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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [736,2,Mod(49,736)] 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("736.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 11, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 736.x (of order \(22\), degree \(10\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.87698958877\)
Analytic rank: \(0\)
Dimension: \(220\)
Relative dimension: \(22\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 593.16
Character \(\chi\) \(=\) 736.593
Dual form 736.2.x.a.561.16

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.38623 + 1.20117i) q^{3} +(2.42814 + 0.349114i) q^{5} +(0.504023 + 1.10366i) q^{7} +(0.0518652 + 0.360730i) q^{9} +(0.0146685 + 0.0499564i) q^{11} +(1.97157 + 0.900388i) q^{13} +(2.94661 + 3.40056i) q^{15} +(-1.80878 + 1.16243i) q^{17} +(0.779449 - 1.21285i) q^{19} +(-0.626992 + 2.13534i) q^{21} +(3.84939 + 2.86045i) q^{23} +(0.976513 + 0.286730i) q^{25} +(2.61359 - 4.06683i) q^{27} +(-3.80111 - 5.91464i) q^{29} +(1.23571 + 1.42608i) q^{31} +(-0.0396724 + 0.0868704i) q^{33} +(0.838537 + 2.85579i) q^{35} +(-9.88119 + 1.42070i) q^{37} +(1.65153 + 3.61634i) q^{39} +(-1.47207 + 10.2385i) q^{41} +(-2.92273 - 2.53256i) q^{43} +0.894010i q^{45} +2.07797 q^{47} +(3.62001 - 4.17771i) q^{49} +(-3.90365 - 0.561261i) q^{51} +(5.87150 - 2.68142i) q^{53} +(0.0181768 + 0.126422i) q^{55} +(2.53733 - 0.745028i) q^{57} +(-10.5443 - 4.81543i) q^{59} +(6.59460 - 5.71425i) q^{61} +(-0.371981 + 0.239058i) q^{63} +(4.47292 + 2.87457i) q^{65} +(-3.35638 + 11.4308i) q^{67} +(1.90024 + 8.58902i) q^{69} +(-11.0242 - 3.23699i) q^{71} +(3.55961 + 2.28762i) q^{73} +(1.00926 + 1.57043i) q^{75} +(-0.0477415 + 0.0413682i) q^{77} +(-1.56885 + 3.43530i) q^{79} +(9.55703 - 2.80620i) q^{81} +(13.0840 - 1.88119i) q^{83} +(-4.79778 + 2.19107i) q^{85} +(1.83531 - 12.7648i) q^{87} +(-9.62643 + 11.1095i) q^{89} +2.62976i q^{91} +3.46117i q^{93} +(2.31603 - 2.67284i) q^{95} +(1.06269 - 7.39115i) q^{97} +(-0.0172600 + 0.00788239i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/736\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(415\) \(645\)
\(\chi(n)\) \(e\left(\frac{6}{11}\right)\) \(1\) \(-1\)

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
\(3\) 1.38623 + 1.20117i 0.800338 + 0.693497i 0.955694 − 0.294361i \(-0.0951069\pi\)
−0.155356 + 0.987859i \(0.549652\pi\)
\(4\) 0 0
\(5\) 2.42814 + 0.349114i 1.08590 + 0.156128i 0.661942 − 0.749555i \(-0.269732\pi\)
