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Properties

Label 6760.2.a.bn
Level $6760$
Weight $2$
Character orbit 6760.a
Self dual yes
Analytic conductor $53.979$
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] = [6760,2,Mod(1,6760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6760, 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("6760.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6760 = 2^{3} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6760.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: \(53.9788717664\)
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: \( x^{9} - x^{8} - 15x^{7} + 12x^{6} + 72x^{5} - 39x^{4} - 127x^{3} + 35x^{2} + 56x - 7 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + q^{5} + ( - \beta_{8} + \beta_{2} - 1) q^{7} + (\beta_{8} + \beta_{7} - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + q^{5} + ( - \beta_{8} + \beta_{2} - 1) q^{7} + (\beta_{8} + \beta_{7} - \beta_{3} + 1) q^{9} + ( - \beta_{7} - \beta_{6} + \beta_{3} - 2) q^{11} + \beta_1 q^{15} + (\beta_{8} + \beta_{7} + \beta_{5} + \cdots + 1) q^{17}+ \cdots + ( - 2 \beta_{8} + \beta_{7} + 2 \beta_{6} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{3} + 9 q^{5} - 7 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{3} + 9 q^{5} - 7 q^{7} + 4 q^{9} - 15 q^{11} + q^{15} - q^{17} - 6 q^{19} - q^{21} - 8 q^{23} + 9 q^{25} + 4 q^{27} - 12 q^{29} - 11 q^{31} - 12 q^{33} - 7 q^{35} - 17 q^{37} - 6 q^{41} + 5 q^{43} + 4 q^{45} - 6 q^{47} + 16 q^{49} - 9 q^{51} + 24 q^{53} - 15 q^{55} + 12 q^{57} - 15 q^{59} - 7 q^{61} - 47 q^{63} - 21 q^{67} - 17 q^{69} - 33 q^{71} + 4 q^{73} + q^{75} + 8 q^{77} - 31 q^{79} - 23 q^{81} - 71 q^{83} - q^{85} - 28 q^{87} - 19 q^{89} - 21 q^{93} - 6 q^{95} + 22 q^{97} - 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 15x^{7} + 12x^{6} + 72x^{5} - 39x^{4} - 127x^{3} + 35x^{2} + 56x - 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{8} + 199\nu^{7} - 39\nu^{6} - 2493\nu^{5} + 509\nu^{4} + 7938\nu^{3} - 940\nu^{2} - 4241\nu + 833 ) / 1202 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{8} + 6\nu^{7} - 125\nu^{6} + 100\nu^{5} + 414\nu^{4} - 1464\nu^{3} + 23\nu^{2} + 3158\nu + 158 ) / 1202 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -17\nu^{8} + 189\nu^{7} - 31\nu^{6} - 2259\nu^{5} + 1622\nu^{4} + 6772\nu^{3} - 3783\nu^{2} - 3895\nu + 770 ) / 1202 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22 \nu^{8} - 215 \nu^{7} + 773 \nu^{6} + 2627 \nu^{5} - 6421 \nu^{4} - 8842 \nu^{3} + 14902 \nu^{2} + \cdots - 5261 ) / 1202 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 50 \nu^{8} + 167 \nu^{7} + 828 \nu^{6} - 2225 \nu^{5} - 4704 \nu^{4} + 8534 \nu^{3} + 10156 \nu^{2} + \cdots - 3816 ) / 1202 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 110 \nu^{8} + 127 \nu^{7} + 1461 \nu^{6} - 1289 \nu^{5} - 5661 \nu^{4} + 2668 \nu^{3} + \cdots - 2265 ) / 1202 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 119 \nu^{8} - 121 \nu^{7} - 1586 \nu^{6} + 1389 \nu^{5} + 6075 \nu^{4} - 4132 \nu^{3} - 5973 \nu^{2} + \cdots - 2385 ) / 1202 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{8} + 8\beta_{7} - \beta_{6} - \beta_{5} - 3\beta_{4} - 8\beta_{3} + 2\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 9\beta_{6} - 10\beta_{5} - 11\beta_{4} + 15\beta_{3} - 9\beta_{2} + 30\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 49\beta_{8} + 60\beta_{7} - 9\beta_{6} - 11\beta_{5} - 37\beta_{4} - 56\beta_{3} + 24\beta_{2} - 2\beta _1 + 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{7} + 73\beta_{6} - 86\beta_{5} - 101\beta_{4} + 149\beta_{3} - 65\beta_{2} + 197\beta _1 - 30 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 356 \beta_{8} + 443 \beta_{7} - 65 \beta_{6} - 101 \beta_{5} - 349 \beta_{4} - 377 \beta_{3} + \cdots + 1012 \) 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
