# Properties

 Label 756.2.b.f Level 756 Weight 2 Character orbit 756.b Analytic conductor 6.037 Analytic rank 0 Dimension 16 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: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{14} q^{2} + \beta_{8} q^{4} + \beta_{10} q^{5} + \beta_{6} q^{7} + \beta_{4} q^{8} +O(q^{10})$$ $$q -\beta_{14} q^{2} + \beta_{8} q^{4} + \beta_{10} q^{5} + \beta_{6} q^{7} + \beta_{4} q^{8} + ( \beta_{2} + \beta_{15} ) q^{10} -\beta_{7} q^{11} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{13} + ( \beta_{4} + \beta_{5} + \beta_{7} + \beta_{13} ) q^{14} + ( \beta_{10} - \beta_{11} ) q^{16} + ( \beta_{3} + \beta_{11} ) q^{17} + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} + \beta_{9} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{19} + ( 2 - \beta_{5} + \beta_{6} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{20} + ( 1 - \beta_{3} + \beta_{9} - \beta_{12} ) q^{22} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{15} ) q^{23} + ( -2 + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{25} + ( 1 - \beta_{3} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{13} ) q^{26} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{15} ) q^{28} + ( -\beta_{1} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{29} + ( \beta_{7} + \beta_{9} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{31} + ( -\beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{32} + ( \beta_{1} + \beta_{5} + \beta_{6} - \beta_{9} - \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{34} + ( 1 - \beta_{3} - \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{35} + ( 1 - \beta_{3} - \beta_{10} + \beta_{11} ) q^{37} + ( 1 + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{10} ) q^{38} + ( \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{7} - \beta_{14} + \beta_{15} ) q^{40} + ( \beta_{5} - \beta_{6} + \beta_{8} - \beta_{12} + \beta_{13} ) q^{41} + ( -\beta_{3} - \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{43} + ( \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{14} - \beta_{15} ) q^{44} + ( -\beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} ) q^{46} + ( 1 - \beta_{3} + 3 \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{47} + ( -1 - \beta_{1} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{49} + ( \beta_{1} - \beta_{2} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{14} + \beta_{15} ) q^{50} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{13} ) q^{52} + ( -\beta_{2} + \beta_{4} ) q^{53} + ( -3 \beta_{2} + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{55} + ( -1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{12} + \beta_{14} + 2 \beta_{15} ) q^{56} + ( 1 + \beta_{3} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{58} + ( 3 - \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} ) q^{59} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{14} - 2 \beta_{15} ) q^{61} + ( -3 + \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{62} + ( 2 + \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{64} + ( -3 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{65} + ( \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{67} + ( -\beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{68} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{70} + ( -\beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{14} - 