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Properties

Label 279.2.o.a.212.18
Level $279$
Weight $2$
Character 279.212
Analytic conductor $2.228$
Analytic rank $0$
Dimension $60$
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] = [279,2,Mod(212,279)] 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("279.212"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(279, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 279 = 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 279.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.22782621639\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 212.18
Character \(\chi\) \(=\) 279.212
Dual form 279.2.o.a.254.18

$q$-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+(0.529444 - 0.305675i) q^{2} +(1.27637 - 1.17084i) q^{3} +(-0.813126 + 1.40838i) q^{4} -1.55166i q^{5} +(0.317871 - 1.01005i) q^{6} +4.38153 q^{7} +2.21691i q^{8} +(0.258252 - 2.98886i) q^{9} +(-0.474304 - 0.821519i) q^{10} +(-2.69238 + 4.66334i) q^{11} +(0.611136 + 2.74965i) q^{12} -5.46593i q^{13} +(2.31978 - 1.33932i) q^{14} +(-1.81675 - 1.98050i) q^{15} +(-0.948599 - 1.64302i) q^{16} +(-0.0887575 + 0.153733i) q^{17} +(-0.776890 - 1.66138i) q^{18} +(0.780073 - 1.35113i) q^{19} +(2.18532 + 1.26170i) q^{20} +(5.59247 - 5.13009i) q^{21} +3.29197i q^{22} +(-1.82782 + 3.16588i) q^{23} +(2.59565 + 2.82960i) q^{24} +2.59234 q^{25} +(-1.67080 - 2.89390i) q^{26} +(-3.16987 - 4.11727i) q^{27} +(-3.56274 + 6.17084i) q^{28} +(0.377407 + 0.653689i) q^{29} +(-1.56726 - 0.493228i) q^{30} +(-2.75532 + 4.83820i) q^{31} +(-4.84426 - 2.79683i) q^{32} +(2.02356 + 9.10451i) q^{33} +0.108524i q^{34} -6.79866i q^{35} +(3.99945 + 2.79404i) q^{36} +(-6.83555 - 3.94651i) q^{37} -0.953794i q^{38} +(-6.39975 - 6.97656i) q^{39} +3.43989 q^{40} +8.98766i q^{41} +(1.39276 - 4.42557i) q^{42} +10.3514i q^{43} +(-4.37849 - 7.58376i) q^{44} +(-4.63771 - 0.400720i) q^{45} +2.23488i q^{46} +(-9.41697 + 5.43689i) q^{47} +(-3.13449 - 0.986447i) q^{48} +12.1978 q^{49} +(1.37250 - 0.792414i) q^{50} +(0.0667091 + 0.300141i) q^{51} +(7.69808 + 4.44449i) q^{52} +(-2.69296 - 4.66434i) q^{53} +(-2.93681 - 1.21092i) q^{54} +(7.23593 + 4.17767i) q^{55} +9.71345i q^{56} +(-0.586293 - 2.63788i) q^{57} +(0.399632 + 0.230728i) q^{58} +(8.84837 - 5.10861i) q^{59} +(4.26654 - 0.948276i) q^{60} +(1.84270 - 1.06388i) q^{61} +(0.0201261 + 3.40379i) q^{62} +(1.13154 - 13.0958i) q^{63} +0.374712 q^{64} -8.48128 q^{65} +(3.85438 + 4.20178i) q^{66} -10.0708 q^{67} +(-0.144342 - 0.250008i) q^{68} +(1.37377 + 6.18093i) q^{69} +(-2.07818 - 3.59951i) q^{70} +(4.87193 - 2.81281i) q^{71} +(6.62603 + 0.572520i) q^{72} +(-2.76988 + 1.59919i) q^{73} -4.82539 q^{74} +(3.30879 - 3.03523i) q^{75} +(1.26859 + 2.19727i) q^{76} +(-11.7967 + 20.4326i) q^{77} +(-5.52087 - 1.73746i) q^{78} -1.24036i q^{79} +(-2.54942 + 1.47191i) q^{80} +(-8.86661 - 1.54376i) q^{81} +(2.74730 + 4.75846i) q^{82} +(1.44061 - 2.49520i) q^{83} +(2.67771 + 12.0477i) q^{84} +(0.238541 + 0.137722i) q^{85} +(3.16417 + 5.48050i) q^{86} +(1.24708 + 0.392465i) q^{87} +(-10.3382 - 5.96876i) q^{88} -4.66152 q^{89} +(-2.57790 + 1.20547i) q^{90} -23.9491i q^{91} +(-2.97250 - 5.14852i) q^{92} +(2.14796 + 9.40140i) q^{93} +(-3.32384 + 5.75706i) q^{94} +(-2.09649 - 1.21041i) q^{95} +(-9.45773 + 2.10207i) q^{96} +(2.43672 + 4.22052i) q^{97} +(6.45807 - 3.72857i) q^{98} +(13.2428 + 9.25147i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 6 q^{2} - 3 q^{3} + 26 q^{4} - 9 q^{6} + q^{9} - 4 q^{10} - 3 q^{11} - 9 q^{12} - 3 q^{14} - 3 q^{15} - 22 q^{16} + 5 q^{18} - 4 q^{19} - 21 q^{20} + 9 q^{21} - 9 q^{23} + 18 q^{24} - 52 q^{25}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/279\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(218\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.529444 − 0.305675i 0.374374 − 0.216145i −0.300994 − 0.953626i \(-0.597318\pi\)
0.675367 + 0.737481i \(0.263985\pi\)
\(3\) 1.27637 − 1.17084i 0.736914 − 0.675987i
\(4\) −0.813126 + 1.40838i −0.406563 + 0.704188i
\(5\) − 1.55166i − 0.693925i −0.937879 − 0.346962i \(-0.887213\pi\)
0.937879 − 0.346962i \(-0.112787\pi\)
\(6\) 0.317871 − 1.01005i 0.129770 − 0.412352i
\(7\) 4.38153 1.65606 0.828032 − 0.560681i \(-0.189461\pi\)
0.828032 + 0.560681i \(0.189461\pi\)
\(8\) 2.21691i 0.783795i
\(9\) 0.258252 − 2.98886i 0.0860840 − 0.996288i
\(10\) −0.474304 − 0.821519i −0.149988 − 0.259787i
\(11\) −2.69238 + 4.66334i −0.811783 + 1.40605i 0.0998322 + 0.995004i \(0.468169\pi\)
−0.911615 + 0.411045i \(0.865164\pi\)
\(12\) 0.611136 + 2.74965i 0.176420 + 0.793757i
\(13\) − 5.46593i − 1.51598i −0.652269 − 0.757988i \(-0.726183\pi\)
0.652269 − 0.757988i \(-0.273817\pi\)
\(14\) 2.31978 − 1.33932i 0.619987 − 0.357949i
\(15\) −1.81675 − 1.98050i −0.469084 − 0.511363i
\(16\) −0.948599 − 1.64302i −0.237150 − 0.410756i
\(17\) −0.0887575 + 0.153733i −0.0215269 + 0.0372856i −0.876588 − 0.481241i \(-0.840186\pi\)
0.855061 + 0.518527i \(0.173519\pi\)
\(18\) −0.776890 − 1.66138i −0.183115 − 0.391590i
\(19\) 0.780073 − 1.35113i 0.178961 − 0.309970i −0.762564 − 0.646913i \(-0.776060\pi\)
0.941525 + 0.336943i \(0.109393\pi\)
\(20\) 2.18532 + 1.26170i 0.488653 + 0.282124i
\(21\) 5.59247 − 5.13009i 1.22038 − 1.11948i
\(22\) 3.29197i 0.701850i
