Properties

Label 378.2.f.b
Level 378
Weight 2
Character orbit 378.f
Analytic conductor 3.018
Analytic rank 0
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.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.00000 + 1.73205i 0 0.500000 0.866025i 1.00000 0 −2.00000
253.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.00000 1.73205i 0 0.500000 + 0.866025i 1.00000 0 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c 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.f.b 2
3.b odd 2 1 126.2.f.b 2
4.b odd 2 1 3024.2.r.c 2
7.b odd 2 1 2646.2.f.b 2
7.c even 3 1 2646.2.e.i 2
7.c even 3 1 2646.2.h.b 2
7.d odd 6 1 2646.2.e.h 2
7.d odd 6 1 2646.2.h.c 2
9.c even 3 1 inner 378.2.f.b 2
9.c even 3 1 1134.2.a.f 1
9.d odd 6 1 126.2.f.b 2
9.d odd 6 1 1134.2.a.c 1
12.b even 2 1 1008.2.r.a 2
21.c even 2 1 882.2.f.f 2
21.g even 6 1 882.2.e.e 2
21.g even 6 1 882.2.h.g 2
21.h odd 6 1 882.2.e.a 2
21.h odd 6 1 882.2.h.h 2
36.f odd 6 1 3024.2.r.c 2
36.f odd 6 1 9072.2.a.f 1
36.h even 6 1 1008.2.r.a 2
36.h even 6 1 9072.2.a.t 1
63.g even 3 1 2646.2.e.i 2
63.h even 3 1 2646.2.h.b 2
63.i even 6 1 882.2.h.g 2
63.j odd 6 1 882.2.h.h 2
63.k odd 6 1 2646.2.e.h 2
63.l odd 6 1 2646.2.f.b 2
63.l odd 6 1 7938.2.a.bb 1
63.n odd 6 1 882.2.e.a 2
63.o even 6 1 882.2.f.f 2
63.o even 6 1 7938.2.a.e 1
63.s even 6 1 882.2.e.e 2
63.t odd 6 1 2646.2.h.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.b 2 3.b odd 2 1
126.2.f.b 2 9.d odd 6 1
378.2.f.b 2 1.a even 1 1 trivial
378.2.f.b 2 9.c even 3 1 inner
882.2.e.a 2 21.h odd 6 1
882.2.e.a 2 63.n odd 6 1
882.2.e.e 2 21.g even 6 1
882.2.e.e 2 63.s even 6 1
882.2.f.f 2 21.c even 2 1
882.2.f.f 2 63.o even 6 1
882.2.h.g 2 21.g even 6 1
882.2.h.g 2 63.i even 6 1
882.2.h.h 2 21.h odd 6 1
882.2.h.h 2 63.j odd 6 1
1008.2.r.a 2 12.b even 2 1
1008.2.r.a 2 36.h even 6 1
1134.2.a.c 1 9.d odd 6 1
1134.2.a.f 1 9.c even 3 1
2646.2.e.h 2 7.d odd 6 1
2646.2.e.h 2 63.k odd 6 1
2646.2.e.i 2 7.c even 3 1
2646.2.e.i 2 63.g even 3 1
2646.2.f.b 2 7.b odd 2 1
2646.2.f.b 2 63.l odd 6 1
2646.2.h.b 2 7.c even 3 1
2646.2.h.b 2 63.h even 3 1
2646.2.h.c 2 7.d odd 6 1
2646.2.h.c 2 63.t odd 6 1
3024.2.r.c 2 4.b odd 2 1
3024.2.r.c 2 36.f odd 6 1
7938.2.a.e 1 63.o even 6 1
7938.2.a.bb 1 63.l odd 6 1
9072.2.a.f 1 36.f odd 6 1
9072.2.a.t 1 36.h even 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( \)
$5$ \( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \)
$13$ \( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 5 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( 1 + 4 T - 13 T^{2} + 116 T^{3} + 841 T^{4} \)
$31$ \( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \)
$43$ \( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 + 12 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 12 T + 83 T^{2} - 732 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 8 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - T + 73 T^{2} )^{2} \)
$79$ \( 1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 16 T + 173 T^{2} - 1328 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \)
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