Properties

Label 968.1.l.c.323.1
Level $968$
Weight $1$
Character 968.323
Analytic conductor $0.483$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -8
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] = [968,1,Mod(3,968)] 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("968.3"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(968, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 5, 8])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 968.l (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,1,3] 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: no
Analytic conductor: \(0.483094932229\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.937024.1

Embedding invariants

Embedding label 323.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 968.323
Dual form 968.1.l.c.3.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.809017 + 0.587785i) q^{2} +(0.190983 - 0.587785i) q^{3} +(0.309017 + 0.951057i) q^{4} +(0.500000 - 0.363271i) q^{6} +(-0.309017 + 0.951057i) q^{8} +(0.500000 + 0.363271i) q^{9} +0.618034 q^{12} +(-0.809017 + 0.587785i) q^{16} +(-1.30902 + 0.951057i) q^{17} +(0.190983 + 0.587785i) q^{18} +(0.500000 - 1.53884i) q^{19} +(0.500000 + 0.363271i) q^{24} +(0.309017 - 0.951057i) q^{25} +(0.809017 - 0.587785i) q^{27} -1.00000 q^{32} -1.61803 q^{34} +(-0.190983 + 0.587785i) q^{36} +(1.30902 - 0.951057i) q^{38} +(-0.190983 + 0.587785i) q^{41} -0.618034 q^{43} +(0.190983 + 0.587785i) q^{48} +(-0.809017 + 0.587785i) q^{49} +(0.809017 - 0.587785i) q^{50} +(0.309017 + 0.951057i) q^{51} +1.00000 q^{54} +(-0.809017 - 0.587785i) q^{57} +(-0.500000 - 1.53884i) q^{59} +(-0.809017 - 0.587785i) q^{64} -1.61803 q^{67} +(-1.30902 - 0.951057i) q^{68} +(-0.500000 + 0.363271i) q^{72} +(-0.190983 - 0.587785i) q^{73} +(-0.500000 - 0.363271i) q^{75} +1.61803 q^{76} +(-0.500000 + 0.363271i) q^{82} +(0.500000 - 0.363271i) q^{83} +(-0.500000 - 0.363271i) q^{86} +0.618034 q^{89} +(-0.190983 + 0.587785i) q^{96} +(-0.500000 - 0.363271i) q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 3 q^{3} - q^{4} + 2 q^{6} + q^{8} + 2 q^{9} - 2 q^{12} - q^{16} - 3 q^{17} + 3 q^{18} + 2 q^{19} + 2 q^{24} - q^{25} + q^{27} - 4 q^{32} - 2 q^{34} - 3 q^{36} + 3 q^{38} - 3 q^{41} + 2 q^{43}+ \cdots - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(727\) \(849\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(3\) 0.190983 − 0.587785i 0.190983 − 0.587785i −0.809017 − 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(4\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(5\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(6\) 0.500000 − 0.363271i 0.500000 − 0.363271i
\(7\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(8\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(9\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(10\) 0 0
\(11\) 0 0
\(12\) 0.618034 0.618034
\(13\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(17\) −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i \(0.600000\pi\)
−1.00000 \(\pi\)
\(18\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(19\) 0.500000 − 1.53884i 0.500000 − 1.53884i −0.309017 − 0.951057i \(-0.600000\pi\)
0.809017 − 0.587785i \(-0.200000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(25\) 0.309017 − 0.951057i 0.309017 − 0.951057i
\(26\) 0 0
\(27\) 0.809017 − 0.587785i 0.809017 − 0.587785i
