	−0.00661365	−	0.999978i	$-0.502105\pi$
$673$	−0.343146	−0.0132273	−0.00661365	+	0.999978i	$0.502105\pi$
$674$	21.3137	0.820973
$675$	3.00000	0.115470
$676$	33.6274	1.29336
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	13.3137	0.511310
$679$	−44.9706	−1.72581
$680$	−2.82843	−0.108465
$681$	−15.3137	−0.586823
$682$	−4.00000	−0.153168
$683$	8.97056	0.343249	0.171625	−	0.985162i	$-0.445098\pi$
$683$	8.97056	0.343249	0.171625	+	0.985162i	$0.445098\pi$
$684$	0	0
$685$	−12.2843	−0.469358
$686$	44.9706	1.71698
$687$	−14.9706	−0.571163
$688$	9.65685	0.368164
$689$	35.3137	1.34535
$690$	24.9706	0.950613
$691$	14.3431	0.545639	0.272819	−	0.962065i	$-0.412044\pi$
$691$	14.3431	0.545639	0.272819	+	0.962065i	$0.412044\pi$
$692$	2.00000	0.0760286
$693$	4.82843	0.183417
$694$	17.6569	0.670245
$695$	−17.9411	−0.680546
$696$	6.00000	0.227429
$697$	−2.00000	−0.0757554
$698$	−9.17157	−0.347149
$699$	14.0000	0.529529
$700$	−14.4853	−0.547492
$701$	−20.6274	−0.779087	−0.389543	−	0.921008i	$-0.627367\pi$
$701$	−20.6274	−0.779087	−0.389543	+	0.921008i	$0.627367\pi$
$702$	6.82843	0.257722
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	18.3431	0.690843
$706$	−18.0000	−0.677439
$707$	−56.2843	−2.11679
$708$	4.00000	0.150329
$709$	2.68629	0.100886	0.0504429	−	0.998727i	$-0.483937\pi$
$709$	2.68629	0.100886	0.0504429	+	0.998727i	$0.483937\pi$
$710$	24.9706	0.937129
$711$	−14.4853	−0.543240
$712$	−10.0000	−0.374766
$713$	35.3137	1.32251
$714$	4.82843	0.180699
$715$	19.3137	0.722292
$716$	−7.31371	−0.273326
$717$	−6.34315	−0.236889
$718$	−6.34315	−0.236724
$719$	−20.1421	−0.751175	−0.375587	−	0.926787i	$-0.622559\pi$
$719$	−20.1421	−0.751175	−0.375587	+	0.926787i	$0.622559\pi$
$720$	2.82843	0.105409
$721$	46.6274	1.73650
$722$	19.0000	0.707107
$723$	7.65685	0.284761
$724$	−13.3137	−0.494800
$725$	−18.0000	−0.668503
$726$	−1.00000	−0.0371135
$727$	−14.3431	−0.531958	−0.265979	−	0.963979i	$-0.585695\pi$
$727$	−14.3431	−0.531958	−0.265979	+	0.963979i	$0.585695\pi$
$728$	−32.9706	−1.22197
$729$	1.00000	0.0370370
$730$	−26.3431	−0.975004
$731$	9.65685	0.357172
$732$	14.1421	0.522708
$733$	−50.1421	−1.85204	−0.926021	−	0.377472i	$-0.876793\pi$
$733$	−50.1421	−1.85204	−0.926021	+	0.377472i	$0.876793\pi$
$734$	−18.6274	−0.687551
$735$	46.1421	1.70198
$736$	8.82843	0.325420
$737$	7.31371	0.269404
$738$	2.00000	0.0736210
$739$	−40.2843	−1.48188	−0.740940	−	0.671571i	$-0.765620\pi$
$739$	−40.2843	−1.48188	−0.740940	+	0.671571i	$0.765620\pi$
$740$	−10.3431	−0.380222
$741$	0	0
$742$	−24.9706	−0.916698
$743$	32.9706	1.20957	0.604786	−	0.796388i	$-0.293259\pi$
