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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 37.14
Character \(\chi\) \(=\) 40.37
Dual form 40.11.i.a.13.14

$q$-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+(-23.6958 - 21.5060i) q^{2} +(119.074 + 119.074i) q^{3} +(98.9850 + 1019.20i) q^{4} +(2966.97 - 981.174i) q^{5} +(-260.753 - 5382.34i) q^{6} +(20153.9 + 20153.9i) q^{7} +(19573.5 - 26279.7i) q^{8} -30692.0i q^{9} +(-91406.0 - 40557.9i) q^{10} -57352.8i q^{11} +(-109574. + 133147. i) q^{12} +(312992. + 312992. i) q^{13} +(-44133.9 - 910993. i) q^{14} +(470120. + 236456. i) q^{15} +(-1.02898e6 + 201772. i) q^{16} +(499000. + 499000. i) q^{17} +(-660061. + 727271. i) q^{18} -4.17967e6 q^{19} +(1.29370e6 + 2.92683e6i) q^{20} +4.79959e6i q^{21} +(-1.23343e6 + 1.35902e6i) q^{22} +(-2.67564e6 + 2.67564e6i) q^{23} +(5.45990e6 - 798531. i) q^{24} +(7.84022e6 - 5.82223e6i) q^{25} +(-685403. - 1.41478e7i) q^{26} +(1.06858e7 - 1.06858e7i) q^{27} +(-1.85460e7 + 2.25359e7i) q^{28} +2.83405e7 q^{29} +(-6.05466e6 - 1.57134e7i) q^{30} -1.38591e6 q^{31} +(2.87218e7 + 1.73481e7i) q^{32} +(6.82921e6 - 6.82921e6i) q^{33} +(-1.09273e6 - 2.25557e7i) q^{34} +(7.95705e7 + 4.00216e7i) q^{35} +(3.12814e7 - 3.03804e6i) q^{36} +(1.65955e7 - 1.65955e7i) q^{37} +(9.90407e7 + 8.98878e7i) q^{38} +7.45381e7i q^{39} +(3.22890e7 - 9.71760e7i) q^{40} +4.47394e7 q^{41} +(1.03220e8 - 1.13730e8i) q^{42} +(7.71456e7 + 7.71456e7i) q^{43} +(5.84543e7 - 5.67707e6i) q^{44} +(-3.01142e7 - 9.10622e7i) q^{45} +(1.20944e8 - 5.85924e6i) q^{46} +(-2.63535e8 - 2.63535e8i) q^{47} +(-1.46550e8 - 9.84986e7i) q^{48} +5.29884e8i q^{49} +(-3.10993e8 - 3.06490e7i) q^{50} +1.18835e8i q^{51} +(-2.88021e8 + 3.49984e8i) q^{52} +(4.81213e8 + 4.81213e8i) q^{53} +(-4.83017e8 + 2.34002e7i) q^{54} +(-5.62731e7 - 1.70164e8i) q^{55} +(9.24120e8 - 1.35156e8i) q^{56} +(-4.97688e8 - 4.97688e8i) q^{57} +(-6.71551e8 - 6.09490e8i) q^{58} +2.22295e8 q^{59} +(-1.94462e8 + 5.02554e8i) q^{60} +6.74164e8i q^{61} +(3.28403e7 + 2.98054e7i) q^{62} +(6.18563e8 - 6.18563e8i) q^{63} +(-3.07500e8 - 1.02877e9i) q^{64} +(1.23574e9 + 6.21538e8i) q^{65} +(-3.08693e8 + 1.49549e7i) q^{66} +(-1.08627e9 + 1.08627e9i) q^{67} +(-4.59190e8 + 5.57976e8i) q^{68} -6.37196e8 q^{69} +(-1.02479e9 - 2.65959e9i) q^{70} -5.34273e8 q^{71} +(-8.06574e8 - 6.00748e8i) q^{72} +(-4.97837e8 + 4.97837e8i) q^{73} +(-7.50147e8 + 3.63416e7i) q^{74} +(1.62684e9 + 2.40289e8i) q^{75} +(-4.13724e8 - 4.25993e9i) q^{76} +(1.15588e9 - 