Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [110,4,Mod(89,110)] 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("110.89"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 110.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.49021010063\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{89})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 45x^{2} + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 89.1
Root \(5.21699i\) of defining polynomial
Character \(\chi\) \(=\) 110.89
Dual form 110.4.b.b.89.3

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +2.00000i q^{3} -4.00000 q^{4} +(-9.43398 - 6.00000i) q^{5} +4.00000 q^{6} +26.4340i q^{7} +8.00000i q^{8} +23.0000 q^{9} +(-12.0000 + 18.8680i) q^{10} +11.0000 q^{11} -8.00000i q^{12} +59.0379i q^{13} +52.8680 q^{14} +(12.0000 - 18.8680i) q^{15} +16.0000 q^{16} +111.038i q^{17} -46.0000i q^{18} -89.2078 q^{19} +(37.7359 + 24.0000i) q^{20} -52.8680 q^{21} -22.0000i q^{22} -107.736i q^{23} -16.0000 q^{24} +(53.0000 + 113.208i) q^{25} +118.076 q^{26} +100.000i q^{27} -105.736i q^{28} +263.548 q^{29} +(-37.7359 - 24.0000i) q^{30} -177.736 q^{31} -32.0000i q^{32} +22.0000i q^{33} +222.076 q^{34} +(158.604 - 249.378i) q^{35} -92.0000 q^{36} -59.8117i q^{37} +178.416i q^{38} -118.076 q^{39} +(48.0000 - 75.4718i) q^{40} -481.284 q^{41} +105.736i q^{42} -99.1136i q^{43} -44.0000 q^{44} +(-216.982 - 138.000i) q^{45} -215.472 q^{46} +353.472i q^{47} +32.0000i q^{48} -355.755 q^{49} +(226.416 - 106.000i) q^{50} -222.076 q^{51} -236.151i q^{52} -340.491i q^{53} +200.000 q^{54} +(-103.774 - 66.0000i) q^{55} -211.472 q^{56} -178.416i q^{57} -527.095i q^{58} -410.264 q^{59} +(-48.0000 + 75.4718i) q^{60} +637.247 q^{61} +355.472i q^{62} +607.982i q^{63} -64.0000 q^{64} +(354.227 - 556.962i) q^{65} +44.0000 q^{66} +218.868i q^{67} -444.151i q^{68} +215.472 q^{69} +(-498.755 - 317.208i) q^{70} +990.944 q^{71} +184.000i q^{72} +509.417i q^{73} -119.623 q^{74} +(-226.416 + 106.000i) q^{75} +356.831 q^{76} +290.774i q^{77} +236.151i q^{78} -266.981 q^{79} +(-150.944 - 96.0000i) q^{80} +421.000 q^{81} +962.567i q^{82} -772.547i q^{83} +211.472 q^{84} +(666.227 - 1047.53i) q^{85} -198.227 q^{86} +527.095i q^{87} +88.0000i q^{88} +458.794 q^{89} +(-276.000 + 433.963i) q^{90} -1560.61 q^{91} +430.944i q^{92} -355.472i q^{93} +706.944 q^{94} +(841.584 + 535.247i) q^{95} +64.0000 q^{96} -404.266i q^{97} +711.511i q^{98} +253.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4} + 16 q^{6} + 92 q^{9} - 48 q^{10} + 44 q^{11} + 136 q^{14} + 48 q^{15} + 64 q^{16} + 96 q^{19} - 136 q^{21} - 64 q^{24} + 212 q^{25} - 56 q^{26} + 224 q^{29} - 560 q^{31} + 360 q^{34} + 408 q^{35}+ \cdots + 1012 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(101\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) 2.00000i 0.384900i 0.981307 + 0.192450i \(0.0616434\pi\)