$743$	32.9706	1.20957	0.604786	+	0.796388i	$0.293259\pi$
$744$	4.00000	0.146647
$745$	5.65685	0.207251
$746$	−23.7990	−0.871343
$747$	−4.00000	−0.146352
$748$	−1.00000	−0.0365636
$749$	19.3137	0.705708
$750$	5.65685	0.206559
$751$	10.6274	0.387800	0.193900	−	0.981021i	$-0.437886\pi$
$751$	10.6274	0.387800	0.193900	+	0.981021i	$0.437886\pi$
$752$	6.48528	0.236494
$753$	−26.6274	−0.970357
$754$	−40.9706	−1.49206
$755$	29.6569	1.07932
$756$	−4.82843	−0.175608
$757$	20.6274	0.749716	0.374858	−	0.927082i	$-0.377691\pi$
$757$	20.6274	0.749716	0.374858	+	0.927082i	$0.377691\pi$
$758$	7.31371	0.265646
$759$	8.82843	0.320452
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	16.8284	0.609630
$763$	−44.2843	−1.60320
$764$	−16.1421	−0.584002
$765$	2.82843	0.102262
$766$	14.4853	0.523374
$767$	−27.3137	−0.986241
$768$	1.00000	0.0360844
$769$	50.2843	1.81330	0.906649	−	0.421887i	$-0.138632\pi$
$769$	50.2843	1.81330	0.906649	+	0.421887i	$0.138632\pi$
$770$	−13.6569	−0.492159
$771$	−20.6274	−0.742878
$772$	−11.6569	−0.419539
$773$	−21.1716	−0.761489	−0.380744	−	0.924680i	$-0.624332\pi$
$773$	−21.1716	−0.761489	−0.380744	+	0.924680i	$0.624332\pi$
$774$	−9.65685	−0.347108
$775$	−12.0000	−0.431053
$776$	−9.31371	−0.334343
$777$	17.6569	0.633436
$778$	−10.8284	−0.388218
$779$	0	0
$780$	−19.3137	−0.691542
$781$	8.82843	0.315906
$782$	8.82843	0.315704
$783$	−6.00000	−0.214423
$784$	16.3137	0.582632
$785$	−32.9706	−1.17677
$786$	9.65685	0.344449
$787$	−20.6863	−0.737387	−0.368693	−	0.929551i	$-0.620195\pi$
$787$	−20.6863	−0.737387	−0.368693	+	0.929551i	$0.620195\pi$
$788$	13.3137	0.474281
$789$	19.3137	0.687586
$790$	40.9706	1.45767
$791$	64.2843	2.28569
$792$	1.00000	0.0355335
$793$	−96.5685	−3.42925
$794$	−1.31371	−0.0466218
$795$	−14.6274	−0.518781
$796$	24.9706	0.885058
$797$	28.7696	1.01907	0.509535	−	0.860450i	$-0.329817\pi$
$797$	28.7696	1.01907	0.509535	+	0.860450i	$0.329817\pi$
$798$	0	0
$799$	6.48528	0.229433
$800$	−3.00000	−0.106066
$801$	10.0000	0.353333
$802$	30.9706	1.09361
$803$	−9.31371	−0.328674
$804$	−7.31371	−0.257935
$805$	120.569	4.24948
$806$	−27.3137	−0.962084
$807$	−0.485281	−0.0170827
$808$	−11.6569	−0.410087
$809$	−27.9411	−0.982358	−0.491179	−	0.871059i	$-0.663434\pi$
$809$	−27.9411	−0.982358	−0.491179	+	0.871059i	$0.663434\pi$
$810$	−2.82843	−0.0993808
$811$	−12.6863	−0.445476	−0.222738	−	0.974878i	$-0.571499\pi$
$811$	−12.6863	−0.445476	−0.222738	+	0.974878i	$0.571499\pi$
$812$	28.9706	1.01667
$813$	16.8284	0.590199
$814$	−3.65685	−0.128173
$815$	11.3137	0.396302
$816$	1.00000	0.0350070
$817$	0	0
$818$	10.9706	0.383577
