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Properties

Label 384.2.v.a.13.10
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 13.10
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.10

$q$-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+(-0.870630 + 1.11445i) q^{2} +(0.290285 + 0.956940i) q^{3} +(-0.484007 - 1.94055i) q^{4} +(-0.328555 - 3.33588i) q^{5} +(-1.31919 - 0.509633i) q^{6} +(-1.53196 + 2.29274i) q^{7} +(2.58404 + 1.15010i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(4.00372 + 2.53816i) q^{10} +(-2.66601 + 1.42501i) q^{11} +(1.71649 - 1.02648i) q^{12} +(-6.16389 - 0.607091i) q^{13} +(-1.22138 - 3.70343i) q^{14} +(3.09686 - 1.28276i) q^{15} +(-3.53147 + 1.87848i) q^{16} +(1.52921 + 0.633419i) q^{17} +(0.104746 - 1.41033i) q^{18} +(-6.31247 + 5.18051i) q^{19} +(-6.31441 + 2.25216i) q^{20} +(-2.63872 - 0.800448i) q^{21} +(0.733000 - 4.21179i) q^{22} +(-0.448825 + 2.25640i) q^{23} +(-0.350469 + 2.80663i) q^{24} +(-6.11620 + 1.21659i) q^{25} +(6.04304 - 6.34081i) q^{26} +(-0.773010 - 0.634393i) q^{27} +(5.19066 + 1.86315i) q^{28} +(1.20106 - 2.24703i) q^{29} +(-1.26664 + 4.56811i) q^{30} +(0.352623 - 0.352623i) q^{31} +(0.981133 - 5.57112i) q^{32} +(-2.13755 - 2.13755i) q^{33} +(-2.03729 + 1.15276i) q^{34} +(8.15164 + 4.35714i) q^{35} +(1.48055 + 1.34461i) q^{36} +(1.37662 - 1.67741i) q^{37} +(-0.277605 - 11.5453i) q^{38} +(-1.20833 - 6.07471i) q^{39} +(2.98759 - 8.99791i) q^{40} +(0.719462 + 0.143110i) q^{41} +(3.18941 - 2.24384i) q^{42} +(0.498187 - 1.64230i) q^{43} +(4.05567 + 4.48381i) q^{44} +(2.12650 + 2.59114i) q^{45} +(-2.12389 - 2.46468i) q^{46} +(5.00956 - 12.0941i) q^{47} +(-2.82273 - 2.83412i) q^{48} +(-0.230978 - 0.557631i) q^{49} +(3.96912 - 7.87540i) q^{50} +(-0.162238 + 1.64723i) q^{51} +(1.80528 + 12.2552i) q^{52} +(1.75774 + 3.28850i) q^{53} +(1.38001 - 0.309161i) q^{54} +(5.62959 + 8.42528i) q^{55} +(-6.59553 + 4.16263i) q^{56} +(-6.78985 - 4.53684i) q^{57} +(1.45853 + 3.29486i) q^{58} +(-3.13374 + 0.308647i) q^{59} +(-3.98816 - 5.38875i) q^{60} +(-9.72455 + 2.94991i) q^{61} +(0.0859772 + 0.699985i) q^{62} -2.75746i q^{63} +(5.35454 + 5.94381i) q^{64} +20.7614i q^{65} +(4.24321 - 0.521182i) q^{66} +(1.59726 - 0.484525i) q^{67} +(0.489034 - 3.27408i) q^{68} +(-2.28952 + 0.225498i) q^{69} +(-11.9529 + 5.29115i) q^{70} +(-11.9303 - 7.97160i) q^{71} +(-2.78751 + 0.479344i) q^{72} +(-2.78270 - 4.16461i) q^{73} +(0.670872 + 2.99458i) q^{74} +(-2.93964 - 5.49968i) q^{75} +(13.1083 + 