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Properties

Label 3800.1.o.a.1101.1
Level $3800$
Weight $1$
Character 3800.1101
Self dual yes
Analytic conductor $1.896$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -152
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] = [3800,1,Mod(1101,3800)] 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("3800.1101"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3800, 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: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,1,1,0,-1,1,-1,0,0,0,1,1,-1,0,1,1,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.152.1
Artin image: $D_6$
Artin field: Galois closure of 6.2.2888000.1
Stark unit: Root of $x^{6} - 52121x^{5} + 8430250x^{4} - 2699842385x^{3} + 8430250x^{2} - 52121x + 1$

Embedding invariants

Embedding label 1101.1
Character \(\chi\) \(=\) 3800.1101

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{19} +1.00000 q^{21} +1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -1.00000 q^{29} -1.00000 q^{32} -1.00000 q^{34} -2.00000 q^{37} -1.00000 q^{38} +1.00000 q^{39} -1.00000 q^{42} -1.00000 q^{46} -2.00000 q^{47} +1.00000 q^{48} +1.00000 q^{51} +1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} +1.00000 q^{57} +1.00000 q^{58} -1.00000 q^{59} +1.00000 q^{64} +1.00000 q^{67} +1.00000 q^{68} +1.00000 q^{69} +1.00000 q^{73} +2.00000 q^{74} +1.00000 q^{76} -1.00000 q^{78} -1.00000 q^{81} +1.00000 q^{84} -1.00000 q^{87} +1.00000 q^{91} +1.00000 q^{92} +2.00000 q^{94} -1.00000 q^{96} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 1.00000 1.00000
\(5\) 0 0
\(6\) −1.00000 −1.00000
\(7\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) −1.00000 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.00000 1.00000
\(13\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) −1.00000 −1.00000
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) 1.00000 1.00000
\(20\) 0 0
\(21\) 1.00000 1.00000
\(22\) 0 0
\(23\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) −1.00000 −1.00000
\(25\) 0 0
\(26\) −1.00000 −1.00000
\(27\) −1.00000 −1.00000
\(28\) 1.00000 1.00000
\(29\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.00000 −1.00000
\(33\) 0 0
\(34\) −1.00000 −1.00000
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(38\) −1.00000 −1.00000
\(39\) 1.00000 1.00000
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −1.00000 −1.00000
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −1.00000
\(47\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(48\) 1.00000 1.00000
\(49\) 0 0
\(50\) 0 0
\(51\) 1.00000 1.00000
\(52\) 1.00000 1.00000
\(53\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 1.00000 1.00000
\(55\) 0 0
\(56\) −1.00000 −1.00000
\(57\) 1.00000 1.00000
\(58\) 1.00000 1.00000
\(59\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 1.00000 1.00000
\(69\) 1.00000 1.00000
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 2.00000 2.00000
\(75\) 0 0
\(76\) 1.00000 1.00000
\(77\) 0 0
\(78\) −1.00000 −1.00000
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 1.00000 1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) −1.00000 −1.00000
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 1.00000 1.00000
\(92\) 1.00000 1.00000
\(93\) 0 0
\(94\) 2.00000 2.00000
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 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 3800.1.o.a.1101.1 1
5.2 odd 4 3800.1.b.b.949.1 2
5.3 odd 4 3800.1.b.b.949.2 2
5.4 even 2 152.1.g.b.37.1 yes 1
8.5 even 2 3800.1.o.b.1101.1 1
15.14 odd 2 1368.1.i.a.37.1 1
