Properties

Label 6975.2.a.by
Level $6975$
Weight $2$
Character orbit 6975.a
Self dual yes
Analytic conductor $55.696$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6975,2,Mod(1,6975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6975, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6975.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6975.a (trivial)

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: yes
Analytic conductor: \(55.6956554098\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.144209.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 6x^{3} - x^{2} + 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 775)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{2} + ( - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{2} + ( - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{8} + ( - \beta_{4} + 2 \beta_1) q^{11} + ( - 2 \beta_{3} - \beta_{2} - \beta_1) q^{13} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{14} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{16} + ( - \beta_{3} + \beta_1 + 4) q^{17} + (2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 3) q^{19} + ( - 3 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{22} + (2 \beta_{4} + \beta_{3} - \beta_1 + 3) q^{23} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{26} + ( - 3 \beta_{4} - \beta_{2} + 2 \beta_1 - 2) q^{28} + (4 \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{29} - q^{31} + ( - 3 \beta_{4} + \beta_{3} + \beta_1) q^{32} + ( - 2 \beta_{4} - 3 \beta_{3} + 6) q^{34} + (2 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 3) q^{37} + ( - \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{38} + ( - \beta_{4} - 3 \beta_{3} - \beta_{2} - 4 \beta_1 + 1) q^{41} + ( - 2 \beta_{4} + \beta_1 - 1) q^{43} + ( - 4 \beta_{4} - \beta_{3} - 4 \beta_{2} + 3 \beta_1 + 1) q^{44} + (4 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{46} + (3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{47} + (\beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{49} + ( - \beta_{4} - 4 \beta_{3} - \beta_{2} + 6) q^{52} + ( - 4 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{53} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 2) q^{56} + (2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 5 \beta_1 - 2) q^{58} + (5 \beta_{4} - \beta_{3} + \beta_1 + 3) q^{59} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{61} + (\beta_{3} - 1) q^{62} + ( - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{64} + (5 \beta_{4} + \beta_{3} - 2 \beta_{2} + 7 \beta_1) q^{67} + ( - 5 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 6) q^{68} + (4 \beta_{4} - 5 \beta_{3} - 4 \beta_{2} - \beta_1 + 3) q^{71} + ( - \beta_{4} + 5 \beta_{3} + 2 \beta_{2} - 6) q^{73} + (5 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 5 \beta_1 - 1) q^{74} + ( - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 2) q^{76} + ( - \beta_{4} - 4 \beta_{3} - 5 \beta_1 + 5) q^{77} + (2 \beta_{4} + 3 \beta_{3} - 1) q^{79} + ( - \beta_{4} - 6 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 7) q^{82} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 4 \beta_1 + 3) q^{83} + ( - 3 \beta_{4} - 2 \beta_{2} + 3 \beta_1 + 1) q^{86} + ( - 6 \beta_{4} - 4 \beta_{3} - 6 \beta_{2} + 1) q^{88} + (\beta_{4} + \beta_{3} + \beta_{2} + 6) q^{89} + ( - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{91} + (2 \beta_{4} + \beta_{3} + 6 \beta_{2} - 6 \beta_1 - 5) q^{92} + (3 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 4) q^{94} + ( - \beta_{4} - \beta_{3} + 5 \beta_{2} - 4 \beta_1 - 2) q^{97} + (\beta_{4} + \beta_{2} - 5 \beta_1 + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{2} + 6 q^{4} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 4 q^{2} + 6 q^{4} - 2 q^{7} + 9 q^{8} + 2 q^{11} - 4 q^{13} - 2 q^{14} + 4 q^{16} + 19 q^{17} + 8 q^{19} + 10 q^{22} + 12 q^{23} + 16 q^{26} - 6 q^{28} - 6 q^{29} - 5 q^{31} + 7 q^{32} + 31 q^{34} - 16 q^{37} + 14 q^{38} + 2 q^{41} - q^{43} + 4 q^{44} - 11 q^{46} + 8 q^{47} - 9 q^{49} + 26 q^{52} + 3 q^{53} + 9 q^{56} - 5 q^{58} + 4 q^{59} - 5 q^{61} - 4 q^{62} - 5 q^{64} - 13 q^{67} + 30 q^{68} - 6 q^{71} - 19 q^{73} - 4 q^{74} - 5 q^{76} + 23 q^{77} - 6 q^{79} + 27 q^{82} + 18 q^{83} + 7 q^{86} + q^{88} + 31 q^{89} + 8 q^{91} - 16 q^{92} - 14 q^{94} + q^{97} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 6x^{3} - x^{2} + 6x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 5\nu^{2} - 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} + 2\beta _1 + 8 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.43848
0.177477
2.32352
−1.93413
0.871612
−1.70816 0 0.917797 0 0 −1.50771 1.84857 0 0
1.2 −0.264183 0 −1.93021 0 0 2.14598 1.03829 0 0
1.3 1.14876 0 −0.680351 0 0 −1.07521 −3.07908 0 0
1.4 2.23959 0 3.01578 0 0 −3.67497 2.27494 0 0
1.5 2.58398 0 4.67698 0 0 2.11190 6.91727 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(31\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6975.2.a.by 5
3.b odd 2 1 775.2.a.h 5
5.b even 2 1 6975.2.a.bp 5
15.d odd 2 1 775.2.a.k yes 5
15.e even 4 2 775.2.b.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.a.h 5 3.b odd 2 1
775.2.a.k yes 5 15.d odd 2 1
775.2.b.g 10 15.e even 4 2
6975.2.a.bp 5 5.b even 2 1
6975.2.a.by 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6975))\):

