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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [864,2,Mod(109,864)] 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("864.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [128,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 109.7
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.17722 + 0.783679i) q^{2} +(0.771696 - 1.84512i) q^{4} +(-3.84162 + 1.59125i) q^{5} +(-2.86540 + 2.86540i) q^{7} +(0.537529 + 2.77688i) q^{8} +(3.27540 - 4.88384i) q^{10} +(1.44931 + 3.49895i) q^{11} +(0.694364 + 0.287615i) q^{13} +(1.12766 - 5.61876i) q^{14} +(-2.80897 - 2.84775i) q^{16} +6.47207i q^{17} +(0.222052 + 0.0919770i) q^{19} +(-0.0285041 + 8.31622i) q^{20} +(-4.44821 - 2.98324i) q^{22} +(-5.63006 - 5.63006i) q^{23} +(8.69041 - 8.69041i) q^{25} +(-1.04282 + 0.205572i) q^{26} +(3.07581 + 7.49824i) q^{28} +(0.607498 - 1.46663i) q^{29} +2.58386 q^{31} +(5.53850 + 1.15110i) q^{32} +(-5.07202 - 7.61905i) q^{34} +(6.44820 - 15.5673i) q^{35} +(-5.21659 + 2.16078i) q^{37} +(-0.333485 + 0.0657403i) q^{38} +(-6.48369 - 9.81236i) q^{40} +(-4.92361 - 4.92361i) q^{41} +(0.733289 + 1.77032i) q^{43} +(7.57442 + 0.0259616i) q^{44} +(11.0400 + 2.21566i) q^{46} -3.13117i q^{47} -9.42105i q^{49} +(-3.42004 + 17.0410i) q^{50} +(1.06652 - 1.05924i) q^{52} +(-2.43120 - 5.86943i) q^{53} +(-11.1354 - 11.1354i) q^{55} +(-9.49711 - 6.41664i) q^{56} +(0.434207 + 2.20263i) q^{58} +(-0.500271 + 0.207219i) q^{59} +(-0.858724 + 2.07314i) q^{61} +(-3.04177 + 2.02492i) q^{62} +(-7.42212 + 2.98531i) q^{64} -3.12515 q^{65} +(-5.48254 + 13.2360i) q^{67} +(11.9418 + 4.99447i) q^{68} +(4.60884 + 23.3795i) q^{70} +(-4.53047 + 4.53047i) q^{71} +(-0.809706 - 0.809706i) q^{73} +(4.44772 - 6.63185i) q^{74} +(0.341066 - 0.338736i) q^{76} +(-14.1788 - 5.87303i) q^{77} -3.87921i q^{79} +(15.3225 + 6.47019i) q^{80} +(9.65471 + 1.93765i) q^{82} +(4.86848 + 2.01659i) q^{83} +(-10.2987 - 24.8632i) q^{85} +(-2.25060 - 1.50939i) q^{86} +(-8.93711 + 5.90535i) q^{88} +(11.2465 - 11.2465i) q^{89} +(-2.81376 + 1.16550i) q^{91} +(-14.7328 + 6.04347i) q^{92} +(2.45383 + 3.68607i) q^{94} -0.999397 q^{95} +8.17436 q^{97} +(7.38308 + 11.0907i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.17722 + 0.783679i −0.832421 + 0.554144i
\(3\) 0 0
\(4\) 0.771696 − 1.84512i 0.385848 − 0.922562i
\(5\) −3.84162 + 1.59125i −1.71802 + 0.711628i −0.718147 + 0.695892i \(0.755009\pi\)
−0.999876 + 0.0157367i \(0.994991\pi\)
\(6\) 0 0
\(7\) −2.86540 + 2.86540i −1.08302 + 1.08302i −0.0867936 + 0.996226i \(0.527662\pi\)
