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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 13.14
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.14

$q$-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 \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −23.6958 + 21.5060i −0.740495 + 0.672062i
\(3\) 119.074 − 119.074i 0.490015 − 0.490015i −0.418296 − 0.908311i \(-0.637373\pi\)
0.908311 + 0.418296i \(0.137373\pi\)
\(4\) 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 0.224712 − 0.974425i \(-0.427856\pi\)
0.974425 − 0.224712i \(-0.0721441\pi\)
\(8\) 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 \(0.0569815\pi\)
−0.984020 + 0.178058i \(0.943019\pi\)
\(12\) −109574. − 133147.i −0.440353 − 0.535087i
\(13\) 312992. − 312992.i 0.842978 − 0.842978i −0.146267 − 0.989245i \(-0.546726\pi\)
0.989245 + 0.146267i \(0.0467260\pi\)
\(14\) −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 −0.509203 − 0.860647i \(-0.670060\pi\)
0.860647 + 0.509203i \(0.170060\pi\)
\(18\) −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 0.468013 − 0.883721i \(-0.344970\pi\)
−0.883721 + 0.468013i \(0.844970\pi\)
\(24\) 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 0.816569 − 0.577248i \(-0.195873\pi\)
−0.577248 + 0.816569i \(0.695873\pi\)
\(38\) 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 −0.394238 − 0.919008i \(-0.628992\pi\)
0.919008 + 0.394238i \(0.128992\pi\)
\(44\) 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 −0.162341 + 0.986735i \(0.551905\pi\)
−0.986735 + 0.162341i \(0.948095\pi\)
\(48\) −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 0.164276 − 0.986414i \(-0.447471\pi\)
0.986414 − 0.164276i \(-0.0525286\pi\)
\(54\) −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 \(-0.869321\pi\)
0.916905 − 0.399104i \(-0.130679\pi\)
\(62\) 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 0.179234 − 0.983806i \(-0.442638\pi\)
−0.983806 + 0.179234i \(0.942638\pi\)
\(68\) −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 0.576765 − 0.816910i \(-0.304315\pi\)
−0.816910 + 0.576765i \(0.804315\pi\)
\(74\) −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 \(-0.874016\pi\)
0.922692 − 0.385538i \(-0.125984\pi\)
\(80\) −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 −0.227311 − 0.973822i \(-0.572993\pi\)
0.973822 + 0.227311i \(0.0729933\pi\)
\(84\) −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 \(0.178832\pi\)
−0.846288 + 0.532726i \(0.821168\pi\)
\(90\) −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 −0.162789 + 0.986661i \(0.552049\pi\)
−0.986661 + 0.162789i \(0.947951\pi\)
\(98\) 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.13.14 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.16 yes 116
8.5 even 2 inner 40.11.i.a.13.16 yes 116
40.37 odd 4 inner 40.11.i.a.37.14 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.14 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.16 yes 116 8.5 even 2 inner
40.11.i.a.37.14 yes 116 40.37 odd 4 inner
40.11.i.a.37.16 yes 116 5.2 odd 4 inner