−0.981307 + 0.192450i \(0.938357\pi\)
\(4\) −4.00000 −0.500000
\(5\) −9.43398 − 6.00000i −0.843801 − 0.536656i
\(6\) 4.00000 0.272166
\(7\) 26.4340i 1.42730i 0.700502 + 0.713650i \(0.252959\pi\)
−0.700502 + 0.713650i \(0.747041\pi\)
\(8\) 8.00000i 0.353553i
\(9\) 23.0000 0.851852
\(10\) −12.0000 + 18.8680i −0.379473 + 0.596657i
\(11\) 11.0000 0.301511
\(12\) − 8.00000i − 0.192450i
\(13\) 59.0379i 1.25955i 0.776777 + 0.629775i \(0.216853\pi\)
−0.776777 + 0.629775i \(0.783147\pi\)
\(14\) 52.8680 1.00925
\(15\) 12.0000 − 18.8680i 0.206559 − 0.324779i
\(16\) 16.0000 0.250000
\(17\) 111.038i 1.58416i 0.610420 + 0.792078i \(0.291001\pi\)
−0.610420 + 0.792078i \(0.708999\pi\)
\(18\) − 46.0000i − 0.602350i
\(19\) −89.2078 −1.07714 −0.538570 − 0.842581i \(-0.681035\pi\)
−0.538570 + 0.842581i \(0.681035\pi\)
\(20\) 37.7359 + 24.0000i 0.421900 + 0.268328i
\(21\) −52.8680 −0.549368
\(22\) − 22.0000i − 0.213201i
\(23\) − 107.736i − 0.976717i −0.872643 − 0.488359i \(-0.837596\pi\)
0.872643 − 0.488359i \(-0.162404\pi\)
\(24\) −16.0000 −0.136083
\(25\) 53.0000 + 113.208i 0.424000 + 0.905662i
\(26\) 118.076 0.890637
\(27\) 100.000i 0.712778i
\(28\) − 105.736i − 0.713650i
\(29\) 263.548 1.68757 0.843785 − 0.536681i \(-0.180322\pi\)
0.843785 + 0.536681i \(0.180322\pi\)
\(30\) −37.7359 − 24.0000i −0.229654 − 0.146059i
\(31\) −177.736 −1.02975 −0.514876 − 0.857265i \(-0.672162\pi\)
−0.514876 + 0.857265i \(0.672162\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 22.0000i 0.116052i
\(34\) 222.076 1.12017
\(35\) 158.604 − 249.378i 0.765970 − 1.20436i
\(36\) −92.0000 −0.425926
\(37\) − 59.8117i − 0.265756i −0.991132 − 0.132878i \(-0.957578\pi\)
0.991132 − 0.132878i \(-0.0424219\pi\)
\(38\) 178.416i 0.761653i
\(39\) −118.076 −0.484801
\(40\) 48.0000 − 75.4718i 0.189737 − 0.298329i
\(41\) −481.284 −1.83326 −0.916632 − 0.399731i \(-0.869103\pi\)
−0.916632 + 0.399731i \(0.869103\pi\)
\(42\) 105.736i 0.388462i
\(43\) − 99.1136i − 0.351504i −0.984434 − 0.175752i \(-0.943764\pi\)
0.984434 − 0.175752i \(-0.0562357\pi\)
\(44\) −44.0000 −0.150756
\(45\) −216.982 − 138.000i −0.718793 − 0.457152i
\(46\) −215.472 −0.690643
\(47\) 353.472i 1.09700i 0.836149 + 0.548502i \(0.184802\pi\)
−0.836149 + 0.548502i \(0.815198\pi\)
\(48\) 32.0000i 0.0962250i
\(49\) −355.755 −1.03719
\(50\) 226.416 − 106.000i 0.640400 − 0.299813i
\(51\) −222.076 −0.609742
\(52\) − 236.151i − 0.629775i
\(53\) − 340.491i − 0.882454i −0.897395 − 0.441227i \(-0.854543\pi\)
