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Properties

Label 384.2.v.a.13.20
Level $384$
Weight $2$
Character 384.13
Analytic conductor $3.066$
Analytic rank $0$
Dimension $512$
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] = [384,2,Mod(13,384)] 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("384.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(32)) chi = DirichletCharacter(H, H._module([0, 15, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.v (of order \(32\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(512\)
Relative dimension: \(32\) over \(\Q(\zeta_{32})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label 13.20
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.20

$q$-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+(0.320440 - 1.37743i) q^{2} +(0.290285 + 0.956940i) q^{3} +(-1.79464 - 0.882769i) q^{4} +(-0.362380 - 3.67931i) q^{5} +(1.41114 - 0.0932050i) q^{6} +(1.08924 - 1.63016i) q^{7} +(-1.79103 + 2.18911i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(-5.18411 - 0.679844i) q^{10} +(-2.84568 + 1.52105i) q^{11} +(0.323802 - 1.97361i) q^{12} +(-0.240556 - 0.0236927i) q^{13} +(-1.89640 - 2.02272i) q^{14} +(3.41568 - 1.41482i) q^{15} +(2.44144 + 3.16850i) q^{16} +(-5.94374 - 2.46198i) q^{17} +(0.498824 + 1.32332i) q^{18} +(6.47447 - 5.31346i) q^{19} +(-2.59764 + 6.92291i) q^{20} +(1.87615 + 0.569125i) q^{21} +(1.18327 + 4.40713i) q^{22} +(-0.594348 + 2.98799i) q^{23} +(-2.61476 - 1.07844i) q^{24} +(-8.50205 + 1.69116i) q^{25} +(-0.109719 + 0.323758i) q^{26} +(-0.773010 - 0.634393i) q^{27} +(-3.39384 + 1.96400i) q^{28} +(2.49129 - 4.66087i) q^{29} +(-0.854298 - 5.15824i) q^{30} +(0.818174 - 0.818174i) q^{31} +(5.14673 - 2.34760i) q^{32} +(-2.28161 - 2.28161i) q^{33} +(-5.29582 + 7.39818i) q^{34} +(-6.39257 - 3.41690i) q^{35} +(1.98263 - 0.263050i) q^{36} +(3.95276 - 4.81646i) q^{37} +(-5.24425 - 10.6208i) q^{38} +(-0.0471573 - 0.237076i) q^{39} +(8.70345 + 5.79645i) q^{40} +(-2.42653 - 0.482668i) q^{41} +(1.38513 - 2.40190i) q^{42} +(-0.861566 + 2.84020i) q^{43} +(6.44969 - 0.217648i) q^{44} +(2.34542 + 2.85790i) q^{45} +(3.92530 + 1.77615i) q^{46} +(3.72788 - 8.99989i) q^{47} +(-2.32335 + 3.25608i) q^{48} +(1.20780 + 2.91590i) q^{49} +(-0.394938 + 12.2529i) q^{50} +(0.630589 - 6.40248i) q^{51} +(0.410796 + 0.254876i) q^{52} +(5.77121 + 10.7972i) q^{53} +(-1.12154 + 0.861484i) q^{54} +(6.62761 + 9.91892i) q^{55} +(1.61775 + 5.30413i) q^{56} +(6.96410 + 4.65326i) q^{57} +(-5.62172 - 4.92511i) q^{58} +(4.74257 - 0.467102i) q^{59} +(-7.37887 - 0.476169i) q^{60} +(-4.34322 + 1.31750i) q^{61} +(-0.864803 - 1.38916i) q^{62} +1.96058i q^{63} +(-1.58444 - 7.84153i) q^{64} +0.893666i