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(8,99)] 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("99.8"); 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([5, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 62.3
Root \(0.752864 + 0.902863i\) of defining polynomial
Character \(\chi\) \(=\) 99.62
Dual form 99.2.j.a.8.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.205228 + 0.149107i) q^{2} +(-0.598148 - 1.84091i) q^{4} +(1.71735 + 2.36373i) q^{5} +(2.58586 - 0.840196i) q^{7} +(0.308515 - 0.949513i) q^{8} +0.741170i q^{10} +(-2.71994 - 1.89788i) q^{11} +(-1.67869 + 2.31052i) q^{13} +(0.655969 + 0.213137i) q^{14} +(-2.92705 + 2.12663i) q^{16} +(-3.60998 + 2.62280i) q^{17} +(-1.81761 - 0.590579i) q^{19} +(3.32418 - 4.57534i) q^{20} +(-0.275220 - 0.795058i) q^{22} +0.816370i q^{23} +(-1.09283 + 3.36340i) q^{25} +(-0.689029 + 0.223879i) q^{26} +(-3.09345 - 4.25777i) q^{28} +(-2.95006 - 9.07936i) q^{29} +(4.84281 + 3.51851i) q^{31} -2.91456 q^{32} -1.13195 q^{34} +(6.42681 + 4.66935i) q^{35} +(1.83750 + 5.65524i) q^{37} +(-0.284966 - 0.392222i) q^{38} +(2.77422 - 0.901398i) q^{40} +(-2.60604 + 8.02058i) q^{41} -11.8763i q^{43} +(-1.86690 + 6.14238i) q^{44} +(-0.121726 + 0.167542i) q^{46} +(7.34041 + 2.38504i) q^{47} +(0.317615 - 0.230761i) q^{49} +(-0.725785 + 0.527314i) q^{50} +(5.25758 + 1.70829i) q^{52} +(6.14564 - 8.45874i) q^{53} +(-0.185009 - 9.68851i) q^{55} -2.71452i q^{56} +(0.748358 - 2.30321i) q^{58} +(-0.0887332 + 0.0288312i) q^{59} +(1.26024 + 1.73457i) q^{61} +(0.469246 + 1.44419i) q^{62} +(5.25595 + 3.81867i) q^{64} -8.34434 q^{65} -1.40778 q^{67} +(6.98766 + 5.07683i) q^{68} +(0.622728 + 1.91656i) q^{70} +(2.12141 + 2.91987i) q^{71} +(-1.67338 + 0.543714i) q^{73} +(-0.466129 + 1.43460i) q^{74} +3.69932i q^{76} +(-8.62796 - 2.62237i) q^{77} +(-4.19260 + 5.77063i) q^{79} +(-10.0535 - 3.26659i) q^{80} +(-1.73075 + 1.25747i) q^{82} +(6.33808 - 4.60488i) q^{83} +(-12.3992 - 4.02874i) q^{85} +(1.77084 - 2.43735i) q^{86} +(-2.64120 + 1.99709i) q^{88} -2.06830i q^{89} +(-2.39957 + 7.38512i) q^{91} +(1.50286 - 0.488310i) q^{92} +(1.15083 + 1.58398i) q^{94} +(-1.72551 - 5.31057i) q^{95} +(-1.35141 - 0.981858i) q^{97} +0.0995913 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} - 20 q^{16} - 48 q^{22} - 32 q^{25} + 40 q^{28} + 16 q^{31} + 40 q^{34} - 12 q^{37} + 60 q^{40} - 40 q^{46} - 24 q^{49} - 40 q^{52} + 16 q^{55} + 12 q^{58} + 36 q^{64} + 96 q^{67} + 76 q^{70}+ \cdots + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(e\left(\frac{7}{10}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.205228 + 0.149107i 0.145118 + 0.105434i 0.657975 − 0.753039i \(-0.271413\pi\)
−0.512857 + 0.858474i \(0.671413\pi\)
\(3\) 0 0
\(4\) −0.598148 − 1.84091i −0.299074 − 0.920456i
\(5\) 1.71735 + 2.36373i 0.768021 + 1.05709i 0.996504 + 0.0835427i \(0.0266235\pi\)
−0.228483 + 0.973548i \(0.573376\pi\)
\(6\) 0 0
\(7\) 2.58586 − 0.840196i 0.977363 − 0.317564i 0.223578 − 0.974686i \(-0.428226\pi\)
0.753785 + 0.657122i \(0.228226\pi\)
