Properties

Label 9405.2.a.t
Level $9405$
Weight $2$
Character orbit 9405.a
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
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] = [9405,2,Mod(1,9405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9405.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.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: \(75.0993031010\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.132889.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 8x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3135)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} - q^{5} + ( - \beta_1 - 1) q^{7} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} - q^{5} + ( - \beta_1 - 1) q^{7} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{8} - \beta_1 q^{10} + q^{11} + ( - \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{2} - \beta_1 - 5) q^{14} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 + 7) q^{16} + ( - \beta_{3} + \beta_1 + 1) q^{17} - q^{19} + ( - \beta_{2} - 3) q^{20} + \beta_1 q^{22} + (\beta_{2} - \beta_1 + 3) q^{23} + q^{25} + ( - \beta_{3} - 4) q^{26} + ( - \beta_{3} - 4 \beta_1 - 2) q^{28} + ( - \beta_{2} + 2 \beta_1) q^{29} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{31} + (2 \beta_{3} - \beta_{2} + 5 \beta_1 - 8) q^{32} + ( - 2 \beta_{3} + \beta_1 + 3) q^{34} + (\beta_1 + 1) q^{35} + (\beta_{3} - 2 \beta_1 + 2) q^{37} - \beta_1 q^{38} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{40} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{41} + (\beta_{2} + \beta_1 - 7) q^{43} + (\beta_{2} + 3) q^{44} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 - 6) q^{46} + (2 \beta_{2} - 2 \beta_1 + 2) q^{47} + (\beta_{2} + 2 \beta_1 - 1) q^{49} + \beta_1 q^{50} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 4) q^{52} + (\beta_{3} + \beta_{2} + 3 \beta_1 - 5) q^{53} - q^{55} + ( - 2 \beta_{3} - 3 \beta_{2} - 12) q^{56} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 + 11) q^{58} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{59} + (\beta_{3} - 2 \beta_{2}) q^{61} + ( - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{62} + (\beta_{3} + 4 \beta_{2} - 5 \beta_1 + 16) q^{64} + (\beta_{2} + \beta_1 - 1) q^{65} + (2 \beta_{3} - 4 \beta_{2} - \beta_1 - 5) q^{67} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{68} + (\beta_{2} + \beta_1 + 5) q^{70} + (\beta_1 - 11) q^{71} + ( - 2 \beta_{3} + 6) q^{73} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 - 8) q^{74} + ( - \beta_{2} - 3) q^{76} + ( - \beta_1 - 1) q^{77} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{79} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 7) q^{80} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{82} + ( - 3 \beta_{3} + 2 \beta_1 - 4) q^{83} + (\beta_{3} - \beta_1 - 1) q^{85} + (\beta_{3} - 6 \beta_1 + 4) q^{86} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{88} + ( - 5 \beta_1 - 3) q^{89} + (\beta_{3} + \beta_{2} + \beta_1 + 3) q^{91} + (5 \beta_{2} - 6 \beta_1 + 18) q^{92} + (2 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 12) q^{94} + q^{95} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 - 7) q^{97} + (\beta_{3} + \beta_{2} + 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 11 q^{4} - 4 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 11 q^{4} - 4 q^{5} - 5 q^{7} - q^{10} + 4 q^{11} + 4 q^{13} - 20 q^{14} + 25 q^{16} + 4 q^{17} - 4 q^{19} - 11 q^{20} + q^{22} + 10 q^{23} + 4 q^{25} - 17 q^{26} - 13 q^{28} + 3 q^{29} - 14 q^{31} - 24 q^{32} + 11 q^{34} + 5 q^{35} + 7 q^{37} - q^{38} - 3 q^{41} - 28 q^{43} + 11 q^{44} - 17 q^{46} + 4 q^{47} - 3 q^{49} + q^{50} - 21 q^{52} - 17 q^{53} - 4 q^{55} - 47 q^{56} + 39 q^{58} - 8 q^{59} + 3 q^{61} - 3 q^{62} + 56 q^{64} - 4 q^{65} - 15 q^{67} - 4 q^{68} + 20 q^{70} - 43 q^{71} + 22 q^{73} - 27 q^{74} - 11 q^{76} - 5 q^{77} - 12 q^{79} - 25 q^{80} - 10 q^{82} - 17 q^{83} - 4 q^{85} + 11 q^{86} - 17 q^{89} + 13 q^{91} + 61 q^{92} - 38 q^{94} + 4 q^{95} - 24 q^{97} + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 9x^{2} + 8x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 6\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 6\beta _1 - 1 \) 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
−2.77177
−0.787040
1.88792
2.67089
−2.77177 0 5.68269 −1.00000 0 1.77177 −10.2076 0 2.77177
1.2 −0.787040 0 −1.38057 −1.00000 0 −0.212960 2.66064 0 0.787040
1.3 1.88792 0 1.56424 −1.00000 0 −2.88792 −0.822677 0 −1.88792
1.4 2.67089 0 5.13364 −1.00000 0 −3.67089 8.36959 0 −2.67089
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)
\(19\) \(1\)

