Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1089,4,Mod(1,1089)] 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("1089.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1089, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,36,-10,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.2530799963\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{26}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 121)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-5.09902\) of defining polynomial
Character \(\chi\) \(=\) 1089.1

$q$-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)
 
\(f(q)\) \(=\) \(q-5.09902 q^{2} +18.0000 q^{4} -5.00000 q^{5} +20.3961 q^{7} -50.9902 q^{8} +25.4951 q^{10} +61.1882 q^{13} -104.000 q^{14} +116.000 q^{16} +20.3961 q^{17} -101.980 q^{19} -90.0000 q^{20} -35.0000 q^{23} -100.000 q^{25} -312.000 q^{26} +367.129 q^{28} +203.961 q^{29} +15.0000 q^{31} -183.565 q^{32} -104.000 q^{34} -101.980 q^{35} -265.000 q^{37} +520.000 q^{38} +254.951 q^{40} +101.980 q^{41} -448.714 q^{43} +178.466 q^{46} -380.000 q^{47} +73.0000 q^{49} +509.902 q^{50} +1101.39 q^{52} -510.000 q^{53} -1040.00 q^{56} -1040.00 q^{58} -21.0000 q^{59} +203.961 q^{61} -76.4853 q^{62} +8.00000 q^{64} -305.941 q^{65} +585.000 q^{67} +367.129 q^{68} +520.000 q^{70} -313.000 q^{71} +469.110 q^{73} +1351.24 q^{74} -1835.65 q^{76} -611.882 q^{79} -580.000 q^{80} -520.000 q^{82} -652.674 q^{83} -101.980 q^{85} +2288.00 q^{86} +185.000 q^{89} +1248.00 q^{91} -630.000 q^{92} +1937.63 q^{94} +509.902 q^{95} +785.000 q^{97} -372.228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 36 q^{4} - 10 q^{5} - 208 q^{14} + 232 q^{16} - 180 q^{20} - 70 q^{23} - 200 q^{25} - 624 q^{26} + 30 q^{31} - 208 q^{34} - 530 q^{37} + 1040 q^{38} - 760 q^{47} + 146 q^{49} - 1020 q^{53} - 2080 q^{56}+ \cdots + 1570 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.09902 −1.80278 −0.901388 − 0.433013i \(-0.857451\pi\)
−0.901388 + 0.433013i \(0.857451\pi\)
\(3\) 0 0
\(4\) 18.0000 2.25000
\(5\) −5.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 20.3961 1.10128 0.550642 − 0.834741i \(-0.314383\pi\)
0.550642 + 0.834741i \(0.314383\pi\)
\(8\) −50.9902 −2.25347
\(9\) 0 0
\(10\) 25.4951 0.806226
\(11\) 0 0
\(12\) 0 0
\(13\) 61.1882 1.30543 0.652714 − 0.757604i \(-0.273630\pi\)
0.652714 + 0.757604i \(0.273630\pi\)
\(14\) −104.000 −1.98537
\(15\) 0 0
\(16\) 116.000 1.81250
\(17\) 20.3961 0.290987 0.145493 − 0.989359i \(-0.453523\pi\)
0.145493 + 0.989359i \(0.453523\pi\)
\(18\) 0 0
\(19\) −101.980 −1.23136 −0.615682 − 0.787995i \(-0.711119\pi\)
−0.615682 + 0.787995i \(0.711119\pi\)
\(20\) −90.0000 −1.00623
\(21\) 0 0
\(22\) 0 0
\(23\) −35.0000 −0.317305 −0.158652 − 0.987335i \(-0.550715\pi\)
−0.158652 + 0.987335i \(0.550715\pi\)
\(24\) 0 0
\(25\) −100.000 −0.800000
\(26\) −312.000 −2.35339
\(27\) 0 0
\(28\) 367.129 2.47789
\(29\) 203.961 1.30602 0.653010 − 0.757349i \(-0.273506\pi\)
0.653010 + 0.757349i \(0.273506\pi\)
\(30\) 0 0
\(31\) 15.0000 0.0869058 0.0434529 − 0.999055i \(-0.486164\pi\)
0.0434529 + 0.999055i \(0.486164\pi\)
\(32\) −183.565 −1.01406
\(33\) 0 0
\(34\) −104.000 −0.524584
\(35\) −101.980 −0.492509
\(36\) 0 0
\(37\) −265.000 −1.17745 −0.588726 − 0.808333i \(-0.700370\pi\)