\(8\) 0.308515 − 0.949513i 0.109077 − 0.335704i
\(9\) 0 0
\(10\) 0.741170i 0.234379i
\(11\) −2.71994 − 1.89788i −0.820092 − 0.572232i
\(12\) 0 0
\(13\) −1.67869 + 2.31052i −0.465586 + 0.640824i −0.975655 − 0.219309i \(-0.929620\pi\)
0.510070 + 0.860133i \(0.329620\pi\)
\(14\) 0.655969 + 0.213137i 0.175315 + 0.0569633i
\(15\) 0 0
\(16\) −2.92705 + 2.12663i −0.731763 + 0.531657i
\(17\) −3.60998 + 2.62280i −0.875549 + 0.636124i −0.932070 − 0.362278i \(-0.881999\pi\)
0.0565211 + 0.998401i \(0.481999\pi\)
\(18\) 0 0
\(19\) −1.81761 − 0.590579i −0.416989 − 0.135488i 0.0930053 − 0.995666i \(-0.470353\pi\)
−0.509995 + 0.860178i \(0.670353\pi\)
\(20\) 3.32418 − 4.57534i 0.743310 − 1.02308i
\(21\) 0 0
\(22\) −0.275220 − 0.795058i −0.0586771 − 0.169507i
\(23\) 0.816370i 0.170225i 0.996371 + 0.0851124i \(0.0271249\pi\)
−0.996371 + 0.0851124i \(0.972875\pi\)
\(24\) 0 0
\(25\) −1.09283 + 3.36340i −0.218567 + 0.672680i
\(26\) −0.689029 + 0.223879i −0.135130 + 0.0439063i
\(27\) 0 0
\(28\) −3.09345 − 4.25777i −0.584608 − 0.804644i
\(29\) −2.95006 − 9.07936i −0.547813 − 1.68599i −0.714207 − 0.699935i \(-0.753212\pi\)
0.166394 − 0.986059i \(-0.446788\pi\)
\(30\) 0 0
\(31\) 4.84281 + 3.51851i 0.869795 + 0.631943i 0.930532 − 0.366211i \(-0.119345\pi\)
−0.0607369 + 0.998154i \(0.519345\pi\)
\(32\) −2.91456 −0.515226
\(33\) 0 0
\(34\) −1.13195 −0.194127
\(35\) 6.42681 + 4.66935i 1.08633 + 0.789265i
\(36\) 0 0
\(37\) 1.83750 + 5.65524i 0.302083 + 0.929717i 0.980750 + 0.195270i \(0.0625583\pi\)
−0.678666 + 0.734447i \(0.737442\pi\)
\(38\) −0.284966 − 0.392222i −0.0462275 − 0.0636267i
\(39\) 0 0
\(40\) 2.77422 − 0.901398i 0.438642 − 0.142524i
\(41\) −2.60604 + 8.02058i −0.406996 + 1.25260i 0.512221 + 0.858853i \(0.328823\pi\)
−0.919217 + 0.393751i \(0.871177\pi\)
\(42\) 0 0
\(43\) − 11.8763i − 1.81112i −0.424214 − 0.905562i \(-0.639450\pi\)
0.424214 − 0.905562i \(-0.360550\pi\)
\(44\) −1.86690 + 6.14238i −0.281446 + 0.925998i
\(45\) 0 0
\(46\) −0.121726 + 0.167542i −0.0179475 + 0.0247027i
\(47\) 7.34041 + 2.38504i 1.07071 + 0.347894i 0.790764 − 0.612122i \(-0.209684\pi\)
0.279945 + 0.960016i \(0.409684\pi\)
\(48\) 0 0
\(49\) 0.317615 − 0.230761i 0.0453735 − 0.0329658i
\(50\) −0.725785 + 0.527314i −0.102641 + 0.0745734i
\(51\) 0 0
\(52\) 5.25758 + 1.70829i 0.729094 + 0.236897i
\(53\) 6.14564 − 8.45874i 0.844168 − 1.16190i −0.140950 − 0.990017i \(-0.545016\pi\)
0.985118 − 0.171881i \(-0.0549845\pi\)
\(54\) 0 0
\(55\) −0.185009 − 9.68851i −0.0249466 − 1.30640i
\(56\) − 2.71452i − 0.362743i
\(57\) 0 0
\(58\) 0.748358 − 2.30321i 0.0982642 − 0.302426i
\(59\) −0.0887332 + 0.0288312i −0.0115521 + 0.00375350i −0.314787 − 0.949162i \(-0.601933\pi\)
0.303235 + 0.952916i \(0.401933\pi\)
\(60\) 0 0
\(61\) 1.26024 + 1.73457i 0.161357 + 0.222089i 0.882038 − 0.471177i \(-0.156171\pi\)
−0.720681 + 0.693267i \(0.756171\pi\)
\(62\) 0.469246 + 1.44419i 0.0595943 + 0.183412i
\(63\) 0 0
\(64\) 5.25595 + 3.81867i 0.656994 + 0.477334i
\(65\) −8.34434 −1.03499
\(66\) 0 0
\(67\) −1.40778 −0.171988 −0.0859941 − 0.996296i \(-0.527407\pi\)
