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Properties

Label 2064.2.d.c.431.16
Level $2064$
Weight $2$
Character 2064.431
Analytic conductor $16.481$
Analytic rank $0$
Dimension $48$
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] = [2064,2,Mod(431,2064)] 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("2064.431"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2064, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2064 = 2^{4} \cdot 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2064.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.4811229772\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 431.16
Character \(\chi\) \(=\) 2064.431
Dual form 2064.2.d.c.431.15

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.897165 + 1.48159i) q^{3} -2.51704i q^{5} -4.79866i q^{7} +(-1.39019 - 2.65845i) q^{9} +4.68233 q^{11} -4.97131 q^{13} +(3.72921 + 2.25820i) q^{15} -3.19989i q^{17} -4.85634i q^{19} +(7.10963 + 4.30519i) q^{21} -3.42079 q^{23} -1.33550 q^{25} +(5.18595 + 0.325391i) q^{27} +7.85098i q^{29} -4.45285i q^{31} +(-4.20082 + 6.93727i) q^{33} -12.0784 q^{35} +2.81475 q^{37} +(4.46009 - 7.36542i) q^{39} +8.68383i q^{41} -1.00000i q^{43} +(-6.69144 + 3.49916i) q^{45} -9.70483 q^{47} -16.0272 q^{49} +(4.74091 + 2.87083i) q^{51} -0.700753i q^{53} -11.7856i q^{55} +(7.19507 + 4.35694i) q^{57} +2.38874 q^{59} -10.2567 q^{61} +(-12.7570 + 6.67104i) q^{63} +12.5130i q^{65} +9.45236i q^{67} +(3.06901 - 5.06819i) q^{69} +9.13633 q^{71} -0.926498 q^{73} +(1.19816 - 1.97865i) q^{75} -22.4689i q^{77} -1.76815i q^{79} +(-5.13475 + 7.39150i) q^{81} -0.807024 q^{83} -8.05426 q^{85} +(-11.6319 - 7.04363i) q^{87} +9.51852i q^{89} +23.8556i q^{91} +(6.59727 + 3.99494i) q^{93} -12.2236 q^{95} -14.8656 q^{97} +(-6.50932 - 12.4478i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{9} - 32 q^{13} - 16 q^{21} - 120 q^{25} - 12 q^{33} + 96 q^{37} + 16 q^{45} - 8 q^{49} + 64 q^{61} + 4 q^{69} - 32 q^{73} - 20 q^{81} - 48 q^{85} + 36 q^{93} - 104 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2064\mathbb{Z}\right)^\times\).

