# Properties

 Label 75.9.c.a Level $75$ Weight $9$ Character orbit 75.c Self dual yes Analytic conductor $30.553$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 75.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.5533957546$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 81 q^{3} + 256 q^{4} + 4273 q^{7} + 6561 q^{9}+O(q^{10})$$ q - 81 * q^3 + 256 * q^4 + 4273 * q^7 + 6561 * q^9 $$q - 81 q^{3} + 256 q^{4} + 4273 q^{7} + 6561 q^{9} - 20736 q^{12} - 56447 q^{13} + 65536 q^{16} + 157967 q^{19} - 346113 q^{21} - 531441 q^{27} + 1093888 q^{28} + 1225967 q^{31} + 1679616 q^{36} - 503522 q^{37} + 4572207 q^{39} + 6837073 q^{43} - 5308416 q^{48} + 12493728 q^{49} - 14450432 q^{52} - 12795327 q^{57} - 307393 q^{61} + 28035153 q^{63} + 16777216 q^{64} + 31874833 q^{67} - 16169282 q^{73} + 40439552 q^{76} - 18887038 q^{79} + 43046721 q^{81} - 88604928 q^{84} - 241198031 q^{91} - 99303327 q^{93} + 82132513 q^{97}+O(q^{100})$$ q - 81 * q^3 + 256 * q^4 + 4273 * q^7 + 6561 * q^9 - 20736 * q^12 - 56447 * q^13 + 65536 * q^16 + 157967 * q^19 - 346113 * q^21 - 531441 * q^27 + 1093888 * q^28 + 1225967 * q^31 + 1679616 * q^36 - 503522 * q^37 + 4572207 * q^39 + 6837073 * q^43 - 5308416 * q^48 + 12493728 * q^49 - 14450432 * q^52 - 12795327 * q^57 - 307393 * q^61 + 28035153 * q^63 + 16777216 * q^64 + 31874833 * q^67 - 16169282 * q^73 + 40439552 * q^76 - 18887038 * q^79 + 43046721 * q^81 - 88604928 * q^84 - 241198031 * q^91 - 99303327 * q^93 + 82132513 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0
0 −81.0000 256.000 0 0 4273.00 0 6561.00 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 $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.9.c.a 1
3.b odd 2 1 CM 75.9.c.a 1
5.b even 2 1 75.9.c.b yes 1
5.c odd 4 2 75.9.d.a 2
15.d odd 2 1 75.9.c.b yes 1
15.e even 4 2 75.9.d.a 2

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(75, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7} - 4273$$ T7 - 4273

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 81$$
$5$ $$T$$
$7$ $$T - 4273$$
$11$ $$T$$
$13$ $$T + 56447$$
$17$ $$T$$
$19$ $$T - 157967$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T - 1225967$$
$37$ $$T + 503522$$
$41$ $$T$$
$43$ $$T - 6837073$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T + 307393$$
$67$ $$T - 31874833$$
$71$ $$T$$
$73$ $$T + 16169282$$
$79$ $$T + 18887038$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T - 82132513$$