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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 78.18
Character \(\chi\) \(=\) 95.78
Dual form 95.4.r.a.67.18

$q$-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+(1.37303 - 0.961404i) q^{2} +(9.11018 - 0.797037i) q^{3} +(-1.77525 + 4.87747i) q^{4} +(-8.75662 + 6.95137i) q^{5} +(11.7422 - 9.85292i) q^{6} +(12.9894 + 3.48050i) q^{7} +(5.72232 + 21.3560i) q^{8} +(55.7702 - 9.83380i) q^{9} +(-5.34001 + 17.9631i) q^{10} +(2.01584 - 3.49154i) q^{11} +(-12.2854 + 45.8496i) q^{12} +(5.39438 - 61.6581i) q^{13} +(21.1810 - 7.70925i) q^{14} +(-74.2339 + 70.3076i) q^{15} +(-3.42058 - 2.87021i) q^{16} +(-60.7074 - 86.6991i) q^{17} +(67.1198 - 67.1198i) q^{18} +(-46.6315 + 68.4434i) q^{19} +(-18.3599 - 55.0506i) q^{20} +(121.110 + 21.3549i) q^{21} +(-0.588976 - 6.73203i) q^{22} +(-39.9564 - 85.6868i) q^{23} +(69.1529 + 189.996i) q^{24} +(28.3569 - 121.741i) q^{25} +(-51.8717 - 89.8444i) q^{26} +(261.738 - 70.1325i) q^{27} +(-40.0356 + 57.1767i) q^{28} +(45.5844 + 258.522i) q^{29} +(-34.3312 + 167.903i) q^{30} +(173.471 - 100.154i) q^{31} +(-183.658 - 16.0680i) q^{32} +(15.5818 - 33.4153i) q^{33} +(-166.706 - 60.6759i) q^{34} +(-137.938 + 59.8168i) q^{35} +(-51.0423 + 289.475i) q^{36} +(34.5169 + 34.5169i) q^{37} +(1.77545 + 138.806i) q^{38} -566.016i q^{39} +(-198.562 - 147.228i) q^{40} +(-56.4531 + 67.2782i) q^{41} +(186.818 - 87.1146i) q^{42} +(-259.360 - 120.942i) q^{43} +(13.4513 + 16.0306i) q^{44} +(-420.001 + 473.791i) q^{45} +(-137.241 - 79.2361i) q^{46} +(13.3727 + 9.36367i) q^{47} +(-33.4498 - 23.4218i) q^{48} +(-140.436 - 81.0807i) q^{49} +(-78.1076 - 194.416i) q^{50} +(-622.157 - 741.458i) q^{51} +(291.159 + 135.770i) q^{52} +(92.2676 - 43.0251i) q^{53} +(291.948 - 347.930i) q^{54} +(6.61903 + 44.5870i) q^{55} +297.318i q^{56} +(-370.269 + 660.699i) q^{57} +(311.132 + 311.132i) q^{58} +(-64.1285 + 363.691i) q^{59} +(-211.139 - 486.888i) q^{60} +(846.506 + 308.103i) q^{61} +(141.893 - 304.289i) q^{62} +(758.649 + 66.3732i) q^{63} +(-236.679 + 136.646i) q^{64} +(381.372 + 577.415i) q^{65} +(-10.7314 - 60.8605i) q^{66} +(-213.204 + 304.486i) q^{67} +(530.644 - 142.186i) q^{68} +(-432.305 - 748.775i) q^{69} +(-131.884 + 214.744i) q^{70} +(-91.5124 - 251.428i) q^{71} +(529.146 + 1134.76i) q^{72} +(-35.2573 - 402.993i) q^{73} +(80.5773 + 14.2080i) q^{74} +(161.304 - 1131.68i) q^{75} +(-251.048 - 348.948i) q^{76} +(38.3369 - 38.3369i) q^{77} +(-544.170 - 777.155i) q^{78} +(-623.040 - 522.793i) q^{79} +(49.9046 + 1.35561i) q^{80} +(891.767 - 324.577i) q^{81} +(-12.8301 + 146.649i) q^{82} +(-39.6239 + 147.878i) q^{83} +(-319.159 + 552.800i) q^{84} +(1134.27 + 337.192i) q^{85} +(-472.383 + 83.2939i) q^{86} +(621.333 + 2318.85i) q^{87} +(86.1007 + 23.0706i) q^{88} +(-890.347 + 747.090i) q^{89} +(-121.168 + 1054.32i) q^{90} +(284.671 - 782.127i) q^{91} +(488.868 - 42.7704i) q^{92} +(1500.53 - 1050.68i) q^{93} +27.3634 q^{94} +(-67.4410 - 923.486i) q^{95} -1685.96 q^{96} +(963.051 - 674.336i) q^{97} +(-270.774 + 23.6896i) q^{98} +(78.0890 - 214.548i) 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 \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.37303 − 0.961404i 0.485438 − 0.339908i −0.305100 − 0.952320i \(-0.598690\pi\)
0.790538 + 0.612413i \(0.209801\pi\)
\(3\) 9.11018 − 0.797037i 1.75325 − 0.153390i 0.835227 − 0.549906i \(-0.185336\pi\)
0.918028 + 0.396516i \(0.129781\pi\)
\(4\) −1.77525 + 4.87747i −0.221907 + 0.609684i
\(5\) −8.75662 + 6.95137i −0.783216 + 0.621749i
\(6\) 11.7422 − 9.85292i 0.798959 − 0.670406i
\(7\) 12.9894 + 3.48050i 0.701362 + 0.187929i 0.591841 − 0.806055i \(-0.298401\pi\)
0.109522 + 0.993984i \(0.465068\pi\)
\(8\) 5.72232 + 21.3560i 0.252893 + 0.943810i
\(9\) 55.7702 − 9.83380i 2.06556 − 0.364215i
\(10\) −5.34001 + 17.9631i −0.168866 + 0.568042i
\(11\) 2.01584 − 3.49154i 0.0552545 − 0.0957037i −0.837075 − 0.547088i \(-0.815736\pi\)
0.892330 + 0.451384i \(0.149070\pi\)
\(12\) −12.2854 + 45.8496i −0.295540 + 1.10297i
\(13\) 5.39438 − 61.6581i 0.115087 − 1.31545i −0.690654 − 0.723185i \(-0.742677\pi\)
0.805742 − 0.592267i \(-0.201767\pi\)
\(14\) 21.1810 − 7.70925i 0.404347 − 0.147170i
\(15\) −74.2339 + 70.3076i −1.27781 + 1.21022i
\(16\) −3.42058 − 2.87021i −0.0534466 − 0.0448470i
\(17\) −60.7074 − 86.6991i −0.866100 − 1.23692i −0.970189 − 0.242351i \(-0.922081\pi\)
0.104088 − 0.994568i \(-0.466807\pi\)
\(18\) 67.1198 − 67.1198i 0.878905 − 0.878905i
\(19\) −46.6315 + 68.4434i −0.563053 + 0.826421i
\(20\) −18.3599 − 55.0506i −0.205270 − 0.615485i
\(21\) 121.110 + 21.3549i 1.25849 + 0.221906i
\(22\) −0.588976 − 6.73203i −0.00570773 − 0.0652397i
\(23\) −39.9564 − 85.6868i −0.362239 − 0.776823i −0.999978 − 0.00657125i \(-0.997908\pi\)
0.637740 − 0.770252i \(-0.279869\pi\)
\(24\) 69.1529 + 189.996i 0.588157 + 1.61595i
\(25\) 28.3569 − 121.741i 0.226855 − 0.973928i
\(26\) −51.8717 − 89.8444i −0.391265 − 0.677690i
\(27\) 261.738 − 70.1325i 1.86561 − 0.499889i
\(28\) −40.0356 + 57.1767i −0.270215 + 0.385907i
