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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 49.7
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.31167 - 0.599018i) q^{3} +(0.865287 + 0.749776i) q^{5} +(-3.05894 + 1.96586i) q^{7} +(-0.602936 - 0.695825i) q^{9} +(4.40813 + 0.633793i) q^{11} +(0.0417306 - 0.0649341i) q^{13} +(-0.685839 - 1.50178i) q^{15} +(-4.08968 - 1.20084i) q^{17} +(-2.28718 - 7.78942i) q^{19} +(5.18989 - 0.746194i) q^{21} +(-4.00119 + 2.64396i) q^{23} +(-0.525016 - 3.65156i) q^{25} +(1.59279 + 5.42455i) q^{27} +(1.37030 - 4.66680i) q^{29} +(-1.87557 - 4.10693i) q^{31} +(-5.40234 - 3.47187i) q^{33} +(-4.12081 - 0.592484i) q^{35} +(-4.86349 + 4.21424i) q^{37} +(-0.0936334 + 0.0601745i) q^{39} +(-3.46066 + 3.99382i) q^{41} +(-6.53195 - 2.98304i) q^{43} -1.05415i q^{45} -1.57240 q^{47} +(2.58459 - 5.65947i) q^{49} +(4.64498 + 4.02489i) q^{51} +(-4.01582 - 6.24874i) q^{53} +(3.33909 + 3.85352i) q^{55} +(-1.66598 + 11.5872i) q^{57} +(-1.53118 + 2.38256i) q^{59} +(-0.127073 + 0.0580325i) q^{61} +(3.21224 + 0.943198i) q^{63} +(0.0847950 - 0.0248981i) q^{65} +(5.22653 - 0.751461i) q^{67} +(6.83200 - 1.07121i) q^{69} +(-1.74104 - 12.1092i) q^{71} +(3.53302 - 1.03739i) q^{73} +(-1.49871 + 5.10413i) q^{75} +(-14.7301 + 6.72703i) q^{77} +(-5.96169 - 3.83134i) q^{79} +(0.767101 - 5.33531i) q^{81} +(1.88610 - 1.63431i) q^{83} +(-2.63839 - 4.10541i) q^{85} +(-4.59287 + 5.30045i) q^{87} +(-4.06591 + 8.90309i) q^{89} +0.280666i q^{91} +6.51043i q^{93} +(3.86125 - 8.45495i) q^{95} +(-5.04770 + 5.82536i) q^{97} +(-2.21681 - 3.44942i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) 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 −0.000931659 − 1.00000i \(-0.500297\pi\)
−0.756359 + 0.654156i \(0.773024\pi\)
\(4\) 0 0
\(5\) 0.865287 + 0.749776i 0.386968 + 0.335310i 0.826520 − 0.562907i \(-0.190317\pi\)
−0.439552 + 0.898217i \(0.644863\pi\)
\(6\) 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 0.770021 − 0.638019i \(-0.220246\pi\)
0.559079 + 0.829114i \(0.311155\pi\)
\(12\) 0 0
\(13\) 0.0417306 − 0.0649341i 0.0115740 − 0.0180095i −0.835418 − 0.549615i \(-0.814775\pi\)
0.846992 + 0.531605i \(0.178411\pi\)
\(14\) 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 \(-0.709647\pi\)
0.0873161 − 0.996181i \(-0.472171\pi\)
\(20\) 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 \(-0.759946\pi\)
0.983311 − 0.181933i \(-0.0582355\pi\)
\(30\) 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 −0.955514 − 0.294947i \(-0.904698\pi\)
0.155961 + 0.987763i \(0.450153\pi\)
\(38\) 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 −0.150445 − 0.988618i \(-0.548071\pi\)
−0.845668 + 0.533709i \(0.820798\pi\)
\(44\) 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 0.447743 − 0.894162i \(-0.352228\pi\)
−0.999358 + 0.0358326i \(0.988592\pi\)
\(54\) 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 −0.926505 − 0.376283i \(-0.877202\pi\)
0.727163 + 0.686465i \(0.240839\pi\)
\(60\) 0 0
\(61\) −0.127073 + 0.0580325i −0.0162701 + 0.00743030i −0.423533 − 0.905880i \(-0.639210\pi\)
0.407263 + 0.913311i \(0.366483\pi\)
\(62\) 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 0.184552 − 0.982823i \(-0.440917\pi\)
0.453970 + 0.891017i \(0.350007\pi\)
\(68\) 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 −0.545176 − 0.838322i \(-0.683537\pi\)
0.752202 + 0.658932i \(0.228992\pi\)
\(84\) 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.49.7 220
4.3 odd 2 184.2.p.a.141.15 yes 220
8.3 odd 2 184.2.p.a.141.10 yes 220
8.5 even 2 inner 736.2.x.a.49.16 220
23.8 even 11 inner 736.2.x.a.721.16 220
92.31 odd 22 184.2.p.a.77.10 ✓ 220
184.77 even 22 inner 736.2.x.a.721.7 220
184.123 odd 22 184.2.p.a.77.15 yes 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.10 ✓ 220 92.31 odd 22
184.2.p.a.77.15 yes 220 184.123 odd 22
184.2.p.a.141.10 yes 220 8.3 odd 2
184.2.p.a.141.15 yes 220 4.3 odd 2
736.2.x.a.49.7 220 1.1 even 1 trivial
736.2.x.a.49.16 220 8.5 even 2 inner
736.2.x.a.721.7 220 184.77 even 22 inner
736.2.x.a.721.16 220 23.8 even 11 inner