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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 2.18
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.18

$q$-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+(0.961404 + 1.37303i) q^{2} +(-0.797037 - 9.11018i) q^{3} +(1.77525 - 4.87747i) q^{4} +(-11.1762 + 0.303590i) q^{5} +(11.7422 - 9.85292i) q^{6} +(-3.48050 + 12.9894i) q^{7} +(21.3560 - 5.72232i) q^{8} +(-55.7702 + 9.83380i) q^{9} +(-11.1617 - 15.0534i) q^{10} +(2.01584 - 3.49154i) q^{11} +(-45.8496 - 12.2854i) q^{12} +(-61.6581 - 5.39438i) q^{13} +(-21.1810 + 7.70925i) q^{14} +(11.6736 + 101.575i) q^{15} +(-3.42058 - 2.87021i) q^{16} +(86.6991 - 60.7074i) 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} +(6.73203 - 0.588976i) q^{22} +(-85.6868 + 39.9564i) q^{23} +(-69.1529 - 189.996i) q^{24} +(124.816 - 6.78598i) q^{25} +(-51.8717 - 89.8444i) q^{26} +(70.1325 + 261.738i) q^{27} +(57.1767 + 40.0356i) q^{28} +(-45.5844 - 258.522i) q^{29} +(-128.243 + 113.683i) q^{30} +(173.471 - 100.154i) q^{31} +(16.0680 - 183.658i) q^{32} +(-33.4153 - 15.5818i) q^{33} +(166.706 + 60.6759i) q^{34} +(34.9554 - 146.229i) q^{35} +(-51.0423 + 289.475i) q^{36} +(-34.5169 + 34.5169i) q^{37} +(138.806 - 1.77545i) q^{38} +566.016i q^{39} +(-236.942 + 70.4374i) q^{40} +(-56.4531 + 67.2782i) q^{41} +(87.1146 + 186.818i) q^{42} +(-120.942 + 259.360i) q^{43} +(-13.4513 - 16.0306i) q^{44} +(620.315 - 126.836i) q^{45} +(-137.241 - 79.2361i) q^{46} +(-9.36367 + 13.3727i) q^{47} +(-23.4218 + 33.4498i) q^{48} +(140.436 + 81.0807i) q^{49} +(129.316 + 164.851i) q^{50} +(-622.157 - 741.458i) q^{51} +(-135.770 + 291.159i) q^{52} +(-43.0251 - 92.2676i) q^{53} +(-291.948 + 347.930i) q^{54} +(-21.4695 + 39.6342i) q^{55} +297.318i q^{56} +(-660.699 - 370.269i) q^{57} +(311.132 - 311.132i) q^{58} +(64.1285 - 363.691i) q^{59} +(516.155 + 123.384i) q^{60} +(846.506 + 308.103i) q^{61} +(304.289 + 141.893i) q^{62} +(66.3732 - 758.649i) q^{63} +(236.679 - 136.646i) q^{64} +(690.742 + 41.5700i) q^{65} +(-10.7314 - 60.8605i) q^{66} +(-304.486 - 213.204i) q^{67} +(-142.186 - 530.644i) q^{68} +(432.305 + 748.775i) q^{69} +(234.383 - 92.5906i) q^{70} +(-91.5124 - 251.428i) q^{71} +(-1134.76 + 529.146i) q^{72} +(-402.993 + 35.2573i) 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} +(-777.155 + 544.170i) q^{78} +(623.040 + 522.793i) q^{79} +(39.1005 + 31.0396i) q^{80} +(891.767 - 324.577i) q^{81} +(-146.649 - 12.8301i) q^{82} +(147.878 + 39.6239i) q^{83} +(319.159 - 552.800i) q^{84} +(-950.538 + 704.800i) q^{85} +(-472.383 + 83.2939i) q^{86} +(-2318.85 + 621.333i) q^{87} +(23.0706 - 86.1007i) q^{88} +(890.347 - 747.090i) q^{89} +(770.523 + 729.769i) q^{90} +(284.671 - 782.127i) q^{91} +(42.7704 + 488.868i) q^{92} +(-1050.68 - 1500.53i) q^{93} -27.3634 q^{94} +(-500.385 + 779.095i) q^{95} -1685.96 q^{96} +(674.336 + 963.051i) q^{97} +(23.6896 + 270.774i) 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 \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.961404 + 1.37303i 0.339908 + 0.485438i 0.952320 − 0.305100i \(-0.0986898\pi\)
−0.612413 + 0.790538i \(0.709801\pi\)
\(3\) −0.797037 − 9.11018i −0.153390 − 1.75325i −0.549906 − 0.835227i \(-0.685336\pi\)
