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Properties

Label 266.2.f.d.197.1
Level $266$
Weight $2$
Character 266.197
Analytic conductor $2.124$
Analytic rank $0$
Dimension $8$
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] = [266,2,Mod(197,266)] 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("266.197"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(266, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 266 = 2 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 266.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.12402069377\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 11x^{6} - 6x^{5} + 104x^{4} - 72x^{3} + 104x^{2} + 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 197.1
Root \(-1.51459 + 2.62334i\) of defining polynomial
Character \(\chi\) \(=\) 266.197
Dual form 266.2.f.d.239.1

$q$-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+(0.500000 + 0.866025i) q^{2} +(-1.51459 - 2.62334i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(1.68446 + 2.91758i) q^{5} +(1.51459 - 2.62334i) q^{6} +1.00000 q^{7} -1.00000 q^{8} +(-3.08794 + 5.34848i) q^{9} +(-1.68446 + 2.91758i) q^{10} +5.02917 q^{11} +3.02917 q^{12} +(2.41807 - 4.18821i) q^{13} +(0.500000 + 0.866025i) q^{14} +(5.10253 - 8.83784i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(0.0733573 + 0.127059i) q^{17} -6.17589 q^{18} +(3.85930 - 2.02629i) q^{19} -3.36893 q^{20} +(-1.51459 - 2.62334i) q^{21} +(2.51459 + 4.35539i) q^{22} +(-2.28098 + 3.95078i) q^{23} +(1.51459 + 2.62334i) q^{24} +(-3.17483 + 5.49897i) q^{25} +4.83613 q^{26} +9.62031 q^{27} +(-0.500000 + 0.866025i) q^{28} +(-3.76278 + 6.51732i) q^{29} +10.2051 q^{30} -1.46721 q^{31} +(0.500000 - 0.866025i) q^{32} +(-7.61712 - 13.1932i) q^{33} +(-0.0733573 + 0.127059i) q^{34} +(1.68446 + 2.91758i) q^{35} +(-3.08794 - 5.34848i) q^{36} -4.73785 q^{37} +(3.68446 + 2.32911i) q^{38} -14.6495 q^{39} +(-1.68446 - 2.91758i) q^{40} +(0.485414 + 0.840761i) q^{41} +(1.51459 - 2.62334i) q^{42} +(-2.26640 - 3.92551i) q^{43} +(-2.51459 + 4.35539i) q^{44} -20.8061 q^{45} -4.56197 q^{46} +(3.76278 - 6.51732i) q^{47} +(-1.51459 + 2.62334i) q^{48} +1.00000 q^{49} -6.34967 q^{50} +(0.222212 - 0.384882i) q^{51} +(2.41807 + 4.18821i) q^{52} +(1.26640 - 2.19346i) q^{53} +(4.81016 + 8.33143i) q^{54} +(8.47146 + 14.6730i) q^{55} -1.00000 q^{56} +(-11.1609 - 7.05526i) q^{57} -7.52555 q^{58} +(-1.27241 - 2.20387i) q^{59} +(5.10253 + 8.83784i) q^{60} +(-6.52060 + 11.2940i) q^{61} +(-0.733604 - 1.27064i) q^{62} +(-3.08794 + 5.34848i) q^{63} +1.00000 q^{64} +16.2926 q^{65} +(7.61712 - 13.1932i) q^{66} +(-0.441229 + 0.764231i) q^{67} -0.146715 q^{68} +13.8190 q^{69} +(-1.68446 + 2.91758i) q^{70} +(-0.645660 - 1.11832i) q^{71} +(3.08794 - 5.34848i) q^{72} +(-4.92770 - 8.53502i) q^{73} +(-2.36893 - 4.10310i) q^{74} +19.2342 q^{75} +(-0.174833 + 4.35539i) q^{76} +5.02917 q^{77} +(-7.32474 - 12.6868i) q^{78} +(-8.20506 - 14.2116i) q^{79} +(1.68446 - 2.91758i) q^{80} +(-5.30696 - 9.19193i) q^{81} +(-0.485414 + 0.840761i) q^{82} -11.2243 q^{83} +3.02917 q^{84} +(-0.247135 + 0.428051i) q^{85} +(2.26640 - 3.92551i) q^{86} +22.7962 q^{87} -5.02917 q^{88} +(-0.970827 + 1.68152i) q^{89} +(-10.4031 - 18.0186i) q^{90} +(2.41807 - 4.18821i) q^{91} +(-2.28098 - 3.95078i) q^{92} +(2.22221 + 3.84898i) q^{93} +7.52555 q^{94} +(12.4127 + 7.84658i) q^{95} -3.02917 q^{96} +(-5.54376 - 9.60207i) q^{97} +(0.500000 + 0.866025i) q^{98} +(-15.5298 + 26.8984i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + q^{3} - 4 q^{4} - q^{5} - q^{6} + 8 q^{7} - 8 q^{8} - 9 q^{9} + q^{10} + 14 q^{11} - 2 q^{12} + 5 q^{13} + 4 q^{14} + 12 q^{15} - 4 q^{16} - 2 q^{17} - 18 q^{18} + 6 q^{19} + 2 q^{20}+ \cdots - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/266\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i