2 \beta_{15} ) q^{71} + ( \beta_{1} - \beta_{5} - \beta_{6} - 2 \beta_{7} - 4 \beta_{14} - 2 \beta_{15} ) q^{73} + ( \beta_{1} + 3 \beta_{2} + \beta_{5} + \beta_{6} - \beta_{9} - \beta_{13} + \beta_{15} ) q^{74} + ( -\beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{76} + ( -\beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{8} - \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{77} + ( -\beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{10} + \beta_{12} - \beta_{13} ) q^{79} + ( -4 + 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{12} ) q^{80} + ( 2 \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{13} + \beta_{15} ) q^{82} + ( -1 + \beta_{3} + \beta_{5} - \beta_{6} - 4 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{13} ) q^{83} + ( 1 - \beta_{3} - 3 \beta_{8} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{85} + ( -3 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{13} - \beta_{14} - 4 \beta_{15} ) q^{86} + ( -2 - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - 3 \beta_{10} - \beta_{12} ) q^{88} + ( -\beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{10} - \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{89} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{91} + ( \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + 2 \beta_{7} + \beta_{14} + 3 \beta_{15} ) q^{92} + ( -\beta_{1} + 3 \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} - \beta_{13} + \beta_{15} ) q^{94} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} + 2 \beta_{15} ) q^{95} + ( -\beta_{1} - 4 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{14} - 4 \beta_{15} ) q^{97} + ( 3 + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + q^{14} + 4q^{16} + 26q^{20} + 10q^{22} - 20q^{25} + 6q^{26} - 11q^{28} + 6q^{35} + 8q^{37} + 20q^{38} - 6q^{46} + 8q^{47} - 14q^{49} - 21q^{56} + 14q^{58} + 44q^{59} - 48q^{62} + 24q^{64} + 2q^{68} - 27q^{70} - 54q^{80} - 4q^{83} + 8q^{85} - 34q^{88} + 47q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{12} - 4 x^{10} - 4 x^{8} - 16 x^{6} - 16 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3}$$ $$\beta_{3}$$ $$=$$ $$\nu^{4}$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{13} - \nu^{9} - 4 \nu^{7} - 4 \nu^{5} - 16 \nu^{3} - 16 \nu$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} - 6 \nu^{14} - 4 \nu^{13} - 24 \nu^{12} - 15 \nu^{11} - 90 \nu^{10} - 56 \nu^{9} + 48 \nu^{8} + 164 \nu^{7} - 168 \nu^{6} + 288 \nu^{5} + 192 \nu^{4} + 400 \nu^{3} + 864 \nu^{2} + 64 \nu + 1920$$$$)/1536$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{15} + 6 \nu^{14} - 4 \nu^{13} + 24 \nu^{12} - 15 \nu^{11} + 90 \nu^{10} - 56 \nu^{9} - 48 \nu^{8} + 164 \nu^{7} + 168 \nu^{6} + 288 \nu^{5} - 192 \nu^{4} + 400 \nu^{3} - 864 \nu^{2} + 64 \nu - 1920$$$$)/1536$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} - 4 \nu^{13} - 15 \nu^{11} + 8 \nu^{9} + 36 \nu^{7} - 32 \nu^{5} + 16 \nu^{3} + 320 \nu$$$$)/256$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{14} + \nu^{10} + 4 \nu^{8} + 4 \nu^{6} + 16 \nu^{4} + 16 \nu^{2}$$$$)/64$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} - 4 \nu^{13} - 4 \nu^{12} + 9 \nu^{11} - 15 \nu^{10} - 8 \nu^{9} - 8 \nu^{8} - 4 \nu^{7} + 