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

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

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 \( T_{5}^{2} + 3T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$89$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$97$ \( T^{2} - T + 1 \) Copy content Toggle raw display
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