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 $8$

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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:

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

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