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Properties

Label 384.2.v.a.13.16
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 13.16
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.16

$q$-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+(-0.162120 - 1.40489i) q^{2} +(-0.290285 - 0.956940i) q^{3} +(-1.94743 + 0.455521i) q^{4} +(-0.344842 - 3.50124i) q^{5} +(-1.29734 + 0.562957i) q^{6} +(1.89335 - 2.83360i) q^{7} +(0.955675 + 2.66208i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(-4.86295 + 1.05208i) q^{10} +(1.05121 - 0.561884i) q^{11} +(1.00122 + 1.73135i) q^{12} +(-2.34506 - 0.230968i) q^{13} +(-4.28785 - 2.20057i) q^{14} +(-3.25037 + 1.34635i) q^{15} +(3.58500 - 1.77419i) q^{16} +(6.04345 + 2.50328i) q^{17} +(0.915313 + 1.07805i) q^{18} +(-5.61059 + 4.60449i) q^{19} +(2.26644 + 6.66134i) q^{20} +(-3.26120 - 0.989275i) q^{21} +(-0.959807 - 1.38574i) q^{22} +(1.35318 - 6.80291i) q^{23} +(2.27004 - 1.68729i) q^{24} +(-7.23581 + 1.43929i) q^{25} +(0.0556954 + 3.33199i) q^{26} +(0.773010 + 0.634393i) q^{27} +(-2.39642 + 6.38072i) q^{28} +(-2.41134 + 4.51129i) q^{29} +(2.41842 + 4.34815i) q^{30} +(-3.51949 + 3.51949i) q^{31} +(-3.07375 - 4.74890i) q^{32} +(-0.842840 - 0.842840i) q^{33} +(2.53707 - 8.89621i) q^{34} +(-10.5740 - 5.65193i) q^{35} +(1.36616 - 1.46069i) q^{36} +(-0.735634 + 0.896373i) q^{37} +(7.37840 + 7.13579i) q^{38} +(0.459712 + 2.31113i) q^{39} +(8.99102 - 4.26404i) q^{40} +(10.2206 + 2.03299i) q^{41} +(-0.861118 + 4.74201i) q^{42} +(1.74021 - 5.73670i) q^{43} +(-1.79121 + 1.57308i) q^{44} +(2.23191 + 2.71959i) q^{45} +(-9.77672 - 0.798188i) q^{46} +(4.27267 - 10.3151i) q^{47} +(-2.73847 - 2.91561i) q^{48} +(-1.76574 - 4.26288i) q^{49} +(3.19511 + 9.93218i) q^{50} +(0.641167 - 6.50988i) q^{51} +(4.67206 - 0.618428i) q^{52} +(1.39597 + 2.61168i) q^{53} +(0.765933 - 1.18884i) q^{54} +(-2.32979 - 3.48678i) q^{55} +(9.35272 + 2.33226i) q^{56} +(6.03490 + 4.03239i) q^{57} +(6.72880 + 2.65629i) q^{58} +(-7.16819 + 0.706005i) q^{59} +(5.71659 - 4.10254i) q^{60} +(10.0521 - 3.04927i) q^{61} +(5.51507 + 4.37391i) q^{62} +3.40795i q^{63} +(-6.17337 + 5.08817i) q^{64} +8.29025i q^{65} +(-1.04746 + 1.32074i) q^{66} +(2.75139 - 0.834626i) q^{67} +(-12.9095 - 2.12205i) q^{68} +(-6.90279 + 0.679865i) q^{69} +(-6.22609 + 15.7716i) q^{70} +(1.42306 + 0.950861i) q^{71} +(-2.27359 - 1.68250i) q^{72} +(0.513291 + 0.768194i) q^{73} +(1.37857 + 0.888166i) q^{74} +(3.47776 + 6.50643i) q^{75} +(8.82882 - 