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Properties

Label 384.2.v.a.13.15
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 13.15
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.15

$q$-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+(-0.229611 - 1.39545i) q^{2} +(0.290285 + 0.956940i) q^{3} +(-1.89456 + 0.640821i) q^{4} +(-0.159671 - 1.62116i) q^{5} +(1.26871 - 0.624802i) q^{6} +(-2.66520 + 3.98875i) q^{7} +(1.32924 + 2.49662i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(-2.22559 + 0.595050i) q^{10} +(3.99771 - 2.13682i) q^{11} +(-1.16319 - 1.62696i) q^{12} +(6.81274 + 0.670996i) q^{13} +(6.17805 + 2.80329i) q^{14} +(1.50501 - 0.623395i) q^{15} +(3.17870 - 2.42815i) q^{16} +(3.66004 + 1.51604i) q^{17} +(0.966185 + 1.03271i) q^{18} +(-3.65067 + 2.99603i) q^{19} +(1.34138 + 2.96907i) q^{20} +(-4.59066 - 1.39256i) q^{21} +(-3.89974 - 5.08796i) q^{22} +(-1.17853 + 5.92485i) q^{23} +(-2.00326 + 1.99674i) q^{24} +(2.30125 - 0.457746i) q^{25} +(-0.627939 - 9.66090i) q^{26} +(-0.773010 - 0.634393i) q^{27} +(2.49329 - 9.26483i) q^{28} +(-0.852848 + 1.59557i) q^{29} +(-1.21548 - 1.95702i) q^{30} +(-1.81765 + 1.81765i) q^{31} +(-4.11822 - 3.87818i) q^{32} +(3.20528 + 3.20528i) q^{33} +(1.27517 - 5.45550i) q^{34} +(6.89197 + 3.68384i) q^{35} +(1.21925 - 1.58538i) q^{36} +(-0.545096 + 0.664201i) q^{37} +(5.01904 + 4.40641i) q^{38} +(1.33553 + 6.71416i) q^{39} +(3.83519 - 2.55356i) q^{40} +(-2.41635 - 0.480641i) q^{41} +(-0.889183 + 6.72578i) q^{42} +(0.604548 - 1.99293i) q^{43} +(-6.20457 + 6.61014i) q^{44} +(1.03343 + 1.25924i) q^{45} +(8.53844 + 0.284162i) q^{46} +(1.28072 - 3.09193i) q^{47} +(3.24632 + 2.33697i) q^{48} +(-6.12806 - 14.7944i) q^{49} +(-1.16715 - 3.10617i) q^{50} +(-0.388305 + 3.94253i) q^{51} +(-13.3371 + 3.09451i) q^{52} +(1.15519 + 2.16120i) q^{53} +(-0.707772 + 1.22436i) q^{54} +(-4.10245 - 6.13975i) q^{55} +(-13.5011 - 1.35196i) q^{56} +(-3.92676 - 2.62378i) q^{57} +(2.42235 + 0.823746i) q^{58} +(7.51336 - 0.740001i) q^{59} +(-2.45184 + 2.14550i) q^{60} +(-11.9569 + 3.62709i) q^{61} +(2.95379 + 2.11909i) q^{62} -4.79723i q^{63} +(-4.46622 + 6.63724i) q^{64} -11.1517i q^{65} +(3.73684 - 5.20877i) q^{66} +(8.08187 - 2.45161i) q^{67} +(-7.90567 - 0.526791i) q^{68} +(-6.01184 + 0.592115i) q^{69} +(3.55813 - 10.4632i) q^{70} +(0.265968 + 0.177714i) q^{71} +(-2.49227 - 1.33737i) q^{72} +(-1.60314 - 2.39927i) q^{73} +(1.05202 + 0.608146i) q^{74} +(1.10605 + 2.06928i) q^{75} +(4.99649 - 