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Properties

Label 36.4.h.b.23.2
Level $36$
Weight $4$
Character 36.23
Analytic conductor $2.124$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [36,4,Mod(11,36)] 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("36.11"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 36.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.12406876021\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 23.2
Character \(\chi\) \(=\) 36.23
Dual form 36.4.h.b.11.2

$q$-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+(-2.52436 - 1.27578i) q^{2} +(5.18398 - 0.355390i) q^{3} +(4.74478 + 6.44105i) q^{4} +(-1.23846 - 0.715028i) q^{5} +(-13.5396 - 5.71649i) q^{6} +(23.8818 - 13.7882i) q^{7} +(-3.76017 - 22.3128i) q^{8} +(26.7474 - 3.68468i) q^{9} +(2.21411 + 3.38499i) q^{10} +(-11.1087 - 19.2409i) q^{11} +(26.8859 + 31.7040i) q^{12} +(-34.5965 + 59.9229i) q^{13} +(-77.8769 + 4.33838i) q^{14} +(-6.67430 - 3.26656i) q^{15} +(-18.9742 + 61.1227i) q^{16} +31.4507i q^{17} +(-72.2209 - 24.8223i) q^{18} -11.4986i q^{19} +(-1.27071 - 11.3697i) q^{20} +(118.903 - 79.9650i) q^{21} +(3.49531 + 62.7433i) q^{22} +(-72.6810 + 125.887i) q^{23} +(-27.4224 - 114.333i) q^{24} +(-61.4775 - 106.482i) q^{25} +(163.782 - 107.129i) q^{26} +(137.349 - 28.6071i) q^{27} +(202.124 + 88.4021i) q^{28} +(-93.6986 + 54.0969i) q^{29} +(12.6809 + 16.7609i) q^{30} +(-102.800 - 59.3514i) q^{31} +(125.877 - 130.089i) q^{32} +(-64.4256 - 95.7966i) q^{33} +(40.1242 - 79.3929i) q^{34} -39.4357 q^{35} +(150.644 + 154.798i) q^{36} -300.439 q^{37} +(-14.6696 + 29.0265i) q^{38} +(-158.052 + 322.935i) q^{39} +(-11.2974 + 30.3222i) q^{40} +(344.853 + 199.101i) q^{41} +(-402.171 + 50.1668i) q^{42} +(173.261 - 100.032i) q^{43} +(71.2231 - 162.846i) q^{44} +(-35.7603 - 14.5618i) q^{45} +(344.077 - 225.060i) q^{46} +(-151.770 - 262.873i) q^{47} +(-76.6395 + 323.602i) q^{48} +(208.727 - 361.526i) q^{49} +(19.3436 + 347.231i) q^{50} +(11.1773 + 163.040i) q^{51} +(-550.119 + 61.4831i) q^{52} +243.342i q^{53} +(-383.213 - 103.012i) q^{54} +31.7722i q^{55} +(-397.452 - 481.024i) q^{56} +(-4.08648 - 59.6085i) q^{57} +(305.545 - 17.0213i) q^{58} +(-41.9197 + 72.6070i) q^{59} +(-10.6280 - 58.4885i) q^{60} +(199.218 + 345.055i) q^{61} +(183.784 + 280.974i) q^{62} +(587.971 - 456.794i) q^{63} +(-483.722 + 167.800i) q^{64} +(85.6931 - 49.4749i) q^{65} +(40.4180 + 324.018i) q^{66} +(-307.763 - 177.687i) q^{67} +(-202.576 + 149.227i) q^{68} +(-332.038 + 678.427i) q^{69} +(99.5499 + 50.3112i) q^{70} +866.235 q^{71} +(-182.790 - 582.954i) q^{72} +64.6645 q^{73} +(758.415 + 383.293i) q^{74} +(-356.541 - 530.153i) q^{75} +(74.0629 - 54.5582i) q^{76} +(-530.594 - 306.338i) q^{77} +(810.973 - 613.564i) q^{78} +(354.896 - 204.899i) q^{79} +(67.2033 - 62.1312i) q^{80} +(701.846 - 197.111i) q^{81} +(-616.524 - 942.559i) q^{82} +(79.8990 + 138.389i) q^{83} +(1079.23 + 386.442i) q^{84} +(22.4881 - 38.9506i) q^{85} +(-564.993 + 31.4747i) q^{86} +(-466.507 + 313.737i) q^{87} +(-387.548 + 320.216i) q^{88} -1493.47i q^{89} +(71.6943 + 82.3815i) q^{90} +1908.09i q^{91} +(-1155.70 + 129.165i) q^{92} +(-554.005 - 271.143i) q^{93} +(47.7536 + 857.209i) q^{94} +(-8.22181 + 14.2406i) q^{95} +(606.310 - 719.113i) q^{96} +(700.115 + 1212.63i) q^{97} +(-988.129 + 646.332i) q^{98} +(-368.026 - 473.712i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 12 q^{4} - 72 q^{5} + 60 q^{6} - 84 q^{9} + 96 q^{10} + 216 q^{12} - 216 q^{13} - 36 q^{14} - 72 q^{16} + 276 q^{18} - 540 q^{20} + 384 q^{21} - 192 q^{22} - 168 q^{24} + 252 q^{25} - 672 q^{28}+ \cdots + 588 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.52436 − 1.27578i −0.892496 − 0.451056i
\(3\) 5.18398 − 0.355390i 0.997658 − 0.0683949i
\(4\) 4.74478 + 6.44105i 0.593097 + 0.805131i
\(5\) −1.23846 − 0.715028i −0.110772 − 0.0639540i 0.443591 − 0.896230i \(-0.353704\pi\)
−0.554362 + 0.832275i \(0.687038\pi\)
\(6\) −13.5396 − 5.71649i −0.921256 − 0.388958i
\(7\) 23.8818 − 13.7882i 1.28950 − 0.744491i 0.310932 − 0.950432i \(-0.399359\pi\)
0.978564 + 0.205941i \(0.0660255\pi\)
\(8\) −3.76017 − 22.3128i −0.166177 − 0.986096i
\(9\) 26.7474 − 3.68468i 0.990644 − 0.136469i
\(10\) 2.21411 + 3.38499i 0.0700164 + 0.107043i
\(11\) −11.1087 − 19.2409i −0.304492 − 0.527396i 0.672656 − 0.739955i \(-0.265153\pi\)
−0.977148 + 0.212560i \(0.931820\pi\)
\(12\) 26.8859 + 31.7040i 0.646775 + 0.762681i
\(13\) −34.5965 + 59.9229i −0.738103 + 1.27843i 0.215245 + 0.976560i \(0.430945\pi\)
−0.953348 + 0.301872i \(0.902388\pi\)
\(14\) −77.8769 + 4.33838i −1.48668 + 0.0828201i
\(15\) −6.67430 − 3.26656i −0.114886 − 0.0562281i
\(16\) −18.9742 + 61.1227i −0.296472 + 0.955042i
\(17\) 31.4507i 0.448701i 0.974509 + 0.224351i \(0.0720261\pi\)
−0.974509 + 0.224351i \(0.927974\pi\)
\(18\) −72.2209 − 24.8223i −0.945701 − 0.325038i
\(19\) − 11.4986i − 0.138840i −0.997588 − 0.0694199i \(-0.977885\pi\)
0.997588 − 0.0694199i \(-0.0221148\pi\)
\(20\) −1.27071 − 11.3697i −0.0142070 − 0.127117i
\(21\) 118.903 − 79.9650i 1.23556 − 0.830943i
\(22\) 3.49531 + 62.7433i 0.0338729 + 0.608041i
\(23\) −72.6810 + 125.887i −0.658914 + 1.14127i 0.321983 + 0.946746i \(0.395651\pi\)
−0.980897 + 0.194528i \(0.937683\pi\)
\(24\) −27.4224 − 114.333i −0.233232 − 0.972421i
\(25\) −61.4775 − 106.482i −0.491820 − 0.851857i
\(26\) 163.782 − 107.129i 1.23540 − 0.808070i
\(27\) 137.349 − 28.6071i 0.978991 − 0.203905i