\(3\) −1.51459 − 2.62334i −0.874447 − 1.51459i −0.857351 − 0.514732i \(-0.827891\pi\)
−0.0170960 − 0.999854i \(-0.505442\pi\)
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 1.68446 + 2.91758i 0.753315 + 1.30478i 0.946208 + 0.323560i \(0.104880\pi\)
−0.192893 + 0.981220i \(0.561787\pi\)
\(6\) 1.51459 − 2.62334i 0.618327 − 1.07097i
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −3.08794 + 5.34848i −1.02931 + 1.78283i
\(10\) −1.68446 + 2.91758i −0.532674 + 0.922618i
\(11\) 5.02917 1.51635 0.758176 − 0.652050i \(-0.226091\pi\)
0.758176 + 0.652050i \(0.226091\pi\)
\(12\) 3.02917 0.874447
\(13\) 2.41807 − 4.18821i 0.670651 − 1.16160i −0.307069 − 0.951687i \(-0.599348\pi\)
0.977720 − 0.209914i \(-0.0673185\pi\)
\(14\) 0.500000 + 0.866025i 0.133631 + 0.231455i
\(15\) 5.10253 − 8.83784i 1.31747 − 2.28192i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 0.0733573 + 0.127059i 0.0177918 + 0.0308162i 0.874784 − 0.484513i \(-0.161003\pi\)
−0.856992 + 0.515329i \(0.827670\pi\)
\(18\) −6.17589 −1.45567
\(19\) 3.85930 − 2.02629i 0.885383 − 0.464862i
\(20\) −3.36893 −0.753315
\(21\) −1.51459 − 2.62334i −0.330510 − 0.572460i
\(22\) 2.51459 + 4.35539i 0.536112 + 0.928573i
\(23\) −2.28098 + 3.95078i −0.475618 + 0.823794i −0.999610 − 0.0279289i \(-0.991109\pi\)
0.523992 + 0.851723i \(0.324442\pi\)
\(24\) 1.51459 + 2.62334i 0.309164 + 0.535487i
\(25\) −3.17483 + 5.49897i −0.634967 + 1.09979i
\(26\) 4.83613 0.948444
\(27\) 9.62031 1.85143
\(28\) −0.500000 + 0.866025i −0.0944911 + 0.163663i
\(29\) −3.76278 + 6.51732i −0.698730 + 1.21024i 0.270177 + 0.962811i \(0.412918\pi\)
−0.968907 + 0.247425i \(0.920416\pi\)
\(30\) 10.2051 1.86318
\(31\) −1.46721 −0.263518 −0.131759 − 0.991282i \(-0.542063\pi\)
−0.131759 + 0.991282i \(0.542063\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) −7.61712 − 13.1932i −1.32597 − 2.29665i
\(34\) −0.0733573 + 0.127059i −0.0125807 + 0.0217904i
\(35\) 1.68446 + 2.91758i 0.284726 + 0.493160i
\(36\) −3.08794 − 5.34848i −0.514657 − 0.891413i
\(37\) −4.73785 −0.778898 −0.389449 − 0.921048i \(-0.627335\pi\)
−0.389449 + 0.921048i \(0.627335\pi\)
\(38\) 3.68446 + 2.32911i 0.597699 + 0.377831i
\(39\) −14.6495 −2.34579
\(40\) −1.68446 − 2.91758i −0.266337 − 0.461309i
\(41\) 0.485414 + 0.840761i 0.0758089 + 0.131305i 0.901438 − 0.432909i \(-0.142513\pi\)
−0.825629 + 0.564214i \(0.809179\pi\)
\(42\) 1.51459 − 2.62334i 0.233706 − 0.404790i
\(43\) −2.26640 − 3.92551i −0.345622 − 0.598635i 0.639844 − 0.768504i \(-0.278999\pi\)
−0.985467 + 0.169869i \(0.945665\pi\)
\(44\) −2.51459 + 4.35539i −0.379088 + 0.656600i
\(45\) −20.8061 −3.10159
\(46\) −4.56197 −0.672625
\(47\) 3.76278 − 6.51732i 0.548857 − 0.950649i −0.449496 − 0.893282i \(-0.648396\pi\)
0.998353 − 0.0573664i \(-0.0182703\pi\)
\(48\) −1.51459 + 2.62334i −0.218612 + 0.378647i
\(49\) 1.00000 0.142857
