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: \(x^{4} - 2 x^{2} + 4\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + 4 \beta_{2}\)\()/3\)

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 \( T_{5}^{4} - 2 T_{5}^{3} + 9 T_{5}^{2} + 10 T_{5} + 25 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ 1
$5$ \( 1 - 2 T - T^{2} + 10 T^{3} - 20 T^{4} + 50 T^{5} - 25 T^{6} - 250 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - T + T^{2} )^{2} \)
$11$ \( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \)
$13$ \( 1 - 2 T^{2} - 165 T^{4} - 338 T^{6} + 28561 T^{8} \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{4} \)
$19$ \( ( 1 - 10 T + 57 T^{2} - 190 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} )^{2} \)
$29$ \( 1 + 4 T - 22 T^{2} - 80 T^{3} + 139 T^{4} - 2320 T^{5} - 18502 T^{6} + 97556 T^{7} + 707281 T^{8} \)
$31$ \( ( 1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 4 T - 18 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( 1 + 14 T^{2} - 1485 T^{4} + 23534 T^{6} + 2825761 T^{8} \)
$43$ \( 1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 3440 T^{5} - 92450 T^{6} + 318028 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 2 T^{2} - 2205 T^{4} + 4418 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 - 12 T + 118 T^{2} - 636 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 2 T - 55 T^{2} + 118 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( 1 + 18 T + 127 T^{2} + 1350 T^{3} + 15324 T^{4} + 82350 T^{5} + 472567 T^{6} + 4085658 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 16 T + 82 T^{2} - 640 T^{3} + 8635 T^{4} - 42880 T^{5} + 368098 T^{6} - 4812208 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 + 10 T + 143 T^{2} + 710 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 4 T + 126 T^{2} + 292 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 + 6 T - 107 T^{2} - 90 T^{3} + 11364 T^{4} - 7110 T^{5} - 667787 T^{6} + 2958234 T^{7} + 38950081 T^{8} \)
$83$ \( ( 1 - 2 T - 79 T^{2} - 166 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 24 T + 298 T^{2} - 2136 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 7760 T^{5} - 1486622 T^{6} - 3650692 T^{7} + 88529281 T^{8} \)
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