\(28\) 202.124 + 88.4021i 1.36421 + 0.596658i
\(29\) −93.6986 + 54.0969i −0.599979 + 0.346398i −0.769033 − 0.639209i \(-0.779262\pi\)
0.169054 + 0.985607i \(0.445929\pi\)
\(30\) 12.6809 + 16.7609i 0.0771736 + 0.102003i
\(31\) −102.800 − 59.3514i −0.595592 − 0.343865i 0.171713 − 0.985147i \(-0.445070\pi\)
−0.767306 + 0.641282i \(0.778403\pi\)
\(32\) 125.877 − 130.089i 0.695377 − 0.718645i
\(33\) −64.4256 − 95.7966i −0.339850 − 0.505335i
\(34\) 40.1242 − 79.3929i 0.202389 − 0.400464i
\(35\) −39.4357 −0.190453
\(36\) 150.644 + 154.798i 0.697424 + 0.716659i
\(37\) −300.439 −1.33491 −0.667457 − 0.744649i \(-0.732617\pi\)
−0.667457 + 0.744649i \(0.732617\pi\)
\(38\) −14.6696 + 29.0265i −0.0626245 + 0.123914i
\(39\) −158.052 + 322.935i −0.648937 + 1.32592i
\(40\) −11.2974 + 30.3222i −0.0446571 + 0.119859i
\(41\) 344.853 + 199.101i 1.31359 + 0.758399i 0.982688 − 0.185268i \(-0.0593153\pi\)
0.330897 + 0.943667i \(0.392649\pi\)
\(42\) −402.171 + 50.1668i −1.47753 + 0.184307i
\(43\) 173.261 − 100.032i 0.614467 − 0.354763i −0.160245 − 0.987077i \(-0.551228\pi\)
0.774712 + 0.632315i \(0.217895\pi\)
\(44\) 71.2231 − 162.846i 0.244029 − 0.557953i
\(45\) −35.7603 − 14.5618i −0.118463 − 0.0482388i
\(46\) 344.077 − 225.060i 1.10286 − 0.721374i
\(47\) −151.770 − 262.873i −0.471018 − 0.815828i 0.528432 − 0.848976i \(-0.322780\pi\)
−0.999450 + 0.0331478i \(0.989447\pi\)
\(48\) −76.6395 + 323.602i −0.230457 + 0.973082i
\(49\) 208.727 − 361.526i 0.608534 − 1.05401i
\(50\) 19.3436 + 347.231i 0.0547119 + 0.982117i
\(51\) 11.1773 + 163.040i 0.0306889 + 0.447650i
\(52\) −550.119 + 61.4831i −1.46707 + 0.163965i
\(53\) 243.342i 0.630673i 0.948980 + 0.315336i \(0.102117\pi\)
−0.948980 + 0.315336i \(0.897883\pi\)
\(54\) −383.213 − 103.012i −0.965717 − 0.259595i
\(55\) 31.7722i 0.0778940i
\(56\) −397.452 − 481.024i −0.948425 − 1.14785i
\(57\) −4.08648 − 59.6085i −0.00949593 − 0.138515i
\(58\) 305.545 − 17.0213i 0.691724 − 0.0385347i
\(59\) −41.9197 + 72.6070i −0.0924996 + 0.160214i −0.908562 − 0.417749i \(-0.862819\pi\)
0.816063 + 0.577963i \(0.196152\pi\)
\(60\) −10.6280 − 58.4885i −0.0228678 − 0.125847i
\(61\) 199.218 + 345.055i 0.418151 + 0.724259i 0.995754 − 0.0920592i \(-0.0293449\pi\)
−0.577602 + 0.816318i \(0.696012\pi\)
\(62\) 183.784 + 280.974i 0.376461 + 0.575544i
\(63\) 587.971 − 456.794i 1.17583 − 0.913503i
\(64\) −483.722 + 167.800i −0.944770 + 0.327734i
\(65\) 85.6931 − 49.4749i 0.163522 − 0.0944094i
\(66\) 40.4180 + 324.018i 0.0753805 + 0.604301i
\(67\) −307.763 − 177.687i −0.561183 − 0.323999i 0.192437 − 0.981309i \(-0.438361\pi\)
−0.753620 + 0.657310i \(0.771694\pi\)
\(68\) −202.576 + 149.227i −0.361263 + 0.266123i
\(69\) −332.038 + 678.427i −0.579314 + 1.18367i
