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Properties

Label 7742.2.a.bj
Level $7742$
Weight $2$
Character orbit 7742.a
Self dual yes
Analytic conductor $61.820$
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] = [7742,2,Mod(1,7742)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7742, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7742.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7742 = 2 \cdot 7^{2} \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7742.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: \(61.8201812449\)
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: \( x^{9} - 2x^{8} - 18x^{7} + 34x^{6} + 105x^{5} - 184x^{4} - 212x^{3} + 342x^{2} + 72x - 108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1106)
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.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} + \beta_{6} q^{5} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} + \beta_{6} q^{5} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 1) q^{9} + \beta_{6} q^{10} + (\beta_{5} + 1) q^{11} - \beta_1 q^{12} + (\beta_{5} - \beta_{4}) q^{13} + ( - \beta_{8} - \beta_{4}) q^{15} + q^{16} + ( - \beta_{8} + \beta_{2} - 1) q^{17} + (\beta_{2} + 1) q^{18} + ( - \beta_{6} + \beta_{4} - \beta_{3} + \cdots - 1) q^{19}+ \cdots + (\beta_{8} - 2 \beta_{7} - 2 \beta_{6} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} - 2 q^{3} + 9 q^{4} - 4 q^{5} - 2 q^{6} + 9 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} - 2 q^{3} + 9 q^{4} - 4 q^{5} - 2 q^{6} + 9 q^{8} + 13 q^{9} - 4 q^{10} + 8 q^{11} - 2 q^{12} - 2 q^{13} - 2 q^{15} + 9 q^{16} - 6 q^{17} + 13 q^{18} - 2 q^{19} - 4 q^{20} + 8 q^{22} - 2 q^{24} + 23 q^{25} - 2 q^{26} - 2 q^{27} + 10 q^{29} - 2 q^{30} + 6 q^{31} + 9 q^{32} + 2 q^{33} - 6 q^{34} + 13 q^{36} + 10 q^{37} - 2 q^{38} - 6 q^{39} - 4 q^{40} - 10 q^{41} + 22 q^{43} + 8 q^{44} + 10 q^{45} + 14 q^{47} - 2 q^{48} + 23 q^{50} + 4 q^{51} - 2 q^{52} + 12 q^{53} - 2 q^{54} + 14 q^{55} + 22 q^{57} + 10 q^{58} + 2 q^{59} - 2 q^{60} - 6 q^{61} + 6 q^{62} + 9 q^{64} - 12 q^{65} + 2 q^{66} + 24 q^{67} - 6 q^{68} + 26 q^{69} + 20 q^{71} + 13 q^{72} + 12 q^{73} + 10 q^{74} - 6 q^{75} - 2 q^{76} - 6 q^{78} + 9 q^{79} - 4 q^{80} - 19 q^{81} - 10 q^{82} + 22 q^{83} + 32 q^{85} + 22 q^{86} + 54 q^{87} + 8 q^{88} - 20 q^{89} + 10 q^{90} - 18 q^{93} + 14 q^{94} - 40 q^{95} - 2 q^{96} + 6 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 18x^{7} + 34x^{6} + 105x^{5} - 184x^{4} - 212x^{3} + 342x^{2} + 72x - 108 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + \nu^{7} - 15\nu^{6} - 11\nu^{5} + 72\nu^{4} + 23\nu^{3} - 134\nu^{2} + 12\nu + 54 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} - 2\nu^{7} + 14\nu^{6} + 26\nu^{5} - 55\nu^{4} - 86\nu^{3} + 60\nu^{2} + 50\nu - 6 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{8} - 2\nu^{7} + 30\nu^{6} + 22\nu^{5} - 135\nu^{4} - 46\nu^{3} + 196\nu^{2} - 24\nu - 45 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{8} - 2\nu^{7} + 46\nu^{6} + 24\nu^{5} - 215\nu^{4} - 66\nu^{3} + 332\nu^{2} + 10\nu - 96 ) / 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{8} + 4\nu^{7} - 60\nu^{6} - 53\nu^{5} + 270\nu^{4} + 191\nu^{3} - 392\nu^{2} - 177\nu + 108 ) / 9 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{8} - 6\nu^{7} + 112\nu^{6} + 74\nu^{5} - 563\nu^{4} - 218\nu^{3} + 976\nu^{2} + 70\nu - 318 ) / 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 2\beta_{3} + 8\beta_{2} + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{7} + 11\beta_{6} - 13\beta_{5} + 11\beta_{4} + 52\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{8} - 2\beta_{6} + 13\beta_{5} - \beta_{4} + 26\beta_{3} + 62\beta_{2} + 177 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{8} + 87\beta_{7} + 104\beta_{6} - 128\beta_{5} + 97\beta_{4} - \beta_{3} + 401\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16\beta_{8} - 36\beta_{6} + 131\beta_{5} - 14\beta_{4} + 256\beta_{3} + 488\beta_{2} - 2\beta _1 + 1329 \) 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
2.84305
2.83497
2.03031
1.36919
0.612322
−0.633348
−1.87839
−2.29544
−2.88266
1.00000 −2.84305 1.00000 −0.961671 −2.84305 0 1.00000 5.08295 −0.961671
1.2 1.00000 −2.83497 1.00000 4.00640 −2.83497 0 1.00000 5.03705 4.00640
1.3 1.00000 −2.03031 1.00000 −2.23822 −2.03031 0 1.00000 1.12214 −2.23822
1.4 1.00000 −1.36919 1.00000 −3.57755 −1.36919 0 1.00000 −1.12533 −3.57755
1.5 1.00000 −0.612322 1.00000 −1.06793 −0.612322 0 1.00000 −2.62506 −1.06793
1.6 1.00000 0.633348 1.00000 2.25697 0.633348 0 1.00000 −2.59887 2.25697
1.7 1.00000 1.87839 1.00000 −3.85604 1.87839 0 1.00000 0.528336 −3.85604
1.8 1.00000 2.29544 1.00000 3.07254 2.29544 0 1.00000 2.26903 3.07254
1.9 1.00000 2.88266 1.00000 −1.63450 2.88266 0 1.00000 5.30976 −1.63450
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(79\) \(-1\)

