# Properties

 Label 1950.2.i.s Level $1950$ Weight $2$ Character orbit 1950.i Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 + 1.73205i −1.00000 −0.500000 0.866025i 0
601.1 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000 1.73205i −1.00000 −0.500000 + 0.866025i 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.i.s 2
5.b even 2 1 390.2.i.a 2
5.c odd 4 2 1950.2.z.h 4
13.c even 3 1 inner 1950.2.i.s 2
15.d odd 2 1 1170.2.i.k 2
65.l even 6 1 5070.2.a.f 1
65.n even 6 1 390.2.i.a 2
65.n even 6 1 5070.2.a.o 1
65.q odd 12 2 1950.2.z.h 4
65.s odd 12 2 5070.2.b.g 2
195.x odd 6 1 1170.2.i.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.a 2 5.b even 2 1
390.2.i.a 2 65.n even 6 1
1170.2.i.k 2 15.d odd 2 1
1170.2.i.k 2 195.x odd 6 1
1950.2.i.s 2 1.a even 1 1 trivial
1950.2.i.s 2 13.c even 3 1 inner
1950.2.z.h 4 5.c odd 4 2
1950.2.z.h 4 65.q odd 12 2
5070.2.a.f 1 65.l even 6 1
5070.2.a.o 1 65.n even 6 1
5070.2.b.g 2 65.s odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$:

 $$T_{7}^{2} - 2 T_{7} + 4$$ $$T_{11}^{2} + T_{11} + 1$$ $$T_{17}^{2} - 2 T_{17} + 4$$ $$T_{19}^{2} + 6 T_{19} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 - 2 T + T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$13 - 2 T + T^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$36 + 6 T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$1 - T + T^{2}$$
$31$ $$( 3 + T )^{2}$$
$37$ $$25 + 5 T + T^{2}$$
$41$ $$100 + 10 T + T^{2}$$
$43$ $$25 - 5 T + T^{2}$$
$47$ $$( 3 + T )^{2}$$
$53$ $$( 14 + T )^{2}$$
$59$ $$25 - 5 T + T^{2}$$
$61$ $$100 - 10 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$16 + 4 T + T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$( -5 + T )^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$100 + 10 T + T^{2}$$
$97$ $$100 + 10 T + T^{2}$$