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Properties

Label 736.2.x.a.657.7
Level $736$
Weight $2$
Character 736.657
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 657.7
Character \(\chi\) \(=\) 736.657
Dual form 736.2.x.a.177.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.603355 + 0.938838i) q^{3} +(1.75955 + 0.803560i) q^{5} +(4.74970 + 1.39464i) q^{7} +(0.728865 + 1.59599i) q^{9} +(1.14564 + 0.992703i) q^{11} +(-1.72609 - 5.87851i) q^{13} +(-1.81605 + 1.16710i) q^{15} +(0.242705 + 1.68805i) q^{17} +(-0.315547 - 0.0453688i) q^{19} +(-4.17509 + 3.61774i) q^{21} +(2.99614 - 3.74475i) q^{23} +(-0.823992 - 0.950937i) q^{25} +(-5.25206 - 0.755132i) q^{27} +(4.12734 - 0.593422i) q^{29} +(-1.74961 + 1.12441i) q^{31} +(-1.62321 + 0.476619i) q^{33} +(7.23667 + 6.27061i) q^{35} +(-9.89680 + 4.51972i) q^{37} +(6.56041 + 1.92631i) q^{39} +(0.599928 - 1.31366i) q^{41} +(-2.59779 + 4.04224i) q^{43} +3.39392i q^{45} +3.84679 q^{47} +(14.7259 + 9.46374i) q^{49} +(-1.73124 - 0.790632i) q^{51} +(1.96998 - 6.70914i) q^{53} +(1.21812 + 2.66730i) q^{55} +(0.232981 - 0.268874i) q^{57} +(0.804164 + 2.73873i) q^{59} +(4.54228 + 7.06793i) q^{61} +(1.23606 + 8.59698i) q^{63} +(1.68660 - 11.7306i) q^{65} +(3.99062 - 3.45789i) q^{67} +(1.70798 + 5.07230i) q^{69} +(-8.15113 - 9.40690i) q^{71} +(-0.932891 + 6.48840i) q^{73} +(1.38994 - 0.199843i) q^{75} +(4.05699 + 6.31280i) q^{77} +(-13.1546 + 3.86253i) q^{79} +(0.430852 - 0.497229i) q^{81} +(-11.8816 + 5.42616i) q^{83} +(-0.929398 + 3.16524i) q^{85} +(-1.93312 + 4.23295i) q^{87} +(-7.53125 - 4.84004i) q^{89} -30.3284i q^{91} -2.32102i q^{93} +(-0.518765 - 0.333390i) q^{95} +(-0.264613 + 0.579421i) q^{97} +(-0.749328 + 2.55198i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(415\) \(645\)
\(\chi(n)\) \(e\left(\frac{7}{11}\right)\) \(1\) \(-1\)

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
\(3\) −0.603355 + 0.938838i −0.348347 + 0.542038i −0.970575 − 0.240797i \(-0.922591\pi\)
0.622228 + 0.782836i \(0.286227\pi\)
\(4\) 0 0
\(5\) 1.75955 + 0.803560i 0.786895 + 0.359363i 0.767993 − 0.640458i \(-0.221256\pi\)
0.0189023 + 0.999821i \(0.493983\pi\)
\(6\) 0 0
\(7\) 4.74970 + 1.39464i 1.79522 + 0.527124i 0.997150 − 0.0754437i \(-0.0240373\pi\)
0.798068 + 0.602567i \(0.205855\pi\)
\(8\) 0 0
\(9\) 0.728865 + 1.59599i 0.242955 + 0.531997i
\(10\) 0 0
