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Properties

Label 1275.2.d.i
Level $1275$
Weight $2$
Character orbit 1275.d
Analytic conductor $10.181$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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] = [1275,2,Mod(424,1275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) 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("1275.424"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1275.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-12,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.1809262577\)
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: \( x^{12} + 20x^{10} + 144x^{8} + 452x^{6} + 604x^{4} + 268x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} - 1) q^{4} - \beta_1 q^{6} + (\beta_{6} - \beta_{3}) q^{7} + (\beta_{10} - \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{8} + q^{9} + ( - \beta_{5} + \beta_{4}) q^{11}+ \cdots + ( - \beta_{5} + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} - 16 q^{4} + 12 q^{9} + 16 q^{12} + 32 q^{16} - 6 q^{17} + 4 q^{19} + 12 q^{22} - 16 q^{23} - 36 q^{26} - 12 q^{27} - 36 q^{28} + 24 q^{34} - 16 q^{36} - 4 q^{37} - 32 q^{48} + 16 q^{49}+ \cdots - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 20x^{10} + 144x^{8} + 452x^{6} + 604x^{4} + 268x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 10\nu^{4} + 22\nu^{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} - 18\nu^{9} - 112\nu^{7} - 286\nu^{5} - 304\nu^{3} - 128\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + 20\nu^{9} + 144\nu^{7} + 446\nu^{5} + 544\nu^{3} + 136\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} + 19\nu^{8} + 128\nu^{6} + 366\nu^{4} + 418\nu^{2} + 102 ) / 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} + 2 \nu^{9} + 22 \nu^{8} + 32 \nu^{7} + 170 \nu^{6} + 166 \nu^{5} + 546 \nu^{4} + 300 \nu^{3} + \cdots + 168 ) / 12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{10} + 2 \nu^{9} - 22 \nu^{8} + 32 \nu^{7} - 170 \nu^{6} + 166 \nu^{5} - 546 \nu^{4} + \cdots - 168 ) / 12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{11} - 58\nu^{9} - 394\nu^{7} - 1106\nu^{5} - 1152\nu^{3} - 184\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{11} - 20\nu^{9} - 142\nu^{7} - 426\nu^{5} - 500\nu^{3} - 124\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -2\nu^{10} - 38\nu^{8} - 253\nu^{6} - 696\nu^{4} - 728\nu^{2} - 156 ) / 6 \) 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
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 2\beta_{6} - \beta_{3} - 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{10} + 10\beta_{9} - 9\beta_{8} - 9\beta_{7} + 8\beta_{5} + 8\beta_{4} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{11} - 20\beta_{6} + 12\beta_{3} + 48\beta_{2} - 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 80\beta_{10} - 78\beta_{9} + 68\beta_{8} + 68\beta_{7} - 52\beta_{5} - 58\beta_{4} - 196\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 80\beta_{11} - 2\beta_{8} + 2\beta_{7} + 158\beta_{6} - 108\beta_{3} - 334\beta_{2} + 664 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -600\beta_{10} + 568\beta_{9} - 488\beta_{8} - 488\beta_{7} + 318\beta_{5} + 414\beta_{4} + 1332\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -606\beta_{11} + 38\beta_{8} - 38\beta_{7} - 1168\beta_{6} + 882\beta_{3} + 2346\beta_{2} - 4520 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4396\beta_{10} - 4044\beta_{9} + 3438\beta_{8} + 3438\beta_{7} - 1884\beta_{5} - 2952\beta_{4} - 9212\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1275\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(751\) \(851\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1
