Show commands: Magma / Pari/GP / SageMath

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 4.18
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.18

$q$-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+(0.692404 - 1.90236i) q^{2} +(6.05130 - 1.06701i) q^{3} +(2.98879 + 2.50789i) q^{4} +(-2.34979 - 10.9306i) q^{5} +(2.16011 - 12.2506i) q^{6} +(-14.4781 - 8.35896i) q^{7} +(20.8662 - 12.0471i) q^{8} +(10.1080 - 3.67902i) q^{9} +(-22.4210 - 3.09824i) q^{10} +(6.40794 + 11.0989i) q^{11} +(20.7620 + 11.9869i) q^{12} +(63.2219 + 11.1477i) q^{13} +(-25.9265 + 21.7549i) q^{14} +(-25.8824 - 63.6372i) q^{15} +(-3.05011 - 17.2981i) q^{16} +(-2.71532 + 7.46027i) q^{17} -21.7765i q^{18} +(-67.7441 + 47.6418i) q^{19} +(20.3898 - 38.5624i) q^{20} +(-96.5306 - 35.1343i) q^{21} +(25.5510 - 4.50533i) q^{22} +(19.1580 - 22.8317i) q^{23} +(113.413 - 95.1650i) q^{24} +(-113.957 + 51.3694i) q^{25} +(64.9822 - 112.552i) q^{26} +(-86.4373 + 49.9046i) q^{27} +(-22.3087 - 61.2928i) q^{28} +(107.092 - 38.9782i) q^{29} +(-138.982 + 5.17501i) q^{30} +(-49.5599 + 85.8402i) q^{31} +(154.806 + 27.2965i) q^{32} +(50.6189 + 60.3253i) q^{33} +(12.3121 + 10.3310i) q^{34} +(-57.3480 + 177.897i) q^{35} +(39.4373 + 14.3540i) q^{36} +165.968i q^{37} +(43.7257 + 161.861i) q^{38} +394.470 q^{39} +(-180.714 - 199.772i) q^{40} +(72.4369 + 410.810i) q^{41} +(-133.676 + 159.309i) q^{42} +(-67.2747 - 80.1748i) q^{43} +(-8.68281 + 49.2426i) q^{44} +(-63.9657 - 101.842i) q^{45} +(-30.1690 - 52.2543i) q^{46} +(-61.9951 - 170.330i) q^{47} +(-36.9143 - 101.421i) q^{48} +(-31.7557 - 55.0025i) q^{49} +(18.8191 + 252.356i) q^{50} +(-8.47103 + 48.0416i) q^{51} +(161.000 + 191.872i) q^{52} +(-310.949 + 370.575i) q^{53} +(35.0872 + 198.989i) q^{54} +(106.260 - 96.1228i) q^{55} -402.805 q^{56} +(-359.106 + 360.578i) q^{57} -230.716i q^{58} +(708.381 + 257.829i) q^{59} +(82.2384 - 255.108i) q^{60} +(-457.247 - 383.676i) q^{61} +(128.984 + 153.717i) q^{62} +(-177.098 - 31.2271i) q^{63} +(229.376 - 397.291i) q^{64} +(-26.7068 - 717.250i) q^{65} +(149.809 - 54.5262i) q^{66} +(-176.681 - 485.426i) q^{67} +(-26.8251 + 15.4875i) q^{68} +(91.5695 - 158.603i) q^{69} +(298.717 + 232.273i) q^{70} +(416.463 - 349.454i) q^{71} +(166.594 - 198.539i) q^{72} +(80.4757 - 14.1900i) q^{73} +(315.732 + 114.917i) q^{74} +(-634.776 + 432.444i) q^{75} +(-321.953 - 27.5037i) q^{76} -214.255i q^{77} +(273.132 - 750.425i) q^{78} +(181.139 + 