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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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 \( T_{5}^{2} - 2T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$17$ \( (T - 5)^{2} \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$61$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$67$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( (T - 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$83$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
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