↑

Properties

Label 36.3.f.c.7.4
Level $36$
Weight $3$
Character 36.7
Analytic conductor $0.981$
Analytic rank $0$
Dimension $16$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [36,3,Mod(7,36)] 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("36.7"); S:= CuspForms(chi, 3); 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, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 36.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.980928951697\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 7.4
Root \(0.186266 - 1.99131i\) of defining polynomial
Character \(\chi\) \(=\) 36.7
Dual form 36.3.f.c.31.4

$q$-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+(-0.186266 + 1.99131i) q^{2} +(-2.67178 + 1.36441i) q^{3} +(-3.93061 - 0.741826i) q^{4} +(3.07403 + 5.32438i) q^{5} +(-2.21930 - 5.57447i) q^{6} +(0.511543 + 0.295340i) q^{7} +(2.20934 - 7.68888i) q^{8} +(5.27677 - 7.29079i) q^{9} +(-11.1751 + 5.12959i) q^{10} +(15.1205 + 8.72982i) q^{11} +(11.5139 - 3.38097i) q^{12} +(-0.892255 - 1.54543i) q^{13} +(-0.683395 + 0.963628i) q^{14} +(-15.4778 - 10.0313i) q^{15} +(14.8994 + 5.83166i) q^{16} -16.9171 q^{17} +(13.5353 + 11.8657i) q^{18} -19.5058i q^{19} +(-8.13306 - 23.2084i) q^{20} +(-1.76969 - 0.0911265i) q^{21} +(-20.2002 + 28.4835i) q^{22} +(-6.86778 + 3.96511i) q^{23} +(4.58791 + 23.5574i) q^{24} +(-6.39933 + 11.0840i) q^{25} +(3.24362 - 1.48889i) q^{26} +(-4.15071 + 26.6790i) q^{27} +(-1.79159 - 1.54034i) q^{28} +(3.17517 - 5.49956i) q^{29} +(22.8584 - 28.9525i) q^{30} +(27.6558 - 15.9671i) q^{31} +(-14.3879 + 28.5830i) q^{32} +(-52.3096 - 2.69357i) q^{33} +(3.15108 - 33.6871i) q^{34} +3.63153i q^{35} +(-26.1494 + 24.7428i) q^{36} +58.2834 q^{37} +(38.8420 + 3.63326i) q^{38} +(4.49251 + 2.91164i) q^{39} +(47.7301 - 11.8725i) q^{40} +(-2.66948 - 4.62368i) q^{41} +(0.511095 - 3.50703i) q^{42} +(-33.9324 - 19.5909i) q^{43} +(-52.9567 - 45.5303i) q^{44} +(55.0399 + 5.68339i) q^{45} +(-6.61653 - 14.4144i) q^{46} +(-9.64117 - 5.56633i) q^{47} +(-47.7646 + 4.74800i) q^{48} +(-24.3255 - 42.1331i) q^{49} +(-20.8796 - 14.8076i) q^{50} +(45.1987 - 23.0819i) q^{51} +(2.36067 + 6.73638i) q^{52} +35.8770 q^{53} +(-52.3530 - 13.2347i) q^{54} +107.343i q^{55} +(3.40100 - 3.28069i) q^{56} +(26.6139 + 52.1151i) q^{57} +(10.3599 + 7.34713i) q^{58} +(-20.8974 + 12.0651i) q^{59} +(53.3955 + 50.9109i) q^{60} +(-37.9460 + 65.7244i) q^{61} +(26.6441 + 58.0454i) q^{62} +(4.85256 - 2.17112i) q^{63} +(-54.2376 - 33.9747i) q^{64} +(5.48564 - 9.50141i) q^{65} +(15.1072 - 103.663i) q^{66} +(-31.8200 + 18.3713i) q^{67} +(66.4945 + 12.5495i) q^{68} +(12.9391 - 19.9644i) q^{69} +(-7.23150 - 0.676431i) q^{70} -87.8370i q^{71} +(-44.3998 - 56.6803i) q^{72} -60.0423 q^{73} +(-10.8562 + 116.060i) q^{74} +(1.97450 - 38.3452i) q^{75} +(-14.4699 + 76.6696i) q^{76} +(5.15652 + 8.93136i) q^{77} +(-6.63478 + 8.40362i) q^{78} +(32.1841 + 18.5815i) q^{79} +(14.7512 + 97.2567i) q^{80} +(-25.3114 - 76.9437i) q^{81} +(9.70439 - 4.45452i) q^{82} +(-66.0281 - 38.1214i) q^{83} +(6.88838 + 1.67099i) q^{84} +(-52.0037 - 90.0730i) q^{85} +(45.3319 - 63.9207i) q^{86} +(-0.979694 + 19.0258i) q^{87} +(100.529 - 96.9724i) q^{88} -27.5873 q^{89} +(-21.5694 + 108.543i) q^{90} -1.05407i q^{91} +(29.9360 - 10.4906i) q^{92} +(-52.1045 + 80.3944i) q^{93} +(12.8801 - 18.1617i) q^{94} +(103.856 - 59.9614i) q^{95} +(-0.557801 - 95.9984i) q^{96} +(13.0585 - 22.6180i) q^{97} +(88.4309 - 40.5917i) q^{98} +(143.435 - 64.1751i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{2} - 5 q^{4} + 6 q^{5} + 9 q^{6} - 54 q^{8} + 18 q^{9} + 20 q^{10} - 36 q^{12} - 46 q^{13} - 12 q^{14} - 17 q^{16} + 12 q^{17} + 48 q^{18} + 36 q^{20} - 66 q^{21} + 33 q^{22} + 129 q^{24} - 30 q^{25}+ \cdots + 1170 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.186266 + 1.99131i −0.0931330 + 0.995654i
\(3\) −2.67178 + 1.36441i −0.890592 + 0.454803i
\(4\) −3.93061 − 0.741826i −0.982652 − 0.185456i
\(5\) 3.07403 + 5.32438i 0.614806 + 1.06488i 0.990418 + 0.138099i \(0.0440991\pi\)
−0.375612 + 0.926777i \(0.622568\pi\)
\(6\) −2.21930 − 5.57447i −0.369883 − 0.929078i
\(7\) 0.511543 + 0.295340i 0.0730776 + 0.0421914i 0.536094 − 0.844159i \(-0.319899\pi\)
−0.463016 + 0.886350i \(0.653233\pi\)
\(8\) 2.20934 − 7.68888i 0.276168 − 0.961109i
\(9\) 5.27677 − 7.29079i 0.586308 − 0.810088i
\(10\) −11.1751 + 5.12959i −1.11751 + 0.512959i
\(11\) 15.1205 + 8.72982i 1.37459 + 0.793620i 0.991502 − 0.130092i \(-0.0415274\pi\)
0.383088 + 0.923712i \(0.374861\pi\)
\(12\) 11.5139 − 3.38097i 0.959489 − 0.281748i
\(13\) −0.892255 − 1.54543i −0.0686350 − 0.118879i 0.829666 − 0.558261i \(-0.188531\pi\)
−0.898301 + 0.439381i \(0.855198\pi\)
\(14\) −0.683395 + 0.963628i −0.0488140 + 0.0688306i
\(15\) −15.4778 − 10.0313i −1.03185 − 0.668754i
\(16\) 14.8994 + 5.83166i 0.931212 + 0.364479i
\(17\) −16.9171 −0.995123 −0.497562 − 0.867429i \(-0.665771\pi\)
−0.497562 + 0.867429i \(0.665771\pi\)
\(18\) 13.5353 + 11.8657i 0.751963 + 0.659206i
\(19\) − 19.5058i − 1.02662i −0.858203 − 0.513310i \(-0.828419\pi\)
0.858203 − 0.513310i \(-0.171581\pi\)
\(20\) −8.13306 − 23.2084i −0.406653 − 1.16042i
\(21\) −1.76969 − 0.0911265i −0.0842711 − 0.00433936i
\(22\) −20.2002 + 28.4835i −0.918190 + 1.29470i
\(23\) −6.86778 + 3.96511i −0.298599 + 0.172396i −0.641813 − 0.766861i \(-0.721818\pi\)
0.343214 + 0.939257i \(0.388484\pi\)
\(24\) 4.58791 + 23.5574i 0.191163 + 0.981558i
\(25\) −6.39933 + 11.0840i −0.255973 + 0.443359i
\(26\) 3.24362 − 1.48889i 0.124755 − 0.0572651i
\(27\) −4.15071 + 26.6790i −0.153730 + 0.988113i