1027.29i) q^{79} +(-181.911 + 73.9865i) q^{80} +(-692.293 + 580.903i) q^{81} +(831.666 + 146.645i) q^{82} +(-861.522 - 497.400i) q^{83} +(-200.397 - 347.097i) q^{84} +(87.9258 + 12.1500i) q^{85} +(-199.103 + 72.4676i) q^{86} +(606.454 - 350.136i) q^{87} +(267.419 + 154.394i) q^{88} +(125.291 - 710.558i) q^{89} +(-238.031 + 51.1702i) q^{90} +(-822.152 - 689.868i) q^{91} +(114.519 - 20.1927i) q^{92} +(-208.310 + 572.326i) q^{93} -366.956 q^{94} +(679.939 + 628.537i) q^{95} +965.904 q^{96} +(448.651 - 1232.66i) q^{97} +(-126.622 + 22.3270i) q^{98} +(105.604 + 88.6127i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.692404 − 1.90236i 0.244802 − 0.672587i −0.755055 − 0.655661i \(-0.772390\pi\)
0.999857 − 0.0169258i \(-0.00538790\pi\)
\(3\) 6.05130 − 1.06701i 1.16457 − 0.205346i 0.442244 − 0.896895i \(-0.354183\pi\)
0.722329 + 0.691549i \(0.243072\pi\)
\(4\) 2.98879 + 2.50789i 0.373599 + 0.313487i
\(5\) −2.34979 − 10.9306i −0.210172 − 0.977664i
\(6\) 2.16011 − 12.2506i 0.146977 − 0.833546i
\(7\) −14.4781 − 8.35896i −0.781746 − 0.451341i 0.0553027 − 0.998470i \(-0.482388\pi\)
−0.837049 + 0.547128i \(0.815721\pi\)
\(8\) 20.8662 − 12.0471i 0.922164 − 0.532412i
\(9\) 10.1080 − 3.67902i 0.374371 − 0.136260i
\(10\) −22.4210 − 3.09824i −0.709015 − 0.0979750i
\(11\) 6.40794 + 11.0989i 0.175642 + 0.304222i 0.940383 − 0.340116i \(-0.110466\pi\)
−0.764741 + 0.644338i \(0.777133\pi\)
\(12\) 20.7620 + 11.9869i 0.499456 + 0.288361i
\(13\) 63.2219 + 11.1477i 1.34882 + 0.237833i 0.800949 − 0.598733i \(-0.204329\pi\)
0.547867 + 0.836565i \(0.315440\pi\)
\(14\) −25.9265 + 21.7549i −0.494939 + 0.415303i
\(15\) −25.8824 − 63.6372i −0.445520 − 1.09540i
\(16\) −3.05011 − 17.2981i −0.0476580 − 0.270282i
\(17\) −2.71532 + 7.46027i −0.0387389 + 0.106434i −0.957554 − 0.288254i \(-0.906925\pi\)
0.918815 + 0.394688i \(0.129147\pi\)
\(18\) − 21.7765i − 0.285154i
\(19\) −67.7441 + 47.6418i −0.817977 + 0.575251i
\(20\) 20.3898 − 38.5624i 0.227965 − 0.431140i
\(21\) −96.5306 − 35.1343i −1.00308 − 0.365092i
\(22\) 25.5510 − 4.50533i 0.247613 − 0.0436609i
\(23\) 19.1580 − 22.8317i 0.173684 − 0.206988i −0.672179 − 0.740389i \(-0.734642\pi\)
0.845863 + 0.533400i \(0.179086\pi\)
\(24\) 113.413 − 95.1650i 0.964599 − 0.809395i
\(25\) −113.957 + 51.3694i −0.911656 + 0.410955i
\(26\) 64.9822 − 112.552i 0.490156 − 0.848975i
\(27\) −86.4373 + 49.9046i −0.616106 + 0.355709i
\(28\) −22.3087 − 61.2928i −0.150570 − 0.413687i
