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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(89,522)] 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("522.89"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([14, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label 143.3
Character \(\chi\) \(=\) 522.143
Dual form 522.2.r.b.449.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.993712 - 0.111964i) q^{2} +(0.974928 + 0.222521i) q^{4} +(2.06579 + 2.59041i) q^{5} +(-0.284820 - 1.24788i) q^{7} +(-0.943883 - 0.330279i) q^{8} +(-1.76276 - 2.80542i) q^{10} +(4.14695 - 1.45108i) q^{11} +(2.33743 - 4.85373i) q^{13} +(0.143311 + 1.27192i) q^{14} +(0.900969 + 0.433884i) q^{16} +(-5.10629 + 5.10629i) q^{17} +(0.584568 - 0.367308i) q^{19} +(1.43757 + 2.98515i) q^{20} +(-4.28334 + 0.977645i) q^{22} +(6.33381 + 5.05105i) q^{23} +(-1.33017 + 5.82784i) q^{25} +(-2.86618 + 4.56150i) q^{26} -1.27997i q^{28} +(0.142631 - 5.38328i) q^{29} +(-0.283525 + 2.51635i) q^{31} +(-0.846724 - 0.532032i) q^{32} +(5.64591 - 4.50246i) q^{34} +(2.64414 - 3.31565i) q^{35} +(-1.56233 + 4.46489i) q^{37} +(-0.622018 + 0.299548i) q^{38} +(-1.09430 - 3.12733i) q^{40} +(-4.44439 - 4.44439i) q^{41} +(7.75652 - 0.873949i) q^{43} +(4.36587 - 0.491915i) q^{44} +(-5.72845 - 5.72845i) q^{46} +(3.81853 + 10.9127i) q^{47} +(4.83071 - 2.32635i) q^{49} +(1.97432 - 5.64227i) q^{50} +(3.35889 - 4.21191i) q^{52} +(-3.85181 + 3.07172i) q^{53} +(12.3256 + 7.74469i) q^{55} +(-0.143311 + 1.27192i) q^{56} +(-0.744469 + 5.33346i) q^{58} -12.1255i q^{59} +(-0.921184 + 1.46606i) q^{61} +(0.563484 - 2.46878i) q^{62} +(0.781831 + 0.623490i) q^{64} +(17.4018 - 3.97185i) q^{65} +(2.78858 + 5.79054i) q^{67} +(-6.11452 + 3.84201i) q^{68} +(-2.99875 + 2.99875i) q^{70} +(-2.54535 - 1.22578i) q^{71} +(0.540042 + 4.79301i) q^{73} +(2.05242 - 4.26189i) q^{74} +(0.651646 - 0.228021i) q^{76} +(-2.99190 - 4.76158i) q^{77} +(3.33878 + 1.16829i) q^{79} +(0.737271 + 3.23019i) q^{80} +(3.91883 + 4.91406i) q^{82} +(0.744303 + 0.169882i) q^{83} +(-23.7759 - 2.67890i) q^{85} -7.80559 q^{86} -4.39350 q^{88} +(-13.4714 - 1.51786i) q^{89} +(-6.72261 - 1.53439i) q^{91} +(5.05105 + 6.33381i) q^{92} +(-2.57268 - 11.2716i) q^{94} +(2.15907 + 0.755493i) q^{95} +(-6.18775 - 9.84774i) q^{97} +(-5.06080 + 1.77085i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 8 q^{7} + 4 q^{10} + 12 q^{16} - 40 q^{19} - 20 q^{25} - 28 q^{31} - 12 q^{37} - 4 q^{40} - 72 q^{43} + 16 q^{46} + 52 q^{49} + 4 q^{52} + 140 q^{55} + 44 q^{58} + 24 q^{61} + 140 q^{67} - 20 q^{70}+ \cdots + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(379\) \(407\)
\(\chi(n)\) \(e\left(\frac{15}{28}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.993712 − 0.111964i −0.702661 − 0.0791708i
