Properties

Label 2100.1.m.c.251.1
Level $2100$
Weight $1$
Character 2100.251
Self dual yes
Analytic conductor $1.048$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -20, -84, 105
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] = [2100,1,Mod(251,2100)] 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("2100.251"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2100.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.04803652653\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-5}, \sqrt{-21})\)
Artin image: $D_4$
Artin field: Galois closure of \(\Q(\sqrt{-20 +2 \sqrt{-5}})\)
Stark unit: Root of $x^{4} - 106244x^{3} + 192006x^{2} - 106244x + 1$

Embedding invariants

Embedding label 251.1
Character \(\chi\) \(=\) 2100.251

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{18} -1.00000 q^{21} -2.00000 q^{23} -1.00000 q^{24} -1.00000 q^{27} +1.00000 q^{28} +1.00000 q^{32} +1.00000 q^{36} +2.00000 q^{41} -1.00000 q^{42} -2.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{54} +1.00000 q^{56} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{69} +1.00000 q^{72} +1.00000 q^{81} +2.00000 q^{82} -1.00000 q^{84} -2.00000 q^{89} -2.00000 q^{92} -1.00000 q^{96} +1.00000 q^{98} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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
\(7\) 1.00000 1.00000
\(8\) 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 \(\pi\)
\(14\) 1.00000 1.00000
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 1.00000
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −1.00000 −1.00000
\(22\) 0 0
\(23\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(24\) −1.00000 −1.00000
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −1.00000
\(28\) 1.00000 1.00000
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(42\) −1.00000 −1.00000
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.00000 −2.00000
\(47\) 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
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −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 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 1.00000 1.00000
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 2.00000 2.00000
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 1.00000
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 2.00000 2.00000
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −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 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.00000 −2.00000
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 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 2100.1.m.c.251.1 1
3.2 odd 2 2100.1.m.a.251.1 1
4.3 odd 2 2100.1.m.b.251.1 1
5.2 odd 4 420.1.o.a.419.2 yes 2
5.3 odd 4 420.1.o.a.419.1 ✓ 2
5.4 even 2 2100.1.m.b.251.1 1
7.6 odd 2 2100.1.m.d.251.1 1
12.11 even 2 2100.1.m.d.251.1 1
15.2 even 4 420.1.o.b.419.1 yes 2
15.8 even 4 420.1.o.b.419.2 yes 2
15.14 odd 2 2100.1.m.d.251.1 1
20.3 even 4 420.1.o.a.419.2 yes 2
20.7 even 4 420.1.o.a.419.1 ✓ 2
20.19 odd 2 CM 2100.1.m.c.251.1 1
21.20 even 2 2100.1.m.b.251.1 1
28.27 even 2 2100.1.m.a.251.1 1
35.2 odd 12 2940.1.be.c.2579.2 4
35.3 even 12 2940.1.be.b.1979.2 4
35.12 even 12 2940.1.be.b.2579.2 4
35.13 even 4 420.1.o.b.419.1 yes 2
35.17 even 12 2940.1.be.b.1979.1 4
35.18 odd 12 2940.1.be.c.1979.2 4
35.23 odd 12 2940.1.be.c.2579.1 4
35.27 even 4 420.1.o.b.419.2 yes 2
35.32 odd 12 2940.1.be.c.1979.1 4
35.33 even 12 2940.1.be.b.2579.1 4
35.34 odd 2 2100.1.m.a.251.1 1
60.23 odd 4 420.1.o.b.419.1 yes 2
