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Properties

Label 384.2.v.a.13.19
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 13.19
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.19

$q$-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+(0.246397 + 1.39258i) q^{2} +(0.290285 + 0.956940i) q^{3} +(-1.87858 + 0.686257i) q^{4} +(0.267357 + 2.71452i) q^{5} +(-1.26109 + 0.640033i) q^{6} +(-0.343422 + 0.513967i) q^{7} +(-1.41855 - 2.44698i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(-3.71432 + 1.04117i) q^{10} +(0.294246 - 0.157278i) q^{11} +(-1.20203 - 1.59848i) q^{12} +(-2.51375 - 0.247583i) q^{13} +(-0.800360 - 0.351603i) q^{14} +(-2.52003 + 1.04383i) q^{15} +(3.05810 - 2.57837i) q^{16} +(4.21613 + 1.74638i) q^{17} +(-0.978550 - 1.02100i) q^{18} +(-1.78752 + 1.46698i) q^{19} +(-2.36511 - 4.91597i) q^{20} +(-0.591525 - 0.179437i) q^{21} +(0.291523 + 0.371009i) q^{22} +(0.582156 - 2.92670i) q^{23} +(1.92983 - 2.06779i) q^{24} +(-2.39323 + 0.476044i) q^{25} +(-0.274602 - 3.56162i) q^{26} +(-0.773010 - 0.634393i) q^{27} +(0.292430 - 1.20120i) q^{28} +(-0.221297 + 0.414017i) q^{29} +(-2.07455 - 3.25215i) q^{30} +(-0.222556 + 0.222556i) q^{31} +(4.34411 + 3.62336i) q^{32} +(0.235920 + 0.235920i) q^{33} +(-1.39313 + 6.30162i) q^{34} +(-1.48699 - 0.794813i) q^{35} +(1.18072 - 1.61428i) q^{36} +(-7.22167 + 8.79964i) q^{37} +(-2.48333 - 2.12781i) q^{38} +(-0.492782 - 2.47738i) q^{39} +(6.26313 - 4.50490i) q^{40} +(3.19749 + 0.636020i) q^{41} +(0.104131 - 0.867961i) q^{42} +(0.347543 - 1.14570i) q^{43} +(-0.444830 + 0.497386i) q^{44} +(-1.73041 - 2.10851i) q^{45} +(4.21911 + 0.0895709i) q^{46} +(2.91400 - 7.03501i) q^{47} +(3.35507 + 2.17796i) q^{48} +(2.53256 + 6.11414i) q^{49} +(-1.25262 - 3.21548i) q^{50} +(-0.447302 + 4.54153i) q^{51} +(4.89219 - 1.25998i) q^{52} +(4.77800 + 8.93902i) q^{53} +(0.692978 - 1.23279i) q^{54} +(0.505602 + 0.756688i) q^{55} +(1.74483 + 0.111261i) q^{56} +(-1.92270 - 1.28471i) q^{57} +(-0.631080 - 0.206161i) q^{58} +(-0.308287 + 0.0303636i) q^{59} +(4.01773 - 3.69030i) q^{60} +(-0.821867 + 0.249311i) q^{61} +(-0.364765 - 0.255090i) q^{62} -0.618143i q^{63} +(-3.97545 + 6.94232i) q^{64} -6.88984i q^{65} +(-0.270409 + 0.386669i) q^{66} +(12.6200 - 3.82822i) q^{67} +(-9.11879 - 0.387355i) q^{68} +(2.96967 - 0.292487i) q^{69} +(0.740453 - 2.26660i) q^{70} +(0.852169 + 0.569401i) q^{71} +(2.53895 + 1.24649i) q^{72} +(3.54645 + 5.30764i) q^{73} +(-14.0336 - 7.88858i) q^{74} +(-1.15026 - 2.15199i) q^{75} +(2.35127 - 