Properties

Label 378.2.f.a
Level 378
Weight 2
Character orbit 378.f
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.f (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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} + q^{8} + 3 q^{10} + ( -6 + 6 \zeta_{6} ) q^{11} -2 \zeta_{6} q^{13} -\zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} -6 q^{17} -7 q^{19} + ( -3 + 3 \zeta_{6} ) q^{20} -6 \zeta_{6} q^{22} + 3 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 2 q^{26} + q^{28} + ( 6 - 6 \zeta_{6} ) q^{29} -2 \zeta_{6} q^{31} -\zeta_{6} q^{32} + ( 6 - 6 \zeta_{6} ) q^{34} + 3 q^{35} + 2 q^{37} + ( 7 - 7 \zeta_{6} ) q^{38} -3 \zeta_{6} q^{40} + ( -2 + 2 \zeta_{6} ) q^{43} + 6 q^{44} -3 q^{46} -\zeta_{6} q^{49} -4 \zeta_{6} q^{50} + ( -2 + 2 \zeta_{6} ) q^{52} -6 q^{53} + 18 q^{55} + ( -1 + \zeta_{6} ) q^{56} + 6 \zeta_{6} q^{58} + ( -5 + 5 \zeta_{6} ) q^{61} + 2 q^{62} + q^{64} + ( -6 + 6 \zeta_{6} ) q^{65} -8 \zeta_{6} q^{67} + 6 \zeta_{6} q^{68} + ( -3 + 3 \zeta_{6} ) q^{70} -3 q^{71} + 2 q^{73} + ( -2 + 2 \zeta_{6} ) q^{74} + 7 \zeta_{6} q^{76} -6 \zeta_{6} q^{77} + ( -5 + 5 \zeta_{6} ) q^{79} + 3 q^{80} + ( 12 - 12 \zeta_{6} ) q^{83} + 18 \zeta_{6} q^{85} -2 \zeta_{6} q^{86} + ( -6 + 6 \zeta_{6} ) q^{88} + 2 q^{91} + ( 3 - 3 \zeta_{6} ) q^{92} + 21 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - 3q^{5} - q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - 3q^{5} - q^{7} + 2q^{8} + 6q^{10} - 6q^{11} - 2q^{13} - q^{14} - q^{16} - 12q^{17} - 14q^{19} - 3q^{20} - 6q^{22} + 3q^{23} - 4q^{25} + 4q^{26} + 2q^{28} + 6q^{29} - 2q^{31} - q^{32} + 6q^{34} + 6q^{35} + 4q^{37} + 7q^{38} - 3q^{40} - 2q^{43} + 12q^{44} - 6q^{46} - q^{49} - 4q^{50} - 2q^{52} - 12q^{53} + 36q^{55} - q^{56} + 6q^{58} - 5q^{61} + 4q^{62} + 2q^{64} - 6q^{65} - 8q^{67} + 6q^{68} - 3q^{70} - 6q^{71} + 4q^{73} - 2q^{74} + 7q^{76} - 6q^{77} - 5q^{79} + 6q^{80} + 12q^{83} + 18q^{85} - 2q^{86} - 6q^{88} + 4q^{91} + 3q^{92} + 21q^{95} - 2q^{97} + 2q^{98} + O(q^{100}) \)

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 −1.50000 2.59808i 0 −0.500000 + 0.866025i 1.00000 0 3.00000
253.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.50000 + 2.59808i 0 −0.500000 0.866025i 1.00000 0 3.00000
\(n\): 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.a 2
3.b odd 2 1 126.2.f.a 2
4.b odd 2 1 3024.2.r.a 2
7.b odd 2 1 2646.2.f.c 2
7.c even 3 1 2646.2.e.f 2
7.c even 3 1 2646.2.h.e 2
7.d odd 6 1 2646.2.e.j 2
7.d odd 6 1 2646.2.h.a 2
9.c even 3 1 inner 378.2.f.a 2
9.c even 3 1 1134.2.a.h 1
9.d odd 6 1 126.2.f.a 2
9.d odd 6 1 1134.2.a.a 1
12.b even 2 1 1008.2.r.d 2
21.c even 2 1 882.2.f.h 2
21.g even 6 1 882.2.e.d 2
21.g even 6 1 882.2.h.f 2
21.h odd 6 1 882.2.e.b 2
21.h odd 6 1 882.2.h.j 2
36.f odd 6 1 3024.2.r.a 2
36.f odd 6 1 9072.2.a.w 1
36.h even 6 1 1008.2.r.d 2
36.h even 6 1 9072.2.a.c 1
63.g even 3 1 2646.2.e.f 2
63.h even 3 1 2646.2.h.e 2
63.i even 6 1 882.2.h.f 2
63.j odd 6 1 882.2.h.j 2
63.k odd 6 1 2646.2.e.j 2
63.l odd 6 1 2646.2.f.c 2
63.l odd 6 1 7938.2.a.u 1
63.n odd 6 1 882.2.e.b 2
63.o even 6 1 882.2.f.h 2
63.o even 6 1 7938.2.a.l 1
63.s even 6 1 882.2.e.d 2
63.t odd 6 1 2646.2.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.a 2 3.b odd 2 1
126.2.f.a 2 9.d odd 6 1
378.2.f.a 2 1.a even 1 1 trivial
378.2.f.a 2 9.c even 3 1 inner
882.2.e.b 2 21.h odd 6 1
882.2.e.b 2 63.n odd 6 1
882.2.e.d 2 21.g even 6 1
882.2.e.d 2 63.s even 6 1
882.2.f.h 2 21.c even 2 1
882.2.f.h 2 63.o even 6 1
882.2.h.f 2 21.g even 6 1
882.2.h.f 2 63.i even 6 1
882.2.h.j 2 21.h odd 6 1
882.2.h.j 2 63.j odd 6 1
1008.2.r.d 2 12.b even 2 1
1008.2.r.d 2 36.h even 6 1
1134.2.a.a 1 9.d odd 6 1
1134.2.a.h 1 9.c even 3 1
2646.2.e.f 2 7.c even 3 1
2646.2.e.f 2 63.g even 3 1
2646.2.e.j 2 7.d odd 6 1
2646.2.e.j 2 63.k odd 6 1
2646.2.f.c 2 7.b odd 2 1
2646.2.f.c 2 63.l odd 6 1
2646.2.h.a 2 7.d odd 6 1
2646.2.h.a 2 63.t odd 6 1
2646.2.h.e 2 7.c even 3 1
2646.2.h.e 2 63.h even 3 1
3024.2.r.a 2 4.b odd 2 1
3024.2.r.a 2 36.f odd 6 1
7938.2.a.l 1 63.o even 6 1
7938.2.a.u 1 63.l odd 6 1
9072.2.a.c 1 36.h even 6 1
9072.2.a.w 1 36.f odd 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

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ 1
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( ( 1 + 6 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \)
$29$ \( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \)
$31$ \( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 41 T^{2} + 1681 T^{4} \)
$43$ \( 1 + 2 T - 39 T^{2} + 86 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 3 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 2 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 89 T^{2} )^{2} \)
$97$ \( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \)
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