Properties

Label 2925.2.c.v
Level $2925$
Weight $2$
Character orbit 2925.c
Analytic conductor $23.356$
Analytic rank $0$
Dimension $4$
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] = [2925,2,Mod(2224,2925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2925, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2925.2224");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.c (of order \(2\), degree \(1\), not 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: \(23.3562425912\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 65)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - q^{4} - 2 \beta_1 q^{7} + \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - q^{4} - 2 \beta_1 q^{7} + \beta_{2} q^{8} + ( - \beta_{3} + 3) q^{11} + \beta_1 q^{13} + 2 \beta_{3} q^{14} - 5 q^{16} + 2 \beta_{2} q^{17} + ( - 3 \beta_{3} + 1) q^{19} + (3 \beta_{2} - 3 \beta_1) q^{22} + ( - \beta_{2} - 3 \beta_1) q^{23} - \beta_{3} q^{26} + 2 \beta_1 q^{28} + ( - 2 \beta_{3} - 6) q^{29} + ( - 3 \beta_{3} + 5) q^{31} - 3 \beta_{2} q^{32} - 6 q^{34} + 4 \beta_1 q^{37} + (\beta_{2} - 9 \beta_1) q^{38} + 2 \beta_{3} q^{41} + (3 \beta_{2} + 5 \beta_1) q^{43} + (\beta_{3} - 3) q^{44} + (3 \beta_{3} + 3) q^{46} + 6 \beta_1 q^{47} + 3 q^{49} - \beta_1 q^{52} + 6 \beta_{2} q^{53} + 2 \beta_{3} q^{56} + ( - 6 \beta_{2} - 6 \beta_1) q^{58} + ( - 7 \beta_{3} - 3) q^{59} + (6 \beta_{3} + 2) q^{61} + (5 \beta_{2} - 9 \beta_1) q^{62} - q^{64} + (6 \beta_{2} + 4 \beta_1) q^{67} - 2 \beta_{2} q^{68} + (\beta_{3} - 3) q^{71} - 4 \beta_1 q^{73} - 4 \beta_{3} q^{74} + (3 \beta_{3} - 1) q^{76} + (2 \beta_{2} - 6 \beta_1) q^{77} + ( - 6 \beta_{3} - 2) q^{79} + 6 \beta_1 q^{82} + 6 \beta_1 q^{83} + ( - 5 \beta_{3} - 9) q^{86} + (3 \beta_{2} - 3 \beta_1) q^{88} + (4 \beta_{3} - 6) q^{89} + 2 q^{91} + (\beta_{2} + 3 \beta_1) q^{92} - 6 \beta_{3} q^{94} - 2 \beta_1 q^{97} + 3 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 12 q^{11} - 20 q^{16} + 4 q^{19} - 24 q^{29} + 20 q^{31} - 24 q^{34} - 12 q^{44} + 12 q^{46} + 12 q^{49} - 12 q^{59} + 8 q^{61} - 4 q^{64} - 12 q^{71} - 4 q^{76} - 8 q^{79} - 36 q^{86} - 24 q^{89} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(2251\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2224.1
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
2224.2 1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
2224.3 1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
2224.4 1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2925.2.c.v 4
3.b odd 2 1 325.2.b.e 4
5.b even 2 1 inner 2925.2.c.v 4
5.c odd 4 1 585.2.a.k 2
5.c odd 4 1 2925.2.a.z 2
15.d odd 2 1 325.2.b.e 4
15.e even 4 1 65.2.a.c 2
15.e even 4 1 325.2.a.g 2
20.e even 4 1 9360.2.a.cm 2
60.l odd 4 1 1040.2.a.h 2
60.l odd 4 1 5200.2.a.ca 2
65.h odd 4 1 7605.2.a.be 2
105.k odd 4 1 3185.2.a.k 2
120.q odd 4 1 4160.2.a.bj 2
120.w even 4 1 4160.2.a.y 2
165.l odd 4 1 7865.2.a.h 2
195.j odd 4 1 845.2.c.e 4
195.s even 4 1 845.2.a.d 2
195.s even 4 1 4225.2.a.w 2
195.u odd 4 1 845.2.c.e 4
195.bc odd 12 1 845.2.m.a 4
195.bc odd 12 1 845.2.m.c 4
195.bf even 12 2 845.2.e.f 4
195.bl even 12 2 845.2.e.e 4
195.bn odd 12 1 845.2.m.a 4
195.bn odd 12 1 845.2.m.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.c 2 15.e even 4 1
325.2.a.g 2 15.e even 4 1
325.2.b.e 4 3.b odd 2 1
325.2.b.e 4 15.d odd 2 1
585.2.a.k 2 5.c odd 4 1
845.2.a.d 2 195.s even 4 1
845.2.c.e 4 195.j odd 4 1
845.2.c.e 4 195.u odd 4 1
845.2.e.e 4 195.bl even 12 2
845.2.e.f 4 195.bf even 12 2
845.2.m.a 4 195.bc odd 12 1
845.2.m.a 4 195.bn odd 12 1
845.2.m.c 4 195.bc odd 12 1
845.2.m.c 4 195.bn odd 12 1
1040.2.a.h 2 60.l odd 4 1
2925.2.a.z 2 5.c odd 4 1
2925.2.c.v 4 1.a even 1 1 trivial
2925.2.c.v 4 5.b even 2 1 inner
3185.2.a.k 2 105.k odd 4 1
4160.2.a.y 2 120.w even 4 1
4160.2.a.bj 2 120.q odd 4 1
4225.2.a.w 2 195.s even 4 1
5200.2.a.ca 2 60.l odd 4 1
7605.2.a.be 2 65.h odd 4 1
7865.2.a.h 2 165.l odd 4 1
9360.2.a.cm 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2925, [\chi])\):

\( T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
$47$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 6 T - 138)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 104)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 248T^{2} + 8464 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 104)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T - 12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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