Properties

Label 378.2.e.a
Level 378
Weight 2
Character orbit 378.e
Analytic conductor 3.018
Analytic rank 1
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} - q^{8} + ( 3 - 3 \zeta_{6} ) q^{10} -3 \zeta_{6} q^{11} -5 \zeta_{6} q^{13} + ( 3 - 2 \zeta_{6} ) q^{14} + q^{16} + ( 3 - 3 \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} + ( -3 + 3 \zeta_{6} ) q^{20} + 3 \zeta_{6} q^{22} + ( -3 + 3 \zeta_{6} ) q^{23} -4 \zeta_{6} q^{25} + 5 \zeta_{6} q^{26} + ( -3 + 2 \zeta_{6} ) q^{28} + ( -3 + 3 \zeta_{6} ) q^{29} -4 q^{31} - q^{32} + ( -3 + 3 \zeta_{6} ) q^{34} + ( 3 - 9 \zeta_{6} ) q^{35} + 7 \zeta_{6} q^{37} + 5 \zeta_{6} q^{38} + ( 3 - 3 \zeta_{6} ) q^{40} -9 \zeta_{6} q^{41} + ( -11 + 11 \zeta_{6} ) q^{43} -3 \zeta_{6} q^{44} + ( 3 - 3 \zeta_{6} ) q^{46} + ( 5 - 8 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{50} -5 \zeta_{6} q^{52} + ( -3 + 3 \zeta_{6} ) q^{53} + 9 q^{55} + ( 3 - 2 \zeta_{6} ) q^{56} + ( 3 - 3 \zeta_{6} ) q^{58} -12 q^{59} + 2 q^{61} + 4 q^{62} + q^{64} + 15 q^{65} -4 q^{67} + ( 3 - 3 \zeta_{6} ) q^{68} + ( -3 + 9 \zeta_{6} ) q^{70} + ( -11 + 11 \zeta_{6} ) q^{73} -7 \zeta_{6} q^{74} -5 \zeta_{6} q^{76} + ( 6 + 3 \zeta_{6} ) q^{77} + 8 q^{79} + ( -3 + 3 \zeta_{6} ) q^{80} + 9 \zeta_{6} q^{82} + ( 3 - 3 \zeta_{6} ) q^{83} + 9 \zeta_{6} q^{85} + ( 11 - 11 \zeta_{6} ) q^{86} + 3 \zeta_{6} q^{88} + 15 \zeta_{6} q^{89} + ( 10 + 5 \zeta_{6} ) q^{91} + ( -3 + 3 \zeta_{6} ) q^{92} + 15 q^{95} + ( 1 - \zeta_{6} ) q^{97} + ( -5 + 8 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 3q^{5} - 4q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 3q^{5} - 4q^{7} - 2q^{8} + 3q^{10} - 3q^{11} - 5q^{13} + 4q^{14} + 2q^{16} + 3q^{17} - 5q^{19} - 3q^{20} + 3q^{22} - 3q^{23} - 4q^{25} + 5q^{26} - 4q^{28} - 3q^{29} - 8q^{31} - 2q^{32} - 3q^{34} - 3q^{35} + 7q^{37} + 5q^{38} + 3q^{40} - 9q^{41} - 11q^{43} - 3q^{44} + 3q^{46} + 2q^{49} + 4q^{50} - 5q^{52} - 3q^{53} + 18q^{55} + 4q^{56} + 3q^{58} - 24q^{59} + 4q^{61} + 8q^{62} + 2q^{64} + 30q^{65} - 8q^{67} + 3q^{68} + 3q^{70} - 11q^{73} - 7q^{74} - 5q^{76} + 15q^{77} + 16q^{79} - 3q^{80} + 9q^{82} + 3q^{83} + 9q^{85} + 11q^{86} + 3q^{88} + 15q^{89} + 25q^{91} - 3q^{92} + 30q^{95} + q^{97} - 2q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-\zeta_{6}\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 0 1.00000 −1.50000 2.59808i 0 −2.00000 1.73205i −1.00000 0 1.50000 + 2.59808i
235.1 −1.00000 0 1.00000 −1.50000 + 2.59808i 0 −2.00000 + 1.73205i −1.00000 0 1.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.e.a 2
3.b odd 2 1 126.2.e.b 2
4.b odd 2 1 3024.2.q.a 2
7.b odd 2 1 2646.2.e.e 2
7.c even 3 1 378.2.h.b 2
7.c even 3 1 2646.2.f.e 2