9.74227i) q^{76} +(0.817038 - 8.29553i) q^{77} +(7.82198 + 3.94219i) q^{78} +(5.23339 + 12.6345i) q^{79} +(7.42666 + 11.1634i) q^{80} +(0.382683 - 0.923880i) q^{81} +(-0.785875 + 0.677210i) q^{82} +(-0.835860 - 1.01850i) q^{83} +(-0.276150 + 5.50800i) q^{84} +(1.61058 - 5.30936i) q^{85} +(1.39653 + 1.98504i) q^{86} +(2.49893 + 0.497068i) q^{87} +(-8.52798 + 0.616113i) q^{88} +(3.31735 + 16.6774i) q^{89} +(-4.73910 + 0.113951i) q^{90} +(10.8348 - 13.2022i) q^{91} +(4.59589 - 0.221143i) q^{92} +(0.439800 + 0.235078i) q^{93} +(9.11687 + 16.1124i) q^{94} +(19.3555 + 19.3555i) q^{95} +(5.61604 - 0.678325i) q^{96} +(3.58469 - 3.58469i) q^{97} +(0.822550 + 0.228076i) q^{98} +(1.42501 - 2.66601i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.870630 + 1.11445i −0.615628 + 0.788037i
\(3\) 0.290285 + 0.956940i 0.167596 + 0.552490i
\(4\) −0.484007 − 1.94055i −0.242003 − 0.970275i
\(5\) −0.328555 − 3.33588i −0.146934 − 1.49185i −0.732246 − 0.681041i \(-0.761528\pi\)
0.585311 − 0.810809i \(-0.300972\pi\)
\(6\) −1.31919 − 0.509633i −0.538559 − 0.208057i
\(7\) −1.53196 + 2.29274i −0.579027 + 0.866575i −0.999164 − 0.0408855i \(-0.986982\pi\)
0.420137 + 0.907461i \(0.361982\pi\)
\(8\) 2.58404 + 1.15010i 0.913597 + 0.406622i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) 4.00372 + 2.53816i 1.26609 + 0.802635i
\(11\) −2.66601 + 1.42501i −0.803831 + 0.429657i −0.821515 − 0.570187i \(-0.806871\pi\)
0.0176837 + 0.999844i \(0.494371\pi\)
\(12\) 1.71649 − 1.02648i 0.495508 − 0.296319i
\(13\) −6.16389 − 0.607091i −1.70956 − 0.168377i −0.804582 − 0.593842i \(-0.797610\pi\)
−0.904975 + 0.425465i \(0.860110\pi\)
\(14\) −1.22138 − 3.70343i −0.326428 − 0.989783i
\(15\) 3.09686 − 1.28276i 0.799606 − 0.331208i
\(16\) −3.53147 + 1.87848i −0.882869 + 0.469620i
\(17\) 1.52921 + 0.633419i 0.370887 + 0.153627i 0.560339 − 0.828264i \(-0.310671\pi\)
−0.189451 + 0.981890i \(0.560671\pi\)
\(18\) 0.104746 − 1.41033i 0.0246889 − 0.332418i
\(19\) −6.31247 + 5.18051i −1.44818 + 1.18849i −0.501993 + 0.864872i \(0.667400\pi\)
−0.946187 + 0.323619i \(0.895100\pi\)
\(20\) −6.31441 + 2.25216i −1.41195 + 0.503599i
\(21\) −2.63872 − 0.800448i −0.575817 − 0.174672i
\(22\) 0.733000 − 4.21179i 0.156276 − 0.897957i
\(23\) −0.448825 + 2.25640i −0.0935865 + 0.470491i 0.905362 + 0.424641i \(0.139600\pi\)
−0.998948 + 0.0458502i \(0.985400\pi\)
\(24\) −0.350469 + 2.80663i −0.0715392 + 0.572901i
\(25\) −6.11620 + 1.21659i −1.22324 + 0.243317i
\(26\) 6.04304 − 6.34081i 1.18514 − 1.24354i
