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Properties

Label 16.2.e.a.13.1
Level $16$
Weight $2$
Character 16.13
Analytic conductor $0.128$
Analytic rank $0$
Dimension $2$
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] = [16,2,Mod(5,16)] 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("16.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(16, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 16.e (of order \(4\), degree \(2\), 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: \(0.127760643234\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 16.13
Dual form 16.2.e.a.5.1

$q$-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+(-1.00000 + 1.00000i) q^{2} +(-1.00000 - 1.00000i) q^{3} -2.00000i q^{4} +(-1.00000 + 1.00000i) q^{5} +2.00000 q^{6} +2.00000i q^{7} +(2.00000 + 2.00000i) q^{8} -1.00000i q^{9} -2.00000i q^{10} +(1.00000 - 1.00000i) q^{11} +(-2.00000 + 2.00000i) q^{12} +(-1.00000 - 1.00000i) q^{13} +(-2.00000 - 2.00000i) q^{14} +2.00000 q^{15} -4.00000 q^{16} -2.00000 q^{17} +(1.00000 + 1.00000i) q^{18} +(3.00000 + 3.00000i) q^{19} +(2.00000 + 2.00000i) q^{20} +(2.00000 - 2.00000i) q^{21} +2.00000i q^{22} -6.00000i q^{23} -4.00000i q^{24} +3.00000i q^{25} +2.00000 q^{26} +(-4.00000 + 4.00000i) q^{27} +4.00000 q^{28} +(3.00000 + 3.00000i) q^{29} +(-2.00000 + 2.00000i) q^{30} -8.00000 q^{31} +(4.00000 - 4.00000i) q^{32} -2.00000 q^{33} +(2.00000 - 2.00000i) q^{34} +(-2.00000 - 2.00000i) q^{35} -2.00000 q^{36} +(3.00000 - 3.00000i) q^{37} -6.00000 q^{38} +2.00000i q^{39} -4.00000 q^{40} +4.00000i q^{42} +(5.00000 - 5.00000i) q^{43} +(-2.00000 - 2.00000i) q^{44} +(1.00000 + 1.00000i) q^{45} +(6.00000 + 6.00000i) q^{46} +8.00000 q^{47} +(4.00000 + 4.00000i) q^{48} +3.00000 q^{49} +(-3.00000 - 3.00000i) q^{50} +(2.00000 + 2.00000i) q^{51} +(-2.00000 + 2.00000i) q^{52} +(-5.00000 + 5.00000i) q^{53} -8.00000i q^{54} +2.00000i q^{55} +(-4.00000 + 4.00000i) q^{56} -6.00000i q^{57} -6.00000 q^{58} +(-3.00000 + 3.00000i) q^{59} -4.00000i q^{60} +(-9.00000 - 9.00000i) q^{61} +(8.00000 - 8.00000i) q^{62} +2.00000 q^{63} +8.00000i q^{64} +2.00000 q^{65} +(2.00000 - 2.00000i) q^{66} +(-5.00000 - 5.00000i) q^{67} +4.00000i q^{68} +(-6.00000 + 6.00000i) q^{69} +4.00000 q^{70} +10.0000i q^{71} +(2.00000 - 2.00000i) q^{72} +4.00000i q^{73} +6.00000i q^{74} +(3.00000 - 3.00000i) q^{75} +(6.00000 - 6.00000i) q^{76} +(2.00000 + 2.00000i) q^{77} +(-2.00000 - 2.00000i) q^{78} +(4.00000 - 4.00000i) q^{80} +5.00000 q^{81} +(-1.00000 - 1.00000i) q^{83} +(-4.00000 - 4.00000i) q^{84} +(2.00000 - 2.00000i) q^{85} +10.0000i q^{86} -6.00000i q^{87} +4.00000 q^{88} -4.00000i q^{89} -2.00000 q^{90} +(2.00000 - 2.00000i) q^{91} -12.0000 q^{92} +(8.00000 + 8.00000i) q^{93} +(-8.00000 + 8.00000i) q^{94} -6.00000 q^{95} -8.00000 q^{96} -2.00000 q^{97} +(-3.00000 + 3.00000i) q^{98} +(-1.00000 - 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{5} + 4 q^{6} + 4 q^{8} + 2 q^{11} - 4 q^{12} - 2 q^{13} - 4 q^{14} + 4 q^{15} - 8 q^{16} - 4 q^{17} + 2 q^{18} + 6 q^{19} + 4 q^{20} + 4 q^{21} + 4 q^{26} - 8 q^{27} + 8 q^{28}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 + 1.00000i −0.707107 + 0.707107i
