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 = [6,0,0,0,3] 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 529.1
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 912.529
Dual form 912.2.bo.d.481.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.939693 + 0.342020i) q^{3} +(-0.613341 + 3.47843i) q^{5} +(1.85844 + 3.21891i) q^{7} +(0.766044 - 0.642788i) q^{9} +(-2.64543 + 4.58202i) q^{11} +(0.213011 + 0.0775297i) q^{13} +(-0.613341 - 3.47843i) q^{15} +(-1.26604 - 1.06234i) q^{17} +(4.17752 + 1.24432i) q^{19} +(-2.84730 - 2.38917i) q^{21} +(-1.50727 - 8.54818i) q^{23} +(-7.02481 - 2.55682i) q^{25} +(-0.500000 + 0.866025i) q^{27} +(0.0923963 - 0.0775297i) q^{29} +(1.56031 + 2.70253i) q^{31} +(0.918748 - 5.21048i) q^{33} +(-12.3366 + 4.49016i) q^{35} +5.12836 q^{37} -0.226682 q^{39} +(-6.67752 + 2.43042i) q^{41} +(0.929892 - 5.27368i) q^{43} +(1.76604 + 3.05888i) q^{45} +(1.92262 - 1.61327i) q^{47} +(-3.40760 + 5.90214i) q^{49} +(1.55303 + 0.565258i) q^{51} +(1.03074 + 5.84564i) q^{53} +(-14.3157 - 12.0123i) q^{55} +(-4.35117 + 0.259515i) q^{57} +(-0.167718 - 0.140732i) q^{59} +(-0.273318 - 1.55007i) q^{61} +(3.49273 + 1.27125i) q^{63} +(-0.400330 + 0.693392i) q^{65} +(-11.8589 + 9.95080i) q^{67} +(4.34002 + 7.51714i) q^{69} +(0.235300 - 1.33445i) q^{71} +(-2.27972 + 0.829748i) q^{73} +7.47565 q^{75} -19.6655 q^{77} +(-2.69207 + 0.979832i) q^{79} +(0.173648 - 0.984808i) q^{81} +(0.960637 + 1.66387i) q^{83} +(4.47178 - 3.75227i) q^{85} +(-0.0603074 + 0.104455i) q^{87} +(11.4226 + 4.15749i) q^{89} +(0.146307 + 0.829748i) q^{91} +(-2.39053 - 2.00589i) q^{93} +(-6.89053 + 13.7680i) q^{95} +(13.4081 + 11.2507i) q^{97} +(0.918748 + 5.21048i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} + 3 q^{7} + 9 q^{13} + 3 q^{15} - 3 q^{17} - 15 q^{21} - 27 q^{23} - 15 q^{25} - 3 q^{27} - 3 q^{29} + 15 q^{31} + 3 q^{33} - 12 q^{35} - 6 q^{37} + 12 q^{39} - 15 q^{41} - 3 q^{43} + 6 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{2}{9}\right)\) \(1\) \(1\) \(1\)

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
\(3\) −0.939693 + 0.342020i −0.542532 + 0.197465i
\(4\) 0 0
\(5\) −0.613341 + 3.47843i −0.274294 + 1.55560i 0.466900 + 0.884310i \(0.345371\pi\)
−0.741194 + 0.671290i \(0.765740\pi\)
\(6\) 0 0
\(7\) 1.85844 + 3.21891i 0.702425 + 1.21664i 0.967613 + 0.252438i \(0.0812325\pi\)
−0.265188 + 0.964197i \(0.585434\pi\)
\(8\) 0 0
\(9\) 0.766044 − 0.642788i 0.255348 − 0.214263i