1.15588e9i) q^{77} +(1.60302e9 - 1.76624e9i) q^{78} +2.37264e9i q^{79} +(-2.85498e9 + 1.60826e9i) q^{80} +7.32459e8 q^{81} +(-1.06014e9 - 9.62165e8i) q^{82} +(2.94054e9 + 2.94054e9i) q^{83} +(-4.89177e9 + 4.75088e8i) q^{84} +(1.97012e9 + 9.90913e8i) q^{85} +(-1.68937e8 - 3.48712e9i) q^{86} +(3.37460e9 + 3.37460e9i) q^{87} +(-1.50721e9 - 1.12259e9i) q^{88} -5.94955e9i q^{89} +(-1.24480e9 + 2.80543e9i) q^{90} +1.26160e10i q^{91} +(-2.99187e9 - 2.46218e9i) q^{92} +(-1.65025e8 - 1.65025e8i) q^{93} +(5.77100e8 + 1.19123e10i) q^{94} +(-1.24009e10 + 4.10098e9i) q^{95} +(1.35431e9 + 5.48571e9i) q^{96} +(-9.87072e9 - 9.87072e9i) q^{97} +(1.13957e10 - 1.25560e10i) q^{98} -1.76027e9 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 \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −23.6958 − 21.5060i −0.740495 − 0.672062i
\(3\) 119.074 + 119.074i 0.490015 + 0.490015i 0.908311 − 0.418296i \(-0.137373\pi\)
−0.418296 + 0.908311i \(0.637373\pi\)
\(4\) 98.9850 + 1019.20i 0.0966650 + 0.995317i
\(5\) 2966.97 − 981.174i 0.949431 − 0.313976i
\(6\) −260.753 − 5382.34i −0.0335330 − 0.692174i
\(7\) 20153.9 + 20153.9i 1.19914 + 1.19914i 0.974425 + 0.224712i \(0.0721441\pi\)
0.224712 + 0.974425i \(0.427856\pi\)
\(8\) 19573.5 − 26279.7i 0.597335 − 0.801992i
\(9\) − 30692.0i − 0.519771i
\(10\) −91406.0 − 40557.9i −0.914060 − 0.405579i
\(11\) − 57352.8i − 0.356116i −0.984020 − 0.178058i \(-0.943019\pi\)
0.984020 − 0.178058i \(-0.0569815\pi\)
\(12\) −109574. + 133147.i −0.440353 + 0.535087i
\(13\) 312992. + 312992.i 0.842978 + 0.842978i 0.989245 − 0.146267i \(-0.0467260\pi\)
−0.146267 + 0.989245i \(0.546726\pi\)
\(14\) −44133.9 − 910993.i −0.0820601 − 1.69385i
\(15\) 470120. + 236456.i 0.619088 + 0.311382i
\(16\) −1.02898e6 + 201772.i −0.981312 + 0.192425i
\(17\) 499000. + 499000.i 0.351444 + 0.351444i 0.860647 − 0.509203i \(-0.170060\pi\)
−0.509203 + 0.860647i \(0.670060\pi\)
\(18\) −660061. + 727271.i −0.349318 + 0.384888i
\(19\) −4.17967e6 −1.68800 −0.844002 − 0.536340i \(-0.819807\pi\)
−0.844002 + 0.536340i \(0.819807\pi\)
\(20\) 1.29370e6 + 2.92683e6i 0.404282 + 0.914634i
\(21\) 4.79959e6i 1.17519i
\(22\) −1.23343e6 + 1.35902e6i −0.239332 + 0.263702i
\(23\) −2.67564e6 + 2.67564e6i −0.415708 + 0.415708i −0.883721 − 0.468013i \(-0.844970\pi\)
0.468013 + 0.883721i \(0.344970\pi\)
\(24\) 5.45990e6 − 798531.i 0.685691 − 0.100285i
\(25\) 7.84022e6 − 5.82223e6i 0.802838 − 0.596197i
\(26\) −685403. − 1.41478e7i −0.0576872 − 1.19075i
\(27\) 1.06858e7 − 1.06858e7i 0.744710 − 0.744710i
