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(175,522)] 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("522.175"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-6,2,-6,-2] 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: \(6\) over \(\Q(\zeta_{3})\)
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: \( x^{12} - 4 x^{11} + 6 x^{10} - 10 x^{9} + 22 x^{8} - 18 x^{7} - 3 x^{6} - 54 x^{5} + 198 x^{4} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 175.5
Root \(-0.452567 + 1.67188i\) of defining polynomial
Character \(\chi\) \(=\) 522.175
Dual form 522.2.e.h.349.5

$q$-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+(-0.500000 - 0.866025i) q^{2} +(1.22161 - 1.22787i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(1.58186 - 2.73987i) q^{5} +(-1.67417 - 0.444006i) q^{6} +(0.567358 + 0.982693i) q^{7} +1.00000 q^{8} +(-0.0153503 - 2.99996i) q^{9} -3.16372 q^{10} +(2.89578 + 5.01564i) q^{11} +(0.452567 + 1.67188i) q^{12} +(2.82732 - 4.89707i) q^{13} +(0.567358 - 0.982693i) q^{14} +(-1.43180 - 5.28937i) q^{15} +(-0.500000 - 0.866025i) q^{16} +2.86528 q^{17} +(-2.59037 + 1.51327i) q^{18} -6.57624 q^{19} +(1.58186 + 2.73987i) q^{20} +(1.89971 + 0.503821i) q^{21} +(2.89578 - 5.01564i) q^{22} +(-1.31392 + 2.27578i) q^{23} +(1.22161 - 1.22787i) q^{24} +(-2.50457 - 4.33805i) q^{25} -5.65465 q^{26} +(-3.70233 - 3.64593i) q^{27} -1.13472 q^{28} +(0.500000 + 0.866025i) q^{29} +(-3.86483 + 3.88466i) q^{30} +(-3.00308 + 5.20149i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(9.69608 + 2.57149i) q^{33} +(-1.43264 - 2.48141i) q^{34} +3.58993 q^{35} +(2.60572 + 1.48669i) q^{36} -0.114074 q^{37} +(3.28812 + 5.69519i) q^{38} +(-2.55911 - 9.45390i) q^{39} +(1.58186 - 2.73987i) q^{40} +(-5.92628 + 10.2646i) q^{41} +(-0.513535 - 1.89711i) q^{42} +(-0.364185 - 0.630787i) q^{43} -5.79156 q^{44} +(-8.24377 - 4.70347i) q^{45} +2.62784 q^{46} +(-5.15230 - 8.92405i) q^{47} +(-1.67417 - 0.444006i) q^{48} +(2.85621 - 4.94710i) q^{49} +(-2.50457 + 4.33805i) q^{50} +(3.50025 - 3.51821i) q^{51} +(2.82732 + 4.89707i) q^{52} +7.56709 q^{53} +(-1.30630 + 5.02927i) q^{54} +18.3229 q^{55} +(0.567358 + 0.982693i) q^{56} +(-8.03358 + 8.07479i) q^{57} +(0.500000 - 0.866025i) q^{58} +(-3.05828 + 5.29710i) q^{59} +(5.29662 + 1.40471i) q^{60} +(-2.82424 - 4.89173i) q^{61} +6.00617 q^{62} +(2.93933 - 1.71714i) q^{63} +1.00000 q^{64} +(-8.94488 - 15.4930i) q^{65} +(-2.62107 - 9.68280i) q^{66} +(-2.86653 + 4.96498i) q^{67} +(-1.43264 + 2.48141i) q^{68} +(1.18927 + 