Properties

Label 756.2.t.e
Level $756$
Weight $2$
Character orbit 756.t
Analytic conductor $6.037$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
Defining polynomial: \(x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{9} \)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + \beta_{10} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + \beta_{10} q^{7} + ( -\beta_{3} - \beta_{5} ) q^{11} + ( \beta_{4} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{13} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{11} ) q^{17} + ( -1 + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{19} + ( -\beta_{2} + \beta_{3} + \beta_{11} ) q^{23} + ( -\beta_{4} + \beta_{8} - \beta_{9} ) q^{25} + ( -4 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{11} ) q^{29} + ( -2 \beta_{4} - 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{31} + ( -\beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} ) q^{35} + ( -4 + \beta_{4} - 4 \beta_{6} + 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{37} + ( -3 \beta_{3} - \beta_{5} + \beta_{11} ) q^{41} + ( -1 + \beta_{8} - \beta_{10} ) q^{43} -\beta_{2} q^{47} + ( -2 \beta_{8} + 3 \beta_{9} ) q^{49} + ( -\beta_{1} + \beta_{3} - 2 \beta_{7} - \beta_{11} ) q^{53} + ( 2 - 2 \beta_{4} + 4 \beta_{6} + \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{55} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} - 3 \beta_{11} ) q^{59} + ( 1 + 2 \beta_{4} - \beta_{6} - 2 \beta_{9} - 2 \beta_{10} ) q^{61} + ( \beta_{2} - 3 \beta_{3} + \beta_{11} ) q^{65} + ( 3 \beta_{4} - 3 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{7} + \beta_{11} ) q^{71} + ( 6 + \beta_{4} + 3 \beta_{6} + \beta_{8} - \beta_{9} ) q^{73} + ( -3 \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{7} + 2 \beta_{11} ) q^{77} + ( 4 + 2 \beta_{4} + 4 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{79} + 5 \beta_{3} q^{83} + ( -1 - 3 \beta_{4} - 4 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} ) q^{85} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{7} + \beta_{11} ) q^{89} + ( -7 \beta_{6} + 3 \beta_{8} - \beta_{9} ) q^{91} + ( -\beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} - \beta_{11} ) q^{95} + ( -2 - 2 \beta_{4} - 4 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + O(q^{10}) \) \( 12 q - 18 q^{19} - 24 q^{37} - 12 q^{43} + 18 q^{61} + 54 q^{73} + 24 q^{79} - 12 q^{85} + 42 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -99 \nu^{11} + 488 \nu^{9} - 1519 \nu^{7} + 1061 \nu^{5} + 1670 \nu^{3} - 3551 \nu \)\()/559\)
\(\beta_{2}\)\(=\)\((\)\( 114 \nu^{11} - 545 \nu^{9} + 2071 \nu^{7} - 2831 \nu^{5} + 3379 \nu^{3} + 464 \nu \)\()/559\)
\(\beta_{3}\)\(=\)\((\)\( -114 \nu^{11} + 545 \nu^{9} - 2071 \nu^{7} + 2831 \nu^{5} - 3379 \nu^{3} + 1213 \nu \)\()/559\)
\(\beta_{4}\)\(=\)\((\)\( 49 \nu^{10} - 298 \nu^{8} + 1356 \nu^{6} - 2987 \nu^{4} + 4419 \nu^{2} - 1811 \)\()/559\)
\(\beta_{5}\)\(=\)\((\)\( -171 \nu^{11} + 538 \nu^{9} - 1709 \nu^{7} - 1064 \nu^{5} + 3037 \nu^{3} - 7963 \nu \)\()/559\)
