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Properties

Label 736.2.x.a.177.8
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 177.8
Character \(\chi\) \(=\) 736.177
Dual form 736.2.x.a.657.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.583943 - 0.908634i) q^{3} +(0.293263 - 0.133929i) q^{5} +(-1.64760 + 0.483779i) q^{7} +(0.761620 - 1.66772i) q^{9} +(-4.72619 + 4.09527i) q^{11} +(-0.960051 + 3.26963i) q^{13} +(-0.292941 - 0.188262i) q^{15} +(-0.348661 + 2.42499i) q^{17} +(-0.170399 + 0.0244997i) q^{19} +(1.40168 + 1.21457i) q^{21} +(4.35783 - 2.00234i) q^{23} +(-3.20624 + 3.70020i) q^{25} +(-5.16739 + 0.742958i) q^{27} +(-1.83955 - 0.264487i) q^{29} +(7.35378 + 4.72599i) q^{31} +(6.48093 + 1.90297i) q^{33} +(-0.418389 + 0.362536i) q^{35} +(-2.57970 - 1.17811i) q^{37} +(3.53151 - 1.03695i) q^{39} +(1.22872 + 2.69052i) q^{41} +(-0.434032 - 0.675367i) q^{43} -0.591082i q^{45} -9.31419 q^{47} +(-3.40823 + 2.19034i) q^{49} +(2.40703 - 1.09925i) q^{51} +(1.61528 + 5.50113i) q^{53} +(-0.837543 + 1.83396i) q^{55} +(0.121765 + 0.140524i) q^{57} +(0.382589 - 1.30298i) q^{59} +(-6.21682 + 9.67356i) q^{61} +(-0.448040 + 3.11618i) q^{63} +(0.156350 + 1.08744i) q^{65} +(9.79045 + 8.48347i) q^{67} +(-4.36411 - 2.79042i) q^{69} +(-3.39541 + 3.91851i) q^{71} +(-1.89100 - 13.1522i) q^{73} +(5.23438 + 0.752591i) q^{75} +(5.80567 - 9.03380i) q^{77} +(6.04520 + 1.77503i) q^{79} +(0.0906820 + 0.104653i) q^{81} +(-4.83071 - 2.20611i) q^{83} +(0.222527 + 0.757856i) q^{85} +(0.833871 + 1.82592i) q^{87} +(7.15465 - 4.59801i) q^{89} -5.85150i q^{91} -9.44160i q^{93} +(-0.0466906 + 0.0300062i) q^{95} +(3.69026 + 8.08055i) q^{97} +(3.23018 + 11.0010i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) 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
\(3\) −0.583943 − 0.908634i −0.337140 − 0.524600i 0.630746 − 0.775989i \(-0.282749\pi\)
−0.967886 + 0.251389i \(0.919113\pi\)
\(4\) 0 0
\(5\) 0.293263 − 0.133929i 0.131151 − 0.0598948i −0.348759 − 0.937213i \(-0.613397\pi\)
0.479910 + 0.877318i \(0.340669\pi\)
\(6\) 0 0
\(7\) −1.64760 + 0.483779i −0.622734 + 0.182851i −0.577853 − 0.816141i \(-0.696109\pi\)
−0.0448816 + 0.998992i \(0.514291\pi\)
\(8\) 0 0
\(9\) 0.761620 − 1.66772i 0.253873 − 0.555905i
\(10\) 0 0
\(11\) −4.72619 + 4.09527i −1.42500 + 1.23477i −0.494146 + 0.869379i \(0.664519\pi\)
−0.930854 + 0.365391i \(0.880935\pi\)
\(12\) 0 0
\(13\) −0.960051 + 3.26963i −0.266270 + 0.906833i 0.712465 + 0.701707i \(0.247579\pi\)
−0.978736 + 0.205126i \(0.934240\pi\)
\(14\) 0 0
\(15\) −0.292941 − 0.188262i −0.0756371 − 0.0486090i
\(16\) 0 0
\(17\) −0.348661 + 2.42499i −0.0845627 + 0.588146i 0.902847 + 0.429962i \(0.141473\pi\)
−0.987410 + 0.158184i \(0.949436\pi\)
\(18\) 0 0
\(19\) −0.170399 + 0.0244997i −0.0390923 + 0.00562062i −0.161833 − 0.986818i \(-0.551741\pi\)
0.122741 + 0.992439i \(0.460832\pi\)
\(20\) 0 0
\(21\) 1.40168 + 1.21457i 0.305872 + 0.265040i
\(22\) 0 0
\(23\) 4.35783 − 2.00234i 0.908669 − 0.417516i
\(24\) 0 0
\(25\) −3.20624 + 3.70020i −0.641247 + 0.740039i
\(26\) 0 0
\(27\) −5.16739 + 0.742958i −0.994464 + 0.142982i
\(28\) 0 0
\(29\) −1.83955 − 0.264487i −0.341596 − 0.0491141i −0.0306178 − 0.999531i \(-0.509747\pi\)
−0.310978 + 0.950417i \(0.600657\pi\)
\(30\) 0 0
\(31\) 7.35378 + 4.72599i 1.32078 + 0.848813i 0.995310 − 0.0967403i \(-0.0308416\pi\)
0.325468 + 0.945553i \(0.394478\pi\)
\(32\) 0 0
\(33\) 6.48093 + 1.90297i 1.12818 + 0.331265i
\(34\) 0 0
\(35\) −0.418389 + 0.362536i −0.0707206 + 0.0612797i
\(36\) 0 0
