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Properties

Label 3800.1.b.a.949.1
Level $3800$
Weight $1$
Character 3800.949
Analytic conductor $1.896$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -152
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] = [3800,1,Mod(949,3800)] 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("3800.949"); 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, 1, 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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,0,-2,0,0,0,0,0,0,0,-2,0,2,0,0,2] 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: no
Analytic conductor: \(1.89644704801\)
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: no (minimal twist has level 152)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.152.1
Artin image: $C_4\times D_6$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

Embedding invariants

Embedding label 949.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3800.949
Dual form 3800.1.b.a.949.2

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

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 1.00000i
\(3\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(4\) −1.00000 −1.00000
\(5\) 0 0
\(6\) −1.00000 −1.00000
\(7\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(8\) 1.00000i 1.00000i
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.00000i 1.00000i
\(13\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(14\) −1.00000 −1.00000
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(18\) 0 0
\(19\) 1.00000 1.00000
\(20\) 0 0
\(21\) −1.00000 −1.00000
\(22\) 0 0
\(23\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 1.00000 1.00000
\(25\) 0 0
\(26\) −1.00000 −1.00000
\(27\) − 1.00000i − 1.00000i
\(28\) 1.00000i 1.00000i
\(29\) −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 \(\pi\)
\(32\) − 1.00000i − 1.00000i
\(33\) 0 0
\(34\) −1.00000 −1.00000
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(38\) − 1.00000i − 1.00000i
\(39\) −1.00000 −1.00000
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 1.00000i 1.00000i
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.00000 1.00000
\(47\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) − 1.00000i − 1.00000i
\(49\) 0 0
\(50\) 0 0
\(51\) −1.00000 −1.00000
\(52\) 1.00000i 1.00000i
\(53\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(54\) −1.00000 −1.00000
\(55\) 0 0
\(56\) 1.00000 1.00000
\(57\) − 1.00000i − 1.00000i
\(58\) 1.00000i 1.00000i
\(59\) −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
\(67\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) 1.00000i 1.00000i
\(69\) 1.00000 1.00000
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) −2.00000 −2.00000
\(75\) 0 0
\(76\) −1.00000 −1.00000
\(77\) 0 0
\(78\) 1.00000i 1.00000i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 1.00000 1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000i 1.00000i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −1.00000 −1.00000
\(92\) − 1.00000i − 1.00000i
\(93\) 0 0
\(94\) 2.00000 2.00000
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 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.b.a.949.1 2
5.2 odd 4 3800.1.o.b.1101.1 1
5.3 odd 4 152.1.g.a.37.1 ✓ 1
5.4 even 2 inner 3800.1.b.a.949.2 2
8.5 even 2 3800.1.b.b.949.2 2
15.8 even 4 1368.1.i.b.37.1 1
19.18 odd 2 3800.1.b.b.949.2 2
20.3 even 4 608.1.g.a.113.1 1
40.3 even 4 608.1.g.b.113.1 1
40.13 odd 4 152.1.g.b.37.1 yes 1
