# Properties

 Label 950.2.h.a Level $950$ Weight $2$ Character orbit 950.h Analytic conductor $7.586$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.h (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + ( 2 - 2 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{3} -\zeta_{10}^{3} q^{4} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{5} + ( -2 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{6} + ( -4 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{7} -\zeta_{10}^{2} q^{8} + ( -4 - \zeta_{10} - 4 \zeta_{10}^{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + ( 2 - 2 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{3} -\zeta_{10}^{3} q^{4} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{5} + ( -2 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{6} + ( -4 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{7} -\zeta_{10}^{2} q^{8} + ( -4 - \zeta_{10} - 4 \zeta_{10}^{2} ) q^{9} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{10} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{11} + ( -2 - 2 \zeta_{10}^{2} ) q^{12} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{13} + ( -4 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{14} + ( 6 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{15} -\zeta_{10} q^{16} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{17} + ( -5 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{18} -\zeta_{10}^{2} q^{19} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{20} + ( -8 + 8 \zeta_{10} + 12 \zeta_{10}^{3} ) q^{21} + 2 \zeta_{10}^{3} q^{22} + ( 6 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{23} + ( -2 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{24} + ( -5 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{25} + ( -3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{26} + ( -12 + 8 \zeta_{10} - 8 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{27} + ( -2 + 2 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{28} + ( -3 + 3 \zeta_{10} - \zeta_{10}^{3} ) q^{29} + ( 6 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{30} + ( -2 \zeta_{10} - 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{31} - q^{32} + ( 4 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{33} + ( -1 - \zeta_{10}^{2} ) q^{34} + ( -6 \zeta_{10} - 2 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{35} + ( -5 + \zeta_{10} - \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{36} + ( -1 - 4 \zeta_{10} - \zeta_{10}^{2} ) q^{37} -\zeta_{10} q^{38} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{39} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{40} + ( 3 - \zeta_{10} + 3 \zeta_{10}^{2} ) q^{41} + ( 8 \zeta_{10} + 4 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{42} + ( -2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{43} + 2 \zeta_{10}^{2} q^{44} + ( 6 - 6 \zeta_{10} - 13 \zeta_{10}^{3} ) q^{45} + ( 4 - 4 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{46} + ( 8 - 8 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{47} + ( -2 + 2 \zeta_{10}^{3} ) q^{48} + ( 13 + 12 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{49} + 5 \zeta_{10}^{3} q^{50} + ( -4 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{51} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{52} + ( 3 - 3 \zeta_{10} ) q^{53} + ( -4 + 4 \zeta_{10} + 12 \zeta_{10}^{3} ) q^{54} + ( -4 + 2 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{55} + ( 2 \zeta_{10} + 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{56} + ( -2 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{57} + ( 3 \zeta_{10} - 4 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{58} + ( 2 + 4 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{59} + ( 2 - 2 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{60} + ( -2 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{61} + ( -2 - 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{62} + ( 18 + 10 \zeta_{10} + 18 \zeta_{10}^{2} ) q^{63} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( -3 + 3 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{65} + ( 4 + 4 \zeta_{10}^{2} ) q^{66} + 2 \zeta_{10}^{2} q^{67} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{68} + ( -12 \zeta_{10} - 8 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{69} + ( -6 - 2 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{70} + ( 12 - 12 \zeta_{10} - 8 \zeta_{10}^{3} ) q^{71} + ( -4 + 4 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{72} + ( 4 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{73} + ( -5 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{74} + ( 10 \zeta_{10} + 10 \zeta_{10}^{3} ) q^{75} - q^{76} + ( 8 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{77} + 