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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 593.7
Character \(\chi\) \(=\) 736.593
Dual form 736.2.x.a.561.7

$q$-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 0.155356 − 0.987859i \(-0.450348\pi\)
−0.955694 + 0.294361i \(0.904893\pi\)
\(4\) 0 0
\(5\) −2.42814 − 0.349114i −1.08590 − 0.156128i −0.423955 − 0.905683i \(-0.639358\pi\)
−0.661942 + 0.749555i \(0.730268\pi\)
\(6\) 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 0.957252 − 0.289255i \(-0.0934076\pi\)
−0.961675 + 0.274193i \(0.911589\pi\)
\(12\) 0 0
\(13\) −1.97157 − 0.900388i −0.546816 − 0.249723i 0.122798 − 0.992432i \(-0.460813\pi\)
−0.669614 + 0.742709i \(0.733541\pi\)
\(14\) 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 −0.919079 − 0.394073i \(-0.871066\pi\)
0.740261 + 0.672320i \(0.234702\pi\)
\(20\) 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 \(0.0445829\pi\)
−0.284359 + 0.958718i \(0.591781\pi\)
\(30\) 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 0.730892 − 0.682493i \(-0.239104\pi\)
0.893566 + 0.448931i \(0.148195\pi\)
\(38\) 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 0.848598 − 0.529039i \(-0.177447\pi\)
−0.402886 + 0.915250i \(0.631993\pi\)
\(44\) 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 −0.775619 − 0.631201i \(-0.782562\pi\)
−0.0308930 + 0.999523i \(0.509835\pi\)
\(54\) 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 0.958983 − 0.283464i \(-0.0914838\pi\)
0.413772 + 0.910381i \(0.364211\pi\)
\(60\) 0 0
\(61\) −6.59460 + 5.71425i −0.844352 + 0.731635i −0.965333 − 0.261020i \(-0.915941\pi\)
0.120982 + 0.992655i \(0.461396\pi\)
\(62\) 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 \(-0.649669\pi\)
0.863111 − 0.505014i \(-0.168513\pi\)
\(68\) 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 −0.816027 − 0.578013i \(-0.803828\pi\)
−0.620127 + 0.784501i \(0.712919\pi\)
\(84\) 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.593.7 220
4.3 odd 2 184.2.p.a.133.22 yes 220
8.3 odd 2 184.2.p.a.133.20 yes 220
8.5 even 2 inner 736.2.x.a.593.16 220
23.9 even 11 inner 736.2.x.a.561.16 220
92.55 odd 22 184.2.p.a.101.20 ✓ 220
184.101 even 22 inner 736.2.x.a.561.7 220
184.147 odd 22 184.2.p.a.101.22 yes 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.101.20 ✓ 220 92.55 odd 22
184.2.p.a.101.22 yes 220 184.147 odd 22
184.2.p.a.133.20 yes 220 8.3 odd 2
184.2.p.a.133.22 yes 220 4.3 odd 2
736.2.x.a.561.7 220 184.101 even 22 inner
736.2.x.a.561.16 220 23.9 even 11 inner
736.2.x.a.593.7 220 1.1 even 1 trivial
736.2.x.a.593.16 220 8.5 even 2 inner