\( T_{2}^{5} - 4T_{2}^{4} + 13T_{2}^{2} - 8T_{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{5} + 2T_{7}^{4} - 11T_{7}^{3} - 13T_{7}^{2} + 25T_{7} + 27 \) Copy content Toggle raw display
\( T_{11}^{5} - 2T_{11}^{4} - 37T_{11}^{3} + 41T_{11}^{2} + 295T_{11} + 233 \) Copy content Toggle raw display
\( T_{13}^{5} + 4T_{13}^{4} - 25T_{13}^{3} - 67T_{13}^{2} + 173T_{13} + 153 \) Copy content Toggle raw display
\( T_{17}^{5} - 19T_{17}^{4} + 129T_{17}^{3} - 365T_{17}^{2} + 346T_{17} + 59 \) Copy content Toggle raw display
\( T_{29}^{5} + 6T_{29}^{4} - 87T_{29}^{3} - 498T_{29}^{2} + 1644T_{29} + 8711 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 4 T^{4} + 13 T^{2} - 8 T - 3 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 2 T^{4} - 11 T^{3} - 13 T^{2} + \cdots + 27 \) Copy content Toggle raw display
$11$ \( T^{5} - 2 T^{4} - 37 T^{3} + 41 T^{2} + \cdots + 233 \) Copy content Toggle raw display
$13$ \( T^{5} + 4 T^{4} - 25 T^{3} - 67 T^{2} + \cdots + 153 \) Copy content Toggle raw display
$17$ \( T^{5} - 19 T^{4} + 129 T^{3} + \cdots + 59 \) Copy content Toggle raw display
$19$ \( T^{5} - 8 T^{4} - 3 T^{3} + 94 T^{2} + \cdots + 59 \) Copy content Toggle raw display
$23$ \( T^{5} - 12 T^{4} + 3 T^{3} + 366 T^{2} + \cdots - 503 \) Copy content Toggle raw display
$29$ \( T^{5} + 6 T^{4} - 87 T^{3} + \cdots + 8711 \) Copy content Toggle raw display
$31$ \( (T + 1)^{5} \) Copy content Toggle raw display
$37$ \( T^{5} + 16 T^{4} - 911 T^{2} + \cdots - 4369 \) Copy content Toggle raw display
$41$ \( T^{5} - 2 T^{4} - 127 T^{3} + \cdots + 507 \) Copy content Toggle raw display
$43$ \( T^{5} + T^{4} - 40 T^{3} - 43 T^{2} + \cdots + 17 \) Copy content Toggle raw display
$47$ \( T^{5} - 8 T^{4} - 91 T^{3} + 578 T^{2} + \cdots + 291 \) Copy content Toggle raw display
$53$ \( T^{5} - 3 T^{4} - 102 T^{3} + \cdots - 389 \) Copy content Toggle raw display
$59$ \( T^{5} - 4 T^{4} - 142 T^{3} + \cdots - 15079 \) Copy content Toggle raw display
$61$ \( T^{5} + 5 T^{4} - 29 T^{3} - 3 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$67$ \( T^{5} + 13 T^{4} - 234 T^{3} + \cdots + 124659 \) Copy content Toggle raw display
$71$ \( T^{5} + 6 T^{4} - 279 T^{3} + \cdots - 57279 \) Copy content Toggle raw display
$73$ \( T^{5} + 19 T^{4} - 11 T^{3} + \cdots + 3291 \) Copy content Toggle raw display
$79$ \( T^{5} + 6 T^{4} - 84 T^{3} + \cdots + 1873 \) Copy content Toggle raw display
$83$ \( T^{5} - 18 T^{4} - 33 T^{3} + \cdots + 1983 \) Copy content Toggle raw display
$89$ \( T^{5} - 31 T^{4} + 361 T^{3} + \cdots - 729 \) Copy content Toggle raw display
$97$ \( T^{5} - T^{4} - 271 T^{3} + \cdots - 25961 \) Copy content Toggle raw display
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