Properties

Label 378.2.f.d
Level $378$
Weight $2$
Character orbit 378.f
Analytic conductor $3.018$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} + ( - \beta_{2} + \beta_1) q^{5} + ( - \beta_1 + 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} + ( - \beta_{2} + \beta_1) q^{5} + ( - \beta_1 + 1) q^{7} - q^{8} + ( - \beta_{3} + 1) q^{10} + ( - 2 \beta_1 + 2) q^{11} - 2 \beta_{2} q^{13} - \beta_1 q^{14} + (\beta_1 - 1) q^{16} - 2 q^{17} + ( - \beta_{3} + 5) q^{19} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{20} - 2 \beta_1 q^{22} - \beta_1 q^{23} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{25} - 2 \beta_{3} q^{26} - q^{28} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{29} - 6 \beta_1 q^{31} + \beta_1 q^{32} + (2 \beta_1 - 2) q^{34} + ( - \beta_{3} + 1) q^{35} + (4 \beta_{3} + 2) q^{37} + ( - \beta_{3} + \beta_{2} - 5 \beta_1 + 5) q^{38} + (\beta_{2} - \beta_1) q^{40} + 4 \beta_{2} q^{41} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{43} - 2 q^{44} - q^{46} + (4 \beta_{3} - 4 \beta_{2}) q^{47} - \beta_1 q^{49} + ( - 2 \beta_{2} + 2 \beta_1) q^{50} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{52} + (2 \beta_{3} + 6) q^{53} + ( - 2 \beta_{3} + 2) q^{55} + (\beta_1 - 1) q^{56} + (2 \beta_{2} + 2 \beta_1) q^{58} - 2 \beta_1 q^{59} + (\beta_{3} - \beta_{2} + 9 \beta_1 - 9) q^{61} - 6 q^{62} + q^{64} + (2 \beta_{3} - 2 \beta_{2} + 12 \beta_1 - 12) q^{65} + (2 \beta_{2} + 8 \beta_1) q^{67} + 2 \beta_1 q^{68} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{70} + (2 \beta_{3} - 5) q^{71} + ( - 2 \beta_{3} - 2) q^{73} + (4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 2) q^{74} + (\beta_{2} - 5 \beta_1) q^{76} - 2 \beta_1 q^{77} + (2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{79} + (\beta_{3} - 1) q^{80} + 4 \beta_{3} q^{82} + ( - 2 \beta_1 + 2) q^{83} + (2 \beta_{2} - 2 \beta_1) q^{85} + (2 \beta_{2} + 2 \beta_1) q^{86} + (2 \beta_1 - 2) q^{88} + (2 \beta_{3} + 12) q^{89} - 2 \beta_{3} q^{91} + (\beta_1 - 1) q^{92} - 4 \beta_{2} q^{94} + ( - 6 \beta_{2} + 11 \beta_1) q^{95} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8} + 4 q^{10} + 4 q^{11} - 2 q^{14} - 2 q^{16} - 8 q^{17} + 20 q^{19} + 2 q^{20} - 4 q^{22} - 2 q^{23} - 4 q^{25} - 4 q^{28} - 4 q^{29} - 12 q^{31} + 2 q^{32} - 4 q^{34} + 4 q^{35} + 8 q^{37} + 10 q^{38} - 2 q^{40} - 4 q^{43} - 8 q^{44} - 4 q^{46} - 2 q^{49} + 4 q^{50} + 24 q^{53} + 8 q^{55} - 2 q^{56} + 4 q^{58} - 4 q^{59} - 18 q^{61} - 24 q^{62} + 4 q^{64} - 24 q^{65} + 16 q^{67} + 4 q^{68} + 2 q^{70} - 20 q^{71} - 8 q^{73} + 4 q^{74} - 10 q^{76} - 4 q^{77} - 6 q^{79} - 4 q^{80} + 4 q^{83} - 4 q^{85} + 4 q^{86} - 4 q^{88} + 48 q^{89} - 2 q^{92} + 22 q^{95} + 4 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 4\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-\beta_{1}\) \(1\)

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
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
0.500000 0.866025i 0 −0.500000 0.866025i −0.724745 1.25529i 0 0.500000 0.866025i −1.00000 0 −1.44949
127.2 0.500000 0.866025i 0 −0.500000 0.866025i 1.72474 + 2.98735i 0 0.500000 0.866025i −1.00000 0 3.44949
253.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.724745 + 1.25529i 0 0.500000 + 0.866025i −1.00000 0 −1.44949
