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Properties

Label 1664.2.b.e.833.4
Level $1664$
Weight $2$
Character 1664.833
Analytic conductor $13.287$
Analytic rank $0$
Dimension $4$
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] = [1664,2,Mod(833,1664)] 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("1664.833"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1664 = 2^{7} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1664.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-4,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.2871068963\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 833.4
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1664.833
Dual form 1664.2.b.e.833.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +3.82843i q^{5} -3.82843 q^{7} +2.00000 q^{9} +4.82843i q^{11} +1.00000i q^{13} -3.82843 q^{15} +6.65685 q^{17} +4.00000i q^{19} -3.82843i q^{21} +3.17157 q^{23} -9.65685 q^{25} +5.00000i q^{27} -3.17157i q^{29} -4.82843 q^{33} -14.6569i q^{35} +3.82843i q^{37} -1.00000 q^{39} -2.82843 q^{41} -3.00000i q^{43} +7.65685i q^{45} -11.4853 q^{47} +7.65685 q^{49} +6.65685i q^{51} +3.17157i q^{53} -18.4853 q^{55} -4.00000 q^{57} -5.17157i q^{59} -10.8284i q^{61} -7.65685 q^{63} -3.82843 q^{65} -3.65685i q^{67} +3.17157i q^{69} +10.1716 q^{71} -5.17157 q^{73} -9.65685i q^{75} -18.4853i q^{77} -7.65685 q^{79} +1.00000 q^{81} -6.00000i q^{83} +25.4853i q^{85} +3.17157 q^{87} +17.6569 q^{89} -3.82843i q^{91} -15.3137 q^{95} -14.4853 q^{97} +9.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7} + 8 q^{9} - 4 q^{15} + 4 q^{17} + 24 q^{23} - 16 q^{25} - 8 q^{33} - 4 q^{39} - 12 q^{47} + 8 q^{49} - 40 q^{55} - 16 q^{57} - 8 q^{63} - 4 q^{65} + 52 q^{71} - 32 q^{73} - 8 q^{79} + 4 q^{81}+ \cdots - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1664\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 0
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) 0 0
\(5\) 3.82843i 1.71212i 0.516873 + 0.856062i \(0.327096\pi\)
−0.516873 + 0.856062i \(0.672904\pi\)
\(6\) 0 0
\(7\) −3.82843 −1.44701 −0.723505 − 0.690319i \(-0.757470\pi\)
−0.723505 + 0.690319i \(0.757470\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 4.82843i 1.45583i 0.685670 + 0.727913i \(0.259509\pi\)
−0.685670 + 0.727913i \(0.740491\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) −3.82843 −0.988496
\(16\) 0 0
\(17\) 6.65685 1.61452 0.807262 − 0.590193i \(-0.200948\pi\)
0.807262 + 0.590193i \(0.200948\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) − 3.82843i − 0.835431i
\(22\) 0 0
\(23\) 3.17157 0.661319 0.330659 − 0.943750i \(-0.392729\pi\)
0.330659 + 0.943750i \(0.392729\pi\)
\(24\) 0 0
\(25\) −9.65685 −1.93137
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) − 3.17157i − 0.588946i −0.955660 − 0.294473i \(-0.904856\pi\)
0.955660 − 0.294473i \(-0.0951442\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −4.82843 −0.840521
