Newspace parameters
Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 675.l (of order \(9\), degree \(6\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.38990213644\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{9})\) |
Coefficient field: | 12.0.1952986685049.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: |
\( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \)
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Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | no (minimal twist has level 27) |
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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu^{10} - 5 \nu^{9} + 22 \nu^{8} - 58 \nu^{7} + 127 \nu^{6} - 199 \nu^{5} + 249 \nu^{4} - 224 \nu^{3} + 145 \nu^{2} - 58 \nu + 9 \)
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\(\beta_{2}\) | \(=\) |
\( 3 \nu^{11} - 16 \nu^{10} + 71 \nu^{9} - 197 \nu^{8} + 445 \nu^{7} - 747 \nu^{6} + 1006 \nu^{5} - 1030 \nu^{4} + 803 \nu^{3} - 445 \nu^{2} + 155 \nu - 25 \)
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\(\beta_{3}\) | \(=\) |
\( - 6 \nu^{11} + 32 \nu^{10} - 140 \nu^{9} + 384 \nu^{8} - 849 \nu^{7} + 1390 \nu^{6} - 1805 \nu^{5} + 1762 \nu^{4} - 1285 \nu^{3} + 649 \nu^{2} - 195 \nu + 25 \)
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\(\beta_{4}\) | \(=\) |
\( - 9 \nu^{11} + 49 \nu^{10} - 216 \nu^{9} + 601 \nu^{8} - 1344 \nu^{7} + 2232 \nu^{6} - 2942 \nu^{5} + 2918 \nu^{4} - 2170 \nu^{3} + 1118 \nu^{2} - 348 \nu + 49 \)
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\(\beta_{5}\) | \(=\) |
\( 9 \nu^{11} - 50 \nu^{10} + 221 \nu^{9} - 623 \nu^{8} + 1402 \nu^{7} - 2360 \nu^{6} + 3144 \nu^{5} - 3178 \nu^{4} + 2411 \nu^{3} - 1286 \nu^{2} + 421 \nu - 62 \)
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\(\beta_{6}\) | \(=\) |
\( 11 \nu^{11} - 60 \nu^{10} + 265 \nu^{9} - 739 \nu^{8} + 1657 \nu^{7} - 2761 \nu^{6} + 3653 \nu^{5} - 3643 \nu^{4} + 2724 \nu^{3} - 1417 \nu^{2} + 442 \nu - 61 \)
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\(\beta_{7}\) | \(=\) |
\( - 16 \nu^{11} + 87 \nu^{10} - 383 \nu^{9} + 1064 \nu^{8} - 2375 \nu^{7} + 3936 \nu^{6} - 5176 \nu^{5} + 5122 \nu^{4} - 3802 \nu^{3} + 1958 \nu^{2} - 610 \nu + 85 \)
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\(\beta_{8}\) | \(=\) |
\( - 16 \nu^{11} + 89 \nu^{10} - 393 \nu^{9} + 1108 \nu^{8} - 2491 \nu^{7} + 4191 \nu^{6} - 5577 \nu^{5} + 5631 \nu^{4} - 4267 \nu^{3} + 2272 \nu^{2} - 742 \nu + 110 \)
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\(\beta_{9}\) | \(=\) |
\( 36 \nu^{11} - 198 \nu^{10} + 873 \nu^{9} - 2443 \nu^{8} + 5472 \nu^{7} - 9134 \nu^{6} + 12076 \nu^{5} - 12058 \nu^{4} + 9024 \nu^{3} - 4708 \nu^{2} + 1486 \nu - 209 \)
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\(\beta_{10}\) | \(=\) |
\( - 36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2444 \nu^{8} - 5476 \nu^{7} + 9150 \nu^{6} - 12110 \nu^{5} + 12120 \nu^{4} - 9096 \nu^{3} + 4772 \nu^{2} - 1519 \nu + 217 \)
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\(\beta_{11}\) | \(=\) |
\( - 42 \nu^{11} + 231 \nu^{10} - 1019 \nu^{9} + 2853 \nu^{8} - 6396 \nu^{7} + 10689 \nu^{6} - 14157 \nu^{5} + 14172 \nu^{4} - 10648 \nu^{3} + 5589 \nu^{2} - 1785 \nu + 257 \)
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\(\nu\) | \(=\) |
