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 91.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 99.91
Dual form 99.2.f.a.37.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.309017 + 0.224514i) q^{2} +(-0.572949 + 1.76336i) q^{4} +(1.30902 + 0.951057i) q^{5} +(-0.309017 + 0.951057i) q^{7} +(-0.454915 - 1.40008i) q^{8} -0.618034 q^{10} +(2.19098 + 2.48990i) q^{11} +(3.42705 - 2.48990i) q^{13} +(-0.118034 - 0.363271i) q^{14} +(-2.54508 - 1.84911i) q^{16} +(-6.35410 - 4.61653i) q^{17} +(-0.263932 - 0.812299i) q^{19} +(-2.42705 + 1.76336i) q^{20} +(-1.23607 - 0.277515i) q^{22} +4.23607 q^{23} +(-0.736068 - 2.26538i) q^{25} +(-0.500000 + 1.53884i) q^{26} +(-1.50000 - 1.08981i) q^{28} +(1.85410 - 5.70634i) q^{29} +(-4.11803 + 2.99193i) q^{31} +4.14590 q^{32} +3.00000 q^{34} +(-1.30902 + 0.951057i) q^{35} +(-0.545085 + 1.67760i) q^{37} +(0.263932 + 0.191758i) q^{38} +(0.736068 - 2.26538i) q^{40} +(1.30902 + 4.02874i) q^{41} +6.70820 q^{43} +(-5.64590 + 2.43690i) q^{44} +(-1.30902 + 0.951057i) q^{46} +(-0.336881 - 1.03681i) q^{47} +(4.85410 + 3.52671i) q^{49} +(0.736068 + 0.534785i) q^{50} +(2.42705 + 7.46969i) q^{52} +(-2.11803 + 1.53884i) q^{53} +(0.500000 + 5.34307i) q^{55} +1.47214 q^{56} +(0.708204 + 2.17963i) q^{58} +(-2.97214 + 9.14729i) q^{59} +(-6.92705 - 5.03280i) q^{61} +(0.600813 - 1.84911i) q^{62} +(3.80902 - 2.76741i) q^{64} +6.85410 q^{65} -4.85410 q^{67} +(11.7812 - 8.55951i) q^{68} +(0.190983 - 0.587785i) q^{70} +(-4.30902 - 3.13068i) q^{71} +(2.38197 - 7.33094i) q^{73} +(-0.208204 - 0.640786i) q^{74} +1.58359 q^{76} +(-3.04508 + 1.31433i) q^{77} +(-8.89919 + 6.46564i) q^{79} +(-1.57295 - 4.84104i) q^{80} +(-1.30902 - 0.951057i) q^{82} +(-6.04508 - 4.39201i) q^{83} +(-3.92705 - 12.0862i) q^{85} +(-2.07295 + 1.50609i) q^{86} +(2.48936 - 4.20025i) q^{88} +3.76393 q^{89} +(1.30902 + 4.02874i) q^{91} +(-2.42705 + 7.46969i) q^{92} +(0.336881 + 0.244758i) q^{94} +(0.427051 - 1.31433i) q^{95} +(-0.927051 + 0.673542i) q^{97} -2.29180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 9 q^{4} + 3 q^{5} + q^{7} - 13 q^{8} + 2 q^{10} + 11 q^{11} + 7 q^{13} + 4 q^{14} + q^{16} - 12 q^{17} - 10 q^{19} - 3 q^{20} + 4 q^{22} + 8 q^{23} + 6 q^{25} - 2 q^{26} - 6 q^{28} - 6 q^{29}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.309017 + 0.224514i −0.218508 + 0.158755i −0.691655 − 0.722228i \(-0.743118\pi\)