−0.588726 + 0.808333i \(0.700370\pi\)
\(38\) 520.000 2.21987
\(39\) 0 0
\(40\) 254.951 1.00778
\(41\) 101.980 0.388455 0.194228 − 0.980956i \(-0.437780\pi\)
0.194228 + 0.980956i \(0.437780\pi\)
\(42\) 0 0
\(43\) −448.714 −1.59135 −0.795677 − 0.605721i \(-0.792885\pi\)
−0.795677 + 0.605721i \(0.792885\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 178.466 0.572029
\(47\) −380.000 −1.17933 −0.589667 − 0.807646i \(-0.700741\pi\)
−0.589667 + 0.807646i \(0.700741\pi\)
\(48\) 0 0
\(49\) 73.0000 0.212828
\(50\) 509.902 1.44222
\(51\) 0 0
\(52\) 1101.39 2.93721
\(53\) −510.000 −1.32177 −0.660886 − 0.750487i \(-0.729819\pi\)
−0.660886 + 0.750487i \(0.729819\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1040.00 −2.48171
\(57\) 0 0
\(58\) −1040.00 −2.35446
\(59\) −21.0000 −0.0463384 −0.0231692 − 0.999732i \(-0.507376\pi\)
−0.0231692 + 0.999732i \(0.507376\pi\)
\(60\) 0 0
\(61\) 203.961 0.428107 0.214053 − 0.976822i \(-0.431333\pi\)
0.214053 + 0.976822i \(0.431333\pi\)
\(62\) −76.4853 −0.156672
\(63\) 0 0
\(64\) 8.00000 0.0156250
\(65\) −305.941 −0.583805
\(66\) 0 0
\(67\) 585.000 1.06670 0.533352 − 0.845894i \(-0.320932\pi\)
0.533352 + 0.845894i \(0.320932\pi\)
\(68\) 367.129 0.654720
\(69\) 0 0
\(70\) 520.000 0.887884
\(71\) −313.000 −0.523187 −0.261593 − 0.965178i \(-0.584248\pi\)
−0.261593 + 0.965178i \(0.584248\pi\)
\(72\) 0 0
\(73\) 469.110 0.752125 0.376063 − 0.926594i \(-0.377278\pi\)
0.376063 + 0.926594i \(0.377278\pi\)
\(74\) 1351.24 2.12268
\(75\) 0 0
\(76\) −1835.65 −2.77057
\(77\) 0 0
\(78\) 0 0
\(79\) −611.882 −0.871420 −0.435710 − 0.900087i \(-0.643503\pi\)
−0.435710 + 0.900087i \(0.643503\pi\)
\(80\) −580.000 −0.810575
\(81\) 0 0
\(82\) −520.000 −0.700297
\(83\) −652.674 −0.863137 −0.431568 − 0.902080i \(-0.642040\pi\)
−0.431568 + 0.902080i \(0.642040\pi\)
\(84\) 0 0
\(85\) −101.980 −0.130133
\(86\) 2288.00 2.86885
\(87\) 0 0
\(88\) 0 0
\(89\) 185.000 0.220337 0.110168 − 0.993913i \(-0.464861\pi\)
0.110168 + 0.993913i \(0.464861\pi\)
\(90\) 0 0
\(91\) 1248.00 1.43765
\(92\) −630.000 −0.713935
\(93\) 0 0
\(94\) 1937.63 2.12607
\(95\) 509.902 0.550682
\(96\) 0 0
\(97\) 785.000 0.821698 0.410849 − 0.911703i \(-0.365232\pi\)
0.410849 + 0.911703i \(0.365232\pi\)
\(98\) −372.228 −0.383681
\(99\) 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 1089.4.a.r.1.1 2
3.2 odd 2 121.4.a.d.1.2 yes 2
11.10 odd 2 inner 1089.4.a.r.1.2 2
12.11 even 2 1936.4.a.ba.1.1 2
33.2 even 10 121.4.c.e.81.2 8
33.5 odd 10 121.4.c.e.3.1 8
33.8 even 10 121.4.c.e.9.1 8
33.14 odd 10 121.4.c.e.9.2 8
33.17 even 10 121.4.c.e.3.2 8
33.20 odd 10 121.4.c.e.81.1 8
33.26 odd 10 121.4.c.e.27.2 8
33.29 even 10 121.4.c.e.27.1 8
33.32 even 2 121.4.a.d.1.1 ✓ 2
132.131 odd 2 1936.4.a.ba.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.a.d.1.1 ✓ 2 33.32 even 2
121.4.a.d.1.2 yes 2 3.2 odd 2
121.4.c.e.3.1 8 33.5 odd 10
121.4.c.e.3.2 8 33.17 even 10
121.4.c.e.9.1 8 33.8 even 10
121.4.c.e.9.2 8 33.14 odd 10
121.4.c.e.27.1 8 33.29 even 10
121.4.c.e.27.2 8 33.26 odd 10
121.4.c.e.81.1 8 33.20 odd 10
121.4.c.e.81.2 8 33.2 even 10
1089.4.a.r.1.1 2 1.1 even 1 trivial
1089.4.a.r.1.2 2 11.10 odd 2 inner
1936.4.a.ba.1.1 2 12.11 even 2
1936.4.a.ba.1.2 2 132.131 odd 2