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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [40,11,Mod(13,40)] 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("40.13"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 40.i (of order \(4\), degree \(2\), 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: \(25.4142901069\)
Analytic rank: \(0\)
Dimension: \(116\)
Relative dimension: \(58\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.12
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.12

$q$-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+(-26.7504 - 17.5617i) q^{2} +(-186.299 + 186.299i) q^{3} +(407.172 + 939.568i) q^{4} +(-1009.16 - 2957.57i) q^{5} +(8255.33 - 1711.85i) q^{6} +(-18272.5 + 18272.5i) q^{7} +(5608.41 - 32284.5i) q^{8} -10365.9i q^{9} +(-24944.5 + 96838.9i) q^{10} +232434. i q^{11} +(-250897. - 99185.0i) q^{12} +(-167904. + 167904. i) q^{13} +(809693. - 167901. i) q^{14} +(739000. + 362987. i) q^{15} +(-716998. + 765131. i) q^{16} +(-1.26403e6 + 1.26403e6i) q^{17} +(-182043. + 277292. i) q^{18} +3.55941e6 q^{19} +(2.36793e6 - 2.15242e6i) q^{20} -6.80830e6i q^{21} +(4.08194e6 - 6.21771e6i) q^{22} +(3.09993e6 + 3.09993e6i) q^{23} +(4.96974e6 + 7.05942e6i) q^{24} +(-7.72880e6 + 5.96934e6i) q^{25} +(7.44020e6 - 1.54282e6i) q^{26} +(-9.06963e6 - 9.06963e6i) q^{27} +(-2.46083e7 - 9.72819e6i) q^{28} -1.50293e7 q^{29} +(-1.33939e7 - 2.26882e7i) q^{30} -4.57072e7 q^{31} +(3.26170e7 - 7.87585e6i) q^{32} +(-4.33023e7 - 4.33023e7i) q^{33} +(5.60119e7 - 1.16148e7i) q^{34} +(7.24820e7 + 3.56022e7i) q^{35} +(9.73946e6 - 4.22070e6i) q^{36} +(-6.65211e7 - 6.65211e7i) q^{37} +(-9.52156e7 - 6.25093e7i) q^{38} -6.25609e7i q^{39} +(-1.01143e8 + 1.59931e7i) q^{40} +1.80176e8 q^{41} +(-1.19566e8 + 1.82125e8i) q^{42} +(1.37288e8 - 1.37288e8i) q^{43} +(-2.18387e8 + 9.46405e7i) q^{44} +(-3.06579e7 + 1.04609e7i) q^{45} +(-2.84844e7 - 1.37365e8i) q^{46} +(-7.78975e7 + 7.78975e7i) q^{47} +(-8.96700e6 - 2.76120e8i) q^{48} -3.85292e8i q^{49} +(3.11581e8 - 2.39513e7i) q^{50} -4.70976e8i q^{51} +(-2.26123e8 - 8.93916e7i) q^{52} +(-8.79989e7 + 8.79989e7i) q^{53} +(8.33382e7 + 4.01895e8i) q^{54} +(6.87440e8 - 2.34564e8i) q^{55} +(4.87438e8 + 6.92397e8i) q^{56} +(-6.63115e8 + 6.63115e8i) q^{57} +(4.02040e8 + 2.63940e8i) q^{58} +2.25905e8 q^{59} +(-4.01508e7 + 8.42138e8i) q^{60} -3.70093e8i q^{61} +(1.22269e9 + 8.02697e8i) q^{62} +(1.89411e8 + 1.89411e8i) q^{63} +(-1.01083e9 - 3.62129e8i) q^{64} +(6.66032e8 + 3.27146e8i) q^{65} +(3.97892e8 + 1.91882e9i) q^{66} +(-4.51767e8 - 4.51767e8i) q^{67} +(-1.70232e9 - 6.72964e8i) q^{68} -1.15503e9 q^{69} +(-1.31369e9 - 2.22528e9i) q^{70} +1.10479e9 