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Properties

Label 99.2.f.a.82.1
Level $99$
Weight $2$
Character 99.82
Analytic conductor $0.791$
Analytic rank $0$
Dimension $4$
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] = [99,2,Mod(37,99)] 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("99.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(99, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,1] 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: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 82.1
Root \(-0.309017 + 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 99.82
Dual form 99.2.f.a.64.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.809017 + 2.48990i) q^{2} +(-3.92705 + 2.85317i) q^{4} +(0.190983 - 0.587785i) q^{5} +(0.809017 - 0.587785i) q^{7} +(-6.04508 - 4.39201i) q^{8} +1.61803 q^{10} +(3.30902 - 0.224514i) q^{11} +(0.0729490 + 0.224514i) q^{13} +(2.11803 + 1.53884i) q^{14} +(3.04508 - 9.37181i) q^{16} +(0.354102 - 1.08981i) q^{17} +(-4.73607 - 3.44095i) q^{19} +(0.927051 + 2.85317i) q^{20} +(3.23607 + 8.05748i) q^{22} -0.236068 q^{23} +(3.73607 + 2.71441i) q^{25} +(-0.500000 + 0.363271i) q^{26} +(-1.50000 + 4.61653i) q^{28} +(-4.85410 + 3.52671i) q^{29} +(-1.88197 - 5.79210i) q^{31} +10.8541 q^{32} +3.00000 q^{34} +(-0.190983 - 0.587785i) q^{35} +(5.04508 - 3.66547i) q^{37} +(4.73607 - 14.5761i) q^{38} +(-3.73607 + 2.71441i) q^{40} +(0.190983 + 0.138757i) q^{41} -6.70820 q^{43} +(-12.3541 + 10.3229i) q^{44} +(-0.190983 - 0.587785i) q^{46} +(-8.16312 - 5.93085i) q^{47} +(-1.85410 + 5.70634i) q^{49} +(-3.73607 + 11.4984i) q^{50} +(-0.927051 - 0.673542i) q^{52} +(0.118034 + 0.363271i) q^{53} +(0.500000 - 1.98787i) q^{55} -7.47214 q^{56} +(-12.7082 - 9.23305i) q^{58} +(5.97214 - 4.33901i) q^{59} +(-3.57295 + 10.9964i) q^{61} +(12.8992 - 9.37181i) q^{62} +(2.69098 + 8.28199i) q^{64} +0.145898 q^{65} +1.85410 q^{67} +(1.71885 + 5.29007i) q^{68} +(1.30902 - 0.951057i) q^{70} +(-3.19098 + 9.82084i) q^{71} +(4.61803 - 3.35520i) q^{73} +(13.2082 + 9.59632i) q^{74} +28.4164 q^{76} +(2.54508 - 2.12663i) q^{77} +(3.39919 + 10.4616i) q^{79} +(-4.92705 - 3.57971i) q^{80} +(-0.190983 + 0.587785i) q^{82} +(-0.454915 + 1.40008i) q^{83} +(-0.572949 - 0.416272i) q^{85} +(-5.42705 - 16.7027i) q^{86} +(-20.9894 - 13.1760i) q^{88} +8.23607 q^{89} +(0.190983 + 0.138757i) q^{91} +(0.927051 - 0.673542i) q^{92} +(8.16312 - 25.1235i) q^{94} +(-2.92705 + 2.12663i) q^{95} +(2.42705 + 7.46969i) q^{97} -15.