# Properties

 Label 950.2.f.a Level $950$ Weight $2$ Character orbit 950.f 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.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} + ( 1 + \zeta_{8}^{2} ) q^{7} + \zeta_{8} q^{8} -3 \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} + ( 1 + \zeta_{8}^{2} ) q^{7} + \zeta_{8} q^{8} -3 \zeta_{8}^{2} q^{9} + 2 q^{11} -6 \zeta_{8} q^{13} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{14} - q^{16} + ( -3 - 3 \zeta_{8}^{2} ) q^{17} + 3 \zeta_{8} q^{18} + ( -3 \zeta_{8} - \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{19} + 2 \zeta_{8}^{3} q^{22} + ( -1 + \zeta_{8}^{2} ) q^{23} + 6 q^{26} + ( 1 - \zeta_{8}^{2} ) q^{28} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{31} -\zeta_{8}^{3} q^{32} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{34} -3 q^{36} + 6 \zeta_{8}^{3} q^{37} + ( 3 + \zeta_{8} - 3 \zeta_{8}^{2} ) q^{38} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{41} + ( 5 - 5 \zeta_{8}^{2} ) q^{43} -2 \zeta_{8}^{2} q^{44} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{46} + ( -7 - 7 \zeta_{8}^{2} ) q^{47} -5 \zeta_{8}^{2} q^{49} + 6 \zeta_{8}^{3} q^{52} -6 \zeta_{8} q^{53} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{56} + 8 q^{61} + ( -6 - 6 \zeta_{8}^{2} ) q^{62} + ( 3 - 3 \zeta_{8}^{2} ) q^{63} + \zeta_{8}^{2} q^{64} -12 \zeta_{8}^{3} q^{67} + ( -3 + 3 \zeta_{8}^{2} ) q^{68} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{71} -3 \zeta_{8}^{3} q^{72} + ( 1 - \zeta_{8}^{2} ) q^{73} -6 \zeta_{8}^{2} q^{74} + ( -1 + 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{76} + ( 2 + 2 \zeta_{8}^{2} ) q^{77} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{79} -9 q^{81} + ( 6 + 6 \zeta_{8}^{2} ) q^{82} + ( -9 + 9 \zeta_{8}^{2} ) q^{83} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{86} + 2 \zeta_{8} q^{88} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{89} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{91} + ( 1 + \zeta_{8}^{2} ) q^{92} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{94} -6 \zeta_{8}^{3} q^{97} + 5 \zeta_{8} q^{98} -6 \zeta_{8}^{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} + O(q^{10})$$ $$4q + 4q^{7} + 8q^{11} - 4q^{16} - 12q^{17} - 4q^{23} + 24q^{26} + 4q^{28} - 12q^{36} + 12q^{38} + 20q^{43} - 28q^{47} + 32q^{61} - 24q^{62} + 12q^{63} - 12q^{68} + 4q^{73} - 4q^{76} + 8q^{77} - 36q^{81} + 24q^{82} - 36q^{83} + 4q^{92} + O(q^{100})$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$\zeta_{8}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
493.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
−0.707107 0.707107i 0 1.00000i 0 0 1.00000 1.00000i 0.707107 0.707107i 3.00000i 0
493.2 0.707107 + 0.707107i 0 1.00000i 0 0 1.00000 1.00000i −0.707107 + 0.707107i 3.00000i 0
607.1 −0.707107 + 0.707107i 0 1.00000i 0 0 1.00000 + 1.00000i 0.707107 + 0.707107i 3.00000i 0
607.2 0.707107 0.707107i 0 1.00000i 0 0 1.00000 + 1.00000i −0.707107 0.707107i 3.00000i 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.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.f.a 4
5.b even 2 1 190.2.f.a 4
5.c odd 4 1 190.2.f.a 4
5.c odd 4 1 inner 950.2.f.a 4
15.d odd 2 1 1710.2.p.a 4
15.e even 4 1 1710.2.p.a 4
19.b odd 2 1 inner 950.2.f.a 4
95.d odd 2 1 190.2.f.a 4
95.g even 4 1 190.2.f.a 4
95.g even 4 1 inner 950.2.f.a 4
285.b even 2 1 1710.2.p.a 4
285.j odd 4 1 1710.2.p.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.f.a 4 5.b even 2 1
190.2.f.a 4 5.c odd 4 1
190.2.f.a 4 95.d odd 2 1
190.2.f.a 4 95.g even 4 1
950.2.f.a 4 1.a even 1 1 trivial
950.2.f.a 4 5.c odd 4 1 inner
950.2.f.a 4 19.b odd 2 1 inner
950.2.f.a 4 95.g even 4 1 inner
1710.2.p.a 4 15.d odd 2 1
1710.2.p.a 4 15.e even 4 1
1710.2.p.a 4 285.b even 2 1
1710.2.p.a 4 285.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 2 - 2 T + T^{2} )^{2}$$
$11$ $$( -2 + T )^{4}$$
$13$ $$1296 + T^{4}$$
$17$ $$( 18 + 6 T + T^{2} )^{2}$$
$19$ $$361 - 34 T^{2} + T^{4}$$
$23$ $$( 2 + 2 T + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 72 + T^{2} )^{2}$$
$37$ $$1296 + T^{4}$$
$41$ $$( 72 + T^{2} )^{2}$$
$43$ $$( 50 - 10 T + T^{2} )^{2}$$
$47$ $$( 98 + 14 T + T^{2} )^{2}$$
$53$ $$1296 + T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -8 + T )^{4}$$
$67$ $$20736 + T^{4}$$
$71$ $$( 72 + T^{2} )^{2}$$
$73$ $$( 2 - 2 T + T^{2} )^{2}$$
$79$ $$( -72 + T^{2} )^{2}$$
$83$ $$( 162 + 18 T + T^{2} )^{2}$$
$89$ $$( -72 + T^{2} )^{2}$$
$97$ $$1296 + T^{4}$$