# Properties

 Label 845.2.c.a Level $845$ Weight $2$ Character orbit 845.c 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.c (of order $$2$$, degree $$1$$, not 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.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
506.1
 − 1.00000i 1.00000i
1.00000i −2.00000 1.00000 1.00000i 2.00000i 4.00000i 3.00000i 1.00000 −1.00000
506.2 1.00000i −2.00000 1.00000 1.00000i 2.00000i 4.00000i 3.00000i 1.00000 −1.00000
 $$n$$: 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.a 2
13.b even 2 1 inner 845.2.c.a 2
13.c even 3 2 845.2.m.b 4
13.d odd 4 1 65.2.a.a 1
13.d odd 4 1 845.2.a.a 1
13.e even 6 2 845.2.m.b 4
13.f odd 12 2 845.2.e.a 2
13.f odd 12 2 845.2.e.b 2
39.f even 4 1 585.2.a.h 1
39.f even 4 1 7605.2.a.f 1
52.f even 4 1 1040.2.a.f 1
65.f even 4 1 325.2.b.b 2
65.g odd 4 1 325.2.a.d 1
65.g odd 4 1 4225.2.a.g 1
65.k even 4 1 325.2.b.b 2
91.i even 4 1 3185.2.a.e 1
104.j odd 4 1 4160.2.a.q 1
104.m even 4 1 4160.2.a.f 1
143.g even 4 1 7865.2.a.c 1
156.l odd 4 1 9360.2.a.ca 1
195.j odd 4 1 2925.2.c.h 2
195.n even 4 1 2925.2.a.f 1
195.u odd 4 1 2925.2.c.h 2
260.u even 4 1 5200.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.a 1 13.d odd 4 1
325.2.a.d 1 65.g odd 4 1
325.2.b.b 2 65.f even 4 1
325.2.b.b 2 65.k even 4 1
585.2.a.h 1 39.f even 4 1
845.2.a.a 1 13.d odd 4 1
845.2.c.a 2 1.a even 1 1 trivial
845.2.c.a 2 13.b even 2 1 inner
845.2.e.a 2 13.f odd 12 2
845.2.e.b 2 13.f odd 12 2
845.2.m.b 4 13.c even 3 2
845.2.m.b 4 13.e even 6 2
1040.2.a.f 1 52.f even 4 1
2925.2.a.f 1 195.n even 4 1
2925.2.c.h 2 195.j odd 4 1
2925.2.c.h 2 195.u odd 4 1
3185.2.a.e 1 91.i even 4 1
4160.2.a.f 1 104.m even 4 1
4160.2.a.q 1 104.j odd 4 1
4225.2.a.g 1 65.g odd 4 1
5200.2.a.d 1 260.u even 4 1
7605.2.a.f 1 39.f even 4 1
7865.2.a.c 1 143.g even 4 1
9360.2.a.ca 1 156.l odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( 2 + T )^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$100 + T^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$36 + T^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$36 + T^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( 12 + T )^{2}$$
$83$ $$256 + T^{2}$$
$89$ $$4 + T^{2}$$
$97$ $$4 + T^{2}$$