60.47 odd 4 420.1.o.b.419.2 yes 2
60.59 even 2 2100.1.m.a.251.1 1
84.83 odd 2 CM 2100.1.m.c.251.1 1
105.2 even 12 2940.1.be.b.2579.1 4
105.17 odd 12 2940.1.be.c.1979.2 4
105.23 even 12 2940.1.be.b.2579.2 4
105.32 even 12 2940.1.be.b.1979.2 4
105.38 odd 12 2940.1.be.c.1979.1 4
105.47 odd 12 2940.1.be.c.2579.1 4
105.53 even 12 2940.1.be.b.1979.1 4
105.62 odd 4 420.1.o.a.419.1 ✓ 2
105.68 odd 12 2940.1.be.c.2579.2 4
105.83 odd 4 420.1.o.a.419.2 yes 2
105.104 even 2 RM 2100.1.m.c.251.1 1
140.3 odd 12 2940.1.be.b.1979.1 4
140.23 even 12 2940.1.be.c.2579.2 4
140.27 odd 4 420.1.o.b.419.1 yes 2
140.47 odd 12 2940.1.be.b.2579.1 4
140.67 even 12 2940.1.be.c.1979.2 4
140.83 odd 4 420.1.o.b.419.2 yes 2
140.87 odd 12 2940.1.be.b.1979.2 4
140.103 odd 12 2940.1.be.b.2579.2 4
140.107 even 12 2940.1.be.c.2579.1 4
140.123 even 12 2940.1.be.c.1979.1 4
140.139 even 2 2100.1.m.d.251.1 1
420.23 odd 12 2940.1.be.b.2579.1 4
420.47 even 12 2940.1.be.c.2579.2 4
420.83 even 4 420.1.o.a.419.1 ✓ 2
420.107 odd 12 2940.1.be.b.2579.2 4
420.143 even 12 2940.1.be.c.1979.2 4
420.167 even 4 420.1.o.a.419.2 yes 2
420.227 even 12 2940.1.be.c.1979.1 4
420.263 odd 12 2940.1.be.b.1979.2 4
420.347 odd 12 2940.1.be.b.1979.1 4
420.383 even 12 2940.1.be.c.2579.1 4
420.419 odd 2 2100.1.m.b.251.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.1.o.a.419.1 ✓ 2 5.3 odd 4
420.1.o.a.419.1 ✓ 2 20.7 even 4
420.1.o.a.419.1 ✓ 2 105.62 odd 4
420.1.o.a.419.1 ✓ 2 420.83 even 4
420.1.o.a.419.2 yes 2 5.2 odd 4
420.1.o.a.419.2 yes 2 20.3 even 4
420.1.o.a.419.2 yes 2 105.83 odd 4
420.1.o.a.419.2 yes 2 420.167 even 4
420.1.o.b.419.1 yes 2 15.2 even 4
420.1.o.b.419.1 yes 2 35.13 even 4
420.1.o.b.419.1 yes 2 60.23 odd 4
420.1.o.b.419.1 yes 2 140.27 odd 4
420.1.o.b.419.2 yes 2 15.8 even 4
420.1.o.b.419.2 yes 2 35.27 even 4
420.1.o.b.419.2 yes 2 60.47 odd 4
420.1.o.b.419.2 yes 2 140.83 odd 4
2100.1.m.a.251.1 1 3.2 odd 2
2100.1.m.a.251.1 1 28.27 even 2
2100.1.m.a.251.1 1 35.34 odd 2
2100.1.m.a.251.1 1 60.59 even 2
2100.1.m.b.251.1 1 4.3 odd 2
2100.1.m.b.251.1 1 5.4 even 2
2100.1.m.b.251.1 1 21.20 even 2
2100.1.m.b.251.1 1 420.419 odd 2
2100.1.m.c.251.1 1 1.1 even 1 trivial
2100.1.m.c.251.1 1 20.19 odd 2 CM
2100.1.m.c.251.1 1 84.83 odd 2 CM
2100.1.m.c.251.1 1 105.104 even 2 RM
2100.1.m.d.251.1 1 7.6 odd 2
2100.1.m.d.251.1 1 12.11 even 2
2100.1.m.d.251.1 1 15.14 odd 2
2100.1.m.d.251.1 1 140.139 even 2
2940.1.be.b.1979.1 4 35.17 even 12
2940.1.be.b.1979.1 4 105.53 even 12
2940.1.be.b.1979.1 4 140.3 odd 12
2940.1.be.b.1979.1 4 420.347 odd 12
2940.1.be.b.1979.2 4 35.3 even 12
2940.1.be.b.1979.2 4 105.32 even 12
2940.1.be.b.1979.2 4 140.87 odd 12
2940.1.be.b.1979.2 4 420.263 odd 12
2940.1.be.b.2579.1 4 35.33 even 12
2940.1.be.b.2579.1 4 105.2 even 12
2940.1.be.b.2579.1 4 140.47 odd 12
2940.1.be.b.2579.1 4 420.23 odd 12
2940.1.be.b.2579.2 4 35.12 even 12
2940.1.be.b.2579.2 4 105.23 even 12
2940.1.be.b.2579.2 4 140.103 odd 12
2940.1.be.b.2579.2 4 420.107 odd 12
2940.1.be.c.1979.1 4 35.32 odd 12
2940.1.be.c.1979.1 4 105.38 odd 12
2940.1.be.c.1979.1 4 140.123 even 12
2940.1.be.c.1979.1 4 420.227 even 12
2940.1.be.c.1979.2 4 35.18 odd 12
2940.1.be.c.1979.2 4 105.17 odd 12
2940.1.be.c.1979.2 4 140.67 even 12
2940.1.be.c.1979.2 4 420.143 even 12
2940.1.be.c.2579.1 4 35.23 odd 12
2940.1.be.c.2579.1 4 105.47 odd 12
2940.1.be.c.2579.1 4 140.107 even 12
2940.1.be.c.2579.1 4 420.383 even 12
2940.1.be.c.2579.2 4 35.2 odd 12
2940.1.be.c.2579.2 4 105.68 odd 12
2940.1.be.c.2579.2 4 140.23 even 12
2940.1.be.c.2579.2 4 420.47 even 12