# Properties

 Label 65.2.d.b 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.

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

## 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.b yes 2
3.b odd 2 1 585.2.h.b 2
4.b odd 2 1 1040.2.f.b 2
5.b even 2 1 65.2.d.a 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.a 2
13.c even 3 2 845.2.l.a 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.b 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.c 2
20.d odd 2 1 1040.2.f.a 2
39.d odd 2 1 585.2.h.c 2
52.b odd 2 1 1040.2.f.a 2
65.d even 2 1 inner 65.2.d.b yes 2
65.f even 4 1 4225.2.a.e 1
65.f even 4 1 4225.2.a.h 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.k 1
65.k even 4 1 4225.2.a.m 1
65.l even 6 2 845.2.l.a 4
65.n even 6 2 845.2.l.b 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.b 2
260.g odd 2 1 1040.2.f.b 2

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

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