# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + q^{3} + ( -2 + \beta_{3} ) q^{4} -\beta_{2} q^{5} + \beta_{1} q^{6} + \beta_{2} q^{7} + 3 \beta_{2} q^{8} -2 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + q^{3} + ( -2 + \beta_{3} ) q^{4} -\beta_{2} q^{5} + \beta_{1} q^{6} + \beta_{2} q^{7} + 3 \beta_{2} q^{8} -2 q^{9} + ( -1 + \beta_{3} ) q^{10} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{11} + ( -2 + \beta_{3} ) q^{12} + ( 1 - \beta_{3} ) q^{14} -\beta_{2} q^{15} + ( -1 - \beta_{3} ) q^{16} + ( -5 + 2 \beta_{3} ) q^{17} -2 \beta_{1} q^{18} + ( 2 \beta_{1} - \beta_{2} ) q^{19} + ( -\beta_{1} + \beta_{2} ) q^{20} + \beta_{2} q^{21} + ( -5 - \beta_{3} ) q^{22} + 3 q^{23} + 3 \beta_{2} q^{24} - q^{25} -5 q^{27} + ( \beta_{1} - \beta_{2} ) q^{28} + ( -1 + 4 \beta_{3} ) q^{29} + ( -1 + \beta_{3} ) q^{30} -4 \beta_{2} q^{31} + ( -\beta_{1} + 3 \beta_{2} ) q^{32} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{33} + ( -5 \beta_{1} + 6 \beta_{2} ) q^{34} + q^{35} + ( 4 - 2 \beta_{3} ) q^{36} + ( 2 \beta_{1} + \beta_{2} ) q^{37} + ( -9 + 3 \beta_{3} ) q^{38} + 3 q^{40} + 3 \beta_{2} q^{41} + ( 1 - \beta_{3} ) q^{42} + ( 5 - 4 \beta_{3} ) q^{43} + ( -\beta_{1} + 3 \beta_{2} ) q^{44} + 2 \beta_{2} q^{45} + 3 \beta_{1} q^{46} + 4 \beta_{1} q^{47} + ( -1 - \beta_{3} ) q^{48} + 6 q^{49} -\beta_{1} q^{50} + ( -5 + 2 \beta_{3} ) q^{51} + ( 2 + 4 \beta_{3} ) q^{53} -5 \beta_{1} q^{54} + ( 1 + 2 \beta_{3} ) q^{55} -3 q^{56} + ( 2 \beta_{1} - \beta_{2} ) q^{57} + ( -\beta_{1} + 12 \beta_{2} ) q^{58} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{59} + ( -\beta_{1} + \beta_{2} ) q^{60} - q^{61} + ( -4 + 4 \beta_{3} ) q^{62} -2 \beta_{2} q^{63} + ( 5 - 6 \beta_{3} ) q^{64} + ( -5 - \beta_{3} ) q^{66} -7 \beta_{2} q^{67} + ( 16 - 7 \beta_{3} ) q^{68} + 3 q^{69} + \beta_{1} q^{70} + ( -6 \beta_{1} - 9 \beta_{2} ) q^{71} -6 \beta_{2} q^{72} + ( 4 \beta_{1} + 10 \beta_{2} ) q^{73} + ( -7 + \beta_{3} ) q^{74} - q^{75} + ( -5 \beta_{1} + 7 \beta_{2} ) q^{76} + ( -1 - 2 \beta_{3} ) q^{77} -4 \beta_{3} q^{79} + ( \beta_{1} + 2 \beta_{2} ) q^{80} + q^{81} + ( 3 - 3 \beta_{3} ) q^{82} + 4 \beta_{1} q^{83} + ( \beta_{1} - \beta_{2} ) q^{84} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{85} + ( 5 \beta_{1} - 12 \beta_{2} ) q^{86} + ( -1 + 4 \beta_{3} ) q^{87} + ( -3 - 6 \beta_{3} ) q^{88} + ( -4 \beta_{1} - 3 \beta_{2} ) q^{89} + ( 2 - 2 \beta_{3} ) q^{90} + ( -6 + 3 \beta_{3} ) q^{92} -4 \beta_{2} q^{93} + ( -16 + 4 \beta_{3} ) q^{94} + ( -3 + 2 \beta_{3} ) q^{95} + ( -\beta_{1} + 3 \beta_{2} ) q^{96} + ( -2 \beta_{1} - 13 \beta_{2} ) q^{97} + 6 \beta_{1} q^{98} + ( -4 \beta_{1} - 6 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} - 6q^{4} - 8q^{9} + O(q^{10})$$ $$4q + 4q^{3} - 6q^{4} - 8q^{9} - 2q^{10} - 6q^{12} + 2q^{14} - 6q^{16} - 16q^{17} - 22q^{22} + 12q^{23} - 4q^{25} - 20q^{27} + 4q^{29} - 2q^{30} + 4q^{35} + 12q^{36} - 30q^{38} + 12q^{40} + 2q^{42} + 12q^{43} - 6q^{48} + 24q^{49} - 16q^{51} + 16q^{53} + 8q^{55} - 12q^{56} - 4q^{61} - 8q^{62} + 8q^{64} - 22q^{66} + 50q^{68} + 12q^{69} - 26q^{74} - 4q^{75} - 8q^{77} - 8q^{79} + 4q^{81} + 6q^{82} + 4q^{87} - 24q^{88} + 4q^{90} - 18q^{92} - 56q^{94} - 8q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} - 4 \beta_{1}$$

