# Properties

 Label 950.2.b.f Level $950$ Weight $2$ Character orbit 950.b Analytic conductor $7.586$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{1} q^{3} - q^{4} + ( 1 - \beta_{3} ) q^{6} -\beta_{1} q^{7} -\beta_{2} q^{8} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + \beta_{1} q^{3} - q^{4} + ( 1 - \beta_{3} ) q^{6} -\beta_{1} q^{7} -\beta_{2} q^{8} + ( -2 + \beta_{3} ) q^{9} + 4 q^{11} -\beta_{1} q^{12} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -1 + \beta_{3} ) q^{14} + q^{16} + ( -\beta_{1} - 6 \beta_{2} ) q^{17} + ( \beta_{1} - \beta_{2} ) q^{18} + q^{19} + ( 5 - \beta_{3} ) q^{21} + 4 \beta_{2} q^{22} -3 \beta_{1} q^{23} + ( -1 + \beta_{3} ) q^{24} + ( -1 + 3 \beta_{3} ) q^{26} + ( \beta_{1} + 4 \beta_{2} ) q^{27} + \beta_{1} q^{28} + ( 1 - 3 \beta_{3} ) q^{29} + ( -2 + 2 \beta_{3} ) q^{31} + \beta_{2} q^{32} + 4 \beta_{1} q^{33} + ( 5 + \beta_{3} ) q^{34} + ( 2 - \beta_{3} ) q^{36} + 6 \beta_{2} q^{37} + \beta_{2} q^{38} + ( 13 - \beta_{3} ) q^{39} + ( 6 - 4 \beta_{3} ) q^{41} + ( -\beta_{1} + 4 \beta_{2} ) q^{42} + ( -2 \beta_{1} - 8 \beta_{2} ) q^{43} -4 q^{44} + ( -3 + 3 \beta_{3} ) q^{46} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{47} + \beta_{1} q^{48} + ( 2 + \beta_{3} ) q^{49} + ( -1 + 5 \beta_{3} ) q^{51} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{52} + ( \beta_{1} - 2 \beta_{2} ) q^{53} + ( -3 - \beta_{3} ) q^{54} + ( 1 - \beta_{3} ) q^{56} + \beta_{1} q^{57} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{58} + ( -1 + \beta_{3} ) q^{59} + ( 8 - 2 \beta_{3} ) q^{61} + 2 \beta_{1} q^{62} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{63} - q^{64} + ( 4 - 4 \beta_{3} ) q^{66} -\beta_{1} q^{67} + ( \beta_{1} + 6 \beta_{2} ) q^{68} + ( 15 - 3 \beta_{3} ) q^{69} + ( 4 - 4 \beta_{3} ) q^{71} + ( -\beta_{1} + \beta_{2} ) q^{72} + ( 3 \beta_{1} + 6 \beta_{2} ) q^{73} -6 q^{74} - q^{76} -4 \beta_{1} q^{77} + ( -\beta_{1} + 12 \beta_{2} ) q^{78} + ( 2 - 2 \beta_{3} ) q^{79} -7 q^{81} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{82} + ( 2 \beta_{1} + 8 \beta_{2} ) q^{83} + ( -5 + \beta_{3} ) q^{84} + ( 6 + 2 \beta_{3} ) q^{86} + ( \beta_{1} - 12 \beta_{2} ) q^{87} -4 \beta_{2} q^{88} -2 q^{89} + ( -13 + \beta_{3} ) q^{91} + 3 \beta_{1} q^{92} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{93} -4 \beta_{3} q^{94} + ( 1 - \beta_{3} ) q^{96} -6 \beta_{2} q^{97} + ( \beta_{1} + 3 \beta_{2} ) q^{98} + ( -8 + 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 2q^{6} - 6q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 2q^{6} - 