Properties

Label 4025.2.a.n
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $0$
Dimension $4$
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] = [4025,2,Mod(1,4025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, 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("4025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.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: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.22545.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (\beta_{3} + 3) q^{4} + ( - \beta_{2} - 3 \beta_1 + 1) q^{6} + q^{7} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{8} + (\beta_{2} + \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (\beta_{3} + 3) q^{4} + ( - \beta_{2} - 3 \beta_1 + 1) q^{6} + q^{7} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{8} + (\beta_{2} + \beta_1 + 4) q^{9} + (\beta_{2} + 1) q^{11} + ( - 2 \beta_{3} + \beta_{2} - 4 \beta_1 - 5) q^{12} + (\beta_1 + 4) q^{13} + \beta_{2} q^{14} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 5) q^{16} + ( - \beta_{2} - 2 \beta_1 + 3) q^{17} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 5) q^{18} + ( - \beta_{2} + \beta_1 + 1) q^{19} + ( - \beta_{3} - \beta_1) q^{21} + (\beta_{3} + \beta_{2} + 5) q^{22} - q^{23} + ( - \beta_{3} - 5 \beta_{2} - 4 \beta_1 + 5) q^{24} + (\beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{26} + ( - 2 \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 1) q^{27} + (\beta_{3} + 3) q^{28} + (2 \beta_1 + 2) q^{29} + ( - \beta_{2} + 3 \beta_1 - 1) q^{31} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 10) q^{32} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 + 1) q^{33} + ( - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1 - 5) q^{34} + (4 \beta_{3} + 5 \beta_{2} + 4 \beta_1 + 10) q^{36} + (\beta_{2} - 2 \beta_1 - 1) q^{37} + (\beta_{2} + 2 \beta_1 - 5) q^{38} + ( - 5 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{39} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{41} + ( - \beta_{2} - 3 \beta_1 + 1) q^{42} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{43} + (4 \beta_{2} + \beta_1 + 2) q^{44} - \beta_{2} q^{46} + (\beta_{2} - \beta_1 + 3) q^{47} + ( - 4 \beta_{3} + 2 \beta_{2} - \beta_1 - 14) q^{48} + q^{49} + ( - \beta_{3} + 5 \beta_{2} + 3) q^{51} + (5 \beta_{3} + \beta_{2} + 3 \beta_1 + 11) q^{52} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 6) q^{53} + ( - 5 \beta_{3} - 3 \beta_{2} - 10 \beta_1 - 13) q^{54} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{56} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{57} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{58} + ( - \beta_1 + 6) q^{59} + ( - \beta_{3} - 2 \beta_{2} - 4) q^{61} + (2 \beta_{3} - \beta_{2} + 6 \beta_1 - 5) q^{62} + (\beta_{2} + \beta_1 + 4) q^{63} + (2 \beta_{3} - 7 \beta_{2} + \beta_1 + 6) q^{64} + ( - 4 \beta_{3} - 9 \beta_1 - 4) q^{66} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{67} + (2 \beta_{3} - 6 \beta_{2} - 7 \beta_1 + 12) q^{68} + (\beta_{3} + \beta_1) q^{69} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 + 5) q^{71} + (\beta_{3} + 6 \beta_{2} + 8 \beta_1 + 11) q^{72} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 4) q^{73} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 + 5) q^{74} + (3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{76} + (\beta_{2} + 1) q^{77} + ( - \beta_{3} - 7 \beta_{2} - 13 \beta_1 - 5) q^{78} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{79} + (3 \beta_{3} + 6 \beta_{2} + 9 \beta_1 + 3) q^{81} + ( - 5 \beta_{3} + 4 \beta_{2} - 5 \beta_1 - 6) q^{82} + (2 \beta_{3} + 5 \beta_1) q^{83} + ( - 2 \beta_{3} + \beta_{2} - 4 \beta_1 - 5) q^{84} + ( - 2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 11) q^{86} + ( - 4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 4) q^{87} + (3 \beta_{3} + 2 \beta_1 + 10) q^{88} + (2 \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{89} + (\beta_1 + 4) q^{91} + ( - \beta_{3} - 3) q^{92} + ( - 2 \beta_{3} - 5 \beta_{2} + 4 \beta_1 - 7) q^{93} + (3 \beta_{2} - 2 \beta_1 + 5) q^{94} + (7 \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 4) q^{96} + ( - 3 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{97} + \beta_{2} q^{98} + (2 \beta_{3} + 5 \beta_{2} + 3 \beta_1 + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 11 q^{4} + 4 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 11 q^{4} + 4 q^{7} + 18 q^{9} + 5 q^{11} - 21 q^{12} + 17 q^{13} + q^{14} + 17 q^{16} + 9 q^{17} + 24 q^{18} + 4 q^{19} + 20 q^{22} - 4 q^{23} + 12 q^{24} + 5 q^{26} - 9 q^{27} + 11 q^{28} + 10 q^{29} - 2 q^{31} - 34 q^{32} - 18 q^{34} + 45 q^{36} - 5 q^{37} - 17 q^{38} - 9 q^{39} + 7 q^{41} + 6 q^{43} + 13 q^{44} - q^{46} + 12 q^{47} - 51 q^{48} + 4 q^{49} + 18 q^{51} + 43 q^{52} - 23 q^{53} - 60 q^{54} - 9 q^{57} + 4 q^{58} + 23 q^{59} - 17 q^{61} - 17 q^{62} + 18 q^{63} + 16 q^{64} - 21 q^{66} - 5 q^{67} + 33 q^{68} + 20 q^{71} + 57 q^{72} + 15 q^{73} + 16 q^{74} + 8 q^{76} + 5 q^{77} - 39 q^{78} + 12 q^{79} + 24 q^{81} - 20 q^{82} + 3 q^{83} - 21 q^{84} - 36 q^{86} - 18 q^{87} + 39 q^{88} - 11 q^{89} + 17 q^{91} - 11 q^{92} - 27 q^{93} + 21 q^{94} + 3 q^{96} - q^{97} + q^{98} + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} + 5x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) 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
−0.519120
1.45106
−2.27060
2.33866
−2.73051 −1.93659 5.45571 0 5.28788 1.00000 −9.43585 0.750366 0
1.2 −0.894434 2.74893 −1.19999 0 −2.45874 1.00000 2.86218 4.55662 0
1.3 2.15561 2.62393 2.64667 0 5.65618 1.00000 1.39396 3.88502 0
1.4 2.46934 −3.43628 4.09762 0 −8.48532 1.00000 5.17972 8.80800 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

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 4025.2.a.n 4
5.b even 2 1 805.2.a.h 4
15.d odd 2 1 7245.2.a.be 4
35.c odd 2 1 5635.2.a.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.h 4 5.b even 2 1
4025.2.a.n 4 1.a even 1 1 trivial
5635.2.a.t 4 35.c odd 2 1
7245.2.a.be 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{4} - T_{2}^{3} - 9T_{2}^{2} + 8T_{2} + 13 \) Copy content Toggle raw display
\( T_{3}^{4} - 15T_{3}^{2} + 3T_{3} + 48 \) Copy content Toggle raw display
\( T_{11}^{4} - 5T_{11}^{3} + 19T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 9 T^{2} + 8 T + 13 \) Copy content Toggle raw display
$3$ \( T^{4} - 15 T^{2} + 3 T + 48 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 5 T^{3} + 19 T - 2 \) Copy content Toggle raw display
$13$ \( T^{4} - 17 T^{3} + 102 T^{2} + \cdots + 208 \) Copy content Toggle raw display
$17$ \( T^{4} - 9 T^{3} - 6 T^{2} + 165 T - 150 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} - 9 T^{2} + 47 T - 32 \) Copy content Toggle raw display
$23$ \( (T + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 10 T^{3} + 12 T^{2} + 80 T - 80 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} - 63 T^{2} + 161 T - 26 \) Copy content Toggle raw display
$37$ \( T^{4} + 5 T^{3} - 24 T^{2} - 169 T - 236 \) Copy content Toggle raw display
$41$ \( T^{4} - 7 T^{3} - 75 T^{2} + \cdots + 1786 \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} - 39 T^{2} + 69 T - 24 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + 39 T^{2} - 39 T + 12 \) Copy content Toggle raw display
$53$ \( T^{4} + 23 T^{3} + 132 T^{2} + \cdots - 1832 \) Copy content Toggle raw display
$59$ \( T^{4} - 23 T^{3} + 192 T^{2} + \cdots + 898 \) Copy content Toggle raw display
$61$ \( T^{4} + 17 T^{3} + 60 T^{2} + \cdots - 158 \) Copy content Toggle raw display
$67$ \( T^{4} + 5 T^{3} - 114 T^{2} + \cdots - 2036 \) Copy content Toggle raw display
$71$ \( T^{4} - 20 T^{3} + 9 T^{2} + \cdots - 2696 \) Copy content Toggle raw display
$73$ \( T^{4} - 15 T^{3} - 102 T^{2} + \cdots + 3396 \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} + 15 T^{2} + \cdots - 474 \) Copy content Toggle raw display
$83$ \( T^{4} - 3 T^{3} - 168 T^{2} + \cdots + 444 \) Copy content Toggle raw display
$89$ \( T^{4} + 11 T^{3} - 150 T^{2} + \cdots + 6922 \) Copy content Toggle raw display
$97$ \( T^{4} + T^{3} - 135 T^{2} - 905 T - 1658 \) Copy content Toggle raw display
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