Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [162,8,Mod(55,162)] 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("162.55"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 162.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-8,0,-64,-210] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.6063741284\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 55.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 162.55
Dual form 162.8.c.a.109.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.00000 + 6.92820i) q^{2} +(-32.0000 - 55.4256i) q^{4} +(-105.000 - 181.865i) q^{5} +(-508.000 + 879.882i) q^{7} +512.000 q^{8} +1680.00 q^{10} +(546.000 - 945.700i) q^{11} +(-691.000 - 1196.85i) q^{13} +(-4064.00 - 7039.05i) q^{14} +(-2048.00 + 3547.24i) q^{16} -14706.0 q^{17} -39940.0 q^{19} +(-6720.00 + 11639.4i) q^{20} +(4368.00 + 7565.60i) q^{22} +(34356.0 + 59506.3i) q^{23} +(17012.5 - 29466.5i) q^{25} +11056.0 q^{26} +65024.0 q^{28} +(-51285.0 + 88828.2i) q^{29} +(-113776. - 197066. i) q^{31} +(-16384.0 - 28377.9i) q^{32} +(58824.0 - 101886. i) q^{34} +213360. q^{35} +160526. q^{37} +(159760. - 276712. i) q^{38} +(-53760.0 - 93115.1i) q^{40} +(5421.00 + 9389.45i) q^{41} +(315374. - 546244. i) q^{43} -69888.0 q^{44} -549696. q^{46} +(236328. - 409332. i) q^{47} +(-104357. - 180751. i) q^{49} +(136100. + 235732. i) q^{50} +(-44224.0 + 76598.2i) q^{52} +1.49402e6 q^{53} -229320. q^{55} +(-260096. + 450499. i) q^{56} +(-410280. - 710626. i) q^{58} +(1.32033e6 + 2.28688e6i) q^{59} +(-413851. + 716811. i) q^{61} +1.82042e6 q^{62} +262144. q^{64} +(-145110. + 251338. i) q^{65} +(63002.0 + 109123. i) q^{67} +(470592. + 815089. i) q^{68} +(-853440. + 1.47820e6i) q^{70} +1.41473e6 q^{71} +980282. q^{73} +(-642104. + 1.11216e6i) q^{74} +(1.27808e6 + 2.21370e6i) q^{76} +(554736. + 960831. i) q^{77} +(1.78340e6 - 3.08894e6i) q^{79} +860160. q^{80} -86736.0 q^{82} +(2.83645e6 - 4.91287e6i) q^{83} +(1.54413e6 + 2.67451e6i) q^{85} +(2.52299e6 + 4.36995e6i) q^{86} +(279552. - 484198. i) q^{88} +1.19512e7 q^{89} +1.40411e6 q^{91} +(2.19878e6 - 3.80841e6i) q^{92} +(1.89062e6 + 3.27466e6i) q^{94} +(4.19370e6 + 7.26370e6i) q^{95} +(-4.34107e6 + 7.51896e6i) q^{97} +1.66970e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 64 q^{4} - 210 q^{5} - 1016 q^{7} + 1024 q^{8} + 3360 q^{10} + 1092 q^{11} - 1382 q^{13} - 8128 q^{14} - 4096 q^{16} - 29412 q^{17} - 79880 q^{19} - 13440 q^{20} + 8736 q^{22} + 68712 q^{23}+ \cdots + 3339408 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 + 6.92820i −0.353553 + 0.612372i
\(3\) 0 0
\(4\) −32.0000 − 55.4256i −0.250000 − 0.433013i
\(5\) −105.000 − 181.865i −0.375659 − 0.650661i 0.614766 − 0.788709i \(-0.289250\pi\)
−0.990425 + 0.138048i \(0.955917\pi\)
\(6\) 0 0
\(7\) −508.000 + 879.882i −0.559784 + 0.969575i 0.437730 + 0.899107i \(0.355783\pi\)
−0.997514 + 0.0704680i \(0.977551\pi\)
\(8\) 512.000 0.353553
\(9\) 0 0
\(10\) 1680.00 0.531263
\(11\) 546.000 − 945.700i 0.123685 − 0.214229i −0.797533 − 0.603275i \(-0.793862\pi\)
0.921218 + 0.389046i \(0.127195\pi\)
\(12\) 0 0
\(13\) −691.000 − 1196.85i −0.0872321 − 0.151090i 0.819108 − 0.573639i \(-0.194469\pi\)
