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 89.2
Root \(-4.21699i\) of defining polynomial
Character \(\chi\) \(=\) 110.89
Dual form 110.4.b.b.89.4

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +2.00000i q^{3} -4.00000 q^{4} +(9.43398 - 6.00000i) q^{5} +4.00000 q^{6} +7.56602i q^{7} +8.00000i q^{8} +23.0000 q^{9} +(-12.0000 - 18.8680i) q^{10} +11.0000 q^{11} -8.00000i q^{12} -73.0379i q^{13} +15.1320 q^{14} +(12.0000 + 18.8680i) q^{15} +16.0000 q^{16} -21.0379i q^{17} -46.0000i q^{18} +137.208 q^{19} +(-37.7359 + 24.0000i) q^{20} -15.1320 q^{21} -22.0000i q^{22} -32.2641i q^{23} -16.0000 q^{24} +(53.0000 - 113.208i) q^{25} -146.076 q^{26} +100.000i q^{27} -30.2641i q^{28} -151.548 q^{29} +(37.7359 - 24.0000i) q^{30} -102.264 q^{31} -32.0000i q^{32} +22.0000i q^{33} -42.0757 q^{34} +(45.3961 + 71.3777i) q^{35} -92.0000 q^{36} +279.812i q^{37} -274.416i q^{38} +146.076 q^{39} +(48.0000 + 75.4718i) q^{40} +9.28351 q^{41} +30.2641i q^{42} +297.114i q^{43} -44.0000 q^{44} +(216.982 - 138.000i) q^{45} -64.5282 q^{46} +202.528i q^{47} +32.0000i q^{48} +285.755 q^{49} +(-226.416 - 106.000i) q^{50} +42.0757 q^{51} +292.151i q^{52} +376.491i q^{53} +200.000 q^{54} +(103.774 - 66.0000i) q^{55} -60.5282 q^{56} +274.416i q^{57} +303.095i q^{58} -485.736 q^{59} +(-48.0000 - 75.4718i) q^{60} -721.247 q^{61} +204.528i q^{62} +174.018i q^{63} -64.0000 q^{64} +(-438.227 - 689.038i) q^{65} +44.0000 q^{66} +181.132i q^{67} +84.1515i q^{68} +64.5282 q^{69} +(142.755 - 90.7922i) q^{70} +689.056 q^{71} +184.000i q^{72} -943.417i q^{73} +559.623 q^{74} +(226.416 + 106.000i) q^{75} -548.831 q^{76} +83.2262i q^{77} -292.151i q^{78} -833.019 q^{79} +(150.944 - 96.0000i) q^{80} +421.000 q^{81} -18.5670i q^{82} -1357.45i q^{83} +60.5282 q^{84} +(-126.227 - 198.471i) q^{85} +594.227 q^{86} -303.095i q^{87} +88.0000i q^{88} -1314.79 q^{89} +(-276.000 - 433.963i) q^{90} +552.606 q^{91} +129.056i q^{92} -204.528i q^{93} +405.056 q^{94} +(1294.42 - 823.247i) q^{95} +64.0000 q^{96} +1520.27i q^{97} -571.511i q^{98} +253.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4} + 16 q^{6} + 92 q^{9} - 48 q^{10} + 44 q^{11} + 136 q^{14} + 48 q^{15} + 64 q^{16} + 96 q^{19} - 136 q^{21} - 64 q^{24} + 212 q^{25} - 56 q^{26} + 224 q^{29} - 560 q^{31} + 360 q^{34} + 408 q^{35}+ \cdots + 1012 q^{99}+O(q^{100}) \) 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
\(7\) 7.56602i 0.408527i 0.978916 + 0.204263i \(0.0654799\pi\)
−0.978916 + 0.204263i \(0.934520\pi\)
\(8\) 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
\(13\) − 73.0379i − 1.55824i −0.626878 − 0.779118i \(-0.715667\pi\)
0.626878 − 0.779118i \(-0.284333\pi\)
\(14\) 15.1320 0.288872
\(15\) 12.0000 + 18.8680i 0.206559 + 0.324779i
\(16\) 16.0000 0.250000
\(17\) − 21.0379i − 0.300143i −0.988675 − 0.150072i \(-0.952050\pi\)
0.988675 − 0.150072i \(-0.0479504\pi\)
\(18\) − 46.0000i − 0.602350i
\(19\) 137.208 1.65672 0.828358 − 0.560198i \(-0.189275\pi\)
0.828358 + 0.560198i \(0.189275\pi\)
\(20\) −37.7359 + 24.0000i −0.421900 + 0.268328i
\(21\) −15.1320 −0.157242
\(22\) − 22.0000i − 0.213201i
\(23\) − 32.2641i − 0.292501i −0.989248 − 0.146251i \(-0.953279\pi\)
0.989248 − 0.146251i \(-0.0467206\pi\)
\(24\) −16.0000 −0.136083
\(25\) 53.0000 − 113.208i 0.424000 − 0.905662i
\(26\) −146.076 −1.10184
\(27\) 100.000i 0.712778i
\(28\) − 30.2641i − 0.204263i
\(29\) −151.548 −0.970403 −0.485201 − 0.874402i \(-0.661254\pi\)
−0.485201 + 0.874402i \(0.661254\pi\)
\(30\) 37.7359 − 24.0000i 0.229654 − 0.146059i
\(31\) −102.264 −0.592489 −0.296245 − 0.955112i \(-0.595734\pi\)
−0.296245 + 0.955112i \(0.595734\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 22.0000i 0.116052i
