# Properties

 Label 950.2.j.e Level $950$ Weight $2$ Character orbit 950.j Analytic conductor $7.586$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 4.00000i 1.00000i −1.00000 1.73205i 0
49.2 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 4.00000i 1.00000i −1.00000 1.73205i 0
349.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 4.00000i 1.00000i −1.00000 + 1.73205i 0
349.2 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 4.00000i 1.00000i −1.00000 + 1.73205i 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
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.e 4
5.b even 2 1 inner 950.2.j.e 4
5.c odd 4 1 38.2.c.a 2
5.c odd 4 1 950.2.e.d 2
15.e even 4 1 342.2.g.b 2
19.c even 3 1 inner 950.2.j.e 4
20.e even 4 1 304.2.i.c 2
40.i odd 4 1 1216.2.i.h 2
40.k even 4 1 1216.2.i.d 2
60.l odd 4 1 2736.2.s.m 2
95.g even 4 1 722.2.c.b 2
95.i even 6 1 inner 950.2.j.e 4
95.l even 12 1 722.2.a.d 1
95.l even 12 1 722.2.c.b 2
95.m odd 12 1 38.2.c.a 2
95.m odd 12 1 722.2.a.c 1
95.m odd 12 1 950.2.e.d 2
95.q odd 36 6 722.2.e.j 6
95.r even 36 6 722.2.e.i 6
285.v even 12 1 342.2.g.b 2
285.v even 12 1 6498.2.a.s 1
285.w odd 12 1 6498.2.a.e 1
380.v even 12 1 304.2.i.c 2
380.v even 12 1 5776.2.a.g 1
380.w odd 12 1 5776.2.a.n 1
760.br odd 12 1 1216.2.i.h 2
760.bw even 12 1 1216.2.i.d 2
1140.bu odd 12 1 2736.2.s.m 2

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

## Hecke kernels

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

 $$T_{3}^{4} - T_{3}^{2} + 1$$ $$T_{7}^{2} + 16$$ $$T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 16 + T^{2} )^{2}$$
$11$ $$( -3 + T )^{4}$$
$13$ $$16 - 4 T^{2} + T^{4}$$
$17$ $$1296 - 36 T^{2} + T^{4}$$
$19$ $$( 19 - 7 T + T^{2} )^{2}$$
$23$ $$1296 - 36 T^{2} + T^{4}$$
$29$ $$T^{4}$$
$31$ $$( -2 + T )^{4}$$
$37$ $$( 100 + T^{2} )^{2}$$
$41$ $$( 81 + 9 T + T^{2} )^{2}$$
$43$ $$256 - 16 T^{2} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$1296 - 36 T^{2} + T^{4}$$
$59$ $$( 81 + 9 T + T^{2} )^{2}$$
$61$ $$( 16 - 4 T + T^{2} )^{2}$$
$67$ $$2401 - 49 T^{2} + T^{4}$$
$71$ $$( 36 - 6 T + T^{2} )^{2}$$
$73$ $$1 - T^{2} + T^{4}$$
$79$ $$( 16 + 4 T + T^{2} )^{2}$$
$83$ $$( 9 + T^{2} )^{2}$$
$89$ $$( 36 - 6 T + T^{2} )^{2}$$
$97$ $$83521 - 289 T^{2} + T^{4}$$