Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [891,2,Mod(136,891)] 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("891.136"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(891, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.n (of order \(15\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-1,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) 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: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 757.1
Root \(0.669131 + 0.743145i\) of defining polynomial
Character \(\chi\) \(=\) 891.757
Dual form 891.2.n.b.379.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.56082 + 0.544320i) q^{2} +(4.43444 - 1.97434i) q^{4} +(-0.604528 - 0.128496i) q^{5} +(0.104528 - 0.994522i) q^{7} +(-6.04508 + 4.39201i) q^{8} +1.61803 q^{10} +(-1.46007 - 2.97795i) q^{11} +(0.157960 + 0.175433i) q^{13} +(0.273659 + 2.60369i) q^{14} +(6.59368 - 7.32302i) q^{16} +(0.354102 + 1.08981i) q^{17} +(-4.73607 + 3.44095i) q^{19} +(-2.93444 + 0.623735i) q^{20} +(5.35995 + 6.83126i) q^{22} +(0.118034 - 0.204441i) q^{23} +(-4.21878 - 1.87832i) q^{25} +(-0.500000 - 0.363271i) q^{26} +(-1.50000 - 4.61653i) q^{28} +(-0.627171 + 5.96713i) q^{29} +(-4.07512 - 4.52588i) q^{31} +(-5.42705 + 9.39993i) q^{32} +(-1.50000 - 2.59808i) q^{34} +(-0.190983 + 0.587785i) q^{35} +(5.04508 + 3.66547i) q^{37} +(10.2553 - 11.3896i) q^{38} +(4.21878 - 1.87832i) q^{40} +(0.0246758 + 0.234775i) q^{41} +(3.35410 + 5.80948i) q^{43} +(-12.3541 - 10.3229i) q^{44} +(-0.190983 + 0.587785i) q^{46} +(9.21783 + 4.10404i) q^{47} +(5.86889 + 1.24747i) q^{49} +(11.8260 + 2.51369i) q^{50} +(1.04683 + 0.466079i) q^{52} +(0.118034 - 0.363271i) q^{53} +(0.500000 + 1.98787i) q^{55} +(3.73607 + 6.47106i) q^{56} +(-1.64195 - 15.6222i) q^{58} +(-6.74376 + 3.00252i) q^{59} +(-7.73669 + 8.59247i) q^{61} +(12.8992 + 9.37181i) q^{62} +(2.69098 - 8.28199i) q^{64} +(-0.0729490 - 0.126351i) q^{65} +(-0.927051 + 1.60570i) q^{67} +(3.72191 + 4.13360i) q^{68} +(0.169131 - 1.60917i) q^{70} +(-3.19098 - 9.82084i) q^{71} +(4.61803 + 3.35520i) q^{73} +(-14.9148 - 6.64048i) q^{74} +(-14.2082 + 24.6093i) q^{76} +(-3.11426 + 1.14079i) q^{77} +(-10.7596 + 2.28703i) q^{79} +(-4.92705 + 3.57971i) q^{80} +(-0.190983 - 0.587785i) q^{82} +(-0.985051 + 1.09401i) q^{83} +(-0.0740275 - 0.704324i) q^{85} +(-11.7515 - 13.0513i) q^{86} +(21.9055 + 11.5893i) q^{88} +8.23607 q^{89} +(0.190983 - 0.138757i) q^{91} +(0.119779 - 1.13962i) q^{92} +(-25.8391 - 5.49228i) q^{94} +(3.30524 - 1.47159i) q^{95} +(-7.68247 + 1.63296i) q^{97} -15.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 9 q^{4} - 3 q^{5} - q^{7} - 26 q^{8} + 4 q^{10} - 11 q^{11} - 7 q^{13} - 4 q^{14} - q^{16} - 24 q^{17} - 20 q^{19} + 3 q^{20} - 4 q^{22} - 8 q^{23} - 6 q^{25} - 4 q^{26} - 12 q^{28} + 6 q^{29}+ \cdots - 72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(650\)