−0.906340 + 0.422549i \(0.861136\pi\)
\(14\) −4064.00 − 7039.05i −0.395827 − 0.685593i
\(15\) 0 0
\(16\) −2048.00 + 3547.24i −0.125000 + 0.216506i
\(17\) −14706.0 −0.725978 −0.362989 − 0.931793i \(-0.618244\pi\)
−0.362989 + 0.931793i \(0.618244\pi\)
\(18\) 0 0
\(19\) −39940.0 −1.33589 −0.667945 − 0.744211i \(-0.732826\pi\)
−0.667945 + 0.744211i \(0.732826\pi\)
\(20\) −6720.00 + 11639.4i −0.187830 + 0.325331i
\(21\) 0 0
\(22\) 4368.00 + 7565.60i 0.0874587 + 0.151483i
\(23\) 34356.0 + 59506.3i 0.588783 + 1.01980i 0.994392 + 0.105755i \(0.0337260\pi\)
−0.405609 + 0.914047i \(0.632941\pi\)
\(24\) 0 0
\(25\) 17012.5 − 29466.5i 0.217760 − 0.377171i
\(26\) 11056.0 0.123365
\(27\) 0 0
\(28\) 65024.0 0.559784
\(29\) −51285.0 + 88828.2i −0.390479 + 0.676329i −0.992513 − 0.122141i \(-0.961024\pi\)
0.602034 + 0.798470i \(0.294357\pi\)
\(30\) 0 0
\(31\) −113776. − 197066.i −0.685938 − 1.18808i −0.973141 − 0.230209i \(-0.926059\pi\)
0.287203 − 0.957870i \(-0.407274\pi\)
\(32\) −16384.0 − 28377.9i −0.0883883 − 0.153093i
\(33\) 0 0
\(34\) 58824.0 − 101886.i 0.256672 − 0.444569i
\(35\) 213360. 0.841153
\(36\) 0 0
\(37\) 160526. 0.521002 0.260501 − 0.965474i \(-0.416112\pi\)
0.260501 + 0.965474i \(0.416112\pi\)
\(38\) 159760. − 276712.i 0.472308 − 0.818062i
\(39\) 0 0
\(40\) −53760.0 − 93115.1i −0.132816 − 0.230043i
\(41\) 5421.00 + 9389.45i 0.0122839 + 0.0212763i 0.872102 − 0.489324i \(-0.162756\pi\)
−0.859818 + 0.510600i \(0.829423\pi\)
\(42\) 0 0
\(43\) 315374. − 546244.i 0.604904 − 1.04772i −0.387163 − 0.922011i \(-0.626545\pi\)
0.992067 − 0.125713i \(-0.0401218\pi\)
\(44\) −69888.0 −0.123685
\(45\) 0 0
\(46\) −549696. −0.832665
\(47\) 236328. − 409332.i 0.332026 − 0.575087i −0.650883 − 0.759178i \(-0.725601\pi\)
0.982909 + 0.184092i \(0.0589343\pi\)
\(48\) 0 0
\(49\) −104357. − 180751.i −0.126717 − 0.219479i
\(50\) 136100. + 235732.i 0.153980 + 0.266700i
\(51\) 0 0
\(52\) −44224.0 + 76598.2i −0.0436160 + 0.0755452i
\(53\) 1.49402e6 1.37845 0.689224 − 0.724548i \(-0.257952\pi\)
0.689224 + 0.724548i \(0.257952\pi\)
\(54\) 0 0
\(55\) −229320. −0.185854
\(56\) −260096. + 450499.i −0.197914 + 0.342796i
\(57\) 0 0
\(58\) −410280. − 710626.i −0.276110 − 0.478237i
\(59\) 1.32033e6 + 2.28688e6i 0.836952 + 1.44964i 0.892431 + 0.451183i \(0.148998\pi\)
−0.0554795 + 0.998460i \(0.517669\pi\)
\(60\) 0 0
\(61\) −413851. + 716811.i −0.233448 + 0.404343i −0.958820 − 0.284013i \(-0.908334\pi\)
0.725373 + 0.688356i \(0.241667\pi\)
\(62\) 1.82042e6 0.970063
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) −145110. + 251338.i −0.0655391 + 0.113517i
\(66\) 0 0
\(67\) 63002.0 + 109123.i 0.0255913 + 0.0443255i 0.878537 − 0.477674i \(-0.158520\pi\)
−0.852946 + 0.521999i \(0.825186\pi\)
\(68\) 470592. + 815089.i 0.181494 + 0.314358i
\(69\) 0 0
\(70\) −853440. + 1.47820e6i −0.297392 + 0.515099i
\(71\) 1.41473e6 0.469104 0.234552 − 0.972104i \(-0.424638\pi\)
0.234552 + 0.972104i \(0.424638\pi\)
\(72\) 0 0
\(73\) 980282. 0.294931 0.147466 − 0.989067i \(-0.452888\pi\)
0.147466 + 0.989067i \(0.452888\pi\)
\(74\) −642104. + 1.11216e6i −0.184202 + 0.319047i
\(75\) 0 0
\(76\) 1.27808e6 + 2.21370e6i 0.333972 + 0.578457i
