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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 13.5
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.5

$q$-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+(-31.1769 + 7.21110i) q^{2} +(179.852 - 179.852i) q^{3} +(920.000 - 449.640i) q^{4} +(-1869.56 - 2504.07i) q^{5} +(-4310.29 + 6904.15i) q^{6} +(532.991 - 532.991i) q^{7} +(-25440.4 + 20652.6i) q^{8} -5644.22i q^{9} +(76344.3 + 64587.6i) q^{10} +316807. i q^{11} +(84595.0 - 246332. i) q^{12} +(-219147. + 219147. i) q^{13} +(-12773.6 + 20460.5i) q^{14} +(-786605. - 114118. i) q^{15} +(644224. - 827337. i) q^{16} +(1.17621e6 - 1.17621e6i) q^{17} +(40701.0 + 175969. i) q^{18} -1.09130e6 q^{19} +(-2.84593e6 - 1.46312e6i) q^{20} -191719. i q^{21} +(-2.28453e6 - 9.87705e6i) q^{22} +(1.11044e6 + 1.11044e6i) q^{23} +(-861086. + 8.28989e6i) q^{24} +(-2.77511e6 + 9.36302e6i) q^{25} +(5.25203e6 - 8.41261e6i) q^{26} +(9.60494e6 + 9.60494e6i) q^{27} +(250698. - 730006. i) q^{28} +2.44182e7 q^{29} +(2.53468e7 - 2.11446e6i) q^{30} -636527. q^{31} +(-1.41189e7 + 3.04394e7i) q^{32} +(5.69782e7 + 5.69782e7i) q^{33} +(-2.81888e7 + 4.51523e7i) q^{34} +(-2.33111e6 - 338188. i) q^{35} +(-2.53787e6 - 5.19268e6i) q^{36} +(-7.91836e6 - 7.91836e6i) q^{37} +(3.40233e7 - 7.86945e6i) q^{38} +7.88278e7i q^{39} +(9.92779e7 + 2.50931e7i) q^{40} +7.56993e7 q^{41} +(1.38250e6 + 5.97720e6i) q^{42} +(-1.18657e8 + 1.18657e8i) q^{43} +(1.42449e8 + 2.91462e8i) q^{44} +(-1.41335e7 + 1.05522e7i) q^{45} +(-4.26275e7 - 2.66125e7i) q^{46} +(2.17964e8 - 2.17964e8i) q^{47} +(-3.29333e7 - 2.64663e8i) q^{48} +2.81907e8i q^{49} +(1.90016e7 - 3.11922e8i) q^{50} -4.23086e8i q^{51} +(-1.03078e8 + 3.00152e8i) q^{52} +(1.58268e8 - 1.58268e8i) q^{53} +(-3.68715e8 - 2.30190e8i) q^{54} +(7.93306e8 - 5.92289e8i) q^{55} +(-2.55183e6 + 2.45671e7i) q^{56} +(-1.96271e8 + 1.96271e8i) q^{57} +(-7.61284e8 + 1.76082e8i) q^{58} -9.61756e8 q^{59} +(-7.74988e8 + 2.48701e8i) q^{60} +1.12071e9i q^{61} +(1.98450e7 - 4.59007e6i) q^{62} +(-3.00832e6 - 3.00832e6i) q^{63} +(2.20682e8 - 1.05082e9i) q^{64} +(9.58467e8 + 1.39051e8i) q^{65} +(-2.18728e9 - 1.36553e9i) q^{66} +(1.60654e9 + 1.60654e9i) q^{67} +(5.53241e8 - 1.61098e9i) q^{68} +3.99428e8 q^{69} +(7.51154e7 - 6.26620e6i) q^{70} -2.10516e8 q^{71} +(1.16568e8 + 1.43591e8i) q^{72} +(-1.06409e9 - 1.06409e9i) q^{73} +(3.03970e8 + 1.89770e8i) q^{74} +(1.18485e9 + 2.18306e9i) q^{75} +(-1.00399e9 + 4.90691e8i) q^{76} +(1.68855e8 + 1.68855e8i) q^{77} +(-5.68435e8 - 2.45761e9i) q^{78} +2.02035e9i q^{79} +(-3.27613e9 - 6.64239e7i) q^{80} +3.78821e9 q^{81} +(-2.36007e9 + 5.45875e8i) q^{82} +(-7.43590e8 + 7.43590e8i) q^{83} +(-8.62044e7 - 1.76381e8i) q^{84} +(-5.14430e9 - 7.46314e8i) q^{85} +(2.84371e9 - 4.55501e9i) q^{86} +(4.39165e9 - 4.39165e9i) q^{87} +(-6.54288e9 - 8.05967e9i) q^{88} +1.26687e9i q^{89} +(3.64546e8 - 4.30903e8i) q^{90} +2.33606e8i q^{91} +(1.52090e9 + 5.22305e8i) q^{92} +(-1.14480e8 + 1.14480e8i) q^{93} +(-5.22369e9 + 8.36721e9i) q^{94} +(2.04025e9 + 2.73268e9i) q^{95} +(2.93527e9 + 8.01388e9i) q^{96} +(6.68998e9 - 6.68998e9i) q^{97} +(-2.03286e9 - 8.78899e9i) q^{98} +1.78813e9 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.1769 + 7.21110i −0.974279 + 0.225347i
\(3\) 179.852 − 179.852i 0.740130 − 0.740130i −0.232473 − 0.972603i \(-0.574682\pi\)
0.972603 + 0.232473i \(0.0746816\pi\)
\(4\) 920.000 − 449.640i 0.898437 − 0.439101i
\(5\) −1869.56 − 2504.07i −0.598259 − 0.801302i
\(6\) −4310.29 + 6904.15i −0.554307 + 0.887879i
\(7\) 532.991 − 532.991i 0.0317124 − 0.0317124i −0.691073 − 0.722785i \(-0.742862\pi\)
0.722785 + 0.691073i \(0.242862\pi\)
\(8\) −25440.4 + 20652.6i −0.776378 + 0.630267i
\(9\) − 5644.22i − 0.0955853i
\(10\) 76344.3 + 64587.6i 0.763443 + 0.645876i
\(11\) 316807.i 1.96712i 0.180582 + 0.983560i \(0.442202\pi\)
−0.180582 + 0.983560i \(0.557798\pi\)
\(12\) 84595.0 − 246332.i 0.339968 − 0.989953i
\(13\) −219147. + 219147.i −0.590226 + 0.590226i −0.937692 − 0.347467i \(-0.887042\pi\)
0.347467 + 0.937692i \(0.387042\pi\)
\(14\) −12773.6 + 20460.5i −0.0237505 + 0.0380431i
\(15\) −786605. − 114118.i −1.03586 − 0.150278i
\(16\) 644224. − 827337.i 0.614380 − 0.789010i
\(17\) 1.17621e6 − 1.17621e6i 0.828399 − 0.828399i −0.158897 − 0.987295i \(-0.550794\pi\)
0.987295 + 0.158897i \(0.0507937\pi\)
\(18\) 40701.0 + 175969.i 0.0215399 + 0.0931267i
\(19\) −1.09130e6 −0.440732 −0.220366 − 0.975417i \(-0.570725\pi\)
−0.220366 + 0.975417i \(0.570725\pi\)
\(20\) −2.84593e6 − 1.46312e6i −0.889352 − 0.457223i
\(21\) − 191719.i − 0.0469427i
\(22\) −2.28453e6 − 9.87705e6i −0.443285 − 1.91652i
\(23\) 1.11044e6 + 1.11044e6i 0.172526 + 0.172526i 0.788088 − 0.615562i \(-0.211071\pi\)
−0.615562 + 0.788088i \(0.711071\pi\)
\(24\) −861086. + 8.28989e6i −0.108141 + 1.04110i
\(25\) −2.77511e6 + 9.36302e6i −0.284171 + 0.958774i
\(26\) 5.25203e6 − 8.41261e6i 0.442039 − 0.708050i
\(27\) 9.60494e6 + 9.60494e6i 0.669385 + 0.669385i
\(28\) 250698. − 730006.i 0.0145667 − 0.0424166i
