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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [592,2,Mod(33,592)] 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("592.33"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(592, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.bc (of order \(9\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 49.1
Root \(-2.20976 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 592.49
Dual form 592.2.bc.d.145.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.972925 - 0.816381i) q^{3} +(-1.43969 + 0.524005i) q^{5} +(1.82850 - 0.665520i) q^{7} +(-0.240839 - 1.36587i) q^{9} +(0.220544 + 0.381994i) q^{11} +(-0.367592 + 2.08472i) q^{13} +(1.82850 + 0.665520i) q^{15} +(-0.664245 - 3.76712i) q^{17} +(-4.55479 - 3.82192i) q^{19} +(-2.32231 - 0.845253i) q^{21} +(2.85558 - 4.94600i) q^{23} +(-2.03209 + 1.70513i) q^{25} +(-2.78585 + 4.82523i) q^{27} +(-1.36712 - 2.36792i) q^{29} -9.87516 q^{31} +(0.0972795 - 0.551699i) q^{33} +(-2.28374 + 1.91629i) q^{35} +(2.18831 - 5.67550i) q^{37} +(2.05956 - 1.72818i) q^{39} +(1.80615 - 10.2432i) q^{41} -3.84681 q^{43} +(1.06246 + 1.84023i) q^{45} +(3.84316 - 6.65655i) q^{47} +(-2.46181 + 2.06571i) q^{49} +(-2.42915 + 4.20740i) q^{51} +(-1.62561 - 0.591673i) q^{53} +(-0.517683 - 0.434387i) q^{55} +(1.31132 + 7.43688i) q^{57} +(1.93027 + 0.702561i) q^{59} +(-1.89894 + 10.7694i) q^{61} +(-1.34939 - 2.33721i) q^{63} +(-0.563183 - 3.19397i) q^{65} +(-1.57261 + 0.572384i) q^{67} +(-6.81608 + 2.48085i) q^{69} +(7.70462 + 6.46494i) q^{71} -9.88570 q^{73} +3.36910 q^{75} +(0.657490 + 0.551699i) q^{77} +(12.5395 - 4.56401i) q^{79} +(2.73975 - 0.997189i) q^{81} +(-0.617999 - 3.50484i) q^{83} +(2.93030 + 5.07543i) q^{85} +(-0.603020 + 3.41989i) q^{87} +(11.1543 + 4.05982i) q^{89} +(0.715278 + 4.05654i) q^{91} +(9.60779 + 8.06190i) q^{93} +(8.56020 + 3.11566i) q^{95} +(0.296748 - 0.513983i) q^{97} +(0.468637 - 0.393233i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 6 q^{5} - 6 q^{7} - 3 q^{9} - 3 q^{11} - 6 q^{13} - 6 q^{15} - 3 q^{17} + 3 q^{19} - 33 q^{21} + 21 q^{23} - 6 q^{25} - 3 q^{27} + 6 q^{29} - 42 q^{31} + 57 q^{33} + 9 q^{35} - 3 q^{37}+ \cdots + 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\) \(1\) \(1\)

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 0
\(3\) −0.972925 − 0.816381i −0.561719 − 0.471338i 0.317168 − 0.948370i \(-0.397268\pi\)
−0.878886 + 0.477032i \(0.841713\pi\)
\(4\) 0 0
\(5\) −1.43969 + 0.524005i −0.643850 + 0.234342i −0.643248 − 0.765658i \(-0.722414\pi\)