\(3\) 0 0
\(4\) 0.974928 + 0.222521i 0.487464 + 0.111260i
\(5\) 2.06579 + 2.59041i 0.923848 + 1.15847i 0.987041 + 0.160467i \(0.0513001\pi\)
−0.0631935 + 0.998001i \(0.520129\pi\)
\(6\) 0 0
\(7\) −0.284820 − 1.24788i −0.107652 − 0.471653i −0.999802 − 0.0199138i \(-0.993661\pi\)
0.892150 − 0.451739i \(-0.149196\pi\)
\(8\) −0.943883 − 0.330279i −0.333713 − 0.116771i
\(9\) 0 0
\(10\) −1.76276 − 2.80542i −0.557435 − 0.887152i
\(11\) 4.14695 − 1.45108i 1.25035 − 0.437517i 0.377731 − 0.925915i \(-0.376704\pi\)
0.872621 + 0.488399i \(0.162419\pi\)
\(12\) 0 0
\(13\) 2.33743 − 4.85373i 0.648287 − 1.34618i −0.274766 − 0.961511i \(-0.588600\pi\)
0.923054 − 0.384671i \(-0.125685\pi\)
\(14\) 0.143311 + 1.27192i 0.0383015 + 0.339935i
\(15\) 0 0
\(16\) 0.900969 + 0.433884i 0.225242 + 0.108471i
\(17\) −5.10629 + 5.10629i −1.23846 + 1.23846i −0.277826 + 0.960631i \(0.589614\pi\)
−0.960631 + 0.277826i \(0.910386\pi\)
\(18\) 0 0
\(19\) 0.584568 − 0.367308i 0.134109 − 0.0842663i −0.463307 − 0.886198i \(-0.653337\pi\)
0.597416 + 0.801932i \(0.296194\pi\)
\(20\) 1.43757 + 2.98515i 0.321451 + 0.667499i
\(21\) 0 0
\(22\) −4.28334 + 0.977645i −0.913211 + 0.208435i
\(23\) 6.33381 + 5.05105i 1.32069 + 1.05322i 0.994147 + 0.108034i \(0.0344554\pi\)
0.326544 + 0.945182i \(0.394116\pi\)
\(24\) 0 0
\(25\) −1.33017 + 5.82784i −0.266033 + 1.16557i
\(26\) −2.86618 + 4.56150i −0.562104 + 0.894584i
\(27\) 0 0
\(28\) − 1.27997i − 0.241891i
\(29\) 0.142631 − 5.38328i 0.0264858 − 0.999649i
\(30\) 0 0
\(31\) −0.283525 + 2.51635i −0.0509225 + 0.451950i 0.942216 + 0.335006i \(0.108738\pi\)
−0.993139 + 0.116944i \(0.962690\pi\)
\(32\) −0.846724 − 0.532032i −0.149681 − 0.0940509i
\(33\) 0 0
\(34\) 5.64591 − 4.50246i 0.968265 − 0.772166i
\(35\) 2.64414 − 3.31565i 0.446941 − 0.560447i
\(36\) 0 0
\(37\) −1.56233 + 4.46489i −0.256846 + 0.734023i 0.741185 + 0.671301i \(0.234264\pi\)
−0.998031 + 0.0627226i \(0.980022\pi\)
\(38\) −0.622018 + 0.299548i −0.100905 + 0.0485931i
\(39\) 0 0
\(40\) −1.09430 − 3.12733i −0.173024 − 0.494475i
\(41\) −4.44439 − 4.44439i −0.694097 − 0.694097i 0.269033 − 0.963131i \(-0.413296\pi\)
−0.963131 + 0.269033i \(0.913296\pi\)
\(42\) 0 0
\(43\) 7.75652 − 0.873949i 1.18286 − 0.133276i 0.501454 − 0.865184i \(-0.332799\pi\)
0.681403 + 0.731908i \(0.261370\pi\)
\(44\) 4.36587 − 0.491915i 0.658180 − 0.0741590i
\(45\) 0 0
\(46\) −5.72845 − 5.72845i −0.844614 − 0.844614i
\(47\) 3.81853 + 10.9127i 0.556989 + 1.59178i 0.785912 + 0.618339i \(0.212194\pi\)
−0.228923 + 0.973445i \(0.573520\pi\)
\(48\) 0 0
\(49\) 4.83071 − 2.32635i 0.690101 − 0.332335i
\(50\) 1.97432 − 5.64227i 0.279210 − 0.797937i
\(51\) 0 0
\(52\) 3.35889 − 4.21191i 0.465794 − 0.584087i
\(53\) −3.85181 + 3.07172i −0.529087 + 0.421933i −0.851257 − 0.524749i \(-0.824159\pi\)
