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Properties

Label 261.2.k.c.136.1
Level $261$
Weight $2$
Character 261.136
Analytic conductor $2.084$
Analytic rank $0$
Dimension $18$
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(82,261)] 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("261.82"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.k (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.08409549276\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} + 18 x^{16} - 37 x^{15} + 71 x^{14} - 83 x^{13} + 225 x^{12} - 237 x^{11} + 485 x^{10} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 136.1
Root \(-1.05678 + 1.32516i\) of defining polynomial
Character \(\chi\) \(=\) 261.136
Dual form 261.2.k.c.190.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.626118 + 0.301523i) q^{2} +(-0.945872 + 1.18609i) q^{4} +(-1.81798 + 0.875492i) q^{5} +(-1.49319 - 1.87240i) q^{7} +(0.543873 - 2.38286i) q^{8} +(0.874288 - 1.09632i) q^{10} +(-0.213150 - 0.933871i) q^{11} +(-1.45377 - 6.36940i) q^{13} +(1.49948 + 0.722111i) q^{14} +(-0.297197 - 1.30211i) q^{16} +3.81642 q^{17} +(-2.69251 + 3.37631i) q^{19} +(0.681165 - 2.98438i) q^{20} +(0.415040 + 0.520444i) q^{22} +(-4.85446 - 2.33778i) q^{23} +(-0.578892 + 0.725907i) q^{25} +(2.83075 + 3.54965i) q^{26} +3.63318 q^{28} +(-5.32202 - 0.822256i) q^{29} +(-2.46382 + 1.18651i) q^{31} +(3.62649 + 4.54747i) q^{32} +(-2.38953 + 1.15074i) q^{34} +(4.35384 + 2.09670i) q^{35} +(0.414041 - 1.81403i) q^{37} +(0.667799 - 2.92582i) q^{38} +(1.09743 + 4.80814i) q^{40} -11.8282 q^{41} +(3.33752 + 1.60727i) q^{43} +(1.30926 + 0.630508i) q^{44} +3.74436 q^{46} +(-0.719344 - 3.15165i) q^{47} +(0.281385 - 1.23283i) q^{49} +(0.143577 - 0.629053i) q^{50} +(8.92974 + 4.30034i) q^{52} +(2.04315 - 0.983927i) q^{53} +(1.20510 + 1.51115i) q^{55} +(-5.27376 + 2.53971i) q^{56} +(3.58014 - 1.08988i) q^{58} +9.30726 q^{59} +(1.95684 + 2.45380i) q^{61} +(1.18488 - 1.48579i) q^{62} +(-1.23512 - 0.594802i) q^{64} +(8.21929 + 10.3067i) q^{65} +(-2.69277 + 11.7978i) q^{67} +(-3.60984 + 4.52660i) q^{68} -3.35822 q^{70} +(-1.02436 - 4.48801i) q^{71} +(-8.19102 - 3.94459i) q^{73} +(0.287733 + 1.26064i) q^{74} +(-1.45781 - 6.38710i) q^{76} +(-1.43030 + 1.79354i) q^{77} +(-0.954127 + 4.18030i) q^{79} +(1.68028 + 2.10701i) q^{80} +(7.40582 - 3.56646i) q^{82} +(-10.0888 + 12.6510i) q^{83} +(-6.93816 + 3.34124i) q^{85} -2.57431 q^{86} -2.34121 q^{88} +(13.9137 - 6.70047i) q^{89} +(-9.75529 + 12.2327i) q^{91} +(7.36451 - 3.54656i) q^{92} +(1.40069 + 1.75641i) q^{94} +(1.93900 - 8.49532i) q^{95} +(9.82817 - 12.3241i) q^{97} +(0.195546 + 0.856741i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{2} - 6 q^{4} + q^{5} - 4 q^{7} + 15 q^{8} - 14 q^{10} - 26 q^{11} + 9 q^{13} + 10 q^{14} - 14 q^{16} - 4 q^{17} - 10 q^{19} + q^{20} - 8 q^{22} + 8 q^{23} + 16 q^{25} - 5 q^{26} + 80 q^{28}+ \cdots - 31 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(118\) \(146\)
