Properties

Label 378.2.u.b
Level $378$
Weight $2$
Character orbit 378.u
Analytic conductor $3.018$
Analytic rank $0$
Dimension $12$
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.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: 12.0.1952986685049.1
Defining polynomial: \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + ( - \beta_{11} - \beta_{10} + \beta_{5} + \beta_{4} - \beta_{2}) q^{3} + \beta_{9} q^{4} + ( - \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} - \beta_1 - 1) q^{5} + (\beta_{10} + \beta_{8} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_1 - 2) q^{6} + \beta_{10} q^{7} + (\beta_{11} - 1) q^{8} + (2 \beta_{11} + \beta_{10} + 2 \beta_{7} + \beta_{6} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + ( - \beta_{11} - \beta_{10} + \beta_{5} + \beta_{4} - \beta_{2}) q^{3} + \beta_{9} q^{4} + ( - \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} - \beta_1 - 1) q^{5} + (\beta_{10} + \beta_{8} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_1 - 2) q^{6} + \beta_{10} q^{7} + (\beta_{11} - 1) q^{8} + (2 \beta_{11} + \beta_{10} + 2 \beta_{7} + \beta_{6} + \beta_1) q^{9} + ( - \beta_{11} + \beta_{4} - \beta_{2}) q^{10} + ( - \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{5} - \beta_{3} + \beta_1 + 1) q^{11} + ( - \beta_{8} + \beta_{6} - \beta_{2} - 1) q^{12} + ( - 2 \beta_{11} + 3 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{3} + \cdots + 1) q^{13}+ \cdots + (\beta_{11} + 2 \beta_{9} - 4 \beta_{8} + 3 \beta_{7} - \beta_{6} - 5 \beta_{5} - 3 \beta_{4} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 6 q^{6} - 6 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{3} - 6 q^{6} - 6 q^{8} + 9 q^{9} + 3 q^{11} - 6 q^{12} + 12 q^{13} + 9 q^{15} - 9 q^{18} + 12 q^{19} - 9 q^{20} + 3 q^{21} + 3 q^{22} + 12 q^{23} + 3 q^{24} - 24 q^{25} - 18 q^{26} + 18 q^{27} + 12 q^{28} - 3 q^{29} + 21 q^{31} - 27 q^{33} + 9 q^{34} + 15 q^{37} + 15 q^{38} - 48 q^{39} + 9 q^{40} + 12 q^{41} + 3 q^{42} + 21 q^{43} + 12 q^{44} - 9 q^{45} - 6 q^{46} + 12 q^{47} + 3 q^{48} + 21 q^{50} - 9 q^{51} + 12 q^{52} - 42 q^{53} - 27 q^{54} - 12 q^{55} + 33 q^{57} - 3 q^{58} + 18 q^{59} - 18 q^{60} + 9 q^{62} + 9 q^{63} - 6 q^{64} - 36 q^{65} - 9 q^{66} - 33 q^{67} - 18 q^{68} + 9 q^{69} + 24 q^{71} - 18 q^{72} + 9 q^{73} - 36 q^{74} - 33 q^{75} + 15 q^{76} + 3 q^{77} - 21 q^{78} + 9 q^{81} - 24 q^{82} + 15 q^{83} + 3 q^{84} - 24 q^{85} + 21 q^{86} + 3 q^{88} - 51 q^{89} + 9 q^{91} - 6 q^{92} - 30 q^{93} - 15 q^{94} - 45 q^{95} + 3 q^{96} + 48 q^{97} - 6 q^{98} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{10} - 5 \nu^{9} + 22 \nu^{8} - 58 \nu^{7} + 127 \nu^{6} - 199 \nu^{5} + 249 \nu^{4} - 224 \nu^{3} + 145 \nu^{2} - 58 \nu + 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3 \nu^{11} - 16 \nu^{10} + 71 \nu^{9} - 197 \nu^{8} + 445 \nu^{7} - 747 \nu^{6} + 1006 \nu^{5} - 1030 \nu^{4} + 803 \nu^{3} - 445 \nu^{2} + 155 \nu - 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( - 6 \nu^{11} + 