↑

Properties

Label 87.2.g.a.82.3
Level $87$
Weight $2$
Character 87.82
Analytic conductor $0.695$
Analytic rank $0$
Dimension $18$
Inner twists $2$

Related objects

Downloads

Learn more

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 82.3
Root \(-0.353498 + 1.54877i\) of defining polynomial
Character \(\chi\) \(=\) 87.82
Dual form 87.2.g.a.52.3

$q$-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+(1.61397 + 2.02385i) q^{2} +(-0.222521 - 0.974928i) q^{3} +(-1.04604 + 4.58301i) q^{4} +(-0.716354 - 0.898279i) q^{5} +(1.61397 - 2.02385i) q^{6} +(-0.615328 - 2.69593i) q^{7} +(-6.29911 + 3.03349i) q^{8} +(-0.900969 + 0.433884i) q^{9} +(0.661812 - 2.89959i) q^{10} +(3.92298 + 1.88921i) q^{11} +4.70087 q^{12} +(-5.07168 - 2.44239i) q^{13} +(4.46304 - 5.59648i) q^{14} +(-0.716354 + 0.898279i) q^{15} +(-7.83521 - 3.77324i) q^{16} +2.41516 q^{17} +(-2.33225 - 1.12315i) q^{18} +(-0.416772 + 1.82600i) q^{19} +(4.86616 - 2.34342i) q^{20} +(-2.49141 + 1.19980i) q^{21} +(2.50809 + 10.9886i) q^{22} +(-5.49948 + 6.89613i) q^{23} +(4.35912 + 5.46616i) q^{24} +(0.818862 - 3.58767i) q^{25} +(-3.24249 - 14.2063i) q^{26} +(0.623490 + 0.781831i) q^{27} +12.9991 q^{28} +(4.53581 - 2.90283i) q^{29} -2.97416 q^{30} +(0.972977 + 1.22007i) q^{31} +(-1.89780 - 8.31482i) q^{32} +(0.968895 - 4.24501i) q^{33} +(3.89799 + 4.88793i) q^{34} +(-1.98091 + 2.48398i) q^{35} +(-1.04604 - 4.58301i) q^{36} +(-6.99449 + 3.36837i) q^{37} +(-4.36821 + 2.10362i) q^{38} +(-1.25260 + 5.48800i) q^{39} +(7.23731 + 3.48530i) q^{40} +3.16072 q^{41} +(-6.44928 - 3.10581i) q^{42} +(-0.912772 + 1.14458i) q^{43} +(-12.7618 + 16.0028i) q^{44} +(1.03516 + 0.498507i) q^{45} -22.8327 q^{46} +(-0.321962 - 0.155049i) q^{47} +(-1.93514 + 8.47839i) q^{48} +(-0.582626 + 0.280578i) q^{49} +(8.58252 - 4.13313i) q^{50} +(-0.537424 - 2.35461i) q^{51} +(16.4987 - 20.6887i) q^{52} +(-1.01940 - 1.27828i) q^{53} +(-0.576019 + 2.52370i) q^{54} +(-1.11320 - 4.87727i) q^{55} +(12.0541 + 15.1154i) q^{56} +1.87296 q^{57} +(13.1955 + 4.49474i) q^{58} -4.85026 q^{59} +(-3.36749 - 4.22269i) q^{60} +(-0.346042 - 1.51611i) q^{61} +(-0.898897 + 3.93832i) q^{62} +(1.72411 + 2.16197i) q^{63} +(2.92070 - 3.66245i) q^{64} +(1.43917 + 6.30540i) q^{65} +(10.1550 - 4.89040i) q^{66} +(8.71192 - 4.19544i) q^{67} +(-2.52636 + 11.0687i) q^{68} +(7.94698 + 3.82706i) q^{69} -8.22432 q^{70} +(7.37090 + 3.54964i) q^{71} +(4.35912 - 5.46616i) q^{72} +(7.09446 - 8.89617i) q^{73} +(-18.1060 - 8.71937i) q^{74} -3.67993 q^{75} +(-7.93260 - 3.82014i) q^{76} +(2.67925 - 