−0.000601695 1.00000i \(0.500192\pi\)
\(6\) 0 0
\(7\) 1.82850 − 0.665520i 0.691108 − 0.251543i 0.0274987 − 0.999622i \(-0.491246\pi\)
0.663610 + 0.748079i \(0.269024\pi\)
\(8\) 0 0
\(9\) −0.240839 − 1.36587i −0.0802798 − 0.455289i
\(10\) 0 0
\(11\) 0.220544 + 0.381994i 0.0664966 + 0.115175i 0.897357 − 0.441306i \(-0.145485\pi\)
−0.830860 + 0.556481i \(0.812151\pi\)
\(12\) 0 0
\(13\) −0.367592 + 2.08472i −0.101952 + 0.578196i 0.890443 + 0.455095i \(0.150395\pi\)
−0.992394 + 0.123100i \(0.960716\pi\)
\(14\) 0 0
\(15\) 1.82850 + 0.665520i 0.472117 + 0.171837i
\(16\) 0 0
\(17\) −0.664245 − 3.76712i −0.161103 − 0.913661i −0.952992 − 0.302996i \(-0.902013\pi\)
0.791889 − 0.610665i \(-0.209098\pi\)
\(18\) 0 0
\(19\) −4.55479 − 3.82192i −1.04494 − 0.876808i −0.0523871 − 0.998627i \(-0.516683\pi\)
−0.992552 + 0.121819i \(0.961127\pi\)
\(20\) 0 0
\(21\) −2.32231 − 0.845253i −0.506770 − 0.184449i
\(22\) 0 0
\(23\) 2.85558 − 4.94600i 0.595429 − 1.03131i −0.398057 − 0.917360i \(-0.630316\pi\)
0.993486 − 0.113952i \(-0.0363511\pi\)
\(24\) 0 0
\(25\) −2.03209 + 1.70513i −0.406418 + 0.341025i
\(26\) 0 0
\(27\) −2.78585 + 4.82523i −0.536136 + 0.928615i
\(28\) 0 0
\(29\) −1.36712 − 2.36792i −0.253867 − 0.439711i 0.710720 − 0.703475i \(-0.248369\pi\)
−0.964587 + 0.263764i \(0.915036\pi\)
\(30\) 0 0
\(31\) −9.87516 −1.77363 −0.886816 − 0.462123i \(-0.847088\pi\)
−0.886816 + 0.462123i \(0.847088\pi\)
\(32\) 0 0
\(33\) 0.0972795 − 0.551699i 0.0169342 − 0.0960386i
\(34\) 0 0
\(35\) −2.28374 + 1.91629i −0.386023 + 0.323912i
\(36\) 0 0
\(37\) 2.18831 − 5.67550i 0.359755 − 0.933047i
\(38\) 0 0
\(39\) 2.05956 − 1.72818i 0.329794 − 0.276730i
\(40\) 0 0
\(41\) 1.80615 − 10.2432i 0.282073 − 1.59972i −0.433482 − 0.901162i \(-0.642715\pi\)
0.715556 − 0.698556i \(-0.246174\pi\)
\(42\) 0 0
\(43\) −3.84681 −0.586633 −0.293317 − 0.956015i \(-0.594759\pi\)
−0.293317 + 0.956015i \(0.594759\pi\)
\(44\) 0 0
\(45\) 1.06246 + 1.84023i 0.158382 + 0.274325i
\(46\) 0 0
\(47\) 3.84316 − 6.65655i 0.560582 − 0.970957i −0.436863 − 0.899528i \(-0.643911\pi\)
0.997446 − 0.0714293i \(-0.0227560\pi\)
\(48\) 0 0
\(49\) −2.46181 + 2.06571i −0.351687 + 0.295101i
\(50\) 0 0
\(51\) −2.42915 + 4.20740i −0.340148 + 0.589154i
\(52\) 0 0
\(53\) −1.62561 − 0.591673i −0.223295 − 0.0812726i 0.227950 − 0.973673i \(-0.426798\pi\)
−0.451245 + 0.892400i \(0.649020\pi\)
\(54\) 0 0
\(55\) −0.517683 − 0.434387i −0.0698043 − 0.0585728i
\(56\) 0 0
\(57\) 1.31132 + 7.43688i 0.173689 + 0.985039i
\(58\) 0 0
