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Properties

Label 925.2.d.f
Level $925$
Weight $2$
Character orbit 925.d
Analytic conductor $7.386$
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] = [925,2,Mod(924,925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("925.924"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,2,0,18,0,0,0,6,-22,0,2,0,20,0,0,30,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
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: \( x^{12} + 21x^{10} + 162x^{8} + 574x^{6} + 985x^{4} + 765x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 185)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{2} q^{3} + ( - \beta_1 + 2) q^{4} + (\beta_{6} + \beta_{4} + \beta_{2}) q^{6} + ( - \beta_{11} - \beta_{4}) q^{7} + ( - \beta_{10} - \beta_{9} - 2 \beta_{3} + 1) q^{8} + (\beta_{9} + \beta_{3} - 2) q^{9}+ \cdots + (\beta_{10} - \beta_{8} - 5 \beta_{3} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 18 q^{4} + 6 q^{8} - 22 q^{9} + 2 q^{11} + 20 q^{13} + 30 q^{16} + 12 q^{17} - 26 q^{18} - 6 q^{21} - 28 q^{22} - 16 q^{23} - 12 q^{26} + 14 q^{32} - 4 q^{34} - 22 q^{36} - 14 q^{37} - 10 q^{41}+ \cdots + 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 21x^{10} + 162x^{8} + 574x^{6} + 985x^{4} + 765x^{2} + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 14\nu^{9} + 36\nu^{7} - 182\nu^{5} - 709\nu^{3} - 488\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 19\nu^{8} + 125\nu^{6} + 337\nu^{4} + 362\nu^{2} + 112 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{11} - 210\nu^{9} - 1376\nu^{7} - 3598\nu^{5} - 3429\nu^{3} - 624\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{11} - 210\nu^{9} - 1376\nu^{7} - 3598\nu^{5} - 3429\nu^{3} - 512\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -11\nu^{11} - 210\nu^{9} - 1376\nu^{7} - 3626\nu^{5} - 3709\nu^{3} - 1100\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{11} + 98\nu^{9} + 670\nu^{7} + 1890\nu^{5} + 2097\nu^{3} + 696\nu ) / 28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - 19\nu^{8} - 124\nu^{6} - 325\nu^{4} - 323\nu^{2} - 80 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{10} - 19\nu^{8} - 124\nu^{6} - 327\nu^{4} - 341\nu^{2} - 104 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -2\nu^{10} - 39\nu^{8} - 265\nu^{6} - 745\nu^{4} - 837\nu^{2} - 264 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -4\nu^{11} - 77\nu^{9} - 515\nu^{7} - 1421\nu^{5} - 1581\nu^{3} - 526\nu ) / 14 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 5\beta_{5} + 7\beta_{4} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + \beta_{8} - 9\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -20\beta_{7} - 4\beta_{6} + 33\beta_{5} - 49\beta_{4} - 20\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{9} - 10\beta_{8} + 4\beta_{3} + 69\beta _1 - 164 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8\beta_{11} + 162\beta_{7} + 56\beta_{6} - 233\beta_{5} + 351\beta_{4} + 166\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -4\beta_{10} - 109\beta_{9} + 79\beta_{8} - 68\beta_{3} - 509\beta _1 + 1168 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 140\beta_{11} - 1236\beta_{7} - 580\beta_{6} + 1673\beta_{5} - 2541\beta_{4} - 1328\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 76\beta_{10} + 908\beta_{9} - 588\beta_{8} + 796\beta_{3} + 3717\beta _1 - 8444 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -1672\beta_{11} + 9250\beta_{7} + 5376\beta_{6} - 12085\beta_{5} + 18495\beta_{4} + 10506\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
924.1
2.59891i
− 2.59891i
− 1.36551i
1.36551i
0.694469i
− 0.694469i
− 1.23687i
1.23687i
1.66705i
− 1.66705i
2.75497i
− 2.75497i
−2.59891 − 0.695617i 4.75431 0 1.80784i 3.94647i −7.15821 2.51612 0
