# Properties

 Label 1890.2.g.a Level 1890 Weight 2 Character orbit 1890.g Analytic conductor 15.092 Analytic rank 0 Dimension 2 CM no Inner twists 2

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1890.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.0917259820$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + ( -2 + i ) q^{5} -i q^{7} -i q^{8} +O(q^{10})$$ $$q + i q^{2} - q^{4} + ( -2 + i ) q^{5} -i q^{7} -i q^{8} + ( -1 - 2 i ) q^{10} -4 q^{11} + 6 i q^{13} + q^{14} + q^{16} + 4 i q^{17} + ( 2 - i ) q^{20} -4 i q^{22} -4 i q^{23} + ( 3 - 4 i ) q^{25} -6 q^{26} + i q^{28} -3 q^{29} + 7 q^{31} + i q^{32} -4 q^{34} + ( 1 + 2 i ) q^{35} + i q^{37} + ( 1 + 2 i ) q^{40} -7 q^{41} -10 i q^{43} + 4 q^{44} + 4 q^{46} -13 i q^{47} - q^{49} + ( 4 + 3 i ) q^{50} -6 i q^{52} -6 i q^{53} + ( 8 - 4 i ) q^{55} - q^{56} -3 i q^{58} + 5 q^{59} -7 q^{61} + 7 i q^{62} - q^{64} + ( -6 - 12 i ) q^{65} -4 i q^{67} -4 i q^{68} + ( -2 + i ) q^{70} -3 q^{71} -7 i q^{73} - q^{74} + 4 i q^{77} -12 q^{79} + ( -2 + i ) q^{80} -7 i q^{82} + 2 i q^{83} + ( -4 - 8 i ) q^{85} + 10 q^{86} + 4 i q^{88} -14 q^{89} + 6 q^{91} + 4 i q^{92} + 13 q^{94} -10 i q^{97} -i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 4q^{5} + O(q^{10})$$ $$2q - 2q^{4} - 4q^{5} - 2q^{10} - 8q^{11} + 2q^{14} + 2q^{16} + 4q^{20} + 6q^{25} - 12q^{26} - 6q^{29} + 14q^{31} - 8q^{34} + 2q^{35} + 2q^{40} - 14q^{41} + 8q^{44} + 8q^{46} - 2q^{49} + 8q^{50} + 16q^{55} - 2q^{56} + 10q^{59} - 14q^{61} - 2q^{64} - 12q^{65} - 4q^{70} - 6q^{71} - 2q^{74} - 24q^{79} - 4q^{80} - 8q^{85} + 20q^{86} - 28q^{89} + 12q^{91} + 26q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1890\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$1081$$ $$1541$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 −2.00000 1.00000i 0 1.00000i 1.00000i 0 −1.00000 + 2.00000i
379.2 1.00000i 0 −1.00000 −2.00000 + 1.00000i 0 1.00000i 1.00000i 0 −1.00000 2.00000i
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1890.2.g.a 2
3.b odd 2 1 1890.2.g.k yes 2
5.b even 2 1 inner 1890.2.g.a 2
5.c odd 4 1 9450.2.a.be 1
5.c odd 4 1 9450.2.a.cc 1
15.d odd 2 1 1890.2.g.k yes 2
15.e even 4 1 9450.2.a.w 1
15.e even 4 1 9450.2.a.dx 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.g.a 2 1.a even 1 1 trivial
1890.2.g.a 2 5.b even 2 1 inner
1890.2.g.k yes 2 3.b odd 2 1
1890.2.g.k yes 2 15.d odd 2 1
9450.2.a.w 1 15.e even 4 1
9450.2.a.be 1 5.c odd 4 1
9450.2.a.cc 1 5.c odd 4 1
9450.2.a.dx 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1890, [\chi])$$:

 $$T_{11} + 4$$ $$T_{13}^{2} + 36$$ $$T_{29} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ 1
$5$ $$1 + 4 T + 5 T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 1 + 4 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$17$ $$1 - 18 T^{2} + 289 T^{4}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$1 - 30 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 3 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )^{2}$$
$37$ $$1 - 73 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + 7 T + 41 T^{2} )^{2}$$
$43$ $$1 + 14 T^{2} + 1849 T^{4}$$
$47$ $$1 + 75 T^{2} + 2209 T^{4}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 - 5 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 7 T + 61 T^{2} )^{2}$$
$67$ $$1 - 118 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 3 T + 71 T^{2} )^{2}$$
$73$ $$1 - 97 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 12 T + 79 T^{2} )^{2}$$
$83$ $$1 - 162 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 14 T + 89 T^{2} )^{2}$$
$97$ $$1 - 94 T^{2} + 9409 T^{4}$$
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