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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 4.4
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.4

$q$-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+(-1.54939 + 4.25691i) q^{2} +(-6.49006 + 1.14437i) q^{3} +(-9.59232 - 8.04891i) q^{4} +(-4.06396 - 10.4156i) q^{5} +(5.18413 - 29.4007i) q^{6} +(13.9343 + 8.04499i) q^{7} +(17.7402 - 10.2423i) q^{8} +(15.4395 - 5.61953i) q^{9} +(50.6348 - 1.16216i) q^{10} +(-2.40072 - 4.15816i) q^{11} +(71.4656 + 41.2607i) q^{12} +(-13.4386 - 2.36959i) q^{13} +(-55.8365 + 46.8524i) q^{14} +(38.2946 + 62.9470i) q^{15} +(-1.28110 - 7.26546i) q^{16} +(14.6029 - 40.1210i) q^{17} +74.4315i q^{18} +(48.3449 + 67.2441i) q^{19} +(-44.8512 + 132.620i) q^{20} +(-99.6411 - 36.2664i) q^{21} +(21.4206 - 3.77702i) q^{22} +(116.814 - 139.214i) q^{23} +(-103.414 + 86.7745i) q^{24} +(-91.9684 + 84.6571i) q^{25} +(30.9088 - 53.5356i) q^{26} +(60.3234 - 34.8277i) q^{27} +(-68.9092 - 189.326i) q^{28} +(148.897 - 54.1940i) q^{29} +(-327.293 + 65.4875i) q^{30} +(43.9641 - 76.1480i) q^{31} +(194.301 + 34.2604i) q^{32} +(20.3393 + 24.2394i) q^{33} +(148.166 + 124.326i) q^{34} +(27.1646 - 177.829i) q^{35} +(-193.332 - 70.3671i) q^{36} +187.590i q^{37} +(-361.157 + 101.612i) q^{38} +89.9292 q^{39} +(-178.775 - 143.150i) q^{40} +(-49.4201 - 280.275i) q^{41} +(308.765 - 367.972i) q^{42} +(-241.257 - 287.519i) q^{43} +(-10.4402 + 59.2096i) q^{44} +(-121.276 - 137.974i) q^{45} +(411.629 + 712.963i) q^{46} +(-178.831 - 491.335i) q^{47} +(16.6288 + 45.6872i) q^{48} +(-42.0562 - 72.8434i) q^{49} +(-217.883 - 522.668i) q^{50} +(-48.8600 + 277.099i) q^{51} +(109.835 + 130.896i) q^{52} +(-116.765 + 139.155i) q^{53} +(54.7941 + 310.753i) q^{54} +(-33.5532 + 41.9035i) q^{55} +329.597 q^{56} +(-390.713 - 381.094i) q^{57} +717.807i q^{58} +(-310.986 - 113.190i) q^{59} +(139.320 - 912.037i) q^{60} +(-187.211 - 157.088i) q^{61} +(256.038 + 305.134i) q^{62} +(260.349 + 45.9065i) q^{63} +(-417.380 + 722.924i) q^{64} +(29.9335 + 149.601i) q^{65} +(-134.698 + 49.0262i) q^{66} +(261.528 + 718.543i) q^{67} +(-463.006 + 267.316i) q^{68} +(-598.818 + 1037.18i) q^{69} +(714.912 + 391.163i) q^{70} +(452.705 - 379.865i) q^{71} +(216.343 - 257.828i) q^{72} +(1060.34 - 186.966i) q^{73} +(-798.552 - 290.649i) q^{74} +(500.001 - 654.675i) q^{75} +(77.5025 - 1034.15i) q^{76} -77.2550i q^{77} +(-139.335 + 382.821i) q^{78} +(-51.7731 - 293.620i) q^{79} +(-70.4676 + 42.8700i) q^{80} +(-691.480 + 580.220i) q^{81} +(1269.68 + 223.878i) q^{82} +(94.4707 + 54.5427i) q^{83} +(663.884 + 1149.88i) q^{84} +(-477.229 + 10.9533i) q^{85} +(1597.74 - 581.531i) q^{86} +(-904.330 + 522.115i) q^{87} +(-85.1784 - 49.1777i) q^{88} +(46.1585 - 261.778i) q^{89} +(775.247 - 302.487i) q^{90} +(-168.195 - 141.132i) q^{91} +(-2241.04 + 395.155i) q^{92} +(-198.188 + 544.516i) q^{93} +2368.65 q^{94} +(503.914 - 776.817i) q^{95} -1300.23 q^{96} +(-64.4414 + 177.051i) q^{97} +(375.249 - 66.1666i) q^{98} +(-60.4329 - 50.7092i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.54939 + 4.25691i −0.547791 + 1.50504i 0.288894 + 0.957361i \(0.406712\pi\)
−0.836686 + 0.547684i \(0.815510\pi\)
\(3\) −6.49006 + 1.14437i −1.24901 + 0.220234i −0.758774 − 0.651354i \(-0.774201\pi\)
−0.490238 + 0.871589i \(0.663090\pi\)
\(4\) −9.59232 − 8.04891i −1.19904 − 1.00611i
\(5\) −4.06396 − 10.4156i −0.363492 − 0.931597i
\(6\) 5.18413 − 29.4007i 0.352735 − 2.00046i
\(7\) 13.9343 + 8.04499i 0.752383 + 0.434389i 0.826554 − 0.562857i \(-0.190298\pi\)
−0.0741710 + 0.997246i \(0.523631\pi\)
\(8\) 17.7402 − 10.2423i 0.784013 − 0.452650i
\(9\) 15.4395 − 5.61953i 0.571835 − 0.208131i
\(10\) 50.6348 − 1.16216i 1.60121 − 0.0367507i
\(11\) −2.40072 − 4.15816i −0.0658039 − 0.113976i 0.831246 − 0.555904i \(-0.187628\pi\)
−0.897050 + 0.441928i \(0.854295\pi\)
\(12\) 71.4656 + 41.2607i 1.71920 + 0.992578i
\(13\) −13.4386 − 2.36959i −0.286708 − 0.0505544i 0.0284444 − 0.999595i \(-0.490945\pi\)
−0.315153 + 0.949041i \(0.602056\pi\)
\(14\) −55.8365 + 46.8524i −1.06592 + 0.894416i
\(15\) 38.2946 + 62.9470i 0.659176 + 1.08352i
\(16\) −1.28110 − 7.26546i −0.0200171 − 0.113523i
\(17\) 14.6029 − 40.1210i 0.208336 − 0.572399i −0.790881 − 0.611970i \(-0.790377\pi\)
0.999217 + 0.0395718i \(0.0125994\pi\)
\(18\) 74.4315i 0.974649i
\(19\) 48.3449 + 67.2441i 0.583741 + 0.811940i
\(20\) −44.8512 + 132.620i −0.501451 + 1.48274i
\(21\) −99.6411 − 36.2664i −1.03540 − 0.376856i
\(22\) 21.4206 − 3.77702i 0.207585 − 0.0366029i
\(23\) 116.814 − 139.214i 1.05902 − 1.26209i 0.0952204 − 0.995456i \(-0.469644\pi\)
0.963798 − 0.266633i \(-0.0859111\pi\)
\(24\) −103.414 + 86.7745i −0.879553 + 0.738032i
\(25\) −91.9684 + 84.6571i −0.735747 + 0.677256i
\(26\) 30.9088 − 53.5356i 0.233143 − 0.403816i
\(27\) 60.3234 − 34.8277i 0.429972 − 0.248244i
\(28\) −68.9092 − 189.326i −0.465093 − 1.27783i