\(10\) 0 0
\(11\) −2.64543 + 4.58202i −0.797627 + 1.38153i 0.123531 + 0.992341i \(0.460578\pi\)
−0.921158 + 0.389190i \(0.872755\pi\)
\(12\) 0 0
\(13\) 0.213011 + 0.0775297i 0.0590786 + 0.0215029i 0.371390 − 0.928477i \(-0.378881\pi\)
−0.312312 + 0.949980i \(0.601103\pi\)
\(14\) 0 0
\(15\) −0.613341 − 3.47843i −0.158364 − 0.898126i
\(16\) 0 0
\(17\) −1.26604 − 1.06234i −0.307061 − 0.257655i 0.476215 − 0.879329i \(-0.342008\pi\)
−0.783276 + 0.621674i \(0.786453\pi\)
\(18\) 0 0
\(19\) 4.17752 + 1.24432i 0.958388 + 0.285467i
\(20\) 0 0
\(21\) −2.84730 − 2.38917i −0.621331 − 0.521359i
\(22\) 0 0
\(23\) −1.50727 − 8.54818i −0.314288 − 1.78242i −0.576182 − 0.817321i \(-0.695458\pi\)
0.261894 − 0.965097i \(-0.415653\pi\)
\(24\) 0 0
\(25\) −7.02481 − 2.55682i −1.40496 − 0.511365i
\(26\) 0 0
\(27\) −0.500000 + 0.866025i −0.0962250 + 0.166667i
\(28\) 0 0
\(29\) 0.0923963 − 0.0775297i 0.0171576 − 0.0143969i −0.634169 − 0.773195i \(-0.718657\pi\)
0.651326 + 0.758798i \(0.274213\pi\)
\(30\) 0 0
\(31\) 1.56031 + 2.70253i 0.280239 + 0.485389i 0.971444 − 0.237271i \(-0.0762528\pi\)
−0.691204 + 0.722660i \(0.742919\pi\)
\(32\) 0 0
\(33\) 0.918748 − 5.21048i 0.159934 − 0.907028i
\(34\) 0 0
\(35\) −12.3366 + 4.49016i −2.08527 + 0.758976i
\(36\) 0 0
\(37\) 5.12836 0.843096 0.421548 − 0.906806i \(-0.361487\pi\)
0.421548 + 0.906806i \(0.361487\pi\)
\(38\) 0 0
\(39\) −0.226682 −0.0362981
\(40\) 0 0
\(41\) −6.67752 + 2.43042i −1.04285 + 0.379568i −0.805961 − 0.591968i \(-0.798351\pi\)
−0.236892 + 0.971536i \(0.576129\pi\)
\(42\) 0 0
\(43\) 0.929892 − 5.27368i 0.141807 − 0.804229i −0.828068 − 0.560627i \(-0.810560\pi\)
0.969875 − 0.243602i \(-0.0783289\pi\)
\(44\) 0 0
\(45\) 1.76604 + 3.05888i 0.263266 + 0.455991i
\(46\) 0 0
\(47\) 1.92262 − 1.61327i 0.280443 − 0.235319i −0.491706 − 0.870761i \(-0.663626\pi\)
0.772149 + 0.635442i \(0.219182\pi\)
\(48\) 0 0
\(49\) −3.40760 + 5.90214i −0.486801 + 0.843163i
\(50\) 0 0
\(51\) 1.55303 + 0.565258i 0.217468 + 0.0791519i
\(52\) 0 0
\(53\) 1.03074 + 5.84564i 0.141584 + 0.802961i 0.970047 + 0.242918i \(0.0781044\pi\)
−0.828463 + 0.560043i \(0.810784\pi\)
\(54\) 0 0
\(55\) −14.3157 − 12.0123i −1.93033 − 1.61974i
\(56\) 0 0
\(57\) −4.35117 + 0.259515i −0.576326 + 0.0343736i
\(58\) 0 0
\(59\) −0.167718 − 0.140732i −0.0218351 − 0.0183218i 0.631805 − 0.775128i \(-0.282314\pi\)
−0.653640 + 0.756806i \(0.726759\pi\)
\(60\) 0 0
\(61\) −0.273318 − 1.55007i −0.0349948 − 0.198466i 0.962298 − 0.271997i \(-0.0876841\pi\)
