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 625.1
Root \(-0.766044 - 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 912.625
Dual form 912.2.bo.d.769.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.766044 + 0.642788i) q^{3} +(2.20574 - 0.802823i) q^{5} +(1.78699 + 3.09516i) q^{7} +(0.173648 + 0.984808i) q^{9} +(1.35844 - 2.35289i) q^{11} +(4.14543 - 3.47843i) q^{13} +(2.20574 + 0.802823i) q^{15} +(-0.673648 + 3.82045i) q^{17} +(-1.01114 - 4.24000i) q^{19} +(-0.620615 + 3.51968i) q^{21} +(-7.73783 - 2.81634i) q^{23} +(0.390530 - 0.327693i) q^{25} +(-0.500000 + 0.866025i) q^{27} +(0.613341 + 3.47843i) q^{29} +(3.26604 + 5.65695i) q^{31} +(2.55303 - 0.929228i) q^{33} +(6.42649 + 5.39246i) q^{35} +0.389185 q^{37} +5.41147 q^{39} +(-1.48886 - 1.24930i) q^{41} +(4.71941 - 1.71772i) q^{43} +(1.17365 + 2.03282i) q^{45} +(-0.518418 - 2.94010i) q^{47} +(-2.88666 + 4.99984i) q^{49} +(-2.97178 + 2.49362i) q^{51} +(-7.80453 - 2.84062i) q^{53} +(1.10741 - 6.28044i) q^{55} +(1.95084 - 3.89798i) q^{57} +(-0.474308 + 2.68993i) q^{59} +(-5.91147 - 2.15160i) q^{61} +(-2.73783 + 2.29731i) q^{63} +(6.35117 - 11.0005i) q^{65} +(2.59374 + 14.7098i) q^{67} +(-4.11721 - 7.13122i) q^{69} +(8.47818 - 3.08580i) q^{71} +(7.88326 + 6.61484i) q^{73} +0.509800 q^{75} +9.71007 q^{77} +(-9.96451 - 8.36121i) q^{79} +(-0.939693 + 0.342020i) q^{81} +(-4.08512 - 7.07564i) q^{83} +(1.58125 + 8.96773i) q^{85} +(-1.76604 + 3.05888i) q^{87} +(8.98158 - 7.53644i) q^{89} +(18.1741 + 6.61484i) q^{91} +(-1.13429 + 6.43285i) q^{93} +(-5.63429 - 8.54055i) q^{95} +(-1.49407 + 8.47329i) q^{97} +(2.55303 + 0.929228i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} + 3 q^{7} + 9 q^{13} + 3 q^{15} - 3 q^{17} - 15 q^{21} - 27 q^{23} - 15 q^{25} - 3 q^{27} - 3 q^{29} + 15 q^{31} + 3 q^{33} - 12 q^{35} - 6 q^{37} + 12 q^{39} - 15 q^{41} - 3 q^{43} + 6 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{5}{9}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 0.766044 + 0.642788i 0.442276 + 0.371114i
\(4\) 0 0
\(5\) 2.20574 − 0.802823i 0.986436 − 0.359033i 0.202097 − 0.979366i \(-0.435225\pi\)
0.784339 + 0.620332i \(0.213002\pi\)
\(6\) 0 0
\(7\) 1.78699 + 3.09516i 0.675418 + 1.16986i 0.976346 + 0.216212i \(0.0693703\pi\)
−0.300928 + 0.953647i \(0.597296\pi\)
\(8\) 0 0
\(9\) 0.173648 + 0.984808i 0.0578827 + 0.328269i
\(10\) 0 0
