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Properties

Label 87.2.g.a.49.3
Level $87$
Weight $2$
Character 87.49
Analytic conductor $0.695$
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] = [87,2,Mod(7,87)] 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("87.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 87.g (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: \(0.694698497585\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 49.3
Root \(-1.05678 + 1.32516i\) of defining polynomial
Character \(\chi\) \(=\) 87.49
Dual form 87.2.g.a.16.3

$q$-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} +(0.623490 + 0.781831i) q^{3} +(-0.945872 + 1.18609i) q^{4} +(1.81798 - 0.875492i) q^{5} +(0.626118 + 0.301523i) q^{6} +(-1.49319 - 1.87240i) q^{7} +(-0.543873 + 2.38286i) q^{8} +(-0.222521 + 0.974928i) q^{9} +(0.874288 - 1.09632i) q^{10} +(0.213150 + 0.933871i) q^{11} -1.51706 q^{12} +(-1.45377 - 6.36940i) q^{13} +(-1.49948 - 0.722111i) q^{14} +(1.81798 + 0.875492i) q^{15} +(-0.297197 - 1.30211i) q^{16} -3.81642 q^{17} +(0.154638 + 0.677515i) q^{18} +(-2.69251 + 3.37631i) q^{19} +(-0.681165 + 2.98438i) q^{20} +(0.532912 - 2.33484i) q^{21} +(0.415040 + 0.520444i) q^{22} +(4.85446 + 2.33778i) q^{23} +(-2.20209 + 1.06047i) q^{24} +(-0.578892 + 0.725907i) q^{25} +(-2.83075 - 3.54965i) q^{26} +(-0.900969 + 0.433884i) q^{27} +3.63318 q^{28} +(5.32202 + 0.822256i) q^{29} +1.40225 q^{30} +(-2.46382 + 1.18651i) q^{31} +(-3.62649 - 4.54747i) q^{32} +(-0.597233 + 0.748906i) q^{33} +(-2.38953 + 1.15074i) q^{34} +(-4.35384 - 2.09670i) q^{35} +(-0.945872 - 1.18609i) q^{36} +(0.414041 - 1.81403i) q^{37} +(-0.667799 + 2.92582i) q^{38} +(4.07339 - 5.10787i) q^{39} +(1.09743 + 4.80814i) q^{40} +11.8282 q^{41} +(-0.370341 - 1.62257i) q^{42} +(3.33752 + 1.60727i) q^{43} +(-1.30926 - 0.630508i) q^{44} +(0.449003 + 1.96721i) q^{45} +3.74436 q^{46} +(0.719344 + 3.15165i) q^{47} +(0.832729 - 1.04421i) q^{48} +(0.281385 - 1.23283i) q^{49} +(-0.143577 + 0.629053i) q^{50} +(-2.37950 - 2.98379i) q^{51} +(8.92974 + 4.30034i) q^{52} +(-2.04315 + 0.983927i) q^{53} +(-0.433287 + 0.543325i) q^{54} +(1.20510 + 1.51115i) q^{55} +(5.27376 - 2.53971i) q^{56} -4.31846 q^{57} +(3.58014 - 1.08988i) q^{58} -9.30726 q^{59} +(-2.75798 + 1.32817i) q^{60} +(1.95684 + 2.45380i) q^{61} +(-1.18488 + 1.48579i) q^{62} +(2.15772 - 1.03910i) q^{63} +(-1.23512 - 0.594802i) q^{64} +(-8.21929 - 10.3067i) q^{65} +(-0.148126 + 0.648983i) q^{66} +(-2.69277 + 11.7978i) q^{67} +(3.60984 - 4.52660i) q^{68} +(1.19895 + 5.25295i) q^{69} -3.35822 q^{70} +(1.02436 + 4.48801i) q^{71} +(-2.20209 - 1.06047i) q^{72} +(-8.19102 - 3.94459i) q^{73} +(-0.287733 - 1.26064i) q^{74} -0.928470 q^{75} +(-1.45781 - 6.38710i) q^{76} +(1.43030 - 1.79354i) q^{77} +(1.01028 - 4.42634i) q^{78} +(-0.954127 + 4.18030i) q^{79} +(-1.68028 - 2.10701i) q^{80} +(-0.900969 - 0.433884i) q^{81} +(7.40582 - 3.56646i) q^{82} +(10.0888 - 12.6510i) q^{83} +(2.26525 + 2.84054i) q^{84} +(-6.93816 + 3.34124i) q^{85} +2.57431 q^{86} +(2.67536 + 4.67359i) q^{87} -2.34121 q^{88} +(-13.9137 + 6.70047i) q^{89} +(0.874288 + 