\(59\) 1.93027 + 0.702561i 0.251300 + 0.0914656i 0.464599 − 0.885521i \(-0.346199\pi\)
−0.213299 + 0.976987i \(0.568421\pi\)
\(60\) 0 0
\(61\) −1.89894 + 10.7694i −0.243134 + 1.37888i 0.581652 + 0.813437i \(0.302406\pi\)
−0.824786 + 0.565444i \(0.808705\pi\)
\(62\) 0 0
\(63\) −1.34939 − 2.33721i −0.170007 − 0.294460i
\(64\) 0 0
\(65\) −0.563183 − 3.19397i −0.0698542 − 0.396163i
\(66\) 0 0
\(67\) −1.57261 + 0.572384i −0.192125 + 0.0699279i −0.436291 − 0.899806i \(-0.643708\pi\)
0.244166 + 0.969734i \(0.421486\pi\)
\(68\) 0 0
\(69\) −6.81608 + 2.48085i −0.820560 + 0.298659i
\(70\) 0 0
\(71\) 7.70462 + 6.46494i 0.914370 + 0.767248i 0.972945 − 0.231035i \(-0.0742113\pi\)
−0.0585751 + 0.998283i \(0.518656\pi\)
\(72\) 0 0
\(73\) −9.88570 −1.15703 −0.578517 − 0.815671i \(-0.696368\pi\)
−0.578517 + 0.815671i \(0.696368\pi\)
\(74\) 0 0
\(75\) 3.36910 0.389030
\(76\) 0 0
\(77\) 0.657490 + 0.551699i 0.0749279 + 0.0628720i
\(78\) 0 0
\(79\) 12.5395 − 4.56401i 1.41081 − 0.513491i 0.479439 − 0.877575i \(-0.340840\pi\)
0.931366 + 0.364084i \(0.118618\pi\)
\(80\) 0 0
\(81\) 2.73975 − 0.997189i 0.304417 − 0.110799i
\(82\) 0 0
\(83\) −0.617999 − 3.50484i −0.0678341 − 0.384707i −0.999757 − 0.0220497i \(-0.992981\pi\)
0.931923 − 0.362657i \(-0.118130\pi\)
\(84\) 0 0
\(85\) 2.93030 + 5.07543i 0.317836 + 0.550508i
\(86\) 0 0
\(87\) −0.603020 + 3.41989i −0.0646505 + 0.366651i
\(88\) 0 0
\(89\) 11.1543 + 4.05982i 1.18235 + 0.430340i 0.857031 − 0.515264i \(-0.172306\pi\)
0.325319 + 0.945604i \(0.394528\pi\)
\(90\) 0 0
\(91\) 0.715278 + 4.05654i 0.0749815 + 0.425241i
\(92\) 0 0
\(93\) 9.60779 + 8.06190i 0.996282 + 0.835980i
\(94\) 0 0
\(95\) 8.56020 + 3.11566i 0.878258 + 0.319660i
\(96\) 0 0
\(97\) 0.296748 − 0.513983i 0.0301302 − 0.0521871i −0.850567 − 0.525867i \(-0.823741\pi\)
0.880697 + 0.473679i \(0.157074\pi\)
\(98\) 0 0
\(99\) 0.468637 − 0.393233i 0.0470998 − 0.0395214i
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 592.2.bc.d.49.1 12
4.3 odd 2 74.2.f.b.49.2 ✓ 12
12.11 even 2 666.2.x.g.271.1 12
37.34 even 9 inner 592.2.bc.d.145.1 12
148.71 odd 18 74.2.f.b.71.2 yes 12
148.95 odd 18 2738.2.a.q.1.4 6
148.127 odd 18 2738.2.a.t.1.4 6
444.71 even 18 666.2.x.g.145.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.49.2 ✓ 12 4.3 odd 2
74.2.f.b.71.2 yes 12 148.71 odd 18
592.2.bc.d.49.1 12 1.1 even 1 trivial
592.2.bc.d.145.1 12 37.34 even 9 inner
666.2.x.g.145.1 12 444.71 even 18
666.2.x.g.271.1 12 12.11 even 2
2738.2.a.q.1.4 6 148.95 odd 18
2738.2.a.t.1.4 6 148.127 odd 18