↑

Properties

Label 2394.2.o.v.505.1
Level $2394$
Weight $2$
Character 2394.505
Analytic conductor $19.116$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2394,2,Mod(505,2394)] 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("2394.505"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2394, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.o (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-4,0,-4,1,0,8,8,0,1,-14,0,5,-4,0,-4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.1161862439\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 266)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 505.1
Root \(-1.51459 - 2.62334i\) of defining polynomial
Character \(\chi\) \(=\) 2394.505
Dual form 2394.2.o.v.1261.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.68446 + 2.91758i) q^{5} +1.00000 q^{7} +1.00000 q^{8} +(-1.68446 - 2.91758i) q^{10} -5.02917 q^{11} +(2.41807 + 4.18821i) q^{13} +(-0.500000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-0.0733573 + 0.127059i) q^{17} +(3.85930 + 2.02629i) q^{19} +3.36893 q^{20} +(2.51459 - 4.35539i) q^{22} +(2.28098 + 3.95078i) q^{23} +(-3.17483 - 5.49897i) q^{25} -4.83613 q^{26} +(-0.500000 - 0.866025i) q^{28} +(3.76278 + 6.51732i) q^{29} -1.46721 q^{31} +(-0.500000 - 0.866025i) q^{32} +(-0.0733573 - 0.127059i) q^{34} +(-1.68446 + 2.91758i) q^{35} -4.73785 q^{37} +(-3.68446 + 2.32911i) q^{38} +(-1.68446 + 2.91758i) q^{40} +(-0.485414 + 0.840761i) q^{41} +(-2.26640 + 3.92551i) q^{43} +(2.51459 + 4.35539i) q^{44} -4.56197 q^{46} +(-3.76278 - 6.51732i) q^{47} +1.00000 q^{49} +6.34967 q^{50} +(2.41807 - 4.18821i) q^{52} +(-1.26640 - 2.19346i) q^{53} +(8.47146 - 14.6730i) q^{55} +1.00000 q^{56} -7.52555 q^{58} +(1.27241 - 2.20387i) q^{59} +(-6.52060 - 11.2940i) q^{61} +(0.733604 - 1.27064i) q^{62} +1.00000 q^{64} -16.2926 q^{65} +(-0.441229 - 0.764231i) q^{67} +0.146715 q^{68} +(-1.68446 - 2.91758i) q^{70} +(0.645660 - 1.11832i) q^{71} +(-4.92770 + 8.53502i) q^{73} +(2.36893 - 4.10310i) q^{74} +(-0.174833 - 4.35539i) q^{76} -5.02917 q^{77} +(-8.20506 + 14.2116i) q^{79} +(-1.68446 - 2.91758i) q^{80} +(-0.485414 - 0.840761i) q^{82} +11.2243 q^{83} +(-0.247135 - 0.428051i) q^{85} +(-2.26640 - 3.92551i) q^{86} -5.02917 q^{88} +(0.970827 + 1.68152i) q^{89} +(2.41807 + 4.18821i) q^{91} +(2.28098 - 3.95078i) q^{92} +7.52555 q^{94} +(-12.4127 + 7.84658i) q^{95} +(-5.54376 + 9.60207i) q^{97} +(-0.500000 + 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} + q^{5} + 8 q^{7} + 8 q^{8} + q^{10} - 14 q^{11} + 5 q^{13} - 4 q^{14} - 4 q^{16} + 2 q^{17} + 6 q^{19} - 2 q^{20} + 7 q^{22} + 5 q^{23} - 15 q^{25} - 10 q^{26} - 4 q^{28} + 4 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.353553 + 0.612372i
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) −1.68446 + 2.91758i −0.753315 + 1.30478i 0.192893 + 0.981220i \(0.438213\pi\)
−0.946208 + 0.323560i \(0.895120\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.68446 − 2.91758i −0.532674 − 0.922618i
\(11\) −5.02917 −1.51635 −0.758176 − 0.652050i \(-0.773909\pi\)
−0.758176 + 0.652050i \(0.773909\pi\)
\(12\) 0 0
\(13\) 2.41807 + 4.18821i 0.670651 + 1.16160i 0.977720 + 0.209914i \(0.0673185\pi\)
−0.307069 + 0.951687i \(0.599348\pi\)
\(14\) −0.500000 + 0.866025i −0.133631 + 0.231455i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −0.0733573 + 0.127059i −0.0177918 + 0.0308162i −0.874784 − 0.484513i \(-0.838997\pi\)
0.856992 + 0.515329i \(0.172330\pi\)
\(18\) 0 0
\(19\) 3.85930 + 2.02629i 0.885383 + 0.464862i
\(20\) 3.36893 0.753315
\(21\) 0 0
\(22\) 2.51459 − 4.35539i 0.536112 − 0.928573i
\(23\) 2.28098 + 3.95078i 0.475618 + 0.823794i 0.999610 − 0.0279289i \(-0.00889121\pi\)
−0.523992 + 0.851723i \(0.675558\pi\)
\(24\) 0 0
\(25\) −3.17483 − 5.49897i −0.634967 − 1.09979i
\(26\) −4.83613 −0.948444
\(27\) 0 0