32 \nu^{10} - 140 \nu^{9} + 384 \nu^{8} - 849 \nu^{7} + 1390 \nu^{6} - 1805 \nu^{5} + 1762 \nu^{4} - 1285 \nu^{3} + 649 \nu^{2} - 195 \nu + 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( - 9 \nu^{11} + 49 \nu^{10} - 216 \nu^{9} + 601 \nu^{8} - 1344 \nu^{7} + 2232 \nu^{6} - 2942 \nu^{5} + 2918 \nu^{4} - 2170 \nu^{3} + 1118 \nu^{2} - 348 \nu + 49 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 9 \nu^{11} - 50 \nu^{10} + 221 \nu^{9} - 623 \nu^{8} + 1402 \nu^{7} - 2360 \nu^{6} + 3144 \nu^{5} - 3178 \nu^{4} + 2411 \nu^{3} - 1286 \nu^{2} + 421 \nu - 62 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 11 \nu^{11} - 60 \nu^{10} + 265 \nu^{9} - 739 \nu^{8} + 1657 \nu^{7} - 2761 \nu^{6} + 3653 \nu^{5} - 3643 \nu^{4} + 2724 \nu^{3} - 1417 \nu^{2} + 442 \nu - 61 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( - 16 \nu^{11} + 87 \nu^{10} - 383 \nu^{9} + 1064 \nu^{8} - 2375 \nu^{7} + 3936 \nu^{6} - 5176 \nu^{5} + 5122 \nu^{4} - 3802 \nu^{3} + 1958 \nu^{2} - 610 \nu + 85 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( - 16 \nu^{11} + 89 \nu^{10} - 393 \nu^{9} + 1108 \nu^{8} - 2491 \nu^{7} + 4191 \nu^{6} - 5577 \nu^{5} + 5631 \nu^{4} - 4267 \nu^{3} + 2272 \nu^{2} - 742 \nu + 110 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 36 \nu^{11} - 198 \nu^{10} + 873 \nu^{9} - 2443 \nu^{8} + 5472 \nu^{7} - 9134 \nu^{6} + 12076 \nu^{5} - 12058 \nu^{4} + 9024 \nu^{3} - 4708 \nu^{2} + 1486 \nu - 209 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( - 36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2444 \nu^{8} - 5476 \nu^{7} + 9150 \nu^{6} - 12110 \nu^{5} + 12120 \nu^{4} - 9096 \nu^{3} + 4772 \nu^{2} - 1519 \nu + 217 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( - 42 \nu^{11} + 231 \nu^{10} - 1019 \nu^{9} + 2853 \nu^{8} - 6396 \nu^{7} + 10689 \nu^{6} - 14157 \nu^{5} + 14172 \nu^{4} - 10648 \nu^{3} + 5589 \nu^{2} - 1785 \nu + 257 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} + \beta_{8} - 8 \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + 10 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 8 \beta _1 - 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11 \beta_{11} + 17 \beta_{10} - 5 \beta_{9} - 20 \beta_{8} + 4 \beta_{7} + 16 \beta_{6} - 14 \beta_{5} + 4 \beta_{4} - 14 \beta_{3} - 8 \beta_{2} + \beta _1 + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19 \beta_{11} - \beta_{10} + 31 \beta_{9} - 32 \beta_{8} + 43 \beta_{7} - 20 \beta_{6} - 41 \beta_{5} - 44 \beta_{4} + 10 \beta_{3} + 25 \beta_{2} + 40 \beta _1 + 87 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 85 \beta_{11} - 97 \beta_{10} + 55 \beta_{9} + 64 \beta_{8} + 10 \beta_{7} - 101 \beta_{6} + 19 \beta_{5} - 62 \beta_{4} + 91 \beta_{3} + 70 \beta_{2} + 31 \beta _1 + 60 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 20 \beta_{11} - 118 \beta_{10} - 134 \beta_{9} + 244 \beta_{8} - 218 \beta_{7} + \beta_{6} + 232 \beta_{5} + 157 \beta_{4} + 52 \beta_{3} - 74 \beta_{2} - 179 \beta _1 - 357 