# Properties

 Label 650.2.c.d Level $650$ Weight $2$ Character orbit 650.c 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.c (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: $$\Z[a_1, a_2, a_3]$$ 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.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
1.00000 1.00000i 1.00000 0 1.00000i 3.00000 1.00000 2.00000 0
649.2 1.00000 1.00000i 1.00000 0 1.00000i 3.00000 1.00000 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
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 650.2.c.d 2
5.b even 2 1 650.2.c.a 2
5.c odd 4 1 26.2.b.a 2
5.c odd 4 1 650.2.d.b 2
13.b even 2 1 650.2.c.a 2
15.e even 4 1 234.2.b.b 2
20.e even 4 1 208.2.f.a 2
35.f even 4 1 1274.2.d.c 2
35.k even 12 2 1274.2.n.c 4
35.l odd 12 2 1274.2.n.d 4
40.i odd 4 1 832.2.f.d 2
40.k even 4 1 832.2.f.b 2
60.l odd 4 1 1872.2.c.f 2
65.d even 2 1 inner 650.2.c.d 2
65.f even 4 1 338.2.a.b 1
65.f even 4 1 8450.2.a.h 1
65.h odd 4 1 26.2.b.a 2
65.h odd 4 1 650.2.d.b 2
65.k even 4 1 338.2.a.d 1
65.k even 4 1 8450.2.a.u 1
65.o even 12 2 338.2.c.b 2
65.q odd 12 2 338.2.e.c 4
65.r odd 12 2 338.2.e.c 4
65.t even 12 2 338.2.c.f 2
195.j odd 4 1 3042.2.a.g 1
195.s even 4 1 234.2.b.b 2
195.u odd 4 1 3042.2.a.j 1
260.l odd 4 1 2704.2.a.k 1
260.p even 4 1 208.2.f.a 2
260.s odd 4 1 2704.2.a.j 1
455.s even 4 1 1274.2.d.c 2
455.cv odd 12 2 1274.2.n.d 4
455.df even 12 2 1274.2.n.c 4
520.bc even 4 1 832.2.f.b 2
520.bg odd 4 1 832.2.f.d 2
780.w odd 4 1 1872.2.c.f 2

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

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7} - 3$$ T7 - 3 $$T_{37} - 3$$ T37 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$(T - 3)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 6T + 13$$
$17$ $$T^{2} + 9$$
$19$ $$T^{2} + 36$$
$23$ $$T^{2} + 36$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$(T - 3)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$(T - 3)^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2} + 36$$
$61$ $$(T + 8)^{2}$$
$67$ $$(T + 12)^{2}$$
$71$ $$T^{2} + 225$$
$73$ $$(T + 6)^{2}$$
$79$ $$(T + 10)^{2}$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 36$$
$97$ $$(T + 12)^{2}$$