Properties

Label 9405.2.a.bd
Level $9405$
Weight $2$
Character orbit 9405.a
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{6} + 4 \beta_{5} - 2 \beta_{4} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} + 7 q^{4} + 7 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{2} + 7 q^{4} + 7 q^{5} - q^{7} - 3 q^{8} - q^{10} - 7 q^{11} + q^{13} - 12 q^{14} + 3 q^{16} - q^{17} - 7 q^{19} + 7 q^{20} + q^{22} + 8 q^{23} + 7 q^{25} + 4 q^{28} - 11 q^{29} + 7 q^{31} - 12 q^{32} - 14 q^{34} - q^{35} - 17 q^{37} + q^{38} - 3 q^{40} - 17 q^{41} - 3 q^{43} - 7 q^{44} + 18 q^{46} - 14 q^{47} + 6 q^{49} - q^{50} - 17 q^{52} - 7 q^{53} - 7 q^{55} - 36 q^{56} - 15 q^{58} - 35 q^{59} + 17 q^{61} - 46 q^{62} + 5 q^{64} + q^{65} + 4 q^{67} + 35 q^{68} - 12 q^{70} - 10 q^{71} + 22 q^{73} + 11 q^{74} - 7 q^{76} + q^{77} + 11 q^{79} + 3 q^{80} - 14 q^{82} - 39 q^{83} - q^{85} + 24 q^{86} + 3 q^{88} - 18 q^{89} - 22 q^{91} + 51 q^{92} + 14 q^{94} - 7 q^{95} - 4 q^{97} + 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{6} + 11\nu^{4} + 2\nu^{3} - 33\nu^{2} - 10\nu + 21 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{6} + \nu^{5} + 20\nu^{4} - 5\nu^{3} - 52\nu^{2} - \nu + 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -3\nu^{6} + \nu^{5} + 30\nu^{4} - 3\nu^{3} - 78\nu^{2} - 10\nu + 41 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 4\nu^{6} - \nu^{5} - 41\nu^{4} + 2\nu^{3} + 111\nu^{2} + 15\nu - 62 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{6} + 7\beta_{5} + 3\beta_{4} + 9\beta_{3} + 28\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{6} - 9\beta_{5} + 11\beta_{4} + 12\beta_{3} + 44\beta_{2} + 11\beta _1 + 76 \) 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
2.58611
1.97792
0.745312
0.719047
−1.08185
−1.54354
−2.40300
−2.58611 0 4.68797 1.00000 0 2.28823 −6.95140 0 −2.58611
1.2 −1.97792 0 1.91218 1.00000 0 1.06653 0.173707 0 −1.97792
1.3 −0.745312 0 −1.44451 1.00000 0 −3.86782 2.56724 0 −0.745312
1.4 −0.719047 0 −1.48297 1.00000 0 1.05678 2.50442 0 −0.719047
1.5 1.08185 0 −0.829591 1.00000 0 4.02863 −3.06121 0 1.08185
1.6 1.54354 0 0.382516 1.00000 0 −3.41636 −2.49665 0 1.54354
1.7 2.40300 0 3.77440 1.00000 0 −2.15598 4.26389 0 2.40300
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.bd 7
3.b odd 2 1 1045.2.a.h 7
15.d odd 2 1 5225.2.a.m 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.h 7 3.b odd 2 1
5225.2.a.m 7 15.d odd 2 1
9405.2.a.bd 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):

\( T_{2}^{7} + T_{2}^{6} - 10T_{2}^{5} - 8T_{2}^{4} + 27T_{2}^{3} + 16T_{2}^{2} - 18T_{2} - 11 \) Copy content Toggle raw display
\( T_{7}^{7} + T_{7}^{6} - 27T_{7}^{5} - 18T_{7}^{4} + 205T_{7}^{3} + 3T_{7}^{2} - 460T_{7} + 296 \) Copy content Toggle raw display
\( T_{13}^{7} - T_{13}^{6} - 49T_{13}^{5} - 55T_{13}^{4} + 526T_{13}^{3} + 1133T_{13}^{2} - 184T_{13} - 1184 \) Copy content Toggle raw display
\( T_{17}^{7} + T_{17}^{6} - 61T_{17}^{5} - 141T_{17}^{4} + 616T_{17}^{3} + 1829T_{17}^{2} + 428T_{17} - 1096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + T^{6} + \cdots - 11 \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( (T - 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + T^{6} + \cdots + 296 \) Copy content Toggle raw display
$11$ \( (T + 1)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} - T^{6} + \cdots - 1184 \) Copy content Toggle raw display
$17$ \( T^{7} + T^{6} + \cdots - 1096 \) Copy content Toggle raw display
$19$ \( (T + 1)^{7} \) Copy content Toggle raw display
$23$ \( T^{7} - 8 T^{6} + \cdots + 7724 \) Copy content Toggle raw display
$29$ \( T^{7} + 11 T^{6} + \cdots + 368 \) Copy content Toggle raw display
$31$ \( T^{7} - 7 T^{6} + \cdots + 8300 \) Copy content Toggle raw display
$37$ \( T^{7} + 17 T^{6} + \cdots - 20896 \) Copy content Toggle raw display
$41$ \( T^{7} + 17 T^{6} + \cdots + 1328 \) Copy content Toggle raw display
$43$ \( T^{7} + 3 T^{6} + \cdots + 2872 \) Copy content Toggle raw display
$47$ \( T^{7} + 14 T^{6} + \cdots - 35972 \) Copy content Toggle raw display
$53$ \( T^{7} + 7 T^{6} + \cdots - 1211488 \) Copy content Toggle raw display
$59$ \( T^{7} + 35 T^{6} + \cdots + 106388 \) Copy content Toggle raw display
$61$ \( T^{7} - 17 T^{6} + \cdots + 1052 \) Copy content Toggle raw display
$67$ \( T^{7} - 4 T^{6} + \cdots - 1748 \) Copy content Toggle raw display
$71$ \( T^{7} + 10 T^{6} + \cdots - 14492 \) Copy content Toggle raw display
$73$ \( T^{7} - 22 T^{6} + \cdots + 85096 \) Copy content Toggle raw display
$79$ \( T^{7} - 11 T^{6} + \cdots + 19664 \) Copy content Toggle raw display
$83$ \( T^{7} + 39 T^{6} + \cdots - 2216 \) Copy content Toggle raw display
$89$ \( T^{7} + 18 T^{6} + \cdots + 20716 \) Copy content Toggle raw display
$97$ \( T^{7} + 4 T^{6} + \cdots - 6460112 \) Copy content Toggle raw display
show more
show less