Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [405,2,Mod(8,405)] 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("405.8"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([2, 27])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.r (of order \(36\), degree \(12\), not 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: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 233.6
Character \(\chi\) \(=\) 405.233
Dual form 405.2.r.a.332.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.343150 + 0.490069i) q^{2} +(0.561625 + 1.54305i) q^{4} +(-2.04155 - 0.912186i) q^{5} +(-0.136472 + 0.0636379i) q^{7} +(-2.10468 - 0.563947i) q^{8} +(1.14759 - 0.687482i) q^{10} +(3.54970 + 4.23037i) q^{11} +(-1.54940 + 1.08490i) q^{13} +(0.0156434 - 0.0887180i) q^{14} +(-1.51722 + 1.27310i) q^{16} +(-4.40907 + 1.18141i) q^{17} +(-6.55178 + 3.78267i) q^{19} +(0.260967 - 3.66252i) q^{20} +(-3.29125 + 0.287947i) q^{22} +(-0.412075 + 0.883698i) q^{23} +(3.33583 + 3.72454i) q^{25} -1.13159i q^{26} +(-0.174843 - 0.174843i) q^{28} +(-0.486461 - 2.75886i) q^{29} +(1.70974 - 0.622294i) q^{31} +(-0.483084 - 5.52167i) q^{32} +(0.934001 - 2.56615i) q^{34} +(0.336664 - 0.00543199i) q^{35} +(-1.53079 - 5.71298i) q^{37} +(0.394473 - 4.50885i) q^{38} +(3.78238 + 3.07118i) q^{40} +(0.682276 + 0.120304i) q^{41} +(1.73456 + 0.151755i) q^{43} +(-4.53407 + 7.85325i) q^{44} +(-0.291669 - 0.505186i) q^{46} +(4.70696 + 10.0941i) q^{47} +(-4.48494 + 5.34494i) q^{49} +(-2.96997 + 0.356711i) q^{50} +(-2.54423 - 1.78149i) q^{52} +(0.204978 - 0.204978i) q^{53} +(-3.38800 - 11.8745i) q^{55} +(0.323118 - 0.0569745i) q^{56} +(1.51896 + 0.708303i) q^{58} +(0.796564 + 0.668396i) q^{59} +(9.12620 + 3.32167i) q^{61} +(-0.281730 + 1.05143i) q^{62} +(-0.558711 - 0.322572i) q^{64} +(4.15280 - 0.801535i) q^{65} +(6.27537 + 8.96216i) q^{67} +(-4.29921 - 6.13991i) q^{68} +(-0.112864 + 0.166852i) q^{70} +(2.53024 + 1.46084i) q^{71} +(2.66706 - 9.95358i) q^{73} +(3.32505 + 1.21022i) q^{74} +(-9.51649 - 7.98529i) q^{76} +(-0.753647 - 0.351431i) q^{77} +(11.9179 - 2.10144i) q^{79} +(4.25878 - 1.21510i) q^{80} +(-0.293080 + 0.293080i) q^{82} +(-5.85978 - 4.10306i) q^{83} +(10.0790 + 1.60999i) q^{85} +(-0.669585 + 0.797981i) q^{86} +(-5.08528 - 10.9054i) q^{88} +(6.74192 + 11.6773i) q^{89} +(0.142409 - 0.246659i) q^{91} +(-1.59502 - 0.139546i) q^{92} +(-6.56201 - 1.15706i) q^{94} +(16.8263 - 1.74606i) q^{95} +(0.311872 - 3.56471i) q^{97} +(-1.08038 - 4.03205i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q + 12 q^{2} + 12 q^{5} - 12 q^{7} + 18 q^{8} - 6 q^{10} + 36 q^{11} - 12 q^{13} - 24 q^{16} + 18 q^{17} - 36 q^{20} - 12 q^{22} + 36 q^{23} - 30 q^{25} - 24 q^{28} - 24 q^{31} + 48 q^{32} - 36 q^{35}+ \cdots - 324 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{17}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.343150 + 0.490069i −0.242644 + 0.346531i −0.921940 − 0.387333i \(-0.873396\pi\)
0.679296 + 0.733864i \(0.262285\pi\)
\(3\) 0 0
\(4\) 0.561625 + 1.54305i 0.280812 + 0.771525i
