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Properties

Label 4025.2.a.u
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $1$
Dimension $8$
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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 10x^{6} + 9x^{5} + 28x^{4} - 22x^{3} - 16x^{2} + 7x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{4} - 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{6}+ \cdots + ( - \beta_{6} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{4} - 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{6}+ \cdots + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} - 4 q^{3} + 5 q^{4} - q^{6} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} - 4 q^{3} + 5 q^{4} - q^{6} + 8 q^{7} + 3 q^{11} - 9 q^{12} - 5 q^{13} - q^{14} - q^{16} - 5 q^{17} - 2 q^{18} - 2 q^{19} - 4 q^{21} - 21 q^{22} - 8 q^{23} - 6 q^{24} + 18 q^{26} - 7 q^{27} + 5 q^{28} - 9 q^{29} - 3 q^{31} - 6 q^{32} - 4 q^{33} - 10 q^{34} + 16 q^{36} - 6 q^{37} - 4 q^{38} - 2 q^{39} - 7 q^{41} - q^{42} - 8 q^{43} + 4 q^{44} + q^{46} - 22 q^{47} - 9 q^{48} + 8 q^{49} - 12 q^{51} - 11 q^{52} - 21 q^{53} - 15 q^{54} - 8 q^{57} - 16 q^{58} + 14 q^{59} + 8 q^{61} - 12 q^{62} - 40 q^{64} + 55 q^{66} - 21 q^{67} - 3 q^{68} + 4 q^{69} + 11 q^{71} + q^{72} - 26 q^{73} - 41 q^{74} + 21 q^{76} + 3 q^{77} - 17 q^{78} - 16 q^{79} - 20 q^{81} + q^{82} - 20 q^{83} - 9 q^{84} + 14 q^{86} + 29 q^{87} - 32 q^{88} + 15 q^{89} - 5 q^{91} - 5 q^{92} - 19 q^{93} + 21 q^{94} + 52 q^{96} - q^{97} - q^{98} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 10x^{6} + 9x^{5} + 28x^{4} - 22x^{3} - 16x^{2} + 7x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - \nu^{5} - 8\nu^{4} + 6\nu^{3} + 17\nu^{2} - 7\nu - 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - \nu^{6} - 9\nu^{5} + 8\nu^{4} + 21\nu^{3} - 17\nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - \nu^{6} - 10\nu^{5} + 9\nu^{4} + 27\nu^{3} - 21\nu^{2} - 10\nu + 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 5\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{4} + 7\beta_{3} + \beta_{2} + 19\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} + 9\beta_{4} + 9\beta_{3} + 24\beta_{2} + 2\beta _1 + 70 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10\beta_{7} + 11\beta_{6} + \beta_{5} + 10\beta_{4} + 43\beta_{3} + 10\beta_{2} + 94\beta _1 + 24 \) Copy content Toggle raw display

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

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
2.39863
2.02014
1.09801
0.651939
−0.333224
−0.571357
−2.07739
−2.18675
−2.39863 −2.12900 3.75344 0 5.10670 1.00000 −4.20585 1.53265 0
1.2 −2.02014 1.91409 2.08098 0 −3.86674 1.00000 −0.163592 0.663753 0
1.3 −1.09801 −0.493654 −0.794374 0 0.542037 1.00000 3.06825 −2.75631 0
1.4 −0.651939 −2.38619 −1.57498 0 1.55565 1.00000 2.33067 2.69390 0
1.5 0.333224 −0.161242 −1.88896 0 −0.0537298 1.00000 −1.29590 −2.97400 0
1.6 0.571357 1.62458 −1.67355 0 0.928216 1.00000 −2.09891 −0.360734 0
1.7 2.07739 0.298210 2.31556 0 0.619500 1.00000 0.655538 −2.91107 0
1.8 2.18675 −2.66680 2.78189 0 −5.83163 1.00000 1.70979 4.11181 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.u ✓ 8
5.b even 2 1 4025.2.a.v yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4025.2.a.u ✓ 8 1.a even 1 1 trivial
4025.2.a.v yes 8 5.b even 2 1

Hecke kernels

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

\( T_{2}^{8} + T_{2}^{7} - 10T_{2}^{6} - 9T_{2}^{5} + 28T_{2}^{4} + 22T_{2}^{3} - 16T_{2}^{2} - 7T_{2} + 3 \) Copy content Toggle raw display
\( T_{3}^{8} + 4T_{3}^{7} - 4T_{3}^{6} - 27T_{3}^{5} - 3T_{3}^{4} + 47T_{3}^{3} + 15T_{3}^{2} - 5T_{3} - 1 \) Copy content Toggle raw display
\( T_{11}^{8} - 3T_{11}^{7} - 30T_{11}^{6} + 43T_{11}^{5} + 297T_{11}^{4} - 28T_{11}^{3} - 982T_{11}^{2} - 908T_{11} - 183 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} - 10 T^{6} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} + \cdots - 183 \) Copy content Toggle raw display
$13$ \( T^{8} + 5 T^{7} + \cdots + 19 \) Copy content Toggle raw display
$17$ \( T^{8} + 5 T^{7} + \cdots + 4083 \) Copy content Toggle raw display
$19$ \( T^{8} + 2 T^{7} + \cdots + 675 \) Copy content Toggle raw display
$23$ \( (T + 1)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 9 T^{7} + \cdots + 7311 \) Copy content Toggle raw display
$31$ \( T^{8} + 3 T^{7} + \cdots + 97 \) Copy content Toggle raw display
$37$ \( T^{8} + 6 T^{7} + \cdots - 210905 \) Copy content Toggle raw display
$41$ \( T^{8} + 7 T^{7} + \cdots + 209409 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + \cdots + 356869 \) Copy content Toggle raw display
$47$ \( T^{8} + 22 T^{7} + \cdots - 269835 \) Copy content Toggle raw display
$53$ \( T^{8} + 21 T^{7} + \cdots + 957 \) Copy content Toggle raw display
$59$ \( T^{8} - 14 T^{7} + \cdots - 2725305 \) Copy content Toggle raw display
$61$ \( T^{8} - 8 T^{7} + \cdots + 23917 \) Copy content Toggle raw display
$67$ \( T^{8} + 21 T^{7} + \cdots + 46337 \) Copy content Toggle raw display
$71$ \( T^{8} - 11 T^{7} + \cdots + 88041 \) Copy content Toggle raw display
$73$ \( T^{8} + 26 T^{7} + \cdots + 40969 \) Copy content Toggle raw display
$79$ \( T^{8} + 16 T^{7} + \cdots - 1052537 \) Copy content Toggle raw display
$83$ \( T^{8} + 20 T^{7} + \cdots + 195159 \) Copy content Toggle raw display
$89$ \( T^{8} - 15 T^{7} + \cdots + 19143423 \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} + \cdots + 124907 \) Copy content Toggle raw display
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