\(29\) 2.44182e7 1.19048 0.595242 − 0.803546i \(-0.297056\pi\)
0.595242 + 0.803546i \(0.297056\pi\)
\(30\) 2.53468e7 − 2.11446e6i 1.04308 − 0.0870147i
\(31\) −636527. −0.0222335 −0.0111168 − 0.999938i \(-0.503539\pi\)
−0.0111168 + 0.999938i \(0.503539\pi\)
\(32\) −1.41189e7 + 3.04394e7i −0.420776 + 0.907165i
\(33\) 5.69782e7 + 5.69782e7i 1.45592 + 1.45592i
\(34\) −2.81888e7 + 4.51523e7i −0.620414 + 0.993768i
\(35\) −2.33111e6 − 338188.i −0.0443835 − 0.00643899i
\(36\) −2.53787e6 − 5.19268e6i −0.0419716 − 0.0858774i
\(37\) −7.91836e6 − 7.91836e6i −0.114190 − 0.114190i 0.647703 − 0.761893i \(-0.275730\pi\)
−0.761893 + 0.647703i \(0.775730\pi\)
\(38\) 3.40233e7 − 7.86945e6i 0.429396 − 0.0993177i
\(39\) 7.88278e7i 0.873688i
\(40\) 9.92779e7 + 2.50931e7i 0.969510 + 0.245050i
\(41\) 7.56993e7 0.653390 0.326695 − 0.945130i \(-0.394065\pi\)
0.326695 + 0.945130i \(0.394065\pi\)
\(42\) 1.38250e6 + 5.97720e6i 0.0105784 + 0.0457352i
\(43\) −1.18657e8 + 1.18657e8i −0.807145 + 0.807145i −0.984201 − 0.177056i \(-0.943343\pi\)
0.177056 + 0.984201i \(0.443343\pi\)
\(44\) 1.42449e8 + 2.91462e8i 0.863765 + 1.76733i
\(45\) −1.41335e7 + 1.05522e7i −0.0765927 + 0.0571848i
\(46\) −4.26275e7 − 2.66125e7i −0.206967 − 0.129210i
\(47\) 2.17964e8 − 2.17964e8i 0.950377 − 0.950377i −0.0484489 − 0.998826i \(-0.515428\pi\)
0.998826 + 0.0484489i \(0.0154278\pi\)
\(48\) −3.29333e7 − 2.64663e8i −0.129249 − 1.03869i
\(49\) 2.81907e8i 0.997989i
\(50\) 1.90016e7 − 3.11922e8i 0.0608051 − 0.998150i
\(51\) − 4.23086e8i − 1.22625i
\(52\) −1.03078e8 + 3.00152e8i −0.271112 + 0.789450i
\(53\) 1.58268e8 − 1.58268e8i 0.378454 − 0.378454i −0.492090 − 0.870544i \(-0.663767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(54\) −3.68715e8 − 2.30190e8i −0.803011 − 0.501323i
\(55\) 7.93306e8 − 5.92289e8i 1.57626 − 1.17685i
\(56\) −2.55183e6 + 2.45671e7i −0.00463353 + 0.0446082i
\(57\) −1.96271e8 + 1.96271e8i −0.326199 + 0.326199i
\(58\) −7.61284e8 + 1.76082e8i −1.15986 + 0.268272i
\(59\) −9.61756e8 −1.34526 −0.672628 − 0.739981i \(-0.734835\pi\)
−0.672628 + 0.739981i \(0.734835\pi\)
\(60\) −7.74988e8 + 2.48701e8i −0.996641 + 0.319831i
\(61\) 1.12071e9i 1.32691i 0.748214 + 0.663457i \(0.230912\pi\)
−0.748214 + 0.663457i \(0.769088\pi\)
\(62\) 1.98450e7 − 4.59007e6i 0.0216617 − 0.00501026i
\(63\) −3.00832e6 − 3.00832e6i −0.00303124 − 0.00303124i
\(64\) 2.20682e8 − 1.05082e9i 0.205526 − 0.978652i
\(65\) 9.58467e8 + 1.39051e8i 0.826057 + 0.119841i
\(66\) −2.18728e9 − 1.36553e9i −1.74656 − 1.09039i
