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 181.2
Root \(-1.56920 + 0.755686i\) of defining polynomial
Character \(\chi\) \(=\) 522.181
Dual form 522.2.k.h.199.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.222521 + 0.974928i) q^{2} +(-0.900969 + 0.433884i) q^{4} +(0.788529 + 3.45477i) q^{5} +(3.72857 + 1.79558i) q^{7} +(-0.623490 - 0.781831i) q^{8} +(-3.19269 + 1.53752i) q^{10} +(1.14832 - 1.43995i) q^{11} +(2.09403 - 2.62583i) q^{13} +(-0.920880 + 4.03464i) q^{14} +(0.623490 - 0.781831i) q^{16} -3.52078 q^{17} +(-2.45556 + 1.18253i) q^{19} +(-2.20941 - 2.77051i) q^{20} +(1.65937 + 0.799109i) q^{22} +(-0.679736 + 2.97812i) q^{23} +(-6.80881 + 3.27895i) q^{25} +(3.02595 + 1.45722i) q^{26} -4.13840 q^{28} +(-0.127372 - 5.38366i) q^{29} +(-0.196643 - 0.861548i) q^{31} +(0.900969 + 0.433884i) q^{32} +(-0.783447 - 3.43250i) q^{34} +(-3.26324 + 14.2972i) q^{35} +(3.04846 + 3.82264i) q^{37} +(-1.69930 - 2.13085i) q^{38} +(2.20941 - 2.77051i) q^{40} +3.01488 q^{41} +(-0.409830 + 1.79558i) q^{43} +(-0.409830 + 1.79558i) q^{44} -3.05470 q^{46} +(-1.25592 + 1.57487i) q^{47} +(6.31365 + 7.91707i) q^{49} +(-4.71184 - 5.90847i) q^{50} +(-0.747349 + 3.27435i) q^{52} +(-1.47479 - 6.46147i) q^{53} +(5.88016 + 2.83174i) q^{55} +(-0.920880 - 4.03464i) q^{56} +(5.22034 - 1.32216i) q^{58} -6.12406 q^{59} +(-1.64476 - 0.792074i) q^{61} +(0.796190 - 0.383425i) q^{62} +(-0.222521 + 0.974928i) q^{64} +(10.7228 + 5.16384i) q^{65} +(-0.0862879 - 0.108202i) q^{67} +(3.17211 - 1.52761i) q^{68} -14.6649 q^{70} +(8.17273 - 10.2483i) q^{71} +(3.42387 - 15.0009i) q^{73} +(-3.04846 + 3.82264i) q^{74} +(1.69930 - 2.13085i) q^{76} +(6.86712 - 3.30703i) q^{77} +(-9.90051 - 12.4148i) q^{79} +(3.19269 + 1.53752i) q^{80} +(0.670875 + 2.93929i) q^{82} +(-0.0422914 + 0.0203665i) q^{83} +(-2.77623 - 12.1635i) q^{85} -1.84176 q^{86} -1.84176 q^{88} +(-0.800961 - 3.50924i) q^{89} +(12.5226 - 6.03056i) q^{91} +(-0.679736 - 2.97812i) q^{92} +(-1.81485 - 0.873987i) q^{94} +(-6.02166 - 7.55092i) q^{95} +(-4.71887 + 2.27249i) q^{97} +(-6.31365 + 7.91707i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 2 q^{4} + q^{7} + 2 q^{8} - 7 q^{10} + 2 q^{11} + q^{13} - q^{14} - 2 q^{16} + 12 q^{17} - 6 q^{19} - 7 q^{20} - 2 q^{22} - 35 q^{23} - 6 q^{25} - 8 q^{26} - 6 q^{28} + 14 q^{29} - 8 q^{31}+ \cdots - 37 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(379\) \(407\)
\(\chi(n)\) \(e\left(\frac{3}{7}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.222521 + 0.974928i 0.157346 + 0.689378i
\(3\) 0 0
\(4\) −0.900969 + 0.433884i −0.450484 + 0.216942i
\(5\) 0.788529 + 3.45477i 0.352641 + 1.54502i 0.771058 + 0.636765i \(0.219728\pi\)
−0.418417 + 0.908255i \(0.637415\pi\)
\(6\) 0 0
