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Properties

Label 925.2.c.c
Level $925$
Weight $2$
Character orbit 925.c
Analytic conductor $7.386$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [925,2,Mod(776,925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(925, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("925.776");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{10} + \beta_{9} - 1) q^{4} + (\beta_{8} - \beta_{7} + \beta_{3} + \cdots - \beta_1) q^{6}+ \cdots + ( - \beta_{11} + \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{10} + \beta_{9} - 1) q^{4} + (\beta_{8} - \beta_{7} + \beta_{3} + \cdots - \beta_1) q^{6}+ \cdots + (2 \beta_{10} + 2 \beta_{9} - 4 \beta_{6} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} - 18 q^{4} + 18 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} - 18 q^{4} + 18 q^{7} + 22 q^{9} + 2 q^{11} + 36 q^{12} + 30 q^{16} - 6 q^{21} - 12 q^{26} - 26 q^{27} - 24 q^{28} + 18 q^{33} + 4 q^{34} - 22 q^{36} - 10 q^{37} - 12 q^{38} - 10 q^{41} + 44 q^{44} - 24 q^{46} - 2 q^{47} - 60 q^{48} + 30 q^{49} + 6 q^{53} + 24 q^{58} - 44 q^{62} + 16 q^{63} - 126 q^{64} - 16 q^{67} + 42 q^{71} + 46 q^{73} + 50 q^{74} - 30 q^{77} - 132 q^{78} - 12 q^{81} + 62 q^{83} - 84 q^{84} - 64 q^{86} - 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{11} + 210\nu^{9} + 1376\nu^{7} + 3626\nu^{5} + 3709\nu^{3} + 1100\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} - 20\nu^{9} - 142\nu^{7} - 432\nu^{5} - 553\nu^{3} - 220\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} + 20\nu^{8} + 142\nu^{6} + 432\nu^{4} + 553\nu^{2} + 220 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{11} - 98\nu^{9} - 670\nu^{7} - 1890\nu^{5} - 2097\nu^{3} - 696\nu ) / 28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{10} - 19\nu^{8} - 125\nu^{6} - 337\nu^{4} - 362\nu^{2} - 112 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{11} - 77\nu^{9} - 515\nu^{7} - 1421\nu^{5} - 1581\nu^{3} - 526\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -19\nu^{11} - 364\nu^{9} - 2406\nu^{7} - 6440\nu^{5} - 6591\nu^{3} - 1620\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{10} - 58\nu^{8} - 388\nu^{6} - 1058\nu^{4} - 1121\nu^{2} - 316 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3\nu^{10} + 58\nu^{8} + 388\nu^{6} + 1058\nu^{4} + 1129\nu^{2} + 340 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{10} - 19\nu^{8} - 124\nu^{6} - 327\nu^{4} - 341\nu^{2} - 104 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} - \beta_{5} + \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - 9\beta_{10} - 8\beta_{9} + \beta_{6} + \beta_{4} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{8} + 9\beta_{7} + 10\beta_{5} - 10\beta_{3} + 2\beta_{2} + 41\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{11} + 69\beta_{10} + 59\beta_{9} - 14\beta_{6} - 10\beta_{4} - 95 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 59\beta_{8} - 75\beta_{7} - 81\beta_{5} + 83\beta_{3} - 28\beta_{2} - 292\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -109\beta_{11} - 509\beta_{10} - 434\beta_{9} + 147\beta_{6} + 83\beta_{4} + 655 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -434\beta_{8} + 619\beta_{7} + 618\beta_{5} - 664\beta_{3} + 290\beta_{2} + 2107\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 908\beta_{11} + 3717\beta_{10} + 3205\beta_{9} - 1384\beta_{6} - 664\beta_{4} - 4651 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3205\beta_{8} - 5065\beta_{7} - 4625\beta_{5} + 5253\beta_{3} - 2688\beta_{2} - 15290\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
776.1
− 2.75497i
− 2.59891i
− 1.66705i
− 1.36551i
− 1.23687i
− 0.694469i
0.694469i
1.23687i
1.36551i
1.66705i
2.59891i
2.75497i
− 2.75497i −2.91885 −5.58988 0 8.04135i −1.45148 9.89002i 5.51969 0
776.2 − 2.59891i 0.695617 −4.75431 0 − 1.80784i 3.94647 7.15821i −2.51612 0
776.3 − 1.66705i −0.792969 −0.779058 0 1.32192i 0.457139 − 2.03537i −2.37120 0
776.4 − 1.36551i −2.79310 0.135372 0 3.81402i 4.67865 − 2.91588i 4.80143 0
776.5 − 1.23687i 2.43200 0.470165 0 − 3.00806i 3.53789 − 3.05526i 2.91464 0
776.6 − 0.694469i 2.37730 1.51771 0 − 1.65096i −2.16868 − 2.44294i 2.65157 0
776.7 0.694469i 2.37730 1.51771 0 1.65096i −2.16868 2.44294i 2.65157 0
776.8 1.23687i 2.43200 0.470165 0 3.00806i 3.53789 3.05526i 2.91464 0
776.9 1.36551i −2.79310 0.135372 0 − 3.81402i 4.67865 2.91588i 4.80143 0
776.10 1.66705i −0.792969 −0.779058 0 − 1.32192i 0.457139 2.03537i −2.37120 0
776.11 2.59891i 0.695617 −4.75431 0 1.80784i 3.94647 − 7.15821i −2.51612 0
776.12 2.75497i −2.91885 −5.58988 0 − 8.04135i −1.45148 − 9.89002i 5.51969 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 776.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{2}^{12} + 21T_{2}^{10} + 162T_{2}^{8} + 574T_{2}^{6} + 985T_{2}^{4} + 765T_{2}^{2} + 196 \) Copy content Toggle raw display
\( T_{3}^{6} + T_{3}^{5} - 14T_{3}^{4} - 8T_{3}^{3} + 54T_{3}^{2} + 8T_{3} - 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 21 T^{10} + \cdots + 196 \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{5} - 14 T^{4} + \cdots - 26)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 9 T^{5} + 12 T^{4} + \cdots + 94)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - T^{5} - 44 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 100 T^{10} + \cdots + 3136 \) Copy content Toggle raw display
$17$ \( T^{12} + 68 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( T^{12} + 120 T^{10} + \cdots + 274576 \) Copy content Toggle raw display
$23$ \( T^{12} + 96 T^{10} + \cdots + 4096 \) 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
$43$ \( T^{12} + 132 T^{10} + \cdots + 50176 \) Copy content Toggle raw display
$47$ \( (T^{6} + T^{5} + \cdots - 41098)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 3 T^{5} + \cdots - 109184)^{2} \) 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
$67$ \( (T^{6} + 8 T^{5} + \cdots + 664)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 21 T^{5} + \cdots - 10624)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 23 T^{5} + \cdots - 13888)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 384316816 \) Copy content Toggle raw display
$83$ \( (T^{6} - 31 T^{5} + \cdots - 158806)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 629407744 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 18183983104 \) Copy content Toggle raw display
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