Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(181,522)] 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("522.181"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.k (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,2,0,-2,0] 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: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 487.2
Root \(0.760453 + 3.33176i\) of defining polynomial
Character \(\chi\) \(=\) 522.487
Dual form 522.2.k.h.343.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.623490 + 0.781831i) q^{2} +(-0.222521 - 0.974928i) q^{4} +(1.85326 - 2.32392i) q^{5} +(-0.115912 + 0.507846i) q^{7} +(0.900969 + 0.433884i) q^{8} +(0.661422 + 2.89788i) q^{10} +(-0.585233 + 0.281833i) q^{11} +(-0.444717 + 0.214164i) q^{13} +(-0.324780 - 0.407261i) q^{14} +(-0.900969 + 0.433884i) q^{16} +7.42032 q^{17} +(-1.47532 - 6.46378i) q^{19} +(-2.67804 - 1.28968i) q^{20} +(0.144541 - 0.633273i) q^{22} +(-4.74970 - 5.95594i) q^{23} +(-0.853408 - 3.73902i) q^{25} +(0.109836 - 0.481223i) q^{26} +0.520906 q^{28} +(4.56917 + 2.85003i) q^{29} +(3.72875 - 4.67571i) q^{31} +(0.222521 - 0.974928i) q^{32} +(-4.62649 + 5.80144i) q^{34} +(0.965376 + 1.21054i) q^{35} +(2.23650 + 1.07704i) q^{37} +(5.97343 + 2.87665i) q^{38} +(2.67804 - 1.28968i) q^{40} -7.82245 q^{41} +(0.404994 + 0.507846i) q^{43} +(0.404994 + 0.507846i) q^{44} +7.61793 q^{46} +(7.92488 - 3.81642i) q^{47} +(6.06231 + 2.91945i) q^{49} +(3.45538 + 1.66402i) q^{50} +(0.307754 + 0.385911i) q^{52} +(0.717766 - 0.900050i) q^{53} +(-0.429633 + 1.88234i) q^{55} +(-0.324780 + 0.407261i) q^{56} +(-5.07708 + 1.79536i) q^{58} +5.31686 q^{59} +(-2.15779 + 9.45389i) q^{61} +(1.33078 + 5.83051i) q^{62} +(0.623490 + 0.781831i) q^{64} +(-0.326477 + 1.43039i) q^{65} +(-4.07320 - 1.96155i) q^{67} +(-1.65118 - 7.23427i) q^{68} -1.54834 q^{70} +(-12.7440 + 6.13719i) q^{71} +(5.44958 + 6.83356i) q^{73} +(-2.23650 + 1.07704i) q^{74} +(-5.97343 + 2.87665i) q^{76} +(-0.0752920 - 0.329876i) q^{77} +(-3.71962 - 1.79127i) q^{79} +(-0.661422 + 2.89788i) q^{80} +(4.87722 - 6.11584i) q^{82} +(0.952729 + 4.17418i) q^{83} +(13.7518 - 17.2442i) q^{85} -0.649559 q^{86} -0.649559 q^{88} +(0.743610 - 0.932457i) q^{89} +(-0.0572142 - 0.250672i) q^{91} +(-4.74970 + 5.95594i) q^{92} +(-1.95728 + 8.57542i) q^{94} +(-17.7555 - 8.55057i) q^{95} +(-2.61081 - 11.4387i) q^{97} +(-6.06231 + 2.91945i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 2 q^{4} + q^{7} + 2 q^{8} - 7 q^{10} + 2 q^{11} + q^{13} - q^{14} - 2 q^{16} + 12 q^{17} - 6 q^{19} - 7 q^{20} - 2 q^{22} - 35 q^{23} - 6 q^{25} - 8 q^{26} - 6 q^{28} + 14 q^{29} - 8 q^{31}+ \cdots - 37 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\)
\(\chi(n)\) \(e\left(\frac{5}{7}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.623490 + 0.781831i −0.440874 + 0.552838i
\(3\) 0 0
\(4\) −0.222521 − 0.974928i −0.111260 − 0.487464i