\(11\) 1.35844 − 2.35289i 0.409585 − 0.709423i −0.585258 − 0.810847i \(-0.699007\pi\)
0.994843 + 0.101425i \(0.0323401\pi\)
\(12\) 0 0
\(13\) 4.14543 − 3.47843i 1.14974 − 0.964742i 0.150022 − 0.988683i \(-0.452066\pi\)
0.999713 + 0.0239402i \(0.00762112\pi\)
\(14\) 0 0
\(15\) 2.20574 + 0.802823i 0.569519 + 0.207288i
\(16\) 0 0
\(17\) −0.673648 + 3.82045i −0.163384 + 0.926595i 0.787332 + 0.616530i \(0.211462\pi\)
−0.950715 + 0.310065i \(0.899649\pi\)
\(18\) 0 0
\(19\) −1.01114 − 4.24000i −0.231972 − 0.972722i
\(20\) 0 0
\(21\) −0.620615 + 3.51968i −0.135429 + 0.768057i
\(22\) 0 0
\(23\) −7.73783 − 2.81634i −1.61345 − 0.587247i −0.631330 − 0.775514i \(-0.717491\pi\)
−0.982118 + 0.188267i \(0.939713\pi\)
\(24\) 0 0
\(25\) 0.390530 − 0.327693i 0.0781059 − 0.0655386i
\(26\) 0 0
\(27\) −0.500000 + 0.866025i −0.0962250 + 0.166667i
\(28\) 0 0
\(29\) 0.613341 + 3.47843i 0.113895 + 0.645928i 0.987292 + 0.158919i \(0.0508009\pi\)
−0.873397 + 0.487009i \(0.838088\pi\)
\(30\) 0 0
\(31\) 3.26604 + 5.65695i 0.586599 + 1.01602i 0.994674 + 0.103071i \(0.0328668\pi\)
−0.408075 + 0.912948i \(0.633800\pi\)
\(32\) 0 0
\(33\) 2.55303 − 0.929228i 0.444426 − 0.161758i
\(34\) 0 0
\(35\) 6.42649 + 5.39246i 1.08627 + 0.911493i
\(36\) 0 0
\(37\) 0.389185 0.0639817 0.0319908 − 0.999488i \(-0.489815\pi\)
0.0319908 + 0.999488i \(0.489815\pi\)
\(38\) 0 0
\(39\) 5.41147 0.866529
\(40\) 0 0
\(41\) −1.48886 − 1.24930i −0.232520 − 0.195108i 0.519082 − 0.854725i \(-0.326274\pi\)
−0.751602 + 0.659617i \(0.770718\pi\)
\(42\) 0 0
\(43\) 4.71941 − 1.71772i 0.719703 − 0.261950i 0.0439033 − 0.999036i \(-0.486021\pi\)
0.675800 + 0.737085i \(0.263798\pi\)
\(44\) 0 0
\(45\) 1.17365 + 2.03282i 0.174957 + 0.303035i
\(46\) 0 0
\(47\) −0.518418 − 2.94010i −0.0756191 − 0.428857i −0.998989 − 0.0449466i \(-0.985688\pi\)
0.923370 − 0.383911i \(-0.125423\pi\)
\(48\) 0 0
\(49\) −2.88666 + 4.99984i −0.412380 + 0.714263i
\(50\) 0 0
\(51\) −2.97178 + 2.49362i −0.416133 + 0.349177i
\(52\) 0 0
\(53\) −7.80453 − 2.84062i −1.07203 − 0.390189i −0.255097 − 0.966915i \(-0.582108\pi\)
−0.816937 + 0.576727i \(0.804330\pi\)
\(54\) 0 0
\(55\) 1.10741 − 6.28044i 0.149323 − 0.846854i
\(56\) 0 0
\(57\) 1.95084 − 3.89798i 0.258395 − 0.516300i
\(58\) 0 0
\(59\) −0.474308 + 2.68993i −0.0617496 + 0.350199i 0.938241 + 0.345981i \(0.112454\pi\)
−0.999991 + 0.00421836i \(0.998657\pi\)
\(60\) 0 0
\(61\) −5.91147 − 2.15160i −0.756887 − 0.275484i −0.0653860 − 0.997860i \(-0.520828\pi\)
