# Properties

 Label 1950.2.e.a 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: 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.

 $$f(q)$$ $$=$$ $$q -i q^{2} -i q^{3} - q^{4} - q^{6} + i q^{7} + i q^{8} - q^{9} +O(q^{10})$$ $$q -i q^{2} -i q^{3} - q^{4} - q^{6} + i q^{7} + i q^{8} - q^{9} -5 q^{11} + i q^{12} -i q^{13} + q^{14} + q^{16} -5 i q^{17} + i q^{18} + q^{21} + 5 i q^{22} + q^{24} - q^{26} + i q^{27} -i q^{28} + 7 q^{29} -9 q^{31} -i q^{32} + 5 i q^{33} -5 q^{34} + q^{36} + 8 i q^{37} - q^{39} -2 q^{41} -i q^{42} + 8 i q^{43} + 5 q^{44} + 9 i q^{47} -i q^{48} + 6 q^{49} -5 q^{51} + i q^{52} + 11 i q^{53} + q^{54} - q^{56} -7 i q^{58} - q^{59} -7 q^{61} + 9 i q^{62} -i q^{63} - q^{64} + 5 q^{66} + 15 i q^{67} + 5 i q^{68} -8 q^{71} -i q^{72} + 4 i q^{73} + 8 q^{74} -5 i q^{77} + i q^{78} + 4 q^{79} + q^{81} + 2 i q^{82} -9 i q^{83} - q^{84} + 8 q^{86} -7 i q^{87} -5 i q^{88} -16 q^{89} + q^{91} + 9 i q^{93} + 9 q^{94} - q^{96} -2 i q^{97} -6 i q^{98} + 5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 10 q^{11} + 2 q^{14} + 2 q^{16} + 2 q^{21} + 2 q^{24} - 2 q^{26} + 14 q^{29} - 18 q^{31} - 10 q^{34} + 2 q^{36} - 2 q^{39} - 4 q^{41} + 10 q^{44} + 12 q^{49} - 10 q^{51} + 2 q^{54} - 2 q^{56} - 2 q^{59} - 14 q^{61} - 2 q^{64} + 10 q^{66} - 16 q^{71} + 16 q^{74} + 8 q^{79} + 2 q^{81} - 2 q^{84} + 16 q^{86} - 32 q^{89} + 2 q^{91} + 18 q^{94} - 2 q^{96} + 10 q^{99} + O(q^{100})$$

## 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 1.00000i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 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.a 2
3.b odd 2 1 5850.2.e.be 2
5.b even 2 1 inner 1950.2.e.a 2
5.c odd 4 1 1950.2.a.l 1
5.c odd 4 1 1950.2.a.p yes 1
15.d odd 2 1 5850.2.e.be 2
15.e even 4 1 5850.2.a.k 1
15.e even 4 1 5850.2.a.bu 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.a.l 1 5.c odd 4 1
1950.2.a.p yes 1 5.c odd 4 1
1950.2.e.a 2 1.a even 1 1 trivial
1950.2.e.a 2 5.b even 2 1 inner
5850.2.a.k 1 15.e even 4 1
5850.2.a.bu 1 15.e even 4 1
5850.2.e.be 2 3.b odd 2 1
5850.2.e.be 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$:

 $$T_{7}^{2} + 1$$ $$T_{11} + 5$$ $$T_{17}^{2} + 25$$ $$T_{31} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 5 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$25 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -7 + T )^{2}$$
$31$ $$( 9 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$81 + T^{2}$$
$53$ $$121 + T^{2}$$
$59$ $$( 1 + T )^{2}$$
$61$ $$( 7 + T )^{2}$$
$67$ $$225 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$16 + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$81 + T^{2}$$
$89$ $$( 16 + T )^{2}$$
$97$ $$4 + T^{2}$$