Newspace parameters
Level: | \( N \) | \(=\) | \( 237 = 3 \cdot 79 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 237.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.89245452790\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 2x^{11} + 11x^{10} + 49x^{8} + 5x^{7} + 137x^{6} + 76x^{5} + 189x^{4} + 14x^{3} + 46x^{2} + 6x + 9 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} - 2x^{11} + 11x^{10} + 49x^{8} + 5x^{7} + 137x^{6} + 76x^{5} + 189x^{4} + 14x^{3} + 46x^{2} + 6x + 9 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 5295523 \nu^{11} + 13452272 \nu^{10} - 22915263 \nu^{9} + 345399173 \nu^{8} - 114273838 \nu^{7} + 1280218292 \nu^{6} - 434295829 \nu^{5} + \cdots + 575507286 ) / 468655901 \) |
\(\beta_{3}\) | \(=\) | \( ( - 8262330 \nu^{11} + 12040263 \nu^{10} - 83511323 \nu^{9} - 40560107 \nu^{8} - 429951151 \nu^{7} - 212803159 \nu^{6} - 1200935436 \nu^{5} + \cdots + 1223523408 ) / 468655901 \) |
\(\beta_{4}\) | \(=\) | \( ( - 27445483 \nu^{11} + 88260248 \nu^{10} - 360056585 \nu^{9} + 345026619 \nu^{8} - 1250336017 \nu^{7} + 1470679732 \nu^{6} + \cdots + 96593619 ) / 1405967703 \) |
\(\beta_{5}\) | \(=\) | \( ( 32855483 \nu^{11} - 132362092 \nu^{10} + 503940199 \nu^{9} - 738156510 \nu^{8} + 1668143813 \nu^{7} - 2994838769 \nu^{6} + \cdots - 2481716196 ) / 1405967703 \) |
\(\beta_{6}\) | \(=\) | \( ( - 11123094 \nu^{11} + 19385424 \nu^{10} - 115008873 \nu^{9} - 31497550 \nu^{8} - 535969049 \nu^{7} - 161633368 \nu^{6} - 1472694087 \nu^{5} + \cdots - 82336449 ) / 468655901 \) |
\(\beta_{7}\) | \(=\) | \( ( 43756787 \nu^{11} - 110676361 \nu^{10} + 518198233 \nu^{9} - 301054383 \nu^{8} + 2135773451 \nu^{7} - 1399337192 \nu^{6} + \cdots - 1872430146 ) / 1405967703 \) |
\(\beta_{8}\) | \(=\) | \( ( - 16322389 \nu^{11} + 68874824 \nu^{10} - 245047712 \nu^{9} + 376524169 \nu^{8} - 714366968 \nu^{7} + 1632313100 \nu^{6} - 1802436980 \nu^{5} + \cdots + 178930068 ) / 468655901 \) |
\(\beta_{9}\) | \(=\) | \( ( - 150127913 \nu^{11} + 350901874 \nu^{10} - 1728005356 \nu^{9} + 509053902 \nu^{8} - 7127972642 \nu^{7} + 1847157203 \nu^{6} + \cdots - 470600175 ) / 1405967703 \) |
\(\beta_{10}\) | \(=\) | \( ( 66912061 \nu^{11} - 158678566 \nu^{10} + 796372108 \nu^{9} - 281923803 \nu^{8} + 3332143262 \nu^{7} - 633415025 \nu^{6} + 9115512771 \nu^{5} + \cdots + 255939213 ) / 468655901 \) |
\(\beta_{11}\) | \(=\) | \( ( 207104953 \nu^{11} - 314859584 \nu^{10} + 2061015383 \nu^{9} + 1096305378 \nu^{8} + 10109466199 \nu^{7} + 5229304208 \nu^{6} + \cdots + 1893702039 ) / 1405967703 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{8} - \beta_{6} + 3\beta_{4} + \beta_1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{9} - 5\beta_{6} + \beta_{5} + 2\beta_{3} - 4 \) |
\(\nu^{4}\) | \(=\) | \( 9\beta_{8} + \beta_{7} + 3\beta_{5} - 19\beta_{4} + 9\beta_{3} - 10\beta _1 - 19 \) |
\(\nu^{5}\) | \(=\) | \( -3\beta_{11} - \beta_{10} - 13\beta_{9} + 25\beta_{8} + 35\beta_{6} - 46\beta_{4} - 35\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( -13\beta_{11} - 2\beta_{10} - 40\beta_{9} - 13\beta_{7} + 93\beta_{6} - 40\beta_{5} - 85\beta_{3} - 2\beta_{2} + 158 \) |
\(\nu^{7}\) | \(=\) | \( -256\beta_{8} - 40\beta_{7} - 136\beta_{5} + 455\beta_{4} - 256\beta_{3} - 11\beta_{2} + 296\beta _1 + 455 \) |
\(\nu^{8}\) | \(=\) | \( 136\beta_{11} + 29\beta_{10} + 421\beta_{9} - 813\beta_{8} - 871\beta_{6} + 1445\beta_{4} + 871\beta_1 \) |
\(\nu^{9}\) | \(=\) | \( 421 \beta_{11} + 107 \beta_{10} + 1341 \beta_{9} + 421 \beta_{7} - 2708 \beta_{6} + 1341 \beta_{5} + 2497 \beta_{3} + 107 \beta_{2} - 4375 \) |
\(\nu^{10}\) | \(=\) | \( 7780 \beta_{8} + 1341 \beta_{7} + 4152 \beta_{5} - 13617 \beta_{4} + 7780 \beta_{3} + 314 \beta_{2} - 8239 \beta _1 - 13617 \) |
\(\nu^{11}\) | \(=\) | \( - 4152 \beta_{11} - 1027 \beta_{10} - 12959 \beta_{9} + 24009 \beta_{8} + 25484 \beta_{6} - 41828 \beta_{4} - 25484 \beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/237\mathbb{Z}\right)^\times\).
