Newspace parameters
Level: | \( N \) | \(=\) | \( 845 = 5 \cdot 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 845.l (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.74735897080\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 12.0.89539436150784.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | no (minimal twist has level 65) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{11} - \nu^{10} + 4 \nu^{9} - 28 \nu^{8} + 18 \nu^{7} - 22 \nu^{6} + 94 \nu^{5} + 146 \nu^{4} - 144 \nu^{3} + 48 \nu^{2} - 48 \nu - 748 ) / 460 \) |
\(\beta_{2}\) | \(=\) | \( ( - 5 \nu^{11} + 5 \nu^{10} - 20 \nu^{9} + 94 \nu^{8} - 44 \nu^{7} + 64 \nu^{6} - 286 \nu^{5} - 454 \nu^{4} + 444 \nu^{3} - 148 \nu^{2} + 148 \nu + 612 ) / 460 \) |
\(\beta_{3}\) | \(=\) | \( ( - 10 \nu^{11} + 32 \nu^{10} - 32 \nu^{9} + 82 \nu^{8} - 123 \nu^{7} - 197 \nu^{6} + 322 \nu^{5} - 84 \nu^{4} + 34 \nu^{3} + 306 \nu^{2} - 10 \nu + 10 ) / 230 \) |
\(\beta_{4}\) | \(=\) | \( ( 12 \nu^{11} - 43 \nu^{10} + 43 \nu^{9} - 103 \nu^{8} + 166 \nu^{7} + 264 \nu^{6} - 414 \nu^{5} + 110 \nu^{4} - 50 \nu^{3} - 1140 \nu^{2} + 12 \nu - 12 ) / 230 \) |
\(\beta_{5}\) | \(=\) | \( ( 39 \nu^{11} - 83 \nu^{10} + 94 \nu^{9} - 328 \nu^{8} + 220 \nu^{7} + 482 \nu^{6} - 330 \nu^{5} + 734 \nu^{4} + 1060 \nu^{3} - 436 \nu^{2} - 156 \nu - 156 ) / 460 \) |
\(\beta_{6}\) | \(=\) | \( ( - 22 \nu^{11} + 75 \nu^{10} - 75 \nu^{9} + 185 \nu^{8} - 289 \nu^{7} - 461 \nu^{6} + 736 \nu^{5} - 194 \nu^{4} + 84 \nu^{3} + 1676 \nu^{2} - 22 \nu + 22 ) / 230 \) |
\(\beta_{7}\) | \(=\) | \( ( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + 86 \nu^{3} - 38 \nu^{2} + 32 \nu - 32 ) / 20 \) |
\(\beta_{8}\) | \(=\) | \( ( - 101 \nu^{11} + 101 \nu^{10} - 36 \nu^{9} + 666 \nu^{8} + 344 \nu^{7} - 1734 \nu^{6} - 846 \nu^{5} - 1774 \nu^{4} - 3856 \nu^{3} - 524 \nu^{2} + 524 \nu + 476 ) / 460 \) |
\(\beta_{9}\) | \(=\) | \( ( - 117 \nu^{11} + 195 \nu^{10} - 195 \nu^{9} + 941 \nu^{8} - 250 \nu^{7} - 1700 \nu^{6} + 92 \nu^{5} - 2510 \nu^{4} - 2790 \nu^{3} + 1294 \nu^{2} - 1060 \nu + 1060 ) / 460 \) |
\(\beta_{10}\) | \(=\) | \( ( - 60 \nu^{11} + 100 \nu^{10} - 100 \nu^{9} + 469 \nu^{8} - 94 \nu^{7} - 906 \nu^{6} + 184 \nu^{5} - 1424 \nu^{4} - 1636 \nu^{3} + 732 \nu^{2} - 612 \nu + 612 ) / 230 \) |
\(\beta_{11}\) | \(=\) | \( ( - 187 \nu^{11} + 293 \nu^{10} - 216 \nu^{9} + 1338 \nu^{8} - 116 \nu^{7} - 3250 \nu^{6} + 174 \nu^{5} - 3050 \nu^{4} - 5552 \nu^{3} + 2560 \nu^{2} + 748 \nu + 748 ) / 460 \) |
\(\nu\) | \(=\) | \( ( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{6} + \beta_{4} - \beta_{3} \) |
\(\nu^{3}\) | \(=\) | \( ( -3\beta_{11} + 4\beta_{8} + 3\beta_{6} - \beta_{5} - \beta_{3} + 2\beta_{2} + 4\beta _1 + 4 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( 5\beta_{10} - \beta_{9} + 7\beta_{7} + \beta_{2} + 5\beta _1 + 7 \) |
\(\nu^{5}\) | \(=\) | \( 8\beta_{10} - 3\beta_{9} + 9\beta_{7} + 6\beta_{6} + 8\beta_{4} - 3\beta_{3} \) |
