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Properties

Label 336.3.bh.f
Level $336$
Weight $3$
Character orbit 336.bh
Analytic conductor $9.155$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{65})\)
Defining polynomial: \( x^{4} - x^{3} + 17x^{2} + 16x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{3} + ( - \beta_{3} - 2 \beta_1 - 1) q^{5} - \beta_{2} q^{7} + ( - 3 \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{3} + ( - \beta_{3} - 2 \beta_1 - 1) q^{5} - \beta_{2} q^{7} + ( - 3 \beta_1 + 3) q^{9} + (\beta_{3} - \beta_{2} - 7 \beta_1) q^{11} + ( - \beta_{2} + \beta_1) q^{13} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{15} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{17} + ( - \beta_{3} + 13 \beta_1 + 14) q^{19} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{21} + (24 \beta_1 - 24) q^{23} + (3 \beta_{3} - 3 \beta_{2} + 32 \beta_1) q^{25} + ( - 6 \beta_1 + 3) q^{27} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{29} + ( - 6 \beta_{3} - 6 \beta_{2} + 5 \beta_1 - 10) q^{31} + (3 \beta_{3} - 6 \beta_1 - 9) q^{33} + (2 \beta_{3} + 3 \beta_{2} - 47 \beta_1 + 47) q^{35} + ( - \beta_{3} - 2 \beta_{2} - 37 \beta_1 + 38) q^{37} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{39} + (4 \beta_{2} - 24 \beta_1 + 10) q^{41} + (6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 22) q^{43} + ( - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 6) q^{45} + ( - 2 \beta_{3} + 14 \beta_1 + 16) q^{47} + (\beta_{2} - 49) q^{49} + ( - 2 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 6) q^{51} + ( - \beta_{3} + \beta_{2} + 49 \beta_1) q^{53} + ( - 3 \beta_{2} - 75 \beta_1 + 39) q^{55} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 42) q^{57} + (\beta_{3} + \beta_{2} + 41 \beta_1 - 82) q^{59} + ( - 8 \beta_{3} - 36 \beta_1 - 28) q^{61} + (3 \beta_{3} + 3 \beta_1 - 3) q^{63} + (2 \beta_{3} + 4 \beta_{2} - 50 \beta_1 + 48) q^{65} + (5 \beta_{3} - 5 \beta_{2} + 6 \beta_1) q^{67} + (48 \beta_1 - 24) q^{69} + ( - 12 \beta_{3} - 6 \beta_{2} - 6 \beta_1 + 60) q^{71} + ( - 7 \beta_{3} - 7 \beta_{2} + 4 \beta_1 - 8) q^{73} + (9 \beta_{3} + 35 \beta_1 + 26) q^{75} + (7 \beta_{3} + 8 \beta_{2} + 56 \beta_1 - 105) q^{77} + (2 \beta_{3} + 4 \beta_{2} + 29 \beta_1 - 31) q^{79} - 9 \beta_1 q^{81} + (17 \beta_{2} + 21 \beta_1 - 19) q^{83} + (12 \beta_{3} + 6 \beta_{2} + 6 \beta_1 + 102) q^{85} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 6) q^{87} + (6 \beta_{3} + 36 \beta_1 + 30) q^{89} + ( - \beta_{3} - \beta_1 - 48) q^{91} + ( - 6 \beta_{3} - 12 \beta_{2} + 21 \beta_1 - 15) q^{93} + ( - 12 \beta_{3} + 12 \beta_{2} - 18 \beta_1) q^{95} + ( - 7 \beta_{2} + 17 \beta_1 - 5) q^{97} + (6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 9 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} - 9 q^{5} - 2 q^{7} + 6 q^{9} - 15 q^{11} - 18 q^{15} - 18 q^{17} + 81 q^{19} - 3 q^{21} - 48 q^{23} + 61 q^{25} + 6 q^{29} - 48 q^{31} - 45 q^{33} + 102 q^{35} + 73 q^{37} + 3 q^{39} - 70 q^{43} - 27 q^{45} + 90 q^{47} - 194 q^{49} - 18 q^{51} + 99 q^{53} + 162 q^{57} - 243 q^{59} - 192 q^{61} - 3 q^{63} + 102 q^{65} + 7 q^{67} + 204 q^{71} - 45 q^{73} + 183 q^{75} - 285 q^{77} - 56 q^{79} - 18 q^{81} + 444 q^{85} + 9 q^{87} + 198 q^{89} - 195 q^{91} - 48 q^{93} - 24 q^{95} - 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 17x^{2} + 16x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 17\nu^{2} - 17\nu + 256 ) / 272 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} + 33\nu + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 17\nu + 33 ) / 17 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 50\beta _1 - 51 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 34\beta_{3} + 17\beta_{2} + 17\beta _1 - 99 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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