− 2.67247i
− 2.60111i
− 1.74480i
− 1.54662i
− 0.632511i
− 0.505687i
0.505687i
0.632511i
1.54662i
1.74480i
2.60111i
2.67247i
− 2.67247i −1.00000 −5.14207 0 2.67247i 3.95896 8.39707i 1.00000 0
424.2 − 2.60111i −1.00000 −4.76576 0 2.60111i 0.892059 7.19404i 1.00000 0
424.3 − 1.74480i −1.00000 −1.04434 0 1.74480i −4.98357 − 1.66744i 1.00000 0
424.4 − 1.54662i −1.00000 −0.392039 0 1.54662i −0.429930 − 2.48691i 1.00000 0
424.5 − 0.632511i −1.00000 1.59993 0 0.632511i −1.76227 − 2.27700i 1.00000 0
424.6 − 0.505687i −1.00000 1.74428 0 0.505687i 2.32475 − 1.89343i 1.00000 0
424.7 0.505687i −1.00000 1.74428 0 − 0.505687i 2.32475 1.89343i 1.00000 0
424.8 0.632511i −1.00000 1.59993 0 − 0.632511i −1.76227 2.27700i 1.00000 0
424.9 1.54662i −1.00000 −0.392039 0 − 1.54662i −0.429930 2.48691i 1.00000 0
424.10 1.74480i −1.00000 −1.04434 0 − 1.74480i −4.98357 1.66744i 1.00000 0
424.11 2.60111i −1.00000 −4.76576 0 − 2.60111i 0.892059 − 7.19404i 1.00000 0
424.12 2.67247i −1.00000 −5.14207 0 − 2.67247i 3.95896 − 8.39707i 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1275.2.d.i 12
5.b even 2 1 1275.2.d.j 12
5.c odd 4 1 1275.2.g.e ✓ 12
5.c odd 4 1 1275.2.g.f yes 12
17.b even 2 1 1275.2.d.j 12
85.c even 2 1 inner 1275.2.d.i 12
85.g odd 4 1 1275.2.g.e ✓ 12
85.g odd 4 1 1275.2.g.f yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1275.2.d.i 12 1.a even 1 1 trivial
1275.2.d.i 12 85.c even 2 1 inner
1275.2.d.j 12 5.b even 2 1
1275.2.d.j 12 17.b even 2 1
1275.2.g.e ✓ 12 5.c odd 4 1
1275.2.g.e ✓ 12 85.g odd 4 1
1275.2.g.f yes 12 5.c odd 4 1
1275.2.g.f yes 12 85.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1275, [\chi])\):

\( T_{2}^{12} + 20T_{2}^{10} + 144T_{2}^{8} + 452T_{2}^{6} + 604T_{2}^{4} + 268T_{2}^{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{6} - 25T_{7}^{4} + 18T_{7}^{3} + 87T_{7}^{2} - 40T_{7} - 31 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 20 T^{10} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 25 T^{4} + \cdots - 31)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 48 T^{10} + \cdots + 14400 \) Copy content Toggle raw display
$13$ \( T^{12} + 78 T^{10} + \cdots + 143641 \) Copy content Toggle raw display
$17$ \( T^{12} + 6 T^{11} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( (T^{6} - 2 T^{5} + \cdots + 645)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 8 T^{5} + \cdots - 382)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 104 T^{10} + \cdots + 3348900 \) Copy content Toggle raw display
$31$ \( T^{12} + 210 T^{10} + \cdots + 1113025 \) Copy content Toggle raw display
$37$ \( (T^{6} + 2 T^{5} + \cdots + 63504)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 2040328900 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 3724538841 \) Copy content Toggle raw display
$47$ \( T^{12} + 212 T^{10} + \cdots + 1623076 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 335549124 \) Copy content Toggle raw display
$59$ \( (T^{6} + 16 T^{5} + \cdots - 840)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 282072025 \) Copy content Toggle raw display
$67$ \( T^{12} + 286 T^{10} + \cdots + 15186609 \) Copy content Toggle raw display
$71$ \( T^{12} + 260 T^{10} + \cdots + 202500 \) Copy content Toggle raw display
$73$ \( (T^{6} - 22 T^{5} + \cdots + 878580)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 114154785424 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 129004164 \) Copy content Toggle raw display
$89$ \( (T^{6} + 2 T^{5} + \cdots + 150)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 8 T^{5} + \cdots - 73551)^{2} \) Copy content Toggle raw display
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