Properties

Label 3800.1.o.g.1101.3
Level $3800$
Weight $1$
Character 3800.1101
Analytic conductor $1.896$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -95
Inner twists $8$

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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 = [8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0] 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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.66724352000.2

Embedding invariants

Embedding label 1101.3
Root \(0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 3800.1101
Dual form 3800.1.o.g.1101.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.382683 - 0.923880i) q^{2} -1.84776 q^{3} +(-0.707107 + 0.707107i) q^{4} +(0.707107 + 1.70711i) q^{6} +(0.923880 + 0.382683i) q^{8} +2.41421 q^{9} -1.41421i q^{11} +(1.30656 - 1.30656i) q^{12} -0.765367 q^{13} -1.00000i q^{16} +(-0.923880 - 2.23044i) q^{18} +1.00000i q^{19} +(-1.30656 + 0.541196i) q^{22} +(-1.70711 - 0.707107i) q^{24} +(0.292893 + 0.707107i) q^{26} -2.61313 q^{27} +(-0.923880 + 0.382683i) q^{32} +2.61313i q^{33} +(-1.70711 + 1.70711i) q^{36} +1.84776 q^{37} +(0.923880 - 0.382683i) q^{38} +1.41421 q^{39} +(1.00000 + 1.00000i) q^{44} +1.84776i q^{48} -1.00000 q^{49} +(0.541196 - 0.541196i) q^{52} +1.84776 q^{53} +(1.00000 + 2.41421i) q^{54} -1.84776i q^{57} +1.41421i q^{61} +(0.707107 + 0.707107i) q^{64} +(2.41421 - 1.00000i) q^{66} -0.765367 q^{67} +(2.23044 + 0.923880i) q^{72} +(-0.707107 - 1.70711i) q^{74} +(-0.707107 - 0.707107i) q^{76} +(-0.541196 - 1.30656i) q^{78} +2.41421 q^{81} +(0.541196 - 1.30656i) q^{88} +(1.70711 - 0.707107i) q^{96} +0.765367i q^{97} +(0.382683 + 0.923880i) q^{98} -3.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9} - 8 q^{24} + 8 q^{26} - 8 q^{36} + 8 q^{44} - 8 q^{49} + 8 q^{54} + 8 q^{66} + 8 q^{81} + 8 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.382683 − 0.923880i −0.382683 − 0.923880i
\(3\) −1.84776 −1.84776 −0.923880 − 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(4\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(5\) 0 0
\(6\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(9\) 2.41421 2.41421
\(10\) 0 0
\(11\) − 1.41421i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(12\) 1.30656 − 1.30656i 1.30656 − 1.30656i
\(13\) −0.765367 −0.765367 −0.382683 − 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) − 1.00000i − 1.00000i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −0.923880 − 2.23044i −0.923880 − 2.23044i
\(19\) 1.00000i 1.00000i
\(20\) 0 0
\(21\) 0 0
\(22\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.70711 − 0.707107i −1.70711 − 0.707107i
\(25\) 0 0
\(26\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(27\) −2.61313 −2.61313
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(33\) 2.61313i 2.61313i
\(34\) 0 0
\(35\) 0 0
\(36\) −1.70711 + 1.70711i −1.70711 + 1.70711i
\(37\) 1.84776 1.84776 0.923880 − 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(38\) 0.923880 − 0.382683i 0.923880 − 0.382683i
\(39\) 1.41421 1.41421
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.84776i 1.84776i
\(49\) −1.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0.541196 − 0.541196i 0.541196 − 0.541196i
\(53\) 1.84776 1.84776 0.923880 − 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(54\) 1.00000 + 2.41421i 1.00000 + 2.41421i
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.84776i − 1.84776i
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(65\) 0 0
\(66\) 2.41421 − 1.00000i 2.41421 − 1.00000i
\(67\) −0.765367 −0.765367 −0.382683 − 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 2.23044 + 0.923880i 2.23044 + 0.923880i
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −0.707107 − 1.70711i −0.707107 − 1.70711i
\(75\) 0 0
\(76\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(77\) 0 0
\(78\) −0.541196 − 1.30656i −0.541196 − 1.30656i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 2.41421 2.41421
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.541196 − 1.30656i 0.541196 − 1.30656i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.70711 − 0.707107i 1.70711 − 0.707107i
\(97\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(98\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(99\) − 3.41421i − 3.41421i
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.g.1101.3 8
5.2 odd 4 760.1.b.a.189.7 yes 8
5.3 odd 4 760.1.b.a.189.2 yes 8
5.4 even 2 inner 3800.1.o.g.1101.6 8
8.5 even 2 inner 3800.1.o.g.1101.5 8
19.18 odd 2 inner 3800.1.o.g.1101.6 8
20.3 even 4 3040.1.b.a.1329.8 8
20.7 even 4 3040.1.b.a.1329.2 8
40.3 even 4 3040.1.b.a.1329.1 8
40.13 odd 4 760.1.b.a.189.1 ✓ 8
40.27 even 4 3040.1.b.a.1329.7 8
40.29 even 2 inner 3800.1.o.g.1101.4 8
40.37 odd 4 760.1.b.a.189.8 yes 8
95.18 even 4 760.1.b.a.189.7 yes 8
95.37 even 4 760.1.b.a.189.2 yes 8
95.94 odd 2 CM 3800.1.o.g.1101.3 8
152.37 odd 2 inner 3800.1.o.g.1101.4 8
380.227 odd 4 3040.1.b.a.1329.8 8
380.303 odd 4 3040.1.b.a.1329.2 8
760.37 even 4 760.1.b.a.189.1 ✓ 8
760.189 odd 2 inner 3800.1.o.g.1101.5 8
760.227 odd 4 3040.1.b.a.1329.1 8
760.493 even 4 760.1.b.a.189.8 yes 8
760.683 odd 4 3040.1.b.a.1329.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.1.b.a.189.1 ✓ 8 40.13 odd 4
760.1.b.a.189.1 ✓ 8 760.37 even 4
760.1.b.a.189.2 yes 8 5.3 odd 4
760.1.b.a.189.2 yes 8 95.37 even 4
760.1.b.a.189.7 yes 8 5.2 odd 4
760.1.b.a.189.7 yes 8 95.18 even 4
760.1.b.a.189.8 yes 8 40.37 odd 4
760.1.b.a.189.8 yes 8 760.493 even 4
3040.1.b.a.1329.1 8 40.3 even 4
3040.1.b.a.1329.1 8 760.227 odd 4
3040.1.b.a.1329.2 8 20.7 even 4
3040.1.b.a.1329.2 8 380.303 odd 4
3040.1.b.a.1329.7 8 40.27 even 4
3040.1.b.a.1329.7 8 760.683 odd 4
3040.1.b.a.1329.8 8 20.3 even 4
3040.1.b.a.1329.8 8 380.227 odd 4
3800.1.o.g.1101.3 8 1.1 even 1 trivial
3800.1.o.g.1101.3 8 95.94 odd 2 CM
3800.1.o.g.1101.4 8 40.29 even 2 inner
3800.1.o.g.1101.4 8 152.37 odd 2 inner
3800.1.o.g.1101.5 8 8.5 even 2 inner
3800.1.o.g.1101.5 8 760.189 odd 2 inner
3800.1.o.g.1101.6 8 5.4 even 2 inner
3800.1.o.g.1101.6 8 19.18 odd 2 inner