q^{65} +(-3.87388 + 2.41164i) q^{66} +(13.4450 - 4.07849i) q^{67} +(8.49349 + 9.66531i) q^{68} +(-3.03186 + 0.298612i) q^{69} +(-6.75499 + 7.71042i) q^{70} +(10.9379 + 7.30845i) q^{71} +(0.272979 - 2.81522i) q^{72} +(3.83429 + 5.73842i) q^{73} +(-5.36772 - 6.98805i) q^{74} +(-4.08636 - 7.64504i) q^{75} +(-16.3099 + 3.82026i) q^{76} +(-0.620071 + 6.29569i) q^{77} +(-0.341667 - 0.0110127i) q^{78} +(0.484546 + 1.16980i) q^{79} +(10.7732 - 10.1310i) q^{80} +(0.382683 - 0.923880i) q^{81} +(-1.44240 + 3.18772i) q^{82} +(-4.93831 - 6.01734i) q^{83} +(-2.86461 - 2.67758i) q^{84} +(-6.90448 + 22.7610i) q^{85} +(3.63611 + 2.09686i) q^{86} +(5.18336 + 1.03103i) q^{87} +(1.76695 - 8.95375i) q^{88} +(-2.97495 - 14.9561i) q^{89} +(4.68814 - 2.31487i) q^{90} +(-0.300646 + 0.366338i) q^{91} +(3.70434 - 4.83768i) q^{92} +(1.02045 + 0.545440i) q^{93} +(-11.2022 - 8.01883i) q^{94} +(-21.8961 - 21.8961i) q^{95} +(3.74053 + 4.24364i) q^{96} +(-7.36441 + 7.36441i) q^{97} +(4.40348 - 0.729296i) q^{98} +(1.52105 - 2.84568i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 512 q - 96 q^{50} - 96 q^{52} - 32 q^{54} - 224 q^{56} - 192 q^{60} - 192 q^{62} - 192 q^{64} - 192 q^{66} - 192 q^{68} - 192 q^{70} - 224 q^{74} - 32 q^{76} - 96 q^{78} - 96 q^{80}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(e\left(\frac{15}{32}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.320440 − 1.37743i 0.226586 − 0.973991i
\(3\) 0.290285 + 0.956940i 0.167596 + 0.552490i
\(4\) −1.79464 − 0.882769i −0.897318 − 0.441385i
\(5\) −0.362380 − 3.67931i −0.162061 − 1.64544i −0.640966 − 0.767569i \(-0.721466\pi\)
0.478905 − 0.877867i \(-0.341034\pi\)
\(6\) 1.41114 − 0.0932050i 0.576095 − 0.0380508i
\(7\) 1.08924 − 1.63016i 0.411693 − 0.616142i −0.566445 − 0.824100i \(-0.691682\pi\)
0.978138 + 0.207957i \(0.0666815\pi\)
\(8\) −1.79103 + 2.18911i −0.633224 + 0.773969i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) −5.18411 − 0.679844i −1.63936 − 0.214986i
\(11\) −2.84568 + 1.52105i −0.858004 + 0.458613i −0.840868 − 0.541240i \(-0.817955\pi\)
−0.0171361 + 0.999853i \(0.505455\pi\)
\(12\) 0.323802 − 1.97361i 0.0934737 − 0.569733i
\(13\) −0.240556 − 0.0236927i −0.0667183 − 0.00657118i 0.0646029 − 0.997911i \(-0.479422\pi\)
−0.131321 + 0.991340i \(0.541922\pi\)
\(14\) −1.89640 − 2.02272i −0.506834 − 0.540594i
\(15\) 3.41568 − 1.41482i 0.881926 − 0.365306i
\(16\) 2.44144 + 3.16850i 0.610359 + 0.792125i
\(17\) −5.94374 − 2.46198i −1.44157 − 0.597117i −0.481391 − 0.876506i \(-0.659868\pi\)
−0.960178 + 0.279389i \(0.909868\pi\)
\(18\) 0.498824 + 1.32332i 0.117574 + 0.311909i
\(19\) 6.47447 − 5.31346i 1.48534 − 1.21899i 0.566689 − 0.823932i \(-0.308224\pi\)
0.918656 − 0.395059i \(-0.129276\pi\)
