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Properties

Label 3380.2.a.s
Level $3380$
Weight $2$
Character orbit 3380.a
Self dual yes
Analytic conductor $26.989$
Analytic rank $0$
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] = [3380,2,Mod(1,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3380.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.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: \(26.9894358832\)
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: \( x^{9} - x^{8} - 19x^{7} + 16x^{6} + 106x^{5} - 87x^{4} - 153x^{3} + 149x^{2} - 26x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{7} + \beta_{5} - \beta_{4} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} + \cdots + 1) q^{7}+ \cdots + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots - 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{3} + 9 q^{5} + q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - q^{3} + 9 q^{5} + q^{7} + 12 q^{9} - 7 q^{11} - q^{15} + 13 q^{17} - 4 q^{19} - 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} + 13 q^{31} + 34 q^{33} + q^{35} + q^{37} - 6 q^{41} + q^{43} + 12 q^{45} - 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} + 38 q^{57} + 15 q^{59} + 21 q^{61} - 17 q^{63} - 7 q^{67} + 15 q^{69} - 7 q^{71} - 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} + 45 q^{83} + 13 q^{85} + 28 q^{87} - 41 q^{89} - 11 q^{93} - 4 q^{95} + 8 q^{97} - 81 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 19x^{7} + 16x^{6} + 106x^{5} - 87x^{4} - 153x^{3} + 149x^{2} - 26x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{8} + 42\nu^{7} - 241\nu^{6} - 394\nu^{5} + 1964\nu^{4} + 214\nu^{3} - 3215\nu^{2} + 1620\nu + 226 ) / 338 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} + 93\nu^{7} - 256\nu^{6} - 969\nu^{5} + 2683\nu^{4} + 1512\nu^{3} - 5103\nu^{2} + 1873\nu - 79 ) / 338 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -33\nu^{8} + 27\nu^{7} + 678\nu^{6} - 543\nu^{5} - 4073\nu^{4} + 3590\nu^{3} + 6347\nu^{2} - 6805\nu + 773 ) / 338 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 84\nu^{8} - 38\nu^{7} - 1649\nu^{6} + 614\nu^{5} + 9369\nu^{4} - 3976\nu^{3} - 13452\nu^{2} + 8288\nu - 1645 ) / 338 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 108\nu^{8} - 73\nu^{7} - 2096\nu^{6} + 1055\nu^{5} + 12070\nu^{4} - 5450\nu^{3} - 19082\nu^{2} + 9473\nu + 82 ) / 338 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 112 \nu^{8} - 107 \nu^{7} - 2086 \nu^{6} + 1551 \nu^{5} + 11478 \nu^{4} - 7780 \nu^{3} - 16922 \nu^{2} + \cdots - 1292 ) / 338 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 323 \nu^{8} + 295 \nu^{7} + 6206 \nu^{6} - 4731 \nu^{5} - 35175 \nu^{4} + 25992 \nu^{3} + \cdots + 3551 ) / 338 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} - 3\beta_{7} + \beta_{5} + 2\beta_{4} - \beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - 10\beta_{7} + 8\beta_{5} - 6\beta_{4} + 3\beta_{3} - 14\beta_{2} - \beta _1 + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{8} - 38\beta_{7} - 3\beta_{6} + 13\beta_{5} + 28\beta_{4} + 5\beta_{3} - 21\beta_{2} + 53\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -21\beta_{8} - 106\beta_{7} - 8\beta_{6} + 74\beta_{5} - 18\beta_{4} + 47\beta_{3} - 162\beta_{2} - 7\beta _1 + 338 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 162 \beta_{8} - 418 \beta_{7} - 55 \beta_{6} + 155 \beta_{5} + 330 \beta_{4} + 105 \beta_{3} + \cdots + 355 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 319 \beta_{8} - 1179 \beta_{7} - 160 \beta_{6} + 747 \beta_{5} + 195 \beta_{4} + 599 \beta_{3} + \cdots + 3428 \) 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
3.40622
2.52382
1.40883
0.715841
0.161999
0.0545075
−1.65879
−2.79462
−2.81781
0 −3.40622 0 1.00000 0 2.33790 0 8.60234 0
1.2 0 −2.52382 0 1.00000 0 −4.54727 0 3.36968 0
1.3 0 −1.40883 0 1.00000 0 −2.17051 0 −1.01520 0
1.4 0 −0.715841 0 1.00000 0 3.76323 0 −2.48757 0
1.5 0 −0.161999 0 1.00000 0 1.89771 0 −2.97376 0
1.6 0 −0.0545075 0 1.00000 0 3.71818 0 −2.99703 0
1.7 0 1.65879 0 1.00000 0 −4.25414 0 −0.248414 0
1.8 0 2.79462 0 1.00000 0 1.22908 0 4.80991 0
1.9 0 2.81781 0 1.00000 0 −0.974186 0 4.94004 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(13\) \(-1\)

