# Properties

 Label 756.2.b.d Level 756 Weight 2 Character orbit 756.b 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.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 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{8} q^{5} + ( \beta_{6} + \beta_{11} ) q^{7} + ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{8} q^{5} + ( \beta_{6} + \beta_{11} ) q^{7} + ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{8} + ( -1 - \beta_{2} + \beta_{6} + \beta_{10} ) q^{10} + ( \beta_{4} + \beta_{8} ) q^{11} + ( -1 - \beta_{2} + 2 \beta_{10} + \beta_{11} ) q^{13} + ( \beta_{7} - \beta_{9} ) q^{14} + ( -1 + \beta_{2} + \beta_{6} + \beta_{10} + 2 \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_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{20} + ( 1 + \beta_{2} - 2 \beta_{10} ) q^{22} + ( -\beta_{4} - \beta_{8} ) q^{23} + ( 1 + \beta_{2} + \beta_{11} ) q^{25} + ( -2 \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} ) q^{26} + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{10} ) q^{28} + ( \beta_{4} + 2 \beta_{9} ) q^{29} + ( -3 \beta_{2} - 3 \beta_{11} ) q^{31} + ( \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{32} + ( 2 - 2 \beta_{2} + \beta_{6} - \beta_{10} ) 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_{1} - \beta_{4} + \beta_{8} - \beta_{9} ) q^{38} + ( -3 - \beta_{2} + 2 \beta_{3} - 2 \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} + 2 \beta_{8} ) q^{44} + ( -1 - \beta_{2} + 2 \beta_{10} ) 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_{5} + \beta_{7} ) q^{50} + ( -2 \beta_{2} - \beta_{6} + 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} + ( -\beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{56} + ( 2 + 2 \beta_{2} - 4 \beta_{3} - \beta_{6} + \beta_{10} ) 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} ) q^{62} + ( -1 - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + 2 \beta_{10} + 2 \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} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{68} + ( -1 - \beta_{2} + 2 \beta_{10} + 4 \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_{1} - 2 \beta_{4} + 2 \beta_{8} - 2 \beta_{9} ) q^{74} + ( -1 - \beta_{2} + 2 \beta_{3} - \beta_{6} - \beta_{10} ) 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} + ( \beta_{1} - 2 \beta_{4} - \beta_{5} - 3 \beta_{7} + 2 \beta_{8} ) q^{80} + ( -3 + \beta_{2} + 3 \beta_{6} - \beta_{10} ) 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} + ( -2 \beta_{1} - \beta_{4} - 4 \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{86} + ( 3 + \beta_{2} + \beta_{6} - 3 \beta_{10} + 2 \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} - 2 \beta_{8} ) q^{92} + ( -4 - 4 \beta_{3} - \beta_{6} + \beta_{10} ) 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} + ( 4 \beta_{1} + 2 \beta_{4} - \beta_{5} + 6 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 8q^{4} + 2q^{7} + O(q^{10})$$ $$12q + 8q^{4} + 2q^{7} + 4q^{10} - 12q^{16} - 12q^{19} - 4q^{22} + 4q^{25} + 20q^{28} + 24q^{31} + 32q^{34} + 12q^{37} - 20q^{40} + 4q^{46} - 18q^{49} + 28q^{52} + 40q^{55} + 8q^{58} + 20q^{64} - 12q^{70} - 16q^{76} - 28q^{82} - 32q^{85} + 12q^{88} + 26q^{91} - 56q^{94} + O(q^{100})$$

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.d yes 12
3.b odd 2 1 inner 756.2.b.d yes 12
4.b odd 2 1 756.2.b.c 12
7.b odd 2 1 756.2.b.c 12
12.b even 2 1 756.2.b.c 12
21.c even 2 1 756.2.b.c 12
28.d even 2 1 inner 756.2.b.d yes 12
84.h odd 2 1 inner 756.2.b.d yes 12

