# Properties

 Label 950.2.b.d Level $950$ Weight $2$ Character orbit 950.b Analytic conductor $7.586$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ 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 190) 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.

 $$f(q)$$ $$=$$ $$q -i q^{2} + i q^{3} - q^{4} + q^{6} -i q^{7} + i q^{8} + 2 q^{9} +O(q^{10})$$ $$q -i q^{2} + i q^{3} - q^{4} + q^{6} -i q^{7} + i q^{8} + 2 q^{9} -i q^{12} + 3 i q^{13} - q^{14} + q^{16} -7 i q^{17} -2 i q^{18} + q^{19} + q^{21} + 5 i q^{23} - q^{24} + 3 q^{26} + 5 i q^{27} + i q^{28} + 5 q^{29} + 10 q^{31} -i q^{32} -7 q^{34} -2 q^{36} + 2 i q^{37} -i q^{38} -3 q^{39} + 2 q^{41} -i q^{42} -6 i q^{43} + 5 q^{46} + i q^{48} + 6 q^{49} + 7 q^{51} -3 i q^{52} -9 i q^{53} + 5 q^{54} + q^{56} + i q^{57} -5 i q^{58} + 7 q^{59} -4 q^{61} -10 i q^{62} -2 i q^{63} - q^{64} + 7 i q^{67} + 7 i q^{68} -5 q^{69} + 2 i q^{72} + 9 i q^{73} + 2 q^{74} - q^{76} + 3 i q^{78} + 10 q^{79} + q^{81} -2 i q^{82} + 2 i q^{83} - q^{84} -6 q^{86} + 5 i q^{87} + 10 q^{89} + 3 q^{91} -5 i q^{92} + 10 i q^{93} + q^{96} -18 i q^{97} -6 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} + 4q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} + 4q^{9} - 2q^{14} + 2q^{16} + 2q^{19} + 2q^{21} - 2q^{24} + 6q^{26} + 10q^{29} + 20q^{31} - 14q^{34} - 4q^{36} - 6q^{39} + 4q^{41} + 10q^{46} + 12q^{49} + 14q^{51} + 10q^{54} + 2q^{56} + 14q^{59} - 8q^{61} - 2q^{64} - 10q^{69} + 4q^{74} - 2q^{76} + 20q^{79} + 2q^{81} - 2q^{84} - 12q^{86} + 20q^{89} + 6q^{91} + 2q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 1.00000i 1.00000i 2.00000 0
799.2 1.00000i 1.00000i −1.00000 0 1.00000 1.00000i 1.00000i 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
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 950.2.b.d 2
5.b even 2 1 inner 950.2.b.d 2
5.c odd 4 1 190.2.a.a 1
5.c odd 4 1 950.2.a.e 1
15.e even 4 1 1710.2.a.r 1
15.e even 4 1 8550.2.a.l 1
20.e even 4 1 1520.2.a.g 1
20.e even 4 1 7600.2.a.g 1
35.f even 4 1 9310.2.a.i 1
40.i odd 4 1 6080.2.a.r 1
40.k even 4 1 6080.2.a.i 1
95.g even 4 1 3610.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.a 1 5.c odd 4 1
950.2.a.e 1 5.c odd 4 1
950.2.b.d 2 1.a even 1 1 trivial
950.2.b.d 2 5.b even 2 1 inner
1520.2.a.g 1 20.e even 4 1
1710.2.a.r 1 15.e even 4 1
3610.2.a.h 1 95.g even 4 1
6080.2.a.i 1 40.k even 4 1
6080.2.a.r 1 40.i odd 4 1
7600.2.a.g 1 20.e even 4 1
8550.2.a.l 1 15.e even 4 1
9310.2.a.i 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{2} + 1$$ $$T_{7}^{2} + 1$$ $$T_{11}$$ $$T_{13}^{2} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$9 + T^{2}$$
$17$ $$49 + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$25 + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$( -10 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$36 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$81 + T^{2}$$
$59$ $$( -7 + T )^{2}$$
$61$ $$( 4 + T )^{2}$$
$67$ $$49 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$81 + T^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$4 + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$324 + T^{2}$$