8.01558i) q^{76} +(-2.13144 + 21.6409i) q^{77} +(9.06262 - 3.40531i) q^{78} +(-0.479447 - 1.15749i) q^{79} +(-4.44397 - 4.76549i) q^{80} +(0.382683 - 0.923880i) q^{81} +(-0.115891 + 3.48225i) q^{82} +(-0.187886 - 0.228940i) q^{83} +(9.58965 - 0.303504i) q^{84} +(1.87335 - 6.17560i) q^{85} +(-2.91984 - 0.386018i) q^{86} +(-1.77443 - 0.352956i) q^{87} +(10.6488 + 7.14040i) q^{88} +(-2.91955 - 14.6776i) q^{89} +(1.51992 - 1.73124i) q^{90} +(-20.8337 + 25.3860i) q^{91} +(-1.56399 - 11.9802i) q^{92} +(-2.26702 - 1.21175i) q^{93} +(-4.60870 - 1.07724i) q^{94} +(5.43997 + 5.43997i) q^{95} +(2.51573 - 5.06666i) q^{96} +(-8.38535 + 8.38535i) q^{97} +(-19.2378 + 11.9484i) q^{98} +(-2.13682 + 3.99771i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.229611 − 1.39545i −0.162360 − 0.986732i
\(3\) 0.290285 + 0.956940i 0.167596 + 0.552490i
\(4\) −1.89456 + 0.640821i −0.947279 + 0.320411i
\(5\) −0.159671 − 1.62116i −0.0714070 − 0.725007i −0.962517 − 0.271220i \(-0.912573\pi\)
0.891110 − 0.453787i \(-0.149927\pi\)
\(6\) 1.26871 − 0.624802i 0.517948 − 0.255074i
\(7\) −2.66520 + 3.98875i −1.00735 + 1.50761i −0.152785 + 0.988259i \(0.548824\pi\)
−0.854564 + 0.519346i \(0.826176\pi\)
\(8\) 1.32924 + 2.49662i 0.469959 + 0.882688i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) −2.22559 + 0.595050i −0.703794 + 0.188171i
\(11\) 3.99771 − 2.13682i 1.20535 − 0.644275i 0.258522 − 0.966005i \(-0.416764\pi\)
0.946831 + 0.321730i \(0.104264\pi\)
\(12\) −1.16319 − 1.62696i −0.335784 − 0.469662i
\(13\) 6.81274 + 0.670996i 1.88951 + 0.186101i 0.975562 − 0.219724i \(-0.0705156\pi\)
0.913951 + 0.405824i \(0.133016\pi\)
\(14\) 6.17805 + 2.80329i 1.65115 + 0.749210i
\(15\) 1.50501 − 0.623395i 0.388591 − 0.160960i
\(16\) 3.17870 − 2.42815i 0.794674 − 0.607036i
\(17\) 3.66004 + 1.51604i 0.887691 + 0.367694i 0.779475 − 0.626434i \(-0.215486\pi\)
0.108216 + 0.994127i \(0.465486\pi\)
\(18\) 0.966185 + 1.03271i 0.227732 + 0.243412i
\(19\) −3.65067 + 2.99603i −0.837522 + 0.687337i −0.952003 − 0.306089i \(-0.900980\pi\)
0.114481 + 0.993425i \(0.463480\pi\)
\(20\) 1.34138 + 2.96907i 0.299942 + 0.663904i
\(21\) −4.59066 − 1.39256i −1.00176 − 0.303882i
\(22\) −3.89974 − 5.08796i −0.831427 − 1.08476i
\(23\) −1.17853 + 5.92485i −0.245740 + 1.23542i 0.638954 + 0.769245i \(0.279367\pi\)
−0.884694 + 0.466173i \(0.845633\pi\)
\(24\) −2.00326 + 1.99674i −0.408913 + 0.407583i
\(25\) 2.30125 − 0.457746i 0.460249 − 0.0915492i
\(26\) −0.627939 − 9.66090i −0.123149 − 1.89466i
