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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 229.15
Character \(\chi\) \(=\) 384.229
Dual form 384.2.v.a.109.15

$q$-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+(-0.127742 + 1.40843i) q^{2} +(0.881921 - 0.471397i) q^{3} +(-1.96736 - 0.359831i) q^{4} +(-1.40713 - 1.71460i) q^{5} +(0.551272 + 1.30234i) q^{6} +(0.0127962 - 0.00855015i) q^{7} +(0.758112 - 2.72493i) q^{8} +(0.555570 - 0.831470i) q^{9} +(2.59464 - 1.76283i) q^{10} +(3.40372 - 1.03251i) q^{11} +(-1.90468 + 0.610066i) q^{12} +(-2.41676 - 1.98339i) q^{13} +(0.0104077 + 0.0191148i) q^{14} +(-2.04924 - 0.848822i) q^{15} +(3.74104 + 1.41584i) q^{16} +(4.96730 - 2.05752i) q^{17} +(1.10010 + 0.888697i) q^{18} +(-0.321304 - 3.26226i) q^{19} +(2.15138 + 3.87957i) q^{20} +(0.00725473 - 0.0135727i) q^{21} +(1.01942 + 4.92581i) q^{22} +(4.07213 - 0.809998i) q^{23} +(-0.615929 - 2.76055i) q^{24} +(0.0156326 - 0.0785902i) q^{25} +(3.10219 - 3.15049i) q^{26} +(0.0980171 - 0.995185i) q^{27} +(-0.0282514 + 0.0122168i) q^{28} +(-0.827213 + 2.72696i) q^{29} +(1.45728 - 2.77778i) q^{30} +(-3.88443 - 3.88443i) q^{31} +(-2.47200 + 5.08814i) q^{32} +(2.51509 - 2.51509i) q^{33} +(2.26335 + 7.25893i) q^{34} +(-0.0326660 - 0.00990914i) q^{35} +(-1.39220 + 1.43589i) q^{36} +(-2.61418 - 0.257475i) q^{37} +(4.63571 - 0.0358090i) q^{38} +(-3.06636 - 0.609936i) q^{39} +(-5.73893 + 2.53449i) q^{40} +(1.20863 + 6.07619i) q^{41} +(0.0181894 + 0.0119516i) q^{42} +(-9.65881 - 5.16274i) q^{43} +(-7.06789 + 0.806554i) q^{44} +(-2.20740 + 0.217410i) q^{45} +(0.620646 + 5.83880i) q^{46} +(2.67337 + 6.45408i) q^{47} +(3.96673 - 0.514858i) q^{48} +(-2.67869 + 6.46694i) q^{49} +(0.108692 + 0.0320567i) q^{50} +(3.41086 - 4.15614i) q^{51} +(4.04097 + 4.77167i) q^{52} +(-0.861628 - 2.84041i) q^{53} +(1.38913 + 0.265177i) q^{54} +(-6.55983 - 4.38314i) q^{55} +(-0.0135976 - 0.0413508i) q^{56} +(-1.82118 - 2.72559i) q^{57} +(-3.73506 - 1.51342i) q^{58} +(5.35526 - 4.39495i) q^{59} +(3.72616 + 2.40732i) q^{60} +(5.71656 + 10.6949i) q^{61} +(5.96715 - 4.97475i) q^{62} -0.0153899i q^{63} +(-6.85053 - 4.13161i) q^{64} +6.93467i q^{65} +(3.22106 + 3.86362i) q^{66} +(2.05001 + 3.83531i) q^{67} +(-10.5128 + 2.26051i) q^{68} +(3.20947 - 2.63395i) q^{69} +(0.0181292 - 0.0447421i) q^{70} +(-1.74600 - 2.61308i) q^{71} +(-1.84452 - 2.14424i) q^{72} +(6.90550 + 4.61411i) q^{73} +(0.696576 - 3.64901i) q^{74} +(-0.0232605 - 0.0766795i) q^{75} +(-0.541739 + 6.53366i) q^{76} +(0.0347266 - 0.0423145i) q^{77} +(1.25076 - 4.24084i) q^{78} +(0.304162 - 0.734312i) q^{79} +(-2.83656 - 8.40666i) q^{80} +(-0.382683 - 0.923880i) q^{81} +(-8.71230 + 0.926090i) q^{82} +(-5.51850 + 0.543524i) q^{83} +(-0.0191566 + 0.0240919i) q^{84} +(-10.5175 - 5.62171i) q^{85} +(8.50520 - 12.9443i) q^{86} +(0.555941 + 2.79491i) q^{87} +(-0.233112 - 10.0577i) q^{88} +(-6.60896 - 1.31460i) q^{89} +(-0.0242301 - 3.13674i) q^{90} +(-0.0478837 - 0.00471613i) q^{91} +(-8.30283 + 0.128280i) q^{92} +(-5.25686 - 1.59465i) q^{93} +(-9.43163 + 2.94080i) q^{94} +(-5.14134 + 5.14134i) q^{95} +(0.218426 + 5.65264i) q^{96} +(10.0236 + 10.0236i) q^{97} +(-8.76606 - 4.59886i) q^{98} +(1.03251 - 3.40372i) 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 \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(e\left(\frac{9}{32}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.127742 + 1.40843i −0.0903270 + 0.995912i
\(3\) 0.881921 − 0.471397i 0.509177 − 0.272161i
\(4\) −1.96736 − 0.359831i −0.983682 − 0.179916i
\(5\) −1.40713 − 1.71460i −0.629289 − 0.766791i 0.356588 − 0.934262i \(-0.383940\pi\)
−0.985877 + 0.167471i \(0.946440\pi\)
\(6\) 0.551272 + 1.30234i 0.225056 + 0.531680i
\(7\) 0.0127962 − 0.00855015i 0.00483651 − 0.00323165i −0.553150 − 0.833082i \(-0.686574\pi\)
0.557986 + 0.829850i \(0.311574\pi\)
\(8\) 0.758112 − 2.72493i 0.268033 − 0.963410i
\(9\) 0.555570 − 0.831470i 0.185190 − 0.277157i
\(10\) 2.59464 − 1.76283i 0.820499 − 0.557455i
\(11\) 3.40372 − 1.03251i 1.02626 − 0.311313i 0.268108 − 0.963389i \(-0.413602\pi\)
0.758154 + 0.652076i \(0.226102\pi\)
\(12\) −1.90468 + 0.610066i −0.549835 + 0.176111i
\(13\) −2.41676 − 1.98339i −0.670290 − 0.550092i 0.236523 − 0.971626i \(-0.423992\pi\)
−0.906813 + 0.421533i \(0.861492\pi\)
\(14\) 0.0104077 + 0.0191148i 0.00278158 + 0.00510865i
\(15\) −2.04924 − 0.848822i −0.529111 − 0.219165i
\(16\) 3.74104 + 1.41584i 0.935261 + 0.353960i
\(17\) 4.96730 − 2.05752i 1.20475 − 0.499022i 0.312217 − 0.950011i \(-0.398928\pi\)
0.892530 + 0.450989i \(0.148928\pi\)
\(18\) 1.10010 + 0.888697i 0.259296 + 0.209468i
\(19\) −0.321304 − 3.26226i −0.0737122 − 0.748413i −0.959007 − 0.283383i \(-0.908543\pi\)
0.885295 − 0.465030i \(-0.153957\pi\)
\(20\) 2.15138 + 3.87957i 0.481063 + 0.867498i
\(21\) 0.00725473 − 0.0135727i 0.00158311 − 0.00296180i
\(22\) 1.01942 + 4.92581i 0.217341 + 1.05019i
\(23\) 4.07213 − 0.809998i 0.849099 − 0.168896i 0.248684 − 0.968585i \(-0.420002\pi\)
0.600415 + 0.799689i \(0.295002\pi\)
\(24\) −0.615929 − 2.76055i −0.125726 − 0.563495i
\(25\) 0.0156326 − 0.0785902i 0.00312651 − 0.0157180i
