↑

Properties

Label 378.2.k.d
Level $378$
Weight $2$
Character orbit 378.k
Analytic conductor $3.018$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{6} + \beta_{5}) q^{5} + (\beta_{7} - 1) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{6} + \beta_{5}) q^{5} + (\beta_{7} - 1) q^{7} + \beta_{3} q^{8} + (\beta_{7} - \beta_{4}) q^{10} + ( - 2 \beta_{6} + \beta_{5}) q^{11} + ( - \beta_{7} - 2 \beta_{2} + 1) q^{13} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{14} + (\beta_{2} - 1) q^{16} - \beta_{5} q^{17} + 2 \beta_{4} q^{19} - \beta_{6} q^{20} + (\beta_{7} - 2 \beta_{4}) q^{22} + 6 \beta_1 q^{23} - \beta_{2} q^{25} + (\beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_1) q^{26} + (\beta_{7} - \beta_{4} - \beta_{2}) q^{28} + ( - \beta_{6} + 2 \beta_{5} + 6 \beta_{3}) q^{29} + ( - 3 \beta_{7} + 3 \beta_{4} + \beta_{2} - 2) q^{31} + (\beta_{3} - \beta_1) q^{32} - \beta_{7} q^{34} + (\beta_{6} - \beta_{5} - 6 \beta_1) q^{35} + ( - 2 \beta_{7} + \beta_{4} + \beta_{2} - 1) q^{37} + 2 \beta_{5} q^{38} - \beta_{4} q^{40} - \beta_{6} q^{41} + 7 q^{43} + ( - \beta_{6} - \beta_{5}) q^{44} + 6 \beta_{2} q^{46} + ( - \beta_{6} + \beta_{5} - 12 \beta_{3} + 6 \beta_1) q^{47} + ( - 2 \beta_{7} - 5) q^{49} - \beta_{3} q^{50} + ( - \beta_{7} + \beta_{4} - \beta_{2} + 2) q^{52} + (4 \beta_{6} - 2 \beta_{5} + 6 \beta_{3} - 6 \beta_1) q^{53} + ( - 12 \beta_{2} + 6) q^{55} + ( - \beta_{6} - \beta_{3}) q^{56} + (2 \beta_{7} - \beta_{4} + 6 \beta_{2} - 6) q^{58} + \beta_{5} q^{59} + ( - \beta_{4} + \beta_{2} + 1) q^{61} + (3 \beta_{6} + \beta_{3} - 2 \beta_1) q^{62} - q^{64} + (\beta_{6} + \beta_{5} + 6 \beta_1) q^{65} + (2 \beta_{7} + 2 \beta_{4} - 5 \beta_{2}) q^{67} + (\beta_{6} - \beta_{5}) q^{68} + ( - \beta_{7} + \beta_{4} - 6 \beta_{2}) q^{70} + (3 \beta_{6} - 6 \beta_{5}) q^{71} + (2 \beta_{7} - 2 \beta_{4} + 6 \beta_{2} - 12) q^{73} + (2 \beta_{6} - \beta_{5} + \beta_{3} - \beta_1) q^{74} + 2 \beta_{7} q^{76} + (2 \beta_{6} - \beta_{5} - 6 \beta_{3} - 6 \beta_1) q^{77} + (2 \beta_{7} - \beta_{4} + 5 \beta_{2} - 5) q^{79} - \beta_{5} q^{80} - \beta_{4} q^{82} + (2 \beta_{6} + 6 \beta_{3} - 12 \beta_1) q^{83} + 6 q^{85} + 7 \beta_1 q^{86} + ( - \beta_{7} - \beta_{4}) q^{88} + (3 \beta_{6} - 3 \beta_{5} - 12 \beta_{3} + 6 \beta_1) q^{89} + (2 \beta_{4} + 2 \beta_{2} + 5) q^{91} + 6 \beta_{3} q^{92} + (\beta_{7} - \beta_{4} - 6 \beta_{2} + 12) q^{94} + (12 \beta_{3} - 12 \beta_1) q^{95} + (2 \beta_{7} + 2 \beta_{2} - 1) q^{97} + (2 \beta_{6} - 2 \beta_{5} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 8 q^{7} - 4 q^{16} - 4 q^{25} - 4 q^{28} - 12 q^{31} - 4 q^{37} + 56 q^{43} + 24 q^{46} - 40 q^{49} + 12 q^{52} - 24 q^{58} + 12 q^{61} - 8 q^{64} - 20 q^{67} - 24 q^{70} - 72 q^{73} - 20 q^{79} + 48 q^{85} + 48 q^{91} + 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.22474 + 2.12132i 0 −1.00000 − 2.44949i − 1.00000i 0 2.12132 − 1.22474i
215.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.22474 − 2.12132i 0 −1.00000 + 2.44949i − 1.00000i 0 −2.12132 + 1.22474i
215.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.22474 + 2.12132i 0 −1.00000 + 2.44949i 1.00000i 0 −2.12132 + 1.22474i
215.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.22474 − 2.12132i 0 −1.00000 − 2.44949i 1.00000i 0 2.12132 − 1.22474i
269.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.22474 − 2.12132i 0 −1.00000 + 2.44949i 1.00000i 0 2.12132 + 1.22474i
269.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.22474 + 2.12132i 0 −1.00000 − 2.44949i 1.00000i 0 −2.12132 − 1.22474i
269.3 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.22474 − 2.12132i 0 −1.00000 − 2.44949i − 1.00000i 0 −2.12132 − 1.22474i
269.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.22474 + 2.12132i 0 −1.00000 + 2.44949i − 1.00000i 0 2.12132 + 1.22474i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.4
Significant digits:
Format:

