# Properties

 Label 75.3.d.a Level $75$ Weight $3$ Character orbit 75.d Analytic conductor $2.044$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.04360198270$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} -4 q^{4} + 11 i q^{7} -9 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} -4 q^{4} + 11 i q^{7} -9 q^{9} -12 i q^{12} + i q^{13} + 16 q^{16} + 37 q^{19} -33 q^{21} -27 i q^{27} -44 i q^{28} -13 q^{31} + 36 q^{36} + 26 i q^{37} -3 q^{39} + 61 i q^{43} + 48 i q^{48} -72 q^{49} -4 i q^{52} + 111 i q^{57} + 47 q^{61} -99 i q^{63} -64 q^{64} -109 i q^{67} + 46 i q^{73} -148 q^{76} + 142 q^{79} + 81 q^{81} + 132 q^{84} -11 q^{91} -39 i q^{93} -169 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{4} - 18q^{9} + O(q^{10})$$ $$2q - 8q^{4} - 18q^{9} + 32q^{16} + 74q^{19} - 66q^{21} - 26q^{31} + 72q^{36} - 6q^{39} - 144q^{49} + 94q^{61} - 128q^{64} - 296q^{76} + 284q^{79} + 162q^{81} + 264q^{84} - 22q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
74.1
 − 1.00000i 1.00000i
0 3.00000i −4.00000 0 0 11.0000i 0 −9.00000 0
74.2 0 3.00000i −4.00000 0 0 11.0000i 0 −9.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.d.a 2
3.b odd 2 1 CM 75.3.d.a 2
4.b odd 2 1 1200.3.c.a 2
5.b even 2 1 inner 75.3.d.a 2
5.c odd 4 1 75.3.c.a 1
5.c odd 4 1 75.3.c.b yes 1
12.b even 2 1 1200.3.c.a 2
15.d odd 2 1 inner 75.3.d.a 2
15.e even 4 1 75.3.c.a 1
15.e even 4 1 75.3.c.b yes 1
20.d odd 2 1 1200.3.c.a 2
20.e even 4 1 1200.3.l.c 1
20.e even 4 1 1200.3.l.d 1
60.h even 2 1 1200.3.c.a 2
60.l odd 4 1 1200.3.l.c 1
60.l odd 4 1 1200.3.l.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.a 1 5.c odd 4 1
75.3.c.a 1 15.e even 4 1
75.3.c.b yes 1 5.c odd 4 1
75.3.c.b yes 1 15.e even 4 1
75.3.d.a 2 1.a even 1 1 trivial
75.3.d.a 2 3.b odd 2 1 CM
75.3.d.a 2 5.b even 2 1 inner
75.3.d.a 2 15.d odd 2 1 inner
1200.3.c.a 2 4.b odd 2 1
1200.3.c.a 2 12.b even 2 1
1200.3.c.a 2 20.d odd 2 1
1200.3.c.a 2 60.h even 2 1
1200.3.l.c 1 20.e even 4 1
1200.3.l.c 1 60.l odd 4 1
1200.3.l.d 1 20.e even 4 1
1200.3.l.d 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{3}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$121 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( -37 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 13 + T )^{2}$$
$37$ $$676 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$3721 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -47 + T )^{2}$$
$67$ $$11881 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$2116 + T^{2}$$
$79$ $$( -142 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$28561 + T^{2}$$