Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [592,2,Mod(65,592)] 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("592.65"); 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, 17])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.bq (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-6] 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 \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 225.2
Root \(0.642788 + 0.766044i\) of defining polynomial
Character \(\chi\) \(=\) 592.225
Dual form 592.2.bq.b.321.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.43969 + 1.20805i) q^{3} +(0.273629 - 0.751790i) q^{5} +(0.138449 + 0.0503913i) q^{7} +(0.0923963 - 0.524005i) q^{9} +(2.40570 - 4.16679i) q^{11} +(1.91858 - 0.338298i) q^{13} +(0.514255 + 1.41290i) q^{15} +(-3.43779 - 0.606175i) q^{17} +(4.33625 + 5.16775i) q^{19} +(-0.260199 + 0.0947047i) q^{21} +(3.61610 - 2.08776i) q^{23} +(3.33991 + 2.80251i) q^{25} +(-2.31908 - 4.01676i) q^{27} +(2.63134 + 1.51921i) q^{29} +8.13740i q^{31} +(1.57021 + 8.90509i) q^{33} +(0.0757674 - 0.0902961i) q^{35} +(6.08227 - 0.0772535i) q^{37} +(-2.35349 + 2.80478i) q^{39} +(0.676822 + 3.83845i) q^{41} -8.10852i q^{43} +(-0.368660 - 0.212846i) q^{45} +(4.16911 + 7.22111i) q^{47} +(-5.34568 - 4.48556i) q^{49} +(5.68164 - 3.28030i) q^{51} +(10.2327 - 3.72440i) q^{53} +(-2.47428 - 2.94874i) q^{55} +(-12.4857 - 2.20157i) q^{57} +(2.96569 + 8.14816i) q^{59} +(-0.346344 + 0.0610698i) q^{61} +(0.0391975 - 0.0678921i) q^{63} +(0.270651 - 1.53494i) q^{65} +(2.25471 + 0.820647i) q^{67} +(-2.68397 + 7.37414i) q^{69} +(-5.34953 + 4.48879i) q^{71} +1.13399 q^{73} -8.19401 q^{75} +(0.543037 - 0.455662i) q^{77} +(-0.646510 + 1.77627i) q^{79} +(9.69119 + 3.52730i) q^{81} +(1.71997 - 9.75442i) q^{83} +(-1.39639 + 2.41863i) q^{85} +(-5.62359 + 0.991591i) q^{87} +(-3.45924 - 9.50418i) q^{89} +(0.282673 + 0.0498429i) q^{91} +(-9.83035 - 11.7154i) q^{93} +(5.07158 - 1.84591i) q^{95} +(5.08812 - 2.93763i) q^{97} +(-1.96114 - 1.64560i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 12 q^{7} - 6 q^{9} + 6 q^{11} + 6 q^{13} + 18 q^{19} - 6 q^{21} - 18 q^{25} + 6 q^{27} + 18 q^{29} - 6 q^{33} - 18 q^{35} + 30 q^{37} - 30 q^{39} + 24 q^{41} - 18 q^{45} - 6 q^{47} + 12 q^{49}+ \cdots - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(1\) \(1\)

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
\(3\) −1.43969 + 1.20805i −0.831207 + 0.697465i −0.955568 − 0.294772i \(-0.904756\pi\)
0.124361 + 0.992237i \(0.460312\pi\)
\(4\) 0 0