0.473147 + 0.880984i \(0.343118\pi\)
\(3\) 0 0
\(4\) −0.572949 + 1.76336i −0.286475 + 0.881678i
\(5\) 1.30902 + 0.951057i 0.585410 + 0.425325i 0.840670 − 0.541547i \(-0.182161\pi\)
−0.255260 + 0.966872i \(0.582161\pi\)
\(6\) 0 0
\(7\) −0.309017 + 0.951057i −0.116797 + 0.359466i −0.992318 − 0.123716i \(-0.960519\pi\)
0.875520 + 0.483181i \(0.160519\pi\)
\(8\) −0.454915 − 1.40008i −0.160837 − 0.495005i
\(9\) 0 0
\(10\) −0.618034 −0.195440
\(11\) 2.19098 + 2.48990i 0.660606 + 0.750733i
\(12\) 0 0
\(13\) 3.42705 − 2.48990i 0.950493 − 0.690574i −0.000430477 − 1.00000i \(-0.500137\pi\)
0.950923 + 0.309426i \(0.100137\pi\)
\(14\) −0.118034 − 0.363271i −0.0315459 − 0.0970883i
\(15\) 0 0
\(16\) −2.54508 − 1.84911i −0.636271 − 0.462278i
\(17\) −6.35410 − 4.61653i −1.54110 − 1.11967i −0.949644 − 0.313332i \(-0.898555\pi\)
−0.591452 − 0.806340i \(-0.701445\pi\)
\(18\) 0 0
\(19\) −0.263932 − 0.812299i −0.0605502 − 0.186354i 0.916206 − 0.400707i \(-0.131236\pi\)
−0.976756 + 0.214353i \(0.931236\pi\)
\(20\) −2.42705 + 1.76336i −0.542705 + 0.394298i
\(21\) 0 0
\(22\) −1.23607 − 0.277515i −0.263531 − 0.0591663i
\(23\) 4.23607 0.883281 0.441641 − 0.897192i \(-0.354397\pi\)
0.441641 + 0.897192i \(0.354397\pi\)
\(24\) 0 0
\(25\) −0.736068 − 2.26538i −0.147214 − 0.453077i
\(26\) −0.500000 + 1.53884i −0.0980581 + 0.301792i
\(27\) 0 0
\(28\) −1.50000 − 1.08981i −0.283473 − 0.205955i
\(29\) 1.85410 − 5.70634i 0.344298 − 1.05964i −0.617660 − 0.786445i \(-0.711919\pi\)
0.961958 − 0.273196i \(-0.0880806\pi\)
\(30\) 0 0
\(31\) −4.11803 + 2.99193i −0.739621 + 0.537366i −0.892592 − 0.450865i \(-0.851116\pi\)
0.152972 + 0.988231i \(0.451116\pi\)
\(32\) 4.14590 0.732898
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) −1.30902 + 0.951057i −0.221264 + 0.160758i
\(36\) 0 0
\(37\) −0.545085 + 1.67760i −0.0896114 + 0.275796i −0.985812 − 0.167854i \(-0.946316\pi\)
0.896201 + 0.443649i \(0.146316\pi\)
\(38\) 0.263932 + 0.191758i 0.0428154 + 0.0311072i
\(39\) 0 0
\(40\) 0.736068 − 2.26538i 0.116383 − 0.358189i
\(41\) 1.30902 + 4.02874i 0.204434 + 0.629183i 0.999736 + 0.0229701i \(0.00731226\pi\)
−0.795302 + 0.606213i \(0.792688\pi\)
\(42\) 0 0
\(43\) 6.70820 1.02299 0.511496 − 0.859286i \(-0.329092\pi\)
0.511496 + 0.859286i \(0.329092\pi\)
\(44\) −5.64590 + 2.43690i −0.851151 + 0.367376i
\(45\) 0 0
\(46\) −1.30902 + 0.951057i −0.193004 + 0.140226i