\( ( \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 3 ) / 3 \)
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\(\nu^{2}\) | \(=\) |
\( ( \beta_{11} - \beta_{10} + \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta _1 - 6 ) / 3 \)
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\(\nu^{3}\) | \(=\) |
\( ( - 5 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} + \beta_{8} - 8 \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + 10 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 8 \beta _1 - 18 ) / 3 \)
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\(\nu^{4}\) | \(=\) |
\( ( - 11 \beta_{11} + 17 \beta_{10} - 5 \beta_{9} - 20 \beta_{8} + 4 \beta_{7} + 16 \beta_{6} - 14 \beta_{5} + 4 \beta_{4} - 14 \beta_{3} - 8 \beta_{2} + \beta _1 + 6 ) / 3 \)
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\(\nu^{5}\) | \(=\) |
\( ( 19 \beta_{11} - \beta_{10} + 31 \beta_{9} - 32 \beta_{8} + 43 \beta_{7} - 20 \beta_{6} - 41 \beta_{5} - 44 \beta_{4} + 10 \beta_{3} + 25 \beta_{2} + 40 \beta _1 + 87 ) / 3 \)
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\(\nu^{6}\) | \(=\) |
\( ( 85 \beta_{11} - 97 \beta_{10} + 55 \beta_{9} + 64 \beta_{8} + 10 \beta_{7} - 101 \beta_{6} + 19 \beta_{5} - 62 \beta_{4} + 91 \beta_{3} + 70 \beta_{2} + 31 \beta _1 + 60 ) / 3 \)
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\(\nu^{7}\) | \(=\) |
\( ( - 20 \beta_{11} - 118 \beta_{10} - 134 \beta_{9} + 244 \beta_{8} - 218 \beta_{7} + \beta_{6} + 232 \beta_{5} + 157 \beta_{4} + 52 \beta_{3} - 74 \beta_{2} - 179 \beta _1 - 357 ) / 3 \)
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\(\nu^{8}\) | \(=\) |
\( ( - 503 \beta_{11} + 386 \beta_{10} - 440 \beta_{9} - 47 \beta_{8} - 233 \beta_{7} + 514 \beta_{6} + 163 \beta_{5} + 466 \beta_{4} - 431 \beta_{3} - 461 \beta_{2} - 329 \beta _1 - 639 ) / 3 \)
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\(\nu^{9}\) | \(=\) |
\( ( - 425 \beta_{11} + 1076 \beta_{10} + 319 \beta_{9} - 1313 \beta_{8} + 955 \beta_{7} + 502 \beta_{6} - 1013 \beta_{5} - 332 \beta_{4} - 743 \beta_{3} - 59 \beta_{2} + 631 \beta _1 + 1164 ) / 3 \)
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\(\nu^{10}\) | \(=\) |
\( ( 2299 \beta_{11} - 862 \beta_{10} + 2725 \beta_{9} - 1193 \beta_{8} + 2104 \beta_{7} - 2135 \beta_{6} - 1907 \beta_{5} - 2705 \beta_{4} + 1495 \beta_{3} + 2425 \beta_{2} + 2344 \beta _1 + 4356 ) / 3 \)
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\(\nu^{11}\) | \(=\) |
\( ( 4708 \beta_{11} - 6628 \beta_{10} + 985 \beta_{9} + 5506 \beta_{8} - 3107 \beta_{7} - 4679 \beta_{6} + 3238 \beta_{5} - 992 \beta_{4} + 5476 \beta_{3} + 2770 \beta_{2} - 1043 \beta _1 - 1698 ) / 3 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).
\(n\) | \(326\) | \(352\) |
\(\chi(n)\) | \(\beta_{10}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
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76.1 |
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0.318266 | − | 0.267057i | −0.159815 | + | 1.72466i | −0.317323 | + | 1.79963i | 0 | 0.409719 | + | 0.591580i | 0.229151 | + | 1.29958i | 0.795075 | + | 1.37711i | −2.94892 | − | 0.551252i | 0 | ||||||||||||||||||||||||||||||||||||||||