19.18 odd 2 3800.1.o.b.1101.1 1
20.19 odd 2 608.1.g.b.113.1 1
40.13 odd 4 3800.1.b.a.949.1 2
40.19 odd 2 608.1.g.a.113.1 1
40.29 even 2 152.1.g.a.37.1 ✓ 1
40.37 odd 4 3800.1.b.a.949.2 2
95.4 even 18 2888.1.s.a.1029.1 6
95.9 even 18 2888.1.s.a.1021.1 6
95.14 odd 18 2888.1.s.b.2293.1 6
95.18 even 4 3800.1.b.a.949.1 2
95.24 even 18 2888.1.s.a.2293.1 6
95.29 odd 18 2888.1.s.b.1021.1 6
95.34 odd 18 2888.1.s.b.1029.1 6
95.37 even 4 3800.1.b.a.949.2 2
95.44 even 18 2888.1.s.a.477.1 6
95.49 even 6 2888.1.l.a.69.1 2
95.54 even 18 2888.1.s.a.333.1 6
95.59 odd 18 2888.1.s.b.2789.1 6
95.64 even 6 2888.1.l.a.293.1 2
95.69 odd 6 2888.1.l.b.293.1 2
95.74 even 18 2888.1.s.a.2789.1 6
95.79 odd 18 2888.1.s.b.333.1 6
95.84 odd 6 2888.1.l.b.69.1 2
95.89 odd 18 2888.1.s.b.477.1 6
95.94 odd 2 152.1.g.a.37.1 ✓ 1
120.29 odd 2 1368.1.i.b.37.1 1
152.37 odd 2 CM 3800.1.o.a.1101.1 1
285.284 even 2 1368.1.i.b.37.1 1
380.379 even 2 608.1.g.a.113.1 1
760.29 odd 18 2888.1.s.a.1021.1 6
760.37 even 4 3800.1.b.b.949.1 2
760.69 odd 6 2888.1.l.a.293.1 2
760.109 odd 18 2888.1.s.a.2293.1 6
760.149 even 18 2888.1.s.b.333.1 6
760.189 odd 2 152.1.g.b.37.1 yes 1
760.269 odd 18 2888.1.s.a.333.1 6
760.309 even 18 2888.1.s.b.2293.1 6
760.349 even 6 2888.1.l.b.293.1 2
760.379 even 2 608.1.g.b.113.1 1
760.389 even 18 2888.1.s.b.1021.1 6
760.429 even 6 2888.1.l.b.69.1 2
760.469 odd 18 2888.1.s.a.477.1 6
760.493 even 4 3800.1.b.b.949.2 2
760.509 odd 18 2888.1.s.a.1029.1 6
760.549 even 18 2888.1.s.b.2789.1 6
760.629 odd 18 2888.1.s.a.2789.1 6
760.669 even 18 2888.1.s.b.1029.1 6
760.709 even 18 2888.1.s.b.477.1 6
760.749 odd 6 2888.1.l.a.69.1 2
2280.1709 even 2 1368.1.i.a.37.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.1.g.a.37.1 ✓ 1 40.29 even 2
152.1.g.a.37.1 ✓ 1 95.94 odd 2
152.1.g.b.37.1 yes 1 5.4 even 2
152.1.g.b.37.1 yes 1 760.189 odd 2
608.1.g.a.113.1 1 40.19 odd 2
608.1.g.a.113.1 1 380.379 even 2
608.1.g.b.113.1 1 20.19 odd 2
608.1.g.b.113.1 1 760.379 even 2
1368.1.i.a.37.1 1 15.14 odd 2
1368.1.i.a.37.1 1 2280.1709 even 2
1368.1.i.b.37.1 1 120.29 odd 2
1368.1.i.b.37.1 1 285.284 even 2
2888.1.l.a.69.1 2 95.49 even 6
2888.1.l.a.69.1 2 760.749 odd 6
2888.1.l.a.293.1 2 95.64 even 6
2888.1.l.a.293.1 2 760.69 odd 6
2888.1.l.b.69.1 2 95.84 odd 6
2888.1.l.b.69.1 2 760.429 even 6
2888.1.l.b.293.1 2 95.69 odd 6
2888.1.l.b.293.1 2 760.349 even 6
2888.1.s.a.333.1 6 95.54 even 18
2888.1.s.a.333.1 6 760.269 odd 18
2888.1.s.a.477.1 6 95.44 even 18
2888.1.s.a.477.1 6 760.469 odd 18
2888.1.s.a.1021.1 6 95.9 even 18
2888.1.s.a.1021.1 6 760.29 odd 18
2888.1.s.a.1029.1 6 95.4 even 18
2888.1.s.a.1029.1 6 760.509 odd 18
2888.1.s.a.2293.1 6 95.24 even 18
2888.1.s.a.2293.1 6 760.109 odd 18
2888.1.s.a.2789.1 6 95.74 even 18
2888.1.s.a.2789.1 6 760.629 odd 18
2888.1.s.b.333.1 6 95.79 odd 18
2888.1.s.b.333.1 6 760.149 even 18
2888.1.s.b.477.1 6 95.89 odd 18
2888.1.s.b.477.1 6 760.709 even 18
2888.1.s.b.1021.1 6 95.29 odd 18
2888.1.s.b.1021.1 6 760.389 even 18
2888.1.s.b.1029.1 6 95.34 odd 18
2888.1.s.b.1029.1 6 760.669 even 18
2888.1.s.b.2293.1 6 95.14 odd 18
2888.1.s.b.2293.1 6 760.309 even 18
2888.1.s.b.2789.1 6 95.59 odd 18
2888.1.s.b.2789.1 6 760.549 even 18
3800.1.b.a.949.1 2 40.13 odd 4
3800.1.b.a.949.1 2 95.18 even 4
3800.1.b.a.949.2 2 40.37 odd 4
3800.1.b.a.949.2 2 95.37 even 4
3800.1.b.b.949.1 2 5.2 odd 4
3800.1.b.b.949.1 2 760.37 even 4
3800.1.b.b.949.2 2 5.3 odd 4
3800.1.b.b.949.2 2 760.493 even 4
3800.1.o.a.1101.1 1 1.1 even 1 trivial
3800.1.o.a.1101.1 1 152.37 odd 2 CM
3800.1.o.b.1101.1 1 8.5 even 2
3800.1.o.b.1101.1 1 19.18 odd 2