# Properties

 Label 845.2.c.b 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: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i −1.41421 −3.82843 1.00000i 3.41421i 4.82843i 4.41421i −1.00000 2.41421
506.2 0.414214i 1.41421 1.82843 1.00000i 0.585786i 0.828427i 1.58579i −1.00000 −0.414214
506.3 0.414214i 1.41421 1.82843 1.00000i 0.585786i 0.828427i 1.58579i −1.00000 −0.414214
506.4 2.41421i −1.41421 −3.82843 1.00000i 3.41421i 4.82843i 4.41421i −1.00000 2.41421
 $$n$$: 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.b 4
13.b even 2 1 inner 845.2.c.b 4
13.c even 3 2 845.2.m.f 8
13.d odd 4 1 65.2.a.b 2
13.d odd 4 1 845.2.a.g 2
13.e even 6 2 845.2.m.f 8
13.f odd 12 2 845.2.e.c 4
13.f odd 12 2 845.2.e.h 4
39.f even 4 1 585.2.a.m 2
39.f even 4 1 7605.2.a.x 2
52.f even 4 1 1040.2.a.j 2
65.f even 4 1 325.2.b.f 4
65.g odd 4 1 325.2.a.i 2
65.g odd 4 1 4225.2.a.r 2
65.k even 4 1 325.2.b.f 4
91.i even 4 1 3185.2.a.j 2
104.j odd 4 1 4160.2.a.bf 2
104.m even 4 1 4160.2.a.z 2
143.g even 4 1 7865.2.a.j 2
156.l odd 4 1 9360.2.a.cd 2
195.j odd 4 1 2925.2.c.r 4
195.n even 4 1 2925.2.a.u 2
195.u odd 4 1 2925.2.c.r 4
260.u even 4 1 5200.2.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 13.d odd 4 1
325.2.a.i 2 65.g odd 4 1
325.2.b.f 4 65.f even 4 1
325.2.b.f 4 65.k even 4 1
585.2.a.m 2 39.f even 4 1
845.2.a.g 2 13.d odd 4 1
845.2.c.b 4 1.a even 1 1 trivial
845.2.c.b 4 13.b even 2 1 inner
845.2.e.c 4 13.f odd 12 2
845.2.e.h 4 13.f odd 12 2
845.2.m.f 8 13.c even 3 2
845.2.m.f 8 13.e even 6 2
1040.2.a.j 2 52.f even 4 1
2925.2.a.u 2 195.n even 4 1
2925.2.c.r 4 195.j odd 4 1
2925.2.c.r 4 195.u odd 4 1
3185.2.a.j 2 91.i even 4 1
4160.2.a.z 2 104.m even 4 1
4160.2.a.bf 2 104.j odd 4 1
4225.2.a.r 2 65.g odd 4 1
5200.2.a.bu 2 260.u even 4 1
7605.2.a.x 2 39.f even 4 1
7865.2.a.j 2 143.g even 4 1
9360.2.a.cd 2 156.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 6 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T^{2} + T^{4}$$
$3$ $$( -2 + T^{2} )^{2}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$16 + 24 T^{2} + T^{4}$$
$11$ $$4 + 12 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -4 - 4 T + T^{2} )^{2}$$
$19$ $$4 + 12 T^{2} + T^{4}$$
$23$ $$( -2 + T^{2} )^{2}$$
$29$ $$( -32 + T^{2} )^{2}$$
$31$ $$324 + 108 T^{2} + T^{4}$$
$37$ $$( 72 + T^{2} )^{2}$$
$41$ $$784 + 88 T^{2} + T^{4}$$
$43$ $$( -34 - 8 T + T^{2} )^{2}$$
$47$ $$16 + 24 T^{2} + T^{4}$$
$53$ $$( -36 + 12 T + T^{2} )^{2}$$
$59$ $$324 + 108 T^{2} + T^{4}$$
$61$ $$( 8 + T )^{4}$$
$67$ $$( 4 + T^{2} )^{2}$$
$71$ $$8836 + 204 T^{2} + T^{4}$$
$73$ $$( 72 + T^{2} )^{2}$$
$79$ $$( -72 + T^{2} )^{2}$$
$83$ $$784 + 88 T^{2} + T^{4}$$
$89$ $$( 36 + T^{2} )^{2}$$
$97$ $$784 + 72 T^{2} + T^{4}$$