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(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 2.1
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.1

$q$-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+(-3.17544 - 4.53500i) q^{2} +(-0.205763 - 2.35188i) q^{3} +(-7.74662 + 21.2837i) q^{4} +(7.01527 + 8.70551i) q^{5} +(-10.0124 + 8.40139i) q^{6} +(2.07008 - 7.72564i) q^{7} +(78.3396 - 20.9910i) q^{8} +(21.1008 - 3.72064i) q^{9} +(17.2029 - 59.4580i) q^{10} +(-20.4947 + 35.4978i) q^{11} +(51.6506 + 13.8397i) q^{12} +(43.6596 + 3.81972i) q^{13} +(-41.6092 + 15.1445i) q^{14} +(19.0308 - 18.2904i) q^{15} +(-205.152 - 172.143i) q^{16} +(-53.4723 + 37.4417i) q^{17} +(-83.8774 - 83.8774i) q^{18} +(82.0879 + 10.9804i) q^{19} +(-239.630 + 81.8724i) q^{20} +(-18.5957 - 3.27893i) q^{21} +(226.062 - 19.7779i) q^{22} +(118.696 - 55.3490i) q^{23} +(-65.4878 - 179.926i) q^{24} +(-26.5719 + 122.143i) q^{25} +(-121.316 - 210.126i) q^{26} +(-29.5903 - 110.432i) q^{27} +(148.394 + 103.906i) q^{28} +(-15.3464 - 87.0339i) q^{29} +(-143.378 - 28.2249i) q^{30} +(188.903 - 109.063i) q^{31} +(-72.6714 + 830.638i) q^{32} +(87.7037 + 40.8969i) q^{33} +(339.596 + 123.603i) q^{34} +(81.7778 - 36.1764i) q^{35} +(-84.2710 + 477.924i) q^{36} +(84.6185 - 84.6185i) q^{37} +(-210.869 - 407.136i) q^{38} -103.468i q^{39} +(732.312 + 534.729i) q^{40} +(-130.455 + 155.470i) q^{41} +(44.1797 + 94.7436i) q^{42} +(-34.5892 + 74.1768i) q^{43} +(-596.758 - 711.189i) q^{44} +(180.418 + 157.592i) q^{45} +(-627.920 - 362.530i) q^{46} +(169.475 - 242.035i) q^{47} +(-362.647 + 517.914i) q^{48} +(241.646 + 139.515i) q^{49} +(638.296 - 267.354i) q^{50} +(99.0611 + 118.056i) q^{51} +(-419.512 + 899.646i) q^{52} +(170.406 + 365.437i) q^{53} +(-406.848 + 484.863i) q^{54} +(-452.802 + 70.6102i) q^{55} -648.677i q^{56} +(8.93387 - 195.320i) q^{57} +(-345.967 + 345.967i) q^{58} +(-54.6150 + 309.737i) q^{59} +(241.861 + 546.734i) q^{60} +(138.819 + 50.5261i) q^{61} +(-1094.45 - 510.351i) q^{62} +(14.9360 - 170.719i) q^{63} +(2142.28 - 1236.85i) q^{64} +(273.032 + 406.876i) q^{65} +(-93.0303 - 527.601i) q^{66} +(-355.488 - 248.915i) q^{67} +(-382.667 - 1428.13i) q^{68} +(-154.597 - 267.771i) q^{69} +(-423.740 - 255.986i) q^{70} +(-78.2726 - 215.052i) q^{71} +(1574.93 - 734.401i) q^{72} +(230.198 - 20.1397i) q^{73} +(-652.445 - 115.044i) q^{74} +(292.734 + 37.3614i) q^{75} +(-869.606 + 1662.07i) q^{76} +(231.818 + 231.818i) q^{77} +(-469.228 + 