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Properties

Label 384.2.v.a.13.3
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 13.3
Character \(\chi\) \(=\) 384.13
Dual form 384.2.v.a.325.3

$q$-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+(-1.34449 + 0.438569i) q^{2} +(-0.290285 - 0.956940i) q^{3} +(1.61531 - 1.17931i) q^{4} +(0.414961 + 4.21317i) q^{5} +(0.809970 + 1.15929i) q^{6} +(-0.841223 + 1.25898i) q^{7} +(-1.65457 + 2.29399i) q^{8} +(-0.831470 + 0.555570i) q^{9} +(-2.40568 - 5.48258i) q^{10} +(0.784308 - 0.419221i) q^{11} +(-1.59743 - 1.20342i) q^{12} +(-5.50473 - 0.542168i) q^{13} +(0.578867 - 2.06162i) q^{14} +(3.91129 - 1.62011i) q^{15} +(1.21848 - 3.80990i) q^{16} +(-5.12904 - 2.12452i) q^{17} +(0.874248 - 1.11162i) q^{18} +(3.70651 - 3.04185i) q^{19} +(5.63890 + 6.31622i) q^{20} +(1.44896 + 0.439538i) q^{21} +(-0.870637 + 0.907612i) q^{22} +(-1.31379 + 6.60489i) q^{23} +(2.67551 + 0.917413i) q^{24} +(-12.6747 + 2.52114i) q^{25} +(7.63883 - 1.68526i) q^{26} +(0.773010 + 0.634393i) q^{27} +(0.125881 + 3.02571i) q^{28} +(-0.0996101 + 0.186357i) q^{29} +(-4.54817 + 3.89360i) q^{30} +(-3.51113 + 3.51113i) q^{31} +(0.0326686 + 5.65676i) q^{32} +(-0.628842 - 0.628842i) q^{33} +(7.82770 + 0.606957i) q^{34} +(-5.65336 - 3.02179i) q^{35} +(-0.687898 + 1.87798i) q^{36} +(-1.41029 + 1.71844i) q^{37} +(-3.64931 + 5.71531i) q^{38} +(1.07911 + 5.42508i) q^{39} +(-10.3516 - 6.01906i) q^{40} +(5.17331 + 1.02903i) q^{41} +(-2.14088 + 0.0445153i) q^{42} +(0.959206 - 3.16208i) q^{43} +(0.772514 - 1.60211i) q^{44} +(-2.68574 - 3.27258i) q^{45} +(-1.13032 - 9.45641i) q^{46} +(0.429316 - 1.03646i) q^{47} +(-3.99955 - 0.0600573i) q^{48} +(1.80141 + 4.34899i) q^{49} +(15.9353 - 8.94837i) q^{50} +(-0.544155 + 5.52490i) q^{51} +(-9.53124 + 5.61598i) q^{52} +(2.70031 + 5.05193i) q^{53} +(-1.31753 - 0.513918i) q^{54} +(2.09171 + 3.13046i) q^{55} +(-1.49623 - 4.01283i) q^{56} +(-3.98681 - 2.66390i) q^{57} +(0.0521943 - 0.294242i) q^{58} +(-11.8882 + 1.17088i) q^{59} +(4.40736 - 7.22959i) q^{60} +(-6.06677 + 1.84033i) q^{61} +(3.18081 - 6.26055i) q^{62} -1.51416i q^{63} +(-2.52480 - 7.59114i) q^{64} -23.4173i q^{65} +(1.12126 + 0.569682i) q^{66} +(12.7691 - 3.87346i) q^{67} +(-10.7905 + 2.61694i) q^{68} +(6.70186 - 0.660076i) q^{69} +(8.92616 + 1.58337i) q^{70} +(0.0488394 + 0.0326335i) q^{71} +(0.101250 - 2.82661i) q^{72} +(6.80173 + 10.1795i) q^{73} +(1.14246 - 2.92894i) q^{74} +(6.09184 + 11.3970i) q^{75} +(2.39990 - 