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Properties

Label 1904.4.a.p
Level $1904$
Weight $4$
Character orbit 1904.a
Self dual yes
Analytic conductor $112.340$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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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] = [1904,4,Mod(1,1904)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1904, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1904.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1904.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,0,-5,0,35] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(112.339636651\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 49x^{5} + 69x^{4} + 753x^{3} - 122x^{2} - 3621x - 2536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: no (minimal twist has level 119)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{3} + ( - \beta_{4} + 5) q^{5} + 7 q^{7} + ( - 2 \beta_{6} + \beta_{5} + \cdots + 19) q^{9} + (\beta_{6} - 2 \beta_{5} + \beta_{4} + \cdots - 7) q^{11} + ( - \beta_{5} + 3 \beta_{4} + \cdots + 11) q^{13}+ \cdots + (56 \beta_{6} - 19 \beta_{5} + \cdots - 1116) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 5 q^{3} + 35 q^{5} + 49 q^{7} + 128 q^{9} - 48 q^{11} + 84 q^{13} + 54 q^{15} - 119 q^{17} - 156 q^{19} - 35 q^{21} - 290 q^{23} + 94 q^{25} + 10 q^{27} + 1074 q^{29} + 291 q^{31} + 26 q^{33} + 245 q^{35}+ \cdots - 7572 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 49x^{5} + 69x^{4} + 753x^{3} - 122x^{2} - 3621x - 2536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 351\nu^{6} - 2656\nu^{5} - 8177\nu^{4} + 74522\nu^{3} + 35729\nu^{2} - 450809\nu - 119398 ) / 20582 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 909\nu^{6} - 5559\nu^{5} - 28037\nu^{4} + 151565\nu^{3} + 232381\nu^{2} - 866402\nu - 593664 ) / 10291 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3121\nu^{6} + 17694\nu^{5} + 101675\nu^{4} - 480324\nu^{3} - 908883\nu^{2} + 2814309\nu + 2552590 ) / 20582 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -1600\nu^{6} + 9615\nu^{5} + 52520\nu^{4} - 277457\nu^{3} - 493352\nu^{2} + 1783563\nu + 1637280 ) / 10291 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2376\nu^{6} - 14021\nu^{5} - 75934\nu^{4} + 380173\nu^{3} + 702578\nu^{2} - 2210143\nu - 2248181 ) / 10291 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 3\beta_{3} + 2\beta_{2} + 3\beta _1 + 60 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{6} - 2\beta_{5} - 5\beta_{4} - 12\beta_{3} + 13\beta_{2} + 45\beta _1 + 148 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 22\beta_{6} + 4\beta_{5} - 35\beta_{4} - 137\beta_{3} + 137\beta_{2} + 142\beta _1 + 1544 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -23\beta_{6} - 42\beta_{5} - 339\beta_{4} - 740\beta_{3} + 747\beta_{2} + 1377\beta _1 + 6722 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 449\beta_{6} + 200\beta_{5} - 2319\beta_{4} - 5938\beta_{3} + 6115\beta_{2} + 6437\beta _1 + 53234 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.93035
3.11723
−2.77278
−4.37776
4.18404
6.60698
−0.827361
0 −9.43251 0 −8.85857 0 7.00000 0 61.9722 0
1.2 0 −8.38428 0 10.3793 0 7.00000 0 43.2962 0
1.3 0 −4.49237 0 19.9390 0 7.00000 0 −6.81863 0
1.4 0 0.191325 0 3.18044 0 7.00000 0 −26.9634 0
1.5 0 1.42027 0 −11.4559 0 7.00000 0 −24.9828 0
1.6 0 5.36988 0 12.6276 0 7.00000 0 1.83559 0
1.7 0 10.3277 0 9.18809 0 7.00000 0 79.6609 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(17\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1904.4.a.p 7
4.b odd 2 1 119.4.a.d ✓ 7
12.b even 2 1 1071.4.a.o 7
28.d even 2 1 833.4.a.f 7
68.d odd 2 1 2023.4.a.g 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.4.a.d ✓ 7 4.b odd 2 1
833.4.a.f 7 28.d even 2 1
1071.4.a.o 7 12.b even 2 1
1904.4.a.p 7 1.a even 1 1 trivial
2023.4.a.g 7 68.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} + 5T_{3}^{6} - 146T_{3}^{5} - 685T_{3}^{4} + 4670T_{3}^{3} + 14223T_{3}^{2} - 30871T_{3} + 5354 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1904))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + 5 T^{6} + \cdots + 5354 \) Copy content Toggle raw display
$5$ \( T^{7} - 35 T^{6} + \cdots - 7749964 \) Copy content Toggle raw display
$7$ \( (T - 7)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots - 34663527936 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots + 13756569984 \) Copy content Toggle raw display
$17$ \( (T + 17)^{7} \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 3100949967360 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 426071334454272 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 394911043338240 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 229799266689312 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 20\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 39\!\cdots\!78 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 26\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 20\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 33\!\cdots\!18 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots + 22\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots + 72\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 52\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 16\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 44\!\cdots\!34 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots + 165111753621760 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots - 11\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots + 36\!\cdots\!26 \) Copy content Toggle raw display
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