\(29\) 148.897 − 54.1940i 0.953429 − 0.347020i 0.181973 − 0.983303i \(-0.441751\pi\)
0.771455 + 0.636284i \(0.219529\pi\)
\(30\) −327.293 + 65.4875i −1.99184 + 0.398544i
\(31\) 43.9641 − 76.1480i 0.254715 − 0.441180i −0.710103 − 0.704098i \(-0.751352\pi\)
0.964818 + 0.262918i \(0.0846849\pi\)
\(32\) 194.301 + 34.2604i 1.07337 + 0.189264i
\(33\) 20.3393 + 24.2394i 0.107291 + 0.127865i
\(34\) 148.166 + 124.326i 0.747361 + 0.627110i
\(35\) 27.1646 − 177.829i 0.131190 − 0.858815i
\(36\) −193.332 − 70.3671i −0.895056 − 0.325774i
\(37\) 187.590i 0.833501i 0.909021 + 0.416751i \(0.136831\pi\)
−0.909021 + 0.416751i \(0.863169\pi\)
\(38\) −361.157 + 101.612i −1.54177 + 0.433782i
\(39\) 89.9292 0.369236
\(40\) −178.775 − 143.150i −0.706670 − 0.565850i
\(41\) −49.4201 − 280.275i −0.188247 − 1.06760i −0.921712 − 0.387874i \(-0.873209\pi\)
0.733465 − 0.679727i \(-0.237902\pi\)
\(42\) 308.765 − 367.972i 1.13437 − 1.35189i
\(43\) −241.257 − 287.519i −0.855614 − 1.01968i −0.999547 − 0.0300932i \(-0.990420\pi\)
0.143934 − 0.989587i \(-0.454025\pi\)
\(44\) −10.4402 + 59.2096i −0.0357711 + 0.202868i
\(45\) −121.276 − 137.974i −0.401751 − 0.457066i
\(46\) 411.629 + 712.963i 1.31938 + 2.28523i
\(47\) −178.831 − 491.335i −0.555004 − 1.52486i −0.826794 − 0.562505i \(-0.809838\pi\)
0.271789 − 0.962357i \(-0.412385\pi\)
\(48\) 16.6288 + 45.6872i 0.0500033 + 0.137383i
\(49\) −42.0562 − 72.8434i −0.122613 − 0.212371i
\(50\) −217.883 − 522.668i −0.616265 − 1.47833i
\(51\) −48.8600 + 277.099i −0.134152 + 0.760815i
\(52\) 109.835 + 130.896i 0.292911 + 0.349078i
\(53\) −116.765 + 139.155i −0.302620 + 0.360648i −0.895828 − 0.444401i \(-0.853417\pi\)
0.593208 + 0.805049i \(0.297861\pi\)
\(54\) 54.7941 + 310.753i 0.138084 + 0.783113i
\(55\) −33.5532 + 41.9035i −0.0822603 + 0.102732i
\(56\) 329.597 0.786505
\(57\) −390.713 − 381.094i −0.907916 − 0.885563i
\(58\) 717.807i 1.62505i
\(59\) −310.986 − 113.190i −0.686218 − 0.249763i −0.0247033 − 0.999695i \(-0.507864\pi\)
−0.661515 + 0.749932i \(0.730086\pi\)
\(60\) 139.320 − 912.037i 0.299769 − 1.96239i
\(61\) −187.211 − 157.088i −0.392949 − 0.329723i 0.424812 − 0.905282i \(-0.360340\pi\)
−0.817761 + 0.575559i \(0.804785\pi\)
\(62\) 256.038 + 305.134i 0.524465 + 0.625033i
\(63\) 260.349 + 45.9065i 0.520649 + 0.0918044i
\(64\) −417.380 + 722.924i −0.815196 + 1.41196i
\(65\) 29.9335 + 149.601i 0.0571198 + 0.285473i
\(66\) −134.698 + 49.0262i −0.251215 + 0.0914350i
