# Properties

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

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## 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: 8.0.1731891456.1 Defining polynomial: $$x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + \beta_{1} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{2} + \beta_{4} - \beta_{5} ) q^{6} + ( \beta_{6} + 3 \beta_{7} ) q^{7} + \beta_{7} q^{8} + ( 1 + 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} + \beta_{1} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{2} + \beta_{4} - \beta_{5} ) q^{6} + ( \beta_{6} + 3 \beta_{7} ) q^{7} + \beta_{7} q^{8} + ( 1 + 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{9} + q^{11} + \beta_{6} q^{12} + ( -2 \beta_{3} - 2 \beta_{7} ) q^{13} + ( 2 \beta_{2} - \beta_{5} ) q^{14} + \beta_{2} q^{16} + 2 \beta_{1} q^{17} + ( -\beta_{6} + \beta_{7} ) q^{18} + ( 2 + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{19} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{21} -\beta_{3} q^{22} + ( -3 \beta_{1} + 3 \beta_{3} + 3 \beta_{6} + 3 \beta_{7} ) q^{23} + ( -\beta_{2} - \beta_{5} ) q^{24} + 2 q^{26} + ( -\beta_{6} - 4 \beta_{7} ) q^{27} + ( -\beta_{1} + 3 \beta_{3} + \beta_{6} + 3 \beta_{7} ) q^{28} + ( 2 + 2 \beta_{2} ) q^{29} -4 \beta_{4} q^{31} + ( \beta_{3} + \beta_{7} ) q^{32} + \beta_{1} q^{33} + ( -2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{34} + ( 2 \beta_{2} + \beta_{5} ) q^{36} + ( 3 \beta_{6} + 3 \beta_{7} ) q^{37} + ( \beta_{1} - 3 \beta_{3} - 2 \beta_{6} - \beta_{7} ) q^{38} + 2 \beta_{4} q^{39} + ( -3 \beta_{2} - 2 \beta_{5} ) q^{41} + ( -2 \beta_{1} + 4 \beta_{3} + 2 \beta_{6} + 4 \beta_{7} ) q^{42} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{43} + ( 1 + \beta_{2} ) q^{44} + ( -3 - 3 \beta_{4} ) q^{46} + ( 2 \beta_{1} - 8 \beta_{3} - 2 \beta_{6} - 8 \beta_{7} ) q^{47} + ( -\beta_{1} + \beta_{6} ) q^{48} + ( -6 - 5 \beta_{4} ) q^{49} + ( 8 + 10 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{51} -2 \beta_{3} q^{52} + ( -\beta_{1} - 5 \beta_{3} + \beta_{6} - 5 \beta_{7} ) q^{53} + ( -3 \beta_{2} + \beta_{5} ) q^{54} + ( -3 - \beta_{4} ) q^{56} + ( 2 \beta_{1} - 4 \beta_{3} + \beta_{6} - 8 \beta_{7} ) q^{57} + 2 \beta_{7} q^{58} + ( -13 \beta_{2} - \beta_{5} ) q^{59} + ( -2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{61} -4 \beta_{1} q^{62} + ( \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{63} - q^{64} + ( -\beta_{2} + \beta_{4} - \beta_{5} ) q^{66} + ( \beta_{1} - 12 \beta_{3} - \beta_{6} - 12 \beta_{7} ) q^{67} + 2 \beta_{6} q^{68} -12 q^{69} + ( 10 \beta_{2} + 4 \beta_{5} ) q^{71} + ( \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{72} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{73} -3 \beta_{5} q^{74} + ( 3 + 3 \beta_{2} + \beta_{4} + \beta_{5} ) q^{76} + ( \beta_{6} + 3 \beta_{7} ) q^{77} + 2 \beta_{1} q^{78} + ( 8 \beta_{2} + 2 \beta_{5} ) q^{79} -7 \beta_{2} q^{81} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{82} + ( -5 \beta_{6} - 2 \beta_{7} ) q^{83} + ( -4 - 2 \beta_{4} ) q^{84} + ( -2 - 2 \beta_{4} + 2 \beta_{5} ) q^{86} + 2 \beta_{6} q^{87} + \beta_{7} q^{88} + ( 5 + 2 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{89} + ( 6 + 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{91} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{92} + ( 4 \beta_{1} + 16 \beta_{3} ) q^{93} + ( 8 + 2 \beta_{4} ) q^{94} -\beta_{4} q^{96} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{97} + ( -5 \beta_{1} + 6 \beta_{3} ) q^{98} + ( 1 + 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} - 2q^{6} + 6q^{9} + O(q^{10})$$ $$8q + 4q^{4} - 2q^{6} + 6q^{9} + 8q^{11} - 10q^{14} - 4q^{16} + 20q^{19} - 12q^{21} + 2q^{24} + 16q^{26} + 8q^{29} + 16q^{31} - 4q^{34} - 6q^{36} - 8q^{39} + 8q^{41} + 4q^{44} - 12q^{46} - 28q^{49} + 36q^{51} + 14q^{54} - 20q^{56} + 50q^{59} - 4q^{61} - 8q^{64} - 2q^{66} - 96q^{69} - 32q^{71} - 6q^{74} + 10q^{76} - 28q^{79} + 28q^{81} - 24q^{84} - 4q^{86} + 14q^{89} + 20q^{91} + 56q^{94} + 4q^{96} + 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$9 \nu^{6} - 65 \nu^{4} + 585 \nu^{2} - 1296$$$$)/1040$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 181 \nu$$$$)/260$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 116$$$$)/65$$ $$\beta_{5}$$ $$=$$ $$($$$$-29 \nu^{6} + 325 \nu^{4} - 1885 \nu^{2} + 4176$$$$)/1040$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{7} - 65 \nu^{5} + 585 \nu^{3} - 256 \nu$$$$)/1040$$ $$\beta_{7}$$ $$=$$ $$($$$$9 \nu^{7} - 65 \nu^{5} + 377 \nu^{3} - 256 \nu$$$$)/832$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + 5 \beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{7} + 5 \beta_{6}$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{5} + 29 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-36 \beta_{7} + 29 \beta_{6} - 36 \beta_{3} - 29 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$65 \beta_{4} - 116$$ $$\nu^{7}$$ $$=$$ $$-260 \beta_{3} - 181 \beta_{1}$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$-1 - \beta_{2}$$