\(\beta_{6}\)\(=\)\((\)\( 114 \nu^{10} - 545 \nu^{8} + 2071 \nu^{6} - 2831 \nu^{4} + 3379 \nu^{2} - 654 \)\()/559\)
\(\beta_{7}\)\(=\)\((\)\( 402 \nu^{11} - 2422 \nu^{9} + 9539 \nu^{7} - 17809 \nu^{5} + 19712 \nu^{3} - 7043 \nu \)\()/559\)
\(\beta_{8}\)\(=\)\((\)\( -187 \nu^{10} + 1046 \nu^{8} - 3863 \nu^{6} + 6414 \nu^{4} - 6038 \nu^{2} + 1926 \)\()/559\)
\(\beta_{9}\)\(=\)\((\)\( 259 \nu^{10} - 1096 \nu^{8} + 4053 \nu^{6} - 4289 \nu^{4} + 4671 \nu^{2} + 809 \)\()/559\)
\(\beta_{10}\)\(=\)\((\)\( -262 \nu^{10} + 1331 \nu^{8} - 4946 \nu^{6} + 6879 \nu^{4} - 6128 \nu^{2} - 527 \)\()/559\)
\(\beta_{11}\)\(=\)\((\)\( -1449 \nu^{11} + 7295 \nu^{9} - 27721 \nu^{7} + 41294 \nu^{5} - 45229 \nu^{3} + 9313 \nu \)\()/559\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{10} - \beta_{9} + 5 \beta_{6} - \beta_{4} + 5\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{11} - \beta_{7} + 3 \beta_{5} - 5 \beta_{3} + 14 \beta_{2} - 2 \beta_{1}\)\()/9\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{10} - 4 \beta_{9} + 3 \beta_{8} + 13 \beta_{6} - 3 \beta_{4}\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(7 \beta_{11} + 5 \beta_{7} - 47 \beta_{3} + 20 \beta_{2} - 5 \beta_{1}\)\()/9\)
\(\nu^{6}\)\(=\)\((\)\(-14 \beta_{10} - 5 \beta_{9} + 14 \beta_{8} + 5 \beta_{4} - 38\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(19 \beta_{11} + 38 \beta_{7} - 42 \beta_{5} - 122 \beta_{3} - 61 \beta_{2} + 19 \beta_{1}\)\()/9\)
\(\nu^{8}\)\(=\)\((\)\(-28 \beta_{10} + 28 \beta_{9} + 19 \beta_{8} - 117 \beta_{6} + 47 \beta_{4} - 117\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-3 \beta_{11} + 22 \beta_{7} - 47 \beta_{5} + 20 \beta_{3} - 128 \beta_{2} + 44 \beta_{1}\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(66 \beta_{10} + 155 \beta_{9} - 89 \beta_{8} - 370 \beta_{6} + 89 \beta_{4}\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-244 \beta_{11} - 221 \beta_{7} + 1472 \beta_{3} - 614 \beta_{2} + 221 \beta_{1}\)\()/9\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1\) \(1 + \beta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
1.56052 + 0.900969i
1.07992 + 0.623490i
0.385418 + 0.222521i
−0.385418 0.222521i
−1.07992 0.623490i
−1.56052 0.900969i
1.56052 0.900969i
1.07992 0.623490i
0.385418 0.222521i
−0.385418 + 0.222521i
−1.07992 + 0.623490i
−1.56052 + 0.900969i
0 0 0 −1.56052 2.70291i 0 2.37047 1.17511i 0 0 0
269.2 0 0 0 −1.07992 1.87047i 0 −0.167563 + 2.64044i 0 0 0
269.3 0 0 0 −0.385418 0.667563i 0 −2.20291 1.46533i 0 0 0
269.4 0 0 0 0.385418 + 0.667563i 0 −2.20291 1.46533i 0 0 0
269.5 0 0 0 1.07992 + 1.87047i 0 −0.167563 + 2.64044i 0 0 0
269.6 0 0 0 1.56052 + 2.70291i 0 2.37047 1.17511i 0 0 0
593.1 0 0 0 −1.56052 + 2.70291i 0 2.37047 + 1.17511i 0 0 0
593.2 0 0 0 −1.07992 + 1.87047i 0 −0.167563 2.64044i 0 0 0
593.3 0 0 0 −0.385418 + 0.667563i 0 −2.20291 + 1.46533i 0 0 0
593.4 0 0 0 0.385418 0.667563i 0 −2.20291 + 1.46533i 0 0 0
593.5 0 0 0 1.07992 1.87047i 0 −0.167563 2.64044i 0 0 0
593.6 0 0 0 1.56052 2.70291i 0 2.37047 + 1.17511i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.t.e 12