## Character values

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

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
506.1
 − 2.30278i − 1.30278i 1.30278i 2.30278i
2.30278i 1.00000 −3.30278 1.00000i 2.30278i 1.00000i 3.00000i −2.00000 −2.30278
506.2 1.30278i 1.00000 0.302776 1.00000i 1.30278i 1.00000i 3.00000i −2.00000 1.30278
506.3 1.30278i 1.00000 0.302776 1.00000i 1.30278i 1.00000i 3.00000i −2.00000 1.30278
506.4 2.30278i 1.00000 −3.30278 1.00000i 2.30278i 1.00000i 3.00000i −2.00000 −2.30278
 $$n$$: 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.c.d 4
13.b even 2 1 inner 845.2.c.d 4
13.c even 3 2 845.2.m.d 8
13.d odd 4 1 845.2.a.c 2
13.d odd 4 1 845.2.a.f 2
13.e even 6 2 845.2.m.d 8
13.f odd 12 2 65.2.e.b 4
13.f odd 12 2 845.2.e.d 4
39.f even 4 1 7605.2.a.bb 2
39.f even 4 1 7605.2.a.bg 2
39.k even 12 2 585.2.j.d 4
52.l even 12 2 1040.2.q.o 4
65.g odd 4 1 4225.2.a.t 2
65.g odd 4 1 4225.2.a.x 2
65.o even 12 2 325.2.o.b 8
65.s odd 12 2 325.2.e.a 4
65.t even 12 2 325.2.o.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.b 4 13.f odd 12 2
325.2.e.a 4 65.s odd 12 2
325.2.o.b 8 65.o even 12 2
325.2.o.b 8 65.t even 12 2
585.2.j.d 4 39.k even 12 2
845.2.a.c 2 13.d odd 4 1
845.2.a.f 2 13.d odd 4 1
845.2.c.d 4 1.a even 1 1 trivial
845.2.c.d 4 13.b even 2 1 inner
845.2.e.d 4 13.f odd 12 2
845.2.m.d 8 13.c even 3 2
845.2.m.d 8 13.e even 6 2
1040.2.q.o 4 52.l even 12 2
4225.2.a.t 2 65.g odd 4 1
4225.2.a.x 2 65.g odd 4 1
7605.2.a.bb 2 39.f even 4 1
7605.2.a.bg 2 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 7 T_{2}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 7 T^{2} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$81 + 34 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( 3 + 8 T + T^{2} )^{2}$$
$19$ $$81 + 34 T^{2} + T^{4}$$
$23$ $$( -3 + T )^{4}$$
$29$ $$( -51 - 2 T + T^{2} )^{2}$$
$31$ $$( 16 + T^{2} )^{2}$$
$37$ $$( 13 + T^{2} )^{2}$$
$41$ $$( 9 + T^{2} )^{2}$$
$43$ $$( -43 - 6 T + T^{2} )^{2}$$
$47$ $$2304 + 112 T^{2} + T^{4}$$
$53$ $$( -36 - 8 T + T^{2} )^{2}$$
$59$ $$( 117 + T^{2} )^{2}$$
$61$ $$( 1 + T )^{4}$$
$67$ $$( 49 + T^{2} )^{2}$$
$71$ $$6561 + 306 T^{2} + T^{4}$$
$73$ $$144 + 232 T^{2} + T^{4}$$
$79$ $$( -48 + 4 T + T^{2} )^{2}$$
$83$ $$2304 + 112 T^{2} + T^{4}$$
$89$ $$2601 + 106 T^{2} + T^{4}$$
$97$ $$17161 + 314 T^{2} + T^{4}$$