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Properties

Label 384.2.v.a.13.17
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 13.17
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.17

$q$-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+(-0.156266 + 1.40555i) q^{2} +(-0.290285 - 0.956940i) q^{3} +(-1.95116 - 0.439279i) q^{4} +(-0.103867 - 1.05457i) q^{5} +(1.39039 - 0.258474i) q^{6} +(-0.0783117 + 0.117202i) q^{7} +(0.922330 - 2.67382i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(1.49849 + 0.0188036i) q^{10} +(-2.29838 + 1.22851i) q^{11} +(0.146029 + 1.99466i) q^{12} +(-6.11996 - 0.602763i) q^{13} +(-0.152496 - 0.128386i) q^{14} +(-0.979014 + 0.405521i) q^{15} +(3.61407 + 1.71421i) q^{16} +(-6.95448 - 2.88064i) q^{17} +(-0.650954 - 1.25549i) q^{18} +(0.671982 - 0.551481i) q^{19} +(-0.260592 + 2.10327i) q^{20} +(0.134888 + 0.0409177i) q^{21} +(-1.36758 - 3.42248i) q^{22} +(1.85915 - 9.34658i) q^{23} +(-2.82642 - 0.106446i) q^{24} +(3.80259 - 0.756382i) q^{25} +(1.80355 - 8.50774i) q^{26} +(0.773010 + 0.634393i) q^{27} +(0.204283 - 0.194279i) q^{28} +(-3.35535 + 6.27741i) q^{29} +(-0.416995 - 1.43943i) q^{30} +(3.67878 - 3.67878i) q^{31} +(-2.97417 + 4.81189i) q^{32} +(1.84280 + 1.84280i) q^{33} +(5.13564 - 9.32475i) q^{34} +(0.131732 + 0.0704121i) q^{35} +(1.86638 - 0.718760i) q^{36} +(-1.42782 + 1.73980i) q^{37} +(0.670129 + 1.03068i) q^{38} +(1.19972 + 6.03141i) q^{39} +(-2.91554 - 0.694945i) q^{40} +(-6.97514 - 1.38744i) q^{41} +(-0.0785904 + 0.183198i) q^{42} +(-0.779364 + 2.56922i) q^{43} +(5.02418 - 1.38739i) q^{44} +(0.672252 + 0.819141i) q^{45} +(12.8466 + 4.07368i) q^{46} +(0.674420 - 1.62819i) q^{47} +(0.591288 - 3.95606i) q^{48} +(2.67118 + 6.44880i) q^{49} +(0.468922 + 5.46294i) q^{50} +(-0.737821 + 7.49122i) q^{51} +(11.6762 + 3.86446i) q^{52} +(-4.02295 - 7.52641i) q^{53} +(-1.01247 + 0.987374i) q^{54} +(1.53428 + 2.29622i) q^{55} +(0.241147 + 0.317490i) q^{56} +(-0.722801 - 0.482960i) q^{57} +(-8.29891 - 5.69706i) q^{58} +(-10.7263 + 1.05645i) q^{59} +(2.08835 - 0.361177i) q^{60} +(6.46233 - 1.96033i) q^{61} +(4.59585 + 5.74558i) q^{62} -0.140957i q^{63} +(-6.29862 - 4.93229i) q^{64} +6.51656i q^{65} +(-2.87812 + 2.30219i) q^{66} +(10.0367 - 3.04460i) q^{67} +(12.3039 + 8.67555i) q^{68} +(-9.48380 + 0.934073i) q^{69} +(-0.119553 + 0.174153i) q^{70} +(9.98888 + 6.67435i) q^{71} +(0.718605 + 2.73562i) q^{72} +(1.28206 + 1.91874i) q^{73} +(-2.22227 - 2.27875i) q^{74} +(-1.82765 - 3.41928i) q^{75} +(-1.55340 + 