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 378.2.k.d ✓ 8
3.b odd 2 1 inner 378.2.k.d ✓ 8
7.c even 3 1 2646.2.d.d 8
7.d odd 6 1 inner 378.2.k.d ✓ 8
7.d odd 6 1 2646.2.d.d 8
9.c even 3 1 1134.2.l.e 8
9.c even 3 1 1134.2.t.f 8
9.d odd 6 1 1134.2.l.e 8
9.d odd 6 1 1134.2.t.f 8
21.g even 6 1 inner 378.2.k.d ✓ 8
21.g even 6 1 2646.2.d.d 8
21.h odd 6 1 2646.2.d.d 8
63.i even 6 1 1134.2.t.f 8
63.k odd 6 1 1134.2.l.e 8
63.s even 6 1 1134.2.l.e 8
63.t odd 6 1 1134.2.t.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.d ✓ 8 1.a even 1 1 trivial
378.2.k.d ✓ 8 3.b odd 2 1 inner
378.2.k.d ✓ 8 7.d odd 6 1 inner
378.2.k.d ✓ 8 21.g even 6 1 inner
1134.2.l.e 8 9.c even 3 1
1134.2.l.e 8 9.d odd 6 1
1134.2.l.e 8 63.k odd 6 1
1134.2.l.e 8 63.s even 6 1
1134.2.t.f 8 9.c even 3 1
1134.2.t.f 8 9.d odd 6 1
1134.2.t.f 8 63.i even 6 1
1134.2.t.f 8 63.t odd 6 1
2646.2.d.d 8 7.c even 3 1
2646.2.d.d 8 7.d odd 6 1
2646.2.d.d 8 21.g even 6 1
2646.2.d.d 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 6T_{5}^{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 18 T^{2} + 9)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 108 T^{2} + 324)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 6 T^{3} - 39 T^{2} - 306 T + 2601)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$43$ \( (T - 7)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} + 228 T^{6} + \cdots + 108243216 \) Copy content Toggle raw display
$53$ \( T^{8} - 216 T^{6} + 45360 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$59$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 6 T^{3} + 9 T^{2} + 18 T + 9)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 10 T^{3} + 147 T^{2} - 470 T + 2209)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 162)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 36 T^{3} + 516 T^{2} + 3024 T + 7056)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} + 93 T^{2} + 70 T + 49)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 264 T^{2} + 7056)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 324 T^{6} + 102060 T^{4} + \cdots + 8503056 \) Copy content Toggle raw display
$97$ \( (T^{4} + 54 T^{2} + 441)^{2} \) Copy content Toggle raw display
show more
show less