924.2 −2.59891 0.695617i 4.75431 0 − 1.80784i − 3.94647i −7.15821 2.51612 0
924.3 −1.36551 − 2.79310i −0.135372 0 3.81402i − 4.67865i 2.91588 −4.80143 0
924.4 −1.36551 2.79310i −0.135372 0 − 3.81402i 4.67865i 2.91588 −4.80143 0
924.5 −0.694469 − 2.37730i −1.51771 0 1.65096i − 2.16868i 2.44294 −2.65157 0
924.6 −0.694469 2.37730i −1.51771 0 − 1.65096i 2.16868i 2.44294 −2.65157 0
924.7 1.23687 − 2.43200i −0.470165 0 − 3.00806i 3.53789i −3.05526 −2.91464 0
924.8 1.23687 2.43200i −0.470165 0 3.00806i − 3.53789i −3.05526 −2.91464 0
924.9 1.66705 − 0.792969i 0.779058 0 − 1.32192i − 0.457139i −2.03537 2.37120 0
924.10 1.66705 0.792969i 0.779058 0 1.32192i 0.457139i −2.03537 2.37120 0
924.11 2.75497 − 2.91885i 5.58988 0 − 8.04135i 1.45148i 9.89002 −5.51969 0
924.12 2.75497 2.91885i 5.58988 0 8.04135i − 1.45148i 9.89002 −5.51969 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 924.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.d.f 12
5.b even 2 1 925.2.d.e 12
5.c odd 4 1 185.2.c.b ✓ 12
5.c odd 4 1 925.2.c.c 12
15.e even 4 1 1665.2.e.e 12
20.e even 4 1 2960.2.p.h 12
37.b even 2 1 925.2.d.e 12
185.d even 2 1 inner 925.2.d.f 12
185.f even 4 1 6845.2.a.h 6
185.h odd 4 1 185.2.c.b ✓ 12
185.h odd 4 1 925.2.c.c 12
185.k even 4 1 6845.2.a.i 6
555.n even 4 1 1665.2.e.e 12
740.m even 4 1 2960.2.p.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.c.b ✓ 12 5.c odd 4 1
185.2.c.b ✓ 12 185.h odd 4 1
925.2.c.c 12 5.c odd 4 1
925.2.c.c 12 185.h odd 4 1
925.2.d.e 12 5.b even 2 1
925.2.d.e 12 37.b even 2 1
925.2.d.f 12 1.a even 1 1 trivial
925.2.d.f 12 185.d even 2 1 inner
1665.2.e.e 12 15.e even 4 1
1665.2.e.e 12 555.n even 4 1
2960.2.p.h 12 20.e even 4 1
2960.2.p.h 12 740.m even 4 1
6845.2.a.h 6 185.f even 4 1
6845.2.a.i 6 185.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\):

\( T_{2}^{6} - T_{2}^{5} - 10T_{2}^{4} + 8T_{2}^{3} + 23T_{2}^{2} - 11T_{2} - 14 \) Copy content Toggle raw display
\( T_{17}^{6} - 6T_{17}^{5} - 16T_{17}^{4} + 92T_{17}^{3} + 76T_{17}^{2} - 316T_{17} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{5} - 10 T^{4} + \cdots - 14)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 29 T^{10} + \cdots + 676 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 57 T^{10} + \cdots + 8836 \) Copy content Toggle raw display
$11$ \( (T^{6} - T^{5} - 44 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 10 T^{5} + \cdots - 56)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 6 T^{5} - 16 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 120 T^{10} + \cdots + 274576 \) Copy content Toggle raw display
$23$ \( (T^{6} + 8 T^{5} - 16 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 451477504 \) Copy content Toggle raw display
$31$ \( T^{12} + 144 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( (T^{6} + 5 T^{5} + \cdots - 1552)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 6 T^{5} + \cdots - 224)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 1689045604 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 11921145856 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 618616384 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 20699152384 \) Copy content Toggle raw display
$67$ \( T^{12} + 232 T^{10} + \cdots + 440896 \) Copy content Toggle raw display
$71$ \( (T^{6} - 21 T^{5} + \cdots - 10624)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 192876544 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 384316816 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 25219345636 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 629407744 \) Copy content Toggle raw display
$97$ \( (T^{6} + 14 T^{5} + \cdots - 134848)^{2} \) Copy content Toggle raw display
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