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Properties

Label 99.4.d.c.98.2
Level $99$
Weight $4$
Character 99.98
Analytic conductor $5.841$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [99,4,Mod(98,99)] 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("99.98"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(99, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 99.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0] 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: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12261951429820416.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 62x^{6} + 1113x^{4} + 5786x^{2} + 5776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 98.2
Root \(-1.14042i\) of defining polynomial
Character \(\chi\) \(=\) 99.98
Dual form 99.4.d.c.98.1

$q$-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)
 
\(f(q)\) \(=\) \(q-5.00866 q^{2} +17.0866 q^{4} +7.07107i q^{5} -10.4716i q^{7} -45.5118 q^{8} -35.4165i q^{10} +(-13.0640 - 34.0636i) q^{11} +43.7271i q^{13} +52.4484i q^{14} +91.2599 q^{16} +115.416 q^{17} +89.3084i q^{19} +120.821i q^{20} +(65.4332 + 170.613i) q^{22} +213.357i q^{23} +75.0000 q^{25} -219.014i q^{26} -178.924i q^{28} +124.132 q^{29} -149.040 q^{31} -92.9950 q^{32} -578.079 q^{34} +74.0451 q^{35} +161.733 q^{37} -447.315i q^{38} -321.817i q^{40} -172.049 q^{41} +258.081i q^{43} +(-223.220 - 582.033i) q^{44} -1068.63i q^{46} +240.416i q^{47} +233.347 q^{49} -375.649 q^{50} +747.149i q^{52} +334.790i q^{53} +(240.866 - 92.3765i) q^{55} +476.579i q^{56} -621.733 q^{58} -382.685i q^{59} +609.752i q^{61} +746.488 q^{62} -264.299 q^{64} -309.197 q^{65} +260.000 q^{67} +1972.07 q^{68} -370.866 q^{70} -17.3487i q^{71} -787.834i q^{73} -810.063 q^{74} +1525.98i q^{76} +(-356.699 + 136.801i) q^{77} -1000.93i q^{79} +645.305i q^{80} +861.733 q^{82} +183.820 q^{83} +816.115i q^{85} -1292.64i q^{86} +(594.567 + 1550.30i) q^{88} +11.1459i q^{89} +457.891 q^{91} +3645.56i q^{92} -1204.16i q^{94} -631.506 q^{95} +1795.20 q^{97} -1168.75 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 44 q^{4} + 452 q^{16} + 60 q^{22} + 600 q^{25} - 80 q^{31} - 2400 q^{34} - 560 q^{37} + 1496 q^{49} + 1000 q^{55} - 3120 q^{58} - 724 q^{64} + 2080 q^{67} - 2040 q^{70} + 5040 q^{82} + 5220 q^{88}+ \cdots + 8800 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.00866 −1.77083 −0.885414 − 0.464804i \(-0.846125\pi\)
−0.885414 + 0.464804i \(0.846125\pi\)
\(3\) 0 0
\(4\) 17.0866 2.13583
\(5\) 7.07107i 0.632456i 0.948683 + 0.316228i \(0.102416\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 0 0
\(7\) − 10.4716i − 0.565411i −0.959207 − 0.282705i \(-0.908768\pi\)
0.959207 − 0.282705i \(-0.0912319\pi\)
\(8\) −45.5118 −2.01136
\(9\) 0 0
\(10\) − 35.4165i − 1.11997i
\(11\) −13.0640 − 34.0636i −0.358086 − 0.933689i
\(12\) 0 0
\(13\) 43.7271i 0.932901i 0.884547 + 0.466451i \(0.154467\pi\)
−0.884547 + 0.466451i \(0.845533\pi\)
\(14\) 52.4484i 1.00124i
\(15\) 0 0
\(16\) 91.2599 1.42594
\(17\) 115.416 1.64662 0.823309 − 0.567594i \(-0.192126\pi\)
0.823309 + 0.567594i \(0.192126\pi\)
\(18\) 0 0
\(19\) 89.3084i 1.07836i 0.842192 + 0.539178i \(0.181265\pi\)
−0.842192 + 0.539178i \(0.818735\pi\)
\(20\) 120.821i 1.35082i
\(21\) 0 0
\(22\) 65.4332 + 170.613i 0.634109 + 1.65340i
\(23\) 213.357i 1.93426i 0.254277 + 0.967131i \(0.418162\pi\)
−0.254277 + 0.967131i \(0.581838\pi\)
\(24\) 0 0
\(25\) 75.0000 0.600000
\(26\) − 219.014i − 1.65201i
