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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,5,Mod(113,288)] 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("288.113"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 5])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 288.n (of order \(6\), degree \(2\), not 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: \(29.7705493681\)
Analytic rank: \(0\)
Dimension: \(92\)
Relative dimension: \(46\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 113.11
Character \(\chi\) \(=\) 288.113
Dual form 288.5.n.a.209.11

$q$-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)
 
\(f(q)\) \(=\) \(q+(-7.20486 + 5.39351i) q^{3} +(21.1127 - 36.5683i) q^{5} +(16.9047 + 29.2798i) q^{7} +(22.8201 - 77.7190i) q^{9} +(28.4273 + 49.2375i) q^{11} +(79.2879 + 45.7769i) q^{13} +(45.1173 + 377.341i) q^{15} +122.909i q^{17} -551.287i q^{19} +(-279.717 - 119.781i) q^{21} +(-595.912 - 344.050i) q^{23} +(-578.994 - 1002.85i) q^{25} +(254.763 + 683.035i) q^{27} +(665.451 + 1152.59i) q^{29} +(-305.568 + 529.260i) q^{31} +(-470.378 - 201.427i) q^{33} +1427.62 q^{35} -1555.96i q^{37} +(-818.156 + 97.8239i) q^{39} +(-978.349 - 564.850i) q^{41} +(2635.89 - 1521.83i) q^{43} +(-2360.26 - 2475.35i) q^{45} +(1412.72 - 815.637i) q^{47} +(628.964 - 1089.40i) q^{49} +(-662.909 - 885.540i) q^{51} +2783.44 q^{53} +2400.71 q^{55} +(2973.37 + 3971.95i) q^{57} +(1220.18 - 2113.42i) q^{59} +(4583.29 - 2646.16i) q^{61} +(2661.36 - 645.648i) q^{63} +(3347.97 - 1932.95i) q^{65} +(-3539.31 - 2043.42i) q^{67} +(6149.10 - 735.226i) q^{69} -1182.69i q^{71} +585.969 q^{73} +(9580.44 + 4102.56i) q^{75} +(-961.108 + 1664.69i) q^{77} +(1960.12 + 3395.02i) q^{79} +(-5519.49 - 3547.11i) q^{81} +(-5665.44 - 9812.83i) q^{83} +(4494.56 + 2594.94i) q^{85} +(-11011.0 - 4715.17i) q^{87} +6300.52i q^{89} +3095.37i q^{91} +(-652.991 - 5461.33i) q^{93} +(-20159.6 - 11639.2i) q^{95} +(373.253 + 646.493i) q^{97} +(4475.40 - 1085.74i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 92 q + 2 q^{7} - 4 q^{9} - 158 q^{15} + 6 q^{23} - 4752 q^{25} + 2 q^{31} - 834 q^{33} - 1178 q^{39} - 3318 q^{41} + 6 q^{47} - 11664 q^{49} - 2492 q^{55} + 880 q^{57} - 24638 q^{63} - 6 q^{65} - 8 q^{73}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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 0
\(3\) −7.20486 + 5.39351i −0.800540 + 0.599279i
\(4\) 0 0
\(5\) 21.1127 − 36.5683i 0.844509 − 1.46273i −0.0415382 − 0.999137i \(-0.513226\pi\)
0.886047 − 0.463595i \(-0.153441\pi\)
\(6\) 0 0
\(7\) 16.9047 + 29.2798i 0.344993 + 0.597546i 0.985353 − 0.170529i \(-0.0545478\pi\)
