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Properties

Label 1664.2.b.k.833.8
Level $1664$
Weight $2$
Character 1664.833
Analytic conductor $13.287$
Analytic rank $0$
Dimension $12$
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] = [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 = [12,0,0,0,0,0,0,0,-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: \(12\)
Coefficient field: 12.0.426337261060096.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 833.8
Root \(-0.760198 + 1.19252i\) of defining polynomial
Character \(\chi\) \(=\) 1664.833
Dual form 1664.2.b.k.833.5

$q$-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)
 
\(f(q)\) \(=\) \(q+0.611393i q^{3} +1.62620i q^{5} +3.10261 q^{7} +2.62620 q^{9} +5.31965i q^{11} +1.00000i q^{13} -0.994247 q^{15} -1.89692 q^{17} -0.885578i q^{19} +1.89692i q^{21} -1.22279 q^{23} +2.35548 q^{25} +3.43982i q^{27} +2.00000i q^{29} +3.71400 q^{31} -3.25240 q^{33} +5.04546i q^{35} -11.1493i q^{37} -0.611393 q^{39} -5.79383 q^{41} +3.43982i q^{43} +4.27072i q^{45} -0.274184 q^{47} +2.62620 q^{49} -1.15976i q^{51} -7.79383i q^{53} -8.65080 q^{55} +0.541436 q^{57} +1.56000i q^{59} -7.04623i q^{61} +8.14807 q^{63} -1.62620 q^{65} +12.7477i q^{67} -0.747604i q^{69} -8.08505 q^{71} +16.2986 q^{73} +1.44012i q^{75} +16.5048i q^{77} -9.41650 q^{79} +5.77551 q^{81} +10.9765i q^{83} -3.08476i q^{85} -1.22279 q^{87} -5.25240 q^{89} +3.10261i q^{91} +2.27072i q^{93} +1.44012 q^{95} -8.50479 q^{97} +13.9704i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{9} - 8 q^{17} - 28 q^{25} + 32 q^{33} - 40 q^{41} - 4 q^{49} + 48 q^{57} + 16 q^{65} + 24 q^{73} - 52 q^{81} + 8 q^{89} + 40 q^{97}+O(q^{100}) \) 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
\(3\) 0.611393i 0.352988i 0.984302 + 0.176494i \(0.0564756\pi\)
−0.984302 + 0.176494i \(0.943524\pi\)
\(4\) 0 0
\(5\) 1.62620i 0.727258i 0.931544 + 0.363629i \(0.118462\pi\)
−0.931544 + 0.363629i \(0.881538\pi\)
\(6\) 0 0
\(7\) 3.10261 1.17268 0.586338 − 0.810066i \(-0.300569\pi\)
0.586338 + 0.810066i \(0.300569\pi\)
\(8\) 0 0
\(9\) 2.62620 0.875399
\(10\) 0 0
\(11\) 5.31965i 1.60393i 0.597369 + 0.801967i \(0.296213\pi\)
−0.597369 + 0.801967i \(0.703787\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) −0.994247 −0.256713
\(16\) 0 0
\(17\) −1.89692 −0.460070 −0.230035 − 0.973182i \(-0.573884\pi\)
−0.230035 + 0.973182i \(0.573884\pi\)
\(18\) 0 0
\(19\) − 0.885578i − 0.203165i −0.994827 − 0.101583i \(-0.967609\pi\)
0.994827 − 0.101583i \(-0.0323907\pi\)
\(20\) 0 0
\(21\) 1.89692i 0.413941i
\(22\) 0 0
\(23\) −1.22279 −0.254969 −0.127484 − 0.991841i \(-0.540690\pi\)
−0.127484 + 0.991841i \(0.540690\pi\)
\(24\) 0 0
\(25\) 2.35548 0.471096
\(26\) 0 0
\(27\) 3.43982i 0.661994i
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 3.71400 0.667055 0.333527 − 0.942740i \(-0.391761\pi\)
0.333527 + 0.942740i \(0.391761\pi\)
\(32\) 0 0
\(33\) −3.25240 −0.566169
\(34\) 0 0
