Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [912,2,Mod(289,912)] 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("912.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bo (of order \(9\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,9] 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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 289.1
Root \(-0.173648 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 912.289
Dual form 912.2.bo.f.385.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.173648 + 0.984808i) q^{3} +(0.907604 - 0.761570i) q^{5} +(0.266044 - 0.460802i) q^{7} +(-0.939693 + 0.342020i) q^{9} +(0.939693 + 1.62760i) q^{11} +(-0.673648 + 3.82045i) q^{13} +(0.907604 + 0.761570i) q^{15} +(-1.09240 - 0.397600i) q^{17} +(3.93969 - 1.86516i) q^{19} +(0.500000 + 0.181985i) q^{21} +(5.13429 + 4.30818i) q^{23} +(-0.624485 + 3.54163i) q^{25} +(-0.500000 - 0.866025i) q^{27} +(3.77972 - 1.37570i) q^{29} +(-0.979055 + 1.69577i) q^{31} +(-1.43969 + 1.20805i) q^{33} +(-0.109470 - 0.620838i) q^{35} +6.88713 q^{37} -3.87939 q^{39} +(-1.56031 - 8.84894i) q^{41} +(-1.85844 + 1.55942i) q^{43} +(-0.592396 + 1.02606i) q^{45} +(1.91875 - 0.698367i) q^{47} +(3.35844 + 5.81699i) q^{49} +(0.201867 - 1.14484i) q^{51} +(9.93629 + 8.33754i) q^{53} +(2.09240 + 0.761570i) q^{55} +(2.52094 + 3.55596i) q^{57} +(-2.51842 - 0.916629i) q^{59} +(-8.69253 - 7.29390i) q^{61} +(-0.0923963 + 0.524005i) q^{63} +(2.29813 + 3.98048i) q^{65} +(-10.4966 + 3.82045i) q^{67} +(-3.35117 + 5.80439i) q^{69} +(-4.65136 + 3.90295i) q^{71} +(-0.0569038 - 0.322718i) q^{73} -3.59627 q^{75} +1.00000 q^{77} +(2.80154 + 15.8883i) q^{79} +(0.766044 - 0.642788i) q^{81} +(5.78699 - 10.0234i) q^{83} +(-1.29426 + 0.471073i) q^{85} +(2.01114 + 3.48340i) q^{87} +(-0.618089 + 3.50535i) q^{89} +(1.58125 + 1.32683i) q^{91} +(-1.84002 - 0.669713i) q^{93} +(2.15523 - 4.69318i) q^{95} +(-5.52481 - 2.01087i) q^{97} +(-1.43969 - 1.20805i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{5} - 3 q^{7} - 3 q^{13} + 9 q^{15} - 3 q^{17} + 18 q^{19} + 3 q^{21} + 21 q^{23} + 9 q^{25} - 3 q^{27} - 3 q^{29} - 9 q^{31} - 3 q^{33} - 18 q^{35} - 18 q^{37} - 12 q^{39} - 15 q^{41} - 3 q^{43}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.173648 + 0.984808i 0.100256 + 0.568579i
\(4\) 0 0
\(5\) 0.907604 − 0.761570i 0.405893 − 0.340584i −0.416873 − 0.908965i \(-0.636874\pi\)
0.822766 + 0.568380i \(0.192430\pi\)
\(6\) 0 0
\(7\) 0.266044 − 0.460802i 0.100555 − 0.174167i −0.811358 − 0.584549i \(-0.801271\pi\)