\(34\) 0 0
\(35\) − 14.6569i − 2.47746i
\(36\) 0 0
\(37\) 3.82843i 0.629390i 0.949193 + 0.314695i \(0.101902\pi\)
−0.949193 + 0.314695i \(0.898098\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −2.82843 −0.441726 −0.220863 − 0.975305i \(-0.570887\pi\)
−0.220863 + 0.975305i \(0.570887\pi\)
\(42\) 0 0
\(43\) − 3.00000i − 0.457496i −0.973486 − 0.228748i \(-0.926537\pi\)
0.973486 − 0.228748i \(-0.0734631\pi\)
\(44\) 0 0
\(45\) 7.65685i 1.14142i
\(46\) 0 0
\(47\) −11.4853 −1.67530 −0.837650 − 0.546207i \(-0.816071\pi\)
−0.837650 + 0.546207i \(0.816071\pi\)
\(48\) 0 0
\(49\) 7.65685 1.09384
\(50\) 0 0
\(51\) 6.65685i 0.932146i
\(52\) 0 0
\(53\) 3.17157i 0.435649i 0.975988 + 0.217825i \(0.0698960\pi\)
−0.975988 + 0.217825i \(0.930104\pi\)
\(54\) 0 0
\(55\) −18.4853 −2.49255
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) − 5.17157i − 0.673281i −0.941633 − 0.336641i \(-0.890709\pi\)
0.941633 − 0.336641i \(-0.109291\pi\)
\(60\) 0 0
\(61\) − 10.8284i − 1.38644i −0.720727 − 0.693219i \(-0.756192\pi\)
0.720727 − 0.693219i \(-0.243808\pi\)
\(62\) 0 0
\(63\) −7.65685 −0.964673
\(64\) 0 0
\(65\) −3.82843 −0.474858
\(66\) 0 0
\(67\) − 3.65685i − 0.446756i −0.974732 − 0.223378i \(-0.928292\pi\)
0.974732 − 0.223378i \(-0.0717084\pi\)
\(68\) 0 0
\(69\) 3.17157i 0.381813i
\(70\) 0 0
\(71\) 10.1716 1.20714 0.603572 − 0.797309i \(-0.293744\pi\)
0.603572 + 0.797309i \(0.293744\pi\)
\(72\) 0 0
\(73\) −5.17157 −0.605287 −0.302643 − 0.953104i \(-0.597869\pi\)
−0.302643 + 0.953104i \(0.597869\pi\)
\(74\) 0 0
\(75\) − 9.65685i − 1.11508i
\(76\) 0 0
\(77\) − 18.4853i − 2.10659i
\(78\) 0 0
\(79\) −7.65685 −0.861463 −0.430732 − 0.902480i \(-0.641744\pi\)
−0.430732 + 0.902480i \(0.641744\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 − 0.329293i \(-0.893190\pi\)
0.944228 − 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 25.4853i 2.76427i
\(86\) 0 0
\(87\) 3.17157 0.340028
\(88\) 0 0
\(89\) 17.6569 1.87162 0.935811 − 0.352501i \(-0.114669\pi\)
0.935811 + 0.352501i \(0.114669\pi\)
\(90\) 0 0
\(91\) − 3.82843i − 0.401328i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −15.3137 −1.57115
\(96\) 0 0
\(97\) −14.4853 −1.47076 −0.735379 − 0.677656i \(-0.762996\pi\)
−0.735379 + 0.677656i \(0.762996\pi\)
\(98\) 0 0
\(99\) 9.65685i 0.970550i
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 1664.2.b.e.833.4 yes 4
4.3 odd 2 1664.2.b.i.833.2 yes 4
8.3 odd 2 1664.2.b.i.833.3 yes 4
8.5 even 2 inner 1664.2.b.e.833.1 ✓ 4
16.3 odd 4 3328.2.a.ba.1.2 2
16.5 even 4 3328.2.a.z.1.1 2
16.11 odd 4 3328.2.a.o.1.1 2
16.13 even 4 3328.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.b.e.833.1 ✓ 4 8.5 even 2 inner
1664.2.b.e.833.4 yes 4 1.1 even 1 trivial
1664.2.b.i.833.2 yes 4 4.3 odd 2
1664.2.b.i.833.3 yes 4 8.3 odd 2
3328.2.a.o.1.1 2 16.11 odd 4
3328.2.a.r.1.2 2 16.13 even 4
3328.2.a.z.1.1 2 16.5 even 4
3328.2.a.ba.1.2 2 16.3 odd 4