\(28\) −1.79159 − 1.54034i −0.0639852 − 0.0550122i
\(29\) 3.17517 − 5.49956i 0.109489 − 0.189640i −0.806075 − 0.591814i \(-0.798412\pi\)
0.915563 + 0.402174i \(0.131745\pi\)
\(30\) 22.8584 − 28.9525i 0.761946 − 0.965083i
\(31\) 27.6558 − 15.9671i 0.892124 − 0.515068i 0.0174873 − 0.999847i \(-0.494433\pi\)
0.874637 + 0.484779i \(0.161100\pi\)
\(32\) −14.3879 + 28.5830i −0.449621 + 0.893219i
\(33\) −52.3096 − 2.69357i −1.58514 − 0.0816233i
\(34\) 3.15108 − 33.6871i 0.0926788 − 0.990798i
\(35\) 3.63153i 0.103758i
\(36\) −26.1494 + 24.7428i −0.726373 + 0.687301i
\(37\) 58.2834 1.57523 0.787614 − 0.616169i \(-0.211316\pi\)
0.787614 + 0.616169i \(0.211316\pi\)
\(38\) 38.8420 + 3.63326i 1.02216 + 0.0956122i
\(39\) 4.49251 + 2.91164i 0.115192 + 0.0746575i
\(40\) 47.7301 − 11.8725i 1.19325 − 0.296812i
\(41\) −2.66948 − 4.62368i −0.0651093 − 0.112773i 0.831633 − 0.555325i \(-0.187406\pi\)
−0.896742 + 0.442553i \(0.854073\pi\)
\(42\) 0.511095 − 3.50703i 0.0121689 − 0.0835007i
\(43\) −33.9324 − 19.5909i −0.789126 − 0.455602i 0.0505290 − 0.998723i \(-0.483909\pi\)
−0.839655 + 0.543121i \(0.817243\pi\)
\(44\) −52.9567 − 45.5303i −1.20356 − 1.03478i
\(45\) 55.0399 + 5.68339i 1.22311 + 0.126298i
\(46\) −6.61653 − 14.4144i −0.143838 − 0.313357i
\(47\) −9.64117 − 5.56633i −0.205131 − 0.118433i 0.393915 − 0.919147i \(-0.371120\pi\)
−0.599047 + 0.800714i \(0.704454\pi\)
\(48\) −47.7646 + 4.74800i −0.995096 + 0.0989166i
\(49\) −24.3255 − 42.1331i −0.496440 − 0.859859i
\(50\) −20.8796 − 14.8076i −0.417592 − 0.296152i
\(51\) 45.1987 − 23.0819i 0.886249 − 0.452585i
\(52\) 2.36067 + 6.73638i 0.0453974 + 0.129546i
\(53\) 35.8770 0.676925 0.338462 − 0.940980i \(-0.390093\pi\)
0.338462 + 0.940980i \(0.390093\pi\)
\(54\) −52.3530 − 13.2347i −0.969501 − 0.245088i
\(55\) 107.343i 1.95169i
\(56\) 3.40100 − 3.28069i 0.0607322 − 0.0585837i
\(57\) 26.6139 + 52.1151i 0.466910 + 0.914299i
\(58\) 10.3599 + 7.34713i 0.178619 + 0.126675i
\(59\) −20.8974 + 12.0651i −0.354194 + 0.204494i −0.666531 − 0.745477i \(-0.732222\pi\)
0.312337 + 0.949971i \(0.398888\pi\)
\(60\) 53.3955 + 50.9109i 0.889926 + 0.848516i
\(61\) −37.9460 + 65.7244i −0.622066 + 1.07745i 0.367034 + 0.930207i \(0.380373\pi\)
−0.989100 + 0.147243i \(0.952960\pi\)
\(62\) 26.6441 + 58.0454i 0.429743 + 0.936216i
\(63\) 4.85256 − 2.17112i 0.0770247 − 0.0344622i
\(64\) −54.2376 − 33.9747i −0.847463 − 0.530855i
\(65\) 5.48564 − 9.50141i 0.0843944 − 0.146175i
\(66\) 15.1072 − 103.663i 0.228897 − 1.57065i
\(67\) −31.8200 + 18.3713i −0.474925 + 0.274198i −0.718299 − 0.695734i \(-0.755079\pi\)
0.243374 + 0.969933i \(0.421746\pi\)
\(68\) 66.4945 + 12.5495i 0.977860 + 0.184552i
\(69\) 12.9391 − 19.9644i 0.187524 − 0.289339i
\(70\) −7.23150 − 0.676431i −0.103307 − 0.00966331i