\(26\) 3.10219 − 3.15049i 0.608389 − 0.617861i
\(27\) 0.0980171 − 0.995185i 0.0188634 − 0.191523i
\(28\) −0.0282514 + 0.0122168i −0.00533901 + 0.00230876i
\(29\) −0.827213 + 2.72696i −0.153610 + 0.506383i −0.999642 − 0.0267482i \(-0.991485\pi\)
0.846033 + 0.533131i \(0.178985\pi\)
\(30\) 1.45728 − 2.77778i 0.266062 − 0.507151i
\(31\) −3.88443 − 3.88443i −0.697663 − 0.697663i 0.266243 − 0.963906i \(-0.414218\pi\)
−0.963906 + 0.266243i \(0.914218\pi\)
\(32\) −2.47200 + 5.08814i −0.436992 + 0.899465i
\(33\) 2.51509 − 2.51509i 0.437822 − 0.437822i
\(34\) 2.26335 + 7.25893i 0.388161 + 1.24490i
\(35\) −0.0326660 − 0.00990914i −0.00552157 − 0.00167495i
\(36\) −1.39220 + 1.43589i −0.232033 + 0.239315i
\(37\) −2.61418 − 0.257475i −0.429769 − 0.0423286i −0.119180 − 0.992873i \(-0.538026\pi\)
−0.310589 + 0.950544i \(0.600526\pi\)
\(38\) 4.63571 − 0.0358090i 0.752012 − 0.00580899i
\(39\) −3.06636 − 0.609936i −0.491010 − 0.0976680i
\(40\) −5.73893 + 2.53449i −0.907405 + 0.400738i
\(41\) 1.20863 + 6.07619i 0.188756 + 0.948942i 0.952759 + 0.303728i \(0.0982315\pi\)
−0.764002 + 0.645213i \(0.776769\pi\)
\(42\) 0.0181894 + 0.0119516i 0.00280669 + 0.00184417i
\(43\) −9.65881 − 5.16274i −1.47295 − 0.787311i −0.477150 − 0.878822i \(-0.658330\pi\)
−0.995804 + 0.0915111i \(0.970830\pi\)
\(44\) −7.06789 + 0.806554i −1.06552 + 0.121593i
\(45\) −2.20740 + 0.217410i −0.329059 + 0.0324095i
\(46\) 0.620646 + 5.83880i 0.0915092 + 0.860884i
\(47\) 2.67337 + 6.45408i 0.389950 + 0.941424i 0.989949 + 0.141423i \(0.0451677\pi\)
−0.599999 + 0.800001i \(0.704832\pi\)
\(48\) 3.96673 − 0.514858i 0.572548 − 0.0743133i
\(49\) −2.67869 + 6.46694i −0.382670 + 0.923848i
\(50\) 0.108692 + 0.0320567i 0.0153714 + 0.00453350i
\(51\) 3.41086 − 4.15614i 0.477615 − 0.581976i
\(52\) 4.04097 + 4.77167i 0.560382 + 0.661712i
\(53\) −0.861628 − 2.84041i −0.118354 − 0.390160i 0.877668 − 0.479269i \(-0.159098\pi\)
−0.996022 + 0.0891089i \(0.971598\pi\)
\(54\) 1.38913 + 0.265177i 0.189037 + 0.0360860i
\(55\) −6.55983 − 4.38314i −0.884527 − 0.591022i
\(56\) −0.0135976 − 0.0413508i −0.00181706 − 0.00552573i
\(57\) −1.82118 − 2.72559i −0.241221 − 0.361013i
\(58\) −3.73506 − 1.51342i −0.490438 − 0.198722i
\(59\) 5.35526 − 4.39495i 0.697196 − 0.572174i −0.217626 − 0.976032i \(-0.569831\pi\)
0.914821 + 0.403858i \(0.132331\pi\)
\(60\) 3.72616 + 2.40732i 0.481046 + 0.310784i
\(61\) 5.71656 + 10.6949i 0.731930 + 1.36934i 0.921851 + 0.387545i \(0.126677\pi\)
−0.189921 + 0.981799i \(0.560823\pi\)
\(62\) 5.96715 − 4.97475i 0.757829 − 0.631794i
\(63\) − 0.0153899i − 0.00193894i