−0.640359 + 0.768076i \(0.721214\pi\)
\(8\) 0 0
\(9\) 22.8201 − 77.7190i 0.281729 − 0.959494i
\(10\) 0 0
\(11\) 28.4273 + 49.2375i 0.234936 + 0.406922i 0.959254 − 0.282545i \(-0.0911785\pi\)
−0.724318 + 0.689466i \(0.757845\pi\)
\(12\) 0 0
\(13\) 79.2879 + 45.7769i 0.469159 + 0.270869i 0.715888 − 0.698216i \(-0.246022\pi\)
−0.246729 + 0.969085i \(0.579356\pi\)
\(14\) 0 0
\(15\) 45.1173 + 377.341i 0.200521 + 1.67707i
\(16\) 0 0
\(17\) 122.909i 0.425289i 0.977130 + 0.212645i \(0.0682077\pi\)
−0.977130 + 0.212645i \(0.931792\pi\)
\(18\) 0 0
\(19\) − 551.287i − 1.52711i −0.645742 − 0.763556i \(-0.723452\pi\)
0.645742 − 0.763556i \(-0.276548\pi\)
\(20\) 0 0
\(21\) −279.717 − 119.781i −0.634278 − 0.271612i
\(22\) 0 0
\(23\) −595.912 − 344.050i −1.12649 − 0.650378i −0.183439 − 0.983031i \(-0.558723\pi\)
−0.943049 + 0.332653i \(0.892056\pi\)
\(24\) 0 0
\(25\) −578.994 − 1002.85i −0.926390 − 1.60456i
\(26\) 0 0
\(27\) 254.763 + 683.035i 0.349469 + 0.936948i
\(28\) 0 0
\(29\) 665.451 + 1152.59i 0.791261 + 1.37050i 0.925186 + 0.379513i \(0.123909\pi\)
−0.133925 + 0.990991i \(0.542758\pi\)
\(30\) 0 0
\(31\) −305.568 + 529.260i −0.317969 + 0.550739i −0.980064 − 0.198681i \(-0.936334\pi\)
0.662095 + 0.749420i \(0.269668\pi\)
\(32\) 0 0
\(33\) −470.378 − 201.427i −0.431935 − 0.184965i
\(34\) 0 0
\(35\) 1427.62 1.16540
\(36\) 0 0
\(37\) − 1555.96i − 1.13657i −0.822832 − 0.568285i \(-0.807607\pi\)
0.822832 − 0.568285i \(-0.192393\pi\)
\(38\) 0 0
\(39\) −818.156 + 97.8239i −0.537907 + 0.0643155i
\(40\) 0 0
\(41\) −978.349 − 564.850i −0.582004 − 0.336020i 0.179925 − 0.983680i \(-0.442414\pi\)
−0.761929 + 0.647660i \(0.775748\pi\)
\(42\) 0 0
\(43\) 2635.89 − 1521.83i 1.42558 − 0.823057i 0.428810 − 0.903395i \(-0.358933\pi\)
0.996768 + 0.0803375i \(0.0255998\pi\)
\(44\) 0 0
\(45\) −2360.26 − 2475.35i −1.16556 − 1.22240i
\(46\) 0 0
\(47\) 1412.72 − 815.637i 0.639531 − 0.369233i −0.144903 − 0.989446i \(-0.546287\pi\)
0.784434 + 0.620212i \(0.212954\pi\)
\(48\) 0 0
\(49\) 628.964 − 1089.40i 0.261959 − 0.453726i
\(50\) 0 0
\(51\) −662.909 − 885.540i −0.254867 − 0.340461i
\(52\) 0 0
\(53\) 2783.44 0.990901 0.495451 − 0.868636i \(-0.335003\pi\)
0.495451 + 0.868636i \(0.335003\pi\)
\(54\) 0 0
\(55\) 2400.71 0.793623
\(56\) 0 0
\(57\) 2973.37 + 3971.95i 0.915166 + 1.22251i
\(58\) 0 0
\(59\) 1220.18 − 2113.42i 0.350526 − 0.607129i −0.635816 − 0.771841i \(-0.719336\pi\)
0.986342 + 0.164712i \(0.0526695\pi\)
\(60\) 0 0
\(61\) 4583.29 − 2646.16i 1.23174 − 0.711143i 0.264344 − 0.964429i \(-0.414845\pi\)
0.967391 + 0.253286i \(0.0815113\pi\)
\(62\) 0 0
\(63\) 2661.36 − 645.648i 0.670537 − 0.162673i
\(64\) 0 0