4.39343i) q^{69} +(-1.79496 - 3.10897i) q^{70} -5.36536 q^{71} +(-0.0153503 - 2.99996i) q^{72} -10.1677 q^{73} +(0.0570372 + 0.0987914i) q^{74} +(-8.38619 - 2.22409i) q^{75} +(3.28812 - 5.69519i) q^{76} +(-3.28589 + 5.69133i) q^{77} +(-6.90776 + 6.94320i) q^{78} +(5.62873 + 9.74924i) q^{79} -3.16372 q^{80} +(-8.99953 + 0.0921003i) q^{81} +11.8526 q^{82} +(3.99660 + 6.92231i) q^{83} +(-1.38618 + 1.39329i) q^{84} +(4.53248 - 7.85049i) q^{85} +(-0.364185 + 0.630787i) q^{86} +(1.67417 + 0.444006i) q^{87} +(2.89578 + 5.01564i) q^{88} +11.5511 q^{89} +(0.0485640 + 9.49105i) q^{90} +6.41642 q^{91} +(-1.31392 - 2.27578i) q^{92} +(2.71819 + 10.0416i) q^{93} +(-5.15230 + 8.92405i) q^{94} +(-10.4027 + 18.0180i) q^{95} +(0.452567 + 1.67188i) q^{96} +(-3.15563 - 5.46572i) q^{97} -5.71242 q^{98} +(15.0023 - 8.76422i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 2 q^{3} - 6 q^{4} - 2 q^{5} + 2 q^{6} + 8 q^{7} + 12 q^{8} - 8 q^{9} + 4 q^{10} - 4 q^{12} - 6 q^{13} + 8 q^{14} + 14 q^{15} - 6 q^{16} + 32 q^{17} + 4 q^{18} + 4 q^{19} - 2 q^{20} - 10 q^{21}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(379\) \(407\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) 1.22161 − 1.22787i 0.705295 − 0.708914i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 1.58186 − 2.73987i 0.707430 − 1.22530i −0.258377 − 0.966044i \(-0.583188\pi\)
0.965807 − 0.259261i \(-0.0834789\pi\)
\(6\) −1.67417 − 0.444006i −0.683479 − 0.181265i
\(7\) 0.567358 + 0.982693i 0.214441 + 0.371423i 0.953100 − 0.302657i \(-0.0978736\pi\)
−0.738658 + 0.674080i \(0.764540\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.0153503 − 2.99996i −0.00511675 − 0.999987i
\(10\) −3.16372 −1.00046
\(11\) 2.89578 + 5.01564i 0.873111 + 1.51227i 0.858761 + 0.512376i \(0.171234\pi\)
0.0143495 + 0.999897i \(0.495432\pi\)
\(12\) 0.452567 + 1.67188i 0.130645 + 0.482630i
\(13\) 2.82732 − 4.89707i 0.784159 − 1.35820i −0.145341 − 0.989382i \(-0.546428\pi\)
0.929500 − 0.368821i \(-0.120239\pi\)
\(14\) 0.567358 − 0.982693i 0.151633 − 0.262636i
\(15\) −1.43180 − 5.28937i −0.369688 − 1.36571i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 2.86528 0.694933 0.347467 − 0.937692i \(-0.387042\pi\)
0.347467 + 0.937692i \(0.387042\pi\)
\(18\) −2.59037 + 1.51327i −0.610555 + 0.356682i
\(19\) −6.57624 −1.50869 −0.754346 − 0.656477i \(-0.772046\pi\)
−0.754346 + 0.656477i \(0.772046\pi\)
\(20\) 1.58186 + 2.73987i 0.353715 + 0.612652i
\(21\) 1.89971 + 0.503821i 0.414551 + 0.109943i
\(22\) 2.89578 − 5.01564i 0.617383 − 1.06934i