\(n\) | \(80\) | \(82\) |
\(\chi(n)\) | \(1\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
55.1 |
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−1.04518 | + | 1.81031i | 0.500000 | − | 0.866025i | −1.18482 | − | 2.05217i | −0.929299 | − | 1.60959i | 1.04518 | + | 1.81031i | −0.641869 | − | 1.11175i | 0.772692 | −0.500000 | − | 0.866025i | 3.88515 | ||||||||||||||||||||||||||||||||||||||||
55.2 | −0.428387 | + | 0.741988i | 0.500000 | − | 0.866025i | 0.632970 | + | 1.09634i | 0.782924 | + | 1.35606i | 0.428387 | + | 0.741988i | 1.24913 | + | 2.16356i | −2.79817 | −0.500000 | − | 0.866025i | −1.34158 | |||||||||||||||||||||||||||||||||||||||||
55.3 | 0.245923 | − | 0.425951i | 0.500000 | − | 0.866025i | 0.879044 | + | 1.52255i | 0.715982 | + | 1.24012i | −0.245923 | − | 0.425951i | −0.761138 | − | 1.31833i | 1.84840 | −0.500000 | − | 0.866025i | 0.704305 | |||||||||||||||||||||||||||||||||||||||||
55.4 | 0.733732 | − | 1.27086i | 0.500000 | − | 0.866025i | −0.0767255 | − | 0.132892i | −1.45453 | − | 2.51933i | −0.733732 | − | 1.27086i | 0.0229355 | + | 0.0397254i | 2.70974 | −0.500000 | − | 0.866025i | −4.26895 | |||||||||||||||||||||||||||||||||||||||||
55.5 | 1.15914 | − | 2.00769i | 0.500000 | − | 0.866025i | −1.68722 | − | 2.92235i | 1.95463 | + | 3.38552i | −1.15914 | − | 2.00769i | −2.04868 | − | 3.54841i | −3.18634 | −0.500000 | − | 0.866025i | 9.06277 | |||||||||||||||||||||||||||||||||||||||||
55.6 | 1.33477 | − | 2.31190i | 0.500000 | − | 0.866025i | −2.56325 | − | 4.43967i | −0.569703 | − | 0.986755i | −1.33477 | − | 2.31190i | 2.17962 | + | 3.77521i | −8.34632 | −0.500000 | − | 0.866025i | −3.04170 | |||||||||||||||||||||||||||||||||||||||||
181.1 | −1.04518 | − | 1.81031i | 0.500000 | + | 0.866025i | −1.18482 | + | 2.05217i | −0.929299 | + | 1.60959i | 1.04518 | − | 1.81031i | −0.641869 | + | 1.11175i | 0.772692 | −0.500000 | + | 0.866025i | 3.88515 | |||||||||||||||||||||||||||||||||||||||||
181.2 | −0.428387 | − | 0.741988i | 0.500000 | + | 0.866025i | 0.632970 | − | 1.09634i | 0.782924 | − | 1.35606i | 0.428387 | − | 0.741988i | 1.24913 | − | 2.16356i | −2.79817 | −0.500000 | + | 0.866025i | −1.34158 | |||||||||||||||||||||||||||||||||||||||||
181.3 | 0.245923 | + | 0.425951i | 0.500000 | + | 0.866025i | 0.879044 | − | 1.52255i | 0.715982 | − | 1.24012i | −0.245923 | + | 0.425951i | −0.761138 | + | 1.31833i | 1.84840 | −0.500000 | + | 0.866025i | 0.704305 | |||||||||||||||||||||||||||||||||||||||||
181.4 | 0.733732 | + | 1.27086i | 0.500000 | + | 0.866025i | −0.0767255 | + | 0.132892i | −1.45453 | + | 2.51933i | −0.733732 | + | 1.27086i | 0.0229355 | − | 0.0397254i | 2.70974 | −0.500000 | + | 0.866025i | −4.26895 | |||||||||||||||||||||||||||||||||||||||||
