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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 4.5
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.5

$q$-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+(-1.40644 + 3.86417i) q^{2} +(0.235273 - 0.0414849i) q^{3} +(-6.82541 - 5.72720i) q^{4} +(-2.66370 + 10.8584i) q^{5} +(-0.170593 + 0.967481i) q^{6} +(14.0326 + 8.10175i) q^{7} +(3.24050 - 1.87090i) q^{8} +(-25.3181 + 9.21502i) q^{9} +(-38.2124 - 25.5647i) q^{10} +(-17.7262 - 30.7026i) q^{11} +(-1.84342 - 1.06430i) q^{12} +(-12.9439 - 2.28236i) q^{13} +(-51.0427 + 42.8299i) q^{14} +(-0.176236 + 2.66519i) q^{15} +(-9.70564 - 55.0434i) q^{16} +(-5.33838 + 14.6671i) q^{17} -110.794i q^{18} +(82.8150 + 0.820328i) q^{19} +(80.3690 - 58.8574i) q^{20} +(3.63760 + 1.32398i) q^{21} +(143.571 - 25.3155i) q^{22} +(-48.4690 + 57.7631i) q^{23} +(0.684786 - 0.574604i) q^{24} +(-110.809 - 57.8469i) q^{25} +(27.0243 - 46.8074i) q^{26} +(-11.1605 + 6.44354i) q^{27} +(-49.3782 - 135.665i) q^{28} +(-181.162 + 65.9377i) q^{29} +(-10.0509 - 4.42944i) q^{30} +(28.9713 - 50.1797i) q^{31} +(255.827 + 45.1093i) q^{32} +(-5.44418 - 6.48812i) q^{33} +(-49.1680 - 41.2568i) q^{34} +(-125.351 + 130.791i) q^{35} +(225.582 + 82.1052i) q^{36} +274.320i q^{37} +(-119.645 + 318.858i) q^{38} -3.14003 q^{39} +(11.6833 + 40.1701i) q^{40} +(56.4082 + 319.907i) q^{41} +(-10.2322 + 12.1942i) q^{42} +(131.311 + 156.490i) q^{43} +(-54.8517 + 311.079i) q^{44} +(-32.6207 - 299.460i) q^{45} +(-155.038 - 268.533i) q^{46} +(174.949 + 480.668i) q^{47} +(-4.56694 - 12.5476i) q^{48} +(-40.2234 - 69.6690i) q^{49} +(379.378 - 346.829i) q^{50} +(-0.647512 + 3.67222i) q^{51} +(75.2758 + 89.7102i) q^{52} +(-106.636 + 127.084i) q^{53} +(-9.20229 - 52.1888i) q^{54} +(380.598 - 110.695i) q^{55} +60.6303 q^{56} +(19.5181 - 3.24257i) q^{57} -792.780i q^{58} +(-115.930 - 42.1950i) q^{59} +(16.4669 - 17.1816i) q^{60} +(483.818 + 405.971i) q^{61} +(153.157 + 182.525i) q^{62} +(-429.937 - 75.8095i) q^{63} +(-310.547 + 537.883i) q^{64} +(59.2613 - 134.470i) q^{65} +(32.7282 - 11.9121i) q^{66} +(-331.720 - 911.394i) q^{67} +(120.438 - 69.5348i) q^{68} +(-9.00713 + 15.6008i) q^{69} +(-329.102 - 668.328i) q^{70} +(1.64708 - 1.38206i) q^{71} +(-64.8027 + 77.2288i) q^{72} +(321.626 - 56.7113i) q^{73} +(-1060.02 - 385.815i) q^{74} +(-28.4702 - 9.01289i) q^{75} +(-560.548 - 479.897i) q^{76} -574.452i q^{77} +(4.41627 - 12.1336i) q^{78} +(183.128 + 