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 9405.2.a.t 4
3.b odd 2 1 3135.2.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3135.2.a.m 4 3.b odd 2 1
9405.2.a.t 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):

\( T_{2}^{4} - T_{2}^{3} - 9T_{2}^{2} + 8T_{2} + 11 \) Copy content Toggle raw display
\( T_{7}^{4} + 5T_{7}^{3} - 19T_{7} - 4 \) Copy content Toggle raw display
\( T_{13}^{4} - 4T_{13}^{3} - 19T_{13}^{2} + 37T_{13} - 14 \) Copy content Toggle raw display
\( T_{17}^{4} - 4T_{17}^{3} - 27T_{17}^{2} - 31T_{17} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 9 T^{2} + 8 T + 11 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 5 T^{3} - 19 T - 4 \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} - 19 T^{2} + 37 T - 14 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} - 27 T^{2} - 31 T - 10 \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 10 T^{3} + 11 T^{2} + 17 T + 4 \) Copy content Toggle raw display
$29$ \( T^{4} - 3 T^{3} - 52 T^{2} + 285 T - 386 \) Copy content Toggle raw display
$31$ \( T^{4} + 14 T^{3} - 35 T^{2} + \cdots - 880 \) Copy content Toggle raw display
$37$ \( T^{4} - 7 T^{3} - 34 T^{2} + 327 T - 558 \) Copy content Toggle raw display
$41$ \( T^{4} + 3 T^{3} - 31 T^{2} - 51 T + 22 \) Copy content Toggle raw display
$43$ \( T^{4} + 28 T^{3} + 269 T^{2} + \cdots + 1240 \) Copy content Toggle raw display
$47$ \( T^{4} - 4 T^{3} - 100 T^{2} + \cdots + 288 \) Copy content Toggle raw display
$53$ \( T^{4} + 17 T^{3} - 63 T^{2} + \cdots - 8302 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} - 121 T^{2} + \cdots + 3220 \) Copy content Toggle raw display
$61$ \( T^{4} - 3 T^{3} - 66 T^{2} - 3 T + 250 \) Copy content Toggle raw display
$67$ \( T^{4} + 15 T^{3} - 182 T^{2} + \cdots - 9756 \) Copy content Toggle raw display
$71$ \( T^{4} + 43 T^{3} + 684 T^{2} + \cdots + 12320 \) Copy content Toggle raw display
$73$ \( T^{4} - 22 T^{3} + 52 T^{2} + \cdots - 3472 \) Copy content Toggle raw display
$79$ \( T^{4} + 12 T^{3} - 171 T^{2} + \cdots - 2592 \) Copy content Toggle raw display
$83$ \( T^{4} + 17 T^{3} - 168 T^{2} + \cdots - 13696 \) Copy content Toggle raw display
$89$ \( T^{4} + 17 T^{3} - 126 T^{2} + \cdots + 2066 \) Copy content Toggle raw display
$97$ \( T^{4} + 24 T^{3} + 35 T^{2} + \cdots - 6754 \) Copy content Toggle raw display
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