0.396516 − 0.918028i \(-0.370219\pi\)
\(4\) 1.77525 − 4.87747i 0.221907 − 0.609684i
\(5\) −11.1762 + 0.303590i −0.999631 + 0.0271539i
\(6\) 11.7422 − 9.85292i 0.798959 − 0.670406i
\(7\) −3.48050 + 12.9894i −0.187929 + 0.701362i 0.806055 + 0.591841i \(0.201599\pi\)
−0.993984 + 0.109522i \(0.965068\pi\)
\(8\) 21.3560 − 5.72232i 0.943810 − 0.252893i
\(9\) −55.7702 + 9.83380i −2.06556 + 0.364215i
\(10\) −11.1617 − 15.0534i −0.352964 − 0.476030i
\(11\) 2.01584 − 3.49154i 0.0552545 − 0.0957037i −0.837075 − 0.547088i \(-0.815736\pi\)
0.892330 + 0.451384i \(0.149070\pi\)
\(12\) −45.8496 − 12.2854i −1.10297 − 0.295540i
\(13\) −61.6581 − 5.39438i −1.31545 − 0.115087i −0.592267 − 0.805742i \(-0.701767\pi\)
−0.723185 + 0.690654i \(0.757323\pi\)
\(14\) −21.1810 + 7.70925i −0.404347 + 0.147170i
\(15\) 11.6736 + 101.575i 0.200941 + 1.74844i
\(16\) −3.42058 − 2.87021i −0.0534466 − 0.0448470i
\(17\) 86.6991 − 60.7074i 1.23692 − 0.866100i 0.242351 − 0.970189i \(-0.422081\pi\)
0.994568 + 0.104088i \(0.0331925\pi\)
\(18\) −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
\(22\) 6.73203 − 0.588976i 0.0652397 − 0.00570773i
\(23\) −85.6868 + 39.9564i −0.776823 + 0.362239i −0.770252 − 0.637740i \(-0.779869\pi\)
−0.00657125 + 0.999978i \(0.502092\pi\)
\(24\) −69.1529 − 189.996i −0.588157 − 1.61595i
\(25\) 124.816 − 6.78598i 0.998525 − 0.0542878i
\(26\) −51.8717 − 89.8444i −0.391265 − 0.677690i
\(27\) 70.1325 + 261.738i 0.499889 + 1.86561i
\(28\) 57.1767 + 40.0356i 0.385907 + 0.270215i
\(29\) −45.5844 − 258.522i −0.291890 − 1.65539i −0.679581 − 0.733600i \(-0.737839\pi\)
0.387692 − 0.921789i \(-0.373273\pi\)
\(30\) −128.243 + 113.683i −0.780460 + 0.691854i
\(31\) 173.471 − 100.154i 1.00504 − 0.580262i 0.0953064 − 0.995448i \(-0.469617\pi\)
0.909736 + 0.415186i \(0.136284\pi\)
\(32\) 16.0680 − 183.658i 0.0887638 − 1.01457i
\(33\) −33.4153 − 15.5818i −0.176268 − 0.0821953i
\(34\) 166.706 + 60.6759i 0.840877 + 0.306054i
\(35\) 34.9554 − 146.229i 0.168815 − 0.706207i
\(36\) −51.0423 + 289.475i −0.236307 + 1.34016i
\(37\) −34.5169 + 34.5169i −0.153366 + 0.153366i −0.779620 − 0.626253i \(-0.784588\pi\)
0.626253 + 0.779620i \(0.284588\pi\)
\(38\) 138.806 − 1.77545i 0.592563 − 0.00757936i
\(39\) 566.016i 2.32398i
\(40\) −236.942 + 70.4374i −0.936595 + 0.278428i
\(41\) −56.4531 + 67.2782i −0.215037 + 0.256271i −0.862770 − 0.505596i \(-0.831273\pi\)
0.647734 + 0.761867i \(0.275717\pi\)
\(42\) 87.1146 + 186.818i 0.320050 + 0.686349i
\(43\) −120.942 + 259.360i −0.428917 + 0.919816i 0.566488 + 0.824070i \(0.308302\pi\)
−0.995406 + 0.0957466i \(0.969476\pi\)
\(44\) −13.4513 − 16.0306i −0.0460876 − 0.0549251i
\(45\) 620.315 − 126.836i 2.05491 − 0.420169i
\(46\) −137.241 − 79.2361i −0.439893 − 0.253972i
\(47\) −9.36367 + 13.3727i −0.0290603 + 0.0415023i −0.833419 − 0.552641i \(-0.813620\pi\)
0.804359 + 0.594144i \(0.202509\pi\)
\(48\) −23.4218 + 33.4498i −0.0704300 + 0.100584i
\(49\) 140.436 + 81.0807i 0.409434 + 0.236387i
\(50\) 129.316 + 164.851i 0.365760 + 0.466270i
\(51\) −622.157 − 741.458i −1.70822 − 2.03578i
\(52\) −135.770 + 291.159i −0.362075 + 0.776472i
