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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 13.7
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.7

$q$-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+(-30.4587 + 9.81171i) q^{2} +(-258.070 + 258.070i) q^{3} +(831.461 - 597.703i) q^{4} +(-2935.38 - 1071.99i) q^{5} +(5328.36 - 10392.6i) q^{6} +(-12430.0 + 12430.0i) q^{7} +(-19460.7 + 26363.3i) q^{8} -74151.3i q^{9} +(99925.8 + 3850.39i) q^{10} -169289. i q^{11} +(-60325.9 + 368824. i) q^{12} +(321675. - 321675. i) q^{13} +(256642. - 500560. i) q^{14} +(1.03418e6 - 480884. i) q^{15} +(334078. - 993933. i) q^{16} +(916510. - 916510. i) q^{17} +(727550. + 2.25855e6i) q^{18} -4.86194e6 q^{19} +(-3.08139e6 + 863165. i) q^{20} -6.41562e6i q^{21} +(1.66101e6 + 5.15632e6i) q^{22} +(3.12357e6 + 3.12357e6i) q^{23} +(-1.78135e6 - 1.18258e7i) q^{24} +(7.46729e6 + 6.29341e6i) q^{25} +(-6.64160e6 + 1.29540e7i) q^{26} +(3.89744e6 + 3.89744e6i) q^{27} +(-2.90561e6 + 1.77645e7i) q^{28} +3.52300e6 q^{29} +(-2.67815e7 + 2.47942e7i) q^{30} -2.51415e7 q^{31} +(-423395. + 3.35518e7i) q^{32} +(4.36884e7 + 4.36884e7i) q^{33} +(-1.89231e7 + 3.69082e7i) q^{34} +(4.98116e7 - 2.31619e7i) q^{35} +(-4.43204e7 - 6.16539e7i) q^{36} +(-5.10891e7 - 5.10891e7i) q^{37} +(1.48088e8 - 4.77039e7i) q^{38} +1.66029e8i q^{39} +(8.53858e7 - 5.65245e7i) q^{40} +7.90174e6 q^{41} +(6.29481e7 + 1.95411e8i) q^{42} +(-1.07733e8 + 1.07733e8i) q^{43} +(-1.01185e8 - 1.40757e8i) q^{44} +(-7.94896e7 + 2.17662e8i) q^{45} +(-1.25787e8 - 6.44921e7i) q^{46} +(-1.38656e8 + 1.38656e8i) q^{47} +(1.70289e8 + 3.42720e8i) q^{48} -2.65342e7i q^{49} +(-2.89193e8 - 1.18422e8i) q^{50} +4.73047e8i q^{51} +(7.51940e7 - 4.59726e8i) q^{52} +(-2.31386e8 + 2.31386e8i) q^{53} +(-1.56951e8 - 8.04703e7i) q^{54} +(-1.81477e8 + 4.96928e8i) q^{55} +(-8.57991e7 - 5.69592e8i) q^{56} +(1.25472e9 - 1.25472e9i) q^{57} +(-1.07306e8 + 3.45666e7i) q^{58} +6.77930e8 q^{59} +(5.72456e8 - 1.01797e9i) q^{60} -1.28489e9i q^{61} +(7.65777e8 - 2.46681e8i) q^{62} +(9.21700e8 + 9.21700e8i) q^{63} +(-3.16304e8 - 1.02610e9i) q^{64} +(-1.28907e9 + 5.99404e8i) q^{65} +(-1.75935e9 - 9.02033e8i) q^{66} +(3.16312e8 + 3.16312e8i) q^{67} +(2.14241e8 - 1.30984e9i) q^{68} -1.61220e9 q^{69} +(-1.28994e9 + 1.19422e9i) q^{70} +1.80367e9 q^{71} +(1.95487e9 + 1.44304e9i) q^{72} +(1.39170e9 + 1.39170e9i) q^{73} +(2.05738e9 + 1.05483e9i) q^{74} +(-3.55122e9 + 3.02941e8i) q^{75} +(-4.04251e9 + 2.90599e9i) q^{76} +(2.10426e9 + 