# Properties

 Label 1960.1.i.d Level $1960$ Weight $1$ Character orbit 1960.i Self dual yes Analytic conductor $0.978$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -40 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1960 = 2^{3} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1960.i (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.978167424761$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Projective image $$D_{3}$$ Projective field Galois closure of 3.1.1960.1 Artin image $S_3$ Artin field Galois closure of 3.1.1960.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{4} + q^{5} + q^{8} + q^{9} + q^{10} - q^{11} - q^{13} + q^{16} + q^{18} - q^{19} + q^{20} - q^{22} - q^{23} + q^{25} - q^{26} + q^{32} + q^{36} - q^{37} - q^{38} + q^{40} - q^{41} - q^{44} + q^{45} - q^{46} - q^{47} + q^{50} - q^{52} - q^{53} - q^{55} + 2q^{59} + q^{64} - q^{65} + q^{72} - q^{74} - q^{76} + q^{80} + q^{81} - q^{82} - q^{88} + 2q^{89} + q^{90} - q^{92} - q^{94} - q^{95} - q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times$$.

 $$n$$ $$981$$ $$1081$$ $$1177$$ $$1471$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 0
1.00000 0 1.00000 1.00000 0 0 1.00000 1.00000 1.00000
 $$n$$: 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
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.1.i.d 1
5.b even 2 1 1960.1.i.a 1
7.b odd 2 1 1960.1.i.c 1
7.c even 3 2 280.1.bi.a 2
7.d odd 6 2 1960.1.bi.a 2
8.d odd 2 1 1960.1.i.a 1
21.h odd 6 2 2520.1.ef.b 2
28.g odd 6 2 1120.1.by.a 2
35.c odd 2 1 1960.1.i.b 1
35.i odd 6 2 1960.1.bi.b 2
35.j even 6 2 280.1.bi.b yes 2
35.l odd 12 4 1400.1.ba.a 4
40.e odd 2 1 CM 1960.1.i.d 1
56.e even 2 1 1960.1.i.b 1
56.k odd 6 2 280.1.bi.b yes 2
56.m even 6 2 1960.1.bi.b 2
56.p even 6 2 1120.1.by.b 2
105.o odd 6 2 2520.1.ef.a 2
140.p odd 6 2 1120.1.by.b 2
168.v even 6 2 2520.1.ef.a 2
280.n even 2 1 1960.1.i.c 1
280.ba even 6 2 1960.1.bi.a 2
280.bf even 6 2 1120.1.by.a 2
280.bi odd 6 2 280.1.bi.a 2
280.br even 12 4 1400.1.ba.a 4
840.cv even 6 2 2520.1.ef.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.bi.a 2 7.c even 3 2
280.1.bi.a 2 280.bi odd 6 2
280.1.bi.b yes 2 35.j even 6 2
280.1.bi.b yes 2 56.k odd 6 2
1120.1.by.a 2 28.g odd 6 2
1120.1.by.a 2 280.bf even 6 2
1120.1.by.b 2 56.p even 6 2
1120.1.by.b 2 140.p odd 6 2
1400.1.ba.a 4 35.l odd 12 4
1400.1.ba.a 4 280.br even 12 4
1960.1.i.a 1 5.b even 2 1
1960.1.i.a 1 8.d odd 2 1
1960.1.i.b 1 35.c odd 2 1
1960.1.i.b 1 56.e even 2 1
1960.1.i.c 1 7.b odd 2 1
1960.1.i.c 1 280.n even 2 1
1960.1.i.d 1 1.a even 1 1 trivial
1960.1.i.d 1 40.e odd 2 1 CM
1960.1.bi.a 2 7.d odd 6 2
1960.1.bi.a 2 280.ba even 6 2
1960.1.bi.b 2 35.i odd 6 2
1960.1.bi.b 2 56.m even 6 2
2520.1.ef.a 2 105.o odd 6 2
2520.1.ef.a 2 168.v even 6 2
2520.1.ef.b 2 21.h odd 6 2
2520.1.ef.b 2 840.cv even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1960, [\chi])$$:

 $$T_{13} + 1$$ $$T_{19} + 1$$

## Hecke characteristic polynomials

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