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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [912,2,Mod(257,912)] 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("912.257"); 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, 9, 17])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.cc (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,-3,0,0,0,0,0,-3,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 641.3
Root \(-0.363139 + 1.69356i\) of defining polynomial
Character \(\chi\) \(=\) 912.641
Dual form 912.2.cc.c.737.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.36678 + 1.06392i) q^{3} +(-2.20556 + 2.62849i) q^{5} +(-1.68651 - 2.92113i) q^{7} +(0.736160 + 2.90828i) q^{9} +(-2.33635 - 1.34889i) q^{11} +(-5.05419 + 0.891189i) q^{13} +(-5.81100 + 1.24602i) q^{15} +(-1.44531 - 3.97095i) q^{17} +(2.73048 - 3.39772i) q^{19} +(0.802750 - 5.78684i) q^{21} +(-1.69398 - 2.01881i) q^{23} +(-1.17620 - 6.67054i) q^{25} +(-2.08800 + 4.75818i) q^{27} +(3.54249 + 1.28936i) q^{29} +(-4.78254 + 2.76120i) q^{31} +(-1.75816 - 4.32933i) q^{33} +(11.3979 + 2.00975i) q^{35} +5.17636i q^{37} +(-7.85610 - 4.15918i) q^{39} +(0.289735 - 1.64317i) q^{41} +(-1.85806 - 1.55910i) q^{43} +(-9.26801 - 4.47940i) q^{45} +(-0.0440069 + 0.120908i) q^{47} +(-2.18866 + 3.79087i) q^{49} +(2.24935 - 6.96509i) q^{51} +(-6.53342 + 5.48219i) q^{53} +(8.69853 - 3.16600i) q^{55} +(7.34685 - 1.73892i) q^{57} +(-3.87665 + 1.41099i) q^{59} +(3.53369 - 2.96512i) q^{61} +(7.25390 - 7.05526i) q^{63} +(8.80484 - 15.2504i) q^{65} +(-3.81629 + 10.4852i) q^{67} +(-0.167450 - 4.56152i) q^{69} +(-9.91131 - 8.31658i) q^{71} +(-0.414656 + 2.35163i) q^{73} +(5.48930 - 10.3685i) q^{75} +9.09972i q^{77} +(-2.22246 - 0.391880i) q^{79} +(-7.91614 + 4.28191i) q^{81} +(-6.27861 + 3.62496i) q^{83} +(13.6253 + 4.95920i) q^{85} +(3.47002 + 5.53118i) q^{87} +(0.209662 + 1.18905i) q^{89} +(11.1272 + 13.2609i) q^{91} +(-9.47436 - 1.31428i) q^{93} +(2.90863 + 14.6709i) q^{95} +(3.13271 + 8.60706i) q^{97} +(2.20303 - 7.78777i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{3} - 3 q^{9} - 12 q^{13} - 18 q^{15} + 6 q^{17} + 6 q^{19} - 18 q^{25} + 6 q^{27} - 6 q^{29} - 24 q^{33} + 24 q^{35} - 6 q^{39} + 3 q^{41} + 6 q^{43} - 54 q^{45} - 30 q^{47} + 21 q^{49} - 42 q^{51}+ \cdots + 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.36678 + 1.06392i 0.789109 + 0.614253i
\(4\) 0 0
\(5\) −2.20556 + 2.62849i −0.986357 + 1.17549i −0.00187711 + 0.999998i \(0.500598\pi\)