\(3\) −1.00000 − 1.00000i −0.577350 − 0.577350i 0.356822 − 0.934172i \(-0.383860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) − 2.00000i − 1.00000i
\(5\) −1.00000 + 1.00000i −0.447214 + 0.447214i −0.894427 − 0.447214i \(-0.852416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 2.00000 0.816497
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 2.00000 + 2.00000i 0.707107 + 0.707107i
\(9\) − 1.00000i − 0.333333i
\(10\) − 2.00000i − 0.632456i
\(11\) 1.00000 − 1.00000i 0.301511 − 0.301511i −0.540094 − 0.841605i \(-0.681611\pi\)
0.841605 + 0.540094i \(0.181611\pi\)
\(12\) −2.00000 + 2.00000i −0.577350 + 0.577350i
\(13\) −1.00000 − 1.00000i −0.277350 − 0.277350i 0.554700 − 0.832050i \(-0.312833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −2.00000 − 2.00000i −0.534522 − 0.534522i
\(15\) 2.00000 0.516398
\(16\) −4.00000 −1.00000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 + 1.00000i 0.235702 + 0.235702i
\(19\) 3.00000 + 3.00000i 0.688247 + 0.688247i 0.961844 − 0.273597i \(-0.0882135\pi\)
−0.273597 + 0.961844i \(0.588214\pi\)
\(20\) 2.00000 + 2.00000i 0.447214 + 0.447214i
\(21\) 2.00000 − 2.00000i 0.436436 − 0.436436i
\(22\) 2.00000i 0.426401i
\(23\) − 6.00000i − 1.25109i −0.780189 − 0.625543i \(-0.784877\pi\)
0.780189 − 0.625543i \(-0.215123\pi\)
\(24\) − 4.00000i − 0.816497i
\(25\) 3.00000i 0.600000i
\(26\) 2.00000 0.392232
\(27\) −4.00000 + 4.00000i −0.769800 + 0.769800i
\(28\) 4.00000 0.755929
\(29\) 3.00000 + 3.00000i 0.557086 + 0.557086i 0.928477 − 0.371391i \(-0.121119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) −2.00000 + 2.00000i −0.365148 + 0.365148i
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 4.00000 − 4.00000i 0.707107 − 0.707107i
\(33\) −2.00000 −0.348155
\(34\) 2.00000 − 2.00000i 0.342997 − 0.342997i
\(35\) −2.00000 − 2.00000i −0.338062 − 0.338062i
\(36\) −2.00000 −0.333333
\(37\) 3.00000 − 3.00000i 0.493197 − 0.493197i −0.416115 − 0.909312i \(-0.636609\pi\)
0.909312 + 0.416115i \(0.136609\pi\)
\(38\) −6.00000 −0.973329
\(39\) 2.00000i 0.320256i
\(40\) −4.00000 −0.632456
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 5.00000 − 5.00000i 0.762493 − 0.762493i −0.214280 − 0.976772i \(-0.568740\pi\)
0.976772 + 0.214280i \(0.0687403\pi\)
\(44\) −2.00000 − 2.00000i −0.301511 − 0.301511i
\(45\) 1.00000 + 1.00000i 0.149071 + 0.149071i
\(46\) 6.00000 + 6.00000i 0.884652 + 0.884652i
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 4.00000 + 4.00000i 0.577350 + 0.577350i