$819$	32.9706	1.15208
$820$	−5.65685	−0.197546
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	4.34315	0.151485
$823$	−42.6274	−1.48590	−0.742949	−	0.669348i	$-0.766574\pi$
$823$	−42.6274	−1.48590	−0.742949	+	0.669348i	$0.766574\pi$
$824$	9.65685	0.336412
$825$	−3.00000	−0.104447
$826$	19.3137	0.672010
$827$	30.3431	1.05513	0.527567	−	0.849513i	$-0.323104\pi$
$827$	30.3431	1.05513	0.527567	+	0.849513i	$0.323104\pi$
$828$	−8.82843	−0.306809
$829$	−27.9411	−0.970435	−0.485218	−	0.874393i	$-0.661260\pi$
$829$	−27.9411	−0.970435	−0.485218	+	0.874393i	$0.661260\pi$
$830$	11.3137	0.392705
$831$	−5.17157	−0.179400
$832$	−6.82843	−0.236733
$833$	16.3137	0.565236
$834$	6.34315	0.219645
$835$	−6.62742	−0.229351
$836$	0	0
$837$	−4.00000	−0.138260
$838$	−24.9706	−0.862594
$839$	−3.85786	−0.133188	−0.0665941	−	0.997780i	$-0.521213\pi$
$839$	−3.85786	−0.133188	−0.0665941	+	0.997780i	$0.521213\pi$
$840$	13.6569	0.471206
$841$	7.00000	0.241379
$842$	5.31371	0.183122
$843$	2.00000	0.0688837
$844$	−11.3137	−0.389434
$845$	95.1127	3.27198
$846$	−6.48528	−0.222969
$847$	−4.82843	−0.165907
$848$	−5.17157	−0.177593
$849$	11.3137	0.388285
$850$	−3.00000	−0.102899
$851$	32.2843	1.10669
$852$	−8.82843	−0.302457
$853$	−1.85786	−0.0636121	−0.0318060	−	0.999494i	$-0.510126\pi$
$853$	−1.85786	−0.0636121	−0.0318060	+	0.999494i	$0.510126\pi$
$854$	68.2843	2.33664
$855$	0	0
$856$	4.00000	0.136717
$857$	−30.6863	−1.04822	−0.524112	−	0.851649i	$-0.675603\pi$
$857$	−30.6863	−1.04822	−0.524112	+	0.851649i	$0.675603\pi$
$858$	−6.82843	−0.233119
$859$	33.6569	1.14836	0.574179	−	0.818730i	$-0.305322\pi$
$859$	33.6569	1.14836	0.574179	+	0.818730i	$0.305322\pi$
$860$	27.3137	0.931390
$861$	9.65685	0.329105
$862$	−23.3137	−0.794068
$863$	6.48528	0.220762	0.110381	−	0.993889i	$-0.464793\pi$
$863$	6.48528	0.220762	0.110381	+	0.993889i	$0.464793\pi$
$864$	−1.00000	−0.0340207
$865$	5.65685	0.192339
$866$	−5.31371	−0.180567
$867$	1.00000	0.0339618
$868$	19.3137	0.655550
$869$	14.4853	0.491380
$870$	16.9706	0.575356
$871$	49.9411	1.69219
$872$	−9.17157	−0.310589
$873$	9.31371	0.315221
$874$	0	0
$875$	27.3137	0.923372
$876$	9.31371	0.314681
$877$	−27.5147	−0.929106	−0.464553	−	0.885545i	$-0.653785\pi$
$877$	−27.5147	−0.929106	−0.464553	+	0.885545i	$0.653785\pi$
$878$	−19.4558	−0.656603
$879$	26.0000	0.876958
$880$	−2.82843	−0.0953463
$881$	−45.3137	−1.52666	−0.763329	−	0.646010i	$-0.776436\pi$
$881$	−45.3137	−1.52666	−0.763329	+	0.646010i	$0.776436\pi$
$882$	−16.3137	−0.549311
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−6.82843	−0.229665
$885$	11.3137	0.380306