−0.984480 + 0.175496i \(0.943847\pi\)
\(6\) 0 0
\(7\) −1.68651 − 2.92113i −0.637442 − 1.10408i −0.985992 − 0.166792i \(-0.946659\pi\)
0.348550 − 0.937290i \(-0.386674\pi\)
\(8\) 0 0
\(9\) 0.736160 + 2.90828i 0.245387 + 0.969425i
\(10\) 0 0
\(11\) −2.33635 − 1.34889i −0.704437 − 0.406707i 0.104561 − 0.994519i \(-0.466656\pi\)
−0.808998 + 0.587811i \(0.799990\pi\)
\(12\) 0 0
\(13\) −5.05419 + 0.891189i −1.40178 + 0.247171i −0.822874 − 0.568224i \(-0.807631\pi\)
−0.578905 + 0.815395i \(0.696520\pi\)
\(14\) 0 0
\(15\) −5.81100 + 1.24602i −1.50039 + 0.321721i
\(16\) 0 0
\(17\) −1.44531 − 3.97095i −0.350538 − 0.963096i −0.982198 − 0.187850i \(-0.939848\pi\)
0.631659 − 0.775246i \(-0.282374\pi\)
\(18\) 0 0
\(19\) 2.73048 − 3.39772i 0.626414 − 0.779490i
\(20\) 0 0
\(21\) 0.802750 − 5.78684i 0.175174 − 1.26279i
\(22\) 0 0
\(23\) −1.69398 − 2.01881i −0.353220 − 0.420951i 0.559952 − 0.828525i \(-0.310819\pi\)
−0.913172 + 0.407574i \(0.866375\pi\)
\(24\) 0 0
\(25\) −1.17620 − 6.67054i −0.235239 − 1.33411i
\(26\) 0 0
\(27\) −2.08800 + 4.75818i −0.401836 + 0.915712i
\(28\) 0 0
\(29\) 3.54249 + 1.28936i 0.657823 + 0.239428i 0.649296 − 0.760536i \(-0.275064\pi\)
0.00852691 + 0.999964i \(0.497286\pi\)
\(30\) 0 0
\(31\) −4.78254 + 2.76120i −0.858970 + 0.495927i −0.863667 − 0.504062i \(-0.831838\pi\)
0.00469717 + 0.999989i \(0.498505\pi\)
\(32\) 0 0
\(33\) −1.75816 − 4.32933i −0.306057 − 0.753639i
\(34\) 0 0
\(35\) 11.3979 + 2.00975i 1.92659 + 0.339710i
\(36\) 0 0
\(37\) 5.17636i 0.850989i 0.904961 + 0.425494i \(0.139900\pi\)
−0.904961 + 0.425494i \(0.860100\pi\)
\(38\) 0 0
\(39\) −7.85610 − 4.15918i −1.25798 − 0.666002i
\(40\) 0 0
\(41\) 0.289735 − 1.64317i 0.0452490 − 0.256620i −0.953789 − 0.300478i \(-0.902854\pi\)
0.999038 + 0.0438581i \(0.0139649\pi\)
\(42\) 0 0
\(43\) −1.85806 − 1.55910i −0.283351 − 0.237760i 0.490023 − 0.871709i \(-0.336988\pi\)
−0.773374 + 0.633950i \(0.781433\pi\)
\(44\) 0 0
\(45\) −9.26801 − 4.47940i −1.38159 − 0.667749i
\(46\) 0 0
\(47\) −0.0440069 + 0.120908i −0.00641906 + 0.0176362i −0.942861 − 0.333187i \(-0.891876\pi\)
0.936442 + 0.350823i \(0.114098\pi\)
\(48\) 0 0
\(49\) −2.18866 + 3.79087i −0.312665 + 0.541552i
\(50\) 0 0
\(51\) 2.24935 − 6.96509i 0.314972 − 0.975307i
\(52\) 0 0
\(53\) −6.53342 + 5.48219i −0.897434 + 0.753036i −0.969687 − 0.244350i \(-0.921425\pi\)
0.0722533 + 0.997386i \(0.476981\pi\)
\(54\) 0 0
\(55\) 8.69853 − 3.16600i 1.17291 − 0.426904i
\(56\) 0 0
\(57\) 7.34685 − 1.73892i 0.973113 − 0.230326i
\(58\) 0 0
