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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 289.2
Root \(0.342020 - 0.939693i\) of defining polynomial
Character \(\chi\) \(=\) 592.289
Dual form 592.2.bq.b.465.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.326352 - 0.118782i) q^{3} +(2.57176 - 0.453471i) q^{5} +(0.361075 + 2.04776i) q^{7} +(-2.20574 - 1.85083i) q^{9} +(2.99810 - 5.19285i) q^{11} +(2.64632 + 3.15377i) q^{13} +(-0.893164 - 0.157489i) q^{15} +(-0.618710 + 0.737350i) q^{17} +(-0.534946 + 1.46975i) q^{19} +(0.125400 - 0.711179i) q^{21} +(5.51705 - 3.18527i) q^{23} +(1.70986 - 0.622339i) q^{25} +(1.02094 + 1.76833i) q^{27} +(-3.51193 - 2.02761i) q^{29} +3.39997i q^{31} +(-1.59525 + 1.33858i) q^{33} +(1.85720 + 5.10261i) q^{35} +(6.07068 - 0.383130i) q^{37} +(-0.489021 - 1.34357i) q^{39} +(7.94502 - 6.66666i) q^{41} +3.76932i q^{43} +(-6.51193 - 3.75967i) q^{45} +(-3.08750 - 5.34771i) q^{47} +(2.51491 - 0.915354i) q^{49} +(0.289501 - 0.167144i) q^{51} +(-1.39401 + 7.90585i) q^{53} +(5.35558 - 14.7143i) q^{55} +(0.349161 - 0.416114i) q^{57} +(-5.02269 - 0.885636i) q^{59} +(-6.25519 - 7.45465i) q^{61} +(2.99362 - 5.18510i) q^{63} +(8.23586 + 6.91071i) q^{65} +(1.83263 + 10.3934i) q^{67} +(-2.17885 + 0.384190i) q^{69} +(10.1503 + 3.69442i) q^{71} +3.55293 q^{73} -0.631940 q^{75} +(11.7162 + 4.26436i) q^{77} +(-2.51098 + 0.442753i) q^{79} +(1.37686 + 7.80856i) q^{81} +(-5.29798 - 4.44553i) q^{83} +(-1.25681 + 2.17686i) q^{85} +(0.905280 + 1.07887i) q^{87} +(-16.0165 - 2.82414i) q^{89} +(-5.50263 + 6.55778i) q^{91} +(0.403856 - 1.10959i) q^{93} +(-0.709264 + 4.02243i) q^{95} +(-14.1175 + 8.15074i) q^{97} +(-16.2241 + 5.90509i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 12 q^{7} - 6 q^{9} + 6 q^{11} + 6 q^{13} + 18 q^{19} - 6 q^{21} - 18 q^{25} + 6 q^{27} + 18 q^{29} - 6 q^{33} - 18 q^{35} + 30 q^{37} - 30 q^{39} + 24 q^{41} - 18 q^{45} - 6 q^{47} + 12 q^{49}+ \cdots - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\) \(1\) \(1\)

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
\(3\) −0.326352 − 0.118782i −0.188419 − 0.0685790i 0.246087 − 0.969248i \(-0.420855\pi\)
−0.434507 + 0.900669i \(0.643077\pi\)
\(4\) 0 0
\(5\) 2.57176 − 0.453471i 1.15013 − 0.202798i 0.434096 − 0.900867i \(-0.357068\pi\)
0.716031 + 0.698068i \(0.245957\pi\)
\(6\) 0 0
\(7\) 0.361075 + 2.04776i 0.136473 + 0.773979i 0.973822 + 0.227311i \(0.0729932\pi\)
