# Properties

 Label 75.7.c.a Level 75 Weight 7 Character orbit 75.c Self dual yes Analytic conductor 17.254 Analytic rank 0 Dimension 1 CM discriminant -3 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 75.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.2540562715$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 27q^{3} + 64q^{4} + 286q^{7} + 729q^{9} + O(q^{10})$$ $$q + 27q^{3} + 64q^{4} + 286q^{7} + 729q^{9} + 1728q^{12} - 506q^{13} + 4096q^{16} - 10582q^{19} + 7722q^{21} + 19683q^{27} + 18304q^{28} + 35282q^{31} + 46656q^{36} + 89206q^{37} - 13662q^{39} - 111386q^{43} + 110592q^{48} - 35853q^{49} - 32384q^{52} - 285714q^{57} - 420838q^{61} + 208494q^{63} + 262144q^{64} - 172874q^{67} - 638066q^{73} - 677248q^{76} - 204622q^{79} + 531441q^{81} + 494208q^{84} - 144716q^{91} + 952614q^{93} + 56446q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0
0 27.0000 64.0000 0 0 286.000 0 729.000 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 75.7.c.a 1
3.b odd 2 1 CM 75.7.c.a 1
5.b even 2 1 3.7.b.a 1
5.c odd 4 2 75.7.d.a 2
15.d odd 2 1 3.7.b.a 1
15.e even 4 2 75.7.d.a 2
20.d odd 2 1 48.7.e.a 1
35.c odd 2 1 147.7.b.a 1
40.e odd 2 1 192.7.e.a 1
40.f even 2 1 192.7.e.b 1
45.h odd 6 2 81.7.d.a 2
45.j even 6 2 81.7.d.a 2
60.h even 2 1 48.7.e.a 1
105.g even 2 1 147.7.b.a 1
120.i odd 2 1 192.7.e.b 1
120.m even 2 1 192.7.e.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.7.b.a 1 5.b even 2 1
3.7.b.a 1 15.d odd 2 1
48.7.e.a 1 20.d odd 2 1
48.7.e.a 1 60.h even 2 1
75.7.c.a 1 1.a even 1 1 trivial
75.7.c.a 1 3.b odd 2 1 CM
75.7.d.a 2 5.c odd 4 2
75.7.d.a 2 15.e even 4 2
81.7.d.a 2 45.h odd 6 2
81.7.d.a 2 45.j even 6 2
147.7.b.a 1 35.c odd 2 1
147.7.b.a 1 105.g even 2 1
192.7.e.a 1 40.e odd 2 1
192.7.e.a 1 120.m even 2 1
192.7.e.b 1 40.f even 2 1
192.7.e.b 1 120.i odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(75, [\chi])$$:

 $$T_{2}$$ $$T_{7} - 286$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 8 T )( 1 + 8 T )$$
$3$ $$1 - 27 T$$
$5$ 1
$7$ $$1 - 286 T + 117649 T^{2}$$
$11$ $$( 1 - 1331 T )( 1 + 1331 T )$$
$13$ $$1 + 506 T + 4826809 T^{2}$$
$17$ $$( 1 - 4913 T )( 1 + 4913 T )$$
$19$ $$1 + 10582 T + 47045881 T^{2}$$
$23$ $$( 1 - 12167 T )( 1 + 12167 T )$$
$29$ $$( 1 - 24389 T )( 1 + 24389 T )$$
$31$ $$1 - 35282 T + 887503681 T^{2}$$
$37$ $$1 - 89206 T + 2565726409 T^{2}$$
$41$ $$( 1 - 68921 T )( 1 + 68921 T )$$
$43$ $$1 + 111386 T + 6321363049 T^{2}$$
$47$ $$( 1 - 103823 T )( 1 + 103823 T )$$
$53$ $$( 1 - 148877 T )( 1 + 148877 T )$$
$59$ $$( 1 - 205379 T )( 1 + 205379 T )$$
$61$ $$1 + 420838 T + 51520374361 T^{2}$$
$67$ $$1 + 172874 T + 90458382169 T^{2}$$
$71$ $$( 1 - 357911 T )( 1 + 357911 T )$$
$73$ $$1 + 638066 T + 151334226289 T^{2}$$
$79$ $$1 + 204622 T + 243087455521 T^{2}$$
$83$ $$( 1 - 571787 T )( 1 + 571787 T )$$
$89$ $$( 1 - 704969 T )( 1 + 704969 T )$$
$97$ $$1 - 56446 T + 832972004929 T^{2}$$