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Properties

Label 1295.2.a.g
Level $1295$
Weight $2$
Character orbit 1295.a
Self dual yes
Analytic conductor $10.341$
Analytic rank $0$
Dimension $12$
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] = [1295,2,Mod(1,1295)] 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("1295.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1295, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1295 = 5 \cdot 7 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1295.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.3406270618\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 19 x^{10} + 17 x^{9} + 131 x^{8} - 103 x^{7} - 395 x^{6} + 263 x^{5} + 495 x^{4} + \cdots + 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{6} + \beta_{2} + 1) q^{6} + q^{7} + ( - \beta_{10} - \beta_{9} - \beta_1) q^{8} + ( - \beta_{7} + \beta_{5} + 2) q^{9}+ \cdots + ( - 2 \beta_{8} + 3 \beta_{7} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 2 q^{3} + 15 q^{4} - 12 q^{5} + 12 q^{6} + 12 q^{7} - 3 q^{8} + 22 q^{9} + q^{10} - 4 q^{11} + 10 q^{13} - q^{14} - 2 q^{15} + 21 q^{16} - 6 q^{18} + 22 q^{19} - 15 q^{20} + 2 q^{21} - 12 q^{22}+ \cdots - 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 19 x^{10} + 17 x^{9} + 131 x^{8} - 103 x^{7} - 395 x^{6} + 263 x^{5} + 495 x^{4} + \cdots + 20 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 351 \nu^{11} - 1188 \nu^{10} + 8082 \nu^{9} + 21376 \nu^{8} - 68507 \nu^{7} - 136200 \nu^{6} + \cdots + 81645 ) / 9565 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 461 \nu^{11} - 647 \nu^{10} - 7917 \nu^{9} + 11384 \nu^{8} + 44822 \nu^{7} - 70215 \nu^{6} + \cdots + 7630 ) / 9565 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 110 \nu^{11} + 78 \nu^{10} - 1748 \nu^{9} - 1674 \nu^{8} + 8836 \nu^{7} + 11667 \nu^{6} - 13632 \nu^{5} + \cdots - 2549 ) / 1913 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 881 \nu^{11} + 697 \nu^{10} + 16852 \nu^{9} - 9514 \nu^{8} - 119602 \nu^{7} + 35755 \nu^{6} + \cdots + 18450 ) / 9565 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1617 \nu^{11} - 1149 \nu^{10} - 29139 \nu^{9} + 16713 \nu^{8} + 186514 \nu^{7} - 81585 \nu^{6} + \cdots - 2845 ) / 9565 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1771 \nu^{11} + 108 \nu^{10} - 34647 \nu^{9} - 6291 \nu^{8} + 247092 \nu^{7} + 73250 \nu^{6} + \cdots - 155245 ) / 9565 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1879 \nu^{11} + 998 \nu^{10} + 36398 \nu^{9} - 15091 \nu^{8} - 255663 \nu^{7} + 73935 \nu^{6} + \cdots + 44985 ) / 9565 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1879 \nu^{11} - 998 \nu^{10} - 36398 \nu^{9} + 15091 \nu^{8} + 255663 \nu^{7} - 73935 \nu^{6} + \cdots - 44985 ) / 9565 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1957 \nu^{11} - 734 \nu^{10} - 38194 \nu^{9} + 9278 \nu^{8} + 269824 \nu^{7} - 30915 \nu^{6} + \cdots - 69505 ) / 9565 \) 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
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{10} + 9\beta_{9} - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 28\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9 \beta_{11} + 11 \beta_{10} + 11 \beta_{9} - 9 \beta_{8} - 11 \beta_{7} - 11 \beta_{6} + 10 \beta_{5} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{11} + 79 \beta_{10} + 66 \beta_{9} - \beta_{8} - 10 \beta_{7} - \beta_{6} + 11 \beta_{5} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 63 \beta_{11} + 95 \beta_{10} + 90 \beta_{9} - 69 \beta_{8} - 93 \beta_{7} - 93 \beta_{6} + 80 \beta_{5} + \cdots + 541 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 36 \beta_{11} + 580 \beta_{10} + 456 \beta_{9} - 14 \beta_{8} - 76 \beta_{7} - 17 \beta_{6} + \cdots + 192 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 403 \beta_{11} + 760 \beta_{10} + 668 \beta_{9} - 510 \beta_{8} - 722 \beta_{7} - 717 \beta_{6} + \cdots + 3506 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 422 \beta_{11} + 4147 \beta_{10} + 3086 \beta_{9} - 147 \beta_{8} - 536 \beta_{7} - 201 \beta_{6} + \cdots + 1818 \) 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