328.557i) q^{78} +(329.380 + 276.383i) q^{79} +(59.3954 - 2993.59i) q^{80} +(289.987 - 105.547i) q^{81} +(1119.31 + 97.9267i) q^{82} +(-891.680 - 238.925i) q^{83} +(213.842 - 370.385i) q^{84} +(-701.073 - 202.840i) q^{85} +(446.227 - 78.6819i) q^{86} +(-201.536 + 54.0013i) q^{87} +(-860.409 + 3211.09i) q^{88} +(-559.634 + 469.588i) q^{89} +(141.773 - 1318.62i) q^{90} +(119.889 - 329.392i) q^{91} +(258.534 + 2955.06i) q^{92} +(-295.373 - 421.836i) q^{93} -1635.78 q^{94} +(480.280 + 791.648i) q^{95} +1968.51 q^{96} +(207.106 + 295.779i) q^{97} +(-134.635 - 1538.89i) q^{98} +(-300.379 + 825.285i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.17544 − 4.53500i −1.12269 − 1.60336i −0.729620 − 0.683853i \(-0.760303\pi\)
−0.393067 − 0.919510i \(-0.628586\pi\)
\(3\) −0.205763 − 2.35188i −0.0395991 − 0.452620i −0.989948 − 0.141433i \(-0.954829\pi\)
0.950349 − 0.311187i \(-0.100727\pi\)
\(4\) −7.74662 + 21.2837i −0.968327 + 2.66046i
\(5\) 7.01527 + 8.70551i 0.627465 + 0.778645i
\(6\) −10.0124 + 8.40139i −0.681256 + 0.571642i
\(7\) 2.07008 − 7.72564i 0.111774 − 0.417145i −0.887252 − 0.461286i \(-0.847388\pi\)
0.999025 + 0.0441405i \(0.0140549\pi\)
\(8\) 78.3396 − 20.9910i 3.46216 − 0.927682i
\(9\) 21.1008 − 3.72064i 0.781511 − 0.137802i
\(10\) 17.2029 − 59.4580i 0.544003 − 1.88023i
\(11\) −20.4947 + 35.4978i −0.561761 + 0.972999i 0.435582 + 0.900149i \(0.356543\pi\)
−0.997343 + 0.0728498i \(0.976791\pi\)
\(12\) 51.6506 + 13.8397i 1.24252 + 0.332932i
\(13\) 43.6596 + 3.81972i 0.931462 + 0.0814923i 0.542776 − 0.839878i \(-0.317373\pi\)
0.388686 + 0.921370i \(0.372929\pi\)
\(14\) −41.6092 + 15.1445i −0.794323 + 0.289110i
\(15\) 19.0308 − 18.2904i 0.327583 − 0.314837i
\(16\) −205.152 − 172.143i −3.20550 − 2.68974i
\(17\) −53.4723 + 37.4417i −0.762879 + 0.534174i −0.889036 − 0.457836i \(-0.848625\pi\)
0.126157 + 0.992010i \(0.459736\pi\)
\(18\) −83.8774 − 83.8774i −1.09834 − 1.09834i
\(19\) 82.0879 + 10.9804i 0.991172 + 0.132583i
\(20\) −239.630 + 81.8724i −2.67914 + 0.915361i
\(21\) −18.5957 − 3.27893i −0.193234 − 0.0340724i
\(22\) 226.062 − 19.7779i 2.19075 − 0.191666i
\(23\) 118.696 − 55.3490i 1.07608 − 0.501785i 0.197954 − 0.980211i \(-0.436570\pi\)
0.878128 + 0.478426i \(0.158793\pi\)
\(24\) −65.4878 − 179.926i −0.556985 − 1.53030i
\(25\) −26.5719 + 122.143i −0.212575 + 0.977145i
\(26\) −121.316 − 210.126i −0.915078 − 1.58496i
\(27\) −29.5903 − 110.432i −0.210913 − 0.787138i
\(28\) 148.394 + 103.906i 1.00156 + 0.701302i
