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Properties

Label 600.1.n.a.101.1
Level $600$
Weight $1$
Character 600.101
Self dual yes
Analytic conductor $0.299$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -15, -24, 40
Inner twists $4$

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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] = [600,1,Mod(101,600)] 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("600.101"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 0])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 600.n (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.299439007580\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-6}, \sqrt{10})\)
Artin image: $D_4$
Artin field: Galois closure of \(\Q(\sqrt{10 -6 \sqrt{-15}})\)
Stark unit: Root of $x^{4} - 34x^{3} - 69x^{2} - 34x + 1$

Embedding invariants

Embedding label 101.1
Character \(\chi\) \(=\) 600.101

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +1.00000 q^{16} -1.00000 q^{18} -1.00000 q^{24} +1.00000 q^{27} -2.00000 q^{31} -1.00000 q^{32} +1.00000 q^{36} +1.00000 q^{48} -1.00000 q^{49} +2.00000 q^{53} -1.00000 q^{54} +2.00000 q^{62} +1.00000 q^{64} -1.00000 q^{72} -2.00000 q^{79} +1.00000 q^{81} -2.00000 q^{83} -2.00000 q^{93} -1.00000 q^{96} +1.00000 q^{98} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −1.00000
\(3\) 1.00000 1.00000
\(4\) 1.00000 1.00000
\(5\) 0 0
\(6\) −1.00000 −1.00000
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −1.00000
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 1.00000
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −1.00000 −1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −1.00000 −1.00000
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(32\) −1.00000 −1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000 1.00000
\(49\) −1.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(54\) −1.00000 −1.00000
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 2.00000 2.00000
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −1.00000 −1.00000
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.00000 −2.00000
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 1.00000 1.00000
\(99\) 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 600.1.n.a.101.1 1
3.2 odd 2 600.1.n.b.101.1 1
4.3 odd 2 2400.1.n.a.401.1 1
5.2 odd 4 120.1.i.a.29.1 ✓ 2
5.3 odd 4 120.1.i.a.29.2 yes 2
5.4 even 2 600.1.n.b.101.1 1
8.3 odd 2 2400.1.n.b.401.1 1
8.5 even 2 600.1.n.b.101.1 1
12.11 even 2 2400.1.n.b.401.1 1
15.2 even 4 120.1.i.a.29.2 yes 2
15.8 even 4 120.1.i.a.29.1 ✓ 2
15.14 odd 2 CM 600.1.n.a.101.1 1
20.3 even 4 480.1.i.a.209.1 2
20.7 even 4 480.1.i.a.209.2 2
20.19 odd 2 2400.1.n.b.401.1 1
24.5 odd 2 CM 600.1.n.a.101.1 1
24.11 even 2 2400.1.n.a.401.1 1
40.3 even 4 480.1.i.a.209.2 2
40.13 odd 4 120.1.i.a.29.1 ✓ 2
40.19 odd 2 2400.1.n.a.401.1 1
40.27 even 4 480.1.i.a.209.1 2
40.29 even 2 RM 600.1.n.a.101.1 1
40.37 odd 4 120.1.i.a.29.2 yes 2
45.2 even 12 3240.1.bh.h.1349.2 4
45.7 odd 12 3240.1.bh.h.1349.1 4
45.13 odd 12 3240.1.bh.h.269.1 4
45.22 odd 12 3240.1.bh.h.269.2 4
45.23 even 12 3240.1.bh.h.269.2 4
45.32 even 12 3240.1.bh.h.269.1 4
45.38 even 12 3240.1.bh.h.1349.1 4
45.43 odd 12 3240.1.bh.h.1349.2 4