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

## 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

$p$ $F_p(T)$
$2$ $$1 - 4 T^{2} + 11 T^{4} - 26 T^{6} + 44 T^{8} - 64 T^{10} + 64 T^{12}$$
$3$ 1
$5$ $$( 1 - 16 T^{2} + 154 T^{4} - 918 T^{6} + 3850 T^{8} - 10000 T^{10} + 15625 T^{12} )^{2}$$
$7$ $$( 1 - T + 5 T^{2} - 2 T^{3} + 35 T^{4} - 49 T^{5} + 343 T^{6} )^{2}$$
$11$ $$( 1 - 40 T^{2} + 722 T^{4} - 8858 T^{6} + 87362 T^{8} - 585640 T^{10} + 1771561 T^{12} )^{2}$$
$13$ $$( 1 - 23 T^{2} + 331 T^{4} - 3102 T^{6} + 55939 T^{8} - 656903 T^{10} + 4826809 T^{12} )^{2}$$
$17$ $$( 1 - 50 T^{2} + 1419 T^{4} - 28884 T^{6} + 410091 T^{8} - 4176050 T^{10} + 24137569 T^{12} )^{2}$$
$19$ $$( 1 + 3 T + 46 T^{2} + 117 T^{3} + 874 T^{4} + 1083 T^{5} + 6859 T^{6} )^{4}$$
$23$ $$( 1 - 112 T^{2} + 5594 T^{4} - 163154 T^{6} + 2959226 T^{8} - 31342192 T^{10} + 148035889 T^{12} )^{2}$$
$29$ $$( 1 + 42 T^{2} + 1155 T^{4} + 14916 T^{6} + 971355 T^{8} + 29705802 T^{10} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 - 6 T + 48 T^{2} - 264 T^{3} + 1488 T^{4} - 5766 T^{5} + 29791 T^{6} )^{4}$$
$37$ $$( 1 - 3 T + 58 T^{2} - 39 T^{3} + 2146 T^{4} - 4107 T^{5} + 50653 T^{6} )^{4}$$
$41$ $$( 1 - 112 T^{2} + 5698 T^{4} - 227382 T^{6} + 9578338 T^{8} - 316485232 T^{10} + 4750104241 T^{12} )^{2}$$
$43$ $$( 1 - 154 T^{2} + 11947 T^{4} - 616020 T^{6} + 22090003 T^{8} - 526495354 T^{10} + 6321363049 T^{12} )^{2}$$
$47$ $$( 1 + 102 T^{2} + 9163 T^{4} + 465404 T^{6} + 20241067 T^{8} + 497727462 T^{10} + 10779215329 T^{12} )^{2}$$
$53$ $$( 1 + 86 T^{2} + 7463 T^{4} + 484852 T^{6} + 20963567 T^{8} + 678581366 T^{10} + 22164361129 T^{12} )^{2}$$
$59$ $$( 1 - 74 T^{2} + 12051 T^{4} - 523684 T^{6} + 41949531 T^{8} - 896684714 T^{10} + 42180533641 T^{12} )^{2}$$
$61$ $$( 1 - 155 T^{2} + 12259 T^{4} - 719586 T^{6} + 45615739 T^{8} - 2146105355 T^{10} + 51520374361 T^{12} )^{2}$$
$67$ $$( 1 - 347 T^{2} + 53251 T^{4} - 4619670 T^{6} + 239043739 T^{8} - 6992438987 T^{10} + 90458382169 T^{12} )^{2}$$
$71$ $$( 1 - 276 T^{2} + 38818 T^{4} - 3386178 T^{6} + 195681538 T^{8} - 7013623956 T^{10} + 128100283921 T^{12} )^{2}$$
$73$ $$( 1 - 247 T^{2} + 30171 T^{4} - 2531358 T^{6} + 160781259 T^{8} - 7014365527 T^{10} + 151334226289 T^{12} )^{2}$$
$79$ $$( 1 - 59 T^{2} + 13987 T^{4} - 673302 T^{6} + 87292867 T^{8} - 2298054779 T^{10} + 243087455521 T^{12} )^{2}$$
$83$ $$( 1 + 102 T^{2} + 16579 T^{4} + 1402844 T^{6} + 114212731 T^{8} + 4840748742 T^{10} + 326940373369 T^{12} )^{2}$$
$89$ $$( 1 - 88 T^{2} + 17938 T^{4} - 1434078 T^{6} + 142086898 T^{8} - 5521317208 T^{10} + 496981290961 T^{12} )^{2}$$
$97$ $$( 1 - 371 T^{2} + 67195 T^{4} - 7803234 T^{6} + 632237755 T^{8} - 32844363251 T^{10} + 832972004929 T^{12} )^{2}$$