\(67\) 1.60654e9 + 1.60654e9i 1.18992 + 1.18992i 0.977090 + 0.212829i \(0.0682676\pi\)
0.212829 + 0.977090i \(0.431732\pi\)
\(68\) 5.53241e8 − 1.61098e9i 0.380513 − 1.10802i
\(69\) 3.99428e8 0.255384
\(70\) 7.51154e7 − 6.26620e6i 0.0446929 − 0.00372833i
\(71\) −2.10516e8 −0.116679 −0.0583396 − 0.998297i \(-0.518581\pi\)
−0.0583396 + 0.998297i \(0.518581\pi\)
\(72\) 1.16568e8 + 1.43591e8i 0.0602443 + 0.0742103i
\(73\) −1.06409e9 − 1.06409e9i −0.513290 − 0.513290i 0.402243 − 0.915533i \(-0.368231\pi\)
−0.915533 + 0.402243i \(0.868231\pi\)
\(74\) 3.03970e8 + 1.89770e8i 0.136985 + 0.0855202i
\(75\) 1.18485e9 + 2.18306e9i 0.499294 + 0.919941i
\(76\) −1.00399e9 + 4.90691e8i −0.395970 + 0.193526i
\(77\) 1.68855e8 + 1.68855e8i 0.0623822 + 0.0623822i
\(78\) −5.68435e8 − 2.45761e9i −0.196883 − 0.851215i
\(79\) 2.02035e9i 0.656586i 0.944576 + 0.328293i \(0.106473\pi\)
−0.944576 + 0.328293i \(0.893527\pi\)
\(80\) −3.27613e9 − 6.64239e7i −0.999795 − 0.0202710i
\(81\) 3.78821e9 1.08645
\(82\) −2.36007e9 + 5.45875e8i −0.636584 + 0.147239i
\(83\) −7.43590e8 + 7.43590e8i −0.188774 + 0.188774i −0.795166 − 0.606392i \(-0.792616\pi\)
0.606392 + 0.795166i \(0.292616\pi\)
\(84\) −8.62044e7 − 1.76381e8i −0.0206126 − 0.0421751i
\(85\) −5.14430e9 − 7.46314e8i −1.15940 − 0.168200i
\(86\) 2.84371e9 − 4.55501e9i 0.604496 − 0.968272i
\(87\) 4.39165e9 − 4.39165e9i 0.881114 − 0.881114i
\(88\) −6.54288e9 − 8.05967e9i −1.23981 − 1.52723i
\(89\) 1.26687e9i 0.226873i 0.993545 + 0.113437i \(0.0361859\pi\)
−0.993545 + 0.113437i \(0.963814\pi\)
\(90\) 3.64546e8 − 4.30903e8i 0.0617362 − 0.0729739i
\(91\) 2.33606e8i 0.0374350i
\(92\) 1.52090e9 + 5.22305e8i 0.230760 + 0.0792474i
\(93\) −1.14480e8 + 1.14480e8i −0.0164557 + 0.0164557i
\(94\) −5.22369e9 + 8.36721e9i −0.711767 + 1.14010i
\(95\) 2.04025e9 + 2.73268e9i 0.263672 + 0.353160i
\(96\) 2.93527e9 + 8.01388e9i 0.359991 + 0.982849i
\(97\) 6.68998e9 − 6.68998e9i 0.779052 − 0.779052i −0.200618 − 0.979670i \(-0.564295\pi\)
0.979670 + 0.200618i \(0.0642950\pi\)
\(98\) −2.03286e9 − 8.78899e9i −0.224894 − 0.972319i
\(99\) 1.78813e9 0.188028
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.13.5 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.26 yes 116
8.5 even 2 inner 40.11.i.a.13.26 yes 116
40.37 odd 4 inner 40.11.i.a.37.5 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.5 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.26 yes 116 8.5 even 2 inner
40.11.i.a.37.5 yes 116 40.37 odd 4 inner
40.11.i.a.37.26 yes 116 5.2 odd 4 inner