Properties

Label 425.2.a.i
Level $425$
Weight $2$
Character orbit 425.a
Self dual yes
Analytic conductor $3.394$
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] = [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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1893456.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{4} + \beta_{2} + \beta_1 + 2) q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{6} + ( - \beta_{4} - \beta_{3}) q^{7} + (\beta_{4} - 2 \beta_{2} - \beta_1 - 1) q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{4} + \beta_{2} + \beta_1 + 2) q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{6} + ( - \beta_{4} - \beta_{3}) q^{7} + (\beta_{4} - 2 \beta_{2} - \beta_1 - 1) q^{8} + (\beta_{2} + 1) q^{9} + (\beta_{3} - \beta_{2} + 1) q^{11} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + \cdots + 3) q^{12}+ \cdots + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} + q^{3} + 11 q^{4} + 3 q^{6} + q^{7} - 9 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{2} + q^{3} + 11 q^{4} + 3 q^{6} + q^{7} - 9 q^{8} + 6 q^{9} + 4 q^{11} + 17 q^{12} - 3 q^{13} - 7 q^{14} + 27 q^{16} + 5 q^{17} - 22 q^{18} + 6 q^{19} - 5 q^{21} + 18 q^{22} + 4 q^{23} - 19 q^{24} - 5 q^{26} - 5 q^{27} - 15 q^{28} + 2 q^{29} + 21 q^{31} - 9 q^{32} + 12 q^{33} - q^{34} + 22 q^{36} - 2 q^{37} - 34 q^{38} + 23 q^{39} - 8 q^{41} + 25 q^{42} - 4 q^{43} - 4 q^{44} - 6 q^{46} - 2 q^{47} + 51 q^{48} + 10 q^{49} + q^{51} - 13 q^{52} + 21 q^{53} - 25 q^{54} - 29 q^{56} - 20 q^{57} + 22 q^{58} + 12 q^{59} - 2 q^{61} + 3 q^{62} + 8 q^{63} + 43 q^{64} - 28 q^{66} - 12 q^{67} + 11 q^{68} + 21 q^{71} - 58 q^{72} + 22 q^{73} - 50 q^{74} - 14 q^{76} - 32 q^{77} - 53 q^{78} + 41 q^{79} - 35 q^{81} + 28 q^{82} - 8 q^{83} - 71 q^{84} - 36 q^{86} - 4 q^{87} + 14 q^{88} - 10 q^{89} + 13 q^{91} + 56 q^{92} + 9 q^{93} - 10 q^{94} - 85 q^{96} - 20 q^{97} + 38 q^{98} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 9\nu^{2} - \nu + 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 9\beta_{2} + \beta _1 + 20 \) 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
2.60789
−2.48887
−1.96189
1.66068
1.18219
−2.80107 2.60789 5.84602 0 −7.30489 −1.33298 −10.7730 3.80107 0
1.2 −2.19447 −2.48887 2.81568 0 5.46174 3.05725 −1.78998 3.19447 0
1.3 0.150980 −1.96189 −1.97720 0 −0.296207 −1.54475 −0.600480 0.849020 0
1.4 1.24214 1.66068 −0.457096 0 2.06279 4.35698 −3.05205 −0.242137 0
1.5 2.60242 1.18219 4.77260 0 3.07656 −3.53650 7.21549 −1.60242 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
\(5\) \( +1 \)
\(17\) \( -1 \)

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.i 5
3.b odd 2 1 3825.2.a.bq 5
4.b odd 2 1 6800.2.a.bz 5
5.b even 2 1 425.2.a.j yes 5
5.c odd 4 2 425.2.b.f 10
15.d odd 2 1 3825.2.a.bl 5
17.b even 2 1 7225.2.a.x 5
20.d odd 2 1 6800.2.a.cd 5
85.c even 2 1 7225.2.a.y 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.a.i 5 1.a even 1 1 trivial
425.2.a.j yes 5 5.b even 2 1
425.2.b.f 10 5.c odd 4 2
3825.2.a.bl 5 15.d odd 2 1
3825.2.a.bq 5 3.b odd 2 1
6800.2.a.bz 5 4.b odd 2 1
6800.2.a.cd 5 20.d odd 2 1
7225.2.a.x 5 17.b even 2 1
7225.2.a.y 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(425))\):

\( T_{2}^{5} + T_{2}^{4} - 10T_{2}^{3} - 6T_{2}^{2} + 21T_{2} - 3 \) Copy content Toggle raw display
\( T_{3}^{5} - T_{3}^{4} - 10T_{3}^{3} + 10T_{3}^{2} + 23T_{3} - 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + T^{4} - 10 T^{3} + \cdots - 3 \) Copy content Toggle raw display
$3$ \( T^{5} - T^{4} + \cdots - 25 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - T^{4} + \cdots + 97 \) Copy content Toggle raw display
$11$ \( T^{5} - 4 T^{4} + \cdots + 60 \) Copy content Toggle raw display
$13$ \( T^{5} + 3 T^{4} + \cdots + 227 \) Copy content Toggle raw display
$17$ \( (T - 1)^{5} \) Copy content Toggle raw display
$19$ \( T^{5} - 6 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$23$ \( T^{5} - 4 T^{4} + \cdots + 108 \) Copy content Toggle raw display
$29$ \( T^{5} - 2 T^{4} + \cdots + 240 \) Copy content Toggle raw display
$31$ \( T^{5} - 21 T^{4} + \cdots + 2151 \) Copy content Toggle raw display
$37$ \( T^{5} + 2 T^{4} + \cdots + 3824 \) Copy content Toggle raw display
$41$ \( T^{5} + 8 T^{4} + \cdots + 48 \) Copy content Toggle raw display
$43$ \( T^{5} + 4 T^{4} + \cdots + 14704 \) Copy content Toggle raw display
$47$ \( T^{5} + 2 T^{4} + \cdots + 480 \) Copy content Toggle raw display
$53$ \( T^{5} - 21 T^{4} + \cdots - 9 \) Copy content Toggle raw display
$59$ \( T^{5} - 12 T^{4} + \cdots - 11760 \) Copy content Toggle raw display
$61$ \( T^{5} + 2 T^{4} + \cdots + 800 \) Copy content Toggle raw display
$67$ \( T^{5} + 12 T^{4} + \cdots + 27008 \) Copy content Toggle raw display
$71$ \( T^{5} - 21 T^{4} + \cdots + 11853 \) Copy content Toggle raw display
$73$ \( T^{5} - 22 T^{4} + \cdots - 41744 \) Copy content Toggle raw display
$79$ \( T^{5} - 41 T^{4} + \cdots + 42575 \) Copy content Toggle raw display
$83$ \( T^{5} + 8 T^{4} + \cdots + 5952 \) Copy content Toggle raw display
$89$ \( T^{5} + 10 T^{4} + \cdots - 18000 \) Copy content Toggle raw display
$97$ \( T^{5} + 20 T^{4} + \cdots - 8000 \) Copy content Toggle raw display
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