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Properties

Label 736.2.x.a.177.5
Level $736$
Weight $2$
Character 736.177
Analytic conductor $5.877$
Analytic rank $0$
Dimension $220$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

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 177.5
Character \(\chi\) \(=\) 736.177
Dual form 736.2.x.a.657.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.16603 - 1.81437i) q^{3} +(1.93519 - 0.883774i) q^{5} +(1.41177 - 0.414532i) q^{7} +(-0.686086 + 1.50232i) q^{9} +(-0.283071 + 0.245282i) q^{11} +(1.76591 - 6.01413i) q^{13} +(-3.85998 - 2.48066i) q^{15} +(-0.532657 + 3.70471i) q^{17} +(3.16421 - 0.454945i) q^{19} +(-2.39827 - 2.07812i) q^{21} +(4.65550 + 1.15165i) q^{23} +(-0.310380 + 0.358198i) q^{25} +(-2.87863 + 0.413885i) q^{27} +(5.13484 + 0.738279i) q^{29} +(-2.19128 - 1.40825i) q^{31} +(0.775102 + 0.227590i) q^{33} +(2.36569 - 2.04988i) q^{35} +(-1.51885 - 0.693637i) q^{37} +(-12.9710 + 3.80862i) q^{39} +(-3.71473 - 8.13412i) q^{41} +(-6.08220 - 9.46409i) q^{43} +3.51362i q^{45} -2.05122 q^{47} +(-4.06753 + 2.61404i) q^{49} +(7.34282 - 3.35335i) q^{51} +(-1.55979 - 5.31217i) q^{53} +(-0.331023 + 0.724840i) q^{55} +(-4.51499 - 5.21058i) q^{57} +(-2.85121 + 9.71032i) q^{59} +(-4.28250 + 6.66369i) q^{61} +(-0.345834 + 2.40533i) q^{63} +(-1.89775 - 13.1992i) q^{65} +(-2.23503 - 1.93667i) q^{67} +(-3.33893 - 9.78967i) q^{69} +(4.88134 - 5.63337i) q^{71} +(-0.758975 - 5.27878i) q^{73} +(1.01182 + 0.145477i) q^{75} +(-0.297953 + 0.463623i) q^{77} +(12.4558 + 3.65736i) q^{79} +(7.35214 + 8.48482i) q^{81} +(8.47618 + 3.87094i) q^{83} +(2.24333 + 7.64008i) q^{85} +(-4.64785 - 10.1774i) q^{87} +(-3.11818 + 2.00394i) q^{89} -9.22257i q^{91} +5.61785i q^{93} +(5.72129 - 3.67685i) q^{95} +(7.61767 + 16.6804i) q^{97} +(-0.174281 - 0.593547i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(415\) \(645\)
\(\chi(n)\) \(e\left(\frac{4}{11}\right)\) \(1\) \(-1\)

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
\(3\) −1.16603 − 1.81437i −0.673206 − 1.04753i −0.994918 − 0.100691i \(-0.967895\pi\)
0.321712 − 0.946838i \(-0.395742\pi\)
\(4\) 0 0
\(5\) 1.93519 − 0.883774i 0.865445 − 0.395236i 0.0673232 − 0.997731i \(-0.478554\pi\)
0.798122 + 0.602496i \(0.205827\pi\)
\(6\) 0 0
\(7\) 1.41177 − 0.414532i 0.533598 − 0.156678i −0.00382311 − 0.999993i \(-0.501217\pi\)
0.537421 + 0.843314i \(0.319399\pi\)
\(8\) 0 0
