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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 49.10
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.302712 - 0.138244i) q^{3} +(-2.42383 - 2.10026i) q^{5} +(3.31871 - 2.13281i) q^{7} +(-1.89206 - 2.18355i) q^{9} +(-1.71129 - 0.246047i) q^{11} +(-2.84392 + 4.42522i) q^{13} +(0.443376 + 0.970857i) q^{15} +(-4.25725 - 1.25004i) q^{17} +(0.717267 + 2.44279i) q^{19} +(-1.29946 + 0.186835i) q^{21} +(1.06992 + 4.67496i) q^{23} +(0.752288 + 5.23228i) q^{25} +(0.552156 + 1.88047i) q^{27} +(0.815234 - 2.77643i) q^{29} +(-0.976875 - 2.13906i) q^{31} +(0.484015 + 0.311058i) q^{33} +(-12.5235 - 1.80060i) q^{35} +(2.53613 - 2.19757i) q^{37} +(1.47265 - 0.946414i) q^{39} +(-5.17915 + 5.97706i) q^{41} +(-7.18551 - 3.28151i) q^{43} +9.26639i q^{45} -8.61069 q^{47} +(3.55708 - 7.78893i) q^{49} +(1.11591 + 0.966942i) q^{51} +(-6.99377 - 10.8825i) q^{53} +(3.63113 + 4.19054i) q^{55} +(0.120575 - 0.838620i) q^{57} +(0.808852 - 1.25860i) q^{59} +(-6.38861 + 2.91758i) q^{61} +(-10.9363 - 3.21119i) q^{63} +(16.1873 - 4.75302i) q^{65} +(3.42719 - 0.492755i) q^{67} +(0.322407 - 1.56308i) q^{69} +(-1.54568 - 10.7504i) q^{71} +(-3.07056 + 0.901599i) q^{73} +(0.495605 - 1.68788i) q^{75} +(-6.20406 + 2.83330i) q^{77} +(11.4777 + 7.37629i) q^{79} +(-1.14073 + 7.93396i) q^{81} +(7.72136 - 6.69059i) q^{83} +(7.69344 + 11.9712i) q^{85} +(-0.630607 + 0.727759i) q^{87} +(0.911256 - 1.99537i) q^{89} +20.7516i q^{91} +0.782567i q^{93} +(3.39196 - 7.42736i) q^{95} +(7.40796 - 8.54924i) q^{97} +(2.70061 + 4.20223i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) 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
\(3\) −0.302712 − 0.138244i −0.174771 − 0.0798153i 0.326108 − 0.945332i \(-0.394263\pi\)
−0.500879 + 0.865517i \(0.666990\pi\)
\(4\) 0 0
\(5\) −2.42383 − 2.10026i −1.08397 − 0.939267i −0.0856012 − 0.996329i \(-0.527281\pi\)
−0.998371 + 0.0570626i \(0.981827\pi\)
\(6\) 0 0
\(7\) 3.31871 − 2.13281i 1.25436 − 0.806125i 0.266854 − 0.963737i \(-0.414016\pi\)
0.987501 + 0.157611i \(0.0503794\pi\)
\(8\) 0 0
\(9\) −1.89206 − 2.18355i −0.630686 − 0.727851i
\(10\) 0 0
\(11\) −1.71129 − 0.246047i −0.515974 − 0.0741859i −0.120591 − 0.992702i \(-0.538479\pi\)
−0.395383 + 0.918516i \(0.629388\pi\)
\(12\) 0 0
\(13\) −2.84392 + 4.42522i −0.788760 + 1.22734i 0.181050 + 0.983474i \(0.442050\pi\)
−0.969810 + 0.243861i \(0.921586\pi\)
\(14\) 0 0
\(15\) 0.443376 + 0.970857i 0.114479 + 0.250674i
\(16\) 0 0
\(17\) −4.25725 − 1.25004i −1.03253 − 0.303179i −0.278793 − 0.960351i \(-0.589934\pi\)
−0.753741 + 0.657172i \(0.771753\pi\)
\(18\) 0 0
\(19\) 0.717267 + 2.44279i 0.164552 + 0.560414i 0.999942 + 0.0107557i \(0.00342372\pi\)
−0.835390 + 0.549658i \(0.814758\pi\)
\(20\) 0 0
\(21\) −1.29946 + 0.186835i −0.283566 + 0.0407707i
\(22\) 0 0
\(23\) 1.06992 + 4.67496i 0.223095 + 0.974797i
\(24\) 0 0
\(25\) 0.752288 + 5.23228i 0.150458 + 1.04646i
\(26\) 0 0
\(27\) 0.552156 + 1.88047i 0.106262 + 0.361897i
\(28\) 0 0
\(29\) 0.815234 − 2.77643i 0.151385 − 0.515570i −0.848523 − 0.529159i \(-0.822507\pi\)
0.999908 + 0.0135889i \(0.00432562\pi\)
\(30\) 0 0
\(31\) −0.976875 − 2.13906i −0.175452 − 0.384186i 0.801392 − 0.598140i \(-0.204093\pi\)
−0.976844 + 0.213954i \(0.931366\pi\)
\(32\) 0 0
\(33\) 0.484015 + 0.311058i 0.0842562 + 0.0541482i
\(34\) 0 0
\(35\) −12.5235 − 1.80060i −2.11685 − 0.304358i
\(36\) 0 0
\(37\) 2.53613 − 2.19757i 0.416937 − 0.361278i −0.420985 − 0.907068i \(-0.638316\pi\)
