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Properties

Label 99.7.c.a.10.1
Level $99$
Weight $7$
Character 99.10
Self dual yes
Analytic conductor $22.775$
Analytic rank $0$
Dimension $1$
CM discriminant -11
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [99,7,Mod(10,99)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("99.10"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(99, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 99.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.7753542784\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 10.1
Character \(\chi\) \(=\) 99.10

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+64.0000 q^{4} -74.0000 q^{5} +1331.00 q^{11} +4096.00 q^{16} -4736.00 q^{20} +12670.0 q^{23} -10149.0 q^{25} +56018.0 q^{31} +87050.0 q^{37} +85184.0 q^{44} +206350. q^{47} +117649. q^{49} -246890. q^{53} -98494.0 q^{55} -107642. q^{59} +262144. q^{64} -428470. q^{67} +341278. q^{71} -303104. q^{80} -1.39234e6 q^{89} +810880. q^{92} -1.82419e6 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 64.0000 1.00000
\(5\) −74.0000 −0.592000 −0.296000 − 0.955188i \(-0.595653\pi\)
−0.296000 + 0.955188i \(0.595653\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1331.00 1.00000
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4096.00 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −4736.00 −0.592000
\(21\) 0 0
\(22\) 0 0
\(23\) 12670.0 1.04134 0.520671 − 0.853758i \(-0.325682\pi\)
0.520671 + 0.853758i \(0.325682\pi\)
\(24\) 0 0
\(25\) −10149.0 −0.649536
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 56018.0 1.88037 0.940183 − 0.340669i \(-0.110654\pi\)
0.940183 + 0.340669i \(0.110654\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 87050.0 1.71856 0.859278 − 0.511509i \(-0.170913\pi\)
0.859278 + 0.511509i \(0.170913\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 85184.0 1.00000
\(45\) 0 0
\(46\) 0 0
\(47\) 206350. 1.98752 0.993759 − 0.111552i \(-0.0355821\pi\)
0.993759 + 0.111552i \(0.0355821\pi\)
\(48\) 0 0
\(49\) 117649. 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −246890. −1.65835 −0.829174 − 0.558990i \(-0.811189\pi\)
−0.829174 + 0.558990i \(0.811189\pi\)
\(54\) 0 0
\(55\) −98494.0 −0.592000
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −107642. −0.524114 −0.262057 − 0.965052i \(-0.584401\pi\)
−0.262057 + 0.965052i \(0.584401\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 262144. 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −428470. −1.42461 −0.712305 − 0.701870i \(-0.752349\pi\)
−0.712305 + 0.701870i \(0.752349\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 341278. 0.953528 0.476764 − 0.879031i \(-0.341810\pi\)
0.476764 + 0.879031i \(0.341810\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −303104. −0.592000
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.39234e6 −1.97503 −0.987517 − 0.157511i \(-0.949653\pi\)
−0.987517 + 0.157511i \(0.949653\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 810880. 1.04134
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.82419e6 −1.99873 −0.999367 − 0.0355838i \(-0.988671\pi\)
−0.999367 + 0.0355838i \(0.988671\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 99.7.c.a.10.1 1
3.2 odd 2 11.7.b.a.10.1 ✓ 1
11.10 odd 2 CM 99.7.c.a.10.1 1
12.11 even 2 176.7.h.a.65.1 1
33.32 even 2 11.7.b.a.10.1 ✓ 1
132.131 odd 2 176.7.h.a.65.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.7.b.a.10.1 ✓ 1 3.2 odd 2
11.7.b.a.10.1 ✓ 1 33.32 even 2
99.7.c.a.10.1 1 1.1 even 1 trivial
99.7.c.a.10.1 1 11.10 odd 2 CM
176.7.h.a.65.1 1 12.11 even 2
176.7.h.a.65.1 1 132.131 odd 2