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Properties

Label 720.2.t.c.181.3
Level $720$
Weight $2$
Character 720.181
Analytic conductor $5.749$
Analytic rank $0$
Dimension $16$
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] = [720,2,Mod(181,720)] 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("720.181"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.t (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 181.3
Root \(-0.966675 - 1.03225i\) of defining polynomial
Character \(\chi\) \(=\) 720.181
Dual form 720.2.t.c.541.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.562546 + 1.29751i) q^{2} +(-1.36708 - 1.45982i) q^{4} +(-0.707107 - 0.707107i) q^{5} -1.73696i q^{7} +(2.66319 - 0.952595i) q^{8} +(1.31526 - 0.519701i) q^{10} +(-0.505430 - 0.505430i) q^{11} +(-1.88750 + 1.88750i) q^{13} +(2.25374 + 0.977122i) q^{14} +(-0.262159 + 3.99140i) q^{16} -4.53524 q^{17} +(-3.22022 + 3.22022i) q^{19} +(-0.0655751 + 1.99892i) q^{20} +(0.940130 - 0.371475i) q^{22} +8.85045i q^{23} +1.00000i q^{25} +(-1.38725 - 3.51086i) q^{26} +(-2.53566 + 2.37458i) q^{28} +(2.44059 - 2.44059i) q^{29} -5.70401 q^{31} +(-5.03142 - 2.58550i) q^{32} +(2.55128 - 5.88454i) q^{34} +(-1.22822 + 1.22822i) q^{35} +(-5.35670 - 5.35670i) q^{37} +(-2.36676 - 5.98979i) q^{38} +(-2.55674 - 1.20957i) q^{40} +10.0343i q^{41} +(-2.10564 - 2.10564i) q^{43} +(-0.0468722 + 1.42880i) q^{44} +(-11.4836 - 4.97878i) q^{46} -4.32303 q^{47} +3.98295 q^{49} +(-1.29751 - 0.562546i) q^{50} +(5.33578 + 0.175041i) q^{52} +(1.37458 + 1.37458i) q^{53} +0.714786i q^{55} +(-1.65462 - 4.62586i) q^{56} +(1.79375 + 4.53964i) q^{58} +(-6.64140 - 6.64140i) q^{59} +(5.26208 - 5.26208i) q^{61} +(3.20877 - 7.40103i) q^{62} +(6.18513 - 5.07388i) q^{64} +2.66933 q^{65} +(-10.5578 + 10.5578i) q^{67} +(6.20006 + 6.62065i) q^{68} +(-0.902702 - 2.28456i) q^{70} -14.0437i q^{71} +6.63830i q^{73} +(9.96378 - 3.93700i) q^{74} +(9.10325 + 0.298634i) q^{76} +(-0.877914 + 0.877914i) q^{77} +4.27297 q^{79} +(3.00772 - 2.63697i) q^{80} +(-13.0196 - 5.64474i) q^{82} +(-9.15483 + 9.15483i) q^{83} +(3.20690 + 3.20690i) q^{85} +(3.91661 - 1.54758i) q^{86} +(-1.82752 - 0.864585i) q^{88} +3.23826i q^{89} +(3.27852 + 3.27852i) q^{91} +(12.9201 - 12.0993i) q^{92} +(2.43190 - 5.60919i) q^{94} +4.55407 q^{95} +1.94129 q^{97} +(-2.24059 + 5.16794i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} + 4 q^{10} + 8 q^{11} - 4 q^{14} + 16 q^{16} - 8 q^{19} - 8 q^{20} - 20 q^{22} + 16 q^{26} - 4 q^{28} + 16 q^{29} + 16 q^{34} - 16 q^{37} - 20 q^{38} + 8 q^{43} - 40 q^{44} - 4 q^{46} + 40 q^{47}+ \cdots - 60 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.562546 + 1.29751i −0.397780 + 0.917481i
\(3\) 0 0
\(4\) −1.36708 − 1.45982i −0.683542 − 0.729911i
\(5\) −0.707107 − 0.707107i −0.316228 − 0.316228i
\(6\) 0 0
\(7\) − 1.73696i − 0.656511i −0.944589 − 0.328255i \(-0.893539\pi\)
0.944589 − 0.328255i \(-0.106461\pi\)
\(8\) 2.66319 − 0.952595i 0.941579 − 0.336793i
\(9\) 0 0
\(10\) 1.31526 − 0.519701i 0.415922 − 0.164344i
\(11\) −0.505430 − 0.505430i −0.152393 − 0.152393i 0.626793 − 0.779186i \(-0.284367\pi\)
