# Properties

 Label 950.2.j.g Level $950$ Weight $2$ Character orbit 950.j Analytic conductor $7.586$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} -\beta_{1} q^{3} + ( 1 + \beta_{3} ) q^{4} + \beta_{7} q^{6} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} ) q^{7} + ( \beta_{2} + \beta_{4} ) q^{8} + ( 4 + 4 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} -\beta_{1} q^{3} + ( 1 + \beta_{3} ) q^{4} + \beta_{7} q^{6} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} ) q^{7} + ( \beta_{2} + \beta_{4} ) q^{8} + ( 4 + 4 \beta_{3} ) q^{9} + ( -2 - \beta_{5} + \beta_{7} ) q^{11} + ( -\beta_{1} - \beta_{6} ) q^{12} -2 \beta_{2} q^{13} + ( -\beta_{3} + \beta_{5} ) q^{14} + \beta_{3} q^{16} + ( 4 \beta_{2} + 4 \beta_{4} ) q^{18} + ( -2 + 2 \beta_{3} - \beta_{7} ) q^{19} + ( 7 \beta_{3} - \beta_{5} ) q^{21} + ( -\beta_{1} - 2 \beta_{4} ) q^{22} + ( -\beta_{2} + \beta_{6} ) q^{23} + \beta_{5} q^{24} + 2 q^{26} + ( -\beta_{1} - \beta_{6} ) q^{27} + ( -\beta_{2} - \beta_{6} ) q^{28} + ( -1 - \beta_{3} - \beta_{7} ) q^{29} + ( -3 - \beta_{5} + \beta_{7} ) q^{31} + \beta_{2} q^{32} + ( 2 \beta_{1} + 7 \beta_{4} ) q^{33} + 4 \beta_{3} q^{36} + ( \beta_{1} - 3 \beta_{2} - 3 \beta_{4} + \beta_{6} ) q^{37} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{38} + ( -2 \beta_{5} + 2 \beta_{7} ) q^{39} + ( 5 \beta_{3} + 2 \beta_{5} ) q^{41} + ( 7 \beta_{2} + \beta_{6} ) q^{42} + ( -2 \beta_{1} - 6 \beta_{4} ) q^{43} + ( -2 - 2 \beta_{3} + \beta_{7} ) q^{44} + ( 1 - \beta_{5} + \beta_{7} ) q^{46} + ( 7 \beta_{2} - \beta_{6} ) q^{47} -\beta_{6} q^{48} + ( -1 - 2 \beta_{5} + 2 \beta_{7} ) q^{49} + 2 \beta_{4} q^{52} + ( 2 \beta_{2} + 4 \beta_{6} ) q^{53} + \beta_{5} q^{54} + ( 1 + \beta_{5} - \beta_{7} ) q^{56} + ( 2 \beta_{1} - 7 \beta_{2} - 7 \beta_{4} - 2 \beta_{6} ) q^{57} + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} ) q^{58} -3 \beta_{5} q^{59} + ( 7 + 7 \beta_{3} + 3 \beta_{7} ) q^{61} + ( -\beta_{1} - 3 \beta_{4} ) q^{62} + ( -4 \beta_{2} - 4 \beta_{6} ) q^{63} - q^{64} + ( 7 + 7 \beta_{3} - 2 \beta_{7} ) q^{66} + ( -2 \beta_{2} + \beta_{6} ) q^{67} + ( 7 - \beta_{5} + \beta_{7} ) q^{69} + ( -8 \beta_{3} - 2 \beta_{5} ) q^{71} + 4 \beta_{2} q^{72} + ( 2 \beta_{1} - 7 \beta_{4} ) q^{73} + ( -3 \beta_{3} - \beta_{5} ) q^{74} + ( -4 - 2 \beta_{3} - \beta_{5} ) q^{76} + ( 3 \beta_{1} + 9 \beta_{2} + 9 \beta_{4} + 3 \beta_{6} ) q^{77} -2 \beta_{1} q^{78} + 4 \beta_{3} q^{79} -5 \beta_{3} q^{81} + ( 5 \beta_{2} - 2 \beta_{6} ) q^{82} + ( 3 \beta_{1} + 3 \beta_{6} ) q^{83} + ( -7 - \beta_{5} + \beta_{7} ) q^{84} + ( -6 - 6 \beta_{3} + 2 \beta_{7} ) q^{86} + ( \beta_{1} - 7 \beta_{2} - 7 \beta_{4} + \beta_{6} ) q^{87} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{88} + ( -2 - 2 \beta_{3} + 2 \beta_{7} ) q^{91} + ( -\beta_{1} + \beta_{4} ) q^{92} + ( 3 \beta_{1} + 7 \beta_{4} ) q^{93} + ( -7 + \beta_{5} - \beta_{7} ) q^{94} + ( \beta_{5} - \beta_{7} ) q^{96} + ( 