# Properties

 Label 950.2.j.d 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 190) 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{7} -\zeta_{12}^{3} q^{8} -3 \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{7} -\zeta_{12}^{3} q^{8} -3 \zeta_{12}^{2} q^{9} + 5 q^{11} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{13} + ( 1 - \zeta_{12}^{2} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 3 \zeta_{12}^{3} q^{18} + ( 3 - 5 \zeta_{12}^{2} ) q^{19} -5 \zeta_{12} q^{22} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{23} + 2 q^{26} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{28} + 6 \zeta_{12}^{2} q^{29} + 4 q^{31} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{32} + ( 3 - 3 \zeta_{12}^{2} ) q^{36} -11 \zeta_{12}^{3} q^{37} + ( -3 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{38} + ( 9 - 9 \zeta_{12}^{2} ) q^{41} + 6 \zeta_{12} q^{43} + 5 \zeta_{12}^{2} q^{44} + q^{46} + 6 q^{49} -2 \zeta_{12} q^{52} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{53} + q^{56} -6 \zeta_{12}^{3} q^{58} -4 \zeta_{12} q^{62} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{63} - q^{64} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{67} + ( -6 + 6 \zeta_{12}^{2} ) q^{71} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{72} + 14 \zeta_{12} q^{73} + ( -11 + 11 \zeta_{12}^{2} ) q^{74} + ( 5 - 2 \zeta_{12}^{2} ) q^{76} + 5 \zeta_{12}^{3} q^{77} + ( 10 - 10 \zeta_{12}^{2} ) q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{82} -14 \zeta_{12}^{3} q^{83} -6 \zeta_{12}^{2} q^{86} -5 \zeta_{12}^{3} q^{88} -7 \zeta_{12}^{2} q^{89} -2 \zeta_{12}^{2} q^{91} -\zeta_{12} q^{92} + 2 \zeta_{12} q^{97} -6 \zeta_{12} q^{98} -15 \zeta_{12}^{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 6q^{9} + O(q^{10})$$ $$4q + 2q^{4} - 6q^{9} + 20q^{11} + 2q^{14} - 2q^{16} + 2q^{19} + 8q^{26} + 12q^{29} + 16q^{31} + 6q^{36} + 18q^{41} + 10q^{44} + 4q^{46} + 24q^{49} + 4q^{56} - 4q^{64} - 12q^{71} - 22q^{74} + 16q^{76} + 20q^{79} - 18q^{81} - 12q^{86} - 14q^{89} - 4q^{91} - 30q^{99} + O(q^{100})$$

## 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 0.500000 + 0.866025i 0 0 1.00000i 1.00000i −1.50000 2.59808i 0
49.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 1.00000i 1.00000i −1.50000 2.59808i 0
349.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 1.00000i 1.00000i −1.50000 + 2.59808i 0
349.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 1.00000i 1.00000i −1.50000 + 2.59808i 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.d 4
5.b even 2 1 inner 950.2.j.d 4
5.c odd 4 1 190.2.e.a 2
5.c odd 4 1 950.2.e.f 2
15.e even 4 1 1710.2.l.h 2
19.c even 3 1 inner 950.2.j.d 4
20.e even 4 1 1520.2.q.f 2
95.i even 6 1 inner 950.2.j.d 4
95.l even 12 1 3610.2.a.c 1
95.m odd 12 1 190.2.e.a 2
95.m odd 12 1 950.2.e.f 2
95.m odd 12 1 3610.2.a.g 1
285.v even 12 1 1710.2.l.h 2
380.v even 12 1 1520.2.q.f 2

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

## Hecke kernels

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

 $$T_{3}$$ $$T_{7}^{2} + 1$$ $$T_{11} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -5 + T )^{4}$$
$13$ $$16 - 4 T^{2} + T^{4}$$
$17$ $$T^{4}$$
$19$ $$( 19 - T + T^{2} )^{2}$$
$23$ $$1 - T^{2} + T^{4}$$
$29$ $$( 36 - 6 T + T^{2} )^{2}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$( 121 + T^{2} )^{2}$$
$41$ $$( 81 - 9 T + T^{2} )^{2}$$
$43$ $$1296 - 36 T^{2} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$625 - 25 T^{2} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$20736 - 144 T^{2} + T^{4}$$
$71$ $$( 36 + 6 T + T^{2} )^{2}$$
$73$ $$38416 - 196 T^{2} + T^{4}$$
$79$ $$( 100 - 10 T + T^{2} )^{2}$$
$83$ $$( 196 + T^{2} )^{2}$$
$89$ $$( 49 + 7 T + T^{2} )^{2}$$
$97$ $$16 - 4 T^{2} + T^{4}$$