0.911914 + 0.410382i \(0.134605\pi\)
\(8\) 0 0
\(9\) −0.939693 + 0.342020i −0.313231 + 0.114007i
\(10\) 0 0
\(11\) 0.939693 + 1.62760i 0.283328 + 0.490738i 0.972202 − 0.234142i \(-0.0752282\pi\)
−0.688874 + 0.724881i \(0.741895\pi\)
\(12\) 0 0
\(13\) −0.673648 + 3.82045i −0.186836 + 1.05960i 0.736737 + 0.676180i \(0.236366\pi\)
−0.923573 + 0.383422i \(0.874745\pi\)
\(14\) 0 0
\(15\) 0.907604 + 0.761570i 0.234342 + 0.196637i
\(16\) 0 0
\(17\) −1.09240 − 0.397600i −0.264945 − 0.0964321i 0.206132 − 0.978524i \(-0.433912\pi\)
−0.471077 + 0.882092i \(0.656135\pi\)
\(18\) 0 0
\(19\) 3.93969 − 1.86516i 0.903827 − 0.427897i
\(20\) 0 0
\(21\) 0.500000 + 0.181985i 0.109109 + 0.0397124i
\(22\) 0 0
\(23\) 5.13429 + 4.30818i 1.07057 + 0.898317i 0.995104 − 0.0988312i \(-0.0315104\pi\)
0.0754683 + 0.997148i \(0.475955\pi\)
\(24\) 0 0
\(25\) −0.624485 + 3.54163i −0.124897 + 0.708326i
\(26\) 0 0
\(27\) −0.500000 − 0.866025i −0.0962250 − 0.166667i
\(28\) 0 0
\(29\) 3.77972 − 1.37570i 0.701875 − 0.255462i 0.0336640 − 0.999433i \(-0.489282\pi\)
0.668211 + 0.743971i \(0.267060\pi\)
\(30\) 0 0
\(31\) −0.979055 + 1.69577i −0.175844 + 0.304570i −0.940453 − 0.339924i \(-0.889599\pi\)
0.764609 + 0.644494i \(0.222932\pi\)
\(32\) 0 0
\(33\) −1.43969 + 1.20805i −0.250618 + 0.210294i
\(34\) 0 0
\(35\) −0.109470 − 0.620838i −0.0185039 − 0.104941i
\(36\) 0 0
\(37\) 6.88713 1.13224 0.566118 − 0.824324i \(-0.308445\pi\)
0.566118 + 0.824324i \(0.308445\pi\)
\(38\) 0 0
\(39\) −3.87939 −0.621199
\(40\) 0 0
\(41\) −1.56031 − 8.84894i −0.243679 − 1.38197i −0.823540 − 0.567258i \(-0.808004\pi\)
0.579861 − 0.814715i \(-0.303107\pi\)
\(42\) 0 0
\(43\) −1.85844 + 1.55942i −0.283410 + 0.237809i −0.773399 − 0.633919i \(-0.781445\pi\)
0.489989 + 0.871728i \(0.337001\pi\)
\(44\) 0 0
\(45\) −0.592396 + 1.02606i −0.0883092 + 0.152956i
\(46\) 0 0
\(47\) 1.91875 − 0.698367i 0.279878 − 0.101867i −0.198267 − 0.980148i \(-0.563531\pi\)
0.478145 + 0.878281i \(0.341309\pi\)
\(48\) 0 0
\(49\) 3.35844 + 5.81699i 0.479777 + 0.830999i
\(50\) 0 0
\(51\) 0.201867 − 1.14484i 0.0282670 − 0.160310i
\(52\) 0 0
\(53\) 9.93629 + 8.33754i 1.36485 + 1.14525i 0.974450 + 0.224605i \(0.0721093\pi\)
0.390404 + 0.920643i \(0.372335\pi\)
\(54\) 0 0
\(55\) 2.09240 + 0.761570i 0.282139 + 0.102690i
\(56\) 0 0
\(57\) 2.52094 + 3.55596i 0.333907 + 0.470998i
\(58\) 0 0
\(59\) −2.51842 − 0.916629i −0.327870 − 0.119335i 0.172840 − 0.984950i \(-0.444706\pi\)
−0.500710 + 0.865615i \(0.666928\pi\)
\(60\) 0 0
