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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [115,2,Mod(4,115)] 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("115.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(115, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.j (of order \(22\), degree \(10\), 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: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 64.4
Character \(\chi\) \(=\) 115.64
Dual form 115.2.j.a.9.4

$q$-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+(-0.320885 + 1.09283i) q^{2} +(1.24484 + 1.07866i) q^{3} +(0.591189 + 0.379934i) q^{4} +(0.429235 - 2.19448i) q^{5} +(-1.57825 + 1.01428i) q^{6} +(1.40950 - 0.643697i) q^{7} +(-2.32646 + 2.01589i) q^{8} +(-0.0408222 - 0.283925i) q^{9} +(2.26047 + 1.17326i) q^{10} +(-4.00579 + 1.17621i) q^{11} +(0.326117 + 1.11065i) q^{12} +(-5.39026 - 2.46165i) q^{13} +(0.251167 + 1.74690i) q^{14} +(2.90144 - 2.26879i) q^{15} +(-0.872642 - 1.91082i) q^{16} +(0.798760 + 1.24289i) q^{17} +(0.323382 + 0.0464953i) q^{18} +(6.07675 + 3.90529i) q^{19} +(1.08752 - 1.13427i) q^{20} +(2.44894 + 0.719074i) q^{21} -4.75509i q^{22} +(-3.89631 - 2.79621i) q^{23} -5.07055 q^{24} +(-4.63151 - 1.88390i) q^{25} +(4.41983 - 5.10075i) q^{26} +(2.92702 - 4.55453i) q^{27} +(1.07784 + 0.154970i) q^{28} +(2.24279 - 1.44135i) q^{29} +(1.54838 + 3.89881i) q^{30} +(0.638526 + 0.736898i) q^{31} +(-3.72580 + 0.535690i) q^{32} +(-6.25531 - 2.85671i) q^{33} +(-1.61459 + 0.474086i) q^{34} +(-0.807576 - 3.36942i) q^{35} +(0.0837390 - 0.183363i) q^{36} +(6.34545 - 0.912338i) q^{37} +(-6.21777 + 5.38773i) q^{38} +(-4.05474 - 8.87865i) q^{39} +(3.42524 + 5.97067i) q^{40} +(-0.434006 + 3.01857i) q^{41} +(-1.57166 + 2.44555i) q^{42} +(7.10485 + 6.15639i) q^{43} +(-2.81506 - 0.826575i) q^{44} +(-0.640591 - 0.0322868i) q^{45} +(4.30606 - 3.36076i) q^{46} +3.03002i q^{47} +(0.974829 - 3.31996i) q^{48} +(-3.01168 + 3.47566i) q^{49} +(3.54497 - 4.45696i) q^{50} +(-0.346334 + 2.40881i) q^{51} +(-2.25140 - 3.50324i) q^{52} +(1.91146 - 0.872932i) q^{53} +(4.03811 + 4.66022i) q^{54} +(0.861738 + 9.29550i) q^{55} +(-1.98152 + 4.33893i) q^{56} +(3.35212 + 11.4163i) q^{57} +(0.855482 + 2.91350i) q^{58} +(-1.42947 + 3.13011i) q^{59} +(2.57729 - 0.238927i) q^{60} +(-1.61183 - 1.86015i) q^{61} +(-1.01020 + 0.461343i) q^{62} +(-0.240301 - 0.373915i) q^{63} +(1.20804 - 8.40212i) q^{64} +(-7.71574 + 10.7722i) q^{65} +(5.12914 - 5.91934i) q^{66} +(1.00941 - 3.43775i) q^{67} +1.03826i q^{68} +(-1.83413 - 7.68365i) q^{69} +(3.94136 + 0.198651i) q^{70} +(-6.31720 - 1.85490i) q^{71} +(0.667332 + 0.578247i) q^{72} +(-4.77788 + 7.43453i) q^{73} +(-1.03913 + 7.22728i) q^{74} +(-3.73342 - 7.34101i) q^{75} +(2.10875 + 4.61753i) q^{76} +(-4.88904 + 4.23637i) q^{77} +(11.0040 - 1.58214i) q^{78} +(2.53011 - 5.54018i) q^{79} +(-4.56783 + 1.09481i) q^{80} +(7.73081 - 2.26997i) q^{81} +(-3.15953 - 1.44291i) q^{82} +(4.51112 - 0.648601i) q^{83} +(1.17459 + 1.35554i) q^{84} +(3.07037 - 1.21937i) q^{85} +(-9.00775 + 5.78893i) q^{86} +(4.34666 + 0.624955i) q^{87} +(6.94820 - 10.8116i) q^{88} +(5.73812 - 6.62215i) q^{89} +(0.240840 - 0.689699i) q^{90} -9.18213 q^{91} +(-1.24108 - 3.13343i) q^{92} +1.60608i q^{93} +(-3.31131 - 0.972287i) q^{94} +(11.1785 - 11.6590i) q^{95} +(-5.21588 - 3.35204i) q^{96} +(-13.8065 - 1.98507i) q^{97} +(-2.83192 - 4.40655i) q^{98} +(0.497479 + 1.08933i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 100 q - 14 q^{4} - 9 q^{5} - 18 q^{6} - 12 q^{9} - 13 q^{10} - 26 q^{11} - 26 q^{14} - 10 q^{15} - 18 q^{16} - 14 q^{19} + 49 q^{20} - 22 q^{21} - 68 q^{24} + 21 q^{25} - 42 q^{26} - 24 q^{29} + 19 q^{30}+ \cdots - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(e\left(\frac{6}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.320885 + 1.09283i −0.226900 + 0.772750i 0.764810 + 0.644256i \(0.222833\pi\)
−0.991710 + 0.128495i \(0.958986\pi\)
\(3\) 1.24484 + 1.07866i 0.718712 + 0.622767i 0.935449 − 0.353463i \(-0.114996\pi\)
−0.216737 + 0.976230i \(0.569541\pi\)
\(4\) 0.591189 + 0.379934i 0.295594 + 0.189967i
\(5\) 0.429235 − 2.19448i 0.191960 − 0.981403i
\(6\) −1.57825 + 1.01428i −0.644319 + 0.414079i
\(7\) 1.40950 − 0.643697i 0.532741 − 0.243295i −0.130834 − 0.991404i \(-0.541766\pi\)
0.663575 + 0.748110i \(0.269038\pi\)
\(8\) −2.32646 + 2.01589i −0.822528 + 0.712725i
\(9\) −0.0408222 − 0.283925i −0.0136074 − 0.0946416i
\(10\) 2.26047 + 1.17326i 0.714823 + 0.371017i
\(11\) −4.00579 + 1.17621i −1.20779 + 0.354639i −0.822827 − 0.568293i \(-0.807604\pi\)
−0.384964 + 0.922932i \(0.625786\pi\)
\(12\) 0.326117 + 1.11065i 0.0941419 + 0.320618i
\(13\) −5.39026 − 2.46165i −1.49499 − 0.682739i −0.510774 − 0.859715i \(-0.670641\pi\)
−0.984215 + 0.176976i \(0.943368\pi\)
\(14\) 0.251167 + 1.74690i 0.0671271 + 0.466879i
\(15\) 2.90144 − 2.26879i 0.749149 − 0.585799i
\(16\) −0.872642 − 1.91082i −0.218161 − 0.477705i
\(17\) 0.798760 + 1.24289i 0.193728 + 0.301446i 0.924505 − 0.381170i \(-0.124479\pi\)
−0.730777 + 0.682616i \(0.760842\pi\)
\(18\) 0.323382 + 0.0464953i 0.0762218 + 0.0109590i
\(19\) 6.07675 + 3.90529i 1.39410 + 0.895935i 0.999735 − 0.0230176i \(-0.00732739\pi\)
0.394367 + 0.918953i \(0.370964\pi\)
\(20\) 1.08752 − 1.13427i 0.243176 − 0.253631i
\(21\) 2.44894 + 0.719074i 0.534403 + 0.156915i
\(22\) − 4.75509i − 1.01379i
\(23\) −3.89631 − 2.79621i −0.812437 − 0.583049i
\(24\) −5.07055 −1.03502