0.423955 + 0.905683i \(0.360642\pi\)
\(6\) 0 0
\(7\) 0.504023 + 1.10366i 0.190503 + 0.417143i 0.980649 − 0.195776i \(-0.0627225\pi\)
−0.790146 + 0.612919i \(0.789995\pi\)
\(8\) 0 0
\(9\) 0.0518652 + 0.360730i 0.0172884 + 0.120243i
\(10\) 0 0
\(11\) 0.0146685 + 0.0499564i 0.00442273 + 0.0150624i 0.961675 − 0.274193i \(-0.0884106\pi\)
−0.957252 + 0.289255i \(0.906592\pi\)
\(12\) 0 0
\(13\) 1.97157 + 0.900388i 0.546816 + 0.249723i 0.669614 − 0.742709i \(-0.266459\pi\)
−0.122798 + 0.992432i \(0.539187\pi\)
\(14\) 0 0
\(15\) 2.94661 + 3.40056i 0.760810 + 0.878022i
\(16\) 0 0
\(17\) −1.80878 + 1.16243i −0.438693 + 0.281931i −0.741286 − 0.671189i \(-0.765784\pi\)
0.302593 + 0.953120i \(0.402148\pi\)
\(18\) 0 0
\(19\) 0.779449 − 1.21285i 0.178818 − 0.278246i −0.740261 − 0.672320i \(-0.765298\pi\)
0.919079 + 0.394073i \(0.128934\pi\)
\(20\) 0 0
\(21\) −0.626992 + 2.13534i −0.136821 + 0.465969i
\(22\) 0 0
\(23\) 3.84939 + 2.86045i 0.802654 + 0.596445i
\(24\) 0 0
\(25\) 0.976513 + 0.286730i 0.195303 + 0.0573460i
\(26\) 0 0
\(27\) 2.61359 − 4.06683i 0.502986 − 0.782662i
\(28\) 0 0
\(29\) −3.80111 − 5.91464i −0.705849 − 1.09832i −0.990207 − 0.139604i \(-0.955417\pi\)
0.284359 − 0.958718i \(-0.408219\pi\)
\(30\) 0 0
\(31\) 1.23571 + 1.42608i 0.221940 + 0.256132i 0.855790 − 0.517324i \(-0.173072\pi\)
−0.633850 + 0.773456i \(0.718526\pi\)
\(32\) 0 0
\(33\) −0.0396724 + 0.0868704i −0.00690607 + 0.0151222i
\(34\) 0 0
\(35\) 0.838537 + 2.85579i 0.141739 + 0.482717i
\(36\) 0 0
\(37\) −9.88119 + 1.42070i −1.62446 + 0.233562i −0.893566 − 0.448931i \(-0.851805\pi\)
−0.730892 + 0.682493i \(0.760896\pi\)
\(38\) 0 0
\(39\) 1.65153 + 3.61634i 0.264456 + 0.579078i
\(40\) 0 0
\(41\) −1.47207 + 10.2385i −0.229899 + 1.59898i 0.468627 + 0.883396i \(0.344749\pi\)
−0.698526 + 0.715585i \(0.746160\pi\)
\(42\) 0 0
\(43\) −2.92273 − 2.53256i −0.445712 − 0.386211i 0.402886 − 0.915250i \(-0.368007\pi\)
−0.848598 + 0.529039i \(0.822553\pi\)
\(44\) 0 0
\(45\) 0.894010i 0.133271i
\(46\) 0 0
\(47\) 2.07797 0.303102 0.151551 − 0.988449i \(-0.451573\pi\)
0.151551 + 0.988449i \(0.451573\pi\)
\(48\) 0 0
\(49\) 3.62001 − 4.17771i 0.517144 − 0.596816i
\(50\) 0 0
\(51\) −3.90365 − 0.561261i −0.546621 − 0.0785922i
\(52\) 0 0
\(53\) 5.87150 − 2.68142i 0.806512 − 0.368322i 0.0308930 − 0.999523i \(-0.490165\pi\)
0.775619 + 0.631201i \(0.217438\pi\)
\(54\) 0 0
\(55\) 0.0181768 + 0.126422i 0.00245095 + 0.0170468i
\(56\) 0 0
\(57\) 2.53733 − 0.745028i 0.336078 − 0.0986813i
\(58\) 0 0
\(59\) −10.5443 − 4.81543i −1.37275 − 0.626916i −0.413772 − 0.910381i \(-0.635789\pi\)