11.5227i) q^{76} +(0.398157 - 4.04256i) q^{77} +(3.17235 - 1.02052i) q^{78} +(-2.54511 - 6.14444i) q^{79} +(-7.44813 - 11.9401i) q^{80} +(0.382683 - 0.923880i) q^{81} +(1.19918 - 14.6883i) q^{82} +(-6.93618 - 8.45176i) q^{83} +(6.80161 + 0.441002i) q^{84} +(6.68053 - 22.0228i) q^{85} +(-8.34156 - 1.51477i) q^{86} +(5.01701 + 0.997946i) q^{87} +(2.50040 + 2.26143i) q^{88} +(-1.91877 - 9.64631i) q^{89} +(3.45889 - 3.57648i) q^{90} +(-5.09450 + 6.20766i) q^{91} +(0.463633 + 13.8646i) q^{92} +(4.38959 + 2.34629i) q^{93} +(-15.1843 - 4.33034i) q^{94} +(18.0562 + 18.0562i) q^{95} +(-3.65215 + 4.31993i) q^{96} +(-0.769901 + 0.769901i) q^{97} +(-5.70261 + 3.17177i) q^{98} +(-0.561884 + 1.05121i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.162120 − 1.40489i −0.114636 − 0.993408i
\(3\) −0.290285 − 0.956940i −0.167596 − 0.552490i
\(4\) −1.94743 + 0.455521i −0.973717 + 0.227760i
\(5\) −0.344842 − 3.50124i −0.154218 − 1.56580i −0.691487 − 0.722389i \(-0.743044\pi\)
0.537269 − 0.843411i \(-0.319456\pi\)
\(6\) −1.29734 + 0.562957i −0.529635 + 0.229826i
\(7\) 1.89335 − 2.83360i 0.715621 − 1.07100i −0.278256 − 0.960507i \(-0.589756\pi\)
0.993877 − 0.110495i \(-0.0352436\pi\)
\(8\) 0.955675 + 2.66208i 0.337882 + 0.941188i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) −4.86295 + 1.05208i −1.53780 + 0.332698i
\(11\) 1.05121 − 0.561884i 0.316952 − 0.169414i −0.305248 − 0.952273i \(-0.598739\pi\)
0.622200 + 0.782859i \(0.286239\pi\)
\(12\) 1.00122 + 1.73135i 0.289026 + 0.499797i
\(13\) −2.34506 − 0.230968i −0.650402 − 0.0640590i −0.232565 − 0.972581i \(-0.574712\pi\)
−0.417837 + 0.908522i \(0.637212\pi\)
\(14\) −4.28785 − 2.20057i −1.14598 − 0.588128i
\(15\) −3.25037 + 1.34635i −0.839242 + 0.347626i
\(16\) 3.58500 − 1.77419i 0.896250 − 0.443549i
\(17\) 6.04345 + 2.50328i 1.46575 + 0.607134i 0.965886 − 0.258969i \(-0.0833828\pi\)
0.499865 + 0.866103i \(0.333383\pi\)
\(18\) 0.915313 + 1.07805i 0.215741 + 0.254100i
\(19\) −5.61059 + 4.60449i −1.28716 + 1.05634i −0.292168 + 0.956367i \(0.594377\pi\)
−0.994990 + 0.0999766i \(0.968123\pi\)
\(20\) 2.26644 + 6.66134i 0.506792 + 1.48952i
\(21\) −3.26120 − 0.989275i −0.711653 − 0.215877i
\(22\) −0.959807 − 1.38574i −0.204632 − 0.295441i
\(23\) 1.35318 − 6.80291i 0.282158 − 1.41850i −0.536343 − 0.844000i \(-0.680195\pi\)
0.818501 − 0.574505i \(-0.194805\pi\)
\(24\) 2.27004 − 1.68729i 0.463369 − 0.344416i
\(25\) −7.23581 + 1.43929i −1.44716 + 0.287858i
\(26\) 0.0556954 + 3.33199i 0.0109228 + 0.653458i
