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 337.2
Root \(-0.984808 + 0.173648i\) of defining polynomial
Character \(\chi\) \(=\) 592.337
Dual form 592.2.bq.b.65.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.266044 + 1.50881i) q^{3} +(-0.247315 - 0.294739i) q^{5} +(2.50048 - 2.09815i) q^{7} +(0.613341 - 0.223238i) q^{9} +(1.29236 - 2.23843i) q^{11} +(-0.466831 + 1.28261i) q^{13} +(0.378909 - 0.451566i) q^{15} +(-1.13965 - 3.13118i) q^{17} +(5.89485 - 1.03942i) q^{19} +(3.83095 + 3.21455i) q^{21} +(-6.53507 + 3.77303i) q^{23} +(0.842535 - 4.77825i) q^{25} +(2.79813 + 4.84651i) q^{27} +(2.78251 + 1.60649i) q^{29} -2.53737i q^{31} +(3.72120 + 1.35440i) q^{33} +(-1.23681 - 0.218083i) q^{35} +(0.543196 + 6.05846i) q^{37} +(-2.05941 - 0.363130i) q^{39} +(7.77046 + 2.82822i) q^{41} +4.33920i q^{43} +(-0.217486 - 0.125565i) q^{45} +(2.61455 + 4.52853i) q^{47} +(0.634616 - 3.59909i) q^{49} +(4.42116 - 2.55256i) q^{51} +(-6.64254 - 5.57375i) q^{53} +(-0.979373 + 0.172690i) q^{55} +(3.13658 + 8.61769i) q^{57} +(-8.33530 + 9.93362i) q^{59} +(-2.39847 + 6.58973i) q^{61} +(1.06526 - 1.84508i) q^{63} +(0.493489 - 0.179615i) q^{65} +(8.60881 - 7.22365i) q^{67} +(-7.43141 - 8.85641i) q^{69} +(-2.60464 - 14.7717i) q^{71} -15.0792 q^{73} +7.43364 q^{75} +(-1.46505 - 8.30870i) q^{77} +(-0.940587 - 1.12095i) q^{79} +(-5.06805 + 4.25260i) q^{81} +(-0.0104473 + 0.00380252i) q^{83} +(-0.641025 + 1.11029i) q^{85} +(-1.68361 + 4.62569i) q^{87} +(-0.612745 + 0.730241i) q^{89} +(1.52380 + 4.18661i) q^{91} +(3.82842 - 0.675054i) q^{93} +(-1.76424 - 1.48038i) q^{95} +(12.8332 - 7.40927i) q^{97} +(0.292954 - 1.66142i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 12 q^{7} - 6 q^{9} + 6 q^{11} + 6 q^{13} + 18 q^{19} - 6 q^{21} - 18 q^{25} + 6 q^{27} + 18 q^{29} - 6 q^{33} - 18 q^{35} + 30 q^{37} - 30 q^{39} + 24 q^{41} - 18 q^{45} - 6 q^{47} + 12 q^{49}+ \cdots - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.266044 + 1.50881i 0.153601 + 0.871114i 0.960054 + 0.279815i \(0.0902733\pi\)
−0.806453 + 0.591298i \(0.798616\pi\)
\(4\) 0 0
\(5\) −0.247315 − 0.294739i −0.110603 − 0.131811i 0.707903 − 0.706310i \(-0.249641\pi\)
−0.818506 + 0.574499i \(0.805197\pi\)
\(6\) 0 0
\(7\) 2.50048 − 2.09815i 0.945091 − 0.793026i −0.0333729 − 0.999443i \(-0.510625\pi\)
0.978464 + 0.206417i \(0.0661805\pi\)
\(8\) 0 0
\(9\) 0.613341 − 0.223238i 0.204447 − 0.0744126i