\(29\) 107.092 − 38.9782i 0.685739 − 0.249589i 0.0244292 − 0.999702i \(-0.492223\pi\)
0.661310 + 0.750113i \(0.270001\pi\)
\(30\) −138.982 + 5.17501i −0.845819 + 0.0314941i
\(31\) −49.5599 + 85.8402i −0.287136 + 0.497334i −0.973125 − 0.230278i \(-0.926037\pi\)
0.685989 + 0.727612i \(0.259370\pi\)
\(32\) 154.806 + 27.2965i 0.855192 + 0.150793i
\(33\) 50.6189 + 60.3253i 0.267019 + 0.318221i
\(34\) 12.3121 + 10.3310i 0.0621029 + 0.0521105i
\(35\) −57.3480 + 177.897i −0.276959 + 0.859144i
\(36\) 39.4373 + 14.3540i 0.182580 + 0.0664537i
\(37\) 165.968i 0.737432i 0.929542 + 0.368716i \(0.120203\pi\)
−0.929542 + 0.368716i \(0.879797\pi\)
\(38\) 43.7257 + 161.861i 0.186664 + 0.690983i
\(39\) 394.470 1.61963
\(40\) −180.714 − 199.772i −0.714333 − 0.789669i
\(41\) 72.4369 + 410.810i 0.275921 + 1.56482i 0.736023 + 0.676957i \(0.236702\pi\)
−0.460102 + 0.887866i \(0.652187\pi\)
\(42\) −133.676 + 159.309i −0.491112 + 0.585285i
\(43\) −67.2747 − 80.1748i −0.238588 − 0.284338i 0.633442 − 0.773790i \(-0.281641\pi\)
−0.872030 + 0.489452i \(0.837197\pi\)
\(44\) −8.68281 + 49.2426i −0.0297496 + 0.168718i
\(45\) −63.9657 − 101.842i −0.211899 − 0.337371i
\(46\) −30.1690 − 52.2543i −0.0966996 − 0.167489i
\(47\) −61.9951 − 170.330i −0.192402 − 0.528621i 0.805554 − 0.592523i \(-0.201868\pi\)
−0.997956 + 0.0639013i \(0.979646\pi\)
\(48\) −36.9143 − 101.421i −0.111002 − 0.304977i
\(49\) −31.7557 − 55.0025i −0.0925822 − 0.160357i
\(50\) 18.8191 + 252.356i 0.0532284 + 0.713770i
\(51\) −8.47103 + 48.0416i −0.0232585 + 0.131905i
\(52\) 161.000 + 191.872i 0.429359 + 0.511690i
\(53\) −310.949 + 370.575i −0.805890 + 0.960422i −0.999788 − 0.0205995i \(-0.993442\pi\)
0.193898 + 0.981022i \(0.437887\pi\)
\(54\) 35.0872 + 198.989i 0.0884216 + 0.501464i
\(55\) 106.260 − 96.1228i 0.260512 − 0.235658i
\(56\) −402.805 −0.961198
\(57\) −359.106 + 360.578i −0.834469 + 0.837890i
\(58\) − 230.716i − 0.522319i
\(59\) 708.381 + 257.829i 1.56311 + 0.568924i 0.971446 − 0.237263i \(-0.0762502\pi\)
0.591661 + 0.806187i \(0.298472\pi\)
\(60\) 82.2384 − 255.108i 0.176949 − 0.548906i
\(61\) −457.247 − 383.676i −0.959746 − 0.805322i 0.0211660 − 0.999776i \(-0.493262\pi\)
−0.980912 + 0.194454i \(0.937707\pi\)
\(62\) 128.984 + 153.717i 0.264209 + 0.314872i
\(63\) −177.098 − 31.2271i −0.354163 − 0.0624484i
\(64\) 229.376 − 397.291i 0.448000 − 0.775959i
\(65\) −26.7068 − 717.250i −0.0509627 − 1.36868i
\(66\) 149.809 − 54.5262i 0.279398 − 0.101693i
