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Properties

Label 87.3.e.a.46.4
Level $87$
Weight $3$
Character 87.46
Analytic conductor $2.371$
Analytic rank $0$
Dimension $8$
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] = [87,3,Mod(46,87)] 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("87.46"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 87.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] 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: \(2.37057829993\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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: \( x^{8} - 4x^{7} - 210x^{6} + 644x^{5} + 17515x^{4} - 36108x^{3} - 683586x^{2} + 701748x + 10556010 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 46.4
Root \(-6.88412 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 87.46
Dual form 87.3.e.a.70.3

$q$-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+(0.224745 - 0.224745i) q^{2} +(1.22474 - 1.22474i) q^{3} +3.89898i q^{4} +9.10887i q^{5} -0.550510i q^{6} -4.65938 q^{7} +(1.77526 + 1.77526i) q^{8} -3.00000i q^{9} +(2.04717 + 2.04717i) q^{10} +(13.5584 - 13.5584i) q^{11} +(4.77526 + 4.77526i) q^{12} -1.10102i q^{13} +(-1.04717 + 1.04717i) q^{14} +(11.1560 + 11.1560i) q^{15} -14.7980 q^{16} +(9.49666 - 9.49666i) q^{17} +(-0.674235 - 0.674235i) q^{18} +(-1.19243 + 1.19243i) q^{19} -35.5153 q^{20} +(-5.70655 + 5.70655i) q^{21} -6.09434i q^{22} +23.0090 q^{23} +4.34847 q^{24} -57.9715 q^{25} +(-0.247449 - 0.247449i) q^{26} +(-3.67423 - 3.67423i) q^{27} -18.1668i q^{28} +(8.79271 + 27.6349i) q^{29} +5.01452 q^{30} +(36.8783 - 36.8783i) q^{31} +(-10.4268 + 10.4268i) q^{32} -33.2111i q^{33} -4.26865i q^{34} -42.4416i q^{35} +11.6969 q^{36} +(-9.49666 - 9.49666i) q^{37} +0.535986i q^{38} +(-1.34847 - 1.34847i) q^{39} +(-16.1706 + 16.1706i) q^{40} +(-1.67948 - 1.67948i) q^{41} +2.56503i q^{42} +(-46.0343 + 46.0343i) q^{43} +(52.8638 + 52.8638i) q^{44} +27.3266 q^{45} +(5.17116 - 5.17116i) q^{46} +(-31.9889 - 31.9889i) q^{47} +(-18.1237 + 18.1237i) q^{48} -27.2902 q^{49} +(-13.0288 + 13.0288i) q^{50} -23.2620i q^{51} +4.29286 q^{52} +12.3575 q^{53} -1.65153 q^{54} +(123.501 + 123.501i) q^{55} +(-8.27158 - 8.27158i) q^{56} +2.92085i q^{57} +(8.18692 + 4.23469i) q^{58} +12.4231 q^{59} +(-43.4972 + 43.4972i) q^{60} +(53.4742 - 53.4742i) q^{61} -16.5764i q^{62} +13.9781i q^{63} -54.5051i q^{64} +10.0290 q^{65} +(-7.46401 - 7.46401i) q^{66} +50.5589i q^{67} +(37.0273 + 37.0273i) q^{68} +(28.1802 - 28.1802i) q^{69} +(-9.53854 - 9.53854i) q^{70} -17.4800i q^{71} +(5.32577 - 5.32577i) q^{72} +(-0.00901400 - 0.00901400i) q^{73} -4.26865 q^{74} +(-71.0002 + 71.0002i) q^{75} +(-4.64927 - 4.64927i) q^{76} +(-63.1735 + 63.1735i) q^{77} -0.606123 q^{78} +(69.5161 - 69.5161i) q^{79} -134.793i q^{80} -9.00000 q^{81} -0.754910 q^{82} -117.205 q^{83} +(-22.2497 - 22.2497i) q^{84} +(86.5038 + 86.5038i) q^{85} +20.6920i q^{86} +(44.6145 + 23.0769i) q^{87} +48.1391 q^{88} +(-27.7333 + 27.7333i) q^{89} +(6.14151 - 6.14151i) q^{90} +5.13007i q^{91} +89.7117i q^{92} -90.3330i q^{93} -14.3787 q^{94} +(-10.8617 - 10.8617i) q^{95} +25.5403i q^{96} +(-83.4159 - 83.4159i) q^{97} +(-6.13333 + 6.13333i) q^{98} +(-40.6751 - 40.6751i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 12 q^{7} + 24 q^{8} + 8 q^{10} + 20 q^{11} + 48 q^{12} + 12 q^{15} - 40 q^{16} + 48 q^{17} + 24 q^{18} - 28 q^{19} - 44 q^{20} + 12 q^{21} - 88 q^{23} - 24 q^{24} - 260 q^{25} + 96 q^{26}+ \cdots - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(31\) \(59\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.224745 − 0.224745i 0.112372 − 0.112372i −0.648685 − 0.761057i \(-0.724681\pi\)
0.761057 + 0.648685i \(0.224681\pi\)
\(3\) 1.22474 − 1.22474i 0.408248 − 0.408248i
\(4\) 3.89898i 0.974745i
\(5\) 9.10887i 1.82177i 0.412657 + 0.910887i \(0.364601\pi\)
−0.412657 + 0.910887i \(0.635399\pi\)
\(6\) − 0.550510i − 0.0917517i
\(7\) −4.65938 −0.665625 −0.332813 − 0.942993i \(-0.607998\pi\)
−0.332813 + 0.942993i \(0.607998\pi\)
\(8\) 1.77526 + 1.77526i 0.221907 + 0.221907i
\(9\) − 3.00000i − 0.333333i
\(10\) 2.04717 + 2.04717i 0.204717 + 0.204717i
\(11\) 13.5584 − 13.5584i 1.23258 − 1.23258i 0.269608 − 0.962970i \(-0.413106\pi\)
0.962970 − 0.269608i \(-0.0868940\pi\)
\(12\) 4.77526 + 4.77526i 0.397938 + 0.397938i
\(13\) − 1.10102i − 0.0846939i −0.999103 − 0.0423469i \(-0.986517\pi\)
0.999103 − 0.0423469i \(-0.0134835\pi\)
\(14\) −1.04717 + 1.04717i −0.0747979 + 0.0747979i
\(15\) 11.1560 + 11.1560i 0.743736 + 0.743736i
\(16\) −14.7980 −0.924872
\(17\) 9.49666 − 9.49666i 0.558627 − 0.558627i −0.370289 − 0.928916i \(-0.620741\pi\)
0.928916 + 0.370289i \(0.120741\pi\)
\(18\) −0.674235 − 0.674235i −0.0374575 − 0.0374575i
\(19\) −1.19243 + 1.19243i −0.0627596 + 0.0627596i −0.737790 − 0.675030i \(-0.764131\pi\)
0.675030 + 0.737790i \(0.264131\pi\)
\(20\) −35.5153 −1.77576
\(21\) −5.70655 + 5.70655i −0.271740 + 0.271740i
\(22\) − 6.09434i − 0.277016i
\(23\) 23.0090 1.00039 0.500196 − 0.865912i \(-0.333261\pi\)
0.500196 + 0.865912i \(0.333261\pi\)
\(24\) 4.34847 0.181186
\(25\) −57.9715 −2.31886
\(26\) −0.247449 − 0.247449i −0.00951726 − 0.00951726i
\(27\) −3.67423 − 3.67423i −0.136083 − 0.136083i
\(28\) − 18.1668i − 0.648815i
\(29\) 8.79271 + 27.6349i 0.303197 + 0.952928i
\(30\) 5.01452 0.167151
\(31\) 36.8783 − 36.8783i 1.18962 − 1.18962i 0.212450 − 0.977172i \(-0.431856\pi\)
0.977172 − 0.212450i \(-0.0681442\pi\)
\(32\) −10.4268 + 10.4268i −0.325837 + 0.325837i
\(33\) − 33.2111i − 1.00640i
\(34\) − 4.26865i − 0.125549i
\(35\) − 42.4416i − 1.21262i
\(36\) 11.6969 0.324915
\(37\) −9.49666 − 9.49666i −0.256667 − 0.256667i 0.567030 − 0.823697i \(-0.308092\pi\)
−0.823697 + 0.567030i \(0.808092\pi\)
\(38\) 0.535986i 0.0141049i
\(39\) −1.34847 − 1.34847i −0.0345761 − 0.0345761i
\(40\) −16.1706 + 16.1706i −0.404264 + 0.404264i
\(41\) −1.67948 − 1.67948i −0.0409630 − 0.0409630i 0.686329 − 0.727292i \(-0.259221\pi\)
−0.727292 + 0.686329i \(0.759221\pi\)
\(42\) 2.56503i 0.0610723i
\(43\) −46.0343 + 46.0343i −1.07057 + 1.07057i −0.0732520 + 0.997313i \(0.523338\pi\)
−0.997313 + 0.0732520i \(0.976662\pi\)
\(44\) 52.8638 + 52.8638i 1.20145 + 1.20145i
\(45\) 27.3266 0.607258
\(46\) 5.17116 − 5.17116i 0.112416 − 0.112416i
\(47\) −31.9889 − 31.9889i −0.680615 − 0.680615i 0.279524 − 0.960139i \(-0.409823\pi\)