181.5 | 1.15914 | + | 2.00769i | 0.500000 | + | 0.866025i | −1.68722 | + | 2.92235i | 1.95463 | − | 3.38552i | −1.15914 | + | 2.00769i | −2.04868 | + | 3.54841i | −3.18634 | −0.500000 | + | 0.866025i | 9.06277 | |||||||||||||||||||||||||||||||||||||||||
181.6 | 1.33477 | + | 2.31190i | 0.500000 | + | 0.866025i | −2.56325 | + | 4.43967i | −0.569703 | + | 0.986755i | −1.33477 | + | 2.31190i | 2.17962 | − | 3.77521i | −8.34632 | −0.500000 | + | 0.866025i | −3.04170 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
79.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 237.2.e.a | ✓ | 12 |
3.b | odd | 2 | 1 | 711.2.f.b | 12 | ||
79.c | even | 3 | 1 | inner | 237.2.e.a | ✓ | 12 |
237.g | odd | 6 | 1 | 711.2.f.b | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
237.2.e.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
237.2.e.a | ✓ | 12 | 79.c | even | 3 | 1 | inner |
711.2.f.b | 12 | 3.b | odd | 2 | 1 | ||
711.2.f.b | 12 | 237.g | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 4 T_{2}^{11} + 18 T_{2}^{10} - 34 T_{2}^{9} + 100 T_{2}^{8} - 153 T_{2}^{7} + 373 T_{2}^{6} - 344 T_{2}^{5} + 475 T_{2}^{4} - 156 T_{2}^{3} + 321 T_{2}^{2} - 120 T_{2} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(237, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - 4 T^{11} + 18 T^{10} - 34 T^{9} + \cdots + 64 \)
$3$
\( (T^{2} - T + 1)^{6} \)
$5$
\( T^{12} - T^{11} + 17 T^{10} + 6 T^{9} + \cdots + 2916 \)
$7$
\( T^{12} + 23 T^{10} + 16 T^{9} + 437 T^{8} + \cdots + 16 \)
$11$
\( T^{12} + T^{11} + 28 T^{10} + \cdots + 169744 \)
$13$
\( T^{12} - 5 T^{11} + 50 T^{10} + \cdots + 522729 \)
$17$
\( (T^{6} + 3 T^{5} - 29 T^{4} - 31 T^{3} + \cdots - 378)^{2} \)
$19$
\( T^{12} - 14 T^{11} + 185 T^{10} + \cdots + 71824 \)
$23$
\( T^{12} - 12 T^{11} + 107 T^{10} + \cdots + 26244 \)
$29$
\( T^{12} + 7 T^{11} + 53 T^{10} + 110 T^{9} + \cdots + 324 \)
$31$
\( T^{12} - T^{11} + 78 T^{10} + \cdots + 2181529 \)
$37$
\( T^{12} + 16 T^{11} + \cdots + 119180889 \)
$41$
\( (T^{6} + 16 T^{5} + 45 T^{4} - 97 T^{3} + \cdots + 224)^{2} \)
$43$
\( T^{12} + 13 T^{11} + 152 T^{10} + \cdots + 37249 \)
$47$
\( T^{12} + 10 T^{11} + 151 T^{10} + \cdots + 7054336 \)
$53$
\( T^{12} + 5 T^{11} + 212 T^{10} + \cdots + 173343556 \)
$59$
\( T^{12} - 14 T^{11} + 218 T^{10} + \cdots + 5531904 \)
$61$
\( (T^{6} + 9 T^{5} - 99 T^{4} - 715 T^{3} + \cdots - 58554)^{2} \)
$67$
\( (T^{6} + 24 T^{5} + 100 T^{4} + \cdots - 66393)^{2} \)
$71$
\( (T^{6} - 22 T^{5} - 23 T^{4} + \cdots + 321198)^{2} \)
$73$
\( T^{12} + 14 T^{11} + \cdots + 579172356 \)
$79$
\( T^{12} - 8 T^{11} + \cdots + 243087455521 \)
$83$
\( T^{12} + 12 T^{11} + \cdots + 1622317284 \)
$89$
\( (T^{6} - 5 T^{5} - 189 T^{4} + 1801 T^{3} + \cdots + 7078)^{2} \)
$97$
\( (T^{6} + 19 T^{5} - 4 T^{4} - 1408 T^{3} + \cdots + 7168)^{2} \)
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