↑

Properties

Label 336.7.f.b
Level $336$
Weight $7$
Character orbit 336.f
Analytic conductor $77.298$
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] = [336,7,Mod(97,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.97");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 336.f (of order \(2\), degree \(1\), not 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: \(77.2981720963\)
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: \( x^{8} + 774x^{6} - 224x^{5} + 454036x^{4} - 86688x^{3} + 112273504x^{2} + 16244480x + 21036601600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{10}\cdot 7 \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + 2 \beta_1) q^{5} + ( - \beta_{3} + 2 \beta_1 + 2) q^{7} - 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} + 2 \beta_1) q^{5} + ( - \beta_{3} + 2 \beta_1 + 2) q^{7} - 243 q^{9} + (\beta_{7} + \beta_{5} - \beta_{4} + \cdots - 125) q^{11}+ \cdots + ( - 243 \beta_{7} - 243 \beta_{5} + \cdots + 30375) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 20 q^{7} - 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 20 q^{7} - 1944 q^{9} - 1008 q^{11} - 4536 q^{15} - 2916 q^{21} - 17856 q^{23} - 46336 q^{25} - 75744 q^{29} - 10440 q^{35} + 228824 q^{37} + 27216 q^{39} + 203224 q^{43} - 198736 q^{49} - 187272 q^{51} - 180720 q^{53} - 316872 q^{57} - 4860 q^{63} + 1878192 q^{65} - 142168 q^{67} + 1196496 q^{71} - 1570176 q^{77} - 2442952 q^{79} + 472392 q^{81} - 1510248 q^{85} + 1973184 q^{91} + 672624 q^{93} - 3446928 q^{95} + 244944 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 774x^{6} - 224x^{5} + 454036x^{4} - 86688x^{3} + 112273504x^{2} + 16244480x + 21036601600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 521645427 \nu^{7} + 511980641865 \nu^{6} - 306002329578 \nu^{5} + 300449720293758 \nu^{4} + \cdots + 35\!\cdots\!20 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 485820791857 \nu^{7} + 1272790708560 \nu^{6} + \cdots + 84\!\cdots\!20 ) / 96\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4580274299681 \nu^{7} - 133382576801180 \nu^{6} + \cdots - 11\!\cdots\!40 ) / 96\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9887973343573 \nu^{7} - 259614421457580 \nu^{6} + \cdots - 66\!\cdots\!60 ) / 96\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2714443235291 \nu^{7} - 25763524686592 \nu^{6} + \cdots + 14\!\cdots\!84 ) / 19\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13965980208487 \nu^{7} + 395527745606280 \nu^{6} + \cdots - 32\!\cdots\!00 ) / 96\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22960869058433 \nu^{7} + 68614696233720 \nu^{6} + \cdots + 18\!\cdots\!40 ) / 96\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} - 7\beta_{5} + 8\beta_{4} - 2\beta_{3} - 51\beta_{2} + 22\beta _1 - 1 ) / 504 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -12\beta_{7} - 26\beta_{6} + 15\beta_{4} + 12\beta_{3} - 23\beta_{2} + 8121\beta _1 - 73137 ) / 378 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2964 \beta_{7} + 2180 \beta_{6} + 8022 \beta_{5} - 3705 \beta_{4} + 7410 \beta_{3} + 1439 \beta_{2} + \cdots + 67209 ) / 756 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2350 \beta_{7} + 544 \beta_{6} - 98 \beta_{5} + 1273 \beta_{4} - 7768 \beta_{3} + 3285 \beta_{2} + \cdots - 4870571 ) / 63 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1069266 \beta_{7} - 919130 \beta_{6} - 790797 \beta_{5} - 532281 \beta_{4} - 1422552 \beta_{3} + \cdots - 21191316 ) / 378 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1149084 \beta_{7} + 5185468 \beta_{6} + 679728 \beta_{5} - 5714712 \beta_{4} + 11429424 \beta_{3} + \cdots + 11999726532 ) / 189 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 85594410 \beta_{7} + 128876306 \beta_{6} - 160632507 \beta_{5} + 336916296 \beta_{4} + \cdots - 8454272445 ) / 189 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
−8.73547 + 15.1303i
−10.8585 + 18.8074i
10.4390 − 18.0809i
9.15488 − 15.8567i
9.15488 + 15.8567i
10.4390 + 18.0809i
−10.8585 − 18.8074i
−8.73547 − 15.1303i
0 − 15.5885i 0 − 245.575i 0 277.848 − 201.121i 0 −243.000 0
97.2 0 − 15.5885i 0 − 71.6282i 0 −323.976 + 112.644i 0 −243.000 0
97.3 0 − 15.5885i 0 33.4741i 0 58.7167 + 337.937i 0 −243.000 0
97.4 0 − 15.5885i 0 138.237i 0 −2.58841 − 342.990i 0 −243.000 0
97.5 0 15.5885i 0 − 138.237i 0 −2.58841 + 342.990i 0 −243.000 0
97.6 0 15.5885i 0 − 33.4741i 0 58.7167 − 337.937i 0 −243.000 0
97.7 0 15.5885i 0 71.6282i 0 −323.976 − 112.644i 0 −243.000 0
97.8 0 15.5885i 0 245.575i 0 277.848 + 201.121i 0 −243.000 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 336.7.f.b 8
4.b odd 2 1 84.7.d.a ✓ 8
7.b odd 2 1 inner 336.7.f.b 8
12.b even 2 1 252.7.d.c 8
28.d even 2 1 84.7.d.a ✓ 8
28.f even 6 1 588.7.m.a 8
28.f even 6 1 588.7.m.d 8
28.g odd 6 1 588.7.m.a 8
28.g odd 6 1 588.7.m.d 8
84.h odd 2 1 252.7.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.7.d.a ✓ 8 4.b odd 2 1
84.7.d.a ✓ 8 28.d even 2 1
252.7.d.c 8 12.b even 2 1
252.7.d.c 8 84.h odd 2 1
336.7.f.b 8 1.a even 1 1 trivial
336.7.f.b 8 7.b odd 2 1 inner
588.7.m.a 8 28.f even 6 1
588.7.m.a 8 28.g odd 6 1
588.7.m.d 8 28.f even 6 1
588.7.m.d 8 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 85668T_{5}^{6} + 1654640100T_{5}^{4} + 7660640880000T_{5}^{2} + 6625308816000000 \) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + 504 T^{3} + \cdots - 56664400824)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 66\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 60\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 78\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots - 45\!\cdots\!32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 15\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 44\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 14\!\cdots\!40)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 15\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 82\!\cdots\!60)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 79\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 16\!\cdots\!68)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
show more
show less