\(\chi(n)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.56082 + 0.544320i −1.81078 + 0.384892i −0.984076 − 0.177750i \(-0.943118\pi\)
−0.826700 + 0.562643i \(0.809785\pi\)
\(3\) 0 0
\(4\) 4.43444 − 1.97434i 2.21722 − 0.987171i
\(5\) −0.604528 − 0.128496i −0.270353 − 0.0574654i 0.0707401 − 0.997495i \(-0.477464\pi\)
−0.341093 + 0.940029i \(0.610797\pi\)
\(6\) 0 0
\(7\) 0.104528 − 0.994522i 0.0395080 − 0.375894i −0.956847 − 0.290592i \(-0.906148\pi\)
0.996355 − 0.0853021i \(-0.0271855\pi\)
\(8\) −6.04508 + 4.39201i −2.13726 + 1.55281i
\(9\) 0 0
\(10\) 1.61803 0.511667
\(11\) −1.46007 − 2.97795i −0.440229 − 0.897886i
\(12\) 0 0
\(13\) 0.157960 + 0.175433i 0.0438103 + 0.0486563i 0.764652 − 0.644444i \(-0.222911\pi\)
−0.720841 + 0.693100i \(0.756244\pi\)
\(14\) 0.273659 + 2.60369i 0.0731385 + 0.695866i
\(15\) 0 0
\(16\) 6.59368 − 7.32302i 1.64842 − 1.83076i
\(17\) 0.354102 + 1.08981i 0.0858823 + 0.264319i 0.984770 − 0.173860i \(-0.0556239\pi\)
−0.898888 + 0.438178i \(0.855624\pi\)
\(18\) 0 0
\(19\) −4.73607 + 3.44095i −1.08653 + 0.789409i −0.978810 − 0.204772i \(-0.934355\pi\)
−0.107719 + 0.994181i \(0.534355\pi\)
\(20\) −2.93444 + 0.623735i −0.656161 + 0.139471i
\(21\) 0 0
\(22\) 5.35995 + 6.83126i 1.14274 + 1.45643i
\(23\) 0.118034 − 0.204441i 0.0246118 − 0.0426289i −0.853457 − 0.521163i \(-0.825498\pi\)
0.878069 + 0.478534i \(0.158832\pi\)
\(24\) 0 0
\(25\) −4.21878 − 1.87832i −0.843757 − 0.375665i
\(26\) −0.500000 − 0.363271i −0.0980581 − 0.0712434i
\(27\) 0 0
\(28\) −1.50000 − 4.61653i −0.283473 − 0.872441i
\(29\) −0.627171 + 5.96713i −0.116463 + 1.10807i 0.767674 + 0.640840i \(0.221414\pi\)
−0.884137 + 0.467228i \(0.845253\pi\)
\(30\) 0 0
\(31\) −4.07512 − 4.52588i −0.731913 − 0.812872i 0.256196 − 0.966625i \(-0.417531\pi\)
−0.988109 + 0.153753i \(0.950864\pi\)
\(32\) −5.42705 + 9.39993i −0.959376 + 1.66169i
\(33\) 0 0
\(34\) −1.50000 − 2.59808i −0.257248 − 0.445566i
\(35\) −0.190983 + 0.587785i −0.0322820 + 0.0993538i
\(36\) 0 0
\(37\) 5.04508 + 3.66547i 0.829407 + 0.602599i 0.919391 − 0.393344i \(-0.128682\pi\)
−0.0899846 + 0.995943i \(0.528682\pi\)
\(38\) 10.2553 − 11.3896i 1.66362 − 1.84764i
\(39\) 0 0
\(40\) 4.21878 − 1.87832i 0.667048 − 0.296989i
\(41\) 0.0246758 + 0.234775i 0.00385372 + 0.0366657i 0.996280 − 0.0861779i \(-0.0274653\pi\)
