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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 49.2
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.46787 - 1.12704i) q^{3} +(-1.04195 - 0.902852i) q^{5} +(2.69045 - 1.72905i) q^{7} +(2.85560 + 3.29554i) q^{9} +(4.70086 + 0.675882i) q^{11} +(-1.57821 + 2.45575i) q^{13} +(1.55384 + 3.40244i) q^{15} +(5.75400 + 1.68953i) q^{17} +(-0.537132 - 1.82930i) q^{19} +(-8.58840 + 1.23483i) q^{21} +(-4.66259 - 1.12262i) q^{23} +(-0.441063 - 3.06766i) q^{25} +(-1.04000 - 3.54192i) q^{27} +(2.30796 - 7.86017i) q^{29} +(-1.31989 - 2.89016i) q^{31} +(-10.8394 - 6.96606i) q^{33} +(-4.36438 - 0.627503i) q^{35} +(3.69422 - 3.20106i) q^{37} +(6.66256 - 4.28176i) q^{39} +(-0.907012 + 1.04675i) q^{41} +(6.38677 + 2.91674i) q^{43} -6.01197i q^{45} -3.02200 q^{47} +(1.34101 - 2.93640i) q^{49} +(-12.2960 - 10.6545i) q^{51} +(-0.831467 - 1.29379i) q^{53} +(-4.28783 - 4.94842i) q^{55} +(-0.736126 + 5.11987i) q^{57} +(6.88570 - 10.7144i) q^{59} +(-5.83507 + 2.66479i) q^{61} +(13.3810 + 3.92902i) q^{63} +(3.86159 - 1.13386i) q^{65} +(-7.19360 + 1.03428i) q^{67} +(10.2414 + 8.02541i) q^{69} +(-1.01497 - 7.05931i) q^{71} +(-9.64965 + 2.83339i) q^{73} +(-2.36889 + 8.06770i) q^{75} +(13.8161 - 6.30959i) q^{77} +(2.90122 + 1.86450i) q^{79} +(0.436452 - 3.03559i) q^{81} +(-2.02691 + 1.75633i) q^{83} +(-4.46997 - 6.95541i) q^{85} +(-14.5545 + 16.7968i) q^{87} +(7.19973 - 15.7652i) q^{89} +9.33586i q^{91} +8.62013i q^{93} +(-1.09193 + 2.39099i) q^{95} +(-4.60156 + 5.31048i) q^{97} +(11.1964 + 17.4220i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) 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
\(3\) −2.46787 − 1.12704i −1.42483 − 0.650697i −0.454116 − 0.890943i \(-0.650045\pi\)
−0.970712 + 0.240245i \(0.922772\pi\)
\(4\) 0 0
\(5\) −1.04195 − 0.902852i −0.465973 − 0.403768i 0.389979 − 0.920824i \(-0.372482\pi\)
−0.855951 + 0.517056i \(0.827028\pi\)
\(6\) 0 0
\(7\) 2.69045 − 1.72905i 1.01689 − 0.653518i 0.0777257 − 0.996975i \(-0.475234\pi\)
0.939169 + 0.343457i \(0.111598\pi\)
\(8\) 0 0
\(9\) 2.85560 + 3.29554i 0.951868 + 1.09851i
\(10\) 0 0
\(11\) 4.70086 + 0.675882i 1.41736 + 0.203786i 0.808036 − 0.589133i \(-0.200530\pi\)
0.609327 + 0.792919i \(0.291439\pi\)
\(12\) 0 0
\(13\) −1.57821 + 2.45575i −0.437717 + 0.681101i −0.988100 − 0.153811i \(-0.950845\pi\)
0.550383 + 0.834912i \(0.314482\pi\)
\(14\) 0 0
\(15\) 1.55384 + 3.40244i 0.401200 + 0.878507i
\(16\) 0 0
\(17\) 5.75400 + 1.68953i 1.39555 + 0.409771i 0.891154 − 0.453701i \(-0.149897\pi\)
0.504397 + 0.863472i \(0.331715\pi\)
\(18\) 0 0
\(19\) −0.537132 − 1.82930i −0.123227 − 0.419671i 0.874654 − 0.484748i \(-0.161089\pi\)
−0.997880 + 0.0650771i \(0.979271\pi\)
\(20\) 0 0
\(21\) −8.58840 + 1.23483i −1.87414 + 0.269461i
\(22\) 0 0
\(23\) −4.66259 − 1.12262i −0.972217 − 0.234083i
\(24\) 0 0
\(25\) −0.441063 − 3.06766i −0.0882126 − 0.613532i
\(26\) 0 0
\(27\) −1.04000 − 3.54192i −0.200148 − 0.681643i
\(28\) 0 0
\(29\) 2.30796 − 7.86017i 0.428577 − 1.45960i −0.408625 − 0.912703i \(-0.633991\pi\)
0.837201 − 0.546895i \(-0.184190\pi\)
\(30\) 0 0
\(31\) −1.31989 − 2.89016i −0.237060 − 0.519088i 0.753289 − 0.657690i \(-0.228466\pi\)
−0.990348 + 0.138602i \(0.955739\pi\)
\(32\) 0 0
\(33\) −10.8394 − 6.96606i −1.88690 − 1.21264i
\(34\) 0 0
\(35\) −4.36438 − 0.627503i −0.737714 − 0.106067i
\(36\) 0 0
\(37\) 3.69422 − 3.20106i 0.607327 − 0.526252i −0.296009 − 0.955185i \(-0.595656\pi\)
