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Properties

Label 4050.2.c.z
Level $4050$
Weight $2$
Character orbit 4050.c
Analytic conductor $32.339$
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] = [4050,2,Mod(649,4050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4050, 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("4050.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4050.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: \(32.3394128186\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 810)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} + (\beta_{2} + 2 \beta_1) q^{7} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{4} + (\beta_{2} + 2 \beta_1) q^{7} + \beta_1 q^{8} + 2 \beta_{3} q^{11} + ( - \beta_{2} - 2 \beta_1) q^{13} + (\beta_{3} + 2) q^{14} + q^{16} - 2 \beta_{2} q^{17} + (4 \beta_{3} + 1) q^{19} - 2 \beta_{2} q^{22} + (2 \beta_{2} - 3 \beta_1) q^{23} + ( - \beta_{3} - 2) q^{26} + ( - \beta_{2} - 2 \beta_1) q^{28} - 4 \beta_{3} q^{29} + (2 \beta_{3} + 2) q^{31} - \beta_1 q^{32} - 2 \beta_{3} q^{34} + 8 \beta_1 q^{37} + ( - 4 \beta_{2} - \beta_1) q^{38} - 3 \beta_{3} q^{41} + (2 \beta_{2} - 2 \beta_1) q^{43} - 2 \beta_{3} q^{44} + (2 \beta_{3} - 3) q^{46} + (2 \beta_{2} + 3 \beta_1) q^{47} - 4 \beta_{3} q^{49} + (\beta_{2} + 2 \beta_1) q^{52} + ( - 2 \beta_{2} - 9 \beta_1) q^{53} + ( - \beta_{3} - 2) q^{56} + 4 \beta_{2} q^{58} + 5 \beta_{3} q^{59} + ( - 6 \beta_{3} + 2) q^{61} + ( - 2 \beta_{2} - 2 \beta_1) q^{62} - q^{64} + 8 \beta_1 q^{67} + 2 \beta_{2} q^{68} + 6 q^{71} + ( - 6 \beta_{2} + 4 \beta_1) q^{73} + 8 q^{74} + ( - 4 \beta_{3} - 1) q^{76} + (4 \beta_{2} + 6 \beta_1) q^{77} + ( - 6 \beta_{3} + 4) q^{79} + 3 \beta_{2} q^{82} + (2 \beta_{2} - 12 \beta_1) q^{83} + (2 \beta_{3} - 2) q^{86} + 2 \beta_{2} q^{88} + 12 q^{89} + (4 \beta_{3} + 7) q^{91} + ( - 2 \beta_{2} + 3 \beta_1) q^{92} + (2 \beta_{3} + 3) q^{94} + ( - 4 \beta_{2} + 8 \beta_1) q^{97} + 4 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{14} + 4 q^{16} + 4 q^{19} - 8 q^{26} + 8 q^{31} - 12 q^{46} - 8 q^{56} + 8 q^{61} - 4 q^{64} + 24 q^{71} + 32 q^{74} - 4 q^{76} + 16 q^{79} - 8 q^{86} + 48 q^{89} + 28 q^{91} + 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) 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

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times\).

\(n\) \(2351\) \(3727\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
− 1.00000i 0 −1.00000 0 0 0.267949i 1.00000i 0 0
649.2 − 1.00000i 0 −1.00000 0 0 3.73205i 1.00000i 0 0
649.3 1.00000i 0 −1.00000 0 0 − 3.73205i − 1.00000i 0 0
649.4 1.00000i 0 −1.00000 0 0 − 0.267949i − 1.00000i 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 4050.2.c.z 4
3.b odd 2 1 4050.2.c.x 4
5.b even 2 1 inner 4050.2.c.z 4
5.c odd 4 1 810.2.a.j ✓ 2
5.c odd 4 1 4050.2.a.bt 2
15.d odd 2 1 4050.2.c.x 4
15.e even 4 1 810.2.a.l yes 2
15.e even 4 1 4050.2.a.bk 2
20.e even 4 1 6480.2.a.bb 2
45.k odd 12 2 810.2.e.n 4
45.l even 12 2 810.2.e.m 4
60.l odd 4 1 6480.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
810.2.a.j ✓ 2 5.c odd 4 1
810.2.a.l yes 2 15.e even 4 1
810.2.e.m 4 45.l even 12 2
810.2.e.n 4 45.k odd 12 2
4050.2.a.bk 2 15.e even 4 1
4050.2.a.bt 2 5.c odd 4 1
4050.2.c.x 4 3.b odd 2 1
4050.2.c.x 4 15.d odd 2 1
4050.2.c.z 4 1.a even 1 1 trivial
4050.2.c.z 4 5.b even 2 1 inner
6480.2.a.bb 2 20.e even 4 1
6480.2.a.bj 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4050, [\chi])\):

\( T_{7}^{4} + 14T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{13}^{4} + 14T_{13}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 12 \) Copy content Toggle raw display
\( T_{19}^{2} - 2T_{19} - 47 \) Copy content Toggle raw display
\( T_{29}^{2} - 48 \) Copy content Toggle raw display
\( T_{41}^{2} - 27 \) Copy content Toggle raw display
\( T_{71} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 14T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 14T^{2} + 1 \) Copy content Toggle raw display
$17$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 47)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 42T^{2} + 9 \) Copy content Toggle raw display
$29$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 42T^{2} + 9 \) Copy content Toggle raw display
$53$ \( T^{4} + 186T^{2} + 4761 \) Copy content Toggle raw display
$59$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 104)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 248T^{2} + 8464 \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T - 92)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 312 T^{2} + 17424 \) Copy content Toggle raw display
$89$ \( (T - 12)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 224T^{2} + 256 \) Copy content Toggle raw display
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