# Properties

 Label 650.2.b.g 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: no (minimal twist has level 130) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - 2 i q^{3} - q^{4} + 2 q^{6} - 4 i q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 - 2*i * q^3 - q^4 + 2 * q^6 - 4*i * q^7 - i * q^8 - q^9 $$q + i q^{2} - 2 i q^{3} - q^{4} + 2 q^{6} - 4 i q^{7} - i q^{8} - q^{9} - 2 q^{11} + 2 i q^{12} + i q^{13} + 4 q^{14} + q^{16} + 2 i q^{17} - i q^{18} - 6 q^{19} - 8 q^{21} - 2 i q^{22} - 6 i q^{23} - 2 q^{24} - q^{26} - 4 i q^{27} + 4 i q^{28} - 2 q^{29} - 6 q^{31} + i q^{32} + 4 i q^{33} - 2 q^{34} + q^{36} - 2 i q^{37} - 6 i q^{38} + 2 q^{39} + 10 q^{41} - 8 i q^{42} + 10 i q^{43} + 2 q^{44} + 6 q^{46} - 12 i q^{47} - 2 i q^{48} - 9 q^{49} + 4 q^{51} - i q^{52} - 2 i q^{53} + 4 q^{54} - 4 q^{56} + 12 i q^{57} - 2 i q^{58} - 10 q^{59} + 2 q^{61} - 6 i q^{62} + 4 i q^{63} - q^{64} - 4 q^{66} - 12 i q^{67} - 2 i q^{68} - 12 q^{69} + 10 q^{71} + i q^{72} - 10 i q^{73} + 2 q^{74} + 6 q^{76} + 8 i q^{77} + 2 i q^{78} + 4 q^{79} - 11 q^{81} + 10 i q^{82} + 8 q^{84} - 10 q^{86} + 4 i q^{87} + 2 i q^{88} + 14 q^{89} + 4 q^{91} + 6 i q^{92} + 12 i q^{93} + 12 q^{94} + 2 q^{96} + 14 i q^{97} - 9 i q^{98} + 2 q^{99} +O(q^{100})$$ q + i * q^2 - 2*i * q^3 - q^4 + 2 * q^6 - 4*i * q^7 - i * q^8 - q^9 - 2 * q^11 + 2*i * q^12 + i * q^13 + 4 * q^14 + q^16 + 2*i * q^17 - i * q^18 - 6 * q^19 - 8 * q^21 - 2*i * q^22 - 6*i * q^23 - 2 * q^24 - q^26 - 4*i * q^27 + 4*i * q^28 - 2 * q^29 - 6 * q^31 + i * q^32 + 4*i * q^33 - 2 * q^34 + q^36 - 2*i * q^37 - 6*i * q^38 + 2 * q^39 + 10 * q^41 - 8*i * q^42 + 10*i * q^43 + 2 * q^44 + 6 * q^46 - 12*i * q^47 - 2*i * q^48 - 9 * q^49 + 4 * q^51 - i * q^52 - 2*i * q^53 + 4 * q^54 - 4 * q^56 + 12*i * q^57 - 2*i * q^58 - 10 * q^59 + 2 * q^61 - 6*i * q^62 + 4*i * q^63 - q^64 - 4 * q^66 - 12*i * q^67 - 2*i * q^68 - 12 * q^69 + 10 * q^71 + i * q^72 - 10*i * q^73 + 2 * q^74 + 6 * q^76 + 8*i * q^77 + 2*i * q^78 + 4 * q^79 - 11 * q^81 + 10*i * q^82 + 8 * q^84 - 10 * q^86 + 4*i * q^87 + 2*i * q^88 + 14 * q^89 + 4 * q^91 + 6*i * q^92 + 12*i * q^93 + 12 * q^94 + 2 * q^96 + 14*i * q^97 - 9*i * q^98 + 2 * 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 $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 4 q^{11} + 8 q^{14} + 2 q^{16} - 12 q^{19} - 16 q^{21} - 4 q^{24} - 2 q^{26} - 4 q^{29} - 12 q^{31} - 4 q^{34} + 2 q^{36} + 4 q^{39} + 20 q^{41} + 4 q^{44} + 12 q^{46} - 18 q^{49} + 8 q^{51} + 