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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,4,Mod(2,95)] 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("95.2"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(36)) 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.r (of order \(36\), degree \(12\), 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: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 2.14
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.14

$q$-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+(-0.382479 - 0.546236i) q^{2} +(-0.0141242 - 0.161440i) q^{3} +(2.58408 - 7.09969i) q^{4} +(5.04018 + 9.97981i) q^{5} +(-0.0827821 + 0.0694624i) q^{6} +(1.96157 - 7.32067i) q^{7} +(-10.0193 + 2.68467i) q^{8} +(26.5639 - 4.68394i) q^{9} +(3.52357 - 6.57020i) q^{10} +(16.0298 - 27.7644i) q^{11} +(-1.18267 - 0.316896i) q^{12} +(-28.3049 - 2.47636i) q^{13} +(-4.74907 + 1.72852i) q^{14} +(1.53995 - 0.954642i) q^{15} +(-41.0031 - 34.4057i) q^{16} +(71.7864 - 50.2654i) q^{17} +(-12.7187 - 12.7187i) q^{18} +(71.5600 - 41.6914i) q^{19} +(83.8778 - 9.99516i) q^{20} +(-1.20955 - 0.213277i) q^{21} +(-21.2970 + 1.86324i) q^{22} +(17.0012 - 7.92779i) q^{23} +(0.574928 + 1.57960i) q^{24} +(-74.1931 + 100.600i) q^{25} +(9.47336 + 16.4083i) q^{26} +(-2.26384 - 8.44875i) q^{27} +(-46.9057 - 32.8437i) q^{28} +(13.6861 + 77.6177i) q^{29} +(-1.11046 - 0.476046i) q^{30} +(-31.6852 + 18.2935i) q^{31} +(-10.3432 + 118.223i) q^{32} +(-4.70869 - 2.19570i) q^{33} +(-54.9135 - 19.9869i) q^{34} +(82.9455 - 17.3214i) q^{35} +(35.3887 - 200.700i) q^{36} +(-297.196 + 297.196i) q^{37} +(-50.1435 - 23.1426i) q^{38} +4.60452i q^{39} +(-77.2919 - 86.4599i) q^{40} +(-36.7541 + 43.8018i) q^{41} +(0.346129 + 0.742276i) q^{42} +(-14.7390 + 31.6080i) q^{43} +(-155.697 - 185.552i) q^{44} +(180.632 + 241.495i) q^{45} +(-10.8330 - 6.25446i) q^{46} +(-243.287 + 347.449i) q^{47} +(-4.97532 + 7.10549i) q^{48} +(247.302 + 142.780i) q^{49} +(83.3287 + 2.04957i) q^{50} +(-9.12875 - 10.8792i) q^{51} +(-90.7235 + 194.557i) q^{52} +(122.590 + 262.896i) q^{53} +(-3.74915 + 4.46806i) q^{54} +(357.876 + 20.0365i) q^{55} +78.6145i q^{56} +(-7.74137 - 10.9638i) q^{57} +(37.1630 - 37.1630i) q^{58} +(65.3901 - 370.846i) q^{59} +(-2.79832 - 13.4000i) q^{60} +(-493.079 - 179.466i) q^{61} +(22.1115 + 10.3108i) q^{62} +(17.8174 - 203.654i) q^{63} +(-302.304 + 174.535i) q^{64} +(-117.948 - 294.959i) q^{65} +(0.601603 + 3.41186i) q^{66} +(673.244 + 471.411i) q^{67} +(-171.367 - 639.551i) q^{68} +(-1.51999 - 2.63270i) q^{69} +(-41.1865 - 38.6828i) q^{70} +(-245.535 - 674.602i) q^{71} +(-253.578 + 118.246i) q^{72} +(-292.650 + 