−0.0859941 + 0.996296i \(0.527407\pi\)
\(68\) 6.98766 + 5.07683i 0.847378 + 0.615656i
\(69\) 0 0
\(70\) 0.622728 + 1.91656i 0.0744303 + 0.229073i
\(71\) 2.12141 + 2.91987i 0.251765 + 0.346525i 0.916128 − 0.400885i \(-0.131297\pi\)
−0.664363 + 0.747410i \(0.731297\pi\)
\(72\) 0 0
\(73\) −1.67338 + 0.543714i −0.195854 + 0.0636369i −0.405302 − 0.914183i \(-0.632833\pi\)
0.209447 + 0.977820i \(0.432833\pi\)
\(74\) −0.466129 + 1.43460i −0.0541863 + 0.166768i
\(75\) 0 0
\(76\) 3.69932i 0.424341i
\(77\) −8.62796 − 2.62237i −0.983247 − 0.298846i
\(78\) 0 0
\(79\) −4.19260 + 5.77063i −0.471705 + 0.649246i −0.976884 − 0.213768i \(-0.931426\pi\)
0.505180 + 0.863014i \(0.331426\pi\)
\(80\) −10.0535 − 3.26659i −1.12402 − 0.365216i
\(81\) 0 0
\(82\) −1.73075 + 1.25747i −0.191130 + 0.138864i
\(83\) 6.33808 − 4.60488i 0.695695 − 0.505452i −0.182833 − 0.983144i \(-0.558527\pi\)
0.878527 + 0.477692i \(0.158527\pi\)
\(84\) 0 0
\(85\) −12.3992 − 4.02874i −1.34488 − 0.436978i
\(86\) 1.77084 − 2.43735i 0.190955 − 0.262826i
\(87\) 0 0
\(88\) −2.64120 + 1.99709i −0.281553 + 0.212890i
\(89\) − 2.06830i − 0.219240i −0.993974 − 0.109620i \(-0.965037\pi\)
0.993974 − 0.109620i \(-0.0349633\pi\)
\(90\) 0 0
\(91\) −2.39957 + 7.38512i −0.251543 + 0.774171i
\(92\) 1.50286 − 0.488310i 0.156684 − 0.0509099i
\(93\) 0 0
\(94\) 1.15083 + 1.58398i 0.118699 + 0.163375i
\(95\) −1.72551 − 5.31057i −0.177034 − 0.544853i
\(96\) 0 0
\(97\) −1.35141 − 0.981858i −0.137215 − 0.0996926i 0.517060 − 0.855949i \(-0.327026\pi\)
−0.654275 + 0.756256i \(0.727026\pi\)
\(98\) 0.0995913 0.0100602
\(99\) 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.j.a.62.3 yes 16
3.2 odd 2 inner 99.2.j.a.62.2 yes 16
4.3 odd 2 1584.2.cd.c.161.3 16
9.2 odd 6 891.2.u.c.458.2 32
9.4 even 3 891.2.u.c.755.2 32
9.5 odd 6 891.2.u.c.755.3 32
9.7 even 3 891.2.u.c.458.3 32
11.5 even 5 1089.2.d.g.1088.7 16
11.6 odd 10 1089.2.d.g.1088.9 16
11.8 odd 10 inner 99.2.j.a.8.2 ✓ 16
12.11 even 2 1584.2.cd.c.161.2 16
33.5 odd 10 1089.2.d.g.1088.10 16
33.8 even 10 inner 99.2.j.a.8.3 yes 16
33.17 even 10 1089.2.d.g.1088.8 16
44.19 even 10 1584.2.cd.c.305.2 16
99.41 even 30 891.2.u.c.107.3 32
99.52 odd 30 891.2.u.c.701.3 32
99.74 even 30 891.2.u.c.701.2 32
99.85 odd 30 891.2.u.c.107.2 32
132.107 odd 10 1584.2.cd.c.305.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.j.a.8.2 ✓ 16 11.8 odd 10 inner
99.2.j.a.8.3 yes 16 33.8 even 10 inner
99.2.j.a.62.2 yes 16 3.2 odd 2 inner
99.2.j.a.62.3 yes 16 1.1 even 1 trivial
891.2.u.c.107.2 32 99.85 odd 30
891.2.u.c.107.3 32 99.41 even 30
891.2.u.c.458.2 32 9.2 odd 6
891.2.u.c.458.3 32 9.7 even 3
891.2.u.c.701.2 32 99.74 even 30
891.2.u.c.701.3 32 99.52 odd 30
891.2.u.c.755.2 32 9.4 even 3
891.2.u.c.755.3 32 9.5 odd 6
1089.2.d.g.1088.7 16 11.5 even 5
1089.2.d.g.1088.8 16 33.17 even 10
1089.2.d.g.1088.9 16 11.6 odd 10
1089.2.d.g.1088.10 16 33.5 odd 10
1584.2.cd.c.161.2 16 12.11 even 2
1584.2.cd.c.161.3 16 4.3 odd 2
1584.2.cd.c.305.2 16 44.19 even 10
1584.2.cd.c.305.3 16 132.107 odd 10