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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 13.16
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.16

$q$-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+(-21.5060 + 23.6958i) 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} +(26279.7 + 19573.5i) q^{8} +30692.0i q^{9} +(87057.4 - 49203.7i) q^{10} -57352.8i q^{11} +(133147. + 109574. 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} +(-727271. - 660061. i) q^{18} +4.17967e6 q^{19} +(-706332. + 3.12107e6i) q^{20} +4.79959e6i q^{21} +(1.35902e6 + 1.23343e6i) 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} +(-2.25359e7 - 1.85460e7i) q^{28} -2.83405e7 q^{29} +(-4.50737e6 + 1.62251e7i) q^{30} -1.38591e6 q^{31} +(1.73481e7 - 2.87218e7i) 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} +(-8.98878e7 + 9.90407e7i) q^{38} -7.45381e7i q^{39} +(-5.87661e7 - 8.38589e7i) q^{40} +4.47394e7 q^{41} +(-1.13730e8 - 1.03220e8i) 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} +(9.84986e7 - 1.46550e8i) q^{48} -5.29884e8i q^{49} +(-3.06574e8 + 6.05676e7i) q^{50} +1.18835e8i q^{51} +(3.49984e8 + 2.88021e8i) 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} +(6.09490e8 - 6.71551e8i) q^{58} -2.22295e8 q^{59} +(-2.87532e8 - 4.55743e8i) q^{60} +6.74164e8i q^{61} +(2.98054e7 - 3.28403e7i) 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} +(-5.57976e8 - 4.59190e8i) q^{68} +6.37196e8 q^{69} +(7.62899e8 - 2.74619e9i) q^{70} -5.34273e8 q^{71} +(-6.00748e8 + 8.06574e8i) 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} +(1.76624e9 + 1.60302e9i) q^{78} -2.37264e9i q^{79} +(3.25093e9 + 4.10957e8i) q^{80} +7.32459e8 q^{81} +(-9.62165e8 + 1.06014e9i) 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} +(1.12259e9 - 1.50721e9i) q^{88} +5.94955e9i q^{89} +(1.51016e9 + 2.67196e9i) q^{90} +1.26160e10i q^{91} +(-2.46218e9 + 2.99187e9i) 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} +(1.25560e10 + 1.13957e10i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −21.5060 + 23.6958i −0.672062 + 0.740495i
\(3\) −119.074 + 119.074i −0.490015 + 0.490015i −0.908311 − 0.418296i \(-0.862627\pi\)
0.418296 + 0.908311i \(0.362627\pi\)
\(4\) −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 \(-0.0721441\pi\)
\(8\) 26279.7 + 19573.5i 0.801992 + 0.597335i
\(9\) 30692.0i 0.519771i
\(10\) 87057.4 − 49203.7i 0.870574 − 0.492037i
\(11\) − 57352.8i − 0.356116i −0.984020 − 0.178058i \(-0.943019\pi\)
0.984020 − 0.178058i \(-0.0569815\pi\)
\(12\) 133147. + 109574.i 0.535087 + 0.440353i
\(13\) −312992. + 312992.i −0.842978 + 0.842978i −0.989245 − 0.146267i \(-0.953274\pi\)
0.146267 + 0.989245i \(0.453274\pi\)
\(14\) 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 \(0.170060\pi\)
\(18\) −727271. − 660061.i −0.384888 − 0.349318i
\(19\) 4.17967e6 1.68800 0.844002 − 0.536340i \(-0.180193\pi\)
0.844002 + 0.536340i \(0.180193\pi\)
\(20\) −706332. + 3.12107e6i −0.220729 + 0.975335i
\(21\) 4.79959e6i 1.17519i
\(22\) 1.35902e6 + 1.23343e6i 0.263702 + 0.239332i
\(23\) −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
\(28\) −2.25359e7 − 1.85460e7i −1.30944 − 1.07761i
\(29\) −2.83405e7 −1.38171 −0.690856 − 0.722993i \(-0.742766\pi\)
−0.690856 + 0.722993i \(0.742766\pi\)
\(30\) −4.50737e6 + 1.62251e7i −0.185489 + 0.667700i