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 503 \beta_{11} + 386 \beta_{10} - 440 \beta_{9} - 47 \beta_{8} - 233 \beta_{7} + 514 \beta_{6} + 163 \beta_{5} + 466 \beta_{4} - 431 \beta_{3} - 461 \beta_{2} - 329 \beta _1 - 639 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 425 \beta_{11} + 1076 \beta_{10} + 319 \beta_{9} - 1313 \beta_{8} + 955 \beta_{7} + 502 \beta_{6} - 1013 \beta_{5} - 332 \beta_{4} - 743 \beta_{3} - 59 \beta_{2} + 631 \beta _1 + 1164 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2299 \beta_{11} - 862 \beta_{10} + 2725 \beta_{9} - 1193 \beta_{8} + 2104 \beta_{7} - 2135 \beta_{6} - 1907 \beta_{5} - 2705 \beta_{4} + 1495 \beta_{3} + 2425 \beta_{2} + 2344 \beta _1 + 4356 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4708 \beta_{11} - 6628 \beta_{10} + 985 \beta_{9} + 5506 \beta_{8} - 3107 \beta_{7} - 4679 \beta_{6} + 3238 \beta_{5} - 992 \beta_{4} + 5476 \beta_{3} + 2770 \beta_{2} - 1043 \beta _1 - 1698 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(\beta_{9}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.500000 + 2.22827i
0.500000 0.258654i
0.500000 + 0.0126039i
0.500000 + 1.27297i
0.500000 0.0126039i
0.500000 1.27297i
0.500000 2.22827i
0.500000 + 0.258654i
0.500000 1.68614i
0.500000 + 1.00210i
0.500000 + 1.68614i
0.500000 1.00210i
0.766044 + 0.642788i −1.67071 0.456871i 0.173648 + 0.984808i −1.78124 0.648319i −0.986166 1.42389i 0.173648 0.984808i −0.500000 + 0.866025i 2.58254 + 1.52660i −0.947779 1.64160i
43.2 0.766044 + 0.642788i 1.23102 + 1.21844i 0.173648 + 0.984808i 0.667901 + 0.243096i 0.159815 + 1.72466i 0.173648 0.984808i −0.500000 + 0.866025i 0.0308031 + 2.99984i 0.355383 + 0.615541i
85.1 −0.939693 0.342020i −1.04508 + 1.38124i 0.766044 + 0.642788i 0.108876 0.617467i 1.45446 0.940501i 0.766044 0.642788i −0.500000 0.866025i −0.815631 2.88700i −0.313496 + 0.542992i
85.2 −0.939693 0.342020i 1.71872 0.214444i 0.766044 + 0.642788i −0.701272 + 3.97711i −1.68842 0.386327i 0.766044 0.642788i −0.500000 0.866025i 2.90803 0.737141i 2.01923 3.49741i
169.1 −0.939693 + 0.342020i −1.04508 1.38124i 0.766044 0.642788i 0.108876 + 0.617467i 1.45446 + 0.940501i 0.766044 + 0.642788i −0.500000 + 0.866025i −0.815631 + 2.88700i −0.313496 0.542992i
169.2 −0.939693 + 0.342020i 1.71872 + 0.214444i 0.766044 0.642788i −0.701272 3.97711i −1.68842 + 0.386327i 0.766044 + 0.642788i −0.500000 + 0.866025i 2.90803 + 0.737141i 2.01923 + 3.49741i
211.1 0.766044 0.642788i −1.67071 + 0.456871i 0.173648 0.984808i −1.78124 + 0.648319i −0.986166 + 1.42389i 0.173648 + 0.984808i −0.500000 0.866025i 2.58254 1.52660i −0.947779 + 1.64160i
211.2 0.766044 0.642788i 1.23102 1.21844i 0.173648 0.984808i 0.667901 0.243096i 0.159815 1.72466i 0.173648 + 0.984808i −0.500000 0.866025i 0.0308031 2.99984i 0.355383 0.615541i
295.1 0.173648 0.984808i −0.390623 1.68743i −0.939693 0.342020i 0.393151 0.329893i −1.72962 + 0.0916693i −0.939693 + 0.342020i −0.500000 + 0.866025i −2.69483 + 1.31830i −0.256611 0.444463i
