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Properties

Label 420.1.o.b.419.2
Level $420$
Weight $1$
Character 420.419
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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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [420,1,Mod(419,420)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("420.419"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
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

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.209607305306\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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

Embedding invariants

Embedding label 419.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 420.419
Dual form 420.1.o.b.419.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{10} +1.00000i q^{12} +1.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -1.00000i q^{18} -1.00000 q^{20} -1.00000 q^{21} +2.00000i q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000i q^{27} +1.00000i q^{28} +1.00000 q^{30} +1.00000i q^{32} -1.00000i q^{35} +1.00000 q^{36} -1.00000i q^{40} -2.00000 q^{41} -1.00000i q^{42} -1.00000 q^{45} -2.00000 q^{46} -1.00000i q^{48} -1.00000 q^{49} +1.00000i q^{50} -1.00000 q^{54} -1.00000 q^{56} +1.00000i q^{60} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{69} +1.00000 q^{70} +1.00000i q^{72} -1.00000i q^{75} +1.00000 q^{80} +1.00000 q^{81} -2.00000i q^{82} +1.00000 q^{84} -2.00000 q^{89} -1.00000i q^{90} -2.00000i q^{92} +1.00000 q^{96} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9} + 2 q^{14} + 2 q^{16} - 2 q^{20} - 2 q^{21} - 2 q^{24} + 2 q^{25} + 2 q^{30} + 2 q^{36} - 4 q^{41} - 2 q^{45} - 4 q^{46} - 2 q^{49} - 2 q^{54} - 2 q^{56} - 2 q^{64}+ \cdots + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 1.00000i
\(3\) − 1.00000i − 1.00000i
\(4\) −1.00000 −1.00000
\(5\) 1.00000 1.00000
\(6\) 1.00000 1.00000
\(7\) − 1.00000i − 1.00000i
\(8\) − 1.00000i − 1.00000i
\(9\) −1.00000 −1.00000
\(10\) 1.00000i 1.00000i
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000i 1.00000i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 1.00000 1.00000
\(15\) − 1.00000i − 1.00000i
\(16\) 1.00000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) − 1.00000i − 1.00000i
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −1.00000
\(21\) −1.00000 −1.00000
\(22\) 0 0
\(23\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −1.00000
\(25\) 1.00000 1.00000
\(26\) 0 0
\(27\) 1.00000i 1.00000i
\(28\) 1.00000i 1.00000i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 1.00000 1.00000
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000i 1.00000i
\(33\) 0 0
\(34\) 0 0
\(35\) − 1.00000i − 1.00000i
\(36\) 1.00000 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) − 1.00000i − 1.00000i
\(41\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(42\) − 1.00000i − 1.00000i
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −1.00000 −1.00000
\(46\) −2.00000 −2.00000
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) − 1.00000i − 1.00000i
\(49\) −1.00000 −1.00000
\(50\) 1.00000i 1.00000i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −1.00000 −1.00000
\(55\) 0 0
\(56\) −1.00000 −1.00000
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 1.00000i 1.00000i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 1.00000i 1.00000i
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 2.00000 2.00000
\(70\) 1.00000 1.00000
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000i 1.00000i
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) − 1.00000i − 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.00000 1.00000
\(81\) 1.00000 1.00000
\(82\) − 2.00000i − 2.00000i
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 1.00000 1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(90\) − 1.00000i − 1.00000i
\(91\) 0 0
\(92\) − 2.00000i − 2.00000i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 1.00000
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) − 1.00000i − 1.00000i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 420.1.o.b.419.2 yes 2
3.2 odd 2 420.1.o.a.419.1 ✓ 2
4.3 odd 2 inner 420.1.o.b.419.1 yes 2
5.2 odd 4 2100.1.m.a.251.1 1