9.28465i) q^{76} +(-0.131987 + 1.34009i) q^{77} +(-3.83013 - 6.82070i) q^{78} +(1.95805 + 4.72715i) q^{79} +(16.5573 + 3.55270i) q^{80} +(0.382683 - 0.923880i) q^{81} +(-7.40677 + 0.885325i) q^{82} +(-0.148807 - 0.181321i) q^{83} +(2.85888 - 0.998777i) q^{84} +(6.82260 - 22.4911i) q^{85} +(0.0971461 + 4.67207i) q^{86} +(0.207248 + 0.0412242i) q^{87} +(-0.336001 + 2.49283i) q^{88} +(2.00111 + 10.0603i) q^{89} +(5.04620 + 3.22207i) q^{90} +(5.31328 - 6.47425i) q^{91} +(5.66699 + 12.2183i) q^{92} +(4.37917 + 2.34071i) q^{93} +(-0.122652 + 1.58180i) q^{94} +(14.3539 + 14.3539i) q^{95} +(5.40370 - 1.67333i) q^{96} +(3.64779 - 3.64779i) q^{97} +(-4.32932 - 5.05714i) q^{98} +(-0.419221 + 0.784308i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34449 + 0.438569i −0.950699 + 0.310115i
\(3\) −0.290285 − 0.956940i −0.167596 − 0.552490i
\(4\) 1.61531 − 1.17931i 0.807657 − 0.589653i
\(5\) 0.414961 + 4.21317i 0.185576 + 1.88419i 0.413497 + 0.910506i \(0.364307\pi\)
−0.227921 + 0.973680i \(0.573193\pi\)
\(6\) 0.809970 + 1.15929i 0.330669 + 0.473277i
\(7\) −0.841223 + 1.25898i −0.317952 + 0.475849i −0.955678 − 0.294414i \(-0.904875\pi\)
0.637725 + 0.770264i \(0.279875\pi\)
\(8\) −1.65457 + 2.29399i −0.584978 + 0.811049i
\(9\) −0.831470 + 0.555570i −0.277157 + 0.185190i
\(10\) −2.40568 − 5.48258i −0.760742 − 1.73374i
\(11\) 0.784308 − 0.419221i 0.236478 − 0.126400i −0.348902 − 0.937159i \(-0.613445\pi\)
0.585380 + 0.810759i \(0.300945\pi\)
\(12\) −1.59743 − 1.20342i −0.461137 − 0.347399i
\(13\) −5.50473 − 0.542168i −1.52674 − 0.150370i −0.700483 − 0.713669i \(-0.747032\pi\)
−0.826253 + 0.563299i \(0.809532\pi\)
\(14\) 0.578867 − 2.06162i 0.154709 − 0.550991i
\(15\) 3.91129 − 1.62011i 1.00989 − 0.418311i
\(16\) 1.21848 − 3.80990i 0.304620 − 0.952474i
\(17\) −5.12904 − 2.12452i −1.24398 − 0.515271i −0.339020 − 0.940779i \(-0.610096\pi\)
−0.904955 + 0.425508i \(0.860096\pi\)
\(18\) 0.874248 − 1.11162i 0.206062 − 0.262010i
\(19\) 3.70651 − 3.04185i 0.850331 − 0.697849i −0.104652 − 0.994509i \(-0.533373\pi\)
0.954983 + 0.296660i \(0.0958728\pi\)
\(20\) 5.63890 + 6.31622i 1.26090 + 1.41235i
\(21\) 1.44896 + 0.439538i 0.316189 + 0.0959150i
\(22\) −0.870637 + 0.907612i −0.185621 + 0.193504i
\(23\) −1.31379 + 6.60489i −0.273945 + 1.37721i 0.561425 + 0.827527i \(0.310253\pi\)
−0.835370 + 0.549687i \(0.814747\pi\)
\(24\) 2.67551 + 0.917413i 0.546136 + 0.187266i
\(25\) −12.6747 + 2.52114i −2.53493 + 0.504229i
\(26\) 7.63883 − 1.68526i 1.49810 − 0.330507i
