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Properties

Label 2275.2.a.ba
Level $2275$
Weight $2$
Character orbit 2275.a
Self dual yes
Analytic conductor $18.166$
Analytic rank $0$
Dimension $10$
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 = [10,-1,4] 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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 16x^{8} + 14x^{7} + 86x^{6} - 67x^{5} - 167x^{4} + 124x^{3} + 62x^{2} - 64x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{9} - \beta_{6} - \beta_{5} + \cdots - 1) q^{6} + q^{7} + ( - \beta_{7} - \beta_{6} - \beta_1 - 1) q^{8} + ( - \beta_{7} - \beta_{3} + 2) q^{9}+ \cdots + ( - 2 \beta_{8} + 2 \beta_{7} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 4 q^{3} + 13 q^{4} + 2 q^{6} + 10 q^{7} - 3 q^{8} + 20 q^{9} + 8 q^{11} + 11 q^{12} + 10 q^{13} - q^{14} + 19 q^{16} + 12 q^{17} - 10 q^{18} + 2 q^{19} + 4 q^{21} - 28 q^{22} + 6 q^{23}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 16x^{8} + 14x^{7} + 86x^{6} - 67x^{5} - 167x^{4} + 124x^{3} + 62x^{2} - 64x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 16\nu^{7} - 2\nu^{6} + 84\nu^{5} + 19\nu^{4} - 152\nu^{3} - 44\nu^{2} + 42\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{9} + \nu^{8} + 32\nu^{7} - 11\nu^{6} - 172\nu^{5} + 38\nu^{4} + 336\nu^{3} - 53\nu^{2} - 139\nu + 43 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{9} - 3\nu^{8} - 114\nu^{7} + 34\nu^{6} + 624\nu^{5} - 125\nu^{4} - 1245\nu^{3} + 194\nu^{2} + 540\nu - 160 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{9} - 4\nu^{8} - 146\nu^{7} + 44\nu^{6} + 796\nu^{5} - 151\nu^{4} - 1578\nu^{3} + 210\nu^{2} + 664\nu - 190 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -9\nu^{9} + 4\nu^{8} + 146\nu^{7} - 44\nu^{6} - 796\nu^{5} + 151\nu^{4} + 1580\nu^{3} - 210\nu^{2} - 674\nu + 188 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 5\nu^{9} - 2\nu^{8} - 81\nu^{7} + 21\nu^{6} + 441\nu^{5} - 67\nu^{4} - 874\nu^{3} + 89\nu^{2} + 371\nu - 97 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\nu^{9} - 5\nu^{8} - 178\nu^{7} + 56\nu^{6} + 968\nu^{5} - 199\nu^{4} - 1917\nu^{3} + 286\nu^{2} + 816\nu - 234 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{5} + \beta_{4} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + \beta_{8} + 9\beta_{7} + 8\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 28\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{9} + 3\beta_{7} + \beta_{6} - 10\beta_{5} + 11\beta_{4} + 47\beta_{2} + 12\beta _1 + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{9} + 12 \beta_{8} + 70 \beta_{7} + 55 \beta_{6} - 13 \beta_{5} + 12 \beta_{4} - 11 \beta_{3} + \cdots + 92 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 108\beta_{9} + 4\beta_{8} + 49\beta_{7} + 15\beta_{6} - 78\beta_{5} + 94\beta_{4} + 318\beta_{2} + 111\beta _1 + 515 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 161 \beta_{9} + 108 \beta_{8} + 522 \beta_{7} + 362 \beta_{6} - 125 \beta_{5} + 111 \beta_{4} + \cdots + 711 \) 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
2.71750
2.38987
1.90093
0.590489
0.557283
0.325442
−0.858932
−1.88895
−2.27890
−2.45474
−2.71750 −2.22684 5.38482 0 6.05144 1.00000 −9.19827 1.95880 0
1.2 −2.38987 3.28504 3.71148 0 −7.85083 1.00000 −4.09022 7.79152 0
1.3 −1.90093 0.527099 1.61353 0 −1.00198 1.00000 0.734648 −2.72217 0
1.4 −0.590489 −2.53915 −1.65132 0 1.49934 1.00000 2.15607 3.44730 0
1.5 −0.557283 3.24228 −1.68944 0 −1.80686 1.00000 2.05606 7.51235 0
1.6 −0.325442 0.168695 −1.89409 0 −0.0549004 1.00000 1.26730 −2.97154 0
1.7 0.858932 −0.269566 −1.26224 0 −0.231539 1.00000 −2.80204 −2.92733 0
1.8 1.88895 −2.55412 1.56814 0 −4.82460 1.00000 −0.815771 3.52350 0
1.9 2.27890 2.83679 3.19338 0 6.46476 1.00000 2.71958 5.04739 0
1.10 2.45474 1.52977 4.02573 0 3.75517 1.00000 4.97264 −0.659817 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.ba 10
5.b even 2 1 2275.2.a.bb 10
5.c odd 4 2 455.2.c.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
455.2.c.c ✓ 20 5.c odd 4 2
2275.2.a.ba 10 1.a even 1 1 trivial
2275.2.a.bb 10 5.b even 2 1