$886$	20.0000	0.671913
$887$	32.9706	1.10704	0.553522	−	0.832835i	$-0.313284\pi$
$887$	32.9706	1.10704	0.553522	+	0.832835i	$0.313284\pi$
$888$	3.65685	0.122716
$889$	81.2548	2.72520
$890$	−28.2843	−0.948091
$891$	−1.00000	−0.0335013
$892$	24.0000	0.803579
$893$	0	0
$894$	−2.00000	−0.0668900
$895$	−20.6863	−0.691466
$896$	4.82843	0.161306
$897$	60.2843	2.01283
$898$	−7.65685	−0.255513
$899$	24.0000	0.800445
$900$	3.00000	0.100000
$901$	−5.17157	−0.172290
$902$	−2.00000	−0.0665927
$903$	−46.6274	−1.55166
$904$	13.3137	0.442807
$905$	−37.6569	−1.25176
$906$	−10.4853	−0.348350
$907$	−6.34315	−0.210621	−0.105310	−	0.994439i	$-0.533584\pi$
$907$	−6.34315	−0.210621	−0.105310	+	0.994439i	$0.533584\pi$
$908$	−15.3137	−0.508203
$909$	11.6569	0.386633
$910$	−93.2548	−3.09137
$911$	41.1127	1.36212	0.681062	−	0.732226i	$-0.261518\pi$
$911$	41.1127	1.36212	0.681062	+	0.732226i	$0.261518\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	28.3431	0.937508
$915$	40.0000	1.32236
$916$	−14.9706	−0.494641
$917$	46.6274	1.53977
$918$	−1.00000	−0.0330049
$919$	18.2010	0.600396	0.300198	−	0.953877i	$-0.402947\pi$
$919$	18.2010	0.600396	0.300198	+	0.953877i	$0.402947\pi$
$920$	24.9706	0.823255
$921$	−27.3137	−0.900017
$922$	14.0000	0.461065
$923$	60.2843	1.98428
$924$	4.82843	0.158844
$925$	−10.9706	−0.360710
$926$	−12.9706	−0.426239
$927$	−9.65685	−0.317173
$928$	6.00000	0.196960
$929$	2.97056	0.0974610	0.0487305	−	0.998812i	$-0.484482\pi$
$929$	2.97056	0.0974610	0.0487305	+	0.998812i	$0.484482\pi$
$930$	11.3137	0.370991
$931$	0	0
$932$	14.0000	0.458585
$933$	13.7990	0.451759
$934$	−16.9706	−0.555294
$935$	−2.82843	−0.0924995
$936$	6.82843	0.223194
$937$	16.6274	0.543194	0.271597	−	0.962411i	$-0.412448\pi$
$937$	16.6274	0.543194	0.271597	+	0.962411i	$0.412448\pi$
$938$	−35.3137	−1.15303
$939$	6.68629	0.218199
$940$	18.3431	0.598287
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	11.6569	0.379801
$943$	17.6569	0.574986
$944$	4.00000	0.130189
$945$	−13.6569	−0.444258
$946$	9.65685	0.313971
$947$	−44.0000	−1.42981	−0.714904	−	0.699223i	$-0.753530\pi$
$947$	−44.0000	−1.42981	−0.714904	+	0.699223i	$0.753530\pi$
$948$	−14.4853	−0.470460
$949$	−63.5980	−2.06448
$950$	0	0
$951$	−26.1421	−0.847717
$952$	4.82843	0.156490
$953$	−16.6274	−0.538615	−0.269307	−	0.963054i	$-0.586795\pi$
$953$	−16.6274	−0.538615	−0.269307	+	0.963054i	$0.586795\pi$
$954$	5.17157	0.167436
$955$	−45.6569	−1.47742
$956$	−6.34315	−0.205152
$957$	6.00000	0.193952
$958$	36.0000	1.16311
$959$	20.9706	0.677175
$960$	2.82843	0.0912871
$961$	−15.0000	−0.483871
$962$	−24.9706	−0.805083
