# Properties

 Label 1456.2.cc.c Level $1456$ Weight $2$ Character orbit 1456.cc Analytic conductor $11.626$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1456 = 2^{4} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1456.cc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.6262185343$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.58891012706304.1 Defining polynomial: $$x^{12} - 5 x^{10} - 2 x^{9} + 15 x^{8} + 2 x^{7} - 30 x^{6} + 4 x^{5} + 60 x^{4} - 16 x^{3} - 80 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{10} ) q^{3} + ( -\beta_{3} - \beta_{5} - \beta_{7} - \beta_{11} ) q^{5} + ( -\beta_{4} - \beta_{7} ) q^{7} + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{10} ) q^{3} + ( -\beta_{3} - \beta_{5} - \beta_{7} - \beta_{11} ) q^{5} + ( -\beta_{4} - \beta_{7} ) q^{7} + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{9} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} ) q^{11} + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{13} + ( -2 - \beta_{1} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{15} + ( -1 + \beta_{1} - \beta_{5} + \beta_{6} + \beta_{9} - \beta_{11} ) q^{17} + ( \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} + 2 \beta_{8} - 2 \beta_{11} ) q^{19} + ( -\beta_{3} + \beta_{8} - \beta_{11} ) q^{21} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{23} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} ) q^{25} + ( -1 - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{27} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{10} + 2 \beta_{11} ) q^{29} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{31} + ( -1 + \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 4 \beta_{10} - \beta_{11} ) q^{33} + ( -1 - \beta_{2} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{35} + ( -4 - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{37} + ( 2 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{39} + ( 4 + \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + \beta_{5} - \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{41} + ( 1 + 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{10} + 2 \beta_{11} ) q^{43} + ( -1 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + 5 \beta_{11} ) q^{45} + ( 3 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{47} + \beta_{9} q^{49} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{10} - \beta_{11} ) q^{51} + ( -3 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{53} + ( 3 \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} + 2 \beta_{7} - 4 \beta_{8} - \beta_{9} + 3 \beta_{10} + 5 \beta_{11} ) q^{55} + ( 1 - 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{5} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{57} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{59} + ( 2 + \beta_{1} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{61} + ( -2 - 2 \beta_{1} - \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{63} + ( 6 + \beta_{1} + 6 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 3 \beta_{11} ) q^{65} + ( 2 + 2 \beta_{1} - \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} ) q^{67} + ( 8 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} + 4 \beta_{8} - 4 \beta_{10} + 2 \beta_{11} ) q^{69} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{71} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 5 \beta_{7} - 4 \beta_{9} + \beta_{10} + \beta_{11} ) q^{73} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} + 4 \beta_{8} - 7 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} ) q^{75} + ( -\beta_{1} + \beta_{2} - \beta_{5} - 2 \beta_{6} + \beta_{8} + 2 \beta_{10} ) q^{77} + ( 4 + 2 \beta_{3} - 4 \beta_{6} + 2 \beta_{8} + 4 \beta_{10} - 2 \beta_{11} ) q^{79} + ( 3 \beta_{2} - 3 \beta_{4} - \beta_{5} - 6 \beta_{7} - \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{81} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + 5 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{83} + ( -4 + \beta_{2} + 3 \beta_{4} - 4 \beta_{5} - \beta_{6} + 3 \beta_{7} - 4 \beta_{9} + \beta_{10} + 