3.b odd 2 1 inner 756.2.t.e 12
7.c even 3 1 5292.2.f.e 12
7.d odd 6 1 inner 756.2.t.e 12
7.d odd 6 1 5292.2.f.e 12
9.c even 3 1 2268.2.w.i 12
9.c even 3 1 2268.2.bm.i 12
9.d odd 6 1 2268.2.w.i 12
9.d odd 6 1 2268.2.bm.i 12
21.g even 6 1 inner 756.2.t.e 12
21.g even 6 1 5292.2.f.e 12
21.h odd 6 1 5292.2.f.e 12
63.i even 6 1 2268.2.bm.i 12
63.k odd 6 1 2268.2.w.i 12
63.s even 6 1 2268.2.w.i 12
63.t odd 6 1 2268.2.bm.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.t.e 12 1.a even 1 1 trivial
756.2.t.e 12 3.b odd 2 1 inner
756.2.t.e 12 7.d odd 6 1 inner
756.2.t.e 12 21.g even 6 1 inner
2268.2.w.i 12 9.c even 3 1
2268.2.w.i 12 9.d odd 6 1
2268.2.w.i 12 63.k odd 6 1
2268.2.w.i 12 63.s even 6 1
2268.2.bm.i 12 9.c even 3 1
2268.2.bm.i 12 9.d odd 6 1
2268.2.bm.i 12 63.i even 6 1
2268.2.bm.i 12 63.t odd 6 1
5292.2.f.e 12 7.c even 3 1
5292.2.f.e 12 7.d odd 6 1
5292.2.f.e 12 21.g even 6 1
5292.2.f.e 12 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):

\( T_{5}^{12} + 15 T_{5}^{10} + 171 T_{5}^{8} + 756 T_{5}^{6} + 2511 T_{5}^{4} + 1458 T_{5}^{2} + 729 \)
\( T_{13}^{6} + 42 T_{13}^{4} + 441 T_{13}^{2} + 1323 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( 729 + 1458 T^{2} + 2511 T^{4} + 756 T^{6} + 171 T^{8} + 15 T^{10} + T^{12} \)
$7$ \( ( 343 - 7 T^{3} + T^{6} )^{2} \)
$11$ \( 531441 - 354294 T^{2} + 203391 T^{4} - 20412 T^{6} + 1539 T^{8} - 45 T^{10} + T^{12} \)
$13$ \( ( 1323 + 441 T^{2} + 42 T^{4} + T^{6} )^{2} \)
$17$ \( 2492305929 + 208927755 T^{2} + 11823003 T^{4} + 377244 T^{6} + 8811 T^{8} + 114 T^{10} + T^{12} \)
$19$ \( ( 27 - 108 T + 171 T^{2} - 108 T^{3} + 15 T^{4} + 9 T^{5} + T^{6} )^{2} \)
$23$ \( 531441 - 295245 T^{2} + 124659 T^{4} - 20412 T^{6} + 2511 T^{8} - 54 T^{10} + T^{12} \)
$29$ \( ( 123201 + 7614 T^{2} + 153 T^{4} + T^{6} )^{2} \)
$31$ \( ( 1323 - 3969 T + 3969 T^{2} - 63 T^{4} + T^{6} )^{2} \)
$37$ \( ( 142129 + 5655 T + 4749 T^{2} + 574 T^{3} + 159 T^{4} + 12 T^{5} + T^{6} )^{2} \)
$41$ \( ( -136107 + 10125 T^{2} - 186 T^{4} + T^{6} )^{2} \)
$43$ \( ( -13 - 18 T + 3 T^{2} + T^{3} )^{4} \)
$47$ \( 729 + 1458 T^{2} + 2511 T^{4} + 756 T^{6} + 171 T^{8} + 15 T^{10} + T^{12} \)
$53$ \( 15178486401 - 1526829993 T^{2} + 124757415 T^{4} - 2653560 T^{6} + 42363 T^{8} - 234 T^{10} + T^{12} \)
$59$ \( 118861526169 + 8024014062 T^{2} + 437215887 T^{4} + 6362496 T^{6} + 68535 T^{8} + 303 T^{10} + T^{12} \)
$61$ \( ( 22707 + 19575 T + 6408 T^{2} + 675 T^{3} - 48 T^{4} - 9 T^{5} + T^{6} )^{2} \)
$67$ \( ( 247009 - 73059 T + 21609 T^{2} - 994 T^{3} + 147 T^{4} + T^{6} )^{2} \)
$71$ \( ( 35721 + 7938 T^{2} + 189 T^{4} + T^{6} )^{2} \)
$73$ \( ( 4563 - 7020 T + 4653 T^{2} - 1620 T^{3} + 303 T^{4} - 27 T^{5} + T^{6} )^{2} \)
$79$ \( ( 142129 - 5655 T + 4749 T^{2} - 574 T^{3} + 159 T^{4} - 12 T^{5} + T^{6} )^{2} \)
$83$ \( ( -421875 + 33750 T^{2} - 375 T^{4} + T^{6} )^{2} \)
$89$ \( ( 729 + 27 T^{2} + T^{4} )^{3} \)
$97$ \( ( 1728 + 1440 T^{2} + 204 T^{4} + T^{6} )^{2} \)
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