\(53\) −43.0251 − 92.2676i −0.111509 − 0.239131i 0.842632 − 0.538490i \(-0.181005\pi\)
−0.954141 + 0.299359i \(0.903227\pi\)
\(54\) −291.948 + 347.930i −0.735723 + 0.876801i
\(55\) −21.4695 + 39.6342i −0.0526354 + 0.0971687i
\(56\) 297.318i 0.709479i
\(57\) −660.699 − 370.269i −1.53529 − 0.860410i
\(58\) 311.132 − 311.132i 0.704374 − 0.704374i
\(59\) 64.1285 − 363.691i 0.141505 − 0.802517i −0.828601 − 0.559839i \(-0.810863\pi\)
0.970107 − 0.242678i \(-0.0780259\pi\)
\(60\) 516.155 + 123.384i 1.11059 + 0.265481i
\(61\) 846.506 + 308.103i 1.77679 + 0.646698i 0.999854 + 0.0171160i \(0.00544847\pi\)
0.776934 + 0.629582i \(0.216774\pi\)
\(62\) 304.289 + 141.893i 0.623303 + 0.290651i
\(63\) 66.3732 − 758.649i 0.132734 − 1.51716i
\(64\) 236.679 − 136.646i 0.462263 − 0.266888i
\(65\) 690.742 + 41.5700i 1.31809 + 0.0793250i
\(66\) −10.7314 − 60.8605i −0.0200142 − 0.113506i
\(67\) −304.486 − 213.204i −0.555208 − 0.388761i 0.262031 − 0.965059i \(-0.415608\pi\)
−0.817239 + 0.576299i \(0.804496\pi\)
\(68\) −142.186 − 530.644i −0.253567 − 0.946323i
\(69\) 432.305 + 748.775i 0.754253 + 1.30640i
\(70\) 234.383 − 92.5906i 0.400202 − 0.158096i
\(71\) −91.5124 − 251.428i −0.152965 − 0.420268i 0.839414 − 0.543493i \(-0.182899\pi\)
−0.992379 + 0.123225i \(0.960676\pi\)
\(72\) −1134.76 + 529.146i −1.85739 + 0.866117i
\(73\) −402.993 + 35.2573i −0.646120 + 0.0565282i −0.405506 − 0.914093i \(-0.632905\pi\)
−0.240615 + 0.970621i \(0.577349\pi\)
\(74\) −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
\(78\) −777.155 + 544.170i −1.12815 + 0.789937i
\(79\) 623.040 + 522.793i 0.887310 + 0.744542i 0.967669 − 0.252224i \(-0.0811621\pi\)
−0.0803586 + 0.996766i \(0.525607\pi\)
\(80\) 39.1005 + 31.0396i 0.0546446 + 0.0433792i
\(81\) 891.767 − 324.577i 1.22327 − 0.445235i
\(82\) −146.649 − 12.8301i −0.197496 − 0.0172787i
\(83\) 147.878 + 39.6239i 0.195563 + 0.0524010i 0.355271 − 0.934763i \(-0.384389\pi\)
−0.159708 + 0.987164i \(0.551055\pi\)
\(84\) 319.159 − 552.800i 0.414561 − 0.718041i
\(85\) −950.538 + 704.800i −1.21295 + 0.899368i
\(86\) −472.383 + 83.2939i −0.592307 + 0.104440i
\(87\) −2318.85 + 621.333i −2.85755 + 0.765677i
\(88\) 23.0706 − 86.1007i 0.0279470 − 0.104300i
\(89\) 890.347 − 747.090i 1.06041 − 0.889791i 0.0662621 − 0.997802i \(-0.478893\pi\)
0.994150 + 0.108011i \(0.0344482\pi\)
\(90\) 770.523 + 729.769i 0.902447 + 0.854715i
\(91\) 284.671 − 782.127i 0.327930 − 0.900980i
\(92\) 42.7704 + 488.868i 0.0484687 + 0.554000i
\(93\) −1050.68 − 1500.53i −1.17151 − 1.67309i
\(94\) −27.3634 −0.0300246
\(95\) −500.385 + 779.095i −0.540405 + 0.841405i
\(96\) −1685.96 −1.79242
\(97\) 674.336 + 963.051i 0.705860 + 1.00807i 0.998652 + 0.0518967i \(0.0165266\pi\)
−0.292792 + 0.956176i \(0.594584\pi\)
\(98\) 23.6896 + 270.774i 0.0244185 + 0.279105i
\(99\) −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.2.18 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.18 yes 336
19.10 odd 18 inner 95.4.r.a.67.18 yes 336
95.48 even 36 inner 95.4.r.a.48.18 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.18 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.18 yes 336 95.48 even 36 inner
95.4.r.a.67.18 yes 336 19.10 odd 18 inner
95.4.r.a.78.18 yes 336 5.3 odd 4 inner