−0.997293 + 0.0735316i \(0.976573\pi\)
\(62\) 0 0
\(63\) 3.49273 + 1.27125i 0.440042 + 0.160162i
\(64\) 0 0
\(65\) −0.400330 + 0.693392i −0.0496548 + 0.0860046i
\(66\) 0 0
\(67\) −11.8589 + 9.95080i −1.44880 + 1.21568i −0.515343 + 0.856984i \(0.672336\pi\)
−0.933453 + 0.358701i \(0.883220\pi\)
\(68\) 0 0
\(69\) 4.34002 + 7.51714i 0.522477 + 0.904957i
\(70\) 0 0
\(71\) 0.235300 − 1.33445i 0.0279249 − 0.158370i −0.967657 − 0.252271i \(-0.918823\pi\)
0.995582 + 0.0939008i \(0.0299337\pi\)
\(72\) 0 0
\(73\) −2.27972 + 0.829748i −0.266820 + 0.0971147i −0.471966 − 0.881617i \(-0.656455\pi\)
0.205145 + 0.978731i \(0.434233\pi\)
\(74\) 0 0
\(75\) 7.47565 0.863214
\(76\) 0 0
\(77\) −19.6655 −2.24109
\(78\) 0 0
\(79\) −2.69207 + 0.979832i −0.302881 + 0.110240i −0.488990 − 0.872289i \(-0.662635\pi\)
0.186109 + 0.982529i \(0.440412\pi\)
\(80\) 0 0
\(81\) 0.173648 − 0.984808i 0.0192942 − 0.109423i
\(82\) 0 0
\(83\) 0.960637 + 1.66387i 0.105444 + 0.182634i 0.913919 − 0.405896i \(-0.133040\pi\)
−0.808476 + 0.588530i \(0.799707\pi\)
\(84\) 0 0
\(85\) 4.47178 − 3.75227i 0.485033 − 0.406991i
\(86\) 0 0
\(87\) −0.0603074 + 0.104455i −0.00646563 + 0.0111988i
\(88\) 0 0
\(89\) 11.4226 + 4.15749i 1.21080 + 0.440693i 0.866980 − 0.498344i \(-0.166058\pi\)
0.343816 + 0.939037i \(0.388280\pi\)
\(90\) 0 0
\(91\) 0.146307 + 0.829748i 0.0153371 + 0.0869813i
\(92\) 0 0
\(93\) −2.39053 − 2.00589i −0.247886 − 0.208001i
\(94\) 0 0
\(95\) −6.89053 + 13.7680i −0.706953 + 1.41257i
\(96\) 0 0
\(97\) 13.4081 + 11.2507i 1.36138 + 1.14234i 0.975552 + 0.219767i \(0.0705296\pi\)
0.385831 + 0.922570i \(0.373915\pi\)
\(98\) 0 0
\(99\) 0.918748 + 5.21048i 0.0923377 + 0.523673i
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.d.529.1 6
4.3 odd 2 114.2.i.c.73.1 yes 6
12.11 even 2 342.2.u.b.73.1 6
19.6 even 9 inner 912.2.bo.d.481.1 6
76.43 odd 18 2166.2.a.r.1.1 3
76.63 odd 18 114.2.i.c.25.1 ✓ 6
76.71 even 18 2166.2.a.p.1.1 3
228.71 odd 18 6498.2.a.bu.1.3 3
228.119 even 18 6498.2.a.bp.1.3 3
228.215 even 18 342.2.u.b.253.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.c.25.1 ✓ 6 76.63 odd 18
114.2.i.c.73.1 yes 6 4.3 odd 2
342.2.u.b.73.1 6 12.11 even 2
342.2.u.b.253.1 6 228.215 even 18
912.2.bo.d.481.1 6 19.6 even 9 inner
912.2.bo.d.529.1 6 1.1 even 1 trivial
2166.2.a.p.1.1 3 76.71 even 18
2166.2.a.r.1.1 3 76.43 odd 18
6498.2.a.bp.1.3 3 228.119 even 18
6498.2.a.bu.1.3 3 228.71 odd 18