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Properties

Label 261.2.o.a.154.1
Level $261$
Weight $2$
Character 261.154
Analytic conductor $2.084$
Analytic rank $0$
Dimension $12$
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] = [261,2,Mod(64,261)] 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("261.64"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(261, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.o (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] 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: \(2.08409549276\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: 12.0.7877952219361.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 154.1
Root \(0.911180 + 1.08155i\) of defining polynomial
Character \(\chi\) \(=\) 261.154
Dual form 261.2.o.a.100.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.0741982 - 0.154074i) q^{2} +(1.22875 - 1.54080i) q^{4} +(2.54740 - 1.22676i) q^{5} +(-1.82432 - 2.28763i) q^{7} +(-0.662012 - 0.151100i) q^{8} +(-0.378025 - 0.301465i) q^{10} +(-3.89257 + 0.888454i) q^{11} +(0.625512 + 2.74055i) q^{13} +(-0.217103 + 0.450819i) q^{14} +(-0.851229 - 3.72948i) q^{16} +0.482650i q^{17} +(2.38432 + 1.90144i) q^{19} +(1.23991 - 5.43242i) q^{20} +(0.425710 + 0.533823i) q^{22} +(4.96829 + 2.39260i) q^{23} +(1.86686 - 2.34097i) q^{25} +(0.375836 - 0.299719i) q^{26} -5.76640 q^{28} +(4.99718 - 2.00704i) q^{29} +(-1.67239 - 3.47275i) q^{31} +(-1.57324 + 1.25462i) q^{32} +(0.0743639 - 0.0358118i) q^{34} +(-7.45366 - 3.58950i) q^{35} +(11.2541 + 2.56868i) q^{37} +(0.116049 - 0.508446i) q^{38} +(-1.87177 + 0.427220i) q^{40} +5.10756i q^{41} +(-3.56577 + 7.40439i) q^{43} +(-3.41405 + 7.08936i) q^{44} -0.943011i q^{46} +(-2.32767 + 0.531276i) q^{47} +(-0.347443 + 1.52225i) q^{49} +(-0.499200 - 0.113939i) q^{50} +(4.99123 + 2.40365i) q^{52} +(-0.401975 + 0.193581i) q^{53} +(-8.82602 + 7.03852i) q^{55} +(0.862063 + 1.79009i) q^{56} +(-0.680015 - 0.621017i) q^{58} -1.24537 q^{59} +(-6.71717 + 5.35677i) q^{61} +(-0.410973 + 0.515344i) q^{62} +(-6.58308 - 3.17024i) q^{64} +(4.95544 + 6.21392i) q^{65} +(-0.210269 + 0.921249i) q^{67} +(0.743667 + 0.593055i) q^{68} +1.41475i q^{70} +(-1.33021 - 5.82802i) q^{71} +(0.209705 - 0.435458i) q^{73} +(-0.439268 - 1.92456i) q^{74} +(5.85946 - 1.33738i) q^{76} +(9.13376 + 7.28393i) q^{77} +(1.80484 + 0.411944i) q^{79} +(-6.74361 - 8.45623i) q^{80} +(0.786943 - 0.378972i) q^{82} +(-2.71744 + 3.40756i) q^{83} +(0.592098 + 1.22950i) q^{85} +1.40540 q^{86} +2.71117 q^{88} +(-6.75011 - 14.0167i) q^{89} +(5.12822 - 6.43058i) q^{91} +(9.79128 - 4.71523i) q^{92} +(0.254565 + 0.319215i) q^{94} +(8.40645 + 1.91872i) q^{95} +(1.88941 + 1.50675i) q^{97} +(0.260319 - 0.0594161i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 7 q^{2} - q^{4} + q^{5} - 11 q^{7} - 14 q^{8} - 7 q^{10} - 7 q^{11} + 9 q^{13} + 7 q^{14} + 9 q^{16} - 7 q^{19} + 11 q^{20} - 4 q^{22} + 5 q^{23} + 13 q^{25} + 21 q^{26} + 12 q^{28} + 15 q^{29}+ \cdots + 42 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).

