Properties

Label 9405.2.a.bl
Level $9405$
Weight $2$
Character orbit 9405.a
Self dual yes
Analytic conductor $75.099$
Analytic rank $0$
Dimension $9$
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: \(0\)
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: \( x^{9} - x^{8} - 15x^{7} + 14x^{6} + 71x^{5} - 59x^{4} - 111x^{3} + 61x^{2} + 46x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{6} + 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{6} + 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + \beta_1 q^{10} + q^{11} + ( - \beta_{8} - \beta_{5} + \beta_{4} + 1) q^{13} + (\beta_{7} - \beta_{4} + \beta_{3} + 2 \beta_1) q^{14} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{16} - \beta_{7} q^{17} - q^{19} + (\beta_{2} + 1) q^{20} + \beta_1 q^{22} + ( - \beta_{8} + \beta_1 - 1) q^{23} + q^{25} + ( - \beta_{7} - \beta_{5} + 2 \beta_1 - 2) q^{26} + (2 \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_{2} + 3) q^{28} + (\beta_{8} + \beta_{4} + 1) q^{29} + (\beta_{8} - \beta_{5} - \beta_{4} + 3) q^{31} + (\beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{32} + ( - 2 \beta_{8} + 2 \beta_{6} - \beta_{5}) q^{34} + ( - \beta_{6} + 1) q^{35} + ( - \beta_{7} + \beta_{4} - \beta_1 + 2) q^{37} - \beta_1 q^{38} + (\beta_{3} + 2 \beta_1) q^{40} + ( - \beta_{8} - \beta_{7} - \beta_{4} - 1) q^{41} + ( - \beta_{8} - 2 \beta_{2} + \beta_1 + 1) q^{43} + (\beta_{2} + 1) q^{44} + (\beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{2} + 3) q^{46} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 1) q^{47} + (\beta_{8} - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_1 + 4) q^{49} + \beta_1 q^{50} + ( - \beta_{8} + 2 \beta_{6} - 3 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 3) q^{52} + (\beta_{8} + \beta_{7} + \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{53} + q^{55} + ( - \beta_{8} + \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} + 7 \beta_1) q^{56} + ( - \beta_{8} + \beta_{7} + 2 \beta_{5} - \beta_{4} - 1) q^{58} + ( - \beta_{8} - \beta_{6} + 2 \beta_{5} + \beta_{4}) q^{59} + ( - 2 \beta_{8} + \beta_{7} - \beta_{5} - 2 \beta_{2}) q^{61} + ( - 2 \beta_{8} + \beta_{7} - \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{62} + (2 \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{64} + ( - \beta_{8} - \beta_{5} + \beta_{4} + 1) q^{65} + (2 \beta_{8} + \beta_{6} + 3) q^{67} + (\beta_{8} - 2 \beta_{7} - 3 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 1) q^{68} + (\beta_{7} - \beta_{4} + \beta_{3} + 2 \beta_1) q^{70} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{71} + (2 \beta_{7} + \beta_{5} - \beta_{4} + \beta_1 + 2) q^{73} + ( - 2 \beta_{8} + 2 \beta_{6} - \beta_{2} + 2 \beta_1 - 4) q^{74} + ( - \beta_{2} - 1) q^{76} + ( - \beta_{6} + 1) q^{77} + ( - \beta_{8} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_1 + 3) q^{79} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{80} + ( - \beta_{8} - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + \beta_{4} + 1) q^{82} + (\beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{3} - 2 \beta_1 + 4) q^{83} - \beta_{7} q^{85} + (\beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 