25.6036i) q^{73} +(276.010 + 48.6681i) q^{74} +(17.2888 + 10.5568i) q^{75} +(-111.079 - 615.788i) q^{76} +(-171.810 - 171.810i) q^{77} +(2.51515 - 1.76113i) q^{78} +(711.706 + 597.192i) q^{79} +(136.699 - 582.614i) q^{80} +(683.038 - 248.605i) q^{81} +(37.9838 + 3.32315i) q^{82} +(-411.531 - 110.269i) q^{83} +(-4.63978 + 8.03633i) q^{84} +(863.455 + 463.068i) q^{85} +(22.9028 - 4.03838i) q^{86} +(12.3373 - 3.30576i) q^{87} +(-86.0695 + 321.216i) q^{88} +(524.051 - 439.731i) q^{89} +(62.8255 - 191.035i) q^{90} +(-73.6506 + 202.353i) q^{91} +(-12.3525 - 141.189i) q^{92} +(3.40082 + 4.85688i) q^{93} +282.842 q^{94} +(776.747 + 504.023i) q^{95} +19.2320 q^{96} +(-252.093 - 360.027i) q^{97} +(-16.5962 - 189.696i) q^{98} +(295.767 - 812.614i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 336 q - 12 q^{2} - 12 q^{3} - 12 q^{5} + 12 q^{6} + 6 q^{7} - 18 q^{8} - 12 q^{10} - 12 q^{11} - 18 q^{12} - 12 q^{13} - 264 q^{15} - 996 q^{16} + 132 q^{17} + 1116 q^{20} - 600 q^{21} + 468 q^{22} + 564 q^{23}+ \cdots + 1290 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.382479 − 0.546236i −0.135227 − 0.193124i 0.745827 − 0.666139i \(-0.232054\pi\)
−0.881054 + 0.473015i \(0.843165\pi\)
\(3\) −0.0141242 − 0.161440i −0.00271819 − 0.0310691i 0.994714 − 0.102680i \(-0.0327417\pi\)
−0.997433 + 0.0716106i \(0.977186\pi\)
\(4\) 2.58408 − 7.09969i 0.323010 − 0.887462i
\(5\) 5.04018 + 9.97981i 0.450808 + 0.892621i
\(6\) −0.0827821 + 0.0694624i −0.00563261 + 0.00472632i
\(7\) 1.96157 − 7.32067i 0.105915 − 0.395279i −0.892533 − 0.450983i \(-0.851074\pi\)
0.998447 + 0.0557038i \(0.0177402\pi\)
\(8\) −10.0193 + 2.68467i −0.442797 + 0.118647i
\(9\) 26.5639 − 4.68394i 0.983850 − 0.173479i
\(10\) 3.52357 − 6.57020i 0.111425 − 0.207768i
\(11\) 16.0298 − 27.7644i 0.439378 − 0.761025i −0.558263 − 0.829664i \(-0.688532\pi\)
0.997642 + 0.0686383i \(0.0218655\pi\)
\(12\) −1.18267 − 0.316896i −0.0284506 − 0.00762333i
\(13\) −28.3049 − 2.47636i −0.603875 − 0.0528322i −0.218881 − 0.975751i \(-0.570241\pi\)
−0.384993 + 0.922919i \(0.625796\pi\)
\(14\) −4.74907 + 1.72852i −0.0906602 + 0.0329976i
\(15\) 1.53995 − 0.954642i 0.0265076 − 0.0164325i
\(16\) −41.0031 − 34.4057i −0.640674 − 0.537589i
\(17\) 71.7864 − 50.2654i 1.02416 − 0.717126i 0.0643951 − 0.997924i \(-0.479488\pi\)
0.959767 + 0.280799i \(0.0905993\pi\)
\(18\) −12.7187 − 12.7187i −0.166546 − 0.166546i
\(19\) 71.5600 − 41.6914i 0.864052 − 0.503403i
\(20\) 83.8778 − 9.99516i 0.937782 − 0.111749i
\(21\) −1.20955 − 0.213277i −0.0125689 − 0.00221623i
\(22\) −21.2970 + 1.86324i −0.206388 + 0.0180566i
\(23\) 17.0012 − 7.92779i 0.154130 − 0.0718721i −0.344023 − 0.938961i \(-0.611790\pi\)
