# Properties

 Label 650.2.b.h Level $650$ Weight $2$ Character orbit 650.b 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

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

 $$f(q)$$ $$=$$ $$q + i q^{2} - 2 i q^{3} - q^{4} + 2 q^{6} + i q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 - 2*i * q^3 - q^4 + 2 * q^6 + i * q^7 - i * q^8 - q^9 $$q + i q^{2} - 2 i q^{3} - q^{4} + 2 q^{6} + i q^{7} - i q^{8} - q^{9} + 3 q^{11} + 2 i q^{12} + i q^{13} - q^{14} + q^{16} - 3 i q^{17} - i q^{18} + 4 q^{19} + 2 q^{21} + 3 i q^{22} - 6 i q^{23} - 2 q^{24} - q^{26} - 4 i q^{27} - i q^{28} + 3 q^{29} - q^{31} + i q^{32} - 6 i q^{33} + 3 q^{34} + q^{36} - 2 i q^{37} + 4 i q^{38} + 2 q^{39} + 2 i q^{42} - 10 i q^{43} - 3 q^{44} + 6 q^{46} + 3 i q^{47} - 2 i q^{48} + 6 q^{49} - 6 q^{51} - i q^{52} + 3 i q^{53} + 4 q^{54} + q^{56} - 8 i q^{57} + 3 i q^{58} + 15 q^{59} - 13 q^{61} - i q^{62} - i q^{63} - q^{64} + 6 q^{66} + 13 i q^{67} + 3 i q^{68} - 12 q^{69} + i q^{72} - 10 i q^{73} + 2 q^{74} - 4 q^{76} + 3 i q^{77} + 2 i q^{78} + 4 q^{79} - 11 q^{81} + 15 i q^{83} - 2 q^{84} + 10 q^{86} - 6 i q^{87} - 3 i q^{88} - 6 q^{89} - q^{91} + 6 i q^{92} + 2 i q^{93} - 3 q^{94} + 2 q^{96} + 4 i q^{97} + 6 i q^{98} - 3 q^{99} +O(q^{100})$$ q + i * q^2 - 2*i * q^3 - q^4 + 2 * q^6 + i * q^7 - i * q^8 - q^9 + 3 * q^11 + 2*i * q^12 + i * q^13 - q^14 + q^16 - 3*i * q^17 - i * q^18 + 4 * q^19 + 2 * q^21 + 3*i * q^22 - 6*i * q^23 - 2 * q^24 - q^26 - 4*i * q^27 - i * q^28 + 3 * q^29 - q^31 + i * q^32 - 6*i * q^33 + 3 * q^34 + q^36 - 2*i * q^37 + 4*i * q^38 + 2 * q^39 + 2*i * q^42 - 10*i * q^43 - 3 * q^44 + 6 * q^46 + 3*i * q^47 - 2*i * q^48 + 6 * q^49 - 6 * q^51 - i * q^52 + 3*i * q^53 + 4 * q^54 + q^56 - 8*i * q^57 + 3*i * q^58 + 15 * q^59 - 13 * q^61 - i * q^62 - i * q^63 - q^64 + 6 * q^66 + 13*i * q^67 + 3*i * q^68 - 12 * q^69 + i * q^72 - 10*i * q^73 + 2 * q^74 - 4 * q^76 + 3*i * q^77 + 2*i * q^78 + 4 * q^79 - 11 * q^81 + 15*i * q^83 - 2 * q^84 + 10 * q^86 - 6*i * q^87 - 3*i * q^88 - 6 * q^89 - q^91 + 6*i * q^92 + 2*i * q^93 - 3 * q^94 + 2 * q^96 + 4*i * q^97 + 6*i * q^98 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 6 q^{11} - 2 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{21} - 4 q^{24} - 2 q^{26} + 6 q^{29} - 2 q^{31} + 6 q^{34} + 2 q^{36} + 4 q^{39} - 6 q^{44} + 12 q^{46} + 12 q^{49} - 12 q^{51} + 8 q^{54} + 2 q^{56} + 30 q^{59} - 26 q^{61} - 2 q^{64} + 12 q^{66} - 24 q^{69} + 4 q^{74} - 8 q^{76} + 8 q^{79} - 22 q^{81} - 4 q^{84} + 20 q^{86} - 12 q^{89} - 2 q^{91} - 6 q^{94} + 4 q^{96} - 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 + 6 * q^11 - 2 * q^14 + 2 * q^16 + 8 * q^19 + 4 * q^21 - 4 * q^24 - 2 * q^26 + 6 * q^29 - 2 * q^31 + 6 * q^34 + 2 * q^36 + 4 * q^39 - 6 * q^44 + 12 * q^46 + 12 * q^49 - 12 * q^51 + 8 * q^54 + 2 * q^56 + 30 * q^59 - 26 * q^61 - 2 * q^64 + 12 * q^66 - 24 * q^69 + 4 * q^74 - 8 * q^76 + 8 * q^79 - 22 * q^81 - 4 * q^84 + 20 * q^86 - 12 * q^89 - 2 * q^91 - 6 * q^94 + 4 * q^96 - 6 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
599.1
 − 1.00000i 1.00000i
1.00000i 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 0
599.2 1.00000i 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 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
5.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.b.h 2
3.b odd 2 1 5850.2.e.j 2
5.b even 2 1 inner 650.2.b.h 2
5.c odd 4 1 650.2.a.b 1
5.c odd 4 1 650.2.a.k yes 1
15.d odd 2 1 5850.2.e.j 2
15.e even 4 1 5850.2.a.q 1
15.e even 4 1 5850.2.a.bm 1
20.e even 4 1 5200.2.a.g 1
20.e even 4 1 5200.2.a.bg 1
65.h odd 4 1 8450.2.a.j 1
65.h odd 4 1 8450.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.a.b 1 5.c odd 4 1
650.2.a.k yes 1 5.c odd 4 1
650.2.b.h 2 1.a even 1 1 trivial
650.2.b.h 2 5.b even 2 1 inner
5200.2.a.g 1 20.e even 4 1
5200.2.a.bg 1 20.e even 4 1
5850.2.a.q 1 15.e even 4 1
5850.2.a.bm 1 15.e even 4 1
5850.2.e.j 2 3.b odd 2 1
5850.2.e.j 2 15.d odd 2 1
8450.2.a.j 1 65.h odd 4 1
8450.2.a.p 1 65.h odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(650, [\chi])$$:

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11} - 3$$ T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 9$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 3)^{2}$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2} + 9$$
$53$ $$T^{2} + 9$$
$59$ $$(T - 15)^{2}$$
$61$ $$(T + 13)^{2}$$
$67$ $$T^{2} + 169$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 225$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 16$$