5.3 odd 4 2100.1.m.d.251.1 1
5.4 even 2 inner 420.1.o.b.419.1 yes 2
7.2 even 3 2940.1.be.b.2579.2 4
7.3 odd 6 2940.1.be.c.1979.1 4
7.4 even 3 2940.1.be.b.1979.1 4
7.5 odd 6 2940.1.be.c.2579.2 4
7.6 odd 2 420.1.o.a.419.2 yes 2
12.11 even 2 420.1.o.a.419.2 yes 2
15.2 even 4 2100.1.m.c.251.1 1
15.8 even 4 2100.1.m.b.251.1 1
15.14 odd 2 420.1.o.a.419.2 yes 2
20.3 even 4 2100.1.m.a.251.1 1
20.7 even 4 2100.1.m.d.251.1 1
20.19 odd 2 CM 420.1.o.b.419.2 yes 2
21.2 odd 6 2940.1.be.c.2579.1 4
21.5 even 6 2940.1.be.b.2579.1 4
21.11 odd 6 2940.1.be.c.1979.2 4
21.17 even 6 2940.1.be.b.1979.2 4
21.20 even 2 inner 420.1.o.b.419.1 yes 2
28.3 even 6 2940.1.be.c.1979.2 4
28.11 odd 6 2940.1.be.b.1979.2 4
28.19 even 6 2940.1.be.c.2579.1 4
28.23 odd 6 2940.1.be.b.2579.1 4
28.27 even 2 420.1.o.a.419.1 ✓ 2
35.4 even 6 2940.1.be.b.1979.2 4
35.9 even 6 2940.1.be.b.2579.1 4
35.13 even 4 2100.1.m.c.251.1 1
35.19 odd 6 2940.1.be.c.2579.1 4
35.24 odd 6 2940.1.be.c.1979.2 4
35.27 even 4 2100.1.m.b.251.1 1
35.34 odd 2 420.1.o.a.419.1 ✓ 2
60.23 odd 4 2100.1.m.c.251.1 1
60.47 odd 4 2100.1.m.b.251.1 1
60.59 even 2 420.1.o.a.419.1 ✓ 2
84.11 even 6 2940.1.be.c.1979.1 4
84.23 even 6 2940.1.be.c.2579.2 4
84.47 odd 6 2940.1.be.b.2579.2 4
84.59 odd 6 2940.1.be.b.1979.1 4
84.83 odd 2 CM 420.1.o.b.419.2 yes 2
105.44 odd 6 2940.1.be.c.2579.2 4
105.59 even 6 2940.1.be.b.1979.1 4
105.62 odd 4 2100.1.m.d.251.1 1
105.74 odd 6 2940.1.be.c.1979.1 4
105.83 odd 4 2100.1.m.a.251.1 1
105.89 even 6 2940.1.be.b.2579.2 4
105.104 even 2 RM 420.1.o.b.419.2 yes 2
140.19 even 6 2940.1.be.c.2579.2 4
140.27 odd 4 2100.1.m.c.251.1 1
140.39 odd 6 2940.1.be.b.1979.1 4
140.59 even 6 2940.1.be.c.1979.1 4
140.79 odd 6 2940.1.be.b.2579.2 4
140.83 odd 4 2100.1.m.b.251.1 1
140.139 even 2 420.1.o.a.419.2 yes 2
420.59 odd 6 2940.1.be.b.1979.2 4
420.83 even 4 2100.1.m.d.251.1 1
420.167 even 4 2100.1.m.a.251.1 1
420.179 even 6 2940.1.be.c.1979.2 4
420.299 odd 6 2940.1.be.b.2579.1 4
420.359 even 6 2940.1.be.c.2579.1 4
420.419 odd 2 inner 420.1.o.b.419.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.1.o.a.419.1 ✓ 2 3.2 odd 2
420.1.o.a.419.1 ✓ 2 28.27 even 2
420.1.o.a.419.1 ✓ 2 35.34 odd 2
420.1.o.a.419.1 ✓ 2 60.59 even 2
420.1.o.a.419.2 yes 2 7.6 odd 2
420.1.o.a.419.2 yes 2 12.11 even 2
420.1.o.a.419.2 yes 2 15.14 odd 2
420.1.o.a.419.2 yes 2 140.139 even 2
420.1.o.b.419.1 yes 2 4.3 odd 2 inner
420.1.o.b.419.1 yes 2 5.4 even 2 inner
420.1.o.b.419.1 yes 2 21.20 even 2 inner
420.1.o.b.419.1 yes 2 420.419 odd 2 inner
420.1.o.b.419.2 yes 2 1.1 even 1 trivial
420.1.o.b.419.2 yes 2 20.19 odd 2 CM
420.1.o.b.419.2 yes 2 84.83 odd 2 CM
420.1.o.b.419.2 yes 2 105.104 even 2 RM
2100.1.m.a.251.1 1 5.2 odd 4
2100.1.m.a.251.1 1 20.3 even 4
2100.1.m.a.251.1 1 105.83 odd 4
2100.1.m.a.251.1 1 420.167 even 4
2100.1.m.b.251.1 1 15.8 even 4
2100.1.m.b.251.1 1 35.27 even 4
2100.1.m.b.251.1 1 60.47 odd 4
2100.1.m.b.251.1 1 140.83 odd 4
2100.1.m.c.251.1 1 15.2 even 4
2100.1.m.c.251.1 1 35.13 even 4
2100.1.m.c.251.1 1 60.23 odd 4
2100.1.m.c.251.1 1 140.27 odd 4
2100.1.m.d.251.1 1 5.3 odd 4
2100.1.m.d.251.1 1 20.7 even 4
2100.1.m.d.251.1 1 105.62 odd 4
2100.1.m.d.251.1 1 420.83 even 4
2940.1.be.b.1979.1 4 7.4 even 3
2940.1.be.b.1979.1 4 84.59 odd 6
2940.1.be.b.1979.1 4 105.59 even 6
2940.1.be.b.1979.1 4 140.39 odd 6
2940.1.be.b.1979.2 4 21.17 even 6
2940.1.be.b.1979.2 4 28.11 odd 6
2940.1.be.b.1979.2 4 35.4 even 6
2940.1.be.b.1979.2 4 420.59 odd 6
2940.1.be.b.2579.1 4 21.5 even 6
2940.1.be.b.2579.1 4 28.23 odd 6
2940.1.be.b.2579.1 4 35.9 even 6
2940.1.be.b.2579.1 4 420.299 odd 6
2940.1.be.b.2579.2 4 7.2 even 3
2940.1.be.b.2579.2 4 84.47 odd 6
2940.1.be.b.2579.2 4 105.89 even 6
2940.1.be.b.2579.2 4 140.79 odd 6
2940.1.be.c.1979.1 4 7.3 odd 6
2940.1.be.c.1979.1 4 84.11 even 6
2940.1.be.c.1979.1 4 105.74 odd 6
2940.1.be.c.1979.1 4 140.59 even 6
2940.1.be.c.1979.2 4 21.11 odd 6
2940.1.be.c.1979.2 4 28.3 even 6
2940.1.be.c.1979.2 4 35.24 odd 6
2940.1.be.c.1979.2 4 420.179 even 6
2940.1.be.c.2579.1 4 21.2 odd 6
2940.1.be.c.2579.1 4 28.19 even 6
2940.1.be.c.2579.1 4 35.19 odd 6
2940.1.be.c.2579.1 4 420.359 even 6
2940.1.be.c.2579.2 4 7.5 odd 6
2940.1.be.c.2579.2 4 84.23 even 6
2940.1.be.c.2579.2 4 105.44 odd 6
2940.1.be.c.2579.2 4 140.19 even 6