20 \nu^{6} + 96 \nu^{4} + 112 \nu^{3} + 208 \nu^{2} + 64 \nu + 256$$$$)/192$$ $$\beta_{10}$$ $$=$$ $$($$$$7 \nu^{14} + 4 \nu^{12} + 9 \nu^{10} + 32 \nu^{8} + 4 \nu^{6} - 192 \nu^{4} - 112 \nu^{2} - 448$$$$)/384$$ $$\beta_{11}$$ $$=$$ $$($$$$7 \nu^{14} + 28 \nu^{12} + 9 \nu^{10} + 8 \nu^{8} - 92 \nu^{6} - 288 \nu^{4} - 496 \nu^{2} - 832$$$$)/384$$ $$\beta_{12}$$ $$=$$ $$($$$$-2 \nu^{15} + \nu^{14} - 8 \nu^{13} + 4 \nu^{12} + 18 \nu^{11} + 15 \nu^{10} - 16 \nu^{9} - 40 \nu^{8} - 8 \nu^{7} - 68 \nu^{6} - 96 \nu^{4} + 224 \nu^{3} + 368 \nu^{2} + 128 \nu - 64$$$$)/384$$ $$\beta_{13}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{14} - 4 \nu^{13} + 4 \nu^{12} + 9 \nu^{11} + 15 \nu^{10} - 8 \nu^{9} + 8 \nu^{8} - 4 \nu^{7} - 20 \nu^{6} - 96 \nu^{4} + 112 \nu^{3} - 208 \nu^{2} + 64 \nu - 256$$$$)/192$$ $$\beta_{14}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{11} + 4 \nu^{9} + 4 \nu^{7} + 16 \nu^{5} + 16 \nu^{3}$$$$)/128$$ $$\beta_{15}$$ $$=$$ $$($$$$-7 \nu^{15} - 28 \nu^{13} - 9 \nu^{11} - 8 \nu^{9} - 100 \nu^{7} + 96 \nu^{5} + 112 \nu^{3} + 448 \nu$$$$)/768$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{13} + \beta_{12} + \beta_{10} + \beta_{8}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{2}$$ $$\nu^{4}$$ $$=$$ $$\beta_{3}$$ $$\nu^{5}$$ $$=$$ $$\beta_{15} + \beta_{14} - \beta_{13} - \beta_{9} - 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-\beta_{13} - \beta_{12} + \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{6} - 2 \beta_{5} - \beta_{3} + 2$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{9} + 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{2} - \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$\beta_{13} - \beta_{12} + 5 \beta_{10} + 5 \beta_{8} - 2 \beta_{6} + 2 \beta_{5} + \beta_{3} + 2$$ $$\nu^{9}$$ $$=$$ $$-\beta_{15} + 7 \beta_{14} - 3 \beta_{13} - 3 \beta_{9} - 2 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} - 7 \beta_{4} + 2 \beta_{2} + \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$\beta_{13} + 5 \beta_{12} - 4 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + 3 \beta_{8} + 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{3} + 10$$ $$\nu^{11}$$ $$=$$ $$-7 \beta_{15} + 5 \beta_{14} + 5 \beta_{13} + 5 \beta_{9} - 6 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} - 6 \beta_{2} + 3 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-11 \beta_{13} + 3 \beta_{12} + 16 \beta_{11} + \beta_{10} + 8 \beta_{9} + 17 \beta_{8} + 6 \beta_{6} - 6 \beta_{5} + \beta_{3} + 26$$ $$\nu^{13}$$ $$=$$ $$-9 \beta_{15} + 15 \beta_{14} - 3 \beta_{13} - 3 \beta_{9} - 2 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 17 \beta_{4} + 10 \beta_{2} + 9 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-7 \beta_{13} + 5 \beta_{12} - 4 \beta_{11} + 35 \beta_{10} + 2 \beta_{9} - 29 \beta_{8} + 2 \beta_{6} - 2 \beta_{5} + 19 \beta_{3} + 26$$ $$\nu^{15}$$ $$=$$ $$-7 \beta_{15} - 75 \beta_{14} - 19 \beta_{13} - 19 \beta_{9} - 38 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 53 \beta_{4} + 10 \beta_{2} + 19 \beta_{1}$$

## 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
 1.41109 − 0.0939769i 1.41109 + 0.0939769i 1.16433 − 0.802711i 1.16433 + 0.802711i 0.748450 − 1.19993i 0.748450 + 1.19993i 0.304958 − 1.38094i 0.304958 + 1.38094i −0.304958 − 1.38094i −0.304958 + 1.38094i −0.748450 − 1.19993i −0.748450 + 1.19993i −1.16433 − 0.802711i −1.16433 + 0.802711i −1.41109 − 0.0939769i −1.41109 + 0.0939769i