0.903336 + 0.428933i \(0.141111\pi\)
\(38\) 0 0
\(39\) 6.66256 − 4.28176i 1.06686 − 0.685631i
\(40\) 0 0
\(41\) −0.907012 + 1.04675i −0.141652 + 0.163475i −0.822142 − 0.569282i \(-0.807221\pi\)
0.680491 + 0.732757i \(0.261767\pi\)
\(42\) 0 0
\(43\) 6.38677 + 2.91674i 0.973974 + 0.444799i 0.837855 − 0.545894i \(-0.183810\pi\)
0.136119 + 0.990692i \(0.456537\pi\)
\(44\) 0 0
\(45\) − 6.01197i − 0.896211i
\(46\) 0 0
\(47\) −3.02200 −0.440805 −0.220402 − 0.975409i \(-0.570737\pi\)
−0.220402 + 0.975409i \(0.570737\pi\)
\(48\) 0 0
\(49\) 1.34101 − 2.93640i 0.191573 − 0.419486i
\(50\) 0 0
\(51\) −12.2960 − 10.6545i −1.72178 − 1.49193i
\(52\) 0 0
\(53\) −0.831467 − 1.29379i −0.114211 − 0.177715i 0.779442 − 0.626474i \(-0.215503\pi\)
−0.893653 + 0.448759i \(0.851866\pi\)
\(54\) 0 0
\(55\) −4.28783 − 4.94842i −0.578170 − 0.667244i
\(56\) 0 0
\(57\) −0.736126 + 5.11987i −0.0975022 + 0.678143i
\(58\) 0 0
\(59\) 6.88570 − 10.7144i 0.896442 − 1.39489i −0.0221809 − 0.999754i \(-0.507061\pi\)
0.918622 − 0.395136i \(-0.129303\pi\)
\(60\) 0 0
\(61\) −5.83507 + 2.66479i −0.747104 + 0.341191i −0.752322 − 0.658795i \(-0.771066\pi\)
0.00521803 + 0.999986i \(0.498339\pi\)
\(62\) 0 0
\(63\) 13.3810 + 3.92902i 1.68585 + 0.495010i
\(64\) 0 0
\(65\) 3.86159 − 1.13386i 0.478971 − 0.140639i
\(66\) 0 0
\(67\) −7.19360 + 1.03428i −0.878838 + 0.126358i −0.566941 − 0.823758i \(-0.691873\pi\)
−0.311897 + 0.950116i \(0.600964\pi\)
\(68\) 0 0
\(69\) 10.2414 + 8.02541i 1.23293 + 0.966146i
\(70\) 0 0
\(71\) −1.01497 − 7.05931i −0.120455 − 0.837785i −0.957042 − 0.289950i \(-0.906361\pi\)
0.836586 − 0.547835i \(-0.184548\pi\)
\(72\) 0 0
\(73\) −9.64965 + 2.83339i −1.12941 + 0.331624i −0.792474 − 0.609905i \(-0.791207\pi\)
−0.336932 + 0.941529i \(0.609389\pi\)
\(74\) 0 0
\(75\) −2.36889 + 8.06770i −0.273536 + 0.931578i
\(76\) 0 0
\(77\) 13.8161 − 6.30959i 1.57449 − 0.719044i
\(78\) 0 0
\(79\) 2.90122 + 1.86450i 0.326412 + 0.209773i 0.693575 − 0.720384i \(-0.256034\pi\)
−0.367163 + 0.930157i \(0.619671\pi\)
\(80\) 0 0
\(81\) 0.436452 − 3.03559i 0.0484947 − 0.337288i
\(82\) 0 0
\(83\) −2.02691 + 1.75633i −0.222482 + 0.192782i −0.758969 − 0.651127i \(-0.774297\pi\)
0.536487 + 0.843909i \(0.319751\pi\)
\(84\) 0 0
\(85\) −4.46997 − 6.95541i −0.484836 − 0.754420i
\(86\) 0 0
\(87\) −14.5545 + 16.7968i −1.56040 + 1.80080i
\(88\) 0 0
\(89\) 7.19973 − 15.7652i 0.763170 − 1.67111i 0.0220320 − 0.999757i \(-0.492986\pi\)
0.741138 − 0.671352i \(-0.234286\pi\)
\(90\) 0 0
\(91\) 9.33586i 0.978664i
\(92\) 0 0
\(93\) 8.62013i 0.893865i
\(94\) 0 0
\(95\) −1.09193 + 2.39099i −0.112029 + 0.245310i
\(96\) 0 0
\(97\) −4.60156 + 5.31048i −0.467217 + 0.539197i −0.939635 − 0.342177i \(-0.888836\pi\)
0.472418 + 0.881374i \(0.343381\pi\)
\(98\) 0 0
\(99\) 11.1964 + 17.4220i 1.12528 + 1.75097i
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.49.2 220
4.3 odd 2 184.2.p.a.141.18 yes 220
8.3 odd 2 184.2.p.a.141.3 yes 220
8.5 even 2 inner 736.2.x.a.49.21 220
23.8 even 11 inner 736.2.x.a.721.21 220
92.31 odd 22 184.2.p.a.77.3 ✓ 220
184.77 even 22 inner 736.2.x.a.721.2 220
184.123 odd 22 184.2.p.a.77.18 yes 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.3 ✓ 220 92.31 odd 22
184.2.p.a.77.18 yes 220 184.123 odd 22
184.2.p.a.141.3 yes 220 8.3 odd 2
184.2.p.a.141.18 yes 220 4.3 odd 2
736.2.x.a.49.2 220 1.1 even 1 trivial
736.2.x.a.49.21 220 8.5 even 2 inner
736.2.x.a.721.2 220 184.77 even 22 inner
736.2.x.a.721.21 220 23.8 even 11 inner