7.d odd 6 1 2646.2.f.i 2
7.d odd 6 1 2646.2.h.f 2
9.c even 3 1 378.2.h.b 2
9.c even 3 1 1134.2.g.f 2
9.d odd 6 1 126.2.h.a yes 2
9.d odd 6 1 1134.2.g.d 2
12.b even 2 1 1008.2.q.e 2
21.c even 2 1 882.2.e.h 2
21.g even 6 1 882.2.f.a 2
21.g even 6 1 882.2.h.e 2
21.h odd 6 1 126.2.h.a yes 2
21.h odd 6 1 882.2.f.e 2
28.g odd 6 1 3024.2.t.f 2
36.f odd 6 1 3024.2.t.f 2
36.h even 6 1 1008.2.t.c 2
63.g even 3 1 1134.2.g.f 2
63.g even 3 1 2646.2.f.e 2
63.h even 3 1 inner 378.2.e.a 2
63.h even 3 1 7938.2.a.o 1
63.i even 6 1 882.2.e.h 2
63.i even 6 1 7938.2.a.bd 1
63.j odd 6 1 126.2.e.b 2
63.j odd 6 1 7938.2.a.r 1
63.k odd 6 1 2646.2.f.i 2
63.l odd 6 1 2646.2.h.f 2
63.n odd 6 1 882.2.f.e 2
63.n odd 6 1 1134.2.g.d 2
63.o even 6 1 882.2.h.e 2
63.s even 6 1 882.2.f.a 2
63.t odd 6 1 2646.2.e.e 2
63.t odd 6 1 7938.2.a.c 1
84.n even 6 1 1008.2.t.c 2
252.u odd 6 1 3024.2.q.a 2
252.bb even 6 1 1008.2.q.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.b 2 3.b odd 2 1
126.2.e.b 2 63.j odd 6 1
126.2.h.a yes 2 9.d odd 6 1
126.2.h.a yes 2 21.h odd 6 1
378.2.e.a 2 1.a even 1 1 trivial
378.2.e.a 2 63.h even 3 1 inner
378.2.h.b 2 7.c even 3 1
378.2.h.b 2 9.c even 3 1
882.2.e.h 2 21.c even 2 1
882.2.e.h 2 63.i even 6 1
882.2.f.a 2 21.g even 6 1
882.2.f.a 2 63.s even 6 1
882.2.f.e 2 21.h odd 6 1
882.2.f.e 2 63.n odd 6 1
882.2.h.e 2 21.g even 6 1
882.2.h.e 2 63.o even 6 1
1008.2.q.e 2 12.b even 2 1
1008.2.q.e 2 252.bb even 6 1
1008.2.t.c 2 36.h even 6 1
1008.2.t.c 2 84.n even 6 1
1134.2.g.d 2 9.d odd 6 1
1134.2.g.d 2 63.n odd 6 1
1134.2.g.f 2 9.c even 3 1
1134.2.g.f 2 63.g even 3 1
2646.2.e.e 2 7.b odd 2 1
2646.2.e.e 2 63.t odd 6 1
2646.2.f.e 2 7.c even 3 1
2646.2.f.e 2 63.g even 3 1
2646.2.f.i 2 7.d odd 6 1
2646.2.f.i 2 63.k odd 6 1
2646.2.h.f 2 7.d odd 6 1
2646.2.h.f 2 63.l odd 6 1
3024.2.q.a 2 4.b odd 2 1
3024.2.q.a 2 252.u odd 6 1
3024.2.t.f 2 28.g odd 6 1
3024.2.t.f 2 36.f odd 6 1
7938.2.a.c 1 63.t odd 6 1
7938.2.a.o 1 63.h even 3 1
7938.2.a.r 1 63.j odd 6 1
7938.2.a.bd 1 63.i even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3 T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ 1
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \)
$19$ \( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \)
$23$ \( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 11 T + 78 T^{2} + 473 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 + 12 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 + 4 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 3 T - 74 T^{2} - 249 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} \)
$97$ \( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \)
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