Properties

Label 378.2.h.c
Level $378$
Weight $2$
Character orbit 378.h
Analytic conductor $3.018$
Analytic rank $0$
Dimension $6$
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.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Defining polynomial: \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{4} ) q^{2} -\beta_{4} q^{4} -\beta_{3} q^{5} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{4} ) q^{2} -\beta_{4} q^{4} -\beta_{3} q^{5} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{7} + q^{8} -\beta_{2} q^{10} + \beta_{3} q^{11} + ( 3 + \beta_{2} - 3 \beta_{4} + 2 \beta_{5} ) q^{13} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{14} + ( -1 + \beta_{4} ) q^{16} + ( 2 - 2 \beta_{4} + 2 \beta_{5} ) q^{17} + ( -3 \beta_{1} - 2 \beta_{4} + 3 \beta_{5} ) q^{19} + ( \beta_{2} + \beta_{3} ) q^{20} + \beta_{2} q^{22} + ( -2 + \beta_{1} ) q^{23} + ( -1 + \beta_{1} - 2 \beta_{3} ) q^{25} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{26} + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{28} + ( -3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{29} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 6 \beta_{4} + \beta_{5} ) q^{31} -\beta_{4} q^{32} + ( 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} ) q^{34} + ( 4 + \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{35} + \beta_{4} q^{37} + ( 2 + 3 \beta_{1} ) q^{38} -\beta_{3} q^{40} + ( -1 + 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{41} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{43} + ( -\beta_{2} - \beta_{3} ) q^{44} + ( 2 - 2 \beta_{4} - \beta_{5} ) q^{46} + ( 3 - 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} ) q^{47} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{49} + ( 1 - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{50} + ( -3 - 2 \beta_{1} + \beta_{3} ) q^{52} + ( -5 + \beta_{2} + 5 \beta_{4} + \beta_{5} ) q^{53} + ( -4 - \beta_{1} + 2 \beta_{3} ) q^{55} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{56} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{58} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{59} + ( 3 - 2 \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{61} + ( -6 + \beta_{1} + \beta_{3} ) q^{62} + q^{64} + ( 2 + 5 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{65} + ( \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{67} + ( -2 - 2 \beta_{1} ) q^{68} + ( -1 - \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{70} + ( -4 - 7 \beta_{1} + 2 \beta_{3} ) q^{71} + ( 8 - 4 \beta_{2} - 8 \beta_{4} + \beta_{5} ) q^{73} - q^{74} + ( -2 + 2 \beta_{4} - 3 \beta_{5} ) q^{76} + ( -4 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{77} + ( \beta_{2} - 4 \beta_{5} ) q^{79} -\beta_{2} q^{80} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{82} + ( 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{83} + ( -2 + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{85} + ( 2 - 3 \beta_{1} + 3 \beta_{3} ) q^{86} + \beta_{3} q^{88} + ( -4 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{89} + ( -8 + \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{91} + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{92} + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{94} + ( 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{95} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 8 \beta_{4} + 2 \beta_{5} ) q^{97} + ( 1 + \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{2} - 3q^{4} + 2q^{5} - 4q^{7} + 6q^{8} + O(q^{10}) \) \( 6q - 3q^{2} - 3q^{4} + 2q^{5} - 4q^{7} + 6q^{8} - q^{10} - 2q^{11} + 8q^{13} + 2q^{14} - 3q^{16} + 4q^{17} - 3q^{19} - q^{20} + q^{22} - 14q^{23} - 4q^{25} + 8q^{26} + 2q^{28} + 5q^{29} + 20q^{31} - 3q^{32} + 4q^{34} + 13q^{35} + 3q^{37} + 6q^{38} + 2q^{40} - 6q^{43} + q^{44} + 7q^{46} + 