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Properties

Label 513.2.h.c.235.6
Level $513$
Weight $2$
Character 513.235
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(235,513)] 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("513.235"); 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([4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,2,0,34,-3] 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 235.6
Character \(\chi\) \(=\) 513.235
Dual form 513.2.h.c.334.6

$q$-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)
 
\(f(q)\) \(=\) \(q-0.791858 q^{2} -1.37296 q^{4} +(1.29546 - 2.24381i) q^{5} +(-0.373088 + 0.646207i) q^{7} +2.67091 q^{8} +(-1.02582 + 1.77678i) q^{10} +(1.28837 - 2.23153i) q^{11} -6.18729 q^{13} +(0.295433 - 0.511704i) q^{14} +0.630945 q^{16} +(-0.119999 - 0.207845i) q^{17} +(3.89399 - 1.95878i) q^{19} +(-1.77862 + 3.08066i) q^{20} +(-1.02021 + 1.76705i) q^{22} -3.86263 q^{23} +(-0.856452 - 1.48342i) q^{25} +4.89946 q^{26} +(0.512235 - 0.887218i) q^{28} +(-3.39869 - 5.88670i) q^{29} +(-3.77423 - 6.53716i) q^{31} -5.84143 q^{32} +(0.0950223 + 0.164583i) q^{34} +(0.966644 + 1.67428i) q^{35} -8.47678 q^{37} +(-3.08349 + 1.55107i) q^{38} +(3.46006 - 5.99300i) q^{40} +(4.07597 - 7.05978i) q^{41} -2.88022 q^{43} +(-1.76889 + 3.06380i) q^{44} +3.05865 q^{46} +(-2.26283 - 3.91933i) q^{47} +(3.22161 + 5.57999i) q^{49} +(0.678188 + 1.17466i) q^{50} +8.49492 q^{52} +(5.57774 - 9.66094i) q^{53} +(-3.33808 - 5.78173i) q^{55} +(-0.996483 + 1.72596i) q^{56} +(2.69128 + 4.66143i) q^{58} +(0.172123 - 0.298126i) q^{59} +(-0.0395099 - 0.0684332i) q^{61} +(2.98865 + 5.17650i) q^{62} +3.36369 q^{64} +(-8.01542 + 13.8831i) q^{65} +9.22770 q^{67} +(0.164754 + 0.285363i) q^{68} +(-0.765444 - 1.32579i) q^{70} +(-2.15288 - 3.72891i) q^{71} +(1.63071 + 2.82448i) q^{73} +6.71241 q^{74} +(-5.34630 + 2.68933i) q^{76} +(0.961354 + 1.66511i) q^{77} -7.14567 q^{79} +(0.817366 - 1.41572i) q^{80} +(-3.22759 + 5.59035i) q^{82} +(-1.78498 + 3.09167i) q^{83} -0.621818 q^{85} +2.28072 q^{86} +(3.44113 - 5.96021i) q^{88} +(-5.21555 + 9.03360i) q^{89} +(2.30841 - 3.99827i) q^{91} +5.30324 q^{92} +(1.79184 + 3.10355i) q^{94} +(0.649404 - 11.2749i) q^{95} -2.41251 q^{97} +(-2.55106 - 4.41856i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 2 q^{2} + 34 q^{4} - 3 q^{5} + q^{7} + 36 q^{8} - 8 q^{10} - 7 q^{11} + 8 q^{13} - q^{14} + 22 q^{16} + 7 q^{17} + 7 q^{19} + 3 q^{20} - 8 q^{22} + 10 q^{23} - 9 q^{25} + 4 q^{26} - 10 q^{28} - 10 q^{29}+ \cdots - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.791858 −0.559928 −0.279964 − 0.960010i \(-0.590323\pi\)
−0.279964 + 0.960010i \(0.590323\pi\)
\(3\) 0 0
\(4\) −1.37296 −0.686481
\(5\) 1.29546 − 2.24381i 0.579349 − 1.00346i −0.416205 − 0.909271i \(-0.636640\pi\)
0.995554 − 0.0941911i \(-0.0300265\pi\)
\(6\) 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
\(13\) −6.18729 −1.71605 −0.858023 − 0.513611i \(-0.828308\pi\)
−0.858023 + 0.513611i \(0.828308\pi\)
\(14\) 0.295433 − 0.511704i 0.0789577 − 0.136759i
\(15\) 0 0
\(16\) 0.630945 0.157736
\(17\) −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
\(22\) −1.02021 + 1.76705i −0.217509 + 0.376737i
\(23\) −3.86263 −0.805414 −0.402707 − 0.915329i \(-0.631931\pi\)
