# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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