Properties

Label 378.2.f.c
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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( -1 + \beta_{1} ) q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( -1 + \beta_{1} ) q^{7} + q^{8} + ( -2 - \beta_{2} ) q^{10} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} -2 \beta_{1} q^{13} -\beta_{1} q^{14} + ( -1 + \beta_{1} ) q^{16} + ( 1 - \beta_{2} ) q^{17} + 5 q^{19} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{20} + ( 2 \beta_{1} - \beta_{3} ) q^{22} + ( -5 \beta_{1} + \beta_{3} ) q^{23} + ( -7 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{25} + 2 q^{26} + q^{28} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{29} -2 \beta_{1} q^{31} -\beta_{1} q^{32} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{34} + ( -2 - \beta_{2} ) q^{35} + 2 q^{37} + ( -5 + 5 \beta_{1} ) q^{38} + ( \beta_{1} + \beta_{3} ) q^{40} + ( 8 \beta_{1} - \beta_{3} ) q^{41} + ( 1 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{43} + ( -1 + \beta_{2} ) q^{44} + ( 4 - \beta_{2} ) q^{46} -\beta_{1} q^{49} + ( -4 \beta_{1} - 3 \beta_{3} ) q^{50} + ( -2 + 2 \beta_{1} ) q^{52} + ( 4 + 2 \beta_{2} ) q^{53} -6 q^{55} + ( -1 + \beta_{1} ) q^{56} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{58} -3 \beta_{3} q^{59} + ( 4 - 7 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{61} + 2 q^{62} + q^{64} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -8 \beta_{1} + 3 \beta_{3} ) q^{67} + ( -2 \beta_{1} + \beta_{3} ) q^{68} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{70} + 3 \beta_{2} q^{71} + ( 5 + 3 \beta_{2} ) q^{73} + ( -2 + 2 \beta_{1} ) q^{74} -5 \beta_{1} q^{76} + ( 2 \beta_{1} - \beta_{3} ) q^{77} + ( -2 + 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{79} + ( -2 - \beta_{2} ) q^{80} + ( -7 + \beta_{2} ) q^{82} + ( 8 - 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{83} -6 \beta_{1} q^{85} + ( -2 \beta_{1} + 3 \beta_{3} ) q^{86} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{88} + ( -10 - 2 \beta_{2} ) q^{89} + 2 q^{91} + ( -4 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{92} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{95} + ( 1 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{4} + 3q^{5} - 2q^{7} + 4q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{4} + 3q^{5} - 2q^{7} + 4q^{8} - 6q^{10} + 3q^{11} - 4q^{13} - 2q^{14} - 2q^{16} + 6q^{17} + 20q^{19} + 3q^{20} + 3q^{22} - 9q^{23} - 11q^{25} + 8q^{26} + 4q^{28} - 6q^{29} - 4q^{31} - 2q^{32} - 3q^{34} - 6q^{35} + 8q^{37} - 10q^{38} + 3q^{40} + 15q^{41} - q^{43} - 6q^{44} + 18q^{46} - 2q^{49} - 11q^{50} - 4q^{52} + 12q^{53} - 24q^{55} - 2q^{56} - 6q^{58} - 3q^{59} + 11q^{61} + 8q^{62} + 4q^{64} + 6q^{65} - 13q^{67} - 3q^{68} + 3q^{70} - 6q^{71} + 14q^{73} - 4q^{74} - 10q^{76} + 3q^{77} - 7q^{79} - 6q^{80} - 30q^{82} + 12q^{83} - 12q^{85} - q^{86} + 3q^{88} - 36q^{89} + 8q^{91} - 9q^{92} + 15q^{95} - q^{97} + 4q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + \nu^{2} + 2 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11\)\()/3\)

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
−1.18614 1.26217i
1.68614 + 0.396143i
−1.18614 + 1.26217i
1.68614 0.396143i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −0.686141 1.18843i 0 −0.500000 + 0.866025i 1.00000 0 1.37228
127.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 2.18614 + 3.78651i 0 −0.500000 + 0.866025i 1.00000 0 −4.37228
253.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.686141 + 1.18843i 0 −0.500000 0.866025i 1.00000 0 1.37228
253.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 2.18614 3.78651i 0 −0.500000 0.866025i 1.00000 0 −4.37228
\(n\): 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.c 4
3.b odd 2 1 126.2.f.d 4
4.b odd 2 1 3024.2.r.f 4
7.b odd 2 1 2646.2.f.j 4
7.c even 3 1 2646.2.e.n 4