−0.691501 + 0.722376i \(0.743050\pi\)
\(62\) 0 0
\(63\) −2.73783 + 2.29731i −0.344934 + 0.289434i
\(64\) 0 0
\(65\) 6.35117 − 11.0005i 0.787765 − 1.36445i
\(66\) 0 0
\(67\) 2.59374 + 14.7098i 0.316876 + 1.79709i 0.561503 + 0.827475i \(0.310223\pi\)
−0.244627 + 0.969617i \(0.578665\pi\)
\(68\) 0 0
\(69\) −4.11721 − 7.13122i −0.495654 − 0.858498i
\(70\) 0 0
\(71\) 8.47818 − 3.08580i 1.00617 − 0.366218i 0.214212 − 0.976787i \(-0.431282\pi\)
0.791963 + 0.610570i \(0.209059\pi\)
\(72\) 0 0
\(73\) 7.88326 + 6.61484i 0.922665 + 0.774208i 0.974486 − 0.224448i \(-0.0720580\pi\)
−0.0518207 + 0.998656i \(0.516502\pi\)
\(74\) 0 0
\(75\) 0.509800 0.0588667
\(76\) 0 0
\(77\) 9.71007 1.10657
\(78\) 0 0
\(79\) −9.96451 − 8.36121i −1.12109 − 0.940710i −0.122435 − 0.992476i \(-0.539070\pi\)
−0.998659 + 0.0517663i \(0.983515\pi\)
\(80\) 0 0
\(81\) −0.939693 + 0.342020i −0.104410 + 0.0380022i
\(82\) 0 0
\(83\) −4.08512 − 7.07564i −0.448400 − 0.776652i 0.549882 − 0.835243i \(-0.314673\pi\)
−0.998282 + 0.0585902i \(0.981339\pi\)
\(84\) 0 0
\(85\) 1.58125 + 8.96773i 0.171511 + 0.972686i
\(86\) 0 0
\(87\) −1.76604 + 3.05888i −0.189340 + 0.327946i
\(88\) 0 0
\(89\) 8.98158 − 7.53644i 0.952046 − 0.798861i −0.0275951 − 0.999619i \(-0.508785\pi\)
0.979641 + 0.200758i \(0.0643405\pi\)
\(90\) 0 0
\(91\) 18.1741 + 6.61484i 1.90516 + 0.693423i
\(92\) 0 0
\(93\) −1.13429 + 6.43285i −0.117620 + 0.667056i
\(94\) 0 0
\(95\) −5.63429 − 8.54055i −0.578065 − 0.876242i
\(96\) 0 0
\(97\) −1.49407 + 8.47329i −0.151700 + 0.860333i 0.810041 + 0.586373i \(0.199445\pi\)
−0.961741 + 0.273960i \(0.911666\pi\)
\(98\) 0 0
\(99\) 2.55303 + 0.929228i 0.256590 + 0.0933909i
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.625.1 6
4.3 odd 2 114.2.i.c.55.1 ✓ 6
12.11 even 2 342.2.u.b.55.1 6
19.9 even 9 inner 912.2.bo.d.769.1 6
76.3 even 18 2166.2.a.p.1.2 3
76.35 odd 18 2166.2.a.r.1.2 3
76.47 odd 18 114.2.i.c.85.1 yes 6
228.35 even 18 6498.2.a.bp.1.2 3
228.47 even 18 342.2.u.b.199.1 6
228.155 odd 18 6498.2.a.bu.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.c.55.1 ✓ 6 4.3 odd 2
114.2.i.c.85.1 yes 6 76.47 odd 18
342.2.u.b.55.1 6 12.11 even 2
342.2.u.b.199.1 6 228.47 even 18
912.2.bo.d.625.1 6 1.1 even 1 trivial
912.2.bo.d.769.1 6 19.9 even 9 inner
2166.2.a.p.1.2 3 76.3 even 18
2166.2.a.r.1.2 3 76.35 odd 18
6498.2.a.bp.1.2 3 228.35 even 18
6498.2.a.bu.1.2 3 228.155 odd 18