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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 81.2
Root \(-2.00752 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 592.81
Dual form 592.2.bc.d.497.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.04963 + 0.746005i) q^{3} +(-0.326352 - 1.85083i) q^{5} +(0.711830 + 4.03699i) q^{7} +(1.34633 + 1.12971i) q^{9} +(1.67088 - 2.89404i) q^{11} +(1.48512 - 1.24616i) q^{13} +(0.711830 - 4.03699i) q^{15} +(3.40626 + 2.85819i) q^{17} +(-0.0932770 - 0.0339500i) q^{19} +(-1.55262 + 8.80536i) q^{21} +(4.76146 + 8.24709i) q^{23} +(1.37939 - 0.502055i) q^{25} +(-1.35504 - 2.34700i) q^{27} +(-0.387288 + 0.670802i) q^{29} -10.3914 q^{31} +(5.58365 - 4.68524i) q^{33} +(7.23948 - 2.63496i) q^{35} +(-3.50302 - 4.97281i) q^{37} +(3.97359 - 1.44627i) q^{39} +(4.36218 - 3.66031i) q^{41} +1.11986 q^{43} +(1.65152 - 2.86052i) q^{45} +(-3.55724 - 6.16132i) q^{47} +(-9.21271 + 3.35315i) q^{49} +(4.84936 + 8.39933i) q^{51} +(0.142188 - 0.806388i) q^{53} +(-5.90168 - 2.14804i) q^{55} +(-0.165857 - 0.139170i) q^{57} +(-1.40642 + 7.97622i) q^{59} +(-8.19737 + 6.87841i) q^{61} +(-3.60225 + 6.23929i) q^{63} +(-2.79111 - 2.34202i) q^{65} +(-2.28630 - 12.9663i) q^{67} +(3.60687 + 20.4556i) q^{69} +(-2.79386 - 1.01688i) q^{71} -2.55740 q^{73} +3.20177 q^{75} +(12.8726 + 4.68524i) q^{77} +(0.943888 + 5.35305i) q^{79} +(-1.94203 - 11.0138i) q^{81} +(-0.504249 - 0.423115i) q^{83} +(4.17840 - 7.23720i) q^{85} +(-1.29422 + 1.08598i) q^{87} +(0.612149 - 3.47167i) q^{89} +(6.08790 + 5.10835i) q^{91} +(-21.2986 - 7.75206i) q^{93} +(-0.0323948 + 0.183720i) q^{95} +(-1.81012 - 3.13522i) q^{97} +(5.51897 - 2.00874i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 6 q^{5} - 6 q^{7} - 3 q^{9} - 3 q^{11} - 6 q^{13} - 6 q^{15} - 3 q^{17} + 3 q^{19} - 33 q^{21} + 21 q^{23} - 6 q^{25} - 3 q^{27} + 6 q^{29} - 42 q^{31} + 57 q^{33} + 9 q^{35} - 3 q^{37}+ \cdots + 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\) \(1\) \(1\)

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
\(3\) 2.04963 + 0.746005i 1.18336 + 0.430706i 0.857386 − 0.514674i \(-0.172087\pi\)
0.325970 + 0.945380i \(0.394309\pi\)
\(4\) 0 0
\(5\) −0.326352 − 1.85083i −0.145949 − 0.827718i −0.966600 − 0.256289i \(-0.917500\pi\)
0.820651 − 0.571429i \(-0.193611\pi\)
\(6\) 0 0
\(7\) 0.711830 + 4.03699i 0.269046 + 1.52584i 0.757260 + 0.653113i \(0.226537\pi\)
