# Properties

 Label 1950.2.i.ba Level $1950$ Weight $2$ Character orbit 1950.i Analytic conductor $15.571$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{10})$$ Defining polynomial: $$x^{4} + 10 x^{2} + 100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 10 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/10$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$10 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$10 \beta_{3}$$

## Character values

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

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$ $$1$$

## 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
 −1.58114 − 2.73861i 1.58114 + 2.73861i −1.58114 + 2.73861i 1.58114 − 2.73861i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i −1.58114 2.73861i 1.00000 −0.500000 0.866025i 0
451.2 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i 1.58114 + 2.73861i 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 −1.58114 + 2.73861i 1.00000 −0.500000 + 0.866025i 0
601.2 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i 1.58114 2.73861i 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.ba 4
5.b even 2 1 1950.2.i.bf yes 4
5.c odd 4 2 1950.2.z.o 8
13.c even 3 1 inner 1950.2.i.ba 4
65.n even 6 1 1950.2.i.bf yes 4
65.q odd 12 2 1950.2.z.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.i.ba 4 1.a even 1 1 trivial
1950.2.i.ba 4 13.c even 3 1 inner
1950.2.i.bf yes 4 5.b even 2 1
1950.2.i.bf yes 4 65.n even 6 1
1950.2.z.o 8 5.c odd 4 2
1950.2.z.o 8 65.q odd 12 2

## Hecke kernels

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

 $$T_{7}^{4} + 10 T_{7}^{2} + 100$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{17}^{4} + 8 T_{17}^{3} + 58 T_{17}^{2} + 48 T_{17} + 36$$ $$T_{19}^{4} + 10 T_{19}^{2} + 100$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$100 + 10 T^{2} + T^{4}$$
$11$ $$( 9 + 3 T + T^{2} )^{2}$$
$13$ $$169 + 78 T + 25 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$36 + 48 T + 58 T^{2} + 8 T^{3} + T^{4}$$
$19$ $$100 + 10 T^{2} + T^{4}$$
$23$ $$81 + 18 T + 13 T^{2} - 2 T^{3} + T^{4}$$
$29$ $$1296 + 144 T + 52 T^{2} - 4 T^{3} + T^{4}$$
$31$ $$( 26 - 12 T + T^{2} )^{2}$$
$37$ $$7921 - 178 T + 93 T^{2} + 2 T^{3} + T^{4}$$
$41$ $$8100 + 90 T^{2} + T^{4}$$
$43$ $$36 - 24 T + 22 T^{2} + 4 T^{3} + T^{4}$$
$47$ $$( -6 + T )^{4}$$
$53$ $$( 6 - 8 T + T^{2} )^{2}$$
$59$ $$( 36 - 6 T + T^{2} )^{2}$$
$61$ $$4225 - 650 T + 165 T^{2} + 10 T^{3} + T^{4}$$
$67$ $$( 16 + 4 T + T^{2} )^{2}$$
$71$ $$12321 - 2442 T + 373 T^{2} - 22 T^{3} + T^{4}$$
$73$ $$( -15 - 10 T + T^{2} )^{2}$$
$79$ $$( 26 - 12 T + T^{2} )^{2}$$
$83$ $$( -159 + 2 T + T^{2} )^{2}$$
$89$ $$36 - 48 T + 58 T^{2} - 8 T^{3} + T^{4}$$
$97$ $$6561 + 1782 T + 403 T^{2} + 22 T^{3} + T^{4}$$