# Properties

 Label 845.2.c.e 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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
1.73205i −0.732051 −1.00000 1.00000i 1.26795i 2.00000i 1.73205i −2.46410 1.73205
506.2 1.73205i 2.73205 −1.00000 1.00000i 4.73205i 2.00000i 1.73205i 4.46410 −1.73205
506.3 1.73205i −0.732051 −1.00000 1.00000i 1.26795i 2.00000i 1.73205i −2.46410 1.73205
506.4 1.73205i 2.73205 −1.00000 1.00000i 4.73205i 2.00000i 1.73205i 4.46410 −1.73205
 $$n$$: 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.e 4
13.b even 2 1 inner 845.2.c.e 4
13.c even 3 1 845.2.m.a 4
13.c even 3 1 845.2.m.c 4
13.d odd 4 1 65.2.a.c 2
13.d odd 4 1 845.2.a.d 2
13.e even 6 1 845.2.m.a 4
13.e even 6 1 845.2.m.c 4
13.f odd 12 2 845.2.e.e 4
13.f odd 12 2 845.2.e.f 4
39.f even 4 1 585.2.a.k 2
39.f even 4 1 7605.2.a.be 2
52.f even 4 1 1040.2.a.h 2
65.f even 4 1 325.2.b.e 4
65.g odd 4 1 325.2.a.g 2
65.g odd 4 1 4225.2.a.w 2
65.k even 4 1 325.2.b.e 4
91.i even 4 1 3185.2.a.k 2
104.j odd 4 1 4160.2.a.y 2
104.m even 4 1 4160.2.a.bj 2
143.g even 4 1 7865.2.a.h 2
156.l odd 4 1 9360.2.a.cm 2
195.j odd 4 1 2925.2.c.v 4
195.n even 4 1 2925.2.a.z 2
195.u odd 4 1 2925.2.c.v 4
260.u even 4 1 5200.2.a.ca 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 3 + T^{2} )^{2}$$
$3$ $$( -2 - 2 T + T^{2} )^{2}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( 4 + T^{2} )^{2}$$
$11$ $$36 + 24 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -12 + T^{2} )^{2}$$
$19$ $$676 + 56 T^{2} + T^{4}$$
$23$ $$( 6 + 6 T + T^{2} )^{2}$$
$29$ $$( 24 + 12 T + T^{2} )^{2}$$
$31$ $$4 + 104 T^{2} + T^{4}$$
$37$ $$( 16 + T^{2} )^{2}$$
$41$ $$( 12 + T^{2} )^{2}$$
$43$ $$( -2 + 10 T + T^{2} )^{2}$$
$47$ $$( 36 + T^{2} )^{2}$$
$53$ $$( -108 + T^{2} )^{2}$$
$59$ $$19044 + 312 T^{2} + T^{4}$$
$61$ $$( -104 - 4 T + T^{2} )^{2}$$
$67$ $$8464 + 248 T^{2} + T^{4}$$
$71$ $$36 + 24 T^{2} + T^{4}$$
$73$ $$( 16 + T^{2} )^{2}$$
$79$ $$( -104 - 4 T + T^{2} )^{2}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$144 + 168 T^{2} + T^{4}$$
$97$ $$( 4 + T^{2} )^{2}$$