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Properties

Label 784.6.f.b
Level $784$
Weight $6$
Character orbit 784.f
Analytic conductor $125.741$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,6,Mod(783,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.783");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.f (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(125.740914733\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 293 x^{10} + 336 x^{9} + 60118 x^{8} + 67616 x^{7} + 5772748 x^{6} + \cdots + 37147165696 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{2}\cdot 7^{8} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{10} + 91) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{10} + 91) q^{9} + (\beta_{5} + \beta_{3}) q^{11} + (\beta_{9} + 4 \beta_{2} + 9 \beta_1) q^{13} + ( - 2 \beta_{6} - \beta_{5} - 2 \beta_{3}) q^{15} + ( - \beta_{9} + 2 \beta_{2} - 14 \beta_1) q^{17} + (3 \beta_{8} + \beta_{7} - 6 \beta_{4}) q^{19} + (7 \beta_{6} - 12 \beta_{5} - 4 \beta_{3}) q^{23} + (2 \beta_{11} + 6 \beta_{10} + 412) q^{25} + ( - 11 \beta_{8} + \cdots + 112 \beta_{4}) q^{27}+ \cdots + ( - 309 \beta_{6} + \cdots + 394 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 1088 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 1088 q^{9} + 4920 q^{25} + 22512 q^{29} + 15876 q^{37} - 45708 q^{53} - 25900 q^{57} + 124440 q^{65} + 186164 q^{81} + 70116 q^{85} + 546364 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 293 x^{10} + 336 x^{9} + 60118 x^{8} + 67616 x^{7} + 5772748 x^{6} + \cdots + 37147165696 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11\!\cdots\!49 \nu^{11} + \cdots - 31\!\cdots\!48 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13\!\cdots\!11 \nu^{11} + \cdots + 83\!\cdots\!28 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 38\!\cdots\!03 \nu^{11} + \cdots - 29\!\cdots\!64 ) / 34\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28\!\cdots\!77 \nu^{11} + \cdots + 22\!\cdots\!76 ) / 14\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 74\!\cdots\!02 \nu^{11} + \cdots - 11\!\cdots\!84 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\!\cdots\!34 \nu^{11} + \cdots - 53\!\cdots\!12 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!73 \nu^{11} + \cdots - 77\!\cdots\!16 ) / 74\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26\!\cdots\!53 \nu^{11} + \cdots - 72\!\cdots\!96 ) / 14\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 40\!\cdots\!27 \nu^{11} + \cdots - 18\!\cdots\!64 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 58\!\cdots\!71 \nu^{11} + \cdots + 49\!\cdots\!48 ) / 21\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 76\!\cdots\!58 \nu^{11} + \cdots - 24\!\cdots\!36 ) / 14\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 3 \beta_{10} - 3 \beta_{9} - \beta_{8} + 2 \beta_{7} - 6 \beta_{6} - 6 \beta_{5} + \cdots + 85 ) / 336 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 5 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 19 \beta_{8} + 2 \beta_{7} - 18 \beta_{6} + \cdots - 16157 ) / 336 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -113\beta_{11} - 447\beta_{10} + 65\beta_{8} - 166\beta_{7} + 10011\beta_{4} - 50801 ) / 168 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 231 \beta_{11} - 913 \beta_{10} + 913 \beta_{9} - 1121 \beta_{8} + 190 \beta_{7} + 2206 \beta_{6} + \cdots - 777519 ) / 112 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15881 \beta_{11} + 85023 \beta_{10} + 85023 \beta_{9} + 5983 \beta_{8} + 10030 \beta_{7} + \cdots + 13627745 ) / 336 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 95725\beta_{11} + 814635\beta_{10} + 597899\beta_{8} - 158026\beta_{7} - 18415623\beta_{4} + 377158069 ) / 168 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2337985 \beta_{11} + 16896471 \beta_{10} - 16896471 \beta_{9} + 3570455 \beta_{8} - 604786 \beta_{7} + \cdots + 