2.10426e9i) q^{77} +(-1.62903e9 - 5.05703e9i) q^{78} -5.21149e9i q^{79} +(-2.04614e9 + 2.55944e9i) q^{80} +2.36693e9 q^{81} +(-2.40677e8 + 7.75296e7i) q^{82} +(6.50415e8 - 6.50415e8i) q^{83} +(-3.83463e9 - 5.33433e9i) q^{84} +(-3.67280e9 + 1.70781e9i) q^{85} +(2.22436e9 - 4.33846e9i) q^{86} +(-9.09180e8 + 9.09180e8i) q^{87} +(4.46302e9 + 3.29448e9i) q^{88} +4.39933e9i q^{89} +(2.85511e8 - 7.40963e9i) q^{90} +7.99683e9i q^{91} +(4.46409e9 + 7.30158e8i) q^{92} +(6.48827e9 - 6.48827e9i) q^{93} +(2.86283e9 - 5.58373e9i) q^{94} +(1.42716e10 + 5.21196e9i) q^{95} +(-8.54944e9 - 8.76797e9i) q^{96} +(-7.71908e9 + 7.71908e9i) q^{97} +(2.60346e8 + 8.08196e8i) q^{98} -1.25530e10 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −30.4587 + 9.81171i −0.951833 + 0.306616i
\(3\) −258.070 + 258.070i −1.06202 + 1.06202i −0.0640712 + 0.997945i \(0.520408\pi\)
−0.997945 + 0.0640712i \(0.979592\pi\)
\(4\) 831.461 − 597.703i 0.811973 − 0.583694i
\(5\) −2935.38 − 1071.99i −0.939322 − 0.343038i
\(6\) 5328.36 − 10392.6i 0.685232 − 1.33649i
\(7\) −12430.0 + 12430.0i −0.739572 + 0.739572i −0.972495 − 0.232923i \(-0.925171\pi\)
0.232923 + 0.972495i \(0.425171\pi\)
\(8\) −19460.7 + 26363.3i −0.593893 + 0.804544i
\(9\) − 74151.3i − 1.25576i
\(10\) 99925.8 + 3850.39i 0.999258 + 0.0385039i
\(11\) − 169289.i − 1.05115i −0.850747 − 0.525576i \(-0.823850\pi\)
0.850747 − 0.525576i \(-0.176150\pi\)
\(12\) −60325.9 + 368824.i −0.242436 + 1.48222i
\(13\) 321675. − 321675.i 0.866363 − 0.866363i −0.125704 − 0.992068i \(-0.540119\pi\)
0.992068 + 0.125704i \(0.0401191\pi\)
\(14\) 256642. − 500560.i 0.477185 − 0.930714i
\(15\) 1.03418e6 − 480884.i 1.36189 − 0.633263i
\(16\) 334078. − 993933.i 0.318602 − 0.947889i
\(17\) 916510. − 916510.i 0.645495 − 0.645495i −0.306406 − 0.951901i \(-0.599127\pi\)
0.951901 + 0.306406i \(0.0991266\pi\)
\(18\) 727550. + 2.25855e6i 0.385035 + 1.19527i
\(19\) −4.86194e6 −1.96355 −0.981773 − 0.190056i \(-0.939133\pi\)
−0.981773 + 0.190056i \(0.939133\pi\)
\(20\) −3.08139e6 + 863165.i −0.962933 + 0.269739i
\(21\) − 6.41562e6i − 1.57088i
\(22\) 1.66101e6 + 5.15632e6i 0.322300 + 1.00052i
\(23\) 3.12357e6 + 3.12357e6i 0.485301 + 0.485301i 0.906820 − 0.421518i \(-0.138503\pi\)
−0.421518 + 0.906820i \(0.638503\pi\)
\(24\) −1.78135e6 − 1.18258e7i −0.223714 − 1.48516i
\(25\) 7.46729e6 + 6.29341e6i 0.764650 + 0.644446i
\(26\) −6.64160e6 + 1.29540e7i −0.558993 + 1.09027i
\(27\) 3.89744e6 + 3.89744e6i 0.271619 + 0.271619i
\(28\) −2.90561e6 + 1.77645e7i −0.168829 + 1.03220i