\(27\) −0.773010 − 0.634393i −0.148766 − 0.122089i
\(28\) 2.49329 − 9.26483i 0.471188 − 1.75089i
\(29\) −0.852848 + 1.59557i −0.158370 + 0.296289i −0.948575 − 0.316554i \(-0.897474\pi\)
0.790205 + 0.612843i \(0.209974\pi\)
\(30\) −1.21548 − 1.95702i −0.221916 − 0.357302i
\(31\) −1.81765 + 1.81765i −0.326460 + 0.326460i −0.851239 − 0.524779i \(-0.824148\pi\)
0.524779 + 0.851239i \(0.324148\pi\)
\(32\) −4.11822 − 3.87818i −0.728005 − 0.685572i
\(33\) 3.20528 + 3.20528i 0.557968 + 0.557968i
\(34\) 1.27517 − 5.45550i 0.218690 − 0.935611i
\(35\) 6.89197 + 3.68384i 1.16496 + 0.622682i
\(36\) 1.21925 − 1.58538i 0.203208 − 0.264231i
\(37\) −0.545096 + 0.664201i −0.0896132 + 0.109194i −0.815887 − 0.578212i \(-0.803751\pi\)
0.726273 + 0.687406i \(0.241251\pi\)
\(38\) 5.01904 + 4.40641i 0.814197 + 0.714814i
\(39\) 1.33553 + 6.71416i 0.213856 + 1.07513i
\(40\) 3.83519 − 2.55356i 0.606397 − 0.403754i
\(41\) −2.41635 − 0.480641i −0.377370 − 0.0750636i 0.00276103 − 0.999996i \(-0.499121\pi\)
−0.380131 + 0.924933i \(0.624121\pi\)
\(42\) −0.889183 + 6.72578i −0.137204 + 1.03781i
\(43\) 0.604548 − 1.99293i 0.0921926 − 0.303918i −0.898702 − 0.438561i \(-0.855488\pi\)
0.990894 + 0.134642i \(0.0429885\pi\)
\(44\) −6.20457 + 6.61014i −0.935374 + 0.996516i
\(45\) 1.03343 + 1.25924i 0.154055 + 0.187717i
\(46\) 8.53844 + 0.284162i 1.25892 + 0.0418975i
\(47\) 1.28072 − 3.09193i 0.186812 − 0.451004i −0.802530 − 0.596611i \(-0.796513\pi\)
0.989343 + 0.145607i \(0.0465134\pi\)
\(48\) 3.24632 + 2.33697i 0.468565 + 0.337312i
\(49\) −6.12806 − 14.7944i −0.875437 − 2.11349i
\(50\) −1.16715 − 3.10617i −0.165060 − 0.439278i
\(51\) −0.388305 + 3.94253i −0.0543736 + 0.552064i
\(52\) −13.3371 + 3.09451i −1.84952 + 0.429131i
\(53\) 1.15519 + 2.16120i 0.158677 + 0.296864i 0.948679 − 0.316240i \(-0.102421\pi\)
−0.790002 + 0.613104i \(0.789921\pi\)
\(54\) −0.707772 + 1.22436i −0.0963156 + 0.166614i
\(55\) −4.10245 − 6.13975i −0.553175 − 0.827884i
\(56\) −13.5011 − 1.35196i −1.80416 − 0.180663i
\(57\) −3.92676 − 2.62378i −0.520112 − 0.347528i
\(58\) 2.42235 + 0.823746i 0.318071 + 0.108163i
\(59\) 7.51336 − 0.740001i 0.978156 − 0.0963400i 0.403714 − 0.914885i \(-0.367719\pi\)
0.574442 + 0.818545i \(0.305219\pi\)
\(60\) −2.45184 + 2.14550i −0.316531 + 0.276983i
\(61\) −11.9569 + 3.62709i −1.53093 + 0.464401i −0.939670 − 0.342082i \(-0.888868\pi\)
−0.591256 + 0.806484i \(0.701368\pi\)
\(62\) 2.95379 + 2.11909i 0.375132 + 0.269124i
\(63\) − 4.79723i − 0.604394i
\(64\) −4.46622 + 6.63724i −0.558277 + 0.829655i