\(9\) −0.686086 + 1.50232i −0.228695 + 0.500773i
\(10\) 0 0
\(11\) −0.283071 + 0.245282i −0.0853491 + 0.0739554i −0.696490 − 0.717566i \(-0.745256\pi\)
0.611141 + 0.791521i \(0.290711\pi\)
\(12\) 0 0
\(13\) 1.76591 − 6.01413i 0.489775 − 1.66802i −0.229512 − 0.973306i \(-0.573713\pi\)
0.719287 − 0.694713i \(-0.244469\pi\)
\(14\) 0 0
\(15\) −3.85998 − 2.48066i −0.996644 − 0.640504i
\(16\) 0 0
\(17\) −0.532657 + 3.70471i −0.129188 + 0.898524i 0.817398 + 0.576073i \(0.195416\pi\)
−0.946586 + 0.322451i \(0.895493\pi\)
\(18\) 0 0
\(19\) 3.16421 − 0.454945i 0.725919 − 0.104371i 0.230558 − 0.973059i \(-0.425945\pi\)
0.495361 + 0.868687i \(0.335036\pi\)
\(20\) 0 0
\(21\) −2.39827 − 2.07812i −0.523346 − 0.453482i
\(22\) 0 0
\(23\) 4.65550 + 1.15165i 0.970740 + 0.240135i
\(24\) 0 0
\(25\) −0.310380 + 0.358198i −0.0620761 + 0.0716396i
\(26\) 0 0
\(27\) −2.87863 + 0.413885i −0.553993 + 0.0796521i
\(28\) 0 0
\(29\) 5.13484 + 0.738279i 0.953516 + 0.137095i 0.601477 − 0.798890i \(-0.294579\pi\)
0.352039 + 0.935985i \(0.385488\pi\)
\(30\) 0 0
\(31\) −2.19128 − 1.40825i −0.393565 − 0.252929i 0.328859 − 0.944379i \(-0.393336\pi\)
−0.722424 + 0.691450i \(0.756972\pi\)
\(32\) 0 0
\(33\) 0.775102 + 0.227590i 0.134928 + 0.0396184i
\(34\) 0 0
\(35\) 2.36569 − 2.04988i 0.399875 − 0.346493i
\(36\) 0 0
\(37\) −1.51885 − 0.693637i −0.249698 − 0.114033i 0.286635 − 0.958040i \(-0.407463\pi\)
−0.536333 + 0.844007i \(0.680191\pi\)
\(38\) 0 0
\(39\) −12.9710 + 3.80862i −2.07702 + 0.609867i
\(40\) 0 0
\(41\) −3.71473 − 8.13412i −0.580143 − 1.27034i −0.941218 − 0.337799i \(-0.890317\pi\)
0.361075 − 0.932537i \(-0.382410\pi\)
\(42\) 0 0
\(43\) −6.08220 − 9.46409i −0.927527 − 1.44326i −0.896149 − 0.443753i \(-0.853647\pi\)
−0.0313780 − 0.999508i \(-0.509990\pi\)
\(44\) 0 0
\(45\) 3.51362i 0.523780i
\(46\) 0 0
\(47\) −2.05122 −0.299202 −0.149601 − 0.988746i \(-0.547799\pi\)
−0.149601 + 0.988746i \(0.547799\pi\)
\(48\) 0 0
\(49\) −4.06753 + 2.61404i −0.581075 + 0.373434i
\(50\) 0 0
\(51\) 7.34282 − 3.35335i 1.02820 − 0.469563i
\(52\) 0 0
\(53\) −1.55979 − 5.31217i −0.214254 − 0.729682i −0.994549 − 0.104270i \(-0.966749\pi\)
0.780295 − 0.625412i \(-0.215069\pi\)
\(54\) 0 0
\(55\) −0.331023 + 0.724840i −0.0446352 + 0.0977374i
\(56\) 0 0
\(57\) −4.51499 − 5.21058i −0.598025 − 0.690158i
\(58\) 0 0
\(59\) −2.85121 + 9.71032i −0.371196 + 1.26418i 0.536269 + 0.844047i \(0.319833\pi\)
−0.907464 + 0.420129i \(0.861985\pi\)
\(60\) 0 0
