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Properties

Label 336.3.bh.g
Level $336$
Weight $3$
Character orbit 336.bh
Analytic conductor $9.155$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.bh (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.15533688251\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.35911766016.9
Defining polynomial: \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + ( - \beta_{6} + \beta_1 - 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{7} + 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + ( - \beta_{6} + \beta_1 - 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{7} + 3 \beta_1 q^{9} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - 6 \beta_1 + 5) q^{11} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} + 3 \beta_1 - 1) q^{13} + ( - \beta_{7} + \beta_{6} - \beta_1 + 2) q^{15} + (\beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 2) q^{17} + (5 \beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 6) q^{19} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{21} + ( - 2 \beta_{7} - \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + \beta_{2} - 10 \beta_1 - 1) q^{23} + (\beta_{7} + 2 \beta_{6} + 5 \beta_{5} - 11 \beta_1 + 10) q^{25} + ( - 6 \beta_1 + 3) q^{27} + (3 \beta_{5} - \beta_{4} + 5 \beta_{2} + 2 \beta_1 + 9) q^{29} + (4 \beta_{7} + 3 \beta_{5} - 6 \beta_{3} + 7 \beta_1 + 3) q^{31} + ( - 3 \beta_{6} - \beta_{5} - \beta_{3} + 7 \beta_1 - 11) q^{33} + ( - 2 \beta_{7} + \beta_{6} + 15 \beta_{5} - \beta_{4} - 5 \beta_{3} + 4 \beta_{2} + \cdots + 4) q^{35}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{6} + 3 \beta_{3} - 3 \beta_1 + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 6 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} - 6 q^{5} - 8 q^{7} + 12 q^{9} + 22 q^{11} + 12 q^{15} + 36 q^{17} - 42 q^{19} + 6 q^{21} - 48 q^{23} + 42 q^{25} + 68 q^{29} + 60 q^{31} - 66 q^{33} + 12 q^{35} - 118 q^{37} + 18 q^{39} + 92 q^{43} - 18 q^{45} + 12 q^{47} - 20 q^{49} - 36 q^{51} + 10 q^{53} + 84 q^{57} + 54 q^{59} + 24 q^{61} + 6 q^{63} - 148 q^{65} - 22 q^{67} + 392 q^{71} - 138 q^{73} - 126 q^{75} - 126 q^{77} - 164 q^{79} - 36 q^{81} + 200 q^{85} - 102 q^{87} - 60 q^{89} - 90 q^{91} - 60 q^{93} + 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4244 \nu^{7} + 873 \nu^{6} - 33756 \nu^{5} - 71462 \nu^{4} + 213594 \nu^{3} + 469674 \nu^{2} - 193196 \nu - 1034839 ) / 658490 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3371 \nu^{7} + 6667 \nu^{6} - 38294 \nu^{5} - 96948 \nu^{4} + 94716 \nu^{3} + 623641 \nu^{2} + 796111 \nu - 1188686 ) / 329245 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 33\nu^{7} + 41\nu^{6} - 222\nu^{5} - 414\nu^{4} + 488\nu^{3} + 1608\nu^{2} - 207\nu + 3637 ) / 2045 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6088 \nu^{7} - 4954 \nu^{6} - 27942 \nu^{5} - 30829 \nu^{4} + 324398 \nu^{3} - 31292 \nu^{2} + 378248 \nu - 816983 ) / 329245 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6546 \nu^{7} - 1812 \nu^{6} + 70064 \nu^{5} + 173218 \nu^{4} - 443336 \nu^{3} - 974856 \nu^{2} + 215444 \nu + 3514676 ) / 329245 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11533 \nu^{7} + 30909 \nu^{6} + 121832 \nu^{5} - 29696 \nu^{4} - 997968 \nu^{3} - 162453 \nu^{2} + 2146717 \nu + 2603393 ) / 329245 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 45244 \nu^{7} - 5583 \nu^{6} + 215876 \nu^{5} + 916372 \nu^{4} - 1365974 \nu^{3} - 3003654 \nu^{2} - 3864944 \nu + 10829159 ) / 658490 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{4} - 2\beta_{3} + 4\beta_{2} - 4\beta _1 + 6 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{5} - 2\beta_{4} - 5\beta_{3} + 3\beta_{2} + 32\beta _1 + 1 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{7} + 7\beta_{6} - 18\beta_{5} + 27\beta_{4} - 6\beta_{3} - 9\beta_{2} + 9\beta _1 + 81 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{7} + 33\beta_{5} + 10\beta_{4} - 10\beta_{3} + 20\beta_{2} + 141\beta _1 - 131 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 49\beta_{6} + 75\beta_{5} + 66\beta_{4} - 108\beta_{3} - 99\beta_{2} + 736\beta _1 - 33 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 91\beta_{7} + 91\beta_{6} - 150\beta_{5} + 225\beta_{4} + 160\beta_{3} - 75\beta_{2} + 75\beta _1 - 494 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -22\beta_{7} + 199\beta_{5} - 8\beta_{4} + 8\beta_{3} - 16\beta_{2} + 718\beta _1 - 726 ) / 2 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
