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 225.1
Root \(-0.642788 - 0.766044i\) of defining polynomial
Character \(\chi\) \(=\) 592.225
Dual form 592.2.bq.b.321.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.43969 + 1.20805i) q^{3} +(-1.45842 + 4.00698i) q^{5} +(3.39364 + 1.23518i) q^{7} +(0.0923963 - 0.524005i) q^{9} +(-1.05840 + 1.83321i) q^{11} +(2.84019 - 0.500802i) q^{13} +(-2.74094 - 7.53066i) q^{15} +(0.0263137 + 0.00463982i) q^{17} +(2.07522 + 2.47315i) q^{19} +(-6.37796 + 2.32139i) q^{21} +(-2.57421 + 1.48622i) q^{23} +(-10.0987 - 8.47380i) q^{25} +(-2.31908 - 4.01676i) q^{27} +(4.96493 + 2.86650i) q^{29} -6.76932i q^{31} +(-0.690823 - 3.91785i) q^{33} +(-9.89872 + 11.7968i) q^{35} +(-4.49375 - 4.09954i) q^{37} +(-3.48401 + 4.15208i) q^{39} +(0.259000 + 1.46886i) q^{41} +5.53737i q^{43} +(1.96493 + 1.13445i) q^{45} +(1.30654 + 2.26300i) q^{47} +(4.62880 + 3.88403i) q^{49} +(-0.0434888 + 0.0251083i) q^{51} +(-1.79389 + 0.652924i) q^{53} +(-5.80203 - 6.91459i) q^{55} +(-5.97536 - 1.05362i) q^{57} +(-1.92380 - 5.28560i) q^{59} +(-5.65366 + 0.996892i) q^{61} +(0.960802 - 1.66416i) q^{63} +(-2.13549 + 12.1110i) q^{65} +(6.50406 + 2.36728i) q^{67} +(1.91065 - 5.24947i) q^{69} +(7.10830 - 5.96457i) q^{71} +16.2707 q^{73} +24.7757 q^{75} +(-5.85619 + 4.91392i) q^{77} +(0.484006 - 1.32980i) q^{79} +(9.69119 + 3.52730i) q^{81} +(0.294580 - 1.67065i) q^{83} +(-0.0569682 + 0.0986718i) q^{85} +(-10.6108 + 1.87098i) q^{87} +(2.82882 + 7.77213i) q^{89} +(10.2572 + 1.80861i) q^{91} +(8.17765 + 9.74574i) q^{93} +(-12.9364 + 4.70847i) q^{95} +(5.65105 - 3.26264i) q^{97} +(0.862818 + 0.723990i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 12 q^{7} - 6 q^{9} + 6 q^{11} + 6 q^{13} + 18 q^{19} - 6 q^{21} - 18 q^{25} + 6 q^{27} + 18 q^{29} - 6 q^{33} - 18 q^{35} + 30 q^{37} - 30 q^{39} + 24 q^{41} - 18 q^{45} - 6 q^{47} + 12 q^{49}+ \cdots - 30 q^{99}+O(q^{100}) \) 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
\(5\) −1.45842 + 4.00698i −0.652226 + 1.79198i −0.0428683 + 0.999081i \(0.513650\pi\)
−0.609358 + 0.792895i \(0.708573\pi\)
\(6\) 0 0
\(7\) 3.39364 + 1.23518i 1.28268 + 0.466856i 0.891316 − 0.453382i \(-0.149783\pi\)
0.391359 + 0.920238i \(0.372005\pi\)
\(8\) 0 0
\(9\) 0.0923963 − 0.524005i 0.0307988 − 0.174668i
\(10\) 0 0
\(11\) −1.05840 + 1.83321i −0.319120 + 0.552733i −0.980305 − 0.197491i \(-0.936721\pi\)
