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Properties

Label 9405.2.a.bn
Level $9405$
Weight $2$
Character orbit 9405.a
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
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: \(1\)
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: \( x^{10} - x^{9} - 11x^{8} + 11x^{7} + 36x^{6} - 31x^{5} - 44x^{4} + 26x^{3} + 22x^{2} - 5x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{4} - q^{5} + (\beta_{8} - 1) q^{7} + (\beta_{8} - \beta_{7} + \cdots + \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{4} - q^{5} + (\beta_{8} - 1) q^{7} + (\beta_{8} - \beta_{7} + \cdots + \beta_{2}) q^{8}+ \cdots + (2 \beta_{9} - \beta_{8} + \cdots + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 3 q^{4} - 10 q^{5} - 8 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} + 3 q^{4} - 10 q^{5} - 8 q^{7} + 3 q^{8} + q^{10} + 10 q^{11} - 2 q^{13} - 4 q^{14} + q^{16} - 8 q^{17} + 10 q^{19} - 3 q^{20} - q^{22} + 20 q^{23} + 10 q^{25} - 4 q^{26} - 15 q^{28} - 8 q^{29} - 8 q^{31} + 20 q^{32} - 22 q^{34} + 8 q^{35} - 10 q^{37} - q^{38} - 3 q^{40} - 14 q^{41} - 28 q^{43} + 3 q^{44} - 7 q^{46} + 20 q^{47} - 10 q^{49} - q^{50} - 14 q^{52} + 4 q^{53} - 10 q^{55} - 3 q^{56} - 16 q^{58} + 2 q^{59} - 16 q^{61} + 22 q^{62} + 3 q^{64} + 2 q^{65} - 22 q^{67} + 4 q^{70} + 8 q^{71} - 12 q^{73} - 21 q^{74} + 3 q^{76} - 8 q^{77} - 18 q^{79} - q^{80} + 20 q^{82} + 18 q^{83} + 8 q^{85} + 33 q^{86} + 3 q^{88} - 6 q^{89} - 2 q^{91} + 16 q^{92} + 52 q^{94} - 10 q^{95} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 11x^{8} + 11x^{7} + 36x^{6} - 31x^{5} - 44x^{4} + 26x^{3} + 22x^{2} - 5x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{9} + \nu^{8} + 10\nu^{7} - 10\nu^{6} - 26\nu^{5} + 21\nu^{4} + 18\nu^{3} - 5\nu^{2} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{9} + 11\nu^{7} - 36\nu^{5} - 4\nu^{4} + 40\nu^{3} + 7\nu^{2} - 13\nu - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{9} - 11\nu^{7} + 36\nu^{5} + 5\nu^{4} - 39\nu^{3} - 13\nu^{2} + 10\nu + 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{9} - 21\nu^{7} + 62\nu^{5} + 10\nu^{4} - 52\nu^{3} - 22\nu^{2} + 5\nu + 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{9} + 2\nu^{8} + 10\nu^{7} - 21\nu^{6} - 25\nu^{5} + 57\nu^{4} + 15\nu^{3} - 42\nu^{2} - 3\nu + 5 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 2\nu^{8} - \nu^{7} - 21\nu^{6} + 11\nu^{5} + 62\nu^{4} - 25\nu^{3} - 56\nu^{2} + 11\nu + 12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 2\nu^{9} - \nu^{8} - 21\nu^{7} + 11\nu^{6} + 62\nu^{5} - 25\nu^{4} - 56\nu^{3} + 11\nu^{2} + 12\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{7} + \beta_{5} - \beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{4} + 7\beta_{2} - \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} - 8\beta_{8} + 9\beta_{7} - \beta_{6} + 7\beta_{5} - \beta_{4} - \beta_{3} - 10\beta_{2} + 21\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} + 10\beta_{8} - 10\beta_{7} - 2\beta_{5} + 9\beta_{4} + \beta_{3} + 45\beta_{2} - 11\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10 \beta_{9} - 54 \beta_{8} + 64 \beta_{7} - 9 \beta_{6} + 42 \beta_{5} - 10 \beta_{4} - 10 \beta_{3} + \cdots - 22 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 10 \beta_{9} + 79 \beta_{8} - 79 \beta_{7} + \beta_{6} - 26 \beta_{5} + 64 \beta_{4} + 11 \beta_{3} + \cdots + 222 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 74 \beta_{9} - 350 \beta_{8} + 424 \beta_{7} - 63 \beta_{6} + 250 \beta_{5} - 79 \beta_{4} - 74 \beta_{3} + \cdots - 189 \) 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
