Newspace parameters
Level: | \( N \) | \(=\) | \( 845 = 5 \cdot 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 845.m (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: | 12.0.17213603549184.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
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 \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559 \) |
\(\beta_{3}\) | \(=\) | \( ( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559 \) |
\(\beta_{4}\) | \(=\) | \( ( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43 \) |
\(\beta_{5}\) | \(=\) | \( ( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559 \) |
\(\beta_{6}\) | \(=\) | \( ( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559 \) |
\(\beta_{7}\) | \(=\) | \( ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559 \) |
\(\beta_{8}\) | \(=\) | \( ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559 \) |
\(\beta_{9}\) | \(=\) | \( ( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559 \) |
\(\beta_{10}\) | \(=\) | \( ( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559 \) |
\(\beta_{11}\) | \(=\) | \( ( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{11} + 3\beta_{8} + \beta_{2} \) |
\(\nu^{4}\) | \(=\) | \( 3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2 \) |
\(\nu^{5}\) | \(=\) | \( 4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( -5\beta_{5} + 9\beta_{3} - 14 \) |
\(\nu^{7}\) | \(=\) | \( -5\beta_{6} - 14\beta_{2} - 28\beta_1 \) |
\(\nu^{8}\) | \(=\) | \( -28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28 \) |
\(\nu^{9}\) | \(=\) | \( -47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2} \) |
\(\nu^{10}\) | \(=\) | \( -89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42 \) |
\(\nu^{11}\) | \(=\) | \( -155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1 \) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
316.1 |
|
−1.56052 | − | 0.900969i | 0.777479 | − | 1.34663i | 0.623490 | + | 1.07992i | − | 1.00000i | −2.42655 | + | 1.40097i | −1.34663 | + | 0.777479i | 1.35690i | 0.291053 | + | 0.504118i | −0.900969 | + | 1.56052i | |||||||||||||||||||||||||||||||||||||||
316.2 | −1.07992 | − | 0.623490i | 0.0990311 | − | 0.171527i | −0.222521 | − | 0.385418i | 1.00000i | −0.213891 | + | 0.123490i | 0.171527 | − | 0.0990311i | 3.04892i | 1.48039 | + | 2.56410i | 0.623490 | − | 1.07992i | |||||||||||||||||||||||||||||||||||||||||
316.3 | −0.385418 | − | 0.222521i | 1.62349 | − | 2.81197i | −0.900969 | − | 1.56052i | − | 1.00000i | −1.25144 | + | 0.722521i | −2.81197 | + | 1.62349i | 1.69202i | −3.77144 | − | 6.53232i | −0.222521 | + | 0.385418i | ||||||||||||||||||||||||||||||||||||||||
316.4 | 0.385418 | + | 0.222521i | 1.62349 | − | 2.81197i | −0.900969 | − | 1.56052i | 1.00000i | 1.25144 | − | 0.722521i | 2.81197 | − | 1.62349i | − | 1.69202i | −3.77144 | − | 6.53232i | −0.222521 | + | 0.385418i | ||||||||||||||||||||||||||||||||||||||||
316.5 | 1.07992 | + | 0.623490i | 0.0990311 | − | 0.171527i | −0.222521 | − | 0.385418i | − | 1.00000i | 0.213891 | − | 0.123490i | −0.171527 | + | 0.0990311i | − | 3.04892i | 1.48039 | + | 2.56410i | 0.623490 | − | 1.07992i | |||||||||||||||||||||||||||||||||||||||
316.6 | 1.56052 | + | 0.900969i | 0.777479 | − | 1.34663i | 0.623490 | + | 1.07992i | 1.00000i | 2.42655 | − | 1.40097i | 1.34663 | − | 0.777479i | − | 1.35690i | 0.291053 | + | 0.504118i | −0.900969 | + | 1.56052i | ||||||||||||||||||||||||||||||||||||||||
361.1 | −1.56052 | + | 0.900969i | 0.777479 | + | 1.34663i | 0.623490 | − | 1.07992i | 1.00000i | −2.42655 | − | 1.40097i | −1.34663 | − | 0.777479i | − | 1.35690i | 0.291053 | − | 0.504118i | −0.900969 | − | 1.56052i | ||||||||||||||||||||||||||||||||||||||||
361.2 | −1.07992 | + | 0.623490i | 0.0990311 | + | 0.171527i | −0.222521 | + | 0.385418i | − | 1.00000i | −0.213891 | − | 0.123490i | 0.171527 | + | 0.0990311i | − | 3.04892i | 1.48039 | − | 2.56410i | 0.623490 | + | 1.07992i | |||||||||||||||||||||||||||||||||||||||