60.23 odd 4 480.1.i.a.209.2 2
60.47 odd 4 480.1.i.a.209.1 2
60.59 even 2 2400.1.n.a.401.1 1
80.3 even 4 3840.1.c.c.3329.1 1
80.13 odd 4 3840.1.c.a.3329.1 1
80.27 even 4 3840.1.c.c.3329.1 1
80.37 odd 4 3840.1.c.a.3329.1 1
80.43 even 4 3840.1.c.b.3329.1 1
80.53 odd 4 3840.1.c.d.3329.1 1
80.67 even 4 3840.1.c.b.3329.1 1
80.77 odd 4 3840.1.c.d.3329.1 1
120.29 odd 2 600.1.n.b.101.1 1
120.53 even 4 120.1.i.a.29.2 yes 2
120.59 even 2 2400.1.n.b.401.1 1
120.77 even 4 120.1.i.a.29.1 ✓ 2
120.83 odd 4 480.1.i.a.209.1 2
120.107 odd 4 480.1.i.a.209.2 2
240.53 even 4 3840.1.c.a.3329.1 1
240.77 even 4 3840.1.c.a.3329.1 1
240.83 odd 4 3840.1.c.b.3329.1 1
240.107 odd 4 3840.1.c.b.3329.1 1
240.173 even 4 3840.1.c.d.3329.1 1
240.197 even 4 3840.1.c.d.3329.1 1
240.203 odd 4 3840.1.c.c.3329.1 1
240.227 odd 4 3840.1.c.c.3329.1 1
360.13 odd 12 3240.1.bh.h.269.2 4
360.77 even 12 3240.1.bh.h.269.2 4
360.133 odd 12 3240.1.bh.h.1349.1 4
360.157 odd 12 3240.1.bh.h.269.1 4
360.173 even 12 3240.1.bh.h.1349.2 4
360.277 odd 12 3240.1.bh.h.1349.2 4
360.293 even 12 3240.1.bh.h.269.1 4
360.317 even 12 3240.1.bh.h.1349.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.1.i.a.29.1 ✓ 2 5.2 odd 4
120.1.i.a.29.1 ✓ 2 15.8 even 4
120.1.i.a.29.1 ✓ 2 40.13 odd 4
120.1.i.a.29.1 ✓ 2 120.77 even 4
120.1.i.a.29.2 yes 2 5.3 odd 4
120.1.i.a.29.2 yes 2 15.2 even 4
120.1.i.a.29.2 yes 2 40.37 odd 4
120.1.i.a.29.2 yes 2 120.53 even 4
480.1.i.a.209.1 2 20.3 even 4
480.1.i.a.209.1 2 40.27 even 4
480.1.i.a.209.1 2 60.47 odd 4
480.1.i.a.209.1 2 120.83 odd 4
480.1.i.a.209.2 2 20.7 even 4
480.1.i.a.209.2 2 40.3 even 4
480.1.i.a.209.2 2 60.23 odd 4
480.1.i.a.209.2 2 120.107 odd 4
600.1.n.a.101.1 1 1.1 even 1 trivial
600.1.n.a.101.1 1 15.14 odd 2 CM
600.1.n.a.101.1 1 24.5 odd 2 CM
600.1.n.a.101.1 1 40.29 even 2 RM
600.1.n.b.101.1 1 3.2 odd 2
600.1.n.b.101.1 1 5.4 even 2
600.1.n.b.101.1 1 8.5 even 2
600.1.n.b.101.1 1 120.29 odd 2
2400.1.n.a.401.1 1 4.3 odd 2
2400.1.n.a.401.1 1 24.11 even 2
2400.1.n.a.401.1 1 40.19 odd 2
2400.1.n.a.401.1 1 60.59 even 2
2400.1.n.b.401.1 1 8.3 odd 2
2400.1.n.b.401.1 1 12.11 even 2
2400.1.n.b.401.1 1 20.19 odd 2
2400.1.n.b.401.1 1 120.59 even 2
3240.1.bh.h.269.1 4 45.13 odd 12
3240.1.bh.h.269.1 4 45.32 even 12
3240.1.bh.h.269.1 4 360.157 odd 12
3240.1.bh.h.269.1 4 360.293 even 12
3240.1.bh.h.269.2 4 45.22 odd 12
3240.1.bh.h.269.2 4 45.23 even 12
3240.1.bh.h.269.2 4 360.13 odd 12
3240.1.bh.h.269.2 4 360.77 even 12
3240.1.bh.h.1349.1 4 45.7 odd 12
3240.1.bh.h.1349.1 4 45.38 even 12
3240.1.bh.h.1349.1 4 360.133 odd 12
3240.1.bh.h.1349.1 4 360.317 even 12
3240.1.bh.h.1349.2 4 45.2 even 12
3240.1.bh.h.1349.2 4 45.43 odd 12
3240.1.bh.h.1349.2 4 360.173 even 12
3240.1.bh.h.1349.2 4 360.277 odd 12
3840.1.c.a.3329.1 1 80.13 odd 4
3840.1.c.a.3329.1 1 80.37 odd 4
3840.1.c.a.3329.1 1 240.53 even 4
3840.1.c.a.3329.1 1 240.77 even 4
3840.1.c.b.3329.1 1 80.43 even 4
3840.1.c.b.3329.1 1 80.67 even 4
3840.1.c.b.3329.1 1 240.83 odd 4
3840.1.c.b.3329.1 1 240.107 odd 4
3840.1.c.c.3329.1 1 80.3 even 4
3840.1.c.c.3329.1 1 80.27 even 4
3840.1.c.c.3329.1 1 240.203 odd 4
3840.1.c.c.3329.1 1 240.227 odd 4
3840.1.c.d.3329.1 1 80.53 odd 4
3840.1.c.d.3329.1 1 80.77 odd 4
3840.1.c.d.3329.1 1 240.173 even 4
3840.1.c.d.3329.1 1 240.197 even 4