# Properties

 Label 1960.1.bi.b Level $1960$ Weight $1$ Character orbit 1960.bi Analytic conductor $0.978$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -40 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1960 = 2^{3} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1960.bi (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.978167424761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - \zeta_{6} q^{5} - q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + z * q^2 + z^2 * q^4 - z * q^5 - q^8 - z * q^9 $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - \zeta_{6} q^{5} - q^{8} - \zeta_{6} q^{9} - \zeta_{6}^{2} q^{10} - \zeta_{6}^{2} q^{11} - q^{13} - \zeta_{6} q^{16} - \zeta_{6}^{2} q^{18} - \zeta_{6} q^{19} + q^{20} + q^{22} - \zeta_{6} q^{23} + \zeta_{6}^{2} q^{25} - \zeta_{6} q^{26} - \zeta_{6}^{2} q^{32} + q^{36} - \zeta_{6} q^{37} - \zeta_{6}^{2} q^{38} + \zeta_{6} q^{40} + q^{41} + \zeta_{6} q^{44} + \zeta_{6}^{2} q^{45} - \zeta_{6}^{2} q^{46} + \zeta_{6} q^{47} - q^{50} - \zeta_{6}^{2} q^{52} + \zeta_{6}^{2} q^{53} - q^{55} - \zeta_{6}^{2} q^{59} + q^{64} + \zeta_{6} q^{65} + \zeta_{6} q^{72} - \zeta_{6}^{2} q^{74} + q^{76} + \zeta_{6}^{2} q^{80} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{82} + \zeta_{6}^{2} q^{88} + \zeta_{6} q^{89} - q^{90} + q^{92} + \zeta_{6}^{2} q^{94} + \zeta_{6}^{2} q^{95} - q^{99} +O(q^{100})$$ q + z * q^2 + z^2 * q^4 - z * q^5 - q^8 - z * q^9 - z^2 * q^10 - z^2 * q^11 - q^13 - z * q^16 - z^2 * q^18 - z * q^19 + q^20 + q^22 - z * q^23 + z^2 * q^25 - z * q^26 - z^2 * q^32 + q^36 - z * q^37 - z^2 * q^38 + z * q^40 + q^41 + z * q^44 + z^2 * q^45 - z^2 * q^46 + z * q^47 - q^50 - z^2 * q^52 + z^2 * q^53 - q^55 - z^2 * q^59 + q^64 + z * q^65 + z * q^72 - z^2 * q^74 + q^76 + z^2 * q^80 + z^2 * q^81 + z * q^82 + z^2 * q^88 + z * q^89 - q^90 + q^92 + z^2 * q^94 + z^2 * q^95 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - q^{5} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 - q^4 - q^5 - 2 * q^8 - q^9 $$2 q + q^{2} - q^{4} - q^{5} - 2 q^{8} - q^{9} + q^{10} + q^{11} - 2 q^{13} - q^{16} + q^{18} - q^{19} + 2 q^{20} + 2 q^{22} - q^{23} - q^{25} - q^{26} + q^{32} + 2 q^{36} - q^{37} + q^{38} + q^{40} + 2 q^{41} + q^{44} - q^{45} + q^{46} + q^{47} - 2 q^{50} + q^{52} - q^{53} - 2 q^{55} + 2 q^{59} + 2 q^{64} + q^{65} + q^{72} + q^{74} + 2 q^{76} - q^{80} - q^{81} + q^{82} - q^{88} + 2 q^{89} - 2 q^{90} + 2 q^{92} - q^{94} - q^{95} - 2 q^{99}+O(q^{100})$$ 2 * q + q^2 - q^4 - q^5 - 2 * q^8 - q^9 + q^10 + q^11 - 2 * q^13 - q^16 + q^18 - q^19 + 2 * q^20 + 2 * q^22 - q^23 - q^25 - q^26 + q^32 + 2 * q^36 - q^37 + q^38 + q^40 + 2 * q^41 + q^44 - q^45 + q^46 + q^47 - 2 * q^50 + q^52 - q^53 - 2 * q^55 + 2 * q^59 + 2 * q^64 + q^65 + q^72 + q^74 + 2 * q^76 - q^80 - q^81 + q^82 - q^88 + 2 * q^89 - 2 * q^90 + 2 * q^92 - q^94 - q^95 - 2 * q^99

## Character values

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

 $$n$$ $$981$$ $$1081$$ $$1177$$ $$1471$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{6}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
459.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 0 −1.00000 −0.500000 0.866025i 0.500000 0.866025i
1059.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 0 −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
 $$n$$: 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 $$\Q(\sqrt{-10})$$
7.c even 3 1 inner
280.bi odd 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1960, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + T + 1$$
$23$ $$T^{2} + T + 1$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + T + 1$$
$41$ $$(T - 1)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} - T + 1$$
$53$ $$T^{2} + T + 1$$
$59$ $$T^{2} - 2T + 4$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 2T + 4$$
$97$ $$T^{2}$$