\(29\) 45.5844 + 258.522i 0.291890 + 1.65539i 0.679581 + 0.733600i \(0.262161\pi\)
−0.387692 + 0.921789i \(0.626727\pi\)
\(30\) −34.3312 + 167.903i −0.208933 + 1.02183i
\(31\) 173.471 − 100.154i 1.00504 − 0.580262i 0.0953064 − 0.995448i \(-0.469617\pi\)
0.909736 + 0.415186i \(0.136284\pi\)
\(32\) −183.658 − 16.0680i −1.01457 − 0.0887638i
\(33\) 15.5818 − 33.4153i 0.0821953 − 0.176268i
\(34\) −166.706 − 60.6759i −0.840877 − 0.306054i
\(35\) −137.938 + 59.8168i −0.666163 + 0.288882i
\(36\) −51.0423 + 289.475i −0.236307 + 1.34016i
\(37\) 34.5169 + 34.5169i 0.153366 + 0.153366i 0.779620 − 0.626253i \(-0.215412\pi\)
−0.626253 + 0.779620i \(0.715412\pi\)
\(38\) 1.77545 + 138.806i 0.00757936 + 0.592563i
\(39\) − 566.016i − 2.32398i
\(40\) −198.562 − 147.228i −0.784884 − 0.581971i
\(41\) −56.4531 + 67.2782i −0.215037 + 0.256271i −0.862770 − 0.505596i \(-0.831273\pi\)
0.647734 + 0.761867i \(0.275717\pi\)
\(42\) 186.818 − 87.1146i 0.686349 − 0.320050i
\(43\) −259.360 − 120.942i −0.919816 − 0.428917i −0.0957466 − 0.995406i \(-0.530524\pi\)
−0.824070 + 0.566488i \(0.808302\pi\)
\(44\) 13.4513 + 16.0306i 0.0460876 + 0.0549251i
\(45\) −420.001 + 473.791i −1.39133 + 1.56952i
\(46\) −137.241 − 79.2361i −0.439893 − 0.253972i
\(47\) 13.3727 + 9.36367i 0.0415023 + 0.0290603i 0.594144 − 0.804359i \(-0.297491\pi\)
−0.552641 + 0.833419i \(0.686380\pi\)
\(48\) −33.4498 − 23.4218i −0.100584 − 0.0704300i
\(49\) −140.436 − 81.0807i −0.409434 − 0.236387i
\(50\) −78.1076 − 194.416i −0.220922 − 0.549892i
\(51\) −622.157 − 741.458i −1.70822 − 2.03578i
\(52\) 291.159 + 135.770i 0.776472 + 0.362075i
\(53\) 92.2676 − 43.0251i 0.239131 − 0.111509i −0.299359 − 0.954141i \(-0.596773\pi\)
0.538490 + 0.842632i \(0.318995\pi\)
\(54\) 291.948 − 347.930i 0.735723 − 0.876801i
\(55\) 6.61903 + 44.5870i 0.0162275 + 0.109311i
\(56\) 297.318i 0.709479i
\(57\) −370.269 + 660.699i −0.860410 + 1.53529i
\(58\) 311.132 + 311.132i 0.704374 + 0.704374i
\(59\) −64.1285 + 363.691i −0.141505 + 0.802517i 0.828601 + 0.559839i \(0.189137\pi\)
−0.970107 + 0.242678i \(0.921974\pi\)
\(60\) −211.139 − 486.888i −0.454299 − 1.04762i
\(61\) 846.506 + 308.103i 1.77679 + 0.646698i 0.999854 + 0.0171160i \(0.00544847\pi\)
0.776934 + 0.629582i \(0.216774\pi\)
\(62\) 141.893 − 304.289i 0.290651 − 0.623303i
\(63\) 758.649 + 66.3732i 1.51716 + 0.132734i
\(64\) −236.679 + 136.646i −0.462263 + 0.266888i
\(65\) 381.372 + 577.415i 0.727744 + 1.10184i
\(66\) −10.7314 − 60.8605i −0.0200142 − 0.113506i
