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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 2.11
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.11

$q$-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+(-0.867729 - 1.23924i) q^{2} +(-0.319403 - 3.65079i) q^{3} +(1.95339 - 5.36688i) q^{4} +(-3.62085 - 10.5778i) q^{5} +(-4.24707 + 3.56372i) q^{6} +(-0.716622 + 2.67447i) q^{7} +(-20.0362 + 5.36869i) q^{8} +(13.3635 - 2.35635i) q^{9} +(-9.96655 + 13.6658i) q^{10} +(-9.82712 + 17.0211i) q^{11} +(-20.2173 - 5.41721i) q^{12} +(9.11241 + 0.797233i) q^{13} +(3.93616 - 1.43264i) q^{14} +(-37.4608 + 16.5975i) q^{15} +(-10.9619 - 9.19811i) q^{16} +(-46.2746 + 32.4018i) q^{17} +(-14.5160 - 14.5160i) q^{18} +(-5.11986 - 82.6607i) q^{19} +(-63.8426 - 1.22983i) q^{20} +(9.99283 + 1.76201i) q^{21} +(29.6206 - 2.59146i) q^{22} +(-5.26306 + 2.45421i) q^{23} +(25.9996 + 71.4333i) q^{24} +(-98.7789 + 76.6010i) q^{25} +(-6.91914 - 11.9843i) q^{26} +(-38.4805 - 143.611i) q^{27} +(12.9537 + 9.07030i) q^{28} +(2.09942 + 11.9064i) q^{29} +(53.0742 + 32.0209i) q^{30} +(220.949 - 127.565i) q^{31} +(-16.3498 + 186.879i) q^{32} +(65.2792 + 30.4402i) q^{33} +(80.3075 + 29.2295i) q^{34} +(30.8847 - 2.10357i) q^{35} +(13.4579 - 76.3235i) q^{36} +(-19.5692 + 19.5692i) q^{37} +(-97.9942 + 78.0718i) q^{38} -33.5222i q^{39} +(129.337 + 192.500i) q^{40} +(333.060 - 396.926i) q^{41} +(-6.48751 - 13.9125i) q^{42} +(132.345 - 283.816i) q^{43} +(72.1540 + 85.9897i) q^{44} +(-73.3123 - 132.825i) q^{45} +(7.60828 + 4.39264i) q^{46} +(-154.709 + 220.947i) q^{47} +(-30.0791 + 42.9574i) q^{48} +(290.407 + 167.667i) q^{49} +(180.641 + 55.9424i) q^{50} +(133.072 + 158.590i) q^{51} +(22.0787 - 47.3480i) q^{52} +(-146.038 - 313.178i) q^{53} +(-144.579 + 172.302i) q^{54} +(215.628 + 42.3185i) q^{55} -57.4336i q^{56} +(-300.142 + 45.0936i) q^{57} +(12.9332 - 12.9332i) q^{58} +(-36.1454 + 204.991i) q^{59} +(15.9017 + 233.469i) q^{60} +(-48.2048 - 17.5451i) q^{61} +(-349.808 - 163.118i) q^{62} +(-3.27461 + 37.4290i) q^{63} +(146.635 - 84.6597i) q^{64} +(-24.5617 - 99.2758i) q^{65} +(-18.9218 - 107.311i) q^{66} +(-524.780 - 367.455i) q^{67} +(83.5046 + 311.643i) q^{68} +(10.6408 + 18.4305i) q^{69} +(-29.4064 - 36.4484i) q^{70} +(-50.1965 - 137.914i) q^{71} +(-255.104 + 118.957i) q^{72} +(793.550 - 69.4266i) q^{73} +(41.2318 + 7.27029i) q^{74} +(311.205 + 336.155i) q^{75} +(-453.631 - 133.991i) q^{76} +(-38.4800 - 38.4800i) q^{77} +(-41.5422 + 29.0881i) q^{78} +(-326.200 - 