\(10\) 0 0
\(11\) 1.29236 − 2.23843i 0.389661 − 0.674912i −0.602743 − 0.797935i \(-0.705926\pi\)
0.992404 + 0.123023i \(0.0392590\pi\)
\(12\) 0 0
\(13\) −0.466831 + 1.28261i −0.129476 + 0.355731i −0.987444 − 0.157972i \(-0.949504\pi\)
0.857968 + 0.513703i \(0.171727\pi\)
\(14\) 0 0
\(15\) 0.378909 − 0.451566i 0.0978339 − 0.116594i
\(16\) 0 0
\(17\) −1.13965 − 3.13118i −0.276407 − 0.759422i −0.997763 − 0.0668568i \(-0.978703\pi\)
0.721356 − 0.692565i \(-0.243519\pi\)
\(18\) 0 0
\(19\) 5.89485 − 1.03942i 1.35237 − 0.238459i 0.549939 − 0.835205i \(-0.314651\pi\)
0.802431 + 0.596745i \(0.203540\pi\)
\(20\) 0 0
\(21\) 3.83095 + 3.21455i 0.835982 + 0.701472i
\(22\) 0 0
\(23\) −6.53507 + 3.77303i −1.36266 + 0.786730i −0.989977 − 0.141230i \(-0.954894\pi\)
−0.372680 + 0.927960i \(0.621561\pi\)
\(24\) 0 0
\(25\) 0.842535 − 4.77825i 0.168507 − 0.955650i
\(26\) 0 0
\(27\) 2.79813 + 4.84651i 0.538501 + 0.932711i
\(28\) 0 0
\(29\) 2.78251 + 1.60649i 0.516700 + 0.298317i 0.735583 − 0.677434i \(-0.236908\pi\)
−0.218883 + 0.975751i \(0.570241\pi\)
\(30\) 0 0
\(31\) − 2.53737i − 0.455726i −0.973693 − 0.227863i \(-0.926826\pi\)
0.973693 − 0.227863i \(-0.0731738\pi\)
\(32\) 0 0
\(33\) 3.72120 + 1.35440i 0.647777 + 0.235772i
\(34\) 0 0
\(35\) −1.23681 − 0.218083i −0.209059 − 0.0368628i
\(36\) 0 0
\(37\) 0.543196 + 6.05846i 0.0893009 + 0.996005i
\(38\) 0 0
\(39\) −2.05941 − 0.363130i −0.329770 − 0.0581473i
\(40\) 0 0
\(41\) 7.77046 + 2.82822i 1.21354 + 0.441693i 0.867931 − 0.496685i \(-0.165449\pi\)
0.345611 + 0.938378i \(0.387672\pi\)
\(42\) 0 0
\(43\) 4.33920i 0.661722i 0.943680 + 0.330861i \(0.107339\pi\)
−0.943680 + 0.330861i \(0.892661\pi\)
\(44\) 0 0
\(45\) −0.217486 − 0.125565i −0.0324208 − 0.0187182i
\(46\) 0 0
\(47\) 2.61455 + 4.52853i 0.381371 + 0.660554i 0.991258 − 0.131934i \(-0.0421188\pi\)
−0.609888 + 0.792488i \(0.708785\pi\)
\(48\) 0 0
\(49\) 0.634616 − 3.59909i 0.0906594 − 0.514155i
\(50\) 0 0
\(51\) 4.42116 − 2.55256i 0.619086 − 0.357430i
\(52\) 0 0
\(53\) −6.64254 − 5.57375i −0.912423 − 0.765614i 0.0601551 − 0.998189i \(-0.480840\pi\)
−0.972578 + 0.232575i \(0.925285\pi\)
\(54\) 0 0
\(55\) −0.979373 + 0.172690i −0.132059 + 0.0232855i
\(56\) 0 0
\(57\) 3.13658 + 8.61769i 0.415450 + 1.14144i
\(58\) 0 0
\(59\) −8.33530 + 9.93362i −1.08516 + 1.29325i −0.131849 + 0.991270i \(0.542091\pi\)
−0.953315 + 0.301978i \(0.902353\pi\)
\(60\) 0 0