0.498153 + 0.867089i \(0.334012\pi\)
\(24\) 0.574928 + 1.57960i 0.00488986 + 0.0134348i
\(25\) −74.1931 + 100.600i −0.593545 + 0.804801i
\(26\) 9.47336 + 16.4083i 0.0714568 + 0.123767i
\(27\) −2.26384 − 8.44875i −0.0161361 − 0.0602209i
\(28\) −46.9057 − 32.8437i −0.316583 − 0.221674i
\(29\) 13.6861 + 77.6177i 0.0876360 + 0.497008i 0.996757 + 0.0804754i \(0.0256438\pi\)
−0.909121 + 0.416533i \(0.863245\pi\)
\(30\) −1.11046 − 0.476046i −0.00675804 − 0.00289712i
\(31\) −31.6852 + 18.2935i −0.183575 + 0.105987i −0.588971 − 0.808154i \(-0.700467\pi\)
0.405396 + 0.914141i \(0.367134\pi\)
\(32\) −10.3432 + 118.223i −0.0571387 + 0.653098i
\(33\) −4.70869 − 2.19570i −0.0248387 − 0.0115825i
\(34\) −54.9135 − 19.9869i −0.276988 − 0.100815i
\(35\) 82.9455 − 17.3214i 0.400582 − 0.0836531i
\(36\) 35.3887 − 200.700i 0.163837 − 0.929165i
\(37\) −297.196 + 297.196i −1.32051 + 1.32051i −0.407140 + 0.913366i \(0.633474\pi\)
−0.913366 + 0.407140i \(0.866526\pi\)
\(38\) −50.1435 − 23.1426i −0.214062 − 0.0987954i
\(39\) 4.60452i 0.0189055i
\(40\) −77.2919 − 86.4599i −0.305523 − 0.341763i
\(41\) −36.7541 + 43.8018i −0.140001 + 0.166846i −0.831489 − 0.555542i \(-0.812511\pi\)
0.691488 + 0.722388i \(0.256955\pi\)
\(42\) 0.346129 + 0.742276i 0.00127164 + 0.00272704i
\(43\) −14.7390 + 31.6080i −0.0522717 + 0.112097i −0.930709 − 0.365761i \(-0.880809\pi\)
0.878437 + 0.477858i \(0.158587\pi\)
\(44\) −155.697 − 185.552i −0.533458 − 0.635750i
\(45\) 180.632 + 241.495i 0.598378 + 0.799999i
\(46\) −10.8330 − 6.25446i −0.0347227 − 0.0200472i
\(47\) −243.287 + 347.449i −0.755043 + 1.07831i 0.239180 + 0.970975i \(0.423121\pi\)
−0.994223 + 0.107338i \(0.965767\pi\)
\(48\) −4.97532 + 7.10549i −0.0149609 + 0.0213664i
\(49\) 247.302 + 142.780i 0.720998 + 0.416268i
\(50\) 83.3287 + 2.04957i 0.235689 + 0.00579706i
\(51\) −9.12875 − 10.8792i −0.0250643 − 0.0298705i
\(52\) −90.7235 + 194.557i −0.241944 + 0.518850i
\(53\) 122.590 + 262.896i 0.317719 + 0.681350i 0.998701 − 0.0509497i \(-0.0162248\pi\)
−0.680983 + 0.732300i \(0.738447\pi\)
\(54\) −3.74915 + 4.46806i −0.00944804 + 0.0112597i
\(55\) 357.876 + 20.0365i 0.877382 + 0.0491222i
\(56\) 78.6145i 0.187595i
\(57\) −7.74137 − 10.9638i −0.0179889 − 0.0254770i
\(58\) 37.1630 − 37.1630i 0.0841334 − 0.0841334i
\(59\) 65.3901 − 370.846i 0.144289 − 0.818305i −0.823646 − 0.567105i \(-0.808063\pi\)
0.967935 − 0.251201i \(-0.0808254\pi\)
\(60\) −2.79832 − 13.4000i −0.00602102 − 0.0288323i
\(61\) −493.079 − 179.466i −1.03496 − 0.376693i −0.231989 − 0.972718i \(-0.574523\pi\)
−0.802966 + 0.596025i \(0.796746\pi\)
\(62\) 22.1115 + 10.3108i 0.0452929 + 0.0211204i
\(63\) 17.8174 − 203.654i 0.0356314 − 0.407269i