\(7\) 3.72857 + 1.79558i 1.40927 + 0.678666i 0.975018 − 0.222127i \(-0.0713001\pi\)
0.434247 + 0.900794i \(0.357014\pi\)
\(8\) −0.623490 − 0.781831i −0.220437 − 0.276419i
\(9\) 0 0
\(10\) −3.19269 + 1.53752i −1.00962 + 0.486206i
\(11\) 1.14832 − 1.43995i 0.346231 − 0.434160i −0.577975 − 0.816055i \(-0.696157\pi\)
0.924206 + 0.381895i \(0.124728\pi\)
\(12\) 0 0
\(13\) 2.09403 − 2.62583i 0.580778 − 0.728273i −0.401467 − 0.915873i \(-0.631500\pi\)
0.982245 + 0.187601i \(0.0600710\pi\)
\(14\) −0.920880 + 4.03464i −0.246115 + 1.07830i
\(15\) 0 0
\(16\) 0.623490 − 0.781831i 0.155872 − 0.195458i
\(17\) −3.52078 −0.853914 −0.426957 − 0.904272i \(-0.640414\pi\)
−0.426957 + 0.904272i \(0.640414\pi\)
\(18\) 0 0
\(19\) −2.45556 + 1.18253i −0.563344 + 0.271292i −0.693807 − 0.720161i \(-0.744068\pi\)
0.130463 + 0.991453i \(0.458354\pi\)
\(20\) −2.20941 − 2.77051i −0.494039 − 0.619505i
\(21\) 0 0
\(22\) 1.65937 + 0.799109i 0.353778 + 0.170371i
\(23\) −0.679736 + 2.97812i −0.141735 + 0.620980i 0.853297 + 0.521425i \(0.174599\pi\)
−0.995032 + 0.0995556i \(0.968258\pi\)
\(24\) 0 0
\(25\) −6.80881 + 3.27895i −1.36176 + 0.655790i
\(26\) 3.02595 + 1.45722i 0.593439 + 0.285785i
\(27\) 0 0
\(28\) −4.13840 −0.782083
\(29\) −0.127372 − 5.38366i −0.0236524 − 0.999720i
\(30\) 0 0
\(31\) −0.196643 − 0.861548i −0.0353181 − 0.154739i 0.954194 − 0.299188i \(-0.0967159\pi\)
−0.989512 + 0.144450i \(0.953859\pi\)
\(32\) 0.900969 + 0.433884i 0.159270 + 0.0767005i
\(33\) 0 0
\(34\) −0.783447 − 3.43250i −0.134360 − 0.588670i
\(35\) −3.26324 + 14.2972i −0.551589 + 2.41667i
\(36\) 0 0
\(37\) 3.04846 + 3.82264i 0.501163 + 0.628439i 0.966491 − 0.256700i \(-0.0826351\pi\)
−0.465328 + 0.885138i \(0.654064\pi\)
\(38\) −1.69930 − 2.13085i −0.275663 − 0.345670i
\(39\) 0 0
\(40\) 2.20941 − 2.77051i 0.349338 − 0.438056i
\(41\) 3.01488 0.470846 0.235423 − 0.971893i \(-0.424353\pi\)
0.235423 + 0.971893i \(0.424353\pi\)
\(42\) 0 0
\(43\) −0.409830 + 1.79558i −0.0624985 + 0.273824i −0.996516 − 0.0834039i \(-0.973421\pi\)
0.934017 + 0.357228i \(0.116278\pi\)
\(44\) −0.409830 + 1.79558i −0.0617842 + 0.270694i
\(45\) 0 0
\(46\) −3.05470 −0.450392
\(47\) −1.25592 + 1.57487i −0.183194 + 0.229718i −0.864946 − 0.501866i \(-0.832647\pi\)
0.681751 + 0.731584i \(0.261219\pi\)
\(48\) 0 0
\(49\) 6.31365 + 7.91707i 0.901950 + 1.13101i
\(50\) −4.71184 − 5.90847i −0.666355 − 0.835583i
\(51\) 0 0
\(52\) −0.747349 + 3.27435i −0.103639 + 0.454071i
\(53\) −1.47479 − 6.46147i −0.202578 − 0.887551i −0.969360 − 0.245644i \(-0.921001\pi\)
0.766783 − 0.641907i \(-0.221856\pi\)
\(54\) 0 0
\(55\) 5.88016 + 2.83174i 0.792881 + 0.381831i
\(56\) −0.920880 − 4.03464i −0.123058 − 0.539151i
\(57\) 0 0
\(58\) 5.22034 − 1.32216i 0.685464 − 0.173607i
\(59\) −6.12406 −0.797285 −0.398642 − 0.917106i \(-0.630519\pi\)
