# Properties

 Label 1900.2.s.a Level $1900$ Weight $2$ Character orbit 1900.s Analytic conductor $15.172$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{3} -2 \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{3} -2 \zeta_{12}^{2} q^{9} -4 q^{11} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{13} + 3 \zeta_{12} q^{17} + ( 5 - 2 \zeta_{12}^{2} ) q^{19} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{23} -5 \zeta_{12}^{3} q^{27} + 7 \zeta_{12}^{2} q^{29} + 4 q^{31} -4 \zeta_{12} q^{33} -10 \zeta_{12}^{3} q^{37} - q^{39} + ( 5 - 5 \zeta_{12}^{2} ) q^{41} + 5 \zeta_{12} q^{43} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{47} + 7 q^{49} + 3 \zeta_{12}^{2} q^{51} + ( 11 \zeta_{12} - 11 \zeta_{12}^{3} ) q^{53} + ( 5 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{57} + ( 3 - 3 \zeta_{12}^{2} ) q^{59} -11 \zeta_{12}^{2} q^{61} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{67} + 5 q^{69} + ( -11 + 11 \zeta_{12}^{2} ) q^{71} -15 \zeta_{12} q^{73} + ( -13 + 13 \zeta_{12}^{2} ) q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} + 7 \zeta_{12}^{3} q^{87} + 3 \zeta_{12}^{2} q^{89} + 4 \zeta_{12} q^{93} -5 \zeta_{12} q^{97} + 8 \zeta_{12}^{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{9} - 16 q^{11} + 16 q^{19} + 14 q^{29} + 16 q^{31} - 4 q^{39} + 10 q^{41} + 28 q^{49} + 6 q^{51} + 6 q^{59} - 22 q^{61} + 20 q^{69} - 22 q^{71} - 26 q^{79} - 2 q^{81} + 6 q^{89} + 16 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 −0.866025 0.500000i 0 0 0 0 0 −1.00000 1.73205i 0
49.2 0 0.866025 + 0.500000i 0 0 0 0 0 −1.00000 1.73205i 0
349.1 0 −0.866025 + 0.500000i 0 0 0 0 0 −1.00000 + 1.73205i 0
349.2 0 0.866025 0.500000i 0 0 0 0 0 −1.00000 + 1.73205i 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
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.s.a 4
5.b even 2 1 inner 1900.2.s.a 4
5.c odd 4 1 76.2.e.a 2
5.c odd 4 1 1900.2.i.a 2
15.e even 4 1 684.2.k.b 2
19.c even 3 1 inner 1900.2.s.a 4
20.e even 4 1 304.2.i.a 2
40.i odd 4 1 1216.2.i.c 2
40.k even 4 1 1216.2.i.g 2
60.l odd 4 1 2736.2.s.g 2
95.g even 4 1 1444.2.e.b 2
95.i even 6 1 inner 1900.2.s.a 4
95.l even 12 1 1444.2.a.c 1
95.l even 12 1 1444.2.e.b 2
95.m odd 12 1 76.2.e.a 2
95.m odd 12 1 1444.2.a.b 1
95.m odd 12 1 1900.2.i.a 2
285.v even 12 1 684.2.k.b 2
380.v even 12 1 304.2.i.a 2
380.v even 12 1 5776.2.a.k 1
380.w odd 12 1 5776.2.a.f 1
760.br odd 12 1 1216.2.i.c 2
760.bw even 12 1 1216.2.i.g 2
1140.bu odd 12 1 2736.2.s.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.e.a 2 5.c odd 4 1
76.2.e.a 2 95.m odd 12 1
304.2.i.a 2 20.e even 4 1
304.2.i.a 2 380.v even 12 1
684.2.k.b 2 15.e even 4 1
684.2.k.b 2 285.v even 12 1
1216.2.i.c 2 40.i odd 4 1
1216.2.i.c 2 760.br odd 12 1
1216.2.i.g 2 40.k even 4 1
1216.2.i.g 2 760.bw even 12 1
1444.2.a.b 1 95.m odd 12 1
1444.2.a.c 1 95.l even 12 1
1444.2.e.b 2 95.g even 4 1
1444.2.e.b 2 95.l even 12 1
1900.2.i.a 2 5.c odd 4 1
1900.2.i.a 2 95.m odd 12 1
1900.2.s.a 4 1.a even 1 1 trivial
1900.2.s.a 4 5.b even 2 1 inner
1900.2.s.a 4 19.c even 3 1 inner
1900.2.s.a 4 95.i even 6 1 inner
2736.2.s.g 2 60.l odd 4 1
2736.2.s.g 2 1140.bu odd 12 1
5776.2.a.f 1 380.w odd 12 1
5776.2.a.k 1 380.v even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 4 + T )^{4}$$
$13$ $$1 - T^{2} + T^{4}$$
$17$ $$81 - 9 T^{2} + T^{4}$$
$19$ $$( 19 - 8 T + T^{2} )^{2}$$
$23$ $$625 - 25 T^{2} + T^{4}$$
$29$ $$( 49 - 7 T + T^{2} )^{2}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$( 100 + T^{2} )^{2}$$
$41$ $$( 25 - 5 T + T^{2} )^{2}$$
$43$ $$625 - 25 T^{2} + T^{4}$$
$47$ $$2401 - 49 T^{2} + T^{4}$$
$53$ $$14641 - 121 T^{2} + T^{4}$$
$59$ $$( 9 - 3 T + T^{2} )^{2}$$
$61$ $$( 121 + 11 T + T^{2} )^{2}$$
$67$ $$81 - 9 T^{2} + T^{4}$$
$71$ $$( 121 + 11 T + T^{2} )^{2}$$
$73$ $$50625 - 225 T^{2} + T^{4}$$
$79$ $$( 169 + 13 T + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$( 9 - 3 T + T^{2} )^{2}$$
$97$ $$625 - 25 T^{2} + T^{4}$$