\(67\) −176.681 − 485.426i −0.322164 − 0.885138i −0.990030 − 0.140857i \(-0.955014\pi\)
0.667866 − 0.744281i \(-0.267208\pi\)
\(68\) −26.8251 + 15.4875i −0.0478385 + 0.0276196i
\(69\) 91.5695 − 158.603i 0.159763 − 0.276718i
\(70\) 298.717 + 232.273i 0.510049 + 0.396599i
\(71\) 416.463 − 349.454i 0.696128 − 0.584121i −0.224541 − 0.974465i \(-0.572088\pi\)
0.920669 + 0.390344i \(0.127644\pi\)
\(72\) 166.594 − 198.539i 0.272685 − 0.324974i
\(73\) 80.4757 − 14.1900i 0.129027 − 0.0227509i −0.108762 − 0.994068i \(-0.534689\pi\)
0.237789 + 0.971317i \(0.423577\pi\)
\(74\) 315.732 + 114.917i 0.495988 + 0.180525i
\(75\) −634.776 + 432.444i −0.977302 + 0.665792i
\(76\) −321.953 − 27.5037i −0.485929 − 0.0415117i
\(77\) − 214.255i − 0.317099i
\(78\) 273.132 − 750.425i 0.396489 − 1.08934i
\(79\) 181.139 + 1027.29i 0.257971 + 1.46302i 0.788331 + 0.615251i \(0.210946\pi\)
−0.530360 + 0.847772i \(0.677943\pi\)
\(80\) −181.911 + 73.9865i −0.254229 + 0.103399i
\(81\) −692.293 + 580.903i −0.949648 + 0.796849i
\(82\) 831.666 + 146.645i 1.12003 + 0.197491i
\(83\) −861.522 − 497.400i −1.13933 − 0.657792i −0.193065 − 0.981186i \(-0.561843\pi\)
−0.946264 + 0.323394i \(0.895176\pi\)
\(84\) −200.397 − 347.097i −0.260299 − 0.450850i
\(85\) 87.9258 + 12.1500i 0.112199 + 0.0155041i
\(86\) −199.103 + 72.4676i −0.249649 + 0.0908649i
\(87\) 606.454 − 350.136i 0.747341 − 0.431478i
\(88\) 267.419 + 154.394i 0.323942 + 0.187028i
\(89\) 125.291 − 710.558i 0.149222 − 0.846281i −0.814657 − 0.579943i \(-0.803075\pi\)
0.963880 − 0.266338i \(-0.0858139\pi\)
\(90\) −238.031 + 51.1702i −0.278785 + 0.0599313i
\(91\) −822.152 − 689.868i −0.947088 − 0.794701i
\(92\) 114.519 − 20.1927i 0.129776 − 0.0228830i
\(93\) −208.310 + 572.326i −0.232266 + 0.638144i
\(94\) −366.956 −0.402644
\(95\) 679.939 + 628.537i 0.734318 + 0.678805i
\(96\) 965.904 1.02690
\(97\) 448.651 − 1232.66i 0.469624 − 1.29028i −0.448426 − 0.893820i \(-0.648015\pi\)
0.918051 − 0.396463i \(-0.129762\pi\)
\(98\) −126.622 + 22.3270i −0.130518 + 0.0230139i
\(99\) 105.604 + 88.6127i 0.107209 + 0.0899587i
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.4.18 yes 168
5.4 even 2 inner 95.4.p.a.4.11 ✓ 168
19.5 even 9 inner 95.4.p.a.24.11 yes 168
95.24 even 18 inner 95.4.p.a.24.18 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.11 ✓ 168 5.4 even 2 inner
95.4.p.a.4.18 yes 168 1.1 even 1 trivial
95.4.p.a.24.11 yes 168 19.5 even 9 inner
95.4.p.a.24.18 yes 168 95.24 even 18 inner