2.31719
1.94604
1.25960
1.14704
0.488821
−0.367713
−0.651497
−0.935628
−1.63771
−2.56615
−2.31719 0 3.36939 −1.00000 0 −0.0584087 −3.17314 0 2.31719
1.2 −1.94604 0 1.78709 −1.00000 0 −2.61382 0.414332 0 1.94604
1.3 −1.25960 0 −0.413415 −1.00000 0 0.764914 3.03993 0 1.25960
1.4 −1.14704 0 −0.684293 −1.00000 0 −3.07105 3.07900 0 1.14704
1.5 −0.488821 0 −1.76105 −1.00000 0 3.63626 1.83848 0 0.488821
1.6 0.367713 0 −1.86479 −1.00000 0 1.63545 −1.42113 0 −0.367713
1.7 0.651497 0 −1.57555 −1.00000 0 −4.63487 −2.32946 0 −0.651497
1.8 0.935628 0 −1.12460 −1.00000 0 −0.498371 −2.92346 0 −0.935628
1.9 1.63771 0 0.682109 −1.00000 0 −1.06651 −2.15833 0 −1.63771
1.10 2.56615 0 4.58511 −1.00000 0 −2.09360 6.63378 0 −2.56615
\(n\): 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.bn ✓ 10
3.b odd 2 1 9405.2.a.bo yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9405.2.a.bn ✓ 10 1.a even 1 1 trivial
9405.2.a.bo yes 10 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):

\( T_{2}^{10} + T_{2}^{9} - 11T_{2}^{8} - 11T_{2}^{7} + 36T_{2}^{6} + 31T_{2}^{5} - 44T_{2}^{4} - 26T_{2}^{3} + 22T_{2}^{2} + 5T_{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{10} + 8 T_{7}^{9} + 2 T_{7}^{8} - 120 T_{7}^{7} - 262 T_{7}^{6} + 168 T_{7}^{5} + 820 T_{7}^{4} + \cdots - 11 \) Copy content Toggle raw display
\( T_{13}^{10} + 2 T_{13}^{9} - 52 T_{13}^{8} - 78 T_{13}^{7} + 937 T_{13}^{6} + 904 T_{13}^{5} + \cdots - 5296 \) Copy content Toggle raw display
\( T_{17}^{10} + 8 T_{17}^{9} - 42 T_{17}^{8} - 406 T_{17}^{7} + 33 T_{17}^{6} + 4646 T_{17}^{5} + \cdots - 1776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + \cdots - 3 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 8 T^{9} + \cdots - 11 \) Copy content Toggle raw display
$11$ \( (T - 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + 2 T^{9} + \cdots - 5296 \) Copy content Toggle raw display
$17$ \( T^{10} + 8 T^{9} + \cdots - 1776 \) Copy content Toggle raw display
$19$ \( (T - 1)^{10} \) Copy content Toggle raw display
$23$ \( T^{10} - 20 T^{9} + \cdots + 2577 \) Copy content Toggle raw display
$29$ \( T^{10} + 8 T^{9} + \cdots - 3504 \) Copy content Toggle raw display
$31$ \( T^{10} + 8 T^{9} + \cdots + 187024 \) Copy content Toggle raw display
$37$ \( T^{10} + 10 T^{9} + \cdots + 1462063 \) Copy content Toggle raw display
$41$ \( T^{10} + 14 T^{9} + \cdots + 2481552 \) Copy content Toggle raw display
$43$ \( T^{10} + 28 T^{9} + \cdots - 8406575 \) Copy content Toggle raw display
$47$ \( T^{10} - 20 T^{9} + \cdots + 4800 \) Copy content Toggle raw display
$53$ \( T^{10} - 4 T^{9} + \cdots + 11520 \) Copy content Toggle raw display
$59$ \( T^{10} - 2 T^{9} + \cdots + 10918119 \) Copy content Toggle raw display
$61$ \( T^{10} + 16 T^{9} + \cdots - 17221328 \) Copy content Toggle raw display
$67$ \( T^{10} + 22 T^{9} + \cdots - 4478009 \) Copy content Toggle raw display
$71$ \( T^{10} - 8 T^{9} + \cdots + 31630467 \) Copy content Toggle raw display
$73$ \( T^{10} + 12 T^{9} + \cdots - 54169600 \) Copy content Toggle raw display
$79$ \( T^{10} + 18 T^{9} + \cdots - 9891541 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 4426774896 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 175923351 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 12582347849 \) Copy content Toggle raw display
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