8 q^{54} - 8 q^{56} - 20 q^{59} + 4 q^{61} - 2 q^{64} - 8 q^{66} - 24 q^{69} + 20 q^{71} + 4 q^{74} + 12 q^{76} + 8 q^{79} - 22 q^{81} + 16 q^{84} - 20 q^{86} + 28 q^{89} + 8 q^{91} + 24 q^{94} + 4 q^{96} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 - 4 * q^11 + 8 * q^14 + 2 * q^16 - 12 * q^19 - 16 * q^21 - 4 * q^24 - 2 * q^26 - 4 * q^29 - 12 * q^31 - 4 * q^34 + 2 * q^36 + 4 * q^39 + 20 * q^41 + 4 * q^44 + 12 * q^46 - 18 * q^49 + 8 * q^51 + 8 * q^54 - 8 * q^56 - 20 * q^59 + 4 * q^61 - 2 * q^64 - 8 * q^66 - 24 * q^69 + 20 * q^71 + 4 * q^74 + 12 * q^76 + 8 * q^79 - 22 * q^81 + 16 * q^84 - 20 * q^86 + 28 * q^89 + 8 * q^91 + 24 * q^94 + 4 * q^96 + 4 * 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 4.00000i 1.00000i −1.00000 0
599.2 1.00000i 2.00000i −1.00000 0 2.00000 4.00000i 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.g 2
3.b odd 2 1 5850.2.e.u 2
5.b even 2 1 inner 650.2.b.g 2
5.c odd 4 1 130.2.a.c 1
5.c odd 4 1 650.2.a.c 1
15.d odd 2 1 5850.2.e.u 2
15.e even 4 1 1170.2.a.d 1
15.e even 4 1 5850.2.a.cb 1
20.e even 4 1 1040.2.a.b 1
20.e even 4 1 5200.2.a.bd 1
35.f even 4 1 6370.2.a.l 1
40.i odd 4 1 4160.2.a.c 1
40.k even 4 1 4160.2.a.t 1
60.l odd 4 1 9360.2.a.by 1
65.f even 4 1 1690.2.d.e 2
65.h odd 4 1 1690.2.a.e 1
65.h odd 4 1 8450.2.a.n 1
65.k even 4 1 1690.2.d.e 2
65.o even 12 2 1690.2.l.a 4
65.q odd 12 2 1690.2.e.a 2
65.r odd 12 2 1690.2.e.g 2
65.t even 12 2 1690.2.l.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.a.c 1 5.c odd 4 1
650.2.a.c 1 5.c odd 4 1
650.2.b.g 2 1.a even 1 1 trivial
650.2.b.g 2 5.b even 2 1 inner
1040.2.a.b 1 20.e even 4 1
1170.2.a.d 1 15.e even 4 1
1690.2.a.e 1 65.h odd 4 1
1690.2.d.e 2 65.f even 4 1
1690.2.d.e 2 65.k even 4 1
1690.2.e.a 2 65.q odd 12 2
1690.2.e.g 2 65.r odd 12 2
1690.2.l.a 4 65.o even 12 2
1690.2.l.a 4 65.t even 12 2
4160.2.a.c 1 40.i odd 4 1
4160.2.a.t 1 40.k even 4 1
5200.2.a.bd 1 20.e even 4 1
5850.2.a.cb 1 15.e even 4 1
5850.2.e.u 2 3.b odd 2 1
5850.2.e.u 2 15.d odd 2 1
6370.2.a.l 1 35.f even 4 1
8450.2.a.n 1 65.h odd 4 1
9360.2.a.by 1 60.l 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 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 4$$
$19$ $$(T + 6)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T + 2)^{2}$$
$31$ $$(T + 6)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 10)^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 4$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 14)^{2}$$
$97$ $$T^{2} + 196$$