−0.837349 + 0.546669i \(0.815896\pi\)
\(8\) 0 0
\(9\) −2.20574 − 1.85083i −0.735246 − 0.616944i
\(10\) 0 0
\(11\) 2.99810 − 5.19285i 0.903960 − 1.56570i 0.0816522 − 0.996661i \(-0.473980\pi\)
0.822308 − 0.569043i \(-0.192686\pi\)
\(12\) 0 0
\(13\) 2.64632 + 3.15377i 0.733958 + 0.874697i 0.995907 − 0.0903843i \(-0.0288095\pi\)
−0.261949 + 0.965082i \(0.584365\pi\)
\(14\) 0 0
\(15\) −0.893164 − 0.157489i −0.230614 − 0.0406634i
\(16\) 0 0
\(17\) −0.618710 + 0.737350i −0.150059 + 0.178834i −0.835838 − 0.548977i \(-0.815018\pi\)
0.685778 + 0.727810i \(0.259462\pi\)
\(18\) 0 0
\(19\) −0.534946 + 1.46975i −0.122725 + 0.337184i −0.985808 − 0.167878i \(-0.946308\pi\)
0.863083 + 0.505063i \(0.168531\pi\)
\(20\) 0 0
\(21\) 0.125400 − 0.711179i 0.0273645 − 0.155192i
\(22\) 0 0
\(23\) 5.51705 − 3.18527i 1.15038 − 0.664174i 0.201403 − 0.979508i \(-0.435450\pi\)
0.948981 + 0.315334i \(0.102117\pi\)
\(24\) 0 0
\(25\) 1.70986 − 0.622339i 0.341973 − 0.124468i
\(26\) 0 0
\(27\) 1.02094 + 1.76833i 0.196481 + 0.340315i
\(28\) 0 0
\(29\) −3.51193 − 2.02761i −0.652149 − 0.376519i 0.137130 − 0.990553i \(-0.456212\pi\)
−0.789279 + 0.614035i \(0.789546\pi\)
\(30\) 0 0
\(31\) 3.39997i 0.610653i 0.952248 + 0.305326i \(0.0987655\pi\)
−0.952248 + 0.305326i \(0.901234\pi\)
\(32\) 0 0
\(33\) −1.59525 + 1.33858i −0.277698 + 0.233016i
\(34\) 0 0
\(35\) 1.85720 + 5.10261i 0.313924 + 0.862498i
\(36\) 0 0
\(37\) 6.07068 − 0.383130i 0.998014 − 0.0629862i
\(38\) 0 0
\(39\) −0.489021 − 1.34357i −0.0783060 − 0.215144i
\(40\) 0 0
\(41\) 7.94502 − 6.66666i 1.24080 − 1.04116i 0.243343 − 0.969940i \(-0.421756\pi\)
0.997461 − 0.0712179i \(-0.0226886\pi\)
\(42\) 0 0
\(43\) 3.76932i 0.574816i 0.957808 + 0.287408i \(0.0927936\pi\)
−0.957808 + 0.287408i \(0.907206\pi\)
\(44\) 0 0
\(45\) −6.51193 − 3.75967i −0.970741 − 0.560458i
\(46\) 0 0
\(47\) −3.08750 − 5.34771i −0.450359 − 0.780044i 0.548050 − 0.836446i \(-0.315371\pi\)
−0.998408 + 0.0564019i \(0.982037\pi\)
\(48\) 0 0
\(49\) 2.51491 − 0.915354i 0.359273 − 0.130765i
\(50\) 0 0
\(51\) 0.289501 − 0.167144i 0.0405383 − 0.0234048i
\(52\) 0 0
\(53\) −1.39401 + 7.90585i −0.191483 + 1.08595i 0.725857 + 0.687846i \(0.241444\pi\)
−0.917339 + 0.398106i \(0.869668\pi\)
\(54\) 0 0
\(55\) 5.35558 − 14.7143i 0.722146 − 1.98408i
\(56\) 0 0
\(57\) 0.349161 − 0.416114i 0.0462475 − 0.0551156i
\(58\) 0 0
\(59\) −5.02269 − 0.885636i −0.653899 − 0.115300i −0.163150 − 0.986601i \(-0.552166\pi\)