\(n\) \(118\) \(146\)
\(\chi(n)\) \(e\left(\frac{5}{14}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0741982 − 0.154074i −0.0524661 − 0.108947i 0.873088 − 0.487563i \(-0.162114\pi\)
−0.925554 + 0.378617i \(0.876400\pi\)
\(3\) 0 0
\(4\) 1.22875 − 1.54080i 0.614373 − 0.770399i
\(5\) 2.54740 − 1.22676i 1.13923 − 0.548626i 0.233450 − 0.972369i \(-0.424998\pi\)
0.905782 + 0.423743i \(0.139284\pi\)
\(6\) 0 0
\(7\) −1.82432 − 2.28763i −0.689529 − 0.864642i 0.306664 − 0.951818i \(-0.400787\pi\)
−0.996193 + 0.0871757i \(0.972216\pi\)
\(8\) −0.662012 − 0.151100i −0.234057 − 0.0534219i
\(9\) 0 0
\(10\) −0.378025 − 0.301465i −0.119542 − 0.0953317i
\(11\) −3.89257 + 0.888454i −1.17365 + 0.267879i −0.764523 − 0.644596i \(-0.777025\pi\)
−0.409132 + 0.912475i \(0.634168\pi\)
\(12\) 0 0
\(13\) 0.625512 + 2.74055i 0.173486 + 0.760091i 0.984546 + 0.175128i \(0.0560339\pi\)
−0.811060 + 0.584963i \(0.801109\pi\)
\(14\) −0.217103 + 0.450819i −0.0580232 + 0.120486i
\(15\) 0 0
\(16\) −0.851229 − 3.72948i −0.212807 − 0.932370i
\(17\) 0.482650i 0.117060i 0.998286 + 0.0585299i \(0.0186413\pi\)
−0.998286 + 0.0585299i \(0.981359\pi\)
\(18\) 0 0
\(19\) 2.38432 + 1.90144i 0.547002 + 0.436219i 0.857596 − 0.514323i \(-0.171957\pi\)
−0.310595 + 0.950542i \(0.600528\pi\)
\(20\) 1.23991 − 5.43242i 0.277253 − 1.21473i
\(21\) 0 0
\(22\) 0.425710 + 0.533823i 0.0907616 + 0.113811i
\(23\) 4.96829 + 2.39260i 1.03596 + 0.498892i 0.872989 − 0.487739i \(-0.162178\pi\)
0.162970 + 0.986631i \(0.447893\pi\)
\(24\) 0 0
\(25\) 1.86686 − 2.34097i 0.373372 − 0.468193i
\(26\) 0.375836 − 0.299719i 0.0737075 − 0.0587797i
\(27\) 0 0
\(28\) −5.76640 −1.08975
\(29\) 4.99718 − 2.00704i 0.927953 − 0.372698i
\(30\) 0 0
\(31\) −1.67239 − 3.47275i −0.300370 − 0.623724i 0.695088 − 0.718924i \(-0.255365\pi\)
−0.995458 + 0.0952000i \(0.969651\pi\)
\(32\) −1.57324 + 1.25462i −0.278112 + 0.221787i
\(33\) 0 0
\(34\) 0.0743639 − 0.0358118i 0.0127533 − 0.00614167i
\(35\) −7.45366 − 3.58950i −1.25990 − 0.606735i
\(36\) 0 0
\(37\) 11.2541 + 2.56868i 1.85016 + 0.422288i 0.995323 − 0.0966033i \(-0.0307978\pi\)
0.854840 + 0.518891i \(0.173655\pi\)
\(38\) 0.116049 − 0.508446i 0.0188257 − 0.0824808i
\(39\) 0 0
\(40\) −1.87177 + 0.427220i −0.295954 + 0.0675495i
\(41\) 5.10756i 0.797667i 0.917023 + 0.398833i \(0.130585\pi\)
−0.917023 + 0.398833i \(0.869415\pi\)
\(42\) 0 0
\(43\) −3.56577 + 7.40439i −0.543775 + 1.12916i 0.430248 + 0.902711i \(0.358426\pi\)
−0.974023 + 0.226449i \(0.927288\pi\)
\(44\) −3.41405 + 7.08936i −0.514688 + 1.06876i
\(45\) 0 0
\(46\) − 0.943011i − 0.139039i
\(47\) −2.32767 + 0.531276i −0.339526 + 0.0774946i −0.388885 − 0.921287i \(-0.627139\pi\)
