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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [592,2,Mod(529,592)] 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("592.529"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

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

Embedding invariants

Embedding label 529.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 592.529
Dual form 592.2.w.d.545.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.366025 + 0.633975i) q^{3} +(-1.50000 - 0.866025i) q^{5} +(2.00000 - 3.46410i) q^{7} +(1.23205 + 2.13397i) q^{9} -4.73205 q^{11} +(-5.19615 - 3.00000i) q^{13} +(1.09808 - 0.633975i) q^{15} +(-1.50000 + 0.866025i) q^{17} +(1.09808 + 0.633975i) q^{19} +(1.46410 + 2.53590i) q^{21} -1.26795i q^{23} +(-1.00000 - 1.73205i) q^{25} -4.00000 q^{27} -4.26795i q^{29} +1.26795i q^{31} +(1.73205 - 3.00000i) q^{33} +(-6.00000 + 3.46410i) q^{35} +(-0.500000 - 6.06218i) q^{37} +(3.80385 - 2.19615i) q^{39} +(-0.232051 + 0.401924i) q^{41} -9.46410i q^{43} -4.26795i q^{45} -11.6603 q^{47} +(-4.50000 - 7.79423i) q^{49} -1.26795i q^{51} +(1.26795 + 2.19615i) q^{53} +(7.09808 + 4.09808i) q^{55} +(-0.803848 + 0.464102i) q^{57} +(2.19615 - 1.26795i) q^{59} +(12.6962 + 7.33013i) q^{61} +9.85641 q^{63} +(5.19615 + 9.00000i) q^{65} +(3.09808 - 5.36603i) q^{67} +(0.803848 + 0.464102i) q^{69} +(1.26795 - 2.19615i) q^{71} -12.3923 q^{73} +1.46410 q^{75} +(-9.46410 + 16.3923i) q^{77} +(7.09808 + 4.09808i) q^{79} +(-2.23205 + 3.86603i) q^{81} +(5.83013 + 10.0981i) q^{83} +3.00000 q^{85} +(2.70577 + 1.56218i) q^{87} +(4.50000 - 2.59808i) q^{89} +(-20.7846 + 12.0000i) q^{91} +(-0.803848 - 0.464102i) q^{93} +(-1.09808 - 1.90192i) q^{95} +5.19615i q^{97} +(-5.83013 - 10.0981i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 6 q^{5} + 8 q^{7} - 2 q^{9} - 12 q^{11} - 6 q^{15} - 6 q^{17} - 6 q^{19} - 8 q^{21} - 4 q^{25} - 16 q^{27} - 24 q^{35} - 2 q^{37} + 36 q^{39} + 6 q^{41} - 12 q^{47} - 18 q^{49} + 12 q^{53}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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
\(3\) −0.366025 + 0.633975i −0.211325 + 0.366025i −0.952129 − 0.305695i \(-0.901111\pi\)
0.740805 + 0.671721i \(0.234444\pi\)
\(4\) 0 0
\(5\) −1.50000 − 0.866025i −0.670820 − 0.387298i 0.125567 − 0.992085i \(-0.459925\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) 2.00000 − 3.46410i 0.755929 − 1.30931i −0.188982 − 0.981981i \(-0.560519\pi\)
0.944911 − 0.327327i \(-0.106148\pi\)
\(8\) 0 0
\(9\) 1.23205 + 2.13397i 0.410684 + 0.711325i
\(10\) 0 0
\(11\) −4.73205 −1.42677 −0.713384 − 0.700774i \(-0.752838\pi\)
−0.713384 + 0.700774i \(0.752838\pi\)
\(12\) 0 0
\(13\) −5.19615 − 3.00000i −1.44115 − 0.832050i −0.443227 − 0.896410i \(-0.646166\pi\)
−0.997927 + 0.0643593i \(0.979500\pi\)
\(14\) 0 0
\(15\) 1.09808 − 0.633975i 0.283522 − 0.163692i
\(16\) 0 0
\(17\) −1.50000 + 0.866025i −0.363803 + 0.210042i −0.670748 − 0.741685i \(-0.734027\pi\)
0.306944 + 0.951727i \(0.400693\pi\)
\(18\) 0 0
\(19\) 1.09808 + 0.633975i 0.251916 + 0.145444i 0.620641 − 0.784095i \(-0.286872\pi\)
−0.368725 + 0.929538i \(0.620206\pi\)
\(20\) 0 0
\(21\) 1.46410 + 2.53590i 0.319493 + 0.553378i
\(22\) 0 0
\(23\) − 1.26795i − 0.264386i −0.991224 − 0.132193i \(-0.957798\pi\)
0.991224 − 0.132193i \(-0.0422018\pi\)
\(24\) 0 0
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) − 4.26795i − 0.792538i −0.918134 − 0.396269i \(-0.870305\pi\)
0.918134 − 0.396269i \(-0.129695\pi\)
\(30\) 0 0