$963$	−4.00000	−0.128898
$964$	7.65685	0.246611
$965$	−32.9706	−1.06136
$966$	−42.6274	−1.37151
$967$	−42.7696	−1.37538	−0.687688	−	0.726006i	$-0.741374\pi$
$967$	−42.7696	−1.37538	−0.687688	+	0.726006i	$0.741374\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−26.3431	−0.845827
$971$	−0.686292	−0.0220241	−0.0110121	−	0.999939i	$-0.503505\pi$
$971$	−0.686292	−0.0220241	−0.0110121	+	0.999939i	$0.503505\pi$
$972$	1.00000	0.0320750
$973$	30.6274	0.981870
$974$	−40.9706	−1.31278
$975$	−20.4853	−0.656054
$976$	14.1421	0.452679
$977$	−7.65685	−0.244964	−0.122482	−	0.992471i	$-0.539085\pi$
$977$	−7.65685	−0.244964	−0.122482	+	0.992471i	$0.539085\pi$
$978$	−4.00000	−0.127906
$979$	−10.0000	−0.319601
$980$	46.1421	1.47396
$981$	9.17157	0.292826
$982$	36.9706	1.17978
$983$	2.48528	0.0792682	0.0396341	−	0.999214i	$-0.487381\pi$
$983$	2.48528	0.0792682	0.0396341	+	0.999214i	$0.487381\pi$
$984$	2.00000	0.0637577
$985$	37.6569	1.19985
$986$	6.00000	0.191079
$987$	−31.3137	−0.996726
$988$	0	0
$989$	−85.2548	−2.71095
$990$	2.82843	0.0898933
$991$	−29.9411	−0.951111	−0.475556	−	0.879686i	$-0.657753\pi$
$991$	−29.9411	−0.951111	−0.475556	+	0.879686i	$0.657753\pi$
$992$	4.00000	0.127000
$993$	25.6569	0.814196
$994$	−42.6274	−1.35206
$995$	70.6274	2.23904
$996$	−4.00000	−0.126745
$997$	−22.8284	−0.722984	−0.361492	−	0.932375i	$-0.617732\pi$
$997$	−22.8284	−0.722984	−0.361492	+	0.932375i	$0.617732\pi$
$998$	−0.686292	−0.0217242
$999$	−3.65685	−0.115698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1122.2.a.q.1.2	✓	2
3.2	odd	2		3366.2.a.w.1.1		2
4.3	odd	2		8976.2.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1122.2.a.q.1.2	✓	2	1.1	even	1	trivial
3366.2.a.w.1.1		2	3.2	odd	2
8976.2.a.bk.1.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1122.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.82843 q^{5} -1.00000 q^{6} -4.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.82843 q^{5} -1.00000 q^{6} -4.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.82843 q^{10} -1.00000 q^{11} +1.00000 q^{12} -6.82843 q^{13} +4.82843 q^{14} +2.82843 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +2.82843 q^{20} -4.82843 q^{21} +1.00000 q^{22} -8.82843 q^{23} -1.00000 q^{24} +3.00000 q^{25} +6.82843 q^{26} +1.00000 q^{27} -4.82843 q^{28} -6.00000 q^{29} -2.82843 q^{30} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} -13.6569 q^{35} +1.00000 q^{36} -3.65685 q^{37} -6.82843 q^{39} -2.82843 q^{40} -2.00000 q^{41} +4.82843 q^{42} +9.65685 q^{43} -1.00000 q^{44} +2.82843 q^{45} +8.82843 q^{46} +6.48528 q^{47} +1.00000 q^{48} +16.3137 