\(25\) −4.63151 − 1.88390i −0.926303 − 0.376780i
\(26\) 4.41983 − 5.10075i 0.866799 − 1.00034i
\(27\) 2.92702 − 4.55453i 0.563305 − 0.876519i
\(28\) 1.07784 + 0.154970i 0.203693 + 0.0292867i
\(29\) 2.24279 − 1.44135i 0.416475 − 0.267653i −0.315578 − 0.948900i \(-0.602198\pi\)
0.732054 + 0.681247i \(0.238562\pi\)
\(30\) 1.54838 + 3.89881i 0.282695 + 0.711823i
\(31\) 0.638526 + 0.736898i 0.114683 + 0.132351i 0.810188 − 0.586170i \(-0.199365\pi\)
−0.695505 + 0.718521i \(0.744819\pi\)
\(32\) −3.72580 + 0.535690i −0.658635 + 0.0946975i
\(33\) −6.25531 − 2.85671i −1.08891 − 0.497289i
\(34\) −1.61459 + 0.474086i −0.276899 + 0.0813050i
\(35\) −0.807576 − 3.36942i −0.136505 − 0.569536i
\(36\) 0.0837390 − 0.183363i 0.0139565 − 0.0305605i
\(37\) 6.34545 − 0.912338i 1.04319 − 0.149987i 0.400641 − 0.916235i \(-0.368788\pi\)
0.642545 + 0.766248i \(0.277878\pi\)
\(38\) −6.21777 + 5.38773i −1.00866 + 0.874005i
\(39\) −4.05474 − 8.87865i −0.649279 − 1.42172i
\(40\) 3.42524 + 5.97067i 0.541578 + 0.944046i
\(41\) −0.434006 + 3.01857i −0.0677803 + 0.471422i 0.927456 + 0.373932i \(0.121991\pi\)
−0.995236 + 0.0974903i \(0.968918\pi\)
\(42\) −1.57166 + 2.44555i −0.242512 + 0.377356i
\(43\) 7.10485 + 6.15639i 1.08348 + 0.938841i 0.998344 − 0.0575286i \(-0.0183220\pi\)
0.0851361 + 0.996369i \(0.472868\pi\)
\(44\) −2.81506 − 0.826575i −0.424386 − 0.124611i
\(45\) −0.640591 − 0.0322868i −0.0954936 − 0.00481303i
\(46\) 4.30606 − 3.36076i 0.634893 − 0.495517i
\(47\) 3.03002i 0.441974i 0.975277 + 0.220987i \(0.0709278\pi\)
−0.975277 + 0.220987i \(0.929072\pi\)
\(48\) 0.974829 − 3.31996i 0.140704 − 0.479195i
\(49\) −3.01168 + 3.47566i −0.430240 + 0.496524i
\(50\) 3.54497 − 4.45696i 0.501335 − 0.630309i
\(51\) −0.346334 + 2.40881i −0.0484964 + 0.337300i
\(52\) −2.25140 − 3.50324i −0.312213 − 0.485812i
\(53\) 1.91146 − 0.872932i 0.262559 − 0.119906i −0.279787 − 0.960062i \(-0.590264\pi\)
0.542345 + 0.840156i \(0.317536\pi\)
\(54\) 4.03811 + 4.66022i 0.549517 + 0.634176i
\(55\) 0.861738 + 9.29550i 0.116197 + 1.25341i
\(56\) −1.98152 + 4.33893i −0.264792 + 0.579814i
\(57\) 3.35212 + 11.4163i 0.443999 + 1.51212i
\(58\) 0.855482 + 2.91350i 0.112330 + 0.382562i
\(59\) −1.42947 + 3.13011i −0.186102 + 0.407506i −0.979569 − 0.201106i \(-0.935546\pi\)
0.793468 + 0.608612i \(0.208274\pi\)
\(60\) 2.57729 − 0.238927i 0.332727 − 0.0308454i
\(61\) −1.61183 − 1.86015i −0.206374 − 0.238168i 0.643122 − 0.765764i \(-0.277639\pi\)
−0.849495 + 0.527596i \(0.823093\pi\)
\(62\) −1.01020 + 0.461343i −0.128296 + 0.0585906i
\(63\) −0.240301 − 0.373915i −0.0302750 − 0.0471088i
\(64\) 1.20804 − 8.40212i 0.151005 − 1.05026i
\(65\) −7.71574 + 10.7722i −0.957019 + 1.33613i
\(66\) 5.12914 − 5.91934i 0.631354 − 0.728621i
