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Properties

Label 513.2.g.c.64.11
Level $513$
Weight $2$
Character 513.64
Analytic conductor $4.096$
Analytic rank $0$
Dimension $32$
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] = [513,2,Mod(64,513)] 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("513.64"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,-1,0,-17,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.09632562369\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 64.11
Character \(\chi\) \(=\) 513.64
Dual form 513.2.g.c.505.11

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.395929 - 0.685769i) q^{2} +(0.686481 + 1.18902i) q^{4} -2.59093 q^{5} +(-0.373088 - 0.646207i) q^{7} +2.67091 q^{8} +(-1.02582 + 1.77678i) q^{10} +(1.28837 + 2.23153i) q^{11} +(3.09365 + 5.35835i) q^{13} -0.590865 q^{14} +(-0.315472 + 0.546414i) q^{16} +(-0.119999 - 0.207845i) q^{17} +(3.89399 - 1.95878i) q^{19} +(-1.77862 - 3.08066i) q^{20} +2.04042 q^{22} +(1.93131 + 3.34513i) q^{23} +1.71290 q^{25} +4.89946 q^{26} +(0.512235 - 0.887218i) q^{28} +6.79737 q^{29} +(-3.77423 + 6.53716i) q^{31} +(2.92071 + 5.05883i) q^{32} -0.190045 q^{34} +(0.966644 + 1.67428i) q^{35} -8.47678 q^{37} +(0.198475 - 3.44592i) q^{38} -6.92012 q^{40} -8.15194 q^{41} +(1.44011 - 2.49434i) q^{43} +(-1.76889 + 3.06380i) q^{44} +3.05865 q^{46} +4.52565 q^{47} +(3.22161 - 5.57999i) q^{49} +(0.678188 - 1.17466i) q^{50} +(-4.24746 + 7.35681i) q^{52} +(5.57774 - 9.66094i) q^{53} +(-3.33808 - 5.78173i) q^{55} +(-0.996483 - 1.72596i) q^{56} +(2.69128 - 4.66143i) q^{58} -0.344246 q^{59} +0.0790199 q^{61} +(2.98865 + 5.17650i) q^{62} +3.36369 q^{64} +(-8.01542 - 13.8831i) q^{65} +(-4.61385 - 7.99142i) q^{67} +(0.164754 - 0.285363i) q^{68} +1.53089 q^{70} +(-2.15288 - 3.72891i) q^{71} +(1.63071 + 2.82448i) q^{73} +(-3.35620 + 5.81312i) q^{74} +(5.00218 + 3.28537i) q^{76} +(0.961354 - 1.66511i) q^{77} +(3.57283 - 6.18833i) q^{79} +(0.817366 - 1.41572i) q^{80} +(-3.22759 + 5.59035i) q^{82} +(-1.78498 - 3.09167i) q^{83} +(0.310909 + 0.538510i) q^{85} +(-1.14036 - 1.97516i) q^{86} +(3.44113 + 5.96021i) q^{88} +(-5.21555 + 9.03360i) q^{89} +(2.30841 - 3.99827i) q^{91} +(-2.65162 + 4.59274i) q^{92} +(1.79184 - 3.10355i) q^{94} +(-10.0891 + 5.07505i) q^{95} +(1.20626 - 2.08930i) q^{97} +(-2.55106 - 4.41856i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - q^{2} - 17 q^{4} + 6 q^{5} + q^{7} + 36 q^{8} - 8 q^{10} - 7 q^{11} - 4 q^{13} + 2 q^{14} - 11 q^{16} + 7 q^{17} + 7 q^{19} + 3 q^{20} + 16 q^{22} - 5 q^{23} + 18 q^{25} + 4 q^{26} - 10 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/513\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.395929 − 0.685769i 0.279964 − 0.484912i −0.691411 − 0.722461i \(-0.743011\pi\)