6 \zeta_{10}^{3} q^{78} + ( -8 + 8 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{79} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{80} + ( 12 \zeta_{10} + 5 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{81} + ( 2 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{82} + ( -2 \zeta_{10} - 8 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{83} + ( 8 + 4 \zeta_{10} + 8 \zeta_{10}^{2} ) q^{84} + ( 3 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{85} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{86} + ( -2 + 6 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{87} + 2 \zeta_{10} q^{88} + ( -7 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{89} + ( -6 \zeta_{10} - 7 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{90} + ( 6 + 6 \zeta_{10}^{2} ) q^{91} + ( -4 \zeta_{10} - 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{92} + ( -12 - 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{93} + ( -8 \zeta_{10} + 2 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{94} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{95} + ( -2 + 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{96} + ( 1 - \zeta_{10} + 5 \zeta_{10}^{3} ) q^{97} + ( 13 - \zeta_{10} + \zeta_{10}^{2} - 13 \zeta_{10}^{3} ) q^{98} + ( 10 + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} + 4q^{3} - q^{4} + 5q^{5} - 4q^{6} - 12q^{7} + q^{8} - 13q^{9} + O(q^{10})$$ $$4q + q^{2} + 4q^{3} - q^{4} + 5q^{5} - 4q^{6} - 12q^{7} + q^{8} - 13q^{9} + 5q^{10} - 2q^{11} - 6q^{12} - 6q^{13} - 8q^{14} + 20q^{15} - q^{16} - 2q^{17} - 12q^{18} + q^{19} - 12q^{21} + 2q^{22} + 14q^{23} - 4q^{24} - 5q^{25} + 6q^{26} - 20q^{27} - 2q^{28} - 10q^{29} + 10q^{30} - 2q^{31} - 4q^{32} + 8q^{33} - 3q^{34} - 10q^{35} - 13q^{36} - 7q^{37} - q^{38} - 6q^{39} + 5q^{40} + 8q^{41} + 12q^{42} + 4q^{43} - 2q^{44} + 5q^{45} + 6q^{46} + 18q^{47} - 6q^{48} + 28q^{49} + 5q^{50} - 12q^{51} - 6q^{52} + 9q^{53} - 10q^{55} + 2q^{56} - 4q^{57} + 10q^{58} + 10q^{59} + 8q^{61} - 8q^{62} + 64q^{63} - q^{64} - 15q^{65} + 12q^{66} - 2q^{67} - 2q^{68} - 16q^{69} - 20q^{70} + 28q^{71} - 7q^{72} - 6q^{73} - 18q^{74} + 20q^{75} - 4q^{76} + 16q^{77} + 6q^{78} - 20q^{79} + 5q^{80} + 19q^{81} + 2q^{82} + 4q^{83} + 28q^{84} + 5q^{85} - 4q^{86} + 2q^{88} - 25q^{89} - 5q^{90} + 18q^{91} - 6q^{92} - 32q^{93} - 18q^{94} + 5q^{95} - 4q^{96} + 8q^{97} + 37q^{98} + 24q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.809017 − 0.587785i −0.309017 + 0.951057i −0.309017 − 0.951057i 0.809017 + 0.587785i
0.809017 + 0.587785i 1.00000 + 3.07768i 0.309017 + 0.951057i 0.690983 2.12663i −1.00000 + 3.07768i −5.23607 −0.309017 + 0.951057i −6.04508 + 4.39201i 1.80902 1.31433i
381.1 −0.309017 0.951057i 1.00000 0.726543i −0.809017 + 0.587785i 1.80902 + 1.31433i −1.00000 0.726543i −0.763932 0.809017 + 0.587785i −0.454915 + 1.40008i 0.690983 2.12663i
571.1 −0.309017 + 0.951057i 1.00000 + 0.726543i −0.809017 0.587785i 1.80902 1.31433i −1.00000 + 0.726543i −0.763932 0.809017 0.587785i −0.454915 1.40008i 0.690983 + 2.12663i
761.1 0.809017 0.587785i 1.00000 3.07768i 0.309017 0.951057i 0.690983 + 2.12663i −1.00000 3.07768i −5.23607 −0.309017 0.951057i −6.04508 4.39201i 1.80902 + 1.31433i
 $$n$$: 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
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.h.a 4
25.d even 5 1 inner 950.2.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.h.a 4 1.a even 1 1 trivial
950.2.h.a 4 25.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 4 T_{3}^{3} + 16 T_{3}^{2} - 24 T_{3} + 16$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$3$ $$16 - 24 T + 16 T^{2} - 4 T^{3} + T^{4}$$
$5$ $$25 - 25 T + 15 T^{2} - 5 T^{3} + T^{4}$$
$7$ $$( 4 + 6 T + T^{2} )^{2}$$
$11$ $$16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$13$ $$81 + 81 T + 36 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$19$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$23$ $$16 - 24 T + 76 T^{2} - 14 T^{3} + T^{4}$$
$29$ $$25 + 25 T + 40 T^{2} + 10 T^{3} + T^{4}$$
$31$ $$16 - 32 T + 24 T^{2} + 2 T^{3} + T^{4}$$
$37$ $$361 + 38 T + 24 T^{2} + 7 T^{3} + T^{4}$$
$41$ $$121 - 77 T + 34 T^{2} - 8 T^{3} + T^{4}$$
$43$ $$( -4 - 2 T + T^{2} )^{2}$$
$47$ $$5776 - 1672 T + 244 T^{2} - 18 T^{3} + T^{4}$$
$53$ $$81 - 54 T + 36 T^{2} - 9 T^{3} + T^{4}$$
$59$ $$400 + 40 T^{2} - 10 T^{3} + T^{4}$$
$61$ $$121 - 187 T + 114 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$71$ $$30976 - 5632 T + 544 T^{2} - 28 T^{3} + T^{4}$$
$73$ $$121 - 209 T + 136 T^{2} + 6 T^{3} + T^{4}$$
$79$ $$6400 + 1600 T + 240 T^{2} + 20 T^{3} + T^{4}$$
$83$ $$5776 - 1064 T + 96 T^{2} - 4 T^{3} + T^{4}$$
$89$ $$9025 + 1900 T + 310 T^{2} + 25 T^{3} + T^{4}$$
$97$ $$841 - 87 T + 34 T^{2} - 8 T^{3} + T^{4}$$