2 \beta_{1} - 9 \beta_{4} ) q^{97} + ( -2 \beta_{1} - \beta_{4} ) q^{98} + ( -8 - 8 \beta_{3} + 4 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + 16q^{9} + O(q^{10})$$ $$8q + 4q^{4} + 16q^{9} - 16q^{11} + 4q^{14} - 4q^{16} - 24q^{19} - 28q^{21} + 16q^{26} - 4q^{29} - 24q^{31} - 16q^{36} - 20q^{41} - 8q^{44} + 8q^{46} - 8q^{49} + 8q^{56} + 28q^{61} - 8q^{64} + 28q^{66} + 56q^{69} + 32q^{71} + 12q^{74} - 24q^{76} - 16q^{79} + 20q^{81} - 56q^{84} - 24q^{86} - 8q^{91} - 56q^{94} - 32q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 15 \nu^{4} + 5 \nu^{2} + 12$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 5 \nu^{5} + 5 \nu^{3} + 12 \nu$$$$)/40$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{6} - 5 \nu^{4} - 15 \nu^{2} - 36$$$$)/20$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 7 \nu$$$$)/10$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 13 \nu$$$$)/10$$ $$\beta_{6}$$ $$=$$ $$($$$$-9 \nu^{6} - 15 \nu^{4} - 5 \nu^{2} - 48$$$$)/20$$ $$\beta_{7}$$ $$=$$ $$($$$$11 \nu^{7} + 25 \nu^{5} + 55 \nu^{3} + 132 \nu$$$$)/40$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - 3 \beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} - \beta_{5} + 5 \beta_{4} + 5 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{7} - 11 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{6} - 5 \beta_{1} - 9$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{5} - 13 \beta_{4}$$$$)/2$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$-1 - \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.228425 + 1.39564i 1.09445 − 0.895644i 0.228425 − 1.39564i −1.09445 + 0.895644i −0.228425 − 1.39564i 1.09445 + 0.895644i 0.228425 + 1.39564i −1.09445 − 0.895644i
−0.866025 0.500000i −2.29129 1.32288i 0.500000 + 0.866025i 0 1.32288 + 2.29129i 1.64575i 1.00000i 2.00000 + 3.46410i 0
49.2 −0.866025 0.500000i 2.29129 + 1.32288i 0.500000 + 0.866025i 0 −1.32288 2.29129i 3.64575i 1.00000i 2.00000 + 3.46410i 0
49.3 0.866025 + 0.500000i −2.29129 1.32288i 0.500000 + 0.866025i 0 −1.32288 2.29129i 3.64575i 1.00000i 2.00000 + 3.46410i 0
49.4 0.866025 + 0.500000i 2.29129 + 1.32288i 0.500000 + 0.866025i 0 1.32288 + 2.29129i 1.64575i 1.00000i 2.00000 + 3.46410i 0
349.1 −0.866025 + 0.500000i −2.29129 + 1.32288i 0.500000 0.866025i 0 1.32288 2.29129i 1.64575i 1.00000i 2.00000 3.46410i 0
349.2 −0.866025 + 0.500000i 2.29129 1.32288i 0.500000 0.866025i 0 −1.32288 + 2.29129i 3.64575i 1.00000i 2.00000 3.46410i 0
349.3 0.866025 0.500000i −2.29129 + 1.32288i 0.500000 0.866025i 0 −1.32288 + 2.29129i 3.64575i 1.00000i 2.00000 3.46410i 0
349.4 0.866025 0.500000i 2.29129 1.32288i 0.500000 0.866025i 0 1.32288 2.29129i 1.64575i 1.00000i 2.00000 3.46410i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.4 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
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.j.g 8
5.b even 2 1 inner 950.2.j.g 8
5.c odd 4 1 38.2.c.b 4