−2.73466
−1.86038
−1.51631
−0.768427
0.120029
0.838737
1.84133
2.25610
2.82358
0 −2.73466 0 1.00000 0 −1.48339 0 4.47835 0
1.2 0 −1.86038 0 1.00000 0 −2.04839 0 0.460998 0
1.3 0 −1.51631 0 1.00000 0 −4.95332 0 −0.700790 0
1.4 0 −0.768427 0 1.00000 0 3.82769 0 −2.40952 0
1.5 0 0.120029 0 1.00000 0 1.00832 0 −2.98559 0
1.6 0 0.838737 0 1.00000 0 2.20703 0 −2.29652 0
1.7 0 1.84133 0 1.00000 0 1.49996 0 0.390486 0
1.8 0 2.25610 0 1.00000 0 −3.09732 0 2.09001 0
1.9 0 2.82358 0 1.00000 0 −3.96058 0 4.97259 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 6760.2.a.bn yes 9
13.b even 2 1 6760.2.a.bm ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6760.2.a.bm ✓ 9 13.b even 2 1
6760.2.a.bn yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6760))\):

\( T_{3}^{9} - T_{3}^{8} - 15T_{3}^{7} + 12T_{3}^{6} + 72T_{3}^{5} - 39T_{3}^{4} - 127T_{3}^{3} + 35T_{3}^{2} + 56T_{3} - 7 \) Copy content Toggle raw display
\( T_{7}^{9} + 7T_{7}^{8} - 15T_{7}^{7} - 168T_{7}^{6} - 30T_{7}^{5} + 1141T_{7}^{4} + 589T_{7}^{3} - 2849T_{7}^{2} - 1022T_{7} + 2359 \) Copy content Toggle raw display
\( T_{11}^{9} + 15 T_{11}^{8} + 66 T_{11}^{7} - 39 T_{11}^{6} - 1148 T_{11}^{5} - 3616 T_{11}^{4} + \cdots + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - T^{8} - 15 T^{7} + \cdots - 7 \) Copy content Toggle raw display
$5$ \( (T - 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 7 T^{8} + \cdots + 2359 \) Copy content Toggle raw display
$11$ \( T^{9} + 15 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + T^{8} + \cdots - 512 \) Copy content Toggle raw display
$19$ \( T^{9} + 6 T^{8} + \cdots - 392512 \) Copy content Toggle raw display
$23$ \( T^{9} + 8 T^{8} + \cdots - 15379 \) Copy content Toggle raw display
$29$ \( T^{9} + 12 T^{8} + \cdots - 2946203 \) Copy content Toggle raw display
$31$ \( T^{9} + 11 T^{8} + \cdots - 35776 \) Copy content Toggle raw display
$37$ \( T^{9} + 17 T^{8} + \cdots + 222656 \) Copy content Toggle raw display
$41$ \( T^{9} + 6 T^{8} + \cdots + 148639 \) Copy content Toggle raw display
$43$ \( T^{9} - 5 T^{8} + \cdots - 651239 \) Copy content Toggle raw display
$47$ \( T^{9} + 6 T^{8} + \cdots - 31157 \) Copy content Toggle raw display
$53$ \( T^{9} - 24 T^{8} + \cdots - 21082048 \) Copy content Toggle raw display
$59$ \( T^{9} + 15 T^{8} + \cdots + 14064128 \) Copy content Toggle raw display
$61$ \( T^{9} + 7 T^{8} + \cdots + 57917 \) Copy content Toggle raw display
$67$ \( T^{9} + 21 T^{8} + \cdots - 1011907 \) Copy content Toggle raw display
$71$ \( T^{9} + 33 T^{8} + \cdots - 2009664 \) Copy content Toggle raw display
$73$ \( T^{9} - 4 T^{8} + \cdots - 10382848 \) Copy content Toggle raw display
$79$ \( T^{9} + 31 T^{8} + \cdots - 443456 \) Copy content Toggle raw display
$83$ \( T^{9} + 71 T^{8} + \cdots - 18194247 \) Copy content Toggle raw display
$89$ \( T^{9} + 19 T^{8} + \cdots - 167929 \) Copy content Toggle raw display
$97$ \( T^{9} - 22 T^{8} + \cdots + 402984448 \) Copy content Toggle raw display
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