\(23\) −1.31392 + 2.27578i −0.273971 + 0.474532i −0.969875 − 0.243603i \(-0.921671\pi\)
0.695904 + 0.718135i \(0.255004\pi\)
\(24\) 1.22161 − 1.22787i 0.249360 − 0.250639i
\(25\) −2.50457 − 4.33805i −0.500915 − 0.867610i
\(26\) −5.65465 −1.10897
\(27\) −3.70233 − 3.64593i −0.712513 − 0.701659i
\(28\) −1.13472 −0.214441
\(29\) 0.500000 + 0.866025i 0.0928477 + 0.160817i
\(30\) −3.86483 + 3.88466i −0.705618 + 0.709238i
\(31\) −3.00308 + 5.20149i −0.539370 + 0.934216i 0.459568 + 0.888142i \(0.348004\pi\)
−0.998938 + 0.0460735i \(0.985329\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 9.69608 + 2.57149i 1.68787 + 0.447639i
\(34\) −1.43264 − 2.48141i −0.245696 − 0.425558i
\(35\) 3.58993 0.606809
\(36\) 2.60572 + 1.48669i 0.434286 + 0.247781i
\(37\) −0.114074 −0.0187537 −0.00937686 − 0.999956i \(-0.502985\pi\)
−0.00937686 + 0.999956i \(0.502985\pi\)
\(38\) 3.28812 + 5.69519i 0.533403 + 0.923882i
\(39\) −2.55911 − 9.45390i −0.409785 − 1.51384i
\(40\) 1.58186 − 2.73987i 0.250114 − 0.433211i
\(41\) −5.92628 + 10.2646i −0.925529 + 1.60306i −0.134822 + 0.990870i \(0.543046\pi\)
−0.790708 + 0.612194i \(0.790287\pi\)
\(42\) −0.513535 − 1.89711i −0.0792401 − 0.292730i
\(43\) −0.364185 − 0.630787i −0.0555377 − 0.0961942i 0.836920 − 0.547325i \(-0.184354\pi\)
−0.892458 + 0.451131i \(0.851021\pi\)
\(44\) −5.79156 −0.873111
\(45\) −8.24377 − 4.70347i −1.22891 − 0.701151i
\(46\) 2.62784 0.387454
\(47\) −5.15230 − 8.92405i −0.751541 − 1.30171i −0.947076 − 0.321010i \(-0.895978\pi\)
0.195535 − 0.980697i \(-0.437356\pi\)
\(48\) −1.67417 − 0.444006i −0.241646 − 0.0640867i
\(49\) 2.85621 − 4.94710i 0.408030 − 0.706729i
\(50\) −2.50457 + 4.33805i −0.354200 + 0.613493i
\(51\) 3.50025 − 3.51821i 0.490133 − 0.492648i
\(52\) 2.82732 + 4.89707i 0.392079 + 0.679101i
\(53\) 7.56709 1.03942 0.519710 − 0.854343i \(-0.326040\pi\)
0.519710 + 0.854343i \(0.326040\pi\)
\(54\) −1.30630 + 5.02927i −0.177765 + 0.684397i
\(55\) 18.3229 2.47066
\(56\) 0.567358 + 0.982693i 0.0758164 + 0.131318i
\(57\) −8.03358 + 8.07479i −1.06407 + 1.06953i
\(58\) 0.500000 − 0.866025i 0.0656532 − 0.113715i
\(59\) −3.05828 + 5.29710i −0.398155 + 0.689624i −0.993498 − 0.113847i \(-0.963683\pi\)
0.595344 + 0.803471i \(0.297016\pi\)
\(60\) 5.29662 + 1.40471i 0.683791 + 0.181348i
\(61\) −2.82424 − 4.89173i −0.361607 − 0.626322i 0.626619 − 0.779326i \(-0.284438\pi\)
−0.988225 + 0.153005i \(0.951105\pi\)
\(62\) 6.00617 0.762784
\(63\) 2.93933 − 1.71714i 0.370321 − 0.216339i
\(64\) 1.00000 0.125000
\(65\) −8.94488 − 15.4930i −1.10948 − 1.92167i