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 9 q^{4} + 3 q^{5} + q^{7} - 13 q^{8} + 2 q^{10} + 11 q^{11} + 7 q^{13} + 4 q^{14} + q^{16} - 12 q^{17} - 10 q^{19} - 3 q^{20} + 4 q^{22} + 8 q^{23} + 6 q^{25} - 2 q^{26} - 6 q^{28} - 6 q^{29}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(e\left(\frac{2}{5}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.809017 + 2.48990i 0.572061 + 1.76062i 0.645974 + 0.763359i \(0.276451\pi\)
−0.0739128 + 0.997265i \(0.523549\pi\)
\(3\) 0 0
\(4\) −3.92705 + 2.85317i −1.96353 + 1.42658i
\(5\) 0.190983 − 0.587785i 0.0854102 − 0.262866i −0.899226 − 0.437485i \(-0.855869\pi\)
0.984636 + 0.174619i \(0.0558694\pi\)
\(6\) 0 0
\(7\) 0.809017 − 0.587785i 0.305780 − 0.222162i −0.424304 − 0.905520i \(-0.639481\pi\)
0.730084 + 0.683358i \(0.239481\pi\)
\(8\) −6.04508 − 4.39201i −2.13726 − 1.55281i
\(9\) 0 0
\(10\) 1.61803 0.511667
\(11\) 3.30902 − 0.224514i 0.997706 − 0.0676935i
\(12\) 0 0
\(13\) 0.0729490 + 0.224514i 0.0202324 + 0.0622690i 0.960663 − 0.277717i \(-0.0895777\pi\)
−0.940431 + 0.339986i \(0.889578\pi\)
\(14\) 2.11803 + 1.53884i 0.566068 + 0.411273i
\(15\) 0 0
\(16\) 3.04508 − 9.37181i 0.761271 − 2.34295i
\(17\) 0.354102 − 1.08981i 0.0858823 − 0.264319i −0.898888 − 0.438178i \(-0.855624\pi\)
0.984770 + 0.173860i \(0.0556239\pi\)
\(18\) 0 0
\(19\) −4.73607 − 3.44095i −1.08653 − 0.789409i −0.107719 − 0.994181i \(-0.534355\pi\)
−0.978810 + 0.204772i \(0.934355\pi\)
\(20\) 0.927051 + 2.85317i 0.207295 + 0.637988i
\(21\) 0 0
\(22\) 3.23607 + 8.05748i 0.689932 + 1.71786i
\(23\) −0.236068 −0.0492236 −0.0246118 − 0.999697i \(-0.507835\pi\)
−0.0246118 + 0.999697i \(0.507835\pi\)
\(24\) 0 0
\(25\) 3.73607 + 2.71441i 0.747214 + 0.542882i
\(26\) −0.500000 + 0.363271i −0.0980581 + 0.0712434i
\(27\) 0 0
\(28\) −1.50000 + 4.61653i −0.283473 + 0.872441i
\(29\) −4.85410 + 3.52671i −0.901384 + 0.654894i −0.938821 − 0.344405i \(-0.888081\pi\)
0.0374370 + 0.999299i \(0.488081\pi\)
\(30\) 0 0
\(31\) −1.88197 − 5.79210i −0.338011 − 1.04029i −0.965220 − 0.261440i \(-0.915803\pi\)
0.627209 − 0.778851i \(-0.284197\pi\)
\(32\) 10.8541 1.91875
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) −0.190983 − 0.587785i −0.0322820 − 0.0993538i
\(36\) 0 0
\(37\) 5.04508 − 3.66547i 0.829407 − 0.602599i −0.0899846 − 0.995943i \(-0.528682\pi\)
0.919391 + 0.393344i \(0.128682\pi\)
\(38\) 4.73607 − 14.5761i 0.768292 − 2.36456i
\(39\) 0 0
\(40\) −3.73607 + 2.71441i −0.590724 + 0.429186i
\(41\) 0.190983 + 0.138757i 0.0298265 + 0.0216702i 0.602599 − 0.798044i \(-0.294132\pi\)