\(29\) −15.3464 − 87.0339i −0.0982676 − 0.557303i −0.993697 − 0.112100i \(-0.964242\pi\)
0.895429 − 0.445203i \(-0.146869\pi\)
\(30\) −143.378 − 28.2249i −0.872571 − 0.171771i
\(31\) 188.903 − 109.063i 1.09445 − 0.631882i 0.159693 − 0.987167i \(-0.448950\pi\)
0.934758 + 0.355285i \(0.115616\pi\)
\(32\) −72.6714 + 830.638i −0.401456 + 4.58867i
\(33\) 87.7037 + 40.8969i 0.462644 + 0.215734i
\(34\) 339.596 + 123.603i 1.71295 + 0.623462i
\(35\) 81.7778 − 36.1764i 0.394942 − 0.174712i
\(36\) −84.2710 + 477.924i −0.390143 + 2.21261i
\(37\) 84.6185 − 84.6185i 0.375978 − 0.375978i −0.493671 − 0.869649i \(-0.664345\pi\)
0.869649 + 0.493671i \(0.164345\pi\)
\(38\) −210.869 − 407.136i −0.900198 − 1.73806i
\(39\) − 103.468i − 0.424825i
\(40\) 732.312 + 534.729i 2.89472 + 2.11370i
\(41\) −130.455 + 155.470i −0.496918 + 0.592204i −0.954963 − 0.296725i \(-0.904105\pi\)
0.458045 + 0.888929i \(0.348550\pi\)
\(42\) 44.1797 + 94.7436i 0.162311 + 0.348078i
\(43\) −34.5892 + 74.1768i −0.122670 + 0.263066i −0.958047 − 0.286611i \(-0.907471\pi\)
0.835377 + 0.549677i \(0.185249\pi\)
\(44\) −596.758 − 711.189i −2.04465 − 2.43672i
\(45\) 180.418 + 157.592i 0.597669 + 0.522054i
\(46\) −627.920 − 362.530i −2.01265 − 1.16200i
\(47\) 169.475 − 242.035i 0.525967 − 0.751158i −0.464993 − 0.885314i \(-0.653943\pi\)
0.990960 + 0.134156i \(0.0428322\pi\)
\(48\) −362.647 + 517.914i −1.09049 + 1.55738i
\(49\) 241.646 + 139.515i 0.704508 + 0.406748i
\(50\) 638.296 − 267.354i 1.80537 − 0.756193i
\(51\) 99.0611 + 118.056i 0.271987 + 0.324141i
\(52\) −419.512 + 899.646i −1.11877 + 2.39920i
\(53\) 170.406 + 365.437i 0.441643 + 0.947106i 0.993571 + 0.113214i \(0.0361145\pi\)
−0.551928 + 0.833892i \(0.686108\pi\)
\(54\) −406.848 + 484.863i −1.02528 + 1.22188i
\(55\) −452.802 + 70.6102i −1.11011 + 0.173111i
\(56\) − 648.677i − 1.54791i
\(57\) 8.93387 − 195.320i 0.0207600 − 0.453874i
\(58\) −345.967 + 345.967i −0.783235 + 0.783235i
\(59\) −54.6150 + 309.737i −0.120513 + 0.683463i 0.863359 + 0.504590i \(0.168356\pi\)
−0.983872 + 0.178873i \(0.942755\pi\)
\(60\) 241.861 + 546.734i 0.520402 + 1.17639i
\(61\) 138.819 + 50.5261i 0.291377 + 0.106052i 0.483573 − 0.875304i \(-0.339339\pi\)
−0.192196 + 0.981357i \(0.561561\pi\)
\(62\) −1094.45 − 510.351i −2.24186 − 1.04540i
\(63\) 14.9360 − 170.719i 0.0298692 − 0.341406i
\(64\) 2142.28 − 1236.85i 4.18414 − 2.41571i
\(65\) 273.032 + 406.876i 0.521006 + 0.776411i
\(66\) −93.0303 − 527.601i −0.173504 − 0.983988i
