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Properties

Label 336.7.bh.g
Level $336$
Weight $7$
Character orbit 336.bh
Analytic conductor $77.298$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(77.2981720963\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 324 q^{3} + 126 q^{5} + 552 q^{7} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 324 q^{3} + 126 q^{5} + 552 q^{7} + 2916 q^{9} + 434 q^{11} - 2268 q^{15} + 3180 q^{17} - 2862 q^{19} - 7614 q^{21} + 3696 q^{23} + 61854 q^{25} + 33964 q^{29} + 62964 q^{31} - 11718 q^{33} - 132924 q^{35} + 69486 q^{37} + 51030 q^{39} - 78732 q^{43} + 30618 q^{45} - 167004 q^{47} + 33348 q^{49} - 28620 q^{51} + 102638 q^{53} + 51516 q^{57} - 12846 q^{59} - 403704 q^{61} + 71442 q^{63} - 19388 q^{65} + 61998 q^{67} - 1551080 q^{71} + 1581570 q^{73} - 1670058 q^{75} + 65718 q^{77} - 1408044 q^{79} - 708588 q^{81} + 1079448 q^{85} - 458514 q^{87} - 540660 q^{89} + 1630242 q^{91} - 566676 q^{93} + 1057536 q^{95} + 210924 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1 0 −13.5000 + 7.79423i 0 −195.955 − 113.135i 0 341.837 + 28.2190i 0 121.500 − 210.444i 0
145.2 0 −13.5000 + 7.79423i 0 −190.240 − 109.835i 0 −15.6818 − 342.641i 0 121.500 − 210.444i 0
145.3 0 −13.5000 + 7.79423i 0 −114.861 − 66.3152i 0 −74.7758 + 334.750i 0 121.500 − 210.444i 0
145.4 0 −13.5000 + 7.79423i 0 −58.1747 − 33.5872i 0 −336.398 − 66.9742i 0 121.500 − 210.444i 0
145.5 0 −13.5000 + 7.79423i 0 −23.8745 − 13.7839i 0 −162.412 + 302.112i 0 121.500 − 210.444i 0
145.6 0 −13.5000 + 7.79423i 0 −17.6311 − 10.1793i 0 342.439 − 19.6179i 0 121.500 − 210.444i 0
145.7 0 −13.5000 + 7.79423i 0 −6.08028 − 3.51045i 0 75.1444 − 334.667i 0 121.500 − 210.444i 0
145.8 0 −13.5000 + 7.79423i 0 102.544 + 59.2040i 0 191.470 + 284.585i 0 121.500 − 210.444i 0
145.9 0 −13.5000 + 7.79423i 0 103.966 + 60.0248i 0 139.570 − 313.320i 0 121.500 − 210.444i 0
145.10 0 −13.5000 + 7.79423i 0 122.266 + 70.5906i 0 −306.427 − 154.115i 0 121.500 − 210.444i 0
145.11 0 −13.5000 + 7.79423i 0 154.657 + 89.2912i 0 337.056 + 63.5791i 0 121.500 − 210.444i 0
145.12 0 −13.5000 + 7.79423i 0 186.383 + 107.608i 0 −255.821 + 228.483i 0 121.500 − 210.444i 0
241.1 0 −13.5000 − 7.79423i 0 −195.955 + 113.135i 0 341.837 − 28.2190i 0 121.500 + 210.444i 0
241.2 0 −13.5000 − 7.79423i 0 −190.240 + 109.835i 0 −15.6818 + 342.641i 0 121.500 + 210.444i 0
241.3 0 −13.5000 − 7.79423i 0 −114.861 + 66.3152i 0 −74.7758 − 334.750i 0 121.500 + 210.444i 0
241.4 0 −13.5000 − 7.79423i 0 −58.1747 + 33.5872i 0 −336.398 + 66.9742i 0 121.500 + 210.444i 0
241.5 0 −13.5000 − 7.79423i 0 −23.8745 + 13.7839i 0 −162.412 − 302.112i 0 121.500 + 210.444i 0
241.6 0 −13.5000 − 7.79423i 0 −17.6311 + 10.1793i 0 342.439 + 19.6179i 0 121.500 + 210.444i 0
241.7 0 −13.5000 − 7.79423i 0 −6.08028 + 3.51045i 0 75.1444 + 334.667i 0 121.500 + 210.444i 0
241.8 0 −13.5000 − 7.79423i 0 102.544 − 59.2040i 0 191.470 − 284.585i 0 121.500 + 210.444i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.12
Significant digits:
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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.g 24
4.b odd 2 1 168.7.z.b ✓ 24
7.d odd 6 1 inner 336.7.bh.g 24
28.f even 6 1 168.7.z.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.7.z.b ✓ 24 4.b odd 2 1
168.7.z.b ✓ 24 28.f even 6 1
336.7.bh.g 24 1.a even 1 1 trivial
336.7.bh.g 24 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 126 T_{5}^{23} - 116739 T_{5}^{22} + 15375906 T_{5}^{21} + 9104072958 T_{5}^{20} - 1367538811338 T_{5}^{19} - 382448561894519 T_{5}^{18} + \cdots + 18\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display