\(31\) 1.26795i 0.227730i 0.993496 + 0.113865i \(0.0363232\pi\)
−0.993496 + 0.113865i \(0.963677\pi\)
\(32\) 0 0
\(33\) 1.73205 − 3.00000i 0.301511 − 0.522233i
\(34\) 0 0
\(35\) −6.00000 + 3.46410i −1.01419 + 0.585540i
\(36\) 0 0
\(37\) −0.500000 − 6.06218i −0.0821995 − 0.996616i
\(38\) 0 0
\(39\) 3.80385 − 2.19615i 0.609103 − 0.351666i
\(40\) 0 0
\(41\) −0.232051 + 0.401924i −0.0362402 + 0.0627700i −0.883577 − 0.468287i \(-0.844871\pi\)
0.847336 + 0.531057i \(0.178205\pi\)
\(42\) 0 0
\(43\) − 9.46410i − 1.44326i −0.692278 − 0.721631i \(-0.743393\pi\)
0.692278 − 0.721631i \(-0.256607\pi\)
\(44\) 0 0
\(45\) − 4.26795i − 0.636228i
\(46\) 0 0
\(47\) −11.6603 −1.70082 −0.850411 − 0.526118i \(-0.823647\pi\)
−0.850411 + 0.526118i \(0.823647\pi\)
\(48\) 0 0
\(49\) −4.50000 − 7.79423i −0.642857 − 1.11346i
\(50\) 0 0
\(51\) − 1.26795i − 0.177548i
\(52\) 0 0
\(53\) 1.26795 + 2.19615i 0.174166 + 0.301665i 0.939872 − 0.341526i \(-0.110944\pi\)
−0.765706 + 0.643191i \(0.777610\pi\)
\(54\) 0 0
\(55\) 7.09808 + 4.09808i 0.957104 + 0.552584i
\(56\) 0 0
\(57\) −0.803848 + 0.464102i −0.106472 + 0.0614718i
\(58\) 0 0
\(59\) 2.19615 − 1.26795i 0.285915 − 0.165073i −0.350183 − 0.936681i \(-0.613881\pi\)
0.636098 + 0.771608i \(0.280547\pi\)
\(60\) 0 0
\(61\) 12.6962 + 7.33013i 1.62558 + 0.938527i 0.985391 + 0.170305i \(0.0544754\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 9.85641 1.24179
\(64\) 0 0
\(65\) 5.19615 + 9.00000i 0.644503 + 1.11631i
\(66\) 0 0
\(67\) 3.09808 − 5.36603i 0.378490 − 0.655564i −0.612353 − 0.790585i \(-0.709777\pi\)
0.990843 + 0.135020i \(0.0431100\pi\)
\(68\) 0 0
\(69\) 0.803848 + 0.464102i 0.0967719 + 0.0558713i
\(70\) 0 0
\(71\) 1.26795 − 2.19615i 0.150478 − 0.260635i −0.780925 − 0.624624i \(-0.785252\pi\)
0.931403 + 0.363989i \(0.118585\pi\)
\(72\) 0 0
\(73\) −12.3923 −1.45041 −0.725205 − 0.688533i \(-0.758255\pi\)
−0.725205 + 0.688533i \(0.758255\pi\)
\(74\) 0 0
\(75\) 1.46410 0.169060
\(76\) 0 0
\(77\) −9.46410 + 16.3923i −1.07853 + 1.86808i
\(78\) 0 0
\(79\) 7.09808 + 4.09808i 0.798596 + 0.461070i 0.842980 − 0.537945i \(-0.180799\pi\)
−0.0443840 + 0.999015i \(0.514133\pi\)
\(80\) 0 0
\(81\) −2.23205 + 3.86603i −0.248006 + 0.429558i
\(82\) 0 0
\(83\) 5.83013 + 10.0981i 0.639940 + 1.10841i 0.985446 + 0.169991i \(0.0543740\pi\)
−0.345506 + 0.938417i \(0.612293\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) 0 0
\(87\) 2.70577 + 1.56218i 0.290089 + 0.167483i
\(88\) 0 0
\(89\) 4.50000 − 2.59808i 0.476999 − 0.275396i −0.242166 − 0.970235i \(-0.577858\pi\)
0.719165 + 0.694839i \(0.244525\pi\)
\(90\) 0 0
\(91\) −20.7846 + 12.0000i −2.17882 + 1.25794i
\(92\) 0 0
\(93\) −0.803848 − 0.464102i −0.0833551 − 0.0481251i
\(94\) 0 0
\(95\) −1.09808 − 1.90192i −0.112660 − 0.195133i
\(96\) 0 0
\(97\) 5.19615i 0.527589i 0.964579 + 0.263795i \(0.0849741\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 0 0
\(99\) −5.83013 − 10.0981i −0.585950 − 1.01489i
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.w.d.529.1 4
4.3 odd 2 74.2.e.a.11.2 ✓ 4
12.11 even 2 666.2.s.e.307.1 4
37.27 even 6 inner 592.2.w.d.545.1 4
148.27 odd 6 74.2.e.a.27.2 yes 4
148.103 even 12 2738.2.a.f.1.2 2
148.119 even 12 2738.2.a.j.1.2 2
444.323 even 6 666.2.s.e.397.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.a.11.2 ✓ 4 4.3 odd 2
74.2.e.a.27.2 yes 4 148.27 odd 6
592.2.w.d.529.1 4 1.1 even 1 trivial
592.2.w.d.545.1 4 37.27 even 6 inner
666.2.s.e.307.1 4 12.11 even 2
666.2.s.e.397.1 4 444.323 even 6
2738.2.a.f.1.2 2 148.103 even 12
2738.2.a.j.1.2 2 148.119 even 12