\(n\) \(433\) \(517\) \(689\) \(1807\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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 0
\(3\) −0.897165 + 1.48159i −0.517979 + 0.855394i
\(4\) 0 0
\(5\) − 2.51704i − 1.12565i −0.826574 − 0.562827i \(-0.809714\pi\)
0.826574 − 0.562827i \(-0.190286\pi\)
\(6\) 0 0
\(7\) − 4.79866i − 1.81372i −0.421428 − 0.906862i \(-0.638471\pi\)
0.421428 − 0.906862i \(-0.361529\pi\)
\(8\) 0 0
\(9\) −1.39019 − 2.65845i −0.463396 − 0.886151i
\(10\) 0 0
\(11\) 4.68233 1.41178 0.705888 − 0.708324i \(-0.250548\pi\)
0.705888 + 0.708324i \(0.250548\pi\)
\(12\) 0 0
\(13\) −4.97131 −1.37879 −0.689396 − 0.724384i \(-0.742124\pi\)
−0.689396 + 0.724384i \(0.742124\pi\)
\(14\) 0 0
\(15\) 3.72921 + 2.25820i 0.962878 + 0.583065i
\(16\) 0 0
\(17\) − 3.19989i − 0.776088i −0.921641 − 0.388044i \(-0.873151\pi\)
0.921641 − 0.388044i \(-0.126849\pi\)
\(18\) 0 0
\(19\) − 4.85634i − 1.11412i −0.830472 − 0.557060i \(-0.811929\pi\)
0.830472 − 0.557060i \(-0.188071\pi\)
\(20\) 0 0
\(21\) 7.10963 + 4.30519i 1.55145 + 0.939470i
\(22\) 0 0
\(23\) −3.42079 −0.713284 −0.356642 − 0.934241i \(-0.616078\pi\)
−0.356642 + 0.934241i \(0.616078\pi\)
\(24\) 0 0
\(25\) −1.33550 −0.267099
\(26\) 0 0
\(27\) 5.18595 + 0.325391i 0.998037 + 0.0626215i
\(28\) 0 0
\(29\) 7.85098i 1.45789i 0.684572 + 0.728945i \(0.259989\pi\)
−0.684572 + 0.728945i \(0.740011\pi\)
\(30\) 0 0
\(31\) − 4.45285i − 0.799755i −0.916569 − 0.399878i \(-0.869053\pi\)
0.916569 − 0.399878i \(-0.130947\pi\)
\(32\) 0 0
\(33\) −4.20082 + 6.93727i −0.731269 + 1.20762i
\(34\) 0 0
\(35\) −12.0784 −2.04163
\(36\) 0 0
\(37\) 2.81475 0.462742 0.231371 − 0.972866i \(-0.425679\pi\)
0.231371 + 0.972866i \(0.425679\pi\)
\(38\) 0 0
\(39\) 4.46009 − 7.36542i 0.714185 − 1.17941i
\(40\) 0 0
\(41\) 8.68383i 1.35619i 0.734976 + 0.678093i \(0.237193\pi\)
−0.734976 + 0.678093i \(0.762807\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i
\(44\) 0 0
\(45\) −6.69144 + 3.49916i −0.997501 + 0.521624i
\(46\) 0 0
\(47\) −9.70483 −1.41560 −0.707798 − 0.706415i \(-0.750311\pi\)
−0.707798 + 0.706415i \(0.750311\pi\)
\(48\) 0 0
\(49\) −16.0272 −2.28959
\(50\) 0 0
\(51\) 4.74091 + 2.87083i 0.663860 + 0.401997i
\(52\) 0 0
\(53\) − 0.700753i − 0.0962558i −0.998841 − 0.0481279i \(-0.984674\pi\)
0.998841 − 0.0481279i \(-0.0153255\pi\)
\(54\) 0 0
\(55\) − 11.7856i − 1.58917i
\(56\) 0 0
\(57\) 7.19507 + 4.35694i 0.953011 + 0.577090i
\(58\) 0 0
\(59\) 2.38874 0.310987 0.155493 − 0.987837i \(-0.450303\pi\)
0.155493 + 0.987837i \(0.450303\pi\)
\(60\) 0 0
\(61\) −10.2567 −1.31323 −0.656617 − 0.754224i \(-0.728013\pi\)
−0.656617 + 0.754224i \(0.728013\pi\)
\(62\) 0 0
\(63\) −12.7570 + 6.67104i −1.60723 + 0.840472i
\(64\) 0 0
\(65\) 12.5130i 1.55205i
\(66\) 0 0
\(67\) 9.45236i 1.15479i 0.816465 + 0.577395i \(0.195931\pi\)
−0.816465 + 0.577395i \(0.804069\pi\)
\(68\) 0 0
\(69\) 3.06901 − 5.06819i 0.369466 − 0.610138i
\(70\) 0 0
\(71\) 9.13633 1.08428 0.542142 − 0.840287i \(-0.317614\pi\)
0.542142 + 0.840287i \(0.317614\pi\)
\(72\) 0 0
\(73\) −0.926498 −0.108438 −0.0542192 − 0.998529i \(-0.517267\pi\)
−0.0542192 + 0.998529i \(0.517267\pi\)
\(74\) 0 0
\(75\) 1.19816 − 1.97865i 0.138352 − 0.228475i
\(76\) 0 0
\(77\) − 22.4689i − 2.56057i
\(78\) 0 0
\(79\) − 1.76815i − 0.198932i −0.995041 − 0.0994660i \(-0.968287\pi\)
0.995041 − 0.0994660i \(-0.0317135\pi\)
\(80\) 0 0
\(81\) −5.13475 + 7.39150i −0.570528 + 0.821278i
\(82\) 0 0
\(83\) −0.807024 −0.0885824 −0.0442912 − 0.999019i \(-0.514103\pi\)
−0.0442912 + 0.999019i \(0.514103\pi\)
\(84\) 0 0
\(85\) −8.05426 −0.873607
\(86\) 0 0
\(87\) −11.6319 − 7.04363i −1.24707 − 0.755156i
\(88\) 0 0
\(89\) 9.51852i 1.00896i 0.863423 + 0.504481i \(0.168316\pi\)
−0.863423 + 0.504481i \(0.831684\pi\)
\(90\) 0 0
\(91\) 23.8556i 2.50075i
\(92\) 0 0
\(93\) 6.59727 + 3.99494i 0.684105 + 0.414256i
\(94\) 0 0
\(95\) −12.2236 −1.25411
\(96\) 0 0
\(97\) −14.8656 −1.50937 −0.754686 − 0.656086i \(-0.772211\pi\)
−0.754686 + 0.656086i \(0.772211\pi\)
\(98\) 0 0
\(99\) −6.50932 − 12.4478i −0.654211 − 1.25105i
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 2064.2.d.c.431.16 yes 48
3.2 odd 2 inner 2064.2.d.c.431.34 yes 48
4.3 odd 2 inner 2064.2.d.c.431.33 yes 48
12.11 even 2 inner 2064.2.d.c.431.15 ✓ 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2064.2.d.c.431.15 ✓ 48 12.11 even 2 inner
2064.2.d.c.431.16 yes 48 1.1 even 1 trivial
2064.2.d.c.431.33 yes 48 4.3 odd 2 inner
2064.2.d.c.431.34 yes 48 3.2 odd 2 inner