# Properties

 Label 845.2.l.a Level $845$ Weight $2$ Character orbit 845.l Analytic conductor $6.747$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.l (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{12}^{2} ) q^{2} -2 \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} + ( 1 - 2 \zeta_{12}^{3} ) q^{5} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{6} -3 q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{12}^{2} ) q^{2} -2 \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} + ( 1 - 2 \zeta_{12}^{3} ) q^{5} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{6} -3 q^{8} + \zeta_{12}^{2} q^{9} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{10} + 2 \zeta_{12} q^{11} -2 \zeta_{12}^{3} q^{12} + ( -4 - 2 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{15} + ( 1 - \zeta_{12}^{2} ) q^{16} - q^{18} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{19} + ( 2 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{20} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{22} -6 \zeta_{12} q^{23} + 6 \zeta_{12} q^{24} + ( -3 - 4 \zeta_{12}^{3} ) q^{25} + 4 \zeta_{12}^{3} q^{27} + ( -6 + 6 \zeta_{12}^{2} ) q^{29} + ( 2 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{30} -6 \zeta_{12}^{3} q^{31} -5 \zeta_{12}^{2} q^{32} -4 \zeta_{12}^{2} q^{33} + ( -1 + \zeta_{12}^{2} ) q^{36} + ( 6 - 6 \zeta_{12}^{2} ) q^{37} + 6 \zeta_{12}^{3} q^{38} + ( -3 + 6 \zeta_{12}^{3} ) q^{40} -8 \zeta_{12} q^{41} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{43} + 2 \zeta_{12}^{3} q^{44} + ( 2 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{45} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{46} + 8 q^{47} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{48} + ( 7 - 7 \zeta_{12}^{2} ) q^{49} + ( 3 + 4 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{50} -12 \zeta_{12}^{3} q^{53} -4 \zeta_{12} q^{54} + ( 4 + 2 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{55} -12 q^{57} -6 \zeta_{12}^{2} q^{58} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{59} + ( -4 - 2 \zeta_{12}^{3} ) q^{60} -6 \zeta_{12}^{2} q^{61} + 6 \zeta_{12} q^{62} + 7 q^{64} + 4 q^{66} + ( 12 - 12 \zeta_{12}^{2} ) q^{67} + 12 \zeta_{12}^{2} q^{69} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{71} -3 \zeta_{12}^{2} q^{72} + 6 q^{73} + 6 \zeta_{12}^{2} q^{74} + ( -8 + 6 \zeta_{12} + 8 \zeta_{12}^{2} ) q^{75} + 6 \zeta_{12} q^{76} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{80} + ( 11 - 11 \zeta_{12}^{2} ) q^{81} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{82} + 4 q^{83} -6 \zeta_{12}^{3} q^{86} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{87} -6 \zeta_{12} q^{88} -8 \zeta_{12} q^{89} + ( -1 + 2 \zeta_{12}^{3} ) q^{90} -6 \zeta_{12}^{3} q^{92} + ( -12 + 12 \zeta_{12}^{2} ) q^{93} + ( -8 + 8 \zeta_{12}^{2} ) q^{94} + ( 6 \zeta_{12} - 12 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{95} + 10 \zeta_{12}^{3} q^{96} -6 \zeta_{12}^{2} q^{97} + 7 \zeta_{12}^{2} q^{98} + 2 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 12 q^{8} + 2 q^{9} + O(q^{10})$$ $$4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 12 q^{8} + 2 q^{9} - 2 q^{10} - 8 q^{15} + 2 q^{16} - 4 q^{18} + 2 q^{20} - 12 q^{25} - 12 q^{29} - 8 q^{30} - 10 q^{32} - 8 q^{33} - 2 q^{36} + 12 q^{37} - 12 q^{40} + 2 q^{45} + 32 q^{47} + 14 q^{49} + 6 q^{50} + 8 q^{55} - 48 q^{57} - 12 q^{58} - 16 q^{60} - 12 q^{61} + 28 q^{64} + 16 q^{66} + 24 q^{67} + 24 q^{69} - 6 q^{72} + 24 q^{73} + 12 q^{74} - 16 q^{75} + 2 q^{80} + 22 q^{81} + 16 q^{83} - 4 q^{90} - 24 q^{93} - 16 q^{94} - 24 q^{95} - 12 q^{97} + 14 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