\(65\) − 11.1517i − 1.38320i
\(66\) 3.73684 − 5.20877i 0.459973 − 0.641156i
\(67\) 8.08187 − 2.45161i 0.987357 − 0.299512i 0.245005 − 0.969522i \(-0.421210\pi\)
0.742352 + 0.670010i \(0.233710\pi\)
\(68\) −7.90567 − 0.526791i −0.958704 − 0.0638828i
\(69\) −6.01184 + 0.592115i −0.723741 + 0.0712822i
\(70\) 3.55813 − 10.4632i 0.425278 − 1.25060i
\(71\) 0.265968 + 0.177714i 0.0315646 + 0.0210908i 0.571252 − 0.820774i \(-0.306458\pi\)
−0.539688 + 0.841865i \(0.681458\pi\)
\(72\) −2.49227 − 1.33737i −0.293717 − 0.157611i
\(73\) −1.60314 − 2.39927i −0.187633 − 0.280813i 0.725712 − 0.687998i \(-0.241510\pi\)
−0.913346 + 0.407185i \(0.866510\pi\)
\(74\) 1.05202 + 0.608146i 0.122295 + 0.0706955i
\(75\) 1.10605 + 2.06928i 0.127716 + 0.238940i
\(76\) 4.99649 − 8.01558i 0.573137 − 0.919450i
\(77\) −2.13144 + 21.6409i −0.242900 + 2.46621i
\(78\) 9.06262 − 3.40531i 1.02614 − 0.385575i
\(79\) −0.479447 − 1.15749i −0.0539421 − 0.130228i 0.894611 − 0.446846i \(-0.147453\pi\)
−0.948553 + 0.316618i \(0.897453\pi\)
\(80\) −4.44397 − 4.76549i −0.496851 − 0.532798i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) −0.115891 + 3.48225i −0.0127980 + 0.384550i
\(83\) −0.187886 − 0.228940i −0.0206232 − 0.0251294i 0.762597 − 0.646874i \(-0.223924\pi\)
−0.783220 + 0.621745i \(0.786424\pi\)
\(84\) 9.58965 − 0.303504i 1.04632 − 0.0331150i
\(85\) 1.87335 − 6.17560i 0.203193 − 0.669838i
\(86\) −2.91984 − 0.386018i −0.314854 − 0.0416253i
\(87\) −1.77443 − 0.352956i −0.190239 − 0.0378409i
\(88\) 10.6488 + 7.14040i 1.13516 + 0.761169i
\(89\) −2.91955 − 14.6776i −0.309471 − 1.55582i −0.752061 − 0.659093i \(-0.770940\pi\)
0.442590 − 0.896724i \(-0.354060\pi\)
\(90\) 1.51992 − 1.73124i 0.160214 − 0.182489i
\(91\) −20.8337 + 25.3860i −2.18397 + 2.66117i
\(92\) −1.56399 − 11.9802i −0.163057 − 1.24902i
\(93\) −2.26702 − 1.21175i −0.235079 − 0.125652i
\(94\) −4.60870 − 1.07724i −0.475351 − 0.111109i
\(95\) 5.43997 + 5.43997i 0.558129 + 0.558129i
\(96\) 2.51573 − 5.06666i 0.256761 − 0.517114i
\(97\) −8.38535 + 8.38535i −0.851403 + 0.851403i −0.990306 − 0.138903i \(-0.955642\pi\)
0.138903 + 0.990306i \(0.455642\pi\)
\(98\) −19.2378 + 11.9484i −1.94331 + 1.20697i
\(99\) −2.13682 + 3.99771i −0.214758 + 0.401785i
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.13.15 ✓ 512
128.69 even 32 inner 384.2.v.a.325.15 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.15 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.15 yes 512 128.69 even 32 inner