−1.41109 0.0939769i 0 1.98234 + 0.265219i 2.91758i 0 1.63542 2.07976i −2.77233 0.560541i 0 −0.274185 + 4.11696i
55.2 −1.41109 + 0.0939769i 0 1.98234 0.265219i 2.91758i 0 1.63542 + 2.07976i −2.77233 + 0.560541i 0 −0.274185 4.11696i
55.3 −1.16433 0.802711i 0 0.711311 + 1.86923i 0.944421i 0 −2.59468 + 0.517335i 0.672257 2.74738i 0 0.758097 1.09961i
55.4 −1.16433 + 0.802711i 0 0.711311 1.86923i 0.944421i 0 −2.59468 0.517335i 0.672257 + 2.74738i 0 0.758097 + 1.09961i
55.5 −0.748450 1.19993i 0 −0.879646 + 1.79617i 3.90968i 0 1.12924 + 2.39266i 2.81364 0.288831i 0 −4.69133 + 2.92620i
55.6 −0.748450 + 1.19993i 0 −0.879646 1.79617i 3.90968i 0 1.12924 2.39266i 2.81364 + 0.288831i 0 −4.69133 2.92620i
55.7 −0.304958 1.38094i 0 −1.81400 + 0.842259i 0.556957i 0 −1.25214 2.33070i 1.71630 + 2.24818i 0 −0.769125 + 0.169848i
55.8 −0.304958 + 1.38094i 0 −1.81400 0.842259i 0.556957i 0 −1.25214 + 2.33070i 1.71630 2.24818i 0 −0.769125 0.169848i
55.9 0.304958 1.38094i 0 −1.81400 0.842259i 0.556957i 0 1.25214 + 2.33070i −1.71630 + 2.24818i 0 0.769125 + 0.169848i
55.10 0.304958 + 1.38094i 0 −1.81400 + 0.842259i 0.556957i 0 1.25214 2.33070i −1.71630 2.24818i 0 0.769125 0.169848i
55.11 0.748450 1.19993i 0 −0.879646 1.79617i 3.90968i 0 −1.12924 2.39266i −2.81364 0.288831i 0 4.69133 + 2.92620i
55.12 0.748450 + 1.19993i 0 −0.879646 + 1.79617i 3.90968i 0 −1.12924 + 2.39266i −2.81364 + 0.288831i 0 4.69133 2.92620i
55.13 1.16433 0.802711i 0 0.711311 1.86923i 0.944421i 0 2.59468 0.517335i −0.672257 2.74738i 0 −0.758097 1.09961i
55.14 1.16433 + 0.802711i 0 0.711311 + 1.86923i 0.944421i 0 2.59468 + 0.517335i −0.672257 + 2.74738i 0 −0.758097 + 1.09961i
55.15 1.41109 0.0939769i 0 1.98234 0.265219i 2.91758i 0 −1.63542 + 2.07976i 2.77233 0.560541i 0 0.274185 + 4.11696i
55.16 1.41109 + 0.0939769i 0 1.98234 + 0.265219i 2.91758i 0 −1.63542 2.07976i 2.77233 + 0.560541i 0 0.274185 4.11696i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 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.f yes 16
3.b odd 2 1 756.2.b.e 16
4.b odd 2 1 756.2.b.e 16
7.b odd 2 1 756.2.b.e 16
12.b even 2 1 inner 756.2.b.f yes 16
21.c even 2 1 inner 756.2.b.f yes 16
28.d even 2 1 inner 756.2.b.f yes 16
84.h odd 2 1 756.2.b.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.b.e 16 3.b odd 2 1
756.2.b.e 16 4.b odd 2 1
756.2.b.e 16 7.b odd 2 1
756.2.b.e 16 84.h odd 2 1
756.2.b.f yes 16 1.a even 1 1 trivial
756.2.b.f yes 16 12.b even 2 1 inner
756.2.b.f yes 16 21.c even 2 1 inner
756.2.b.f yes 16 28.d even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{8} + 25 T_{5}^{6} + 159 T_{5}^{4} + 163 T_{5}^{2} + 36$$ $$T_{19}^{8} - 68 T_{19}^{6} + 1424 T_{19}^{4} - 9280 T_{19}^{2} + 9216$$ $$T_{47}^{4} - 2 T_{47}^{3} - 115 T_{47}^{2} + 408 T_{47} - 324$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} - 4 T^{6} - 4 T^{8} - 16 T^{10} - 16 T^{12} + 256 T^{16}$$
$3$ 1
$5$ $$( 1 - 15 T^{2} + 109 T^{4} - 642 T^{6} + 3506 T^{8} - 16050 T^{10} + 68125 T^{12} - 234375 T^{14} + 390625 T^{16} )^{2}$$
$7$ $$1 + 7 T^{2} + 62 T^{4} - 343 T^{6} - 1662 T^{8} - 16807 T^{10} + 148862 T^{12} + 823543 T^{14} + 5764801 T^{16}$$