0.780842i) q^{76} +(0.0360066 - 0.365581i) q^{77} +(-8.66494 + 0.743772i) q^{78} +(-5.10188 - 12.3170i) q^{79} +(1.43238 - 3.98935i) q^{80} +(0.382683 - 0.923880i) q^{81} +(3.04010 - 9.58713i) q^{82} +(6.80230 + 8.28863i) q^{83} +(-0.245213 - 0.139090i) q^{84} +(-2.31551 + 7.63321i) q^{85} +(-3.48939 - 1.49692i) q^{86} +(6.98111 + 1.38863i) q^{87} +(1.16495 + 7.27856i) q^{88} +(-1.23326 - 6.20000i) q^{89} +(-1.25640 + 0.816883i) q^{90} +(0.549909 - 0.670066i) q^{91} +(-7.73326 + 17.4200i) q^{92} +(-4.58826 - 2.45248i) q^{93} +(2.18312 + 1.20236i) q^{94} +(-0.651374 - 0.651374i) q^{95} +(5.46805 + 1.44928i) q^{96} +(-12.5043 + 12.5043i) q^{97} +(-9.48155 + 2.74676i) q^{98} +(1.22851 - 2.29838i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.156266 + 1.40555i −0.110496 + 0.993877i
\(3\) −0.290285 − 0.956940i −0.167596 − 0.552490i
\(4\) −1.95116 − 0.439279i −0.975581 − 0.219640i
\(5\) −0.103867 − 1.05457i −0.0464505 − 0.471620i −0.990069 − 0.140584i \(-0.955102\pi\)
0.943618 − 0.331036i \(-0.107398\pi\)
\(6\) 1.39039 − 0.258474i 0.567625 − 0.105522i
\(7\) −0.0783117 + 0.117202i −0.0295990 + 0.0442981i −0.845974 − 0.533224i \(-0.820980\pi\)
0.816375 + 0.577522i \(0.195980\pi\)
\(8\) 0.922330 − 2.67382i 0.326093 − 0.945338i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) 1.49849 + 0.0188036i 0.473865 + 0.00594624i
\(11\) −2.29838 + 1.22851i −0.692989 + 0.370410i −0.779985 − 0.625798i \(-0.784773\pi\)
0.0869960 + 0.996209i \(0.472273\pi\)
\(12\) 0.146029 + 1.99466i 0.0421548 + 0.575809i
\(13\) −6.11996 − 0.602763i −1.69737 − 0.167176i −0.797496 − 0.603324i \(-0.793842\pi\)
−0.899875 + 0.436148i \(0.856342\pi\)
\(14\) −0.152496 − 0.128386i −0.0407562 − 0.0343126i
\(15\) −0.979014 + 0.405521i −0.252780 + 0.104705i
\(16\) 3.61407 + 1.71421i 0.903517 + 0.428552i
\(17\) −6.95448 − 2.88064i −1.68671 − 0.698658i −0.687098 − 0.726565i \(-0.741116\pi\)
−0.999611 + 0.0279073i \(0.991116\pi\)
\(18\) −0.650954 − 1.25549i −0.153431 − 0.295922i
\(19\) 0.671982 − 0.551481i 0.154163 − 0.126519i −0.554150 − 0.832417i \(-0.686957\pi\)
0.708313 + 0.705898i \(0.249457\pi\)
\(20\) −0.260592 + 2.10327i −0.0582702 + 0.470306i
\(21\) 0.134888 + 0.0409177i 0.0294349 + 0.00892898i
\(22\) −1.36758 − 3.42248i −0.291569 − 0.729674i
\(23\) 1.85915 − 9.34658i 0.387660 − 1.94890i 0.0825156 − 0.996590i \(-0.473705\pi\)
0.305144 − 0.952306i \(-0.401295\pi\)
\(24\) −2.82642 − 0.106446i −0.576941 − 0.0217282i
\(25\) 3.80259 − 0.756382i 0.760518 − 0.151276i
\(26\) 1.80355 − 8.50774i 0.353706 − 1.66850i
