Properties

Label 2200.1.d.b.901.1
Level $2200$
Weight $1$
Character 2200.901
Self dual yes
Analytic conductor $1.098$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -88
Inner twists $2$

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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] = [2200,1,Mod(901,2200)] 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("2200.901"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2200.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,0,1,0,0,0,-1,1,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.09794302779\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2200.1
Artin image: $D_6$
Artin field: Galois closure of 6.2.24200000.1
Stark unit: Root of $x^{6} - 486x^{5} + 15355x^{4} - 206460x^{3} + 15355x^{2} - 486x + 1$

Embedding invariants

Embedding label 901.1
Character \(\chi\) \(=\) 2200.901

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.00000 q^{16} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{22} +1.00000 q^{23} -1.00000 q^{26} -1.00000 q^{29} -1.00000 q^{31} -1.00000 q^{32} +1.00000 q^{36} +1.00000 q^{38} +1.00000 q^{43} +1.00000 q^{44} -1.00000 q^{46} -2.00000 q^{47} +1.00000 q^{49} +1.00000 q^{52} +1.00000 q^{58} +2.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} -1.00000 q^{71} -1.00000 q^{72} -1.00000 q^{76} +1.00000 q^{81} +1.00000 q^{83} -1.00000 q^{86} -1.00000 q^{88} -1.00000 q^{89} +1.00000 q^{92} +2.00000 q^{94} +1.00000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(551\) \(1101\) \(1201\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.00000 −1.00000
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 1.00000 1.00000
\(12\) 0 0
\(13\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −1.00000 −1.00000
\(19\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 −1.00000
\(23\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −1.00000
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) −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 \(\pi\)
\(38\) 1.00000 1.00000
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 1.00000 1.00000
\(45\) 0 0
\(46\) −1.00000 −1.00000
\(47\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000 1.00000
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 1.00000 1.00000
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(62\) 1.00000 1.00000
\(63\) 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
\(71\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(72\) −1.00000 −1.00000
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.00000 −1.00000
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 −1.00000
\(87\) 0 0
\(88\) −1.00000 −1.00000
\(89\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 1.00000
\(93\) 0 0
\(94\) 2.00000 2.00000
\(95\) 0 0
\(96\) 0 0
\(97\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) −1.00000 −1.00000
\(99\) 1.00000 1.00000
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 2200.1.d.b.901.1 yes 1
5.2 odd 4 2200.1.o.b.549.1 2
5.3 odd 4 2200.1.o.b.549.2 2
5.4 even 2 2200.1.d.d.901.1 yes 1
8.5 even 2 2200.1.d.c.901.1 yes 1
11.10 odd 2 2200.1.d.c.901.1 yes 1
40.13 odd 4 2200.1.o.a.549.1 2
40.29 even 2 2200.1.d.a.901.1 ✓ 1
40.37 odd 4 2200.1.o.a.549.2 2
55.32 even 4 2200.1.o.a.549.2 2
55.43 even 4 2200.1.o.a.549.1 2
55.54 odd 2 2200.1.d.a.901.1 ✓ 1
88.21 odd 2 CM 2200.1.d.b.901.1 yes 1
440.109 odd 2 2200.1.d.d.901.1 yes 1
440.197 even 4 2200.1.o.b.549.1 2
440.373 even 4 2200.1.o.b.549.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2200.1.d.a.901.1 ✓ 1 40.29 even 2
2200.1.d.a.901.1 ✓ 1 55.54 odd 2
2200.1.d.b.901.1 yes 1 1.1 even 1 trivial
2200.1.d.b.901.1 yes 1 88.21 odd 2 CM
2200.1.d.c.901.1 yes 1 8.5 even 2
2200.1.d.c.901.1 yes 1 11.10 odd 2
2200.1.d.d.901.1 yes 1 5.4 even 2
2200.1.d.d.901.1 yes 1 440.109 odd 2
2200.1.o.a.549.1 2 40.13 odd 4
2200.1.o.a.549.1 2 55.43 even 4
2200.1.o.a.549.2 2 40.37 odd 4
2200.1.o.a.549.2 2 55.32 even 4
2200.1.o.b.549.1 2 5.2 odd 4
2200.1.o.b.549.1 2 440.197 even 4
2200.1.o.b.549.2 2 5.3 odd 4
2200.1.o.b.549.2 2 440.373 even 4