\(\nu^{6}\) | \(=\) | \( -17\beta_{11} + 22\beta_{8} + 17\beta_{6} - 11\beta_{5} - 11\beta_{3} \) |
\(\nu^{7}\) | \(=\) | \( - 25 \beta_{11} + 33 \beta_{10} - 11 \beta_{9} + 33 \beta_{8} + 39 \beta_{7} - 14 \beta_{5} - 33 \beta_{4} + 11 \beta_{2} + 33 \beta _1 + 39 \) |
\(\nu^{8}\) | \(=\) | \( 94\beta_{10} - 28\beta_{9} + 116\beta_{7} \) |
\(\nu^{9}\) | \(=\) | \( -105\beta_{11} + 138\beta_{8} + 105\beta_{6} - 61\beta_{5} - 61\beta_{3} - 44\beta_{2} - 138\beta _1 - 166 \) |
\(\nu^{10}\) | \(=\) | \( -304\beta_{11} + 398\beta_{8} - 182\beta_{5} - 398\beta_{4} \) |
\(\nu^{11}\) | \(=\) | \( 580\beta_{10} - 182\beta_{9} + 702\beta_{7} - 442\beta_{6} - 580\beta_{4} + 260\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).
\(n\) | \(171\) | \(677\) |
\(\chi(n)\) | \(-\beta_{7}\) | \(-1\) |
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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654.1 |
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−0.607160 | + | 1.05163i | −1.13545 | − | 0.655554i | 0.262714 | + | 0.455034i | −0.311108 | + | 2.21432i | 1.37880 | − | 0.796052i | −1.45161 | − | 2.51426i | −3.06668 | −0.640498 | − | 1.10938i | −2.13976 | − | 1.67162i | ||||||||||||||||||||||||||||||||||||||
654.2 | −0.607160 | + | 1.05163i | 1.13545 | + | 0.655554i | 0.262714 | + | 0.455034i | −0.311108 | − | 2.21432i | −1.37880 | + | 0.796052i | −1.45161 | − | 2.51426i | −3.06668 | −0.640498 | − | 1.10938i | 2.51754 | + | 1.01728i | |||||||||||||||||||||||||||||||||||||||
654.3 | 0.769594 | − | 1.33298i | −2.74538 | − | 1.58504i | −0.184551 | − | 0.319652i | −2.17009 | − | 0.539189i | −4.22565 | + | 2.43968i | 0.854638 | + | 1.48028i | 2.51026 | 3.52472 | + | 6.10500i | −2.38881 | + | 2.47772i | |||||||||||||||||||||||||||||||||||||||
654.4 | 0.769594 | − | 1.33298i | 2.74538 | + | 1.58504i | −0.184551 | − | 0.319652i | −2.17009 | + | 0.539189i | 4.22565 | − | 2.43968i | 0.854638 | + | 1.48028i | 2.51026 | 3.52472 | + | 6.10500i | −0.951360 | + | 3.30763i | |||||||||||||||||||||||||||||||||||||||
654.5 | 1.33757 | − | 2.31673i | −0.416726 | − | 0.240597i | −2.57816 | − | 4.46551i | 1.48119 | + | 1.67513i | −1.11480 | + | 0.643629i | −0.403032 | − | 0.698071i | −8.44358 | −1.38423 | − | 2.39755i | 5.86202 | − | 1.19093i | |||||||||||||||||||||||||||||||||||||||
654.6 | 1.33757 | − | 2.31673i | 0.416726 | + | 0.240597i | −2.57816 | − | 4.46551i | 1.48119 | − | 1.67513i | 1.11480 | − | 0.643629i | −0.403032 | − | 0.698071i | −8.44358 | −1.38423 | − | 2.39755i | −1.89963 | − | 5.67213i | |||||||||||||||||||||||||||||||||||||||
699.1 | −0.607160 | − | 1.05163i | −1.13545 | + | 0.655554i | 0.262714 | − | 0.455034i | −0.311108 | − | 2.21432i | 1.37880 | + | 0.796052i | −1.45161 | + | 2.51426i | −3.06668 | −0.640498 | + | 1.10938i | −2.13976 | + | 1.67162i | |||||||||||||||||||||||||||||||||||||||