\(28\) −1.85460e7 + 2.25359e7i −1.07761 + 1.30944i
\(29\) 2.83405e7 1.38171 0.690856 − 0.722993i \(-0.257234\pi\)
0.690856 + 0.722993i \(0.257234\pi\)
\(30\) −6.05466e6 − 1.57134e7i −0.249163 − 0.646643i
\(31\) −1.38591e6 −0.0484091 −0.0242045 − 0.999707i \(-0.507705\pi\)
−0.0242045 + 0.999707i \(0.507705\pi\)
\(32\) 2.87218e7 + 1.73481e7i 0.855978 + 0.517013i
\(33\) 6.82921e6 − 6.82921e6i 0.174502 − 0.174502i
\(34\) −1.09273e6 − 2.25557e7i −0.0240502 − 0.496434i
\(35\) 7.95705e7 + 4.00216e7i 1.51500 + 0.761998i
\(36\) 3.12814e7 − 3.03804e6i 0.517337 − 0.0502437i
\(37\) 1.65955e7 − 1.65955e7i 0.239322 − 0.239322i −0.577248 − 0.816569i \(-0.695873\pi\)
0.816569 + 0.577248i \(0.195873\pi\)
\(38\) 9.90407e7 + 8.98878e7i 1.24996 + 1.13444i
\(39\) 7.45381e7i 0.826143i
\(40\) 3.22890e7 − 9.71760e7i 0.315322 − 0.948985i
\(41\) 4.47394e7 0.386163 0.193082 − 0.981183i \(-0.438152\pi\)
0.193082 + 0.981183i \(0.438152\pi\)
\(42\) 1.03220e8 − 1.13730e8i 0.789801 − 0.870222i
\(43\) 7.71456e7 + 7.71456e7i 0.524770 + 0.524770i 0.919008 − 0.394238i \(-0.128992\pi\)
−0.394238 + 0.919008i \(0.628992\pi\)
\(44\) 5.84543e7 − 5.67707e6i 0.354448 − 0.0344240i
\(45\) −3.01142e7 − 9.10622e7i −0.163196 − 0.493487i
\(46\) 1.20944e8 − 5.85924e6i 0.587212 − 0.0284480i
\(47\) −2.63535e8 − 2.63535e8i −1.14908 − 1.14908i −0.986735 − 0.162341i \(-0.948095\pi\)
−0.162341 − 0.986735i \(-0.551905\pi\)
\(48\) −1.46550e8 − 9.84986e7i −0.575148 − 0.386566i
\(49\) 5.29884e8i 1.87586i
\(50\) −3.10993e8 − 3.06490e7i −0.995179 − 0.0980767i
\(51\) 1.18835e8i 0.344425i
\(52\) −2.88021e8 + 3.49984e8i −0.757544 + 0.920517i
\(53\) 4.81213e8 + 4.81213e8i 1.15069 + 1.15069i 0.986414 + 0.164276i \(0.0525286\pi\)
0.164276 + 0.986414i \(0.447471\pi\)
\(54\) −4.83017e8 + 2.34002e7i −1.05195 + 0.0509625i
\(55\) −5.62731e7 − 1.70164e8i −0.111812 − 0.338108i
\(56\) 9.24120e8 − 1.35156e8i 1.67798 − 0.245412i
\(57\) −4.97688e8 − 4.97688e8i −0.827147 − 0.827147i
\(58\) −6.71551e8 − 6.09490e8i −1.02315 − 0.928596i
\(59\) 2.22295e8 0.310935 0.155467 − 0.987841i \(-0.450312\pi\)
0.155467 + 0.987841i \(0.450312\pi\)
\(60\) −1.94462e8 + 5.02554e8i −0.250080 + 0.646289i
\(61\) 6.74164e8i 0.798209i 0.916905 + 0.399104i \(0.130679\pi\)
−0.916905 + 0.399104i \(0.869321\pi\)
\(62\) 3.28403e7 + 2.98054e7i 0.0358467 + 0.0325339i
\(63\) 6.18563e8 − 6.18563e8i 0.623277 − 0.623277i
\(64\) −3.07500e8 − 1.02877e9i −0.286382 − 0.958116i
\(65\) 1.23574e9 + 6.21538e8i 1.06502 + 0.535675i
\(66\) −3.08693e8 + 1.49549e7i −0.246494 + 0.0119416i