3.98254i) q^{76} +(-0.0202149 + 0.205245i) q^{77} +(3.32854 - 1.29666i) q^{78} +(2.11907 + 5.11588i) q^{79} +(7.81667 + 7.61194i) q^{80} +(0.382683 - 0.923880i) q^{81} +(-0.0978583 + 4.60948i) q^{82} +(-1.75844 - 2.14267i) q^{83} +(1.23437 - 0.0688521i) q^{84} +(-3.61337 + 11.9117i) q^{85} +(1.68111 + 0.201686i) q^{86} +(-0.460429 - 0.0915849i) q^{87} +(-0.802257 - 0.496908i) q^{88} +(-2.44979 - 12.3159i) q^{89} +(2.50991 - 2.92927i) q^{90} +(0.990527 - 1.20696i) q^{91} +(0.914843 + 5.89754i) q^{92} +(-0.277577 - 0.148368i) q^{93} +(10.5148 + 2.32458i) q^{94} +(-4.46006 - 4.46006i) q^{95} +(-2.20631 + 5.20886i) q^{96} +(5.61426 - 5.61426i) q^{97} +(-7.89044 + 5.03331i) q^{98} +(-0.157278 + 0.294246i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.246397 + 1.39258i 0.174229 + 0.984705i
\(3\) 0.290285 + 0.956940i 0.167596 + 0.552490i
\(4\) −1.87858 + 0.686257i −0.939288 + 0.343129i
\(5\) 0.267357 + 2.71452i 0.119566 + 1.21397i 0.848428 + 0.529311i \(0.177550\pi\)
−0.728862 + 0.684661i \(0.759950\pi\)
\(6\) −1.26109 + 0.640033i −0.514839 + 0.261292i
\(7\) −0.343422 + 0.513967i −0.129801 + 0.194261i −0.890677 − 0.454636i \(-0.849769\pi\)
0.760876 + 0.648897i \(0.224769\pi\)
\(8\) −1.41855 − 2.44698i −0.501532 − 0.865139i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) −3.71432 + 1.04117i −1.17457 + 0.329246i
\(11\) 0.294246 − 0.157278i 0.0887184 − 0.0474210i −0.426441 − 0.904515i \(-0.640233\pi\)
0.515159 + 0.857094i \(0.327733\pi\)
\(12\) −1.20203 − 1.59848i −0.346996 − 0.461440i
\(13\) −2.51375 − 0.247583i −0.697190 − 0.0686672i −0.256788 − 0.966468i \(-0.582664\pi\)
−0.440402 + 0.897801i \(0.645164\pi\)
\(14\) −0.800360 − 0.351603i −0.213905 − 0.0939699i
\(15\) −2.52003 + 1.04383i −0.650668 + 0.269516i
\(16\) 3.05810 − 2.57837i 0.764525 − 0.644594i
\(17\) 4.21613 + 1.74638i 1.02256 + 0.423559i 0.830022 − 0.557730i \(-0.188328\pi\)
0.192540 + 0.981289i \(0.438328\pi\)
\(18\) −0.978550 − 1.02100i −0.230646 − 0.240652i
\(19\) −1.78752 + 1.46698i −0.410086 + 0.336549i −0.816711 − 0.577046i \(-0.804205\pi\)
0.406626 + 0.913595i \(0.366705\pi\)
\(20\) −2.36511 − 4.91597i −0.528855 − 1.09924i
\(21\) −0.591525 − 0.179437i −0.129081 − 0.0391564i
\(22\) 0.291523 + 0.371009i 0.0621530 + 0.0790993i
\(23\) 0.582156 − 2.92670i 0.121388 − 0.610259i −0.871420 − 0.490538i \(-0.836800\pi\)
0.992808 − 0.119720i \(-0.0381998\pi\)
\(24\) 1.92983 − 2.06779i 0.393926 − 0.422085i
\(25\) −2.39323 + 0.476044i −0.478647 + 0.0952088i
\(26\) −0.274602 − 3.56162i −0.0538539 − 0.698490i