\(\chi(n)\) \(e\left(\frac{6}{7}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.626118 + 0.301523i −0.442732 + 0.213209i −0.641950 − 0.766747i \(-0.721874\pi\)
0.199218 + 0.979955i \(0.436160\pi\)
\(3\) 0 0
\(4\) −0.945872 + 1.18609i −0.472936 + 0.593043i
\(5\) −1.81798 + 0.875492i −0.813024 + 0.391532i −0.793721 − 0.608282i \(-0.791859\pi\)
−0.0193032 + 0.999814i \(0.506145\pi\)
\(6\) 0 0
\(7\) −1.49319 − 1.87240i −0.564371 − 0.707699i 0.414988 − 0.909827i \(-0.363786\pi\)
−0.979359 + 0.202128i \(0.935214\pi\)
\(8\) 0.543873 − 2.38286i 0.192288 − 0.842469i
\(9\) 0 0
\(10\) 0.874288 − 1.09632i 0.276474 − 0.346688i
\(11\) −0.213150 − 0.933871i −0.0642671 − 0.281573i 0.932576 − 0.360975i \(-0.117556\pi\)
−0.996843 + 0.0794022i \(0.974699\pi\)
\(12\) 0 0
\(13\) −1.45377 − 6.36940i −0.403205 − 1.76655i −0.614283 − 0.789086i \(-0.710555\pi\)
0.211078 − 0.977469i \(-0.432302\pi\)
\(14\) 1.49948 + 0.722111i 0.400753 + 0.192992i
\(15\) 0 0
\(16\) −0.297197 − 1.30211i −0.0742994 − 0.325527i
\(17\) 3.81642 0.925617 0.462808 − 0.886458i \(-0.346842\pi\)
0.462808 + 0.886458i \(0.346842\pi\)
\(18\) 0 0
\(19\) −2.69251 + 3.37631i −0.617705 + 0.774578i −0.988020 − 0.154329i \(-0.950678\pi\)
0.370315 + 0.928906i \(0.379250\pi\)
\(20\) 0.681165 − 2.98438i 0.152313 − 0.667328i
\(21\) 0 0
\(22\) 0.415040 + 0.520444i 0.0884868 + 0.110959i
\(23\) −4.85446 − 2.33778i −1.01222 − 0.487462i −0.147155 − 0.989113i \(-0.547012\pi\)
−0.865070 + 0.501652i \(0.832726\pi\)
\(24\) 0 0
\(25\) −0.578892 + 0.725907i −0.115778 + 0.145181i
\(26\) 2.83075 + 3.54965i 0.555156 + 0.696144i
\(27\) 0 0
\(28\) 3.63318 0.686607
\(29\) −5.32202 − 0.822256i −0.988274 − 0.152689i
\(30\) 0 0
\(31\) −2.46382 + 1.18651i −0.442515 + 0.213104i −0.641855 − 0.766826i \(-0.721835\pi\)
0.199339 + 0.979931i \(0.436120\pi\)
\(32\) 3.62649 + 4.54747i 0.641079 + 0.803887i
\(33\) 0 0
\(34\) −2.38953 + 1.15074i −0.409800 + 0.197349i
\(35\) 4.35384 + 2.09670i 0.735934 + 0.354407i
\(36\) 0 0
\(37\) 0.414041 − 1.81403i 0.0680679 − 0.298225i −0.929423 − 0.369015i \(-0.879695\pi\)
0.997491 + 0.0707903i \(0.0225521\pi\)
\(38\) 0.667799 − 2.92582i 0.108331 − 0.474631i
\(39\) 0 0
\(40\) 1.09743 + 4.80814i 0.173519 + 0.760234i
\(41\) −11.8282 −1.84725 −0.923624 − 0.383300i \(-0.874788\pi\)
−0.923624 + 0.383300i \(0.874788\pi\)
\(42\) 0 0
\(43\) 3.33752 + 1.60727i 0.508967 + 0.245106i 0.670700 − 0.741729i \(-0.265994\pi\)
−0.161733 + 0.986835i \(0.551708\pi\)
\(44\) 1.30926 + 0.630508i 0.197379 + 0.0950526i
\(45\) 0 0
\(46\) 3.74436 0.552075
\(47\) −0.719344 − 3.15165i −0.104927 − 0.459716i −0.999907 − 0.0136100i \(-0.995668\pi\)
0.894980 − 0.446106i \(-0.147189\pi\)
\(48\) 0 0
\(49\) 0.281385 − 1.23283i 0.0401979 − 0.176119i
\(50\) 0.143577 − 0.629053i 0.0203049 − 0.0889615i
\(51\) 0 0
\(52\) 8.92974 + 4.30034i 1.23833 + 0.596350i