\(67\) −1.08627e9 + 1.08627e9i −0.804573 + 0.804573i −0.983806 − 0.179234i \(-0.942638\pi\)
0.179234 + 0.983806i \(0.442638\pi\)
\(68\) −4.59190e8 + 5.57976e8i −0.315826 + 0.383770i
\(69\) −6.37196e8 −0.407406
\(70\) −1.02479e9 − 2.65959e9i −0.609738 − 1.58243i
\(71\) −5.34273e8 −0.296122 −0.148061 − 0.988978i \(-0.547303\pi\)
−0.148061 + 0.988978i \(0.547303\pi\)
\(72\) −8.06574e8 − 6.00748e8i −0.416852 − 0.310477i
\(73\) −4.97837e8 + 4.97837e8i −0.240145 + 0.240145i −0.816910 − 0.576765i \(-0.804315\pi\)
0.576765 + 0.816910i \(0.304315\pi\)
\(74\) −7.50147e8 + 3.63416e7i −0.338055 + 0.0163774i
\(75\) 1.62684e9 + 2.40289e8i 0.685548 + 0.101257i
\(76\) −4.13724e8 − 4.25993e9i −0.163171 − 1.68010i
\(77\) 1.15588e9 − 1.15588e9i 0.427032 − 0.427032i
\(78\) 1.60302e9 − 1.76624e9i 0.555220 − 0.611755i
\(79\) 2.37264e9i 0.771075i 0.922692 + 0.385538i \(0.125984\pi\)
−0.922692 + 0.385538i \(0.874016\pi\)
\(80\) −2.85498e9 + 1.60826e9i −0.871271 + 0.490802i
\(81\) 7.32459e8 0.210067
\(82\) −1.06014e9 − 9.62165e8i −0.285952 − 0.259526i
\(83\) 2.94054e9 + 2.94054e9i 0.746512 + 0.746512i 0.973822 − 0.227311i \(-0.0729933\pi\)
−0.227311 + 0.973822i \(0.572993\pi\)
\(84\) −4.89177e9 + 4.75088e8i −1.16969 + 0.113600i
\(85\) 1.97012e9 + 9.90913e8i 0.444016 + 0.223327i
\(86\) −1.68937e8 − 3.48712e9i −0.0359114 − 0.741267i
\(87\) 3.37460e9 + 3.37460e9i 0.677059 + 0.677059i
\(88\) −1.50721e9 − 1.12259e9i −0.285602 − 0.212721i
\(89\) − 5.94955e9i − 1.06545i −0.846288 − 0.532726i \(-0.821168\pi\)
0.846288 − 0.532726i \(-0.178832\pi\)
\(90\) −1.24480e9 + 2.80543e9i −0.210808 + 0.475102i
\(91\) 1.26160e10i 2.02169i
\(92\) −2.99187e9 − 2.46218e9i −0.453946 − 0.373577i
\(93\) −1.65025e8 − 1.65025e8i −0.0237212 − 0.0237212i
\(94\) 5.77100e8 + 1.19123e10i 0.0786343 + 1.62314i
\(95\) −1.24009e10 + 4.10098e9i −1.60264 + 0.529992i
\(96\) 1.35431e9 + 5.48571e9i 0.166098 + 0.672786i
\(97\) −9.87072e9 − 9.87072e9i −1.14945 − 1.14945i −0.986661 − 0.162789i \(-0.947951\pi\)
−0.162789 − 0.986661i \(-0.552049\pi\)
\(98\) 1.13957e10 − 1.25560e10i 1.26069 − 1.38906i
\(99\) −1.76027e9 −0.185099
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.37.14 yes 116
5.3 odd 4 inner 40.11.i.a.13.16 yes 116
8.5 even 2 inner 40.11.i.a.37.16 yes 116
40.13 odd 4 inner 40.11.i.a.13.14 ✓ 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.14 ✓ 116 40.13 odd 4 inner
40.11.i.a.13.16 yes 116 5.3 odd 4 inner
40.11.i.a.37.14 yes 116 1.1 even 1 trivial
40.11.i.a.37.16 yes 116 8.5 even 2 inner