9q^{47} - 12q^{49} + 2q^{50} - 16q^{52} - 15q^{53} - 26q^{55} - 4q^{56} - 10q^{58} + 14q^{59} + 8q^{61} - 40q^{62} + 6q^{64} + 12q^{65} + q^{67} - 8q^{68} + 10q^{70} - 14q^{71} + 19q^{73} - 6q^{74} - 3q^{76} - 13q^{77} + 5q^{79} - q^{80} - 2q^{83} - 2q^{85} + 12q^{86} - 2q^{88} + 9q^{89} - 46q^{91} + 7q^{92} + 9q^{94} + 4q^{95} + 28q^{97} + 12q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 2 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 4\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} - 10\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(16 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} + 5\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-14 \beta_{5} - 16 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 23 \beta_{1} + 47\)\()/3\)

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 + 0.224437i
0.500000 + 2.05195i
0.500000 1.41036i
0.500000 0.224437i
0.500000 2.05195i
0.500000 + 1.41036i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −1.58836 0 −2.64400 + 0.0963576i 1.00000 0 0.794182 + 1.37556i
289.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.593579 0 −0.0665372 2.64491i 1.00000 0 0.296790 + 0.514055i
289.3 −0.500000 0.866025i 0 −0.500000 + 0.866025i 3.18194 0 0.710533 + 2.54856i 1.00000 0 −1.59097 2.75564i
361.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.58836 0 −2.64400 0.0963576i 1.00000 0 0.794182 1.37556i
361.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.593579 0 −0.0665372 + 2.64491i 1.00000 0 0.296790 0.514055i
361.3 −0.500000 + 0.866025i 0 −0.500000 0.866025i 3.18194 0 0.710533 2.54856i 1.00000 0 −1.59097 + 2.75564i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g 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.h.c 6
3.b odd 2 1 126.2.h.d yes 6
4.b odd 2 1 3024.2.t.h 6
7.b odd 2 1 2646.2.h.o 6
7.c even 3 1 378.2.e.d 6
7.c even 3 1 2646.2.f.l 6
7.d odd 6 1 2646.2.e.p 6
7.d odd 6 1 2646.2.f.m 6
9.c even 3 1 378.2.e.d 6
9.c even 3 1 1134.2.g.l 6
9.d odd 6 1 126.2.e.c 6
9.d odd 6 1 1134.2.g.m 6
12.b even 2 1 1008.2.t.h 6
21.c even 2 1 882.2.h.p 6
21.g even 6 1 882.2.e.o 6
21.g even 6 1 882.2.f.o 6
21.h odd 6 1 126.2.e.c 6
21.h odd 6 1 882.2.f.n 6
28.g odd 6 1 3024.2.q.g 6
36.f odd 6 1 3024.2.q.g 6
36.h even 6 1 1008.2.q.g 6
63.g even 3 1 inner 378.2.h.c 6
63.g even 3 1 7938.2.a.ca 3
63.h even 3 1 1134.2.g.l 6
63.h even 3 1 2646.2.f.l 6
63.i even 6 1 882.2.f.o 6
63.j odd 6 1 882.2.f.n 6
63.j odd 6 1 1134.2.g.m 6
63.k odd 6 1 2646.2.h.o 6
63.k odd 6 1 7938.2.a.bz 3
63.l odd 6 1 2646.2.e.p 6
63.n odd 6 1 126.2.h.d yes 6
63.n odd 6 1 7938.2.a.bv 3
63.o even 6 1 882.2.e.o 6
63.s even 6 1 882.2.h.p 6
63.s even 6 1 7938.2.a.bw 3
63.t odd 6 1 2646.2.f.m 6
84.n even 6 1 1008.2.q.g 6
252.o even 6 1 1008.2.t.h 6
252.bl odd 6 1 3024.2.t.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 9.d odd 6 1
126.2.e.c 6 21.h odd 6 1
126.2.h.d yes 6 3.b odd 2 1
126.2.h.d yes 6 63.n odd 6 1
378.2.e.d 6 7.c even 3 1
378.2.e.d 6 9.c even 3 1
378.2.h.c 6 1.a even 1 1 trivial
378.2.h.c 6 63.g even 3 1 inner
882.2.e.o 6 21.g even 6 1
882.2.e.o 6 63.o even 6 1
882.2.f.n 6 21.h odd 6 1
882.2.f.n 6 63.j odd 6 1
882.2.f.o 6 21.g even 6 1
882.2.f.o 6 63.i even 6 1
882.2.h.p 6 21.c even 2 1
882.2.h.p 6 63.s even 6 1
1008.2.q.g 6 36.h even 6 1
1008.2.q.g 6 84.n even 6 1
1008.2.t.h 6 12.b even 2 1
1008.2.t.h 6 252.o even 6 1
1134.2.g.l 6 9.c even 3 1
1134.2.g.l 6 63.h even 3 1
1134.2.g.m 6 9.d odd 6 1
1134.2.g.m 6 63.j odd 6 1
2646.2.e.p 6 7.d odd 6 1
2646.2.e.p 6 63.l odd 6 1
2646.2.f.l 6 7.c even 3 1
2646.2.f.l 6 63.h even 3 1
2646.2.f.m 6 7.d odd 6 1
2646.2.f.m 6 63.t odd 6 1
2646.2.h.o 6 7.b odd 2 1
2646.2.h.o 6 63.k odd 6 1
3024.2.q.g 6 28.g odd 6 1
3024.2.q.g 6 36.f odd 6 1
3024.2.t.h 6 4.b odd 2 1
3024.2.t.h 6 252.bl odd 6 1
7938.2.a.bv 3 63.n odd 6 1