−0.960139 + 0.279524i \(0.909823\pi\)
\(48\) −18.1237 + 18.1237i −0.377578 + 0.377578i
\(49\) −27.2902 −0.556943
\(50\) −13.0288 + 13.0288i −0.260576 + 0.260576i
\(51\) − 23.2620i − 0.456117i
\(52\) 4.29286 0.0825549
\(53\) 12.3575 0.233160 0.116580 − 0.993181i \(-0.462807\pi\)
0.116580 + 0.993181i \(0.462807\pi\)
\(54\) −1.65153 −0.0305839
\(55\) 123.501 + 123.501i 2.24548 + 2.24548i
\(56\) −8.27158 − 8.27158i −0.147707 − 0.147707i
\(57\) 2.92085i 0.0512430i
\(58\) 8.18692 + 4.23469i 0.141154 + 0.0730119i
\(59\) 12.4231 0.210561 0.105280 − 0.994443i \(-0.466426\pi\)
0.105280 + 0.994443i \(0.466426\pi\)
\(60\) −43.4972 + 43.4972i −0.724953 + 0.724953i
\(61\) 53.4742 − 53.4742i 0.876626 − 0.876626i −0.116558 − 0.993184i \(-0.537186\pi\)
0.993184 + 0.116558i \(0.0371860\pi\)
\(62\) − 16.5764i − 0.267361i
\(63\) 13.9781i 0.221875i
\(64\) − 54.5051i − 0.851642i
\(65\) 10.0290 0.154293
\(66\) −7.46401 − 7.46401i −0.113091 − 0.113091i
\(67\) 50.5589i 0.754610i 0.926089 + 0.377305i \(0.123149\pi\)
−0.926089 + 0.377305i \(0.876851\pi\)
\(68\) 37.0273 + 37.0273i 0.544519 + 0.544519i
\(69\) 28.1802 − 28.1802i 0.408408 − 0.408408i
\(70\) −9.53854 − 9.53854i −0.136265 − 0.136265i
\(71\) − 17.4800i − 0.246197i −0.992394 − 0.123099i \(-0.960717\pi\)
0.992394 − 0.123099i \(-0.0392832\pi\)
\(72\) 5.32577 − 5.32577i 0.0739690 − 0.0739690i
\(73\) −0.00901400 − 0.00901400i −0.000123479 − 0.000123479i 0.707045 − 0.707169i \(-0.250028\pi\)
−0.707169 + 0.707045i \(0.750028\pi\)
\(74\) −4.26865 −0.0576845
\(75\) −71.0002 + 71.0002i −0.946670 + 0.946670i
\(76\) −4.64927 − 4.64927i −0.0611745 − 0.0611745i
\(77\) −63.1735 + 63.1735i −0.820435 + 0.820435i
\(78\) −0.606123 −0.00777081
\(79\) 69.5161 − 69.5161i 0.879951 − 0.879951i −0.113578 − 0.993529i \(-0.536231\pi\)
0.993529 + 0.113578i \(0.0362313\pi\)
\(80\) − 134.793i − 1.68491i
\(81\) −9.00000 −0.111111
\(82\) −0.754910 −0.00920622
\(83\) −117.205 −1.41210 −0.706052 − 0.708160i \(-0.749526\pi\)
−0.706052 + 0.708160i \(0.749526\pi\)
\(84\) −22.2497 − 22.2497i −0.264878 − 0.264878i
\(85\) 86.5038 + 86.5038i 1.01769 + 1.01769i
\(86\) 20.6920i 0.240604i
\(87\) 44.6145 + 23.0769i 0.512811 + 0.265252i
\(88\) 48.1391 0.547035
\(89\) −27.7333 + 27.7333i −0.311610 + 0.311610i −0.845533 − 0.533923i \(-0.820717\pi\)
0.533923 + 0.845533i \(0.320717\pi\)
\(90\) 6.14151 − 6.14151i 0.0682390 − 0.0682390i
\(91\) 5.13007i 0.0563744i
\(92\) 89.7117i 0.975127i
\(93\) − 90.3330i − 0.971322i
\(94\) −14.3787 −0.152965
\(95\) −10.8617 − 10.8617i −0.114334 − 0.114334i
\(96\) 25.5403i 0.266045i
\(97\) −83.4159 − 83.4159i −0.859958 − 0.859958i 0.131375 − 0.991333i \(-0.458061\pi\)
−0.991333 + 0.131375i \(0.958061\pi\)
\(98\) −6.13333 + 6.13333i −0.0625850 + 0.0625850i
\(99\) −40.6751 − 40.6751i −0.410859 − 0.410859i
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 87.3.e.a.46.4 ✓ 8
3.2 odd 2 261.3.f.b.46.1 8
29.12 odd 4 inner 87.3.e.a.70.3 yes 8
87.41 even 4 261.3.f.b.244.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.3.e.a.46.4 ✓ 8 1.1 even 1 trivial
87.3.e.a.70.3 yes 8 29.12 odd 4 inner
261.3.f.b.46.1 8 3.2 odd 2
261.3.f.b.244.2 8 87.41 even 4