11.7386i) q^{77} +(-13.1286 + 6.32238i) q^{78} +(7.32556 - 3.52780i) q^{79} +(2.22336 + 9.74119i) q^{80} +(0.623490 - 0.781831i) q^{81} +(5.10130 + 6.39683i) q^{82} +(0.107218 - 0.469754i) q^{83} +(-2.89258 - 12.6732i) q^{84} +(-1.73011 - 2.16949i) q^{85} -3.78965 q^{86} +(-3.83936 - 3.77615i) q^{87} -30.4421 q^{88} +(-1.24461 - 1.56070i) q^{89} +(0.661812 + 2.89959i) q^{90} +(-3.46377 + 15.1758i) q^{91} +(-25.8523 - 32.4178i) q^{92} +(0.972977 - 1.22007i) q^{93} +(-0.205841 - 0.901847i) q^{94} +(1.93881 - 0.933683i) q^{95} +(-7.68405 + 3.70044i) q^{96} +(-1.03142 + 4.51895i) q^{97} +(-1.50819 - 0.726305i) q^{98} -4.35418 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 \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(59\)
\(\chi(n)\) \(e\left(\frac{2}{7}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61397 + 2.02385i 1.14125 + 1.43108i 0.885685 + 0.464287i \(0.153689\pi\)
0.255563 + 0.966792i \(0.417739\pi\)
\(3\) −0.222521 − 0.974928i −0.128473 − 0.562875i
\(4\) −1.04604 + 4.58301i −0.523021 + 2.29150i
\(5\) −0.716354 − 0.898279i −0.320363 − 0.401723i 0.595408 − 0.803424i \(-0.296991\pi\)
−0.915771 + 0.401701i \(0.868419\pi\)
\(6\) 1.61397 − 2.02385i 0.658900 − 0.826234i
\(7\) −0.615328 − 2.69593i −0.232572 − 1.01897i −0.947497 − 0.319764i \(-0.896396\pi\)
0.714925 − 0.699201i \(-0.246461\pi\)
\(8\) −6.29911 + 3.03349i −2.22707 + 1.07250i
\(9\) −0.900969 + 0.433884i −0.300323 + 0.144628i
\(10\) 0.661812 − 2.89959i 0.209283 − 0.916930i
\(11\) 3.92298 + 1.88921i 1.18282 + 0.569617i 0.918732 − 0.394881i \(-0.129214\pi\)
0.264090 + 0.964498i \(0.414929\pi\)
\(12\) 4.70087 1.35702
\(13\) −5.07168 − 2.44239i −1.40663 − 0.677397i −0.432135 − 0.901809i \(-0.642240\pi\)
−0.974494 + 0.224411i \(0.927954\pi\)
\(14\) 4.46304 − 5.59648i 1.19280 − 1.49572i
\(15\) −0.716354 + 0.898279i −0.184962 + 0.231935i
\(16\) −7.83521 − 3.77324i −1.95880 − 0.943310i
\(17\) 2.41516 0.585763 0.292881 − 0.956149i \(-0.405386\pi\)
0.292881 + 0.956149i \(0.405386\pi\)
\(18\) −2.33225 − 1.12315i −0.549717 − 0.264730i
\(19\) −0.416772 + 1.82600i −0.0956141 + 0.418913i −0.999969 − 0.00786845i \(-0.997495\pi\)
0.904355 + 0.426781i \(0.140353\pi\)
\(20\) 4.86616 − 2.34342i 1.08811 − 0.524004i
\(21\) −2.49141 + 1.19980i −0.543671 + 0.261818i
\(22\) 2.50809 + 10.9886i 0.534726 + 2.34279i
\(23\) −5.49948 + 6.89613i −1.14672 + 1.43794i −0.266210 + 0.963915i \(0.585771\pi\)
−0.880510 + 0.474027i \(0.842800\pi\)
\(24\) 4.35912 + 5.46616i 0.889801 + 1.11577i
\(25\) 0.818862 − 3.58767i 0.163772 − 0.717534i