7.c even 3 1 2646.2.h.k 4
7.d odd 6 1 2646.2.e.m 4
7.d odd 6 1 2646.2.h.l 4
9.c even 3 1 inner 378.2.f.c 4
9.c even 3 1 1134.2.a.n 2
9.d odd 6 1 126.2.f.d 4
9.d odd 6 1 1134.2.a.k 2
12.b even 2 1 1008.2.r.f 4
21.c even 2 1 882.2.f.k 4
21.g even 6 1 882.2.e.k 4
21.g even 6 1 882.2.h.n 4
21.h odd 6 1 882.2.e.l 4
21.h odd 6 1 882.2.h.m 4
36.f odd 6 1 3024.2.r.f 4
36.f odd 6 1 9072.2.a.bb 2
36.h even 6 1 1008.2.r.f 4
36.h even 6 1 9072.2.a.bm 2
63.g even 3 1 2646.2.e.n 4
63.h even 3 1 2646.2.h.k 4
63.i even 6 1 882.2.h.n 4
63.j odd 6 1 882.2.h.m 4
63.k odd 6 1 2646.2.e.m 4
63.l odd 6 1 2646.2.f.j 4
63.l odd 6 1 7938.2.a.bs 2
63.n odd 6 1 882.2.e.l 4
63.o even 6 1 882.2.f.k 4
63.o even 6 1 7938.2.a.bh 2
63.s even 6 1 882.2.e.k 4
63.t odd 6 1 2646.2.h.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 3.b odd 2 1
126.2.f.d 4 9.d odd 6 1
378.2.f.c 4 1.a even 1 1 trivial
378.2.f.c 4 9.c even 3 1 inner
882.2.e.k 4 21.g even 6 1
882.2.e.k 4 63.s even 6 1
882.2.e.l 4 21.h odd 6 1
882.2.e.l 4 63.n odd 6 1
882.2.f.k 4 21.c even 2 1
882.2.f.k 4 63.o even 6 1
882.2.h.m 4 21.h odd 6 1
882.2.h.m 4 63.j odd 6 1
882.2.h.n 4 21.g even 6 1
882.2.h.n 4 63.i even 6 1
1008.2.r.f 4 12.b even 2 1
1008.2.r.f 4 36.h even 6 1
1134.2.a.k 2 9.d odd 6 1
1134.2.a.n 2 9.c even 3 1
2646.2.e.m 4 7.d odd 6 1
2646.2.e.m 4 63.k odd 6 1
2646.2.e.n 4 7.c even 3 1
2646.2.e.n 4 63.g even 3 1
2646.2.f.j 4 7.b odd 2 1
2646.2.f.j 4 63.l odd 6 1
2646.2.h.k 4 7.c even 3 1
2646.2.h.k 4 63.h even 3 1
2646.2.h.l 4 7.d odd 6 1
2646.2.h.l 4 63.t odd 6 1
3024.2.r.f 4 4.b odd 2 1
3024.2.r.f 4 36.f odd 6 1
7938.2.a.bh 2 63.o even 6 1
7938.2.a.bs 2 63.l odd 6 1
9072.2.a.bb 2 36.f odd 6 1
9072.2.a.bm 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 3 T_{5}^{3} + 15 T_{5}^{2} + 18 T_{5} + 36 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( \)
$5$ \( ( 1 - 3 T + 5 T^{2} )^{2}( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} ) \)
$7$ \( ( 1 + T + T^{2} )^{2} \)
$11$ \( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 198 T^{5} - 847 T^{6} - 3993 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 3 T + 28 T^{2} - 51 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 5 T + 19 T^{2} )^{4} \)
$23$ \( ( 1 + 9 T + 23 T^{2} )^{2}( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} ) \)
$29$ \( 1 + 6 T + 2 T^{2} - 144 T^{3} - 729 T^{4} - 4176 T^{5} + 1682 T^{6} + 146334 T^{7} + 707281 T^{8} \)
$31$ \( ( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{4} \)
$41$ \( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 29520 T^{5} + 159695 T^{6} - 1033815 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 3182 T^{5} - 20339 T^{6} + 79507 T^{7} + 3418801 T^{8} \)
$47$ \( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 6 T + 82 T^{2} - 318 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 + 3 T - 37 T^{2} - 216 T^{3} - 1896 T^{4} - 12744 T^{5} - 128797 T^{6} + 616137 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 11 T + 43 T^{2} + 484 T^{3} - 5018 T^{4} + 29524 T^{5} + 160003 T^{6} - 2496791 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 + 13 T + 67 T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} ) \)
$71$ \( ( 1 + 3 T + 70 T^{2} + 213 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 7 T + 84 T^{2} - 511 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 + 7 T - 47 T^{2} - 434 T^{3} - 896 T^{4} - 34286 T^{5} - 293327 T^{6} + 3451273 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 12 T + 74 T^{2} + 1152 T^{3} - 13941 T^{4} + 95616 T^{5} + 509786 T^{6} - 6861444 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 + 18 T + 226 T^{2} + 1602 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 + T - 119 T^{2} - 74 T^{3} + 4894 T^{4} - 7178 T^{5} - 1119671 T^{6} + 912673 T^{7} + 88529281 T^{8} \)
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