\(64\) −6.85053 − 4.13161i −0.856316 − 0.516452i
\(65\) 6.93467i 0.860140i
\(66\) 3.22106 + 3.86362i 0.396485 + 0.475579i
\(67\) 2.05001 + 3.83531i 0.250449 + 0.468557i 0.975459 − 0.220181i \(-0.0706648\pi\)
−0.725010 + 0.688738i \(0.758165\pi\)
\(68\) −10.5128 + 2.26051i −1.27487 + 0.274127i
\(69\) 3.20947 − 2.63395i 0.386375 − 0.317090i
\(70\) 0.0181292 − 0.0447421i 0.00216685 − 0.00534770i
\(71\) −1.74600 − 2.61308i −0.207212 − 0.310115i 0.713277 − 0.700882i \(-0.247210\pi\)
−0.920490 + 0.390766i \(0.872210\pi\)
\(72\) −1.84452 − 2.14424i −0.217378 − 0.252701i
\(73\) 6.90550 + 4.61411i 0.808228 + 0.540040i 0.889652 − 0.456639i \(-0.150947\pi\)
−0.0814244 + 0.996680i \(0.525947\pi\)
\(74\) 0.696576 − 3.64901i 0.0809753 − 0.424189i
\(75\) −0.0232605 − 0.0766795i −0.00268589 − 0.00885419i
\(76\) −0.541739 + 6.53366i −0.0621418 + 0.749462i
\(77\) 0.0347266 − 0.0423145i 0.00395747 − 0.00482219i
\(78\) 1.25076 − 4.24084i 0.141620 − 0.480181i
\(79\) 0.304162 − 0.734312i 0.0342209 − 0.0826166i −0.905845 − 0.423609i \(-0.860763\pi\)
0.940066 + 0.340992i \(0.110763\pi\)
\(80\) −2.83656 − 8.40666i −0.317137 − 0.939893i
\(81\) −0.382683 − 0.923880i −0.0425204 − 0.102653i
\(82\) −8.71230 + 0.926090i −0.962112 + 0.102270i
\(83\) −5.51850 + 0.543524i −0.605734 + 0.0596595i −0.396235 − 0.918149i \(-0.629683\pi\)
−0.209499 + 0.977809i \(0.567183\pi\)
\(84\) −0.0191566 + 0.0240919i −0.00209015 + 0.00262864i
\(85\) −10.5175 − 5.62171i −1.14078 − 0.609760i
\(86\) 8.50520 − 12.9443i 0.917140 − 1.39582i
\(87\) 0.555941 + 2.79491i 0.0596032 + 0.299645i
\(88\) −0.233112 − 10.0577i −0.0248498 − 1.07215i
\(89\) −6.60896 − 1.31460i −0.700548 − 0.139348i −0.168050 − 0.985779i \(-0.553747\pi\)
−0.532499 + 0.846431i \(0.678747\pi\)
\(90\) −0.0242301 − 3.13674i −0.00255407 − 0.330642i
\(91\) −0.0478837 − 0.00471613i −0.00501957 − 0.000494385i
\(92\) −8.30283 + 0.128280i −0.865630 + 0.0133741i
\(93\) −5.25686 − 1.59465i −0.545111 − 0.165358i
\(94\) −9.43163 + 2.94080i −0.972798 + 0.303320i
\(95\) −5.14134 + 5.14134i −0.527490 + 0.527490i
\(96\) 0.218426 + 5.65264i 0.0222930 + 0.576920i
\(97\) 10.0236 + 10.0236i 1.01775 + 1.01775i 0.999840 + 0.0179066i \(0.00570016\pi\)
0.0179066 + 0.999840i \(0.494300\pi\)
\(98\) −8.76606 − 4.59886i −0.885506 − 0.464555i
\(99\) 1.03251 − 3.40372i 0.103771 − 0.342087i
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.229.15 yes 512
128.109 even 32 inner 384.2.v.a.109.15 ✓ 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.109.15 ✓ 512 128.109 even 32 inner
384.2.v.a.229.15 yes 512 1.1 even 1 trivial