# Properties

 Label 1650.2.c.g Level $1650$ Weight $2$ Character orbit 1650.c Analytic conductor $13.175$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1650.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.1753163335$$ 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 330) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} - q^{4} - q^{6} -i q^{8} - q^{9} +O(q^{10})$$ $$q + i q^{2} + i q^{3} - q^{4} - q^{6} -i q^{8} - q^{9} + q^{11} -i q^{12} -6 i q^{13} + q^{16} + 2 i q^{17} -i q^{18} + 4 q^{19} + i q^{22} + q^{24} + 6 q^{26} -i q^{27} + 10 q^{29} + i q^{32} + i q^{33} -2 q^{34} + q^{36} + 6 i q^{37} + 4 i q^{38} + 6 q^{39} + 2 q^{41} -4 i q^{43} - q^{44} -8 i q^{47} + i q^{48} + 7 q^{49} -2 q^{51} + 6 i q^{52} + 10 i q^{53} + q^{54} + 4 i q^{57} + 10 i q^{58} + 4 q^{59} -2 q^{61} - q^{64} - q^{66} -4 i q^{67} -2 i q^{68} -8 q^{71} + i q^{72} -2 i q^{73} -6 q^{74} -4 q^{76} + 6 i q^{78} + 8 q^{79} + q^{81} + 2 i q^{82} + 12 i q^{83} + 4 q^{86} + 10 i q^{87} -i q^{88} + 6 q^{89} + 8 q^{94} - q^{96} + 18 i q^{97} + 7 i q^{98} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + 2q^{11} + 2q^{16} + 8q^{19} + 2q^{24} + 12q^{26} + 20q^{29} - 4q^{34} + 2q^{36} + 12q^{39} + 4q^{41} - 2q^{44} + 14q^{49} - 4q^{51} + 2q^{54} + 8q^{59} - 4q^{61} - 2q^{64} - 2q^{66} - 16q^{71} - 12q^{74} - 8q^{76} + 16q^{79} + 2q^{81} + 8q^{86} + 12q^{89} + 16q^{94} - 2q^{96} - 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$727$$ $$1201$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 0 1.00000i −1.00000 0
199.2 1.00000i 1.00000i −1.00000 0 −1.00000 0 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 1650.2.c.g 2
3.b odd 2 1 4950.2.c.j 2
5.b even 2 1 inner 1650.2.c.g 2
5.c odd 4 1 330.2.a.d 1
5.c odd 4 1 1650.2.a.h 1
15.d odd 2 1 4950.2.c.j 2
15.e even 4 1 990.2.a.b 1
15.e even 4 1 4950.2.a.bg 1
20.e even 4 1 2640.2.a.t 1
55.e even 4 1 3630.2.a.f 1
60.l odd 4 1 7920.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.a.d 1 5.c odd 4 1
990.2.a.b 1 15.e even 4 1
1650.2.a.h 1 5.c odd 4 1
1650.2.c.g 2 1.a even 1 1 trivial
1650.2.c.g 2 5.b even 2 1 inner
2640.2.a.t 1 20.e even 4 1
3630.2.a.f 1 55.e even 4 1
4950.2.a.bg 1 15.e even 4 1
4950.2.c.j 2 3.b odd 2 1
4950.2.c.j 2 15.d odd 2 1
7920.2.a.m 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 $$S_{2}^{\mathrm{new}}(1650, [\chi])$$:

 $$T_{7}$$ $$T_{13}^{2} + 36$$ $$T_{17}^{2} + 4$$ $$T_{19} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -10 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$324 + T^{2}$$