\(27\) 0.773010 + 0.634393i 0.148766 + 0.122089i
\(28\) 0.125881 + 3.02571i 0.0237893 + 0.571805i
\(29\) −0.0996101 + 0.186357i −0.0184971 + 0.0346057i −0.891000 − 0.454003i \(-0.849995\pi\)
0.872503 + 0.488609i \(0.162495\pi\)
\(30\) −4.54817 + 3.89360i −0.830378 + 0.710870i
\(31\) −3.51113 + 3.51113i −0.630617 + 0.630617i −0.948223 − 0.317606i \(-0.897121\pi\)
0.317606 + 0.948223i \(0.397121\pi\)
\(32\) 0.0326686 + 5.65676i 0.00577504 + 0.999983i
\(33\) −0.628842 − 0.628842i −0.109467 − 0.109467i
\(34\) 7.82770 + 0.606957i 1.34244 + 0.104092i
\(35\) −5.65336 − 3.02179i −0.955593 − 0.510775i
\(36\) −0.687898 + 1.87798i −0.114650 + 0.312996i
\(37\) −1.41029 + 1.71844i −0.231850 + 0.282510i −0.875920 − 0.482457i \(-0.839745\pi\)
0.644070 + 0.764967i \(0.277245\pi\)
\(38\) −3.64931 + 5.71531i −0.591996 + 0.927145i
\(39\) 1.07911 + 5.42508i 0.172797 + 0.868707i
\(40\) −10.3516 − 6.01906i −1.63672 − 0.951697i
\(41\) 5.17331 + 1.02903i 0.807935 + 0.160708i 0.581741 − 0.813374i \(-0.302372\pi\)
0.226194 + 0.974082i \(0.427372\pi\)
\(42\) −2.14088 + 0.0445153i −0.330346 + 0.00686887i
\(43\) 0.959206 − 3.16208i 0.146278 − 0.482212i −0.852930 − 0.522026i \(-0.825176\pi\)
0.999207 + 0.0398135i \(0.0126764\pi\)
\(44\) 0.772514 − 1.60211i 0.116461 − 0.241527i
\(45\) −2.68574 − 3.27258i −0.400366 − 0.487847i
\(46\) −1.13032 − 9.45641i −0.166656 − 1.39427i
\(47\) 0.429316 − 1.03646i 0.0626222 − 0.151183i −0.889471 − 0.456992i \(-0.848927\pi\)
0.952093 + 0.305809i \(0.0989268\pi\)
\(48\) −3.99955 − 0.0600573i −0.577285 − 0.00866853i
\(49\) 1.80141 + 4.34899i 0.257345 + 0.621285i
\(50\) 15.9353 − 8.94837i 2.25359 − 1.26549i
\(51\) −0.544155 + 5.52490i −0.0761970 + 0.773641i
\(52\) −9.53124 + 5.61598i −1.32175 + 0.778796i
\(53\) 2.70031 + 5.05193i 0.370917 + 0.693936i 0.996259 − 0.0864141i \(-0.0275408\pi\)
−0.625343 + 0.780350i \(0.715041\pi\)
\(54\) −1.31753 − 0.513918i −0.179293 − 0.0699353i
\(55\) 2.09171 + 3.13046i 0.282045 + 0.422111i
\(56\) −1.49623 − 4.01283i −0.199942 − 0.536237i
\(57\) −3.98681 − 2.66390i −0.528067 − 0.352843i
\(58\) 0.0521943 − 0.294242i 0.00685345 − 0.0386358i
\(59\) −11.8882 + 1.17088i −1.54771 + 0.152436i −0.835484 − 0.549515i \(-0.814813\pi\)
−0.712226 + 0.701951i \(0.752313\pi\)
\(60\) 4.40736 − 7.22959i 0.568988 − 0.933336i
\(61\) −6.06677 + 1.84033i −0.776770 + 0.235631i −0.653686 − 0.756766i \(-0.726778\pi\)
−0.123084 + 0.992396i \(0.539278\pi\)
\(62\) 3.18081 − 6.26055i 0.403963 − 0.795091i
\(63\) − 1.51416i − 0.190766i