1038.57i) q^{79} +(623.536 + 41.2313i) q^{80} +(554.907 - 465.623i) q^{81} +(-1315.51 - 231.960i) q^{82} +(880.682 + 508.462i) q^{83} +(-17.2454 - 29.8699i) q^{84} +(-145.041 - 97.0348i) q^{85} +(-789.387 + 287.313i) q^{86} +(-39.8871 + 23.0288i) q^{87} +(-114.883 - 66.3278i) q^{88} +(110.514 - 626.754i) q^{89} +(1203.04 + 295.121i) q^{90} +(-163.146 - 136.896i) q^{91} +(661.641 - 116.665i) q^{92} +(4.73444 - 13.0078i) q^{93} -2103.44 q^{94} +(-229.502 + 897.053i) q^{95} +62.0606 q^{96} +(125.046 - 343.560i) q^{97} +(325.785 - 57.4447i) q^{98} +(731.718 + 613.984i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40644 + 3.86417i −0.497253 + 1.36619i 0.396666 + 0.917963i \(0.370167\pi\)
−0.893919 + 0.448229i \(0.852055\pi\)
\(3\) 0.235273 − 0.0414849i 0.0452783 − 0.00798378i −0.150963 − 0.988539i \(-0.548237\pi\)
0.196241 + 0.980556i \(0.437126\pi\)
\(4\) −6.82541 − 5.72720i −0.853176 − 0.715899i
\(5\) −2.66370 + 10.8584i −0.238248 + 0.971204i
\(6\) −0.170593 + 0.967481i −0.0116074 + 0.0658288i
\(7\) 14.0326 + 8.10175i 0.757691 + 0.437453i 0.828466 − 0.560039i \(-0.189214\pi\)
−0.0707750 + 0.997492i \(0.522547\pi\)
\(8\) 3.24050 − 1.87090i 0.143211 − 0.0826829i
\(9\) −25.3181 + 9.21502i −0.937706 + 0.341297i
\(10\) −38.2124 − 25.5647i −1.20838 − 0.808427i
\(11\) −17.7262 − 30.7026i −0.485877 − 0.841563i 0.513992 − 0.857795i \(-0.328166\pi\)
−0.999868 + 0.0162322i \(0.994833\pi\)
\(12\) −1.84342 − 1.06430i −0.0443459 − 0.0256031i
\(13\) −12.9439 − 2.28236i −0.276153 − 0.0486932i 0.0338565 − 0.999427i \(-0.489221\pi\)
−0.310009 + 0.950733i \(0.600332\pi\)
\(14\) −51.0427 + 42.8299i −0.974410 + 0.817627i
\(15\) −0.176236 + 2.66519i −0.00303359 + 0.0458766i
\(16\) −9.70564 − 55.0434i −0.151651 − 0.860053i
\(17\) −5.33838 + 14.6671i −0.0761615 + 0.209252i −0.971931 − 0.235267i \(-0.924403\pi\)
0.895769 + 0.444520i \(0.146626\pi\)
\(18\) − 110.794i − 1.45080i
\(19\) 82.8150 + 0.820328i 0.999951 + 0.00990506i
\(20\) 80.3690 − 58.8574i 0.898552 − 0.658046i
\(21\) 3.63760 + 1.32398i 0.0377995 + 0.0137579i
\(22\) 143.571 − 25.3155i 1.39134 − 0.245331i
\(23\) −48.4690 + 57.7631i −0.439412 + 0.523671i −0.939613 − 0.342238i \(-0.888815\pi\)
0.500201 + 0.865909i \(0.333259\pi\)
\(24\) 0.684786 − 0.574604i 0.00582422 − 0.00488710i
\(25\) −110.809 − 57.8469i −0.886475 − 0.462776i
\(26\) 27.0243 − 46.8074i 0.203842 − 0.353065i
\(27\) −11.1605 + 6.44354i −0.0795499 + 0.0459281i
\(28\) −49.3782 − 135.665i −0.333271 − 0.915655i