\(59\) −3.87665 + 1.41099i −0.504697 + 0.183695i −0.581805 − 0.813328i \(-0.697653\pi\)
0.0771085 + 0.997023i \(0.475431\pi\)
\(60\) 0 0
\(61\) 3.53369 − 2.96512i 0.452443 − 0.379645i −0.387898 − 0.921702i \(-0.626799\pi\)
0.840342 + 0.542057i \(0.182354\pi\)
\(62\) 0 0
\(63\) 7.25390 − 7.05526i 0.913906 − 0.888880i
\(64\) 0 0
\(65\) 8.80484 − 15.2504i 1.09211 − 1.89158i
\(66\) 0 0
\(67\) −3.81629 + 10.4852i −0.466234 + 1.28097i 0.454490 + 0.890752i \(0.349821\pi\)
−0.920724 + 0.390215i \(0.872401\pi\)
\(68\) 0 0
\(69\) −0.167450 − 4.56152i −0.0201586 − 0.549143i
\(70\) 0 0
\(71\) −9.91131 − 8.31658i −1.17626 − 0.986996i −0.999996 − 0.00265261i \(-0.999156\pi\)
−0.176260 − 0.984344i \(-0.556400\pi\)
\(72\) 0 0
\(73\) −0.414656 + 2.35163i −0.0485318 + 0.275238i −0.999411 − 0.0343255i \(-0.989072\pi\)
0.950879 + 0.309563i \(0.100183\pi\)
\(74\) 0 0
\(75\) 5.48930 − 10.3685i 0.633850 − 1.19725i
\(76\) 0 0
\(77\) 9.09972i 1.03701i
\(78\) 0 0
\(79\) −2.22246 − 0.391880i −0.250046 − 0.0440899i 0.0472200 − 0.998885i \(-0.484964\pi\)
−0.297266 + 0.954795i \(0.596075\pi\)
\(80\) 0 0
\(81\) −7.91614 + 4.28191i −0.879571 + 0.475768i
\(82\) 0 0
\(83\) −6.27861 + 3.62496i −0.689167 + 0.397891i −0.803300 − 0.595575i \(-0.796924\pi\)
0.114133 + 0.993465i \(0.463591\pi\)
\(84\) 0 0
\(85\) 13.6253 + 4.95920i 1.47787 + 0.537901i
\(86\) 0 0
\(87\) 3.47002 + 5.53118i 0.372025 + 0.593005i
\(88\) 0 0
\(89\) 0.209662 + 1.18905i 0.0222241 + 0.126039i 0.993901 − 0.110273i \(-0.0351724\pi\)
−0.971677 + 0.236312i \(0.924061\pi\)
\(90\) 0 0
\(91\) 11.1272 + 13.2609i 1.16645 + 1.39012i
\(92\) 0 0
\(93\) −9.47436 − 1.31428i −0.982446 − 0.136285i
\(94\) 0 0
\(95\) 2.90863 + 14.6709i 0.298419 + 1.50520i
\(96\) 0 0
\(97\) 3.13271 + 8.60706i 0.318079 + 0.873914i 0.990959 + 0.134163i \(0.0428346\pi\)
−0.672880 + 0.739751i \(0.734943\pi\)
\(98\) 0 0
\(99\) 2.20303 − 7.78777i 0.221413 − 0.782700i
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.cc.c.641.3 18
3.2 odd 2 912.2.cc.d.641.3 18
4.3 odd 2 114.2.l.b.71.1 yes 18
12.11 even 2 114.2.l.a.71.1 yes 18
19.15 odd 18 912.2.cc.d.737.3 18
57.53 even 18 inner 912.2.cc.c.737.3 18
76.15 even 18 114.2.l.a.53.1 ✓ 18
228.167 odd 18 114.2.l.b.53.1 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.l.a.53.1 ✓ 18 76.15 even 18
114.2.l.a.71.1 yes 18 12.11 even 2
114.2.l.b.53.1 yes 18 228.167 odd 18
114.2.l.b.71.1 yes 18 4.3 odd 2
912.2.cc.c.641.3 18 1.1 even 1 trivial
912.2.cc.c.737.3 18 57.53 even 18 inner
912.2.cc.d.641.3 18 3.2 odd 2
912.2.cc.d.737.3 18 19.15 odd 18