0.322170 + 0.946682i \(0.395588\pi\)
\(54\) 0 0
\(55\) 12.3256 + 7.74469i 1.66198 + 1.04429i
\(56\) −0.143311 + 1.27192i −0.0191507 + 0.169968i
\(57\) 0 0
\(58\) −0.744469 + 5.33346i −0.0977536 + 0.700317i
\(59\) − 12.1255i − 1.57861i −0.614001 − 0.789305i \(-0.710441\pi\)
0.614001 − 0.789305i \(-0.289559\pi\)
\(60\) 0 0
\(61\) −0.921184 + 1.46606i −0.117946 + 0.187709i −0.900479 − 0.434899i \(-0.856784\pi\)
0.782534 + 0.622608i \(0.213927\pi\)
\(62\) 0.563484 − 2.46878i 0.0715625 − 0.313536i
\(63\) 0 0
\(64\) 0.781831 + 0.623490i 0.0977289 + 0.0779362i
\(65\) 17.4018 − 3.97185i 2.15843 − 0.492647i
\(66\) 0 0
\(67\) 2.78858 + 5.79054i 0.340679 + 0.707427i 0.998972 − 0.0453350i \(-0.0144355\pi\)
−0.658293 + 0.752762i \(0.728721\pi\)
\(68\) −6.11452 + 3.84201i −0.741495 + 0.465912i
\(69\) 0 0
\(70\) −2.99875 + 2.99875i −0.358419 + 0.358419i
\(71\) −2.54535 − 1.22578i −0.302077 − 0.145473i 0.276705 − 0.960955i \(-0.410758\pi\)
−0.578782 + 0.815482i \(0.696472\pi\)
\(72\) 0 0
\(73\) 0.540042 + 4.79301i 0.0632072 + 0.560979i 0.984908 + 0.173078i \(0.0553713\pi\)
−0.921701 + 0.387901i \(0.873200\pi\)
\(74\) 2.05242 − 4.26189i 0.238589 − 0.495435i
\(75\) 0 0
\(76\) 0.651646 − 0.228021i 0.0747489 − 0.0261558i
\(77\) −2.99190 − 4.76158i −0.340959 − 0.542633i
\(78\) 0 0
\(79\) 3.33878 + 1.16829i 0.375642 + 0.131443i 0.511495 − 0.859286i \(-0.329092\pi\)
−0.135853 + 0.990729i \(0.543377\pi\)
\(80\) 0.737271 + 3.23019i 0.0824294 + 0.361147i
\(81\) 0 0
\(82\) 3.91883 + 4.91406i 0.432763 + 0.542667i
\(83\) 0.744303 + 0.169882i 0.0816979 + 0.0186470i 0.263174 − 0.964748i \(-0.415231\pi\)
−0.181477 + 0.983395i \(0.558088\pi\)
\(84\) 0 0
\(85\) −23.7759 − 2.67890i −2.57886 − 0.290568i
\(86\) −7.80559 −0.841699
\(87\) 0 0
\(88\) −4.39350 −0.468348
\(89\) −13.4714 − 1.51786i −1.42797 − 0.160893i −0.636109 − 0.771599i \(-0.719457\pi\)
−0.791857 + 0.610706i \(0.790886\pi\)
\(90\) 0 0
\(91\) −6.72261 − 1.53439i −0.704721 − 0.160848i
\(92\) 5.05105 + 6.33381i 0.526608 + 0.660345i
\(93\) 0 0
\(94\) −2.57268 − 11.2716i −0.265351 − 1.16258i
\(95\) 2.15907 + 0.755493i 0.221516 + 0.0775119i
\(96\) 0 0
\(97\) −6.18775 − 9.84774i −0.628270 − 0.999887i −0.997508 − 0.0705517i \(-0.977524\pi\)
0.369238 − 0.929335i \(-0.379619\pi\)
\(98\) −5.06080 + 1.77085i −0.511218 + 0.178883i
\(99\) 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.r.b.143.3 ✓ 72
3.2 odd 2 inner 522.2.r.b.143.4 yes 72
29.14 odd 28 inner 522.2.r.b.449.4 yes 72
87.14 even 28 inner 522.2.r.b.449.3 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.r.b.143.3 ✓ 72 1.1 even 1 trivial
522.2.r.b.143.4 yes 72 3.2 odd 2 inner
522.2.r.b.449.3 yes 72 87.14 even 28 inner
522.2.r.b.449.4 yes 72 29.14 odd 28 inner