−0.402707 + 0.915329i \(0.631931\pi\)
\(24\) 0 0
\(25\) −0.856452 − 1.48342i −0.171290 − 0.296684i
\(26\) 4.89946 0.960863
\(27\) 0 0
\(28\) 0.512235 − 0.887218i 0.0968034 − 0.167668i
\(29\) −3.39869 − 5.88670i −0.631120 − 1.09313i −0.987323 − 0.158724i \(-0.949262\pi\)
0.356203 − 0.934409i \(-0.384071\pi\)
\(30\) 0 0
\(31\) −3.77423 − 6.53716i −0.677872 − 1.17411i −0.975621 − 0.219464i \(-0.929569\pi\)
0.297749 − 0.954644i \(-0.403764\pi\)
\(32\) −5.84143 −1.03263
\(33\) 0 0
\(34\) 0.0950223 + 0.164583i 0.0162962 + 0.0282258i
\(35\) 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 \(0.745388\pi\)
\(38\) −3.08349 + 1.55107i −0.500208 + 0.251617i
\(39\) 0 0
\(40\) 3.46006 − 5.99300i 0.547084 − 0.947577i
\(41\) 4.07597 − 7.05978i 0.636559 − 1.10255i −0.349623 − 0.936890i \(-0.613690\pi\)
0.986182 − 0.165663i \(-0.0529763\pi\)
\(42\) 0 0
\(43\) −2.88022 −0.439229 −0.219615 − 0.975587i \(-0.570480\pi\)
−0.219615 + 0.975587i \(0.570480\pi\)
\(44\) −1.76889 + 3.06380i −0.266670 + 0.461886i
\(45\) 0 0
\(46\) 3.05865 0.450974
\(47\) −2.26283 − 3.91933i −0.330067 − 0.571693i 0.652458 − 0.757825i \(-0.273738\pi\)
−0.982525 + 0.186132i \(0.940405\pi\)
\(48\) 0 0
\(49\) 3.22161 + 5.57999i 0.460230 + 0.797142i
\(50\) 0.678188 + 1.17466i 0.0959103 + 0.166122i
\(51\) 0 0
\(52\) 8.49492 1.17803
\(53\) 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
\(59\) 0.172123 − 0.298126i 0.0224085 − 0.0388127i −0.854604 − 0.519281i \(-0.826200\pi\)
0.877012 + 0.480468i \(0.159533\pi\)
\(60\) 0 0
\(61\) −0.0395099 − 0.0684332i −0.00505873 − 0.00876198i 0.863485 − 0.504375i \(-0.168277\pi\)
−0.868544 + 0.495613i \(0.834944\pi\)
\(62\) 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
\(67\) 9.22770 1.12734 0.563671 − 0.825999i \(-0.309388\pi\)
0.563671 + 0.825999i \(0.309388\pi\)
\(68\) 0.164754 + 0.285363i 0.0199794 + 0.0346053i
\(69\) 0 0
\(70\) −0.765444 − 1.32579i −0.0914881 − 0.158462i
\(71\) −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 \(0.772206\pi\)
\(74\) 6.71241 0.780302
\(75\) 0 0
\(76\) −5.34630 + 2.68933i −0.613263 + 0.308487i
\(77\) 0.961354 + 1.66511i 0.109556 + 0.189757i
\(78\) 0 0
\(79\) −7.14567 −0.803950 −0.401975 − 0.915651i \(-0.631676\pi\)
−0.401975 + 0.915651i \(0.631676\pi\)
\(80\) 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
\(85\) −0.621818 −0.0674457
\(86\) 2.28072 0.245937
\(87\) 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
\(92\) 5.30324 0.552901
\(93\) 0 0
\(94\) 1.79184 + 3.10355i 0.184814 + 0.320107i
\(95\) 0.649404 − 11.2749i 0.0666274 − 1.15678i
\(96\) 0 0
\(97\) −2.41251 −0.244954 −0.122477 − 0.992471i \(-0.539084\pi\)
−0.122477 + 0.992471i \(0.539084\pi\)
\(98\) −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.h.c.235.6 32
3.2 odd 2 171.2.h.c.7.11 yes 32
9.4 even 3 513.2.g.c.64.11 32
9.5 odd 6 171.2.g.c.121.6 yes 32
19.11 even 3 513.2.g.c.505.11 32
57.11 odd 6 171.2.g.c.106.6 ✓ 32
171.49 even 3 inner 513.2.h.c.334.6 32
171.68 odd 6 171.2.h.c.49.11 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.c.106.6 ✓ 32 57.11 odd 6
171.2.g.c.121.6 yes 32 9.5 odd 6
171.2.h.c.7.11 yes 32 3.2 odd 2
171.2.h.c.49.11 yes 32 171.68 odd 6
513.2.g.c.64.11 32 9.4 even 3
513.2.g.c.505.11 32 19.11 even 3
513.2.h.c.235.6 32 1.1 even 1 trivial
513.2.h.c.334.6 32 171.49 even 3 inner