# Properties

 Label 420.1.o.b Level $420$ Weight $1$ Character orbit 420.o Analytic conductor $0.210$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -20, -84, 105 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 420.o (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.209607305306$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-5}, \sqrt{-21})$$ Artin image: $D_4:C_2$ Artin field: Galois closure of 8.0.70560000.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + i q^{2} -i q^{3} - q^{4} + q^{5} + q^{6} -i q^{7} -i q^{8} - q^{9} +O(q^{10})$$ $$q + i q^{2} -i q^{3} - q^{4} + q^{5} + q^{6} -i q^{7} -i q^{8} - q^{9} + i q^{10} + i q^{12} + q^{14} -i q^{15} + q^{16} -i q^{18} - q^{20} - q^{21} + 2 i q^{23} - q^{24} + q^{25} + i q^{27} + i q^{28} + q^{30} + i q^{32} -i q^{35} + q^{36} -i q^{40} -2 q^{41} -i q^{42} - q^{45} -2 q^{46} -i q^{48} - q^{49} + i q^{50} - q^{54} - q^{56} + i q^{60} + i q^{63} - q^{64} + 2 q^{69} + q^{70} + i q^{72} -i q^{75} + q^{80} + q^{81} -2 i q^{82} + q^{84} -2 q^{89} -i q^{90} -2 i q^{92} + q^{96} -i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9} + 2 q^{14} + 2 q^{16} - 2 q^{20} - 2 q^{21} - 2 q^{24} + 2 q^{25} + 2 q^{30} + 2 q^{36} - 4 q^{41} - 2 q^{45} - 4 q^{46} - 2 q^{49} - 2 q^{54} - 2 q^{56} - 2 q^{64} + 4 q^{69} + 2 q^{70} + 2 q^{80} + 2 q^{81} + 2 q^{84} - 4 q^{89} + 2 q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$241$$ $$281$$ $$337$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
419.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 1.00000 1.00000 1.00000i 1.00000i −1.00000 1.00000i
419.2 1.00000i 1.00000i −1.00000 1.00000 1.00000 1.00000i 1.00000i −1.00000 1.00000i
 $$n$$: 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
84.h odd 2 1 CM by $$\Q(\sqrt{-21})$$
105.g even 2 1 RM by $$\Q(\sqrt{105})$$
4.b odd 2 1 inner
5.b even 2 1 inner
21.c even 2 1 inner
420.o odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.1.o.b yes 2
3.b odd 2 1 420.1.o.a 2
4.b odd 2 1 inner 420.1.o.b yes 2
5.b even 2 1 inner 420.1.o.b yes 2
5.c odd 4 1 2100.1.m.a 1
5.c odd 4 1 2100.1.m.d 1
7.b odd 2 1 420.1.o.a 2
7.c even 3 2 2940.1.be.b 4
7.d odd 6 2 2940.1.be.c 4
12.b even 2 1 420.1.o.a 2
15.d odd 2 1 420.1.o.a 2
15.e even 4 1 2100.1.m.b 1
15.e even 4 1 2100.1.m.c 1
20.d odd 2 1 CM 420.1.o.b yes 2
20.e even 4 1 2100.1.m.a 1
20.e even 4 1 2100.1.m.d 1
21.c even 2 1 inner 420.1.o.b yes 2
21.g even 6 2 2940.1.be.b 4
21.h odd 6 2 2940.1.be.c 4
28.d even 2 1 420.1.o.a 2
28.f even 6 2 2940.1.be.c 4
28.g odd 6 2 2940.1.be.b 4
35.c odd 2 1 420.1.o.a 2
35.f even 4 1 2100.1.m.b 1
35.f even 4 1 2100.1.m.c 1
35.i odd 6 2 2940.1.be.c 4
35.j even 6 2 2940.1.be.b 4
60.h even 2 1 420.1.o.a 2
60.l odd 4 1 2100.1.m.b 1
60.l odd 4 1 2100.1.m.c 1
84.h odd 2 1 CM 420.1.o.b yes 2
84.j odd 6 2 2940.1.be.b 4
84.n even 6 2 2940.1.be.c 4
105.g even 2 1 RM 420.1.o.b yes 2
105.k odd 4 1 2100.1.m.a 1
105.k odd 4 1 2100.1.m.d 1
105.o odd 6 2 2940.1.be.c 4
105.p even 6 2 2940.1.be.b 4
140.c even 2 1 420.1.o.a 2
140.j odd 4 1 2100.1.m.b 1
140.j odd 4 1 2100.1.m.c 1
140.p odd 6 2 2940.1.be.b 4
140.s even 6 2 2940.1.be.c 4
420.o odd 2 1 inner 420.1.o.b yes 2
420.w even 4 1 2100.1.m.a 1
420.w even 4 1 2100.1.m.d 1
420.ba even 6 2 2940.1.be.c 4
420.be odd 6 2 2940.1.be.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.1.o.a 2 3.b odd 2 1
420.1.o.a 2 7.b odd 2 1
420.1.o.a 2 12.b even 2 1
420.1.o.a 2 15.d odd 2 1
420.1.o.a 2 28.d even 2 1
420.1.o.a 2 35.c odd 2 1
420.1.o.a 2 60.h even 2 1
420.1.o.a 2 140.c even 2 1
420.1.o.b yes 2 1.a even 1 1 trivial
420.1.o.b yes 2 4.b odd 2 1 inner
420.1.o.b yes 2 5.b even 2 1 inner
420.1.o.b yes 2 20.d odd 2 1 CM
420.1.o.b yes 2 21.c even 2 1 inner
420.1.o.b yes 2 84.h odd 2 1 CM
420.1.o.b yes 2 105.g even 2 1 RM
420.1.o.b yes 2 420.o odd 2 1 inner
2100.1.m.a 1 5.c odd 4 1
2100.1.m.a 1 20.e even 4 1
2100.1.m.a 1 105.k odd 4 1
2100.1.m.a 1 420.w even 4 1
2100.1.m.b 1 15.e even 4 1
2100.1.m.b 1 35.f even 4 1
2100.1.m.b 1 60.l odd 4 1
2100.1.m.b 1 140.j odd 4 1
2100.1.m.c 1 15.e even 4 1
2100.1.m.c 1 35.f even 4 1
2100.1.m.c 1 60.l odd 4 1
2100.1.m.c 1 140.j odd 4 1
2100.1.m.d 1 5.c odd 4 1
2100.1.m.d 1 20.e even 4 1
2100.1.m.d 1 105.k odd 4 1
2100.1.m.d 1 420.w even 4 1
2940.1.be.b 4 7.c even 3 2
2940.1.be.b 4 21.g even 6 2
2940.1.be.b 4 28.g odd 6 2
2940.1.be.b 4 35.j even 6 2
2940.1.be.b 4 84.j odd 6 2
2940.1.be.b 4 105.p even 6 2
2940.1.be.b 4 140.p odd 6 2
2940.1.be.b 4 420.be odd 6 2
2940.1.be.c 4 7.d odd 6 2
2940.1.be.c 4 21.h odd 6 2
2940.1.be.c 4 28.f even 6 2
2940.1.be.c 4 35.i odd 6 2
2940.1.be.c 4 84.n even 6 2
2940.1.be.c 4 105.o odd 6 2
2940.1.be.c 4 140.s even 6 2
2940.1.be.c 4 420.ba even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{41} + 2$$ acting on $$S_{1}^{\mathrm{new}}(420, [\chi])$$.

## Hecke characteristic polynomials

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