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Properties

Label 912.2.cc.c.545.3
Level $912$
Weight $2$
Character 912.545
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 545.3
Root \(1.40849 + 1.00804i\) of defining polynomial
Character \(\chi\) \(=\) 912.545
Dual form 912.2.cc.c.497.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.748148 + 1.56214i) q^{3} +(-0.262261 + 0.0462437i) q^{5} +(-0.604656 + 1.04730i) q^{7} +(-1.88055 + 2.33742i) q^{9} +(-2.03630 + 1.17566i) q^{11} +(1.01749 + 2.79553i) q^{13} +(-0.268449 - 0.375091i) q^{15} +(0.576470 + 0.687011i) q^{17} +(-1.97979 - 3.88335i) q^{19} +(-2.08839 - 0.161024i) q^{21} +(-5.53770 - 0.976446i) q^{23} +(-4.63182 + 1.68584i) q^{25} +(-5.05830 - 1.18894i) q^{27} +(1.92487 + 1.61516i) q^{29} +(8.98131 + 5.18536i) q^{31} +(-3.36000 - 2.30141i) q^{33} +(0.110147 - 0.302626i) q^{35} +3.95916i q^{37} +(-3.60577 + 3.68093i) q^{39} +(-10.4227 - 3.79356i) q^{41} +(0.834031 + 4.73003i) q^{43} +(0.385104 - 0.699978i) q^{45} +(-1.24341 + 1.48183i) q^{47} +(2.76878 + 4.79567i) q^{49} +(-0.641920 + 1.41451i) q^{51} +(0.998339 - 5.66186i) q^{53} +(0.479676 - 0.402496i) q^{55} +(4.58516 - 5.99803i) q^{57} +(9.78136 - 8.20754i) q^{59} +(-0.153642 + 0.871345i) q^{61} +(-1.31088 - 3.38283i) q^{63} +(-0.396123 - 0.686106i) q^{65} +(-3.28864 + 3.91925i) q^{67} +(-2.61768 - 9.38117i) q^{69} +(1.64669 + 9.33885i) q^{71} +(0.320853 + 0.116781i) q^{73} +(-6.09881 - 5.97428i) q^{75} -2.84348i q^{77} +(-2.33914 + 6.42674i) q^{79} +(-1.92708 - 8.79127i) q^{81} +(-12.2240 - 7.05752i) q^{83} +(-0.182956 - 0.153518i) q^{85} +(-1.08301 + 4.21529i) q^{87} +(-11.5580 + 4.20677i) q^{89} +(-3.54297 - 0.624722i) q^{91} +(-1.38090 + 17.9095i) q^{93} +(0.698802 + 0.926900i) q^{95} +(3.88456 + 4.62944i) q^{97} +(1.08135 - 6.97057i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{3} - 3 q^{9} - 12 q^{13} - 18 q^{15} + 6 q^{17} + 6 q^{19} - 18 q^{25} + 6 q^{27} - 6 q^{29} - 24 q^{33} + 24 q^{35} - 6 q^{39} + 3 q^{41} + 6 q^{43} - 54 q^{45} - 30 q^{47} + 21 q^{49} - 42 q^{51}+ \cdots + 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(1\) \(-1\) \(1\)

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
\(3\) 0.748148 + 1.56214i 0.431944 + 0.901901i
\(4\) 0 0
\(5\) −0.262261 + 0.0462437i −0.117287 + 0.0206808i −0.231983 − 0.972720i \(-0.574522\pi\)
0.114697 + 0.993401i \(0.463410\pi\)
\(6\) 0 0
\(7\) −0.604656 + 1.04730i −0.228539 + 0.395840i −0.957375 − 0.288847i \(-0.906728\pi\)
