# Properties

 Label 75.4.e.a Level $75$ Weight $4$ Character orbit 75.e Analytic conductor $4.425$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.42514325043$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -8 \beta_{2} q^{4} -6 \beta_{3} q^{7} + 27 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -8 \beta_{2} q^{4} -6 \beta_{3} q^{7} + 27 \beta_{2} q^{9} -8 \beta_{3} q^{12} + 12 \beta_{1} q^{13} -64 q^{16} -56 \beta_{2} q^{19} + 162 q^{21} + 27 \beta_{3} q^{27} -48 \beta_{1} q^{28} -308 q^{31} + 216 q^{36} + 84 \beta_{3} q^{37} + 324 \beta_{2} q^{39} + 42 \beta_{1} q^{43} -64 \beta_{1} q^{48} -629 \beta_{2} q^{49} -96 \beta_{3} q^{52} -56 \beta_{3} q^{57} + 182 q^{61} + 162 \beta_{1} q^{63} + 512 \beta_{2} q^{64} -126 \beta_{3} q^{67} + 72 \beta_{1} q^{73} -448 q^{76} + 884 \beta_{2} q^{79} -729 q^{81} -1296 \beta_{2} q^{84} + 1944 q^{91} -308 \beta_{1} q^{93} + 264 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 256q^{16} + 648q^{21} - 1232q^{31} + 864q^{36} + 728q^{61} - 1792q^{76} - 2916q^{81} + 7776q^{91} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/3$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}$$

## 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 −3.67423 + 3.67423i 8.00000i 0 0 −22.0454 22.0454i 0 27.0000i 0
32.2 0 3.67423 3.67423i 8.00000i 0 0 22.0454 + 22.0454i 0 27.0000i 0
68.1 0 −3.67423 3.67423i 8.00000i 0 0 −22.0454 + 22.0454i 0 27.0000i 0
68.2 0 3.67423 + 3.67423i 8.00000i 0 0 22.0454 22.0454i 0 27.0000i 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 75.4.e.a 4
3.b odd 2 1 CM 75.4.e.a 4
5.b even 2 1 inner 75.4.e.a 4
5.c odd 4 2 inner 75.4.e.a 4
15.d odd 2 1 inner 75.4.e.a 4
15.e even 4 2 inner 75.4.e.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.e.a 4 1.a even 1 1 trivial
75.4.e.a 4 3.b odd 2 1 CM
75.4.e.a 4 5.b even 2 1 inner
75.4.e.a 4 5.c odd 4 2 inner
75.4.e.a 4 15.d odd 2 1 inner
75.4.e.a 4 15.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{4}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$729 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$944784 + T^{4}$$
$11$ $$T^{4}$$
$13$ $$15116544 + T^{4}$$
$17$ $$T^{4}$$
$19$ $$( 3136 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$( 308 + T )^{4}$$
$37$ $$36294822144 + T^{4}$$
$41$ $$T^{4}$$
$43$ $$2268426384 + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -182 + T )^{4}$$
$67$ $$183742537104 + T^{4}$$
$71$ $$T^{4}$$
$73$ $$19591041024 + T^{4}$$
$79$ $$( 781456 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$3541141131264 + T^{4}$$