\(53\) 2.04315 − 0.983927i 0.280648 − 0.135153i −0.288265 − 0.957551i \(-0.593078\pi\)
0.568913 + 0.822398i \(0.307364\pi\)
\(54\) 0 0
\(55\) 1.20510 + 1.51115i 0.162495 + 0.203763i
\(56\) −5.27376 + 2.53971i −0.704736 + 0.339383i
\(57\) 0 0
\(58\) 3.58014 − 1.08988i 0.470096 − 0.143108i
\(59\) 9.30726 1.21170 0.605851 − 0.795578i \(-0.292833\pi\)
0.605851 + 0.795578i \(0.292833\pi\)
\(60\) 0 0
\(61\) 1.95684 + 2.45380i 0.250547 + 0.314176i 0.891161 − 0.453687i \(-0.149891\pi\)
−0.640614 + 0.767863i \(0.721320\pi\)
\(62\) 1.18488 − 1.48579i 0.150480 − 0.188696i
\(63\) 0 0
\(64\) −1.23512 − 0.594802i −0.154390 − 0.0743503i
\(65\) 8.21929 + 10.3067i 1.01948 + 1.27838i
\(66\) 0 0
\(67\) −2.69277 + 11.7978i −0.328975 + 1.44133i 0.492113 + 0.870531i \(0.336225\pi\)
−0.821088 + 0.570801i \(0.806633\pi\)
\(68\) −3.60984 + 4.52660i −0.437757 + 0.548930i
\(69\) 0 0
\(70\) −3.35822 −0.401384
\(71\) −1.02436 − 4.48801i −0.121569 − 0.532629i −0.998634 − 0.0522567i \(-0.983359\pi\)
0.877065 − 0.480373i \(-0.159499\pi\)
\(72\) 0 0
\(73\) −8.19102 − 3.94459i −0.958687 − 0.461679i −0.111963 − 0.993712i \(-0.535714\pi\)
−0.846723 + 0.532033i \(0.821428\pi\)
\(74\) 0.287733 + 1.26064i 0.0334483 + 0.146547i
\(75\) 0 0
\(76\) −1.45781 − 6.38710i −0.167223 − 0.732651i
\(77\) −1.43030 + 1.79354i −0.162998 + 0.204393i
\(78\) 0 0
\(79\) −0.954127 + 4.18030i −0.107348 + 0.470321i 0.892468 + 0.451111i \(0.148972\pi\)
−0.999815 + 0.0192099i \(0.993885\pi\)
\(80\) 1.68028 + 2.10701i 0.187861 + 0.235571i
\(81\) 0 0
\(82\) 7.40582 − 3.56646i 0.817836 − 0.393849i
\(83\) −10.0888 + 12.6510i −1.10739 + 1.38863i −0.194270 + 0.980948i \(0.562234\pi\)
−0.913124 + 0.407681i \(0.866338\pi\)
\(84\) 0 0
\(85\) −6.93816 + 3.34124i −0.752549 + 0.362408i
\(86\) −2.57431 −0.277595
\(87\) 0 0
\(88\) −2.34121 −0.249574
\(89\) 13.9137 − 6.70047i 1.47485 − 0.710248i 0.488140 − 0.872765i \(-0.337675\pi\)
0.986706 + 0.162517i \(0.0519612\pi\)
\(90\) 0 0
\(91\) −9.75529 + 12.2327i −1.02263 + 1.28234i
\(92\) 7.36451 − 3.54656i 0.767803 − 0.369754i
\(93\) 0 0
\(94\) 1.40069 + 1.75641i 0.144470 + 0.181160i
\(95\) 1.93900 − 8.49532i 0.198937 − 0.871602i
\(96\) 0 0
\(97\) 9.82817 − 12.3241i 0.997899 − 1.25133i 0.0301150 − 0.999546i \(-0.490413\pi\)
0.967785 − 0.251780i \(-0.0810159\pi\)
\(98\) 0.195546 + 0.856741i 0.0197531 + 0.0865439i
\(99\) 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.k.c.136.1 18
3.2 odd 2 87.2.g.a.49.3 yes 18
29.4 even 14 7569.2.a.bm.1.3 9
29.16 even 7 inner 261.2.k.c.190.1 18
29.25 even 7 7569.2.a.bj.1.7 9
87.62 odd 14 2523.2.a.o.1.7 9
87.74 odd 14 87.2.g.a.16.3 ✓ 18
87.83 odd 14 2523.2.a.r.1.3 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.g.a.16.3 ✓ 18 87.74 odd 14
87.2.g.a.49.3 yes 18 3.2 odd 2
261.2.k.c.136.1 18 1.1 even 1 trivial
261.2.k.c.190.1 18 29.16 even 7 inner
2523.2.a.o.1.7 9 87.62 odd 14
2523.2.a.r.1.3 9 87.83 odd 14
7569.2.a.bj.1.7 9 29.25 even 7
7569.2.a.bm.1.3 9 29.4 even 14