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 7742.2.a.bj 9
7.b odd 2 1 1106.2.a.l ✓ 9
21.c even 2 1 9954.2.a.bn 9
28.d even 2 1 8848.2.a.t 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1106.2.a.l ✓ 9 7.b odd 2 1
7742.2.a.bj 9 1.a even 1 1 trivial
8848.2.a.t 9 28.d even 2 1
9954.2.a.bn 9 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7742))\):

\( T_{3}^{9} + 2T_{3}^{8} - 18T_{3}^{7} - 34T_{3}^{6} + 105T_{3}^{5} + 184T_{3}^{4} - 212T_{3}^{3} - 342T_{3}^{2} + 72T_{3} + 108 \) Copy content Toggle raw display
\( T_{5}^{9} + 4T_{5}^{8} - 26T_{5}^{7} - 120T_{5}^{6} + 140T_{5}^{5} + 1056T_{5}^{4} + 552T_{5}^{3} - 2512T_{5}^{2} - 3680T_{5} - 1440 \) Copy content Toggle raw display
\( T_{11}^{9} - 8 T_{11}^{8} - 22 T_{11}^{7} + 232 T_{11}^{6} + 221 T_{11}^{5} - 2088 T_{11}^{4} + \cdots + 1536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 2 T^{8} + \cdots + 108 \) Copy content Toggle raw display
$5$ \( T^{9} + 4 T^{8} + \cdots - 1440 \) Copy content Toggle raw display
$7$ \( T^{9} \) Copy content Toggle raw display
$11$ \( T^{9} - 8 T^{8} + \cdots + 1536 \) Copy content Toggle raw display
$13$ \( T^{9} + 2 T^{8} + \cdots + 138872 \) Copy content Toggle raw display
$17$ \( T^{9} + 6 T^{8} + \cdots + 384 \) Copy content Toggle raw display
$19$ \( T^{9} + 2 T^{8} + \cdots - 5632 \) Copy content Toggle raw display
$23$ \( T^{9} - 152 T^{7} + \cdots + 746496 \) Copy content Toggle raw display
$29$ \( T^{9} - 10 T^{8} + \cdots + 110592 \) Copy content Toggle raw display
$31$ \( T^{9} - 6 T^{8} + \cdots + 7240 \) Copy content Toggle raw display
$37$ \( T^{9} - 10 T^{8} + \cdots - 3963424 \) Copy content Toggle raw display
$41$ \( T^{9} + 10 T^{8} + \cdots - 31296 \) Copy content Toggle raw display
$43$ \( T^{9} - 22 T^{8} + \cdots - 256 \) Copy content Toggle raw display
$47$ \( T^{9} - 14 T^{8} + \cdots + 153600 \) Copy content Toggle raw display
$53$ \( T^{9} - 12 T^{8} + \cdots + 469800 \) Copy content Toggle raw display
$59$ \( T^{9} - 2 T^{8} + \cdots + 1824468 \) Copy content Toggle raw display
$61$ \( T^{9} + 6 T^{8} + \cdots - 2560 \) Copy content Toggle raw display
$67$ \( T^{9} - 24 T^{8} + \cdots + 56138176 \) Copy content Toggle raw display
$71$ \( T^{9} - 20 T^{8} + \cdots + 16390656 \) Copy content Toggle raw display
$73$ \( T^{9} - 12 T^{8} + \cdots + 185857792 \) Copy content Toggle raw display
$79$ \( (T - 1)^{9} \) Copy content Toggle raw display
$83$ \( T^{9} - 22 T^{8} + \cdots + 460800 \) Copy content Toggle raw display
$89$ \( T^{9} + 20 T^{8} + \cdots - 99702528 \) Copy content Toggle raw display
$97$ \( T^{9} - 6 T^{8} + \cdots - 428032 \) Copy content Toggle raw display
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