q^{71} +(-3.34658e8 - 5.81362e7i) q^{72} +(3.56592e8 + 3.56592e8i) q^{73} +(6.11243e8 + 2.94769e9i) q^{74} +(3.27787e8 - 2.55196e9i) q^{75} +(1.44929e9 + 3.34430e9i) q^{76} +(-4.24715e9 - 4.24715e9i) q^{77} +(-1.09868e9 + 1.67353e9i) q^{78} +1.55014e9i q^{79} +(2.98650e9 + 1.34843e9i) q^{80} +3.99143e9 q^{81} +(-4.81979e9 - 3.16420e9i) q^{82} +(-4.52551e9 + 4.52551e9i) q^{83} +(6.39686e9 - 2.77215e9i) q^{84} +(5.01407e9 + 2.46284e9i) q^{85} +(-6.08354e9 + 1.26150e9i) q^{86} +(2.79995e9 - 2.79995e9i) q^{87} +(7.50401e9 + 1.30358e9i) q^{88} +1.95095e9i q^{89} +(1.00382e9 + 2.58572e8i) q^{90} -6.13606e9i q^{91} +(-1.65039e9 + 4.17480e9i) q^{92} +(8.51522e9 - 8.51522e9i) q^{93} +(3.45180e9 - 7.15777e8i) q^{94} +(-3.59202e9 - 1.05272e10i) q^{95} +(-4.60927e9 + 7.54380e9i) q^{96} +(-4.99131e8 + 4.99131e8i) q^{97} +(-6.76639e9 + 1.03067e10i) q^{98} +2.40939e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 116 q - 2 q^{2} - 4864 q^{6} - 4 q^{7} + 69124 q^{8} + 256166 q^{10} + 502036 q^{12} - 4 q^{15} + 3502536 q^{16} - 905772 q^{17} - 5688470 q^{18} + 4385256 q^{20} + 9808012 q^{22} - 4 q^{23} - 1476988 q^{25}+ \cdots - 65612488734 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −26.7504 − 17.5617i −0.835951 − 0.548804i
\(3\) −186.299 + 186.299i −0.766664 + 0.766664i −0.977518 − 0.210854i \(-0.932376\pi\)
0.210854 + 0.977518i \(0.432376\pi\)
\(4\) 407.172 + 939.568i 0.397629 + 0.917546i
\(5\) −1009.16 − 2957.57i −0.322932 − 0.946422i
\(6\) 8255.33 − 1711.85i 1.06164 − 0.220145i
\(7\) −18272.5 + 18272.5i −1.08719 + 1.08719i −0.0913783 + 0.995816i \(0.529127\pi\)
−0.995816 + 0.0913783i \(0.970873\pi\)
\(8\) 5608.41 − 32284.5i 0.171155 − 0.985244i
\(9\) − 10365.9i − 0.175547i
\(10\) −24944.5 + 96838.9i −0.249445 + 0.968389i
\(11\) 232434.i 1.44323i 0.692293 + 0.721616i \(0.256600\pi\)
−0.692293 + 0.721616i \(0.743400\pi\)
\(12\) −250897. − 99185.0i −1.00830 − 0.398602i
\(13\) −167904. + 167904.i −0.452215 + 0.452215i −0.896089 − 0.443874i \(-0.853604\pi\)
0.443874 + 0.896089i \(0.353604\pi\)
\(14\) 809693. − 167901.i 1.50550 − 0.312185i
\(15\) 739000. + 362987.i 0.973168 + 0.478007i
\(16\) −716998. + 765131.i −0.683783 + 0.729685i
\(17\) −1.26403e6 + 1.26403e6i −0.890251 + 0.890251i −0.994546 − 0.104295i \(-0.966741\pi\)
0.104295 + 0.994546i \(0.466741\pi\)
\(18\) −182043. + 277292.i −0.0963411 + 0.146749i
\(19\) 3.55941e6 1.43751 0.718753 − 0.695266i \(-0.244713\pi\)
0.718753 + 0.695266i \(0.244713\pi\)
\(20\) 2.36793e6 − 2.15242e6i 0.739979 − 0.672630i
\(21\) − 6.80830e6i − 1.66703i
\(22\) 4.08194e6 − 6.21771e6i 0.792051 − 1.20647i
\(23\) 3.09993e6 + 3.09993e6i 0.481630 + 0.481630i 0.905652 − 0.424022i \(-0.139382\pi\)
