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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,4,Mod(4,95)] 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("95.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.p (of order \(18\), degree \(6\), 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: \(5.60518145055\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(28\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 4.19
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.19

$q$-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+(0.811338 - 2.22913i) q^{2} +(8.50321 - 1.49935i) q^{3} +(1.81760 + 1.52514i) q^{4} +(-7.21702 + 8.53901i) q^{5} +(3.55674 - 20.1713i) q^{6} +(5.98741 + 3.45683i) q^{7} +(21.3095 - 12.3030i) q^{8} +(44.6849 - 16.2640i) q^{9} +(13.1791 + 23.0157i) q^{10} +(-3.49409 - 6.05194i) q^{11} +(17.7421 + 10.2434i) q^{12} +(-59.7409 - 10.5339i) q^{13} +(12.5635 - 10.5421i) q^{14} +(-48.5649 + 83.4298i) q^{15} +(-6.83977 - 38.7902i) q^{16} +(-19.0179 + 52.2513i) q^{17} -112.804i q^{18} +(-42.8048 - 70.8995i) q^{19} +(-26.1408 + 4.51348i) q^{20} +(56.0952 + 20.4170i) q^{21} +(-16.3255 + 2.87862i) q^{22} +(46.0941 - 54.9328i) q^{23} +(162.752 - 136.565i) q^{24} +(-20.8294 - 123.252i) q^{25} +(-71.9515 + 124.624i) q^{26} +(153.684 - 88.7298i) q^{27} +(5.61053 + 15.4148i) q^{28} +(-195.497 + 71.1551i) q^{29} +(146.574 + 175.947i) q^{30} +(-43.6469 + 75.5987i) q^{31} +(101.840 + 17.9571i) q^{32} +(-38.7849 - 46.2221i) q^{33} +(101.045 + 84.7869i) q^{34} +(-72.7292 + 26.1785i) q^{35} +(106.024 + 38.5896i) q^{36} -90.2882i q^{37} +(-192.774 + 37.8942i) q^{38} -523.783 q^{39} +(-48.7351 + 270.753i) q^{40} +(-13.9198 - 78.9434i) q^{41} +(91.0243 - 108.479i) q^{42} +(151.002 + 179.958i) q^{43} +(2.87924 - 16.3290i) q^{44} +(-183.613 + 498.942i) q^{45} +(-85.0546 - 147.319i) q^{46} +(209.567 + 575.779i) q^{47} +(-116.320 - 319.587i) q^{48} +(-147.601 - 255.652i) q^{49} +(-291.645 - 53.5679i) q^{50} +(-83.3707 + 472.819i) q^{51} +(-92.5190 - 110.260i) q^{52} +(74.1754 - 88.3988i) q^{53} +(-73.1004 - 414.573i) q^{54} +(76.8945 + 13.8409i) q^{55} +170.118 q^{56} +(-470.282 - 538.695i) q^{57} +493.520i q^{58} +(700.002 + 254.780i) q^{59} +(-215.514 + 77.5732i) q^{60} +(227.794 + 191.142i) q^{61} +(133.107 + 158.631i) q^{62} +(323.769 + 57.0891i) q^{63} +(280.210 - 485.338i) q^{64} +(521.100 - 434.104i) q^{65} +(-134.503 + 48.9550i) q^{66} +(-54.1306 - 148.723i) q^{67} +(-114.258 + 65.9667i) q^{68} +(309.585 - 536.216i) q^{69} +(-0.652511 + 183.363i) q^{70} +(-569.897 + 478.200i) q^{71} +(752.115 - 