Properties

Label 7581.2.a.v
Level $7581$
Weight $2$
Character orbit 7581.a
Self dual yes
Analytic conductor $60.535$
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] = [7581,2,Mod(1,7581)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7581, 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("7581.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7581 = 3 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7581.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: \(60.5345897723\)
Analytic rank: \(0\)
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: 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{2} + q^{3} + ( - \beta_{2} - \beta_1 + 1) q^{4} + (\beta_{3} - \beta_{2} + 1) q^{5} + ( - \beta_{3} + 1) q^{6} - q^{7} - \beta_{2} q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{2} + q^{3} + ( - \beta_{2} - \beta_1 + 1) q^{4} + (\beta_{3} - \beta_{2} + 1) q^{5} + ( - \beta_{3} + 1) q^{6} - q^{7} - \beta_{2} q^{8} + q^{9} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{10} - 2 \beta_{3} q^{11} + ( - \beta_{2} - \beta_1 + 1) q^{12} + (2 \beta_1 - 2) q^{13} + (\beta_{3} - 1) q^{14} + (\beta_{3} - \beta_{2} + 1) q^{15} + (2 \beta_1 - 1) q^{16} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{17} + ( - \beta_{3} + 1) q^{18} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{20} - q^{21} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{22} + (2 \beta_{2} + 2) q^{23} - \beta_{2} q^{24} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{25} + (4 \beta_{3} - 2 \beta_{2} - 2) q^{26} + q^{27} + (\beta_{2} + \beta_1 - 1) q^{28} + (\beta_{3} + \beta_{2} + 1) q^{29} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{30} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{31} + (3 \beta_{3} - 1) q^{32} - 2 \beta_{3} q^{33} + (4 \beta_{3} - 5 \beta_{2} - \beta_1 + 2) q^{34} + ( - \beta_{3} + \beta_{2} - 1) q^{35} + ( - \beta_{2} - \beta_1 + 1) q^{36} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{37} + (2 \beta_1 - 2) q^{39} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 4) q^{40} + (2 \beta_1 + 4) q^{41} + (\beta_{3} - 1) q^{42} + ( - 2 \beta_{3} + 4 \beta_{2} + 2) q^{43} + ( - 4 \beta_{3} + 2 \beta_1 + 2) q^{44} + (\beta_{3} - \beta_{2} + 1) q^{45} + ( - 2 \beta_{3} + 4 \beta_{2}) q^{46} + ( - \beta_{3} - \beta_{2} + 3) q^{47} + (2 \beta_1 - 1) q^{48} + q^{49} + ( - 5 \beta_{3} + 2 \beta_1 - 1) q^{50} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{51} + ( - 2 \beta_{3} - 4) q^{52} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{53} + ( - \beta_{3} + 1) q^{54} + ( - 6 \beta_{3} + 2 \beta_1 - 2) q^{55} + \beta_{2} q^{56} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{58} + (2 \beta_{3} - 6 \beta_{2} - 4 \beta_1 + 2) q^{59} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{60} + ( - 2 \beta_{3} - 2 \beta_1 + 4) q^{61} + ( - 6 \beta_{3} + 2 \beta_1) q^{62} - q^{63} + ( - 2 \beta_{3} + 3 \beta_{2} - \beta_1 - 5) q^{64} + ( - 6 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 2) q^{65} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{66} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{67} + ( - 5 \beta_{3} - 3 \beta_{2} + 1) q^{68} + (2 \beta_{2} + 2) q^{69} + (2 \beta_{3} + \beta_{2} - \beta_1) q^{70} + ( - 3 \beta_{3} - 3 \beta_{2} - 6 \beta_1 + 5) q^{71} - \beta_{2} q^{72} + (4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{73} + ( - 2 \beta_{3} - 2 \beta_1 + 6) q^{74} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{75} + 2 \beta_{3} q^{77} + (4 \beta_{3} - 2 \beta_{2} - 2) q^{78} + ( - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 6) q^{79} + ( - 5 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{80} + q^{81} + ( - 2 \beta_{3} - 2 \beta_{2} + 4) q^{82} + (\beta_{3} + 3 \beta_{2} + 2 \beta_1 - 7) q^{83} + (\beta_{2} + \beta_1 - 1) q^{84} + ( - 10 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 6) q^{85} + (6 \beta_{2} - 2 \beta_1 + 2) q^{86} + (\beta_{3} + \beta_{2} + 1) q^{87} + ( - 2 \beta_{2} + 2) q^{88} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{89} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{90} + ( - 2 \beta_1 + 2) q^{91} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{92} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{93} + ( - 2 \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{94} + (3 \beta_{3} - 1) q^{96} + ( - 2 \beta_{3} - 2 \beta_1) q^{97} + ( - \beta_{3} + 1) q^{98} - 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} + 6 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} + 6 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9} - 4 q^{10} - 4 q^{11} + 4 q^{12} - 8 q^{13} - 2 q^{14} + 6 q^{15} - 4 q^{16} - 6 q^{17} + 2 q^{18} + 14 q^{20} - 4 q^{21} + 20 q^{22} + 8 q^{23} + 8 q^{25} + 4 q^{27} - 4 q^{28} + 6 q^{29} - 4 q^{30} + 12 q^{31} + 2 q^{32} - 4 q^{33} + 16 q^{34} - 6 q^{35} + 4 q^{36} + 4 q^{37} - 8 q^{39} + 12 q^{40} + 16 q^{41} - 2 q^{42} + 4 q^{43} + 6 q^{45} - 4 q^{46} + 10 q^{47} - 4 q^{48} + 4 q^{49} - 14 q^{50} - 6 q^{51} - 20 q^{52} + 10 q^{53} + 2 q^{54} - 20 q^{55} - 12 q^{58} + 12 q^{59} + 14 q^{60} + 12 q^{61} - 12 q^{62} - 4 q^{63} - 24 q^{64} - 4 q^{65} + 20 q^{66} - 6 q^{68} + 8 q^{69} + 4 q^{70} + 14 q^{71} + 20 q^{74} + 8 q^{75} + 4 q^{77} + 16 q^{79} + 2 q^{80} + 4 q^{81} + 12 q^{82} - 26 q^{83} - 4 q^{84} + 4 q^{85} + 8 q^{86} + 6 q^{87} + 8 q^{88} + 12 q^{89} - 4 q^{90} + 8 q^{91} - 12 q^{92} + 12 q^{93} + 20 q^{94} + 2 q^{96} - 4 q^{97} + 2 q^{98} - 4 q^{99}+O(q^{100}) \) 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
−1.37033
2.44579
−1.87228
0.796815
−2.03032 1.00000 2.12221 3.78220 −2.03032 −1.00000 −0.248119 1.00000 −7.67909
1.2 0.134632 1.00000 −1.98187 1.32928 0.134632 −1.00000 −0.536087 1.00000 0.178964
1.3 1.57942 1.00000 0.494582 −1.95712 1.57942 −1.00000 −2.37769 1.00000 −3.09112
1.4 2.31627 1.00000 3.36509 2.84564 2.31627 −1.00000 3.16190 1.00000 6.59125
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(19\) \(1\)