76.2 | 1.62143 | − | 1.36054i | 0.986166 | − | 1.42389i | 0.430663 | − | 2.44241i | 0 | −0.338267 | − | 3.65046i | −0.168844 | − | 0.957561i | −0.508086 | − | 0.880031i | −1.05495 | − | 2.80839i | 0 | |||||||||||||||||||||||||||||||||||||||||
151.1 | 0.318266 | + | 0.267057i | −0.159815 | − | 1.72466i | −0.317323 | − | 1.79963i | 0 | 0.409719 | − | 0.591580i | 0.229151 | − | 1.29958i | 0.795075 | − | 1.37711i | −2.94892 | + | 0.551252i | 0 | |||||||||||||||||||||||||||||||||||||||||
151.2 | 1.62143 | + | 1.36054i | 0.986166 | + | 1.42389i | 0.430663 | + | 2.44241i | 0 | −0.338267 | + | 3.65046i | −0.168844 | + | 0.957561i | −0.508086 | + | 0.880031i | −1.05495 | + | 2.80839i | 0 | |||||||||||||||||||||||||||||||||||||||||
301.1 | −0.753189 | − | 0.274138i | 1.68842 | + | 0.386327i | −1.03995 | − | 0.872619i | 0 | −1.16579 | − | 0.753837i | −1.82076 | + | 1.52780i | 1.34559 | + | 2.33062i | 2.70150 | + | 1.30456i | 0 | |||||||||||||||||||||||||||||||||||||||||
301.2 | 1.57954 | + | 0.574906i | −1.45446 | + | 0.940501i | 0.632343 | + | 0.530599i | 0 | −2.83808 | + | 0.649381i | 2.99441 | − | 2.51261i | −0.987144 | − | 1.70978i | 1.23092 | − | 2.73584i | 0 | |||||||||||||||||||||||||||||||||||||||||
376.1 | −0.183082 | + | 1.03831i | 1.72962 | − | 0.0916693i | 0.834822 | + | 0.303850i | 0 | −0.221481 | + | 1.81266i | 2.31094 | − | 0.841112i | −1.52266 | + | 2.63732i | 2.98319 | − | 0.317107i | 0 | |||||||||||||||||||||||||||||||||||||||||
376.2 | 0.417037 | − | 2.36514i | 0.210069 | + | 1.71926i | −3.54056 | − | 1.28866i | 0 | 4.15390 | + | 0.220155i | −0.544891 | + | 0.198324i | −2.12277 | + | 3.67675i | −2.91174 | + | 0.722330i | 0 | |||||||||||||||||||||||||||||||||||||||||
526.1 | −0.183082 | − | 1.03831i | 1.72962 | + | 0.0916693i | 0.834822 | − | 0.303850i | 0 | −0.221481 | − | 1.81266i | 2.31094 | + | 0.841112i | −1.52266 | − | 2.63732i | 2.98319 | + | 0.317107i | 0 | |||||||||||||||||||||||||||||||||||||||||
526.2 | 0.417037 | + | 2.36514i | 0.210069 | − | 1.71926i | −3.54056 | + | 1.28866i | 0 | 4.15390 | − | 0.220155i | −0.544891 | − | 0.198324i | −2.12277 | − | 3.67675i | −2.91174 | − | 0.722330i | 0 | |||||||||||||||||||||||||||||||||||||||||
601.1 | −0.753189 | + | 0.274138i | 1.68842 | − | 0.386327i | −1.03995 | + | 0.872619i | 0 | −1.16579 | + | 0.753837i | −1.82076 | − | 1.52780i | 1.34559 | − | 2.33062i | 2.70150 | − | 1.30456i | 0 | |||||||||||||||||||||||||||||||||||||||||
601.2 | 1.57954 | − | 0.574906i | −1.45446 | − | 0.940501i | 0.632343 | − | 0.530599i | 0 | −2.83808 | − | 0.649381i | 2.99441 | + | 2.51261i | −0.987144 | + | 1.70978i | 1.23092 | + | 2.73584i | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
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1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
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Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 675.2.l.c | 12 | |
5.b | even | 2 | 1 | 27.2.e.a | ✓ | 12 | |
5.c | odd | 4 | 2 | 675.2.u.b | 24 | ||
15.d | odd | 2 | 1 | 81.2.e.a | 12 | ||
20.d | odd | 2 | 1 | 432.2.u.c | 12 | ||
27.e | even | 9 | 1 | inner | 675.2.l.c | 12 | |
45.h | odd | 6 | 1 | 243.2.e.a | 12 | ||
45.h | odd | 6 | 1 | 243.2.e.b | 12 | ||
45.j | even | 6 | 1 | 243.2.e.c | 12 | ||
45.j | even | 6 | 1 | 243.2.e.d | 12 | ||