\(27\) 0.773010 + 0.634393i 0.148766 + 0.122089i
\(28\) 0.204283 − 0.194279i 0.0386059 − 0.0367152i
\(29\) −3.35535 + 6.27741i −0.623072 + 1.16569i 0.350340 + 0.936623i \(0.386066\pi\)
−0.973412 + 0.229063i \(0.926434\pi\)
\(30\) −0.416995 − 1.43943i −0.0761325 − 0.262802i
\(31\) 3.67878 − 3.67878i 0.660728 − 0.660728i −0.294824 − 0.955552i \(-0.595261\pi\)
0.955552 + 0.294824i \(0.0952610\pi\)
\(32\) −2.97417 + 4.81189i −0.525764 + 0.850631i
\(33\) 1.84280 + 1.84280i 0.320790 + 0.320790i
\(34\) 5.13564 − 9.32475i 0.880755 − 1.59918i
\(35\) 0.131732 + 0.0704121i 0.0222667 + 0.0119018i
\(36\) 1.86638 − 0.718760i 0.311064 − 0.119793i
\(37\) −1.42782 + 1.73980i −0.234732 + 0.286022i −0.877032 − 0.480431i \(-0.840480\pi\)
0.642300 + 0.766453i \(0.277980\pi\)
\(38\) 0.670129 + 1.03068i 0.108709 + 0.167199i
\(39\) 1.19972 + 6.03141i 0.192109 + 0.965798i
\(40\) −2.91554 − 0.694945i −0.460987 − 0.109880i
\(41\) −6.97514 − 1.38744i −1.08933 − 0.216682i −0.382425 − 0.923987i \(-0.624911\pi\)
−0.706909 + 0.707305i \(0.749911\pi\)
\(42\) −0.0785904 + 0.183198i −0.0121268 + 0.0282680i
\(43\) −0.779364 + 2.56922i −0.118852 + 0.391802i −0.996100 − 0.0882357i \(-0.971877\pi\)
0.877248 + 0.480038i \(0.159377\pi\)
\(44\) 5.02418 − 1.38739i 0.757424 − 0.209157i
\(45\) 0.672252 + 0.819141i 0.100213 + 0.122110i
\(46\) 12.8466 + 4.07368i 1.89413 + 0.600632i
\(47\) 0.674420 − 1.62819i 0.0983742 − 0.237496i −0.867029 − 0.498257i \(-0.833973\pi\)
0.965403 + 0.260761i \(0.0839735\pi\)
\(48\) 0.591288 − 3.95606i 0.0853450 − 0.571007i
\(49\) 2.67118 + 6.44880i 0.381597 + 0.921257i
\(50\) 0.468922 + 5.46294i 0.0663156 + 0.772576i
\(51\) −0.737821 + 7.49122i −0.103316 + 1.04898i
\(52\) 11.6762 + 3.86446i 1.61920 + 0.535904i
\(53\) −4.02295 − 7.52641i −0.552595 − 1.03383i −0.990848 − 0.134980i \(-0.956903\pi\)
0.438254 − 0.898851i \(-0.355597\pi\)
\(54\) −1.01247 + 0.987374i −0.137780 + 0.134365i
\(55\) 1.53428 + 2.29622i 0.206883 + 0.309622i
\(56\) 0.241147 + 0.317490i 0.0322246 + 0.0424264i
\(57\) −0.722801 − 0.482960i −0.0957373 − 0.0639696i
\(58\) −8.29891 − 5.69706i −1.08970 − 0.748061i
\(59\) −10.7263 + 1.05645i −1.39645 + 0.137538i −0.768067 − 0.640369i \(-0.778781\pi\)
−0.628379 + 0.777907i \(0.716281\pi\)
\(60\) 2.08835 − 0.361177i 0.269605 − 0.0466277i
\(61\) 6.46233 − 1.96033i 0.827417 − 0.250994i 0.151953 − 0.988388i \(-0.451444\pi\)
0.675463 + 0.737393i \(0.263944\pi\)
\(62\) 4.59585 + 5.74558i 0.583674 + 0.729690i
\(63\) − 0.140957i − 0.0177589i