\(23\) −1.82782 + 3.16588i −0.381127 + 0.660131i −0.991224 − 0.132196i \(-0.957797\pi\)
0.610097 + 0.792327i \(0.291131\pi\)
\(24\) 2.59565 + 2.82960i 0.529835 + 0.577589i
\(25\) 2.59234 0.518468
\(26\) −1.67080 − 2.89390i −0.327670 − 0.567541i
\(27\) −3.16987 − 4.11727i −0.610041 − 0.792370i
\(28\) −3.56274 + 6.17084i −0.673294 + 1.16618i
\(29\) 0.377407 + 0.653689i 0.0700828 + 0.121387i 0.898937 − 0.438077i \(-0.144340\pi\)
−0.828855 + 0.559464i \(0.811007\pi\)
\(30\) −1.56726 − 0.493228i −0.286141 − 0.0900507i
\(31\) −2.75532 + 4.83820i −0.494871 + 0.868967i
\(32\) −4.84426 − 2.79683i −0.856352 − 0.494415i
\(33\) 2.02356 + 9.10451i 0.352257 + 1.58489i
\(34\) 0.108524i 0.0186117i
\(35\) − 6.79866i − 1.14918i
\(36\) 3.99945 + 2.79404i 0.666575 + 0.465673i
\(37\) −6.83555 − 3.94651i −1.12376 − 0.648801i −0.181400 − 0.983409i \(-0.558063\pi\)
−0.942357 + 0.334608i \(0.891396\pi\)
\(38\) − 0.953794i − 0.154726i
\(39\) −6.39975 − 6.97656i −1.02478 − 1.11714i
\(40\) 3.43989 0.543895
\(41\) 8.98766i 1.40364i 0.712356 + 0.701818i \(0.247628\pi\)
−0.712356 + 0.701818i \(0.752372\pi\)
\(42\) 1.39276 − 4.42557i 0.214908 − 0.682881i
\(43\) 10.3514i 1.57858i 0.614022 + 0.789289i \(0.289551\pi\)
−0.614022 + 0.789289i \(0.710449\pi\)
\(44\) −4.37849 − 7.58376i −0.660082 − 1.14330i
\(45\) −4.63771 − 0.400720i −0.691349 − 0.0597358i
\(46\) 2.23488i 0.329514i
\(47\) −9.41697 + 5.43689i −1.37361 + 0.793052i −0.991380 − 0.131016i \(-0.958176\pi\)
−0.382227 + 0.924069i \(0.624843\pi\)
\(48\) −3.13449 − 0.986447i −0.452424 − 0.142381i
\(49\) 12.1978 1.74255
\(50\) 1.37250 − 0.792414i 0.194101 − 0.112064i
\(51\) 0.0667091 + 0.300141i 0.00934114 + 0.0420282i
\(52\) 7.69808 + 4.44449i 1.06753 + 0.616340i
\(53\) −2.69296 − 4.66434i −0.369906 − 0.640696i 0.619645 − 0.784883i \(-0.287277\pi\)
−0.989551 + 0.144187i \(0.953943\pi\)
\(54\) −2.93681 − 1.21092i −0.399650 − 0.164785i
\(55\) 7.23593 + 4.17767i 0.975692 + 0.563316i
\(56\) 9.71345i 1.29801i
\(57\) −0.586293 − 2.63788i −0.0776564 − 0.349396i
\(58\) 0.399632 + 0.230728i 0.0524743 + 0.0302960i
\(59\) 8.84837 − 5.10861i 1.15196 − 0.665084i 0.202596 − 0.979262i \(-0.435062\pi\)
0.949364 + 0.314178i \(0.101729\pi\)
\(60\) 4.26654 − 0.948276i 0.550807 − 0.122422i
\(61\) 1.84270 − 1.06388i 0.235934 − 0.136216i −0.377373 − 0.926062i \(-0.623172\pi\)
0.613306 + 0.789845i \(0.289839\pi\)
\(62\) 0.0201261 + 3.40379i 0.00255602 + 0.432282i
\(63\) 1.13154 − 13.0958i 0.142561 − 1.64992i
\(64\) 0.374712 0.0468390
\(65\) −8.48128 −1.05197
\(66\) 3.85438 + 4.20178i 0.474441 + 0.517203i
\(67\) −10.0708 −1.23035 −0.615173 − 0.788392i \(-0.710914\pi\)