\(27\) 0 0
\(28\) − 178.924i − 1.20762i
\(29\) 124.132 0.794851 0.397425 − 0.917635i \(-0.369904\pi\)
0.397425 + 0.917635i \(0.369904\pi\)
\(30\) 0 0
\(31\) −149.040 −0.863493 −0.431747 − 0.901995i \(-0.642103\pi\)
−0.431747 + 0.901995i \(0.642103\pi\)
\(32\) −92.9950 −0.513729
\(33\) 0 0
\(34\) −578.079 −2.91588
\(35\) 74.0451 0.357597
\(36\) 0 0
\(37\) 161.733 0.718613 0.359306 − 0.933220i \(-0.383013\pi\)
0.359306 + 0.933220i \(0.383013\pi\)
\(38\) − 447.315i − 1.90958i
\(39\) 0 0
\(40\) − 321.817i − 1.27209i
\(41\) −172.049 −0.655353 −0.327677 − 0.944790i \(-0.606266\pi\)
−0.327677 + 0.944790i \(0.606266\pi\)
\(42\) 0 0
\(43\) 258.081i 0.915277i 0.889138 + 0.457639i \(0.151305\pi\)
−0.889138 + 0.457639i \(0.848695\pi\)
\(44\) −223.220 − 582.033i −0.764811 − 1.99420i
\(45\) 0 0
\(46\) − 1068.63i − 3.42525i
\(47\) 240.416i 0.746135i 0.927804 + 0.373067i \(0.121694\pi\)
−0.927804 + 0.373067i \(0.878306\pi\)
\(48\) 0 0
\(49\) 233.347 0.680311
\(50\) −375.649 −1.06250
\(51\) 0 0
\(52\) 747.149i 1.99252i
\(53\) 334.790i 0.867679i 0.900990 + 0.433840i \(0.142842\pi\)
−0.900990 + 0.433840i \(0.857158\pi\)
\(54\) 0 0
\(55\) 240.866 − 92.3765i 0.590516 − 0.226474i
\(56\) 476.579i 1.13724i
\(57\) 0 0
\(58\) −621.733 −1.40754
\(59\) − 382.685i − 0.844429i −0.906496 − 0.422214i \(-0.861253\pi\)
0.906496 − 0.422214i \(-0.138747\pi\)
\(60\) 0 0
\(61\) 609.752i 1.27985i 0.768438 + 0.639924i \(0.221034\pi\)
−0.768438 + 0.639924i \(0.778966\pi\)
\(62\) 746.488 1.52910
\(63\) 0 0
\(64\) −264.299 −0.516210
\(65\) −309.197 −0.590019
\(66\) 0 0
\(67\) 260.000 0.474090 0.237045 − 0.971499i \(-0.423821\pi\)
0.237045 + 0.971499i \(0.423821\pi\)
\(68\) 1972.07 3.51689
\(69\) 0 0
\(70\) −370.866 −0.633243
\(71\) − 17.3487i − 0.0289988i −0.999895 − 0.0144994i \(-0.995385\pi\)
0.999895 − 0.0144994i \(-0.00461546\pi\)
\(72\) 0 0
\(73\) − 787.834i − 1.26314i −0.775320 − 0.631569i \(-0.782411\pi\)
0.775320 − 0.631569i \(-0.217589\pi\)
\(74\) −810.063 −1.27254
\(75\) 0 0
\(76\) 1525.98i 2.30318i
\(77\) −356.699 + 136.801i −0.527918 + 0.202466i
\(78\) 0 0
\(79\) − 1000.93i − 1.42549i −0.701422 − 0.712746i \(-0.747451\pi\)
0.701422 − 0.712746i \(-0.252549\pi\)
\(80\) 645.305i 0.901841i
\(81\) 0 0
\(82\) 861.733 1.16052
\(83\) 183.820 0.243095 0.121548 − 0.992586i \(-0.461214\pi\)
0.121548 + 0.992586i \(0.461214\pi\)
\(84\) 0 0
\(85\) 816.115i 1.04141i
\(86\) − 1292.64i − 1.62080i
\(87\) 0 0
\(88\) 594.567 + 1550.30i 0.720239 + 1.87798i
\(89\) 11.1459i 0.0132748i 0.999978 + 0.00663741i \(0.00211277\pi\)
−0.999978 + 0.00663741i \(0.997887\pi\)
\(90\) 0 0
\(91\) 457.891 0.527473
\(92\) 3645.56i 4.13125i
\(93\) 0 0
\(94\) − 1204.16i − 1.32128i
\(95\) −631.506 −0.682012
\(96\) 0 0
\(97\) 1795.20 1.87912 0.939560 − 0.342383i \(-0.111234\pi\)
0.939560 + 0.342383i \(0.111234\pi\)
\(98\) −1168.75 −1.20471
\(99\) 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 99.4.d.c.98.2 yes 8
3.2 odd 2 inner 99.4.d.c.98.7 yes 8
4.3 odd 2 1584.4.b.g.593.7 8
11.10 odd 2 inner 99.4.d.c.98.8 yes 8
12.11 even 2 1584.4.b.g.593.3 8
33.32 even 2 inner 99.4.d.c.98.1 ✓ 8
44.43 even 2 1584.4.b.g.593.6 8
132.131 odd 2 1584.4.b.g.593.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.4.d.c.98.1 ✓ 8 33.32 even 2 inner
99.4.d.c.98.2 yes 8 1.1 even 1 trivial
99.4.d.c.98.7 yes 8 3.2 odd 2 inner
99.4.d.c.98.8 yes 8 11.10 odd 2 inner
1584.4.b.g.593.2 8 132.131 odd 2
1584.4.b.g.593.3 8 12.11 even 2
1584.4.b.g.593.6 8 44.43 even 2
1584.4.b.g.593.7 8 4.3 odd 2