\(29\) 3.52300e6 0.171760 0.0858801 − 0.996305i \(-0.472630\pi\)
0.0858801 + 0.996305i \(0.472630\pi\)
\(30\) −2.67815e7 + 2.47942e7i −1.10212 + 1.02034i
\(31\) −2.51415e7 −0.878178 −0.439089 − 0.898443i \(-0.644699\pi\)
−0.439089 + 0.898443i \(0.644699\pi\)
\(32\) −423395. + 3.35518e7i −0.0126182 + 0.999920i
\(33\) 4.36884e7 + 4.36884e7i 1.11634 + 1.11634i
\(34\) −1.89231e7 + 3.69082e7i −0.416484 + 0.812322i
\(35\) 4.98116e7 − 2.31619e7i 0.948398 − 0.440995i
\(36\) −4.43204e7 − 6.16539e7i −0.732979 − 1.01964i
\(37\) −5.10891e7 − 5.10891e7i −0.736749 − 0.736749i 0.235198 − 0.971947i \(-0.424426\pi\)
−0.971947 + 0.235198i \(0.924426\pi\)
\(38\) 1.48088e8 − 4.77039e7i 1.86897 − 0.602054i
\(39\) 1.66029e8i 1.84018i
\(40\) 8.53858e7 − 5.65245e7i 0.833846 − 0.551997i
\(41\) 7.90174e6 0.0682030 0.0341015 − 0.999418i \(-0.489143\pi\)
0.0341015 + 0.999418i \(0.489143\pi\)
\(42\) 6.29481e7 + 1.95411e8i 0.481655 + 1.49521i
\(43\) −1.07733e8 + 1.07733e8i −0.732837 + 0.732837i −0.971181 − 0.238344i \(-0.923396\pi\)
0.238344 + 0.971181i \(0.423396\pi\)
\(44\) −1.01185e8 − 1.40757e8i −0.613551 − 0.853507i
\(45\) −7.94896e7 + 2.17662e8i −0.430772 + 1.17956i
\(46\) −1.25787e8 − 6.44921e7i −0.610727 − 0.313125i
\(47\) −1.38656e8 + 1.38656e8i −0.604574 + 0.604574i −0.941523 − 0.336949i \(-0.890605\pi\)
0.336949 + 0.941523i \(0.390605\pi\)
\(48\) 1.70289e8 + 3.42720e8i 0.668313 + 1.34503i
\(49\) − 2.65342e7i − 0.0939345i
\(50\) −2.89193e8 − 1.18422e8i −0.925417 − 0.378951i
\(51\) 4.73047e8i 1.37105i
\(52\) 7.51940e7 − 4.59726e8i 0.197773 − 1.20916i
\(53\) −2.31386e8 + 2.31386e8i −0.553296 + 0.553296i −0.927391 − 0.374095i \(-0.877954\pi\)
0.374095 + 0.927391i \(0.377954\pi\)
\(54\) −1.56951e8 − 8.04703e7i −0.341819 − 0.175254i
\(55\) −1.81477e8 + 4.96928e8i −0.360585 + 0.987370i
\(56\) −8.57991e7 − 5.69592e8i −0.155791 − 1.03425i
\(57\) 1.25472e9 − 1.25472e9i 2.08532 − 2.08532i
\(58\) −1.07306e8 + 3.45666e7i −0.163487 + 0.0526644i
\(59\) 6.77930e8 0.948255 0.474127 − 0.880456i \(-0.342764\pi\)
0.474127 + 0.880456i \(0.342764\pi\)
\(60\) 5.72456e8 − 1.01797e9i 0.736184 − 1.30912i
\(61\) − 1.28489e9i − 1.52130i −0.649160 − 0.760652i \(-0.724880\pi\)
0.649160 − 0.760652i \(-0.275120\pi\)
\(62\) 7.65777e8 − 2.46681e8i 0.835880 − 0.269263i
\(63\) 9.21700e8 + 9.21700e8i 0.928724 + 0.928724i
\(64\) −3.16304e8 − 1.02610e9i −0.294581 − 0.955627i
\(65\) −1.28907e9 + 5.99404e8i −1.11099 + 0.516598i
\(66\) −1.75935e9 − 9.02033e8i −1.40486 − 0.720283i