7938.2.a.bw 3 63.s even 6 1
7938.2.a.bz 3 63.k odd 6 1
7938.2.a.ca 3 63.g even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - T_{5}^{2} - 6 T_{5} - 3 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{3} \)
$3$ 1
$5$ \( ( 1 - T + 9 T^{2} - 13 T^{3} + 45 T^{4} - 25 T^{5} + 125 T^{6} )^{2} \)
$7$ \( 1 + 4 T + 14 T^{2} + 55 T^{3} + 98 T^{4} + 196 T^{5} + 343 T^{6} \)
$11$ \( ( 1 + T + 27 T^{2} + 25 T^{3} + 297 T^{4} + 121 T^{5} + 1331 T^{6} )^{2} \)
$13$ \( 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 18304 T^{7} - 5408 T^{8} - 92274 T^{9} + 685464 T^{10} - 2970344 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 4 T - 23 T^{2} + 68 T^{3} + 410 T^{4} - 220 T^{5} - 8111 T^{6} - 3740 T^{7} + 118490 T^{8} + 334084 T^{9} - 1920983 T^{10} - 5679428 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 1026 T^{7} - 55233 T^{8} - 459553 T^{9} - 1563852 T^{10} + 7428297 T^{11} + 47045881 T^{12} \)
$23$ \( ( 1 + 7 T + 81 T^{2} + 325 T^{3} + 1863 T^{4} + 3703 T^{5} + 12167 T^{6} )^{2} \)
$29$ \( 1 - 5 T + 4 T^{2} - 251 T^{3} + 197 T^{4} + 3418 T^{5} + 20293 T^{6} + 99122 T^{7} + 165677 T^{8} - 6121639 T^{9} + 2829124 T^{10} - 102555745 T^{11} + 594823321 T^{12} \)
$31$ \( 1 - 20 T + 186 T^{2} - 1398 T^{3} + 10342 T^{4} - 62234 T^{5} + 331987 T^{6} - 1929254 T^{7} + 9938662 T^{8} - 41647818 T^{9} + 171774906 T^{10} - 572583020 T^{11} + 887503681 T^{12} \)
$37$ \( ( 1 - 11 T + 37 T^{2} )^{3}( 1 + 10 T + 37 T^{2} )^{3} \)
$41$ \( 1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 33210 T^{7} + 7413210 T^{8} - 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12} \)
$43$ \( ( 1 - 12 T - 6 T^{2} + 547 T^{3} - 258 T^{4} - 22188 T^{5} + 79507 T^{6} )( 1 + 18 T + 198 T^{2} + 1519 T^{3} + 8514 T^{4} + 33282 T^{5} + 79507 T^{6} ) \)
$47$ \( 1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 161586 T^{7} - 5374497 T^{8} + 55130013 T^{9} - 29278086 T^{10} - 2064105063 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 15 T + 33 T^{3} + 13635 T^{4} + 60360 T^{5} - 225155 T^{6} + 3199080 T^{7} + 38300715 T^{8} + 4912941 T^{9} + 6272932395 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 14 T - 20 T^{2} + 154 T^{3} + 11666 T^{4} - 35126 T^{5} - 499301 T^{6} - 2072434 T^{7} + 40609346 T^{8} + 31628366 T^{9} - 242347220 T^{10} - 10008940186 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 8 T - 114 T^{2} + 342 T^{3} + 13762 T^{4} - 13214 T^{5} - 937217 T^{6} - 806054 T^{7} + 51208402 T^{8} + 77627502 T^{9} - 1578425874 T^{10} - 6756770408 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 - T - 88 T^{2} - 243 T^{3} + 2035 T^{4} + 14290 T^{5} + 72259 T^{6} + 957430 T^{7} + 9135115 T^{8} - 73085409 T^{9} - 1773298648 T^{10} - 1350125107 T^{11} + 90458382169 T^{12} \)
$71$ \( ( 1 + 7 T + 15 T^{2} - 599 T^{3} + 1065 T^{4} + 35287 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 3064978 T^{7} - 30689711 T^{8} - 10503459 T^{9} + 3805364294 T^{10} - 39388360267 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 - 5 T - 138 T^{2} + 123 T^{3} + 11347 T^{4} + 21118 T^{5} - 1048937 T^{6} + 1668322 T^{7} + 70816627 T^{8} + 60643797 T^{9} - 5375111178 T^{10} - 15385281995 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 + 2 T - 182 T^{2} + 2 T^{3} + 18788 T^{4} - 13564 T^{5} - 1721225 T^{6} - 1125812 T^{7} + 129430532 T^{8} + 1143574 T^{9} - 8637414422 T^{10} + 7878081286 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 9 T - 144 T^{2} + 1197 T^{3} + 16101 T^{4} - 73314 T^{5} - 1141967 T^{6} - 6524946 T^{7} + 127536021 T^{8} + 843847893 T^{9} - 9034882704 T^{10} - 50256535041 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 37655012 T^{7} + 425079802 T^{8} - 2486121252 T^{9} + 24876727961 T^{10} - 240445527196 T^{11} + 832972004929 T^{12} \)
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