−0.992426 + 0.122844i \(0.960799\pi\)
\(42\) 0 0
\(43\) 3.35410 + 5.80948i 0.511496 + 0.885937i 0.999911 + 0.0133254i \(0.00424174\pi\)
−0.488415 + 0.872611i \(0.662425\pi\)
\(44\) −12.3541 − 10.3229i −1.86245 − 1.55623i
\(45\) 0 0
\(46\) −0.190983 + 0.587785i −0.0281589 + 0.0866642i
\(47\) 9.21783 + 4.10404i 1.34456 + 0.598636i 0.947676 − 0.319233i \(-0.103425\pi\)
0.396882 + 0.917869i \(0.370092\pi\)
\(48\) 0 0
\(49\) 5.86889 + 1.24747i 0.838412 + 0.178210i
\(50\) 11.8260 + 2.51369i 1.67244 + 0.355489i
\(51\) 0 0
\(52\) 1.04683 + 0.466079i 0.145169 + 0.0646335i
\(53\) 0.118034 − 0.363271i 0.0162132 − 0.0498991i −0.942623 − 0.333860i \(-0.891649\pi\)
0.958836 + 0.283961i \(0.0916486\pi\)
\(54\) 0 0
\(55\) 0.500000 + 1.98787i 0.0674200 + 0.268044i
\(56\) 3.73607 + 6.47106i 0.499253 + 0.864732i
\(57\) 0 0
\(58\) −1.64195 − 15.6222i −0.215599 − 2.05129i
\(59\) −6.74376 + 3.00252i −0.877963 + 0.390894i −0.795682 − 0.605715i \(-0.792887\pi\)
−0.0822812 + 0.996609i \(0.526221\pi\)
\(60\) 0 0
\(61\) −7.73669 + 8.59247i −0.990582 + 1.10015i 0.00438910 + 0.999990i \(0.498603\pi\)
−0.994971 + 0.100162i \(0.968064\pi\)
\(62\) 12.8992 + 9.37181i 1.63820 + 1.19022i
\(63\) 0 0
\(64\) 2.69098 − 8.28199i 0.336373 − 1.03525i
\(65\) −0.0729490 − 0.126351i −0.00904821 − 0.0156720i
\(66\) 0 0
\(67\) −0.927051 + 1.60570i −0.113257 + 0.196167i −0.917082 − 0.398699i \(-0.869462\pi\)
0.803824 + 0.594867i \(0.202795\pi\)
\(68\) 3.72191 + 4.13360i 0.451348 + 0.501272i
\(69\) 0 0
\(70\) 0.169131 − 1.60917i 0.0202150 − 0.192333i
\(71\) −3.19098 − 9.82084i −0.378700 − 1.16552i −0.940948 − 0.338550i \(-0.890063\pi\)
0.562248 − 0.826968i \(-0.309937\pi\)
\(72\) 0 0
\(73\) 4.61803 + 3.35520i 0.540500 + 0.392696i 0.824271 − 0.566196i \(-0.191585\pi\)
−0.283771 + 0.958892i \(0.591585\pi\)
\(74\) −14.9148 − 6.64048i −1.73381 − 0.771940i
\(75\) 0 0
\(76\) −14.2082 + 24.6093i −1.62979 + 2.82288i
\(77\) −3.11426 + 1.14079i −0.354902 + 0.130006i
\(78\) 0 0
\(79\) −10.7596 + 2.28703i −1.21055 + 0.257311i −0.768601 − 0.639728i \(-0.779047\pi\)
−0.441951 + 0.897039i \(0.645714\pi\)
\(80\) −4.92705 + 3.57971i −0.550861 + 0.400224i
\(81\) 0 0
\(82\) −0.190983 − 0.587785i −0.0210905 − 0.0649100i
\(83\) −0.985051 + 1.09401i −0.108123 + 0.120083i −0.794777 − 0.606901i \(-0.792412\pi\)
0.686654 + 0.726985i \(0.259079\pi\)
\(84\) 0 0
\(85\) −0.0740275 − 0.704324i −0.00802941 − 0.0763947i
\(86\) −11.7515 − 13.0513i −1.26719 − 1.40736i
\(87\) 0 0
\(88\) 21.9055 + 11.5893i 2.33513 + 1.23542i
\(89\) 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