253.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.72474 2.98735i 0 0.500000 + 0.866025i −1.00000 0 3.44949
\(n\): 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.d 4
3.b odd 2 1 126.2.f.c 4
4.b odd 2 1 3024.2.r.e 4
7.b odd 2 1 2646.2.f.k 4
7.c even 3 1 2646.2.e.l 4
7.c even 3 1 2646.2.h.m 4
7.d odd 6 1 2646.2.e.k 4
7.d odd 6 1 2646.2.h.n 4
9.c even 3 1 inner 378.2.f.d 4
9.c even 3 1 1134.2.a.i 2
9.d odd 6 1 126.2.f.c 4
9.d odd 6 1 1134.2.a.p 2
12.b even 2 1 1008.2.r.e 4
21.c even 2 1 882.2.f.j 4
21.g even 6 1 882.2.e.n 4
21.g even 6 1 882.2.h.l 4
21.h odd 6 1 882.2.e.m 4
21.h odd 6 1 882.2.h.k 4
36.f odd 6 1 3024.2.r.e 4
36.f odd 6 1 9072.2.a.bd 2
36.h even 6 1 1008.2.r.e 4
36.h even 6 1 9072.2.a.bk 2
63.g even 3 1 2646.2.e.l 4
63.h even 3 1 2646.2.h.m 4
63.i even 6 1 882.2.h.l 4
63.j odd 6 1 882.2.h.k 4
63.k odd 6 1 2646.2.e.k 4
63.l odd 6 1 2646.2.f.k 4
63.l odd 6 1 7938.2.a.bm 2
63.n odd 6 1 882.2.e.m 4
63.o even 6 1 882.2.f.j 4
63.o even 6 1 7938.2.a.bn 2
63.s even 6 1 882.2.e.n 4
63.t odd 6 1 2646.2.h.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 3.b odd 2 1
126.2.f.c 4 9.d odd 6 1
378.2.f.d 4 1.a even 1 1 trivial
378.2.f.d 4 9.c even 3 1 inner
882.2.e.m 4 21.h odd 6 1
882.2.e.m 4 63.n odd 6 1
882.2.e.n 4 21.g even 6 1
882.2.e.n 4 63.s even 6 1
882.2.f.j 4 21.c even 2 1
882.2.f.j 4 63.o even 6 1
882.2.h.k 4 21.h odd 6 1
882.2.h.k 4 63.j odd 6 1
882.2.h.l 4 21.g even 6 1
882.2.h.l 4 63.i even 6 1
1008.2.r.e 4 12.b even 2 1
1008.2.r.e 4 36.h even 6 1
1134.2.a.i 2 9.c even 3 1
1134.2.a.p 2 9.d odd 6 1
2646.2.e.k 4 7.d odd 6 1
2646.2.e.k 4 63.k odd 6 1
2646.2.e.l 4 7.c even 3 1
2646.2.e.l 4 63.g even 3 1
2646.2.f.k 4 7.b odd 2 1
2646.2.f.k 4 63.l odd 6 1
2646.2.h.m 4 7.c even 3 1
2646.2.h.m 4 63.h even 3 1
2646.2.h.n 4 7.d odd 6 1
2646.2.h.n 4 63.t odd 6 1
3024.2.r.e 4 4.b odd 2 1
3024.2.r.e 4 36.f odd 6 1
7938.2.a.bm 2 63.l odd 6 1
7938.2.a.bn 2 63.o even 6 1
9072.2.a.bd 2 36.f odd 6 1
9072.2.a.bk 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 2T_{5}^{3} + 9T_{5}^{2} + 10T_{5} + 25 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + 9 T^{2} + 10 T + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$17$ \( (T + 2)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 10 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400 \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T - 92)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400 \) Copy content Toggle raw display
$47$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 12)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 18 T^{3} + 249 T^{2} + \cdots + 5625 \) Copy content Toggle raw display
$67$ \( T^{4} - 16 T^{3} + 216 T^{2} + \cdots + 1600 \) Copy content Toggle raw display
$71$ \( (T^{2} + 10 T + 1)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4 T - 20)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + 51 T^{2} - 90 T + 225 \) Copy content Toggle raw display
$83$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 24 T + 120)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400 \) Copy content Toggle raw display
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