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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1116,2,Mod(991,1116)] 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("1116.991"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1116, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1116 = 2^{2} \cdot 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1116.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-8,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.91130486557\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{31})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 32x^{2} + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 991.4
Root \(-4.64411i\) of defining polynomial
Character \(\chi\) \(=\) 1116.991
Dual form 1116.2.g.c.991.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} -2.82843i q^{8} +4.00000 q^{16} +7.87401i q^{17} -5.00000 q^{25} +7.87401i q^{29} -5.56776 q^{31} +5.65685i q^{32} -11.1355 q^{34} -11.1355 q^{43} -1.41421i q^{47} +7.00000 q^{49} -7.07107i q^{50} +7.87401i q^{53} -11.1355 q^{58} +7.07107i q^{59} -7.87401i q^{62} -8.00000 q^{64} -15.7480i q^{68} -9.89949i q^{71} -11.1355 q^{79} -15.7480i q^{86} +7.87401i q^{89} +2.00000 q^{94} -4.00000 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 28 q^{49} - 32 q^{64} + 8 q^{94} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1116\mathbb{Z}\right)^\times\).

\(n\) \(497\) \(559\) \(685\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 2.82843i − 1.00000i
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 7.87401i 1.90973i 0.297044 + 0.954864i \(0.403999\pi\)
−0.297044 + 0.954864i \(0.596001\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.87401i 1.46217i 0.682288 + 0.731083i \(0.260985\pi\)
−0.682288 + 0.731083i \(0.739015\pi\)
\(30\) 0 0
\(31\) −5.56776 −1.00000
\(32\) 5.65685i 1.00000i
\(33\) 0 0
\(34\) −11.1355 −1.90973
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −11.1355 −1.69815 −0.849076 − 0.528271i \(-0.822841\pi\)
−0.849076 + 0.528271i \(0.822841\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.41421i − 0.206284i −0.994667 − 0.103142i \(-0.967110\pi\)
0.994667 − 0.103142i \(-0.0328896\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) − 7.07107i − 1.00000i
\(51\) 0 0
\(52\) 0 0
\(53\) 7.87401i 1.08158i 0.841158 + 0.540789i \(0.181874\pi\)
−0.841158 + 0.540789i \(0.818126\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −11.1355 −1.46217
\(59\) 7.07107i 0.920575i 0.887770 + 0.460287i \(0.152254\pi\)
−0.887770 + 0.460287i \(0.847746\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) − 7.87401i − 1.00000i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) − 15.7480i − 1.90973i
\(69\) 0 0
\(70\) 0 0
\(71\) − 9.89949i − 1.17485i −0.809277 − 0.587427i \(-0.800141\pi\)
0.809277 − 0.587427i \(-0.199859\pi\)
\(72\) 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
\(79\) −11.1355 −1.25284 −0.626422 − 0.779484i \(-0.715481\pi\)
−0.626422 + 0.779484i \(0.715481\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 15.7480i − 1.69815i
\(87\) 0 0
\(88\) 0 0
\(89\) 7.87401i 0.834643i 0.908759 + 0.417322i \(0.137031\pi\)
−0.908759 + 0.417322i \(0.862969\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 2.00000 0.206284
\(95\) 0 0
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 − 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 9.89949i 1.00000i
\(99\) 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 1116.2.g.c.991.4 yes 4
3.2 odd 2 inner 1116.2.g.c.991.1 ✓ 4
4.3 odd 2 inner 1116.2.g.c.991.2 yes 4
12.11 even 2 inner 1116.2.g.c.991.3 yes 4
31.30 odd 2 inner 1116.2.g.c.991.3 yes 4
93.92 even 2 inner 1116.2.g.c.991.2 yes 4
124.123 even 2 inner 1116.2.g.c.991.1 ✓ 4
372.371 odd 2 CM 1116.2.g.c.991.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1116.2.g.c.991.1 ✓ 4 3.2 odd 2 inner
1116.2.g.c.991.1 ✓ 4 124.123 even 2 inner
1116.2.g.c.991.2 yes 4 4.3 odd 2 inner
1116.2.g.c.991.2 yes 4 93.92 even 2 inner
1116.2.g.c.991.3 yes 4 12.11 even 2 inner
1116.2.g.c.991.3 yes 4 31.30 odd 2 inner
1116.2.g.c.991.4 yes 4 1.1 even 1 trivial
1116.2.g.c.991.4 yes 4 372.371 odd 2 CM