\(49\) 3.00000 0.428571
\(50\) −3.00000 − 3.00000i −0.424264 − 0.424264i
\(51\) 2.00000 + 2.00000i 0.280056 + 0.280056i
\(52\) −2.00000 + 2.00000i −0.277350 + 0.277350i
\(53\) −5.00000 + 5.00000i −0.686803 + 0.686803i −0.961524 − 0.274721i \(-0.911414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) − 8.00000i − 1.08866i
\(55\) 2.00000i 0.269680i
\(56\) −4.00000 + 4.00000i −0.534522 + 0.534522i
\(57\) − 6.00000i − 0.794719i
\(58\) −6.00000 −0.787839
\(59\) −3.00000 + 3.00000i −0.390567 + 0.390567i −0.874889 − 0.484323i \(-0.839066\pi\)
0.484323 + 0.874889i \(0.339066\pi\)
\(60\) − 4.00000i − 0.516398i
\(61\) −9.00000 − 9.00000i −1.15233 − 1.15233i −0.986084 − 0.166248i \(-0.946835\pi\)
−0.166248 − 0.986084i \(-0.553165\pi\)
\(62\) 8.00000 − 8.00000i 1.01600 − 1.01600i
\(63\) 2.00000 0.251976
\(64\) 8.00000i 1.00000i
\(65\) 2.00000 0.248069
\(66\) 2.00000 − 2.00000i 0.246183 − 0.246183i
\(67\) −5.00000 − 5.00000i −0.610847 − 0.610847i 0.332320 − 0.943167i \(-0.392169\pi\)
−0.943167 + 0.332320i \(0.892169\pi\)
\(68\) 4.00000i 0.485071i
\(69\) −6.00000 + 6.00000i −0.722315 + 0.722315i
\(70\) 4.00000 0.478091
\(71\) 10.0000i 1.18678i 0.804914 + 0.593391i \(0.202211\pi\)
−0.804914 + 0.593391i \(0.797789\pi\)
\(72\) 2.00000 − 2.00000i 0.235702 − 0.235702i
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 6.00000i 0.697486i
\(75\) 3.00000 − 3.00000i 0.346410 − 0.346410i
\(76\) 6.00000 − 6.00000i 0.688247 − 0.688247i
\(77\) 2.00000 + 2.00000i 0.227921 + 0.227921i
\(78\) −2.00000 − 2.00000i −0.226455 − 0.226455i
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 4.00000 − 4.00000i 0.447214 − 0.447214i
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) −1.00000 − 1.00000i −0.109764 − 0.109764i 0.650092 − 0.759856i \(-0.274731\pi\)
−0.759856 + 0.650092i \(0.774731\pi\)
\(84\) −4.00000 − 4.00000i −0.436436 − 0.436436i
\(85\) 2.00000 − 2.00000i 0.216930 − 0.216930i
\(86\) 10.0000i 1.07833i
\(87\) − 6.00000i − 0.643268i
\(88\) 4.00000 0.426401
\(89\) − 4.00000i − 0.423999i −0.977270 − 0.212000i \(-0.932002\pi\)
0.977270 − 0.212000i \(-0.0679975\pi\)
\(90\) −2.00000 −0.210819
\(91\) 2.00000 − 2.00000i 0.209657 − 0.209657i
\(92\) −12.0000 −1.25109
\(93\) 8.00000 + 8.00000i 0.829561 + 0.829561i
\(94\) −8.00000 + 8.00000i −0.825137 + 0.825137i
\(95\) −6.00000 −0.615587
\(96\) −8.00000 −0.816497
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −3.00000 + 3.00000i −0.303046 + 0.303046i
\(99\) −1.00000 − 1.00000i −0.100504 − 0.100504i
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 16.2.e.a.13.1 yes 2
3.2 odd 2 144.2.k.a.109.1 2
4.3 odd 2 64.2.e.a.17.1 2
5.2 odd 4 400.2.q.a.349.1 2
5.3 odd 4 400.2.q.b.349.1 2
5.4 even 2 400.2.l.c.301.1 2
7.2 even 3 784.2.x.f.557.1 4
7.3 odd 6 784.2.x.c.765.1 4
7.4 even 3 784.2.x.f.765.1 4