361.3 | −0.385418 | + | 0.222521i | 1.62349 | + | 2.81197i | −0.900969 | + | 1.56052i | 1.00000i | −1.25144 | − | 0.722521i | −2.81197 | − | 1.62349i | − | 1.69202i | −3.77144 | + | 6.53232i | −0.222521 | − | 0.385418i | ||||||||||||||||||||||||||||||||||||||||
361.4 | 0.385418 | − | 0.222521i | 1.62349 | + | 2.81197i | −0.900969 | + | 1.56052i | − | 1.00000i | 1.25144 | + | 0.722521i | 2.81197 | + | 1.62349i | 1.69202i | −3.77144 | + | 6.53232i | −0.222521 | − | 0.385418i | ||||||||||||||||||||||||||||||||||||||||
361.5 | 1.07992 | − | 0.623490i | 0.0990311 | + | 0.171527i | −0.222521 | + | 0.385418i | 1.00000i | 0.213891 | + | 0.123490i | −0.171527 | − | 0.0990311i | 3.04892i | 1.48039 | − | 2.56410i | 0.623490 | + | 1.07992i | |||||||||||||||||||||||||||||||||||||||||
361.6 | 1.56052 | − | 0.900969i | 0.777479 | + | 1.34663i | 0.623490 | − | 1.07992i | − | 1.00000i | 2.42655 | + | 1.40097i | 1.34663 | + | 0.777479i | 1.35690i | 0.291053 | − | 0.504118i | −0.900969 | − | 1.56052i | ||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
13.c | even | 3 | 1 | inner |
13.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 845.2.m.i | 12 | |
13.b | even | 2 | 1 | inner | 845.2.m.i | 12 | |
13.c | even | 3 | 1 | 845.2.c.f | 6 | ||
13.c | even | 3 | 1 | inner | 845.2.m.i | 12 | |
13.d | odd | 4 | 1 | 845.2.e.j | 6 | ||
13.d | odd | 4 | 1 | 845.2.e.l | 6 | ||
13.e | even | 6 | 1 | 845.2.c.f | 6 | ||
13.e | even | 6 | 1 | inner | 845.2.m.i | 12 | |
13.f | odd | 12 | 1 | 845.2.a.h | ✓ | 3 | |
13.f | odd | 12 | 1 | 845.2.a.j | yes | 3 | |
13.f | odd | 12 | 1 | 845.2.e.j | 6 | ||
13.f | odd | 12 | 1 | 845.2.e.l | 6 | ||
39.k | even | 12 | 1 | 7605.2.a.br | 3 | ||
39.k | even | 12 | 1 | 7605.2.a.by | 3 | ||
65.s | odd | 12 | 1 | 4225.2.a.bd | 3 | ||
65.s | odd | 12 | 1 | 4225.2.a.bf | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
845.2.a.h | ✓ | 3 | 13.f | odd | 12 | 1 | |
845.2.a.j | yes | 3 | 13.f | odd | 12 | 1 | |
845.2.c.f | 6 | 13.c | even | 3 | 1 | ||
845.2.c.f | 6 | 13.e | even | 6 | 1 | ||
845.2.e.j | 6 | 13.d | odd | 4 | 1 | ||
845.2.e.j | 6 | 13.f | odd | 12 | 1 | ||
845.2.e.l | 6 | 13.d | odd | 4 | 1 | ||
845.2.e.l | 6 | 13.f | odd | 12 | 1 | ||
845.2.m.i | 12 | 1.a | even | 1 | 1 | trivial | |
845.2.m.i | 12 | 13.b | even | 2 | 1 | inner | |
845.2.m.i | 12 | 13.c | even | 3 | 1 | inner | |
845.2.m.i | 12 | 13.e | even | 6 | 1 | inner | |
4225.2.a.bd | 3 | 65.s | odd | 12 | 1 | ||
4225.2.a.bf | 3 | 65.s | odd | 12 | 1 | ||
7605.2.a.br | 3 | 39.k | even | 12 | 1 | ||
7605.2.a.by | 3 | 39.k | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 5T_{2}^{10} + 19T_{2}^{8} - 28T_{2}^{6} + 31T_{2}^{4} - 6T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - 5 T^{10} + 19 T^{8} - 28 T^{6} + \cdots + 1 \)
$3$
\( (T^{6} - 5 T^{5} + 19 T^{4} - 28 T^{3} + \cdots + 1)^{2} \)
$5$
\( (T^{2} + 1)^{6} \)
$7$
\( T^{12} - 13 T^{10} + 143 T^{8} - 336 T^{6} + \cdots + 1 \)
$11$
\( T^{12} - 17 T^{10} + 279 T^{8} - 168 T^{6} + \cdots + 1 \)
$13$
\( T^{12} \)
$17$
\( (T^{6} + 3 T^{5} + 27 T^{4} + 405 T^{2} + \cdots + 729)^{2} \)
$19$
\( T^{12} - 82 T^{10} + 6179 T^{8} + \cdots + 707281 \)
$23$
\( (T^{6} + 2 T^{5} + 61 T^{4} + 28 T^{3} + \cdots + 5041)^{2} \)
$29$
\( (T^{6} - 6 T^{5} + 87 T^{4} - 308 T^{3} + \cdots + 94249)^{2} \)
$31$
\( (T^{6} + 101 T^{4} + 2110 T^{2} + 1)^{2} \)
$37$
\( T^{12} - 27 T^{10} + 598 T^{8} + \cdots + 28561 \)
$41$
\( T^{12} - 194 T^{10} + 28747 T^{8} + \cdots + 3418801 \)
$43$
\( (T^{6} + 23 T^{5} + 355 T^{4} + \cdots + 187489)^{2} \)
$47$
\( (T^{6} + 146 T^{4} + 1713 T^{2} + \cdots + 1681)^{2} \)
$53$
\( (T^{3} + 2 T^{2} - T - 1)^{4} \)
$59$
\( T^{12} - 59 T^{10} + 2806 T^{8} + \cdots + 28561 \)
$61$
\( (T^{6} - 17 T^{5} + 195 T^{4} + \cdots + 28561)^{2} \)
$67$
\( T^{12} - 243 T^{10} + \cdots + 15178486401 \)
$71$
\( T^{12} - 229 T^{10} + \cdots + 130466162401 \)
$73$
\( (T^{6} + 440 T^{4} + 56656 T^{2} + \cdots + 2096704)^{2} \)
$79$
\( (T^{3} + 37 T^{2} + 412 T + 1217)^{4} \)
$83$
\( (T^{6} + 369 T^{4} + 21650 T^{2} + \cdots + 344569)^{2} \)
$89$
\( T^{12} - 395 T^{10} + \cdots + 80706559921 \)
$97$
\( T^{12} - 146 T^{10} + \cdots + 1073283121 \)
show more
show less