−0.424022 + 0.905652i \(0.639382\pi\)
\(24\) 4.96974e6 + 7.05942e6i 0.624133 + 0.886570i
\(25\) −7.72880e6 + 5.96934e6i −0.791430 + 0.611260i
\(26\) 7.44020e6 − 1.54282e6i 0.626207 − 0.129852i
\(27\) −9.06963e6 − 9.06963e6i −0.632078 − 0.632078i
\(28\) −2.46083e7 − 9.72819e6i −1.42985 − 0.565252i
\(29\) −1.50293e7 −0.732738 −0.366369 − 0.930470i \(-0.619399\pi\)
−0.366369 + 0.930470i \(0.619399\pi\)
\(30\) −1.33939e7 − 2.26882e7i −0.551189 − 0.933669i
\(31\) −4.57072e7 −1.59653 −0.798263 − 0.602309i \(-0.794247\pi\)
−0.798263 + 0.602309i \(0.794247\pi\)
\(32\) 3.26170e7 − 7.87585e6i 0.972063 − 0.234719i
\(33\) −4.33023e7 − 4.33023e7i −1.10647 − 1.10647i
\(34\) 5.60119e7 − 1.16148e7i 1.23278 − 0.255633i
\(35\) 7.24820e7 + 3.56022e7i 1.38004 + 0.677855i
\(36\) 9.73946e6 − 4.22070e6i 0.161073 − 0.0698027i
\(37\) −6.65211e7 − 6.65211e7i −0.959291 − 0.959291i 0.0399117 − 0.999203i \(-0.487292\pi\)
−0.999203 + 0.0399117i \(0.987292\pi\)
\(38\) −9.52156e7 − 6.25093e7i −1.20168 − 0.788908i
\(39\) − 6.25609e7i − 0.693394i
\(40\) −1.01143e8 + 1.59931e7i −0.987728 + 0.156182i
\(41\) 1.80176e8 1.55517 0.777584 − 0.628779i \(-0.216445\pi\)
0.777584 + 0.628779i \(0.216445\pi\)
\(42\) −1.19566e8 + 1.82125e8i −0.914870 + 1.39355i
\(43\) 1.37288e8 − 1.37288e8i 0.933881 − 0.933881i −0.0640649 − 0.997946i \(-0.520406\pi\)
0.997946 + 0.0640649i \(0.0204065\pi\)
\(44\) −2.18387e8 + 9.46405e7i −1.32423 + 0.573870i
\(45\) −3.06579e7 + 1.04609e7i −0.166142 + 0.0566899i
\(46\) −2.84844e7 − 1.37365e8i −0.138299 − 0.666939i
\(47\) −7.78975e7 + 7.78975e7i −0.339652 + 0.339652i −0.856236 − 0.516584i \(-0.827203\pi\)
0.516584 + 0.856236i \(0.327203\pi\)
\(48\) −8.96700e6 − 2.76120e8i −0.0351918 − 1.08366i
\(49\) − 3.85292e8i − 1.36398i
\(50\) 3.11581e8 − 2.39513e7i 0.997059 − 0.0766442i
\(51\) − 4.70976e8i − 1.36505i
\(52\) −2.26123e8 − 8.93916e7i −0.594742 − 0.235115i
\(53\) −8.79989e7 + 8.79989e7i −0.210425 + 0.210425i −0.804448 − 0.594023i \(-0.797539\pi\)
0.594023 + 0.804448i \(0.297539\pi\)
\(54\) 8.33382e7 + 4.01895e8i 0.181500 + 0.875273i
\(55\) 6.87440e8 − 2.34564e8i 1.36591 − 0.466066i
\(56\) 4.87438e8 + 6.92397e8i 0.885073 + 1.25723i
\(57\) −6.63115e8 + 6.63115e8i −1.10208 + 1.10208i
\(58\) 4.02040e8 + 2.63940e8i 0.612533 + 0.402129i
\(59\) 2.25905e8 0.315984 0.157992 − 0.987440i \(-0.449498\pi\)
0.157992 + 0.987440i \(0.449498\pi\)
\(60\) −4.01508e7 + 8.42138e8i −0.0516343 + 1.08300i
\(61\) − 3.70093e8i − 0.438189i −0.975704 − 0.219094i \(-0.929690\pi\)
0.975704 − 0.219094i \(-0.0703103\pi\)
\(62\) 1.22269e9 + 8.02697e8i 1.33462 + 0.876179i
\(63\) 1.89411e8 + 1.89411e8i 0.190854 + 0.190854i
