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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 49.16
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.16

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.31167 + 0.599018i) q^{3} +(-0.865287 - 0.749776i) q^{5} +(-3.05894 + 1.96586i) q^{7} +(-0.602936 - 0.695825i) q^{9} +(-4.40813 - 0.633793i) q^{11} +(-0.0417306 + 0.0649341i) q^{13} +(-0.685839 - 1.50178i) q^{15} +(-4.08968 - 1.20084i) q^{17} +(2.28718 + 7.78942i) q^{19} +(-5.18989 + 0.746194i) q^{21} +(-4.00119 + 2.64396i) q^{23} +(-0.525016 - 3.65156i) q^{25} +(-1.59279 - 5.42455i) q^{27} +(-1.37030 + 4.66680i) q^{29} +(-1.87557 - 4.10693i) q^{31} +(-5.40234 - 3.47187i) q^{33} +(4.12081 + 0.592484i) q^{35} +(4.86349 - 4.21424i) q^{37} +(-0.0936334 + 0.0601745i) q^{39} +(-3.46066 + 3.99382i) q^{41} +(6.53195 + 2.98304i) q^{43} +1.05415i q^{45} -1.57240 q^{47} +(2.58459 - 5.65947i) q^{49} +(-4.64498 - 4.02489i) q^{51} +(4.01582 + 6.24874i) q^{53} +(3.33909 + 3.85352i) q^{55} +(-1.66598 + 11.5872i) q^{57} +(1.53118 - 2.38256i) q^{59} +(0.127073 - 0.0580325i) q^{61} +(3.21224 + 0.943198i) q^{63} +(0.0847950 - 0.0248981i) q^{65} +(-5.22653 + 0.751461i) q^{67} +(-6.83200 + 1.07121i) q^{69} +(-1.74104 - 12.1092i) q^{71} +(3.53302 - 1.03739i) q^{73} +(1.49871 - 5.10413i) q^{75} +(14.7301 - 6.72703i) q^{77} +(-5.96169 - 3.83134i) q^{79} +(0.767101 - 5.33531i) q^{81} +(-1.88610 + 1.63431i) q^{83} +(2.63839 + 4.10541i) q^{85} +(-4.59287 + 5.30045i) q^{87} +(-4.06591 + 8.90309i) q^{89} -0.280666i q^{91} -6.51043i q^{93} +(3.86125 - 8.45495i) q^{95} +(-5.04770 + 5.82536i) q^{97} +(2.21681 + 3.44942i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(415\) \(645\)
\(\chi(n)\) \(e\left(\frac{8}{11}\right)\) \(1\) \(-1\)

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
\(3\) 1.31167 + 0.599018i 0.757291 + 0.345843i 0.756359 − 0.654156i \(-0.226976\pi\)
0.000931659 1.00000i \(0.499703\pi\)
\(4\) 0 0
\(5\) −0.865287 − 0.749776i −0.386968 − 0.335310i 0.439552 − 0.898217i \(-0.355137\pi\)
−0.826520 + 0.562907i \(0.809683\pi\)
\(6\) 0 0
\(7\) −3.05894 + 1.96586i −1.15617 + 0.743025i −0.970858 − 0.239654i \(-0.922966\pi\)
−0.185312 + 0.982680i \(0.559330\pi\)
\(8\) 0 0
\(9\) −0.602936 − 0.695825i −0.200979 − 0.231942i
\(10\) 0 0
\(11\) −4.40813 − 0.633793i −1.32910 − 0.191096i −0.559079 − 0.829114i \(-0.688845\pi\)
−0.770021 + 0.638019i \(0.779754\pi\)
\(12\) 0 0
\(13\) −0.0417306 + 0.0649341i −0.0115740 + 0.0180095i −0.846992 − 0.531605i \(-0.821589\pi\)
0.835418 + 0.549615i \(0.185225\pi\)
\(14\) 0 0
\(15\) −0.685839 − 1.50178i −0.177083 − 0.387757i
\(16\) 0 0
\(17\) −4.08968 − 1.20084i −0.991894 − 0.291246i −0.254768 − 0.967002i \(-0.581999\pi\)
−0.737125 + 0.675756i \(0.763817\pi\)
\(18\) 0 0
\(19\) 2.28718 + 7.78942i 0.524715 + 1.78701i 0.612031 + 0.790834i \(0.290353\pi\)
−0.0873161 + 0.996181i \(0.527829\pi\)
\(20\) 0 0
\(21\) −5.18989 + 0.746194i −1.13253 + 0.162833i
\(22\) 0 0
\(23\) −4.00119 + 2.64396i −0.834305 + 0.551303i
\(24\) 0 0
\(25\) −0.525016 − 3.65156i −0.105003 − 0.730313i
\(26\) 0 0
\(27\) −1.59279 − 5.42455i −0.306533 − 1.04396i
\(28\) 0 0
\(29\) −1.37030 + 4.66680i −0.254458 + 0.866603i 0.728853 + 0.684670i \(0.240054\pi\)
−0.983311 + 0.181933i \(0.941765\pi\)
\(30\) 0 0
\(31\) −1.87557 − 4.10693i −0.336863 − 0.737627i 0.663078 − 0.748551i \(-0.269250\pi\)
−0.999940 + 0.0109241i \(0.996523\pi\)
\(32\) 0 0
\(33\) −5.40234 − 3.47187i −0.940426 − 0.604375i
\(34\) 0 0
\(35\) 4.12081 + 0.592484i 0.696545 + 0.100148i
\(36\) 0 0
