# Properties

 Label 75.2.e.b Level $75$ Weight $2$ Character orbit 75.e Analytic conductor $0.599$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 75.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.598878015160$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - 2 \beta_{2} q^{4} + \beta_{3} q^{7} + 3 \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 - 2*b2 * q^4 + b3 * q^7 + 3*b2 * q^9 $$q + \beta_1 q^{3} - 2 \beta_{2} q^{4} + \beta_{3} q^{7} + 3 \beta_{2} q^{9} - 2 \beta_{3} q^{12} - 3 \beta_1 q^{13} - 4 q^{16} + \beta_{2} q^{19} - 3 q^{21} + 3 \beta_{3} q^{27} + 2 \beta_1 q^{28} + 7 q^{31} + 6 q^{36} - 4 \beta_{3} q^{37} - 9 \beta_{2} q^{39} + 7 \beta_1 q^{43} - 4 \beta_1 q^{48} + 4 \beta_{2} q^{49} + 6 \beta_{3} q^{52} + \beta_{3} q^{57} - 13 q^{61} - 3 \beta_1 q^{63} + 8 \beta_{2} q^{64} - 9 \beta_{3} q^{67} - 8 \beta_1 q^{73} + 2 q^{76} - 4 \beta_{2} q^{79} - 9 q^{81} + 6 \beta_{2} q^{84} + 9 q^{91} + 7 \beta_1 q^{93} + 11 \beta_{3} q^{97}+O(q^{100})$$ q + b1 * q^3 - 2*b2 * q^4 + b3 * q^7 + 3*b2 * q^9 - 2*b3 * q^12 - 3*b1 * q^13 - 4 * q^16 + b2 * q^19 - 3 * q^21 + 3*b3 * q^27 + 2*b1 * q^28 + 7 * q^31 + 6 * q^36 - 4*b3 * q^37 - 9*b2 * q^39 + 7*b1 * q^43 - 4*b1 * q^48 + 4*b2 * q^49 + 6*b3 * q^52 + b3 * q^57 - 13 * q^61 - 3*b1 * q^63 + 8*b2 * q^64 - 9*b3 * q^67 - 8*b1 * q^73 + 2 * q^76 - 4*b2 * q^79 - 9 * q^81 + 6*b2 * q^84 + 9 * q^91 + 7*b1 * q^93 + 11*b3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 16 q^{16} - 12 q^{21} + 28 q^{31} + 24 q^{36} - 52 q^{61} + 8 q^{76} - 36 q^{81} + 36 q^{91}+O(q^{100})$$ 4 * q - 16 * q^16 - 12 * q^21 + 28 * q^31 + 24 * q^36 - 52 * q^61 + 8 * q^76 - 36 * q^81 + 36 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 −1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
0 −1.22474 + 1.22474i 2.00000i 0 0 1.22474 + 1.22474i 0 3.00000i 0
32.2 0 1.22474 1.22474i 2.00000i 0 0 −1.22474 1.22474i 0 3.00000i 0
68.1 0 −1.22474 1.22474i 2.00000i 0 0 1.22474 1.22474i 0 3.00000i 0
68.2 0 1.22474 + 1.22474i 2.00000i 0 0 −1.22474 + 1.22474i 0 3.00000i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.2.e.b 4
3.b odd 2 1 CM 75.2.e.b 4
4.b odd 2 1 1200.2.v.f 4
5.b even 2 1 inner 75.2.e.b 4
5.c odd 4 2 inner 75.2.e.b 4
12.b even 2 1 1200.2.v.f 4
15.d odd 2 1 inner 75.2.e.b 4
15.e even 4 2 inner 75.2.e.b 4
20.d odd 2 1 1200.2.v.f 4
20.e even 4 2 1200.2.v.f 4
60.h even 2 1 1200.2.v.f 4
60.l odd 4 2 1200.2.v.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.e.b 4 1.a even 1 1 trivial
75.2.e.b 4 3.b odd 2 1 CM
75.2.e.b 4 5.b even 2 1 inner
75.2.e.b 4 5.c odd 4 2 inner
75.2.e.b 4 15.d odd 2 1 inner
75.2.e.b 4 15.e even 4 2 inner
1200.2.v.f 4 4.b odd 2 1
1200.2.v.f 4 12.b even 2 1
1200.2.v.f 4 20.d odd 2 1
1200.2.v.f 4 20.e even 4 2
1200.2.v.f 4 60.h even 2 1
1200.2.v.f 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 9$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 729$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 1)^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$(T - 7)^{4}$$
$37$ $$T^{4} + 2304$$
$41$ $$T^{4}$$
$43$ $$T^{4} + 21609$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T + 13)^{4}$$
$67$ $$T^{4} + 59049$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 36864$$
$79$ $$(T^{2} + 16)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 131769$$