\(77\) 554736. + 960831.i 0.138474 + 0.239844i
\(78\) 0 0
\(79\) 1.78340e6 − 3.08894e6i 0.406962 − 0.704879i −0.587586 − 0.809162i \(-0.699921\pi\)
0.994548 + 0.104283i \(0.0332548\pi\)
\(80\) 860160. 0.187830
\(81\) 0 0
\(82\) −86736.0 −0.0173720
\(83\) 2.83645e6 − 4.91287e6i 0.544504 − 0.943109i −0.454134 − 0.890934i \(-0.650051\pi\)
0.998638 − 0.0521754i \(-0.0166155\pi\)
\(84\) 0 0
\(85\) 1.54413e6 + 2.67451e6i 0.272720 + 0.472366i
\(86\) 2.52299e6 + 4.36995e6i 0.427732 + 0.740853i
\(87\) 0 0
\(88\) 279552. − 484198.i 0.0437294 − 0.0757415i
\(89\) 1.19512e7 1.79699 0.898496 − 0.438982i \(-0.144661\pi\)
0.898496 + 0.438982i \(0.144661\pi\)
\(90\) 0 0
\(91\) 1.40411e6 0.195325
\(92\) 2.19878e6 − 3.80841e6i 0.294391 − 0.509901i
\(93\) 0 0
\(94\) 1.89062e6 + 3.27466e6i 0.234778 + 0.406648i
\(95\) 4.19370e6 + 7.26370e6i 0.501839 + 0.869211i
\(96\) 0 0
\(97\) −4.34107e6 + 7.51896e6i −0.482943 + 0.836482i −0.999808 − 0.0195848i \(-0.993766\pi\)
0.516865 + 0.856067i \(0.327099\pi\)
\(98\) 1.66970e6 0.179204
\(99\) 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 162.8.c.a.55.1 2
3.2 odd 2 162.8.c.l.55.1 2
9.2 odd 6 2.8.a.a.1.1 ✓ 1
9.4 even 3 inner 162.8.c.a.109.1 2
9.5 odd 6 162.8.c.l.109.1 2
9.7 even 3 18.8.a.b.1.1 1
36.7 odd 6 144.8.a.i.1.1 1
36.11 even 6 16.8.a.b.1.1 1
45.2 even 12 50.8.b.c.49.1 2
45.7 odd 12 450.8.c.g.199.2 2
45.29 odd 6 50.8.a.g.1.1 1
45.34 even 6 450.8.a.c.1.1 1
45.38 even 12 50.8.b.c.49.2 2
45.43 odd 12 450.8.c.g.199.1 2
63.2 odd 6 98.8.c.d.67.1 2
63.11 odd 6 98.8.c.d.79.1 2
63.20 even 6 98.8.a.a.1.1 1
63.38 even 6 98.8.c.e.79.1 2
63.47 even 6 98.8.c.e.67.1 2
72.11 even 6 64.8.a.e.1.1 1
72.29 odd 6 64.8.a.c.1.1 1
72.43 odd 6 576.8.a.f.1.1 1
72.61 even 6 576.8.a.g.1.1 1
99.65 even 6 242.8.a.e.1.1 1
117.38 odd 6 338.8.a.d.1.1 1
117.47 even 12 338.8.b.d.337.1 2
117.83 even 12 338.8.b.d.337.2 2
144.11 even 12 256.8.b.f.129.2 2
144.29 odd 12 256.8.b.b.129.2 2
144.83 even 12 256.8.b.f.129.1 2
144.101 odd 12 256.8.b.b.129.1 2
153.101 odd 6 578.8.a.b.1.1 1
180.47 odd 12 400.8.c.j.49.2 2
180.83 odd 12 400.8.c.j.49.1 2
180.119 even 6 400.8.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 ✓ 1 9.2 odd 6
16.8.a.b.1.1 1 36.11 even 6
18.8.a.b.1.1 1 9.7 even 3
50.8.a.g.1.1 1 45.29 odd 6
50.8.b.c.49.1 2 45.2 even 12
50.8.b.c.49.2 2 45.38 even 12
64.8.a.c.1.1 1 72.29 odd 6
64.8.a.e.1.1 1 72.11 even 6
98.8.a.a.1.1 1 63.20 even 6
98.8.c.d.67.1 2 63.2 odd 6
98.8.c.d.79.1 2 63.11 odd 6
98.8.c.e.67.1 2 63.47 even 6
98.8.c.e.79.1 2 63.38 even 6
144.8.a.i.1.1 1 36.7 odd 6
162.8.c.a.55.1 2 1.1 even 1 trivial
162.8.c.a.109.1 2 9.4 even 3 inner
162.8.c.l.55.1 2 3.2 odd 2
162.8.c.l.109.1 2 9.5 odd 6
242.8.a.e.1.1 1 99.65 even 6
256.8.b.b.129.1 2 144.101 odd 12
256.8.b.b.129.2 2 144.29 odd 12
256.8.b.f.129.1 2 144.83 even 12
256.8.b.f.129.2 2 144.11 even 12
338.8.a.d.1.1 1 117.38 odd 6
338.8.b.d.337.1 2 117.47 even 12
338.8.b.d.337.2 2 117.83 even 12
400.8.a.l.1.1 1 180.119 even 6
400.8.c.j.49.1 2 180.83 odd 12
400.8.c.j.49.2 2 180.47 odd 12
450.8.a.c.1.1 1 45.34 even 6
450.8.c.g.199.1 2 45.43 odd 12
450.8.c.g.199.2 2 45.7 odd 12
576.8.a.f.1.1 1 72.43 odd 6
576.8.a.g.1.1 1 72.61 even 6
578.8.a.b.1.1 1 153.101 odd 6