Hecke kernels

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

\( T_{2}^{10} + T_{2}^{9} - 16T_{2}^{8} - 14T_{2}^{7} + 86T_{2}^{6} + 67T_{2}^{5} - 167T_{2}^{4} - 124T_{2}^{3} + 62T_{2}^{2} + 64T_{2} + 12 \) Copy content Toggle raw display
\( T_{3}^{10} - 4 T_{3}^{9} - 17 T_{3}^{8} + 72 T_{3}^{7} + 91 T_{3}^{6} - 424 T_{3}^{5} - 110 T_{3}^{4} + \cdots + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + \cdots + 12 \) Copy content Toggle raw display
$3$ \( T^{10} - 4 T^{9} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T - 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 8 T^{9} + \cdots - 256 \) Copy content Toggle raw display
$13$ \( (T - 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} - 12 T^{9} + \cdots + 48 \) Copy content Toggle raw display
$19$ \( T^{10} - 2 T^{9} + \cdots + 6860 \) Copy content Toggle raw display
$23$ \( T^{10} - 6 T^{9} + \cdots + 10304 \) Copy content Toggle raw display
$29$ \( T^{10} - 30 T^{9} + \cdots + 235740 \) Copy content Toggle raw display
$31$ \( T^{10} - 2 T^{9} + \cdots + 265372 \) Copy content Toggle raw display
$37$ \( T^{10} + 2 T^{9} + \cdots + 172560 \) Copy content Toggle raw display
$41$ \( T^{10} - 32 T^{9} + \cdots + 25968640 \) Copy content Toggle raw display
$43$ \( T^{10} + 6 T^{9} + \cdots - 846528 \) Copy content Toggle raw display
$47$ \( T^{10} + 2 T^{9} + \cdots - 7442432 \) Copy content Toggle raw display
$53$ \( T^{10} + 4 T^{9} + \cdots + 313600 \) Copy content Toggle raw display
$59$ \( T^{10} - 4 T^{9} + \cdots - 10829760 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 263961600 \) Copy content Toggle raw display
$67$ \( T^{10} + 10 T^{9} + \cdots - 920320 \) Copy content Toggle raw display
$71$ \( T^{10} - 36 T^{9} + \cdots + 37056512 \) Copy content Toggle raw display
$73$ \( T^{10} + 6 T^{9} + \cdots - 14714816 \) Copy content Toggle raw display
$79$ \( T^{10} + 24 T^{9} + \cdots - 15997760 \) Copy content Toggle raw display
$83$ \( T^{10} - 18 T^{9} + \cdots - 768 \) Copy content Toggle raw display
$89$ \( T^{10} - 14 T^{9} + \cdots + 48683600 \) Copy content Toggle raw display
$97$ \( T^{10} + 2 T^{9} + \cdots - 159168 \) Copy content Toggle raw display
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