896.335i) q^{72} +(-700.391 + 123.498i) q^{73} +(-201.264 - 73.2542i) q^{74} +(-361.914 - 1016.81i) q^{75} +(30.3301 - 194.150i) q^{76} -48.3139i q^{77} +(-424.965 + 1167.58i) q^{78} +(-181.078 - 1026.94i) q^{79} +(380.593 + 221.545i) q^{80} +(190.234 - 159.625i) q^{81} +(-187.269 - 33.0206i) q^{82} +(972.154 + 561.273i) q^{83} +(70.8196 + 122.663i) q^{84} +(-308.922 - 539.493i) q^{85} +(523.663 - 190.598i) q^{86} +(-1555.67 + 898.165i) q^{87} +(-148.914 - 85.9757i) q^{88} +(233.952 - 1326.81i) q^{89} +(963.235 + 814.109i) q^{90} +(-321.279 - 269.585i) q^{91} +(167.561 - 29.5455i) q^{92} +(-257.791 + 708.274i) q^{93} +1453.52 q^{94} +(914.335 + 146.172i) q^{95} +892.890 q^{96} +(-168.557 + 463.105i) q^{97} +(-689.635 + 121.601i) q^{98} +(-254.561 - 213.602i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 168 q + 6 q^{4} - 6 q^{5} + 6 q^{6} - 12 q^{9} + 3 q^{10} - 90 q^{11} + 162 q^{14} - 72 q^{15} + 426 q^{16} - 444 q^{19} - 966 q^{20} + 114 q^{21} + 108 q^{24} + 270 q^{25} - 642 q^{26} - 60 q^{29} + 396 q^{30}+ \cdots + 22692 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.811338 − 2.22913i 0.286851 − 0.788117i −0.709651 − 0.704553i \(-0.751147\pi\)
0.996502 − 0.0835640i \(-0.0266303\pi\)
\(3\) 8.50321 − 1.49935i 1.63644 − 0.288549i 0.721586 − 0.692325i \(-0.243413\pi\)
0.914858 + 0.403776i \(0.132302\pi\)
\(4\) 1.81760 + 1.52514i 0.227200 + 0.190643i
\(5\) −7.21702 + 8.53901i −0.645510 + 0.763752i
\(6\) 3.55674 − 20.1713i 0.242005 − 1.37248i
\(7\) 5.98741 + 3.45683i 0.323290 + 0.186651i 0.652858 − 0.757480i \(-0.273570\pi\)
−0.329568 + 0.944132i \(0.606903\pi\)
\(8\) 21.3095 − 12.3030i 0.941754 − 0.543722i
\(9\) 44.6849 − 16.2640i 1.65500 − 0.602369i
\(10\) 13.1791 + 23.0157i 0.416761 + 0.727820i
\(11\) −3.49409 − 6.05194i −0.0957734 − 0.165884i 0.814158 − 0.580644i \(-0.197199\pi\)
−0.909931 + 0.414759i \(0.863866\pi\)
\(12\) 17.7421 + 10.2434i 0.426809 + 0.246418i
\(13\) −59.7409 − 10.5339i −1.27455 − 0.224737i −0.504885 − 0.863187i \(-0.668465\pi\)
−0.769664 + 0.638449i \(0.779576\pi\)
\(14\) 12.5635 − 10.5421i 0.239839 − 0.201249i
\(15\) −48.5649 + 83.4298i −0.835960 + 1.43610i
\(16\) −6.83977 − 38.7902i −0.106871 − 0.606098i
\(17\) −19.0179 + 52.2513i −0.271325 + 0.745459i 0.726947 + 0.686694i \(0.240939\pi\)
−0.998272 + 0.0587656i \(0.981284\pi\)
\(18\) − 112.804i − 1.47712i
\(19\) −42.8048 − 70.8995i −0.516848 − 0.856077i
\(20\) −26.1408 + 4.51348i −0.292264 + 0.0504622i
\(21\) 56.0952 + 20.4170i 0.582904 + 0.212160i
\(22\) −16.3255 + 2.87862i −0.158209 + 0.0278965i
\(23\) 46.0941 − 54.9328i 0.417882 − 0.498012i −0.515504 − 0.856887i \(-0.672395\pi\)