\(67\) 1.00941 − 3.43775i 0.123320 − 0.419988i −0.874572 − 0.484896i \(-0.838857\pi\)
0.997891 + 0.0649084i \(0.0206755\pi\)
\(68\) 1.03826i 0.125908i
\(69\) −1.83413 − 7.68365i −0.220804 − 0.925003i
\(70\) 3.94136 + 0.198651i 0.471082 + 0.0237433i
\(71\) −6.31720 − 1.85490i −0.749714 − 0.220136i −0.115515 − 0.993306i \(-0.536852\pi\)
−0.634199 + 0.773170i \(0.718670\pi\)
\(72\) 0.667332 + 0.578247i 0.0786459 + 0.0681470i
\(73\) −4.77788 + 7.43453i −0.559209 + 0.870146i −0.999618 − 0.0276515i \(-0.991197\pi\)
0.440409 + 0.897797i \(0.354833\pi\)
\(74\) −1.03913 + 7.22728i −0.120796 + 0.840154i
\(75\) −3.73342 − 7.34101i −0.431099 − 0.847667i
\(76\) 2.10875 + 4.61753i 0.241891 + 0.529667i
\(77\) −4.88904 + 4.23637i −0.557157 + 0.482780i
\(78\) 11.0040 − 1.58214i 1.24596 − 0.179142i
\(79\) 2.53011 − 5.54018i 0.284660 − 0.623319i −0.712245 − 0.701931i \(-0.752322\pi\)
0.996905 + 0.0786121i \(0.0250489\pi\)
\(80\) −4.56783 + 1.09481i −0.510699 + 0.122403i
\(81\) 7.73081 − 2.26997i 0.858979 − 0.252219i
\(82\) −3.15953 − 1.44291i −0.348912 − 0.159343i
\(83\) 4.51112 − 0.648601i 0.495159 − 0.0711932i 0.109789 − 0.993955i \(-0.464983\pi\)
0.385371 + 0.922762i \(0.374074\pi\)
\(84\) 1.17459 + 1.35554i 0.128158 + 0.147902i
\(85\) 3.07037 − 1.21937i 0.333028 − 0.132259i
\(86\) −9.00775 + 5.78893i −0.971331 + 0.624236i
\(87\) 4.34666 + 0.624955i 0.466011 + 0.0670023i
\(88\) 6.94820 − 10.8116i 0.740681 − 1.15252i
\(89\) 5.73812 − 6.62215i 0.608240 − 0.701946i −0.365190 − 0.930933i \(-0.618996\pi\)
0.973429 + 0.228987i \(0.0735413\pi\)
\(90\) 0.240840 − 0.689699i 0.0253868 − 0.0727006i
\(91\) −9.18213 −0.962548
\(92\) −1.24108 − 3.13343i −0.129392 − 0.326682i
\(93\) 1.60608i 0.166543i
\(94\) −3.31131 − 0.972287i −0.341535 − 0.100284i
\(95\) 11.1785 − 11.6590i 1.14688 − 1.19619i
\(96\) −5.21588 − 3.35204i −0.532343 − 0.342116i
\(97\) −13.8065 − 1.98507i −1.40183 − 0.201553i −0.600430 − 0.799678i \(-0.705004\pi\)
−0.801403 + 0.598124i \(0.795913\pi\)
\(98\) −2.83192 − 4.40655i −0.286067 − 0.445129i
\(99\) 0.497479 + 1.08933i 0.0499985 + 0.109481i
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 115.2.j.a.64.4 yes 100
5.2 odd 4 575.2.k.g.501.4 100
5.3 odd 4 575.2.k.g.501.7 100
5.4 even 2 inner 115.2.j.a.64.7 yes 100
23.9 even 11 inner 115.2.j.a.9.7 yes 100
115.9 even 22 inner 115.2.j.a.9.4 ✓ 100
115.32 odd 44 575.2.k.g.101.4 100
115.78 odd 44 575.2.k.g.101.7 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.j.a.9.4 ✓ 100 115.9 even 22 inner
115.2.j.a.9.7 yes 100 23.9 even 11 inner
115.2.j.a.64.4 yes 100 1.1 even 1 trivial
115.2.j.a.64.7 yes 100 5.4 even 2 inner
575.2.k.g.101.4 100 115.32 odd 44
575.2.k.g.101.7 100 115.78 odd 44
575.2.k.g.501.4 100 5.2 odd 4
575.2.k.g.501.7 100 5.3 odd 4