\(67\) −355.488 − 248.915i −0.648205 − 0.453878i 0.202669 − 0.979247i \(-0.435038\pi\)
−0.850874 + 0.525369i \(0.823927\pi\)
\(68\) −382.667 − 1428.13i −0.682430 − 2.54686i
\(69\) −154.597 − 267.771i −0.269730 − 0.467186i
\(70\) −423.740 − 255.986i −0.723523 − 0.437089i
\(71\) −78.2726 − 215.052i −0.130834 − 0.359465i 0.856927 − 0.515438i \(-0.172371\pi\)
−0.987761 + 0.155973i \(0.950149\pi\)
\(72\) 1574.93 − 734.401i 2.57788 − 1.20208i
\(73\) 230.198 − 20.1397i 0.369078 − 0.0322901i 0.0988916 − 0.995098i \(-0.468470\pi\)
0.270186 + 0.962808i \(0.412915\pi\)
\(74\) −652.445 − 115.044i −1.02493 − 0.180724i
\(75\) 292.734 + 37.3614i 0.450693 + 0.0575216i
\(76\) −869.606 + 1662.07i −1.31251 + 2.50859i
\(77\) 231.818 + 231.818i 0.343092 + 0.343092i
\(78\) −469.228 + 328.557i −0.681149 + 0.476946i
\(79\) 329.380 + 276.383i 0.469090 + 0.393614i 0.846463 − 0.532448i \(-0.178728\pi\)
−0.377372 + 0.926062i \(0.623172\pi\)
\(80\) 59.3954 − 2993.59i 0.0830076 − 4.18366i
\(81\) 289.987 − 105.547i 0.397787 − 0.144783i
\(82\) 1119.31 + 97.9267i 1.50740 + 0.131880i
\(83\) −891.680 − 238.925i −1.17921 − 0.315969i −0.384598 − 0.923084i \(-0.625660\pi\)
−0.794614 + 0.607115i \(0.792327\pi\)
\(84\) 213.842 − 370.385i 0.277762 − 0.481099i
\(85\) −701.073 − 202.840i −0.894612 − 0.258836i
\(86\) 446.227 − 78.6819i 0.559511 − 0.0986569i
\(87\) −201.536 + 54.0013i −0.248355 + 0.0665465i
\(88\) −860.409 + 3211.09i −1.04227 + 3.88981i
\(89\) −559.634 + 469.588i −0.666529 + 0.559284i −0.912036 − 0.410111i \(-0.865490\pi\)
0.245507 + 0.969395i \(0.421046\pi\)
\(90\) 141.773 − 1318.62i 0.166046 − 1.54438i
\(91\) 119.889 − 329.392i 0.138107 − 0.379446i
\(92\) 258.534 + 2955.06i 0.292979 + 3.34876i
\(93\) −295.373 − 421.836i −0.329341 − 0.470348i
\(94\) −1635.78 −1.79488
\(95\) 480.280 + 791.648i 0.518691 + 0.854962i
\(96\) 1968.51 2.09282
\(97\) 207.106 + 295.779i 0.216788 + 0.309606i 0.912786 − 0.408438i \(-0.133926\pi\)
−0.695998 + 0.718044i \(0.745038\pi\)
\(98\) −134.635 − 1538.89i −0.138777 − 1.58623i
\(99\) −300.379 + 825.285i −0.304942 + 0.837821i
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.2.1 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.1 yes 336
19.10 odd 18 inner 95.4.r.a.67.1 yes 336
95.48 even 36 inner 95.4.r.a.48.1 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.1 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.1 yes 336 95.48 even 36 inner
95.4.r.a.67.1 yes 336 19.10 odd 18 inner
95.4.r.a.78.1 yes 336 5.3 odd 4 inner