# Properties

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

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 756.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.60771337450861625344.1 Defining polynomial: $$x^{12} - 4 x^{10} + 11 x^{8} - 26 x^{6} + 44 x^{4} - 64 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + ( 1 + \beta_{11} ) q^{4} -\beta_{8} q^{5} + ( -\beta_{6} - \beta_{11} ) q^{7} + ( \beta_{1} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{8} +O(q^{10})$$ $$q + \beta_{7} q^{2} + ( 1 + \beta_{11} ) q^{4} -\beta_{8} q^{5} + ( -\beta_{6} - \beta_{11} ) q^{7} + ( \beta_{1} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{8} + ( \beta_{2} - \beta_{3} + \beta_{10} + \beta_{11} ) q^{10} + ( -\beta_{4} - \beta_{8} ) q^{11} + ( -1 - \beta_{2} + 2 \beta_{10} + \beta_{11} ) q^{13} + ( -\beta_{4} + \beta_{5} ) q^{14} + ( \beta_{2} + \beta_{3} - \beta_{10} + \beta_{11} ) q^{16} + ( 2 \beta_{1} + \beta_{4} + 2 \beta_{7} ) q^{17} + ( 2 + \beta_{2} - \beta_{3} - \beta_{6} ) q^{19} + ( -\beta_{1} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{20} + ( -1 + 2 \beta_{10} + \beta_{11} ) q^{22} + ( \beta_{4} + \beta_{8} ) q^{23} + ( 1 + \beta_{2} + \beta_{11} ) q^{25} + ( \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{8} ) q^{26} + ( -2 - 3 \beta_{2} + \beta_{3} + \beta_{10} - \beta_{11} ) q^{28} + ( \beta_{4} + 2 \beta_{9} ) q^{29} + ( 3 \beta_{2} + 3 \beta_{11} ) q^{31} + ( -\beta_{1} - \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{32} + ( -1 + \beta_{2} - \beta_{3} - \beta_{10} + 2 \beta_{11} ) q^{34} + ( -2 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{35} + ( -1 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} ) q^{37} + ( \beta_{7} + \beta_{8} + \beta_{9} ) q^{38} + ( 3 + 2 \beta_{2} - 2 \beta_{6} - \beta_{11} ) q^{40} + ( -2 \beta_{1} + 2 \beta_{4} - 2 \beta_{7} - \beta_{8} ) q^{41} + ( 1 - \beta_{2} + \beta_{3} - \beta_{6} - 2 \beta_{10} ) q^{43} + ( -\beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{44} + ( 1 - 2 \beta_{10} - \beta_{11} ) q^{46} + ( -2 \beta_{1} - \beta_{4} + 2 \beta_{7} - 2 \beta_{9} ) q^{47} + ( 1 + \beta_{2} - \beta_{6} - 2 \beta_{10} + 2 \beta_{11} ) q^{49} + ( -\beta_{1} - \beta_{5} + \beta_{8} ) q^{50} + ( -3 + \beta_{2} + \beta_{3} + 3 \beta_{10} + 2 \beta_{11} ) q^{52} + ( 4 \beta_{5} - 2 \beta_{8} ) q^{53} + ( -3 - \beta_{3} - \beta_{6} - \beta_{11} ) q^{55} + ( 3 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{56} + ( 3 + \beta_{2} - \beta_{3} - 4 \beta_{6} - \beta_{10} - 2 \beta_{11} ) q^{58} + ( 2 \beta_{1} - \beta_{4} + 4 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{59} + ( 2 \beta_{2} + \beta_{3} - \beta_{6} - 3 \beta_{11} ) q^{61} + ( -3 \beta_{1} - 3 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} ) q^{62} + ( 1 + 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{10} - \beta_{11} ) q^{64} + ( 2 \beta_{1} + \beta_{4} - 2 \beta_{7} + 2 \beta_{9} ) q^{65} + ( -1 - \beta_{2} + 2 \beta_{10} + \beta_{11} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{68} + ( 3 - 2 \beta_{2} - 2 \beta_{3} - \beta_{11} ) q^{70} + ( 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} - \beta_{8} ) q^{71} + ( -1 + \beta_{2} - 2 \beta_{3} + 2 \beta_{6} + 2 \beta_{10} + \beta_{11} ) q^{73} + ( \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{74} + ( 2 - \beta_{2} + \beta_{3} - 2 \beta_{6} - \beta_{10} - \beta_{11} ) q^{76} + ( 