654.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.500000 + 0.866025i −1.73205 1.00000i 0.500000 + 0.866025i 1.00000 2.00000i 1.73205 1.00000i 0 −3.00000 0.500000 + 0.866025i 1.23205 + 1.86603i
654.2 −0.500000 + 0.866025i 1.73205 + 1.00000i 0.500000 + 0.866025i 1.00000 + 2.00000i −1.73205 + 1.00000i 0 −3.00000 0.500000 + 0.866025i −2.23205 0.133975i
699.1 −0.500000 0.866025i −1.73205 + 1.00000i 0.500000 0.866025i 1.00000 + 2.00000i 1.73205 + 1.00000i 0 −3.00000 0.500000 0.866025i 1.23205 1.86603i
699.2 −0.500000 0.866025i 1.73205 1.00000i 0.500000 0.866025i 1.00000 2.00000i −1.73205 1.00000i 0 −3.00000 0.500000 0.866025i −2.23205 + 0.133975i
 $$n$$: e.g. 2-40 or 990-1000 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.c even 3 1 inner
65.d even 2 1 inner
65.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.l.a 4
5.b even 2 1 845.2.l.b 4
13.b even 2 1 845.2.l.b 4
13.c even 3 1 65.2.d.b yes 2
13.c even 3 1 inner 845.2.l.a 4
13.d odd 4 1 845.2.n.a 4
13.d odd 4 1 845.2.n.b 4
13.e even 6 1 65.2.d.a 2
13.e even 6 1 845.2.l.b 4
13.f odd 12 1 845.2.b.a 2
13.f odd 12 1 845.2.b.b 2
13.f odd 12 1 845.2.n.a 4
13.f odd 12 1 845.2.n.b 4
39.h odd 6 1 585.2.h.c 2
39.i odd 6 1 585.2.h.b 2
52.i odd 6 1 1040.2.f.a 2
52.j odd 6 1 1040.2.f.b 2
65.d even 2 1 inner 845.2.l.a 4
65.g odd 4 1 845.2.n.a 4
65.g odd 4 1 845.2.n.b 4
65.l even 6 1 65.2.d.b yes 2
65.l even 6 1 inner 845.2.l.a 4
65.n even 6 1 65.2.d.a 2
65.n even 6 1 845.2.l.b 4
65.o even 12 1 4225.2.a.k 1
65.o even 12 1 4225.2.a.m 1
65.q odd 12 1 325.2.c.b 2
65.q odd 12 1 325.2.c.e 2
65.r odd 12 1 325.2.c.b 2
65.r odd 12 1 325.2.c.e 2
65.s odd 12 1 845.2.b.a 2
65.s odd 12 1 845.2.b.b 2
65.s odd 12 1 845.2.n.a 4
65.s odd 12 1 845.2.n.b 4
65.t even 12 1 4225.2.a.e 1
65.t even 12 1 4225.2.a.h 1
195.x odd 6 1 585.2.h.c 2
195.y odd 6 1 585.2.h.b 2
260.v odd 6 1 1040.2.f.a 2
260.w odd 6 1 1040.2.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.d.a 2 13.e even 6 1
65.2.d.a 2 65.n even 6 1
65.2.d.b yes 2 13.c even 3 1
65.2.d.b yes 2 65.l even 6 1
325.2.c.b 2 65.q odd 12 1
325.2.c.b 2 65.r odd 12 1
325.2.c.e 2 65.q odd 12 1
325.2.c.e 2 65.r odd 12 1
585.2.h.b 2 39.i odd 6 1
585.2.h.b 2 195.y odd 6 1
585.2.h.c 2 39.h odd 6 1
585.2.h.c 2 195.x odd 6 1
845.2.b.a 2 13.f odd 12 1
845.2.b.a 2 65.s odd 12 1
845.2.b.b 2 13.f odd 12 1
845.2.b.b 2 65.s odd 12 1
845.2.l.a 4 1.a even 1 1 trivial
845.2.l.a 4 13.c even 3 1 inner
845.2.l.a 4 65.d even 2 1 inner
845.2.l.a 4 65.l even 6 1 inner
845.2.l.b 4 5.b even 2 1
845.2.l.b 4 13.b even 2 1
845.2.l.b 4 13.e even 6 1
845.2.l.b 4 65.n even 6 1
845.2.n.a 4 13.d odd 4 1
845.2.n.a 4 13.f odd 12 1
845.2.n.a 4 65.g odd 4 1
845.2.n.a 4 65.s odd 12 1
845.2.n.b 4 13.d odd 4 1
845.2.n.b 4 13.f odd 12 1
845.2.n.b 4 65.g odd 4 1
845.2.n.b 4 65.s odd 12 1
1040.2.f.a 2 52.i odd 6 1
1040.2.f.a 2 260.v odd 6 1
1040.2.f.b 2 52.j odd 6 1
1040.2.f.b 2 260.w odd 6 1
4225.2.a.e 1 65.t even 12 1
4225.2.a.h 1 65.t even 12 1
4225.2.a.k 1 65.o even 12 1
4225.2.a.m 1 65.o even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$16 - 4 T^{2} + T^{4}$$
$5$ $$( 5 - 2 T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$16 - 4 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$1296 - 36 T^{2} + T^{4}$$
$23$ $$1296 - 36 T^{2} + T^{4}$$
$29$ $$( 36 + 6 T + T^{2} )^{2}$$
$31$ $$( 36 + T^{2} )^{2}$$
$37$ $$( 36 - 6 T + T^{2} )^{2}$$
$41$ $$4096 - 64 T^{2} + T^{4}$$
$43$ $$1296 - 36 T^{2} + T^{4}$$
$47$ $$( -8 + T )^{4}$$
$53$ $$( 144 + T^{2} )^{2}$$
$59$ $$16 - 4 T^{2} + T^{4}$$
$61$ $$( 36 + 6 T + T^{2} )^{2}$$
$67$ $$( 144 - 12 T + T^{2} )^{2}$$
$71$ $$16 - 4 T^{2} + T^{4}$$
$73$ $$( -6 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$( -4 + T )^{4}$$
$89$ $$4096 - 64 T^{2} + T^{4}$$
$97$ $$( 36 + 6 T + T^{2} )^{2}$$