5.c odd 4 1 950.2.e.k 4
15.e even 4 1 342.2.g.f 4
19.c even 3 1 inner 950.2.j.g 8
20.e even 4 1 304.2.i.e 4
40.i odd 4 1 1216.2.i.l 4
40.k even 4 1 1216.2.i.k 4
60.l odd 4 1 2736.2.s.v 4
95.g even 4 1 722.2.c.j 4
95.i even 6 1 inner 950.2.j.g 8
95.l even 12 1 722.2.a.g 2
95.l even 12 1 722.2.c.j 4
95.m odd 12 1 38.2.c.b 4
95.m odd 12 1 722.2.a.j 2
95.m odd 12 1 950.2.e.k 4
95.q odd 36 6 722.2.e.n 12
95.r even 36 6 722.2.e.o 12
285.v even 12 1 342.2.g.f 4
285.v even 12 1 6498.2.a.ba 2
285.w odd 12 1 6498.2.a.bg 2
380.v even 12 1 304.2.i.e 4
380.v even 12 1 5776.2.a.ba 2
380.w odd 12 1 5776.2.a.z 2
760.br odd 12 1 1216.2.i.l 4
760.bw even 12 1 1216.2.i.k 4
1140.bu odd 12 1 2736.2.s.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.b 4 5.c odd 4 1
38.2.c.b 4 95.m odd 12 1
304.2.i.e 4 20.e even 4 1
304.2.i.e 4 380.v even 12 1
342.2.g.f 4 15.e even 4 1
342.2.g.f 4 285.v even 12 1
722.2.a.g 2 95.l even 12 1
722.2.a.j 2 95.m odd 12 1
722.2.c.j 4 95.g even 4 1
722.2.c.j 4 95.l even 12 1
722.2.e.n 12 95.q odd 36 6
722.2.e.o 12 95.r even 36 6
950.2.e.k 4 5.c odd 4 1
950.2.e.k 4 95.m odd 12 1
950.2.j.g 8 1.a even 1 1 trivial
950.2.j.g 8 5.b even 2 1 inner
950.2.j.g 8 19.c even 3 1 inner
950.2.j.g 8 95.i even 6 1 inner
1216.2.i.k 4 40.k even 4 1
1216.2.i.k 4 760.bw even 12 1
1216.2.i.l 4 40.i odd 4 1
1216.2.i.l 4 760.br odd 12 1
2736.2.s.v 4 60.l odd 4 1
2736.2.s.v 4 1140.bu odd 12 1
5776.2.a.z 2 380.w odd 12 1
5776.2.a.ba 2 380.v even 12 1
6498.2.a.ba 2 285.v even 12 1
6498.2.a.bg 2 285.w odd 12 1

## Hecke kernels

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

 $$T_{3}^{4} - 7 T_{3}^{2} + 49$$ $$T_{7}^{4} + 16 T_{7}^{2} + 36$$ $$T_{11}^{2} + 4 T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$( 49 - 7 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 36 + 16 T^{2} + T^{4} )^{2}$$
$11$ $$( -3 + 4 T + T^{2} )^{4}$$
$13$ $$( 16 - 4 T^{2} + T^{4} )^{2}$$
$17$ $$T^{8}$$
$19$ $$( 361 + 228 T + 67 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$23$ $$1296 - 576 T^{2} + 220 T^{4} - 16 T^{6} + T^{8}$$
$29$ $$( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$31$ $$( 2 + 6 T + T^{2} )^{4}$$
$37$ $$( 4 + 32 T^{2} + T^{4} )^{2}$$
$41$ $$( 9 - 30 T + 103 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$43$ $$4096 - 8192 T^{2} + 16320 T^{4} - 128 T^{6} + T^{8}$$
$47$ $$3111696 - 197568 T^{2} + 10780 T^{4} - 112 T^{6} + T^{8}$$
$53$ $$136048896 - 2706048 T^{2} + 42160 T^{4} - 232 T^{6} + T^{8}$$
$59$ $$( 3969 + 63 T^{2} + T^{4} )^{2}$$
$61$ $$( 196 + 196 T + 210 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$67$ $$81 - 198 T^{2} + 475 T^{4} - 22 T^{6} + T^{8}$$
$71$ $$( 1296 - 576 T + 220 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$73$ $$194481 - 67914 T^{2} + 23275 T^{4} - 154 T^{6} + T^{8}$$
$79$ $$( 16 + 4 T + T^{2} )^{4}$$
$83$ $$( 63 + T^{2} )^{4}$$
$89$ $$T^{8}$$
$97$ $$7890481 - 612362 T^{2} + 44715 T^{4} - 218 T^{6} + T^{8}$$