0.0493584 + 0.998781i \(0.484282\pi\)
\(48\) 0 0
\(49\) −0.347443 + 1.52225i −0.0496347 + 0.217464i
\(50\) −0.499200 − 0.113939i −0.0705975 − 0.0161134i
\(51\) 0 0
\(52\) 4.99123 + 2.40365i 0.692159 + 0.333326i
\(53\) −0.401975 + 0.193581i −0.0552156 + 0.0265904i −0.461288 − 0.887251i \(-0.652612\pi\)
0.406072 + 0.913841i \(0.366898\pi\)
\(54\) 0 0
\(55\) −8.82602 + 7.03852i −1.19010 + 0.949074i
\(56\) 0.862063 + 1.79009i 0.115198 + 0.239211i
\(57\) 0 0
\(58\) −0.680015 − 0.621017i −0.0892904 − 0.0815436i
\(59\) −1.24537 −0.162133 −0.0810664 − 0.996709i \(-0.525833\pi\)
−0.0810664 + 0.996709i \(0.525833\pi\)
\(60\) 0 0
\(61\) −6.71717 + 5.35677i −0.860046 + 0.685864i −0.950731 − 0.310016i \(-0.899666\pi\)
0.0906856 + 0.995880i \(0.471094\pi\)
\(62\) −0.410973 + 0.515344i −0.0521936 + 0.0654487i
\(63\) 0 0
\(64\) −6.58308 − 3.17024i −0.822885 − 0.396280i
\(65\) 4.95544 + 6.21392i 0.614646 + 0.770742i
\(66\) 0 0
\(67\) −0.210269 + 0.921249i −0.0256885 + 0.112548i −0.986147 − 0.165876i \(-0.946955\pi\)
0.960458 + 0.278424i \(0.0898121\pi\)
\(68\) 0.743667 + 0.593055i 0.0901829 + 0.0719184i
\(69\) 0 0
\(70\) 1.41475i 0.169095i
\(71\) −1.33021 − 5.82802i −0.157867 − 0.691659i −0.990463 − 0.137779i \(-0.956004\pi\)
0.832597 − 0.553880i \(-0.186853\pi\)
\(72\) 0 0
\(73\) 0.209705 − 0.435458i 0.0245442 − 0.0509665i −0.888336 − 0.459193i \(-0.848138\pi\)
0.912880 + 0.408227i \(0.133853\pi\)
\(74\) −0.439268 − 1.92456i −0.0510639 − 0.223725i
\(75\) 0 0
\(76\) 5.85946 − 1.33738i 0.672126 − 0.153408i
\(77\) 9.13376 + 7.28393i 1.04089 + 0.830081i
\(78\) 0 0
\(79\) 1.80484 + 0.411944i 0.203061 + 0.0463473i 0.322841 − 0.946453i \(-0.395362\pi\)
−0.119781 + 0.992800i \(0.538219\pi\)
\(80\) −6.74361 − 8.45623i −0.753959 − 0.945435i
\(81\) 0 0
\(82\) 0.786943 − 0.378972i 0.0869033 − 0.0418504i
\(83\) −2.71744 + 3.40756i −0.298277 + 0.374028i −0.908274 − 0.418376i \(-0.862599\pi\)
0.609996 + 0.792404i \(0.291171\pi\)
\(84\) 0 0
\(85\) 0.592098 + 1.22950i 0.0642220 + 0.133358i
\(86\) 1.40540 0.151548
\(87\) 0 0
\(88\) 2.71117 0.289012
\(89\) −6.75011 − 14.0167i −0.715510 − 1.48577i −0.867525 − 0.497394i \(-0.834290\pi\)
0.152015 − 0.988378i \(-0.451424\pi\)
\(90\) 0 0
\(91\) 5.12822 − 6.43058i 0.537583 − 0.674108i
\(92\) 9.79128 − 4.71523i 1.02081 − 0.491597i
\(93\) 0 0
\(94\) 0.254565 + 0.319215i 0.0262564 + 0.0329245i
\(95\) 8.40645 + 1.91872i 0.862483 + 0.196856i
\(96\) 0 0
\(97\) 1.88941 + 1.50675i 0.191840 + 0.152988i 0.714700 − 0.699431i \(-0.246563\pi\)
−0.522859 + 0.852419i \(0.675135\pi\)
\(98\) 0.260319 − 0.0594161i 0.0262962 − 0.00600193i
\(99\) 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 261.2.o.a.154.1 12
