# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{7}^{\mathrm{new}}(336, [\chi])$$.