# 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: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - x^{12} - 4 x^{10} - 4 x^{8} - 16 x^{6} - 16 x^{4} + 256$$ 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + O(q^{10})$$ $$16 q + q^{14} + 4 q^{16} + 26 q^{20} + 10 q^{22} - 20 q^{25} + 6 q^{26} - 11 q^{28} + 6 q^{35} + 8 q^{37} + 20 q^{38} - 6 q^{46} + 8 q^{47} - 14 q^{49} - 21 q^{56} + 14 q^{58} + 44 q^{59} - 48 q^{62} + 24 q^{64} + 2 q^{68} - 27 q^{70} - 54 q^{80} - 4 q^{83} + 8 q^{85} - 34 q^{88} + 47 q^{98} + O(q^{100})$$

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

$p$ $F_p(T)$
$2$ $$256 - 16 T^{4} - 16 T^{6} - 4 T^{8} - 4 T^{10} - T^{12} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 36 + 163 T^{2} + 159 T^{4} + 25 T^{6} + T^{8} )^{2}$$
$7$ $$5764801 + 823543 T^{2} + 148862 T^{4} - 16807 T^{6} - 1662 T^{8} - 343 T^{10} + 62 T^{12} + 7 T^{14} + T^{16}$$
$11$ $$( 256 + 1808 T^{2} + 520 T^{4} + 43 T^{6} + T^{8} )^{2}$$
$13$ $$( 5184 + 4752 T^{2} + 1076 T^{4} + 61 T^{6} + T^{8} )^{2}$$
$17$ $$( 86436 + 25627 T^{2} + 2287 T^{4} + 81 T^{6} + T^{8} )^{2}$$
$19$ $$( 9216 - 9280 T^{2} + 1424 T^{4} - 68 T^{6} + T^{8} )^{2}$$
$23$ $$( 65536 + 22208 T^{2} + 2416 T^{4} + 97 T^{6} + T^{8} )^{2}$$
$29$ $$( 166464 - 42928 T^{2} + 3300 T^{4} - 99 T^{6} + T^{8} )^{2}$$
$31$ $$( 11664 - 51192 T^{2} + 4635 T^{4} - 124 T^{6} + T^{8} )^{2}$$
$37$ $$( 1152 + 88 T - 71 T^{2} - 2 T^{3} + T^{4} )^{4}$$
$41$ $$( 36864 + 24448 T^{2} + 3897 T^{4} + 118 T^{6} + T^{8} )^{2}$$
$43$ $$( 459684 + 176579 T^{2} + 10335 T^{4} + 185 T^{6} + T^{8} )^{2}$$
$47$ $$( -324 + 408 T - 115 T^{2} - 2 T^{3} + T^{4} )^{4}$$
$53$ $$( 20736 - 14560 T^{2} + 1843 T^{4} - 76 T^{6} + T^{8} )^{2}$$
$59$ $$( -1044 + 843 T - 77 T^{2} - 11 T^{3} + T^{4} )^{4}$$
$61$ $$( 2985984 + 453312 T^{2} + 16400 T^{4} + 220 T^{6} + T^{8} )^{2}$$
$67$ $$( 419904 + 152208 T^{2} + 11132 T^{4} + 211 T^{6} + T^{8} )^{2}$$
$71$ $$( 16384 + 48640 T^{2} + 6220 T^{4} + 149 T^{6} + T^{8} )^{2}$$
$73$ $$( 15116544 + 1938816 T^{2} + 65468 T^{4} + 479 T^{6} + T^{8} )^{2}$$
$79$ $$( 3779136 + 483660 T^{2} + 17153 T^{4} + 226 T^{6} + T^{8} )^{2}$$
$83$ $$( 1764 - 441 T - 157 T^{2} + T^{3} + T^{4} )^{4}$$
$89$ $$( 244859904 + 9691264 T^{2} + 124356 T^{4} + 619 T^{6} + T^{8} )^{2}$$
$97$ $$( 1035809856 + 26617392 T^{2} + 232916 T^{4} + 823 T^{6} + T^{8} )^{2}$$