−0.779186 + 0.626793i \(0.784367\pi\)
\(12\) 0 0
\(13\) −1.88750 + 1.88750i −0.523498 + 0.523498i −0.918626 − 0.395128i \(-0.870700\pi\)
0.395128 + 0.918626i \(0.370700\pi\)
\(14\) 2.25374 + 0.977122i 0.602336 + 0.261147i
\(15\) 0 0
\(16\) −0.262159 + 3.99140i −0.0655399 + 0.997850i
\(17\) −4.53524 −1.09996 −0.549979 − 0.835178i \(-0.685364\pi\)
−0.549979 + 0.835178i \(0.685364\pi\)
\(18\) 0 0
\(19\) −3.22022 + 3.22022i −0.738768 + 0.738768i −0.972340 − 0.233571i \(-0.924959\pi\)
0.233571 + 0.972340i \(0.424959\pi\)
\(20\) −0.0655751 + 1.99892i −0.0146630 + 0.446973i
\(21\) 0 0
\(22\) 0.940130 − 0.371475i 0.200436 − 0.0791987i
\(23\) 8.85045i 1.84545i 0.385463 + 0.922723i \(0.374042\pi\)
−0.385463 + 0.922723i \(0.625958\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) −1.38725 − 3.51086i −0.272062 − 0.688536i
\(27\) 0 0
\(28\) −2.53566 + 2.37458i −0.479195 + 0.448753i
\(29\) 2.44059 − 2.44059i 0.453205 − 0.453205i −0.443212 − 0.896417i \(-0.646161\pi\)
0.896417 + 0.443212i \(0.146161\pi\)
\(30\) 0 0
\(31\) −5.70401 −1.02447 −0.512235 − 0.858845i \(-0.671182\pi\)
−0.512235 + 0.858845i \(0.671182\pi\)
\(32\) −5.03142 − 2.58550i −0.889438 − 0.457056i
\(33\) 0 0
\(34\) 2.55128 − 5.88454i 0.437541 − 1.00919i
\(35\) −1.22822 + 1.22822i −0.207607 + 0.207607i
\(36\) 0 0
\(37\) −5.35670 − 5.35670i −0.880636 − 0.880636i 0.112963 − 0.993599i \(-0.463966\pi\)
−0.993599 + 0.112963i \(0.963966\pi\)
\(38\) −2.36676 − 5.98979i −0.383939 − 0.971673i
\(39\) 0 0
\(40\) −2.55674 − 1.20957i −0.404257 − 0.191250i
\(41\) 10.0343i 1.56709i 0.621335 + 0.783545i \(0.286591\pi\)
−0.621335 + 0.783545i \(0.713409\pi\)
\(42\) 0 0
\(43\) −2.10564 − 2.10564i −0.321107 − 0.321107i 0.528085 − 0.849192i \(-0.322910\pi\)
−0.849192 + 0.528085i \(0.822910\pi\)
\(44\) −0.0468722 + 1.42880i −0.00706625 + 0.215400i
\(45\) 0 0
\(46\) −11.4836 − 4.97878i −1.69316 − 0.734082i
\(47\) −4.32303 −0.630578 −0.315289 − 0.948996i \(-0.602101\pi\)
−0.315289 + 0.948996i \(0.602101\pi\)
\(48\) 0 0
\(49\) 3.98295 0.568993
\(50\) −1.29751 − 0.562546i −0.183496 − 0.0795560i
\(51\) 0 0
\(52\) 5.33578 + 0.175041i 0.739940 + 0.0242739i
\(53\) 1.37458 + 1.37458i 0.188814 + 0.188814i 0.795183 − 0.606369i \(-0.207375\pi\)
−0.606369 + 0.795183i \(0.707375\pi\)
\(54\) 0 0
\(55\) 0.714786i 0.0963817i
\(56\) −1.65462 − 4.62586i −0.221108 − 0.618157i
\(57\) 0 0
\(58\) 1.79375 + 4.53964i 0.235531 + 0.596083i
\(59\) −6.64140 − 6.64140i −0.864637 − 0.864637i 0.127236 − 0.991872i \(-0.459389\pi\)
−0.991872 + 0.127236i \(0.959389\pi\)
\(60\) 0 0
\(61\) 5.26208 − 5.26208i 0.673741 − 0.673741i −0.284836 − 0.958576i \(-0.591939\pi\)
0.958576 + 0.284836i \(0.0919391\pi\)
\(62\) 3.20877 − 7.40103i 0.407514 − 0.939932i
\(63\) 0 0
\(64\) 6.18513 − 5.07388i 0.773141 − 0.634234i
\(65\) 2.66933 0.331089
\(66\) 0 0
\(67\) −10.5578 + 10.5578i −1.28984 + 1.28984i −0.354954 + 0.934884i \(0.615503\pi\)
−0.934884 + 0.354954i \(0.884497\pi\)
\(68\) 6.20006 + 6.62065i 0.751868 + 0.802871i
\(69\) 0 0
\(70\) −0.902702 − 2.28456i −0.107894 − 0.273057i
\(71\) − 14.0437i − 1.66668i −0.552764 − 0.833338i \(-0.686427\pi\)
0.552764 − 0.833338i \(-0.313573\pi\)