−0.488214 + 0.872724i \(0.662351\pi\)
\(8\) 0 0
\(9\) 1.34633 + 1.12971i 0.448778 + 0.376569i
\(10\) 0 0
\(11\) 1.67088 − 2.89404i 0.503788 − 0.872586i −0.496203 − 0.868207i \(-0.665273\pi\)
0.999990 − 0.00437929i \(-0.00139397\pi\)
\(12\) 0 0
\(13\) 1.48512 − 1.24616i 0.411898 − 0.345623i −0.413173 − 0.910653i \(-0.635580\pi\)
0.825071 + 0.565029i \(0.191135\pi\)
\(14\) 0 0
\(15\) 0.711830 − 4.03699i 0.183794 − 1.04235i
\(16\) 0 0
\(17\) 3.40626 + 2.85819i 0.826140 + 0.693214i 0.954401 − 0.298527i \(-0.0964952\pi\)
−0.128261 + 0.991740i \(0.540940\pi\)
\(18\) 0 0
\(19\) −0.0932770 − 0.0339500i −0.0213992 − 0.00778868i 0.331298 − 0.943526i \(-0.392513\pi\)
−0.352698 + 0.935737i \(0.614736\pi\)
\(20\) 0 0
\(21\) −1.55262 + 8.80536i −0.338810 + 1.92149i
\(22\) 0 0
\(23\) 4.76146 + 8.24709i 0.992833 + 1.71964i 0.599913 + 0.800065i \(0.295202\pi\)
0.392920 + 0.919573i \(0.371465\pi\)
\(24\) 0 0
\(25\) 1.37939 − 0.502055i 0.275877 − 0.100411i
\(26\) 0 0
\(27\) −1.35504 − 2.34700i −0.260777 − 0.451680i
\(28\) 0 0
\(29\) −0.387288 + 0.670802i −0.0719175 + 0.124565i −0.899742 − 0.436423i \(-0.856245\pi\)
0.827824 + 0.560988i \(0.189579\pi\)
\(30\) 0 0
\(31\) −10.3914 −1.86636 −0.933179 − 0.359413i \(-0.882977\pi\)
−0.933179 + 0.359413i \(0.882977\pi\)
\(32\) 0 0
\(33\) 5.58365 − 4.68524i 0.971988 − 0.815595i
\(34\) 0 0
\(35\) 7.23948 − 2.63496i 1.22370 − 0.445389i
\(36\) 0 0
\(37\) −3.50302 − 4.97281i −0.575894 − 0.817525i
\(38\) 0 0
\(39\) 3.97359 − 1.44627i 0.636284 − 0.231588i
\(40\) 0 0
\(41\) 4.36218 − 3.66031i 0.681259 − 0.571644i −0.235115 − 0.971968i \(-0.575547\pi\)
0.916374 + 0.400324i \(0.131102\pi\)
\(42\) 0 0
\(43\) 1.11986 0.170777 0.0853884 − 0.996348i \(-0.472787\pi\)
0.0853884 + 0.996348i \(0.472787\pi\)
\(44\) 0 0
\(45\) 1.65152 − 2.86052i 0.246194 − 0.426421i
\(46\) 0 0
\(47\) −3.55724 − 6.16132i −0.518877 − 0.898721i −0.999759 − 0.0219357i \(-0.993017\pi\)
0.480883 − 0.876785i \(-0.340316\pi\)
\(48\) 0 0
\(49\) −9.21271 + 3.35315i −1.31610 + 0.479022i
\(50\) 0 0
\(51\) 4.84936 + 8.39933i 0.679046 + 1.17614i
\(52\) 0 0
\(53\) 0.142188 − 0.806388i 0.0195310 − 0.110766i −0.973484 − 0.228756i \(-0.926534\pi\)
0.993015 + 0.117990i \(0.0376452\pi\)
\(54\) 0 0
\(55\) −5.90168 − 2.14804i −0.795782 − 0.289641i
\(56\) 0 0
\(57\) −0.165857 − 0.139170i −0.0219682 − 0.0184335i
\(58\) 0 0
\(59\) −1.40642 + 7.97622i −0.183101 + 1.03842i 0.745271 + 0.666762i \(0.232320\pi\)
