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Properties

Label 891.2.n.b.460.1
Level $891$
Weight $2$
Character 891.460
Analytic conductor $7.115$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [891,2,Mod(136,891)] 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("891.136"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(891, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.n (of order \(15\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-1,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 460.1
Root \(-0.978148 + 0.207912i\) of defining polynomial
Character \(\chi\) \(=\) 891.460
Dual form 891.2.n.b.676.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.75181 + 1.94558i) q^{2} +(-0.507392 + 4.82751i) q^{4} +(0.413545 - 0.459289i) q^{5} +(-0.913545 + 0.406737i) q^{7} +(-6.04508 + 4.39201i) q^{8} +1.61803 q^{10} +(-1.84894 + 2.75344i) q^{11} +(-0.230909 + 0.0490813i) q^{13} +(-2.39169 - 1.06485i) q^{14} +(-9.63877 - 2.04878i) q^{16} +(0.354102 + 1.08981i) q^{17} +(-4.73607 + 3.44095i) q^{19} +(2.00739 + 2.22943i) q^{20} +(-8.59602 + 1.22622i) q^{22} +(0.118034 + 0.204441i) q^{23} +(0.482716 + 4.59274i) q^{25} +(-0.500000 - 0.363271i) q^{26} +(-1.50000 - 4.61653i) q^{28} +(5.48127 - 2.44042i) q^{29} +(5.95709 - 1.26622i) q^{31} +(-5.42705 - 9.39993i) q^{32} +(-1.50000 + 2.59808i) q^{34} +(-0.190983 + 0.587785i) q^{35} +(5.04508 + 3.66547i) q^{37} +(-14.9913 - 3.18650i) q^{38} +(-0.482716 + 4.59274i) q^{40} +(-0.215659 - 0.0960175i) q^{41} +(3.35410 - 5.80948i) q^{43} +(-12.3541 - 10.3229i) q^{44} +(-0.190983 + 0.587785i) q^{46} +(-1.05471 - 10.0349i) q^{47} +(-4.01478 + 4.45887i) q^{49} +(-8.08990 + 8.98475i) q^{50} +(-0.119779 - 1.13962i) q^{52} +(0.118034 - 0.363271i) q^{53} +(0.500000 + 1.98787i) q^{55} +(3.73607 - 6.47106i) q^{56} +(14.3502 + 6.38910i) q^{58} +(0.771626 - 7.34153i) q^{59} +(11.3096 + 2.40394i) q^{61} +(12.8992 + 9.37181i) q^{62} +(2.69098 - 8.28199i) q^{64} +(-0.0729490 + 0.126351i) q^{65} +(-0.927051 - 1.60570i) q^{67} +(-5.44076 + 1.15647i) q^{68} +(-1.47815 + 0.658114i) q^{70} +(-3.19098 - 9.82084i) q^{71} +(4.61803 + 3.35520i) q^{73} +(1.70656 + 16.2368i) q^{74} +(-14.2082 - 24.6093i) q^{76} +(0.569171 - 3.26742i) q^{77} +(7.36044 + 8.17459i) q^{79} +(-4.92705 + 3.57971i) q^{80} +(-0.190983 - 0.587785i) q^{82} +(1.43997 + 0.306074i) q^{83} +(0.646976 + 0.288052i) q^{85} +(17.1785 - 3.65141i) q^{86} +(-0.916102 - 24.7653i) q^{88} +8.23607 q^{89} +(0.190983 - 0.138757i) q^{91} +(-1.04683 + 0.466079i) q^{92} +(17.6760 - 19.6312i) q^{94} +(-0.378188 + 3.59821i) q^{95} +(5.25542 + 5.83674i) q^{97} -15.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 9 q^{4} - 3 q^{5} - q^{7} - 26 q^{8} + 4 q^{10} - 11 q^{11} - 7 q^{13} - 4 q^{14} - q^{16} - 24 q^{17} - 20 q^{19} + 3 q^{20} - 4 q^{22} - 8 q^{23} - 6 q^{25} - 4 q^{26} - 12 q^{28} + 6 q^{29}+ \cdots - 72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(650\)
