# Properties

 Label 1900.3.g.a Level $1900$ Weight $3$ Character orbit 1900.g Analytic conductor $51.771$ Analytic rank $0$ Dimension $4$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1900.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$51.7712502285$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{57})$$ Defining polynomial: $$x^{4} + 29 x^{2} + 196$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{1} + \beta_{2} ) q^{7} -9 q^{9} +O(q^{10})$$ $$q + ( 2 \beta_{1} + \beta_{2} ) q^{7} -9 q^{9} + ( -2 + \beta_{3} ) q^{11} + ( 6 \beta_{1} + \beta_{2} ) q^{17} + 19 q^{19} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{23} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{43} + ( -6 \beta_{1} - 7 \beta_{2} ) q^{47} + ( -87 + 3 \beta_{3} ) q^{49} + ( -50 - 3 \beta_{3} ) q^{61} + ( -18 \beta_{1} - 9 \beta_{2} ) q^{63} + ( 26 \beta_{1} + 7 \beta_{2} ) q^{73} + ( 9 \beta_{1} - 26 \beta_{2} ) q^{77} + 81 q^{81} + ( 9 \beta_{1} - 9 \beta_{2} ) q^{83} + ( 18 - 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 36q^{9} + O(q^{10})$$ $$4q - 36q^{9} - 6q^{11} + 76q^{19} - 342q^{49} - 206q^{61} + 324q^{81} + 54q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 29 x^{2} + 196$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{3} + 59 \nu$$$$)/14$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - 13 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$5 \nu^{2} + 73$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{2} + 7 \beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 73$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$-59 \beta_{2} - 91 \beta_{1}$$$$)/10$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
949.1
 − 4.27492i − 3.27492i 3.27492i 4.27492i
0 0 0 0 0 13.8248i 0 −9.00000 0
949.2 0 0 0 0 0 8.82475i 0 −9.00000 0
949.3 0 0 0 0 0 8.82475i 0 −9.00000 0
949.4 0 0 0 0 0 13.8248i 0 −9.00000 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
5.b even 2 1 inner
95.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.3.g.a 4
5.b even 2 1 inner 1900.3.g.a 4
5.c odd 4 1 76.3.c.b 2
5.c odd 4 1 1900.3.e.a 2
15.e even 4 1 684.3.h.a 2
19.b odd 2 1 CM 1900.3.g.a 4
20.e even 4 1 304.3.e.e 2
40.i odd 4 1 1216.3.e.f 2
40.k even 4 1 1216.3.e.e 2
60.l odd 4 1 2736.3.o.c 2
95.d odd 2 1 inner 1900.3.g.a 4
95.g even 4 1 76.3.c.b 2
95.g even 4 1 1900.3.e.a 2
285.j odd 4 1 684.3.h.a 2
380.j odd 4 1 304.3.e.e 2
760.t even 4 1 1216.3.e.f 2
760.y odd 4 1 1216.3.e.e 2
1140.w even 4 1 2736.3.o.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.b 2 5.c odd 4 1
76.3.c.b 2 95.g even 4 1
304.3.e.e 2 20.e even 4 1
304.3.e.e 2 380.j odd 4 1
684.3.h.a 2 15.e even 4 1
684.3.h.a 2 285.j odd 4 1
1216.3.e.e 2 40.k even 4 1
1216.3.e.e 2 760.y odd 4 1
1216.3.e.f 2 40.i odd 4 1
1216.3.e.f 2 760.t even 4 1
1900.3.e.a 2 5.c odd 4 1
1900.3.e.a 2 95.g even 4 1
1900.3.g.a 4 1.a even 1 1 trivial
1900.3.g.a 4 5.b even 2 1 inner
1900.3.g.a 4 19.b odd 2 1 CM
1900.3.g.a 4 95.d odd 2 1 inner
2736.3.o.c 2 60.l odd 4 1
2736.3.o.c 2 1140.w even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$14884 + 269 T^{2} + T^{4}$$
$11$ $$( -354 + 3 T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$412164 + 1509 T^{2} + T^{4}$$
$19$ $$( -19 + T )^{4}$$
$23$ $$( 900 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$2815684 + 3869 T^{2} + T^{4}$$
$47$ $$1004004 + 7629 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -554 + 103 T + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$235991044 + 31349 T^{2} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 8100 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$