−0.928372 + 0.371653i \(0.878791\pi\)
\(60\) 0 0
\(61\) −8.19737 + 6.87841i −1.04957 + 0.880691i −0.993048 − 0.117711i \(-0.962444\pi\)
−0.0565183 + 0.998402i \(0.518000\pi\)
\(62\) 0 0
\(63\) −3.60225 + 6.23929i −0.453841 + 0.786076i
\(64\) 0 0
\(65\) −2.79111 − 2.34202i −0.346195 − 0.290492i
\(66\) 0 0
\(67\) −2.28630 − 12.9663i −0.279317 − 1.58408i −0.724907 − 0.688846i \(-0.758117\pi\)
0.445591 − 0.895237i \(-0.352994\pi\)
\(68\) 0 0
\(69\) 3.60687 + 20.4556i 0.434216 + 2.46256i
\(70\) 0 0
\(71\) −2.79386 − 1.01688i −0.331570 − 0.120682i 0.170870 − 0.985294i \(-0.445342\pi\)
−0.502440 + 0.864612i \(0.667564\pi\)
\(72\) 0 0
\(73\) −2.55740 −0.299321 −0.149660 − 0.988737i \(-0.547818\pi\)
−0.149660 + 0.988737i \(0.547818\pi\)
\(74\) 0 0
\(75\) 3.20177 0.369708
\(76\) 0 0
\(77\) 12.8726 + 4.68524i 1.46697 + 0.533932i
\(78\) 0 0
\(79\) 0.943888 + 5.35305i 0.106196 + 0.602265i 0.990736 + 0.135802i \(0.0433610\pi\)
−0.884540 + 0.466464i \(0.845528\pi\)
\(80\) 0 0
\(81\) −1.94203 − 11.0138i −0.215781 − 1.22375i
\(82\) 0 0
\(83\) −0.504249 − 0.423115i −0.0553485 − 0.0464429i 0.614694 − 0.788766i \(-0.289280\pi\)
−0.670042 + 0.742323i \(0.733724\pi\)
\(84\) 0 0
\(85\) 4.17840 − 7.23720i 0.453211 − 0.784985i
\(86\) 0 0
\(87\) −1.29422 + 1.08598i −0.138755 + 0.116429i
\(88\) 0 0
\(89\) 0.612149 − 3.47167i 0.0648877 − 0.367996i −0.935022 − 0.354589i \(-0.884621\pi\)
0.999910 − 0.0134078i \(-0.00426798\pi\)
\(90\) 0 0
\(91\) 6.08790 + 5.10835i 0.638185 + 0.535501i
\(92\) 0 0
\(93\) −21.2986 − 7.75206i −2.20856 − 0.803852i
\(94\) 0 0
\(95\) −0.0323948 + 0.183720i −0.00332363 + 0.0188493i
\(96\) 0 0
\(97\) −1.81012 − 3.13522i −0.183790 − 0.318334i 0.759378 − 0.650650i \(-0.225503\pi\)
−0.943168 + 0.332316i \(0.892170\pi\)
\(98\) 0 0
\(99\) 5.51897 − 2.00874i 0.554678 − 0.201886i
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.81.2 12
4.3 odd 2 74.2.f.b.7.1 ✓ 12
12.11 even 2 666.2.x.g.451.1 12
37.16 even 9 inner 592.2.bc.d.497.2 12
148.107 odd 18 2738.2.a.t.1.5 6
148.115 odd 18 2738.2.a.q.1.5 6
148.127 odd 18 74.2.f.b.53.1 yes 12
444.275 even 18 666.2.x.g.127.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.7.1 ✓ 12 4.3 odd 2
74.2.f.b.53.1 yes 12 148.127 odd 18
592.2.bc.d.81.2 12 1.1 even 1 trivial
592.2.bc.d.497.2 12 37.16 even 9 inner
666.2.x.g.127.1 12 444.275 even 18
666.2.x.g.451.1 12 12.11 even 2
2738.2.a.q.1.5 6 148.115 odd 18
2738.2.a.t.1.5 6 148.107 odd 18