699.2 | −0.607160 | − | 1.05163i | 1.13545 | − | 0.655554i | 0.262714 | − | 0.455034i | −0.311108 | + | 2.21432i | −1.37880 | − | 0.796052i | −1.45161 | + | 2.51426i | −3.06668 | −0.640498 | + | 1.10938i | 2.51754 | − | 1.01728i | |||||||||||||||||||||||||||||||||||||||
699.3 | 0.769594 | + | 1.33298i | −2.74538 | + | 1.58504i | −0.184551 | + | 0.319652i | −2.17009 | + | 0.539189i | −4.22565 | − | 2.43968i | 0.854638 | − | 1.48028i | 2.51026 | 3.52472 | − | 6.10500i | −2.38881 | − | 2.47772i | |||||||||||||||||||||||||||||||||||||||
699.4 | 0.769594 | + | 1.33298i | 2.74538 | − | 1.58504i | −0.184551 | + | 0.319652i | −2.17009 | − | 0.539189i | 4.22565 | + | 2.43968i | 0.854638 | − | 1.48028i | 2.51026 | 3.52472 | − | 6.10500i | −0.951360 | − | 3.30763i | |||||||||||||||||||||||||||||||||||||||
699.5 | 1.33757 | + | 2.31673i | −0.416726 | + | 0.240597i | −2.57816 | + | 4.46551i | 1.48119 | − | 1.67513i | −1.11480 | − | 0.643629i | −0.403032 | + | 0.698071i | −8.44358 | −1.38423 | + | 2.39755i | 5.86202 | + | 1.19093i | |||||||||||||||||||||||||||||||||||||||
699.6 | 1.33757 | + | 2.31673i | 0.416726 | − | 0.240597i | −2.57816 | + | 4.46551i | 1.48119 | + | 1.67513i | 1.11480 | + | 0.643629i | −0.403032 | + | 0.698071i | −8.44358 | −1.38423 | + | 2.39755i | −1.89963 | + | 5.67213i | |||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
65.d | even | 2 | 1 | inner |
65.l | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 3T_{2}^{5} + 10T_{2}^{4} - 7T_{2}^{3} + 16T_{2}^{2} - 5T_{2} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{6} - 3 T^{5} + 10 T^{4} - 7 T^{3} + \cdots + 25)^{2} \)
$3$
\( T^{12} - 12 T^{10} + 124 T^{8} + \cdots + 16 \)
$5$
\( (T^{6} + 2 T^{5} + 3 T^{4} + 12 T^{3} + \cdots + 125)^{2} \)
$7$
\( (T^{6} + 2 T^{5} + 8 T^{4} + 24 T^{2} + \cdots + 16)^{2} \)
$11$
\( T^{12} - 20 T^{10} + 312 T^{8} + \cdots + 16 \)
$13$
\( T^{12} \)
$17$
\( T^{12} - 44 T^{10} + 1824 T^{8} + \cdots + 4096 \)
$19$
\( T^{12} - 8 T^{10} + 48 T^{8} - 120 T^{6} + \cdots + 16 \)
$23$
\( T^{12} - 72 T^{10} + 3748 T^{8} + \cdots + 54700816 \)
$29$
\( (T^{6} - 6 T^{5} + 72 T^{4} + 1944 T^{2} + \cdots + 11664)^{2} \)
$31$
\( (T^{6} + 60 T^{4} + 920 T^{2} + 676)^{2} \)
$37$
\( (T^{6} + 28 T^{4} + 104 T^{3} + 784 T^{2} + \cdots + 2704)^{2} \)
$41$
\( T^{12} - 80 T^{10} + 5632 T^{8} + \cdots + 1048576 \)
$43$
\( T^{12} - 128 T^{10} + \cdots + 5972816656 \)
$47$
\( (T^{3} - 10 T^{2} + 28 T - 20)^{4} \)
$53$
\( (T^{6} + 144 T^{4} + 6464 T^{2} + \cdots + 92416)^{2} \)
$59$
\( T^{12} - 144 T^{10} + \cdots + 4711998736 \)
$61$
\( (T^{6} + 6 T^{5} + 52 T^{4} - 104 T^{3} + \cdots + 16)^{2} \)
$67$
\( (T^{6} - 10 T^{5} + 160 T^{4} + \cdots + 364816)^{2} \)
$71$
\( T^{12} - 320 T^{10} + \cdots + 323210442256 \)
$73$
\( (T^{3} - 24 T^{2} + 164 T - 236)^{4} \)
$79$
\( (T^{3} - 16 T^{2} + 24 T + 16)^{4} \)
$83$
\( (T^{3} - 22 T^{2} + 152 T - 316)^{4} \)
$89$
\( T^{12} - 204 T^{10} + \cdots + 1600000000 \)
$97$
\( (T^{6} + 14 T^{5} + 280 T^{4} + \cdots + 40000)^{2} \)
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