# Properties

 Label 650.2.b.b 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 2 i q^{3} - q^{4} - 2 q^{6} + 5 i q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 + 2*i * q^3 - q^4 - 2 * q^6 + 5*i * q^7 - i * q^8 - q^9 $$q + i q^{2} + 2 i q^{3} - q^{4} - 2 q^{6} + 5 i q^{7} - i q^{8} - q^{9} - 3 q^{11} - 2 i q^{12} - i q^{13} - 5 q^{14} + q^{16} + 3 i q^{17} - i q^{18} + 4 q^{19} - 10 q^{21} - 3 i q^{22} - 6 i q^{23} + 2 q^{24} + q^{26} + 4 i q^{27} - 5 i q^{28} - 9 q^{29} + 5 q^{31} + i q^{32} - 6 i q^{33} - 3 q^{34} + q^{36} + 2 i q^{37} + 4 i q^{38} + 2 q^{39} - 10 i q^{42} - 2 i q^{43} + 3 q^{44} + 6 q^{46} - 9 i q^{47} + 2 i q^{48} - 18 q^{49} - 6 q^{51} + i q^{52} + 9 i q^{53} - 4 q^{54} + 5 q^{56} + 8 i q^{57} - 9 i q^{58} + 9 q^{59} - q^{61} + 5 i q^{62} - 5 i q^{63} - q^{64} + 6 q^{66} + 5 i q^{67} - 3 i q^{68} + 12 q^{69} + i q^{72} - 14 i q^{73} - 2 q^{74} - 4 q^{76} - 15 i q^{77} + 2 i q^{78} + 16 q^{79} - 11 q^{81} + 15 i q^{83} + 10 q^{84} + 2 q^{86} - 18 i q^{87} + 3 i q^{88} + 6 q^{89} + 5 q^{91} + 6 i q^{92} + 10 i q^{93} + 9 q^{94} - 2 q^{96} + 8 i q^{97} - 18 i q^{98} + 3 q^{99} +O(q^{100})$$ q + i * q^2 + 2*i * q^3 - q^4 - 2 * q^6 + 5*i * q^7 - i * q^8 - q^9 - 3 * q^11 - 2*i * q^12 - i * q^13 - 5 * q^14 + q^16 + 3*i * q^17 - i * q^18 + 4 * q^19 - 10 * q^21 - 3*i * q^22 - 6*i * q^23 + 2 * q^24 + q^26 + 4*i * q^27 - 5*i * q^28 - 9 * q^29 + 5 * 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 - 10*i * q^42 - 2*i * q^43 + 3 * q^44 + 6 * q^46 - 9*i * q^47 + 2*i * q^48 - 18 * q^49 - 6 * q^51 + i * q^52 + 9*i * q^53 - 4 * q^54 + 5 * q^56 + 8*i * q^57 - 9*i * q^58 + 9 * q^59 - q^61 + 5*i * q^62 - 5*i * q^63 - q^64 + 6 * q^66 + 5*i * q^67 - 3*i * q^68 + 12 * q^69 + i * q^72 - 14*i * q^73 - 2 * q^74 - 4 * q^76 - 15*i * q^77 + 2*i * q^78 + 16 * q^79 - 11 * q^81 + 15*i * q^83 + 10 * q^84 + 2 * q^86 - 18*i * q^87 + 3*i * q^88 + 6 * q^89 + 5 * q^91 + 6*i * q^92 + 10*i * q^93 + 9 * q^94 - 2 * q^96 + 8*i * q^97 - 18*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 $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 6 q^{11} - 10 q^{14} + 2 q^{16} + 8 q^{19} - 20 q^{21} + 4 q^{24} + 2 q^{26} - 18 q^{29} + 10 q^{31} - 6 q^{34} + 2 q^{36} + 4 q^{39} + 6 q^{44} + 12 q^{46} - 36 q^{49} - 12 q^{51} - 8 q^{54} + 10 q^{56} + 18 q^{59} - 2 q^{61} - 2 q^{64} + 12 q^{66} + 24 q^{69} - 4 q^{74} - 8 q^{76} + 32 q^{79} - 22 q^{81} + 20 q^{84} + 4 q^{86} + 12 q^{89} + 10 q^{91} + 18 q^{94} - 4 q^{96} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 - 6 * q^11 - 10 * q^14 + 2 * q^16 + 8 * q^19 - 20 * q^21 + 4 * q^24 + 2 * q^26 - 18 * q^29 + 10 * q^31 - 6 * q^34 + 2 * q^36 + 4 * q^39 + 6 * q^44 + 12 * q^46 - 36 * q^49 - 12 * q^51 - 8 * q^54 + 10 * q^56 + 18 * q^59 - 2 * q^61 - 2 * q^64 + 12 * q^66 + 24 * q^69 - 4 * q^74 - 8 * q^76 + 32 * q^79 - 22 * q^81 + 20 * q^84 + 4 * q^86 + 12 * q^89 + 10 * q^91 + 18 * 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 5.00000i 1.00000i −1.00000 0
599.2 1.00000i 2.00000i −1.00000 0 −2.00000 5.00000i 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.b 2
3.b odd 2 1 5850.2.e.ba 2
5.b even 2 1 inner 650.2.b.b 2
5.c odd 4 1 650.2.a.f 1
5.c odd 4 1 650.2.a.h yes 1
15.d odd 2 1 5850.2.e.ba 2
15.e even 4 1 5850.2.a.bb 1
15.e even 4 1 5850.2.a.bc 1
20.e even 4 1 5200.2.a.i 1
20.e even 4 1 5200.2.a.bc 1
65.h odd 4 1 8450.2.a.a 1
65.h odd 4 1 8450.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.a.f 1 5.c odd 4 1
650.2.a.h yes 1 5.c odd 4 1
650.2.b.b 2 1.a even 1 1 trivial
650.2.b.b 2 5.b even 2 1 inner
5200.2.a.i 1 20.e even 4 1
5200.2.a.bc 1 20.e even 4 1
5850.2.a.bb 1 15.e even 4 1
5850.2.a.bc 1 15.e even 4 1
5850.2.e.ba 2 3.b odd 2 1
5850.2.e.ba 2 15.d odd 2 1
8450.2.a.a 1 65.h odd 4 1
8450.2.a.x 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 $$T_{7}^{2} + 25$$ T7^2 + 25 $$T_{11} + 3$$ T11 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 25$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 9$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T + 9)^{2}$$
$31$ $$(T - 5)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 4$$
$47$ $$T^{2} + 81$$
$53$ $$T^{2} + 81$$
$59$ $$(T - 9)^{2}$$
$61$ $$(T + 1)^{2}$$
$67$ $$T^{2} + 25$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T - 16)^{2}$$
$83$ $$T^{2} + 225$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 64$$