295.2 0.173648 0.984808i 1.65667 0.505425i −0.939693 0.342020i 1.31259 1.10139i −0.210069 1.71926i −0.939693 + 0.342020i −0.500000 + 0.866025i 2.48909 1.67464i −0.856730 1.48390i
337.1 0.173648 + 0.984808i −0.390623 + 1.68743i −0.939693 + 0.342020i 0.393151 + 0.329893i −1.72962 0.0916693i −0.939693 0.342020i −0.500000 0.866025i −2.69483 1.31830i −0.256611 + 0.444463i
337.2 0.173648 + 0.984808i 1.65667 + 0.505425i −0.939693 + 0.342020i 1.31259 + 1.10139i −0.210069 + 1.71926i −0.939693 0.342020i −0.500000 0.866025i 2.48909 + 1.67464i −0.856730 + 1.48390i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.u.b 12
27.e even 9 1 inner 378.2.u.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.u.b 12 1.a even 1 1 trivial
378.2.u.b 12 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 12 T_{5}^{10} - 18 T_{5}^{9} - 27 T_{5}^{8} + 117 T_{5}^{7} - 3 T_{5}^{6} - 243 T_{5}^{5} + 360 T_{5}^{4} - 297 T_{5}^{3} + 162 T_{5}^{2} - 54 T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 3 T^{11} + 3 T^{9} + 9 T^{8} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 12 T^{10} - 18 T^{9} - 27 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{11} - 15 T^{10} - 6 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{12} - 12 T^{11} + 93 T^{10} + \cdots + 5041 \) Copy content Toggle raw display
$17$ \( T^{12} + 57 T^{10} - 18 T^{9} + \cdots + 3249 \) Copy content Toggle raw display
$19$ \( T^{12} - 12 T^{11} + 126 T^{10} + \cdots + 7284601 \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{11} + 57 T^{10} - 78 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{12} + 3 T^{11} + 30 T^{10} + \cdots + 3249 \) Copy content Toggle raw display
$31$ \( T^{12} - 21 T^{11} + 255 T^{10} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( T^{12} - 15 T^{11} + \cdots + 1506138481 \) Copy content Toggle raw display
$41$ \( T^{12} - 12 T^{11} + 39 T^{10} + \cdots + 25735329 \) Copy content Toggle raw display
$43$ \( T^{12} - 21 T^{11} + \cdots + 248094001 \) Copy content Toggle raw display
$47$ \( T^{12} - 12 T^{11} - 72 T^{10} + \cdots + 23409 \) Copy content Toggle raw display
$53$ \( (T^{6} + 21 T^{5} + 132 T^{4} + 303 T^{3} + \cdots - 537)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 18 T^{11} + \cdots + 4766383521 \) Copy content Toggle raw display
$61$ \( T^{12} + 189 T^{10} + \cdots + 18342097489 \) Copy content Toggle raw display
$67$ \( T^{12} + 33 T^{11} + \cdots + 7307343289 \) Copy content Toggle raw display
$71$ \( T^{12} - 24 T^{11} + \cdots + 2961862929 \) Copy content Toggle raw display
$73$ \( T^{12} - 9 T^{11} + \cdots + 103800596761 \) Copy content Toggle raw display
$79$ \( T^{12} - 135 T^{10} + \cdots + 9204675481 \) Copy content Toggle raw display
$83$ \( T^{12} - 15 T^{11} + \cdots + 1232220609 \) Copy content Toggle raw display
$89$ \( T^{12} + 51 T^{11} + \cdots + 252147592449 \) Copy content Toggle raw display
$97$ \( T^{12} - 48 T^{11} + \cdots + 12171164329 \) Copy content Toggle raw display
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