# Properties

 Label 950.3.c.a Level $950$ Weight $3$ Character orbit 950.c Analytic conductor $25.886$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 950.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.8856251142$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 2 \beta q^{3} -2 q^{4} -4 q^{6} -5 q^{7} -2 \beta q^{8} + q^{9} +O(q^{10})$$ $$q + \beta q^{2} + 2 \beta q^{3} -2 q^{4} -4 q^{6} -5 q^{7} -2 \beta q^{8} + q^{9} + 5 q^{11} -4 \beta q^{12} -12 \beta q^{13} -5 \beta q^{14} + 4 q^{16} + 25 q^{17} + \beta q^{18} + 19 q^{19} -10 \beta q^{21} + 5 \beta q^{22} + 10 q^{23} + 8 q^{24} + 24 q^{26} + 20 \beta q^{27} + 10 q^{28} -30 \beta q^{29} + 30 \beta q^{31} + 4 \beta q^{32} + 10 \beta q^{33} + 25 \beta q^{34} -2 q^{36} -18 \beta q^{37} + 19 \beta q^{38} + 48 q^{39} + 30 \beta q^{41} + 20 q^{42} -5 q^{43} -10 q^{44} + 10 \beta q^{46} -5 q^{47} + 8 \beta q^{48} -24 q^{49} + 50 \beta q^{51} + 24 \beta q^{52} -18 \beta q^{53} -40 q^{54} + 10 \beta q^{56} + 38 \beta q^{57} + 60 q^{58} -60 \beta q^{59} + 95 q^{61} -60 q^{62} -5 q^{63} -8 q^{64} -20 q^{66} + 78 \beta q^{67} -50 q^{68} + 20 \beta q^{69} -2 \beta q^{72} + 25 q^{73} + 36 q^{74} -38 q^{76} -25 q^{77} + 48 \beta q^{78} -30 \beta q^{79} -71 q^{81} -60 q^{82} + 130 q^{83} + 20 \beta q^{84} -5 \beta q^{86} + 120 q^{87} -10 \beta q^{88} + 90 \beta q^{89} + 60 \beta q^{91} -20 q^{92} -120 q^{93} -5 \beta q^{94} -16 q^{96} + 12 \beta q^{97} -24 \beta q^{98} + 5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} - 8q^{6} - 10q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 4q^{4} - 8q^{6} - 10q^{7} + 2q^{9} + 10q^{11} + 8q^{16} + 50q^{17} + 38q^{19} + 20q^{23} + 16q^{24} + 48q^{26} + 20q^{28} - 4q^{36} + 96q^{39} + 40q^{42} - 10q^{43} - 20q^{44} - 10q^{47} - 48q^{49} - 80q^{54} + 120q^{58} + 190q^{61} - 120q^{62} - 10q^{63} - 16q^{64} - 40q^{66} - 100q^{68} + 50q^{73} + 72q^{74} - 76q^{76} - 50q^{77} - 142q^{81} - 120q^{82} + 260q^{83} + 240q^{87} - 40q^{92} - 240q^{93} - 32q^{96} + 10q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 − 1.41421i 1.41421i
1.41421i 2.82843i −2.00000 0 −4.00000 −5.00000 2.82843i 1.00000 0
151.2 1.41421i 2.82843i −2.00000 0 −4.00000 −5.00000 2.82843i 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
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.3.c.a 2
5.b even 2 1 38.3.b.a 2
5.c odd 4 2 950.3.d.a 4
15.d odd 2 1 342.3.d.a 2
19.b odd 2 1 inner 950.3.c.a 2
20.d odd 2 1 304.3.e.c 2
40.e odd 2 1 1216.3.e.i 2
40.f even 2 1 1216.3.e.j 2
60.h even 2 1 2736.3.o.h 2
95.d odd 2 1 38.3.b.a 2
95.g even 4 2 950.3.d.a 4
285.b even 2 1 342.3.d.a 2
380.d even 2 1 304.3.e.c 2
760.b odd 2 1 1216.3.e.j 2
760.p even 2 1 1216.3.e.i 2
1140.p odd 2 1 2736.3.o.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a 2 5.b even 2 1
38.3.b.a 2 95.d odd 2 1
304.3.e.c 2 20.d odd 2 1
304.3.e.c 2 380.d even 2 1
342.3.d.a 2 15.d odd 2 1
342.3.d.a 2 285.b even 2 1
950.3.c.a 2 1.a even 1 1 trivial
950.3.c.a 2 19.b odd 2 1 inner
950.3.d.a 4 5.c odd 4 2
950.3.d.a 4 95.g even 4 2
1216.3.e.i 2 40.e odd 2 1
1216.3.e.i 2 760.p even 2 1
1216.3.e.j 2 40.f even 2 1
1216.3.e.j 2 760.b odd 2 1
2736.3.o.h 2 60.h even 2 1
2736.3.o.h 2 1140.p odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{2} + 8$$ $$T_{7} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T^{2}$$
$3$ $$8 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 5 + T )^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$288 + T^{2}$$
$17$ $$( -25 + T )^{2}$$
$19$ $$( -19 + T )^{2}$$
$23$ $$( -10 + T )^{2}$$
$29$ $$1800 + T^{2}$$
$31$ $$1800 + T^{2}$$
$37$ $$648 + T^{2}$$
$41$ $$1800 + T^{2}$$
$43$ $$( 5 + T )^{2}$$
$47$ $$( 5 + T )^{2}$$
$53$ $$648 + T^{2}$$
$59$ $$7200 + T^{2}$$
$61$ $$( -95 + T )^{2}$$
$67$ $$12168 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -25 + T )^{2}$$
$79$ $$1800 + T^{2}$$
$83$ $$( -130 + T )^{2}$$
$89$ $$16200 + T^{2}$$
$97$ $$288 + T^{2}$$