# Properties

 Label 1950.2.bc.e Level $1950$ Weight $2$ Character orbit 1950.bc Analytic conductor $15.571$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.bc (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$1 - \zeta_{24}^{4}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
751.1
 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i
−0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i 0.417738 + 0.241181i 1.00000i −0.500000 + 0.866025i 0
751.2 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i 1.31431 + 0.758819i 1.00000i −0.500000 + 0.866025i 0
751.3 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i −2.53906 1.46593i 1.00000i −0.500000 + 0.866025i 0
751.4 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i 0.807007 + 0.465926i 1.00000i −0.500000 + 0.866025i 0
901.1 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i 0.417738 0.241181i 1.00000i −0.500000 0.866025i 0
901.2 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i 1.31431 0.758819i 1.00000i −0.500000 0.866025i 0
901.3 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i −2.53906 + 1.46593i 1.00000i −0.500000 0.866025i 0
901.4 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i 0.807007 0.465926i 1.00000i −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.bc.e 8
5.b even 2 1 1950.2.bc.f yes 8
5.c odd 4 1 1950.2.y.i 8
5.c odd 4 1 1950.2.y.l 8
13.e even 6 1 inner 1950.2.bc.e 8
65.l even 6 1 1950.2.bc.f yes 8
65.r odd 12 1 1950.2.y.i 8
65.r odd 12 1 1950.2.y.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.y.i 8 5.c odd 4 1
1950.2.y.i 8 65.r odd 12 1
1950.2.y.l 8 5.c odd 4 1
1950.2.y.l 8 65.r odd 12 1
1950.2.bc.e 8 1.a even 1 1 trivial
1950.2.bc.e 8 13.e even 6 1 inner
1950.2.bc.f yes 8 5.b even 2 1
1950.2.bc.f yes 8 65.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 6 T_{7}^{6} + 38 T_{7}^{4} - 72 T_{7}^{3} + 60 T_{7}^{2} - 24 T_{7} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$( 1 + T + T^{2} )^{4}$$
$5$ $$T^{8}$$
$7$ $$4 - 24 T + 60 T^{2} - 72 T^{3} + 38 T^{4} - 6 T^{6} + T^{8}$$
$11$ $$529 + 828 T + 294 T^{2} - 216 T^{3} - 85 T^{4} + 72 T^{5} + 54 T^{6} + 12 T^{7} + T^{8}$$
$13$ $$28561 + 191 T^{4} + T^{8}$$
$17$ $$2116 + 3128 T + 6004 T^{2} - 1672 T^{3} + 1126 T^{4} - 16 T^{5} + 46 T^{6} - 4 T^{7} + T^{8}$$
$19$ $$324 - 648 T + 324 T^{2} + 216 T^{3} - 90 T^{4} - 72 T^{5} + 54 T^{6} - 12 T^{7} + T^{8}$$
$23$ $$58081 + 47236 T + 50948 T^{2} - 8264 T^{3} + 3247 T^{4} - 184 T^{5} + 68 T^{6} - 4 T^{7} + T^{8}$$
$29$ $$341056 - 270976 T + 166240 T^{2} - 43648 T^{3} + 9496 T^{4} - 592 T^{5} + 100 T^{6} - 4 T^{7} + T^{8}$$
$31$ $$145924 + 171672 T^{2} + 12128 T^{4} + 204 T^{6} + T^{8}$$
$37$ $$16056049 + 8078112 T + 1499004 T^{2} + 72576 T^{3} - 10825 T^{4} - 864 T^{5} + 156 T^{6} + 24 T^{7} + T^{8}$$
$41$ $$21316 - 61320 T + 73692 T^{2} - 42840 T^{3} + 10550 T^{4} - 102 T^{6} + T^{8}$$
$43$ $$386884 - 47272 T + 64244 T^{2} + 2168 T^{3} + 8518 T^{4} - 224 T^{5} + 110 T^{6} + 4 T^{7} + T^{8}$$
$47$ $$2972176 + 364896 T^{2} + 14360 T^{4} + 216 T^{6} + T^{8}$$
$53$ $$( -98 - 140 T - 30 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$59$ $$5740816 + 4542816 T + 1466624 T^{2} + 212352 T^{3} + 7356 T^{4} - 1344 T^{5} - 64 T^{6} + 12 T^{7} + T^{8}$$
$61$ $$8567329 + 3606064 T + 1412452 T^{2} + 138016 T^{3} + 23935 T^{4} + 1888 T^{5} + 292 T^{6} + 16 T^{7} + T^{8}$$
$67$ $$20647936 - 11341824 T + 2367488 T^{2} - 159744 T^{3} - 11328 T^{4} + 1536 T^{5} + 128 T^{6} - 24 T^{7} + T^{8}$$
$71$ $$6115729 - 7329972 T + 3907740 T^{2} - 1173744 T^{3} + 213623 T^{4} - 23760 T^{5} + 1596 T^{6} - 60 T^{7} + T^{8}$$
$73$ $$4977361 + 494708 T^{2} + 15990 T^{4} + 212 T^{6} + T^{8}$$
$79$ $$( 622 - 212 T - 102 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$83$ $$69169 + 78780 T^{2} + 7094 T^{4} + 156 T^{6} + T^{8}$$
$89$ $$5080516 + 2461368 T + 275772 T^{2} - 58968 T^{3} - 8074 T^{4} + 1296 T^{5} + 138 T^{6} - 24 T^{7} + T^{8}$$
$97$ $$61606801 + 9512988 T - 1001662 T^{2} - 230280 T^{3} + 23403 T^{4} + 2280 T^{5} - 142 T^{6} - 12 T^{7} + T^{8}$$