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Properties

Label 336.7.bh.b
Level $336$
Weight $7$
Character orbit 336.bh
Analytic conductor $77.298$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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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: \( x^{8} - x^{7} + 212x^{6} - 787x^{5} + 38792x^{4} - 92833x^{3} + 1563109x^{2} + 3107772x + 38787984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + (9 \beta_{2} - 9) q^{3} + ( - \beta_{6} + \beta_{3} - 3 \beta_{2} - 7) q^{5} + ( - 2 \beta_{6} - \beta_{5} + \cdots - 75) q^{7}+ \cdots - 243 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (9 \beta_{2} - 9) q^{3} + ( - \beta_{6} + \beta_{3} - 3 \beta_{2} - 7) q^{5} + ( - 2 \beta_{6} - \beta_{5} + \cdots - 75) q^{7}+ \cdots + ( - 243 \beta_{7} + 243 \beta_{6} + \cdots + 65853) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 108 q^{3} - 42 q^{5} - 748 q^{7} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 108 q^{3} - 42 q^{5} - 748 q^{7} + 972 q^{9} + 1070 q^{11} + 756 q^{15} + 7212 q^{17} + 24606 q^{19} + 8154 q^{21} + 15224 q^{23} + 22274 q^{25} + 32524 q^{29} - 40200 q^{31} - 28890 q^{33} + 242436 q^{35} - 45670 q^{37} - 93366 q^{39} + 445660 q^{43} - 10206 q^{45} - 82884 q^{47} + 24116 q^{49} - 64908 q^{51} - 13034 q^{53} - 442908 q^{57} - 1810362 q^{59} - 392856 q^{61} - 38394 q^{63} - 389004 q^{65} - 384094 q^{67} - 225688 q^{71} + 903078 q^{73} - 601398 q^{75} - 327674 q^{77} + 559592 q^{79} - 236196 q^{81} + 1953576 q^{85} - 439074 q^{87} - 1770036 q^{89} + 2960718 q^{91} + 361800 q^{93} - 1160112 q^{95} + 520020 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 212x^{6} - 787x^{5} + 38792x^{4} - 92833x^{3} + 1563109x^{2} + 3107772x + 38787984 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1585013359 \nu^{7} - 18232571539 \nu^{6} + 303603349712 \nu^{5} - 4245938445433 \nu^{4} + \cdots - 22\!\cdots\!68 ) / 23\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1585013359 \nu^{7} - 18232571539 \nu^{6} + 303603349712 \nu^{5} - 4245938445433 \nu^{4} + \cdots - 22\!\cdots\!68 ) / 23\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13858370275 \nu^{7} - 1233059081317 \nu^{6} - 15146056834333 \nu^{5} + \cdots + 31\!\cdots\!42 ) / 11\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 32076220 \nu^{7} + 62465284 \nu^{6} + 5777520100 \nu^{5} - 8215312820 \nu^{4} + \cdots + 129923180639535 ) / 11465622241311 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 85151344841 \nu^{7} - 799568126245 \nu^{6} - 32979536564113 \nu^{5} + \cdots - 17\!\cdots\!58 ) / 11\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41883584669 \nu^{7} + 456963608136 \nu^{6} + 4283641236335 \nu^{5} + 19286324891880 \nu^{4} + \cdots - 44\!\cdots\!44 ) / 39\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 661077962011 \nu^{7} - 1518878968019 \nu^{6} - 138652672166732 \nu^{5} + \cdots + 17\!\cdots\!28 ) / 23\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - 2\beta_{6} - 3\beta_{4} + 2\beta_{3} - 423\beta_{2} - 3\beta _1 - 423 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{7} - 8\beta_{6} + 16\beta_{5} + 157\beta_{4} - 16\beta_{3} + 8\beta_{2} + 883 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 414\beta_{7} + 406\beta_{6} - 414\beta_{5} - 8\beta_{3} + 62983\beta_{2} + 975\beta _1 - 406 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3968 \beta_{7} - 2272 \beta_{6} - 1696 \beta_{5} - 27421 \beta_{4} + 2272 \beta_{3} - 337267 \beta_{2} + \cdots - 337267 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 608\beta_{7} - 608\beta_{6} + 77226\beta_{5} + 237963\beta_{4} - 77226\beta_{3} + 608\beta_{2} + 10825873 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 559536 \beta_{7} + 862968 \beta_{6} - 559536 \beta_{5} + 303432 \beta_{3} + 86136107 \beta_{2} + \cdots - 862968 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{2}\)

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
5.73828 − 9.93899i
−7.08935 + 12.2791i
−2.30325 + 3.98935i
4.15432 − 7.19549i
5.73828 + 9.93899i
−7.08935 − 12.2791i
−2.30325 − 3.98935i
4.15432 + 7.19549i
0 −13.5000 + 7.79423i 0 −175.367 − 101.248i 0 −284.280 + 191.921i 0 121.500 − 210.444i 0
145.2 0 −13.5000 + 7.79423i 0 −68.9069 − 39.7834i 0 −284.244 − 191.975i 0 121.500 − 210.444i 0
145.3 0 −13.5000 + 7.79423i 0 71.9311 + 41.5295i 0 −77.0894 + 334.225i 0 121.500 − 210.444i 0
145.4 0 −13.5000 + 7.79423i 0 151.343 + 87.3778i 0 271.614 − 209.463i 0 121.500 − 210.444i 0
241.1 0 −13.5000 − 7.79423i 0 −175.367 + 101.248i 0 −284.280 − 191.921i 0 121.500 + 210.444i 0
241.2 0 −13.5000 − 7.79423i 0 −68.9069 + 39.7834i 0 −284.244 + 191.975i 0 121.500 + 210.444i 0
241.3 0 −13.5000 − 7.79423i 0 71.9311 − 41.5295i 0 −77.0894 − 334.225i 0 121.500 + 210.444i 0
241.4 0 −13.5000 − 7.79423i 0 151.343 − 87.3778i 0 271.614 + 209.463i 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.b 8
4.b odd 2 1 21.7.f.b ✓ 8
7.d odd 6 1 inner 336.7.bh.b 8
12.b even 2 1 63.7.m.c 8
28.d even 2 1 147.7.f.a 8
28.f even 6 1 21.7.f.b ✓ 8
28.f even 6 1 147.7.d.a 8
28.g odd 6 1 147.7.d.a 8
28.g odd 6 1 147.7.f.a 8
84.j odd 6 1 63.7.m.c 8
84.j odd 6 1 441.7.d.d 8
84.n even 6 1 441.7.d.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.b ✓ 8 4.b odd 2 1
21.7.f.b ✓ 8 28.f even 6 1
63.7.m.c 8 12.b even 2 1
63.7.m.c 8 84.j odd 6 1
147.7.d.a 8 28.f even 6 1
147.7.d.a 8 28.g odd 6 1
147.7.f.a 8 28.d even 2 1
147.7.f.a 8 28.g odd 6 1
336.7.bh.b 8 1.a even 1 1 trivial
336.7.bh.b 8 7.d odd 6 1 inner
441.7.d.d 8 84.j odd 6 1
441.7.d.d 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 42 T_{5}^{7} - 41505 T_{5}^{6} - 1767906 T_{5}^{5} + 1536505749 T_{5}^{4} + \cdots + 54\!\cdots\!00 \) 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
$11$ \( T^{8} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 42\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 53\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 14\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 25\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 58\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
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