\(64\) −1.01083e9 − 3.62129e8i −0.941412 − 0.337259i
\(65\) 6.66032e8 + 3.27146e8i 0.574021 + 0.281952i
\(66\) 3.97892e8 + 1.91882e9i 0.317721 + 1.53220i
\(67\) −4.51767e8 − 4.51767e8i −0.334611 − 0.334611i 0.519723 − 0.854335i \(-0.326035\pi\)
−0.854335 + 0.519723i \(0.826035\pi\)
\(68\) −1.70232e9 − 6.72964e8i −1.17084 − 0.462858i
\(69\) −1.15503e9 −0.738497
\(70\) −1.31369e9 − 2.22528e9i −0.781633 − 1.32402i
\(71\) 1.10479e9 0.612334 0.306167 − 0.951978i \(-0.400953\pi\)
0.306167 + 0.951978i \(0.400953\pi\)
\(72\) −3.34658e8 − 5.81362e7i −0.172957 − 0.0300458i
\(73\) 3.56592e8 + 3.56592e8i 0.172012 + 0.172012i 0.787863 − 0.615851i \(-0.211188\pi\)
−0.615851 + 0.787863i \(0.711188\pi\)
\(74\) 6.11243e8 + 2.94769e9i 0.275458 + 1.32838i
\(75\) 3.27787e8 − 2.55196e9i 0.138129 − 1.07539i
\(76\) 1.44929e9 + 3.34430e9i 0.571593 + 1.31898i
\(77\) −4.24715e9 − 4.24715e9i −1.56907 − 1.56907i
\(78\) −1.09868e9 + 1.67353e9i −0.380537 + 0.579644i
\(79\) 1.55014e9i 0.503774i 0.967757 + 0.251887i \(0.0810511\pi\)
−0.967757 + 0.251887i \(0.918949\pi\)
\(80\) 2.98650e9 + 1.34843e9i 0.911406 + 0.411508i
\(81\) 3.99143e9 1.14473
\(82\) −4.81979e9 − 3.16420e9i −1.30005 − 0.853483i
\(83\) −4.52551e9 + 4.52551e9i −1.14889 + 1.14889i −0.162113 + 0.986772i \(0.551831\pi\)
−0.986772 + 0.162113i \(0.948169\pi\)
\(84\) 6.39686e9 − 2.77215e9i 1.52957 − 0.662857i
\(85\) 5.01407e9 + 2.46284e9i 1.13004 + 0.555063i
\(86\) −6.08354e9 + 1.26150e9i −1.29320 + 0.268161i
\(87\) 2.79995e9 − 2.79995e9i 0.561764 − 0.561764i
\(88\) 7.50401e9 + 1.30358e9i 1.42194 + 0.247016i
\(89\) 1.95095e9i 0.349379i 0.984624 + 0.174690i \(0.0558922\pi\)
−0.984624 + 0.174690i \(0.944108\pi\)
\(90\) 1.00382e9 + 2.58572e8i 0.169998 + 0.0437893i
\(91\) − 6.13606e9i − 0.983292i
\(92\) −1.65039e9 + 4.17480e9i −0.250408 + 0.633428i
\(93\) 8.51522e9 − 8.51522e9i 1.22400 − 1.22400i
\(94\) 3.45180e9 − 7.15777e8i 0.470335 − 0.0975301i
\(95\) −3.59202e9 − 1.05272e10i −0.464217 − 1.36049i
\(96\) −4.60927e9 + 7.54380e9i −0.565296 + 0.925196i
\(97\) −4.99131e8 + 4.99131e8i −0.0581241 + 0.0581241i −0.735571 − 0.677447i \(-0.763086\pi\)
0.677447 + 0.735571i \(0.263086\pi\)
\(98\) −6.76639e9 + 1.03067e10i −0.748560 + 1.14022i
\(99\) 2.40939e9 0.253356
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 40.11.i.a.13.12 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.40 yes 116
8.5 even 2 inner 40.11.i.a.13.40 yes 116
40.37 odd 4 inner 40.11.i.a.37.12 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.12 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.40 yes 116 8.5 even 2 inner
40.11.i.a.37.12 yes 116 40.37 odd 4 inner
40.11.i.a.37.40 yes 116 5.2 odd 4 inner