0.933386 + 0.358875i \(0.116840\pi\)
\(24\) 162.752 − 136.565i 1.38424 − 1.16151i
\(25\) −20.8294 − 123.252i −0.166635 − 0.986019i
\(26\) −71.9515 + 124.624i −0.542725 + 0.940028i
\(27\) 153.684 − 88.7298i 1.09543 − 0.632446i
\(28\) 5.61053 + 15.4148i 0.0378675 + 0.104040i
\(29\) −195.497 + 71.1551i −1.25182 + 0.455627i −0.881018 − 0.473083i \(-0.843141\pi\)
−0.370806 + 0.928710i \(0.620919\pi\)
\(30\) 146.574 + 175.947i 0.892018 + 1.07078i
\(31\) −43.6469 + 75.5987i −0.252878 + 0.437998i −0.964317 − 0.264750i \(-0.914711\pi\)
0.711439 + 0.702748i \(0.248044\pi\)
\(32\) 101.840 + 17.9571i 0.562591 + 0.0992000i
\(33\) −38.7849 − 46.2221i −0.204594 − 0.243825i
\(34\) 101.045 + 84.7869i 0.509679 + 0.427672i
\(35\) −72.7292 + 26.1785i −0.351242 + 0.126428i
\(36\) 106.024 + 38.5896i 0.490852 + 0.178655i
\(37\) − 90.2882i − 0.401170i −0.979676 − 0.200585i \(-0.935716\pi\)
0.979676 − 0.200585i \(-0.0642843\pi\)
\(38\) −192.774 + 37.8942i −0.822948 + 0.161770i
\(39\) −523.783 −2.15058
\(40\) −48.7351 + 270.753i −0.192642 + 1.07024i
\(41\) −13.9198 − 78.9434i −0.0530223 − 0.300704i 0.946752 − 0.321965i \(-0.104343\pi\)
−0.999774 + 0.0212603i \(0.993232\pi\)
\(42\) 91.0243 − 108.479i 0.334413 − 0.398538i
\(43\) 151.002 + 179.958i 0.535527 + 0.638216i 0.964179 − 0.265254i \(-0.0854558\pi\)
−0.428652 + 0.903470i \(0.641011\pi\)
\(44\) 2.87924 − 16.3290i 0.00986503 − 0.0559474i
\(45\) −183.613 + 498.942i −0.608255 + 1.65284i
\(46\) −85.0546 − 147.319i −0.272622 − 0.472195i
\(47\) 209.567 + 575.779i 0.650392 + 1.78694i 0.616291 + 0.787519i \(0.288635\pi\)
0.0341012 + 0.999418i \(0.489143\pi\)
\(48\) −116.320 − 319.587i −0.349778 − 0.961007i
\(49\) −147.601 − 255.652i −0.430322 − 0.745340i
\(50\) −291.645 − 53.5679i −0.824898 − 0.151513i
\(51\) −83.3707 + 472.819i −0.228906 + 1.29819i
\(52\) −92.5190 − 110.260i −0.246732 − 0.294044i
\(53\) 74.1754 − 88.3988i 0.192241 − 0.229104i −0.661311 − 0.750112i \(-0.729999\pi\)
0.853552 + 0.521008i \(0.174444\pi\)
\(54\) −73.1004 − 414.573i −0.184217 − 1.04474i
\(55\) 76.8945 + 13.8409i 0.188517 + 0.0339328i
\(56\) 170.118 0.405946
\(57\) −470.282 − 538.695i −1.09281 − 1.25179i
\(58\) 493.520i 1.11728i
\(59\) 700.002 + 254.780i 1.54462 + 0.562195i 0.967147 − 0.254216i \(-0.0818174\pi\)
0.577471 + 0.816411i \(0.304040\pi\)
\(60\) −215.514 + 77.5732i −0.463712 + 0.166911i
\(61\) 227.794 + 191.142i 0.478132 + 0.401200i 0.849750 − 0.527185i \(-0.176753\pi\)
−0.371618 + 0.928386i \(0.621197\pi\)
\(62\) 133.107 + 158.631i 0.272655 + 0.324938i
\(63\) 323.769 + 57.0891i 0.647476 + 0.114168i