−0.398642 + 0.917106i \(0.630519\pi\)
\(60\) 0 0
\(61\) −1.64476 − 0.792074i −0.210590 − 0.101415i 0.325616 − 0.945502i \(-0.394428\pi\)
−0.536205 + 0.844088i \(0.680143\pi\)
\(62\) 0.796190 − 0.383425i 0.101116 − 0.0486950i
\(63\) 0 0
\(64\) −0.222521 + 0.974928i −0.0278151 + 0.121866i
\(65\) 10.7228 + 5.16384i 1.33000 + 0.640495i
\(66\) 0 0
\(67\) −0.0862879 − 0.108202i −0.0105417 − 0.0132189i 0.776533 − 0.630077i \(-0.216977\pi\)
−0.787074 + 0.616858i \(0.788405\pi\)
\(68\) 3.17211 − 1.52761i 0.384675 − 0.185250i
\(69\) 0 0
\(70\) −14.6649 −1.75279
\(71\) 8.17273 − 10.2483i 0.969925 − 1.21625i −0.00640935 − 0.999979i \(-0.502040\pi\)
0.976334 − 0.216268i \(-0.0693884\pi\)
\(72\) 0 0
\(73\) 3.42387 − 15.0009i 0.400733 − 1.75573i −0.223709 − 0.974656i \(-0.571817\pi\)
0.624442 − 0.781071i \(-0.285326\pi\)
\(74\) −3.04846 + 3.82264i −0.354376 + 0.444373i
\(75\) 0 0
\(76\) 1.69930 − 2.13085i 0.194923 − 0.244426i
\(77\) 6.86712 − 3.30703i 0.782581 − 0.376871i
\(78\) 0 0
\(79\) −9.90051 − 12.4148i −1.11389 − 1.39678i −0.908390 − 0.418123i \(-0.862688\pi\)
−0.205504 − 0.978656i \(-0.565883\pi\)
\(80\) 3.19269 + 1.53752i 0.356953 + 0.171900i
\(81\) 0 0
\(82\) 0.670875 + 2.93929i 0.0740857 + 0.324591i
\(83\) −0.0422914 + 0.0203665i −0.00464209 + 0.00223551i −0.436203 − 0.899848i \(-0.643677\pi\)
0.431561 + 0.902084i \(0.357963\pi\)
\(84\) 0 0
\(85\) −2.77623 − 12.1635i −0.301125 − 1.31931i
\(86\) −1.84176 −0.198602
\(87\) 0 0
\(88\) −1.84176 −0.196332
\(89\) −0.800961 − 3.50924i −0.0849017 − 0.371979i 0.914572 − 0.404423i \(-0.132528\pi\)
−0.999474 + 0.0324449i \(0.989671\pi\)
\(90\) 0 0
\(91\) 12.5226 − 6.03056i 1.31272 − 0.632175i
\(92\) −0.679736 − 2.97812i −0.0708673 − 0.310490i
\(93\) 0 0
\(94\) −1.81485 − 0.873987i −0.187188 − 0.0901449i
\(95\) −6.02166 − 7.55092i −0.617809 − 0.774708i
\(96\) 0 0
\(97\) −4.71887 + 2.27249i −0.479129 + 0.230736i −0.657829 − 0.753167i \(-0.728525\pi\)
0.178700 + 0.983904i \(0.442811\pi\)
\(98\) −6.31365 + 7.91707i −0.637775 + 0.799745i
\(99\) 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.181.2 12
3.2 odd 2 58.2.d.b.7.1 ✓ 12
12.11 even 2 464.2.u.h.65.2 12
29.25 even 7 inner 522.2.k.h.199.2 12
87.2 even 28 1682.2.b.i.1681.2 12
87.5 odd 14 1682.2.a.q.1.2 6
87.53 odd 14 1682.2.a.t.1.5 6
87.56 even 28 1682.2.b.i.1681.11 12
87.83 odd 14 58.2.d.b.25.1 yes 12
348.83 even 14 464.2.u.h.257.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.7.1 ✓ 12 3.2 odd 2
58.2.d.b.25.1 yes 12 87.83 odd 14
464.2.u.h.65.2 12 12.11 even 2
464.2.u.h.257.2 12 348.83 even 14
522.2.k.h.181.2 12 1.1 even 1 trivial
522.2.k.h.199.2 12 29.25 even 7 inner
1682.2.a.q.1.2 6 87.5 odd 14
1682.2.a.t.1.5 6 87.53 odd 14
1682.2.b.i.1681.2 12 87.2 even 28
1682.2.b.i.1681.11 12 87.56 even 28