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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 2.13
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.13

$q$-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+(-0.555654 - 0.793556i) q^{2} +(0.639977 + 7.31497i) q^{3} +(2.41518 - 6.63566i) q^{4} +(-2.22293 - 10.9571i) q^{5} +(5.44923 - 4.57245i) q^{6} +(3.52152 - 13.1425i) q^{7} +(-14.0937 + 3.77640i) q^{8} +(-26.5094 + 4.67433i) q^{9} +(-7.45991 + 7.85238i) q^{10} +(15.1143 - 26.1788i) q^{11} +(50.0853 + 13.4203i) q^{12} +(39.7297 + 3.47590i) q^{13} +(-12.3860 + 4.50815i) q^{14} +(78.7284 - 23.2730i) q^{15} +(-32.4475 - 27.2267i) q^{16} +(107.570 - 75.3214i) q^{17} +(18.4394 + 18.4394i) q^{18} +(-61.1994 + 55.7999i) q^{19} +(-78.0765 - 11.7128i) q^{20} +(98.3905 + 17.3489i) q^{21} +(-29.1727 + 2.55228i) q^{22} +(3.34212 - 1.55845i) q^{23} +(-36.6439 - 100.678i) q^{24} +(-115.117 + 48.7139i) q^{25} +(-19.3176 - 33.4591i) q^{26} +(0.155101 + 0.578844i) q^{27} +(-78.7039 - 55.1090i) q^{28} +(-31.0494 - 176.090i) q^{29} +(-62.2141 - 49.5437i) q^{30} +(52.4961 - 30.3086i) q^{31} +(-13.7498 + 157.160i) q^{32} +(201.170 + 93.8071i) q^{33} +(-119.543 - 43.5103i) q^{34} +(-151.832 - 9.37087i) q^{35} +(-33.0079 + 187.197i) q^{36} +(86.1552 - 86.1552i) q^{37} +(78.2860 + 17.5598i) q^{38} +292.846i q^{39} +(72.7078 + 146.032i) q^{40} +(-196.349 + 233.999i) q^{41} +(-40.9037 - 87.7183i) q^{42} +(-222.228 + 476.570i) q^{43} +(-137.210 - 163.520i) q^{44} +(110.146 + 280.077i) q^{45} +(-3.09378 - 1.78619i) q^{46} +(150.602 - 215.081i) q^{47} +(178.397 - 254.777i) q^{48} +(136.723 + 78.9371i) q^{49} +(102.622 + 64.2838i) q^{50} +(619.816 + 738.668i) q^{51} +(119.019 - 255.238i) q^{52} +(-86.5959 - 185.705i) q^{53} +(0.373163 - 0.444718i) q^{54} +(-320.442 - 107.416i) q^{55} +198.525i q^{56} +(-447.341 - 411.962i) q^{57} +(-122.484 + 122.484i) q^{58} +(-44.8058 + 254.106i) q^{59} +(35.7119 - 578.623i) q^{60} +(-495.159 - 180.223i) q^{61} +(-53.2213 - 24.8175i) q^{62} +(-31.9211 + 364.860i) q^{63} +(-161.104 + 93.0132i) q^{64} +(-50.2306 - 443.050i) q^{65} +(-37.3397 - 211.764i) q^{66} +(133.951 + 93.7936i) q^{67} +(-240.006 - 895.713i) q^{68} +(13.5389 + 23.4501i) q^{69} +(76.9296 + 125.694i) q^{70} +(-8.56300 - 23.5266i) q^{71} +(355.964 - 165.989i) q^{72} +(149.605 - 13.0887i) q^{73} +(-116.241 - 20.4965i) q^{74} +(-430.013 - 810.903i) q^{75} +(222.461 + 540.865i) q^{76} +(-290.829 - 290.829i) q^{77} +(232.390 - 162.721i) q^{78} +(936.285 + 785.637i) q^{79} +(-226.198 + 416.054i) q^{80} +(-687.100 + 250.084i) q^{81} +(294.793 + 25.7911i) q^{82} +(1194.22 + 319.991i) q^{83} +(352.752 - 610.985i) q^{84} +(-1064.43 - 1011.22i) q^{85} +(501.667 - 88.4575i) q^{86} +(1268.22 - 339.819i) q^{87} +(-114.155 + 426.034i) q^{88} +(222.333 - 186.560i) q^{89} +(161.053 - 243.032i) q^{90} +(185.591 - 509.906i) q^{91} +(-2.26955 - 25.9411i) q^{92} +(255.303 + 364.611i) q^{93} -254.361 q^{94} +(747.448 + 546.531i) q^{95} -1158.42 q^{96} +(614.696 + 877.876i) q^{97} +(-13.3297 - 152.359i) q^{98} +(-278.304 + 764.634i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.555654 − 0.793556i −0.196453 − 0.280564i 0.708824 − 0.705385i \(-0.249226\pi\)
−0.905277 + 0.424821i \(0.860337\pi\)
\(3\) 0.639977 + 7.31497i 0.123164 + 1.40777i 0.766530 + 0.642209i \(0.221982\pi\)
−0.643366 + 0.765558i \(0.722463\pi\)
\(4\) 2.41518 − 6.63566i 0.301898 − 0.829457i
\(5\) −2.22293 − 10.9571i −0.198825 − 0.980035i
\(6\) 5.44923 − 4.57245i 0.370773 − 0.311116i
\(7\) 3.52152 − 13.1425i 0.190144 − 0.709627i −0.803327 − 0.595539i \(-0.796939\pi\)
0.993471 − 0.114088i \(-0.0363947\pi\)
\(8\) −14.0937 + 3.77640i −0.622860 + 0.166895i
\(9\) −26.5094 + 4.67433i −0.981831 + 0.173123i
\(10\) −7.45991 + 7.85238i −0.235903 + 0.248314i
\(11\) 15.1143 − 26.1788i 0.414286 − 0.717564i −0.581068 − 0.813855i \(-0.697365\pi\)
0.995353 + 0.0962916i \(0.0306981\pi\)
\(12\) 50.0853 + 13.4203i 1.20487 + 0.322843i
\(13\) 39.7297 + 3.47590i 0.847619 + 0.0741570i 0.502685 − 0.864470i \(-0.332346\pi\)
0.344934 + 0.938627i \(0.387901\pi\)
\(14\) −12.3860 + 4.50815i −0.236450 + 0.0860609i
\(15\) 78.7284 − 23.2730i 1.35517 − 0.400604i
\(16\) −32.4475 − 27.2267i −0.506992 − 0.425417i
\(17\) 107.570 − 75.3214i 1.53468 − 1.07460i 0.566590 − 0.824000i \(-0.308262\pi\)
0.968092 − 0.250596i \(-0.0806264\pi\)
\(18\) 18.4394 + 18.4394i 0.241456 + 0.241456i
\(19\) −61.1994 + 55.7999i −0.738953 + 0.673756i
\(20\) −78.0765 − 11.7128i −0.872922 − 0.130954i
\(21\) 98.3905 + 17.3489i 1.02241 + 0.180278i
\(22\) −29.1727 + 2.55228i −0.282710 + 0.0247340i
\(23\) 3.34212 − 1.55845i 0.0302991 − 0.0141287i −0.407410 − 0.913246i \(-0.633568\pi\)
0.437709 + 0.899117i \(0.355790\pi\)
\(24\) −36.6439 − 100.678i −0.311663 − 0.856286i
\(25\) −115.117 + 48.7139i −0.920937 + 0.389711i
\(26\) −19.3176 − 33.4591i −0.145712 − 0.252380i
\(27\) 0.155101 + 0.578844i 0.00110553 + 0.00412588i
\(28\) −78.7039 − 55.1090i −0.531201 − 0.371951i
