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Properties

Label 4275.2.a.bx
Level $4275$
Weight $2$
Character orbit 4275.a
Self dual yes
Analytic conductor $34.136$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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] = [4275,2,Mod(1,4275)] 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("4275.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,16,0,0,0,0,0,0,0,0,0] 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: \(34.1360468641\)
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: \( x^{12} - 19x^{10} + 115x^{8} - 288x^{6} + 295x^{4} - 111x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 855)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{6} + 1) q^{4} + \beta_{3} q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{8} - \beta_1 q^{11} + ( - \beta_{9} + \beta_{3}) q^{13} + ( - \beta_{8} + \beta_1) q^{14} + ( - \beta_{11} + 2 \beta_{6} + 3) q^{16}+ \cdots + ( - 5 \beta_{5} - 5 \beta_{4} - \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{4} + 40 q^{16} - 12 q^{19} + 24 q^{31} + 56 q^{34} + 80 q^{46} + 40 q^{49} + 44 q^{61} + 72 q^{64} - 16 q^{76} + 48 q^{79} + 72 q^{91} + 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 19x^{10} + 115x^{8} - 288x^{6} + 295x^{4} - 111x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{11} + 33\nu^{9} - 121\nu^{7} - 177\nu^{5} + 1167\nu^{3} - 756\nu ) / 53 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{10} + 56\nu^{8} - 11\nu^{6} - 363\nu^{4} - 236\nu^{2} + 336 ) / 159 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8\nu^{11} + 185\nu^{9} - 1385\nu^{7} + 3585\nu^{5} - 2063\nu^{3} - 798\nu ) / 159 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} + 10\nu^{8} - 337\nu^{6} + 1493\nu^{4} - 1776\nu^{2} + 378 ) / 53 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -52\nu^{10} + 964\nu^{8} - 5425\nu^{6} + 10821\nu^{4} - 4585\nu^{2} - 894 ) / 477 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -11\nu^{10} + 208\nu^{8} - 1222\nu^{6} + 2710\nu^{4} - 1929\nu^{2} + 400 ) / 53 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -65\nu^{11} + 1205\nu^{9} - 6980\nu^{7} + 16428\nu^{5} - 14993\nu^{3} + 3573\nu ) / 159 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 29\nu^{11} - 505\nu^{9} + 2576\nu^{7} - 4880\nu^{5} + 2821\nu^{3} - 9\nu ) / 53 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -299\nu^{11} + 5543\nu^{9} - 31790\nu^{7} + 70926\nu^{5} - 54149\nu^{3} + 9885\nu ) / 477 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 131\nu^{11} - 2453\nu^{9} + 14312\nu^{7} - 32688\nu^{5} + 26567\nu^{3} - 6609\nu ) / 159 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39\nu^{10} - 670\nu^{8} + 3340\nu^{6} - 6274\nu^{4} + 4035\nu^{2} - 416 ) / 53 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + 3\beta_{9} + \beta_{8} - \beta_{7} - 2\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - 3\beta_{6} + \beta_{4} - 3\beta_{2} + 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{10} + 21\beta_{9} + 9\beta_{8} + 3\beta_{7} - 14\beta_{3} + 8\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{11} - 13\beta_{6} - 9\beta_{5} + 4\beta_{4} - 21\beta_{2} + 50 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 105\beta_{10} + 183\beta_{9} + 93\beta_{8} + 63\beta_{7} - 142\beta_{3} + 108\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -127\beta_{11} - 243\beta_{6} - 246\beta_{5} + 67\beta_{4} - 475\beta_{2} + 902 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1041\beta_{10} + 1767\beta_{9} + 959\beta_{8} + 741\beta_{7} - 1462\beta_{3} + 1204\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -657\beta_{11} - 1202\beta_{6} - 1365\beta_{5} + 316\beta_{4} - 2519\beta_{2} + 4426 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 10547\beta_{10} + 17757\beta_{9} + 9871\beta_{8} + 7917\beta_{7} - 15022\beta_{3} + 12704\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -13519\beta_{11} - 24361\beta_{6} - 28728\beta_{5} + 6303\beta_{4} - 52317\beta_{2} + 89506 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 107793\beta_{10} + 181011\beta_{9} + 101495\beta_{8} + 82353\beta_{7} - 154258\beta_{3} + 131672\beta_1 ) / 8 \) 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
1.05007
−1.05007
−1.82216
1.82216
1.89447
−1.89447
3.20488
−3.20488
0.779075
−0.779075
0.331466
−0.331466
−2.67125 0 5.13555 0 0 −2.04966 −8.37582 0 0
1.2 −2.67125 0 5.13555 0 0 2.04966 −8.37582 0 0
1.3 −1.62885 0 0.653165 0 0 −4.97181 2.19380 0 0
1.4 −1.62885 0 0.653165 0 0 4.97181 2.19380 0 0
1.5 −0.459657 0 −1.78872 0 0 −1.44222 1.74151 0 0
1.6 −0.459657 0 −1.78872 0 0 1.44222 1.74151 0 0
1.7 0.459657 0 −1.78872 0 0 −1.44222 −1.74151 0 0
1.8 0.459657 0 −1.78872 0 0 1.44222 −1.74151 0 0
1.9 1.62885 0 0.653165 0 0 −4.97181 −2.19380 0 0
1.10 1.62885 0 0.653165 0 0 4.97181 −2.19380 0 0
1.11 2.67125 0 5.13555 0 0 −2.04966 8.37582 0 0
1.12 2.67125 0 5.13555 0 0 2.04966 8.37582 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(19\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4275.2.a.bx 12
3.b odd 2 1 inner 4275.2.a.bx 12
5.b even 2 1 inner 4275.2.a.bx 12
5.c odd 4 2 855.2.c.f ✓ 12
15.d odd 2 1 inner 4275.2.a.bx 12
15.e even 4 2 855.2.c.f ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.c.f ✓ 12 5.c odd 4 2
855.2.c.f ✓ 12 15.e even 4 2
4275.2.a.bx 12 1.a even 1 1 trivial
4275.2.a.bx 12 3.b odd 2 1 inner
4275.2.a.bx 12 5.b even 2 1 inner
4275.2.a.bx 12 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4275))\):

\( T_{2}^{6} - 10T_{2}^{4} + 21T_{2}^{2} - 4 \) Copy content Toggle raw display
\( T_{7}^{6} - 31T_{7}^{4} + 164T_{7}^{2} - 216 \) Copy content Toggle raw display
\( T_{11}^{6} - 51T_{11}^{4} + 676T_{11}^{2} - 1944 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 10 T^{4} + 21 T^{2} - 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 31 T^{4} + \cdots - 216)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 51 T^{4} + \cdots - 1944)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 64 T^{4} + \cdots - 96)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 53 T^{4} + \cdots - 1296)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{12} \) Copy content Toggle raw display
$23$ \( (T^{6} - 96 T^{4} + \cdots - 16384)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 96 T^{4} + \cdots - 7776)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} - 32 T - 16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} - 116 T^{4} + \cdots - 55296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 292 T^{4} + \cdots - 812544)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 175 T^{4} + \cdots - 190104)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 37 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 60 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 144 T^{4} + \cdots - 24576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 11 T^{2} + 28 T + 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 112 T^{4} + \cdots - 24576)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 144 T^{4} + \cdots - 24576)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 251 T^{4} + \cdots - 46464)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 12 T^{2} + \cdots + 1408)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 96 T^{4} + \cdots - 4096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 256 T^{4} + \cdots - 161376)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 64 T^{4} + \cdots - 96)^{2} \) Copy content Toggle raw display
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