\(47\) −0.336881 − 1.03681i −0.0491391 − 0.151235i 0.923476 − 0.383656i \(-0.125335\pi\)
−0.972615 + 0.232421i \(0.925335\pi\)
\(48\) 0 0
\(49\) 4.85410 + 3.52671i 0.693443 + 0.503816i
\(50\) 0.736068 + 0.534785i 0.104096 + 0.0756300i
\(51\) 0 0
\(52\) 2.42705 + 7.46969i 0.336571 + 1.03586i
\(53\) −2.11803 + 1.53884i −0.290934 + 0.211376i −0.723673 − 0.690143i \(-0.757547\pi\)
0.432738 + 0.901520i \(0.357547\pi\)
\(54\) 0 0
\(55\) 0.500000 + 5.34307i 0.0674200 + 0.720459i
\(56\) 1.47214 0.196722
\(57\) 0 0
\(58\) 0.708204 + 2.17963i 0.0929917 + 0.286199i
\(59\) −2.97214 + 9.14729i −0.386939 + 1.19088i 0.548125 + 0.836397i \(0.315342\pi\)
−0.935064 + 0.354480i \(0.884658\pi\)
\(60\) 0 0
\(61\) −6.92705 − 5.03280i −0.886918 − 0.644384i 0.0481546 − 0.998840i \(-0.484666\pi\)
−0.935073 + 0.354456i \(0.884666\pi\)
\(62\) 0.600813 − 1.84911i 0.0763033 − 0.234838i
\(63\) 0 0
\(64\) 3.80902 − 2.76741i 0.476127 − 0.345927i
\(65\) 6.85410 0.850147
\(66\) 0 0
\(67\) −4.85410 −0.593023 −0.296511 − 0.955029i \(-0.595823\pi\)
−0.296511 + 0.955029i \(0.595823\pi\)
\(68\) 11.7812 − 8.55951i 1.42867 − 1.03799i
\(69\) 0 0
\(70\) 0.190983 − 0.587785i 0.0228268 − 0.0702538i
\(71\) −4.30902 − 3.13068i −0.511386 − 0.371544i 0.301963 − 0.953320i \(-0.402358\pi\)
−0.813349 + 0.581776i \(0.802358\pi\)
\(72\) 0 0
\(73\) 2.38197 − 7.33094i 0.278788 − 0.858021i −0.709404 − 0.704802i \(-0.751036\pi\)
0.988192 − 0.153219i \(-0.0489641\pi\)
\(74\) −0.208204 − 0.640786i −0.0242032 − 0.0744898i
\(75\) 0 0
\(76\) 1.58359 0.181650
\(77\) −3.04508 + 1.31433i −0.347020 + 0.149782i
\(78\) 0 0
\(79\) −8.89919 + 6.46564i −1.00124 + 0.727441i −0.962353 − 0.271803i \(-0.912380\pi\)
−0.0388837 + 0.999244i \(0.512380\pi\)
\(80\) −1.57295 − 4.84104i −0.175861 − 0.541245i
\(81\) 0 0
\(82\) −1.30902 − 0.951057i −0.144557 − 0.105027i
\(83\) −6.04508 − 4.39201i −0.663534 − 0.482086i 0.204320 − 0.978904i \(-0.434502\pi\)
−0.867855 + 0.496818i \(0.834502\pi\)
\(84\) 0 0
\(85\) −3.92705 − 12.0862i −0.425948 − 1.31093i
\(86\) −2.07295 + 1.50609i −0.223532 + 0.162405i
\(87\) 0 0
\(88\) 2.48936 − 4.20025i 0.265366 − 0.447749i
\(89\) 3.76393 0.398976 0.199488 − 0.979900i \(-0.436072\pi\)
0.199488 + 0.979900i \(0.436072\pi\)
\(90\) 0 0
\(91\) 1.30902 + 4.02874i 0.137222 + 0.422327i
\(92\) −2.42705 + 7.46969i −0.253038 + 0.778770i
\(93\) 0 0
\(94\) 0.336881 + 0.244758i 0.0347466 + 0.0252449i
\(95\) 0.427051 − 1.31433i 0.0438145 − 0.134847i
