# Properties

 Label 84.12.k.a Level $84$ Weight $12$ Character orbit 84.k Analytic conductor $64.541$ Analytic rank $0$ Dimension $2$ CM discriminant -3 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (243 \zeta_{6} + 243) q^{3} + ( - 25807 \zeta_{6} + 51346) q^{7} + 177147 \zeta_{6} q^{9}+O(q^{10})$$ q + (243*z + 243) * q^3 + (-25807*z + 51346) * q^7 + 177147*z * q^9 $$q + (243 \zeta_{6} + 243) q^{3} + ( - 25807 \zeta_{6} + 51346) q^{7} + 177147 \zeta_{6} q^{9} + (1678242 \zeta_{6} - 839121) q^{13} + (3757055 \zeta_{6} - 7514110) q^{19} + ( - 65124 \zeta_{6} + 18748179) q^{21} + ( - 48828125 \zeta_{6} + 48828125) q^{25} + (86093442 \zeta_{6} - 43046721) q^{27} + (182070569 \zeta_{6} + 182070569) q^{31} + 119374607 \zeta_{6} q^{37} + (611719209 \zeta_{6} - 611719209) q^{39} + 218924719 q^{43} + ( - 1984171195 \zeta_{6} + 1970410467) q^{49} - 2738893095 q^{57} + (7174548484 \zeta_{6} - 14349096968) q^{61} + (4524157233 \zeta_{6} + 4571632629) q^{63} + (5951291615 \zeta_{6} - 5951291615) q^{67} + (18381400487 \zeta_{6} + 18381400487) q^{73} + ( - 11865234375 \zeta_{6} + 23730468750) q^{75} + 54296224537 \zeta_{6} q^{79} + (31381059609 \zeta_{6} - 31381059609) q^{81} + (64515818085 \zeta_{6} + 224884428) q^{91} + 132729444801 \zeta_{6} q^{93} + (145716229808 \zeta_{6} - 72858114904) q^{97}+O(q^{100})$$ q + (243*z + 243) * q^3 + (-25807*z + 51346) * q^7 + 177147*z * q^9 + (1678242*z - 839121) * q^13 + (3757055*z - 7514110) * q^19 + (-65124*z + 18748179) * q^21 + (-48828125*z + 48828125) * q^25 + (86093442*z - 43046721) * q^27 + (182070569*z + 182070569) * q^31 + 119374607*z * q^37 + (611719209*z - 611719209) * q^39 + 218924719 * q^43 + (-1984171195*z + 1970410467) * q^49 - 2738893095 * q^57 + (7174548484*z - 14349096968) * q^61 + (4524157233*z + 4571632629) * q^63 + (5951291615*z - 5951291615) * q^67 + (18381400487*z + 18381400487) * q^73 + (-11865234375*z + 23730468750) * q^75 + 54296224537*z * q^79 + (31381059609*z - 31381059609) * q^81 + (64515818085*z + 224884428) * q^91 + 132729444801*z * q^93 + (145716229808*z - 72858114904) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 729 q^{3} + 76885 q^{7} + 177147 q^{9}+O(q^{10})$$ 2 * q + 729 * q^3 + 76885 * q^7 + 177147 * q^9 $$2 q + 729 q^{3} + 76885 q^{7} + 177147 q^{9} - 11271165 q^{19} + 37431234 q^{21} + 48828125 q^{25} + 546211707 q^{31} + 119374607 q^{37} - 611719209 q^{39} + 437849438 q^{43} + 1956649739 q^{49} - 5477786190 q^{57} - 21523645452 q^{61} + 13667422491 q^{63} - 5951291615 q^{67} + 55144201461 q^{73} + 35595703125 q^{75} + 54296224537 q^{79} - 31381059609 q^{81} + 64965586941 q^{91} + 132729444801 q^{93}+O(q^{100})$$ 2 * q + 729 * q^3 + 76885 * q^7 + 177147 * q^9 - 11271165 * q^19 + 37431234 * q^21 + 48828125 * q^25 + 546211707 * q^31 + 119374607 * q^37 - 611719209 * q^39 + 437849438 * q^43 + 1956649739 * q^49 - 5477786190 * q^57 - 21523645452 * q^61 + 13667422491 * q^63 - 5951291615 * q^67 + 55144201461 * q^73 + 35595703125 * q^75 + 54296224537 * q^79 - 31381059609 * q^81 + 64965586941 * q^91 + 132729444801 * q^93

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$43$$ $$73$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 364.500 210.444i 0 0 0 38442.5 + 22349.5i 0 88573.5 153414.i 0
17.1 0 364.500 + 210.444i 0 0 0 38442.5 22349.5i 0 88573.5 + 153414.i 0
 $$n$$: e.g. 2-40 or 990-1000 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 CM by $$\Q(\sqrt{-3})$$
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.a 2
3.b odd 2 1 CM 84.12.k.a 2
7.d odd 6 1 inner 84.12.k.a 2
21.g even 6 1 inner 84.12.k.a 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 729T + 177147$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 76885 T + 1977326743$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 2112372157923$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 11271165 T + 42346386819075$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 546211707 T + 99\!\cdots\!83$$
$37$ $$T^{2} - 119374607 T + 14\!\cdots\!49$$
$41$ $$T^{2}$$
$43$ $$(T - 218924719)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 21523645452 T + 15\!\cdots\!68$$
$67$ $$T^{2} + 5951291615 T + 35\!\cdots\!25$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 55144201461 T + 10\!\cdots\!07$$
$79$ $$T^{2} - 54296224537 T + 29\!\cdots\!69$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 15\!\cdots\!48$$