# Properties

 Label 65.2.d.a Level $65$ Weight $2$ Character orbit 65.d Analytic conductor $0.519$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$65 = 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 65.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.519027613138$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + 2 i q^{3} - q^{4} + ( -1 + 2 i ) q^{5} -2 i q^{6} + 3 q^{8} - q^{9} +O(q^{10})$$ $$q - q^{2} + 2 i q^{3} - q^{4} + ( -1 + 2 i ) q^{5} -2 i q^{6} + 3 q^{8} - q^{9} + ( 1 - 2 i ) q^{10} + 2 i q^{11} -2 i q^{12} + ( 3 - 2 i ) q^{13} + ( -4 - 2 i ) q^{15} - q^{16} + q^{18} -6 i q^{19} + ( 1 - 2 i ) q^{20} -2 i q^{22} + 6 i q^{23} + 6 i q^{24} + ( -3 - 4 i ) q^{25} + ( -3 + 2 i ) q^{26} + 4 i q^{27} + 6 q^{29} + ( 4 + 2 i ) q^{30} + 6 i q^{31} -5 q^{32} -4 q^{33} + q^{36} + 6 q^{37} + 6 i q^{38} + ( 4 + 6 i ) q^{39} + ( -3 + 6 i ) q^{40} -8 i q^{41} -6 i q^{43} -2 i q^{44} + ( 1 - 2 i ) q^{45} -6 i q^{46} -8 q^{47} -2 i q^{48} -7 q^{49} + ( 3 + 4 i ) q^{50} + ( -3 + 2 i ) q^{52} -12 i q^{53} -4 i q^{54} + ( -4 - 2 i ) q^{55} + 12 q^{57} -6 q^{58} + 2 i q^{59} + ( 4 + 2 i ) q^{60} + 6 q^{61} -6 i q^{62} + 7 q^{64} + ( 1 + 8 i ) q^{65} + 4 q^{66} + 12 q^{67} -12 q^{69} -2 i q^{71} -3 q^{72} -6 q^{73} -6 q^{74} + ( 8 - 6 i ) q^{75} + 6 i q^{76} + ( -4 - 6 i ) q^{78} + ( 1 - 2 i ) q^{80} -11 q^{81} + 8 i q^{82} -4 q^{83} + 6 i q^{86} + 12 i q^{87} + 6 i q^{88} -8 i q^{89} + ( -1 + 2 i ) q^{90} -6 i q^{92} -12 q^{93} + 8 q^{94} + ( 12 + 6 i ) q^{95} -10 i q^{96} -6 q^{97} + 7 q^{98} -2 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{4} - 2 q^{5} + 6 q^{8} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} - 2 q^{4} - 2 q^{5} + 6 q^{8} - 2 q^{9} + 2 q^{10} + 6 q^{13} - 8 q^{15} - 2 q^{16} + 2 q^{18} + 2 q^{20} - 6 q^{25} - 6 q^{26} + 12 q^{29} + 8 q^{30} - 10 q^{32} - 8 q^{33} + 2 q^{36} + 12 q^{37} + 8 q^{39} - 6 q^{40} + 2 q^{45} - 16 q^{47} - 14 q^{49} + 6 q^{50} - 6 q^{52} - 8 q^{55} + 24 q^{57} - 12 q^{58} + 8 q^{60} + 12 q^{61} + 14 q^{64} + 2 q^{65} + 8 q^{66} + 24 q^{67} - 24 q^{69} - 6 q^{72} - 12 q^{73} - 12 q^{74} + 16 q^{75} - 8 q^{78} + 2 q^{80} - 22 q^{81} - 8 q^{83} - 2 q^{90} - 24 q^{93} + 16 q^{94} + 24 q^{95} - 12 q^{97} + 14 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/65\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 − 1.00000i 1.00000i
−1.00000 2.00000i −1.00000 −1.00000 2.00000i 2.00000i 0 3.00000 −1.00000 1.00000 + 2.00000i
64.2 −1.00000 2.00000i −1.00000 −1.00000 + 2.00000i 2.00000i 0 3.00000 −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
65.d even 2 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.d.a 2 1.a even 1 1 trivial
65.2.d.a 2 65.d even 2 1 inner
65.2.d.b yes 2 5.b even 2 1
65.2.d.b yes 2 13.b even 2 1
325.2.c.b 2 5.c odd 4 1
325.2.c.b 2 65.h odd 4 1
325.2.c.e 2 5.c odd 4 1
325.2.c.e 2 65.h odd 4 1
585.2.h.b 2 15.d odd 2 1
585.2.h.b 2 39.d odd 2 1
585.2.h.c 2 3.b odd 2 1
585.2.h.c 2 195.e odd 2 1
845.2.b.a 2 13.d odd 4 1
845.2.b.a 2 65.g odd 4 1
845.2.b.b 2 13.d odd 4 1
845.2.b.b 2 65.g odd 4 1
845.2.l.a 4 13.e even 6 2
845.2.l.a 4 65.n even 6 2
845.2.l.b 4 13.c even 3 2
845.2.l.b 4 65.l even 6 2
845.2.n.a 4 13.f odd 12 2
845.2.n.a 4 65.s odd 12 2
845.2.n.b 4 13.f odd 12 2
845.2.n.b 4 65.s odd 12 2
1040.2.f.a 2 4.b odd 2 1
1040.2.f.a 2 260.g odd 2 1
1040.2.f.b 2 20.d odd 2 1
1040.2.f.b 2 52.b odd 2 1
4225.2.a.e 1 65.k even 4 1
4225.2.a.h 1 65.k even 4 1
4225.2.a.k 1 65.f even 4 1
4225.2.a.m 1 65.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(65, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$13 - 6 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$36 + T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$36 + T^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$144 + T^{2}$$
$59$ $$4 + T^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$( -12 + T )^{2}$$
$71$ $$4 + T^{2}$$
$73$ $$( 6 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$64 + T^{2}$$
$97$ $$( 6 + T )^{2}$$