↑

Properties

Label 736.2.x.a.177.18
Level $736$
Weight $2$
Character 736.177
Analytic conductor $5.877$
Analytic rank $0$
Dimension $220$
Inner twists $4$

Related objects

Downloads

Learn more

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 177.18
Character \(\chi\) \(=\) 736.177
Dual form 736.2.x.a.657.18

$q$-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 \(0.0321054\pi\)
−0.321712 + 0.946838i \(0.604258\pi\)
\(4\) 0 0
\(5\) −1.93519 + 0.883774i −0.865445 + 0.395236i −0.798122 − 0.602496i \(-0.794173\pi\)
−0.0673232 + 0.997731i \(0.521446\pi\)
\(6\) 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 −0.611141 − 0.791521i \(-0.709289\pi\)
0.696490 + 0.717566i \(0.254744\pi\)
\(12\) 0 0
\(13\) −1.76591 + 6.01413i −0.489775 + 1.66802i 0.229512 + 0.973306i \(0.426287\pi\)
−0.719287 + 0.694713i \(0.755531\pi\)
\(14\) 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 −0.495361 − 0.868687i \(-0.664964\pi\)
−0.230558 + 0.973059i \(0.574055\pi\)
\(20\) 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 −0.352039 − 0.935985i \(-0.614512\pi\)
−0.601477 + 0.798890i \(0.705421\pi\)
\(30\) 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 0.536333 − 0.844007i \(-0.319809\pi\)
−0.286635 + 0.958040i \(0.592537\pi\)
\(38\) 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 \(0.146353\pi\)
0.0313780 + 0.999508i \(0.490010\pi\)
\(44\) 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 \(0.0332508\pi\)
−0.780295 + 0.625412i \(0.784931\pi\)
\(54\) 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 \(-0.680167\pi\)
0.907464 − 0.420129i \(-0.138015\pi\)
\(60\) 0 0
\(61\) 4.28250 − 6.66369i 0.548317 − 0.853198i −0.450907 − 0.892571i \(-0.648899\pi\)
0.999225 + 0.0393724i \(0.0125359\pi\)
\(62\) 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 0.780613 − 0.625015i \(-0.214907\pi\)
−0.507560 + 0.861616i \(0.669452\pi\)
\(68\) 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 −0.108208 − 0.994128i \(-0.534511\pi\)
−0.822173 + 0.569237i \(0.807239\pi\)
\(84\) 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.177.18 220
4.3 odd 2 184.2.p.a.85.3 yes 220
8.3 odd 2 184.2.p.a.85.15 yes 220
8.5 even 2 inner 736.2.x.a.177.5 220
23.13 even 11 inner 736.2.x.a.657.5 220
92.59 odd 22 184.2.p.a.13.15 yes 220
184.13 even 22 inner 736.2.x.a.657.18 220
184.59 odd 22 184.2.p.a.13.3 ✓ 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.13.3 ✓ 220 184.59 odd 22
184.2.p.a.13.15 yes 220 92.59 odd 22
184.2.p.a.85.3 yes 220 4.3 odd 2
184.2.p.a.85.15 yes 220 8.3 odd 2
736.2.x.a.177.5 220 8.5 even 2 inner
736.2.x.a.177.18 220 1.1 even 1 trivial
736.2.x.a.657.5 220 23.13 even 11 inner
736.2.x.a.657.18 220 184.13 even 22 inner