0.971375 + 0.237549i \(0.0763441\pi\)
\(3\) 0 0
\(4\) 0.686481 + 1.18902i 0.343240 + 0.594510i
\(5\) −2.59093 −1.15870 −0.579349 − 0.815080i \(-0.696693\pi\)
−0.579349 + 0.815080i \(0.696693\pi\)
\(6\) 0 0
\(7\) −0.373088 − 0.646207i −0.141014 − 0.244243i 0.786865 − 0.617126i \(-0.211703\pi\)
−0.927879 + 0.372882i \(0.878370\pi\)
\(8\) 2.67091 0.944308
\(9\) 0 0
\(10\) −1.02582 + 1.77678i −0.324394 + 0.561866i
\(11\) 1.28837 + 2.23153i 0.388460 + 0.672832i 0.992243 − 0.124317i \(-0.0396740\pi\)
−0.603783 + 0.797149i \(0.706341\pi\)
\(12\) 0 0
\(13\) 3.09365 + 5.35835i 0.858023 + 1.48614i 0.873812 + 0.486265i \(0.161641\pi\)
−0.0157882 + 0.999875i \(0.505026\pi\)
\(14\) −0.590865 −0.157915
\(15\) 0 0
\(16\) −0.315472 + 0.546414i −0.0788681 + 0.136604i
\(17\) −0.119999 − 0.207845i −0.0291041 − 0.0504097i 0.851107 − 0.524993i \(-0.175932\pi\)
−0.880211 + 0.474583i \(0.842599\pi\)
\(18\) 0 0
\(19\) 3.89399 − 1.95878i 0.893343 − 0.449375i
\(20\) −1.77862 − 3.08066i −0.397712 − 0.688857i
\(21\) 0 0
\(22\) 2.04042 0.435019
\(23\) 1.93131 + 3.34513i 0.402707 + 0.697509i 0.994052 − 0.108910i \(-0.0347361\pi\)
−0.591345 + 0.806419i \(0.701403\pi\)
\(24\) 0 0
\(25\) 1.71290 0.342581
\(26\) 4.89946 0.960863
\(27\) 0 0
\(28\) 0.512235 − 0.887218i 0.0968034 − 0.167668i
\(29\) 6.79737 1.26224 0.631120 − 0.775685i \(-0.282595\pi\)
0.631120 + 0.775685i \(0.282595\pi\)
\(30\) 0 0
\(31\) −3.77423 + 6.53716i −0.677872 + 1.17411i 0.297749 + 0.954644i \(0.403764\pi\)
−0.975621 + 0.219464i \(0.929569\pi\)
\(32\) 2.92071 + 5.05883i 0.516314 + 0.894283i
\(33\) 0 0
\(34\) −0.190045 −0.0325924
\(35\) 0.966644 + 1.67428i 0.163393 + 0.283004i
\(36\) 0 0
\(37\) −8.47678 −1.39357 −0.696787 − 0.717278i \(-0.745388\pi\)
−0.696787 + 0.717278i \(0.745388\pi\)
\(38\) 0.198475 − 3.44592i 0.0321970 − 0.559002i
\(39\) 0 0
\(40\) −6.92012 −1.09417
\(41\) −8.15194 −1.27312 −0.636559 − 0.771228i \(-0.719643\pi\)
−0.636559 + 0.771228i \(0.719643\pi\)
\(42\) 0 0
\(43\) 1.44011 − 2.49434i 0.219615 − 0.380384i −0.735076 − 0.677985i \(-0.762853\pi\)
0.954690 + 0.297602i \(0.0961867\pi\)
\(44\) −1.76889 + 3.06380i −0.266670 + 0.461886i
\(45\) 0 0
\(46\) 3.05865 0.450974
\(47\) 4.52565 0.660134 0.330067 − 0.943957i \(-0.392929\pi\)
0.330067 + 0.943957i \(0.392929\pi\)
\(48\) 0 0
\(49\) 3.22161 − 5.57999i 0.460230 − 0.797142i
\(50\) 0.678188 − 1.17466i 0.0959103 − 0.166122i
\(51\) 0 0
\(52\) −4.24746 + 7.35681i −0.589016 + 1.02021i
\(53\) 5.57774 − 9.66094i 0.766162 − 1.32703i −0.173468 − 0.984840i \(-0.555497\pi\)
0.939630 − 0.342192i \(-0.111169\pi\)
\(54\) 0 0
\(55\) −3.33808 − 5.78173i −0.450107 − 0.779609i
\(56\) −0.996483 − 1.72596i −0.133161 − 0.230641i
\(57\) 0 0
