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Properties

Label 912.2.cc.c.737.3
Level $912$
Weight $2$
Character 912.737
Analytic conductor $7.282$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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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(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 737.3
Root \(-0.363139 - 1.69356i\) of defining polynomial
Character \(\chi\) \(=\) 912.737
Dual form 912.2.cc.c.641.3

$q$-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 \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{11}{18}\right)\) \(1\) \(-1\) \(1\)

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 −0.984480 − 0.175496i \(-0.943847\pi\)
−0.00187711 − 0.999998i \(-0.500598\pi\)
\(6\) 0 0
\(7\) −1.68651 + 2.92113i −0.637442 + 1.10408i 0.348550 + 0.937290i \(0.386674\pi\)
−0.985992 + 0.166792i \(0.946659\pi\)
\(8\) 0 0
\(9\) 0.736160 − 2.90828i 0.245387 − 0.969425i
\(10\) 0 0
\(11\) −2.33635 + 1.34889i −0.704437 + 0.406707i −0.808998 − 0.587811i \(-0.799990\pi\)
0.104561 + 0.994519i \(0.466656\pi\)
\(12\) 0 0
\(13\) −5.05419 − 0.891189i −1.40178 − 0.247171i −0.578905 − 0.815395i \(-0.696520\pi\)
−0.822874 + 0.568224i \(0.807631\pi\)
\(14\) 0 0
\(15\) −5.81100 − 1.24602i −1.50039 − 0.321721i
\(16\) 0 0
\(17\) −1.44531 + 3.97095i −0.350538 + 0.963096i 0.631659 + 0.775246i \(0.282374\pi\)
−0.982198 + 0.187850i \(0.939848\pi\)
\(18\) 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 −0.913172 − 0.407574i \(-0.866375\pi\)
0.559952 + 0.828525i \(0.310819\pi\)
\(24\) 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 0.00852691 − 0.999964i \(-0.497286\pi\)
0.649296 + 0.760536i \(0.275064\pi\)
\(30\) 0 0
\(31\) −4.78254 − 2.76120i −0.858970 − 0.495927i 0.00469717 − 0.999989i \(-0.498505\pi\)
−0.863667 + 0.504062i \(0.831838\pi\)
\(32\) 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 \(-0.860100\pi\)
0.904961 − 0.425494i \(-0.139900\pi\)
\(38\) 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 \(-0.0139649\pi\)
−0.953789 + 0.300478i \(0.902854\pi\)
\(42\) 0 0
\(43\) −1.85806 + 1.55910i −0.283351 + 0.237760i −0.773374 − 0.633950i \(-0.781433\pi\)
0.490023 + 0.871709i \(0.336988\pi\)
\(44\) 0 0
\(45\) −9.26801 + 4.47940i −1.38159 + 0.667749i
\(46\) 0 0
\(47\) −0.0440069 − 0.120908i −0.00641906 − 0.0176362i 0.936442 − 0.350823i \(-0.114098\pi\)
−0.942861 + 0.333187i \(0.891876\pi\)
\(48\) 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 0.0722533 − 0.997386i \(-0.476981\pi\)
−0.969687 + 0.244350i \(0.921425\pi\)
\(54\) 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 0.0771085 − 0.997023i \(-0.475431\pi\)
−0.581805 + 0.813328i \(0.697653\pi\)
\(60\) 0 0
\(61\) 3.53369 + 2.96512i 0.452443 + 0.379645i 0.840342 − 0.542057i \(-0.182354\pi\)
−0.387898 + 0.921702i \(0.626799\pi\)
\(62\) 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 −0.920724 − 0.390215i \(-0.872401\pi\)
0.454490 − 0.890752i \(-0.349821\pi\)
\(68\) 0 0
\(69\) −0.167450 + 4.56152i −0.0201586 + 0.549143i
\(70\) 0 0
\(71\) −9.91131 + 8.31658i −1.17626 + 0.986996i −0.176260 + 0.984344i \(0.556400\pi\)
−0.999996 + 0.00265261i \(0.999156\pi\)
\(72\) 0 0
\(73\) −0.414656 − 2.35163i −0.0485318 − 0.275238i 0.950879 − 0.309563i \(-0.100183\pi\)
−0.999411 + 0.0343255i \(0.989072\pi\)
\(74\) 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 −0.297266 − 0.954795i \(-0.596075\pi\)
0.0472200 + 0.998885i \(0.484964\pi\)
\(80\) 0 0
\(81\) −7.91614 − 4.28191i −0.879571 − 0.475768i
\(82\) 0 0
\(83\) −6.27861 − 3.62496i −0.689167 − 0.397891i 0.114133 − 0.993465i \(-0.463591\pi\)
−0.803300 + 0.595575i \(0.796924\pi\)
\(84\) 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 −0.971677 − 0.236312i \(-0.924061\pi\)
0.993901 + 0.110273i \(0.0351724\pi\)
\(90\) 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 −0.672880 − 0.739751i \(-0.734943\pi\)
0.990959 − 0.134163i \(-0.0428346\pi\)
\(98\) 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.737.3 18
3.2 odd 2 912.2.cc.d.737.3 18
4.3 odd 2 114.2.l.b.53.1 yes 18
12.11 even 2 114.2.l.a.53.1 ✓ 18
19.14 odd 18 912.2.cc.d.641.3 18
57.14 even 18 inner 912.2.cc.c.641.3 18
76.71 even 18 114.2.l.a.71.1 yes 18
228.71 odd 18 114.2.l.b.71.1 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.l.a.53.1 ✓ 18 12.11 even 2
114.2.l.a.71.1 yes 18 76.71 even 18
114.2.l.b.53.1 yes 18 4.3 odd 2
114.2.l.b.71.1 yes 18 228.71 odd 18
912.2.cc.c.641.3 18 57.14 even 18 inner
912.2.cc.c.737.3 18 1.1 even 1 trivial
912.2.cc.d.641.3 18 19.14 odd 18
912.2.cc.d.737.3 18 3.2 odd 2