−0.996226 + 0.0867936i \(0.972338\pi\)
\(8\) 0.537529 + 2.77688i 0.190045 + 0.981775i
\(9\) 0 0
\(10\) 3.27540 − 4.88384i 1.03577 − 1.54441i
\(11\) 1.44931 + 3.49895i 0.436984 + 1.05497i 0.976985 + 0.213308i \(0.0684238\pi\)
−0.540001 + 0.841664i \(0.681576\pi\)
\(12\) 0 0
\(13\) 0.694364 + 0.287615i 0.192582 + 0.0797701i 0.476890 − 0.878963i \(-0.341764\pi\)
−0.284308 + 0.958733i \(0.591764\pi\)
\(14\) 1.12766 − 5.61876i 0.301378 − 1.50168i
\(15\) 0 0
\(16\) −2.80897 − 2.84775i −0.702243 − 0.711937i
\(17\) 6.47207i 1.56971i 0.619681 + 0.784854i \(0.287262\pi\)
−0.619681 + 0.784854i \(0.712738\pi\)
\(18\) 0 0
\(19\) 0.222052 + 0.0919770i 0.0509422 + 0.0211010i 0.408009 − 0.912978i \(-0.366223\pi\)
−0.357067 + 0.934079i \(0.616223\pi\)
\(20\) −0.0285041 + 8.31622i −0.00637371 + 1.85956i
\(21\) 0 0
\(22\) −4.44821 − 2.98324i −0.948362 − 0.636029i
\(23\) −5.63006 − 5.63006i −1.17395 − 1.17395i −0.981260 − 0.192688i \(-0.938280\pi\)
−0.192688 − 0.981260i \(-0.561720\pi\)
\(24\) 0 0
\(25\) 8.69041 − 8.69041i 1.73808 − 1.73808i
\(26\) −1.04282 + 0.205572i −0.204513 + 0.0403160i
\(27\) 0 0
\(28\) 3.07581 + 7.49824i 0.581273 + 1.41703i
\(29\) 0.607498 − 1.46663i 0.112809 − 0.272346i −0.857384 − 0.514677i \(-0.827912\pi\)
0.970194 + 0.242331i \(0.0779119\pi\)
\(30\) 0 0
\(31\) 2.58386 0.464075 0.232038 − 0.972707i \(-0.425461\pi\)
0.232038 + 0.972707i \(0.425461\pi\)
\(32\) 5.53850 + 1.15110i 0.979078 + 0.203487i
\(33\) 0 0
\(34\) −5.07202 − 7.61905i −0.869845 − 1.30666i
\(35\) 6.44820 − 15.5673i 1.08995 − 2.63136i
\(36\) 0 0
\(37\) −5.21659 + 2.16078i −0.857602 + 0.355230i −0.767769 − 0.640727i \(-0.778633\pi\)
−0.0898327 + 0.995957i \(0.528633\pi\)
\(38\) −0.333485 + 0.0657403i −0.0540983 + 0.0106645i
\(39\) 0 0
\(40\) −6.48369 − 9.81236i −1.02516 − 1.55147i
\(41\) −4.92361 − 4.92361i −0.768939 − 0.768939i 0.208981 − 0.977920i \(-0.432985\pi\)
−0.977920 + 0.208981i \(0.932985\pi\)
\(42\) 0 0
\(43\) 0.733289 + 1.77032i 0.111825 + 0.269971i 0.969878 − 0.243591i \(-0.0783256\pi\)
−0.858052 + 0.513562i \(0.828326\pi\)
\(44\) 7.57442 + 0.0259616i 1.14189 + 0.00391385i
\(45\) 0 0
\(46\) 11.0400 + 2.21566i 1.62775 + 0.326682i
\(47\) − 3.13117i − 0.456728i −0.973576 − 0.228364i \(-0.926662\pi\)
0.973576 − 0.228364i \(-0.0733376\pi\)
\(48\) 0 0
\(49\) − 9.42105i − 1.34586i
\(50\) −3.42004 + 17.0410i −0.483666 + 2.40996i
\(51\) 0 0
\(52\) 1.06652 − 1.05924i 0.147900 − 0.146890i
\(53\) −2.43120 − 5.86943i −0.333950 − 0.806228i −0.998271 − 0.0587798i \(-0.981279\pi\)
0.664321 − 0.747448i \(-0.268721\pi\)
\(54\) 0 0
\(55\) −11.1354 − 11.1354i −1.50150 − 1.50150i
