# Properties

 Label 84.12.f.b Level $84$ Weight $12$ Character orbit 84.f Analytic conductor $64.541$ Analytic rank $0$ Dimension $28$ 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.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.5408271670$$ Analytic rank: $$0$$ Dimension: $$28$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 9632 q^{7} + 267660 q^{9}+O(q^{10})$$ 28 * q + 9632 * q^7 + 267660 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 9632 q^{7} + 267660 q^{9} - 3434160 q^{15} - 18804156 q^{21} + 397876900 q^{25} - 2059460504 q^{37} + 2276313936 q^{39} + 607100560 q^{43} + 1145242588 q^{49} + 1424787216 q^{51} - 32512522344 q^{57} + 16390616256 q^{63} - 48876957136 q^{67} - 1293110368 q^{79} + 82706814108 q^{81} + 197440859760 q^{85} - 329206232880 q^{91} - 243855044280 q^{93} - 81383696064 q^{99}+O(q^{100})$$ 28 * q + 9632 * q^7 + 267660 * q^9 - 3434160 * q^15 - 18804156 * q^21 + 397876900 * q^25 - 2059460504 * q^37 + 2276313936 * q^39 + 607100560 * q^43 + 1145242588 * q^49 + 1424787216 * q^51 - 32512522344 * q^57 + 16390616256 * q^63 - 48876957136 * q^67 - 1293110368 * q^79 + 82706814108 * q^81 + 197440859760 * q^85 - 329206232880 * q^91 - 243855044280 * q^93 - 81383696064 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 0 −419.400 35.3635i 0 9428.81 0 19241.0 + 40088.8i 0 174646. + 29662.9i 0
41.2 0 −419.400 + 35.3635i 0 9428.81 0 19241.0 40088.8i 0 174646. 29662.9i 0
41.3 0 −402.361 123.503i 0 −10055.0 0 27333.5 + 35074.3i 0 146641. + 99385.4i 0
41.4 0 −402.361 + 123.503i 0 −10055.0 0 27333.5 35074.3i 0 146641. 99385.4i 0
41.5 0 −384.586 170.999i 0 2406.66 0 −39402.8 + 20609.4i 0 118665. + 131528.i 0
41.6 0 −384.586 + 170.999i 0 2406.66 0 −39402.8 20609.4i 0 118665. 131528.i 0
41.7 0 −300.501 294.696i 0 −5023.98 0 −28355.3 34253.5i 0 3454.97 + 177113.i 0
41.8 0 −300.501 + 294.696i 0 −5023.98 0 −28355.3 + 34253.5i 0 3454.97 177113.i 0
41.9 0 −240.448 345.444i 0 113.306 0 44008.9 + 6367.04i 0 −61516.6 + 166123.i 0
41.10 0 −240.448 + 345.444i 0 113.306 0 44008.9 6367.04i 0 −61516.6 166123.i 0
41.11 0 −138.135 397.575i 0 8832.29 0 16667.3 41225.3i 0 −138985. + 109838.i 0
41.12 0 −138.135 + 397.575i 0 8832.29 0 16667.3 + 41225.3i 0 −138985. 109838.i 0
41.13 0 −24.0407 420.201i 0 11925.0 0 −37084.7 + 24536.8i 0 −175991. + 20203.9i 0
41.14 0 −24.0407 + 420.201i 0 11925.0 0 −37084.7 24536.8i 0 −175991. 20203.9i 0
41.15 0 24.0407 420.201i 0 −11925.0 0 −37084.7 + 24536.8i 0 −175991. 20203.9i 0
41.16 0 24.0407 + 420.201i 0 −11925.0 0 −37084.7 24536.8i 0 −175991. + 20203.9i 0
41.17 0 138.135 397.575i 0 −8832.29 0 16667.3 41225.3i 0 −138985. 109838.i 0
41.18 0 138.135 + 397.575i 0 −8832.29 0 16667.3 + 41225.3i 0 −138985. + 109838.i 0
41.19 0 240.448 345.444i 0 −113.306 0 44008.9 + 6367.04i 0 −61516.6 166123.i 0
41.20 0 240.448 + 345.444i 0 −113.306 0 44008.9 6367.04i 0 −61516.6 + 166123.i 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 41.28 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.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.12.f.b 28
3.b odd 2 1 inner 84.12.f.b 28
7.b odd 2 1 inner 84.12.f.b 28
21.c even 2 1 inner 84.12.f.b 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.12.f.b 28 1.a even 1 1 trivial
84.12.f.b 28 3.b odd 2 1 inner
84.12.f.b 28 7.b odd 2 1 inner
84.12.f.b 28 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{14} - 441266100 T_{5}^{12} + \cdots - 18\!\cdots\!00$$ acting on $$S_{12}^{\mathrm{new}}(84, [\chi])$$.