# Properties

 Label 75.9.d.a Level $75$ Weight $9$ Character orbit 75.d Analytic conductor $30.553$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 75.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.5533957546$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + 81 i q^{3} - 256 q^{4} + 4273 i q^{7} - 6561 q^{9}+O(q^{10})$$ q + 81*i * q^3 - 256 * q^4 + 4273*i * q^7 - 6561 * q^9 $$q + 81 i q^{3} - 256 q^{4} + 4273 i q^{7} - 6561 q^{9} - 20736 i q^{12} + 56447 i q^{13} + 65536 q^{16} - 157967 q^{19} - 346113 q^{21} - 531441 i q^{27} - 1093888 i q^{28} + 1225967 q^{31} + 1679616 q^{36} - 503522 i q^{37} - 4572207 q^{39} - 6837073 i q^{43} + 5308416 i q^{48} - 12493728 q^{49} - 14450432 i q^{52} - 12795327 i q^{57} - 307393 q^{61} - 28035153 i q^{63} - 16777216 q^{64} + 31874833 i q^{67} + 16169282 i q^{73} + 40439552 q^{76} + 18887038 q^{79} + 43046721 q^{81} + 88604928 q^{84} - 241198031 q^{91} + 99303327 i q^{93} + 82132513 i q^{97} +O(q^{100})$$ q + 81*i * q^3 - 256 * q^4 + 4273*i * q^7 - 6561 * q^9 - 20736*i * q^12 + 56447*i * q^13 + 65536 * q^16 - 157967 * q^19 - 346113 * q^21 - 531441*i * q^27 - 1093888*i * q^28 + 1225967 * q^31 + 1679616 * q^36 - 503522*i * q^37 - 4572207 * q^39 - 6837073*i * q^43 + 5308416*i * q^48 - 12493728 * q^49 - 14450432*i * q^52 - 12795327*i * q^57 - 307393 * q^61 - 28035153*i * q^63 - 16777216 * q^64 + 31874833*i * q^67 + 16169282*i * q^73 + 40439552 * q^76 + 18887038 * q^79 + 43046721 * q^81 + 88604928 * q^84 - 241198031 * q^91 + 99303327*i * q^93 + 82132513*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 512 q^{4} - 13122 q^{9}+O(q^{10})$$ 2 * q - 512 * q^4 - 13122 * q^9 $$2 q - 512 q^{4} - 13122 q^{9} + 131072 q^{16} - 315934 q^{19} - 692226 q^{21} + 2451934 q^{31} + 3359232 q^{36} - 9144414 q^{39} - 24987456 q^{49} - 614786 q^{61} - 33554432 q^{64} + 80879104 q^{76} + 37774076 q^{79} + 86093442 q^{81} + 177209856 q^{84} - 482396062 q^{91}+O(q^{100})$$ 2 * q - 512 * q^4 - 13122 * q^9 + 131072 * q^16 - 315934 * q^19 - 692226 * q^21 + 2451934 * q^31 + 3359232 * q^36 - 9144414 * q^39 - 24987456 * q^49 - 614786 * q^61 - 33554432 * q^64 + 80879104 * q^76 + 37774076 * q^79 + 86093442 * q^81 + 177209856 * q^84 - 482396062 * q^91

## 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 81.0000i −256.000 0 0 4273.00i 0 −6561.00 0
74.2 0 81.0000i −256.000 0 0 4273.00i 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 $$\Q(\sqrt{-3})$$
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 75.9.d.a 2
3.b odd 2 1 CM 75.9.d.a 2
5.b even 2 1 inner 75.9.d.a 2
5.c odd 4 1 75.9.c.a 1
5.c odd 4 1 75.9.c.b yes 1
15.d odd 2 1 inner 75.9.d.a 2
15.e even 4 1 75.9.c.a 1
15.e even 4 1 75.9.c.b yes 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{9}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 6561$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 18258529$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 3186263809$$
$17$ $$T^{2}$$
$19$ $$(T + 157967)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T - 1225967)^{2}$$
$37$ $$T^{2} + 253534404484$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 46745567207329$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 307393)^{2}$$
$67$ $$T^{2} + 10\!\cdots\!89$$
$71$ $$T^{2}$$
$73$ $$T^{2} + \cdots + 261445680395524$$
$79$ $$(T - 18887038)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 67\!\cdots\!69$$