\(96\) 0 0
\(97\) −0.927051 + 0.673542i −0.0941278 + 0.0683878i −0.633854 − 0.773453i \(-0.718528\pi\)
0.539726 + 0.841841i \(0.318528\pi\)
\(98\) −2.29180 −0.231506
\(99\) 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.91.1 4
3.2 odd 2 33.2.e.b.25.1 yes 4
9.2 odd 6 891.2.n.c.190.1 8
9.4 even 3 891.2.n.b.784.1 8
9.5 odd 6 891.2.n.c.784.1 8
9.7 even 3 891.2.n.b.190.1 8
11.2 odd 10 1089.2.a.l.1.2 2
11.4 even 5 inner 99.2.f.a.37.1 4
11.9 even 5 1089.2.a.t.1.1 2
12.11 even 2 528.2.y.b.289.1 4
15.2 even 4 825.2.bx.d.124.2 8
15.8 even 4 825.2.bx.d.124.1 8
15.14 odd 2 825.2.n.c.751.1 4
33.2 even 10 363.2.a.i.1.1 2
33.5 odd 10 363.2.e.k.130.1 4
33.8 even 10 363.2.e.b.148.1 4
33.14 odd 10 363.2.e.k.148.1 4
33.17 even 10 363.2.e.b.130.1 4
33.20 odd 10 363.2.a.d.1.2 2
33.26 odd 10 33.2.e.b.4.1 ✓ 4
33.29 even 10 363.2.e.f.202.1 4
33.32 even 2 363.2.e.f.124.1 4
99.4 even 15 891.2.n.b.136.1 8
99.59 odd 30 891.2.n.c.136.1 8
99.70 even 15 891.2.n.b.433.1 8
99.92 odd 30 891.2.n.c.433.1 8
132.35 odd 10 5808.2.a.ci.1.2 2
132.59 even 10 528.2.y.b.433.1 4
132.119 even 10 5808.2.a.cj.1.2 2
165.59 odd 10 825.2.n.c.301.1 4
165.92 even 20 825.2.bx.d.499.1 8
165.119 odd 10 9075.2.a.cb.1.1 2
165.134 even 10 9075.2.a.u.1.2 2
165.158 even 20 825.2.bx.d.499.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.4.1 ✓ 4 33.26 odd 10
33.2.e.b.25.1 yes 4 3.2 odd 2
99.2.f.a.37.1 4 11.4 even 5 inner
99.2.f.a.91.1 4 1.1 even 1 trivial
363.2.a.d.1.2 2 33.20 odd 10
363.2.a.i.1.1 2 33.2 even 10
363.2.e.b.130.1 4 33.17 even 10
363.2.e.b.148.1 4 33.8 even 10
363.2.e.f.124.1 4 33.32 even 2
363.2.e.f.202.1 4 33.29 even 10
363.2.e.k.130.1 4 33.5 odd 10
363.2.e.k.148.1 4 33.14 odd 10
528.2.y.b.289.1 4 12.11 even 2
528.2.y.b.433.1 4 132.59 even 10
825.2.n.c.301.1 4 165.59 odd 10
825.2.n.c.751.1 4 15.14 odd 2
825.2.bx.d.124.1 8 15.8 even 4
825.2.bx.d.124.2 8 15.2 even 4
825.2.bx.d.499.1 8 165.92 even 20
825.2.bx.d.499.2 8 165.158 even 20
891.2.n.b.136.1 8 99.4 even 15
891.2.n.b.190.1 8 9.7 even 3
891.2.n.b.433.1 8 99.70 even 15
891.2.n.b.784.1 8 9.4 even 3
891.2.n.c.136.1 8 99.59 odd 30
891.2.n.c.190.1 8 9.2 odd 6
891.2.n.c.433.1 8 99.92 odd 30
891.2.n.c.784.1 8 9.5 odd 6
1089.2.a.l.1.2 2 11.2 odd 10
1089.2.a.t.1.1 2 11.9 even 5
5808.2.a.ci.1.2 2 132.35 odd 10
5808.2.a.cj.1.2 2 132.119 even 10
9075.2.a.u.1.2 2 165.134 even 10
9075.2.a.cb.1.1 2 165.119 odd 10