\(61\) −4.28250 + 6.66369i −0.548317 + 0.853198i −0.999225 − 0.0393724i \(-0.987464\pi\)
0.450907 + 0.892571i \(0.351101\pi\)
\(62\) 0 0
\(63\) −0.345834 + 2.40533i −0.0435710 + 0.303043i
\(64\) 0 0
\(65\) −1.89775 − 13.1992i −0.235388 − 1.63716i
\(66\) 0 0
\(67\) −2.23503 − 1.93667i −0.273053 − 0.236602i 0.507560 − 0.861616i \(-0.330548\pi\)
−0.780613 + 0.625015i \(0.785093\pi\)
\(68\) 0 0
\(69\) −3.33893 − 9.78967i −0.401960 − 1.17854i
\(70\) 0 0
\(71\) 4.88134 − 5.63337i 0.579309 − 0.668558i −0.388147 − 0.921597i \(-0.626885\pi\)
0.967456 + 0.253039i \(0.0814302\pi\)
\(72\) 0 0
\(73\) −0.758975 − 5.27878i −0.0888313 − 0.617835i −0.984797 − 0.173711i \(-0.944424\pi\)
0.895965 − 0.444124i \(-0.146485\pi\)
\(74\) 0 0
\(75\) 1.01182 + 0.145477i 0.116835 + 0.0167983i
\(76\) 0 0
\(77\) −0.297953 + 0.463623i −0.0339548 + 0.0528348i
\(78\) 0 0
\(79\) 12.4558 + 3.65736i 1.40139 + 0.411485i 0.893161 − 0.449736i \(-0.148482\pi\)
0.508229 + 0.861222i \(0.330300\pi\)
\(80\) 0 0
\(81\) 7.35214 + 8.48482i 0.816904 + 0.942758i
\(82\) 0 0
\(83\) 8.47618 + 3.87094i 0.930381 + 0.424891i 0.822173 − 0.569237i \(-0.192761\pi\)
0.108208 + 0.994128i \(0.465489\pi\)
\(84\) 0 0
\(85\) 2.24333 + 7.64008i 0.243323 + 0.828683i
\(86\) 0 0
\(87\) −4.64785 − 10.1774i −0.498302 − 1.09113i
\(88\) 0 0
\(89\) −3.11818 + 2.00394i −0.330527 + 0.212417i −0.695369 − 0.718653i \(-0.744759\pi\)
0.364842 + 0.931069i \(0.381123\pi\)
\(90\) 0 0
\(91\) − 9.22257i − 0.966788i
\(92\) 0 0
\(93\) 5.61785i 0.582544i
\(94\) 0 0
\(95\) 5.72129 − 3.67685i 0.586992 − 0.377237i
\(96\) 0 0
\(97\) 7.61767 + 16.6804i 0.773457 + 1.69363i 0.718890 + 0.695124i \(0.244651\pi\)
0.0545672 + 0.998510i \(0.482622\pi\)
\(98\) 0 0
\(99\) −0.174281 − 0.593547i −0.0175159 − 0.0596537i
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.177.5 220
4.3 odd 2 184.2.p.a.85.15 yes 220
8.3 odd 2 184.2.p.a.85.3 yes 220
8.5 even 2 inner 736.2.x.a.177.18 220
23.13 even 11 inner 736.2.x.a.657.18 220
92.59 odd 22 184.2.p.a.13.3 ✓ 220
184.13 even 22 inner 736.2.x.a.657.5 220
184.59 odd 22 184.2.p.a.13.15 yes 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.13.3 ✓ 220 92.59 odd 22
184.2.p.a.13.15 yes 220 184.59 odd 22
184.2.p.a.85.3 yes 220 8.3 odd 2
184.2.p.a.85.15 yes 220 4.3 odd 2
736.2.x.a.177.5 220 1.1 even 1 trivial
736.2.x.a.177.18 220 8.5 even 2 inner
736.2.x.a.657.5 220 184.13 even 22 inner
736.2.x.a.657.18 220 23.13 even 11 inner