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + \beta_{8} ) q^{77} + ( -3 - \beta_{2} - 2 \beta_{3} + 2 \beta_{6} + 6 \beta_{10} + 3 \beta_{11} ) q^{79} + ( -3 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} ) q^{80} + ( 4 + 3 \beta_{2} - 3 \beta_{3} - \beta_{10} - \beta_{11} ) q^{82} + ( -\beta_{4} - 4 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} ) q^{83} + ( 1 + 5 \beta_{2} - \beta_{3} - \beta_{6} + 4 \beta_{11} ) q^{85} + ( 4 \beta_{1} + 2 \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{86} + ( \beta_{2} + \beta_{3} + 3 \beta_{10} + \beta_{11} ) q^{88} + ( -4 \beta_{1} + 4 \beta_{4} - 4 \beta_{7} + \beta_{8} ) q^{89} + ( -3 - 4 \beta_{2} + \beta_{3} - 4 \beta_{10} - 4 \beta_{11} ) q^{91} + ( \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} ) q^{92} + ( 3 - \beta_{2} + \beta_{3} + 4 \beta_{6} + \beta_{10} + 4 \beta_{11} ) q^{94} + ( -4 \beta_{1} + \beta_{4} - 4 \beta_{7} - \beta_{8} ) q^{95} + ( -2 \beta_{2} - \beta_{3} + \beta_{6} + 3 \beta_{11} ) q^{97} + ( 3 \beta_{1} + \beta_{4} - 2 \beta_{5} + 5 \beta_{8} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 8 q^{4} - 2 q^{7} + O(q^{10})$$ $$12 q + 8 q^{4} - 2 q^{7} - 4 q^{10} - 12 q^{16} + 12 q^{19} - 4 q^{22} + 4 q^{25} - 24 q^{31} - 32 q^{34} + 12 q^{37} + 20 q^{40} + 4 q^{46} - 18 q^{49} - 28 q^{52} - 40 q^{55} + 8 q^{58} + 20 q^{64} + 44 q^{70} + 16 q^{76} + 28 q^{82} - 32 q^{85} + 12 q^{88} - 26 q^{91} + 56 q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{10} + 11 x^{8} - 26 x^{6} + 44 x^{4} - 64 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{10} + 4 \nu^{8} - 3 \nu^{6} + 10 \nu^{4} - 4 \nu^{2}$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{11} + 11 \nu^{7} - 14 \nu^{5} + 20 \nu^{3} - 48 \nu$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{11} + 4 \nu^{9} - 11 \nu^{7} + 26 \nu^{5} - 12 \nu^{3} + 32 \nu$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{8} + 2 \nu^{6} - 3 \nu^{4} + 8 \nu^{2} - 4$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{11} - 4 \nu^{9} + 11 \nu^{7} - 26 \nu^{5} + 44 \nu^{3} - 64 \nu$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{11} - 2 \nu^{9} + 3 \nu^{7} - 4 \nu^{5} + 8 \nu^{3} - 8 \nu$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{11} - 4 \nu^{9} + 5 \nu^{7} + 2 \nu^{5} + 20 \nu^{3}$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{10} - 2 \nu^{8} + 7 \nu^{6} - 12 \nu^{4} + 20 \nu^{2} - 24$$$$)/8$$ $$\beta_{11}$$ $$=$$ $$($$$$-\nu^{10} + 4 \nu^{8} - 11 \nu^{6} + 26 \nu^{4} - 44 \nu^{2} + 48$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{5} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{11} + \beta_{10} + \beta_{6} + \beta_{2} - 1$$ $$\nu^{5}$$ $$=$$ $$\beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{11} + 2 \beta_{10} + 2 \beta_{6} + 2 \beta_{3} - 3 \beta_{2} - 1$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{9} - 3 \beta_{7} - \beta_{5} + 4 \beta_{4} + \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-2 \beta_{11} + \beta_{10} - 3 \beta_{6} + 4 \beta_{3} - \beta_{2} + 5$$ $$\nu^{9}$$ $$=$$ $$-3 \beta_{9} - 3 \beta_{8} + \beta_{7} + 3 \beta_{5} + 5 \beta_{4} + 5 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$6 \beta_{11} + 8 \beta_{10} - 8 \beta_{6} - 6 \beta_{3} + 11 \beta_{2} + 9$$ $$\nu^{11}$$ $$=$$ $$-8 \beta_{9} + 14 \beta_{8} - \beta_{7} + 5 \beta_{5} + 2 \beta_{4} + 3 \beta_{1}$$