\(5\) −2.04155 − 0.912186i −0.913008 − 0.407942i
\(6\) 0 0
\(7\) −0.136472 + 0.0636379i −0.0515816 + 0.0240529i −0.448238 − 0.893914i \(-0.647948\pi\)
0.396656 + 0.917967i \(0.370170\pi\)
\(8\) −2.10468 − 0.563947i −0.744117 − 0.199385i
\(9\) 0 0
\(10\) 1.14759 − 0.687482i 0.362900 − 0.217401i
\(11\) 3.54970 + 4.23037i 1.07027 + 1.27550i 0.959515 + 0.281657i \(0.0908840\pi\)
0.110760 + 0.993847i \(0.464672\pi\)
\(12\) 0 0
\(13\) −1.54940 + 1.08490i −0.429725 + 0.300897i −0.768356 − 0.640022i \(-0.778925\pi\)
0.338631 + 0.940919i \(0.390036\pi\)
\(14\) 0.0156434 − 0.0887180i 0.00418087 − 0.0237109i
\(15\) 0 0
\(16\) −1.51722 + 1.27310i −0.379305 + 0.318275i
\(17\) −4.40907 + 1.18141i −1.06936 + 0.286533i −0.750228 − 0.661179i \(-0.770056\pi\)
−0.319127 + 0.947712i \(0.603390\pi\)
\(18\) 0 0
\(19\) −6.55178 + 3.78267i −1.50308 + 0.867804i −0.503087 + 0.864236i \(0.667803\pi\)
−0.999994 + 0.00356839i \(0.998864\pi\)
\(20\) 0.260967 − 3.66252i 0.0583539 − 0.818964i
\(21\) 0 0
\(22\) −3.29125 + 0.287947i −0.701697 + 0.0613906i
\(23\) −0.412075 + 0.883698i −0.0859236 + 0.184264i −0.944575 − 0.328297i \(-0.893525\pi\)
0.858651 + 0.512561i \(0.171303\pi\)
\(24\) 0 0
\(25\) 3.33583 + 3.72454i 0.667166 + 0.744909i
\(26\) − 1.13159i − 0.221924i
\(27\) 0 0
\(28\) −0.174843 − 0.174843i −0.0330421 − 0.0330421i
\(29\) −0.486461 − 2.75886i −0.0903336 − 0.512307i −0.996078 − 0.0884824i \(-0.971798\pi\)
0.905744 − 0.423825i \(-0.139313\pi\)
\(30\) 0 0
\(31\) 1.70974 − 0.622294i 0.307078 − 0.111767i −0.183885 − 0.982948i \(-0.558867\pi\)
0.490963 + 0.871181i \(0.336645\pi\)
\(32\) −0.483084 − 5.52167i −0.0853979 − 0.976103i
\(33\) 0 0
\(34\) 0.934001 − 2.56615i 0.160180 − 0.440090i
\(35\) 0.336664 − 0.00543199i 0.0569065 − 0.000918174i
\(36\) 0 0
\(37\) −1.53079 − 5.71298i −0.251660 − 0.939209i −0.969918 − 0.243432i \(-0.921727\pi\)
0.718258 − 0.695777i \(-0.244940\pi\)
\(38\) 0.394473 − 4.50885i 0.0639920 − 0.731431i
\(39\) 0 0
\(40\) 3.78238 + 3.07118i 0.598047 + 0.485597i
\(41\) 0.682276 + 0.120304i 0.106554 + 0.0187883i 0.226671 − 0.973971i \(-0.427216\pi\)
−0.120117 + 0.992760i \(0.538327\pi\)
\(42\) 0 0
\(43\) 1.73456 + 0.151755i 0.264518 + 0.0231424i 0.218643 − 0.975805i \(-0.429837\pi\)
0.0458750 + 0.998947i \(0.485392\pi\)
\(44\) −4.53407 + 7.85325i −0.683537 + 1.18392i
\(45\) 0 0
\(46\) −0.291669 − 0.505186i −0.0430043 − 0.0744856i
\(47\) 4.70696 + 10.0941i 0.686581 + 1.47238i 0.871892 + 0.489698i \(0.162893\pi\)
−0.185311 + 0.982680i \(0.559329\pi\)
\(48\) 0 0
\(49\) −4.48494 + 5.34494i −0.640705 + 0.763563i
\(50\) −2.96997 + 0.356711i −0.420018 + 0.0504466i
\(51\) 0 0
\(52\) −2.54423 − 1.78149i −0.352822 − 0.247049i
\(53\) 0.204978 − 0.204978i 0.0281559 − 0.0281559i −0.692889 − 0.721045i \(-0.743662\pi\)
0.721045 + 0.692889i \(0.243662\pi\)
\(54\) 0 0
\(55\) −3.38800 − 11.8745i −0.456838 − 1.60116i
\(56\) 0.323118 − 0.0569745i 0.0431785 − 0.00761353i
\(57\) 0 0
\(58\) 1.51896 + 0.708303i 0.199449 + 0.0930047i
\(59\) 0.796564 + 0.668396i 0.103704 + 0.0870178i 0.693165 − 0.720779i \(-0.256216\pi\)