3123366217 ) / 336 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4942071 \beta_{11} - 65498769 \beta_{10} - 65498769 \beta_{9} - 37268945 \beta_{8} + \cdots - 21902795663 ) / 112 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 358072537 \beta_{11} - 3416402847 \beta_{10} - 1011361439 \beta_{8} + 424373890 \beta_{7} + \cdots - 675850502977 ) / 168 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2568429725 \beta_{11} - 43754613627 \beta_{10} + 43754613627 \beta_{9} - 21693921019 \beta_{8} + \cdots - 12117101140325 ) / 336 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 57560474225 \beta_{11} + 696512316039 \beta_{10} + 696512316039 \beta_{9} + 240739597447 \beta_{8} + \cdots + 142492369240793 ) / 336 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
783.1
7.16500 − 12.4101i
7.16500 + 12.4101i
−1.91464 − 3.31625i
−1.91464 + 3.31625i
5.50778 − 9.53976i
5.50778 + 9.53976i
−5.11675 + 8.86248i
−5.11675 − 8.86248i
−5.54593 − 9.60583i
−5.54593 + 9.60583i
1.40454 − 2.43273i
1.40454 + 2.43273i
0 −27.4261 0 − 1.90654i 0 0 0 509.189 0
783.2 0 −27.4261 0 1.90654i 0 0 0 509.189 0
783.3 0 −12.6060 0 − 73.4149i 0 0 0 −84.0890 0
783.4 0 −12.6060 0 73.4149i 0 0 0 −84.0890 0
783.5 0 −9.48156 0 − 52.4558i 0 0 0 −153.100 0
783.6 0 −9.48156 0 52.4558i 0 0 0 −153.100 0
783.7 0 9.48156 0 − 52.4558i 0 0 0 −153.100 0
783.8 0 9.48156 0 52.4558i 0 0 0 −153.100 0
783.9 0 12.6060 0 − 73.4149i 0 0 0 −84.0890 0
783.10 0 12.6060 0 73.4149i 0 0 0 −84.0890 0
783.11 0 27.4261 0 − 1.90654i 0 0 0 509.189 0
783.12 0 27.4261 0 1.90654i 0 0 0 509.189 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 783.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.6.f.b 12
4.b odd 2 1 inner 784.6.f.b 12
7.b odd 2 1 inner 784.6.f.b 12
7.c even 3 1 112.6.p.a ✓ 12
7.d odd 6 1 112.6.p.a ✓ 12
28.d even 2 1 inner 784.6.f.b 12
28.f even 6 1 112.6.p.a ✓ 12
28.g odd 6 1 112.6.p.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.6.p.a ✓ 12 7.c even 3 1
112.6.p.a ✓ 12 7.d odd 6 1
112.6.p.a ✓ 12 28.f even 6 1
112.6.p.a ✓ 12 28.g odd 6 1
784.6.f.b 12 1.a even 1 1 trivial
784.6.f.b 12 4.b odd 2 1 inner
784.6.f.b 12 7.b odd 2 1 inner
784.6.f.b 12 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 1001T_{3}^{4} + 201439T_{3}^{2} - 10745847 \) acting on \(S_{6}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 1001 T^{4} + \cdots - 10745847)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 8145 T^{4} + \cdots + 53907363)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 2221798369581)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 532528090841088)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 22963272466707)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 33\!\cdots\!63)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 18\!\cdots\!01)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 5628 T^{2} + \cdots + 189142218720)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 11\!\cdots\!87)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 3969 T^{2} + \cdots - 64581517987)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 27\!\cdots\!12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 36\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 57\!\cdots\!83)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 11427 T^{2} + \cdots - 436397468751)^{4} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 71\!\cdots\!27)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 44\!\cdots\!67)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 12\!\cdots\!25)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 55\!\cdots\!67)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 33\!\cdots\!81)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 76\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 29\!\cdots\!87)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
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