# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
339.1
 − 1.00000i 1.00000i
1.00000i 2.00000i 1.00000 2.00000 1.00000i −2.00000 0 3.00000i −1.00000 −1.00000 2.00000i
339.2 1.00000i 2.00000i 1.00000 2.00000 + 1.00000i −2.00000 0 3.00000i −1.00000 −1.00000 + 2.00000i
 $$n$$: 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
5.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.b.b 2
5.b even 2 1 inner 845.2.b.b 2
5.c odd 4 1 4225.2.a.h 1
5.c odd 4 1 4225.2.a.k 1
13.b even 2 1 845.2.b.a 2
13.c even 3 2 845.2.n.b 4
13.d odd 4 1 65.2.d.a 2
13.d odd 4 1 65.2.d.b yes 2
13.e even 6 2 845.2.n.a 4
13.f odd 12 2 845.2.l.a 4
13.f odd 12 2 845.2.l.b 4
39.f even 4 1 585.2.h.b 2
39.f even 4 1 585.2.h.c 2
52.f even 4 1 1040.2.f.a 2
52.f even 4 1 1040.2.f.b 2
65.d even 2 1 845.2.b.a 2
65.f even 4 1 325.2.c.b 2
65.f even 4 1 325.2.c.e 2
65.g odd 4 1 65.2.d.a 2
65.g odd 4 1 65.2.d.b yes 2
65.h odd 4 1 4225.2.a.e 1
65.h odd 4 1 4225.2.a.m 1
65.k even 4 1 325.2.c.b 2
65.k even 4 1 325.2.c.e 2
65.l even 6 2 845.2.n.a 4
65.n even 6 2 845.2.n.b 4
65.s odd 12 2 845.2.l.a 4
65.s odd 12 2 845.2.l.b 4
195.n even 4 1 585.2.h.b 2
195.n even 4 1 585.2.h.c 2
260.u even 4 1 1040.2.f.a 2
260.u even 4 1 1040.2.f.b 2

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$:

 $$T_{2}^{2} + 1$$ $$T_{11} + 2$$

## Hecke characteristic polynomials

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