$11$ $$( 1 - 45 T^{2} + 1070 T^{4} - 17563 T^{6} + 218210 T^{8} - 2125123 T^{10} + 15665870 T^{12} - 79720245 T^{14} + 214358881 T^{16} )^{2}$$
$13$ $$( 1 - 43 T^{2} + 1050 T^{4} - 19597 T^{6} + 291626 T^{8} - 3311893 T^{10} + 29989050 T^{12} - 207552787 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 - 55 T^{2} + 2117 T^{4} - 53882 T^{6} + 1068186 T^{8} - 15571898 T^{10} + 176813957 T^{12} - 1327566295 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 + 84 T^{2} + 3780 T^{4} + 114828 T^{6} + 2535190 T^{8} + 41452908 T^{10} + 492613380 T^{12} + 3951854004 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 87 T^{2} + 3842 T^{4} - 111721 T^{6} + 2697242 T^{8} - 59100409 T^{10} + 1075149122 T^{12} - 12879122343 T^{14} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 + 133 T^{2} + 9622 T^{4} + 456771 T^{6} + 15547890 T^{8} + 384144411 T^{10} + 6805457782 T^{12} + 79111501693 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 + 124 T^{2} + 8479 T^{4} + 404384 T^{6} + 14327960 T^{8} + 388613024 T^{10} + 7830534559 T^{12} + 110050456444 T^{14} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 - 2 T + 77 T^{2} - 134 T^{3} + 4112 T^{4} - 4958 T^{5} + 105413 T^{6} - 101306 T^{7} + 1874161 T^{8} )^{4}$$
$41$ $$( 1 - 210 T^{2} + 21937 T^{4} - 1498866 T^{6} + 72486980 T^{8} - 2519593746 T^{10} + 61988719057 T^{12} - 997521890610 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 - 159 T^{2} + 14377 T^{4} - 922458 T^{6} + 45070550 T^{8} - 1705624842 T^{10} + 49152101977 T^{12} - 1005096724791 T^{14} + 11688200277601 T^{16} )^{2}$$
$47$ $$( 1 - 2 T + 73 T^{2} + 126 T^{3} + 2120 T^{4} + 5922 T^{5} + 161257 T^{6} - 207646 T^{7} + 4879681 T^{8} )^{4}$$
$53$ $$( 1 + 348 T^{2} + 56327 T^{4} + 5511008 T^{6} + 355579928 T^{8} + 15480421472 T^{10} + 444447123287 T^{12} + 7713197672892 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 - 11 T + 159 T^{2} - 1104 T^{3} + 10756 T^{4} - 65136 T^{5} + 553479 T^{6} - 2259169 T^{7} + 12117361 T^{8} )^{4}$$
$61$ $$( 1 - 268 T^{2} + 40068 T^{4} - 3979924 T^{6} + 284320790 T^{8} - 14809297204 T^{10} + 554775157188 T^{12} - 13807460328748 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 - 325 T^{2} + 52002 T^{4} - 5466211 T^{6} + 421211930 T^{8} - 24537821179 T^{10} + 1047898594242 T^{12} - 29398974204925 T^{14} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 - 419 T^{2} + 83894 T^{4} - 10494221 T^{6} + 893482514 T^{8} - 52901368061 T^{10} + 2131887565814 T^{12} - 53674018962899 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 - 105 T^{2} + 4878 T^{4} - 673927 T^{6} + 86417250 T^{8} - 3591356983 T^{10} + 138526619598 T^{12} - 15890093760345 T^{14} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 - 406 T^{2} + 84777 T^{4} - 11389882 T^{6} + 1067641484 T^{8} - 71084253562 T^{10} + 3302071016937 T^{12} - 98693506941526 T^{14} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 + T + 175 T^{2} - 192 T^{3} + 17036 T^{4} - 15936 T^{5} + 1205575 T^{6} + 571787 T^{7} + 47458321 T^{8} )^{4}$$
$89$ $$( 1 - 93 T^{2} + 15598 T^{4} - 511251 T^{6} + 94398818 T^{8} - 4049619171 T^{10} + 978653475118 T^{12} - 46219260059373 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 + 47 T^{2} + 17382 T^{4} + 1290401 T^{6} + 195527762 T^{8} + 12141383009 T^{10} + 1538815962342 T^{12} + 39149684231663 T^{14} + 7837433594376961 T^{16} )^{2}$$