\(92\) 0.119779 − 1.13962i 0.0124878 − 0.118814i
\(93\) 0 0
\(94\) −25.8391 − 5.49228i −2.66510 − 0.566485i
\(95\) 3.30524 − 1.47159i 0.339110 − 0.150982i
\(96\) 0 0
\(97\) −7.68247 + 1.63296i −0.780037 + 0.165802i −0.580687 − 0.814127i \(-0.697216\pi\)
−0.199349 + 0.979928i \(0.563883\pi\)
\(98\) −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 891.2.n.b.757.1 8
3.2 odd 2 891.2.n.c.757.1 8
9.2 odd 6 891.2.n.c.460.1 8
9.4 even 3 99.2.f.a.64.1 4
9.5 odd 6 33.2.e.b.31.1 yes 4
9.7 even 3 inner 891.2.n.b.460.1 8
11.5 even 5 inner 891.2.n.b.676.1 8
33.5 odd 10 891.2.n.c.676.1 8
36.23 even 6 528.2.y.b.97.1 4
45.14 odd 6 825.2.n.c.526.1 4
45.23 even 12 825.2.bx.d.724.2 8
45.32 even 12 825.2.bx.d.724.1 8
99.4 even 15 1089.2.a.t.1.2 2
99.5 odd 30 33.2.e.b.16.1 ✓ 4
99.14 odd 30 363.2.e.k.202.1 4
99.16 even 15 inner 891.2.n.b.379.1 8
99.32 even 6 363.2.e.f.130.1 4
99.38 odd 30 891.2.n.c.379.1 8
99.40 odd 30 1089.2.a.l.1.1 2
99.41 even 30 363.2.e.b.202.1 4
99.49 even 15 99.2.f.a.82.1 4
99.50 even 30 363.2.e.f.148.1 4
99.59 odd 30 363.2.a.d.1.1 2
99.68 even 30 363.2.e.b.124.1 4
99.86 odd 30 363.2.e.k.124.1 4
99.95 even 30 363.2.a.i.1.2 2
396.59 even 30 5808.2.a.cj.1.1 2
396.95 odd 30 5808.2.a.ci.1.1 2
396.203 even 30 528.2.y.b.49.1 4
495.59 odd 30 9075.2.a.cb.1.2 2
495.104 odd 30 825.2.n.c.676.1 4
495.194 even 30 9075.2.a.u.1.1 2
495.203 even 60 825.2.bx.d.49.1 8
495.302 even 60 825.2.bx.d.49.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.16.1 ✓ 4 99.5 odd 30
33.2.e.b.31.1 yes 4 9.5 odd 6
99.2.f.a.64.1 4 9.4 even 3
99.2.f.a.82.1 4 99.49 even 15
363.2.a.d.1.1 2 99.59 odd 30
363.2.a.i.1.2 2 99.95 even 30
363.2.e.b.124.1 4 99.68 even 30
363.2.e.b.202.1 4 99.41 even 30
363.2.e.f.130.1 4 99.32 even 6
363.2.e.f.148.1 4 99.50 even 30
363.2.e.k.124.1 4 99.86 odd 30
363.2.e.k.202.1 4 99.14 odd 30
528.2.y.b.49.1 4 396.203 even 30
528.2.y.b.97.1 4 36.23 even 6
825.2.n.c.526.1 4 45.14 odd 6
825.2.n.c.676.1 4 495.104 odd 30
825.2.bx.d.49.1 8 495.203 even 60
825.2.bx.d.49.2 8 495.302 even 60
825.2.bx.d.724.1 8 45.32 even 12
825.2.bx.d.724.2 8 45.23 even 12
891.2.n.b.379.1 8 99.16 even 15 inner
891.2.n.b.460.1 8 9.7 even 3 inner
891.2.n.b.676.1 8 11.5 even 5 inner
891.2.n.b.757.1 8 1.1 even 1 trivial
891.2.n.c.379.1 8 99.38 odd 30
891.2.n.c.460.1 8 9.2 odd 6
891.2.n.c.676.1 8 33.5 odd 10
891.2.n.c.757.1 8 3.2 odd 2
1089.2.a.l.1.1 2 99.40 odd 30
1089.2.a.t.1.2 2 99.4 even 15
5808.2.a.ci.1.1 2 396.95 odd 30
5808.2.a.cj.1.1 2 396.59 even 30
9075.2.a.u.1.1 2 495.194 even 30
9075.2.a.cb.1.2 2 495.59 odd 30