1.83172 − 0.480194i
−1.90015 + 1.67440i
2.40015 − 0.808379i
−1.33172 + 1.34622i
1.83172 + 0.480194i
−1.90015 − 1.67440i
2.40015 + 0.808379i
−1.33172 − 1.34622i
0 −1.50000 + 0.866025i 0 −6.80550 − 3.92916i 0 −6.99187 − 0.337312i 0 1.50000 − 2.59808i 0
145.2 0 −1.50000 + 0.866025i 0 −4.68140 − 2.70281i 0 6.12873 + 3.38210i 0 1.50000 − 2.59808i 0
145.3 0 −1.50000 + 0.866025i 0 3.18140 + 1.83678i 0 −2.47188 − 6.54903i 0 1.50000 − 2.59808i 0
145.4 0 −1.50000 + 0.866025i 0 5.30550 + 3.06313i 0 −0.664986 + 6.96834i 0 1.50000 − 2.59808i 0
241.1 0 −1.50000 − 0.866025i 0 −6.80550 + 3.92916i 0 −6.99187 + 0.337312i 0 1.50000 + 2.59808i 0
241.2 0 −1.50000 − 0.866025i 0 −4.68140 + 2.70281i 0 6.12873 − 3.38210i 0 1.50000 + 2.59808i 0
241.3 0 −1.50000 − 0.866025i 0 3.18140 − 1.83678i 0 −2.47188 + 6.54903i 0 1.50000 + 2.59808i 0
241.4 0 −1.50000 − 0.866025i 0 5.30550 − 3.06313i 0 −0.664986 − 6.96834i 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.4
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 336.3.bh.g 8
3.b odd 2 1 1008.3.cg.p 8
4.b odd 2 1 168.3.z.b ✓ 8
7.c even 3 1 2352.3.f.g 8
7.d odd 6 1 inner 336.3.bh.g 8
7.d odd 6 1 2352.3.f.g 8
12.b even 2 1 504.3.by.c 8
21.g even 6 1 1008.3.cg.p 8
28.d even 2 1 1176.3.z.c 8
28.f even 6 1 168.3.z.b ✓ 8
28.f even 6 1 1176.3.f.c 8
28.g odd 6 1 1176.3.f.c 8
28.g odd 6 1 1176.3.z.c 8
84.j odd 6 1 504.3.by.c 8
84.j odd 6 1 3528.3.f.b 8
84.n even 6 1 3528.3.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.z.b ✓ 8 4.b odd 2 1
168.3.z.b ✓ 8 28.f even 6 1
336.3.bh.g 8 1.a even 1 1 trivial
336.3.bh.g 8 7.d odd 6 1 inner
504.3.by.c 8 12.b even 2 1
504.3.by.c 8 84.j odd 6 1
1008.3.cg.p 8 3.b odd 2 1
1008.3.cg.p 8 21.g even 6 1
1176.3.f.c 8 28.f even 6 1
1176.3.f.c 8 28.g odd 6 1
1176.3.z.c 8 28.d even 2 1
1176.3.z.c 8 28.g odd 6 1
2352.3.f.g 8 7.c even 3 1
2352.3.f.g 8 7.d odd 6 1
3528.3.f.b 8 84.j odd 6 1
3528.3.f.b 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 6T_{5}^{7} - 53T_{5}^{6} - 390T_{5}^{5} + 2861T_{5}^{4} + 13260T_{5}^{3} - 48268T_{5}^{2} - 195024T_{5} + 913936 \) acting on \(S_{3}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} - 53 T^{6} + \cdots + 913936 \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + 42 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{8} - 22 T^{7} + 527 T^{6} + \cdots + 17875984 \) Copy content Toggle raw display
$13$ \( T^{8} + 262 T^{6} + 19817 T^{4} + \cdots + 2408704 \) Copy content Toggle raw display
$17$ \( T^{8} - 36 T^{7} + \cdots + 3470623744 \) Copy content Toggle raw display
$19$ \( T^{8} + 42 T^{7} + 43 T^{6} + \cdots + 17272336 \) Copy content Toggle raw display
$23$ \( T^{8} + 48 T^{7} + \cdots + 22620160000 \) Copy content Toggle raw display
$29$ \( (T^{4} - 34 T^{3} - 1063 T^{2} + \cdots + 224128)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 60 T^{7} + \cdots + 25912950625 \) Copy content Toggle raw display
$37$ \( T^{8} + 118 T^{7} + \cdots + 17069945104 \) Copy content Toggle raw display
$41$ \( T^{8} + 7280 T^{6} + \cdots + 580010189056 \) Copy content Toggle raw display
$43$ \( (T^{4} - 46 T^{3} - 3723 T^{2} + \cdots + 1658308)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots + 52408029184 \) Copy content Toggle raw display
$53$ \( T^{8} - 10 T^{7} + \cdots + 1491466217536 \) Copy content Toggle raw display
$59$ \( T^{8} - 54 T^{7} + \cdots + 1048985640000 \) Copy content Toggle raw display
$61$ \( T^{8} - 24 T^{7} + \cdots + 43785853599744 \) Copy content Toggle raw display
$67$ \( T^{8} + 22 T^{7} + \cdots + 95387087104 \) Copy content Toggle raw display
$71$ \( (T^{4} - 196 T^{3} + 2460 T^{2} + \cdots - 13209344)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 622497709609216 \) Copy content Toggle raw display
$79$ \( T^{8} + 164 T^{7} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + 4430 T^{6} + \cdots + 839297841424 \) Copy content Toggle raw display
$89$ \( T^{8} + 60 T^{7} + \cdots + 723343446016 \) Copy content Toggle raw display
$97$ \( T^{8} + 65270 T^{6} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
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