\(11\) 1.14564 + 0.992703i 0.345424 + 0.299311i 0.810243 − 0.586094i \(-0.199335\pi\)
−0.464819 + 0.885406i \(0.653881\pi\)
\(12\) 0 0
\(13\) −1.72609 − 5.87851i −0.478730 − 1.63041i −0.745400 − 0.666618i \(-0.767741\pi\)
0.266670 − 0.963788i \(-0.414077\pi\)
\(14\) 0 0
\(15\) −1.81605 + 1.16710i −0.468901 + 0.301344i
\(16\) 0 0
\(17\) 0.242705 + 1.68805i 0.0588646 + 0.409412i 0.997855 + 0.0654680i \(0.0208540\pi\)
−0.938990 + 0.343944i \(0.888237\pi\)
\(18\) 0 0
\(19\) −0.315547 − 0.0453688i −0.0723914 − 0.0104083i 0.106024 − 0.994364i \(-0.466188\pi\)
−0.178415 + 0.983955i \(0.557097\pi\)
\(20\) 0 0
\(21\) −4.17509 + 3.61774i −0.911080 + 0.789455i
\(22\) 0 0
\(23\) 2.99614 − 3.74475i 0.624738 − 0.780834i
\(24\) 0 0
\(25\) −0.823992 − 0.950937i −0.164798 − 0.190187i
\(26\) 0 0
\(27\) −5.25206 − 0.755132i −1.01076 − 0.145325i
\(28\) 0 0
\(29\) 4.12734 − 0.593422i 0.766428 − 0.110196i 0.251998 − 0.967728i \(-0.418912\pi\)
0.514431 + 0.857532i \(0.328003\pi\)
\(30\) 0 0
\(31\) −1.74961 + 1.12441i −0.314239 + 0.201949i −0.688247 − 0.725476i \(-0.741620\pi\)
0.374008 + 0.927425i \(0.377983\pi\)
\(32\) 0 0
\(33\) −1.62321 + 0.476619i −0.282565 + 0.0829687i
\(34\) 0 0
\(35\) 7.23667 + 6.27061i 1.22322 + 1.05993i
\(36\) 0 0
\(37\) −9.89680 + 4.51972i −1.62702 + 0.743037i −0.999371 − 0.0354574i \(-0.988711\pi\)
−0.627652 + 0.778494i \(0.715984\pi\)
\(38\) 0 0
\(39\) 6.56041 + 1.92631i 1.05051 + 0.308457i
\(40\) 0 0
\(41\) 0.599928 − 1.31366i 0.0936930 − 0.205159i −0.856983 − 0.515345i \(-0.827664\pi\)
0.950676 + 0.310186i \(0.100391\pi\)
\(42\) 0 0
\(43\) −2.59779 + 4.04224i −0.396160 + 0.616436i −0.980837 − 0.194829i \(-0.937585\pi\)
0.584678 + 0.811266i \(0.301221\pi\)
\(44\) 0 0
\(45\) 3.39392i 0.505935i
\(46\) 0 0
\(47\) 3.84679 0.561112 0.280556 − 0.959838i \(-0.409481\pi\)
0.280556 + 0.959838i \(0.409481\pi\)
\(48\) 0 0
\(49\) 14.7259 + 9.46374i 2.10370 + 1.35196i
\(50\) 0 0
\(51\) −1.73124 − 0.790632i −0.242422 − 0.110711i
\(52\) 0 0
\(53\) 1.96998 − 6.70914i 0.270598 − 0.921572i −0.706309 − 0.707904i \(-0.749641\pi\)
0.976906 − 0.213668i \(-0.0685409\pi\)
\(54\) 0 0
\(55\) 1.21812 + 2.66730i 0.164251 + 0.359659i
\(56\) 0 0
\(57\) 0.232981 − 0.268874i 0.0308590 − 0.0356132i
\(58\) 0 0
\(59\) 0.804164 + 2.73873i 0.104693 + 0.356552i 0.995132 − 0.0985505i \(-0.0314206\pi\)
−0.890439 + 0.455103i \(0.849602\pi\)
\(60\) 0 0
\(61\) 4.54228 + 7.06793i 0.581579 + 0.904955i 0.999995 − 0.00330065i \(-0.00105063\pi\)