\(72\) 0 0
\(73\) 6.63830i 0.776954i 0.921458 + 0.388477i \(0.126999\pi\)
−0.921458 + 0.388477i \(0.873001\pi\)
\(74\) 9.96378 − 3.93700i 1.15827 − 0.457667i
\(75\) 0 0
\(76\) 9.10325 + 0.298634i 1.04421 + 0.0342557i
\(77\) −0.877914 + 0.877914i −0.100048 + 0.100048i
\(78\) 0 0
\(79\) 4.27297 0.480746 0.240373 − 0.970681i \(-0.422730\pi\)
0.240373 + 0.970681i \(0.422730\pi\)
\(80\) 3.00772 − 2.63697i 0.336273 − 0.294822i
\(81\) 0 0
\(82\) −13.0196 − 5.64474i −1.43778 − 0.623357i
\(83\) −9.15483 + 9.15483i −1.00487 + 1.00487i −0.00488547 + 0.999988i \(0.501555\pi\)
−0.999988 + 0.00488547i \(0.998445\pi\)
\(84\) 0 0
\(85\) 3.20690 + 3.20690i 0.347837 + 0.347837i
\(86\) 3.91661 − 1.54758i 0.422339 − 0.166879i
\(87\) 0 0
\(88\) −1.82752 − 0.864585i −0.194815 − 0.0921650i
\(89\) 3.23826i 0.343255i 0.985162 + 0.171627i \(0.0549025\pi\)
−0.985162 + 0.171627i \(0.945097\pi\)
\(90\) 0 0
\(91\) 3.27852 + 3.27852i 0.343682 + 0.343682i
\(92\) 12.9201 − 12.0993i 1.34701 − 1.26144i
\(93\) 0 0
\(94\) 2.43190 − 5.60919i 0.250831 − 0.578543i
\(95\) 4.55407 0.467238
\(96\) 0 0
\(97\) 1.94129 0.197108 0.0985541 − 0.995132i \(-0.468578\pi\)
0.0985541 + 0.995132i \(0.468578\pi\)
\(98\) −2.24059 + 5.16794i −0.226334 + 0.522041i
\(99\) 0 0
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 720.2.t.c.181.3 16
3.2 odd 2 80.2.l.a.21.6 ✓ 16
4.3 odd 2 2880.2.t.c.2161.4 16
12.11 even 2 320.2.l.a.241.4 16
15.2 even 4 400.2.q.g.149.7 16
15.8 even 4 400.2.q.h.149.2 16
15.14 odd 2 400.2.l.h.101.3 16
16.3 odd 4 2880.2.t.c.721.1 16
16.13 even 4 inner 720.2.t.c.541.3 16
24.5 odd 2 640.2.l.b.481.4 16
24.11 even 2 640.2.l.a.481.5 16
48.5 odd 4 640.2.l.b.161.4 16
48.11 even 4 640.2.l.a.161.5 16
48.29 odd 4 80.2.l.a.61.6 yes 16
48.35 even 4 320.2.l.a.81.4 16
60.23 odd 4 1600.2.q.g.49.4 16
60.47 odd 4 1600.2.q.h.49.5 16
60.59 even 2 1600.2.l.i.1201.5 16
96.29 odd 8 5120.2.a.s.1.6 8
96.35 even 8 5120.2.a.u.1.3 8
96.77 odd 8 5120.2.a.v.1.3 8
96.83 even 8 5120.2.a.t.1.6 8
240.29 odd 4 400.2.l.h.301.3 16
240.77 even 4 400.2.q.h.349.2 16
240.83 odd 4 1600.2.q.h.849.5 16
240.173 even 4 400.2.q.g.349.7 16
240.179 even 4 1600.2.l.i.401.5 16
240.227 odd 4 1600.2.q.g.849.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.6 ✓ 16 3.2 odd 2
80.2.l.a.61.6 yes 16 48.29 odd 4
320.2.l.a.81.4 16 48.35 even 4
320.2.l.a.241.4 16 12.11 even 2
400.2.l.h.101.3 16 15.14 odd 2
400.2.l.h.301.3 16 240.29 odd 4
400.2.q.g.149.7 16 15.2 even 4
400.2.q.g.349.7 16 240.173 even 4
400.2.q.h.149.2 16 15.8 even 4
400.2.q.h.349.2 16 240.77 even 4
640.2.l.a.161.5 16 48.11 even 4
640.2.l.a.481.5 16 24.11 even 2
640.2.l.b.161.4 16 48.5 odd 4
640.2.l.b.481.4 16 24.5 odd 2
720.2.t.c.181.3 16 1.1 even 1 trivial
720.2.t.c.541.3 16 16.13 even 4 inner
1600.2.l.i.401.5 16 240.179 even 4
1600.2.l.i.1201.5 16 60.59 even 2
1600.2.q.g.49.4 16 60.23 odd 4
1600.2.q.g.849.4 16 240.227 odd 4
1600.2.q.h.49.5 16 60.47 odd 4
1600.2.q.h.849.5 16 240.83 odd 4
2880.2.t.c.721.1 16 16.3 odd 4
2880.2.t.c.2161.4 16 4.3 odd 2
5120.2.a.s.1.6 8 96.29 odd 8
5120.2.a.t.1.6 8 96.83 even 8
5120.2.a.u.1.3 8 96.35 even 8
5120.2.a.v.1.3 8 96.77 odd 8