\(67\) −213.204 + 304.486i −0.388761 + 0.555208i −0.965059 − 0.262031i \(-0.915608\pi\)
0.576299 + 0.817239i \(0.304496\pi\)
\(68\) 530.644 − 142.186i 0.946323 − 0.253567i
\(69\) −432.305 − 748.775i −0.754253 − 1.30640i
\(70\) −131.884 + 214.744i −0.225188 + 0.366669i
\(71\) −91.5124 − 251.428i −0.152965 − 0.420268i 0.839414 − 0.543493i \(-0.182899\pi\)
−0.992379 + 0.123225i \(0.960676\pi\)
\(72\) 529.146 + 1134.76i 0.866117 + 1.85739i
\(73\) −35.2573 − 402.993i −0.0565282 − 0.646120i −0.970621 − 0.240615i \(-0.922651\pi\)
0.914093 − 0.405506i \(-0.132905\pi\)
\(74\) 80.5773 + 14.2080i 0.126580 + 0.0223195i
\(75\) 161.304 − 1131.68i 0.248344 − 1.74234i
\(76\) −251.048 − 348.948i −0.378910 − 0.526673i
\(77\) 38.3369 − 38.3369i 0.0567390 − 0.0567390i
\(78\) −544.170 − 777.155i −0.789937 − 1.12815i
\(79\) −623.040 − 522.793i −0.887310 − 0.744542i 0.0803586 − 0.996766i \(-0.474393\pi\)
−0.967669 + 0.252224i \(0.918838\pi\)
\(80\) 49.9046 + 1.35561i 0.0697438 + 0.00189452i
\(81\) 891.767 − 324.577i 1.22327 − 0.445235i
\(82\) −12.8301 + 146.649i −0.0172787 + 0.197496i
\(83\) −39.6239 + 147.878i −0.0524010 + 0.195563i −0.987164 − 0.159708i \(-0.948945\pi\)
0.934763 + 0.355271i \(0.115611\pi\)
\(84\) −319.159 + 552.800i −0.414561 + 0.718041i
\(85\) 1134.27 + 337.192i 1.44740 + 0.430278i
\(86\) −472.383 + 83.2939i −0.592307 + 0.104440i
\(87\) 621.333 + 2318.85i 0.765677 + 2.85755i
\(88\) 86.1007 + 23.0706i 0.104300 + 0.0279470i
\(89\) −890.347 + 747.090i −1.06041 + 0.889791i −0.994150 − 0.108011i \(-0.965552\pi\)
−0.0662621 + 0.997802i \(0.521107\pi\)
\(90\) −121.168 + 1054.32i −0.141914 + 1.23483i
\(91\) 284.671 − 782.127i 0.327930 − 0.900980i
\(92\) 488.868 − 42.7704i 0.554000 − 0.0484687i
\(93\) 1500.53 − 1050.68i 1.67309 − 1.17151i
\(94\) 27.3634 0.0300246
\(95\) −67.4410 − 923.486i −0.0728348 − 0.997344i
\(96\) −1685.96 −1.79242
\(97\) 963.051 − 674.336i 1.00807 − 0.705860i 0.0518967 − 0.998652i \(-0.483473\pi\)
0.956176 + 0.292792i \(0.0945845\pi\)
\(98\) −270.774 + 23.6896i −0.279105 + 0.0244185i
\(99\) 78.0890 − 214.548i 0.0792751 − 0.217807i
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.78.18 yes 336
5.2 odd 4 inner 95.4.r.a.2.18 ✓ 336
19.10 odd 18 inner 95.4.r.a.48.18 yes 336
95.67 even 36 inner 95.4.r.a.67.18 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.18 ✓ 336 5.2 odd 4 inner
95.4.r.a.48.18 yes 336 19.10 odd 18 inner
95.4.r.a.67.18 yes 336 95.67 even 36 inner
95.4.r.a.78.18 yes 336 1.1 even 1 trivial