# Properties

 Label 950.2.j.h 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: $$3^{2}$$ Twist minimal: yes 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} + 5 \nu^{5} - 5 \nu^{3} - 12 \nu$$$$)/40$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{5} + 15 \nu^{3} + 42 \nu$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu$$$$)/40$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 5 \nu^{4} + 5 \nu^{2} - 2$$$$)/10$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{6} - 5 \nu^{4} - 15 \nu^{2} - 16$$$$)/20$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 3 \nu^{4} + \nu^{2} + 4$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-9 \nu^{7} - 15 \nu^{5} - 25 \nu^{3} - 8 \nu$$$$)/40$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - 2 \beta_{3} + \beta_{2} - \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{6} - 4 \beta_{5} + \beta_{4} - 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} + 7 \beta_{3} + \beta_{2} - \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{3} + 2 \beta_{2} + 16 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$5 \beta_{6} - 5 \beta_{5} - 10 \beta_{4} - 11$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{7} - 16 \beta_{3} - 7 \beta_{2} - 23 \beta_{1}$$$$)/3$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$-1 + \beta_{5}$$

## 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 −1.09445 − 0.895644i 0.228425 + 1.39564i −0.228425 + 1.39564i 1.09445 − 0.895644i −1.09445 + 0.895644i 0.228425 − 1.39564i
−0.866025 0.500000i −2.41733 1.39564i 0.500000 + 0.866025i 0 1.39564 + 2.41733i 1.00000i 1.00000i 2.39564 + 4.14938i 0
49.2 −0.866025 0.500000i 1.55130 + 0.895644i 0.500000 + 0.866025i 0 −0.895644 1.55130i 1.00000i 1.00000i 0.104356 + 0.180750i 0
49.3 0.866025 + 0.500000i −1.55130 0.895644i 0.500000 + 0.866025i 0 −0.895644 1.55130i 1.00000i 1.00000i 0.104356 + 0.180750i 0
49.4 0.866025 + 0.500000i 2.41733 + 1.39564i 0.500000 + 0.866025i 0 1.39564 + 2.41733i 1.00000i 1.00000i 2.39564 + 4.14938i 0
349.1 −0.866025 + 0.500000i −2.41733 + 1.39564i 0.500000 0.866025i 0 1.39564 2.41733i 1.00000i 1.00000i 2.39564 4.14938i 0
349.2 −0.866025 + 0.500000i 1.55130 0.895644i 0.500000 0.866025i 0 −0.895644 + 1.55130i 1.00000i 1.00000i 0.104356 0.180750i 0
349.3 0.866025 0.500000i −1.55130 + 0.895644i 0.500000 0.866025i 0 −0.895644 + 1.55130i 1.00000i 1.00000i 0.104356 0.180750i 0
349.4 0.866025 0.500000i 2.41733 1.39564i 0.500000 0.866025i 0 1.39564 2.41733i 1.00000i 1.00000i 2.39564 4.14938i 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.h 8
5.b even 2 1 inner 950.2.j.h 8
5.c odd 4 1 950.2.e.i 4
5.c odd 4 1 950.2.e.j yes 4
19.c even 3 1 inner 950.2.j.h 8
95.i even 6 1 inner 950.2.j.h 8
95.m odd 12 1 950.2.e.i 4
95.m odd 12 1 950.2.e.j yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.e.i 4 5.c odd 4 1
950.2.e.i 4 95.m odd 12 1
950.2.e.j yes 4 5.c odd 4 1
950.2.e.j yes 4 95.m odd 12 1
950.2.j.h 8 1.a even 1 1 trivial
950.2.j.h 8 5.b even 2 1 inner
950.2.j.h 8 19.c even 3 1 inner
950.2.j.h 8 95.i even 6 1 inner

## Hecke kernels

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

 $$T_{3}^{8} - 11 T_{3}^{6} + 96 T_{3}^{4} - 275 T_{3}^{2} + 625$$ $$T_{7}^{2} + 1$$ $$T_{11}^{2} + 3 T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$625 - 275 T^{2} + 96 T^{4} - 11 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$( -3 + 3 T + T^{2} )^{4}$$
$13$ $$1 - 23 T^{2} + 528 T^{4} - 23 T^{6} + T^{8}$$
$17$ $$81 - 135 T^{2} + 216 T^{4} - 15 T^{6} + T^{8}$$
$19$ $$( 19 - 7 T + T^{2} )^{4}$$
$23$ $$( 441 - 21 T^{2} + T^{4} )^{2}$$
$29$ $$( 225 - 135 T + 66 T^{2} - 9 T^{3} + T^{4} )^{2}$$
$31$ $$( 1 + 5 T + T^{2} )^{4}$$
$37$ $$( 400 + 44 T^{2} + T^{4} )^{2}$$
$41$ $$( 729 + 243 T + 108 T^{2} - 9 T^{3} + T^{4} )^{2}$$
$43$ $$2401 - 1715 T^{2} + 1176 T^{4} - 35 T^{6} + T^{8}$$
$47$ $$31640625 - 1046250 T^{2} + 28971 T^{4} - 186 T^{6} + T^{8}$$
$53$ $$( 441 - 21 T^{2} + T^{4} )^{2}$$
$59$ $$( 441 + 21 T^{2} + T^{4} )^{2}$$
$61$ $$( 289 + 187 T + 138 T^{2} - 11 T^{3} + T^{4} )^{2}$$
$67$ $$100000000 - 2840000 T^{2} + 70656 T^{4} - 284 T^{6} + T^{8}$$
$71$ $$( 441 + 21 T^{2} + T^{4} )^{2}$$
$73$ $$625 - 275 T^{2} + 96 T^{4} - 11 T^{6} + T^{8}$$
$79$ $$( 14161 + 833 T + 168 T^{2} - 7 T^{3} + T^{4} )^{2}$$
$83$ $$( 9 + 15 T^{2} + T^{4} )^{2}$$
$89$ $$( 441 + 21 T^{2} + T^{4} )^{2}$$
$97$ $$83521 - 44795 T^{2} + 23736 T^{4} - 155 T^{6} + T^{8}$$