# Properties

 Label 650.2.d.b Level $650$ Weight $2$ Character orbit 650.d Analytic conductor $5.190$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$650 = 2 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 650.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.19027613138$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{3} - q^{4} + i q^{6} + 3 i q^{7} -i q^{8} -2 q^{9} +O(q^{10})$$ $$q + i q^{2} + q^{3} - q^{4} + i q^{6} + 3 i q^{7} -i q^{8} -2 q^{9} - q^{12} + ( -2 + 3 i ) q^{13} -3 q^{14} + q^{16} -3 q^{17} -2 i q^{18} + 6 i q^{19} + 3 i q^{21} + 6 q^{23} -i q^{24} + ( -3 - 2 i ) q^{26} -5 q^{27} -3 i q^{28} + i q^{32} -3 i q^{34} + 2 q^{36} + 3 i q^{37} -6 q^{38} + ( -2 + 3 i ) q^{39} -3 q^{42} + q^{43} + 6 i q^{46} + 3 i q^{47} + q^{48} -2 q^{49} -3 q^{51} + ( 2 - 3 i ) q^{52} + 6 q^{53} -5 i q^{54} + 3 q^{56} + 6 i q^{57} + 6 i q^{59} -8 q^{61} -6 i q^{63} - q^{64} -12 i q^{67} + 3 q^{68} + 6 q^{69} -15 i q^{71} + 2 i q^{72} + 6 i q^{73} -3 q^{74} -6 i q^{76} + ( -3 - 2 i ) q^{78} + 10 q^{79} + q^{81} + 6 i q^{83} -3 i q^{84} + i q^{86} + 6 i q^{89} + ( -9 - 6 i ) q^{91} -6 q^{92} -3 q^{94} + i q^{96} -12 i q^{97} -2 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{4} - 4q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{4} - 4q^{9} - 2q^{12} - 4q^{13} - 6q^{14} + 2q^{16} - 6q^{17} + 12q^{23} - 6q^{26} - 10q^{27} + 4q^{36} - 12q^{38} - 4q^{39} - 6q^{42} + 2q^{43} + 2q^{48} - 4q^{49} - 6q^{51} + 4q^{52} + 12q^{53} + 6q^{56} - 16q^{61} - 2q^{64} + 6q^{68} + 12q^{69} - 6q^{74} - 6q^{78} + 20q^{79} + 2q^{81} - 18q^{91} - 12q^{92} - 6q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 − 1.00000i 1.00000i
1.00000i 1.00000 −1.00000 0 1.00000i 3.00000i 1.00000i −2.00000 0
51.2 1.00000i 1.00000 −1.00000 0 1.00000i 3.00000i 1.00000i −2.00000 0
 $$n$$: 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 650.2.d.b 2
5.b even 2 1 26.2.b.a 2
5.c odd 4 1 650.2.c.a 2
5.c odd 4 1 650.2.c.d 2
13.b even 2 1 inner 650.2.d.b 2
13.d odd 4 1 8450.2.a.h 1
13.d odd 4 1 8450.2.a.u 1
15.d odd 2 1 234.2.b.b 2
20.d odd 2 1 208.2.f.a 2
35.c odd 2 1 1274.2.d.c 2
35.i odd 6 2 1274.2.n.c 4
35.j even 6 2 1274.2.n.d 4
40.e odd 2 1 832.2.f.b 2
40.f even 2 1 832.2.f.d 2
60.h even 2 1 1872.2.c.f 2
65.d even 2 1 26.2.b.a 2
65.g odd 4 1 338.2.a.b 1
65.g odd 4 1 338.2.a.d 1
65.h odd 4 1 650.2.c.a 2
65.h odd 4 1 650.2.c.d 2
65.l even 6 2 338.2.e.c 4
65.n even 6 2 338.2.e.c 4
65.s odd 12 2 338.2.c.b 2
65.s odd 12 2 338.2.c.f 2
195.e odd 2 1 234.2.b.b 2
195.n even 4 1 3042.2.a.g 1
195.n even 4 1 3042.2.a.j 1
260.g odd 2 1 208.2.f.a 2
260.u even 4 1 2704.2.a.j 1
260.u even 4 1 2704.2.a.k 1
455.h odd 2 1 1274.2.d.c 2
455.bf odd 6 2 1274.2.n.c 4
455.bh even 6 2 1274.2.n.d 4
520.b odd 2 1 832.2.f.b 2
520.p even 2 1 832.2.f.d 2
780.d even 2 1 1872.2.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 5.b even 2 1
26.2.b.a 2 65.d even 2 1
208.2.f.a 2 20.d odd 2 1
208.2.f.a 2 260.g odd 2 1
234.2.b.b 2 15.d odd 2 1
234.2.b.b 2 195.e odd 2 1
338.2.a.b 1 65.g odd 4 1
338.2.a.d 1 65.g odd 4 1
338.2.c.b 2 65.s odd 12 2
338.2.c.f 2 65.s odd 12 2
338.2.e.c 4 65.l even 6 2
338.2.e.c 4 65.n even 6 2
650.2.c.a 2 5.c odd 4 1
650.2.c.a 2 65.h odd 4 1
650.2.c.d 2 5.c odd 4 1
650.2.c.d 2 65.h odd 4 1
650.2.d.b 2 1.a even 1 1 trivial
650.2.d.b 2 13.b even 2 1 inner
832.2.f.b 2 40.e odd 2 1
832.2.f.b 2 520.b odd 2 1
832.2.f.d 2 40.f even 2 1
832.2.f.d 2 520.p even 2 1
1274.2.d.c 2 35.c odd 2 1
1274.2.d.c 2 455.h odd 2 1
1274.2.n.c 4 35.i odd 6 2
1274.2.n.c 4 455.bf odd 6 2
1274.2.n.d 4 35.j even 6 2
1274.2.n.d 4 455.bh even 6 2
1872.2.c.f 2 60.h even 2 1
1872.2.c.f 2 780.d even 2 1
2704.2.a.j 1 260.u even 4 1
2704.2.a.k 1 260.u even 4 1
3042.2.a.g 1 195.n even 4 1
3042.2.a.j 1 195.n even 4 1
8450.2.a.h 1 13.d odd 4 1
8450.2.a.u 1 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{2}^{\mathrm{new}}(650, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$9 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$13 + 4 T + T^{2}$$
$17$ $$( 3 + T )^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$9 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -1 + T )^{2}$$
$47$ $$9 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$( 8 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$225 + T^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$36 + T^{2}$$
$97$ $$144 + T^{2}$$