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Properties

Label 1664.2.b.c.833.1
Level $1664$
Weight $2$
Character 1664.833
Analytic conductor $13.287$
Analytic rank $0$
Dimension $2$
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 = [2,0,0,0,0,0,2,0,4] 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: \(2\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 833.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1664.833
Dual form 1664.2.b.c.833.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.00000i q^{5} +1.00000 q^{7} +2.00000 q^{9} +2.00000i q^{11} +1.00000i q^{13} +1.00000 q^{15} +1.00000 q^{17} +8.00000i q^{19} -1.00000i q^{21} -6.00000 q^{23} +4.00000 q^{25} -5.00000i q^{27} +6.00000i q^{29} -8.00000 q^{31} +2.00000 q^{33} +1.00000i q^{35} +1.00000i q^{37} +1.00000 q^{39} +4.00000 q^{41} -5.00000i q^{43} +2.00000i q^{45} +11.0000 q^{47} -6.00000 q^{49} -1.00000i q^{51} +6.00000i q^{53} -2.00000 q^{55} +8.00000 q^{57} +8.00000i q^{59} +4.00000i q^{61} +2.00000 q^{63} -1.00000 q^{65} -2.00000i q^{67} +6.00000i q^{69} +3.00000 q^{71} +8.00000 q^{73} -4.00000i q^{75} +2.00000i q^{77} -10.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +1.00000i q^{85} +6.00000 q^{87} +16.0000 q^{89} +1.00000i q^{91} +8.00000i q^{93} -8.00000 q^{95} -2.00000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} + 4 q^{9} + 2 q^{15} + 2 q^{17} - 12 q^{23} + 8 q^{25} - 16 q^{31} + 4 q^{33} + 2 q^{39} + 8 q^{41} + 22 q^{47} - 12 q^{49} - 4 q^{55} + 16 q^{57} + 4 q^{63} - 2 q^{65} + 6 q^{71} + 16 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) \) 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 \(-0.906785\pi\)
0.957427 − 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i 0.974679 + 0.223607i \(0.0717831\pi\)
−0.974679 + 0.223607i \(0.928217\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 − 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 8.00000i 1.83533i 0.397360 + 0.917663i \(0.369927\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) − 1.00000i − 0.218218i
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 1.00000i 0.169031i
\(36\) 0 0
\(37\) 1.00000i 0.164399i 0.996616 + 0.0821995i \(0.0261945\pi\)
−0.996616 + 0.0821995i \(0.973806\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) − 5.00000i − 0.762493i −0.924473 − 0.381246i \(-0.875495\pi\)
0.924473 − 0.381246i \(-0.124505\pi\)
\(44\) 0 0
\(45\) 2.00000i 0.298142i
\(46\) 0 0
\(47\) 11.0000 1.60451 0.802257 − 0.596978i \(-0.203632\pi\)
0.802257 + 0.596978i \(0.203632\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) − 1.00000i − 0.140028i
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) 8.00000i 1.04151i 0.853706 + 0.520756i \(0.174350\pi\)
−0.853706 + 0.520756i \(0.825650\pi\)
\(60\) 0 0
\(61\) 4.00000i 0.512148i 0.966657 + 0.256074i \(0.0824290\pi\)
−0.966657 + 0.256074i \(0.917571\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) − 2.00000i − 0.244339i −0.992509 − 0.122169i \(-0.961015\pi\)
0.992509 − 0.122169i \(-0.0389851\pi\)
\(68\) 0 0
\(69\) 6.00000i 0.722315i
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 − 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) − 4.00000i − 0.461880i
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 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
\(85\) 1.00000i 0.108465i
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 16.0000 1.69600 0.847998 − 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) 1.00000i 0.104828i
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 4.00000i 0.402015i
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.c.833.1 yes 2
4.3 odd 2 1664.2.b.b.833.2 yes 2
8.3 odd 2 1664.2.b.b.833.1 ✓ 2
8.5 even 2 inner 1664.2.b.c.833.2 yes 2
16.3 odd 4 3328.2.a.e.1.1 1
16.5 even 4 3328.2.a.d.1.1 1
16.11 odd 4 3328.2.a.h.1.1 1
16.13 even 4 3328.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.b.b.833.1 ✓ 2 8.3 odd 2
1664.2.b.b.833.2 yes 2 4.3 odd 2
1664.2.b.c.833.1 yes 2 1.1 even 1 trivial
1664.2.b.c.833.2 yes 2 8.5 even 2 inner
3328.2.a.d.1.1 1 16.5 even 4
3328.2.a.e.1.1 1 16.3 odd 4
3328.2.a.h.1.1 1 16.11 odd 4
3328.2.a.i.1.1 1 16.13 even 4