\(26\) −3.24249 − 14.2063i −0.635904 − 2.78608i
\(27\) 0.623490 + 0.781831i 0.119991 + 0.150464i
\(28\) 12.9991 2.45660
\(29\) 4.53581 − 2.90283i 0.842279 − 0.539042i
\(30\) −2.97416 −0.543004
\(31\) 0.972977 + 1.22007i 0.174752 + 0.219132i 0.861492 − 0.507771i \(-0.169530\pi\)
−0.686740 + 0.726903i \(0.740959\pi\)
\(32\) −1.89780 − 8.31482i −0.335488 − 1.46987i
\(33\) 0.968895 − 4.24501i 0.168663 − 0.738961i
\(34\) 3.89799 + 4.88793i 0.668500 + 0.838273i
\(35\) −1.98091 + 2.48398i −0.334834 + 0.419869i
\(36\) −1.04604 − 4.58301i −0.174340 − 0.763835i
\(37\) −6.99449 + 3.36837i −1.14989 + 0.553756i −0.909000 − 0.416797i \(-0.863153\pi\)
−0.240887 + 0.970553i \(0.577438\pi\)
\(38\) −4.36821 + 2.10362i −0.708617 + 0.341252i
\(39\) −1.25260 + 5.48800i −0.200577 + 0.878784i
\(40\) 7.23731 + 3.48530i 1.14432 + 0.551075i
\(41\) 3.16072 0.493622 0.246811 − 0.969064i \(-0.420617\pi\)
0.246811 + 0.969064i \(0.420617\pi\)
\(42\) −6.44928 − 3.10581i −0.995146 − 0.479237i
\(43\) −0.912772 + 1.14458i −0.139196 + 0.174547i −0.846543 − 0.532320i \(-0.821320\pi\)
0.707347 + 0.706867i \(0.249892\pi\)
\(44\) −12.7618 + 16.0028i −1.92392 + 2.41252i
\(45\) 1.03516 + 0.498507i 0.154313 + 0.0743131i
\(46\) −22.8327 −3.36650
\(47\) −0.321962 − 0.155049i −0.0469630 − 0.0226162i 0.410255 − 0.911971i \(-0.365440\pi\)
−0.457218 + 0.889355i \(0.651154\pi\)
\(48\) −1.93514 + 8.47839i −0.279313 + 1.22375i
\(49\) −0.582626 + 0.280578i −0.0832323 + 0.0400826i
\(50\) 8.58252 − 4.13313i 1.21375 − 0.584512i
\(51\) −0.537424 − 2.35461i −0.0752544 − 0.329711i
\(52\) 16.4987 − 20.6887i 2.28796 − 2.86901i
\(53\) −1.01940 − 1.27828i −0.140025 − 0.175586i 0.706874 − 0.707339i \(-0.250105\pi\)
−0.846899 + 0.531754i \(0.821533\pi\)
\(54\) −0.576019 + 2.52370i −0.0783862 + 0.343432i
\(55\) −1.11320 − 4.87727i −0.150104 − 0.657651i
\(56\) 12.0541 + 15.1154i 1.61080 + 2.01987i
\(57\) 1.87296 0.248079
\(58\) 13.1955 + 4.49474i 1.73266 + 0.590188i
\(59\) −4.85026 −0.631450 −0.315725 − 0.948851i \(-0.602248\pi\)
−0.315725 + 0.948851i \(0.602248\pi\)
\(60\) −3.36749 − 4.22269i −0.434741 − 0.545147i
\(61\) −0.346042 − 1.51611i −0.0443062 − 0.194118i 0.947931 − 0.318474i \(-0.103171\pi\)
−0.992238 + 0.124356i \(0.960313\pi\)
\(62\) −0.898897 + 3.93832i −0.114160 + 0.500168i
\(63\) 1.72411 + 2.16197i 0.217218 + 0.272382i
\(64\) 2.92070 − 3.66245i 0.365088 − 0.457806i
\(65\) 1.43917 + 6.30540i 0.178507 + 0.782088i
\(66\) 10.1550 − 4.89040i 1.25000 − 0.601967i
\(67\) 8.71192 − 4.19544i 1.06433 − 0.512554i 0.182055 − 0.983288i \(-0.441725\pi\)