\(27\) −0.773010 − 0.634393i −0.148766 − 0.122089i
\(28\) 5.19066 + 1.86315i 0.980943 + 0.352102i
\(29\) 1.20106 − 2.24703i 0.223032 − 0.417264i −0.745372 − 0.666648i \(-0.767728\pi\)
0.968405 + 0.249385i \(0.0802283\pi\)
\(30\) −1.26664 + 4.56811i −0.231256 + 0.834019i
\(31\) 0.352623 − 0.352623i 0.0633329 − 0.0633329i −0.674731 − 0.738064i \(-0.735740\pi\)
0.738064 + 0.674731i \(0.235740\pi\)
\(32\) 0.981133 − 5.57112i 0.173441 − 0.984844i
\(33\) −2.13755 − 2.13755i −0.372100 − 0.372100i
\(34\) −2.03729 + 1.15276i −0.349392 + 0.197696i
\(35\) 8.15164 + 4.35714i 1.37788 + 0.736491i
\(36\) 1.48055 + 1.34461i 0.246758 + 0.224102i
\(37\) 1.37662 − 1.67741i 0.226314 − 0.275765i −0.647457 − 0.762102i \(-0.724167\pi\)
0.873771 + 0.486337i \(0.161667\pi\)
\(38\) −0.277605 − 11.5453i −0.0450334 − 1.87289i
\(39\) −1.20833 − 6.07471i −0.193488 − 0.972732i
\(40\) 2.98759 − 8.99791i 0.472379 − 1.42270i
\(41\) 0.719462 + 0.143110i 0.112361 + 0.0223500i 0.250951 − 0.968000i \(-0.419257\pi\)
−0.138589 + 0.990350i \(0.544257\pi\)
\(42\) 3.18941 − 2.24384i 0.492137 − 0.346231i
\(43\) 0.498187 − 1.64230i 0.0759729 − 0.250449i −0.910725 − 0.413013i \(-0.864476\pi\)
0.986698 + 0.162564i \(0.0519764\pi\)
\(44\) 4.05567 + 4.48381i 0.611415 + 0.675959i
\(45\) 2.12650 + 2.59114i 0.316999 + 0.386265i
\(46\) −2.12389 − 2.46468i −0.313150 − 0.363397i
\(47\) 5.00956 − 12.0941i 0.730719 − 1.76411i 0.0905304 − 0.995894i \(-0.471144\pi\)
0.640189 − 0.768218i \(-0.278856\pi\)
\(48\) −2.82273 − 2.83412i −0.407425 − 0.409070i
\(49\) −0.230978 − 0.557631i −0.0329969 − 0.0796616i
\(50\) 3.96912 − 7.87540i 0.561318 − 1.11375i
\(51\) −0.162238 + 1.64723i −0.0227179 + 0.230659i
\(52\) 1.80528 + 12.2552i 0.250347 + 1.69949i
\(53\) 1.75774 + 3.28850i 0.241444 + 0.451711i 0.973233 − 0.229821i \(-0.0738141\pi\)
−0.731788 + 0.681532i \(0.761314\pi\)
\(54\) 1.38001 − 0.309161i 0.187795 − 0.0420715i
\(55\) 5.62959 + 8.42528i 0.759094 + 1.13606i
\(56\) −6.59553 + 4.16263i −0.881365 + 0.556255i
\(57\) −6.78985 − 4.53684i −0.899338 − 0.600919i
\(58\) 1.45853 + 3.29486i 0.191514 + 0.432637i
\(59\) −3.13374 + 0.308647i −0.407979 + 0.0401824i −0.299925 − 0.953963i \(-0.596962\pi\)
−0.108053 + 0.994145i \(0.534462\pi\)
\(60\) −3.98816 − 5.38875i −0.514870 − 0.695685i
\(61\) −9.72455 + 2.94991i −1.24510 + 0.377697i −0.843008 − 0.537901i \(-0.819217\pi\)
−0.402093 + 0.915599i \(0.631717\pi\)
\(62\) 0.0859772 + 0.699985i 0.0109191 + 0.0888982i
\(63\) − 2.75746i − 0.347407i