4 \beta_1 + 3) q^{86} + (\beta_{3} + 2 \beta_1) q^{88} + ( - \beta_{6} - 2 \beta_{4} - 3) q^{89} + ( - \beta_{8} - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{2} + 3) q^{91} + ( - 2 \beta_{8} + \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + \beta_{3} + 3 \beta_1) q^{92} + ( - \beta_{8} + \beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_{3} - 6 \beta_1 - 3) q^{94} - q^{95} + ( - \beta_{8} - \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_1 + 4) q^{97} + (3 \beta_{7} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 5 \beta_1 + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} + 13 q^{4} + 9 q^{5} + 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{2} + 13 q^{4} + 9 q^{5} + 7 q^{7} + q^{10} + 9 q^{11} + 10 q^{13} + 21 q^{16} - 9 q^{19} + 13 q^{20} + q^{22} - 4 q^{23} + 9 q^{25} - 19 q^{26} + 30 q^{28} + 5 q^{29} + 20 q^{31} + q^{32} + 9 q^{34} + 7 q^{35} + 17 q^{37} - q^{38} - 5 q^{41} + 6 q^{43} + 13 q^{44} + 24 q^{46} - 22 q^{47} + 32 q^{49} + q^{50} + 41 q^{52} - 11 q^{53} + 9 q^{55} + 16 q^{56} + q^{58} + 8 q^{59} - 3 q^{61} + 7 q^{62} + 16 q^{64} + 10 q^{65} + 21 q^{67} - 18 q^{68} - 19 q^{71} + 22 q^{73} - 26 q^{74} - 13 q^{76} + 7 q^{77} + 28 q^{79} + 21 q^{80} + 8 q^{82} + 39 q^{83} + 24 q^{86} - 29 q^{89} + 25 q^{91} + 13 q^{92} - 24 q^{94} - 9 q^{95} + 46 q^{97} + 35 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 15x^{7} + 14x^{6} + 71x^{5} - 59x^{4} - 111x^{3} + 61x^{2} + 46x + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - 2\nu^{5} - 9\nu^{4} + 16\nu^{3} + 18\nu^{2} - 23\nu - 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{7} - 2\nu^{6} - 9\nu^{5} + 16\nu^{4} + 18\nu^{3} - 23\nu^{2} - 8\nu + 1 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 24\nu^{4} + 3\nu^{3} - 34\nu^{2} + 9\nu + 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{8} + 3\nu^{7} + 8\nu^{6} - 26\nu^{5} - 12\nu^{4} + 49\nu^{3} + 9\nu^{2} - 22\nu - 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{8} - 3\nu^{7} - 8\nu^{6} + 27\nu^{5} + 11\nu^{4} - 57\nu^{3} - 3\nu^{2} + 32\nu + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 9\beta_{3} + \beta_{2} + 38\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{8} + 2\beta_{7} - 11\beta_{6} + 11\beta_{5} - 10\beta_{4} + 11\beta_{3} + 47\beta_{2} + 3\beta _1 + 98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{8} + 13\beta_{7} - 15\beta_{6} + 16\beta_{5} - 13\beta_{4} + 69\beta_{3} + 14\beta_{2} + 248\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 29\beta_{8} + 28\beta_{7} - 95\beta_{6} + 98\beta_{5} - 81\beta_{4} + 98\beta_{3} + 317\beta_{2} + 52\beta _1 + 635 \) 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
−2.50437
−2.49902
−1.31781
−0.353963
−0.141469
1.19030
1.68660
2.20601
2.73372
−2.50437 0 4.27185 1.00000 0 4.23556 −5.68954 0 −2.50437
1.2 −2.49902 0 4.24509 1.00000 0 −0.263615 −5.61052 0 −2.49902
1.3 −1.31781 0 −0.263370 1.00000 0 −2.81599 2.98270 0 −1.31781
1.4 −0.353963 0 −1.87471 1.00000 0 4.16954 1.37151 0 −0.353963
1.5 −0.141469 0 −1.97999 1.00000 0 −1.04782 0.563044 0 −0.141469
1.6 1.19030 0 −0.583181 1.00000 0 3.09540 −3.07477 0 1.19030