−0.958983 + 0.283464i \(0.908516\pi\)
\(60\) 0 0
\(61\) 6.59460 − 5.71425i 0.844352 − 0.731635i −0.120982 − 0.992655i \(-0.538604\pi\)
0.965333 + 0.261020i \(0.0840588\pi\)
\(62\) 0 0
\(63\) −0.371981 + 0.239058i −0.0468652 + 0.0301185i
\(64\) 0 0
\(65\) 4.47292 + 2.87457i 0.554797 + 0.356546i
\(66\) 0 0
\(67\) −3.35638 + 11.4308i −0.410047 + 1.39649i 0.453064 + 0.891478i \(0.350331\pi\)
−0.863111 + 0.505014i \(0.831487\pi\)
\(68\) 0 0
\(69\) 1.90024 + 8.58902i 0.228762 + 1.03400i
\(70\) 0 0
\(71\) −11.0242 − 3.23699i −1.30833 − 0.384160i −0.448062 − 0.894002i \(-0.647886\pi\)
−0.860268 + 0.509842i \(0.829704\pi\)
\(72\) 0 0
\(73\) 3.55961 + 2.28762i 0.416621 + 0.267746i 0.732114 − 0.681182i \(-0.238534\pi\)
−0.315494 + 0.948928i \(0.602170\pi\)
\(74\) 0 0
\(75\) 1.00926 + 1.57043i 0.116539 + 0.181338i
\(76\) 0 0
\(77\) −0.0477415 + 0.0413682i −0.00544065 + 0.00471435i
\(78\) 0 0
\(79\) −1.56885 + 3.43530i −0.176509 + 0.386501i −0.977122 − 0.212681i \(-0.931780\pi\)
0.800612 + 0.599182i \(0.204508\pi\)
\(80\) 0 0
\(81\) 9.55703 − 2.80620i 1.06189 − 0.311800i
\(82\) 0 0
\(83\) 13.0840 − 1.88119i 1.43615 − 0.206488i 0.620127 − 0.784501i \(-0.287081\pi\)
0.816027 + 0.578013i \(0.196172\pi\)
\(84\) 0 0
\(85\) −4.79778 + 2.19107i −0.520393 + 0.237655i
\(86\) 0 0
\(87\) 1.83531 − 12.7648i 0.196765 − 1.36853i
\(88\) 0 0
\(89\) −9.62643 + 11.1095i −1.02040 + 1.17760i −0.0364178 + 0.999337i \(0.511595\pi\)
−0.983982 + 0.178268i \(0.942951\pi\)
\(90\) 0 0
\(91\) 2.62976i 0.275673i
\(92\) 0 0
\(93\) 3.46117i 0.358907i
\(94\) 0 0
\(95\) 2.31603 − 2.67284i 0.237620 − 0.274228i
\(96\) 0 0
\(97\) 1.06269 − 7.39115i 0.107899 − 0.750457i −0.861992 − 0.506921i \(-0.830783\pi\)
0.969892 − 0.243536i \(-0.0783074\pi\)
\(98\) 0 0
\(99\) −0.0172600 + 0.00788239i −0.00173470 + 0.000792210i
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 736.2.x.a.593.16 220
4.3 odd 2 184.2.p.a.133.20 yes 220
8.3 odd 2 184.2.p.a.133.22 yes 220
8.5 even 2 inner 736.2.x.a.593.7 220
23.9 even 11 inner 736.2.x.a.561.7 220
92.55 odd 22 184.2.p.a.101.22 yes 220
184.101 even 22 inner 736.2.x.a.561.16 220
184.147 odd 22 184.2.p.a.101.20 ✓ 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.101.20 ✓ 220 184.147 odd 22
184.2.p.a.101.22 yes 220 92.55 odd 22
184.2.p.a.133.20 yes 220 4.3 odd 2
184.2.p.a.133.22 yes 220 8.3 odd 2
736.2.x.a.561.7 220 23.9 even 11 inner
736.2.x.a.561.16 220 184.101 even 22 inner
736.2.x.a.593.7 220 8.5 even 2 inner
736.2.x.a.593.16 220 1.1 even 1 trivial