\(27\) 0.773010 + 0.634393i 0.148766 + 0.122089i
\(28\) −2.39642 + 6.38072i −0.452880 + 1.20584i
\(29\) −2.41134 + 4.51129i −0.447774 + 0.837726i 0.552218 + 0.833700i \(0.313782\pi\)
−0.999992 + 0.00402629i \(0.998718\pi\)
\(30\) 2.41842 + 4.34815i 0.441541 + 0.793859i
\(31\) −3.51949 + 3.51949i −0.632118 + 0.632118i −0.948599 − 0.316481i \(-0.897499\pi\)
0.316481 + 0.948599i \(0.397499\pi\)
\(32\) −3.07375 − 4.74890i −0.543367 − 0.839495i
\(33\) −0.842840 − 0.842840i −0.146720 − 0.146720i
\(34\) 2.53707 − 8.89621i 0.435104 − 1.52569i
\(35\) −10.5740 − 5.65193i −1.78734 − 0.955351i
\(36\) 1.36616 − 1.46069i 0.227693 − 0.243448i
\(37\) −0.735634 + 0.896373i −0.120937 + 0.147363i −0.829960 − 0.557823i \(-0.811637\pi\)
0.709022 + 0.705186i \(0.249137\pi\)
\(38\) 7.37840 + 7.13579i 1.19693 + 1.15758i
\(39\) 0.459712 + 2.31113i 0.0736128 + 0.370076i
\(40\) 8.99102 − 4.26404i 1.42161 − 0.674204i
\(41\) 10.2206 + 2.03299i 1.59618 + 0.317500i 0.911487 − 0.411329i \(-0.134935\pi\)
0.684695 + 0.728829i \(0.259935\pi\)
\(42\) −0.861118 + 4.74201i −0.132873 + 0.731708i
\(43\) 1.74021 − 5.73670i 0.265379 − 0.874839i −0.718427 − 0.695603i \(-0.755137\pi\)
0.983806 − 0.179236i \(-0.0573626\pi\)
\(44\) −1.79121 + 1.57308i −0.270036 + 0.237151i
\(45\) 2.23191 + 2.71959i 0.332713 + 0.405412i
\(46\) −9.77672 − 0.798188i −1.44150 − 0.117686i
\(47\) 4.27267 − 10.3151i 0.623233 − 1.50462i −0.224654 − 0.974439i \(-0.572125\pi\)
0.847886 − 0.530178i \(-0.177875\pi\)
\(48\) −2.73847 − 2.91561i −0.395264 − 0.420832i
\(49\) −1.76574 − 4.26288i −0.252249 − 0.608982i
\(50\) 3.19511 + 9.93218i 0.451857 + 1.40462i
\(51\) 0.641167 − 6.50988i 0.0897814 − 0.911566i
\(52\) 4.67206 − 0.618428i 0.647898 − 0.0857605i
\(53\) 1.39597 + 2.61168i 0.191751 + 0.358742i 0.959364 − 0.282173i \(-0.0910552\pi\)
−0.767612 + 0.640915i \(0.778555\pi\)
\(54\) 0.765933 − 1.18884i 0.104230 − 0.161781i
\(55\) −2.32979 − 3.48678i −0.314149 − 0.470157i
\(56\) 9.35272 + 2.33226i 1.24981 + 0.311662i
\(57\) 6.03490 + 4.03239i 0.799341 + 0.534103i
\(58\) 6.72880 + 2.65629i 0.883534 + 0.348788i
\(59\) −7.16819 + 0.706005i −0.933219 + 0.0919141i −0.553185 − 0.833058i \(-0.686588\pi\)
−0.380034 + 0.924972i \(0.624088\pi\)
\(60\) 5.71659 − 4.10254i 0.738009 − 0.529635i
\(61\) 10.0521 − 3.04927i 1.28704 − 0.390418i 0.428673 − 0.903460i \(-0.358981\pi\)
0.858364 + 0.513041i \(0.171481\pi\)
\(62\) 5.51507 + 4.37391i 0.700415 + 0.555488i
\(63\) 3.40795i 0.429361i
\(64\) −6.17337 + 5.08817i −0.771672 + 0.636021i