1.7 1.68660 0 0.844624 1.00000 0 −4.29390 −1.94866 0 1.68660
1.8 2.20601 0 2.86646 1.00000 0 −0.766522 1.91141 0 2.20601
1.9 2.73372 0 5.47323 1.00000 0 4.68734 9.49483 0 2.73372
\(n\): 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.bl 9
3.b odd 2 1 3135.2.a.t 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3135.2.a.t 9 3.b odd 2 1
9405.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):

\( T_{2}^{9} - T_{2}^{8} - 15T_{2}^{7} + 14T_{2}^{6} + 71T_{2}^{5} - 59T_{2}^{4} - 111T_{2}^{3} + 61T_{2}^{2} + 46T_{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{9} - 7T_{7}^{8} - 23T_{7}^{7} + 216T_{7}^{6} + 79T_{7}^{5} - 1858T_{7}^{4} - 106T_{7}^{3} + 4667T_{7}^{2} + 3692T_{7} + 656 \) Copy content Toggle raw display
\( T_{13}^{9} - 10 T_{13}^{8} - 29 T_{13}^{7} + 553 T_{13}^{6} - 924 T_{13}^{5} - 5060 T_{13}^{4} + 11896 T_{13}^{3} + 7120 T_{13}^{2} - 7008 T_{13} - 1344 \) Copy content Toggle raw display
\( T_{17}^{9} - 73T_{17}^{7} + 5T_{17}^{6} + 1812T_{17}^{5} - 136T_{17}^{4} - 17896T_{17}^{3} + 3728T_{17}^{2} + 55680T_{17} - 45312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - T^{8} - 15 T^{7} + 14 T^{6} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( (T - 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 7 T^{8} - 23 T^{7} + 216 T^{6} + \cdots + 656 \) Copy content Toggle raw display
$11$ \( (T - 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} - 10 T^{8} - 29 T^{7} + \cdots - 1344 \) Copy content Toggle raw display
$17$ \( T^{9} - 73 T^{7} + 5 T^{6} + \cdots - 45312 \) Copy content Toggle raw display
$19$ \( (T + 1)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} + 4 T^{8} - 44 T^{7} + \cdots - 1284 \) Copy content Toggle raw display
$29$ \( T^{9} - 5 T^{8} - 64 T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$31$ \( T^{9} - 20 T^{8} + 21 T^{7} + \cdots + 3364544 \) Copy content Toggle raw display
$37$ \( T^{9} - 17 T^{8} + 31 T^{7} + \cdots + 263700 \) Copy content Toggle raw display
$41$ \( T^{9} + 5 T^{8} - 131 T^{7} + \cdots + 9984 \) Copy content Toggle raw display
$43$ \( T^{9} - 6 T^{8} - 126 T^{7} + \cdots - 38448 \) Copy content Toggle raw display
$47$ \( T^{9} + 22 T^{8} + 3 T^{7} + \cdots - 2095136 \) Copy content Toggle raw display
$53$ \( T^{9} + 11 T^{8} - 243 T^{7} + \cdots + 631872 \) Copy content Toggle raw display
$59$ \( T^{9} - 8 T^{8} - 264 T^{7} + \cdots + 8621644 \) Copy content Toggle raw display
$61$ \( T^{9} + 3 T^{8} - 324 T^{7} + \cdots + 8036800 \) Copy content Toggle raw display
$67$ \( T^{9} - 21 T^{8} - 19 T^{7} + \cdots + 7488 \) Copy content Toggle raw display
$71$ \( T^{9} + 19 T^{8} - 101 T^{7} + \cdots - 763100 \) Copy content Toggle raw display
$73$ \( T^{9} - 22 T^{8} - 91 T^{7} + \cdots - 6010880 \) Copy content Toggle raw display
$79$ \( T^{9} - 28 T^{8} + 54 T^{7} + \cdots - 43340864 \) Copy content Toggle raw display
$83$ \( T^{9} - 39 T^{8} + 234 T^{7} + \cdots - 4321536 \) Copy content Toggle raw display
$89$ \( T^{9} + 29 T^{8} + 171 T^{7} + \cdots - 72332 \) Copy content Toggle raw display
$97$ \( T^{9} - 46 T^{8} + \cdots - 161210684 \) Copy content Toggle raw display
show more
show less