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Properties

Label 612.1.j.a.89.2
Level $612$
Weight $1$
Character 612.89
Analytic conductor $0.305$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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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] = [612,1,Mod(89,612)] 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("612.89"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(612, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 612 = 2^{2} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 612.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.305427787731\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.530604.1

Embedding invariants

Embedding label 89.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 612.89
Dual form 612.1.j.a.557.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.707107 + 0.707107i) q^{5} +(-0.707107 + 0.707107i) q^{11} +1.00000 q^{13} +(-0.707107 + 0.707107i) q^{17} -1.00000i q^{19} +(0.707107 - 0.707107i) q^{23} +(-1.00000 + 1.00000i) q^{31} +(0.707107 - 0.707107i) q^{41} +1.00000i q^{43} -1.41421i q^{47} -1.00000i q^{49} -1.41421 q^{53} -1.00000 q^{55} +1.41421 q^{59} +(0.707107 + 0.707107i) q^{65} +(-1.41421 - 1.41421i) q^{71} +(-1.00000 + 1.00000i) q^{73} +(-1.00000 - 1.00000i) q^{79} -1.41421 q^{83} -1.00000 q^{85} +(0.707107 - 0.707107i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{13} - 4 q^{31} - 4 q^{55} - 4 q^{73} - 4 q^{79} - 4 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/612\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(137\) \(307\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 − 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(6\) 0 0
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 − 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(12\) 0 0
\(13\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(18\) 0 0
\(19\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.707107 − 0.707107i 0.707107 − 0.707107i −0.258819 − 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 0 0
\(31\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.707107 − 0.707107i 0.707107 − 0.707107i −0.258819 − 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(42\) 0 0
\(43\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.41421i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(48\) 0 0
\(49\) − 1.00000i − 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.41421 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) −1.00000 −1.00000
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.41421 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) 0 0
\(61\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.41421 − 1.41421i −1.41421 − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 − 0.707107i \(-0.750000\pi\)
\(72\) 0 0
\(73\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.00000 − 1.00000i −1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.41421 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) −1.00000 −1.00000
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(96\) 0 0
\(97\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) 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 612.1.j.a.89.2 yes 4
3.2 odd 2 inner 612.1.j.a.89.1 ✓ 4
4.3 odd 2 2448.1.bd.a.1313.2 4
12.11 even 2 2448.1.bd.a.1313.1 4
17.13 even 4 inner 612.1.j.a.557.1 yes 4
51.47 odd 4 inner 612.1.j.a.557.2 yes 4
68.47 odd 4 2448.1.bd.a.1169.1 4
204.47 even 4 2448.1.bd.a.1169.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
612.1.j.a.89.1 ✓ 4 3.2 odd 2 inner
612.1.j.a.89.2 yes 4 1.1 even 1 trivial
612.1.j.a.557.1 yes 4 17.13 even 4 inner
612.1.j.a.557.2 yes 4 51.47 odd 4 inner
2448.1.bd.a.1169.1 4 68.47 odd 4
2448.1.bd.a.1169.2 4 204.47 even 4
2448.1.bd.a.1313.1 4 12.11 even 2
2448.1.bd.a.1313.2 4 4.3 odd 2