4 \beta_{11} ) q^{85} + ( 1 + 5 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{87} + ( 3 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} - 3 \beta_{10} ) q^{89} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{91} + ( -4 + 5 \beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 5 \beta_{6} + 3 \beta_{8} + 2 \beta_{9} - 4 \beta_{10} + \beta_{11} ) q^{93} + ( 5 - \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - 4 \beta_{8} - 5 \beta_{9} + \beta_{10} ) q^{95} + ( 1 - \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} - \beta_{5} + 5 \beta_{6} + 5 \beta_{7} - 3 \beta_{8} + \beta_{9} - 4 \beta_{10} + \beta_{11} ) q^{97} + ( 3 - \beta_{1} - \beta_{2} - 4 \beta_{3} - 2 \beta_{6} - \beta_{7} + 4 \beta_{8} - 6 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 4 q^{9} + O(q^{10})$$ $$12 q - 4 q^{9} - 6 q^{11} + 4 q^{13} - 6 q^{15} - 4 q^{17} + 12 q^{23} - 20 q^{25} - 12 q^{27} + 8 q^{29} - 30 q^{33} - 6 q^{35} - 42 q^{37} + 4 q^{39} + 30 q^{41} - 2 q^{43} + 6 q^{49} - 52 q^{51} - 44 q^{53} + 6 q^{55} - 18 q^{59} + 14 q^{61} - 12 q^{63} + 60 q^{65} + 24 q^{67} + 4 q^{69} + 24 q^{71} - 46 q^{75} + 8 q^{77} + 56 q^{79} + 2 q^{81} - 48 q^{85} + 2 q^{87} - 12 q^{89} - 14 q^{91} - 18 q^{93} + 22 q^{95} + 6 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5 x^{10} - 2 x^{9} + 15 x^{8} + 2 x^{7} - 30 x^{6} + 4 x^{5} + 60 x^{4} - 16 x^{3} - 80 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} - 5 \nu^{9} - 2 \nu^{8} + 15 \nu^{7} + 2 \nu^{6} - 30 \nu^{5} + 4 \nu^{4} + 60 \nu^{3} - 16 \nu^{2} - 80 \nu$$$$)/32$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} - 4 \nu^{3} + 16 \nu^{2} - 48 \nu + 96$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{11} + 2 \nu^{10} - 11 \nu^{9} - 8 \nu^{8} + 21 \nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 84 \nu^{3} + 24 \nu^{2} - 64 \nu - 64$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{11} + 3 \nu^{10} - 3 \nu^{9} - 13 \nu^{8} + 7 \nu^{7} + 23 \nu^{6} - 26 \nu^{5} - 38 \nu^{4} + 60 \nu^{3} + 68 \nu^{2} - 56 \nu - 64$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 10 \nu^{5} - 68 \nu^{4} - 20 \nu^{3} + 48 \nu^{2} - 32 \nu + 64$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} - 68 \nu^{3} + 120 \nu^{2} + 16 \nu - 32$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{11} + \nu^{10} + 5 \nu^{9} - 3 \nu^{8} - 13 \nu^{7} + 13 \nu^{6} + 20 \nu^{5} - 34 \nu^{4} - 28 \nu^{3} + 52 \nu^{2} + 24 \nu - 32$$$$)/8$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + 8 \nu^{3} + 108 \nu^{2} - 16 \nu - 80$$$$)/16$$ $$\beta_{10}$$ $$=$$ $$($$$$-3 \nu^{11} + 15 \nu^{9} - 2 \nu^{8} - 37 \nu^{7} + 18 \nu^{6} + 66 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + 96 \nu^{2} + 96 \nu - 64$$$$)/16$$ $$\beta_{11}$$ $$=$$ $$($$$$-\nu^{11} + 4 \nu^{10} + 9 \nu^{9} - 18 \nu^{8} - 27 \nu^{7} + 50 \nu^{6} + 34 \nu^{5} - 116 \nu^{4} - 20 \nu^{3} + 192 \nu^{2} + 16 \nu - 160$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{9} + \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{11} + 4 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{1} + 1$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} - 2$$ $$\nu^{7}$$ $$=$$ $$-5 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} - 3 \beta_{3} + \beta_{2} + 1$$ $$\nu^{8}$$ $$=$$ $$-3 \beta_{11} + 7 \beta_{8} + \beta_{7} - 5 \beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 10 \beta_{2} + 8 \beta_{1} - 3$$ $$\nu^{9}$$ $$=$$ $$-7 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} - 11 \beta_{6} + 12 \beta_{5} - 17 \beta_{4} + 7 \beta_{3} + 7 \beta_{2} + \beta_{1} - 8$$ $$\nu^{10}$$ $$=$$ $$-17 \beta_{11} + 4 \beta_{10} + 13 \beta_{9} + 18 \beta_{8} - 9 \beta_{7} - 8 \beta_{6} + 3 \beta_{5} + 7 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} - 6 \beta_{1} - 11$$ $$\nu^{11}$$ $$=$$ $$18 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} - 13 \beta_{8} - 21 \beta_{7} - 11 \beta_{6} - \beta_{5} - 26 \beta_{4} + 22 \beta_{3} + 14 \beta_{2} + 3 \beta_{1} - 7$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$$.