\(64\) −302.304 + 174.535i −0.590437 + 0.340889i
\(65\) −117.948 − 294.959i −0.225072 − 0.562849i
\(66\) 0.601603 + 3.41186i 0.00112200 + 0.00636320i
\(67\) 673.244 + 471.411i 1.22761 + 0.859582i 0.993634 − 0.112661i \(-0.0359373\pi\)
0.233976 + 0.972242i \(0.424826\pi\)
\(68\) −171.367 − 639.551i −0.305608 − 1.14054i
\(69\) −1.51999 − 2.63270i −0.00265196 − 0.00459332i
\(70\) −41.1865 − 38.6828i −0.0703247 − 0.0660497i
\(71\) −245.535 − 674.602i −0.410418 − 1.12761i −0.956970 − 0.290188i \(-0.906282\pi\)
0.546552 − 0.837425i \(-0.315940\pi\)
\(72\) −253.578 + 118.246i −0.415063 + 0.193547i
\(73\) −292.650 + 25.6036i −0.469207 + 0.0410503i −0.319308 − 0.947651i \(-0.603450\pi\)
−0.149899 + 0.988701i \(0.547895\pi\)
\(74\) 276.010 + 48.6681i 0.433589 + 0.0764534i
\(75\) 17.2888 + 10.5568i 0.0266178 + 0.0162533i
\(76\) −111.079 − 615.788i −0.167654 − 0.929417i
\(77\) −171.810 − 171.810i −0.254281 − 0.254281i
\(78\) 2.51515 − 1.76113i 0.00365109 − 0.00255652i
\(79\) 711.706 + 597.192i 1.01358 + 0.850499i 0.988808 − 0.149194i \(-0.0476680\pi\)
0.0247770 + 0.999693i \(0.492112\pi\)
\(80\) 136.699 − 582.614i 0.191043 − 0.814228i
\(81\) 683.038 − 248.605i 0.936951 − 0.341022i
\(82\) 37.9838 + 3.32315i 0.0511538 + 0.00447537i
\(83\) −411.531 − 110.269i −0.544233 − 0.145827i −0.0237802 − 0.999717i \(-0.507570\pi\)
−0.520453 + 0.853890i \(0.674237\pi\)
\(84\) −4.63978 + 8.03633i −0.00602668 + 0.0104385i
\(85\) 863.455 + 463.068i 1.10182 + 0.590903i
\(86\) 22.9028 − 4.03838i 0.0287171 − 0.00506360i
\(87\) 12.3373 − 3.30576i 0.0152034 − 0.00407374i
\(88\) −86.0695 + 321.216i −0.104262 + 0.389110i
\(89\) 524.051 − 439.731i 0.624149 − 0.523724i −0.274955 − 0.961457i \(-0.588663\pi\)
0.899105 + 0.437733i \(0.144219\pi\)
\(90\) 62.8255 − 191.035i 0.0735821 − 0.223742i
\(91\) −73.6506 + 202.353i −0.0848427 + 0.233103i
\(92\) −12.3525 − 141.189i −0.0139982 − 0.160000i
\(93\) 3.40082 + 4.85688i 0.00379192 + 0.00541542i
\(94\) 282.842 0.310350
\(95\) 776.747 + 504.023i 0.838869 + 0.544333i
\(96\) 19.2320 0.0204465
\(97\) −252.093 − 360.027i −0.263878 − 0.376857i 0.665246 − 0.746624i \(-0.268327\pi\)
−0.929125 + 0.369767i \(0.879438\pi\)
\(98\) −16.5962 − 189.696i −0.0171069 − 0.195532i
\(99\) 295.767 − 812.614i 0.300260 − 0.824958i
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.r.a.2.14 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.14 yes 336
19.10 odd 18 inner 95.4.r.a.67.14 yes 336
95.48 even 36 inner 95.4.r.a.48.14 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.14 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.14 yes 336 95.48 even 36 inner
95.4.r.a.67.14 yes 336 19.10 odd 18 inner
95.4.r.a.78.14 yes 336 5.3 odd 4 inner