−0.490749 + 0.871301i \(0.663277\pi\)
\(60\) 0 0
\(61\) −6.25519 − 7.45465i −0.800895 − 0.954470i 0.198778 − 0.980045i \(-0.436303\pi\)
−0.999673 + 0.0255750i \(0.991858\pi\)
\(62\) 0 0
\(63\) 2.99362 − 5.18510i 0.377161 − 0.653262i
\(64\) 0 0
\(65\) 8.23586 + 6.91071i 1.02153 + 0.857168i
\(66\) 0 0
\(67\) 1.83263 + 10.3934i 0.223892 + 1.26975i 0.864792 + 0.502130i \(0.167450\pi\)
−0.640901 + 0.767624i \(0.721439\pi\)
\(68\) 0 0
\(69\) −2.17885 + 0.384190i −0.262303 + 0.0462511i
\(70\) 0 0
\(71\) 10.1503 + 3.69442i 1.20462 + 0.438447i 0.864835 − 0.502056i \(-0.167423\pi\)
0.339787 + 0.940502i \(0.389645\pi\)
\(72\) 0 0
\(73\) 3.55293 0.415839 0.207920 − 0.978146i \(-0.433331\pi\)
0.207920 + 0.978146i \(0.433331\pi\)
\(74\) 0 0
\(75\) −0.631940 −0.0729701
\(76\) 0 0
\(77\) 11.7162 + 4.26436i 1.33519 + 0.485969i
\(78\) 0 0
\(79\) −2.51098 + 0.442753i −0.282507 + 0.0498136i −0.313106 − 0.949718i \(-0.601370\pi\)
0.0305991 + 0.999532i \(0.490258\pi\)
\(80\) 0 0
\(81\) 1.37686 + 7.80856i 0.152984 + 0.867617i
\(82\) 0 0
\(83\) −5.29798 − 4.44553i −0.581529 − 0.487960i 0.303920 − 0.952698i \(-0.401704\pi\)
−0.885449 + 0.464737i \(0.846149\pi\)
\(84\) 0 0
\(85\) −1.25681 + 2.17686i −0.136320 + 0.236113i
\(86\) 0 0
\(87\) 0.905280 + 1.07887i 0.0970562 + 0.115667i
\(88\) 0 0
\(89\) −16.0165 − 2.82414i −1.69774 − 0.299358i −0.760838 − 0.648942i \(-0.775212\pi\)
−0.936905 + 0.349584i \(0.886323\pi\)
\(90\) 0 0
\(91\) −5.50263 + 6.55778i −0.576832 + 0.687442i
\(92\) 0 0
\(93\) 0.403856 − 1.10959i 0.0418780 − 0.115059i
\(94\) 0 0
\(95\) −0.709264 + 4.02243i −0.0727689 + 0.412693i
\(96\) 0 0
\(97\) −14.1175 + 8.15074i −1.43342 + 0.827583i −0.997380 − 0.0723469i \(-0.976951\pi\)
−0.436036 + 0.899929i \(0.643618\pi\)
\(98\) 0 0
\(99\) −16.2241 + 5.90509i −1.63058 + 0.593484i
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.289.2 12
4.3 odd 2 74.2.h.a.67.1 yes 12
12.11 even 2 666.2.bj.c.289.2 12
37.21 even 18 inner 592.2.bq.b.465.2 12
148.87 even 36 2738.2.a.s.1.4 6
148.95 odd 18 74.2.h.a.21.1 ✓ 12
148.135 even 36 2738.2.a.r.1.3 6
444.95 even 18 666.2.bj.c.613.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.21.1 ✓ 12 148.95 odd 18
74.2.h.a.67.1 yes 12 4.3 odd 2
592.2.bq.b.289.2 12 1.1 even 1 trivial
592.2.bq.b.465.2 12 37.21 even 18 inner
666.2.bj.c.289.2 12 12.11 even 2
666.2.bj.c.613.2 12 444.95 even 18
2738.2.a.r.1.3 6 148.135 even 36
2738.2.a.s.1.4 6 148.87 even 36