7.5 odd 6 784.2.x.c.557.1 4
7.6 odd 2 784.2.m.b.589.1 2
8.3 odd 2 128.2.e.a.33.1 2
8.5 even 2 128.2.e.b.33.1 2
12.11 even 2 576.2.k.a.145.1 2
16.3 odd 4 128.2.e.a.97.1 2
16.5 even 4 inner 16.2.e.a.5.1 ✓ 2
16.11 odd 4 64.2.e.a.49.1 2
16.13 even 4 128.2.e.b.97.1 2
20.3 even 4 1600.2.q.a.849.1 2
20.7 even 4 1600.2.q.b.849.1 2
20.19 odd 2 1600.2.l.a.401.1 2
24.5 odd 2 1152.2.k.b.289.1 2
24.11 even 2 1152.2.k.a.289.1 2
32.3 odd 8 1024.2.b.b.513.2 2
32.5 even 8 1024.2.a.b.1.2 2
32.11 odd 8 1024.2.a.e.1.2 2
32.13 even 8 1024.2.b.e.513.2 2
32.19 odd 8 1024.2.b.b.513.1 2
32.21 even 8 1024.2.a.b.1.1 2
32.27 odd 8 1024.2.a.e.1.1 2
32.29 even 8 1024.2.b.e.513.1 2
48.5 odd 4 144.2.k.a.37.1 2
48.11 even 4 576.2.k.a.433.1 2
48.29 odd 4 1152.2.k.b.865.1 2
48.35 even 4 1152.2.k.a.865.1 2
80.27 even 4 1600.2.q.a.49.1 2
80.37 odd 4 400.2.q.b.149.1 2
80.43 even 4 1600.2.q.b.49.1 2
80.53 odd 4 400.2.q.a.149.1 2
80.59 odd 4 1600.2.l.a.1201.1 2
80.69 even 4 400.2.l.c.101.1 2
96.5 odd 8 9216.2.a.d.1.2 2
96.11 even 8 9216.2.a.s.1.1 2
96.53 odd 8 9216.2.a.d.1.1 2
96.59 even 8 9216.2.a.s.1.2 2
112.5 odd 12 784.2.x.c.165.1 4
112.37 even 12 784.2.x.f.165.1 4
112.53 even 12 784.2.x.f.373.1 4
112.69 odd 4 784.2.m.b.197.1 2
112.101 odd 12 784.2.x.c.373.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.2.e.a.5.1 ✓ 2 16.5 even 4 inner
16.2.e.a.13.1 yes 2 1.1 even 1 trivial
64.2.e.a.17.1 2 4.3 odd 2
64.2.e.a.49.1 2 16.11 odd 4
128.2.e.a.33.1 2 8.3 odd 2
128.2.e.a.97.1 2 16.3 odd 4
128.2.e.b.33.1 2 8.5 even 2
128.2.e.b.97.1 2 16.13 even 4
144.2.k.a.37.1 2 48.5 odd 4
144.2.k.a.109.1 2 3.2 odd 2
400.2.l.c.101.1 2 80.69 even 4
400.2.l.c.301.1 2 5.4 even 2
400.2.q.a.149.1 2 80.53 odd 4
400.2.q.a.349.1 2 5.2 odd 4
400.2.q.b.149.1 2 80.37 odd 4
400.2.q.b.349.1 2 5.3 odd 4
576.2.k.a.145.1 2 12.11 even 2
576.2.k.a.433.1 2 48.11 even 4
784.2.m.b.197.1 2 112.69 odd 4
784.2.m.b.589.1 2 7.6 odd 2
784.2.x.c.165.1 4 112.5 odd 12
784.2.x.c.373.1 4 112.101 odd 12
784.2.x.c.557.1 4 7.5 odd 6
784.2.x.c.765.1 4 7.3 odd 6
784.2.x.f.165.1 4 112.37 even 12
784.2.x.f.373.1 4 112.53 even 12
784.2.x.f.557.1 4 7.2 even 3
784.2.x.f.765.1 4 7.4 even 3
1024.2.a.b.1.1 2 32.21 even 8
1024.2.a.b.1.2 2 32.5 even 8
1024.2.a.e.1.1 2 32.27 odd 8
1024.2.a.e.1.2 2 32.11 odd 8
1024.2.b.b.513.1 2 32.19 odd 8
1024.2.b.b.513.2 2 32.3 odd 8
1024.2.b.e.513.1 2 32.29 even 8
1024.2.b.e.513.2 2 32.13 even 8
1152.2.k.a.289.1 2 24.11 even 2
1152.2.k.a.865.1 2 48.35 even 4
1152.2.k.b.289.1 2 24.5 odd 2
1152.2.k.b.865.1 2 48.29 odd 4
1600.2.l.a.401.1 2 20.19 odd 2
1600.2.l.a.1201.1 2 80.59 odd 4
1600.2.q.a.49.1 2 80.27 even 4
1600.2.q.a.849.1 2 20.3 even 4
1600.2.q.b.49.1 2 80.43 even 4
1600.2.q.b.849.1 2 20.7 even 4
9216.2.a.d.1.1 2 96.53 odd 8
9216.2.a.d.1.2 2 96.5 odd 8
9216.2.a.s.1.1 2 96.11 even 8
9216.2.a.s.1.2 2 96.59 even 8