↑

Properties

Label 336.7.bh.f
Level $336$
Weight $7$
Character orbit 336.bh
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(145,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.145");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 336.bh (of order \(6\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 193x^{6} + 306x^{5} + 29845x^{4} + 16988x^{3} + 1125468x^{2} + 214128x + 35378704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 9 \beta_1 + 9) q^{3} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots + 77) q^{5}+ \cdots - 243 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 9 \beta_1 + 9) q^{3} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots + 77) q^{5}+ \cdots + ( - 3402 \beta_{7} - 2916 \beta_{6} + \cdots - 110565) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 108 q^{3} + 462 q^{5} - 580 q^{7} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 108 q^{3} + 462 q^{5} - 580 q^{7} + 972 q^{9} - 1806 q^{11} + 8316 q^{15} + 9564 q^{17} - 23022 q^{19} - 1350 q^{21} - 2400 q^{23} + 32762 q^{25} - 2484 q^{29} - 148416 q^{31} - 48762 q^{33} - 11412 q^{35} - 84046 q^{37} + 75222 q^{39} - 92972 q^{43} + 112266 q^{45} + 323124 q^{47} - 8644 q^{49} + 86076 q^{51} + 358086 q^{53} - 414396 q^{57} - 719382 q^{59} + 421536 q^{61} + 104490 q^{63} - 322740 q^{65} - 267010 q^{67} + 464664 q^{71} + 1944486 q^{73} + 884574 q^{75} - 1713498 q^{77} + 685904 q^{79} - 236196 q^{81} - 3876168 q^{85} - 33534 q^{87} + 4130604 q^{89} + 484266 q^{91} - 1335744 q^{93} + 2105232 q^{95} - 877716 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 193x^{6} + 306x^{5} + 29845x^{4} + 16988x^{3} + 1125468x^{2} + 214128x + 35378704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1636073 \nu^{7} - 177960445 \nu^{6} + 557057579 \nu^{5} - 27650057549 \nu^{4} + \cdots - 197665427324196 ) / 164920217026180 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5696588781 \nu^{7} - 629632122285 \nu^{6} + 1970894968827 \nu^{5} + \cdots - 69\!\cdots\!48 ) / 14\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 31448254527 \nu^{7} + 1968223271760 \nu^{6} - 10845607382811 \nu^{5} + \cdots + 35\!\cdots\!84 ) / 28\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 369495651653 \nu^{7} + 2826191904405 \nu^{6} - 143960676182679 \nu^{5} + \cdots - 94\!\cdots\!24 ) / 57\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 474632212627 \nu^{7} - 1876757118200 \nu^{6} + 111593476518171 \nu^{5} + \cdots - 27\!\cdots\!04 ) / 57\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 156913428428 \nu^{7} - 201842164189 \nu^{6} + 21775570970562 \nu^{5} + \cdots + 13\!\cdots\!12 ) / 11\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 580833884751 \nu^{7} - 646807832245 \nu^{6} + 69581678069333 \nu^{5} + \cdots + 12\!\cdots\!38 ) / 28\!\cdots\!30 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4\beta_{7} - 10\beta_{6} + 10\beta_{5} + 4\beta_{4} + \beta_{2} - 82\beta _1 + 4 ) / 168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -8\beta_{7} + 6\beta_{6} + 12\beta_{5} + 16\beta_{4} - 143\beta_{3} - 16\beta_{2} - 8026\beta _1 - 8018 ) / 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -248\beta_{7} + 300\beta_{6} + 150\beta_{5} + 124\beta_{4} - 287\beta_{3} - 411\beta_{2} - 124\beta _1 - 6266 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -1154\beta_{7} + 1002\beta_{6} - 1002\beta_{5} - 1154\beta_{4} - 13319\beta_{2} + 528644\beta _1 - 1154 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 148916 \beta_{7} - 146554 \beta_{6} - 293108 \beta_{5} - 297832 \beta_{4} + 455449 \beta_{3} + \cdots + 11289986 ) / 168 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 149120 \beta_{7} - 134492 \beta_{6} - 67246 \beta_{5} - 74560 \beta_{4} + 719931 \beta_{3} + \cdots + 23310626 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 24903764 \beta_{7} - 23191906 \beta_{6} + 23191906 \beta_{5} + 24903764 \beta_{4} + 67300521 \beta_{2} + \cdots + 24903764 ) / 168 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−5.50376 + 9.53280i
−3.27778 + 5.67728i
3.07067 − 5.31856i
6.71087 − 11.6236i
−5.50376 − 9.53280i
−3.27778 − 5.67728i
3.07067 + 5.31856i
6.71087 + 11.6236i
0 13.5000 − 7.79423i 0 −124.321 − 71.7768i 0 −342.924 + 7.21089i 0 121.500 − 210.444i 0
145.2 0 13.5000 − 7.79423i 0 63.2658 + 36.5265i 0 271.352 − 209.802i 0 121.500 − 210.444i 0
145.3 0 13.5000 − 7.79423i 0 77.6900 + 44.8543i 0 −204.220 + 275.578i 0 121.500 − 210.444i 0
145.4 0 13.5000 − 7.79423i 0 214.365 + 123.764i 0 −14.2078 + 342.706i 0 121.500 − 210.444i 0
241.1 0 13.5000 + 7.79423i 0 −124.321 + 71.7768i 0 −342.924 − 7.21089i 0 121.500 + 210.444i 0
241.2 0 13.5000 + 7.79423i 0 63.2658 − 36.5265i 0 271.352 + 209.802i 0 121.500 + 210.444i 0
241.3 0 13.5000 + 7.79423i 0 77.6900 − 44.8543i 0 −204.220 − 275.578i 0 121.500 + 210.444i 0
241.4 0 13.5000 + 7.79423i 0 214.365 − 123.764i 0 −14.2078 − 342.706i 0 121.500 + 210.444i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.7.bh.f 8
4.b odd 2 1 42.7.g.a ✓ 8
7.d odd 6 1 inner 336.7.bh.f 8
12.b even 2 1 126.7.n.a 8
28.d even 2 1 294.7.g.d 8
28.f even 6 1 42.7.g.a ✓ 8
28.f even 6 1 294.7.c.b 8
28.g odd 6 1 294.7.c.b 8
28.g odd 6 1 294.7.g.d 8
84.j odd 6 1 126.7.n.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.7.g.a ✓ 8 4.b odd 2 1
42.7.g.a ✓ 8 28.f even 6 1
126.7.n.a 8 12.b even 2 1
126.7.n.a 8 84.j odd 6 1
294.7.c.b 8 28.f even 6 1
294.7.c.b 8 28.g odd 6 1
294.7.g.d 8 28.d even 2 1
294.7.g.d 8 28.g odd 6 1
336.7.bh.f 8 1.a even 1 1 trivial
336.7.bh.f 8 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 462 T_{5}^{7} + 59091 T_{5}^{6} + 5570334 T_{5}^{5} - 982673451 T_{5}^{4} + \cdots + 54\!\cdots\!00 \) 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
$3$ \( (T^{2} - 27 T + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 23\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 37\!\cdots\!49 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 14\!\cdots\!12)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 68\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 16\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 59\!\cdots\!49 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
show more
show less