\(27\) −0.773010 − 0.634393i −0.148766 − 0.122089i
\(28\) 0.292430 − 1.20120i 0.0552641 − 0.227006i
\(29\) −0.221297 + 0.414017i −0.0410938 + 0.0768810i −0.901630 − 0.432508i \(-0.857629\pi\)
0.860536 + 0.509389i \(0.170129\pi\)
\(30\) −2.07455 − 3.25215i −0.378759 − 0.593759i
\(31\) −0.222556 + 0.222556i −0.0399722 + 0.0399722i −0.726810 − 0.686838i \(-0.758998\pi\)
0.686838 + 0.726810i \(0.258998\pi\)
\(32\) 4.34411 + 3.62336i 0.767937 + 0.640525i
\(33\) 0.235920 + 0.235920i 0.0410684 + 0.0410684i
\(34\) −1.39313 + 6.30162i −0.238921 + 1.08072i
\(35\) −1.48699 − 0.794813i −0.251347 − 0.134348i
\(36\) 1.18072 − 1.61428i 0.196786 − 0.269047i
\(37\) −7.22167 + 8.79964i −1.18724 + 1.44665i −0.320955 + 0.947094i \(0.604004\pi\)
−0.866281 + 0.499557i \(0.833496\pi\)
\(38\) −2.48333 − 2.12781i −0.402850 − 0.345177i
\(39\) −0.492782 − 2.47738i −0.0789083 − 0.396699i
\(40\) 6.26313 − 4.50490i 0.990289 − 0.712287i
\(41\) 3.19749 + 0.636020i 0.499363 + 0.0993296i 0.438344 − 0.898807i \(-0.355565\pi\)
0.0610189 + 0.998137i \(0.480565\pi\)
\(42\) 0.104131 − 0.867961i 0.0160678 − 0.133929i
\(43\) 0.347543 − 1.14570i 0.0529998 − 0.174717i −0.926445 − 0.376431i \(-0.877151\pi\)
0.979444 + 0.201714i \(0.0646512\pi\)
\(44\) −0.444830 + 0.497386i −0.0670607 + 0.0749838i
\(45\) −1.73041 − 2.10851i −0.257954 − 0.314318i
\(46\) 4.21911 + 0.0895709i 0.622074 + 0.0132065i
\(47\) 2.91400 − 7.03501i 0.425050 − 1.02616i −0.555786 − 0.831326i \(-0.687583\pi\)
0.980836 − 0.194836i \(-0.0624175\pi\)
\(48\) 3.35507 + 2.17796i 0.484263 + 0.314361i
\(49\) 2.53256 + 6.11414i 0.361794 + 0.873449i
\(50\) −1.25262 − 3.21548i −0.177147 − 0.454738i
\(51\) −0.447302 + 4.54153i −0.0626348 + 0.635942i
\(52\) 4.89219 − 1.25998i 0.678424 − 0.174728i
\(53\) 4.77800 + 8.93902i 0.656309 + 1.22787i 0.961298 + 0.275512i \(0.0888475\pi\)
−0.304988 + 0.952356i \(0.598653\pi\)
\(54\) 0.692978 − 1.23279i 0.0943023 − 0.167762i
\(55\) 0.505602 + 0.756688i 0.0681754 + 0.102032i
\(56\) 1.74483 + 0.111261i 0.233162 + 0.0148678i
\(57\) −1.92270 − 1.28471i −0.254668 − 0.170164i
\(58\) −0.631080 − 0.206161i −0.0828649 − 0.0270703i
\(59\) −0.308287 + 0.0303636i −0.0401356 + 0.00395301i −0.118065 − 0.993006i \(-0.537669\pi\)
0.0779294 + 0.996959i \(0.475169\pi\)
\(60\) 4.01773 − 3.69030i 0.518687 − 0.476416i
\(61\) −0.821867 + 0.249311i −0.105229 + 0.0319210i −0.342460 − 0.939532i \(-0.611260\pi\)
0.237231 + 0.971453i \(0.423760\pi\)
\(62\) −0.364765 − 0.255090i −0.0463252 − 0.0323965i
\(63\) − 0.618143i − 0.0778786i