q^{49} -3.00000 q^{50} +1.00000 q^{51} -6.82843 q^{52} -5.17157 q^{53} -1.00000 q^{54} -2.82843 q^{55} +4.82843 q^{56} +6.00000 q^{58} +4.00000 q^{59} +2.82843 q^{60} +14.1421 q^{61} +4.00000 q^{62} -4.82843 q^{63} +1.00000 q^{64} -19.3137 q^{65} +1.00000 q^{66} -7.31371 q^{67} +1.00000 q^{68} -8.82843 q^{69} +13.6569 q^{70} -8.82843 q^{71} -1.00000 q^{72} +9.31371 q^{73} +3.65685 q^{74} +3.00000 q^{75} +4.82843 q^{77} +6.82843 q^{78} -14.4853 q^{79} +2.82843 q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} -4.82843 q^{84} +2.82843 q^{85} -9.65685 q^{86} -6.00000 q^{87} +1.00000 q^{88} +10.0000 q^{89} -2.82843 q^{90} +32.9706 q^{91} -8.82843 q^{92} -4.00000 q^{93} -6.48528 q^{94} -1.00000 q^{96} +9.31371 q^{97} -16.3137 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{11} + 2 q^{12} - 8 q^{13} + 4 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 4 q^{21} + 2 q^{22} - 12 q^{23} - 2 q^{24} + 6 q^{25} + 8 q^{26} + 2 q^{27} - 4 q^{28} - 12 q^{29} - 8 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} - 16 q^{35} + 2 q^{36} + 4 q^{37} - 8 q^{39} - 4 q^{41} + 4 q^{42} + 8 q^{43} - 2 q^{44} + 12 q^{46} - 4 q^{47} + 2 q^{48} + 10 q^{49} - 6 q^{50} + 2 q^{51} - 8 q^{52} - 16 q^{53} - 2 q^{54} + 4 q^{56} + 12 q^{58} + 8 q^{59} + 8 q^{62} - 4 q^{63} + 2 q^{64} - 16 q^{65} + 2 q^{66} + 8 q^{67} + 2 q^{68} - 12 q^{69} + 16 q^{70} - 12 q^{71} - 2 q^{72} - 4 q^{73} - 4 q^{74} + 6 q^{75} + 4 q^{77} + 8 q^{78} - 12 q^{79} + 2 q^{81} + 4 q^{82} - 8 q^{83} - 4 q^{84} - 8 q^{86} - 12 q^{87} + 2 q^{88} + 20 q^{89} + 32 q^{91} - 12 q^{92} - 8 q^{93} + 4 q^{94} - 2 q^{96} - 4 q^{97} - 10 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^11 + 2 * q^12 - 8 * q^13 + 4 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^18 - 4 * q^21 + 2 * q^22 - 12 * q^23 - 2 * q^24 + 6 * q^25 + 8 * q^26 + 2 * q^27 - 4 * q^28 - 12 * q^29 - 8 * q^31 - 2 * q^32 - 2 * q^33 - 2 * q^34 - 16 * q^35 + 2 * q^36 + 4 * q^37 - 8 * q^39 - 4 * q^41 + 4 * q^42 + 8 * q^43 - 2 * q^44 + 12 * q^46 - 4 * q^47 + 2 * q^48 + 10 * q^49 - 6 * q^50 + 2 * q^51 - 8 * q^52 - 16 * q^53 - 2 * q^54 + 4 * q^56 + 12 * q^58 + 8 * q^59 + 8 * q^62 - 4 * q^63 + 2 * q^64 - 16 * q^65 + 2 * q^66 + 8 * q^67 + 2 * q^68 - 12 * q^69 + 16 * q^70 - 12 * q^71 - 2 * q^72 - 4 * q^73 - 4 * q^74 + 6 * q^75 + 4 * q^77 + 8 * q^78 - 12 * q^79 + 2 * q^81 + 4 * q^82 - 8 * q^83 - 4 * q^84 - 8 * q^86 - 12 * q^87 + 2 * q^88 + 20 * q^89 + 32 * q^91 - 12 * q^92 - 8 * q^93 + 4 * q^94 - 2 * q^96 - 4 * q^97 - 10 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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