0.897395 − 0.441227i \(-0.145457\pi\)
\(54\) 200.000 0.504010
\(55\) −103.774 − 66.0000i −0.254416 − 0.161808i
\(56\) −211.472 −0.504627
\(57\) − 178.416i − 0.414592i
\(58\) − 527.095i − 1.19329i
\(59\) −410.264 −0.905285 −0.452643 − 0.891692i \(-0.649519\pi\)
−0.452643 + 0.891692i \(0.649519\pi\)
\(60\) −48.0000 + 75.4718i −0.103280 + 0.162390i
\(61\) 637.247 1.33756 0.668779 − 0.743461i \(-0.266817\pi\)
0.668779 + 0.743461i \(0.266817\pi\)
\(62\) 355.472i 0.728145i
\(63\) 607.982i 1.21585i
\(64\) −64.0000 −0.125000
\(65\) 354.227 − 556.962i 0.675946 − 1.06281i
\(66\) 44.0000 0.0820610
\(67\) 218.868i 0.399089i 0.979889 + 0.199545i \(0.0639463\pi\)
−0.979889 + 0.199545i \(0.936054\pi\)
\(68\) − 444.151i − 0.792078i
\(69\) 215.472 0.375939
\(70\) −498.755 − 317.208i −0.851610 − 0.541623i
\(71\) 990.944 1.65639 0.828193 − 0.560443i \(-0.189369\pi\)
0.828193 + 0.560443i \(0.189369\pi\)
\(72\) 184.000i 0.301175i
\(73\) 509.417i 0.816749i 0.912814 + 0.408375i \(0.133904\pi\)
−0.912814 + 0.408375i \(0.866096\pi\)
\(74\) −119.623 −0.187918
\(75\) −226.416 + 106.000i −0.348590 + 0.163198i
\(76\) 356.831 0.538570
\(77\) 290.774i 0.430347i
\(78\) 236.151i 0.342806i
\(79\) −266.981 −0.380224 −0.190112 − 0.981762i \(-0.560885\pi\)
−0.190112 + 0.981762i \(0.560885\pi\)
\(80\) −150.944 − 96.0000i −0.210950 − 0.134164i
\(81\) 421.000 0.577503
\(82\) 962.567i 1.29631i
\(83\) − 772.547i − 1.02166i −0.859681 − 0.510831i \(-0.829338\pi\)
0.859681 − 0.510831i \(-0.170662\pi\)
\(84\) 211.472 0.274684
\(85\) 666.227 − 1047.53i 0.850147 − 1.33671i
\(86\) −198.227 −0.248551
\(87\) 527.095i 0.649546i
\(88\) 88.0000i 0.106600i
\(89\) 458.794 0.546428 0.273214 − 0.961953i \(-0.411913\pi\)
0.273214 + 0.961953i \(0.411913\pi\)
\(90\) −276.000 + 433.963i −0.323255 + 0.508264i
\(91\) −1560.61 −1.79776
\(92\) 430.944i 0.488359i
\(93\) − 355.472i − 0.396352i
\(94\) 706.944 0.775699
\(95\) 841.584 + 535.247i 0.908892 + 0.578054i
\(96\) 64.0000 0.0680414
\(97\) − 404.266i − 0.423165i −0.977360 − 0.211582i \(-0.932138\pi\)
0.977360 − 0.211582i \(-0.0678617\pi\)
\(98\) 711.511i 0.733402i
\(99\) 253.000 0.256843
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 110.4.b.b.89.1 ✓ 4
3.2 odd 2 990.4.c.e.199.4 4
4.3 odd 2 880.4.b.d.529.1 4
5.2 odd 4 550.4.a.x.1.1 2
5.3 odd 4 550.4.a.p.1.2 2
5.4 even 2 inner 110.4.b.b.89.3 yes 4
15.14 odd 2 990.4.c.e.199.2 4
20.19 odd 2 880.4.b.d.529.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.4.b.b.89.1 ✓ 4 1.1 even 1 trivial
110.4.b.b.89.3 yes 4 5.4 even 2 inner
550.4.a.p.1.2 2 5.3 odd 4
550.4.a.x.1.1 2 5.2 odd 4
880.4.b.d.529.1 4 4.3 odd 2
880.4.b.d.529.3 4 20.19 odd 2
990.4.c.e.199.2 4 15.14 odd 2
990.4.c.e.199.4 4 3.2 odd 2