\(5\) 1.85326 − 2.32392i 0.828804 − 1.03929i −0.169747 − 0.985488i \(-0.554295\pi\)
0.998552 − 0.0538002i \(-0.0171334\pi\)
\(6\) 0 0
\(7\) −0.115912 + 0.507846i −0.0438108 + 0.191948i −0.992098 − 0.125466i \(-0.959957\pi\)
0.948287 + 0.317414i \(0.102814\pi\)
\(8\) 0.900969 + 0.433884i 0.318541 + 0.153401i
\(9\) 0 0
\(10\) 0.661422 + 2.89788i 0.209160 + 0.916390i
\(11\) −0.585233 + 0.281833i −0.176454 + 0.0849759i −0.520026 − 0.854151i \(-0.674078\pi\)
0.343571 + 0.939127i \(0.388363\pi\)
\(12\) 0 0
\(13\) −0.444717 + 0.214164i −0.123342 + 0.0593985i −0.494537 − 0.869156i \(-0.664662\pi\)
0.371195 + 0.928555i \(0.378948\pi\)
\(14\) −0.324780 − 0.407261i −0.0868010 − 0.108845i
\(15\) 0 0
\(16\) −0.900969 + 0.433884i −0.225242 + 0.108471i
\(17\) 7.42032 1.79969 0.899846 − 0.436208i \(-0.143679\pi\)
0.899846 + 0.436208i \(0.143679\pi\)
\(18\) 0 0
\(19\) −1.47532 − 6.46378i −0.338461 − 1.48289i −0.802272 − 0.596959i \(-0.796375\pi\)
0.463811 − 0.885934i \(-0.346482\pi\)
\(20\) −2.67804 − 1.28968i −0.598828 − 0.288381i
\(21\) 0 0
\(22\) 0.144541 − 0.633273i 0.0308161 − 0.135014i
\(23\) −4.74970 − 5.95594i −0.990381 − 1.24190i −0.970251 − 0.242101i \(-0.922163\pi\)
−0.0201303 − 0.999797i \(-0.506408\pi\)
\(24\) 0 0
\(25\) −0.853408 − 3.73902i −0.170682 − 0.747805i
\(26\) 0.109836 − 0.481223i 0.0215406 − 0.0943755i
\(27\) 0 0
\(28\) 0.520906 0.0984420
\(29\) 4.56917 + 2.85003i 0.848474 + 0.529237i
\(30\) 0 0
\(31\) 3.72875 − 4.67571i 0.669703 − 0.839782i −0.324657 − 0.945832i \(-0.605249\pi\)
0.994361 + 0.106050i \(0.0338204\pi\)
\(32\) 0.222521 − 0.974928i 0.0393365 − 0.172345i
\(33\) 0 0
\(34\) −4.62649 + 5.80144i −0.793437 + 0.994938i
\(35\) 0.965376 + 1.21054i 0.163178 + 0.204619i
\(36\) 0 0
\(37\) 2.23650 + 1.07704i 0.367678 + 0.177065i 0.608595 − 0.793481i \(-0.291734\pi\)
−0.240916 + 0.970546i \(0.577448\pi\)
\(38\) 5.97343 + 2.87665i 0.969019 + 0.466655i
\(39\) 0 0
\(40\) 2.67804 − 1.28968i 0.423436 − 0.203916i
\(41\) −7.82245 −1.22166 −0.610830 − 0.791761i \(-0.709164\pi\)
−0.610830 + 0.791761i \(0.709164\pi\)
\(42\) 0 0
\(43\) 0.404994 + 0.507846i 0.0617609 + 0.0774458i 0.811752 − 0.584002i \(-0.198514\pi\)
−0.749991 + 0.661447i \(0.769942\pi\)
\(44\) 0.404994 + 0.507846i 0.0610551 + 0.0765606i
\(45\) 0 0
\(46\) 7.61793 1.12320
\(47\) 7.92488 − 3.81642i 1.15596 − 0.556682i 0.245143 − 0.969487i \(-0.421165\pi\)
0.910819 + 0.412805i \(0.135451\pi\)
\(48\) 0 0
\(49\) 6.06231 + 2.91945i 0.866044 + 0.417065i
\(50\) 3.45538 + 1.66402i 0.488664 + 0.235328i
\(51\) 0 0
\(52\) 0.307754 + 0.385911i 0.0426777 + 0.0535162i
\(53\) 0.717766 − 0.900050i 0.0985927 − 0.123631i −0.730088 − 0.683354i \(-0.760521\pi\)
0.828680 + 0.559722i \(0.189092\pi\)
\(54\) 0 0
\(55\) −0.429633 + 1.88234i −0.0579317 + 0.253815i
\(56\) −0.324780 + 0.407261i −0.0434005 + 0.0544225i
\(57\) 0 0