\(64\) −6.29862 − 4.93229i −0.787327 − 0.616536i
\(65\) 6.51656i 0.808279i
\(66\) −2.87812 + 2.30219i −0.354272 + 0.283379i
\(67\) 10.0367 − 3.04460i 1.22618 − 0.371957i 0.390206 − 0.920728i \(-0.372404\pi\)
0.835973 + 0.548770i \(0.184904\pi\)
\(68\) 12.3039 + 8.67555i 1.49207 + 1.05206i
\(69\) −9.48380 + 0.934073i −1.14172 + 0.112449i
\(70\) −0.119553 + 0.174153i −0.0142893 + 0.0208153i
\(71\) 9.98888 + 6.67435i 1.18546 + 0.792100i 0.982349 − 0.187055i \(-0.0598944\pi\)
0.203112 + 0.979155i \(0.434894\pi\)
\(72\) 0.718605 + 2.73562i 0.0846884 + 0.322396i
\(73\) 1.28206 + 1.91874i 0.150054 + 0.224571i 0.898880 − 0.438195i \(-0.144382\pi\)
−0.748826 + 0.662767i \(0.769382\pi\)
\(74\) −2.22227 − 2.27875i −0.258333 − 0.264899i
\(75\) −1.82765 − 3.41928i −0.211038 − 0.394825i
\(76\) −1.55340 + 0.780842i −0.178187 + 0.0895687i
\(77\) 0.0360066 − 0.365581i 0.00410333 − 0.0416618i
\(78\) −8.66494 + 0.743772i −0.981111 + 0.0842156i
\(79\) −5.10188 − 12.3170i −0.574006 − 1.38577i −0.898118 − 0.439754i \(-0.855066\pi\)
0.324112 − 0.946019i \(-0.394934\pi\)
\(80\) 1.43238 − 3.98935i 0.160145 − 0.446023i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) 3.04010 − 9.58713i 0.335723 − 1.05872i
\(83\) 6.80230 + 8.28863i 0.746649 + 0.909795i 0.998315 − 0.0580192i \(-0.0184785\pi\)
−0.251666 + 0.967814i \(0.580978\pi\)
\(84\) −0.245213 − 0.139090i −0.0267550 − 0.0151760i
\(85\) −2.31551 + 7.63321i −0.251152 + 0.827938i
\(86\) −3.48939 − 1.49692i −0.376270 − 0.161417i
\(87\) 6.98111 + 1.38863i 0.748454 + 0.148877i
\(88\) 1.16495 + 7.27856i 0.124184 + 0.775897i
\(89\) −1.23326 − 6.20000i −0.130725 − 0.657198i −0.989463 − 0.144784i \(-0.953751\pi\)
0.858738 − 0.512414i \(-0.171249\pi\)
\(90\) −1.25640 + 0.816883i −0.132436 + 0.0861070i
\(91\) 0.549909 − 0.670066i 0.0576461 − 0.0702420i
\(92\) −7.73326 + 17.4200i −0.806248 + 1.81616i
\(93\) −4.58826 − 2.45248i −0.475781 − 0.254310i
\(94\) 2.18312 + 1.20236i 0.225172 + 0.124014i
\(95\) −0.651374 − 0.651374i −0.0668296 − 0.0668296i
\(96\) 5.46805 + 1.44928i 0.558081 + 0.147917i
\(97\) −12.5043 + 12.5043i −1.26962 + 1.26962i −0.323336 + 0.946284i \(0.604805\pi\)
−0.946284 + 0.323336i \(0.895195\pi\)
\(98\) −9.48155 + 2.74676i −0.957781 + 0.277465i
\(99\) 1.22851 − 2.29838i 0.123470 − 0.230996i
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.13.17 ✓ 512
128.69 even 32 inner 384.2.v.a.325.17 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.17 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.17 yes 512 128.69 even 32 inner