−1.76556 − 3.05805i
2.26556 + 3.92407i
−1.76556 + 3.05805i
2.26556 − 3.92407i
0 1.50000 − 0.866025i 0 −8.29669 − 4.79010i 0 −0.500000 + 6.98212i 0 1.50000 − 2.59808i 0
145.2 0 1.50000 − 0.866025i 0 3.79669 + 2.19202i 0 −0.500000 − 6.98212i 0 1.50000 − 2.59808i 0
241.1 0 1.50000 + 0.866025i 0 −8.29669 + 4.79010i 0 −0.500000 − 6.98212i 0 1.50000 + 2.59808i 0
241.2 0 1.50000 + 0.866025i 0 3.79669 − 2.19202i 0 −0.500000 + 6.98212i 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
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.f 4
3.b odd 2 1 1008.3.cg.m 4
4.b odd 2 1 84.3.m.b ✓ 4
7.c even 3 1 2352.3.f.f 4
7.d odd 6 1 inner 336.3.bh.f 4
7.d odd 6 1 2352.3.f.f 4
12.b even 2 1 252.3.z.e 4
20.d odd 2 1 2100.3.bd.f 4
20.e even 4 2 2100.3.be.d 8
21.g even 6 1 1008.3.cg.m 4
28.d even 2 1 588.3.m.d 4
28.f even 6 1 84.3.m.b ✓ 4
28.f even 6 1 588.3.d.b 4
28.g odd 6 1 588.3.d.b 4
28.g odd 6 1 588.3.m.d 4
84.h odd 2 1 1764.3.z.h 4
84.j odd 6 1 252.3.z.e 4
84.j odd 6 1 1764.3.d.f 4
84.n even 6 1 1764.3.d.f 4
84.n even 6 1 1764.3.z.h 4
140.s even 6 1 2100.3.bd.f 4
140.x odd 12 2 2100.3.be.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.m.b ✓ 4 4.b odd 2 1
84.3.m.b ✓ 4 28.f even 6 1
252.3.z.e 4 12.b even 2 1
252.3.z.e 4 84.j odd 6 1
336.3.bh.f 4 1.a even 1 1 trivial
336.3.bh.f 4 7.d odd 6 1 inner
588.3.d.b 4 28.f even 6 1
588.3.d.b 4 28.g odd 6 1
588.3.m.d 4 28.d even 2 1
588.3.m.d 4 28.g odd 6 1
1008.3.cg.m 4 3.b odd 2 1
1008.3.cg.m 4 21.g even 6 1
1764.3.d.f 4 84.j odd 6 1
1764.3.d.f 4 84.n even 6 1
1764.3.z.h 4 84.h odd 2 1
1764.3.z.h 4 84.n even 6 1
2100.3.bd.f 4 20.d odd 2 1
2100.3.bd.f 4 140.s even 6 1
2100.3.be.d 8 20.e even 4 2
2100.3.be.d 8 140.x odd 12 2
2352.3.f.f 4 7.c even 3 1
2352.3.f.f 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 9T_{5}^{3} - 15T_{5}^{2} - 378T_{5} + 1764 \) acting on \(S_{3}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 9 T^{3} - 15 T^{2} + \cdots + 1764 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 15 T^{3} + 315 T^{2} + \cdots + 8100 \) Copy content Toggle raw display
$13$ \( T^{4} + 99T^{2} + 2304 \) Copy content Toggle raw display
$17$ \( T^{4} + 18 T^{3} - 60 T^{2} + \cdots + 28224 \) Copy content Toggle raw display
$19$ \( T^{4} - 81 T^{3} + 2685 T^{2} + \cdots + 248004 \) Copy content Toggle raw display
$23$ \( (T^{2} + 24 T + 576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T - 144)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 48 T^{3} - 795 T^{2} + \cdots + 2442969 \) Copy content Toggle raw display
$37$ \( T^{4} - 73 T^{3} + 4143 T^{2} + \cdots + 1406596 \) Copy content Toggle raw display
$41$ \( T^{4} + 2424 T^{2} + 121104 \) Copy content Toggle raw display
$43$ \( (T^{2} + 35 T - 1010)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 90 T^{3} + 3180 T^{2} + \cdots + 230400 \) Copy content Toggle raw display
$53$ \( T^{4} - 99 T^{3} + 7497 T^{2} + \cdots + 5308416 \) Copy content Toggle raw display
$59$ \( T^{4} + 243 T^{3} + \cdots + 23736384 \) Copy content Toggle raw display
$61$ \( T^{4} + 192 T^{3} + 12240 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$67$ \( T^{4} - 7 T^{3} + 3693 T^{2} + \cdots + 13278736 \) Copy content Toggle raw display
$71$ \( (T^{2} - 102 T - 2664)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 45 T^{3} - 1545 T^{2} + \cdots + 4928400 \) Copy content Toggle raw display
$79$ \( T^{4} + 56 T^{3} + 2937 T^{2} + \cdots + 39601 \) Copy content Toggle raw display
$83$ \( T^{4} + 28839 T^{2} + \cdots + 189282564 \) Copy content Toggle raw display
$89$ \( T^{4} - 198 T^{3} + 14580 T^{2} + \cdots + 2286144 \) Copy content Toggle raw display
$97$ \( T^{4} + 5211 T^{2} + \cdots + 4717584 \) Copy content Toggle raw display
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