\(35\) 5.04546i 0.852839i
\(36\) 0 0
\(37\) − 11.1493i − 1.83294i −0.400108 − 0.916468i \(-0.631028\pi\)
0.400108 − 0.916468i \(-0.368972\pi\)
\(38\) 0 0
\(39\) −0.611393 −0.0979013
\(40\) 0 0
\(41\) −5.79383 −0.904845 −0.452422 − 0.891804i \(-0.649440\pi\)
−0.452422 + 0.891804i \(0.649440\pi\)
\(42\) 0 0
\(43\) 3.43982i 0.524568i 0.964991 + 0.262284i \(0.0844757\pi\)
−0.964991 + 0.262284i \(0.915524\pi\)
\(44\) 0 0
\(45\) 4.27072i 0.636641i
\(46\) 0 0
\(47\) −0.274184 −0.0399939 −0.0199970 − 0.999800i \(-0.506366\pi\)
−0.0199970 + 0.999800i \(0.506366\pi\)
\(48\) 0 0
\(49\) 2.62620 0.375171
\(50\) 0 0
\(51\) − 1.15976i − 0.162399i
\(52\) 0 0
\(53\) − 7.79383i − 1.07057i −0.844673 − 0.535283i \(-0.820205\pi\)
0.844673 − 0.535283i \(-0.179795\pi\)
\(54\) 0 0
\(55\) −8.65080 −1.16647
\(56\) 0 0
\(57\) 0.541436 0.0717150
\(58\) 0 0
\(59\) 1.56000i 0.203094i 0.994831 + 0.101547i \(0.0323793\pi\)
−0.994831 + 0.101547i \(0.967621\pi\)
\(60\) 0 0
\(61\) − 7.04623i − 0.902177i −0.892479 − 0.451089i \(-0.851036\pi\)
0.892479 − 0.451089i \(-0.148964\pi\)
\(62\) 0 0
\(63\) 8.14807 1.02656
\(64\) 0 0
\(65\) −1.62620 −0.201705
\(66\) 0 0
\(67\) 12.7477i 1.55737i 0.627413 + 0.778687i \(0.284114\pi\)
−0.627413 + 0.778687i \(0.715886\pi\)
\(68\) 0 0
\(69\) − 0.747604i − 0.0900009i
\(70\) 0 0
\(71\) −8.08505 −0.959519 −0.479759 − 0.877400i \(-0.659276\pi\)
−0.479759 + 0.877400i \(0.659276\pi\)
\(72\) 0 0
\(73\) 16.2986 1.90761 0.953805 − 0.300427i \(-0.0971291\pi\)
0.953805 + 0.300427i \(0.0971291\pi\)
\(74\) 0 0
\(75\) 1.44012i 0.166291i
\(76\) 0 0
\(77\) 16.5048i 1.88090i
\(78\) 0 0
\(79\) −9.41650 −1.05944 −0.529720 − 0.848173i \(-0.677703\pi\)
−0.529720 + 0.848173i \(0.677703\pi\)
\(80\) 0 0
\(81\) 5.77551 0.641723
\(82\) 0 0
\(83\) 10.9765i 1.20483i 0.798184 + 0.602414i \(0.205794\pi\)
−0.798184 + 0.602414i \(0.794206\pi\)
\(84\) 0 0
\(85\) − 3.08476i − 0.334589i
\(86\) 0 0
\(87\) −1.22279 −0.131097
\(88\) 0 0
\(89\) −5.25240 −0.556753 −0.278376 − 0.960472i \(-0.589796\pi\)
−0.278376 + 0.960472i \(0.589796\pi\)
\(90\) 0 0
\(91\) 3.10261i 0.325242i
\(92\) 0 0
\(93\) 2.27072i 0.235463i
\(94\) 0 0
\(95\) 1.44012 0.147754
\(96\) 0 0
\(97\) −8.50479 −0.863531 −0.431765 − 0.901986i \(-0.642109\pi\)
−0.431765 + 0.901986i \(0.642109\pi\)
\(98\) 0 0
\(99\) 13.9704i 1.40408i
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.k.833.8 yes 12
4.3 odd 2 inner 1664.2.b.k.833.6 yes 12
8.3 odd 2 inner 1664.2.b.k.833.7 yes 12
8.5 even 2 inner 1664.2.b.k.833.5 ✓ 12
16.3 odd 4 3328.2.a.bo.1.4 6
16.5 even 4 3328.2.a.bp.1.4 6
16.11 odd 4 3328.2.a.bp.1.3 6
16.13 even 4 3328.2.a.bo.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1664.2.b.k.833.5 ✓ 12 8.5 even 2 inner
1664.2.b.k.833.6 yes 12 4.3 odd 2 inner
1664.2.b.k.833.7 yes 12 8.3 odd 2 inner
1664.2.b.k.833.8 yes 12 1.1 even 1 trivial
3328.2.a.bo.1.3 6 16.13 even 4
3328.2.a.bo.1.4 6 16.3 odd 4
3328.2.a.bp.1.3 6 16.11 odd 4
3328.2.a.bp.1.4 6 16.5 even 4