# Properties

 Label 84.12.k.b Level $84$ Weight $12$ Character orbit 84.k Analytic conductor $64.541$ Analytic rank $0$ Dimension $56$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 84.k (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.5408271670$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$28$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56 q - 729 q^{3} - 27194 q^{7} - 172347 q^{9}+O(q^{10})$$ 56 * q - 729 * q^3 - 27194 * q^7 - 172347 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$56 q - 729 q^{3} - 27194 q^{7} - 172347 q^{9} - 4853058 q^{15} + 28700520 q^{19} - 11325429 q^{21} - 316601194 q^{25} - 1368416388 q^{31} + 40874949 q^{33} - 87435712 q^{37} + 1177474410 q^{39} - 3055078348 q^{43} + 4109921793 q^{45} - 742582522 q^{49} - 694793715 q^{51} + 14605100370 q^{57} + 72584834058 q^{61} - 7310837811 q^{63} + 6131679148 q^{67} - 74402605464 q^{73} - 161115157854 q^{75} + 52181713528 q^{79} + 44948282337 q^{81} + 4658488716 q^{85} + 243101263104 q^{87} - 85311757146 q^{91} - 256628211777 q^{93} + 157345775874 q^{99}+O(q^{100})$$ 56 * q - 729 * q^3 - 27194 * q^7 - 172347 * q^9 - 4853058 * q^15 + 28700520 * q^19 - 11325429 * q^21 - 316601194 * q^25 - 1368416388 * q^31 + 40874949 * q^33 - 87435712 * q^37 + 1177474410 * q^39 - 3055078348 * q^43 + 4109921793 * q^45 - 742582522 * q^49 - 694793715 * q^51 + 14605100370 * q^57 + 72584834058 * q^61 - 7310837811 * q^63 + 6131679148 * q^67 - 74402605464 * q^73 - 161115157854 * q^75 + 52181713528 * q^79 + 44948282337 * q^81 + 4658488716 * q^85 + 243101263104 * q^87 - 85311757146 * q^91 - 256628211777 * q^93 + 157345775874 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 0 −418.334 + 46.3016i 0 −4424.05 + 7662.67i 0 6943.00 + 43921.8i 0 172859. 38739.1i 0
5.2 0 −416.405 + 61.2712i 0 3914.60 6780.29i 0 −44460.5 768.569i 0 169639. 51027.2i 0
5.3 0 −409.317 98.0135i 0 3554.32 6156.27i 0 42315.2 + 13665.8i 0 157934. + 80237.1i 0
5.4 0 −408.062 + 103.113i 0 −2288.66 + 3964.07i 0 14878.7 41904.1i 0 155882. 84153.0i 0
5.5 0 −377.843 185.422i 0 2528.83 4380.06i 0 −41471.0 + 16046.3i 0 108384. + 140121.i 0
5.6 0 −310.222 284.445i 0 −3956.77 + 6853.32i 0 −29669.5 33121.7i 0 15328.9 + 176483.i 0
5.7 0 −293.330 + 301.836i 0 2288.66 3964.07i 0 14878.7 41904.1i 0 −5062.60 177075.i 0
5.8 0 −289.505 305.506i 0 −1893.47 + 3279.58i 0 44455.1 + 1035.29i 0 −9520.78 + 176891.i 0
5.9 0 −261.265 + 329.981i 0 −3914.60 + 6780.29i 0 −44460.5 768.569i 0 −40628.4 172425.i 0
5.10 0 −249.265 + 339.137i 0 4424.05 7662.67i 0 6943.00 + 43921.8i 0 −52880.7 169070.i 0
5.11 0 −202.466 368.991i 0 5758.66 9974.29i 0 6342.31 44012.5i 0 −95162.3 + 149416.i 0
5.12 0 −119.776 + 403.486i 0 −3554.32 + 6156.27i 0 42315.2 + 13665.8i 0 −148454. 96656.0i 0
5.13 0 −95.9923 409.796i 0 −1583.42 + 2742.56i 0 −21857.5 + 38724.3i 0 −158718. + 78674.5i 0
5.14 0 −28.3414 + 419.933i 0 −2528.83 + 4380.06i 0 −41471.0 + 16046.3i 0 −175541. 23802.9i 0
5.15 0 −3.64668 420.873i 0 3597.09 6230.34i 0 19546.0 + 39941.0i 0 −177120. + 3069.57i 0
5.16 0 72.5029 414.597i 0 −6175.61 + 10696.5i 0 36613.7 25234.1i 0 −166634. 60118.9i 0
5.17 0 91.2256 + 410.883i 0 3956.77 6853.32i 0 −29669.5 33121.7i 0 −160503. + 74966.1i 0
5.18 0 119.823 + 403.472i 0 1893.47 3279.58i 0 44455.1 + 1035.29i 0 −148432. + 96690.7i 0
5.19 0 180.754 380.099i 0 832.414 1441.78i 0 −35902.5 26236.2i 0 −111803. 137409.i 0
5.20 0 204.376 367.937i 0 1631.25 2825.41i 0 25977.5 36090.1i 0 −93607.5 150395.i 0
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.12.k.b 56
3.b odd 2 1 inner 84.12.k.b 56
7.d odd 6 1 inner 84.12.k.b 56
21.g even 6 1 inner 84.12.k.b 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.12.k.b 56 1.a even 1 1 trivial
84.12.k.b 56 3.b odd 2 1 inner
84.12.k.b 56 7.d odd 6 1 inner
84.12.k.b 56 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{56} + 841894347 T_{5}^{54} + \cdots + 36\!\cdots\!00$$ acting on $$S_{12}^{\mathrm{new}}(84, [\chi])$$.