\(20\) −2.59764 + 6.92291i −0.580850 + 1.54801i
\(21\) 1.87615 + 0.569125i 0.409410 + 0.124193i
\(22\) 1.18327 + 4.40713i 0.252273 + 0.939603i
\(23\) −0.594348 + 2.98799i −0.123930 + 0.623039i 0.868033 + 0.496507i \(0.165384\pi\)
−0.991963 + 0.126531i \(0.959616\pi\)
\(24\) −2.61476 − 1.07844i −0.533735 − 0.220136i
\(25\) −8.50205 + 1.69116i −1.70041 + 0.338233i
\(26\) −0.109719 + 0.323758i −0.0215177 + 0.0634941i
\(27\) −0.773010 − 0.634393i −0.148766 − 0.122089i
\(28\) −3.39384 + 1.96400i −0.641375 + 0.371161i
\(29\) 2.49129 − 4.66087i 0.462621 − 0.865502i −0.537137 − 0.843495i \(-0.680494\pi\)
0.999758 − 0.0220074i \(-0.00700573\pi\)
\(30\) −0.854298 − 5.15824i −0.155973 − 0.941761i
\(31\) 0.818174 − 0.818174i 0.146948 − 0.146948i −0.629805 − 0.776753i \(-0.716865\pi\)
0.776753 + 0.629805i \(0.216865\pi\)
\(32\) 5.14673 − 2.34760i 0.909821 − 0.415000i
\(33\) −2.28161 − 2.28161i −0.397177 − 0.397177i
\(34\) −5.29582 + 7.39818i −0.908226 + 1.26878i
\(35\) −6.39257 − 3.41690i −1.08054 − 0.577562i
\(36\) 1.98263 − 0.263050i 0.330438 − 0.0438417i
\(37\) 3.95276 − 4.81646i 0.649831 − 0.791821i −0.339013 − 0.940782i \(-0.610093\pi\)
0.988843 + 0.148961i \(0.0475929\pi\)
\(38\) −5.24425 − 10.6208i −0.850729 − 1.72292i
\(39\) −0.0471573 − 0.237076i −0.00755121 − 0.0379625i
\(40\) 8.70345 + 5.79645i 1.37614 + 0.916499i
\(41\) −2.42653 − 0.482668i −0.378961 − 0.0753800i 0.00193451 − 0.999998i \(-0.499384\pi\)
−0.380895 + 0.924618i \(0.624384\pi\)
\(42\) 1.38513 − 2.40190i 0.213730 − 0.370622i
\(43\) −0.861566 + 2.84020i −0.131388 + 0.433127i −0.997807 − 0.0661900i \(-0.978916\pi\)
0.866419 + 0.499317i \(0.166416\pi\)
\(44\) 6.44969 − 0.217648i 0.972327 − 0.0328116i
\(45\) 2.34542 + 2.85790i 0.349635 + 0.426031i
\(46\) 3.92530 + 1.77615i 0.578753 + 0.261878i
\(47\) 3.72788 − 8.99989i 0.543767 − 1.31277i −0.378280 − 0.925691i \(-0.623484\pi\)
0.922047 − 0.387078i \(-0.126516\pi\)
\(48\) −2.32335 + 3.25608i −0.335347 + 0.469974i
\(49\) 1.20780 + 2.91590i 0.172543 + 0.416556i
\(50\) −0.394938 + 12.2529i −0.0558527 + 1.73282i
\(51\) 0.630589 − 6.40248i 0.0883002 − 0.896527i
\(52\) 0.410796 + 0.254876i 0.0569671 + 0.0353449i
\(53\) 5.77121 + 10.7972i 0.792737 + 1.48311i 0.873741 + 0.486392i \(0.161687\pi\)
−0.0810041 + 0.996714i \(0.525813\pi\)
\(54\) −1.12154 + 0.861484i −0.152622 + 0.117233i
\(55\) 6.62761 + 9.91892i 0.893667 + 1.33747i
\(56\) 1.61775 + 5.30413i 0.216181 + 0.708794i
\(57\) 6.96410 + 4.65326i 0.922418 + 0.616340i
\(58\) −5.62172 − 4.92511i −0.738169 − 0.646699i
\(59\) 4.74257 − 0.467102i 0.617429 − 0.0608115i 0.215533 − 0.976497i \(-0.430851\pi\)
0.401897 + 0.915685i \(0.368351\pi\)
\(60\) −7.37887 − 0.476169i −0.952608 − 0.0614732i