\(28\) 0 0
\(29\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(32\) −1.00000 −1.00000
\(33\) 0 0
\(34\) −1.61803 −1.61803
\(35\) 0 0
\(36\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(37\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(38\) 1.30902 − 0.951057i 1.30902 − 0.951057i
\(39\) 0 0
\(40\) 0 0
\(41\) −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i \(0.200000\pi\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −0.618034 −0.618034 −0.309017 − 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.309017 − 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(48\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(49\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(50\) 0.809017 − 0.587785i 0.809017 − 0.587785i
\(51\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(52\) 0 0
\(53\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(54\) 1.00000 1.00000
\(55\) 0 0
\(56\) 0 0
\(57\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(58\) 0 0
\(59\) −0.500000 − 1.53884i −0.500000 − 1.53884i −0.809017 − 0.587785i \(-0.800000\pi\)
0.309017 − 0.951057i \(-0.400000\pi\)
\(60\) 0 0
\(61\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(65\) 0 0
\(66\) 0 0
\(67\) −1.61803 −1.61803 −0.809017 − 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(68\) −1.30902 − 0.951057i −1.30902 − 0.951057i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(72\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(73\) −0.190983 − 0.587785i −0.190983 − 0.587785i 0.809017 − 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −0.500000 − 0.363271i −0.500000 − 0.363271i
\(76\) 1.61803 1.61803
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(83\) 0.500000 − 0.363271i 0.500000 − 0.363271i −0.309017 − 0.951057i \(-0.600000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.500000 − 0.363271i −0.500000 − 0.363271i
\(87\) 0 0
\(88\) 0 0
\(89\) 0.618034 0.618034 0.309017 − 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(97\) −0.500000 − 0.363271i −0.500000 − 0.363271i 0.309017 − 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(98\) −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 968.1.l.c.323.1 4
4.3 odd 2 3872.1.t.a.1775.1 4
8.3 odd 2 CM 968.1.l.c.323.1 4
8.5 even 2 3872.1.t.a.1775.1 4
11.2 odd 10 88.1.l.a.27.1 ✓ 4
11.3 even 5 inner 968.1.l.c.3.1 4
11.4 even 5 968.1.l.b.251.1 4
11.5 even 5 968.1.f.a.243.2 2
11.6 odd 10 968.1.f.b.243.2 2
11.7 odd 10 88.1.l.a.75.1 yes 4
11.8 odd 10 968.1.l.a.3.1 4
11.9 even 5 968.1.l.b.27.1 4
11.10 odd 2 968.1.l.a.323.1 4
33.2 even 10 792.1.bu.a.379.1 4
33.29 even 10 792.1.bu.a.163.1 4
44.3 odd 10 3872.1.t.a.1455.1 4
44.7 even 10 352.1.t.a.207.1 4
44.15 odd 10 3872.1.t.c.2671.1 4
44.19 even 10 3872.1.t.b.1455.1 4
44.27 odd 10 3872.1.f.b.3631.1 2
44.31 odd 10 3872.1.t.c.2447.1 4
44.35 even 10 352.1.t.a.335.1 4
44.39 even 10 3872.1.f.a.3631.1 2
44.43 even 2 3872.1.t.b.1775.1 4
55.2 even 20 2200.1.dd.a.1699.1 8
55.7 even 20 2200.1.dd.a.2099.2 8
55.13 even 20 2200.1.dd.a.1699.2 8
55.18 even 20 2200.1.dd.a.2099.1 8
55.24 odd 10 2200.1.cl.a.2051.1 4
55.29 odd 10 2200.1.cl.a.251.1 4
88.3 odd 10 inner 968.1.l.c.3.1 4
88.5 even 10 3872.1.f.b.3631.1 2
88.13 odd 10 352.1.t.a.335.1 4
88.19 even 10 968.1.l.a.3.1 4
88.21 odd 2 3872.1.t.b.1775.1 4
88.27 odd 10 968.1.f.a.243.2 2
88.29 odd 10 352.1.t.a.207.1 4
