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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 = [18,0,0,0,0,0,0,0,0,0,0,0,-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.d.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)
 
\(f(q)\) \(=\) \(q+(1.64823 - 0.532290i) q^{3} +(2.20556 - 2.62849i) q^{5} +(-1.68651 - 2.92113i) q^{7} +(2.43333 - 1.75467i) q^{9} +(2.33635 + 1.34889i) q^{11} +(-5.05419 + 0.891189i) q^{13} +(2.23616 - 5.50635i) q^{15} +(1.44531 + 3.97095i) q^{17} +(2.73048 - 3.39772i) q^{19} +(-4.33465 - 3.91698i) q^{21} +(1.69398 + 2.01881i) q^{23} +(-1.17620 - 6.67054i) q^{25} +(3.07670 - 4.18735i) q^{27} +(-3.54249 - 1.28936i) q^{29} +(-4.78254 + 2.76120i) q^{31} +(4.56886 + 0.979673i) q^{33} +(-11.3979 - 2.00975i) q^{35} +5.17636i q^{37} +(-7.85610 + 4.15918i) q^{39} +(-0.289735 + 1.64317i) q^{41} +(-1.85806 - 1.55910i) q^{43} +(0.754733 - 10.2660i) q^{45} +(0.0440069 - 0.120908i) q^{47} +(-2.18866 + 3.79087i) q^{49} +(4.49590 + 5.77572i) q^{51} +(6.53342 - 5.48219i) q^{53} +(8.69853 - 3.16600i) q^{55} +(2.69188 - 7.05363i) q^{57} +(3.87665 - 1.41099i) q^{59} +(3.53369 - 2.96512i) q^{61} +(-9.22948 - 4.14880i) q^{63} +(-8.80484 + 15.2504i) q^{65} +(-3.81629 + 10.4852i) q^{67} +(3.86667 + 2.42578i) q^{69} +(9.91131 + 8.31658i) q^{71} +(-0.414656 + 2.35163i) q^{73} +(-5.48930 - 10.3685i) q^{75} -9.09972i q^{77} +(-2.22246 - 0.391880i) q^{79} +(2.84224 - 8.53942i) q^{81} +(6.27861 - 3.62496i) q^{83} +(13.6253 + 4.95920i) q^{85} +(-6.52515 - 0.239533i) q^{87} +(-0.209662 - 1.18905i) q^{89} +(11.1272 + 13.2609i) q^{91} +(-6.41298 + 7.09680i) q^{93} +(-2.90863 - 14.6709i) q^{95} +(3.13271 + 8.60706i) q^{97} +(8.05200 - 0.817228i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 12 q^{13} + 24 q^{15} - 6 q^{17} + 6 q^{19} - 18 q^{25} + 3 q^{27} + 6 q^{29} - 27 q^{33} - 24 q^{35} - 6 q^{39} - 3 q^{41} + 6 q^{43} + 54 q^{45} + 30 q^{47} + 21 q^{49} + 33 q^{51} + 60 q^{53}+ \cdots + 30 q^{99}+O(q^{100}) \) 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
\(3\) 1.64823 − 0.532290i 0.951607 − 0.307318i
\(4\) 0 0
\(5\) 2.20556 − 2.62849i 0.986357 − 1.17549i 0.00187711 − 0.999998i \(-0.499402\pi\)
0.984480 − 0.175496i \(-0.0561531\pi\)
\(6\) 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
\(9\) 2.43333 − 1.75467i 0.811112 − 0.584891i
\(10\) 0 0
\(11\) 2.33635 + 1.34889i 0.704437 + 0.406707i 0.808998 − 0.587811i \(-0.200010\pi\)
−0.104561 + 0.994519i \(0.533344\pi\)
\(12\) 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
\(15\) 2.23616 − 5.50635i 0.577374 − 1.42173i
\(16\) 0 0
\(17\) 1.44531 + 3.97095i 0.350538 + 0.963096i 0.982198 + 0.187850i \(0.0601520\pi\)
−0.631659 + 0.775246i \(0.717626\pi\)
\(18\) 0 0
\(19\) 2.73048 − 3.39772i 0.626414 − 0.779490i
\(20\) 0 0
\(21\) −4.33465 − 3.91698i −0.945899 − 0.854755i
\(22\) 0 0
\(23\) 1.69398 + 2.01881i 0.353220 + 0.420951i 0.913172 − 0.407574i \(-0.133625\pi\)
−0.559952 + 0.828525i \(0.689181\pi\)
\(24\) 0 0