0.728837 + 0.684688i \(0.240061\pi\)
\(8\) 0 0
\(9\) −1.88055 + 2.33742i −0.626849 + 0.779140i
\(10\) 0 0
\(11\) −2.03630 + 1.17566i −0.613968 + 0.354474i −0.774517 − 0.632553i \(-0.782007\pi\)
0.160549 + 0.987028i \(0.448674\pi\)
\(12\) 0 0
\(13\) 1.01749 + 2.79553i 0.282201 + 0.775340i 0.997099 + 0.0761119i \(0.0242506\pi\)
−0.714899 + 0.699228i \(0.753527\pi\)
\(14\) 0 0
\(15\) −0.268449 − 0.375091i −0.0693133 − 0.0968480i
\(16\) 0 0
\(17\) 0.576470 + 0.687011i 0.139815 + 0.166625i 0.831408 − 0.555663i \(-0.187535\pi\)
−0.691593 + 0.722287i \(0.743091\pi\)
\(18\) 0 0
\(19\) −1.97979 − 3.88335i −0.454194 − 0.890903i
\(20\) 0 0
\(21\) −2.08839 − 0.161024i −0.455724 − 0.0351383i
\(22\) 0 0
\(23\) −5.53770 − 0.976446i −1.15469 − 0.203603i −0.436668 − 0.899623i \(-0.643841\pi\)
−0.718022 + 0.696020i \(0.754952\pi\)
\(24\) 0 0
\(25\) −4.63182 + 1.68584i −0.926364 + 0.337169i
\(26\) 0 0
\(27\) −5.05830 − 1.18894i −0.973471 − 0.228811i
\(28\) 0 0
\(29\) 1.92487 + 1.61516i 0.357440 + 0.299928i 0.803769 − 0.594941i \(-0.202825\pi\)
−0.446329 + 0.894869i \(0.647269\pi\)
\(30\) 0 0
\(31\) 8.98131 + 5.18536i 1.61309 + 0.931319i 0.988648 + 0.150249i \(0.0480075\pi\)
0.624443 + 0.781070i \(0.285326\pi\)
\(32\) 0 0
\(33\) −3.36000 − 2.30141i −0.584900 − 0.400625i
\(34\) 0 0
\(35\) 0.110147 − 0.302626i 0.0186182 − 0.0511532i
\(36\) 0 0
\(37\) 3.95916i 0.650882i 0.945562 + 0.325441i \(0.105513\pi\)
−0.945562 + 0.325441i \(0.894487\pi\)
\(38\) 0 0
\(39\) −3.60577 + 3.68093i −0.577385 + 0.589420i
\(40\) 0 0
\(41\) −10.4227 − 3.79356i −1.62776 − 0.592455i −0.642920 − 0.765934i \(-0.722277\pi\)
−0.984838 + 0.173479i \(0.944499\pi\)
\(42\) 0 0
\(43\) 0.834031 + 4.73003i 0.127189 + 0.721322i 0.979984 + 0.199077i \(0.0637944\pi\)
−0.852795 + 0.522245i \(0.825095\pi\)
\(44\) 0 0
\(45\) 0.385104 − 0.699978i 0.0574079 − 0.104347i
\(46\) 0 0
\(47\) −1.24341 + 1.48183i −0.181369 + 0.216147i −0.849067 − 0.528285i \(-0.822835\pi\)
0.667698 + 0.744432i \(0.267280\pi\)
\(48\) 0 0
\(49\) 2.76878 + 4.79567i 0.395540 + 0.685096i
\(50\) 0 0
\(51\) −0.641920 + 1.41451i −0.0898868 + 0.198071i
\(52\) 0 0
\(53\) 0.998339 − 5.66186i 0.137132 − 0.777717i −0.836219 − 0.548396i \(-0.815239\pi\)
0.973351 − 0.229320i \(-0.0736504\pi\)
\(54\) 0 0
\(55\) 0.479676 − 0.402496i 0.0646794 − 0.0542725i
\(56\) 0 0
\(57\) 4.58516 − 5.99803i 0.607319 − 0.794458i
\(58\) 0 0
\(59\) 9.78136 − 8.20754i 1.27342 − 1.06853i 0.279310 − 0.960201i \(-0.409894\pi\)