\(67\) 3.16312e8 + 3.16312e8i 0.234284 + 0.234284i 0.814478 − 0.580194i \(-0.197023\pi\)
−0.580194 + 0.814478i \(0.697023\pi\)
\(68\) 2.14241e8 − 1.30984e9i 0.147353 − 0.900896i
\(69\) −1.61220e9 −1.03080
\(70\) −1.28994e9 + 1.19422e9i −0.767500 + 0.710547i
\(71\) 1.80367e9 0.999688 0.499844 − 0.866115i \(-0.333391\pi\)
0.499844 + 0.866115i \(0.333391\pi\)
\(72\) 1.95487e9 + 1.44304e9i 1.01031 + 0.745787i
\(73\) 1.39170e9 + 1.39170e9i 0.671323 + 0.671323i 0.958021 − 0.286698i \(-0.0925578\pi\)
−0.286698 + 0.958021i \(0.592558\pi\)
\(74\) 2.05738e9 + 1.05483e9i 0.927161 + 0.475363i
\(75\) −3.55122e9 + 3.02941e8i −1.49648 + 0.127659i
\(76\) −4.04251e9 + 2.90599e9i −1.59435 + 1.14611i
\(77\) 2.10426e9 + 2.10426e9i 0.777403 + 0.777403i
\(78\) −1.62903e9 − 5.05703e9i −0.564230 − 1.75155i
\(79\) − 5.21149e9i − 1.69366i −0.531863 − 0.846831i \(-0.678508\pi\)
0.531863 − 0.846831i \(-0.321492\pi\)
\(80\) −2.04614e9 + 2.55944e9i −0.624431 + 0.781080i
\(81\) 2.36693e9 0.678830
\(82\) −2.40677e8 + 7.75296e7i −0.0649179 + 0.0209121i
\(83\) 6.50415e8 − 6.50415e8i 0.165120 − 0.165120i −0.619710 − 0.784831i \(-0.712750\pi\)
0.784831 + 0.619710i \(0.212750\pi\)
\(84\) −3.83463e9 − 5.33433e9i −0.916911 − 1.27551i
\(85\) −3.67280e9 + 1.70781e9i −0.827756 + 0.384898i
\(86\) 2.22436e9 − 4.33846e9i 0.472839 − 0.922238i
\(87\) −9.09180e8 + 9.09180e8i −0.182412 + 0.182412i
\(88\) 4.46302e9 + 3.29448e9i 0.845698 + 0.624272i
\(89\) 4.39933e9i 0.787837i 0.919145 + 0.393919i \(0.128881\pi\)
−0.919145 + 0.393919i \(0.871119\pi\)
\(90\) 2.85511e8 − 7.40963e9i 0.0483516 − 1.25483i
\(91\) 7.99683e9i 1.28148i
\(92\) 4.46409e9 + 7.30158e8i 0.677319 + 0.110784i
\(93\) 6.48827e9 − 6.48827e9i 0.932640 − 0.932640i
\(94\) 2.86283e9 − 5.58373e9i 0.390082 − 0.760826i
\(95\) 1.42716e10 + 5.21196e9i 1.84440 + 0.673571i
\(96\) −8.54944e9 − 8.76797e9i −1.04853 − 1.07533i
\(97\) −7.71908e9 + 7.71908e9i −0.898891 + 0.898891i −0.995338 − 0.0964474i \(-0.969252\pi\)
0.0964474 + 0.995338i \(0.469252\pi\)
\(98\) 2.60346e8 + 8.08196e8i 0.0288018 + 0.0894100i
\(99\) −1.25530e10 −1.31999
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.13.7 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.23 yes 116
8.5 even 2 inner 40.11.i.a.13.23 yes 116
40.37 odd 4 inner 40.11.i.a.37.7 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.7 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.23 yes 116 8.5 even 2 inner
40.11.i.a.37.7 yes 116 40.37 odd 4 inner
40.11.i.a.37.23 yes 116 5.2 odd 4 inner