## Character values

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

 $$n$$ $$29$$ $$325$$ $$379$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 1.40141 − 0.189886i 1.40141 + 0.189886i 1.15343 − 0.818285i 1.15343 + 0.818285i 0.840028 − 1.13770i 0.840028 + 1.13770i −0.840028 − 1.13770i −0.840028 + 1.13770i −1.15343 − 0.818285i −1.15343 + 0.818285i −1.40141 − 0.189886i −1.40141 + 0.189886i
−1.40141 0.189886i 0 1.92789 + 0.532217i 1.46432i 0 −2.07772 1.63801i −2.60069 1.11193i 0 0.278054 2.05211i
55.2 −1.40141 + 0.189886i 0 1.92789 0.532217i 1.46432i 0 −2.07772 + 1.63801i −2.60069 + 1.11193i 0 0.278054 + 2.05211i
55.3 −1.15343 0.818285i 0 0.660819 + 1.88768i 2.16295i 0 1.94827 + 1.79004i 0.782447 2.71805i 0 1.76991 2.49482i
55.4 −1.15343 + 0.818285i 0 0.660819 1.88768i 2.16295i 0 1.94827 1.79004i 0.782447 + 2.71805i 0 1.76991 + 2.49482i
55.5 −0.840028 1.13770i 0 −0.588705 + 1.91139i 2.67907i 0 −0.370556 2.61967i 2.66911 0.935858i 0 −3.04797 + 2.25049i
55.6 −0.840028 + 1.13770i 0 −0.588705 1.91139i 2.67907i 0 −0.370556 + 2.61967i 2.66911 + 0.935858i 0 −3.04797 2.25049i
55.7 0.840028 1.13770i 0 −0.588705 1.91139i 2.67907i 0 −0.370556 + 2.61967i −2.66911 0.935858i 0 −3.04797 2.25049i
55.8 0.840028 + 1.13770i 0 −0.588705 + 1.91139i 2.67907i 0 −0.370556 2.61967i −2.66911 + 0.935858i 0 −3.04797 + 2.25049i
55.9 1.15343 0.818285i 0 0.660819 1.88768i 2.16295i 0 1.94827 1.79004i −0.782447 2.71805i 0 1.76991 + 2.49482i
55.10 1.15343 + 0.818285i 0 0.660819 + 1.88768i 2.16295i 0 1.94827 + 1.79004i −0.782447 + 2.71805i 0 1.76991 2.49482i
55.11 1.40141 0.189886i 0 1.92789 0.532217i 1.46432i 0 −2.07772 + 1.63801i 2.60069 1.11193i 0 0.278054 + 2.05211i
55.12 1.40141 + 0.189886i 0 1.92789 + 0.532217i 1.46432i 0 −2.07772 1.63801i 2.60069 + 1.11193i 0 0.278054 2.05211i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.12 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
28.d even 2 1 inner
84.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.b.c 12
3.b odd 2 1 inner 756.2.b.c 12
4.b odd 2 1 756.2.b.d yes 12
7.b odd 2 1 756.2.b.d yes 12
12.b even 2 1 756.2.b.d yes 12
21.c even 2 1 756.2.b.d yes 12
28.d even 2 1 inner 756.2.b.c 12
84.h odd 2 1 inner 756.2.b.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.b.c 12 1.a even 1 1 trivial
756.2.b.c 12 3.b odd 2 1 inner
756.2.b.c 12 28.d even 2 1 inner
756.2.b.c 12 84.h odd 2 1 inner
756.2.b.d yes 12 4.b odd 2 1
756.2.b.d yes 12 7.b odd 2 1
756.2.b.d yes 12 12.b even 2 1
756.2.b.d yes 12 21.c even 2 1

