Properties

Label 336.7.bh.h
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 324 q^{3} - 126 q^{5} + 12 q^{7} + 2916 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 324 q^{3} - 126 q^{5} + 12 q^{7} + 2916 q^{9} + 1190 q^{11} - 2268 q^{15} - 1500 q^{17} + 13446 q^{19} - 2106 q^{21} - 21504 q^{23} + 22542 q^{25} - 85484 q^{29} - 6264 q^{31} + 32130 q^{33} - 32268 q^{35} - 46938 q^{37} + 17010 q^{39} + 19548 q^{43} - 30618 q^{45} + 167004 q^{47} + 250644 q^{49} - 13500 q^{51} - 258982 q^{53} + 242028 q^{57} - 744834 q^{59} - 390096 q^{61} - 59778 q^{63} - 19388 q^{65} - 62742 q^{67} + 1102984 q^{71} - 663534 q^{73} + 608634 q^{75} + 404298 q^{77} + 271032 q^{79} - 708588 q^{81} + 2540040 q^{85} - 1154034 q^{87} - 433740 q^{89} + 2142270 q^{91} - 56376 q^{93} - 2205360 q^{95} + 578340 q^{99} + O(q^{100}) \)

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 −160.491 92.6595i 0 338.827 + 53.3419i 0 121.500 210.444i 0
145.2 0 13.5000 7.79423i 0 −151.084 87.2284i 0 −180.229 + 291.833i 0 121.500 210.444i 0
145.3 0 13.5000 7.79423i 0 −126.269 72.9015i 0 −114.286 323.400i 0 121.500 210.444i 0
145.4 0 13.5000 7.79423i 0 −118.587 68.4663i 0 302.168 162.306i 0 121.500 210.444i 0
145.5 0 13.5000 7.79423i 0 −56.9387 32.8735i 0 −298.061 + 169.732i 0 121.500 210.444i 0
145.6 0 13.5000 7.79423i 0 −51.6596 29.8257i 0 −161.216 302.751i 0 121.500 210.444i 0
145.7 0 13.5000 7.79423i 0 44.9687 + 25.9627i 0 120.755 + 321.041i 0 121.500 210.444i 0
145.8 0 13.5000 7.79423i 0 59.9565 + 34.6159i 0 −9.46147 342.869i 0 121.500 210.444i 0
145.9 0 13.5000 7.79423i 0 76.6020 + 44.2262i 0 298.004 + 169.831i 0 121.500 210.444i 0
145.10 0 13.5000 7.79423i 0 112.001 + 64.6636i 0 −289.805 + 183.473i 0 121.500 210.444i 0
145.11 0 13.5000 7.79423i 0 127.654 + 73.7013i 0 −327.920 100.586i 0 121.500 210.444i 0
145.12 0 13.5000 7.79423i 0 180.847 + 104.412i 0 327.223 102.830i 0 121.500 210.444i 0
241.1 0 13.5000 + 7.79423i 0 −160.491 + 92.6595i 0 338.827 53.3419i 0 121.500 + 210.444i 0
241.2 0 13.5000 + 7.79423i 0 −151.084 + 87.2284i 0 −180.229 291.833i 0 121.500 + 210.444i 0
241.3 0 13.5000 + 7.79423i 0 −126.269 + 72.9015i 0 −114.286 + 323.400i 0 121.500 + 210.444i 0
241.4 0 13.5000 + 7.79423i 0 −118.587 + 68.4663i 0 302.168 + 162.306i 0 121.500 + 210.444i 0
241.5 0 13.5000 + 7.79423i 0 −56.9387 + 32.8735i 0 −298.061 169.732i 0 121.500 + 210.444i 0
241.6 0 13.5000 + 7.79423i 0 −51.6596 + 29.8257i 0 −161.216 + 302.751i 0 121.500 + 210.444i 0
241.7 0 13.5000 + 7.79423i 0 44.9687 25.9627i 0 120.755 321.041i 0 121.500 + 210.444i 0
241.8 0 13.5000 + 7.79423i 0 59.9565 34.6159i 0 −9.46147 + 342.869i 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:
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.h 24
4.b odd 2 1 168.7.z.a 24
7.d odd 6 1 inner 336.7.bh.h 24
28.f even 6 1 168.7.z.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.7.z.a 24 4.b odd 2 1
168.7.z.a 24 28.f even 6 1
336.7.bh.h 24 1.a even 1 1 trivial
336.7.bh.h 24 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(22\!\cdots\!03\)\( T_{5}^{18} - \)\(32\!\cdots\!94\)\( T_{5}^{17} + \)\(62\!\cdots\!86\)\( T_{5}^{16} + \)\(87\!\cdots\!90\)\( T_{5}^{15} - \)\(12\!\cdots\!75\)\( T_{5}^{14} - \)\(16\!\cdots\!50\)\( T_{5}^{13} + \)\(18\!\cdots\!25\)\( T_{5}^{12} + \)\(20\!\cdots\!00\)\( T_{5}^{11} - \)\(19\!\cdots\!00\)\( T_{5}^{10} - \)\(18\!\cdots\!00\)\( T_{5}^{9} + \)\(16\!\cdots\!00\)\( T_{5}^{8} + \)\(10\!\cdots\!00\)\( T_{5}^{7} - \)\(85\!\cdots\!00\)\( T_{5}^{6} - \)\(39\!\cdots\!00\)\( T_{5}^{5} + \)\(32\!\cdots\!00\)\( T_{5}^{4} + \)\(91\!\cdots\!00\)\( T_{5}^{3} - \)\(69\!\cdots\!00\)\( T_{5}^{2} - \)\(12\!\cdots\!00\)\( T_{5} + \)\(10\!\cdots\!00\)\( \)">\(T_{5}^{24} + \cdots\) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\).