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Properties

Label 9405.2.a.bk
Level $9405$
Weight $2$
Character orbit 9405.a
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 11x^{7} + 21x^{6} + 34x^{5} - 61x^{4} - 24x^{3} + 45x^{2} - 7x - 2 \) 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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 11x^{7} + 21x^{6} + 34x^{5} - 61x^{4} - 24x^{3} + 45x^{2} - 7x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 6\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{8} - \nu^{7} - 11\nu^{6} + 10\nu^{5} + 35\nu^{4} - 26\nu^{3} - 31\nu^{2} + 13\nu + 1 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{8} - \nu^{7} - 12\nu^{6} + 10\nu^{5} + 44\nu^{4} - 26\nu^{3} - 50\nu^{2} + 14\nu + 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{8} + 2\nu^{7} + 12\nu^{6} - 20\nu^{5} - 44\nu^{4} + 52\nu^{3} + 50\nu^{2} - 27\nu - 6 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 2\nu^{8} - 3\nu^{7} - 23\nu^{6} + 31\nu^{5} + 78\nu^{4} - 86\nu^{3} - 75\nu^{2} + 53\nu + 3 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + 8\beta_{3} + 27\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{6} + \beta_{5} + 9\beta_{4} + 35\beta_{2} + \beta _1 + 83 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10\beta_{8} + 11\beta_{7} + \beta_{6} - 10\beta_{5} + 10\beta_{4} + 54\beta_{3} + 153\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{7} - 10\beta_{6} + 12\beta_{5} + 64\beta_{4} + 206\beta_{2} + 11\beta _1 + 480 \) 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
2.49579
2.40870
1.59430
0.639726
0.444071
−0.144742
−1.10022
−1.86705
−2.47056
−2.49579 0 4.22895 −1.00000 0 2.90021 −5.56299 0 2.49579
1.2 −2.40870 0 3.80182 −1.00000 0 −2.71774 −4.34003 0 2.40870
1.3 −1.59430 0 0.541783 −1.00000 0 1.64098 2.32483 0 1.59430
1.4 −0.639726 0 −1.59075 −1.00000 0 −4.71082 2.29710 0 0.639726
1.5 −0.444071 0 −1.80280 −1.00000 0 1.87005 1.68871 0 0.444071
1.6 0.144742 0 −1.97905 −1.00000 0 3.02352 −0.575935 0 −0.144742
1.7 1.10022 0 −0.789518 −1.00000 0 −4.13560 −3.06908 0 −1.10022
1.8 1.86705 0 1.48589 −1.00000 0 1.01335 −0.959879 0 −1.86705
1.9 2.47056 0 4.10368 −1.00000 0 −1.88395 5.19728 0 −2.47056
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.bk 9
3.b odd 2 1 3135.2.a.u ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3135.2.a.u ✓ 9 3.b odd 2 1
9405.2.a.bk 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):

\( T_{2}^{9} + 2T_{2}^{8} - 11T_{2}^{7} - 21T_{2}^{6} + 34T_{2}^{5} + 61T_{2}^{4} - 24T_{2}^{3} - 45T_{2}^{2} - 7T_{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{9} + 3T_{7}^{8} - 33T_{7}^{7} - 62T_{7}^{6} + 419T_{7}^{5} + 296T_{7}^{4} - 2230T_{7}^{3} + 363T_{7}^{2} + 3954T_{7} - 2720 \) Copy content Toggle raw display
\( T_{13}^{9} - T_{13}^{8} - 67 T_{13}^{7} + 106 T_{13}^{6} + 1447 T_{13}^{5} - 2998 T_{13}^{4} + \cdots - 46624 \) Copy content Toggle raw display
\( T_{17}^{9} + T_{17}^{8} - 87 T_{17}^{7} - 146 T_{17}^{6} + 2153 T_{17}^{5} + 3938 T_{17}^{4} + \cdots + 38432 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 2 T^{8} + \cdots + 2 \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 3 T^{8} + \cdots - 2720 \) Copy content Toggle raw display
$11$ \( (T + 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} - T^{8} + \cdots - 46624 \) Copy content Toggle raw display
$17$ \( T^{9} + T^{8} + \cdots + 38432 \) Copy content Toggle raw display
$19$ \( (T - 1)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 6 T^{8} + \cdots - 88000 \) Copy content Toggle raw display
$29$ \( T^{9} + 21 T^{8} + \cdots + 80896 \) Copy content Toggle raw display
$31$ \( T^{9} - 2 T^{8} + \cdots - 27648 \) Copy content Toggle raw display
$37$ \( T^{9} - 11 T^{8} + \cdots - 38160 \) Copy content Toggle raw display
$41$ \( T^{9} + 17 T^{8} + \cdots - 2304 \) Copy content Toggle raw display
$43$ \( T^{9} + 6 T^{8} + \cdots - 40128 \) Copy content Toggle raw display
$47$ \( T^{9} - 6 T^{8} + \cdots + 68352 \) Copy content Toggle raw display
$53$ \( T^{9} + 4 T^{8} + \cdots + 3036128 \) Copy content Toggle raw display
$59$ \( T^{9} + 17 T^{8} + \cdots + 130564 \) Copy content Toggle raw display
$61$ \( T^{9} - 11 T^{8} + \cdots + 4558592 \) Copy content Toggle raw display
$67$ \( T^{9} + T^{8} + \cdots - 10171680 \) Copy content Toggle raw display
$71$ \( T^{9} + 10 T^{8} + \cdots + 276008 \) Copy content Toggle raw display
$73$ \( T^{9} - 2 T^{8} + \cdots - 212182432 \) Copy content Toggle raw display
$79$ \( T^{9} - 23 T^{8} + \cdots + 880 \) Copy content Toggle raw display
$83$ \( T^{9} - 4 T^{8} + \cdots + 11693888 \) Copy content Toggle raw display
$89$ \( T^{9} + 44 T^{8} + \cdots + 58230 \) Copy content Toggle raw display
$97$ \( T^{9} - 8 T^{8} + \cdots - 2982584 \) Copy content Toggle raw display
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