Properties

Label 12.13.c.a
Level $12$
Weight $13$
Character orbit 12.c
Analytic conductor $10.968$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [12,13,Mod(5,12)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(12, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("12.5");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 12.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: \(10.9679258073\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4105x^{2} + 385000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{7} \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 75) q^{3} + (\beta_{2} - 3 \beta_1) q^{5} + ( - \beta_{3} + 28 \beta_1 + 3950) q^{7} + ( - 12 \beta_{3} - 27 \beta_{2} + \cdots + 24201) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 75) q^{3} + (\beta_{2} - 3 \beta_1) q^{5} + ( - \beta_{3} + 28 \beta_1 + 3950) q^{7} + ( - 12 \beta_{3} - 27 \beta_{2} + \cdots + 24201) q^{9}+ \cdots + ( - 37353663 \beta_{3} + \cdots + 318016504080) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 300 q^{3} + 15800 q^{7} + 96804 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 300 q^{3} + 15800 q^{7} + 96804 q^{9} + 3432200 q^{13} + 6613920 q^{15} - 2050024 q^{19} + 59979336 q^{21} - 437451260 q^{25} + 508358700 q^{27} - 2519008264 q^{31} + 3795184800 q^{33} - 7466711800 q^{37} + 14132878296 q^{39} - 26119930600 q^{43} + 35876727360 q^{45} - 52127844660 q^{49} + 54522702720 q^{51} - 96029271360 q^{55} + 68929593000 q^{57} - 9307235704 q^{61} + 18020676600 q^{63} + 89055584600 q^{67} - 143365584960 q^{69} + 464142475400 q^{73} - 487877005140 q^{75} + 567022026488 q^{79} - 929051518716 q^{81} + 1015432485120 q^{85} - 976838637600 q^{87} + 762832207984 q^{91} - 1392739649400 q^{93} + 862958525000 q^{97} + 1272066016320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4105x^{2} + 385000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 20\nu^{2} + 4505\nu - 41050 ) / 75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{3} - 20\nu^{2} + 35155\nu - 41050 ) / 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28\nu^{3} + 520\nu^{2} + 126140\nu + 1067300 ) / 75 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 5\beta_{3} - 9\beta_{2} + 157\beta_1 ) / 5184 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{3} - 140\beta _1 - 147780 ) / 72 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -15325\beta_{3} + 40545\beta_{2} - 520085\beta_1 ) / 5184 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
9.79973i
9.79973i
63.3164i
63.3164i
0 −446.724 576.089i 0 11638.0i 0 −24223.1 0 −132316. + 514706.i 0
5.2 0 −446.724 + 576.089i 0 11638.0i 0 −24223.1 0 −132316. 514706.i 0
5.3 0 596.724 418.762i 0 23907.4i 0 32123.1 0 180718. 499770.i 0
5.4 0 596.724 + 418.762i 0 23907.4i 0 32123.1 0 180718. + 499770.i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.13.c.a 4
3.b odd 2 1 inner 12.13.c.a 4
4.b odd 2 1 48.13.e.d 4
8.b even 2 1 192.13.e.f 4
8.d odd 2 1 192.13.e.g 4
12.b even 2 1 48.13.e.d 4
24.f even 2 1 192.13.e.g 4
24.h odd 2 1 192.13.e.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.13.c.a 4 1.a even 1 1 trivial
12.13.c.a 4 3.b odd 2 1 inner
48.13.e.d 4 4.b odd 2 1
48.13.e.d 4 12.b even 2 1
192.13.e.f 4 8.b even 2 1
192.13.e.f 4 24.h odd 2 1
192.13.e.g 4 8.d odd 2 1
192.13.e.g 4 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(12, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7900 T - 778121036)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots - 43470976258556)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{2} + \cdots - 10\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots + 63\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 53\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots + 42\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 23\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 32\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 16\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots - 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 25\!\cdots\!24)^{2} \) Copy content Toggle raw display
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