\(29\) −31.0494 − 176.090i −0.198818 − 1.12755i −0.906876 − 0.421399i \(-0.861539\pi\)
0.708057 − 0.706155i \(-0.249572\pi\)
\(30\) −62.2141 − 49.5437i −0.378623 − 0.301513i
\(31\) 52.4961 − 30.3086i 0.304148 − 0.175600i −0.340157 − 0.940369i \(-0.610480\pi\)
0.644305 + 0.764769i \(0.277147\pi\)
\(32\) −13.7498 + 157.160i −0.0759574 + 0.868197i
\(33\) 201.170 + 93.8071i 1.06119 + 0.494840i
\(34\) −119.543 − 43.5103i −0.602986 − 0.219469i
\(35\) −151.832 − 9.37087i −0.733265 − 0.0452562i
\(36\) −33.0079 + 187.197i −0.152814 + 0.866652i
\(37\) 86.1552 − 86.1552i 0.382806 − 0.382806i −0.489306 − 0.872112i \(-0.662750\pi\)
0.872112 + 0.489306i \(0.162750\pi\)
\(38\) 78.2860 + 17.5598i 0.334202 + 0.0749623i
\(39\) 292.846i 1.20238i
\(40\) 72.7078 + 146.032i 0.287403 + 0.577242i
\(41\) −196.349 + 233.999i −0.747915 + 0.891331i −0.997020 − 0.0771460i \(-0.975419\pi\)
0.249105 + 0.968477i \(0.419864\pi\)
\(42\) −40.9037 − 87.7183i −0.150276 − 0.322267i
\(43\) −222.228 + 476.570i −0.788128 + 1.69015i −0.0665820 + 0.997781i \(0.521209\pi\)
−0.721546 + 0.692366i \(0.756568\pi\)
\(44\) −137.210 − 163.520i −0.470116 − 0.560263i
\(45\) 110.146 + 280.077i 0.364879 + 0.927808i
\(46\) −3.09378 − 1.78619i −0.00991637 − 0.00572522i
\(47\) 150.602 − 215.081i 0.467394 − 0.667507i −0.514380 − 0.857563i \(-0.671978\pi\)
0.981773 + 0.190055i \(0.0608667\pi\)
\(48\) 178.397 − 254.777i 0.536445 − 0.766123i
\(49\) 136.723 + 78.9371i 0.398610 + 0.230137i
\(50\) 102.622 + 64.2838i 0.290260 + 0.181822i
\(51\) 619.816 + 738.668i 1.70180 + 2.02812i
\(52\) 119.019 − 255.238i 0.317404 − 0.680676i
\(53\) −86.5959 − 185.705i −0.224431 − 0.481295i 0.761559 − 0.648095i \(-0.224434\pi\)
−0.985991 + 0.166801i \(0.946656\pi\)
\(54\) 0.373163 − 0.444718i 0.000940390 − 0.00112071i
\(55\) −320.442 − 107.416i −0.785608 − 0.263345i
\(56\) 198.525i 0.473732i
\(57\) −447.341 − 411.962i −1.03950 − 0.957292i
\(58\) −122.484 + 122.484i −0.277293 + 0.277293i
\(59\) −44.8058 + 254.106i −0.0988681 + 0.560709i 0.894625 + 0.446818i \(0.147443\pi\)
−0.993493 + 0.113891i \(0.963668\pi\)
\(60\) 35.7119 − 578.623i 0.0768398 − 1.24500i
\(61\) −495.159 − 180.223i −1.03932 − 0.378282i −0.234698 − 0.972068i \(-0.575410\pi\)
−0.804623 + 0.593787i \(0.797632\pi\)
\(62\) −53.2213 − 24.8175i −0.109018 − 0.0508359i
\(63\) −31.9211 + 364.860i −0.0638363 + 0.729652i
\(64\) −161.104 + 93.0132i −0.314656 + 0.181666i
\(65\) −50.2306 − 443.050i −0.0958513 − 0.845440i
\(66\) −37.3397 − 211.764i −0.0696393 − 0.394944i