\(\chi(n)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.75181 + 1.94558i 1.23871 + 1.37573i 0.900613 + 0.434622i \(0.143118\pi\)
0.338101 + 0.941110i \(0.390215\pi\)
\(3\) 0 0
\(4\) −0.507392 + 4.82751i −0.253696 + 2.41376i
\(5\) 0.413545 − 0.459289i 0.184943 − 0.205400i −0.643543 − 0.765410i \(-0.722536\pi\)
0.828486 + 0.560010i \(0.189203\pi\)
\(6\) 0 0
\(7\) −0.913545 + 0.406737i −0.345288 + 0.153732i −0.572051 − 0.820218i \(-0.693852\pi\)
0.226764 + 0.973950i \(0.427186\pi\)
\(8\) −6.04508 + 4.39201i −2.13726 + 1.55281i
\(9\) 0 0
\(10\) 1.61803 0.511667
\(11\) −1.84894 + 2.75344i −0.557477 + 0.830192i
\(12\) 0 0
\(13\) −0.230909 + 0.0490813i −0.0640427 + 0.0136127i −0.239822 − 0.970817i \(-0.577089\pi\)
0.175779 + 0.984430i \(0.443756\pi\)
\(14\) −2.39169 − 1.06485i −0.639207 − 0.284593i
\(15\) 0 0
\(16\) −9.63877 − 2.04878i −2.40969 − 0.512196i
\(17\) 0.354102 + 1.08981i 0.0858823 + 0.264319i 0.984770 − 0.173860i \(-0.0556239\pi\)
−0.898888 + 0.438178i \(0.855624\pi\)
\(18\) 0 0
\(19\) −4.73607 + 3.44095i −1.08653 + 0.789409i −0.978810 − 0.204772i \(-0.934355\pi\)
−0.107719 + 0.994181i \(0.534355\pi\)
\(20\) 2.00739 + 2.22943i 0.448866 + 0.498517i
\(21\) 0 0
\(22\) −8.59602 + 1.22622i −1.83268 + 0.261432i
\(23\) 0.118034 + 0.204441i 0.0246118 + 0.0426289i 0.878069 − 0.478534i \(-0.158832\pi\)
−0.853457 + 0.521163i \(0.825498\pi\)
\(24\) 0 0
\(25\) 0.482716 + 4.59274i 0.0965432 + 0.918547i
\(26\) −0.500000 − 0.363271i −0.0980581 − 0.0712434i
\(27\) 0 0
\(28\) −1.50000 − 4.61653i −0.283473 − 0.872441i
\(29\) 5.48127 − 2.44042i 1.01785 − 0.453175i 0.171147 − 0.985246i \(-0.445253\pi\)
0.846700 + 0.532071i \(0.178586\pi\)
\(30\) 0 0
\(31\) 5.95709 − 1.26622i 1.06992 − 0.227419i 0.360901 − 0.932604i \(-0.382469\pi\)
0.709023 + 0.705185i \(0.249136\pi\)
\(32\) −5.42705 − 9.39993i −0.959376 − 1.66169i
\(33\) 0 0
\(34\) −1.50000 + 2.59808i −0.257248 + 0.445566i
\(35\) −0.190983 + 0.587785i −0.0322820 + 0.0993538i
\(36\) 0 0
\(37\) 5.04508 + 3.66547i 0.829407 + 0.602599i 0.919391 − 0.393344i \(-0.128682\pi\)
−0.0899846 + 0.995943i \(0.528682\pi\)
\(38\) −14.9913 − 3.18650i −2.43191 − 0.516919i
\(39\) 0 0
\(40\) −0.482716 + 4.59274i −0.0763241 + 0.726175i
\(41\) −0.215659 − 0.0960175i −0.0336803 − 0.0149954i 0.389827 − 0.920888i \(-0.372535\pi\)