\(37\) −2.57970 − 1.17811i −0.424100 − 0.193680i 0.191920 − 0.981411i \(-0.438529\pi\)
−0.616020 + 0.787731i \(0.711256\pi\)
\(38\) 0 0
\(39\) 3.53151 − 1.03695i 0.565495 − 0.166044i
\(40\) 0 0
\(41\) 1.22872 + 2.69052i 0.191894 + 0.420188i 0.980984 − 0.194089i \(-0.0621750\pi\)
−0.789090 + 0.614277i \(0.789448\pi\)
\(42\) 0 0
\(43\) −0.434032 − 0.675367i −0.0661893 − 0.102993i 0.806581 − 0.591123i \(-0.201315\pi\)
−0.872771 + 0.488131i \(0.837679\pi\)
\(44\) 0 0
\(45\) − 0.591082i − 0.0881133i
\(46\) 0 0
\(47\) −9.31419 −1.35861 −0.679307 − 0.733854i \(-0.737719\pi\)
−0.679307 + 0.733854i \(0.737719\pi\)
\(48\) 0 0
\(49\) −3.40823 + 2.19034i −0.486890 + 0.312905i
\(50\) 0 0
\(51\) 2.40703 − 1.09925i 0.337051 − 0.153926i
\(52\) 0 0
\(53\) 1.61528 + 5.50113i 0.221875 + 0.755638i 0.992912 + 0.118848i \(0.0379201\pi\)
−0.771037 + 0.636790i \(0.780262\pi\)
\(54\) 0 0
\(55\) −0.837543 + 1.83396i −0.112934 + 0.247292i
\(56\) 0 0
\(57\) 0.121765 + 0.140524i 0.0161281 + 0.0186129i
\(58\) 0 0
\(59\) 0.382589 − 1.30298i 0.0498088 − 0.169633i −0.930832 − 0.365446i \(-0.880916\pi\)
0.980641 + 0.195813i \(0.0627346\pi\)
\(60\) 0 0
\(61\) −6.21682 + 9.67356i −0.795982 + 1.23857i 0.171388 + 0.985204i \(0.445175\pi\)
−0.967370 + 0.253369i \(0.918461\pi\)
\(62\) 0 0
\(63\) −0.448040 + 3.11618i −0.0564477 + 0.392602i
\(64\) 0 0
\(65\) 0.156350 + 1.08744i 0.0193929 + 0.134880i
\(66\) 0 0
\(67\) 9.79045 + 8.48347i 1.19609 + 1.03642i 0.998422 + 0.0561529i \(0.0178834\pi\)
0.197672 + 0.980268i \(0.436662\pi\)
\(68\) 0 0
\(69\) −4.36411 − 2.79042i −0.525378 − 0.335927i
\(70\) 0 0
\(71\) −3.39541 + 3.91851i −0.402961 + 0.465042i −0.920571 − 0.390575i \(-0.872276\pi\)
0.517610 + 0.855617i \(0.326822\pi\)
\(72\) 0 0
\(73\) −1.89100 − 13.1522i −0.221324 − 1.53934i −0.733037 − 0.680188i \(-0.761898\pi\)
0.511713 − 0.859156i \(-0.329011\pi\)
\(74\) 0 0
\(75\) 5.23438 + 0.752591i 0.604414 + 0.0869017i
\(76\) 0 0
\(77\) 5.80567 − 9.03380i 0.661618 − 1.02950i
\(78\) 0 0
\(79\) 6.04520 + 1.77503i 0.680138 + 0.199707i 0.603513 − 0.797353i \(-0.293767\pi\)
0.0766255 + 0.997060i \(0.475585\pi\)
\(80\) 0 0
\(81\) 0.0906820 + 0.104653i 0.0100758 + 0.0116281i
\(82\) 0 0
\(83\) −4.83071 − 2.20611i −0.530240 − 0.242152i 0.132259 − 0.991215i \(-0.457777\pi\)
−0.662499 + 0.749063i \(0.730504\pi\)
\(84\) 0 0
\(85\) 0.222527 + 0.757856i 0.0241364 + 0.0822010i
\(86\) 0 0
\(87\) 0.833871 + 1.82592i 0.0894003 + 0.195759i
\(88\) 0 0
\(89\) 7.15465 − 4.59801i 0.758391 − 0.487388i −0.103407 − 0.994639i \(-0.532975\pi\)
0.861799 + 0.507251i \(0.169338\pi\)
\(90\) 0 0
\(91\) − 5.85150i − 0.613404i
\(92\) 0 0
\(93\) − 9.44160i − 0.979048i
\(94\) 0 0
\(95\) −0.0466906 + 0.0300062i −0.00479036 + 0.00307857i
\(96\) 0 0
\(97\) 3.69026 + 8.08055i 0.374690 + 0.820456i 0.999221 + 0.0394562i \(0.0125626\pi\)
−0.624532 + 0.780999i \(0.714710\pi\)
\(98\) 0 0
\(99\) 3.23018 + 11.0010i 0.324645 + 1.10564i
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.177.8 220
4.3 odd 2 184.2.p.a.85.10 yes 220
8.3 odd 2 184.2.p.a.85.22 yes 220
8.5 even 2 inner 736.2.x.a.177.15 220
23.13 even 11 inner 736.2.x.a.657.15 220
92.59 odd 22 184.2.p.a.13.22 yes 220
184.13 even 22 inner 736.2.x.a.657.8 220
184.59 odd 22 184.2.p.a.13.10 ✓ 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.13.10 ✓ 220 184.59 odd 22
184.2.p.a.13.22 yes 220 92.59 odd 22
184.2.p.a.85.10 yes 220 4.3 odd 2
184.2.p.a.85.22 yes 220 8.3 odd 2
736.2.x.a.177.8 220 1.1 even 1 trivial
736.2.x.a.177.15 220 8.5 even 2 inner
736.2.x.a.657.8 220 184.13 even 22 inner
736.2.x.a.657.15 220 23.13 even 11 inner