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 7581.2.a.v yes 4
19.b odd 2 1 7581.2.a.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7581.2.a.q 4 19.b odd 2 1
7581.2.a.v yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7581))\):

\( T_{2}^{4} - 2T_{2}^{3} - 4T_{2}^{2} + 8T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 6T_{5}^{3} + 4T_{5}^{2} + 24T_{5} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 4 T^{2} + 8 T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} + 4 T^{2} + 24 T - 28 \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} - 16 T^{2} - 16 T + 32 \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} - 80 T - 32 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} - 44 T^{2} - 176 T + 580 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} - 8 T^{2} + 144 T - 224 \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} - 4 T^{2} + 72 T - 100 \) Copy content Toggle raw display
$31$ \( T^{4} - 12 T^{3} + 8 T^{2} + 96 T + 16 \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} - 32 T^{2} + 64 T + 128 \) Copy content Toggle raw display
$41$ \( T^{4} - 16 T^{3} + 72 T^{2} - 80 T + 16 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} - 112 T^{2} + \cdots + 752 \) Copy content Toggle raw display
$47$ \( T^{4} - 10 T^{3} + 20 T^{2} - 8 T - 4 \) Copy content Toggle raw display
$53$ \( T^{4} - 10 T^{3} - 4 T^{2} + 64 T + 28 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} - 176 T^{2} + \cdots + 10048 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + 16 T^{2} + 112 T - 80 \) Copy content Toggle raw display
$67$ \( T^{4} - 104 T^{2} + 544 T - 752 \) Copy content Toggle raw display
$71$ \( T^{4} - 14 T^{3} - 156 T^{2} + \cdots - 6956 \) Copy content Toggle raw display
$73$ \( T^{4} - 120 T^{2} + 544 T - 304 \) Copy content Toggle raw display
$79$ \( T^{4} - 16 T^{3} - 88 T^{2} + \cdots - 2864 \) Copy content Toggle raw display
$83$ \( T^{4} + 26 T^{3} + 180 T^{2} + \cdots - 148 \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} - 72 T^{2} + 896 T + 80 \) Copy content Toggle raw display
$97$ \( T^{4} + 4 T^{3} - 32 T^{2} - 80 T + 112 \) Copy content Toggle raw display
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