\(70\) 99.5499 + 50.3112i 0.169978 + 0.0859049i
\(71\) 866.235 1.44793 0.723966 − 0.689836i \(-0.242317\pi\)
0.723966 + 0.689836i \(0.242317\pi\)
\(72\) −182.790 − 582.954i −0.299195 − 0.954192i
\(73\) 64.6645 0.103677 0.0518384 − 0.998655i \(-0.483492\pi\)
0.0518384 + 0.998655i \(0.483492\pi\)
\(74\) 758.415 + 383.293i 1.19140 + 0.602121i
\(75\) −356.541 − 530.153i −0.548931 − 0.816224i
\(76\) 74.0629 − 54.5582i 0.111784 − 0.0823455i
\(77\) −530.594 − 306.338i −0.785283 − 0.453383i
\(78\) 810.973 − 613.564i 1.17724 − 0.890672i
\(79\) 354.896 − 204.899i 0.505429 − 0.291809i −0.225524 − 0.974238i \(-0.572409\pi\)
0.730953 + 0.682428i \(0.239076\pi\)
\(80\) 67.2033 − 62.1312i 0.0939194 − 0.0868310i
\(81\) 701.846 − 197.111i 0.962752 − 0.270385i
\(82\) −616.524 − 942.559i −0.830289 − 1.26937i
\(83\) 79.8990 + 138.389i 0.105663 + 0.183014i 0.914009 − 0.405694i \(-0.132970\pi\)
−0.808346 + 0.588708i \(0.799637\pi\)
\(84\) 1079.23 + 386.442i 1.40182 + 0.501956i
\(85\) 22.4881 − 38.9506i 0.0286962 − 0.0497034i
\(86\) −564.993 + 31.4747i −0.708427 + 0.0394652i
\(87\) −466.507 + 313.737i −0.574882 + 0.386623i
\(88\) −387.548 + 320.216i −0.469463 + 0.387900i
\(89\) − 1493.47i − 1.77873i −0.457195 − 0.889366i \(-0.651146\pi\)
0.457195 − 0.889366i \(-0.348854\pi\)
\(90\) 71.6943 + 82.3815i 0.0839694 + 0.0964863i
\(91\) 1908.09i 2.19805i
\(92\) −1155.70 + 129.165i −1.30967 + 0.146373i
\(93\) −554.005 − 271.143i −0.617716 − 0.302325i
\(94\) 47.7536 + 857.209i 0.0523979 + 0.940578i
\(95\) −8.22181 + 14.2406i −0.00887936 + 0.0153795i
\(96\) 606.310 − 719.113i 0.644597 − 0.764523i
\(97\) 700.115 + 1212.63i 0.732844 + 1.26932i 0.955663 + 0.294463i \(0.0951408\pi\)
−0.222819 + 0.974860i \(0.571526\pi\)
\(98\) −988.129 + 646.332i −1.01853 + 0.666218i
\(99\) −368.026 − 473.712i −0.373617 − 0.480908i
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 36.4.h.b.23.2 yes 24
3.2 odd 2 108.4.h.b.71.11 24
4.3 odd 2 inner 36.4.h.b.23.6 yes 24
9.2 odd 6 inner 36.4.h.b.11.6 yes 24
9.4 even 3 324.4.b.c.323.19 24
9.5 odd 6 324.4.b.c.323.6 24
9.7 even 3 108.4.h.b.35.7 24
12.11 even 2 108.4.h.b.71.7 24
36.7 odd 6 108.4.h.b.35.11 24
36.11 even 6 inner 36.4.h.b.11.2 ✓ 24
36.23 even 6 324.4.b.c.323.20 24
36.31 odd 6 324.4.b.c.323.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.4.h.b.11.2 ✓ 24 36.11 even 6 inner
36.4.h.b.11.6 yes 24 9.2 odd 6 inner
36.4.h.b.23.2 yes 24 1.1 even 1 trivial
36.4.h.b.23.6 yes 24 4.3 odd 2 inner
108.4.h.b.35.7 24 9.7 even 3
108.4.h.b.35.11 24 36.7 odd 6
108.4.h.b.71.7 24 12.11 even 2
108.4.h.b.71.11 24 3.2 odd 2
324.4.b.c.323.5 24 36.31 odd 6
324.4.b.c.323.6 24 9.5 odd 6
324.4.b.c.323.19 24 9.4 even 3
324.4.b.c.323.20 24 36.23 even 6