\(67\) 261.528 + 718.543i 0.476877 + 1.31021i 0.912130 + 0.409900i \(0.134436\pi\)
−0.435254 + 0.900308i \(0.643341\pi\)
\(68\) −463.006 + 267.316i −0.825701 + 0.476719i
\(69\) −598.818 + 1037.18i −1.04477 + 1.80960i
\(70\) 714.912 + 391.163i 1.22069 + 0.667899i
\(71\) 452.705 − 379.865i 0.756707 − 0.634952i −0.180560 − 0.983564i \(-0.557791\pi\)
0.937267 + 0.348611i \(0.113347\pi\)
\(72\) 216.343 − 257.828i 0.354115 − 0.422018i
\(73\) 1060.34 − 186.966i 1.70004 − 0.299763i 0.762332 − 0.647186i \(-0.224054\pi\)
0.937709 + 0.347423i \(0.112943\pi\)
\(74\) −798.552 − 290.649i −1.25446 − 0.456585i
\(75\) 500.001 − 654.675i 0.769802 − 1.00794i
\(76\) 77.5025 − 1034.15i 0.116976 − 1.56086i
\(77\) − 77.2550i − 0.114338i
\(78\) −139.335 + 382.821i −0.202264 + 0.555716i
\(79\) −51.7731 − 293.620i −0.0737333 − 0.418163i −0.999224 − 0.0393917i \(-0.987458\pi\)
0.925491 − 0.378771i \(-0.123653\pi\)
\(80\) −70.4676 + 42.8700i −0.0984815 + 0.0599126i
\(81\) −691.480 + 580.220i −0.948532 + 0.795913i
\(82\) 1269.68 + 223.878i 1.70991 + 0.301503i
\(83\) 94.4707 + 54.5427i 0.124934 + 0.0721306i 0.561164 − 0.827704i \(-0.310354\pi\)
−0.436231 + 0.899835i \(0.643687\pi\)
\(84\) 663.884 + 1149.88i 0.862330 + 1.49360i
\(85\) −477.229 + 10.9533i −0.608973 + 0.0139770i
\(86\) 1597.74 − 581.531i 2.00336 − 0.729164i
\(87\) −904.330 + 522.115i −1.11442 + 0.643409i
\(88\) −85.1784 − 49.1777i −0.103182 − 0.0595723i
\(89\) 46.1585 − 261.778i 0.0549752 − 0.311780i −0.944904 − 0.327349i \(-0.893845\pi\)
0.999879 + 0.0155689i \(0.00495595\pi\)
\(90\) 775.247 − 302.487i 0.907980 − 0.354277i
\(91\) −168.195 − 141.132i −0.193754 − 0.162579i
\(92\) −2241.04 + 395.155i −2.53961 + 0.447802i
\(93\) −198.188 + 544.516i −0.220980 + 0.607136i
\(94\) 2368.65 2.59901
\(95\) 503.914 − 776.817i 0.544216 − 0.838945i
\(96\) −1300.23 −1.38233
\(97\) −64.4414 + 177.051i −0.0674539 + 0.185328i −0.968840 − 0.247689i \(-0.920329\pi\)
0.901386 + 0.433017i \(0.142551\pi\)
\(98\) 375.249 − 66.1666i 0.386795 − 0.0682024i
\(99\) −60.4329 − 50.7092i −0.0613508 − 0.0514795i
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.4.4 ✓ 168
5.4 even 2 inner 95.4.p.a.4.25 yes 168
19.5 even 9 inner 95.4.p.a.24.25 yes 168
95.24 even 18 inner 95.4.p.a.24.4 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.4 ✓ 168 1.1 even 1 trivial
95.4.p.a.4.25 yes 168 5.4 even 2 inner
95.4.p.a.24.4 yes 168 95.24 even 18 inner
95.4.p.a.24.25 yes 168 19.5 even 9 inner