−0.418415 + 0.908256i \(0.637414\pi\)
\(62\) 0 0
\(63\) 1.23606 + 8.59698i 0.155729 + 1.08312i
\(64\) 0 0
\(65\) 1.68660 − 11.7306i 0.209197 − 1.45500i
\(66\) 0 0
\(67\) 3.99062 − 3.45789i 0.487532 − 0.422448i −0.376095 − 0.926581i \(-0.622733\pi\)
0.863626 + 0.504133i \(0.168188\pi\)
\(68\) 0 0
\(69\) 1.70798 + 5.07230i 0.205616 + 0.610633i
\(70\) 0 0
\(71\) −8.15113 − 9.40690i −0.967361 − 1.11639i −0.993164 − 0.116727i \(-0.962760\pi\)
0.0258031 − 0.999667i \(-0.491786\pi\)
\(72\) 0 0
\(73\) −0.932891 + 6.48840i −0.109187 + 0.759410i 0.859502 + 0.511132i \(0.170774\pi\)
−0.968689 + 0.248278i \(0.920135\pi\)
\(74\) 0 0
\(75\) 1.38994 − 0.199843i 0.160496 − 0.0230758i
\(76\) 0 0
\(77\) 4.05699 + 6.31280i 0.462337 + 0.719410i
\(78\) 0 0
\(79\) −13.1546 + 3.86253i −1.48000 + 0.434568i −0.919336 − 0.393473i \(-0.871273\pi\)
−0.560667 + 0.828041i \(0.689455\pi\)
\(80\) 0 0
\(81\) 0.430852 − 0.497229i 0.0478724 − 0.0552477i
\(82\) 0 0
\(83\) −11.8816 + 5.42616i −1.30418 + 0.595598i −0.941719 − 0.336401i \(-0.890790\pi\)
−0.362459 + 0.932000i \(0.618063\pi\)
\(84\) 0 0
\(85\) −0.929398 + 3.16524i −0.100807 + 0.343318i
\(86\) 0 0
\(87\) −1.93312 + 4.23295i −0.207253 + 0.453820i
\(88\) 0 0
\(89\) −7.53125 − 4.84004i −0.798310 − 0.513043i 0.0767536 − 0.997050i \(-0.475545\pi\)
−0.875064 + 0.484007i \(0.839181\pi\)
\(90\) 0 0
\(91\) − 30.3284i − 3.17928i
\(92\) 0 0
\(93\) − 2.32102i − 0.240678i
\(94\) 0 0
\(95\) −0.518765 − 0.333390i −0.0532241 − 0.0342051i
\(96\) 0 0
\(97\) −0.264613 + 0.579421i −0.0268674 + 0.0588313i −0.922590 − 0.385782i \(-0.873932\pi\)
0.895723 + 0.444613i \(0.146659\pi\)
\(98\) 0 0
\(99\) −0.749328 + 2.55198i −0.0753103 + 0.256483i
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.657.7 220
4.3 odd 2 184.2.p.a.13.20 yes 220
8.3 odd 2 184.2.p.a.13.14 ✓ 220
8.5 even 2 inner 736.2.x.a.657.16 220
23.16 even 11 inner 736.2.x.a.177.16 220
92.39 odd 22 184.2.p.a.85.14 yes 220
184.85 even 22 inner 736.2.x.a.177.7 220
184.131 odd 22 184.2.p.a.85.20 yes 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.13.14 ✓ 220 8.3 odd 2
184.2.p.a.13.20 yes 220 4.3 odd 2
184.2.p.a.85.14 yes 220 92.39 odd 22
184.2.p.a.85.20 yes 220 184.131 odd 22
736.2.x.a.177.7 220 184.85 even 22 inner
736.2.x.a.177.16 220 23.16 even 11 inner
736.2.x.a.657.7 220 1.1 even 1 trivial
736.2.x.a.657.16 220 8.5 even 2 inner