\(64\) 5.35454 + 5.94381i 0.669318 + 0.742976i
\(65\) 20.7614i 2.57514i
\(66\) 4.24321 − 0.521182i 0.522304 − 0.0641531i
\(67\) 1.59726 − 0.484525i 0.195137 − 0.0591941i −0.191203 − 0.981550i \(-0.561239\pi\)
0.386340 + 0.922356i \(0.373739\pi\)
\(68\) 0.489034 − 3.27408i 0.0593041 − 0.397041i
\(69\) −2.28952 + 0.225498i −0.275626 + 0.0271468i
\(70\) −11.9529 + 5.29115i −1.42864 + 0.632414i
\(71\) −11.9303 − 7.97160i −1.41587 − 0.946055i −0.999317 − 0.0369447i \(-0.988237\pi\)
−0.416555 − 0.909111i \(-0.636763\pi\)
\(72\) −2.78751 + 0.479344i −0.328512 + 0.0564912i
\(73\) −2.78270 − 4.16461i −0.325691 − 0.487431i 0.632104 − 0.774884i \(-0.282192\pi\)
−0.957795 + 0.287453i \(0.907192\pi\)
\(74\) 0.670872 + 2.99458i 0.0779872 + 0.348113i
\(75\) −2.93964 − 5.49968i −0.339440 − 0.635048i
\(76\) 13.1083 + 9.74227i 1.50363 + 1.11751i
\(77\) 0.817038 − 8.29553i 0.0931102 − 0.945363i
\(78\) 7.82198 + 3.94219i 0.885665 + 0.446365i
\(79\) 5.23339 + 12.6345i 0.588802 + 1.42149i 0.884649 + 0.466258i \(0.154398\pi\)
−0.295847 + 0.955235i \(0.595602\pi\)
\(80\) 7.42666 + 11.1634i 0.830326 + 1.24810i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) −0.785875 + 0.677210i −0.0867853 + 0.0747854i
\(83\) −0.835860 − 1.01850i −0.0917475 − 0.111795i 0.725113 − 0.688630i \(-0.241787\pi\)
−0.816861 + 0.576835i \(0.804287\pi\)
\(84\) −0.276150 + 5.50800i −0.0301304 + 0.600972i
\(85\) 1.61058 − 5.30936i 0.174692 − 0.575881i
\(86\) 1.39653 + 1.98504i 0.150592 + 0.214053i
\(87\) 2.49893 + 0.497068i 0.267913 + 0.0532913i
\(88\) −8.52798 + 0.616113i −0.909085 + 0.0656779i
\(89\) 3.31735 + 16.6774i 0.351638 + 1.76781i 0.600855 + 0.799358i \(0.294827\pi\)
−0.249217 + 0.968448i \(0.580173\pi\)
\(90\) −4.73910 + 0.113951i −0.499545 + 0.0120115i
\(91\) 10.8348 − 13.2022i 1.13579 − 1.38396i
\(92\) 4.59589 − 0.221143i 0.479154 − 0.0230558i
\(93\) 0.439800 + 0.235078i 0.0456052 + 0.0243765i
\(94\) 9.11687 + 16.1124i 0.940333 + 1.66187i
\(95\) 19.3555 + 19.3555i 1.98584 + 1.98584i
\(96\) 5.61604 − 0.678325i 0.573184 − 0.0692313i
\(97\) 3.58469 − 3.58469i 0.363971 − 0.363971i −0.501302 − 0.865272i \(-0.667145\pi\)
0.865272 + 0.501302i \(0.167145\pi\)
\(98\) 0.822550 + 0.228076i 0.0830901 + 0.0230392i
\(99\) 1.42501 − 2.66601i 0.143219 − 0.267944i
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.13.10 ✓ 512
128.69 even 32 inner 384.2.v.a.325.10 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.10 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.10 yes 512 128.69 even 32 inner