−0.615173 + 0.788392i \(0.710914\pi\)
\(68\) −0.144342 − 0.250008i −0.0175041 − 0.0303179i
\(69\) 1.37377 + 6.18093i 0.165382 + 0.744097i
\(70\) −2.07818 − 3.59951i −0.248390 − 0.430224i
\(71\) 4.87193 − 2.81281i 0.578191 − 0.333819i −0.182223 − 0.983257i \(-0.558329\pi\)
0.760414 + 0.649438i \(0.224996\pi\)
\(72\) 6.62603 + 0.572520i 0.780886 + 0.0674722i
\(73\) −2.76988 + 1.59919i −0.324190 + 0.187171i −0.653259 − 0.757135i \(-0.726599\pi\)
0.329069 + 0.944306i \(0.393265\pi\)
\(74\) −4.82539 −0.560940
\(75\) 3.30879 − 3.03523i 0.382067 − 0.350478i
\(76\) 1.26859 + 2.19727i 0.145518 + 0.252044i
\(77\) −11.7967 + 20.4326i −1.34436 + 2.32851i
\(78\) −5.52087 − 1.73746i −0.625115 − 0.196728i
\(79\) − 1.24036i − 0.139551i −0.997563 − 0.0697755i \(-0.977772\pi\)
0.997563 − 0.0697755i \(-0.0222283\pi\)
\(80\) −2.54942 + 1.47191i −0.285033 + 0.164564i
\(81\) −8.86661 − 1.54376i −0.985179 − 0.171529i
\(82\) 2.74730 + 4.75846i 0.303389 + 0.525484i
\(83\) 1.44061 − 2.49520i 0.158127 − 0.273884i −0.776066 − 0.630651i \(-0.782788\pi\)
0.934193 + 0.356767i \(0.116121\pi\)
\(84\) 2.67771 + 12.0477i 0.292162 + 1.31451i
\(85\) 0.238541 + 0.137722i 0.0258734 + 0.0149380i
\(86\) 3.16417 + 5.48050i 0.341201 + 0.590978i
\(87\) 1.24708 + 0.392465i 0.133701 + 0.0420767i
\(88\) −10.3382 − 5.96876i −1.10205 − 0.636271i
\(89\) −4.66152 −0.494120 −0.247060 − 0.969000i \(-0.579464\pi\)
−0.247060 + 0.969000i \(0.579464\pi\)
\(90\) −2.57790 + 1.20547i −0.271734 + 0.127068i
\(91\) − 23.9491i − 2.51055i
\(92\) −2.97250 − 5.14852i −0.309904 − 0.536770i
\(93\) 2.14796 + 9.40140i 0.222733 + 0.974880i
\(94\) −3.32384 + 5.75706i −0.342828 + 0.593796i
\(95\) −2.09649 − 1.21041i −0.215096 − 0.124185i
\(96\) −9.45773 + 2.10207i −0.965275 + 0.214541i
\(97\) 2.43672 + 4.22052i 0.247411 + 0.428529i 0.962807 − 0.270191i \(-0.0870868\pi\)
−0.715396 + 0.698720i \(0.753753\pi\)
\(98\) 6.45807 − 3.72857i 0.652364 − 0.376642i
\(99\) 13.2428 + 9.25147i 1.33095 + 0.929808i
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 279.2.o.a.212.18 ✓ 60
3.2 odd 2 837.2.o.a.584.13 60
9.2 odd 6 279.2.r.a.119.18 yes 60
9.7 even 3 837.2.r.a.305.13 60
31.6 odd 6 279.2.r.a.68.18 yes 60
93.68 even 6 837.2.r.a.719.13 60
279.223 odd 6 837.2.o.a.440.13 60
279.254 even 6 inner 279.2.o.a.254.18 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.18 ✓ 60 1.1 even 1 trivial
279.2.o.a.254.18 yes 60 279.254 even 6 inner
279.2.r.a.68.18 yes 60 31.6 odd 6
279.2.r.a.119.18 yes 60 9.2 odd 6
837.2.o.a.440.13 60 279.223 odd 6
837.2.o.a.584.13 60 3.2 odd 2
837.2.r.a.305.13 60 9.7 even 3
837.2.r.a.719.13 60 93.68 even 6