1.09632i) q^{90} +(-9.75529 + 12.2327i) q^{91} +(-7.36451 + 3.54656i) q^{92} +(-2.46382 - 1.18651i) q^{93} +(1.40069 + 1.75641i) q^{94} +(-1.93900 + 8.49532i) q^{95} +(1.29428 - 5.67061i) q^{96} +(9.82817 - 12.3241i) q^{97} +(-0.195546 - 0.856741i) q^{98} -0.957887 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 4 q^{2} - 3 q^{3} - 6 q^{4} - q^{5} - 4 q^{6} - 4 q^{7} - 15 q^{8} - 3 q^{9} - 14 q^{10} + 26 q^{11} + 22 q^{12} + 9 q^{13} - 10 q^{14} - q^{15} - 14 q^{16} + 4 q^{17} - 4 q^{18} - 10 q^{19} - q^{20}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(59\)
\(\chi(n)\) \(e\left(\frac{6}{7}\right)\) \(1\)

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 −0.199218 − 0.979955i \(-0.563840\pi\)
0.641950 + 0.766747i \(0.278126\pi\)
\(3\) 0.623490 + 0.781831i 0.359972 + 0.451391i
\(4\) −0.945872 + 1.18609i −0.472936 + 0.593043i
\(5\) 1.81798 − 0.875492i 0.813024 − 0.391532i 0.0193032 − 0.999814i \(-0.493855\pi\)
0.793721 + 0.608282i \(0.208141\pi\)
\(6\) 0.626118 + 0.301523i 0.255612 + 0.123096i
\(7\) −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
\(9\) −0.222521 + 0.974928i −0.0741736 + 0.324976i
\(10\) 0.874288 − 1.09632i 0.276474 − 0.346688i
\(11\) 0.213150 + 0.933871i 0.0642671 + 0.281573i 0.996843 − 0.0794022i \(-0.0253012\pi\)
−0.932576 + 0.360975i \(0.882444\pi\)
\(12\) −1.51706 −0.437938
\(13\) −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
\(15\) 1.81798 + 0.875492i 0.469400 + 0.226051i
\(16\) −0.297197 − 1.30211i −0.0742994 − 0.325527i
\(17\) −3.81642 −0.925617 −0.462808 − 0.886458i \(-0.653158\pi\)
−0.462808 + 0.886458i \(0.653158\pi\)
\(18\) 0.154638 + 0.677515i 0.0364486 + 0.159692i
\(19\) −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
\(21\) 0.532912 − 2.33484i 0.116291 − 0.509504i
\(22\) 0.415040 + 0.520444i 0.0884868 + 0.110959i
\(23\) 4.85446 + 2.33778i 1.01222 + 0.487462i 0.865070 − 0.501652i \(-0.167274\pi\)
0.147155 + 0.989113i \(0.452988\pi\)
\(24\) −2.20209 + 1.06047i −0.449501 + 0.216468i
\(25\) −0.578892 + 0.725907i −0.115778 + 0.145181i
\(26\) −2.83075 − 3.54965i −0.555156 − 0.696144i
\(27\) −0.900969 + 0.433884i −0.173392 + 0.0835010i
\(28\) 3.63318 0.686607
\(29\) 5.32202 + 0.822256i 0.988274 + 0.152689i
\(30\) 1.40225 0.256015
\(31\) −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
\(33\) −0.597233 + 0.748906i −0.103965 + 0.130368i
\(34\) −2.38953 + 1.15074i −0.409800 + 0.197349i
\(35\) −4.35384 − 2.09670i −0.735934 − 0.354407i
\(36\) −0.945872 − 1.18609i −0.157645 − 0.197681i
\(37\) 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
\(39\) 4.07339 − 5.10787i 0.652264 − 0.817913i
\(40\) 1.09743 + 4.80814i 0.173519 + 0.760234i
\(41\) 11.8282 1.84725 0.923624 − 0.383300i \(-0.125212\pi\)
0.923624 + 0.383300i \(0.125212\pi\)
\(42\) −0.370341 − 1.62257i −0.0571448 − 0.250368i
\(43\) 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
\(45\) 0.449003 + 1.96721i 0.0669335 + 0.293255i
\(46\) 3.74436 0.552075
\(47\) 0.719344 + 3.15165i 0.104927 + 0.459716i 0.999907 + 0.0136100i \(0.00433232\pi\)
−0.894980 + 0.446106i \(0.852811\pi\)
\(48\) 0.832729 − 1.04421i 0.120194 − 0.150719i