6q^{9} + 16q^{11} - 2q^{14} + 4q^{16} + 4q^{19} + 18q^{21} - 2q^{24} + 2q^{26} - 2q^{29} - 4q^{31} + 22q^{34} + 6q^{36} + 50q^{39} + 16q^{41} - 16q^{44} - 6q^{46} + 10q^{49} + 6q^{51} - 14q^{54} + 2q^{56} - 2q^{59} + 28q^{61} - 4q^{64} + 8q^{66} + 54q^{69} + 8q^{71} - 24q^{74} - 4q^{76} + 4q^{79} - 28q^{81} - 18q^{84} + 28q^{86} - 8q^{89} - 50q^{91} - 8q^{94} + 2q^{96} - 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 5 \beta_{1}$$

## 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
 − 1.56155i 2.56155i − 2.56155i 1.56155i
1.00000i 1.56155i −1.00000 0 −1.56155 1.56155i 1.00000i 0.561553 0
799.2 1.00000i 2.56155i −1.00000 0 2.56155 2.56155i 1.00000i −3.56155 0
799.3 1.00000i 2.56155i −1.00000 0 2.56155 2.56155i 1.00000i −3.56155 0
799.4 1.00000i 1.56155i −1.00000 0 −1.56155 1.56155i 1.00000i 0.561553 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.f 4
5.b even 2 1 inner 950.2.b.f 4
5.c odd 4 1 190.2.a.d 2
5.c odd 4 1 950.2.a.h 2
15.e even 4 1 1710.2.a.w 2
15.e even 4 1 8550.2.a.br 2
20.e even 4 1 1520.2.a.n 2
20.e even 4 1 7600.2.a.y 2
35.f even 4 1 9310.2.a.bc 2
40.i odd 4 1 6080.2.a.bh 2
40.k even 4 1 6080.2.a.bb 2
95.g even 4 1 3610.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.d 2 5.c odd 4 1
950.2.a.h 2 5.c odd 4 1
950.2.b.f 4 1.a even 1 1 trivial
950.2.b.f 4 5.b even 2 1 inner
1520.2.a.n 2 20.e even 4 1
1710.2.a.w 2 15.e even 4 1
3610.2.a.t 2 95.g even 4 1
6080.2.a.bb 2 40.k even 4 1
6080.2.a.bh 2 40.i odd 4 1
7600.2.a.y 2 20.e even 4 1
8550.2.a.br 2 15.e even 4 1
9310.2.a.bc 2 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 $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{4} + 9 T_{3}^{2} + 16$$ $$T_{7}^{4} + 9 T_{7}^{2} + 16$$ $$T_{11} - 4$$ $$T_{13}^{4} + 77 T_{13}^{2} + 1444$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$16 + 9 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$16 + 9 T^{2} + T^{4}$$
$11$ $$( -4 + T )^{4}$$
$13$ $$1444 + 77 T^{2} + T^{4}$$
$17$ $$676 + 69 T^{2} + T^{4}$$
$19$ $$( -1 + T )^{4}$$
$23$ $$1296 + 81 T^{2} + T^{4}$$
$29$ $$( -38 + T + T^{2} )^{2}$$
$31$ $$( -16 + 2 T + T^{2} )^{2}$$
$37$ $$( 36 + T^{2} )^{2}$$
$41$ $$( -52 - 8 T + T^{2} )^{2}$$
$43$ $$1024 + 132 T^{2} + T^{4}$$
$47$ $$4096 + 144 T^{2} + T^{4}$$
$53$ $$4 + 21 T^{2} + T^{4}$$
$59$ $$( -4 + T + T^{2} )^{2}$$
$61$ $$( 32 - 14 T + T^{2} )^{2}$$
$67$ $$16 + 9 T^{2} + T^{4}$$
$71$ $$( -64 - 4 T + T^{2} )^{2}$$
$73$ $$324 + 117 T^{2} + T^{4}$$
$79$ $$( -16 - 2 T + T^{2} )^{2}$$
$83$ $$1024 + 132 T^{2} + T^{4}$$
$89$ $$( 2 + T )^{4}$$
$97$ $$( 36 + T^{2} )^{2}$$