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Properties

Label 513.2.g.c.505.11
Level $513$
Weight $2$
Character 513.505
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 505.11
Character \(\chi\) \(=\) 513.505
Dual form 513.2.g.c.64.11

$q$-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 \(\left(\mathbb{Z}/513\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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 0.971375 − 0.237549i \(-0.0763441\pi\)
−0.691411 + 0.722461i \(0.743011\pi\)
\(3\) 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 −0.927879 − 0.372882i \(-0.878370\pi\)
0.786865 + 0.617126i \(0.211703\pi\)
\(8\) 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 −0.603783 − 0.797149i \(-0.706341\pi\)
0.992243 + 0.124317i \(0.0396740\pi\)
\(12\) 0 0
\(13\) 3.09365 − 5.35835i 0.858023 − 1.48614i −0.0157882 − 0.999875i \(-0.505026\pi\)
0.873812 − 0.486265i \(-0.161641\pi\)
\(14\) −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 −0.880211 − 0.474583i \(-0.842599\pi\)
0.851107 + 0.524993i \(0.175932\pi\)
\(18\) 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 −0.591345 − 0.806419i \(-0.701403\pi\)
0.994052 + 0.108910i \(0.0347361\pi\)
\(24\) 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 −0.975621 − 0.219464i \(-0.929569\pi\)
0.297749 − 0.954644i \(-0.403764\pi\)
\(32\) 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 0.954690 − 0.297602i \(-0.0961867\pi\)
−0.735076 + 0.677985i \(0.762853\pi\)
\(44\) −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 0.939630 + 0.342192i \(0.111169\pi\)
−0.173468 + 0.984840i \(0.555497\pi\)
\(54\) 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 0.433501 + 0.901153i \(0.357278\pi\)
−0.997172 + 0.0751540i \(0.976055\pi\)
\(68\) 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 −0.965031 − 0.262135i \(-0.915574\pi\)
0.709531 + 0.704674i \(0.248907\pi\)
\(72\) 0 0
\(73\) 1.63071 − 2.82448i 0.190861 − 0.330580i −0.754675 − 0.656099i \(-0.772206\pi\)
0.945536 + 0.325518i \(0.105539\pi\)
\(74\) −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 0.993964 − 0.109705i \(-0.0349905\pi\)
−0.591989 + 0.805946i \(0.701657\pi\)
\(80\) 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 −0.947204 − 0.320632i \(-0.896105\pi\)
0.751277 + 0.659987i \(0.229438\pi\)
\(84\) 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 −0.998068 − 0.0621388i \(-0.980208\pi\)
0.445220 − 0.895421i \(-0.353125\pi\)
\(90\) 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 0.920744 − 0.390168i \(-0.127583\pi\)
−0.798267 + 0.602304i \(0.794250\pi\)
\(98\) −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.505.11 32
3.2 odd 2 171.2.g.c.106.6 ✓ 32
9.4 even 3 513.2.h.c.334.6 32
9.5 odd 6 171.2.h.c.49.11 yes 32
19.7 even 3 513.2.h.c.235.6 32
57.26 odd 6 171.2.h.c.7.11 yes 32
171.121 even 3 inner 513.2.g.c.64.11 32
171.140 odd 6 171.2.g.c.121.6 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.c.106.6 ✓ 32 3.2 odd 2
171.2.g.c.121.6 yes 32 171.140 odd 6
171.2.h.c.7.11 yes 32 57.26 odd 6
171.2.h.c.49.11 yes 32 9.5 odd 6
513.2.g.c.64.11 32 171.121 even 3 inner
513.2.g.c.505.11 32 1.1 even 1 trivial
513.2.h.c.235.6 32 19.7 even 3
513.2.h.c.334.6 32 9.4 even 3