\(31\) −1.38591e6 −0.0484091 −0.0242045 − 0.999707i \(-0.507705\pi\)
−0.0242045 + 0.999707i \(0.507705\pi\)
\(32\) 1.73481e7 − 2.87218e7i 0.517013 − 0.855978i
\(33\) 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 \(-0.304127\pi\)
−0.816569 + 0.577248i \(0.804127\pi\)
\(38\) −8.98878e7 + 9.90407e7i −1.13444 + 1.24996i
\(39\) − 7.45381e7i − 0.826143i
\(40\) −5.87661e7 − 8.38589e7i −0.573887 − 0.818934i
\(41\) 4.47394e7 0.386163 0.193082 − 0.981183i \(-0.438152\pi\)
0.193082 + 0.981183i \(0.438152\pi\)
\(42\) −1.13730e8 − 1.03220e8i −0.870222 − 0.789801i
\(43\) −7.71456e7 + 7.71456e7i −0.524770 + 0.524770i −0.919008 − 0.394238i \(-0.871008\pi\)
0.394238 + 0.919008i \(0.371008\pi\)
\(44\) −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 \(0.948095\pi\)
\(48\) 9.84986e7 − 1.46550e8i 0.386566 − 0.575148i
\(49\) − 5.29884e8i − 1.87586i
\(50\) −3.06574e8 + 6.05676e7i −0.981038 + 0.193816i
\(51\) 1.18835e8i 0.344425i
\(52\) 3.49984e8 + 2.88021e8i 0.920517 + 0.757544i
\(53\) −4.81213e8 + 4.81213e8i −1.15069 + 1.15069i −0.164276 + 0.986414i \(0.552529\pi\)
−0.986414 + 0.164276i \(0.947471\pi\)
\(54\) 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
\(58\) 6.09490e8 − 6.71551e8i 0.928596 − 1.02315i
\(59\) −2.22295e8 −0.310935 −0.155467 − 0.987841i \(-0.549688\pi\)
−0.155467 + 0.987841i \(0.549688\pi\)
\(60\) −2.87532e8 − 4.55743e8i −0.369768 − 0.586089i
\(61\) 6.74164e8i 0.798209i 0.916905 + 0.399104i \(0.130679\pi\)
−0.916905 + 0.399104i \(0.869321\pi\)
\(62\) 2.98054e7 − 3.28403e7i 0.0325339 − 0.0358467i
\(63\) 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 \(-0.0573619\pi\)
−0.179234 + 0.983806i \(0.557362\pi\)
\(68\) −5.57976e8 − 4.59190e8i −0.383770 − 0.315826i
\(69\) 6.37196e8 0.407406
\(70\) 7.62899e8 − 2.74619e9i 0.453917 − 1.63396i
\(71\) −5.34273e8 −0.296122 −0.148061 − 0.988978i \(-0.547303\pi\)
−0.148061 + 0.988978i \(0.547303\pi\)
\(72\) −6.00748e8 + 8.06574e8i −0.310477 + 0.416852i
\(73\) −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
\(78\) 1.76624e9 + 1.60302e9i 0.611755 + 0.555220i
\(79\) − 2.37264e9i − 0.771075i −0.922692 − 0.385538i \(-0.874016\pi\)
0.922692 − 0.385538i \(-0.125984\pi\)
\(80\) 3.25093e9 + 4.10957e8i 0.992104 + 0.125414i
\(81\) 7.32459e8 0.210067
\(82\) −9.62165e8 + 1.06014e9i −0.259526 + 0.285952i
\(83\) −2.94054e9 + 2.94054e9i −0.746512 + 0.746512i −0.973822 − 0.227311i \(-0.927007\pi\)
0.227311 + 0.973822i \(0.427007\pi\)
\(84\) 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
\(88\) 1.12259e9 − 1.50721e9i 0.212721 − 0.285602i
\(89\) 5.94955e9i 1.06545i 0.846288 + 0.532726i \(0.178832\pi\)
−0.846288 + 0.532726i \(0.821168\pi\)
\(90\) 1.51016e9 + 2.67196e9i 0.255747 + 0.452499i
\(91\) 1.26160e10i 2.02169i
\(92\) −2.46218e9 + 2.99187e9i −0.373577 + 0.453946i
\(93\) 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 \(0.947951\pi\)
\(98\) 1.25560e10 + 1.13957e10i 1.38906 + 1.26069i
\(99\) 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.13.16 yes 116
5.2 odd 4 inner 40.11.i.a.37.14 yes 116
8.5 even 2 inner 40.11.i.a.13.14 ✓ 116
40.37 odd 4 inner 40.11.i.a.37.16 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.14 ✓ 116 8.5 even 2 inner
40.11.i.a.13.16 yes 116 1.1 even 1 trivial
40.11.i.a.37.14 yes 116 5.2 odd 4 inner
40.11.i.a.37.16 yes 116 40.37 odd 4 inner