\(56\) −9.49711 − 6.41664i −1.26911 − 0.857459i
\(57\) 0 0
\(58\) 0.434207 + 2.20263i 0.0570142 + 0.289219i
\(59\) −0.500271 + 0.207219i −0.0651297 + 0.0269776i −0.415010 − 0.909817i \(-0.636222\pi\)
0.349881 + 0.936794i \(0.386222\pi\)
\(60\) 0 0
\(61\) −0.858724 + 2.07314i −0.109948 + 0.265439i −0.969271 − 0.245995i \(-0.920885\pi\)
0.859323 + 0.511434i \(0.170885\pi\)
\(62\) −3.04177 + 2.02492i −0.386306 + 0.257165i
\(63\) 0 0
\(64\) −7.42212 + 2.98531i −0.927766 + 0.373164i
\(65\) −3.12515 −0.387627
\(66\) 0 0
\(67\) −5.48254 + 13.2360i −0.669799 + 1.61704i 0.112148 + 0.993692i \(0.464227\pi\)
−0.781947 + 0.623345i \(0.785773\pi\)
\(68\) 11.9418 + 4.99447i 1.44815 + 0.605668i
\(69\) 0 0
\(70\) 4.60884 + 23.3795i 0.550861 + 2.79439i
\(71\) −4.53047 + 4.53047i −0.537667 + 0.537667i −0.922843 − 0.385176i \(-0.874141\pi\)
0.385176 + 0.922843i \(0.374141\pi\)
\(72\) 0 0
\(73\) −0.809706 − 0.809706i −0.0947690 − 0.0947690i 0.658133 − 0.752902i \(-0.271347\pi\)
−0.752902 + 0.658133i \(0.771347\pi\)
\(74\) 4.44772 − 6.63185i 0.517036 − 0.770936i
\(75\) 0 0
\(76\) 0.341066 − 0.338736i 0.0391229 − 0.0388556i
\(77\) −14.1788 − 5.87303i −1.61582 − 0.669294i
\(78\) 0 0
\(79\) − 3.87921i − 0.436445i −0.975899 − 0.218223i \(-0.929974\pi\)
0.975899 − 0.218223i \(-0.0700259\pi\)
\(80\) 15.3225 + 6.47019i 1.71310 + 0.723389i
\(81\) 0 0
\(82\) 9.65471 + 1.93765i 1.06618 + 0.213977i
\(83\) 4.86848 + 2.01659i 0.534385 + 0.221350i 0.633523 − 0.773724i \(-0.281608\pi\)
−0.0991375 + 0.995074i \(0.531608\pi\)
\(84\) 0 0
\(85\) −10.2987 − 24.8632i −1.11705 − 2.69679i
\(86\) −2.25060 − 1.50939i −0.242689 − 0.162762i
\(87\) 0 0
\(88\) −8.93711 + 5.90535i −0.952699 + 0.629513i
\(89\) 11.2465 − 11.2465i 1.19212 − 1.19212i 0.215652 − 0.976470i \(-0.430812\pi\)
0.976470 − 0.215652i \(-0.0691877\pi\)
\(90\) 0 0
\(91\) −2.81376 + 1.16550i −0.294963 + 0.122178i
\(92\) −14.7328 + 6.04347i −1.53601 + 0.630075i
\(93\) 0 0
\(94\) 2.45383 + 3.68607i 0.253093 + 0.380190i
\(95\) −0.999397 −0.102536
\(96\) 0 0
\(97\) 8.17436 0.829980 0.414990 − 0.909826i \(-0.363785\pi\)
0.414990 + 0.909826i \(0.363785\pi\)
\(98\) 7.38308 + 11.0907i 0.745803 + 1.12033i
\(99\) 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 864.2.v.a.109.7 ✓ 128
3.2 odd 2 inner 864.2.v.a.109.26 yes 128
32.5 even 8 inner 864.2.v.a.325.7 yes 128
96.5 odd 8 inner 864.2.v.a.325.26 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.7 ✓ 128 1.1 even 1 trivial
864.2.v.a.109.26 yes 128 3.2 odd 2 inner
864.2.v.a.325.7 yes 128 32.5 even 8 inner
864.2.v.a.325.26 yes 128 96.5 odd 8 inner