# Properties

 Label 950.2.b.c 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 38) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} - q^{4} - q^{6} + 3 i q^{7} -i q^{8} + 2 q^{9} +O(q^{10})$$ $$q + i q^{2} + i q^{3} - q^{4} - q^{6} + 3 i q^{7} -i q^{8} + 2 q^{9} + 2 q^{11} -i q^{12} + i q^{13} -3 q^{14} + q^{16} + 3 i q^{17} + 2 i q^{18} + q^{19} -3 q^{21} + 2 i q^{22} + i q^{23} + q^{24} - q^{26} + 5 i q^{27} -3 i q^{28} + 5 q^{29} -8 q^{31} + i q^{32} + 2 i q^{33} -3 q^{34} -2 q^{36} -2 i q^{37} + i q^{38} - q^{39} -8 q^{41} -3 i q^{42} -4 i q^{43} -2 q^{44} - q^{46} + 8 i q^{47} + i q^{48} -2 q^{49} -3 q^{51} -i q^{52} + i q^{53} -5 q^{54} + 3 q^{56} + i q^{57} + 5 i q^{58} -15 q^{59} + 2 q^{61} -8 i q^{62} + 6 i q^{63} - q^{64} -2 q^{66} + 3 i q^{67} -3 i q^{68} - q^{69} + 2 q^{71} -2 i q^{72} -9 i q^{73} + 2 q^{74} - q^{76} + 6 i q^{77} -i q^{78} + 10 q^{79} + q^{81} -8 i q^{82} + 6 i q^{83} + 3 q^{84} + 4 q^{86} + 5 i q^{87} -2 i q^{88} -3 q^{91} -i q^{92} -8 i q^{93} -8 q^{94} - q^{96} -2 i q^{97} -2 i q^{98} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} + 4q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} + 4q^{9} + 4q^{11} - 6q^{14} + 2q^{16} + 2q^{19} - 6q^{21} + 2q^{24} - 2q^{26} + 10q^{29} - 16q^{31} - 6q^{34} - 4q^{36} - 2q^{39} - 16q^{41} - 4q^{44} - 2q^{46} - 4q^{49} - 6q^{51} - 10q^{54} + 6q^{56} - 30q^{59} + 4q^{61} - 2q^{64} - 4q^{66} - 2q^{69} + 4q^{71} + 4q^{74} - 2q^{76} + 20q^{79} + 2q^{81} + 6q^{84} + 8q^{86} - 6q^{91} - 16q^{94} - 2q^{96} + 8q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 3.00000i 1.00000i 2.00000 0
799.2 1.00000i 1.00000i −1.00000 0 −1.00000 3.00000i 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.c 2
5.b even 2 1 inner 950.2.b.c 2
5.c odd 4 1 38.2.a.b 1
5.c odd 4 1 950.2.a.b 1
15.e even 4 1 342.2.a.d 1
15.e even 4 1 8550.2.a.u 1
20.e even 4 1 304.2.a.d 1
20.e even 4 1 7600.2.a.h 1
35.f even 4 1 1862.2.a.f 1
40.i odd 4 1 1216.2.a.n 1
40.k even 4 1 1216.2.a.g 1
55.e even 4 1 4598.2.a.a 1
60.l odd 4 1 2736.2.a.w 1
65.h odd 4 1 6422.2.a.b 1
95.g even 4 1 722.2.a.b 1
95.l even 12 2 722.2.c.f 2
95.m odd 12 2 722.2.c.d 2
95.q odd 36 6 722.2.e.c 6
95.r even 36 6 722.2.e.d 6
285.j odd 4 1 6498.2.a.y 1
380.j odd 4 1 5776.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 5.c odd 4 1
304.2.a.d 1 20.e even 4 1
342.2.a.d 1 15.e even 4 1
722.2.a.b 1 95.g even 4 1
722.2.c.d 2 95.m odd 12 2
722.2.c.f 2 95.l even 12 2
722.2.e.c 6 95.q odd 36 6
722.2.e.d 6 95.r even 36 6
950.2.a.b 1 5.c odd 4 1
950.2.b.c 2 1.a even 1 1 trivial
950.2.b.c 2 5.b even 2 1 inner
1216.2.a.g 1 40.k even 4 1
1216.2.a.n 1 40.i odd 4 1
1862.2.a.f 1 35.f even 4 1
2736.2.a.w 1 60.l odd 4 1
4598.2.a.a 1 55.e even 4 1
5776.2.a.d 1 380.j odd 4 1
6422.2.a.b 1 65.h odd 4 1
6498.2.a.y 1 285.j odd 4 1
7600.2.a.h 1 20.e even 4 1
8550.2.a.u 1 15.e even 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$9 + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$1 + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 8 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$1 + T^{2}$$
$59$ $$( 15 + T )^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$9 + T^{2}$$
$71$ $$( -2 + T )^{2}$$
$73$ $$81 + T^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$4 + T^{2}$$