# Properties

 Label 950.2.b.h Level $950$ Weight $2$ Character orbit 950.b Analytic conductor $7.586$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.63107136.1 Defining polynomial: $$x^{6} + 13 x^{4} + 42 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} - q^{4} + ( 1 - \beta_{2} ) q^{6} + ( \beta_{3} - \beta_{5} ) q^{7} -\beta_{3} q^{8} + ( -2 + 2 \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} - q^{4} + ( 1 - \beta_{2} ) q^{6} + ( \beta_{3} - \beta_{5} ) q^{7} -\beta_{3} q^{8} + ( -2 + 2 \beta_{2} + \beta_{4} ) q^{9} + 2 \beta_{2} q^{11} + ( -\beta_{1} + \beta_{3} ) q^{12} + ( -\beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{13} + ( -1 - \beta_{4} ) q^{14} + q^{16} + ( -\beta_{1} + 4 \beta_{3} + \beta_{5} ) q^{17} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{18} - q^{19} + ( 2 + \beta_{2} ) q^{21} + 2 \beta_{1} q^{22} + ( \beta_{1} + \beta_{3} - 2 \beta_{5} ) q^{23} + ( -1 + \beta_{2} ) q^{24} + ( -2 + \beta_{2} + \beta_{4} ) q^{26} + ( -3 \beta_{1} + 6 \beta_{3} + 2 \beta_{5} ) q^{27} + ( -\beta_{3} + \beta_{5} ) q^{28} + ( -2 - \beta_{2} - \beta_{4} ) q^{29} + ( 2 + 2 \beta_{2} ) q^{31} + \beta_{3} q^{32} + ( -2 \beta_{1} + 8 \beta_{3} + 2 \beta_{5} ) q^{33} + ( -4 + \beta_{2} + \beta_{4} ) q^{34} + ( 2 - 2 \beta_{2} - \beta_{4} ) q^{36} + ( 2 \beta_{1} - \beta_{3} ) q^{37} -\beta_{3} q^{38} + ( 5 - 5 \beta_{2} - \beta_{4} ) q^{39} + ( \beta_{1} + 2 \beta_{3} ) q^{42} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{43} -2 \beta_{2} q^{44} + ( -1 - \beta_{2} - 2 \beta_{4} ) q^{46} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{5} ) q^{47} + ( \beta_{1} - \beta_{3} ) q^{48} + ( -3 - 3 \beta_{2} - 3 \beta_{4} ) q^{49} + ( 7 - 7 \beta_{2} - \beta_{4} ) q^{51} + ( \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{52} + ( -4 \beta_{1} + 5 \beta_{3} - \beta_{5} ) q^{53} + ( -6 + 3 \beta_{2} + 2 \beta_{4} ) q^{54} + ( 1 + \beta_{4} ) q^{56} + ( -\beta_{1} + \beta_{3} ) q^{57} + ( -\beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{58} + ( -1 + \beta_{2} + 4 \beta_{4} ) q^{59} + 2 \beta_{4} q^{61} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{62} + ( \beta_{1} + 5 \beta_{3} - 2 \beta_{5} ) q^{63} - q^{64} + ( -8 + 2 \beta_{2} + 2 \beta_{4} ) q^{66} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{67} + ( \beta_{1} - 4 \beta_{3} - \beta_{5} ) q^{68} + ( -1 + 4 \beta_{2} + \beta_{4} ) q^{69} + ( -4 + 4 \beta_{2} - 2 \beta_{4} ) q^{71} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{72} + ( -\beta_{1} + 5 \beta_{3} - 2 \beta_{5} ) q^{73} + ( 1 - 2 \beta_{2} ) q^{74} + q^{76} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{77} + ( -5 \beta_{1} + 5 \beta_{3} + \beta_{5} ) q^{78} + ( -8 + 4 \beta_{2} ) q^{79} + ( 10 - 7 \beta_{2} ) q^{81} + ( -4 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{83} + ( -2 - \beta_{2} ) q^{84} + ( -4 - 2 \beta_{2} ) q^{86} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{87} -2 \beta_{1} q^{88} + ( -6 + 4 \beta_{2} ) q^{89} + ( 6 + 2 \beta_{2} + \beta_{4} ) q^{91} + ( -\beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{92} + ( 6 \beta_{3} + 2 \beta_{5} ) q^{93} + ( -1 - 2 \beta_{2} - 2 \beta_{4} ) q^{94} + ( 1 - \beta_{2} ) q^{96} + ( -6 \beta_{1} - 8 \beta_{3} + 2 \beta_{5} ) q^{97} + ( -3 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} ) q^{98} + ( 14 - 8 \beta_{2} - 2 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{4} + 4q^{6} - 10q^{9} + O(q^{10})$$ $$6q - 6q^{4} + 