# 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 $2$

# 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.

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

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