\(34\) −42.0757 −0.212233
\(35\) 45.3961 + 71.3777i 0.219238 + 0.344715i
\(36\) −92.0000 −0.425926
\(37\) 279.812i 1.24326i 0.783310 + 0.621632i \(0.213530\pi\)
−0.783310 + 0.621632i \(0.786470\pi\)
\(38\) − 274.416i − 1.17148i
\(39\) 146.076 0.599765
\(40\) 48.0000 + 75.4718i 0.189737 + 0.298329i
\(41\) 9.28351 0.0353620 0.0176810 − 0.999844i \(-0.494372\pi\)
0.0176810 + 0.999844i \(0.494372\pi\)
\(42\) 30.2641i 0.111187i
\(43\) 297.114i 1.05371i 0.849956 + 0.526853i \(0.176628\pi\)
−0.849956 + 0.526853i \(0.823372\pi\)
\(44\) −44.0000 −0.150756
\(45\) 216.982 − 138.000i 0.718793 − 0.457152i
\(46\) −64.5282 −0.206829
\(47\) 202.528i 0.628548i 0.949332 + 0.314274i \(0.101761\pi\)
−0.949332 + 0.314274i \(0.898239\pi\)
\(48\) 32.0000i 0.0962250i
\(49\) 285.755 0.833106
\(50\) −226.416 − 106.000i −0.640400 − 0.299813i
\(51\) 42.0757 0.115525
\(52\) 292.151i 0.779118i
\(53\) 376.491i 0.975756i 0.872912 + 0.487878i \(0.162229\pi\)
−0.872912 + 0.487878i \(0.837771\pi\)
\(54\) 200.000 0.504010
\(55\) 103.774 − 66.0000i 0.254416 − 0.161808i
\(56\) −60.5282 −0.144436
\(57\) 274.416i 0.637671i
\(58\) 303.095i 0.686178i
\(59\) −485.736 −1.07182 −0.535910 − 0.844275i \(-0.680032\pi\)
−0.535910 + 0.844275i \(0.680032\pi\)
\(60\) −48.0000 − 75.4718i −0.103280 − 0.162390i
\(61\) −721.247 −1.51387 −0.756936 − 0.653489i \(-0.773304\pi\)
−0.756936 + 0.653489i \(0.773304\pi\)
\(62\) 204.528i 0.418953i
\(63\) 174.018i 0.348004i
\(64\) −64.0000 −0.125000
\(65\) −438.227 − 689.038i −0.836237 − 1.31484i
\(66\) 44.0000 0.0820610
\(67\) 181.132i 0.330281i 0.986270 + 0.165140i \(0.0528077\pi\)
−0.986270 + 0.165140i \(0.947192\pi\)
\(68\) 84.1515i 0.150072i
\(69\) 64.5282 0.112584
\(70\) 142.755 − 90.7922i 0.243750 − 0.155025i
\(71\) 689.056 1.15177 0.575887 − 0.817529i \(-0.304657\pi\)
0.575887 + 0.817529i \(0.304657\pi\)
\(72\) 184.000i 0.301175i
\(73\) − 943.417i − 1.51258i −0.654235 − 0.756291i \(-0.727009\pi\)
0.654235 − 0.756291i \(-0.272991\pi\)
\(74\) 559.623 0.879120
\(75\) 226.416 + 106.000i 0.348590 + 0.163198i
\(76\) −548.831 −0.828358
\(77\) 83.2262i 0.123175i
\(78\) − 292.151i − 0.424098i
\(79\) −833.019 −1.18635 −0.593177 − 0.805072i \(-0.702127\pi\)
−0.593177 + 0.805072i \(0.702127\pi\)
\(80\) 150.944 − 96.0000i 0.210950 − 0.134164i
\(81\) 421.000 0.577503
\(82\) − 18.5670i − 0.0250047i
\(83\) − 1357.45i − 1.79518i −0.440832 − 0.897590i \(-0.645317\pi\)
0.440832 − 0.897590i \(-0.354683\pi\)
\(84\) 60.5282 0.0786210
\(85\) −126.227 − 198.471i −0.161074 − 0.253261i
\(86\) 594.227 0.745083
\(87\) − 303.095i − 0.373508i
\(88\) 88.0000i 0.106600i
\(89\) −1314.79 −1.56593 −0.782966 − 0.622065i \(-0.786294\pi\)
−0.782966 + 0.622065i \(0.786294\pi\)
\(90\) −276.000 − 433.963i −0.323255 − 0.508264i
\(91\) 552.606 0.636581
\(92\) 129.056i 0.146251i
\(93\) − 204.528i − 0.228049i
\(94\) 405.056 0.444451
\(95\) 1294.42 − 823.247i 1.39794 − 0.889088i
\(96\) 64.0000 0.0680414
\(97\) 1520.27i 1.59134i 0.605733 + 0.795668i \(0.292880\pi\)
−0.605733 + 0.795668i \(0.707120\pi\)
\(98\) − 571.511i − 0.589095i
\(99\) 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.89.2 ✓ 4
3.2 odd 2 990.4.c.e.199.3 4
4.3 odd 2 880.4.b.d.529.2 4
5.2 odd 4 550.4.a.x.1.2 2
5.3 odd 4 550.4.a.p.1.1 2
5.4 even 2 inner 110.4.b.b.89.4 yes 4
15.14 odd 2 990.4.c.e.199.1 4
20.19 odd 2 880.4.b.d.529.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.4.b.b.89.2 ✓ 4 1.1 even 1 trivial
110.4.b.b.89.4 yes 4 5.4 even 2 inner
550.4.a.p.1.1 2 5.3 odd 4
550.4.a.x.1.2 2 5.2 odd 4
880.4.b.d.529.2 4 4.3 odd 2
880.4.b.d.529.4 4 20.19 odd 2
990.4.c.e.199.1 4 15.14 odd 2
990.4.c.e.199.3 4 3.2 odd 2