−0.572772 + 0.819715i \(0.694132\pi\)
\(42\) 0 0
\(43\) −6.70820 −1.02299 −0.511496 − 0.859286i \(-0.670908\pi\)
−0.511496 + 0.859286i \(0.670908\pi\)
\(44\) −12.3541 + 10.3229i −1.86245 + 1.55623i
\(45\) 0 0
\(46\) −0.190983 − 0.587785i −0.0281589 − 0.0866642i
\(47\) −8.16312 − 5.93085i −1.19071 − 0.865104i −0.197374 − 0.980328i \(-0.563241\pi\)
−0.993339 + 0.115224i \(0.963241\pi\)
\(48\) 0 0
\(49\) −1.85410 + 5.70634i −0.264872 + 0.815191i
\(50\) −3.73607 + 11.4984i −0.528360 + 1.62612i
\(51\) 0 0
\(52\) −0.927051 − 0.673542i −0.128559 − 0.0934035i
\(53\) 0.118034 + 0.363271i 0.0162132 + 0.0498991i 0.958836 − 0.283961i \(-0.0916486\pi\)
−0.942623 + 0.333860i \(0.891649\pi\)
\(54\) 0 0
\(55\) 0.500000 − 1.98787i 0.0674200 − 0.268044i
\(56\) −7.47214 −0.998506
\(57\) 0 0
\(58\) −12.7082 − 9.23305i −1.66867 − 1.21236i
\(59\) 5.97214 − 4.33901i 0.777506 − 0.564891i −0.126724 − 0.991938i \(-0.540446\pi\)
0.904229 + 0.427047i \(0.140446\pi\)
\(60\) 0 0
\(61\) −3.57295 + 10.9964i −0.457469 + 1.40795i 0.410742 + 0.911751i \(0.365270\pi\)
−0.868212 + 0.496194i \(0.834730\pi\)
\(62\) 12.8992 − 9.37181i 1.63820 − 1.19022i
\(63\) 0 0
\(64\) 2.69098 + 8.28199i 0.336373 + 1.03525i
\(65\) 0.145898 0.0180964
\(66\) 0 0
\(67\) 1.85410 0.226515 0.113257 − 0.993566i \(-0.463872\pi\)
0.113257 + 0.993566i \(0.463872\pi\)
\(68\) 1.71885 + 5.29007i 0.208441 + 0.641515i
\(69\) 0 0
\(70\) 1.30902 − 0.951057i 0.156457 − 0.113673i
\(71\) −3.19098 + 9.82084i −0.378700 + 1.16552i 0.562248 + 0.826968i \(0.309937\pi\)
−0.940948 + 0.338550i \(0.890063\pi\)
\(72\) 0 0
\(73\) 4.61803 − 3.35520i 0.540500 − 0.392696i −0.283771 − 0.958892i \(-0.591585\pi\)
0.824271 + 0.566196i \(0.191585\pi\)
\(74\) 13.2082 + 9.59632i 1.53542 + 1.11555i
\(75\) 0 0
\(76\) 28.4164 3.25959
\(77\) 2.54508 − 2.12663i 0.290039 − 0.242352i
\(78\) 0 0
\(79\) 3.39919 + 10.4616i 0.382438 + 1.17702i 0.938322 + 0.345764i \(0.112380\pi\)
−0.555883 + 0.831260i \(0.687620\pi\)
\(80\) −4.92705 − 3.57971i −0.550861 − 0.400224i
\(81\) 0 0
\(82\) −0.190983 + 0.587785i −0.0210905 + 0.0649100i
\(83\) −0.454915 + 1.40008i −0.0499334 + 0.153679i −0.972914 − 0.231167i \(-0.925746\pi\)
0.922981 + 0.384846i \(0.125746\pi\)
\(84\) 0 0
\(85\) −0.572949 − 0.416272i −0.0621450 − 0.0451510i
\(86\) −5.42705 − 16.7027i −0.585214 − 1.80110i
\(87\) 0 0
\(88\) −20.9894 − 13.1760i −2.23747 − 1.40457i
\(89\) 8.23607 0.873021 0.436511 − 0.899699i \(-0.356214\pi\)
0.436511 + 0.899699i \(0.356214\pi\)
\(90\) 0 0
\(91\) 0.190983 + 0.138757i 0.0200205 + 0.0145457i
