# Properties

 Label 950.3.d.a Level $950$ Weight $3$ Character orbit 950.d Analytic conductor $25.886$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 950.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.8856251142$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} + 2 q^{4} -4 q^{6} + 5 \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} - q^{9} +O(q^{10})$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} + 2 q^{4} -4 q^{6} + 5 \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} - q^{9} + 5 q^{11} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{12} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{13} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{14} + 4 q^{16} -25 \zeta_{8}^{2} q^{17} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{18} -19 q^{19} + ( -10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{21} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{22} + 10 \zeta_{8}^{2} q^{23} -8 q^{24} + 24 q^{26} + ( 20 \zeta_{8} - 20 \zeta_{8}^{3} ) q^{27} + 10 \zeta_{8}^{2} q^{28} + ( 30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{29} + ( 30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{31} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{33} + ( -25 \zeta_{8} - 25 \zeta_{8}^{3} ) q^{34} -2 q^{36} + ( -18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{37} + ( -19 \zeta_{8} + 19 \zeta_{8}^{3} ) q^{38} -48 q^{39} + ( 30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{41} -20 \zeta_{8}^{2} q^{42} -5 \zeta_{8}^{2} q^{43} + 10 q^{44} + ( 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{46} + 5 \zeta_{8}^{2} q^{47} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{48} + 24 q^{49} + ( 50 \zeta_{8} + 50 \zeta_{8}^{3} ) q^{51} + ( 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{52} + ( 18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{53} + 40 q^{54} + ( 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{56} + ( 38 \zeta_{8} - 38 \zeta_{8}^{3} ) q^{57} + 60 \zeta_{8}^{2} q^{58} + ( 60 \zeta_{8} + 60 \zeta_{8}^{3} ) q^{59} + 95 q^{61} + 60 \zeta_{8}^{2} q^{62} -5 \zeta_{8}^{2} q^{63} + 8 q^{64} -20 q^{66} + ( 78 \zeta_{8} - 78 \zeta_{8}^{3} ) q^{67} -50 \zeta_{8}^{2} q^{68} + ( -20 \zeta_{8} - 20 \zeta_{8}^{3} ) q^{69} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{72} + 25 \zeta_{8}^{2} q^{73} -36 q^{74} -38 q^{76} + 25 \zeta_{8}^{2} q^{77} + ( -48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{78} + ( 30 \zeta_{8} + 30 \zeta_{8}^{3} ) q^{79} -71 q^{81} + 60 \zeta_{8}^{2} q^{82} + 130 \zeta_{8}^{2} q^{83} + ( -20 \zeta_{8} - 20 \zeta_{8}^{3} ) q^{84} + ( -5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{86} -120 \zeta_{8}^{2} q^{87} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{88} + ( -90 \zeta_{8} - 90 \zeta_{8}^{3} ) q^{89} + ( 60 \zeta_{8} + 60 \zeta_{8}^{3} ) q^{91} + 20 \zeta_{8}^{2} q^{92} -120 \zeta_{8}^{2} q^{93} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{94} -16 q^{96} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{97} + ( 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{98} -5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} - 16q^{6} - 4q^{9} + O(q^{10})$$ $$4q + 8q^{4} - 16q^{6} - 4q^{9} + 20q^{11} + 16q^{16} - 76q^{19} - 32q^{24} + 96q^{26} - 8q^{36} - 192q^{39} + 40q^{44} + 96q^{49} + 160q^{54} + 380q^{61} + 32q^{64} - 80q^{66} - 144q^{74} - 152q^{76} - 284q^{81} - 64q^{96} - 20q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
949.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
−1.41421 2.82843 2.00000 0 −4.00000 5.00000i −2.82843 −1.00000 0
949.2 −1.41421 2.82843 2.00000 0 −4.00000 5.00000i −2.82843 −1.00000 0
949.3 1.41421 −2.82843 2.00000 0 −4.00000 5.00000i 2.82843 −1.00000 0
949.4 1.41421 −2.82843 2.00000 0 −4.00000 5.00000i 2.82843 −1.00000 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.b odd 2 1 inner
95.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.3.d.a 4
5.b even 2 1 inner 950.3.d.a 4
5.c odd 4 1 38.3.b.a 2
5.c odd 4 1 950.3.c.a 2
15.e even 4 1 342.3.d.a 2
19.b odd 2 1 inner 950.3.d.a 4
20.e even 4 1 304.3.e.c 2
40.i odd 4 1 1216.3.e.j 2
40.k even 4 1 1216.3.e.i 2
60.l odd 4 1 2736.3.o.h 2
95.d odd 2 1 inner 950.3.d.a 4
95.g even 4 1 38.3.b.a 2
95.g even 4 1 950.3.c.a 2
285.j odd 4 1 342.3.d.a 2
380.j odd 4 1 304.3.e.c 2
760.t even 4 1 1216.3.e.j 2
760.y odd 4 1 1216.3.e.i 2
1140.w even 4 1 2736.3.o.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a 2 5.c odd 4 1
38.3.b.a 2 95.g even 4 1
304.3.e.c 2 20.e even 4 1
304.3.e.c 2 380.j odd 4 1
342.3.d.a 2 15.e even 4 1
342.3.d.a 2 285.j odd 4 1
950.3.c.a 2 5.c odd 4 1
950.3.c.a 2 95.g even 4 1
950.3.d.a 4 1.a even 1 1 trivial
950.3.d.a 4 5.b even 2 1 inner
950.3.d.a 4 19.b odd 2 1 inner
950.3.d.a 4 95.d odd 2 1 inner
1216.3.e.i 2 40.k even 4 1
1216.3.e.i 2 760.y odd 4 1
1216.3.e.j 2 40.i odd 4 1
1216.3.e.j 2 760.t even 4 1
2736.3.o.h 2 60.l odd 4 1
2736.3.o.h 2 1140.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 8$$ acting on $$S_{3}^{\mathrm{new}}(950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$( -8 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 25 + T^{2} )^{2}$$
$11$ $$( -5 + T )^{4}$$
$13$ $$( -288 + T^{2} )^{2}$$
$17$ $$( 625 + T^{2} )^{2}$$
$19$ $$( 19 + T )^{4}$$
$23$ $$( 100 + T^{2} )^{2}$$
$29$ $$( 1800 + T^{2} )^{2}$$
$31$ $$( 1800 + T^{2} )^{2}$$
$37$ $$( -648 + T^{2} )^{2}$$
$41$ $$( 1800 + T^{2} )^{2}$$
$43$ $$( 25 + T^{2} )^{2}$$
$47$ $$( 25 + T^{2} )^{2}$$
$53$ $$( -648 + T^{2} )^{2}$$
$59$ $$( 7200 + T^{2} )^{2}$$
$61$ $$( -95 + T )^{4}$$
$67$ $$( -12168 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( 625 + T^{2} )^{2}$$
$79$ $$( 1800 + T^{2} )^{2}$$
$83$ $$( 16900 + T^{2} )^{2}$$
$89$ $$( 16200 + T^{2} )^{2}$$
$97$ $$( -288 + T^{2} )^{2}$$