# Properties

 Label 1900.3.g.b Level $1900$ Weight $3$ Character orbit 1900.g Analytic conductor $51.771$ 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: $$k$$ $$=$$ $$3$$ 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: $$\Q(i, \sqrt{29})$$ Defining polynomial: $$x^{4} + 15 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 76) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} -\beta_{1} q^{7} + 20 q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} -\beta_{1} q^{7} + 20 q^{9} + 14 q^{11} + 3 \beta_{3} q^{13} + 23 \beta_{1} q^{17} + ( -10 - 3 \beta_{2} ) q^{19} -\beta_{2} q^{21} + \beta_{1} q^{23} + 11 \beta_{3} q^{27} + 9 \beta_{2} q^{29} -6 \beta_{2} q^{31} + 14 \beta_{3} q^{33} -6 \beta_{3} q^{37} + 87 q^{39} + 6 \beta_{2} q^{41} -68 \beta_{1} q^{43} + 26 \beta_{1} q^{47} + 48 q^{49} + 23 \beta_{2} q^{51} -15 \beta_{3} q^{53} + ( -87 \beta_{1} - 10 \beta_{3} ) q^{57} -3 \beta_{2} q^{59} -40 q^{61} -20 \beta_{1} q^{63} + 3 \beta_{3} q^{67} + \beta_{2} q^{69} -6 \beta_{2} q^{71} + 7 \beta_{1} q^{73} -14 \beta_{1} q^{77} -18 \beta_{2} q^{79} + 139 q^{81} -32 \beta_{1} q^{83} + 261 \beta_{1} q^{87} + 24 \beta_{2} q^{89} -3 \beta_{2} q^{91} -174 \beta_{1} q^{93} + 18 \beta_{3} q^{97} + 280 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 80q^{9} + O(q^{10})$$ $$4q + 80q^{9} + 56q^{11} - 40q^{19} + 348q^{39} + 192q^{49} - 160q^{61} + 556q^{81} + 1120q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 15 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 8 \nu$$$$)/7$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 22 \nu$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 15$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 15$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{2} + 11 \beta_{1}$$

## 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
 − 3.19258i 3.19258i 2.19258i − 2.19258i
0 −5.38516 0 0 0 1.00000i 0 20.0000 0
949.2 0 −5.38516 0 0 0 1.00000i 0 20.0000 0
949.3 0 5.38516 0 0 0 1.00000i 0 20.0000 0
949.4 0 5.38516 0 0 0 1.00000i 0 20.0000 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.b odd 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.b 4
5.b even 2 1 inner 1900.3.g.b 4
5.c odd 4 1 76.3.c.a 2
5.c odd 4 1 1900.3.e.b 2
15.e even 4 1 684.3.h.c 2
19.b odd 2 1 inner 1900.3.g.b 4
20.e even 4 1 304.3.e.b 2
40.i odd 4 1 1216.3.e.k 2
40.k even 4 1 1216.3.e.l 2
60.l odd 4 1 2736.3.o.i 2
95.d odd 2 1 inner 1900.3.g.b 4
95.g even 4 1 76.3.c.a 2
95.g even 4 1 1900.3.e.b 2
285.j odd 4 1 684.3.h.c 2
380.j odd 4 1 304.3.e.b 2
760.t even 4 1 1216.3.e.k 2
760.y odd 4 1 1216.3.e.l 2
1140.w even 4 1 2736.3.o.i 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 29$$ acting on $$S_{3}^{\mathrm{new}}(1900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -29 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -14 + T )^{4}$$
$13$ $$( -261 + T^{2} )^{2}$$
$17$ $$( 529 + T^{2} )^{2}$$
$19$ $$( 361 + 20 T + T^{2} )^{2}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( 2349 + T^{2} )^{2}$$
$31$ $$( 1044 + T^{2} )^{2}$$
$37$ $$( -1044 + T^{2} )^{2}$$
$41$ $$( 1044 + T^{2} )^{2}$$
$43$ $$( 4624 + T^{2} )^{2}$$
$47$ $$( 676 + T^{2} )^{2}$$
$53$ $$( -6525 + T^{2} )^{2}$$
$59$ $$( 261 + T^{2} )^{2}$$
$61$ $$( 40 + T )^{4}$$
$67$ $$( -261 + T^{2} )^{2}$$
$71$ $$( 1044 + T^{2} )^{2}$$
$73$ $$( 49 + T^{2} )^{2}$$
$79$ $$( 9396 + T^{2} )^{2}$$
$83$ $$( 1024 + T^{2} )^{2}$$
$89$ $$( 16704 + T^{2} )^{2}$$
$97$ $$( -9396 + T^{2} )^{2}$$