\(61\) −2.39847 + 6.58973i −0.307092 + 0.843728i 0.686128 + 0.727481i \(0.259309\pi\)
−0.993220 + 0.116248i \(0.962913\pi\)
\(62\) 0 0
\(63\) 1.06526 − 1.84508i 0.134210 − 0.232458i
\(64\) 0 0
\(65\) 0.493489 − 0.179615i 0.0612098 − 0.0222785i
\(66\) 0 0
\(67\) 8.60881 − 7.22365i 1.05173 − 0.882509i 0.0584586 − 0.998290i \(-0.481381\pi\)
0.993275 + 0.115781i \(0.0369370\pi\)
\(68\) 0 0
\(69\) −7.43141 − 8.85641i −0.894636 − 1.06619i
\(70\) 0 0
\(71\) −2.60464 − 14.7717i −0.309114 − 1.75307i −0.603477 − 0.797380i \(-0.706219\pi\)
0.294363 − 0.955694i \(-0.404893\pi\)
\(72\) 0 0
\(73\) −15.0792 −1.76489 −0.882445 − 0.470416i \(-0.844104\pi\)
−0.882445 + 0.470416i \(0.844104\pi\)
\(74\) 0 0
\(75\) 7.43364 0.858363
\(76\) 0 0
\(77\) −1.46505 − 8.30870i −0.166958 − 0.946864i
\(78\) 0 0
\(79\) −0.940587 − 1.12095i −0.105824 − 0.126117i 0.710532 − 0.703664i \(-0.248454\pi\)
−0.816357 + 0.577548i \(0.804010\pi\)
\(80\) 0 0
\(81\) −5.06805 + 4.25260i −0.563116 + 0.472511i
\(82\) 0 0
\(83\) −0.0104473 + 0.00380252i −0.00114674 + 0.000417381i −0.342593 − 0.939484i \(-0.611305\pi\)
0.341447 + 0.939901i \(0.389083\pi\)
\(84\) 0 0
\(85\) −0.641025 + 1.11029i −0.0695290 + 0.120428i
\(86\) 0 0
\(87\) −1.68361 + 4.62569i −0.180502 + 0.495926i
\(88\) 0 0
\(89\) −0.612745 + 0.730241i −0.0649509 + 0.0774054i −0.797542 − 0.603264i \(-0.793867\pi\)
0.732591 + 0.680669i \(0.238311\pi\)
\(90\) 0 0
\(91\) 1.52380 + 4.18661i 0.159738 + 0.438876i
\(92\) 0 0
\(93\) 3.82842 − 0.675054i 0.396989 − 0.0699998i
\(94\) 0 0
\(95\) −1.76424 − 1.48038i −0.181008 − 0.151883i
\(96\) 0 0
\(97\) 12.8332 − 7.40927i 1.30302 − 0.752297i 0.322097 − 0.946707i \(-0.395612\pi\)
0.980920 + 0.194410i \(0.0622791\pi\)
\(98\) 0 0
\(99\) 0.292954 − 1.66142i 0.0294430 − 0.166979i
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.337.2 12
4.3 odd 2 74.2.h.a.41.2 ✓ 12
12.11 even 2 666.2.bj.c.559.1 12
37.28 even 18 inner 592.2.bq.b.65.2 12
148.19 even 36 2738.2.a.s.1.6 6
148.55 even 36 2738.2.a.r.1.5 6
148.139 odd 18 74.2.h.a.65.2 yes 12
444.287 even 18 666.2.bj.c.361.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.41.2 ✓ 12 4.3 odd 2
74.2.h.a.65.2 yes 12 148.139 odd 18
592.2.bq.b.65.2 12 37.28 even 18 inner
592.2.bq.b.337.2 12 1.1 even 1 trivial
666.2.bj.c.361.1 12 444.287 even 18
666.2.bj.c.559.1 12 12.11 even 2
2738.2.a.r.1.5 6 148.55 even 36
2738.2.a.s.1.6 6 148.19 even 36