\(61\) −4.34322 + 1.31750i −0.556093 + 0.168689i −0.555807 − 0.831311i \(-0.687591\pi\)
−0.000285186 1.00000i \(0.500091\pi\)
\(62\) −0.864803 − 1.38916i −0.109830 − 0.176423i
\(63\) 1.96058i 0.247009i
\(64\) −1.58444 − 7.84153i −0.198055 − 0.980191i
\(65\) 0.893666i 0.110846i
\(66\) −3.87388 + 2.41164i −0.476841 + 0.296852i
\(67\) 13.4450 − 4.07849i 1.64257 − 0.498267i 0.672294 − 0.740284i \(-0.265309\pi\)
0.970272 + 0.242017i \(0.0778091\pi\)
\(68\) 8.49349 + 9.66531i 1.02999 + 1.17209i
\(69\) −3.03186 + 0.298612i −0.364993 + 0.0359486i
\(70\) −6.75499 + 7.71042i −0.807375 + 0.921572i
\(71\) 10.9379 + 7.30845i 1.29809 + 0.867353i 0.996293 − 0.0860202i \(-0.0274150\pi\)
0.301793 + 0.953374i \(0.402415\pi\)
\(72\) 0.272979 − 2.81522i 0.0321709 − 0.331777i
\(73\) 3.83429 + 5.73842i 0.448770 + 0.671631i 0.985022 − 0.172426i \(-0.0551606\pi\)
−0.536253 + 0.844057i \(0.680161\pi\)
\(74\) −5.36772 − 6.98805i −0.623984 − 0.812344i
\(75\) −4.08636 − 7.64504i −0.471852 − 0.882773i
\(76\) −16.3099 + 3.82026i −1.87087 + 0.438214i
\(77\) −0.620071 + 6.29569i −0.0706637 + 0.717460i
\(78\) −0.341667 − 0.0110127i −0.0386861 − 0.00124694i
\(79\) 0.484546 + 1.16980i 0.0545156 + 0.131612i 0.948791 − 0.315905i \(-0.102308\pi\)
−0.894275 + 0.447518i \(0.852308\pi\)
\(80\) 10.7732 − 10.1310i 1.20447 − 1.13268i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) −1.44240 + 3.18772i −0.159287 + 0.352025i
\(83\) −4.93831 − 6.01734i −0.542049 − 0.660489i 0.427891 − 0.903830i \(-0.359257\pi\)
−0.969941 + 0.243341i \(0.921757\pi\)
\(84\) −2.86461 − 2.67758i −0.312554 − 0.292148i
\(85\) −6.90448 + 22.7610i −0.748896 + 2.46878i
\(86\) 3.63611 + 2.09686i 0.392091 + 0.226111i
\(87\) 5.18336 + 1.03103i 0.555714 + 0.110538i
\(88\) 1.76695 − 8.95375i 0.188357 − 0.954473i
\(89\) −2.97495 − 14.9561i −0.315345 − 1.58534i −0.735263 − 0.677782i \(-0.762941\pi\)
0.419918 − 0.907562i \(-0.362059\pi\)
\(90\) 4.68814 − 2.31487i 0.494173 − 0.244009i
\(91\) −0.300646 + 0.366338i −0.0315163 + 0.0384027i
\(92\) 3.70434 − 4.83768i 0.386204 − 0.504363i
\(93\) 1.02045 + 0.545440i 0.105815 + 0.0565595i
\(94\) −11.2022 − 8.01883i −1.15542 − 0.827079i
\(95\) −21.8961 − 21.8961i −2.24649 − 2.24649i
\(96\) 3.74053 + 4.24364i 0.381766 + 0.433114i
\(97\) −7.36441 + 7.36441i −0.747742 + 0.747742i −0.974055 − 0.226312i \(-0.927333\pi\)
0.226312 + 0.974055i \(0.427333\pi\)
\(98\) 4.40348 − 0.729296i 0.444818 − 0.0736700i
\(99\) 1.52105 − 2.84568i 0.152871 − 0.286001i
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 384.2.v.a.13.20 ✓ 512
128.69 even 32 inner 384.2.v.a.325.20 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.20 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.20 yes 512 128.69 even 32 inner