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 $4$

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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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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
0.500000 0.866025i
0.500000 + 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:

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 \(S_{12}^{\mathrm{new}}(84, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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