4q^{6} - 10q^{9} + 4q^{11} - 4q^{14} + 6q^{16} - 6q^{19} + 14q^{21} - 4q^{24} - 12q^{26} - 12q^{29} + 16q^{31} - 24q^{34} + 10q^{36} + 22q^{39} - 4q^{44} - 4q^{46} - 18q^{49} + 30q^{51} - 34q^{54} + 4q^{56} - 12q^{59} - 4q^{61} - 6q^{64} - 48q^{66} - 12q^{71} + 2q^{74} + 6q^{76} - 40q^{79} + 46q^{81} - 14q^{84} - 28q^{86} - 28q^{89} + 38q^{91} - 6q^{94} + 4q^{96} + 72q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 13 x^{4} + 42 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} + 7 \nu^{2} + 3$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 10 \nu^{3} + 21 \nu$$$$)/9$$ $$\beta_{4}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} - 19 \nu^{3} - 75 \nu$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - 4$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} - \beta_{3} - 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{4} + 3 \beta_{2} + 25$$ $$\nu^{5}$$ $$=$$ $$10 \beta_{5} + 19 \beta_{3} + 39 \beta_{1}$$

## 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
 − 2.77339i − 0.480031i 2.25342i − 2.25342i 0.480031i 2.77339i
1.00000i 1.77339i −1.00000 0 −1.77339 2.69168i 1.00000i −0.144903 0
799.2 1.00000i 0.519969i −1.00000 0 0.519969 4.76957i 1.00000i 2.72963 0
799.3 1.00000i 3.25342i −1.00000 0 3.25342 0.0778929i 1.00000i −7.58473 0
799.4 1.00000i 3.25342i −1.00000 0 3.25342 0.0778929i 1.00000i −7.58473 0
799.5 1.00000i 0.519969i −1.00000 0 0.519969 4.76957i 1.00000i 2.72963 0
799.6 1.00000i 1.77339i −1.00000 0 −1.77339 2.69168i 1.00000i −0.144903 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 799.6 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.h 6
5.b even 2 1 inner 950.2.b.h 6
5.c odd 4 1 950.2.a.j 3
5.c odd 4 1 950.2.a.l yes 3
15.e even 4 1 8550.2.a.ci 3
15.e even 4 1 8550.2.a.cp 3
20.e even 4 1 7600.2.a.bk 3
20.e even 4 1 7600.2.a.bz 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.j 3 5.c odd 4 1
950.2.a.l yes 3 5.c odd 4 1
950.2.b.h 6 1.a even 1 1 trivial
950.2.b.h 6 5.b even 2 1 inner
7600.2.a.bk 3 20.e even 4 1
7600.2.a.bz 3 20.e even 4 1
8550.2.a.ci 3 15.e even 4 1
8550.2.a.cp 3 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 $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{6} + 14 T_{3}^{4} + 37 T_{3}^{2} + 9$$ $$T_{7}^{6} + 30 T_{7}^{4} + 165 T_{7}^{2} + 1$$ $$T_{11}^{3} - 2 T_{11}^{2} - 24 T_{11} + 24$$ $$T_{13}^{6} + 42 T_{13}^{4} + 429 T_{13}^{2} + 1225$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$9 + 37 T^{2} + 14 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$1 + 165 T^{2} + 30 T^{4} + T^{6}$$
$11$ $$( 24 - 24 T - 2 T^{2} + T^{3} )^{2}$$
$13$ $$1225 + 429 T^{2} + 42 T^{4} + T^{6}$$
$17$ $$81 + 1305 T^{2} + 78 T^{4} + T^{6}$$
$19$ $$( 1 + T )^{6}$$
$23$ $$12321 + 2157 T^{2} + 106 T^{4} + T^{6}$$
$29$ $$( -9 - 3 T + 6 T^{2} + T^{3} )^{2}$$
$31$ $$( 56 - 4 T - 8 T^{2} + T^{3} )^{2}$$
$37$ $$1 + 627 T^{2} + 51 T^{4} + T^{6}$$
$41$ $$T^{6}$$
$43$ $$576 + 2272 T^{2} + 116 T^{4} + T^{6}$$
$47$ $$2025 + 2979 T^{2} + 123 T^{4} + T^{6}$$
$53$ $$751689 + 28365 T^{2} + 310 T^{4} + T^{6}$$
$59$ $$( -1431 - 201 T + 6 T^{2} + T^{3} )^{2}$$
$61$ $$( -120 - 56 T + 2 T^{2} + T^{3} )^{2}$$
$67$ $$5625 + 1129 T^{2} + 62 T^{4} + T^{6}$$
$71$ $$( -1512 - 192 T + 6 T^{2} + T^{3} )^{2}$$
$73$ $$100489 + 8337 T^{2} + 198 T^{4} + T^{6}$$
$79$ $$( -320 + 32 T + 20 T^{2} + T^{3} )^{2}$$
$83$ $$28224 + 10320 T^{2} + 232 T^{4} + T^{6}$$
$89$ $$( -312 - 36 T + 14 T^{2} + T^{3} )^{2}$$
$97$ $$5953600 + 138576 T^{2} + 696 T^{4} + T^{6}$$