\(29\) −181.162 + 65.9377i −1.16003 + 0.422218i −0.849110 − 0.528217i \(-0.822861\pi\)
−0.310924 + 0.950435i \(0.600639\pi\)
\(30\) −10.0509 − 4.42944i −0.0611677 − 0.0269567i
\(31\) 28.9713 − 50.1797i 0.167851 − 0.290727i −0.769813 − 0.638270i \(-0.779650\pi\)
0.937664 + 0.347543i \(0.112984\pi\)
\(32\) 255.827 + 45.1093i 1.41326 + 0.249196i
\(33\) −5.44418 − 6.48812i −0.0287185 − 0.0342254i
\(34\) −49.1680 − 41.2568i −0.248007 − 0.208103i
\(35\) −125.351 + 130.791i −0.605375 + 0.631650i
\(36\) 225.582 + 82.1052i 1.04436 + 0.380117i
\(37\) 274.320i 1.21886i 0.792839 + 0.609431i \(0.208602\pi\)
−0.792839 + 0.609431i \(0.791398\pi\)
\(38\) −119.645 + 318.858i −0.510761 + 1.36120i
\(39\) −3.14003 −0.0128925
\(40\) 11.6833 + 40.1701i 0.0461822 + 0.158786i
\(41\) 56.4082 + 319.907i 0.214865 + 1.21856i 0.881140 + 0.472855i \(0.156777\pi\)
−0.666275 + 0.745706i \(0.732112\pi\)
\(42\) −10.2322 + 12.1942i −0.0375918 + 0.0448002i
\(43\) 131.311 + 156.490i 0.465691 + 0.554989i 0.946863 − 0.321637i \(-0.104233\pi\)
−0.481172 + 0.876626i \(0.659789\pi\)
\(44\) −54.8517 + 311.079i −0.187936 + 1.06584i
\(45\) −32.6207 − 299.460i −0.108062 − 0.992018i
\(46\) −155.038 − 268.533i −0.496936 − 0.860719i
\(47\) 174.949 + 480.668i 0.542956 + 1.49176i 0.843042 + 0.537847i \(0.180762\pi\)
−0.300087 + 0.953912i \(0.597016\pi\)
\(48\) −4.56694 − 12.5476i −0.0137329 − 0.0377310i
\(49\) −40.2234 − 69.6690i −0.117269 − 0.203117i
\(50\) 379.378 − 346.829i 1.07304 − 0.980979i
\(51\) −0.647512 + 3.67222i −0.00177784 + 0.0100826i
\(52\) 75.2758 + 89.7102i 0.200748 + 0.239242i
\(53\) −106.636 + 127.084i −0.276369 + 0.329364i −0.886318 − 0.463077i \(-0.846745\pi\)
0.609949 + 0.792441i \(0.291190\pi\)
\(54\) −9.20229 − 52.1888i −0.0231902 − 0.131518i
\(55\) 380.598 − 110.695i 0.933089 − 0.271384i
\(56\) 60.6303 0.144680
\(57\) 19.5181 − 3.24257i 0.0453551 − 0.00753490i
\(58\) − 792.780i − 1.79478i
\(59\) −115.930 − 42.1950i −0.255810 − 0.0931071i 0.210932 − 0.977501i \(-0.432350\pi\)
−0.466742 + 0.884394i \(0.654572\pi\)
\(60\) 16.4669 − 17.1816i 0.0354312 − 0.0369690i
\(61\) 483.818 + 405.971i 1.01552 + 0.852119i 0.989058 − 0.147530i \(-0.0471325\pi\)
0.0264589 + 0.999650i \(0.491577\pi\)
\(62\) 153.157 + 182.525i 0.313724 + 0.373882i
\(63\) −429.937 − 75.8095i −0.859793 − 0.151605i
\(64\) −310.547 + 537.883i −0.606537 + 1.05055i
\(65\) 59.2613 − 134.470i 0.113084 − 0.256600i
\(66\) 32.7282 − 11.9121i 0.0610388 − 0.0222163i
