Properties

Label 7406.2.a.l
Level $7406$
Weight $2$
Character orbit 7406.a
Self dual yes
Analytic conductor $59.137$
Analytic rank $1$
Dimension $2$
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] = [7406,2,Mod(1,7406)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7406, 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("7406.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7406.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: \(59.1372077370\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta q^{3} + q^{4} + ( - \beta + 1) q^{5} + \beta q^{6} + q^{7} - q^{8} + \beta q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta q^{3} + q^{4} + ( - \beta + 1) q^{5} + \beta q^{6} + q^{7} - q^{8} + \beta q^{9} + (\beta - 1) q^{10} + ( - \beta - 1) q^{11} - \beta q^{12} + ( - \beta + 4) q^{13} - q^{14} + 3 q^{15} + q^{16} + (2 \beta - 5) q^{17} - \beta q^{18} + (2 \beta - 5) q^{19} + ( - \beta + 1) q^{20} - \beta q^{21} + (\beta + 1) q^{22} + \beta q^{24} + ( - \beta - 1) q^{25} + (\beta - 4) q^{26} + (2 \beta - 3) q^{27} + q^{28} + (2 \beta + 2) q^{29} - 3 q^{30} + ( - 2 \beta - 3) q^{31} - q^{32} + (2 \beta + 3) q^{33} + ( - 2 \beta + 5) q^{34} + ( - \beta + 1) q^{35} + \beta q^{36} + ( - 4 \beta + 5) q^{37} + ( - 2 \beta + 5) q^{38} + ( - 3 \beta + 3) q^{39} + (\beta - 1) q^{40} + (5 \beta - 6) q^{41} + \beta q^{42} + (4 \beta - 5) q^{43} + ( - \beta - 1) q^{44} - 3 q^{45} + 3 \beta q^{47} - \beta q^{48} + q^{49} + (\beta + 1) q^{50} + (3 \beta - 6) q^{51} + ( - \beta + 4) q^{52} + ( - 2 \beta - 4) q^{53} + ( - 2 \beta + 3) q^{54} + (\beta + 2) q^{55} - q^{56} + (3 \beta - 6) q^{57} + ( - 2 \beta - 2) q^{58} + ( - 2 \beta + 6) q^{59} + 3 q^{60} + (2 \beta - 8) q^{61} + (2 \beta + 3) q^{62} + \beta q^{63} + q^{64} + ( - 4 \beta + 7) q^{65} + ( - 2 \beta - 3) q^{66} + (4 \beta - 4) q^{67} + (2 \beta - 5) q^{68} + (\beta - 1) q^{70} + (\beta + 3) q^{71} - \beta q^{72} - 9 q^{73} + (4 \beta - 5) q^{74} + (2 \beta + 3) q^{75} + (2 \beta - 5) q^{76} + ( - \beta - 1) q^{77} + (3 \beta - 3) q^{78} + (4 \beta - 3) q^{79} + ( - \beta + 1) q^{80} + ( - 2 \beta - 6) q^{81} + ( - 5 \beta + 6) q^{82} + ( - 7 \beta + 5) q^{83} - \beta q^{84} + (5 \beta - 11) q^{85} + ( - 4 \beta + 5) q^{86} + ( - 4 \beta - 6) q^{87} + (\beta + 1) q^{88} + ( - 2 \beta + 9) q^{89} + 3 q^{90} + ( - \beta + 4) q^{91} + (5 \beta + 6) q^{93} - 3 \beta q^{94} + (5 \beta - 11) q^{95} + \beta q^{96} + ( - 2 \beta - 9) q^{97} - q^{98} + ( - 2 \beta - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + q^{5} + q^{6} + 2 q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + q^{5} + q^{6} + 2 q^{7} - 2 q^{8} + q^{9} - q^{10} - 3 q^{11} - q^{12} + 7 q^{13} - 2 q^{14} + 6 q^{15} + 2 q^{16} - 8 q^{17} - q^{18} - 8 q^{19} + q^{20} - q^{21} + 3 q^{22} + q^{24} - 3 q^{25} - 7 q^{26} - 4 q^{27} + 2 q^{28} + 6 q^{29} - 6 q^{30} - 8 q^{31} - 2 q^{32} + 8 q^{33} + 8 q^{34} + q^{35} + q^{36} + 6 q^{37} + 8 q^{38} + 3 q^{39} - q^{40} - 7 q^{41} + q^{42} - 6 q^{43} - 3 q^{44} - 6 q^{45} + 3 q^{47} - q^{48} + 2 q^{49} + 3 q^{50} - 9 q^{51} + 7 q^{52} - 10 q^{53} + 4 q^{54} + 5 q^{55} - 2 q^{56} - 9 q^{57} - 6 q^{58} + 10 q^{59} + 6 q^{60} - 14 q^{61} + 8 q^{62} + q^{63} + 2 q^{64} + 10 q^{65} - 8 q^{66} - 4 q^{67} - 8 q^{68} - q^{70} + 7 q^{71} - q^{72} - 18 q^{73} - 6 q^{74} + 8 q^{75} - 8 q^{76} - 3 q^{77} - 3 q^{78} - 2 q^{79} + q^{80} - 14 q^{81} + 7 q^{82} + 3 q^{83} - q^{84} - 17 q^{85} + 6 q^{86} - 16 q^{87} + 3 q^{88} + 16 q^{89} + 6 q^{90} + 7 q^{91} + 17 q^{93} - 3 q^{94} - 17 q^{95} + q^{96} - 20 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
−1.00000 −2.30278 1.00000 −1.30278 2.30278 1.00000 −1.00000 2.30278 1.30278
1.2 −1.00000 1.30278 1.00000 2.30278 −1.30278 1.00000 −1.00000 −1.30278 −2.30278
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(23\) \(-1\)

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 7406.2.a.l yes 2
23.b odd 2 1 7406.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7406.2.a.k 2 23.b odd 2 1
7406.2.a.l yes 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7406))\):

\( T_{3}^{2} + T_{3} - 3 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$5$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 7T + 9 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 3 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 3 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 4 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 3 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T - 43 \) Copy content Toggle raw display
$41$ \( T^{2} + 7T - 69 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 43 \) Copy content Toggle raw display
$47$ \( T^{2} - 3T - 27 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 12 \) Copy content Toggle raw display
$59$ \( T^{2} - 10T + 12 \) Copy content Toggle raw display
$61$ \( T^{2} + 14T + 36 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$71$ \( T^{2} - 7T + 9 \) Copy content Toggle raw display
$73$ \( (T + 9)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 51 \) Copy content Toggle raw display
$83$ \( T^{2} - 3T - 157 \) Copy content Toggle raw display
$89$ \( T^{2} - 16T + 51 \) Copy content Toggle raw display
$97$ \( T^{2} + 20T + 87 \) Copy content Toggle raw display
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