88.35 even 10 88.1.l.a.27.1 ✓ 4
88.37 even 10 3872.1.t.c.2671.1 4
88.43 even 2 968.1.l.a.323.1 4
88.51 even 10 88.1.l.a.75.1 yes 4
88.53 even 10 3872.1.t.c.2447.1 4
88.59 odd 10 968.1.l.b.251.1 4
88.61 odd 10 3872.1.f.a.3631.1 2
88.69 even 10 3872.1.t.a.1455.1 4
88.75 odd 10 968.1.l.b.27.1 4
88.83 even 10 968.1.f.b.243.2 2
88.85 odd 10 3872.1.t.b.1455.1 4
132.35 odd 10 3168.1.ck.a.3151.1 4
132.95 odd 10 3168.1.ck.a.559.1 4
176.13 odd 20 2816.1.v.c.511.2 8
176.29 odd 20 2816.1.v.c.1791.1 8
176.35 even 20 2816.1.v.c.511.1 8
176.51 even 20 2816.1.v.c.1791.2 8
176.101 odd 20 2816.1.v.c.511.1 8
176.117 odd 20 2816.1.v.c.1791.2 8
176.123 even 20 2816.1.v.c.511.2 8
176.139 even 20 2816.1.v.c.1791.1 8
264.29 even 10 3168.1.ck.a.559.1 4
264.35 odd 10 792.1.bu.a.379.1 4
264.101 even 10 3168.1.ck.a.3151.1 4
264.227 odd 10 792.1.bu.a.163.1 4
440.123 odd 20 2200.1.dd.a.1699.2 8
440.139 even 10 2200.1.cl.a.251.1 4
440.227 odd 20 2200.1.dd.a.2099.2 8
440.299 even 10 2200.1.cl.a.2051.1 4
440.387 odd 20 2200.1.dd.a.1699.1 8
440.403 odd 20 2200.1.dd.a.2099.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
88.1.l.a.27.1 ✓ 4 11.2 odd 10
88.1.l.a.27.1 ✓ 4 88.35 even 10
88.1.l.a.75.1 yes 4 11.7 odd 10
88.1.l.a.75.1 yes 4 88.51 even 10
352.1.t.a.207.1 4 44.7 even 10
352.1.t.a.207.1 4 88.29 odd 10
352.1.t.a.335.1 4 44.35 even 10
352.1.t.a.335.1 4 88.13 odd 10
792.1.bu.a.163.1 4 33.29 even 10
792.1.bu.a.163.1 4 264.227 odd 10
792.1.bu.a.379.1 4 33.2 even 10
792.1.bu.a.379.1 4 264.35 odd 10
968.1.f.a.243.2 2 11.5 even 5
968.1.f.a.243.2 2 88.27 odd 10
968.1.f.b.243.2 2 11.6 odd 10
968.1.f.b.243.2 2 88.83 even 10
968.1.l.a.3.1 4 11.8 odd 10
968.1.l.a.3.1 4 88.19 even 10
968.1.l.a.323.1 4 11.10 odd 2
968.1.l.a.323.1 4 88.43 even 2
968.1.l.b.27.1 4 11.9 even 5
968.1.l.b.27.1 4 88.75 odd 10
968.1.l.b.251.1 4 11.4 even 5
968.1.l.b.251.1 4 88.59 odd 10
968.1.l.c.3.1 4 11.3 even 5 inner
968.1.l.c.3.1 4 88.3 odd 10 inner
968.1.l.c.323.1 4 1.1 even 1 trivial
968.1.l.c.323.1 4 8.3 odd 2 CM
2200.1.cl.a.251.1 4 55.29 odd 10
2200.1.cl.a.251.1 4 440.139 even 10
2200.1.cl.a.2051.1 4 55.24 odd 10
2200.1.cl.a.2051.1 4 440.299 even 10
2200.1.dd.a.1699.1 8 55.2 even 20
2200.1.dd.a.1699.1 8 440.387 odd 20
2200.1.dd.a.1699.2 8 55.13 even 20
2200.1.dd.a.1699.2 8 440.123 odd 20
2200.1.dd.a.2099.1 8 55.18 even 20
2200.1.dd.a.2099.1 8 440.403 odd 20
2200.1.dd.a.2099.2 8 55.7 even 20
2200.1.dd.a.2099.2 8 440.227 odd 20
2816.1.v.c.511.1 8 176.35 even 20
2816.1.v.c.511.1 8 176.101 odd 20
2816.1.v.c.511.2 8 176.13 odd 20
2816.1.v.c.511.2 8 176.123 even 20
2816.1.v.c.1791.1 8 176.29 odd 20
2816.1.v.c.1791.1 8 176.139 even 20
2816.1.v.c.1791.2 8 176.51 even 20
2816.1.v.c.1791.2 8 176.117 odd 20
3168.1.ck.a.559.1 4 132.95 odd 10
3168.1.ck.a.559.1 4 264.29 even 10
3168.1.ck.a.3151.1 4 132.35 odd 10
3168.1.ck.a.3151.1 4 264.101 even 10
3872.1.f.a.3631.1 2 44.39 even 10
3872.1.f.a.3631.1 2 88.61 odd 10
3872.1.f.b.3631.1 2 44.27 odd 10
3872.1.f.b.3631.1 2 88.5 even 10
3872.1.t.a.1455.1 4 44.3 odd 10
3872.1.t.a.1455.1 4 88.69 even 10
3872.1.t.a.1775.1 4 4.3 odd 2
3872.1.t.a.1775.1 4 8.5 even 2
3872.1.t.b.1455.1 4 44.19 even 10
3872.1.t.b.1455.1 4 88.85 odd 10
3872.1.t.b.1775.1 4 44.43 even 2
3872.1.t.b.1775.1 4 88.21 odd 2
3872.1.t.c.2447.1 4 44.31 odd 10
3872.1.t.c.2447.1 4 88.53 even 10
3872.1.t.c.2671.1 4 44.15 odd 10
3872.1.t.c.2671.1 4 88.37 even 10