\(5\) 0.273629 − 0.751790i 0.122371 − 0.336211i −0.863349 − 0.504608i \(-0.831637\pi\)
0.985719 + 0.168397i \(0.0538592\pi\)
\(6\) 0 0
\(7\) 0.138449 + 0.0503913i 0.0523288 + 0.0190461i 0.368052 − 0.929805i \(-0.380025\pi\)
−0.315723 + 0.948851i \(0.602247\pi\)
\(8\) 0 0
\(9\) 0.0923963 − 0.524005i 0.0307988 − 0.174668i
\(10\) 0 0
\(11\) 2.40570 − 4.16679i 0.725346 − 1.25634i −0.233486 − 0.972360i \(-0.575013\pi\)
0.958832 − 0.283975i \(-0.0916533\pi\)
\(12\) 0 0
\(13\) 1.91858 − 0.338298i 0.532119 − 0.0938270i 0.0988686 − 0.995100i \(-0.468478\pi\)
0.433251 + 0.901274i \(0.357367\pi\)
\(14\) 0 0
\(15\) 0.514255 + 1.41290i 0.132780 + 0.364810i
\(16\) 0 0
\(17\) −3.43779 − 0.606175i −0.833786 − 0.147019i −0.259572 − 0.965724i \(-0.583581\pi\)
−0.574214 + 0.818705i \(0.694692\pi\)
\(18\) 0 0
\(19\) 4.33625 + 5.16775i 0.994805 + 1.18556i 0.982619 + 0.185636i \(0.0594346\pi\)
0.0121861 + 0.999926i \(0.496121\pi\)
\(20\) 0 0
\(21\) −0.260199 + 0.0947047i −0.0567801 + 0.0206663i
\(22\) 0 0
\(23\) 3.61610 − 2.08776i 0.754009 − 0.435327i −0.0731316 − 0.997322i \(-0.523299\pi\)
0.827141 + 0.561995i \(0.189966\pi\)
\(24\) 0 0
\(25\) 3.33991 + 2.80251i 0.667981 + 0.560503i
\(26\) 0 0
\(27\) −2.31908 − 4.01676i −0.446307 − 0.773026i
\(28\) 0 0
\(29\) 2.63134 + 1.51921i 0.488628 + 0.282109i 0.724005 − 0.689795i \(-0.242299\pi\)
−0.235377 + 0.971904i \(0.575633\pi\)
\(30\) 0 0
\(31\) 8.13740i 1.46152i 0.682634 + 0.730760i \(0.260834\pi\)
−0.682634 + 0.730760i \(0.739166\pi\)
\(32\) 0 0
\(33\) 1.57021 + 8.90509i 0.273338 + 1.55018i
\(34\) 0 0
\(35\) 0.0757674 − 0.0902961i 0.0128070 − 0.0152628i
\(36\) 0 0
\(37\) 6.08227 − 0.0772535i 0.999919 − 0.0127004i
\(38\) 0 0
\(39\) −2.35349 + 2.80478i −0.376860 + 0.449124i
\(40\) 0 0
\(41\) 0.676822 + 3.83845i 0.105702 + 0.599465i 0.990938 + 0.134322i \(0.0428858\pi\)
−0.885236 + 0.465142i \(0.846003\pi\)
\(42\) 0 0
\(43\) − 8.10852i − 1.23654i −0.785966 − 0.618269i \(-0.787834\pi\)
0.785966 − 0.618269i \(-0.212166\pi\)
\(44\) 0 0
\(45\) −0.368660 − 0.212846i −0.0549565 − 0.0317292i
\(46\) 0 0
\(47\) 4.16911 + 7.22111i 0.608127 + 1.05331i 0.991549 + 0.129734i \(0.0414122\pi\)
−0.383422 + 0.923573i \(0.625254\pi\)
\(48\) 0 0
\(49\) −5.34568 − 4.48556i −0.763669 − 0.640794i
\(50\) 0 0
\(51\) 5.68164 − 3.28030i 0.795589 − 0.459334i
\(52\) 0 0
\(53\) 10.2327 − 3.72440i 1.40557 − 0.511586i 0.475744 − 0.879584i \(-0.342179\pi\)
0.929827 + 0.367998i \(0.119957\pi\)
\(54\) 0 0
\(55\) −2.47428 − 2.94874i −0.333632 − 0.397607i
\(56\) 0 0
\(57\) −12.4857 − 2.20157i −1.65378 − 0.291606i