\(61\) −8.69253 − 7.29390i −1.11296 − 0.933888i −0.114737 − 0.993396i \(-0.536603\pi\)
−0.998228 + 0.0595075i \(0.981047\pi\)
\(62\) 0 0
\(63\) −0.0923963 + 0.524005i −0.0116408 + 0.0660185i
\(64\) 0 0
\(65\) 2.29813 + 3.98048i 0.285048 + 0.493718i
\(66\) 0 0
\(67\) −10.4966 + 3.82045i −1.28236 + 0.466742i −0.891213 − 0.453585i \(-0.850145\pi\)
−0.391150 + 0.920327i \(0.627923\pi\)
\(68\) 0 0
\(69\) −3.35117 + 5.80439i −0.403433 + 0.698767i
\(70\) 0 0
\(71\) −4.65136 + 3.90295i −0.552015 + 0.463195i −0.875623 − 0.482996i \(-0.839549\pi\)
0.323608 + 0.946191i \(0.395104\pi\)
\(72\) 0 0
\(73\) −0.0569038 − 0.322718i −0.00666009 − 0.0377712i 0.981297 − 0.192502i \(-0.0616601\pi\)
−0.987957 + 0.154731i \(0.950549\pi\)
\(74\) 0 0
\(75\) −3.59627 −0.415261
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 2.80154 + 15.8883i 0.315198 + 1.78757i 0.571109 + 0.820874i \(0.306513\pi\)
−0.255912 + 0.966700i \(0.582376\pi\)
\(80\) 0 0
\(81\) 0.766044 − 0.642788i 0.0851160 − 0.0714208i
\(82\) 0 0
\(83\) 5.78699 − 10.0234i 0.635205 − 1.10021i −0.351267 − 0.936275i \(-0.614249\pi\)
0.986472 − 0.163931i \(-0.0524175\pi\)
\(84\) 0 0
\(85\) −1.29426 + 0.471073i −0.140383 + 0.0510951i
\(86\) 0 0
\(87\) 2.01114 + 3.48340i 0.215617 + 0.373460i
\(88\) 0 0
\(89\) −0.618089 + 3.50535i −0.0655173 + 0.371567i 0.934366 + 0.356314i \(0.115967\pi\)
−0.999884 + 0.0152532i \(0.995145\pi\)
\(90\) 0 0
\(91\) 1.58125 + 1.32683i 0.165760 + 0.139089i
\(92\) 0 0
\(93\) −1.84002 − 0.669713i −0.190801 − 0.0694460i
\(94\) 0 0
\(95\) 2.15523 − 4.69318i 0.221122 − 0.481510i
\(96\) 0 0
\(97\) −5.52481 − 2.01087i −0.560960 − 0.204173i 0.0459494 − 0.998944i \(-0.485369\pi\)
−0.606909 + 0.794771i \(0.707591\pi\)
\(98\) 0 0
\(99\) −1.43969 − 1.20805i −0.144695 − 0.121413i
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 912.2.bo.f.289.1 6
4.3 odd 2 114.2.i.d.61.1 yes 6
12.11 even 2 342.2.u.a.289.1 6
19.5 even 9 inner 912.2.bo.f.385.1 6
76.43 odd 18 114.2.i.d.43.1 ✓ 6
76.47 odd 18 2166.2.a.o.1.2 3
76.67 even 18 2166.2.a.u.1.2 3
228.47 even 18 6498.2.a.bs.1.2 3
228.119 even 18 342.2.u.a.271.1 6
228.143 odd 18 6498.2.a.bn.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.43.1 ✓ 6 76.43 odd 18
114.2.i.d.61.1 yes 6 4.3 odd 2
342.2.u.a.271.1 6 228.119 even 18
342.2.u.a.289.1 6 12.11 even 2
912.2.bo.f.289.1 6 1.1 even 1 trivial
912.2.bo.f.385.1 6 19.5 even 9 inner
2166.2.a.o.1.2 3 76.47 odd 18
2166.2.a.u.1.2 3 76.67 even 18
6498.2.a.bn.1.2 3 228.143 odd 18
6498.2.a.bs.1.2 3 228.47 even 18