\(64\) 280.210 − 485.338i 0.547285 − 0.947925i
\(65\) 521.100 − 434.104i 0.994377 − 0.828370i
\(66\) −134.503 + 48.9550i −0.250851 + 0.0913022i
\(67\) −54.1306 − 148.723i −0.0987030 − 0.271184i 0.880507 − 0.474033i \(-0.157202\pi\)
−0.979210 + 0.202849i \(0.934980\pi\)
\(68\) −114.258 + 65.9667i −0.203762 + 0.117642i
\(69\) 309.585 − 536.216i 0.540139 − 0.935548i
\(70\) −0.652511 + 183.363i −0.00111414 + 0.313086i
\(71\) −569.897 + 478.200i −0.952596 + 0.799323i −0.979733 − 0.200309i \(-0.935805\pi\)
0.0271369 + 0.999632i \(0.491361\pi\)
\(72\) 752.115 − 896.335i 1.23108 − 1.46714i
\(73\) −700.391 + 123.498i −1.12294 + 0.198004i −0.704131 − 0.710070i \(-0.748663\pi\)
−0.418808 + 0.908075i \(0.637552\pi\)
\(74\) −201.264 − 73.2542i −0.316169 − 0.115076i
\(75\) −361.914 − 1016.81i −0.557203 − 1.56548i
\(76\) 30.3301 − 194.150i 0.0457777 − 0.293034i
\(77\) − 48.3139i − 0.0715050i
\(78\) −424.965 + 1167.58i −0.616895 + 1.69491i
\(79\) −181.078 − 1026.94i −0.257884 − 1.46253i −0.788560 − 0.614958i \(-0.789173\pi\)
0.530676 − 0.847575i \(-0.321938\pi\)
\(80\) 380.593 + 221.545i 0.531895 + 0.309619i
\(81\) 190.234 − 159.625i 0.260952 − 0.218965i
\(82\) −187.269 − 33.0206i −0.252200 − 0.0444696i
\(83\) 972.154 + 561.273i 1.28564 + 0.742262i 0.977873 − 0.209201i \(-0.0670863\pi\)
0.307763 + 0.951463i \(0.400420\pi\)
\(84\) 70.8196 + 122.663i 0.0919887 + 0.159329i
\(85\) −308.922 − 539.493i −0.394203 − 0.688426i
\(86\) 523.663 − 190.598i 0.656605 − 0.238985i
\(87\) −1555.67 + 898.165i −1.91707 + 1.10682i
\(88\) −148.914 − 85.9757i −0.180390 − 0.104148i
\(89\) 233.952 − 1326.81i 0.278639 − 1.58024i −0.448523 − 0.893771i \(-0.648050\pi\)
0.727161 − 0.686467i \(-0.240839\pi\)
\(90\) 963.235 + 814.109i 1.12815 + 0.953495i
\(91\) −321.279 − 269.585i −0.370101 − 0.310552i
\(92\) 167.561 − 29.5455i 0.189885 − 0.0334819i
\(93\) −257.791 + 708.274i −0.287437 + 0.789727i
\(94\) 1453.52 1.59488
\(95\) 914.335 + 146.172i 0.987461 + 0.157863i
\(96\) 892.890 0.949273
\(97\) −168.557 + 463.105i −0.176436 + 0.484755i −0.996114 − 0.0880697i \(-0.971930\pi\)
0.819678 + 0.572825i \(0.194152\pi\)
\(98\) −689.635 + 121.601i −0.710854 + 0.125343i
\(99\) −254.561 − 213.602i −0.258428 − 0.216847i
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 95.4.p.a.4.19 yes 168
5.4 even 2 inner 95.4.p.a.4.10 ✓ 168
19.5 even 9 inner 95.4.p.a.24.10 yes 168
95.24 even 18 inner 95.4.p.a.24.19 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.10 ✓ 168 5.4 even 2 inner
95.4.p.a.4.19 yes 168 1.1 even 1 trivial
95.4.p.a.24.10 yes 168 19.5 even 9 inner
95.4.p.a.24.19 yes 168 95.24 even 18 inner