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Properties

Label 4950.2.f.d
Level $4950$
Weight $2$
Character orbit 4950.f
Analytic conductor $39.526$
Analytic rank $0$
Dimension $8$
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] = [4950,2,Mod(4949,4950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4950.4949");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4950.f (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: \(39.5259490005\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4328587264.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 109x^{4} + 260x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 990)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} + (\beta_{4} + \beta_{3}) q^{7} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{4} + (\beta_{4} + \beta_{3}) q^{7} + \beta_1 q^{8} + (\beta_{6} + \beta_{3} + \cdots - 2 \beta_1) q^{11}+ \cdots + (\beta_{7} + \beta_{5} + \cdots - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 8 q^{13} + 8 q^{16} - 12 q^{22} + 8 q^{23} + 8 q^{29} + 8 q^{31} - 16 q^{34} + 16 q^{38} - 8 q^{41} - 16 q^{43} + 32 q^{47} + 16 q^{49} + 8 q^{52} + 24 q^{53} - 8 q^{64} - 24 q^{73} + 24 q^{74} + 28 q^{77} + 12 q^{88} + 24 q^{91} - 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 109x^{4} + 260x^{2} + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{7} - 40\nu^{5} - 131\nu^{3} - 66\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 18\nu^{5} - 95\nu^{3} - 134\nu ) / 28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} + 29\nu^{5} + 7\nu^{4} + 127\nu^{3} + 63\nu^{2} + 198\nu + 98 ) / 28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} - 29\nu^{5} + 7\nu^{4} - 127\nu^{3} + 63\nu^{2} - 198\nu + 98 ) / 28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} - 14\nu^{5} - 26\nu^{4} - 55\nu^{3} - 84\nu^{2} - 58\nu - 48 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 55\nu^{2} + 58 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 14\nu^{5} + 26\nu^{4} - 55\nu^{3} + 84\nu^{2} - 58\nu + 48 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} - 7\beta_{5} + 5\beta_{4} - 5\beta_{3} + 2\beta_{2} + 18\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{7} - 18\beta_{6} - 9\beta_{5} + 13\beta_{4} + 13\beta_{3} + 62 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 53\beta_{7} + 53\beta_{5} - 31\beta_{4} + 31\beta_{3} - 34\beta_{2} - 142\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -71\beta_{7} + 150\beta_{6} + 71\beta_{5} - 127\beta_{4} - 127\beta_{3} - 434 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -423\beta_{7} - 423\beta_{5} + 217\beta_{4} - 217\beta_{3} + 366\beta_{2} + 1114\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4950\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2377\) \(4501\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4949.1
− 2.18398i
1.87996i
1.18398i
− 2.87996i
2.18398i
− 1.87996i
− 1.18398i
2.87996i
− 1.00000i 0 −1.00000 0 0 −3.08861 1.00000i 0 0
4949.2 − 1.00000i 0 −1.00000 0 0 −2.65867 1.00000i 0 0
4949.3 − 1.00000i 0 −1.00000 0 0 1.67440 1.00000i 0 0
4949.4 − 1.00000i 0 −1.00000 0 0 4.07288 1.00000i 0 0
4949.5 1.00000i 0 −1.00000 0 0 −3.08861 − 1.00000i 0 0
4949.6 1.00000i 0 −1.00000 0 0 −2.65867 − 1.00000i 0 0
4949.7 1.00000i 0 −1.00000 0 0 1.67440 − 1.00000i 0 0
4949.8 1.00000i 0 −1.00000 0 0 4.07288 − 1.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4949.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4950.2.f.d 8
3.b odd 2 1 4950.2.f.c 8
5.b even 2 1 4950.2.f.e 8
5.c odd 4 1 990.2.d.a ✓ 8
5.c odd 4 1 4950.2.d.m 8
11.b odd 2 1 4950.2.f.f 8
15.d odd 2 1 4950.2.f.f 8
15.e even 4 1 990.2.d.b yes 8
15.e even 4 1 4950.2.d.h 8
20.e even 4 1 7920.2.f.a 8
33.d even 2 1 4950.2.f.e 8
55.d odd 2 1 4950.2.f.c 8
55.e even 4 1 990.2.d.b yes 8
55.e even 4 1 4950.2.d.h 8
60.l odd 4 1 7920.2.f.b 8
165.d even 2 1 inner 4950.2.f.d 8
165.l odd 4 1 990.2.d.a ✓ 8
165.l odd 4 1 4950.2.d.m 8
220.i odd 4 1 7920.2.f.b 8
660.q even 4 1 7920.2.f.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
990.2.d.a ✓ 8 5.c odd 4 1
990.2.d.a ✓ 8 165.l odd 4 1
990.2.d.b yes 8 15.e even 4 1
990.2.d.b yes 8 55.e even 4 1
4950.2.d.h 8 15.e even 4 1
4950.2.d.h 8 55.e even 4 1
4950.2.d.m 8 5.c odd 4 1
4950.2.d.m 8 165.l odd 4 1
4950.2.f.c 8 3.b odd 2 1
4950.2.f.c 8 55.d odd 2 1
4950.2.f.d 8 1.a even 1 1 trivial
4950.2.f.d 8 165.d even 2 1 inner
4950.2.f.e 8 5.b even 2 1
4950.2.f.e 8 33.d even 2 1
4950.2.f.f 8 11.b odd 2 1
4950.2.f.f 8 15.d odd 2 1
7920.2.f.a 8 20.e even 4 1
7920.2.f.a 8 660.q even 4 1
7920.2.f.b 8 60.l odd 4 1
7920.2.f.b 8 220.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4950, [\chi])\):

\( T_{7}^{4} - 18T_{7}^{2} - 8T_{7} + 56 \) Copy content Toggle raw display
\( T_{23}^{4} - 4T_{23}^{3} - 62T_{23}^{2} + 224T_{23} - 128 \) Copy content Toggle raw display
\( T_{29}^{4} - 4T_{29}^{3} - 28T_{29}^{2} + 64T_{29} + 224 \) Copy content Toggle raw display
\( T_{41}^{4} + 4T_{41}^{3} - 54T_{41}^{2} + 104T_{41} - 56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 18 T^{2} + \cdots + 56)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 18 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} + \cdots + 188)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 24 T^{2} + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 68 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} + \cdots - 128)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} + \cdots + 224)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + \cdots + 224)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 172 T^{6} + \cdots + 602176 \) Copy content Toggle raw display
$41$ \( (T^{4} + 4 T^{3} - 54 T^{2} + \cdots - 56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{3} + 6 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 16 T^{3} + \cdots + 56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 12 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 140 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$61$ \( T^{8} + 196 T^{6} + \cdots + 1032256 \) Copy content Toggle raw display
$67$ \( (T^{4} + 272 T^{2} + 16448)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 236 T^{6} + \cdots + 200704 \) Copy content Toggle raw display
$73$ \( (T^{4} + 12 T^{3} + \cdots + 32)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 136 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$83$ \( T^{8} + 392 T^{6} + \cdots + 4194304 \) Copy content Toggle raw display
$89$ \( T^{8} + 424 T^{6} + \cdots + 36529936 \) Copy content Toggle raw display
$97$ \( T^{8} + 616 T^{6} + \cdots + 20647936 \) Copy content Toggle raw display
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