\(64\) −3.97545 + 6.94232i −0.496931 + 0.867790i
\(65\) − 6.88984i − 0.854580i
\(66\) −0.270409 + 0.386669i −0.0332850 + 0.0475956i
\(67\) 12.6200 − 3.82822i 1.54177 − 0.467692i 0.598894 − 0.800828i \(-0.295607\pi\)
0.942879 + 0.333136i \(0.108107\pi\)
\(68\) −9.11879 − 0.387355i −1.10582 − 0.0469737i
\(69\) 2.96967 − 0.292487i 0.357506 − 0.0352112i
\(70\) 0.740453 − 2.26660i 0.0885011 − 0.270910i
\(71\) 0.852169 + 0.569401i 0.101134 + 0.0675755i 0.605109 − 0.796143i \(-0.293130\pi\)
−0.503975 + 0.863718i \(0.668130\pi\)
\(72\) 2.53895 + 1.24649i 0.299218 + 0.146900i
\(73\) 3.54645 + 5.30764i 0.415081 + 0.621213i 0.978816 − 0.204743i \(-0.0656359\pi\)
−0.563735 + 0.825956i \(0.690636\pi\)
\(74\) −14.0336 − 7.88858i −1.63138 − 0.917029i
\(75\) −1.15026 − 2.15199i −0.132821 − 0.248491i
\(76\) 2.35127 − 3.98254i 0.269709 − 0.456828i
\(77\) −0.0202149 + 0.205245i −0.00230370 + 0.0233898i
\(78\) 3.32854 − 1.29666i 0.376883 − 0.146818i
\(79\) 2.11907 + 5.11588i 0.238414 + 0.575581i 0.997119 − 0.0758525i \(-0.0241678\pi\)
−0.758705 + 0.651434i \(0.774168\pi\)
\(80\) 7.81667 + 7.61194i 0.873930 + 0.851041i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) −0.0978583 + 4.60948i −0.0108066 + 0.509032i
\(83\) −1.75844 − 2.14267i −0.193014 − 0.235188i 0.667507 − 0.744604i \(-0.267362\pi\)
−0.860521 + 0.509416i \(0.829862\pi\)
\(84\) 1.23437 − 0.0688521i 0.134680 − 0.00751238i
\(85\) −3.61337 + 11.9117i −0.391925 + 1.29200i
\(86\) 1.68111 + 0.201686i 0.181279 + 0.0217484i
\(87\) −0.460429 − 0.0915849i −0.0493631 − 0.00981894i
\(88\) −0.802257 − 0.496908i −0.0855209 − 0.0529706i
\(89\) −2.44979 − 12.3159i −0.259677 − 1.30548i −0.861869 − 0.507132i \(-0.830706\pi\)
0.602192 − 0.798351i \(-0.294294\pi\)
\(90\) 2.50991 − 2.92927i 0.264567 − 0.308772i
\(91\) 0.990527 − 1.20696i 0.103835 − 0.126524i
\(92\) 0.914843 + 5.89754i 0.0953790 + 0.614861i
\(93\) −0.277577 − 0.148368i −0.0287834 − 0.0153851i
\(94\) 10.5148 + 2.32458i 1.08452 + 0.239762i
\(95\) −4.46006 − 4.46006i −0.457593 − 0.457593i
\(96\) −2.20631 + 5.20886i −0.225180 + 0.531627i
\(97\) 5.61426 − 5.61426i 0.570042 − 0.570042i −0.362098 − 0.932140i \(-0.617939\pi\)
0.932140 + 0.362098i \(0.117939\pi\)
\(98\) −7.89044 + 5.03331i −0.797054 + 0.508441i
\(99\) −0.157278 + 0.294246i −0.0158070 + 0.0295728i
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.13.19 ✓ 512
128.69 even 32 inner 384.2.v.a.325.19 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.19 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.19 yes 512 128.69 even 32 inner