# Properties

 Label 1075.2.b.b Level 1075 Weight 2 Character orbit 1075.b Analytic conductor 8.584 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1075 = 5^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1075.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.58391821729$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 43) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} -2 i q^{3} -2 q^{4} + 4 q^{6} - q^{9} +O(q^{10})$$ $$q + 2 i q^{2} -2 i q^{3} -2 q^{4} + 4 q^{6} - q^{9} + 3 q^{11} + 4 i q^{12} -5 i q^{13} -4 q^{16} + 3 i q^{17} -2 i q^{18} + 2 q^{19} + 6 i q^{22} -i q^{23} + 10 q^{26} -4 i q^{27} + 6 q^{29} - q^{31} -8 i q^{32} -6 i q^{33} -6 q^{34} + 2 q^{36} + 4 i q^{38} -10 q^{39} + 5 q^{41} -i q^{43} -6 q^{44} + 2 q^{46} -4 i q^{47} + 8 i q^{48} + 7 q^{49} + 6 q^{51} + 10 i q^{52} -5 i q^{53} + 8 q^{54} -4 i q^{57} + 12 i q^{58} + 12 q^{59} + 2 q^{61} -2 i q^{62} + 8 q^{64} + 12 q^{66} + 3 i q^{67} -6 i q^{68} -2 q^{69} + 2 q^{71} + 2 i q^{73} -4 q^{76} -20 i q^{78} + 8 q^{79} -11 q^{81} + 10 i q^{82} + 15 i q^{83} + 2 q^{86} -12 i q^{87} + 4 q^{89} + 2 i q^{92} + 2 i q^{93} + 8 q^{94} -16 q^{96} -7 i q^{97} + 14 i q^{98} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} + 8q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 4q^{4} + 8q^{6} - 2q^{9} + 6q^{11} - 8q^{16} + 4q^{19} + 20q^{26} + 12q^{29} - 2q^{31} - 12q^{34} + 4q^{36} - 20q^{39} + 10q^{41} - 12q^{44} + 4q^{46} + 14q^{49} + 12q^{51} + 16q^{54} + 24q^{59} + 4q^{61} + 16q^{64} + 24q^{66} - 4q^{69} + 4q^{71} - 8q^{76} + 16q^{79} - 22q^{81} + 4q^{86} + 8q^{89} + 16q^{94} - 32q^{96} - 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1075\mathbb{Z}\right)^\times$$.

 $$n$$ $$302$$ $$476$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
474.1
 − 1.00000i 1.00000i
2.00000i 2.00000i −2.00000 0 4.00000 0 0 −1.00000 0
474.2 2.00000i 2.00000i −2.00000 0 4.00000 0 0 −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
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 1075.2.b.b 2
5.b even 2 1 inner 1075.2.b.b 2
5.c odd 4 1 43.2.a.a 1
5.c odd 4 1 1075.2.a.h 1
15.e even 4 1 387.2.a.e 1
15.e even 4 1 9675.2.a.b 1
20.e even 4 1 688.2.a.b 1
35.f even 4 1 2107.2.a.a 1
40.i odd 4 1 2752.2.a.f 1
40.k even 4 1 2752.2.a.b 1
55.e even 4 1 5203.2.a.a 1
60.l odd 4 1 6192.2.a.ba 1
65.h odd 4 1 7267.2.a.a 1
215.g even 4 1 1849.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.a.a 1 5.c odd 4 1
387.2.a.e 1 15.e even 4 1
688.2.a.b 1 20.e even 4 1
1075.2.a.h 1 5.c odd 4 1
1075.2.b.b 2 1.a even 1 1 trivial
1075.2.b.b 2 5.b even 2 1 inner
1849.2.a.d 1 215.g even 4 1
2107.2.a.a 1 35.f even 4 1
2752.2.a.b 1 40.k even 4 1
2752.2.a.f 1 40.i odd 4 1
5203.2.a.a 1 55.e even 4 1
6192.2.a.ba 1 60.l odd 4 1
7267.2.a.a 1 65.h odd 4 1
9675.2.a.b 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 $$S_{2}^{\mathrm{new}}(1075, [\chi])$$:

 $$T_{2}^{2} + 4$$ $$T_{3}^{2} + 4$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$3$ $$1 - 2 T^{2} + 9 T^{4}$$
$5$ 1
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$( 1 - 3 T + 11 T^{2} )^{2}$$
$13$ $$1 - T^{2} + 169 T^{4}$$
$17$ $$1 - 25 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 2 T + 19 T^{2} )^{2}$$
$23$ $$1 - 45 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 37 T^{2} )^{2}$$
$41$ $$( 1 - 5 T + 41 T^{2} )^{2}$$
$43$ $$1 + T^{2}$$
$47$ $$1 - 78 T^{2} + 2209 T^{4}$$
$53$ $$1 - 81 T^{2} + 2809 T^{4}$$
$59$ $$( 1 - 12 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 2 T + 61 T^{2} )^{2}$$
$67$ $$1 - 125 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 2 T + 71 T^{2} )^{2}$$
$73$ $$1 - 142 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}$$
$83$ $$1 + 59 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 4 T + 89 T^{2} )^{2}$$
$97$ $$1 - 145 T^{2} + 9409 T^{4}$$