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Properties

Label 2275.2.a.x
Level $2275$
Weight $2$
Character orbit 2275.a
Self dual yes
Analytic conductor $18.166$
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] = [2275,2,Mod(1,2275)] 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("2275.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2275 = 5^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.1659664598\)
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: \( x^{7} - 15x^{5} - 2x^{4} + 66x^{3} + 17x^{2} - 72x - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 455)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{5} + \beta_{4} + \beta_1 + 2) q^{4} + ( - \beta_{5} - \beta_{3}) q^{6} - q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{8} + ( - \beta_{5} + \beta_{2} + 3) q^{9}+ \cdots + (2 \beta_{6} - 4 \beta_{5} - 6 \beta_{4} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 16 q^{4} + 2 q^{6} - 7 q^{7} + 6 q^{8} + 21 q^{9} + 6 q^{11} + 5 q^{12} + 7 q^{13} + 34 q^{16} + 2 q^{17} + 15 q^{18} + 6 q^{19} - 26 q^{22} - 4 q^{23} + q^{24} + 6 q^{27} - 16 q^{28} + 16 q^{29}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 15x^{5} - 2x^{4} + 66x^{3} + 17x^{2} - 72x - 19 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 6\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{6} - \nu^{5} - 26\nu^{4} + 12\nu^{3} + 26\nu^{2} - 37\nu + 39 ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 2\nu^{5} + 11\nu^{4} + 17\nu^{3} - 25\nu^{2} - 25\nu + 1 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 2\nu^{5} - 11\nu^{4} - 17\nu^{3} + 32\nu^{2} + 18\nu - 29 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 5\nu^{5} + 18\nu^{4} - 46\nu^{3} - 88\nu^{2} + 73\nu + 71 ) / 14 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 8\beta_{5} + 9\beta_{4} + \beta_{3} + 9\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 9\beta_{2} + 40\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{6} + 61\beta_{5} + 69\beta_{4} + 13\beta_{3} - \beta_{2} + 71\beta _1 + 160 \) 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
−2.70161
−2.26348
−1.33646
−0.264180
1.24464
2.57143
2.74966
−2.70161 0.784452 5.29870 0 −2.11928 −1.00000 −8.91179 −2.38464 0
1.2 −2.26348 −2.09164 3.12334 0 4.73438 −1.00000 −2.54265 1.37496 0
1.3 −1.33646 3.10904 −0.213862 0 −4.15513 −1.00000 2.95875 6.66616 0
1.4 −0.264180 −3.32159 −1.93021 0 0.877498 −1.00000 1.03828 8.03297 0
1.5 1.24464 0.624550 −0.450880 0 0.777337 −1.00000 −3.05045 −2.60994 0
1.6 2.57143 3.23325 4.61227 0 8.31409 −1.00000 6.71728 7.45393 0
1.7 2.74966 −2.33807 5.56065 0 −6.42890 −1.00000 9.79059 2.46656 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
\(5\) \( +1 \)
\(7\) \( +1 \)
\(13\) \( -1 \)

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 2275.2.a.x 7
5.b even 2 1 455.2.a.f ✓ 7
15.d odd 2 1 4095.2.a.bo 7
20.d odd 2 1 7280.2.a.cg 7
35.c odd 2 1 3185.2.a.u 7
65.d even 2 1 5915.2.a.ba 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
455.2.a.f ✓ 7 5.b even 2 1
2275.2.a.x 7 1.a even 1 1 trivial
3185.2.a.u 7 35.c odd 2 1
4095.2.a.bo 7 15.d odd 2 1
5915.2.a.ba 7 65.d even 2 1
7280.2.a.cg 7 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2275))\):

\( T_{2}^{7} - 15T_{2}^{5} - 2T_{2}^{4} + 66T_{2}^{3} + 17T_{2}^{2} - 72T_{2} - 19 \) Copy content Toggle raw display
\( T_{3}^{7} - 21T_{3}^{5} - 2T_{3}^{4} + 127T_{3}^{3} + 16T_{3}^{2} - 184T_{3} + 80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 15 T^{5} + \cdots - 19 \) Copy content Toggle raw display
$3$ \( T^{7} - 21 T^{5} + \cdots + 80 \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( (T + 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} - 6 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( (T - 1)^{7} \) Copy content Toggle raw display
$17$ \( T^{7} - 2 T^{6} + \cdots + 50056 \) Copy content Toggle raw display
$19$ \( T^{7} - 6 T^{6} + \cdots - 5808 \) Copy content Toggle raw display
$23$ \( T^{7} + 4 T^{6} + \cdots + 19200 \) Copy content Toggle raw display
$29$ \( T^{7} - 16 T^{6} + \cdots - 4952 \) Copy content Toggle raw display
$31$ \( T^{7} + 6 T^{6} + \cdots - 120784 \) Copy content Toggle raw display
$37$ \( T^{7} - 6 T^{6} + \cdots - 1000 \) Copy content Toggle raw display
$41$ \( T^{7} - 14 T^{6} + \cdots + 6024 \) Copy content Toggle raw display
$43$ \( T^{7} + 22 T^{6} + \cdots + 108800 \) Copy content Toggle raw display
$47$ \( T^{7} - 10 T^{6} + \cdots - 8192 \) Copy content Toggle raw display
$53$ \( T^{7} - 20 T^{6} + \cdots - 70016 \) Copy content Toggle raw display
$59$ \( T^{7} + 22 T^{6} + \cdots - 912 \) Copy content Toggle raw display
$61$ \( T^{7} - 10 T^{6} + \cdots + 776576 \) Copy content Toggle raw display
$67$ \( T^{7} + 12 T^{6} + \cdots + 153152 \) Copy content Toggle raw display
$71$ \( T^{7} - 12 T^{6} + \cdots + 43776 \) Copy content Toggle raw display
$73$ \( T^{7} + 28 T^{6} + \cdots + 117888 \) Copy content Toggle raw display
$79$ \( T^{7} - 28 T^{6} + \cdots - 75648 \) Copy content Toggle raw display
$83$ \( T^{7} + 6 T^{6} + \cdots + 74752 \) Copy content Toggle raw display
$89$ \( T^{7} + 14 T^{6} + \cdots + 858568 \) Copy content Toggle raw display
$97$ \( T^{7} + 8 T^{6} + \cdots + 47744 \) Copy content Toggle raw display
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