\(64\) −2.52480 − 7.59114i −0.315600 − 0.948892i
\(65\) − 23.4173i − 2.90456i
\(66\) 1.12126 + 0.569682i 0.138018 + 0.0701230i
\(67\) 12.7691 − 3.87346i 1.55999 − 0.473219i 0.611827 − 0.790992i \(-0.290435\pi\)
0.948167 + 0.317773i \(0.102935\pi\)
\(68\) −10.7905 + 2.61694i −1.30854 + 0.317351i
\(69\) 6.70186 − 0.660076i 0.806809 − 0.0794638i
\(70\) 8.92616 + 1.58337i 1.06688 + 0.189249i
\(71\) 0.0488394 + 0.0326335i 0.00579617 + 0.00387288i 0.558465 − 0.829528i \(-0.311391\pi\)
−0.552669 + 0.833401i \(0.686391\pi\)
\(72\) 0.101250 − 2.82661i 0.0119324 − 0.333120i
\(73\) 6.80173 + 10.1795i 0.796083 + 1.19142i 0.978102 + 0.208127i \(0.0667368\pi\)
−0.182019 + 0.983295i \(0.558263\pi\)
\(74\) 1.14246 − 2.92894i 0.132809 − 0.340482i
\(75\) 6.09184 + 11.3970i 0.703425 + 1.31602i
\(76\) 2.39990 − 9.28465i 0.275288 − 1.06502i
\(77\) −0.131987 + 1.34009i −0.0150413 + 0.152717i
\(78\) −3.83013 − 6.82070i −0.433677 − 0.772292i
\(79\) 1.95805 + 4.72715i 0.220298 + 0.531846i 0.994930 − 0.100565i \(-0.0320651\pi\)
−0.774632 + 0.632412i \(0.782065\pi\)
\(80\) 16.5573 + 3.55270i 1.85117 + 0.397204i
\(81\) 0.382683 − 0.923880i 0.0425204 − 0.102653i
\(82\) −7.40677 + 0.885325i −0.817941 + 0.0977678i
\(83\) −0.148807 − 0.181321i −0.0163336 − 0.0199026i 0.764780 − 0.644292i \(-0.222848\pi\)
−0.781113 + 0.624389i \(0.785348\pi\)
\(84\) 2.85888 − 0.998777i 0.311929 − 0.108975i
\(85\) 6.82260 − 22.4911i 0.740015 − 2.43950i
\(86\) 0.0971461 + 4.67207i 0.0104755 + 0.503802i
\(87\) 0.207248 + 0.0412242i 0.0222193 + 0.00441970i
\(88\) −0.336001 + 2.49283i −0.0358178 + 0.265736i
\(89\) 2.00111 + 10.0603i 0.212117 + 1.06639i 0.929251 + 0.369450i \(0.120454\pi\)
−0.717133 + 0.696936i \(0.754546\pi\)
\(90\) 5.04620 + 3.22207i 0.531916 + 0.339636i
\(91\) 5.31328 − 6.47425i 0.556983 − 0.678686i
\(92\) 5.66699 + 12.2183i 0.590825 + 1.27385i
\(93\) 4.37917 + 2.34071i 0.454098 + 0.242721i
\(94\) −0.122652 + 1.58180i −0.0126506 + 0.163150i
\(95\) 14.3539 + 14.3539i 1.47268 + 1.47268i
\(96\) 5.40370 − 1.67333i 0.551513 − 0.170784i
\(97\) 3.64779 − 3.64779i 0.370376 − 0.370376i −0.497238 − 0.867614i \(-0.665652\pi\)
0.867614 + 0.497238i \(0.165652\pi\)
\(98\) −4.32932 − 5.05714i −0.437327 − 0.510848i
\(99\) −0.419221 + 0.784308i −0.0421333 + 0.0788259i
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.13.3 ✓ 512
128.69 even 32 inner 384.2.v.a.325.3 yes 512
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.v.a.13.3 ✓ 512 1.1 even 1 trivial
384.2.v.a.325.3 yes 512 128.69 even 32 inner