−0.423508 + 0.905893i \(0.639201\pi\)
\(42\) 0 0
\(43\) 3.35410 − 5.80948i 0.511496 − 0.885937i −0.488415 − 0.872611i \(-0.662425\pi\)
0.999911 − 0.0133254i \(-0.00424174\pi\)
\(44\) −12.3541 − 10.3229i −1.86245 − 1.55623i
\(45\) 0 0
\(46\) −0.190983 + 0.587785i −0.0281589 + 0.0866642i
\(47\) −1.05471 − 10.0349i −0.153845 − 1.46374i −0.750302 − 0.661095i \(-0.770092\pi\)
0.596457 − 0.802645i \(-0.296575\pi\)
\(48\) 0 0
\(49\) −4.01478 + 4.45887i −0.573541 + 0.636981i
\(50\) −8.08990 + 8.98475i −1.14409 + 1.27064i
\(51\) 0 0
\(52\) −0.119779 − 1.13962i −0.0166104 − 0.158037i
\(53\) 0.118034 − 0.363271i 0.0162132 − 0.0498991i −0.942623 − 0.333860i \(-0.891649\pi\)
0.958836 + 0.283961i \(0.0916486\pi\)
\(54\) 0 0
\(55\) 0.500000 + 1.98787i 0.0674200 + 0.268044i
\(56\) 3.73607 − 6.47106i 0.499253 − 0.864732i
\(57\) 0 0
\(58\) 14.3502 + 6.38910i 1.88427 + 0.838930i
\(59\) 0.771626 − 7.34153i 0.100457 − 0.955785i −0.821948 − 0.569562i \(-0.807113\pi\)
0.922405 − 0.386223i \(-0.126221\pi\)
\(60\) 0 0
\(61\) 11.3096 + 2.40394i 1.44805 + 0.307793i 0.863823 − 0.503796i \(-0.168064\pi\)
0.584229 + 0.811589i \(0.301397\pi\)
\(62\) 12.8992 + 9.37181i 1.63820 + 1.19022i
\(63\) 0 0
\(64\) 2.69098 − 8.28199i 0.336373 − 1.03525i
\(65\) −0.0729490 + 0.126351i −0.00904821 + 0.0156720i
\(66\) 0 0
\(67\) −0.927051 − 1.60570i −0.113257 − 0.196167i 0.803824 − 0.594867i \(-0.202795\pi\)
−0.917082 + 0.398699i \(0.869462\pi\)
\(68\) −5.44076 + 1.15647i −0.659789 + 0.140242i
\(69\) 0 0
\(70\) −1.47815 + 0.658114i −0.176672 + 0.0786596i
\(71\) −3.19098 − 9.82084i −0.378700 − 1.16552i −0.940948 − 0.338550i \(-0.890063\pi\)
0.562248 − 0.826968i \(-0.309937\pi\)
\(72\) 0 0
\(73\) 4.61803 + 3.35520i 0.540500 + 0.392696i 0.824271 − 0.566196i \(-0.191585\pi\)
−0.283771 + 0.958892i \(0.591585\pi\)
\(74\) 1.70656 + 16.2368i 0.198383 + 1.88749i
\(75\) 0 0
\(76\) −14.2082 − 24.6093i −1.62979 − 2.82288i
\(77\) 0.569171 − 3.26742i 0.0648630 − 0.372357i
\(78\) 0 0
\(79\) 7.36044 + 8.17459i 0.828114 + 0.919714i 0.997834 − 0.0657787i \(-0.0209531\pi\)
−0.169720 + 0.985492i \(0.554286\pi\)
\(80\) −4.92705 + 3.57971i −0.550861 + 0.400224i
\(81\) 0 0
\(82\) −0.190983 − 0.587785i −0.0210905 − 0.0649100i
\(83\) 1.43997 + 0.306074i 0.158057 + 0.0335960i 0.286260 − 0.958152i \(-0.407588\pi\)
−0.128204 + 0.991748i \(0.540921\pi\)
\(84\) 0 0
\(85\) 0.646976 + 0.288052i 0.0701745 + 0.0312437i
\(86\) 17.1785 − 3.65141i 1.85241 − 0.393742i
\(87\) 0 0
\(88\) −0.916102 − 24.7653i −0.0976568 − 2.63999i
\(89\) 8.23607 0.873021 0.436511 − 0.899699i \(-0.356214\pi\)
0.436511 + 0.899699i \(0.356214\pi\)
\(90\) 0 0