40.29 even 2 3800.1.b.b.949.1 2
40.37 odd 4 3800.1.o.a.1101.1 1
95.3 even 36 2888.1.s.a.333.1 6
95.8 even 12 2888.1.l.a.69.1 2
95.13 even 36 2888.1.s.a.477.1 6
95.18 even 4 152.1.g.b.37.1 yes 1
95.23 odd 36 2888.1.s.b.1029.1 6
95.28 odd 36 2888.1.s.b.1021.1 6
95.33 even 36 2888.1.s.a.2293.1 6
95.37 even 4 3800.1.o.a.1101.1 1
95.43 odd 36 2888.1.s.b.2293.1 6
95.48 even 36 2888.1.s.a.1021.1 6
95.53 even 36 2888.1.s.a.1029.1 6
95.63 odd 36 2888.1.s.b.477.1 6
95.68 odd 12 2888.1.l.b.69.1 2
95.73 odd 36 2888.1.s.b.333.1 6
95.78 even 36 2888.1.s.a.2789.1 6
95.83 odd 12 2888.1.l.b.293.1 2
95.88 even 12 2888.1.l.a.293.1 2
95.93 odd 36 2888.1.s.b.2789.1 6
95.94 odd 2 3800.1.b.b.949.1 2
120.53 even 4 1368.1.i.a.37.1 1
152.37 odd 2 CM 3800.1.b.a.949.1 2
285.113 odd 4 1368.1.i.a.37.1 1
380.303 odd 4 608.1.g.b.113.1 1
760.13 even 36 2888.1.s.b.477.1 6
760.37 even 4 3800.1.o.b.1101.1 1
760.53 even 36 2888.1.s.b.1029.1 6
760.93 odd 36 2888.1.s.a.2789.1 6
760.173 even 36 2888.1.s.b.2789.1 6
760.189 odd 2 inner 3800.1.b.a.949.2 2
760.213 odd 36 2888.1.s.a.1029.1 6
760.253 odd 36 2888.1.s.a.477.1 6
760.293 even 12 2888.1.l.b.69.1 2
760.333 even 36 2888.1.s.b.1021.1 6
760.373 even 12 2888.1.l.b.293.1 2
760.413 even 36 2888.1.s.b.2293.1 6
760.453 odd 36 2888.1.s.a.333.1 6
760.493 even 4 152.1.g.a.37.1 ✓ 1
760.573 even 36 2888.1.s.b.333.1 6
760.613 odd 36 2888.1.s.a.2293.1 6
760.653 odd 12 2888.1.l.a.293.1 2
760.683 odd 4 608.1.g.a.113.1 1
760.693 odd 36 2888.1.s.a.1021.1 6
760.733 odd 12 2888.1.l.a.69.1 2
2280.1253 odd 4 1368.1.i.b.37.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.1.g.a.37.1 ✓ 1 5.3 odd 4
152.1.g.a.37.1 ✓ 1 760.493 even 4
152.1.g.b.37.1 yes 1 40.13 odd 4
152.1.g.b.37.1 yes 1 95.18 even 4
608.1.g.a.113.1 1 20.3 even 4
608.1.g.a.113.1 1 760.683 odd 4
608.1.g.b.113.1 1 40.3 even 4
608.1.g.b.113.1 1 380.303 odd 4
1368.1.i.a.37.1 1 120.53 even 4
1368.1.i.a.37.1 1 285.113 odd 4
1368.1.i.b.37.1 1 15.8 even 4
1368.1.i.b.37.1 1 2280.1253 odd 4
2888.1.l.a.69.1 2 95.8 even 12
2888.1.l.a.69.1 2 760.733 odd 12
2888.1.l.a.293.1 2 95.88 even 12
2888.1.l.a.293.1 2 760.653 odd 12
2888.1.l.b.69.1 2 95.68 odd 12
2888.1.l.b.69.1 2 760.293 even 12
2888.1.l.b.293.1 2 95.83 odd 12
2888.1.l.b.293.1 2 760.373 even 12
2888.1.s.a.333.1 6 95.3 even 36
2888.1.s.a.333.1 6 760.453 odd 36
2888.1.s.a.477.1 6 95.13 even 36
2888.1.s.a.477.1 6 760.253 odd 36
2888.1.s.a.1021.1 6 95.48 even 36
2888.1.s.a.1021.1 6 760.693 odd 36
2888.1.s.a.1029.1 6 95.53 even 36
2888.1.s.a.1029.1 6 760.213 odd 36
2888.1.s.a.2293.1 6 95.33 even 36
2888.1.s.a.2293.1 6 760.613 odd 36
2888.1.s.a.2789.1 6 95.78 even 36
2888.1.s.a.2789.1 6 760.93 odd 36
2888.1.s.b.333.1 6 95.73 odd 36
2888.1.s.b.333.1 6 760.573 even 36
2888.1.s.b.477.1 6 95.63 odd 36
2888.1.s.b.477.1 6 760.13 even 36
2888.1.s.b.1021.1 6 95.28 odd 36
2888.1.s.b.1021.1 6 760.333 even 36
2888.1.s.b.1029.1 6 95.23 odd 36
2888.1.s.b.1029.1 6 760.53 even 36
2888.1.s.b.2293.1 6 95.43 odd 36
2888.1.s.b.2293.1 6 760.413 even 36
2888.1.s.b.2789.1 6 95.93 odd 36
2888.1.s.b.2789.1 6 760.173 even 36
3800.1.b.a.949.1 2 1.1 even 1 trivial
3800.1.b.a.949.1 2 152.37 odd 2 CM
3800.1.b.a.949.2 2 5.4 even 2 inner
3800.1.b.a.949.2 2 760.189 odd 2 inner
3800.1.b.b.949.1 2 40.29 even 2
3800.1.b.b.949.1 2 95.94 odd 2
3800.1.b.b.949.2 2 8.5 even 2
3800.1.b.b.949.2 2 19.18 odd 2
3800.1.o.a.1101.1 1 40.37 odd 4
3800.1.o.a.1101.1 1 95.37 even 4
3800.1.o.b.1101.1 1 5.2 odd 4
3800.1.o.b.1101.1 1 760.37 even 4