# Properties

 Label 1950.2.e.k Level $1950$ Weight $2$ Character orbit 1950.e Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ 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 390) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} -i q^{3} - q^{4} + q^{6} + 2 i q^{7} -i q^{8} - q^{9} +O(q^{10})$$ $$q + i q^{2} -i q^{3} - q^{4} + q^{6} + 2 i q^{7} -i q^{8} - q^{9} + i q^{12} -i q^{13} -2 q^{14} + q^{16} -i q^{18} -2 q^{19} + 2 q^{21} + 6 i q^{23} - q^{24} + q^{26} + i q^{27} -2 i q^{28} -4 q^{31} + i q^{32} + q^{36} + 2 i q^{37} -2 i q^{38} - q^{39} -6 q^{41} + 2 i q^{42} + 4 i q^{43} -6 q^{46} -i q^{48} + 3 q^{49} + i q^{52} + 6 i q^{53} - q^{54} + 2 q^{56} + 2 i q^{57} -10 q^{61} -4 i q^{62} -2 i q^{63} - q^{64} + 8 i q^{67} + 6 q^{69} + i q^{72} -8 i q^{73} -2 q^{74} + 2 q^{76} -i q^{78} -8 q^{79} + q^{81} -6 i q^{82} + 12 i q^{83} -2 q^{84} -4 q^{86} -6 q^{89} + 2 q^{91} -6 i q^{92} + 4 i q^{93} + q^{96} + 8 i q^{97} + 3 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} - 4q^{14} + 2q^{16} - 4q^{19} + 4q^{21} - 2q^{24} + 2q^{26} - 8q^{31} + 2q^{36} - 2q^{39} - 12q^{41} - 12q^{46} + 6q^{49} - 2q^{54} + 4q^{56} - 20q^{61} - 2q^{64} + 12q^{69} - 4q^{74} + 4q^{76} - 16q^{79} + 2q^{81} - 4q^{84} - 8q^{86} - 12q^{89} + 4q^{91} + 2q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 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 1950.2.e.k 2
3.b odd 2 1 5850.2.e.r 2
5.b even 2 1 inner 1950.2.e.k 2
5.c odd 4 1 390.2.a.g 1
5.c odd 4 1 1950.2.a.b 1
15.d odd 2 1 5850.2.e.r 2
15.e even 4 1 1170.2.a.g 1
15.e even 4 1 5850.2.a.bk 1
20.e even 4 1 3120.2.a.b 1
60.l odd 4 1 9360.2.a.bg 1
65.f even 4 1 5070.2.b.n 2
65.h odd 4 1 5070.2.a.k 1
65.k even 4 1 5070.2.b.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.g 1 5.c odd 4 1
1170.2.a.g 1 15.e even 4 1
1950.2.a.b 1 5.c odd 4 1
1950.2.e.k 2 1.a even 1 1 trivial
1950.2.e.k 2 5.b even 2 1 inner
3120.2.a.b 1 20.e even 4 1
5070.2.a.k 1 65.h odd 4 1
5070.2.b.n 2 65.f even 4 1
5070.2.b.n 2 65.k even 4 1
5850.2.a.bk 1 15.e even 4 1
5850.2.e.r 2 3.b odd 2 1
5850.2.e.r 2 15.d odd 2 1
9360.2.a.bg 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 $$S_{2}^{\mathrm{new}}(1950, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{11}$$ $$T_{17}$$ $$T_{31} + 4$$

## Hecke characteristic polynomials

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