0.661184 + 0.750223i \(0.270054\pi\)
\(12\) 0 0
\(13\) 2.84019 − 0.500802i 0.787726 − 0.138897i 0.234707 − 0.972066i \(-0.424587\pi\)
0.553019 + 0.833169i \(0.313476\pi\)
\(14\) 0 0
\(15\) −2.74094 − 7.53066i −0.707707 − 1.94441i
\(16\) 0 0
\(17\) 0.0263137 + 0.00463982i 0.00638202 + 0.00112532i 0.176838 − 0.984240i \(-0.443413\pi\)
−0.170456 + 0.985365i \(0.554524\pi\)
\(18\) 0 0
\(19\) 2.07522 + 2.47315i 0.476088 + 0.567380i 0.949623 − 0.313395i \(-0.101466\pi\)
−0.473534 + 0.880775i \(0.657022\pi\)
\(20\) 0 0
\(21\) −6.37796 + 2.32139i −1.39178 + 0.506568i
\(22\) 0 0
\(23\) −2.57421 + 1.48622i −0.536760 + 0.309899i −0.743765 − 0.668441i \(-0.766962\pi\)
0.207005 + 0.978340i \(0.433628\pi\)
\(24\) 0 0
\(25\) −10.0987 − 8.47380i −2.01974 − 1.69476i
\(26\) 0 0
\(27\) −2.31908 − 4.01676i −0.446307 − 0.773026i
\(28\) 0 0
\(29\) 4.96493 + 2.86650i 0.921964 + 0.532296i 0.884261 − 0.466993i \(-0.154663\pi\)
0.0377027 + 0.999289i \(0.487996\pi\)
\(30\) 0 0
\(31\) − 6.76932i − 1.21581i −0.794011 − 0.607903i \(-0.792011\pi\)
0.794011 − 0.607903i \(-0.207989\pi\)
\(32\) 0 0
\(33\) −0.690823 − 3.91785i −0.120257 − 0.682011i
\(34\) 0 0
\(35\) −9.89872 + 11.7968i −1.67319 + 1.99403i
\(36\) 0 0
\(37\) −4.49375 − 4.09954i −0.738767 − 0.673961i
\(38\) 0 0
\(39\) −3.48401 + 4.15208i −0.557887 + 0.664864i
\(40\) 0 0
\(41\) 0.259000 + 1.46886i 0.0404490 + 0.229398i 0.998330 − 0.0577674i \(-0.0183982\pi\)
−0.957881 + 0.287165i \(0.907287\pi\)
\(42\) 0 0
\(43\) 5.53737i 0.844441i 0.906493 + 0.422221i \(0.138749\pi\)
−0.906493 + 0.422221i \(0.861251\pi\)
\(44\) 0 0
\(45\) 1.96493 + 1.13445i 0.292914 + 0.169114i
\(46\) 0 0
\(47\) 1.30654 + 2.26300i 0.190579 + 0.330092i 0.945442 − 0.325790i \(-0.105630\pi\)
−0.754863 + 0.655882i \(0.772297\pi\)
\(48\) 0 0
\(49\) 4.62880 + 3.88403i 0.661257 + 0.554861i
\(50\) 0 0
\(51\) −0.0434888 + 0.0251083i −0.00608965 + 0.00351586i
\(52\) 0 0
\(53\) −1.79389 + 0.652924i −0.246410 + 0.0896860i −0.462273 − 0.886738i \(-0.652966\pi\)
0.215862 + 0.976424i \(0.430744\pi\)
\(54\) 0 0
\(55\) −5.80203 − 6.91459i −0.782345 − 0.932363i
\(56\) 0 0
\(57\) −5.97536 − 1.05362i −0.791456 − 0.139555i
\(58\) 0 0
\(59\) −1.92380 − 5.28560i −0.250457 − 0.688126i −0.999667 − 0.0257935i \(-0.991789\pi\)
0.749210 − 0.662333i \(-0.230433\pi\)
\(60\) 0 0