 $$n$$ $$561$$ $$911$$ $$1093$$ $$1249$$ $$\chi(n)$$ $$1 - \beta_{9}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
225.1
 0.759479 + 1.19298i 1.40744 − 0.138282i 1.34408 − 0.439820i −1.12906 + 0.851598i −1.30089 − 0.554694i −1.08105 − 0.911778i 0.759479 − 1.19298i 1.40744 + 0.138282i 1.34408 + 0.439820i −1.12906 − 0.851598i −1.30089 + 0.554694i −1.08105 + 0.911778i
0 −1.41289 + 2.44719i 0 0.518957i 0 0.866025 0.500000i 0 −2.49250 4.31714i 0
225.2 0 −0.583963 + 1.01145i 0 1.81487i 0 0.866025 0.500000i 0 0.817975 + 1.41677i 0
225.3 0 −0.291146 + 0.504280i 0 1.68817i 0 −0.866025 + 0.500000i 0 1.33047 + 2.30444i 0
225.4 0 −0.172975 + 0.299601i 0 3.25812i 0 −0.866025 + 0.500000i 0 1.44016 + 2.49443i 0
225.5 0 1.13082 1.95864i 0 3.60178i 0 0.866025 0.500000i 0 −1.05753 1.83169i 0
225.6 0 1.33015 2.30388i 0 3.16209i 0 −0.866025 + 0.500000i 0 −2.03858 3.53092i 0
673.1 0 −1.41289 2.44719i 0 0.518957i 0 0.866025 + 0.500000i 0 −2.49250 + 4.31714i 0
673.2 0 −0.583963 1.01145i 0 1.81487i 0 0.866025 + 0.500000i 0 0.817975 1.41677i 0
673.3 0 −0.291146 0.504280i 0 1.68817i 0 −0.866025 0.500000i 0 1.33047 2.30444i 0
673.4 0 −0.172975 0.299601i 0 3.25812i 0 −0.866025 0.500000i 0 1.44016 2.49443i 0
673.5 0 1.13082 + 1.95864i 0 3.60178i 0 0.866025 + 0.500000i 0 −1.05753 + 1.83169i 0
673.6 0 1.33015 + 2.30388i 0 3.16209i 0 −0.866025 0.500000i 0 −2.03858 + 3.53092i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 673.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.2.cc.c 12
4.b odd 2 1 91.2.q.a 12
12.b even 2 1 819.2.ct.a 12
13.e even 6 1 inner 1456.2.cc.c 12
28.d even 2 1 637.2.q.h 12
28.f even 6 1 637.2.k.g 12
28.f even 6 1 637.2.u.i 12
28.g odd 6 1 637.2.k.h 12
28.g odd 6 1 637.2.u.h 12
52.i odd 6 1 91.2.q.a 12
52.i odd 6 1 1183.2.c.i 12
52.j odd 6 1 1183.2.c.i 12
52.l even 12 1 1183.2.a.m 6
52.l even 12 1 1183.2.a.p 6
156.r even 6 1 819.2.ct.a 12
364.s odd 6 1 637.2.k.h 12
364.w even 6 1 637.2.u.i 12
364.bc even 6 1 637.2.q.h 12
364.bk odd 6 1 637.2.u.h 12
364.bp even 6 1 637.2.k.g 12
364.bv odd 12 1 8281.2.a.by 6
364.bv odd 12 1 8281.2.a.ch 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a 12 4.b odd 2 1
