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Properties

Label 5950.2.a.cc
Level $5950$
Weight $2$
Character orbit 5950.a
Self dual yes
Analytic conductor $47.511$
Analytic rank $1$
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] = [5950,2,Mod(1,5950)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5950.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5950.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,7,-2,7,0,-2,-7,7,7,0,-7,-2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.5109892027\)
Analytic rank: \(1\)
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: \( x^{7} - 2x^{6} - 12x^{5} + 17x^{4} + 40x^{3} - 32x^{2} - 40x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1190)
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.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} - q^{7} + q^{8} + (\beta_{4} - \beta_{3} + \beta_{2} + \cdots + 1) q^{9} + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots - 1) q^{11} - \beta_1 q^{12}+ \cdots + ( - 2 \beta_{6} - 2 \beta_{5} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} - 2 q^{3} + 7 q^{4} - 2 q^{6} - 7 q^{7} + 7 q^{8} + 7 q^{9} - 7 q^{11} - 2 q^{12} + q^{13} - 7 q^{14} + 7 q^{16} - 7 q^{17} + 7 q^{18} - 2 q^{19} + 2 q^{21} - 7 q^{22} - 27 q^{23} - 2 q^{24}+ \cdots - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 12x^{5} + 17x^{4} + 40x^{3} - 32x^{2} - 40x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + \nu^{5} + 10\nu^{4} - 3\nu^{3} - 9\nu^{2} - 8\nu - 30 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 4\nu^{4} - 6\nu^{3} + 29\nu^{2} + 2\nu - 30 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 14\nu^{4} - 3\nu^{3} + 46\nu^{2} + 2\nu - 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} + 10\nu^{4} - 13\nu^{3} - 20\nu^{2} + 10\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{5} - 4\nu^{4} - 34\nu^{3} + 23\nu^{2} + 78\nu - 10 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + 7\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + 2\beta_{5} + 12\beta_{4} - 13\beta_{3} + 8\beta_{2} + 13\beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{6} + 14\beta_{5} + 31\beta_{4} - 21\beta_{3} + 3\beta_{2} + 63\beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{6} + 31\beta_{5} + 136\beta_{4} - 139\beta_{3} + 66\beta_{2} + 155\beta _1 + 249 \) 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
3.34903
2.07687
1.58141
0.0938154
−0.944189
−1.61515
−2.54179
1.00000 −3.34903 1.00000 0 −3.34903 −1.00000 1.00000 8.21603 0
1.2 1.00000 −2.07687 1.00000 0 −2.07687 −1.00000 1.00000 1.31339 0
1.3 1.00000 −1.58141 1.00000 0 −1.58141 −1.00000 1.00000 −0.499141 0
1.4 1.00000 −0.0938154 1.00000 0 −0.0938154 −1.00000 1.00000 −2.99120 0
1.5 1.00000 0.944189 1.00000 0 0.944189 −1.00000 1.00000 −2.10851 0
1.6 1.00000 1.61515 1.00000 0 1.61515 −1.00000 1.00000 −0.391285 0
1.7 1.00000 2.54179 1.00000 0 2.54179 −1.00000 1.00000 3.46070 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
\(2\) \( -1 \)
\(5\) \( -1 \)
\(7\) \( +1 \)
\(17\) \( +1 \)

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 5950.2.a.cc 7
5.b even 2 1 5950.2.a.cb 7
5.c odd 4 2 1190.2.e.g ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1190.2.e.g ✓ 14 5.c odd 4 2
5950.2.a.cb 7 5.b even 2 1
5950.2.a.cc 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5950))\):

\( T_{3}^{7} + 2T_{3}^{6} - 12T_{3}^{5} - 17T_{3}^{4} + 40T_{3}^{3} + 32T_{3}^{2} - 40T_{3} - 4 \) Copy content Toggle raw display
\( T_{11}^{7} + 7T_{11}^{6} - 31T_{11}^{5} - 194T_{11}^{4} + 448T_{11}^{3} + 1360T_{11}^{2} - 2376T_{11} - 1264 \) Copy content Toggle raw display
\( T_{13}^{7} - T_{13}^{6} - 56T_{13}^{5} - 36T_{13}^{4} + 884T_{13}^{3} + 1548T_{13}^{2} - 1792T_{13} - 3072 \) Copy content Toggle raw display
\( T_{19}^{7} + 2T_{19}^{6} - 74T_{19}^{5} - 153T_{19}^{4} + 1464T_{19}^{3} + 1956T_{19}^{2} - 9544T_{19} - 2116 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + 2 T^{6} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( (T + 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} + 7 T^{6} + \cdots - 1264 \) Copy content Toggle raw display
$13$ \( T^{7} - T^{6} + \cdots - 3072 \) Copy content Toggle raw display
$17$ \( (T + 1)^{7} \) Copy content Toggle raw display
$19$ \( T^{7} + 2 T^{6} + \cdots - 2116 \) Copy content Toggle raw display
$23$ \( T^{7} + 27 T^{6} + \cdots + 26496 \) Copy content Toggle raw display
$29$ \( T^{7} + 10 T^{6} + \cdots - 3144 \) Copy content Toggle raw display
$31$ \( T^{7} + 18 T^{6} + \cdots + 457056 \) Copy content Toggle raw display
$37$ \( T^{7} + 12 T^{6} + \cdots - 557056 \) Copy content Toggle raw display
$41$ \( T^{7} - 17 T^{6} + \cdots + 457984 \) Copy content Toggle raw display
$43$ \( T^{7} + 7 T^{6} + \cdots - 13056 \) Copy content Toggle raw display
$47$ \( T^{7} + 8 T^{6} + \cdots - 848832 \) Copy content Toggle raw display
$53$ \( T^{7} + 20 T^{6} + \cdots - 155472 \) Copy content Toggle raw display
$59$ \( T^{7} - 326 T^{5} + \cdots + 324468 \) Copy content Toggle raw display
$61$ \( T^{7} - T^{6} + \cdots - 1282048 \) Copy content Toggle raw display
$67$ \( T^{7} + 7 T^{6} + \cdots - 11776 \) Copy content Toggle raw display
$71$ \( T^{7} + 11 T^{6} + \cdots - 1330944 \) Copy content Toggle raw display
$73$ \( T^{7} - 7 T^{6} + \cdots - 260352 \) Copy content Toggle raw display
$79$ \( T^{7} + 26 T^{6} + \cdots + 8699392 \) Copy content Toggle raw display
$83$ \( T^{7} + 22 T^{6} + \cdots - 334912 \) Copy content Toggle raw display
$89$ \( T^{7} - 21 T^{6} + \cdots - 1418432 \) Copy content Toggle raw display
$97$ \( T^{7} - 19 T^{6} + \cdots + 528064 \) Copy content Toggle raw display
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