# Properties

 Label 275.1.c.a Level $275$ Weight $1$ Character orbit 275.c Self dual yes Analytic conductor $0.137$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -11, -55, 5 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 275.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.137242878474$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{-11})$$ Artin image: $D_4$ Artin field: Galois closure of 4.2.1375.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{4} - q^{9}+O(q^{10})$$ q + q^4 - q^9 $$q + q^{4} - q^{9} - q^{11} + q^{16} - 2 q^{31} - q^{36} - q^{44} + q^{49} - 2 q^{59} + q^{64} + 2 q^{71} + q^{81} + 2 q^{89} + q^{99}+O(q^{100})$$ q + q^4 - q^9 - q^11 + q^16 - 2 * q^31 - q^36 - q^44 + q^49 - 2 * q^59 + q^64 + 2 * q^71 + q^81 + 2 * q^89 + q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 0
0 0 1.00000 0 0 0 0 −1.00000 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
5.b even 2 1 RM by $$\Q(\sqrt{5})$$
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
55.d odd 2 1 CM by $$\Q(\sqrt{-55})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.1.c.a 1
3.b odd 2 1 2475.1.b.a 1
5.b even 2 1 RM 275.1.c.a 1
5.c odd 4 2 55.1.d.a 1
11.b odd 2 1 CM 275.1.c.a 1
11.c even 5 4 3025.1.x.a 4
11.d odd 10 4 3025.1.x.a 4
15.d odd 2 1 2475.1.b.a 1
15.e even 4 2 495.1.h.a 1
20.e even 4 2 880.1.i.a 1
33.d even 2 1 2475.1.b.a 1
35.f even 4 2 2695.1.g.c 1
35.k even 12 4 2695.1.q.b 2
35.l odd 12 4 2695.1.q.c 2
40.i odd 4 2 3520.1.i.b 1
40.k even 4 2 3520.1.i.a 1
55.d odd 2 1 CM 275.1.c.a 1
55.e even 4 2 55.1.d.a 1
55.h odd 10 4 3025.1.x.a 4
55.j even 10 4 3025.1.x.a 4
55.k odd 20 8 605.1.h.a 4
55.l even 20 8 605.1.h.a 4
165.d even 2 1 2475.1.b.a 1
165.l odd 4 2 495.1.h.a 1
220.i odd 4 2 880.1.i.a 1
385.l odd 4 2 2695.1.g.c 1
385.bc even 12 4 2695.1.q.c 2
385.bf odd 12 4 2695.1.q.b 2
440.t even 4 2 3520.1.i.b 1
440.w odd 4 2 3520.1.i.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.1.d.a 1 5.c odd 4 2
55.1.d.a 1 55.e even 4 2
275.1.c.a 1 1.a even 1 1 trivial
275.1.c.a 1 5.b even 2 1 RM
275.1.c.a 1 11.b odd 2 1 CM
275.1.c.a 1 55.d odd 2 1 CM
495.1.h.a 1 15.e even 4 2
495.1.h.a 1 165.l odd 4 2
605.1.h.a 4 55.k odd 20 8
605.1.h.a 4 55.l even 20 8
880.1.i.a 1 20.e even 4 2
880.1.i.a 1 220.i odd 4 2
2475.1.b.a 1 3.b odd 2 1
2475.1.b.a 1 15.d odd 2 1
2475.1.b.a 1 33.d even 2 1
2475.1.b.a 1 165.d even 2 1
2695.1.g.c 1 35.f even 4 2
2695.1.g.c 1 385.l odd 4 2
2695.1.q.b 2 35.k even 12 4
2695.1.q.b 2 385.bf odd 12 4
2695.1.q.c 2 35.l odd 12 4
2695.1.q.c 2 385.bc even 12 4
3025.1.x.a 4 11.c even 5 4
3025.1.x.a 4 11.d odd 10 4
3025.1.x.a 4 55.h odd 10 4
3025.1.x.a 4 55.j even 10 4
3520.1.i.a 1 40.k even 4 2
3520.1.i.a 1 440.w odd 4 2
3520.1.i.b 1 40.i odd 4 2
3520.1.i.b 1 440.t even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(275, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 2$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T + 2$$
$61$ $$T$$
$67$ $$T$$
$71$ $$T - 2$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 2$$
$97$ $$T$$