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 109.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 162.109
Dual form 162.8.c.l.55.1

$q$-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 \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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 \(-0.710750\pi\)
0.990425 + 0.138048i \(0.0440829\pi\)
\(6\) 0 0
\(7\) −508.000 − 879.882i −0.559784 − 0.969575i −0.997514 − 0.0704680i \(-0.977551\pi\)
0.437730 − 0.899107i \(-0.355783\pi\)
\(8\) −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 \(-0.206138\pi\)
−0.921218 + 0.389046i \(0.872805\pi\)
\(12\) 0 0
\(13\) −691.000 + 1196.85i −0.0872321 + 0.151090i −0.906340 − 0.422549i \(-0.861136\pi\)
0.819108 + 0.573639i \(0.194469\pi\)
\(14\) 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 \(-0.381756\pi\)
0.362989 + 0.931793i \(0.381756\pi\)
\(18\) 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 0.405609 + 0.914047i \(0.367059\pi\)
−0.994392 + 0.105755i \(0.966274\pi\)
\(24\) 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 \(-0.0389762\pi\)
−0.602034 + 0.798470i \(0.705643\pi\)
\(30\) 0 0
\(31\) −113776. + 197066.i −0.685938 + 1.18808i 0.287203 + 0.957870i \(0.407274\pi\)
−0.973141 + 0.230209i \(0.926059\pi\)
\(32\) 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 \(-0.837244\pi\)
0.859818 + 0.510600i \(0.170577\pi\)
\(42\) 0 0
\(43\) 315374. + 546244.i 0.604904 + 1.04772i 0.992067 + 0.125713i \(0.0401218\pi\)
−0.387163 + 0.922011i \(0.626545\pi\)
\(44\) 69888.0 0.123685
\(45\) 0 0
\(46\) −549696. −0.832665
\(47\) −236328. − 409332.i −0.332026 − 0.575087i 0.650883 − 0.759178i \(-0.274399\pi\)
−0.982909 + 0.184092i \(0.941066\pi\)
\(48\) 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 \(-0.742048\pi\)
−0.689224 + 0.724548i \(0.742048\pi\)
\(54\) 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 0.0554795 + 0.998460i \(0.482331\pi\)
−0.892431 + 0.451183i \(0.851002\pi\)
\(60\) 0 0
\(61\) −413851. − 716811.i −0.233448 − 0.404343i 0.725373 − 0.688356i \(-0.241667\pi\)
−0.958820 + 0.284013i \(0.908334\pi\)
\(62\) −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 −0.852946 − 0.521999i \(-0.825186\pi\)
0.878537 + 0.477674i \(0.158520\pi\)
\(68\) −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 \(-0.575362\pi\)
−0.234552 + 0.972104i \(0.575362\pi\)
\(72\) 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 0.994548 − 0.104283i \(-0.0332548\pi\)
−0.587586 + 0.809162i \(0.699921\pi\)
\(80\) −860160. −0.187830
\(81\) 0 0
\(82\) −86736.0 −0.0173720
\(83\) −2.83645e6 − 4.91287e6i −0.544504 − 0.943109i −0.998638 − 0.0521754i \(-0.983385\pi\)
0.454134 − 0.890934i \(-0.349949\pi\)
\(84\) 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 \(-0.855339\pi\)
−0.898496 + 0.438982i \(0.855339\pi\)
\(90\) 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 0.516865 − 0.856067i \(-0.327099\pi\)
−0.999808 + 0.0195848i \(0.993766\pi\)
\(98\) −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.l.109.1 2
3.2 odd 2 162.8.c.a.109.1 2
9.2 odd 6 162.8.c.a.55.1 2
9.4 even 3 2.8.a.a.1.1 ✓ 1
9.5 odd 6 18.8.a.b.1.1 1
9.7 even 3 inner 162.8.c.l.55.1 2
36.23 even 6 144.8.a.i.1.1 1
36.31 odd 6 16.8.a.b.1.1 1
45.4 even 6 50.8.a.g.1.1 1
45.13 odd 12 50.8.b.c.49.2 2
45.14 odd 6 450.8.a.c.1.1 1
45.22 odd 12 50.8.b.c.49.1 2
45.23 even 12 450.8.c.g.199.1 2
45.32 even 12 450.8.c.g.199.2 2
63.4 even 3 98.8.c.d.79.1 2
63.13 odd 6 98.8.a.a.1.1 1
63.31 odd 6 98.8.c.e.79.1 2
63.40 odd 6 98.8.c.e.67.1 2
63.58 even 3 98.8.c.d.67.1 2
72.5 odd 6 576.8.a.g.1.1 1
72.13 even 6 64.8.a.c.1.1 1
72.59 even 6 576.8.a.f.1.1 1
72.67 odd 6 64.8.a.e.1.1 1
99.76 odd 6 242.8.a.e.1.1 1
117.31 odd 12 338.8.b.d.337.2 2
117.103 even 6 338.8.a.d.1.1 1
117.112 odd 12 338.8.b.d.337.1 2
144.13 even 12 256.8.b.b.129.2 2
144.67 odd 12 256.8.b.f.129.1 2
144.85 even 12 256.8.b.b.129.1 2
144.139 odd 12 256.8.b.f.129.2 2
153.67 even 6 578.8.a.b.1.1 1
180.67 even 12 400.8.c.j.49.2 2
180.103 even 12 400.8.c.j.49.1 2
180.139 odd 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 9.4 even 3
16.8.a.b.1.1 1 36.31 odd 6
18.8.a.b.1.1 1 9.5 odd 6
50.8.a.g.1.1 1 45.4 even 6
50.8.b.c.49.1 2 45.22 odd 12
50.8.b.c.49.2 2 45.13 odd 12
64.8.a.c.1.1 1 72.13 even 6
64.8.a.e.1.1 1 72.67 odd 6
98.8.a.a.1.1 1 63.13 odd 6
98.8.c.d.67.1 2 63.58 even 3
98.8.c.d.79.1 2 63.4 even 3
98.8.c.e.67.1 2 63.40 odd 6
98.8.c.e.79.1 2 63.31 odd 6
144.8.a.i.1.1 1 36.23 even 6
162.8.c.a.55.1 2 9.2 odd 6
162.8.c.a.109.1 2 3.2 odd 2
162.8.c.l.55.1 2 9.7 even 3 inner
162.8.c.l.109.1 2 1.1 even 1 trivial
242.8.a.e.1.1 1 99.76 odd 6
256.8.b.b.129.1 2 144.85 even 12
256.8.b.b.129.2 2 144.13 even 12
256.8.b.f.129.1 2 144.67 odd 12
256.8.b.f.129.2 2 144.139 odd 12
338.8.a.d.1.1 1 117.103 even 6
338.8.b.d.337.1 2 117.112 odd 12
338.8.b.d.337.2 2 117.31 odd 12
400.8.a.l.1.1 1 180.139 odd 6
400.8.c.j.49.1 2 180.103 even 12
400.8.c.j.49.2 2 180.67 even 12
450.8.a.c.1.1 1 45.14 odd 6
450.8.c.g.199.1 2 45.23 even 12
450.8.c.g.199.2 2 45.32 even 12
576.8.a.f.1.1 1 72.59 even 6
576.8.a.g.1.1 1 72.5 odd 6
578.8.a.b.1.1 1 153.67 even 6