91.2.q.a 12 52.i odd 6 1
637.2.k.g 12 28.f even 6 1
637.2.k.g 12 364.bp even 6 1
637.2.k.h 12 28.g odd 6 1
637.2.k.h 12 364.s odd 6 1
637.2.q.h 12 28.d even 2 1
637.2.q.h 12 364.bc even 6 1
637.2.u.h 12 28.g odd 6 1
637.2.u.h 12 364.bk odd 6 1
637.2.u.i 12 28.f even 6 1
637.2.u.i 12 364.w even 6 1
819.2.ct.a 12 12.b even 2 1
819.2.ct.a 12 156.r even 6 1
1183.2.a.m 6 52.l even 12 1
1183.2.a.p 6 52.l even 12 1
1183.2.c.i 12 52.i odd 6 1
1183.2.c.i 12 52.j odd 6 1
1456.2.cc.c 12 1.a even 1 1 trivial
1456.2.cc.c 12 13.e even 6 1 inner
8281.2.a.by 6 364.bv odd 12 1
8281.2.a.ch 6 364.bv odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1456, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$16 + 80 T + 300 T^{2} + 516 T^{3} + 709 T^{4} + 390 T^{5} + 287 T^{6} + 42 T^{7} + 96 T^{8} + 4 T^{9} + 11 T^{10} + T^{12}$$
$5$ $$3481 + 16148 T^{2} + 13040 T^{4} + 4146 T^{6} + 600 T^{8} + 40 T^{10} + T^{12}$$
$7$ $$( 1 - T^{2} + T^{4} )^{3}$$
$11$ $$256 - 768 T - 80 T^{2} + 2544 T^{3} + 2025 T^{4} - 1494 T^{5} - 771 T^{6} + 522 T^{7} + 248 T^{8} - 114 T^{9} - 7 T^{10} + 6 T^{11} + T^{12}$$
$13$ $$4826809 - 1485172 T + 599781 T^{2} - 70304 T^{3} - 23998 T^{4} + 12012 T^{5} - 6587 T^{6} + 924 T^{7} - 142 T^{8} - 32 T^{9} + 21 T^{10} - 4 T^{11} + T^{12}$$
$17$ $$241081 - 109984 T + 132173 T^{2} - 21512 T^{3} + 31018 T^{4} - 2576 T^{5} + 5229 T^{6} + 148 T^{7} + 514 T^{8} + 36 T^{9} + 37 T^{10} + 4 T^{11} + T^{12}$$
$19$ $$55696 + 138768 T + 137196 T^{2} + 54684 T^{3} - 4499 T^{4} - 9486 T^{5} + 299 T^{6} + 2370 T^{7} + 748 T^{8} - 29 T^{10} + T^{12}$$
$23$ $$38539264 - 9138176 T + 12198912 T^{2} - 5170176 T^{3} + 3630592 T^{4} - 1115904 T^{5} + 332096 T^{6} - 52416 T^{7} + 9312 T^{8} - 976 T^{9} + 164 T^{10} - 12 T^{11} + T^{12}$$
$29$ $$10042561 - 1064784 T + 4587524 T^{2} - 3112876 T^{3} + 2323356 T^{4} - 854112 T^{5} + 261878 T^{6} - 47832 T^{7} + 7876 T^{8} - 780 T^{9} + 108 T^{10} - 8 T^{11} + T^{12}$$
$31$ $$913936 + 1285560 T^{2} + 568225 T^{4} + 96896 T^{6} + 5854 T^{8} + 136 T^{10} + T^{12}$$