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 3380.2.a.s yes 9
13.b even 2 1 3380.2.a.r ✓ 9
13.d odd 4 2 3380.2.f.j 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3380.2.a.r ✓ 9 13.b even 2 1
3380.2.a.s yes 9 1.a even 1 1 trivial
3380.2.f.j 18 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3380))\):

\( T_{3}^{9} + T_{3}^{8} - 19T_{3}^{7} - 16T_{3}^{6} + 106T_{3}^{5} + 87T_{3}^{4} - 153T_{3}^{3} - 149T_{3}^{2} - 26T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{9} - T_{7}^{8} - 41T_{7}^{7} + 60T_{7}^{6} + 518T_{7}^{5} - 951T_{7}^{4} - 1877T_{7}^{3} + 3909T_{7}^{2} + 1126T_{7} - 3121 \) Copy content Toggle raw display
\( T_{19}^{9} + 4 T_{19}^{8} - 87 T_{19}^{7} - 253 T_{19}^{6} + 2116 T_{19}^{5} + 3412 T_{19}^{4} + \cdots - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + T^{8} - 19 T^{7} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - T^{8} + \cdots - 3121 \) Copy content Toggle raw display
$11$ \( T^{9} + 7 T^{8} + \cdots - 94016 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - 13 T^{8} + \cdots + 86528 \) Copy content Toggle raw display
$19$ \( T^{9} + 4 T^{8} + \cdots - 64 \) Copy content Toggle raw display
$23$ \( T^{9} - 12 T^{8} + \cdots + 46171 \) Copy content Toggle raw display
$29$ \( T^{9} - 16 T^{8} + \cdots - 96559 \) Copy content Toggle raw display
$31$ \( T^{9} - 13 T^{8} + \cdots - 105664 \) Copy content Toggle raw display
$37$ \( T^{9} - T^{8} + \cdots + 18752 \) Copy content Toggle raw display
$41$ \( T^{9} + 6 T^{8} + \cdots + 23863181 \) Copy content Toggle raw display
$43$ \( T^{9} - T^{8} + \cdots - 9484117 \) Copy content Toggle raw display
$47$ \( T^{9} + 2 T^{8} + \cdots + 25493819 \) Copy content Toggle raw display
$53$ \( T^{9} - 30 T^{8} + \cdots + 118208 \) Copy content Toggle raw display
$59$ \( T^{9} - 15 T^{8} + \cdots - 512 \) Copy content Toggle raw display
$61$ \( T^{9} - 21 T^{8} + \cdots + 1253057 \) Copy content Toggle raw display
$67$ \( T^{9} + 7 T^{8} + \cdots + 4851601 \) Copy content Toggle raw display
$71$ \( T^{9} + 7 T^{8} + \cdots + 110282432 \) Copy content Toggle raw display
$73$ \( T^{9} + 28 T^{8} + \cdots - 7046656 \) Copy content Toggle raw display
$79$ \( T^{9} - 31 T^{8} + \cdots + 342428864 \) Copy content Toggle raw display
$83$ \( T^{9} - 45 T^{8} + \cdots - 65550407 \) Copy content Toggle raw display
$89$ \( T^{9} + 41 T^{8} + \cdots + 54990949 \) Copy content Toggle raw display
$97$ \( T^{9} - 8 T^{8} + \cdots + 7849472 \) Copy content Toggle raw display
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