\(28\) −0.500000 − 0.866025i −0.0944911 − 0.163663i
\(29\) 3.76278 + 6.51732i 0.698730 + 1.21024i 0.968907 + 0.247425i \(0.0795845\pi\)
−0.270177 + 0.962811i \(0.587082\pi\)
\(30\) 0 0
\(31\) −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
\(33\) 0 0
\(34\) −0.0733573 − 0.127059i −0.0125807 − 0.0217904i
\(35\) −1.68446 + 2.91758i −0.284726 + 0.493160i
\(36\) 0 0
\(37\) −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
\(39\) 0 0
\(40\) −1.68446 + 2.91758i −0.266337 + 0.461309i
\(41\) −0.485414 + 0.840761i −0.0758089 + 0.131305i −0.901438 − 0.432909i \(-0.857487\pi\)
0.825629 + 0.564214i \(0.190821\pi\)
\(42\) 0 0
\(43\) −2.26640 + 3.92551i −0.345622 + 0.598635i −0.985467 − 0.169869i \(-0.945665\pi\)
0.639844 + 0.768504i \(0.278999\pi\)
\(44\) 2.51459 + 4.35539i 0.379088 + 0.656600i
\(45\) 0 0
\(46\) −4.56197 −0.672625
\(47\) −3.76278 − 6.51732i −0.548857 − 0.950649i −0.998353 − 0.0573664i \(-0.981730\pi\)
0.449496 − 0.893282i \(-0.351604\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 6.34967 0.897978
\(51\) 0 0
\(52\) 2.41807 − 4.18821i 0.335326 − 0.580801i
\(53\) −1.26640 − 2.19346i −0.173953 − 0.301295i 0.765845 − 0.643025i \(-0.222321\pi\)
−0.939798 + 0.341729i \(0.888987\pi\)
\(54\) 0 0
\(55\) 8.47146 − 14.6730i 1.14229 − 1.97851i
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −7.52555 −0.988153
\(59\) 1.27241 − 2.20387i 0.165653 − 0.286920i −0.771234 − 0.636552i \(-0.780360\pi\)
0.936887 + 0.349632i \(0.113693\pi\)
\(60\) 0 0
\(61\) −6.52060 − 11.2940i −0.834877 − 1.44605i −0.894131 − 0.447806i \(-0.852205\pi\)
0.0592538 − 0.998243i \(-0.481128\pi\)
\(62\) 0.733604 − 1.27064i 0.0931677 − 0.161371i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −16.2926 −2.02085
\(66\) 0 0
\(67\) −0.441229 − 0.764231i −0.0539047 − 0.0933657i 0.837814 − 0.545956i \(-0.183833\pi\)
−0.891719 + 0.452590i \(0.850500\pi\)
\(68\) 0.146715 0.0177918
\(69\) 0 0
\(70\) −1.68446 − 2.91758i −0.201332 − 0.348717i
\(71\) 0.645660 − 1.11832i 0.0766257 − 0.132720i −0.825166 − 0.564890i \(-0.808919\pi\)
0.901792 + 0.432170i \(0.142252\pi\)
\(72\) 0 0
\(73\) −4.92770 + 8.53502i −0.576743 + 0.998949i 0.419106 + 0.907937i \(0.362343\pi\)
−0.995850 + 0.0910117i \(0.970990\pi\)
\(74\) 2.36893 − 4.10310i 0.275382 − 0.476976i
\(75\) 0 0
\(76\) −0.174833 − 4.35539i −0.0200547 − 0.499598i
\(77\) −5.02917 −0.573127
\(78\) 0 0
\(79\) −8.20506 + 14.2116i −0.923141 + 1.59893i −0.128618 + 0.991694i \(0.541054\pi\)
−0.794523 + 0.607234i \(0.792279\pi\)
\(80\) −1.68446 − 2.91758i −0.188329 − 0.326195i
\(81\) 0 0
\(82\) −0.485414 − 0.840761i −0.0536050 − 0.0928465i
\(83\) 11.2243 1.23203 0.616015 − 0.787735i \(-0.288746\pi\)
0.616015 + 0.787735i \(0.288746\pi\)
\(84\) 0 0
\(85\) −0.247135 − 0.428051i −0.0268056 − 0.0464286i
\(86\) −2.26640 − 3.92551i −0.244392 − 0.423299i
\(87\) 0 0
\(88\) −5.02917 −0.536112
\(89\) 0.970827 + 1.68152i 0.102907 + 0.178241i 0.912881 − 0.408225i \(-0.133852\pi\)
−0.809974 + 0.586466i \(0.800519\pi\)
\(90\) 0 0
\(91\) 2.41807 + 4.18821i 0.253482 + 0.439044i
\(92\) 2.28098 − 3.95078i 0.237809 − 0.411897i
\(93\) 0 0
\(94\) 7.52555 0.776201
\(95\) −12.4127 + 7.84658i −1.27351 + 0.805043i
\(96\) 0 0
\(97\) −5.54376 + 9.60207i −0.562883 + 0.974943i 0.434360 + 0.900740i \(0.356975\pi\)
−0.997243 + 0.0742032i \(0.976359\pi\)
\(98\) −0.500000 + 0.866025i −0.0505076 + 0.0874818i
\(99\) 0 0
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 2394.2.o.v.505.1 8
3.2 odd 2 266.2.f.d.239.1 yes 8
19.7 even 3 inner 2394.2.o.v.1261.1 8
57.8 even 6 5054.2.a.x.1.1 4
57.11 odd 6 5054.2.a.w.1.4 4
57.26 odd 6 266.2.f.d.197.1 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.f.d.197.1 ✓ 8 57.26 odd 6
266.2.f.d.239.1 yes 8 3.2 odd 2
2394.2.o.v.505.1 8 1.1 even 1 trivial
2394.2.o.v.1261.1 8 19.7 even 3 inner
5054.2.a.w.1.4 4 57.11 odd 6
5054.2.a.x.1.1 4 57.8 even 6