0.882275 + 0.470734i \(0.156011\pi\)
\(68\) −2.52636 + 11.0687i −0.306366 + 1.34228i
\(69\) 7.94698 + 3.82706i 0.956703 + 0.460724i
\(70\) −8.22432 −0.982994
\(71\) 7.37090 + 3.54964i 0.874765 + 0.421264i 0.816710 − 0.577049i \(-0.195796\pi\)
0.0580551 + 0.998313i \(0.481510\pi\)
\(72\) 4.35912 − 5.46616i 0.513727 − 0.644193i
\(73\) 7.09446 − 8.89617i 0.830344 − 1.04122i −0.168117 − 0.985767i \(-0.553769\pi\)
0.998461 − 0.0554515i \(-0.0176598\pi\)
\(74\) −18.1060 − 8.71937i −2.10477 − 1.01361i
\(75\) −3.67993 −0.424922
\(76\) −7.93260 − 3.82014i −0.909932 − 0.438200i
\(77\) 2.67925 − 11.7386i 0.305329 − 1.33773i
\(78\) −13.1286 + 6.32238i −1.48652 + 0.715869i
\(79\) 7.32556 − 3.52780i 0.824189 − 0.396909i 0.0262567 − 0.999655i \(-0.491641\pi\)
0.797933 + 0.602747i \(0.205927\pi\)
\(80\) 2.22336 + 9.74119i 0.248579 + 1.08910i
\(81\) 0.623490 − 0.781831i 0.0692766 − 0.0868702i
\(82\) 5.10130 + 6.39683i 0.563345 + 0.706412i
\(83\) 0.107218 − 0.469754i 0.0117687 − 0.0515622i −0.968703 − 0.248224i \(-0.920153\pi\)
0.980471 + 0.196662i \(0.0630102\pi\)
\(84\) −2.89258 − 12.6732i −0.315606 − 1.38276i
\(85\) −1.73011 − 2.16949i −0.187657 − 0.235314i
\(86\) −3.78965 −0.408648
\(87\) −3.83936 − 3.77615i −0.411623 − 0.404846i
\(88\) −30.4421 −3.24514
\(89\) −1.24461 − 1.56070i −0.131929 − 0.165434i 0.711479 − 0.702707i \(-0.248026\pi\)
−0.843408 + 0.537274i \(0.819454\pi\)
\(90\) 0.661812 + 2.89959i 0.0697611 + 0.305643i
\(91\) −3.46377 + 15.1758i −0.363101 + 1.59085i
\(92\) −25.8523 − 32.4178i −2.69529 − 3.37979i
\(93\) 0.972977 − 1.22007i 0.100893 − 0.126516i
\(94\) −0.205841 − 0.901847i −0.0212309 − 0.0930184i
\(95\) 1.93881 − 0.933683i 0.198918 − 0.0957938i
\(96\) −7.68405 + 3.70044i −0.784250 + 0.377675i
\(97\) −1.03142 + 4.51895i −0.104725 + 0.458830i 0.895189 + 0.445687i \(0.147041\pi\)
−0.999914 + 0.0131424i \(0.995817\pi\)
\(98\) −1.50819 − 0.726305i −0.152350 − 0.0733679i
\(99\) −4.35418 −0.437611
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.82.3 yes 18
3.2 odd 2 261.2.k.c.82.1 18
29.9 even 14 2523.2.a.o.1.2 9
29.20 even 7 2523.2.a.r.1.8 9
29.23 even 7 inner 87.2.g.a.52.3 ✓ 18
87.20 odd 14 7569.2.a.bj.1.2 9
87.23 odd 14 261.2.k.c.226.1 18
87.38 odd 14 7569.2.a.bm.1.8 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.g.a.52.3 ✓ 18 29.23 even 7 inner
87.2.g.a.82.3 yes 18 1.1 even 1 trivial
261.2.k.c.82.1 18 3.2 odd 2
261.2.k.c.226.1 18 87.23 odd 14
2523.2.a.o.1.2 9 29.9 even 14
2523.2.a.r.1.8 9 29.20 even 7
7569.2.a.bj.1.2 9 87.20 odd 14
7569.2.a.bm.1.8 9 87.38 odd 14