\(58\) −5.07708 + 1.79536i −0.666653 + 0.235742i
\(59\) 5.31686 0.692196 0.346098 − 0.938198i \(-0.387507\pi\)
0.346098 + 0.938198i \(0.387507\pi\)
\(60\) 0 0
\(61\) −2.15779 + 9.45389i −0.276277 + 1.21045i 0.626184 + 0.779675i \(0.284616\pi\)
−0.902461 + 0.430772i \(0.858241\pi\)
\(62\) 1.33078 + 5.83051i 0.169009 + 0.740475i
\(63\) 0 0
\(64\) 0.623490 + 0.781831i 0.0779362 + 0.0977289i
\(65\) −0.326477 + 1.43039i −0.0404945 + 0.177418i
\(66\) 0 0
\(67\) −4.07320 − 1.96155i −0.497620 − 0.239641i 0.168201 − 0.985753i \(-0.446204\pi\)
−0.665821 + 0.746112i \(0.731919\pi\)
\(68\) −1.65118 − 7.23427i −0.200234 − 0.877285i
\(69\) 0 0
\(70\) −1.54834 −0.185062
\(71\) −12.7440 + 6.13719i −1.51243 + 0.728350i −0.992081 − 0.125601i \(-0.959914\pi\)
−0.520353 + 0.853951i \(0.674200\pi\)
\(72\) 0 0
\(73\) 5.44958 + 6.83356i 0.637825 + 0.799808i 0.990729 − 0.135852i \(-0.0433772\pi\)
−0.352904 + 0.935660i \(0.614806\pi\)
\(74\) −2.23650 + 1.07704i −0.259988 + 0.125204i
\(75\) 0 0
\(76\) −5.97343 + 2.87665i −0.685200 + 0.329975i
\(77\) −0.0752920 − 0.329876i −0.00858032 − 0.0375928i
\(78\) 0 0
\(79\) −3.71962 − 1.79127i −0.418490 − 0.201534i 0.212775 − 0.977101i \(-0.431750\pi\)
−0.631265 + 0.775567i \(0.717464\pi\)
\(80\) −0.661422 + 2.89788i −0.0739492 + 0.323993i
\(81\) 0 0
\(82\) 4.87722 − 6.11584i 0.538598 − 0.675381i
\(83\) 0.952729 + 4.17418i 0.104576 + 0.458176i 0.999918 + 0.0127974i \(0.00407365\pi\)
−0.895343 + 0.445378i \(0.853069\pi\)
\(84\) 0 0
\(85\) 13.7518 − 17.2442i 1.49159 − 1.87040i
\(86\) −0.649559 −0.0700438
\(87\) 0 0
\(88\) −0.649559 −0.0692432
\(89\) 0.743610 − 0.932457i 0.0788225 − 0.0988403i −0.740857 − 0.671663i \(-0.765580\pi\)
0.819679 + 0.572823i \(0.194152\pi\)
\(90\) 0 0
\(91\) −0.0572142 − 0.250672i −0.00599768 − 0.0262775i
\(92\) −4.74970 + 5.95594i −0.495191 + 0.620949i
\(93\) 0 0
\(94\) −1.95728 + 8.57542i −0.201878 + 0.884487i
\(95\) −17.7555 − 8.55057i −1.82167 − 0.877270i
\(96\) 0 0
\(97\) −2.61081 − 11.4387i −0.265088 − 1.16143i −0.915651 − 0.401974i \(-0.868324\pi\)
0.650563 − 0.759452i \(-0.274533\pi\)
\(98\) −6.06231 + 2.91945i −0.612386 + 0.294909i
\(99\) 0 0
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 522.2.k.h.487.2 12
3.2 odd 2 58.2.d.b.23.2 ✓ 12
12.11 even 2 464.2.u.h.81.1 12
29.24 even 7 inner 522.2.k.h.343.2 12
87.11 even 28 1682.2.b.i.1681.4 12
87.47 even 28 1682.2.b.i.1681.9 12
87.53 odd 14 58.2.d.b.53.2 yes 12
87.71 odd 14 1682.2.a.q.1.4 6
87.74 odd 14 1682.2.a.t.1.3 6
348.227 even 14 464.2.u.h.401.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.23.2 ✓ 12 3.2 odd 2
58.2.d.b.53.2 yes 12 87.53 odd 14
464.2.u.h.81.1 12 12.11 even 2
464.2.u.h.401.1 12 348.227 even 14
522.2.k.h.343.2 12 29.24 even 7 inner
522.2.k.h.487.2 12 1.1 even 1 trivial
1682.2.a.q.1.4 6 87.71 odd 14
1682.2.a.t.1.3 6 87.74 odd 14
1682.2.b.i.1681.4 12 87.11 even 28
1682.2.b.i.1681.9 12 87.47 even 28