\(92\) 0.927051 − 0.673542i 0.0966517 − 0.0702216i
\(93\) 0 0
\(94\) 8.16312 − 25.1235i 0.841961 − 2.59129i
\(95\) −2.92705 + 2.12663i −0.300309 + 0.218187i
\(96\) 0 0
\(97\) 2.42705 + 7.46969i 0.246430 + 0.758433i 0.995398 + 0.0958268i \(0.0305495\pi\)
−0.748968 + 0.662606i \(0.769451\pi\)
\(98\) −15.7082 −1.58677
\(99\) 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 99.2.f.a.82.1 4
3.2 odd 2 33.2.e.b.16.1 ✓ 4
9.2 odd 6 891.2.n.c.676.1 8
9.4 even 3 891.2.n.b.379.1 8
9.5 odd 6 891.2.n.c.379.1 8
9.7 even 3 891.2.n.b.676.1 8
11.3 even 5 1089.2.a.t.1.2 2
11.8 odd 10 1089.2.a.l.1.1 2
11.9 even 5 inner 99.2.f.a.64.1 4
12.11 even 2 528.2.y.b.49.1 4
15.2 even 4 825.2.bx.d.49.2 8
15.8 even 4 825.2.bx.d.49.1 8
15.14 odd 2 825.2.n.c.676.1 4
33.2 even 10 363.2.e.f.130.1 4
33.5 odd 10 363.2.e.k.202.1 4
33.8 even 10 363.2.a.i.1.2 2
33.14 odd 10 363.2.a.d.1.1 2
33.17 even 10 363.2.e.b.202.1 4
33.20 odd 10 33.2.e.b.31.1 yes 4
33.26 odd 10 363.2.e.k.124.1 4
33.29 even 10 363.2.e.b.124.1 4
33.32 even 2 363.2.e.f.148.1 4
99.20 odd 30 891.2.n.c.757.1 8
99.31 even 15 891.2.n.b.460.1 8
99.86 odd 30 891.2.n.c.460.1 8
99.97 even 15 891.2.n.b.757.1 8
132.47 even 10 5808.2.a.cj.1.1 2
132.107 odd 10 5808.2.a.ci.1.1 2
132.119 even 10 528.2.y.b.97.1 4
165.14 odd 10 9075.2.a.cb.1.2 2
165.53 even 20 825.2.bx.d.724.2 8
165.74 even 10 9075.2.a.u.1.1 2
165.119 odd 10 825.2.n.c.526.1 4
165.152 even 20 825.2.bx.d.724.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.16.1 ✓ 4 3.2 odd 2
33.2.e.b.31.1 yes 4 33.20 odd 10
99.2.f.a.64.1 4 11.9 even 5 inner
99.2.f.a.82.1 4 1.1 even 1 trivial
363.2.a.d.1.1 2 33.14 odd 10
363.2.a.i.1.2 2 33.8 even 10
363.2.e.b.124.1 4 33.29 even 10
363.2.e.b.202.1 4 33.17 even 10
363.2.e.f.130.1 4 33.2 even 10
363.2.e.f.148.1 4 33.32 even 2
363.2.e.k.124.1 4 33.26 odd 10
363.2.e.k.202.1 4 33.5 odd 10
528.2.y.b.49.1 4 12.11 even 2
528.2.y.b.97.1 4 132.119 even 10
825.2.n.c.526.1 4 165.119 odd 10
825.2.n.c.676.1 4 15.14 odd 2
825.2.bx.d.49.1 8 15.8 even 4
825.2.bx.d.49.2 8 15.2 even 4
825.2.bx.d.724.1 8 165.152 even 20
825.2.bx.d.724.2 8 165.53 even 20
891.2.n.b.379.1 8 9.4 even 3
891.2.n.b.460.1 8 99.31 even 15
891.2.n.b.676.1 8 9.7 even 3
891.2.n.b.757.1 8 99.97 even 15
891.2.n.c.379.1 8 9.5 odd 6
891.2.n.c.460.1 8 99.86 odd 30
891.2.n.c.676.1 8 9.2 odd 6
891.2.n.c.757.1 8 99.20 odd 30
1089.2.a.l.1.1 2 11.8 odd 10
1089.2.a.t.1.2 2 11.3 even 5
5808.2.a.ci.1.1 2 132.107 odd 10
5808.2.a.cj.1.1 2 132.47 even 10
9075.2.a.u.1.1 2 165.74 even 10
9075.2.a.cb.1.2 2 165.14 odd 10