3.2 odd 2 29.2.e.a.9.2 ✓ 12
12.11 even 2 464.2.y.d.241.2 12
15.2 even 4 725.2.p.a.299.2 24
15.8 even 4 725.2.p.a.299.3 24
15.14 odd 2 725.2.q.a.676.1 12
29.10 odd 28 7569.2.a.bp.1.6 12
29.13 even 14 inner 261.2.o.a.100.1 12
29.19 odd 28 7569.2.a.bp.1.7 12
87.2 even 28 841.2.d.k.645.3 24
87.5 odd 14 841.2.e.e.196.2 12
87.8 even 28 841.2.d.k.605.2 24
87.11 even 28 841.2.d.m.190.3 24
87.14 even 28 841.2.d.l.778.2 24
87.17 even 4 841.2.d.m.571.3 24
87.20 odd 14 841.2.e.e.236.2 12
87.23 odd 14 841.2.e.a.63.2 12
87.26 even 28 841.2.d.l.574.3 24
87.32 even 28 841.2.d.l.574.2 24
87.35 odd 14 841.2.e.h.63.1 12
87.38 odd 14 841.2.e.f.236.1 12
87.41 even 4 841.2.d.m.571.2 24
87.44 even 28 841.2.d.l.778.3 24
87.47 even 28 841.2.d.m.190.2 24
87.50 even 28 841.2.d.k.605.3 24
87.53 odd 14 841.2.e.f.196.1 12
87.56 even 28 841.2.d.k.645.2 24
87.62 odd 14 841.2.b.e.840.7 12
87.65 odd 14 841.2.e.h.267.1 12
87.68 even 28 841.2.a.k.1.7 12
87.71 odd 14 29.2.e.a.13.2 yes 12
87.74 odd 14 841.2.e.i.651.1 12
87.77 even 28 841.2.a.k.1.6 12
87.80 odd 14 841.2.e.a.267.2 12
87.83 odd 14 841.2.b.e.840.6 12
87.86 odd 2 841.2.e.i.270.1 12
348.71 even 14 464.2.y.d.129.2 12
435.158 even 28 725.2.p.a.274.2 24
435.332 even 28 725.2.p.a.274.3 24
435.419 odd 14 725.2.q.a.651.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.2.e.a.9.2 ✓ 12 3.2 odd 2
29.2.e.a.13.2 yes 12 87.71 odd 14
261.2.o.a.100.1 12 29.13 even 14 inner
261.2.o.a.154.1 12 1.1 even 1 trivial
464.2.y.d.129.2 12 348.71 even 14
464.2.y.d.241.2 12 12.11 even 2
725.2.p.a.274.2 24 435.158 even 28
725.2.p.a.274.3 24 435.332 even 28
725.2.p.a.299.2 24 15.2 even 4
725.2.p.a.299.3 24 15.8 even 4
725.2.q.a.651.1 12 435.419 odd 14
725.2.q.a.676.1 12 15.14 odd 2
841.2.a.k.1.6 12 87.77 even 28
841.2.a.k.1.7 12 87.68 even 28
841.2.b.e.840.6 12 87.83 odd 14
841.2.b.e.840.7 12 87.62 odd 14
841.2.d.k.605.2 24 87.8 even 28
841.2.d.k.605.3 24 87.50 even 28
841.2.d.k.645.2 24 87.56 even 28
841.2.d.k.645.3 24 87.2 even 28
841.2.d.l.574.2 24 87.32 even 28
841.2.d.l.574.3 24 87.26 even 28
841.2.d.l.778.2 24 87.14 even 28
841.2.d.l.778.3 24 87.44 even 28
841.2.d.m.190.2 24 87.47 even 28
841.2.d.m.190.3 24 87.11 even 28
841.2.d.m.571.2 24 87.41 even 4
841.2.d.m.571.3 24 87.17 even 4
841.2.e.a.63.2 12 87.23 odd 14
841.2.e.a.267.2 12 87.80 odd 14
841.2.e.e.196.2 12 87.5 odd 14
841.2.e.e.236.2 12 87.20 odd 14
841.2.e.f.196.1 12 87.53 odd 14
841.2.e.f.236.1 12 87.38 odd 14
841.2.e.h.63.1 12 87.35 odd 14
841.2.e.h.267.1 12 87.65 odd 14
841.2.e.i.270.1 12 87.86 odd 2
841.2.e.i.651.1 12 87.74 odd 14
7569.2.a.bp.1.6 12 29.10 odd 28
7569.2.a.bp.1.7 12 29.19 odd 28