## 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
 −1.35234 − 0.780776i 2.21837 + 1.28078i −2.21837 − 1.28078i 1.35234 + 0.780776i −1.35234 + 0.780776i 2.21837 − 1.28078i −2.21837 + 1.28078i 1.35234 − 0.780776i
−0.866025 0.500000i −1.35234 0.780776i 0.500000 + 0.866025i 0 0.780776 + 1.35234i 4.56155i 1.00000i −0.280776 0.486319i 0
49.2 −0.866025 0.500000i 2.21837 + 1.28078i 0.500000 + 0.866025i 0 −1.28078 2.21837i 0.438447i 1.00000i 1.78078 + 3.08440i 0
49.3 0.866025 + 0.500000i −2.21837 1.28078i 0.500000 + 0.866025i 0 −1.28078 2.21837i 0.438447i 1.00000i 1.78078 + 3.08440i 0
49.4 0.866025 + 0.500000i 1.35234 + 0.780776i 0.500000 + 0.866025i 0 0.780776 + 1.35234i 4.56155i 1.00000i −0.280776 0.486319i 0
349.1 −0.866025 + 0.500000i −1.35234 + 0.780776i 0.500000 0.866025i 0 0.780776 1.35234i 4.56155i 1.00000i −0.280776 + 0.486319i 0
349.2 −0.866025 + 0.500000i 2.21837 1.28078i 0.500000 0.866025i 0 −1.28078 + 2.21837i 0.438447i 1.00000i 1.78078 3.08440i 0
349.3 0.866025 0.500000i −2.21837 + 1.28078i 0.500000 0.866025i 0 −1.28078 + 2.21837i 0.438447i 1.00000i 1.78078 3.08440i 0
349.4 0.866025 0.500000i 1.35234 0.780776i 0.500000 0.866025i 0 0.780776 1.35234i 4.56155i 1.00000i −0.280776 + 0.486319i 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.f 8
5.b even 2 1 inner 950.2.j.f 8
5.c odd 4 1 190.2.e.c 4
5.c odd 4 1 950.2.e.h 4
15.e even 4 1 1710.2.l.m 4
19.c even 3 1 inner 950.2.j.f 8
20.e even 4 1 1520.2.q.h 4
95.i even 6 1 inner 950.2.j.f 8
95.l even 12 1 3610.2.a.u 2
95.m odd 12 1 190.2.e.c 4
95.m odd 12 1 950.2.e.h 4
95.m odd 12 1 3610.2.a.k 2
285.v even 12 1 1710.2.l.m 4
380.v even 12 1 1520.2.q.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.e.c 4 5.c odd 4 1
190.2.e.c 4 95.m odd 12 1
950.2.e.h 4 5.c odd 4 1
950.2.e.h 4 95.m odd 12 1
950.2.j.f 8 1.a even 1 1 trivial
950.2.j.f 8 5.b even 2 1 inner
950.2.j.f 8 19.c even 3 1 inner
950.2.j.f 8 95.i even 6 1 inner
1520.2.q.h 4 20.e even 4 1
1520.2.q.h 4 380.v even 12 1
1710.2.l.m 4 15.e even 4 1
1710.2.l.m 4 285.v even 12 1
3610.2.a.k 2 95.m odd 12 1
3610.2.a.u 2 95.l even 12 1