\(91\) 0.190983 − 0.138757i 0.0200205 − 0.0145457i
\(92\) −1.04683 + 0.466079i −0.109140 + 0.0485921i
\(93\) 0 0
\(94\) 17.6760 − 19.6312i 1.82314 − 2.02481i
\(95\) −0.378188 + 3.59821i −0.0388012 + 0.369169i
\(96\) 0 0
\(97\) 5.25542 + 5.83674i 0.533607 + 0.592631i 0.948318 − 0.317323i \(-0.102784\pi\)
−0.414711 + 0.909953i \(0.636117\pi\)
\(98\) −15.7082 −1.58677
\(99\) 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 891.2.n.b.460.1 8
3.2 odd 2 891.2.n.c.460.1 8
9.2 odd 6 33.2.e.b.31.1 yes 4
9.4 even 3 inner 891.2.n.b.757.1 8
9.5 odd 6 891.2.n.c.757.1 8
9.7 even 3 99.2.f.a.64.1 4
11.5 even 5 inner 891.2.n.b.379.1 8
33.5 odd 10 891.2.n.c.379.1 8
36.11 even 6 528.2.y.b.97.1 4
45.2 even 12 825.2.bx.d.724.1 8
45.29 odd 6 825.2.n.c.526.1 4
45.38 even 12 825.2.bx.d.724.2 8
99.2 even 30 363.2.e.b.124.1 4
99.5 odd 30 891.2.n.c.676.1 8
99.7 odd 30 1089.2.a.l.1.1 2
99.16 even 15 99.2.f.a.82.1 4
99.20 odd 30 363.2.e.k.124.1 4
99.29 even 30 363.2.a.i.1.2 2
99.38 odd 30 33.2.e.b.16.1 ✓ 4
99.47 odd 30 363.2.e.k.202.1 4
99.49 even 15 inner 891.2.n.b.676.1 8
99.65 even 6 363.2.e.f.130.1 4
99.70 even 15 1089.2.a.t.1.2 2
99.74 even 30 363.2.e.b.202.1 4
99.83 even 30 363.2.e.f.148.1 4
99.92 odd 30 363.2.a.d.1.1 2
396.191 even 30 5808.2.a.cj.1.1 2
396.227 odd 30 5808.2.a.ci.1.1 2
396.335 even 30 528.2.y.b.49.1 4
495.29 even 30 9075.2.a.u.1.1 2
495.38 even 60 825.2.bx.d.49.1 8
495.137 even 60 825.2.bx.d.49.2 8
495.389 odd 30 9075.2.a.cb.1.2 2
495.434 odd 30 825.2.n.c.676.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.16.1 ✓ 4 99.38 odd 30
33.2.e.b.31.1 yes 4 9.2 odd 6
99.2.f.a.64.1 4 9.7 even 3
99.2.f.a.82.1 4 99.16 even 15
363.2.a.d.1.1 2 99.92 odd 30
363.2.a.i.1.2 2 99.29 even 30
363.2.e.b.124.1 4 99.2 even 30
363.2.e.b.202.1 4 99.74 even 30
363.2.e.f.130.1 4 99.65 even 6
363.2.e.f.148.1 4 99.83 even 30
363.2.e.k.124.1 4 99.20 odd 30
363.2.e.k.202.1 4 99.47 odd 30
528.2.y.b.49.1 4 396.335 even 30
528.2.y.b.97.1 4 36.11 even 6
825.2.n.c.526.1 4 45.29 odd 6
825.2.n.c.676.1 4 495.434 odd 30
825.2.bx.d.49.1 8 495.38 even 60
825.2.bx.d.49.2 8 495.137 even 60
825.2.bx.d.724.1 8 45.2 even 12
825.2.bx.d.724.2 8 45.38 even 12
891.2.n.b.379.1 8 11.5 even 5 inner
891.2.n.b.460.1 8 1.1 even 1 trivial
891.2.n.b.676.1 8 99.49 even 15 inner
891.2.n.b.757.1 8 9.4 even 3 inner
891.2.n.c.379.1 8 33.5 odd 10
891.2.n.c.460.1 8 3.2 odd 2
891.2.n.c.676.1 8 99.5 odd 30
891.2.n.c.757.1 8 9.5 odd 6
1089.2.a.l.1.1 2 99.7 odd 30
1089.2.a.t.1.2 2 99.70 even 15
5808.2.a.ci.1.1 2 396.227 odd 30
5808.2.a.cj.1.1 2 396.191 even 30
9075.2.a.u.1.1 2 495.29 even 30
9075.2.a.cb.1.2 2 495.389 odd 30