\(65\) 8.29025i 1.02828i
\(66\) −1.04746 + 1.32074i −0.128933 + 0.162572i
\(67\) 2.75139 − 0.834626i 0.336136 − 0.101966i −0.117704 − 0.993049i \(-0.537554\pi\)
0.453841 + 0.891083i \(0.350054\pi\)
\(68\) −12.9095 − 2.12205i −1.56551 − 0.257337i
\(69\) −6.90279 + 0.679865i −0.830998 + 0.0818461i
\(70\) −6.22609 + 15.7716i −0.744160 + 1.88507i
\(71\) 1.42306 + 0.950861i 0.168887 + 0.112846i 0.637141 − 0.770747i \(-0.280117\pi\)
−0.468254 + 0.883594i \(0.655117\pi\)
\(72\) −2.27359 − 1.68250i −0.267945 − 0.198284i
\(73\) 0.513291 + 0.768194i 0.0600761 + 0.0899103i 0.860291 − 0.509802i \(-0.170282\pi\)
−0.800215 + 0.599713i \(0.795282\pi\)
\(74\) 1.37857 + 0.888166i 0.160255 + 0.103247i
\(75\) 3.47776 + 6.50643i 0.401577 + 0.751298i
\(76\) 8.82882 − 11.5227i 1.01273 − 1.32174i
\(77\) 0.398157 − 4.04256i 0.0453743 − 0.460693i
\(78\) 3.17235 − 1.02052i 0.359198 − 0.115552i
\(79\) −2.54511 − 6.14444i −0.286347 − 0.691304i 0.713610 − 0.700543i \(-0.247059\pi\)
−0.999957 + 0.00923943i \(0.997059\pi\)
\(80\) −7.44813 − 11.9401i −0.832726 − 1.33495i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) 1.19918 − 14.6883i 0.132427 − 1.62206i
\(83\) −6.93618 − 8.45176i −0.761344 − 0.927701i 0.237703 − 0.971338i \(-0.423605\pi\)
−0.999047 + 0.0436372i \(0.986105\pi\)
\(84\) 6.80161 + 0.441002i 0.742117 + 0.0481173i
\(85\) 6.68053 − 22.0228i 0.724605 − 2.38870i
\(86\) −8.34156 − 1.51477i −0.899494 − 0.163342i
\(87\) 5.01701 + 0.997946i 0.537880 + 0.106991i
\(88\) 2.50040 + 2.26143i 0.266543 + 0.241069i
\(89\) −1.91877 − 9.64631i −0.203389 − 1.02251i −0.938689 − 0.344765i \(-0.887959\pi\)
0.735300 − 0.677742i \(-0.237041\pi\)
\(90\) 3.45889 − 3.57648i 0.364599 − 0.376995i
\(91\) −5.09450 + 6.20766i −0.534048 + 0.650740i
\(92\) 0.463633 + 13.8646i 0.0483371 + 1.44549i
\(93\) 4.38959 + 2.34629i 0.455179 + 0.243298i
\(94\) −15.1843 − 4.33034i −1.56614 − 0.446641i
\(95\) 18.0562 + 18.0562i 1.85253 + 1.85253i
\(96\) −3.65215 + 4.31993i −0.372746 + 0.440901i
\(97\) −0.769901 + 0.769901i −0.0781716 + 0.0781716i −0.745112 − 0.666940i \(-0.767604\pi\)
0.666940 + 0.745112i \(0.267604\pi\)
\(98\) −5.70261 + 3.17177i −0.576051 + 0.320397i
\(99\) −0.561884 + 1.05121i −0.0564715 + 0.105651i
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.13.16 ✓ 512
128.69 even 32 inner 384.2.v.a.325.16 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.16 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.16 yes 512 128.69 even 32 inner