## Hecke kernels

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

 $$T_{3}^{8} - 9 T_{3}^{6} + 65 T_{3}^{4} - 144 T_{3}^{2} + 256$$ $$T_{7}^{4} + 21 T_{7}^{2} + 4$$ $$T_{11} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$256 - 144 T^{2} + 65 T^{4} - 9 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 4 + 21 T^{2} + T^{4} )^{2}$$
$11$ $$( -1 + T )^{8}$$
$13$ $$( 16 - 4 T^{2} + T^{4} )^{2}$$
$17$ $$65536 - 9216 T^{2} + 1040 T^{4} - 36 T^{6} + T^{8}$$
$19$ $$( 19 - 5 T + T^{2} )^{4}$$
$23$ $$1679616 - 104976 T^{2} + 5265 T^{4} - 81 T^{6} + T^{8}$$
$29$ $$( 4 - 2 T + T^{2} )^{4}$$
$31$ $$( -64 - 4 T + T^{2} )^{4}$$
$37$ $$( 1296 + 81 T^{2} + T^{4} )^{2}$$
$41$ $$( 169 + 52 T + 29 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$43$ $$65536 - 9216 T^{2} + 1040 T^{4} - 36 T^{6} + T^{8}$$
$47$ $$1048576 - 135168 T^{2} + 16400 T^{4} - 132 T^{6} + T^{8}$$
$53$ $$456976 - 46644 T^{2} + 4085 T^{4} - 69 T^{6} + T^{8}$$
$59$ $$( 23104 - 3800 T + 473 T^{2} - 25 T^{3} + T^{4} )^{2}$$
$61$ $$( 256 - 32 T + 20 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$67$ $$268435456 - 4472832 T^{2} + 58145 T^{4} - 273 T^{6} + T^{8}$$
$71$ $$( 16 - 64 T + 260 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$73$ $$104976 - 37908 T^{2} + 13365 T^{4} - 117 T^{6} + T^{8}$$
$79$ $$( 1024 + 448 T + 164 T^{2} + 14 T^{3} + T^{4} )^{2}$$
$83$ $$( 11236 + 213 T^{2} + T^{4} )^{2}$$
$89$ $$( 676 + 182 T + 75 T^{2} - 7 T^{3} + T^{4} )^{2}$$
$97$ $$104976 - 37908 T^{2} + 13365 T^{4} - 117 T^{6} + T^{8}$$
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