$37$ $$1755945216 - 633588480 T - 260200512 T^{2} + 121383360 T^{3} + 65323584 T^{4} + 8644320 T^{5} - 396036 T^{6} - 159840 T^{7} + 8109 T^{8} + 5670 T^{9} + 723 T^{10} + 42 T^{11} + T^{12}$$
$41$ $$884705536 - 421175040 T - 8238656 T^{2} + 35739840 T^{3} - 872384 T^{4} - 2742240 T^{5} + 452252 T^{6} + 35760 T^{7} - 11651 T^{8} - 450 T^{9} + 315 T^{10} - 30 T^{11} + T^{12}$$
$43$ $$2408704 - 5860352 T + 10500784 T^{2} - 8862336 T^{3} + 5690569 T^{4} - 1037954 T^{5} + 282645 T^{6} - 3650 T^{7} + 9640 T^{8} - 38 T^{9} + 113 T^{10} + 2 T^{11} + T^{12}$$
$47$ $$9461776 + 47561752 T^{2} + 10113609 T^{4} + 722232 T^{6} + 21782 T^{8} + 272 T^{10} + T^{12}$$
$53$ $$( -2339 + 7302 T - 3353 T^{2} - 700 T^{3} + 91 T^{4} + 22 T^{5} + T^{6} )^{2}$$
$59$ $$4571923456 + 3919023360 T + 1190986848 T^{2} + 61031880 T^{3} - 34227911 T^{4} - 3548430 T^{5} + 1035110 T^{6} + 185580 T^{7} - 2237 T^{8} - 1980 T^{9} - 2 T^{10} + 18 T^{11} + T^{12}$$
$61$ $$5607424 - 3788800 T + 7030784 T^{2} - 3685376 T^{3} + 6036160 T^{4} - 2984960 T^{5} + 1823136 T^{6} - 177656 T^{7} + 29281 T^{8} - 1614 T^{9} + 283 T^{10} - 14 T^{11} + T^{12}$$
$67$ $$613651984 + 1248211536 T + 790406444 T^{2} - 113725716 T^{3} - 37845559 T^{4} + 4849356 T^{5} + 1364502 T^{6} - 185988 T^{7} - 11949 T^{8} + 2352 T^{9} + 94 T^{10} - 24 T^{11} + T^{12}$$
$71$ $$46895104 - 26296320 T - 3083264 T^{2} + 4485120 T^{3} + 367104 T^{4} - 620544 T^{5} + 82752 T^{6} + 17280 T^{7} - 3808 T^{8} - 480 T^{9} + 212 T^{10} - 24 T^{11} + T^{12}$$
$73$ $$1386221824 + 513361280 T^{2} + 55965104 T^{4} + 2238456 T^{6} + 40473 T^{8} + 334 T^{10} + T^{12}$$
$79$ $$( -512 - 1664 T - 1584 T^{2} - 192 T^{3} + 212 T^{4} - 28 T^{5} + T^{6} )^{2}$$
$83$ $$141324544 + 454322976 T^{2} + 48190849 T^{4} + 1905008 T^{6} + 35086 T^{8} + 304 T^{10} + T^{12}$$
$89$ $$1834580224 + 1271596416 T - 422744080 T^{2} - 496650552 T^{3} + 291555473 T^{4} - 41341974 T^{5} - 2811261 T^{6} + 792798 T^{7} + 66288 T^{8} - 3660 T^{9} - 257 T^{10} + 12 T^{11} + T^{12}$$
$97$ $$53465344 + 190755456 T + 211235504 T^{2} - 55750056 T^{3} - 14421439 T^{4} + 4289964 T^{5} + 938031 T^{6} - 349092 T^{7} + 28032 T^{8} + 1110 T^{9} - 173 T^{10} - 6 T^{11} + T^{12}$$