# Properties

 Label 845.2.c.c 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: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ 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 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2 \beta_{1}$$

## 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
 − 1.61803i − 0.618034i 0.618034i 1.61803i
1.61803i −2.23607 −0.618034 1.00000i 3.61803i 0.236068i 2.23607i 2.00000 −1.61803
506.2 0.618034i 2.23607 1.61803 1.00000i 1.38197i 4.23607i 2.23607i 2.00000 0.618034
506.3 0.618034i 2.23607 1.61803 1.00000i 1.38197i 4.23607i 2.23607i 2.00000 0.618034
506.4 1.61803i −2.23607 −0.618034 1.00000i 3.61803i 0.236068i 2.23607i 2.00000 −1.61803
 $$n$$: 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.c 4
13.b even 2 1 inner 845.2.c.c 4
13.c even 3 2 845.2.m.e 8
13.d odd 4 1 845.2.a.b 2
13.d odd 4 1 845.2.a.e 2
13.e even 6 2 845.2.m.e 8
13.f odd 12 2 65.2.e.a 4
13.f odd 12 2 845.2.e.g 4
39.f even 4 1 7605.2.a.ba 2
39.f even 4 1 7605.2.a.bf 2
39.k even 12 2 585.2.j.e 4
52.l even 12 2 1040.2.q.n 4
65.g odd 4 1 4225.2.a.u 2
65.g odd 4 1 4225.2.a.y 2
65.o even 12 2 325.2.o.a 8
65.s odd 12 2 325.2.e.b 4
65.t even 12 2 325.2.o.a 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T^{2} + T^{4}$$
$3$ $$( -5 + T^{2} )^{2}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$1 + 18 T^{2} + T^{4}$$
$11$ $$1 + 18 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -19 + 2 T + T^{2} )^{2}$$
$19$ $$1 + 18 T^{2} + T^{4}$$
$23$ $$( 31 + 12 T + T^{2} )^{2}$$
$29$ $$( -11 + 6 T + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 9 + T^{2} )^{2}$$
$41$ $$5041 + 178 T^{2} + T^{4}$$
$43$ $$( 11 - 8 T + T^{2} )^{2}$$
$47$ $$4096 + 192 T^{2} + T^{4}$$
$53$ $$( -6 + T )^{4}$$
$59$ $$81 + 162 T^{2} + T^{4}$$
$61$ $$( -179 - 2 T + T^{2} )^{2}$$
$67$ $$841 + 122 T^{2} + T^{4}$$
$71$ $$121 + 42 T^{2} + T^{4}$$
$73$ $$( 36 + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$( 80 + T^{2} )^{2}$$
$89$ $$( 81 + T^{2} )^{2}$$
$97$ $$361 + 42 T^{2} + T^{4}$$