273.715i) q^{79} +(-57.6043 + 149.257i) q^{80} +(-167.718 + 61.0443i) q^{81} +(-780.894 - 68.3194i) q^{82} +(-810.670 - 217.218i) q^{83} +(28.9763 - 50.1885i) q^{84} +(510.292 + 372.160i) q^{85} +(-466.557 + 82.2666i) q^{86} +(42.7972 - 11.4675i) q^{87} +(105.517 - 393.797i) q^{88} +(-682.192 + 572.427i) q^{89} +(-100.987 + 206.108i) q^{90} +(-8.66233 + 23.7996i) q^{91} +(2.89065 + 33.0403i) q^{92} +(-536.284 - 765.893i) q^{93} +408.053 q^{94} +(-855.828 + 353.458i) q^{95} +687.477 q^{96} +(147.802 + 211.084i) q^{97} +(-44.2146 - 505.375i) q^{98} +(-91.2175 + 250.618i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.867729 − 1.23924i −0.306788 − 0.438139i 0.635921 − 0.771754i \(-0.280620\pi\)
−0.942709 + 0.333615i \(0.891731\pi\)
\(3\) −0.319403 − 3.65079i −0.0614691 − 0.702595i −0.962933 − 0.269741i \(-0.913062\pi\)
0.901464 − 0.432854i \(-0.142494\pi\)
\(4\) 1.95339 − 5.36688i 0.244173 − 0.670861i
\(5\) −3.62085 − 10.5778i −0.323858 − 0.946106i
\(6\) −4.24707 + 3.56372i −0.288977 + 0.242480i
\(7\) −0.716622 + 2.67447i −0.0386940 + 0.144408i −0.982570 − 0.185891i \(-0.940483\pi\)
0.943876 + 0.330299i \(0.107150\pi\)
\(8\) −20.0362 + 5.36869i −0.885484 + 0.237265i
\(9\) 13.3635 − 2.35635i 0.494946 − 0.0872723i
\(10\) −9.96655 + 13.6658i −0.315170 + 0.432149i
\(11\) −9.82712 + 17.0211i −0.269363 + 0.466550i −0.968697 − 0.248245i \(-0.920146\pi\)
0.699335 + 0.714794i \(0.253480\pi\)
\(12\) −20.2173 − 5.41721i −0.486353 − 0.130318i
\(13\) 9.11241 + 0.797233i 0.194410 + 0.0170087i 0.183945 − 0.982937i \(-0.441113\pi\)
0.0104651 + 0.999945i \(0.496669\pi\)
\(14\) 3.93616 − 1.43264i 0.0751416 − 0.0273493i
\(15\) −37.4608 + 16.5975i −0.644822 + 0.285698i
\(16\) −10.9619 − 9.19811i −0.171279 − 0.143720i
\(17\) −46.2746 + 32.4018i −0.660190 + 0.462270i −0.855039 − 0.518563i \(-0.826467\pi\)
0.194849 + 0.980833i \(0.437578\pi\)
\(18\) −14.5160 − 14.5160i −0.190081 − 0.190081i
\(19\) −5.11986 − 82.6607i −0.0618198 − 0.998087i
\(20\) −63.8426 − 1.22983i −0.713782 − 0.0137499i
\(21\) 9.99283 + 1.76201i 0.103839 + 0.0183096i
\(22\) 29.6206 − 2.59146i 0.287051 − 0.0251137i
\(23\) −5.26306 + 2.45421i −0.0477141 + 0.0222495i −0.446329 − 0.894869i \(-0.647269\pi\)
0.398615 + 0.917118i \(0.369491\pi\)
\(24\) 25.9996 + 71.4333i 0.221131 + 0.607552i
\(25\) −98.7789 + 76.6010i −0.790232 + 0.612808i
\(26\) −6.91914 − 11.9843i −0.0521905 − 0.0903967i
\(27\) −38.4805 − 143.611i −0.274281 − 1.02363i
\(28\) 12.9537 + 9.07030i 0.0874295 + 0.0612188i
