# Properties

 Label 1950.2.f.l Level $1950$ Weight $2$ Character orbit 1950.f 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.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu + 5$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 4 \nu^{2} + 9 \nu - 20$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} + \beta_{1} - 10$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{3} - 5 \beta_{2} + 13 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.56155i 2.56155i 1.56155i − 2.56155i
−1.00000 1.00000i 1.00000 0 1.00000i −5.12311 −1.00000 −1.00000 0
649.2 −1.00000 1.00000i 1.00000 0 1.00000i 3.12311 −1.00000 −1.00000 0
649.3 −1.00000 1.00000i 1.00000 0 1.00000i −5.12311 −1.00000 −1.00000 0
649.4 −1.00000 1.00000i 1.00000 0 1.00000i 3.12311 −1.00000 −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
65.d 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.f.l 4
5.b even 2 1 1950.2.f.o 4
5.c odd 4 1 390.2.b.d 4
5.c odd 4 1 1950.2.b.h 4
13.b even 2 1 1950.2.f.o 4
15.e even 4 1 1170.2.b.f 4
20.e even 4 1 3120.2.g.o 4
65.d even 2 1 inner 1950.2.f.l 4
65.f even 4 1 5070.2.a.bh 2
65.h odd 4 1 390.2.b.d 4
65.h odd 4 1 1950.2.b.h 4
65.k even 4 1 5070.2.a.bd 2
195.s even 4 1 1170.2.b.f 4
260.p even 4 1 3120.2.g.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.d 4 5.c odd 4 1
390.2.b.d 4 65.h odd 4 1
1170.2.b.f 4 15.e even 4 1
1170.2.b.f 4 195.s even 4 1
1950.2.b.h 4 5.c odd 4 1
1950.2.b.h 4 65.h odd 4 1
1950.2.f.l 4 1.a even 1 1 trivial
1950.2.f.l 4 65.d even 2 1 inner
1950.2.f.o 4 5.b even 2 1
1950.2.f.o 4 13.b even 2 1
3120.2.g.o 4 20.e even 4 1
3120.2.g.o 4 260.p even 4 1
5070.2.a.bd 2 65.k even 4 1
5070.2.a.bh 2 65.f even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 2 T_{7} - 16$$ $$T_{11}^{4} + 36 T_{11}^{2} + 256$$ $$T_{19}^{2} + 36$$ $$T_{37}^{2} + 2 T_{37} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( -16 + 2 T + T^{2} )^{2}$$
$11$ $$256 + 36 T^{2} + T^{4}$$
$13$ $$169 + 78 T + 18 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$( 4 + T^{2} )^{2}$$
$19$ $$( 36 + T^{2} )^{2}$$
$23$ $$256 + 36 T^{2} + T^{4}$$
$29$ $$( 2 + T )^{4}$$
$31$ $$256 + 36 T^{2} + T^{4}$$
$37$ $$( -16 + 2 T + T^{2} )^{2}$$
$41$ $$64 + 84 T^{2} + T^{4}$$
$43$ $$4096 + 144 T^{2} + T^{4}$$
$47$ $$( -64 - 4 T + T^{2} )^{2}$$
$53$ $$23104 + 308 T^{2} + T^{4}$$
$59$ $$64 + 52 T^{2} + T^{4}$$
$61$ $$( -10 + T )^{4}$$
$67$ $$( 64 - 18 T + T^{2} )^{2}$$
$71$ $$4096 + 144 T^{2} + T^{4}$$
$73$ $$( 64 - 18 T + T^{2} )^{2}$$
$79$ $$( 8 + T )^{4}$$
$83$ $$( -64 + 4 T + T^{2} )^{2}$$
$89$ $$256 + 36 T^{2} + T^{4}$$
$97$ $$( 64 + 18 T + T^{2} )^{2}$$