# Properties

 Label 1950.2.b.k Level $1950$ Weight $2$ Character orbit 1950.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 10 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{2} + 5 \beta_{1}$$

## Character values

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

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 − 1.30278i 2.30278i − 2.30278i 1.30278i
1.00000i 1.00000 −1.00000 0 1.00000i 2.60555i 1.00000i 1.00000 0
1351.2 1.00000i 1.00000 −1.00000 0 1.00000i 4.60555i 1.00000i 1.00000 0
1351.3 1.00000i 1.00000 −1.00000 0 1.00000i 4.60555i 1.00000i 1.00000 0
1351.4 1.00000i 1.00000 −1.00000 0 1.00000i 2.60555i 1.00000i 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.b.k 4
5.b even 2 1 390.2.b.c 4
5.c odd 4 1 1950.2.f.m 4
5.c odd 4 1 1950.2.f.n 4
13.b even 2 1 inner 1950.2.b.k 4
15.d odd 2 1 1170.2.b.d 4
20.d odd 2 1 3120.2.g.q 4
65.d even 2 1 390.2.b.c 4
65.g odd 4 1 5070.2.a.z 2
65.g odd 4 1 5070.2.a.bf 2
65.h odd 4 1 1950.2.f.m 4
65.h odd 4 1 1950.2.f.n 4
195.e odd 2 1 1170.2.b.d 4
260.g odd 2 1 3120.2.g.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.c 4 5.b even 2 1
390.2.b.c 4 65.d even 2 1
1170.2.b.d 4 15.d odd 2 1
1170.2.b.d 4 195.e odd 2 1
1950.2.b.k 4 1.a even 1 1 trivial
1950.2.b.k 4 13.b even 2 1 inner
1950.2.f.m 4 5.c odd 4 1
1950.2.f.m 4 65.h odd 4 1
1950.2.f.n 4 5.c odd 4 1
1950.2.f.n 4 65.h odd 4 1
3120.2.g.q 4 20.d odd 2 1
3120.2.g.q 4 260.g odd 2 1
5070.2.a.z 2 65.g odd 4 1
5070.2.a.bf 2 65.g odd 4 1

## Hecke kernels

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

 $$T_{7}^{4} + 28 T_{7}^{2} + 144$$ $$T_{11}$$ $$T_{17}^{2} + 2 T_{17} - 12$$ $$T_{19}^{4} + 28 T_{19}^{2} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$144 + 28 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$( -13 + T^{2} )^{2}$$
$17$ $$( -12 + 2 T + T^{2} )^{2}$$
$19$ $$144 + 28 T^{2} + T^{4}$$
$23$ $$( 12 - 10 T + T^{2} )^{2}$$
$29$ $$( -12 - 2 T + T^{2} )^{2}$$
$31$ $$( 36 + T^{2} )^{2}$$
$37$ $$2304 + 112 T^{2} + T^{4}$$
$41$ $$1296 + 136 T^{2} + T^{4}$$
$43$ $$( 8 + T )^{4}$$
$47$ $$2304 + 112 T^{2} + T^{4}$$
$53$ $$( 6 + T )^{4}$$
$59$ $$2304 + 112 T^{2} + T^{4}$$
$61$ $$( -36 + 8 T + T^{2} )^{2}$$
$67$ $$1296 + 136 T^{2} + T^{4}$$
$71$ $$2304 + 112 T^{2} + T^{4}$$
$73$ $$144 + 76 T^{2} + T^{4}$$
$79$ $$( -208 + T^{2} )^{2}$$
$83$ $$2304 + 304 T^{2} + T^{4}$$
$89$ $$144 + 232 T^{2} + T^{4}$$
$97$ $$144 + 76 T^{2} + T^{4}$$