Properties

Label 9405.2.a.bm
Level $9405$
Weight $2$
Character orbit 9405.a
Self dual yes
Analytic conductor $75.099$
Analytic rank $0$
Dimension $10$
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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 10x^{8} + 55x^{7} + 5x^{6} - 232x^{5} + 166x^{4} + 276x^{3} - 337x^{2} + 63x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + \beta_{7} q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + \beta_{7} q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{8} - \beta_1 q^{10} - q^{11} - \beta_{6} q^{13} + (\beta_{8} + \beta_{5} - \beta_{3} - \beta_1) q^{14} + (\beta_{4} + \beta_{2} + \beta_1 + 2) q^{16} + (\beta_{7} - \beta_{6} + \beta_{5} - 2) q^{17} + q^{19} + (\beta_{2} + 2) q^{20} + \beta_1 q^{22} + ( - \beta_{9} - \beta_{8} + \beta_{3} + \beta_{2} + 1) q^{23} + q^{25} + ( - \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2) q^{26} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{2} + 4) q^{28} + ( - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{3} - \beta_1 - 1) q^{29} - \beta_{6} q^{31} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{32} + ( - \beta_{9} + \beta_{7} + 2 \beta_{6} - \beta_{3} + \beta_1 + 1) q^{34} + \beta_{7} q^{35} + ( - \beta_{8} + \beta_{5} + \beta_{2} - \beta_1 + 2) q^{37} - \beta_1 q^{38} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{40} + (\beta_{9} - \beta_{5} + \beta_{3} - \beta_1 - 1) q^{41} + ( - \beta_{9} + \beta_{8} - \beta_{3} + \beta_{2} + 5) q^{43} + ( - \beta_{2} - 2) q^{44} + (\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 - 2) q^{46} + (\beta_{9} + \beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{47} + ( - \beta_{9} + \beta_{6} - \beta_{4} + \beta_{2} + 4) q^{49} - \beta_1 q^{50} + (\beta_{9} + 2 \beta_{8} - \beta_{7} - 3 \beta_{6} - \beta_{3} - \beta_1 + 1) q^{52} + (\beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3}) q^{53} - q^{55} + (\beta_{9} + \beta_{8} + \beta_{7} - 3 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_1 - 2) q^{56} + (\beta_{9} + 2 \beta_{6} + \beta_{5} - \beta_{3} + \beta_1 + 3) q^{58} + ( - \beta_{8} - \beta_{7} + \beta_{4} - \beta_{3} + 2 \beta_1) q^{59} + (\beta_{9} + 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{61} + ( - \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2) q^{62} + (\beta_{9} - 2 \beta_{7} + 2 \beta_{2} + 3 \beta_1 + 3) q^{64} - \beta_{6} q^{65} + (\beta_{9} - \beta_{8} - \beta_{6} + \beta_{4} + 2 \beta_{3} - \beta_1 + 3) q^{67} + (\beta_{9} + 3 \beta_{8} + \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_1 - 3) q^{68} + (\beta_{8} + \beta_{5} - \beta_{3} - \beta_1) q^{70} + ( - \beta_{9} - 2 \beta_{8} - 2 \beta_{6} - \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{71} + (\beta_{9} + \beta_{8} + \beta_{6} - \beta_{3} - \beta_{2} + 3) q^{73} + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{2} - 3 \beta_1 + 4) q^{74} + (\beta_{2} + 2) q^{76} - \beta_{7} q^{77} + (3 \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{79} + (\beta_{4} + \beta_{2} + \beta_1 + 2) q^{80} + ( - 3 \beta_{7} - \beta_{5} + 4) q^{82} + (\beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 2) q^{83} + (\beta_{7} - \beta_{6} + \beta_{5} - 2) q^{85} + (\beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} - 5 \beta_1) q^{86} + (\beta_{3} + \beta_{2} + \beta_1) q^{88} + (\beta_{9} + \beta_{8} - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_1 + 1) q^{89} + (\beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2) q^{91} + (\beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{92} + ( - \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 3) q^{94} + q^{95} + ( - \beta_{9} - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_1 + 7) q^{97} + (\beta_{9} + \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} + 16 q^{4} + 10 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{2} + 16 q^{4} + 10 q^{5} + 5 q^{7} - 3 q^{8} - 4 q^{10} - 10 q^{11} + 4 q^{13} - 6 q^{14} + 20 q^{16} - 10 q^{17} + 10 q^{19} + 16 q^{20} + 4 q^{22} + 6 q^{23} + 10 q^{25} + 3 q^{26} + 34 q^{28} - 5 q^{29} + 4 q^{31} - 30 q^{32} + 5 q^{34} + 5 q^{35} + 13 q^{37} - 4 q^{38} - 3 q^{40} - 9 q^{41} + 40 q^{43} - 16 q^{44} - 12 q^{46} + 24 q^{47} + 29 q^{49} - 4 q^{50} + 13 q^{52} - 13 q^{53} - 10 q^{55} - 8 q^{56} + 27 q^{58} - 5 q^{61} + 3 q^{62} + 27 q^{64} + 4 q^{65} + 39 q^{67} - 16 q^{68} - 6 q^{70} - 11 q^{71} + 30 q^{73} + 30 q^{74} + 16 q^{76} - 5 q^{77} + 4 q^{79} + 20 q^{80} + 24 q^{82} - 19 q^{83} - 10 q^{85} - 12 q^{86} + 3 q^{88} + 15 q^{89} - 17 q^{91} + 19 q^{92} + 2 q^{94} + 10 q^{95} + 58 q^{97} - 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} - 10x^{8} + 55x^{7} + 5x^{6} - 232x^{5} + 166x^{4} + 276x^{3} - 337x^{2} + 63x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} - \nu + 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 9\nu^{3} - \nu^{2} + 16\nu + 1 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} - 4\nu^{8} - 13\nu^{7} + 55\nu^{6} + 47\nu^{5} - 235\nu^{4} - 14\nu^{3} + 309\nu^{2} - 103\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{9} + 10\nu^{8} + 55\nu^{7} - 139\nu^{6} - 227\nu^{5} + 604\nu^{4} + 227\nu^{3} - 810\nu^{2} + 169\nu + 27 ) / 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 2\nu^{9} - 5\nu^{8} - 27\nu^{7} + 69\nu^{6} + 107\nu^{5} - 297\nu^{4} - 88\nu^{3} + 393\nu^{2} - 113\nu - 9 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8 \nu^{9} + 20 \nu^{8} + 110 \nu^{7} - 275 \nu^{6} - 454 \nu^{5} + 1178 \nu^{4} + 454 \nu^{3} - 1554 \nu^{2} + 329 \nu + 45 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{2} + \beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 9\beta_{3} + 10\beta_{2} + 29\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} - 2\beta_{7} + 10\beta_{4} + 48\beta_{2} + 13\beta _1 + 135 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{9} + 2\beta_{8} + \beta_{7} + 13\beta_{5} + 66\beta_{3} + 81\beta_{2} + 182\beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14\beta_{9} + \beta_{8} - 27\beta_{7} - 2\beta_{6} + 79\beta_{4} + 3\beta_{3} + 331\beta_{2} + 124\beta _1 + 874 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 14 \beta_{9} + 30 \beta_{8} + 15 \beta_{7} - 5 \beta_{6} + 122 \beta_{5} + \beta_{4} + 461 \beta_{3} + 617 \beta_{2} + 1192 \beta _1 + 406 \) 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
2.69746
2.63729
1.91754
1.44753
1.35956
0.398675
−0.0931385
−1.60160
−2.19410
−2.56922
−2.69746 0 5.27631 1.00000 0 −1.40612 −8.83771 0 −2.69746
1.2 −2.63729 0 4.95528 1.00000 0 3.89727 −7.79391 0 −2.63729
1.3 −1.91754 0 1.67697 1.00000 0 4.35488 0.619418 0 −1.91754
1.4 −1.44753 0 0.0953458 1.00000 0 2.30708 2.75705 0 −1.44753
1.5 −1.35956 0 −0.151601 1.00000 0 −4.37842 2.92523 0 −1.35956