\(58\) 2.69128 − 4.66143i 0.353382 − 0.612075i
\(59\) −0.344246 −0.0448170 −0.0224085 − 0.999749i \(-0.507133\pi\)
−0.0224085 + 0.999749i \(0.507133\pi\)
\(60\) 0 0
\(61\) 0.0790199 0.0101175 0.00505873 − 0.999987i \(-0.498390\pi\)
0.00505873 + 0.999987i \(0.498390\pi\)
\(62\) 2.98865 + 5.17650i 0.379559 + 0.657416i
\(63\) 0 0
\(64\) 3.36369 0.420462
\(65\) −8.01542 − 13.8831i −0.994190 − 1.72199i
\(66\) 0 0
\(67\) −4.61385 − 7.99142i −0.563671 − 0.976307i −0.997172 − 0.0751540i \(-0.976055\pi\)
0.433501 − 0.901153i \(-0.357278\pi\)
\(68\) 0.164754 − 0.285363i 0.0199794 − 0.0346053i
\(69\) 0 0
\(70\) 1.53089 0.182976
\(71\) −2.15288 − 3.72891i −0.255500 − 0.442540i 0.709531 − 0.704674i \(-0.248907\pi\)
−0.965031 + 0.262135i \(0.915574\pi\)
\(72\) 0 0
\(73\) 1.63071 + 2.82448i 0.190861 + 0.330580i 0.945536 − 0.325518i \(-0.105539\pi\)
−0.754675 + 0.656099i \(0.772206\pi\)
\(74\) −3.35620 + 5.81312i −0.390151 + 0.675761i
\(75\) 0 0
\(76\) 5.00218 + 3.28537i 0.573789 + 0.376858i
\(77\) 0.961354 − 1.66511i 0.109556 − 0.189757i
\(78\) 0 0
\(79\) 3.57283 − 6.18833i 0.401975 − 0.696241i −0.591989 − 0.805946i \(-0.701657\pi\)
0.993964 + 0.109705i \(0.0349905\pi\)
\(80\) 0.817366 − 1.41572i 0.0913843 − 0.158282i
\(81\) 0 0
\(82\) −3.22759 + 5.59035i −0.356427 + 0.617350i
\(83\) −1.78498 − 3.09167i −0.195927 − 0.339355i 0.751277 − 0.659987i \(-0.229438\pi\)
−0.947204 + 0.320632i \(0.896105\pi\)
\(84\) 0 0
\(85\) 0.310909 + 0.538510i 0.0337228 + 0.0584096i
\(86\) −1.14036 − 1.97516i −0.122968 − 0.212987i
\(87\) 0 0
\(88\) 3.44113 + 5.96021i 0.366825 + 0.635360i
\(89\) −5.21555 + 9.03360i −0.552848 + 0.957560i 0.445220 + 0.895421i \(0.353125\pi\)
−0.998068 + 0.0621388i \(0.980208\pi\)
\(90\) 0 0
\(91\) 2.30841 − 3.99827i 0.241987 − 0.419133i
\(92\) −2.65162 + 4.59274i −0.276450 + 0.478826i
\(93\) 0 0
\(94\) 1.79184 − 3.10355i 0.184814 − 0.320107i
\(95\) −10.0891 + 5.07505i −1.03512 + 0.520689i
\(96\) 0 0
\(97\) 1.20626 − 2.08930i 0.122477 − 0.212136i −0.798267 − 0.602304i \(-0.794250\pi\)
0.920744 + 0.390168i \(0.127583\pi\)
\(98\) −2.55106 − 4.41856i −0.257696 − 0.446342i
\(99\) 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 513.2.g.c.64.11 32
3.2 odd 2 171.2.g.c.121.6 yes 32
9.2 odd 6 171.2.h.c.7.11 yes 32
9.7 even 3 513.2.h.c.235.6 32
19.11 even 3 513.2.h.c.334.6 32
57.11 odd 6 171.2.h.c.49.11 yes 32
171.11 odd 6 171.2.g.c.106.6 ✓ 32
171.106 even 3 inner 513.2.g.c.505.11 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.c.106.6 ✓ 32 171.11 odd 6
171.2.g.c.121.6 yes 32 3.2 odd 2
171.2.h.c.7.11 yes 32 9.2 odd 6
171.2.h.c.49.11 yes 32 57.11 odd 6
513.2.g.c.64.11 32 1.1 even 1 trivial
513.2.g.c.505.11 32 171.106 even 3 inner
513.2.h.c.235.6 32 9.7 even 3
513.2.h.c.334.6 32 19.11 even 3