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

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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: \(\Q(\zeta_{24})\)
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 primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + ( -2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{5} + ( -1 + \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{7} + \zeta_{24}^{6} q^{8} +O(q^{10})\) \( q + \zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + ( -2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{5} + ( -1 + \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{7} + \zeta_{24}^{6} q^{8} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{10} + ( -3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{11} + ( 1 - \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{13} + ( -2 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{14} + ( -1 + \zeta_{24}^{4} ) q^{16} + ( \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{17} + ( 4 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{19} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{20} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{22} + 6 \zeta_{24}^{2} q^{23} -\zeta_{24}^{4} q^{25} + ( 2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{26} + ( -\zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{28} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{29} + ( -2 + 3 \zeta_{24} + 6 \zeta_{24}^{3} + \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{31} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{32} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{34} + ( 2 \zeta_{24} - 6 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{35} + ( -1 + 3 \zeta_{24}^{3} + \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{37} + ( -2 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{38} + ( -2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{40} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{41} + 7 q^{43} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{44} + 6 \zeta_{24}^{4} q^{46} + ( -2 \zeta_{24} + 6 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 12 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{47} + ( -5 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{49} -\zeta_{24}^{6} q^{50} + ( 2 + \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{52} + ( 6 \zeta_{24} - 6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{53} + ( 6 - 12 \zeta_{24}^{4} ) q^{55} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{56} + ( -6 - 3 \zeta_{24}^{3} + 6 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{58} + ( -\zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{59} + ( 1 - 2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{61} + ( 3 \zeta_{24} - 2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{62} - q^{64} + ( 6 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{65} + ( 6 \zeta_{24} - 5 \zeta_{24}^{4} + 6 \zeta_{24}^{7} ) q^{67} + ( 2 \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{68} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{70} + ( 9 \zeta_{24} - 9 \zeta_{24}^{3} - 9 \zeta_{24}^{5} ) q^{71} + ( -12 - 2 \zeta_{24} - 4 \zeta_{24}^{3} + 6 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{73} + ( 3 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{74} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{76} + ( 3 \zeta_{24} - 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{77} + ( -5 - 3 \zeta_{24}^{3} + 5 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{79} + ( \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{80} + ( -2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{82} + ( 2 \zeta_{24} - 12 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{83} + 6 q^{85} + 7 \zeta_{24}^{2} q^{86} + ( -3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{88} + ( 6 \zeta_{24} + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 12 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{89} + ( 5 + 4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{91} + 6 \zeta_{24}^{6} q^{92} + ( 12 - \zeta_{24} - 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{94} + ( -12 \zeta_{24}^{2} + 12 \zeta_{24}^{6} ) q^{95} + ( -1 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{97} + ( 4 \zeta_{24} - 5 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} - 8q^{7} + O(q^{10}) \) \( 8q + 4q^{4} - 8q^{7} - 4q^{16} - 4q^{25} - 4q^{28} - 12q^{31} - 4q^{37} + 56q^{43} + 24q^{46} - 40q^{49} + 12q^{52} - 24q^{58} + 12q^{61} - 8q^{64} - 20q^{67} - 24q^{70} - 72q^{73} - 20q^{79} + 48q^{85} + 48q^{91} + 72q^{94} + O(q^{100}) \)

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{24}^{2}\)

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 \( T_{5}^{4} + 6 T_{5}^{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( \)
$5$ \( ( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 + 2 T + 7 T^{2} )^{4} \)
$11$ \( ( 1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 34 T^{2} + 555 T^{4} - 5746 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 28 T^{2} + 495 T^{4} - 8092 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 14 T^{2} - 165 T^{4} + 5054 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 8 T^{2} - 894 T^{4} - 6728 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 + 6 T + 23 T^{2} + 66 T^{3} - 468 T^{4} + 2046 T^{5} + 22103 T^{6} + 178746 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 2 T - 53 T^{2} - 34 T^{3} + 1732 T^{4} - 1258 T^{5} - 72557 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 + 76 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 7 T + 43 T^{2} )^{8} \)
$47$ \( 1 + 40 T^{2} - 626 T^{4} - 87680 T^{6} - 3459005 T^{8} - 193685120 T^{10} - 3054680306 T^{12} + 431168613160 T^{14} + 23811286661761 T^{16} \)
$53$ \( 1 - 4 T^{2} + 4762 T^{4} + 41456 T^{6} + 14589091 T^{8} + 116449904 T^{10} + 37574470522 T^{12} - 88657444516 T^{14} + 62259690411361 T^{16} \)
$59$ \( ( 1 - 112 T^{2} + 9063 T^{4} - 389872 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 6 T + 131 T^{2} - 714 T^{3} + 11172 T^{4} - 43554 T^{5} + 487451 T^{6} - 1361886 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 10 T + 13 T^{2} - 470 T^{3} - 3620 T^{4} - 31490 T^{5} + 58357 T^{6} + 3007630 T^{7} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 20 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 + 36 T + 662 T^{2} + 8280 T^{3} + 79107 T^{4} + 604440 T^{5} + 3527798 T^{6} + 14004612 T^{7} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 + 10 T - 65 T^{2} + 70 T^{3} + 13084 T^{4} + 5530 T^{5} - 405665 T^{6} + 4930390 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 68 T^{2} + 4566 T^{4} + 468452 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( 1 - 32 T^{2} + 8254 T^{4} + 738304 T^{6} - 18580445 T^{8} + 5848105984 T^{10} + 517874457214 T^{12} - 15903401310752 T^{14} + 3936588805702081 T^{16} \)
$97$ \( ( 1 - 334 T^{2} + 46419 T^{4} - 3142606 T^{6} + 88529281 T^{8} )^{2} \)
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