\(37\) 4.86349 − 4.21424i 0.799552 − 0.692816i −0.155961 − 0.987763i \(-0.549847\pi\)
0.955514 + 0.294947i \(0.0953020\pi\)
\(38\) 0 0
\(39\) −0.0936334 + 0.0601745i −0.0149933 + 0.00963563i
\(40\) 0 0
\(41\) −3.46066 + 3.99382i −0.540465 + 0.623729i −0.958635 − 0.284639i \(-0.908126\pi\)
0.418170 + 0.908369i \(0.362672\pi\)
\(42\) 0 0
\(43\) 6.53195 + 2.98304i 0.996113 + 0.454910i 0.845668 − 0.533709i \(-0.179202\pi\)
0.150445 + 0.988618i \(0.451929\pi\)
\(44\) 0 0
\(45\) 1.05415i 0.157144i
\(46\) 0 0
\(47\) −1.57240 −0.229358 −0.114679 − 0.993403i \(-0.536584\pi\)
−0.114679 + 0.993403i \(0.536584\pi\)
\(48\) 0 0
\(49\) 2.58459 − 5.65947i 0.369228 − 0.808496i
\(50\) 0 0
\(51\) −4.64498 − 4.02489i −0.650427 − 0.563598i
\(52\) 0 0
\(53\) 4.01582 + 6.24874i 0.551615 + 0.858330i 0.999358 − 0.0358326i \(-0.0114083\pi\)
−0.447743 + 0.894162i \(0.647772\pi\)
\(54\) 0 0
\(55\) 3.33909 + 3.85352i 0.450243 + 0.519608i
\(56\) 0 0
\(57\) −1.66598 + 11.5872i −0.220665 + 1.53476i
\(58\) 0 0
\(59\) 1.53118 − 2.38256i 0.199342 − 0.310182i −0.727163 − 0.686465i \(-0.759161\pi\)
0.926505 + 0.376283i \(0.122798\pi\)
\(60\) 0 0
\(61\) 0.127073 − 0.0580325i 0.0162701 − 0.00743030i −0.407263 − 0.913311i \(-0.633517\pi\)
0.423533 + 0.905880i \(0.360790\pi\)
\(62\) 0 0
\(63\) 3.21224 + 0.943198i 0.404704 + 0.118832i
\(64\) 0 0
\(65\) 0.0847950 − 0.0248981i 0.0105175 − 0.00308822i
\(66\) 0 0
\(67\) −5.22653 + 0.751461i −0.638522 + 0.0918056i −0.453970 − 0.891017i \(-0.649993\pi\)
−0.184552 + 0.982823i \(0.559083\pi\)
\(68\) 0 0
\(69\) −6.83200 + 1.07121i −0.822476 + 0.128958i
\(70\) 0 0
\(71\) −1.74104 − 12.1092i −0.206623 − 1.43710i −0.784073 − 0.620669i \(-0.786861\pi\)
0.577450 − 0.816426i \(-0.304048\pi\)
\(72\) 0 0
\(73\) 3.53302 − 1.03739i 0.413509 − 0.121417i −0.0683593 − 0.997661i \(-0.521776\pi\)
0.481869 + 0.876243i \(0.339958\pi\)
\(74\) 0 0
\(75\) 1.49871 − 5.10413i 0.173056 − 0.589374i
\(76\) 0 0
\(77\) 14.7301 − 6.72703i 1.67865 − 0.766616i
\(78\) 0 0
\(79\) −5.96169 − 3.83134i −0.670742 − 0.431060i 0.160451 − 0.987044i \(-0.448705\pi\)
−0.831193 + 0.555984i \(0.812342\pi\)
\(80\) 0 0
\(81\) 0.767101 − 5.33531i 0.0852335 − 0.592812i
\(82\) 0 0
\(83\) −1.88610 + 1.63431i −0.207026 + 0.179389i −0.752202 − 0.658932i \(-0.771008\pi\)
0.545176 + 0.838322i \(0.316463\pi\)
\(84\) 0 0
\(85\) 2.63839 + 4.10541i 0.286174 + 0.445295i
\(86\) 0 0
\(87\) −4.59287 + 5.30045i −0.492407 + 0.568268i
\(88\) 0 0
\(89\) −4.06591 + 8.90309i −0.430985 + 0.943726i 0.562181 + 0.827014i \(0.309963\pi\)
−0.993166 + 0.116711i \(0.962765\pi\)
\(90\) 0 0
\(91\) − 0.280666i − 0.0294218i
\(92\) 0 0
\(93\) − 6.51043i − 0.675100i
\(94\) 0 0
\(95\) 3.86125 − 8.45495i 0.396156 − 0.867460i
\(96\) 0 0
\(97\) −5.04770 + 5.82536i −0.512517 + 0.591476i −0.951741 − 0.306902i \(-0.900708\pi\)
0.439225 + 0.898377i \(0.355253\pi\)
\(98\) 0 0
\(99\) 2.21681 + 3.44942i 0.222798 + 0.346680i
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.49.16 220
4.3 odd 2 184.2.p.a.141.10 yes 220
8.3 odd 2 184.2.p.a.141.15 yes 220
8.5 even 2 inner 736.2.x.a.49.7 220
23.8 even 11 inner 736.2.x.a.721.7 220
92.31 odd 22 184.2.p.a.77.15 yes 220
184.77 even 22 inner 736.2.x.a.721.16 220
184.123 odd 22 184.2.p.a.77.10 ✓ 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.10 ✓ 220 184.123 odd 22
184.2.p.a.77.15 yes 220 92.31 odd 22
184.2.p.a.141.10 yes 220 4.3 odd 2
184.2.p.a.141.15 yes 220 8.3 odd 2
736.2.x.a.49.7 220 8.5 even 2 inner
736.2.x.a.49.16 220 1.1 even 1 trivial
736.2.x.a.721.7 220 23.8 even 11 inner
736.2.x.a.721.16 220 184.77 even 22 inner