\(66\) −2.62107 − 9.68280i −0.322631 − 1.19187i
\(67\) −2.86653 + 4.96498i −0.350202 + 0.606568i −0.986285 − 0.165053i \(-0.947221\pi\)
0.636082 + 0.771621i \(0.280554\pi\)
\(68\) −1.43264 + 2.48141i −0.173733 + 0.300915i
\(69\) 1.18927 + 4.39343i 0.143172 + 0.528907i
\(70\) −1.79496 − 3.10897i −0.214539 − 0.371593i
\(71\) −5.36536 −0.636751 −0.318375 − 0.947965i \(-0.603137\pi\)
−0.318375 + 0.947965i \(0.603137\pi\)
\(72\) −0.0153503 − 2.99996i −0.00180904 − 0.353549i
\(73\) −10.1677 −1.19004 −0.595019 − 0.803711i \(-0.702856\pi\)
−0.595019 + 0.803711i \(0.702856\pi\)
\(74\) 0.0570372 + 0.0987914i 0.00663044 + 0.0114843i
\(75\) −8.38619 − 2.22409i −0.968353 − 0.256816i
\(76\) 3.28812 − 5.69519i 0.377173 − 0.653283i
\(77\) −3.28589 + 5.69133i −0.374462 + 0.648587i
\(78\) −6.90776 + 6.94320i −0.782150 + 0.786162i
\(79\) 5.62873 + 9.74924i 0.633281 + 1.09688i 0.986877 + 0.161477i \(0.0516257\pi\)
−0.353595 + 0.935398i \(0.615041\pi\)
\(80\) −3.16372 −0.353715
\(81\) −8.99953 + 0.0921003i −0.999948 + 0.0102334i
\(82\) 11.8526 1.30890
\(83\) 3.99660 + 6.92231i 0.438684 + 0.759822i 0.997588 − 0.0694095i \(-0.0221115\pi\)
−0.558905 + 0.829232i \(0.688778\pi\)
\(84\) −1.38618 + 1.39329i −0.151244 + 0.152020i
\(85\) 4.53248 − 7.85049i 0.491617 − 0.851505i
\(86\) −0.364185 + 0.630787i −0.0392711 + 0.0680195i
\(87\) 1.67417 + 0.444006i 0.179490 + 0.0476024i
\(88\) 2.89578 + 5.01564i 0.308691 + 0.534669i
\(89\) 11.5511 1.22442 0.612209 − 0.790696i \(-0.290281\pi\)
0.612209 + 0.790696i \(0.290281\pi\)
\(90\) 0.0485640 + 9.49105i 0.00511909 + 1.00044i
\(91\) 6.41642 0.672624
\(92\) −1.31392 − 2.27578i −0.136986 − 0.237266i
\(93\) 2.71819 + 10.0416i 0.281863 + 1.04126i
\(94\) −5.15230 + 8.92405i −0.531420 + 0.920446i
\(95\) −10.4027 + 18.0180i −1.06729 + 1.84861i
\(96\) 0.452567 + 1.67188i 0.0461899 + 0.170636i
\(97\) −3.15563 − 5.46572i −0.320406 − 0.554959i 0.660166 − 0.751120i \(-0.270486\pi\)
−0.980572 + 0.196161i \(0.937153\pi\)
\(98\) −5.71242 −0.577041
\(99\) 15.0023 − 8.76422i 1.50779 − 0.880837i
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.e.h.175.5 ✓ 12
3.2 odd 2 1566.2.e.h.523.1 12
9.2 odd 6 1566.2.e.h.1045.1 12
9.4 even 3 4698.2.a.bh.1.1 6
9.5 odd 6 4698.2.a.be.1.6 6
9.7 even 3 inner 522.2.e.h.349.5 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.e.h.175.5 ✓ 12 1.1 even 1 trivial
522.2.e.h.349.5 yes 12 9.7 even 3 inner
1566.2.e.h.523.1 12 3.2 odd 2
1566.2.e.h.1045.1 12 9.2 odd 6
4698.2.a.be.1.6 6 9.5 odd 6
4698.2.a.bh.1.1 6 9.4 even 3