0.837922 + 0.545789i \(0.183770\pi\)
\(38\) 0 0
\(39\) 1.47265 − 0.946414i 0.235813 − 0.151548i
\(40\) 0 0
\(41\) −5.17915 + 5.97706i −0.808847 + 0.933460i −0.998832 − 0.0483275i \(-0.984611\pi\)
0.189984 + 0.981787i \(0.439156\pi\)
\(42\) 0 0
\(43\) −7.18551 − 3.28151i −1.09578 − 0.500426i −0.216283 − 0.976331i \(-0.569393\pi\)
−0.879497 + 0.475905i \(0.842121\pi\)
\(44\) 0 0
\(45\) 9.26639i 1.38135i
\(46\) 0 0
\(47\) −8.61069 −1.25600 −0.627999 − 0.778214i \(-0.716126\pi\)
−0.627999 + 0.778214i \(0.716126\pi\)
\(48\) 0 0
\(49\) 3.55708 − 7.78893i 0.508155 − 1.11270i
\(50\) 0 0
\(51\) 1.11591 + 0.966942i 0.156259 + 0.135399i
\(52\) 0 0
\(53\) −6.99377 − 10.8825i −0.960668 − 1.49483i −0.866442 − 0.499278i \(-0.833599\pi\)
−0.0942263 − 0.995551i \(-0.530038\pi\)
\(54\) 0 0
\(55\) 3.63113 + 4.19054i 0.489621 + 0.565053i
\(56\) 0 0
\(57\) 0.120575 − 0.838620i 0.0159706 − 0.111078i
\(58\) 0 0
\(59\) 0.808852 − 1.25860i 0.105304 − 0.163855i −0.784603 − 0.619999i \(-0.787133\pi\)
0.889906 + 0.456143i \(0.150770\pi\)
\(60\) 0 0
\(61\) −6.38861 + 2.91758i −0.817978 + 0.373558i −0.780046 − 0.625722i \(-0.784804\pi\)
−0.0379320 + 0.999280i \(0.512077\pi\)
\(62\) 0 0
\(63\) −10.9363 − 3.21119i −1.37784 − 0.404571i
\(64\) 0 0
\(65\) 16.1873 − 4.75302i 2.00779 − 0.589540i
\(66\) 0 0
\(67\) 3.42719 − 0.492755i 0.418698 − 0.0601996i 0.0702535 − 0.997529i \(-0.477619\pi\)
0.348444 + 0.937330i \(0.386710\pi\)
\(68\) 0 0
\(69\) 0.322407 − 1.56308i 0.0388132 − 0.188173i
\(70\) 0 0
\(71\) −1.54568 − 10.7504i −0.183438 − 1.27584i −0.848558 − 0.529103i \(-0.822529\pi\)
0.665120 − 0.746737i \(-0.268381\pi\)
\(72\) 0 0
\(73\) −3.07056 + 0.901599i −0.359382 + 0.105524i −0.456439 − 0.889755i \(-0.650875\pi\)
0.0970567 + 0.995279i \(0.469057\pi\)
\(74\) 0 0
\(75\) 0.495605 − 1.68788i 0.0572276 − 0.194899i
\(76\) 0 0
\(77\) −6.20406 + 2.83330i −0.707018 + 0.322884i
\(78\) 0 0
\(79\) 11.4777 + 7.37629i 1.29135 + 0.829898i 0.992242 − 0.124323i \(-0.0396758\pi\)
0.299104 + 0.954220i \(0.403312\pi\)
\(80\) 0 0
\(81\) −1.14073 + 7.93396i −0.126748 + 0.881551i
\(82\) 0 0
\(83\) 7.72136 − 6.69059i 0.847529 − 0.734388i −0.118463 − 0.992958i \(-0.537797\pi\)
0.965992 + 0.258570i \(0.0832514\pi\)
\(84\) 0 0
\(85\) 7.69344 + 11.9712i 0.834471 + 1.29846i
\(86\) 0 0
\(87\) −0.630607 + 0.727759i −0.0676081 + 0.0780239i
\(88\) 0 0
\(89\) 0.911256 − 1.99537i 0.0965929 − 0.211509i −0.855167 − 0.518352i \(-0.826546\pi\)
0.951760 + 0.306844i \(0.0992728\pi\)
\(90\) 0 0
\(91\) 20.7516i 2.17535i
\(92\) 0 0
\(93\) 0.782567i 0.0811484i
\(94\) 0 0
\(95\) 3.39196 − 7.42736i 0.348008 − 0.762031i
\(96\) 0 0
\(97\) 7.40796 − 8.54924i 0.752164 − 0.868044i −0.242612 − 0.970123i \(-0.578004\pi\)
0.994776 + 0.102080i \(0.0325496\pi\)
\(98\) 0 0
\(99\) 2.70061 + 4.20223i 0.271422 + 0.422340i
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.49.10 220
4.3 odd 2 184.2.p.a.141.2 yes 220
8.3 odd 2 184.2.p.a.141.22 yes 220
8.5 even 2 inner 736.2.x.a.49.13 220
23.8 even 11 inner 736.2.x.a.721.13 220
92.31 odd 22 184.2.p.a.77.22 yes 220
184.77 even 22 inner 736.2.x.a.721.10 220
184.123 odd 22 184.2.p.a.77.2 ✓ 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.2 ✓ 220 184.123 odd 22
184.2.p.a.77.22 yes 220 92.31 odd 22
184.2.p.a.141.2 yes 220 4.3 odd 2
184.2.p.a.141.22 yes 220 8.3 odd 2
736.2.x.a.49.10 220 1.1 even 1 trivial
736.2.x.a.49.13 220 8.5 even 2 inner
736.2.x.a.721.10 220 184.77 even 22 inner
736.2.x.a.721.13 220 23.8 even 11 inner