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Properties

Label 5696.2.a.bq
Level $5696$
Weight $2$
Character orbit 5696.a
Self dual yes
Analytic conductor $45.483$
Analytic rank $1$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5696,2,Mod(1,5696)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5696, 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("5696.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5696 = 2^{6} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5696.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: \(45.4827889913\)
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: \( x^{10} - 16x^{8} + 91x^{6} - 235x^{4} + 278x^{2} - 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 2848)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{7} - 1) q^{5} - \beta_{5} q^{7} + ( - \beta_{9} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{7} - 1) q^{5} - \beta_{5} q^{7} + ( - \beta_{9} + 1) q^{9} - \beta_{2} q^{11} + (\beta_{6} - 2) q^{13} + (\beta_{8} + \beta_{5} + \beta_{3}) q^{15} + ( - \beta_{7} + \beta_{4} + 1) q^{17} + \beta_1 q^{19} + (\beta_{9} - \beta_{7} - 1) q^{21} + ( - \beta_{8} + \beta_{3} + \beta_{2}) q^{23} + (\beta_{9} - 2 \beta_{7} - \beta_{4} + 1) q^{25} + ( - \beta_{8} + \beta_{5} + \cdots - \beta_1) q^{27}+ \cdots + (2 \beta_{8} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{5} + 8 q^{9} - 16 q^{13} + 6 q^{17} - 12 q^{21} + 4 q^{25} - 24 q^{29} + 8 q^{33} - 16 q^{37} + 8 q^{41} - 24 q^{45} - 2 q^{49} - 42 q^{53} - 6 q^{57} - 20 q^{61} + 28 q^{65} - 30 q^{69} + 2 q^{73} - 28 q^{77} + 2 q^{81} - 42 q^{85} + 10 q^{89} - 54 q^{93} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 16x^{8} + 91x^{6} - 235x^{4} + 278x^{2} - 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} - 16\nu^{7} + 91\nu^{5} - 224\nu^{3} + 179\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{9} - 37\nu^{7} + 130\nu^{5} - 133\nu^{3} - 13\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{9} + 32\nu^{7} - 171\nu^{5} + 349\nu^{3} - 215\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{6} + 12\nu^{4} - 40\nu^{2} + 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{9} - 48\nu^{7} + 262\nu^{5} - 573\nu^{3} + 416\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{8} - 13\nu^{6} + 52\nu^{4} - 79\nu^{2} + 41 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{8} - 14\nu^{6} + 63\nu^{4} - 110\nu^{2} + 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -18\nu^{9} + 255\nu^{7} - 1176\nu^{5} + 2140\nu^{3} - 1297\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 2\nu^{8} - 28\nu^{6} + 127\nu^{4} - 227\nu^{2} + 135 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{7} - \beta_{6} - \beta_{4} + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} + 7\beta_{5} + 4\beta_{3} + 3\beta_{2} - 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{9} - 11\beta_{7} - 7\beta_{6} - 7\beta_{4} + 35 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{8} + 50\beta_{5} + 25\beta_{3} + 27\beta_{2} - 19\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 34\beta_{9} - 46\beta_{7} - 22\beta_{6} - 23\beta_{4} + 107 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 68\beta_{8} + 357\beta_{5} + 179\beta_{3} + 206\beta_{2} - 107\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 495\beta_{9} - 703\beta_{7} - 285\beta_{6} - 313\beta_{4} + 1433 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 493\beta_{8} + 2551\beta_{5} + 1306\beta_{3} + 1511\beta_{2} - 678\beta_1 ) / 2 \) 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
1.31184
−1.85444
1.59572
2.67856
1.05789
−1.05789
−2.67856
−1.59572
1.85444
−1.31184
0 −2.94848 0 −2.30430 0 1.48864 0 5.69352 0
1.2 0 −2.59505 0 0.256005 0 1.92300 0 3.73426 0
1.3 0 −1.33129 0 2.28408 0 1.54468 0 −1.22766 0
1.4 0 −1.30755 0 −3.98141 0 −3.26699 0 −1.29032 0
1.5 0 −0.300321 0 0.745631 0 −3.87644 0 −2.90981 0
1.6 0 0.300321 0 0.745631 0 3.87644 0 −2.90981 0
1.7 0 1.30755 0 −3.98141 0 3.26699 0 −1.29032 0
1.8 0 1.33129 0 2.28408 0 −1.54468 0 −1.22766 0
1.9 0 2.59505 0 0.256005 0 −1.92300 0 3.73426 0
1.10 0 2.94848 0 −2.30430 0 −1.48864 0 5.69352 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(89\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5696.2.a.bq 10
4.b odd 2 1 inner 5696.2.a.bq 10
8.b even 2 1 2848.2.a.o ✓ 10
8.d odd 2 1 2848.2.a.o ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2848.2.a.o ✓ 10 8.b even 2 1
2848.2.a.o ✓ 10 8.d odd 2 1
5696.2.a.bq 10 1.a even 1 1 trivial
5696.2.a.bq 10 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5696))\):

\( T_{3}^{10} - 19T_{3}^{8} + 117T_{3}^{6} - 261T_{3}^{4} + 200T_{3}^{2} - 16 \) Copy content Toggle raw display
\( T_{5}^{5} + 3T_{5}^{4} - 9T_{5}^{3} - 15T_{5}^{2} + 20T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{10} - 34T_{7}^{8} + 396T_{7}^{6} - 1924T_{7}^{4} + 4080T_{7}^{2} - 3136 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 19 T^{8} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( (T^{5} + 3 T^{4} - 9 T^{3} + \cdots - 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} - 34 T^{8} + \cdots - 3136 \) Copy content Toggle raw display
$11$ \( T^{10} - 40 T^{8} + \cdots - 4096 \) Copy content Toggle raw display
$13$ \( (T^{5} + 8 T^{4} + 4 T^{3} + \cdots - 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} - 3 T^{4} - 29 T^{3} + \cdots + 28)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} - 37 T^{8} + \cdots - 784 \) Copy content Toggle raw display
$23$ \( T^{10} - 105 T^{8} + \cdots - 15376 \) Copy content Toggle raw display
$29$ \( (T^{5} + 12 T^{4} + \cdots - 272)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 103 T^{8} + \cdots - 71824 \) Copy content Toggle raw display
$37$ \( (T^{5} + 8 T^{4} + \cdots + 3296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} - 4 T^{4} + \cdots + 272)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 339591184 \) Copy content Toggle raw display
$47$ \( T^{10} - 232 T^{8} + \cdots - 1183744 \) Copy content Toggle raw display
$53$ \( (T^{5} + 21 T^{4} + \cdots + 5108)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 362 T^{8} + \cdots - 2849344 \) Copy content Toggle raw display
$61$ \( (T^{5} + 10 T^{4} + \cdots - 1232)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 371101696 \) Copy content Toggle raw display
$71$ \( T^{10} - 184 T^{8} + \cdots - 65536 \) Copy content Toggle raw display
$73$ \( (T^{5} - T^{4} - 169 T^{3} + \cdots - 9748)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} - 368 T^{8} + \cdots - 16777216 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 10043246656 \) Copy content Toggle raw display
$89$ \( (T - 1)^{10} \) Copy content Toggle raw display
$97$ \( (T^{5} - T^{4} - 121 T^{3} + \cdots + 2012)^{2} \) Copy content Toggle raw display
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