↑

Properties

Label 392.3.c.b
Level $392$
Weight $3$
Character orbit 392.c
Analytic conductor $10.681$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,3,Mod(97,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.c (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: \(10.6812263629\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 30x^{6} + 84x^{5} + 597x^{4} - 384x^{3} - 4236x^{2} + 688x + 13378 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 7^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_1) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{4}) q^{5} + (\beta_{3} + 3 \beta_{2} - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_1) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{4}) q^{5} + (\beta_{3} + 3 \beta_{2} - 8) q^{9} + ( - \beta_{3} + \beta_{2} - 3) q^{11} + (\beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{13} + ( - \beta_{7} + 13 \beta_{2} + 5) q^{15} + ( - \beta_{6} + 4 \beta_{5} + \cdots - 3 \beta_1) q^{17}+ \cdots + ( - 3 \beta_{7} + 5 \beta_{3} + \cdots - 110) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 64 q^{9} - 24 q^{11} + 40 q^{15} - 136 q^{23} - 80 q^{25} + 64 q^{29} - 64 q^{37} - 376 q^{39} + 136 q^{43} - 408 q^{51} - 104 q^{53} - 240 q^{57} + 224 q^{65} - 16 q^{67} + 352 q^{71} - 112 q^{79} + 584 q^{81} + 200 q^{85} + 784 q^{93} + 120 q^{95} - 880 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 30x^{6} + 84x^{5} + 597x^{4} - 384x^{3} - 4236x^{2} + 688x + 13378 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3054 \nu^{7} - 601373 \nu^{6} + 3030837 \nu^{5} + 13338118 \nu^{4} - 48570170 \nu^{3} + \cdots + 984126746 ) / 127090173 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 338 \nu^{7} + 1586 \nu^{6} + 9042 \nu^{5} - 49669 \nu^{4} - 92314 \nu^{3} + 418959 \nu^{2} + \cdots - 3364486 ) / 1366561 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 172306 \nu^{7} + 733427 \nu^{6} + 5457332 \nu^{5} - 18552200 \nu^{4} - 105520554 \nu^{3} + \cdots - 420888611 ) / 127090173 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 179646 \nu^{7} - 1008143 \nu^{6} - 2754222 \nu^{5} + 14939683 \nu^{4} + 53220904 \nu^{3} + \cdots + 126358352 ) / 127090173 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 181978 \nu^{7} - 778811 \nu^{6} - 4319477 \nu^{5} + 12990521 \nu^{4} + 87213225 \nu^{3} + \cdots + 186171584 ) / 127090173 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 201365 \nu^{7} + 989918 \nu^{6} + 3388375 \nu^{5} - 13493871 \nu^{4} - 65356001 \nu^{3} + \cdots + 249460552 ) / 127090173 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 477774 \nu^{7} - 2572240 \nu^{6} - 14264394 \nu^{5} + 75222167 \nu^{4} + 222597134 \nu^{3} + \cdots + 1866439591 ) / 127090173 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 4\beta_{5} - 2\beta_{4} + 4\beta_{3} + 5\beta_{2} + 7 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{7} - 28\beta_{6} - 8\beta_{5} - 24\beta_{4} + 6\beta_{3} + 39\beta_{2} + 133 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15\beta_{7} - 84\beta_{6} + 110\beta_{5} - 174\beta_{4} + 32\beta_{3} + 229\beta_{2} + 42\beta _1 + 301 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 67\beta_{7} - 840\beta_{6} + 152\beta_{5} - 1168\beta_{4} + 86\beta_{3} + 111\beta_{2} + 224\beta _1 + 427 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 273 \beta_{7} - 3920 \beta_{6} + 1892 \beta_{5} - 6014 \beta_{4} - 378 \beta_{3} - 553 \beta_{2} + \cdots - 1253 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 401 \beta_{7} - 2800 \beta_{6} + 1152 \beta_{5} - 4580 \beta_{4} - 568 \beta_{3} - 4651 \beta_{2} + \cdots - 7103 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 27285 \beta_{7} - 85162 \beta_{6} + 29894 \beta_{5} - 123776 \beta_{4} - 38972 \beta_{3} + \cdots - 379743 ) / 14 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−2.61548 + 0.765367i
−2.75518 − 1.84776i
2.20126 − 0.765367i
5.16940 + 1.84776i
5.16940 − 1.84776i
2.20126 + 0.765367i
−2.75518 + 1.84776i
−2.61548 − 0.765367i
0 − 5.91517i 0 − 5.38029i 0 0 0 −25.9892 0
97.2 0 − 4.72185i 0 8.02107i 0 0 0 −13.2959 0
97.3 0 − 2.98501i 0 1.69372i 0 0 0 0.0896893 0
97.4 0 − 1.34336i 0 6.62165i 0 0 0 7.19538 0
97.5 0 1.34336i 0 − 6.62165i 0 0 0 7.19538 0
97.6 0 2.98501i 0 − 1.69372i 0 0 0 0.0896893 0
97.7 0 4.72185i 0 − 8.02107i 0 0 0 −13.2959 0
97.8 0 5.91517i 0 5.38029i 0 0 0 −25.9892 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.c.b ✓ 8
3.b odd 2 1 3528.3.f.e 8
4.b odd 2 1 784.3.c.g 8
7.b odd 2 1 inner 392.3.c.b ✓ 8
7.c even 3 2 392.3.o.d 16
7.d odd 6 2 392.3.o.d 16
21.c even 2 1 3528.3.f.e 8
28.d even 2 1 784.3.c.g 8
28.f even 6 2 784.3.s.k 16
28.g odd 6 2 784.3.s.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.3.c.b ✓ 8 1.a even 1 1 trivial
392.3.c.b ✓ 8 7.b odd 2 1 inner
392.3.o.d 16 7.c even 3 2
392.3.o.d 16 7.d odd 6 2
784.3.c.g 8 4.b odd 2 1
784.3.c.g 8 28.d even 2 1
784.3.s.k 16 28.f even 6 2
784.3.s.k 16 28.g odd 6 2
3528.3.f.e 8 3.b odd 2 1
3528.3.f.e 8 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 68T_{3}^{6} + 1410T_{3}^{4} + 9280T_{3}^{2} + 12544 \) acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 68 T^{6} + \cdots + 12544 \) Copy content Toggle raw display
$5$ \( T^{8} + 140 T^{6} + \cdots + 234256 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 12 T^{3} + \cdots + 15584)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 620 T^{6} + \cdots + 49056016 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 2868030916 \) Copy content Toggle raw display
$19$ \( T^{8} + 980 T^{6} + \cdots + 944701696 \) Copy content Toggle raw display
$23$ \( (T^{4} + 68 T^{3} + \cdots - 680512)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 32 T^{3} + \cdots - 14704)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 107206475776 \) Copy content Toggle raw display
$37$ \( (T^{4} + 32 T^{3} + \cdots + 1088)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 307783067524 \) Copy content Toggle raw display
$43$ \( (T^{4} - 68 T^{3} + \cdots - 103904)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 6406325469184 \) Copy content Toggle raw display
$53$ \( (T^{4} + 52 T^{3} + \cdots + 2742016)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 223071068416 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 4830869909776 \) Copy content Toggle raw display
$67$ \( (T^{4} + 8 T^{3} + \cdots + 1891328)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 176 T^{3} + \cdots - 3110912)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 931437072458116 \) Copy content Toggle raw display
$79$ \( (T^{4} + 56 T^{3} + \cdots - 30349568)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 6466157322496 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 510878521274884 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 62\!\cdots\!96 \) Copy content Toggle raw display
show more
show less