\(67\) −331.720 − 911.394i −0.604867 − 1.66186i −0.741271 − 0.671205i \(-0.765777\pi\)
0.136404 − 0.990653i \(-0.456445\pi\)
\(68\) 120.438 − 69.5348i 0.214783 − 0.124005i
\(69\) −9.00713 + 15.6008i −0.0157150 + 0.0272191i
\(70\) −329.102 − 668.328i −0.561931 − 1.14115i
\(71\) 1.64708 − 1.38206i 0.00275313 − 0.00231015i −0.641410 − 0.767198i \(-0.721650\pi\)
0.644163 + 0.764888i \(0.277206\pi\)
\(72\) −64.8027 + 77.2288i −0.106070 + 0.126410i
\(73\) 321.626 − 56.7113i 0.515664 − 0.0909254i 0.0902406 − 0.995920i \(-0.471236\pi\)
0.425423 + 0.904995i \(0.360125\pi\)
\(74\) −1060.02 − 385.815i −1.66520 − 0.606083i
\(75\) −28.4702 − 9.01289i −0.0438328 − 0.0138762i
\(76\) −560.548 − 479.897i −0.846043 − 0.724315i
\(77\) − 574.452i − 0.850193i
\(78\) 4.41627 − 12.1336i 0.00641083 − 0.0176136i
\(79\) 183.128 + 1038.57i 0.260804 + 1.47909i 0.780723 + 0.624877i \(0.214851\pi\)
−0.519919 + 0.854215i \(0.674038\pi\)
\(80\) 623.536 + 41.2313i 0.871418 + 0.0576225i
\(81\) 554.907 − 465.623i 0.761190 − 0.638714i
\(82\) −1315.51 − 231.960i −1.77163 − 0.312386i
\(83\) 880.682 + 508.462i 1.16467 + 0.672421i 0.952418 − 0.304795i \(-0.0985878\pi\)
0.212249 + 0.977216i \(0.431921\pi\)
\(84\) −17.2454 − 29.8699i −0.0224003 − 0.0387985i
\(85\) −145.041 − 97.0348i −0.185081 − 0.123822i
\(86\) −789.387 + 287.313i −0.989788 + 0.360253i
\(87\) −39.8871 + 23.0288i −0.0491534 + 0.0283787i
\(88\) −114.883 − 66.3278i −0.139166 − 0.0803474i
\(89\) 110.514 − 626.754i 0.131623 − 0.746470i −0.845529 − 0.533929i \(-0.820715\pi\)
0.977152 − 0.212541i \(-0.0681738\pi\)
\(90\) 1203.04 + 295.121i 1.40902 + 0.345650i
\(91\) −163.146 − 136.896i −0.187938 − 0.157698i
\(92\) 661.641 − 116.665i 0.749792 − 0.132209i
\(93\) 4.73444 − 13.0078i 0.00527891 − 0.0145037i
\(94\) −2103.44 −2.30802
\(95\) −229.502 + 897.053i −0.247856 + 0.968797i
\(96\) 62.0606 0.0659795
\(97\) 125.046 − 343.560i 0.130891 − 0.359621i −0.856883 − 0.515511i \(-0.827602\pi\)
0.987774 + 0.155890i \(0.0498245\pi\)
\(98\) 325.785 − 57.4447i 0.335809 − 0.0592122i
\(99\) 731.718 + 613.984i 0.742833 + 0.623311i
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.4.5 ✓ 168
5.4 even 2 inner 95.4.p.a.4.24 yes 168
19.5 even 9 inner 95.4.p.a.24.24 yes 168
95.24 even 18 inner 95.4.p.a.24.5 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.5 ✓ 168 1.1 even 1 trivial
95.4.p.a.4.24 yes 168 5.4 even 2 inner
95.4.p.a.24.5 yes 168 95.24 even 18 inner
95.4.p.a.24.24 yes 168 19.5 even 9 inner