\(29\) 2.09942 + 11.9064i 0.0134432 + 0.0762400i 0.990791 − 0.135398i \(-0.0432313\pi\)
−0.977348 + 0.211638i \(0.932120\pi\)
\(30\) 53.0742 + 32.0209i 0.322999 + 0.194873i
\(31\) 220.949 − 127.565i 1.28012 − 0.739075i 0.303246 − 0.952912i \(-0.401930\pi\)
0.976869 + 0.213838i \(0.0685963\pi\)
\(32\) −16.3498 + 186.879i −0.0903205 + 1.03237i
\(33\) 65.2792 + 30.4402i 0.344353 + 0.160574i
\(34\) 80.3075 + 29.2295i 0.405077 + 0.147436i
\(35\) 30.8847 − 2.10357i 0.149156 − 0.0101591i
\(36\) 13.4579 − 76.3235i 0.0623050 − 0.353349i
\(37\) −19.5692 + 19.5692i −0.0869503 + 0.0869503i −0.749244 − 0.662294i \(-0.769583\pi\)
0.662294 + 0.749244i \(0.269583\pi\)
\(38\) −97.9942 + 78.0718i −0.418336 + 0.333287i
\(39\) − 33.5222i − 0.137637i
\(40\) 129.337 + 192.500i 0.511249 + 0.760921i
\(41\) 333.060 − 396.926i 1.26867 − 1.51194i 0.510492 − 0.859882i \(-0.329463\pi\)
0.758173 − 0.652054i \(-0.226092\pi\)
\(42\) −6.48751 − 13.9125i −0.0238344 − 0.0511130i
\(43\) 132.345 − 283.816i 0.469360 − 1.00655i −0.519051 − 0.854744i \(-0.673714\pi\)
0.988411 − 0.151803i \(-0.0485079\pi\)
\(44\) 72.1540 + 85.9897i 0.247219 + 0.294624i
\(45\) −73.3123 − 132.825i −0.242861 − 0.440007i
\(46\) 7.60828 + 4.39264i 0.0243865 + 0.0140796i
\(47\) −154.709 + 220.947i −0.480140 + 0.685711i −0.984019 − 0.178063i \(-0.943017\pi\)
0.503879 + 0.863774i \(0.331906\pi\)
\(48\) −30.0791 + 42.9574i −0.0904489 + 0.129174i
\(49\) 290.407 + 167.667i 0.846669 + 0.488825i
\(50\) 180.641 + 55.9424i 0.510929 + 0.158229i
\(51\) 133.072 + 158.590i 0.365370 + 0.435431i
\(52\) 22.0787 − 47.3480i 0.0588801 − 0.126269i
\(53\) −146.038 − 313.178i −0.378487 − 0.811667i −0.999609 − 0.0279539i \(-0.991101\pi\)
0.621122 − 0.783713i \(-0.286677\pi\)
\(54\) −144.579 + 172.302i −0.364346 + 0.434211i
\(55\) 215.628 + 42.3185i 0.528641 + 0.103749i
\(56\) − 57.4336i − 0.137052i
\(57\) −300.142 + 45.0936i −0.697451 + 0.104786i
\(58\) 12.9332 − 12.9332i 0.0292795 − 0.0292795i
\(59\) −36.1454 + 204.991i −0.0797581 + 0.452331i 0.918607 + 0.395172i \(0.129315\pi\)
−0.998365 + 0.0571584i \(0.981796\pi\)
\(60\) 15.9017 + 233.469i 0.0342149 + 0.502345i
\(61\) −48.2048 − 17.5451i −0.101180 − 0.0368265i 0.290934 − 0.956743i \(-0.406034\pi\)
−0.392114 + 0.919917i \(0.628256\pi\)
\(62\) −349.808 − 163.118i −0.716542 − 0.334129i
\(63\) −3.27461 + 37.4290i −0.00654861 + 0.0748510i
\(64\) 146.635 − 84.6597i 0.286396 − 0.165351i
\(65\) −24.5617 − 99.2758i −0.0468693 − 0.189441i
\(66\) −18.9218 − 107.311i −0.0352896 − 0.200137i