\(25\) −1.17620 − 6.67054i −0.235239 − 1.33411i
\(26\) 0 0
\(27\) 3.07670 − 4.18735i 0.592112 − 0.805856i
\(28\) 0 0
\(29\) −3.54249 − 1.28936i −0.657823 − 0.239428i −0.00852691 − 0.999964i \(-0.502714\pi\)
−0.649296 + 0.760536i \(0.724936\pi\)
\(30\) 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
\(33\) 4.56886 + 0.979673i 0.795336 + 0.170539i
\(34\) 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 −0.999038 − 0.0438581i \(-0.986035\pi\)
0.953789 + 0.300478i \(0.0971462\pi\)
\(42\) 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
\(45\) 0.754733 − 10.2660i 0.112509 − 1.53037i
\(46\) 0 0
\(47\) 0.0440069 − 0.120908i 0.00641906 − 0.0176362i −0.936442 − 0.350823i \(-0.885902\pi\)
0.942861 + 0.333187i \(0.108124\pi\)
\(48\) 0 0
\(49\) −2.18866 + 3.79087i −0.312665 + 0.541552i
\(50\) 0 0
\(51\) 4.49590 + 5.77572i 0.629551 + 0.808763i
\(52\) 0 0
\(53\) 6.53342 − 5.48219i 0.897434 − 0.753036i −0.0722533 − 0.997386i \(-0.523019\pi\)
0.969687 + 0.244350i \(0.0785746\pi\)
\(54\) 0 0
\(55\) 8.69853 − 3.16600i 1.17291 − 0.426904i
\(56\) 0 0
\(57\) 2.69188 − 7.05363i 0.356549 − 0.934277i
\(58\) 0 0
\(59\) 3.87665 − 1.41099i 0.504697 − 0.183695i −0.0771085 − 0.997023i \(-0.524569\pi\)
0.581805 + 0.813328i \(0.302347\pi\)
\(60\) 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
\(63\) −9.22948 − 4.14880i −1.16281 − 0.522700i
\(64\) 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
\(69\) 3.86667 + 2.42578i 0.465492 + 0.292029i
\(70\) 0 0
\(71\) 9.91131 + 8.31658i 1.17626 + 0.986996i 0.999996 + 0.00265261i \(0.000844352\pi\)
0.176260 + 0.984344i \(0.443600\pi\)
\(72\) 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
\(81\) 2.84224 − 8.53942i 0.315804 − 0.948824i
\(82\) 0 0
\(83\) 6.27861 − 3.62496i 0.689167 − 0.397891i −0.114133 − 0.993465i \(-0.536409\pi\)
0.803300 + 0.595575i \(0.203076\pi\)
\(84\) 0 0
\(85\) 13.6253 + 4.95920i 1.47787 + 0.537901i
\(86\) 0 0
\(87\) −6.52515 − 0.239533i −0.699570 − 0.0256807i
\(88\) 0 0
\(89\) −0.209662 − 1.18905i −0.0222241 − 0.126039i 0.971677 − 0.236312i \(-0.0759387\pi\)
−0.993901 + 0.110273i \(0.964828\pi\)
\(90\) 0 0
\(91\) 11.1272 + 13.2609i 1.16645 + 1.39012i
\(92\) 0 0
\(93\) −6.41298 + 7.09680i −0.664995 + 0.735904i
\(94\) 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
\(99\) 8.05200 − 0.817228i 0.809257 − 0.0821345i
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.d.641.3 18
3.2 odd 2 912.2.cc.c.641.3 18
4.3 odd 2 114.2.l.a.71.1 yes 18
12.11 even 2 114.2.l.b.71.1 yes 18
19.15 odd 18 912.2.cc.c.737.3 18
57.53 even 18 inner 912.2.cc.d.737.3 18
76.15 even 18 114.2.l.b.53.1 yes 18
228.167 odd 18 114.2.l.a.53.1 ✓ 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.l.a.53.1 ✓ 18 228.167 odd 18
114.2.l.a.71.1 yes 18 4.3 odd 2
114.2.l.b.53.1 yes 18 76.15 even 18
114.2.l.b.71.1 yes 18 12.11 even 2
912.2.cc.c.641.3 18 3.2 odd 2
912.2.cc.c.737.3 18 19.15 odd 18
912.2.cc.d.641.3 18 1.1 even 1 trivial
912.2.cc.d.737.3 18 57.53 even 18 inner