Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1881,2,Mod(208,1881)] 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("1881.208"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1881, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1881.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.0198606202\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-11}, \sqrt{19})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 31x^{2} + 32x + 465 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 208.4
Root \(4.85890 + 1.65831i\) of defining polynomial
Character \(\chi\) \(=\) 1881.208
Dual form 1881.2.h.b.208.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} +3.31662i q^{11} +4.35890 q^{13} +4.00000 q^{16} -3.31662i q^{17} -4.35890 q^{19} -5.00000 q^{25} -6.63325i q^{44} +7.00000 q^{49} -8.71780 q^{52} +14.4568i q^{53} +14.4568i q^{59} -8.00000 q^{64} +6.63325i q^{68} +14.4568i q^{71} +8.71780 q^{76} +4.35890 q^{79} +16.5831i q^{83} +14.4568i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 28 q^{49} - 32 q^{64}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1881\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(496\) \(1046\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 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 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.31662i 1.00000i
\(12\) 0 0
\(13\) 4.35890 1.20894 0.604471 − 0.796628i \(-0.293385\pi\)
0.604471 + 0.796628i \(0.293385\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) − 3.31662i − 0.804400i −0.915552 − 0.402200i \(-0.868246\pi\)
0.915552 − 0.402200i \(-0.131754\pi\)
\(18\) 0 0
\(19\) −4.35890 −1.00000
\(20\) 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
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 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
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) − 6.63325i − 1.00000i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −8.71780 −1.20894
\(53\) 14.4568i 1.98580i 0.118958 + 0.992899i \(0.462045\pi\)
−0.118958 + 0.992899i \(0.537955\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.4568i 1.88212i 0.338241 + 0.941060i \(0.390168\pi\)
−0.338241 + 0.941060i \(0.609832\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 6.63325i 0.804400i
\(69\) 0 0
\(70\) 0 0
\(71\) 14.4568i 1.71571i 0.513892 + 0.857855i \(0.328203\pi\)
−0.513892 + 0.857855i \(0.671797\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 8.71780 1.00000
\(77\) 0 0
\(78\) 0 0
\(79\) 4.35890 0.490414 0.245207 − 0.969471i \(-0.421144\pi\)
0.245207 + 0.969471i \(0.421144\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.5831i 1.82023i 0.414351 + 0.910117i \(0.364009\pi\)
−0.414351 + 0.910117i \(0.635991\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.4568i 1.53242i 0.642590 + 0.766211i \(0.277860\pi\)
−0.642590 + 0.766211i \(0.722140\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 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 1881.2.h.b.208.4 yes 4
3.2 odd 2 inner 1881.2.h.b.208.2 yes 4
11.10 odd 2 inner 1881.2.h.b.208.1 ✓ 4
19.18 odd 2 inner 1881.2.h.b.208.3 yes 4
33.32 even 2 inner 1881.2.h.b.208.3 yes 4
57.56 even 2 inner 1881.2.h.b.208.1 ✓ 4
209.208 even 2 inner 1881.2.h.b.208.2 yes 4
627.626 odd 2 CM 1881.2.h.b.208.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1881.2.h.b.208.1 ✓ 4 11.10 odd 2 inner
1881.2.h.b.208.1 ✓ 4 57.56 even 2 inner
1881.2.h.b.208.2 yes 4 3.2 odd 2 inner
1881.2.h.b.208.2 yes 4 209.208 even 2 inner
1881.2.h.b.208.3 yes 4 19.18 odd 2 inner
1881.2.h.b.208.3 yes 4 33.32 even 2 inner
1881.2.h.b.208.4 yes 4 1.1 even 1 trivial
1881.2.h.b.208.4 yes 4 627.626 odd 2 CM