\(58\) 0 0
\(59\) 2.96569 + 8.14816i 0.386100 + 1.06080i 0.968742 + 0.248072i \(0.0797968\pi\)
−0.582642 + 0.812729i \(0.697981\pi\)
\(60\) 0 0
\(61\) −0.346344 + 0.0610698i −0.0443448 + 0.00781919i −0.195777 − 0.980649i \(-0.562723\pi\)
0.151432 + 0.988468i \(0.451612\pi\)
\(62\) 0 0
\(63\) 0.0391975 − 0.0678921i 0.00493842 − 0.00855360i
\(64\) 0 0
\(65\) 0.270651 − 1.53494i 0.0335702 − 0.190386i
\(66\) 0 0
\(67\) 2.25471 + 0.820647i 0.275457 + 0.100258i 0.476055 − 0.879416i \(-0.342066\pi\)
−0.200598 + 0.979674i \(0.564289\pi\)
\(68\) 0 0
\(69\) −2.68397 + 7.37414i −0.323112 + 0.887742i
\(70\) 0 0
\(71\) −5.34953 + 4.48879i −0.634873 + 0.532721i −0.902439 − 0.430818i \(-0.858225\pi\)
0.267566 + 0.963539i \(0.413781\pi\)
\(72\) 0 0
\(73\) 1.13399 0.132724 0.0663618 − 0.997796i \(-0.478861\pi\)
0.0663618 + 0.997796i \(0.478861\pi\)
\(74\) 0 0
\(75\) −8.19401 −0.946162
\(76\) 0 0
\(77\) 0.543037 − 0.455662i 0.0618848 − 0.0519275i
\(78\) 0 0
\(79\) −0.646510 + 1.77627i −0.0727380 + 0.199846i −0.970734 − 0.240158i \(-0.922801\pi\)
0.897996 + 0.440004i \(0.145023\pi\)
\(80\) 0 0
\(81\) 9.69119 + 3.52730i 1.07680 + 0.391923i
\(82\) 0 0
\(83\) 1.71997 − 9.75442i 0.188791 − 1.07069i −0.732196 − 0.681094i \(-0.761505\pi\)
0.920987 − 0.389593i \(-0.127384\pi\)
\(84\) 0 0
\(85\) −1.39639 + 2.41863i −0.151460 + 0.262337i
\(86\) 0 0
\(87\) −5.62359 + 0.991591i −0.602912 + 0.106310i
\(88\) 0 0
\(89\) −3.45924 − 9.50418i −0.366679 − 1.00744i −0.976616 − 0.214992i \(-0.931028\pi\)
0.609937 − 0.792450i \(-0.291195\pi\)
\(90\) 0 0
\(91\) 0.282673 + 0.0498429i 0.0296322 + 0.00522496i
\(92\) 0 0
\(93\) −9.83035 − 11.7154i −1.01936 − 1.21483i
\(94\) 0 0
\(95\) 5.07158 − 1.84591i 0.520333 − 0.189386i
\(96\) 0 0
\(97\) 5.08812 − 2.93763i 0.516620 − 0.298271i −0.218930 − 0.975740i \(-0.570257\pi\)
0.735551 + 0.677470i \(0.236923\pi\)
\(98\) 0 0
\(99\) −1.96114 − 1.64560i −0.197102 − 0.165389i
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.bq.b.225.2 12
4.3 odd 2 74.2.h.a.3.1 ✓ 12
12.11 even 2 666.2.bj.c.595.2 12
37.25 even 18 inner 592.2.bq.b.321.2 12
148.79 even 36 2738.2.a.s.1.2 6
148.99 odd 18 74.2.h.a.25.1 yes 12
148.143 even 36 2738.2.a.r.1.1 6
444.395 even 18 666.2.bj.c.469.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.3.1 ✓ 12 4.3 odd 2
74.2.h.a.25.1 yes 12 148.99 odd 18
592.2.bq.b.225.2 12 1.1 even 1 trivial
592.2.bq.b.321.2 12 37.25 even 18 inner
666.2.bj.c.469.2 12 444.395 even 18
666.2.bj.c.595.2 12 12.11 even 2
2738.2.a.r.1.1 6 148.143 even 36
2738.2.a.s.1.2 6 148.79 even 36