\(67\) −524.780 − 367.455i −0.956896 − 0.670026i −0.0128773 − 0.999917i \(-0.504099\pi\)
−0.944019 + 0.329891i \(0.892988\pi\)
\(68\) 83.5046 + 311.643i 0.148918 + 0.555769i
\(69\) 10.6408 + 18.4305i 0.0185653 + 0.0321561i
\(70\) −29.4064 − 36.4484i −0.0502106 − 0.0622346i
\(71\) −50.1965 − 137.914i −0.0839047 − 0.230526i 0.890644 − 0.454701i \(-0.150254\pi\)
−0.974549 + 0.224174i \(0.928031\pi\)
\(72\) −255.104 + 118.957i −0.417560 + 0.194711i
\(73\) 793.550 − 69.4266i 1.27230 − 0.111312i 0.569079 − 0.822283i \(-0.307300\pi\)
0.703222 + 0.710971i \(0.251744\pi\)
\(74\) 41.2318 + 7.27029i 0.0647717 + 0.0114210i
\(75\) 311.205 + 336.155i 0.479131 + 0.517544i
\(76\) −453.631 − 133.991i −0.684672 − 0.202234i
\(77\) −38.4800 − 38.4800i −0.0569507 − 0.0569507i
\(78\) −41.5422 + 29.0881i −0.0603042 + 0.0422254i
\(79\) −326.200 − 273.715i −0.464562 − 0.389814i 0.380244 − 0.924886i \(-0.375840\pi\)
−0.844806 + 0.535072i \(0.820284\pi\)
\(80\) −57.6043 + 149.257i −0.0805045 + 0.208593i
\(81\) −167.718 + 61.0443i −0.230066 + 0.0837370i
\(82\) −780.894 − 68.3194i −1.05165 − 0.0920075i
\(83\) −810.670 − 217.218i −1.07208 − 0.287263i −0.320732 − 0.947170i \(-0.603929\pi\)
−0.751347 + 0.659907i \(0.770596\pi\)
\(84\) 28.9763 − 50.1885i 0.0376378 − 0.0651906i
\(85\) 510.292 + 372.160i 0.651164 + 0.474899i
\(86\) −466.557 + 82.2666i −0.585002 + 0.103152i
\(87\) 42.7972 − 11.4675i 0.0527395 − 0.0141315i
\(88\) 105.517 − 393.797i 0.127820 − 0.477032i
\(89\) −682.192 + 572.427i −0.812496 + 0.681765i −0.951202 − 0.308568i \(-0.900150\pi\)
0.138706 + 0.990334i \(0.455706\pi\)
\(90\) −100.987 + 206.108i −0.118277 + 0.241396i
\(91\) −8.66233 + 23.7996i −0.00997867 + 0.0274162i
\(92\) 2.89065 + 33.0403i 0.00327577 + 0.0374422i
\(93\) −536.284 − 765.893i −0.597958 − 0.853973i
\(94\) 408.053 0.447738
\(95\) −855.828 + 353.458i −0.924275 + 0.381727i
\(96\) 687.477 0.730889
\(97\) 147.802 + 211.084i 0.154712 + 0.220952i 0.889024 − 0.457860i \(-0.151384\pi\)
−0.734312 + 0.678812i \(0.762495\pi\)
\(98\) −44.2146 − 505.375i −0.0455750 − 0.520925i
\(99\) −91.2175 + 250.618i −0.0926031 + 0.254425i
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.2.11 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.11 yes 336
19.10 odd 18 inner 95.4.r.a.67.11 yes 336
95.48 even 36 inner 95.4.r.a.48.11 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.11 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.11 yes 336 95.48 even 36 inner
95.4.r.a.67.11 yes 336 19.10 odd 18 inner
95.4.r.a.78.11 yes 336 5.3 odd 4 inner