\(61\) −5.65366 + 0.996892i −0.723876 + 0.127639i −0.523434 − 0.852066i \(-0.675349\pi\)
−0.200443 + 0.979705i \(0.564238\pi\)
\(62\) 0 0
\(63\) 0.960802 − 1.66416i 0.121050 − 0.209664i
\(64\) 0 0
\(65\) −2.13549 + 12.1110i −0.264875 + 1.50218i
\(66\) 0 0
\(67\) 6.50406 + 2.36728i 0.794597 + 0.289210i 0.707246 − 0.706968i \(-0.249937\pi\)
0.0873516 + 0.996178i \(0.472160\pi\)
\(68\) 0 0
\(69\) 1.91065 − 5.24947i 0.230015 − 0.631962i
\(70\) 0 0
\(71\) 7.10830 − 5.96457i 0.843600 − 0.707865i −0.114770 − 0.993392i \(-0.536613\pi\)
0.958371 + 0.285527i \(0.0921688\pi\)
\(72\) 0 0
\(73\) 16.2707 1.90434 0.952169 − 0.305571i \(-0.0988473\pi\)
0.952169 + 0.305571i \(0.0988473\pi\)
\(74\) 0 0
\(75\) 24.7757 2.86085
\(76\) 0 0
\(77\) −5.85619 + 4.91392i −0.667374 + 0.559993i
\(78\) 0 0
\(79\) 0.484006 − 1.32980i 0.0544549 − 0.149614i −0.909483 − 0.415742i \(-0.863522\pi\)
0.963938 + 0.266128i \(0.0857443\pi\)
\(80\) 0 0
\(81\) 9.69119 + 3.52730i 1.07680 + 0.391923i
\(82\) 0 0
\(83\) 0.294580 − 1.67065i 0.0323343 − 0.183377i −0.964363 − 0.264583i \(-0.914766\pi\)
0.996697 + 0.0812054i \(0.0258770\pi\)
\(84\) 0 0
\(85\) −0.0569682 + 0.0986718i −0.00617907 + 0.0107025i
\(86\) 0 0
\(87\) −10.6108 + 1.87098i −1.13760 + 0.200590i
\(88\) 0 0
\(89\) 2.82882 + 7.77213i 0.299855 + 0.823844i 0.994523 + 0.104514i \(0.0333288\pi\)
−0.694669 + 0.719330i \(0.744449\pi\)
\(90\) 0 0
\(91\) 10.2572 + 1.80861i 1.07524 + 0.189594i
\(92\) 0 0
\(93\) 8.17765 + 9.74574i 0.847983 + 1.01059i
\(94\) 0 0
\(95\) −12.9364 + 4.70847i −1.32725 + 0.483079i
\(96\) 0 0
\(97\) 5.65105 − 3.26264i 0.573777 − 0.331270i −0.184879 − 0.982761i \(-0.559189\pi\)
0.758657 + 0.651491i \(0.225856\pi\)
\(98\) 0 0
\(99\) 0.862818 + 0.723990i 0.0867164 + 0.0727637i
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.225.1 12
4.3 odd 2 74.2.h.a.3.2 ✓ 12
12.11 even 2 666.2.bj.c.595.1 12
37.25 even 18 inner 592.2.bq.b.321.1 12
148.79 even 36 2738.2.a.r.1.2 6
148.99 odd 18 74.2.h.a.25.2 yes 12
148.143 even 36 2738.2.a.s.1.1 6
444.395 even 18 666.2.bj.c.469.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.3.2 ✓ 12 4.3 odd 2
74.2.h.a.25.2 yes 12 148.99 odd 18
592.2.bq.b.225.1 12 1.1 even 1 trivial
592.2.bq.b.321.1 12 37.25 even 18 inner
666.2.bj.c.469.1 12 444.395 even 18
666.2.bj.c.595.1 12 12.11 even 2
2738.2.a.r.1.2 6 148.79 even 36
2738.2.a.s.1.1 6 148.143 even 36