−0.589462 + 0.807796i \(0.700660\pi\)
\(60\) 0 0
\(61\) 9.12620 + 3.32167i 1.16849 + 0.425296i 0.852123 − 0.523341i \(-0.175315\pi\)
0.316367 + 0.948637i \(0.397537\pi\)
\(62\) −0.281730 + 1.05143i −0.0357797 + 0.133532i
\(63\) 0 0
\(64\) −0.558711 − 0.322572i −0.0698389 − 0.0403215i
\(65\) 4.15280 − 0.801535i 0.515091 − 0.0994182i
\(66\) 0 0
\(67\) 6.27537 + 8.96216i 0.766659 + 1.09490i 0.992679 + 0.120780i \(0.0385394\pi\)
−0.226021 + 0.974122i \(0.572572\pi\)
\(68\) −4.29921 − 6.13991i −0.521356 − 0.744573i
\(69\) 0 0
\(70\) −0.112864 + 0.166852i −0.0134898 + 0.0199427i
\(71\) 2.53024 + 1.46084i 0.300285 + 0.173369i 0.642571 − 0.766226i \(-0.277868\pi\)
−0.342286 + 0.939596i \(0.611201\pi\)
\(72\) 0 0
\(73\) 2.66706 − 9.95358i 0.312155 − 1.16498i −0.614454 − 0.788953i \(-0.710624\pi\)
0.926609 − 0.376026i \(-0.122710\pi\)
\(74\) 3.32505 + 1.21022i 0.386529 + 0.140685i
\(75\) 0 0
\(76\) −9.51649 − 7.98529i −1.09162 − 0.915975i
\(77\) −0.753647 − 0.351431i −0.0858860 − 0.0400493i
\(78\) 0 0
\(79\) 11.9179 − 2.10144i 1.34086 − 0.236430i 0.543236 − 0.839580i \(-0.317199\pi\)
0.797628 + 0.603150i \(0.206088\pi\)
\(80\) 4.25878 − 1.21510i 0.476146 − 0.135853i
\(81\) 0 0
\(82\) −0.293080 + 0.293080i −0.0323653 + 0.0323653i
\(83\) −5.85978 − 4.10306i −0.643194 − 0.450370i 0.205927 − 0.978567i \(-0.433979\pi\)
−0.849121 + 0.528198i \(0.822868\pi\)
\(84\) 0 0
\(85\) 10.0790 + 1.60999i 1.09322 + 0.174628i
\(86\) −0.669585 + 0.797981i −0.0722033 + 0.0860485i
\(87\) 0 0
\(88\) −5.08528 − 10.9054i −0.542092 − 1.16252i
\(89\) 6.74192 + 11.6773i 0.714642 + 1.23780i 0.963097 + 0.269153i \(0.0867437\pi\)
−0.248455 + 0.968643i \(0.579923\pi\)
\(90\) 0 0
\(91\) 0.142409 − 0.246659i 0.0149285 − 0.0258569i
\(92\) −1.59502 − 0.139546i −0.166293 − 0.0145487i
\(93\) 0 0
\(94\) −6.56201 − 1.15706i −0.676819 − 0.119341i
\(95\) 16.8263 − 1.74606i 1.72634 − 0.179142i
\(96\) 0 0
\(97\) 0.311872 − 3.56471i 0.0316658 − 0.361941i −0.963786 − 0.266676i \(-0.914075\pi\)
0.995452 − 0.0952651i \(-0.0303699\pi\)
\(98\) −1.08038 − 4.03205i −0.109135 − 0.407298i
\(99\) 0 0
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 405.2.r.a.233.6 192
3.2 odd 2 135.2.q.a.68.11 yes 192
5.2 odd 4 inner 405.2.r.a.152.6 192
15.2 even 4 135.2.q.a.122.11 yes 192
15.8 even 4 675.2.ba.b.257.6 192
15.14 odd 2 675.2.ba.b.68.6 192
27.2 odd 18 inner 405.2.r.a.8.6 192
27.25 even 9 135.2.q.a.83.11 yes 192
135.2 even 36 inner 405.2.r.a.332.6 192
135.52 odd 36 135.2.q.a.2.11 ✓ 192
135.79 even 18 675.2.ba.b.218.6 192
135.133 odd 36 675.2.ba.b.407.6 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.q.a.2.11 ✓ 192 135.52 odd 36
135.2.q.a.68.11 yes 192 3.2 odd 2
135.2.q.a.83.11 yes 192 27.25 even 9
135.2.q.a.122.11 yes 192 15.2 even 4
405.2.r.a.8.6 192 27.2 odd 18 inner
405.2.r.a.152.6 192 5.2 odd 4 inner
405.2.r.a.233.6 192 1.1 even 1 trivial
405.2.r.a.332.6 192 135.2 even 36 inner
675.2.ba.b.68.6 192 15.14 odd 2
675.2.ba.b.218.6 192 135.79 even 18
675.2.ba.b.257.6 192 15.8 even 4
675.2.ba.b.407.6 192 135.133 odd 36