2.69428
2.53700
1.85280
1.65073
0.732505
0.608939
−0.199539
−0.733681
−1.19445
−2.06512
−2.29053
−2.59293
−2.69428 −2.99539 5.25913 −1.00000 8.07042 1.00000 −8.78101 5.97237 2.69428
1.2 −2.53700 0.443718 4.43635 −1.00000 −1.12571 1.00000 −6.18100 −2.80311 2.53700
1.3 −1.85280 2.35440 1.43288 −1.00000 −4.36225 1.00000 1.05076 2.54322 1.85280
1.4 −1.65073 −1.25174 0.724919 −1.00000 2.06629 1.00000 2.10482 −1.43314 1.65073
1.5 −0.732505 −2.96923 −1.46344 −1.00000 2.17497 1.00000 2.53699 5.81630 0.732505
1.6 −0.608939 3.37289 −1.62919 −1.00000 −2.05389 1.00000 2.20996 8.37640 0.608939
1.7 0.199539 −0.673680 −1.96018 −1.00000 −0.134426 1.00000 −0.790212 −2.54616 −0.199539
1.8 0.733681 2.10472 −1.46171 −1.00000 1.54420 1.00000 −2.53979 1.42986 −0.733681
1.9 1.19445 −0.725794 −0.573278 −1.00000 −0.866927 1.00000 −3.07366 −2.47322 −1.19445
1.10 2.06512 −2.27426 2.26473 −1.00000 −4.69663 1.00000 0.546691 2.17227 −2.06512
1.11 2.29053 1.92049 3.24653 −1.00000 4.39895 1.00000 2.85523 0.688291 −2.29053
1.12 2.59293 2.69387 4.72327 −1.00000 6.98500 1.00000 7.06123 4.25692 −2.59293
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( -1 \)
\(37\) \( +1 \)

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 1295.2.a.g ✓ 12
5.b even 2 1 6475.2.a.s 12
7.b odd 2 1 9065.2.a.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1295.2.a.g ✓ 12 1.a even 1 1 trivial
6475.2.a.s 12 5.b even 2 1
9065.2.a.l 12 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + T_{2}^{11} - 19 T_{2}^{10} - 17 T_{2}^{9} + 131 T_{2}^{8} + 103 T_{2}^{7} - 395 T_{2}^{6} + \cdots + 20 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1295))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots + 20 \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 475 \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots + 502960 \) Copy content Toggle raw display
$13$ \( T^{12} - 10 T^{11} + \cdots + 1280 \) Copy content Toggle raw display
$17$ \( T^{12} - 142 T^{10} + \cdots + 60160 \) Copy content Toggle raw display
$19$ \( T^{12} - 22 T^{11} + \cdots - 91220 \) Copy content Toggle raw display
$23$ \( T^{12} - 2 T^{11} + \cdots + 474356 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 1185930496 \) Copy content Toggle raw display
$31$ \( T^{12} - 40 T^{11} + \cdots + 92657344 \) Copy content Toggle raw display
$37$ \( (T + 1)^{12} \) Copy content Toggle raw display
$41$ \( T^{12} - 20 T^{11} + \cdots + 4505600 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 119260480 \) Copy content Toggle raw display
$47$ \( T^{12} + 8 T^{11} + \cdots - 16653125 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 10152412160 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 192068500 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 53297905280 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 1395192832 \) Copy content Toggle raw display
$71$ \( T^{12} + 16 T^{11} + \cdots + 26870000 \) Copy content Toggle raw display
$73$ \( T^{12} - 28 T^{11} + \cdots - 3593792 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 1105662976 \) Copy content Toggle raw display
$83$ \( T^{12} + 34 T^{11} + \cdots + 81503680 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 68758508288 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 4980915200 \) Copy content Toggle raw display
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