\(50\) −6.34967 −0.897978
\(51\) 0.222212 − 0.384882i 0.0311159 − 0.0538943i
\(52\) 2.41807 + 4.18821i 0.335326 + 0.580801i
\(53\) 1.26640 − 2.19346i 0.173953 − 0.301295i −0.765845 − 0.643025i \(-0.777679\pi\)
0.939798 + 0.341729i \(0.111013\pi\)
\(54\) 4.81016 + 8.33143i 0.654579 + 1.13376i
\(55\) 8.47146 + 14.6730i 1.14229 + 1.97851i
\(56\) −1.00000 −0.133631
\(57\) −11.1609 − 7.05526i −1.47829 − 0.934492i
\(58\) −7.52555 −0.988153
\(59\) −1.27241 − 2.20387i −0.165653 − 0.286920i 0.771234 − 0.636552i \(-0.219640\pi\)
−0.936887 + 0.349632i \(0.886307\pi\)
\(60\) 5.10253 + 8.83784i 0.658734 + 1.14096i
\(61\) −6.52060 + 11.2940i −0.834877 + 1.44605i 0.0592538 + 0.998243i \(0.481128\pi\)
−0.894131 + 0.447806i \(0.852205\pi\)
\(62\) −0.733604 − 1.27064i −0.0931677 − 0.161371i
\(63\) −3.08794 + 5.34848i −0.389044 + 0.673845i
\(64\) 1.00000 0.125000
\(65\) 16.2926 2.02085
\(66\) 7.61712 − 13.1932i 0.937602 − 1.62397i
\(67\) −0.441229 + 0.764231i −0.0539047 + 0.0933657i −0.891719 − 0.452590i \(-0.850500\pi\)
0.837814 + 0.545956i \(0.183833\pi\)
\(68\) −0.146715 −0.0177918
\(69\) 13.8190 1.66361
\(70\) −1.68446 + 2.91758i −0.201332 + 0.348717i
\(71\) −0.645660 − 1.11832i −0.0766257 − 0.132720i 0.825166 − 0.564890i \(-0.191081\pi\)
−0.901792 + 0.432170i \(0.857748\pi\)
\(72\) 3.08794 − 5.34848i 0.363918 − 0.630324i
\(73\) −4.92770 − 8.53502i −0.576743 − 0.998949i −0.995850 − 0.0910117i \(-0.970990\pi\)
0.419106 − 0.907937i \(-0.362343\pi\)
\(74\) −2.36893 − 4.10310i −0.275382 − 0.476976i
\(75\) 19.2342 2.22098
\(76\) −0.174833 + 4.35539i −0.0200547 + 0.499598i
\(77\) 5.02917 0.573127
\(78\) −7.32474 − 12.6868i −0.829364 − 1.43650i
\(79\) −8.20506 − 14.2116i −0.923141 − 1.59893i −0.794523 − 0.607234i \(-0.792279\pi\)
−0.128618 − 0.991694i \(-0.541054\pi\)
\(80\) 1.68446 − 2.91758i 0.188329 − 0.326195i
\(81\) −5.30696 − 9.19193i −0.589662 − 1.02133i
\(82\) −0.485414 + 0.840761i −0.0536050 + 0.0928465i
\(83\) −11.2243 −1.23203 −0.616015 − 0.787735i \(-0.711254\pi\)
−0.616015 + 0.787735i \(0.711254\pi\)
\(84\) 3.02917 0.330510
\(85\) −0.247135 + 0.428051i −0.0268056 + 0.0464286i
\(86\) 2.26640 − 3.92551i 0.244392 − 0.423299i
\(87\) 22.7962 2.44401
\(88\) −5.02917 −0.536112
\(89\) −0.970827 + 1.68152i −0.102907 + 0.178241i −0.912881 − 0.408225i \(-0.866148\pi\)
0.809974 + 0.586466i \(0.199481\pi\)
\(90\) −10.4031 − 18.0186i −1.09658 − 1.89933i
\(91\) 2.41807 − 4.18821i 0.253482 − 0.439044i
\(92\) −2.28098 − 3.95078i −0.237809 − 0.411897i
\(93\) 2.22221 + 3.84898i 0.230433 + 0.399121i
\(94\) 7.52555 0.776201
\(95\) 12.4127 + 7.84658i 1.27351 + 0.805043i
\(96\) −3.02917 −0.309164
\(97\) −5.54376 − 9.60207i −0.562883 − 0.974943i −0.997243 − 0.0742032i \(-0.976359\pi\)
0.434360 − 0.900740i \(-0.356975\pi\)
\(98\) 0.500000 + 0.866025i 0.0505076 + 0.0874818i
\(99\) −15.5298 + 26.8984i −1.56080 + 2.70339i
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 266.2.f.d.197.1 ✓ 8
3.2 odd 2 2394.2.o.v.1261.1 8
19.7 even 3 5054.2.a.w.1.4 4
19.11 even 3 inner 266.2.f.d.239.1 yes 8
19.12 odd 6 5054.2.a.x.1.1 4
57.11 odd 6 2394.2.o.v.505.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.f.d.197.1 ✓ 8 1.1 even 1 trivial
266.2.f.d.239.1 yes 8 19.11 even 3 inner
2394.2.o.v.505.1 8 57.11 odd 6
2394.2.o.v.1261.1 8 3.2 odd 2
5054.2.a.w.1.4 4 19.7 even 3
5054.2.a.x.1.1 4 19.12 odd 6