\(65\) 3347.97 − 1932.95i 0.792418 − 0.457503i
\(66\) 0 0
\(67\) −3539.31 − 2043.42i −0.788441 − 0.455207i 0.0509724 − 0.998700i \(-0.483768\pi\)
−0.839413 + 0.543493i \(0.817101\pi\)
\(68\) 0 0
\(69\) 6149.10 − 735.226i 1.29156 − 0.154427i
\(70\) 0 0
\(71\) − 1182.69i − 0.234615i −0.993096 − 0.117307i \(-0.962574\pi\)
0.993096 − 0.117307i \(-0.0374262\pi\)
\(72\) 0 0
\(73\) 585.969 0.109959 0.0549793 − 0.998487i \(-0.482491\pi\)
0.0549793 + 0.998487i \(0.482491\pi\)
\(74\) 0 0
\(75\) 9580.44 + 4102.56i 1.70319 + 0.729345i
\(76\) 0 0
\(77\) −961.108 + 1664.69i −0.162103 + 0.280771i
\(78\) 0 0
\(79\) 1960.12 + 3395.02i 0.314071 + 0.543986i 0.979240 − 0.202707i \(-0.0649738\pi\)
−0.665169 + 0.746693i \(0.731640\pi\)
\(80\) 0 0
\(81\) −5519.49 − 3547.11i −0.841257 − 0.540635i
\(82\) 0 0
\(83\) −5665.44 − 9812.83i −0.822390 − 1.42442i −0.903898 − 0.427748i \(-0.859307\pi\)
0.0815082 − 0.996673i \(-0.474026\pi\)
\(84\) 0 0
\(85\) 4494.56 + 2594.94i 0.622085 + 0.359161i
\(86\) 0 0
\(87\) −11011.0 − 4715.17i −1.45475 − 0.622958i
\(88\) 0 0
\(89\) 6300.52i 0.795419i 0.917511 + 0.397710i \(0.130195\pi\)
−0.917511 + 0.397710i \(0.869805\pi\)
\(90\) 0 0
\(91\) 3095.37i 0.373792i
\(92\) 0 0
\(93\) −652.991 − 5461.33i −0.0754990 − 0.631441i
\(94\) 0 0
\(95\) −20159.6 − 11639.2i −2.23376 − 1.28966i
\(96\) 0 0
\(97\) 373.253 + 646.493i 0.0396697 + 0.0687100i 0.885179 − 0.465251i \(-0.154036\pi\)
−0.845509 + 0.533961i \(0.820703\pi\)
\(98\) 0 0
\(99\) 4475.40 − 1085.74i 0.456627 − 0.110778i
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 288.5.n.a.113.11 92
3.2 odd 2 864.5.n.a.17.2 92
4.3 odd 2 72.5.j.a.5.27 yes 92
8.3 odd 2 72.5.j.a.5.12 ✓ 92
8.5 even 2 inner 288.5.n.a.113.36 92
9.2 odd 6 inner 288.5.n.a.209.36 92
9.7 even 3 864.5.n.a.305.45 92
12.11 even 2 216.5.j.a.125.20 92
24.5 odd 2 864.5.n.a.17.45 92
24.11 even 2 216.5.j.a.125.35 92
36.7 odd 6 216.5.j.a.197.35 92
36.11 even 6 72.5.j.a.29.12 yes 92
72.11 even 6 72.5.j.a.29.27 yes 92
72.29 odd 6 inner 288.5.n.a.209.11 92
72.43 odd 6 216.5.j.a.197.20 92
72.61 even 6 864.5.n.a.305.2 92
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.5.j.a.5.12 ✓ 92 8.3 odd 2
72.5.j.a.5.27 yes 92 4.3 odd 2
72.5.j.a.29.12 yes 92 36.11 even 6
72.5.j.a.29.27 yes 92 72.11 even 6
216.5.j.a.125.20 92 12.11 even 2
216.5.j.a.125.35 92 24.11 even 2
216.5.j.a.197.20 92 72.43 odd 6
216.5.j.a.197.35 92 36.7 odd 6
288.5.n.a.113.11 92 1.1 even 1 trivial
288.5.n.a.113.36 92 8.5 even 2 inner
288.5.n.a.209.11 92 72.29 odd 6 inner
288.5.n.a.209.36 92 9.2 odd 6 inner
864.5.n.a.17.2 92 3.2 odd 2
864.5.n.a.17.45 92 24.5 odd 2
864.5.n.a.305.2 92 72.61 even 6
864.5.n.a.305.45 92 9.7 even 3