## Hecke kernels

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

 $$T_{5}^{6} + 14 T_{5}^{4} + 59 T_{5}^{2} + 72$$ $$T_{19}^{3} - 3 T_{19}^{2} - 11 T_{19} - 3$$ $$T_{47}^{6} - 180 T_{47}^{4} + 9868 T_{47}^{2} - 152928$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 - 64 T^{2} + 44 T^{4} - 26 T^{6} + 11 T^{8} - 4 T^{10} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$( 72 + 59 T^{2} + 14 T^{4} + T^{6} )^{2}$$
$7$ $$( 343 + 49 T + 35 T^{2} + 2 T^{3} + 5 T^{4} + T^{5} + T^{6} )^{2}$$
$11$ $$( 8 + 51 T^{2} + 26 T^{4} + T^{6} )^{2}$$
$13$ $$( 2124 + 656 T^{2} + 55 T^{4} + T^{6} )^{2}$$
$17$ $$( 288 + 620 T^{2} + 52 T^{4} + T^{6} )^{2}$$
$19$ $$( -3 - 11 T - 3 T^{2} + T^{3} )^{4}$$
$23$ $$( 8 + 51 T^{2} + 26 T^{4} + T^{6} )^{2}$$
$29$ $$( -30208 + 3852 T^{2} - 132 T^{4} + T^{6} )^{2}$$
$31$ $$( -108 - 45 T + 6 T^{2} + T^{3} )^{4}$$
$37$ $$( 183 - 53 T - 3 T^{2} + T^{3} )^{4}$$
$41$ $$( 1152 + 2459 T^{2} + 134 T^{4} + T^{6} )^{2}$$
$43$ $$( 944 + 2100 T^{2} + 104 T^{4} + T^{6} )^{2}$$
$47$ $$( -152928 + 9868 T^{2} - 180 T^{4} + T^{6} )^{2}$$
$53$ $$( -120832 + 14512 T^{2} - 232 T^{4} + T^{6} )^{2}$$
$59$ $$( -2871648 + 60844 T^{2} - 428 T^{4} + T^{6} )^{2}$$
$61$ $$( 76464 + 7928 T^{2} + 211 T^{4} + T^{6} )^{2}$$
$67$ $$( 2124 + 656 T^{2} + 55 T^{4} + T^{6} )^{2}$$
$71$ $$( 59168 + 5803 T^{2} + 150 T^{4} + T^{6} )^{2}$$
$73$ $$( 19116 + 6008 T^{2} + 191 T^{4} + T^{6} )^{2}$$
$79$ $$( 1786284 + 51512 T^{2} + 415 T^{4} + T^{6} )^{2}$$
$83$ $$( -1087488 + 44716 T^{2} - 396 T^{4} + T^{6} )^{2}$$
$89$ $$( 1774728 + 57899 T^{2} + 446 T^{4} + T^{6} )^{2}$$
$97$ $$( 76464 + 7928 T^{2} + 211 T^{4} + T^{6} )^{2}$$
show more
show less