1.6 −0.398675 0 −1.84106 1.00000 0 −2.49168 1.53133 0 −0.398675
1.7 0.0931385 0 −1.99133 1.00000 0 1.36553 −0.371746 0 0.0931385
1.8 1.60160 0 0.565109 1.00000 0 −3.36659 −2.29812 0 1.60160
1.9 2.19410 0 2.81407 1.00000 0 4.33879 1.78616 0 2.19410
1.10 2.56922 0 4.60090 1.00000 0 0.379271 6.68229 0 2.56922
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.bm 10
3.b odd 2 1 3135.2.a.x 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3135.2.a.x 10 3.b odd 2 1
9405.2.a.bm 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):

\( T_{2}^{10} + 4T_{2}^{9} - 10T_{2}^{8} - 55T_{2}^{7} + 5T_{2}^{6} + 232T_{2}^{5} + 166T_{2}^{4} - 276T_{2}^{3} - 337T_{2}^{2} - 63T_{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{10} - 5 T_{7}^{9} - 37 T_{7}^{8} + 194 T_{7}^{7} + 417 T_{7}^{6} - 2404 T_{7}^{5} - 1412 T_{7}^{4} + 10699 T_{7}^{3} - 804 T_{7}^{2} - 13092 T_{7} + 4544 \) Copy content Toggle raw display
\( T_{13}^{10} - 4 T_{13}^{9} - 61 T_{13}^{8} + 269 T_{13}^{7} + 822 T_{13}^{6} - 3456 T_{13}^{5} - 4952 T_{13}^{4} + 13120 T_{13}^{3} + 12704 T_{13}^{2} - 8192 T_{13} - 6144 \) Copy content Toggle raw display
\( T_{17}^{10} + 10 T_{17}^{9} - 55 T_{17}^{8} - 727 T_{17}^{7} + 644 T_{17}^{6} + 18512 T_{17}^{5} + 11016 T_{17}^{4} - 186288 T_{17}^{3} - 250240 T_{17}^{2} + 559488 T_{17} + 940032 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 4 T^{9} - 10 T^{8} - 55 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 5 T^{9} - 37 T^{8} + \cdots + 4544 \) Copy content Toggle raw display
$11$ \( (T + 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 4 T^{9} - 61 T^{8} + \cdots - 6144 \) Copy content Toggle raw display
$17$ \( T^{10} + 10 T^{9} - 55 T^{8} + \cdots + 940032 \) Copy content Toggle raw display
$19$ \( (T - 1)^{10} \) Copy content Toggle raw display
$23$ \( T^{10} - 6 T^{9} - 94 T^{8} + \cdots - 195792 \) Copy content Toggle raw display
$29$ \( T^{10} + 5 T^{9} - 174 T^{8} + \cdots - 589824 \) Copy content Toggle raw display
$31$ \( T^{10} - 4 T^{9} - 61 T^{8} + \cdots - 6144 \) Copy content Toggle raw display
$37$ \( T^{10} - 13 T^{9} - 103 T^{8} + \cdots + 622584 \) Copy content Toggle raw display
$41$ \( T^{10} + 9 T^{9} - 175 T^{8} + \cdots - 352512 \) Copy content Toggle raw display
$43$ \( T^{10} - 40 T^{9} + 476 T^{8} + \cdots - 13025808 \) Copy content Toggle raw display
$47$ \( T^{10} - 24 T^{9} - 21 T^{8} + \cdots + 15929472 \) Copy content Toggle raw display
$53$ \( T^{10} + 13 T^{9} - 169 T^{8} + \cdots - 7860096 \) Copy content Toggle raw display
$59$ \( T^{10} - 264 T^{8} + \cdots + 30678048 \) Copy content Toggle raw display
$61$ \( T^{10} + 5 T^{9} - 350 T^{8} + \cdots - 76434048 \) Copy content Toggle raw display
$67$ \( T^{10} - 39 T^{9} + \cdots - 6716483584 \) Copy content Toggle raw display
$71$ \( T^{10} + 11 T^{9} - 389 T^{8} + \cdots - 12060864 \) Copy content Toggle raw display
$73$ \( T^{10} - 30 T^{9} + 277 T^{8} + \cdots + 82688 \) Copy content Toggle raw display
$79$ \( T^{10} - 4 T^{9} + \cdots + 8438083072 \) Copy content Toggle raw display
$83$ \( T^{10} + 19 T^{9} - 200 T^{8} + \cdots - 2087424 \) Copy content Toggle raw display
$89$ \( T^{10} - 15 T^{9} + \cdots + 590259528 \) Copy content Toggle raw display
$97$ \( T^{10} - 58 T^{9} + \cdots + 773139288 \) Copy content Toggle raw display
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