135.n | odd | 18 | 1 | 81.2.e.a | 12 | ||
135.n | odd | 18 | 1 | 243.2.e.a | 12 | ||
135.n | odd | 18 | 1 | 243.2.e.b | 12 | ||
135.n | odd | 18 | 1 | 729.2.a.d | 6 | ||
135.n | odd | 18 | 2 | 729.2.c.b | 12 | ||
135.p | even | 18 | 1 | 27.2.e.a | ✓ | 12 | |
135.p | even | 18 | 1 | 243.2.e.c | 12 | ||
135.p | even | 18 | 1 | 243.2.e.d | 12 | ||
135.p | even | 18 | 1 | 729.2.a.a | 6 | ||
135.p | even | 18 | 2 | 729.2.c.e | 12 | ||
135.r | odd | 36 | 2 | 675.2.u.b | 24 | ||
540.bf | odd | 18 | 1 | 432.2.u.c | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
27.2.e.a | ✓ | 12 | 5.b | even | 2 | 1 | |
27.2.e.a | ✓ | 12 | 135.p | even | 18 | 1 | |
81.2.e.a | 12 | 15.d | odd | 2 | 1 | ||
81.2.e.a | 12 | 135.n | odd | 18 | 1 | ||
243.2.e.a | 12 | 45.h | odd | 6 | 1 | ||
243.2.e.a | 12 | 135.n | odd | 18 | 1 | ||
243.2.e.b | 12 | 45.h | odd | 6 | 1 | ||
243.2.e.b | 12 | 135.n | odd | 18 | 1 | ||
243.2.e.c | 12 | 45.j | even | 6 | 1 | ||
243.2.e.c | 12 | 135.p | even | 18 | 1 | ||
243.2.e.d | 12 | 45.j | even | 6 | 1 | ||
243.2.e.d | 12 | 135.p | even | 18 | 1 | ||
432.2.u.c | 12 | 20.d | odd | 2 | 1 | ||
432.2.u.c | 12 | 540.bf | odd | 18 | 1 | ||
675.2.l.c | 12 | 1.a | even | 1 | 1 | trivial | |
675.2.l.c | 12 | 27.e | even | 9 | 1 | inner | |
675.2.u.b | 24 | 5.c | odd | 4 | 2 | ||
675.2.u.b | 24 | 135.r | odd | 36 | 2 | ||
729.2.a.a | 6 | 135.p | even | 18 | 1 | ||
729.2.a.d | 6 | 135.n | odd | 18 | 1 | ||
729.2.c.b | 12 | 135.n | odd | 18 | 2 | ||
729.2.c.e | 12 | 135.p | even | 18 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 6 T_{2}^{11} + 21 T_{2}^{10} - 48 T_{2}^{9} + 72 T_{2}^{8} - 54 T_{2}^{7} + 6 T_{2}^{6} + 9 T_{2}^{5} - 18 T_{2}^{4} + 45 T_{2}^{3} + 27 T_{2}^{2} - 27 T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - 6 T^{11} + 21 T^{10} - 48 T^{9} + \cdots + 9 \)
$3$
\( T^{12} - 6 T^{11} + 18 T^{10} - 39 T^{9} + \cdots + 729 \)
$5$
\( T^{12} \)
$7$
\( T^{12} - 6 T^{11} + 12 T^{10} + 11 T^{9} + \cdots + 289 \)
$11$
\( T^{12} - 3 T^{11} - 15 T^{10} - 6 T^{9} + \cdots + 9 \)
$13$
\( T^{12} - 6 T^{11} + 48 T^{10} - 214 T^{9} + \cdots + 1 \)
$17$
\( T^{12} + 9 T^{11} + 72 T^{10} + 189 T^{9} + \cdots + 729 \)
$19$
\( T^{12} + 3 T^{11} + 39 T^{10} - 14 T^{9} + \cdots + 361 \)
$23$
\( T^{12} - 12 T^{11} + 48 T^{10} + \cdots + 106929 \)
$29$
\( T^{12} + 6 T^{11} + 21 T^{10} + \cdots + 45369 \)
$31$
\( T^{12} - 3 T^{11} + 84 T^{10} + \cdots + 26569 \)
$37$
\( T^{12} - 3 T^{11} + 66 T^{10} + \cdots + 24334489 \)
$41$
\( T^{12} - 15 T^{11} + 93 T^{10} + \cdots + 11229201 \)
$43$
\( T^{12} + 3 T^{11} - 60 T^{10} + \cdots + 3308761 \)
$47$
\( T^{12} - 15 T^{11} + 111 T^{10} + \cdots + 42732369 \)
$53$
\( (T^{6} - 9 T^{5} - 108 T^{4} + 513 T^{3} + \cdots - 12393)^{2} \)
$59$
\( T^{12} + 12 T^{11} + \cdots + 176384961 \)
$61$
\( T^{12} - 12 T^{11} + \cdots + 273670849 \)
$67$
\( T^{12} - 15 T^{11} + 255 T^{10} + \cdots + 8288641 \)
$71$
\( T^{12} - 27 T^{11} + 504 T^{10} + \cdots + 729 \)
$73$
\( T^{12} + 6 T^{11} + 210 T^{10} + \cdots + 185761 \)
$79$
\( T^{12} + 42 T^{11} + 813 T^{10} + \cdots + 3508129 \)
$83$
\( T^{12} + 39 T^{11} + \cdots + 6951057129 \)
$89$
\( T^{12} - 9 T^{11} + \cdots + 1062042921 \)
$97$
\( T^{12} + 3 T^{11} + 102 T^{10} + \cdots + 66765241 \)
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