0.994115 − 0.108329i \(-0.0345501\pi\)
\(60\) 0 0
\(61\) −0.153642 + 0.871345i −0.0196718 + 0.111564i −0.993063 − 0.117587i \(-0.962484\pi\)
0.973391 + 0.229152i \(0.0735951\pi\)
\(62\) 0 0
\(63\) −1.31088 − 3.38283i −0.165156 − 0.426196i
\(64\) 0 0
\(65\) −0.396123 − 0.686106i −0.0491331 − 0.0851010i
\(66\) 0 0
\(67\) −3.28864 + 3.91925i −0.401771 + 0.478812i −0.928559 − 0.371184i \(-0.878952\pi\)
0.526788 + 0.849997i \(0.323396\pi\)
\(68\) 0 0
\(69\) −2.61768 − 9.38117i −0.315131 − 1.12936i
\(70\) 0 0
\(71\) 1.64669 + 9.33885i 0.195426 + 1.10832i 0.911810 + 0.410612i \(0.134685\pi\)
−0.716384 + 0.697706i \(0.754204\pi\)
\(72\) 0 0
\(73\) 0.320853 + 0.116781i 0.0375530 + 0.0136682i 0.360728 − 0.932671i \(-0.382528\pi\)
−0.323175 + 0.946339i \(0.604750\pi\)
\(74\) 0 0
\(75\) −6.09881 − 5.97428i −0.704230 − 0.689850i
\(76\) 0 0
\(77\) − 2.84348i − 0.324044i
\(78\) 0 0
\(79\) −2.33914 + 6.42674i −0.263174 + 0.723064i 0.735775 + 0.677226i \(0.236818\pi\)
−0.998949 + 0.0458383i \(0.985404\pi\)
\(80\) 0 0
\(81\) −1.92708 − 8.79127i −0.214120 − 0.976807i
\(82\) 0 0
\(83\) −12.2240 − 7.05752i −1.34176 − 0.774663i −0.354691 − 0.934984i \(-0.615414\pi\)
−0.987065 + 0.160320i \(0.948747\pi\)
\(84\) 0 0
\(85\) −0.182956 − 0.153518i −0.0198443 − 0.0166514i
\(86\) 0 0
\(87\) −1.08301 + 4.21529i −0.116111 + 0.451927i
\(88\) 0 0
\(89\) −11.5580 + 4.20677i −1.22515 + 0.445917i −0.871933 − 0.489626i \(-0.837133\pi\)
−0.353213 + 0.935543i \(0.614911\pi\)
\(90\) 0 0
\(91\) −3.54297 − 0.624722i −0.371405 − 0.0654887i
\(92\) 0 0
\(93\) −1.38090 + 17.9095i −0.143192 + 1.85713i
\(94\) 0 0
\(95\) 0.698802 + 0.926900i 0.0716956 + 0.0950979i
\(96\) 0 0
\(97\) 3.88456 + 4.62944i 0.394418 + 0.470049i 0.926309 − 0.376764i \(-0.122963\pi\)
−0.531892 + 0.846812i \(0.678519\pi\)
\(98\) 0 0
\(99\) 1.08135 − 6.97057i 0.108680 − 0.700569i
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.545.3 18
3.2 odd 2 912.2.cc.d.545.1 18
4.3 odd 2 114.2.l.b.89.1 yes 18
12.11 even 2 114.2.l.a.89.3 yes 18
19.3 odd 18 912.2.cc.d.497.1 18
57.41 even 18 inner 912.2.cc.c.497.3 18
76.3 even 18 114.2.l.a.41.3 ✓ 18
228.155 odd 18 114.2.l.b.41.1 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.l.a.41.3 ✓ 18 76.3 even 18
114.2.l.a.89.3 yes 18 12.11 even 2
114.2.l.b.41.1 yes 18 228.155 odd 18
114.2.l.b.89.1 yes 18 4.3 odd 2
912.2.cc.c.497.3 18 57.41 even 18 inner
912.2.cc.c.545.3 18 1.1 even 1 trivial
912.2.cc.d.497.1 18 19.3 odd 18
912.2.cc.d.545.1 18 3.2 odd 2