Properties

Label 425.2.a.g
Level $425$
Weight $2$
Character orbit 425.a
Self dual yes
Analytic conductor $3.394$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,2,Mod(1,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("425.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.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: \(3.39364208590\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.6224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 2x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{2} + (\beta_1 - 1) q^{3} + ( - \beta_{2} - \beta_1 + 1) q^{4} + ( - 2 \beta_{3} + \beta_{2} + 1) q^{6} + ( - \beta_{3} - 2) q^{7} + \beta_{2} q^{8} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{2} + (\beta_1 - 1) q^{3} + ( - \beta_{2} - \beta_1 + 1) q^{4} + ( - 2 \beta_{3} + \beta_{2} + 1) q^{6} + ( - \beta_{3} - 2) q^{7} + \beta_{2} q^{8} + (\beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} - 2) q^{12} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{13} + ( - 3 \beta_{3} + \beta_{2} + \beta_1) q^{14} + (2 \beta_1 - 1) q^{16} + q^{17} + (2 \beta_{3} - 3 \beta_{2}) q^{18} + (2 \beta_{3} + 2 \beta_1) q^{19} + (2 \beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{21} + (2 \beta_{2} + \beta_1 - 3) q^{22} + ( - 2 \beta_{2} - \beta_1 - 1) q^{23} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{24} + ( - 3 \beta_{2} + \beta_1) q^{26} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{27} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 1) q^{28}+ \cdots + (3 \beta_{2} + 2 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 4 q^{3} + 4 q^{4} - 10 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 4 q^{3} + 4 q^{4} - 10 q^{7} + 4 q^{9} - 2 q^{11} - 10 q^{12} - 6 q^{13} - 6 q^{14} - 4 q^{16} + 4 q^{17} + 4 q^{18} + 4 q^{19} + 12 q^{21} - 12 q^{22} - 4 q^{23} - 6 q^{24} - 10 q^{27} - 8 q^{28} - 4 q^{29} - 12 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} - 12 q^{37} + 16 q^{38} - 22 q^{39} - 6 q^{41} + 18 q^{42} - 18 q^{43} + 16 q^{44} - 4 q^{46} - 6 q^{47} + 28 q^{48} + 8 q^{49} - 4 q^{51} - 2 q^{52} - 8 q^{53} + 10 q^{54} - 4 q^{56} + 16 q^{57} - 12 q^{58} - 8 q^{59} + 6 q^{61} + 14 q^{62} - 16 q^{63} - 24 q^{64} + 12 q^{66} - 6 q^{67} + 4 q^{68} + 4 q^{69} - 10 q^{71} + 22 q^{72} - 2 q^{73} + 20 q^{74} - 12 q^{76} + 18 q^{77} + 30 q^{78} - 12 q^{79} - 4 q^{81} + 34 q^{82} + 14 q^{83} + 36 q^{84} + 20 q^{86} - 4 q^{87} - 14 q^{88} + 24 q^{89} + 18 q^{91} + 22 q^{92} + 18 q^{93} + 8 q^{96} + 4 q^{97} + 60 q^{98} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 6x^{2} - 2x + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) 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
0.796815
−1.87228
2.44579
−1.37033
−2.31627 −0.203185 3.36509 0 0.470630 −0.683735 −3.16190 −2.95872 0
1.2 −1.57942 −2.87228 0.494582 0 4.53654 −1.42058 2.37769 5.24997 0
1.3 −0.134632 1.44579 −1.98187 0 −0.194649 −2.86537 0.536087 −0.909700 0
1.4 2.03032 −2.37033 2.12221 0 −4.81252 −5.03032 0.248119 2.61845 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(17\) \( -1 \)

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 425.2.a.g 4
3.b odd 2 1 3825.2.a.bj 4
4.b odd 2 1 6800.2.a.bw 4
5.b even 2 1 425.2.a.h 4
5.c odd 4 2 85.2.b.a 8
15.d odd 2 1 3825.2.a.bh 4
15.e even 4 2 765.2.b.c 8
17.b even 2 1 7225.2.a.v 4
20.d odd 2 1 6800.2.a.bt 4
20.e even 4 2 1360.2.e.d 8
85.c even 2 1 7225.2.a.w 4
85.g odd 4 2 1445.2.b.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.b.a 8 5.c odd 4 2
425.2.a.g 4 1.a even 1 1 trivial
425.2.a.h 4 5.b even 2 1
765.2.b.c 8 15.e even 4 2
1360.2.e.d 8 20.e even 4 2
1445.2.b.e 8 85.g odd 4 2
3825.2.a.bh 4 15.d odd 2 1
3825.2.a.bj 4 3.b odd 2 1
6800.2.a.bt 4 20.d odd 2 1
6800.2.a.bw 4 4.b odd 2 1
7225.2.a.v 4 17.b even 2 1
7225.2.a.w 4 85.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(425))\):

\( T_{2}^{4} + 2T_{2}^{3} - 4T_{2}^{2} - 8T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{4} + 4T_{3}^{3} - 10T_{3} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 10 T^{3} + \cdots + 14 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + \cdots - 164 \) Copy content Toggle raw display
$17$ \( (T - 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} + \cdots + 112 \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + \cdots - 10 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + \cdots - 80 \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + \cdots + 74 \) Copy content Toggle raw display
$37$ \( T^{4} + 12 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} + \cdots + 392 \) Copy content Toggle raw display
$43$ \( T^{4} + 18 T^{3} + \cdots - 316 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} + \cdots - 164 \) Copy content Toggle raw display
$53$ \( T^{4} + 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( (T + 2)^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} + \cdots + 1880 \) Copy content Toggle raw display
$67$ \( T^{4} + 6 T^{3} + \cdots - 52 \) Copy content Toggle raw display
$71$ \( T^{4} + 10 T^{3} + \cdots + 242 \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + \cdots + 1256 \) Copy content Toggle raw display
$79$ \( T^{4} + 12 T^{3} + \cdots - 526 \) Copy content Toggle raw display
$83$ \( T^{4} - 14 T^{3} + \cdots - 956 \) Copy content Toggle raw display
$89$ \( T^{4} - 24 T^{3} + \cdots - 484 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + \cdots + 2000 \) Copy content Toggle raw display
show more
show less