\(49\) 0.281385 − 1.23283i 0.0401979 − 0.176119i
\(50\) −0.143577 + 0.629053i −0.0203049 + 0.0889615i
\(51\) −2.37950 − 2.98379i −0.333196 − 0.417815i
\(52\) 8.92974 + 4.30034i 1.23833 + 0.596350i
\(53\) −2.04315 + 0.983927i −0.280648 + 0.135153i −0.568913 − 0.822398i \(-0.692636\pi\)
0.288265 + 0.957551i \(0.406922\pi\)
\(54\) −0.433287 + 0.543325i −0.0589629 + 0.0739371i
\(55\) 1.20510 + 1.51115i 0.162495 + 0.203763i
\(56\) 5.27376 − 2.53971i 0.704736 − 0.339383i
\(57\) −4.31846 −0.571994
\(58\) 3.58014 − 1.08988i 0.470096 − 0.143108i
\(59\) −9.30726 −1.21170 −0.605851 − 0.795578i \(-0.707167\pi\)
−0.605851 + 0.795578i \(0.707167\pi\)
\(60\) −2.75798 + 1.32817i −0.356054 + 0.171467i
\(61\) 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
\(63\) 2.15772 − 1.03910i 0.271847 − 0.130914i
\(64\) −1.23512 − 0.594802i −0.154390 − 0.0743503i
\(65\) −8.21929 − 10.3067i −1.01948 − 1.27838i
\(66\) −0.148126 + 0.648983i −0.0182331 + 0.0798843i
\(67\) −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
\(69\) 1.19895 + 5.25295i 0.144337 + 0.632381i
\(70\) −3.35822 −0.401384
\(71\) 1.02436 + 4.48801i 0.121569 + 0.532629i 0.998634 + 0.0522567i \(0.0166414\pi\)
−0.877065 + 0.480373i \(0.840501\pi\)
\(72\) −2.20209 − 1.06047i −0.259519 − 0.124978i
\(73\) −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
\(75\) −0.928470 −0.107211
\(76\) −1.45781 − 6.38710i −0.167223 − 0.732651i
\(77\) 1.43030 − 1.79354i 0.162998 − 0.204393i
\(78\) 1.01028 − 4.42634i 0.114392 − 0.501185i
\(79\) −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
\(81\) −0.900969 − 0.433884i −0.100108 − 0.0482093i
\(82\) 7.40582 − 3.56646i 0.817836 − 0.393849i
\(83\) 10.0888 − 12.6510i 1.10739 − 1.38863i 0.194270 − 0.980948i \(-0.437766\pi\)
0.913124 − 0.407681i \(-0.133662\pi\)
\(84\) 2.26525 + 2.84054i 0.247159 + 0.309928i
\(85\) −6.93816 + 3.34124i −0.752549 + 0.362408i
\(86\) 2.57431 0.277595
\(87\) 2.67536 + 4.67359i 0.286829 + 0.501062i
\(88\) −2.34121 −0.249574
\(89\) −13.9137 + 6.70047i −1.47485 + 0.710248i −0.986706 − 0.162517i \(-0.948039\pi\)
−0.488140 + 0.872765i \(0.662325\pi\)
\(90\) 0.874288 + 1.09632i 0.0921581 + 0.115563i
\(91\) −9.75529 + 12.2327i −1.02263 + 1.28234i
\(92\) −7.36451 + 3.54656i −0.767803 + 0.369754i
\(93\) −2.46382 − 1.18651i −0.255486 − 0.123036i
\(94\) 1.40069 + 1.75641i 0.144470 + 0.181160i
\(95\) −1.93900 + 8.49532i −0.198937 + 0.871602i
\(96\) 1.29428 − 5.67061i 0.132097 − 0.578754i
\(97\) 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
\(99\) −0.957887 −0.0962713
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 87.2.g.a.49.3 yes 18
3.2 odd 2 261.2.k.c.136.1 18
29.4 even 14 2523.2.a.o.1.7 9
29.16 even 7 inner 87.2.g.a.16.3 ✓ 18
29.25 even 7 2523.2.a.r.1.3 9
87.62 odd 14 7569.2.a.bm.1.3 9
87.74 odd 14 261.2.k.c.190.1 18
87.83 odd 14 7569.2.a.bj.1.7 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.g.a.16.3 ✓ 18 29.16 even 7 inner
87.2.g.a.49.3 yes 18 1.1 even 1 trivial
261.2.k.c.136.1 18 3.2 odd 2
261.2.k.c.190.1 18 87.74 odd 14
2523.2.a.o.1.7 9 29.4 even 14
2523.2.a.r.1.3 9 29.25 even 7
7569.2.a.bj.1.7 9 87.83 odd 14
7569.2.a.bm.1.3 9 87.62 odd 14