\(67\) 133.951 + 93.7936i 0.244250 + 0.171025i 0.689291 − 0.724485i \(-0.257922\pi\)
−0.445041 + 0.895510i \(0.646811\pi\)
\(68\) −240.006 − 895.713i −0.428014 − 1.59737i
\(69\) 13.5389 + 23.4501i 0.0236217 + 0.0409139i
\(70\) 76.9296 + 125.694i 0.131355 + 0.214619i
\(71\) −8.56300 − 23.5266i −0.0143133 − 0.0393253i 0.932330 − 0.361608i \(-0.117772\pi\)
−0.946644 + 0.322283i \(0.895550\pi\)
\(72\) 355.964 − 165.989i 0.582650 − 0.271694i
\(73\) 149.605 − 13.0887i 0.239862 − 0.0209852i 0.0334093 − 0.999442i \(-0.489364\pi\)
0.206453 + 0.978457i \(0.433808\pi\)
\(74\) −116.241 − 20.4965i −0.182605 − 0.0321982i
\(75\) −430.013 − 810.903i −0.662048 − 1.24847i
\(76\) 222.461 + 540.865i 0.335764 + 0.816336i
\(77\) −290.829 − 290.829i −0.430429 − 0.430429i
\(78\) 232.390 − 162.721i 0.337346 − 0.236212i
\(79\) 936.285 + 785.637i 1.33342 + 1.11887i 0.983265 + 0.182183i \(0.0583162\pi\)
0.350157 + 0.936691i \(0.386128\pi\)
\(80\) −226.198 + 416.054i −0.316121 + 0.581453i
\(81\) −687.100 + 250.084i −0.942524 + 0.343051i
\(82\) 294.793 + 25.7911i 0.397006 + 0.0347335i
\(83\) 1194.22 + 319.991i 1.57931 + 0.423176i 0.938713 − 0.344699i \(-0.112019\pi\)
0.640600 + 0.767875i \(0.278686\pi\)
\(84\) 352.752 − 610.985i 0.458196 − 0.793618i
\(85\) −1064.43 − 1011.22i −1.35827 − 1.29039i
\(86\) 501.667 − 88.4575i 0.629025 − 0.110914i
\(87\) 1268.22 − 339.819i 1.56285 − 0.418763i
\(88\) −114.155 + 426.034i −0.138284 + 0.516084i
\(89\) 222.333 − 186.560i 0.264801 − 0.222194i −0.500713 − 0.865613i \(-0.666929\pi\)
0.765514 + 0.643419i \(0.222485\pi\)
\(90\) 161.053 − 243.032i 0.188628 − 0.284643i
\(91\) 185.591 − 509.906i 0.213793 − 0.587393i
\(92\) −2.26955 − 25.9411i −0.00257192 − 0.0293972i
\(93\) 255.303 + 364.611i 0.284664 + 0.406542i
\(94\) −254.361 −0.279100
\(95\) 747.448 + 546.531i 0.807227 + 0.590241i
\(96\) −1158.42 −1.23157
\(97\) 614.696 + 877.876i 0.643432 + 0.918916i 0.999881 − 0.0153987i \(-0.00490174\pi\)
−0.356450 + 0.934315i \(0.616013\pi\)
\(98\) −13.3297 − 152.359i −0.0137398 − 0.157047i
\(99\) −278.304 + 764.634i −0.282531 + 0.776249i
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.2.13 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.13 yes 336
19.10 odd 18 inner 95.4.r.a.67.13 yes 336
95.48 even 36 inner 95.4.r.a.48.13 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.13 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.13 yes 336 95.48 even 36 inner
95.4.r.a.67.13 yes 336 19.10 odd 18 inner
95.4.r.a.78.13 yes 336 5.3 odd 4 inner