\(71\) − 87.8370i − 1.23714i −0.785730 − 0.618570i \(-0.787712\pi\)
0.785730 − 0.618570i \(-0.212288\pi\)
\(72\) −44.3998 − 56.6803i −0.616664 − 0.787226i
\(73\) −60.0423 −0.822498 −0.411249 − 0.911523i \(-0.634907\pi\)
−0.411249 + 0.911523i \(0.634907\pi\)
\(74\) −10.8562 + 116.060i −0.146706 + 1.56838i
\(75\) 1.97450 − 38.3452i 0.0263267 − 0.511269i
\(76\) −14.4699 + 76.6696i −0.190393 + 1.00881i
\(77\) 5.15652 + 8.93136i 0.0669678 + 0.115992i
\(78\) −6.63478 + 8.40362i −0.0850613 + 0.107739i
\(79\) 32.1841 + 18.5815i 0.407394 + 0.235209i 0.689669 − 0.724124i \(-0.257756\pi\)
−0.282275 + 0.959333i \(0.591089\pi\)
\(80\) 14.7512 + 97.2567i 0.184391 + 1.21571i
\(81\) −25.3114 − 76.9437i −0.312486 − 0.949922i
\(82\) 9.70439 − 4.45452i 0.118346 − 0.0543234i
\(83\) −66.0281 − 38.1214i −0.795520 − 0.459294i 0.0463824 − 0.998924i \(-0.485231\pi\)
−0.841902 + 0.539630i \(0.818564\pi\)
\(84\) 6.88838 + 1.67099i 0.0820045 + 0.0198927i
\(85\) −52.0037 − 90.0730i −0.611808 − 1.05968i
\(86\) 45.3319 − 63.9207i 0.527115 − 0.743264i
\(87\) −0.979694 + 19.0258i −0.0112609 + 0.218688i
\(88\) 100.529 − 96.9724i 1.14237 − 1.10196i
\(89\) −27.5873 −0.309969 −0.154985 − 0.987917i \(-0.549533\pi\)
−0.154985 + 0.987917i \(0.549533\pi\)
\(90\) −21.5694 + 108.543i −0.239660 + 1.20603i
\(91\) − 1.05407i − 0.0115832i
\(92\) 29.9360 − 10.4906i 0.325391 − 0.114028i
\(93\) −52.1045 + 80.3944i −0.560264 + 0.864456i
\(94\) 12.8801 − 18.1617i 0.137022 − 0.193210i
\(95\) 103.856 − 59.9614i 1.09322 − 0.631172i
\(96\) −0.557801 − 95.9984i −0.00581043 − 0.999983i
\(97\) 13.0585 − 22.6180i 0.134624 − 0.233176i −0.790830 − 0.612036i \(-0.790351\pi\)
0.925454 + 0.378861i \(0.123684\pi\)
\(98\) 88.4309 − 40.5917i 0.902357 − 0.414201i
\(99\) 143.435 − 64.1751i 1.44883 − 0.648234i
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 36.3.f.c.7.4 yes 16
3.2 odd 2 108.3.f.c.19.5 16
4.3 odd 2 inner 36.3.f.c.7.3 ✓ 16
8.3 odd 2 576.3.o.g.511.2 16
8.5 even 2 576.3.o.g.511.7 16
9.2 odd 6 324.3.d.g.163.2 8
9.4 even 3 inner 36.3.f.c.31.3 yes 16
9.5 odd 6 108.3.f.c.91.6 16
9.7 even 3 324.3.d.i.163.7 8
12.11 even 2 108.3.f.c.19.6 16
24.5 odd 2 1728.3.o.g.127.8 16
24.11 even 2 1728.3.o.g.127.7 16
36.7 odd 6 324.3.d.i.163.8 8
36.11 even 6 324.3.d.g.163.1 8
36.23 even 6 108.3.f.c.91.5 16
36.31 odd 6 inner 36.3.f.c.31.4 yes 16
72.5 odd 6 1728.3.o.g.1279.7 16
72.13 even 6 576.3.o.g.319.2 16
72.59 even 6 1728.3.o.g.1279.8 16
72.67 odd 6 576.3.o.g.319.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.f.c.7.3 ✓ 16 4.3 odd 2 inner
36.3.f.c.7.4 yes 16 1.1 even 1 trivial
36.3.f.c.31.3 yes 16 9.4 even 3 inner
36.3.f.c.31.4 yes 16 36.31 odd 6 inner
108.3.f.c.19.5 16 3.2 odd 2
108.3.f.c.19.6 16 12.11 even 2
108.3.f.c.91.5 16 36.23 even 6
108.3.f.c.91.6 16 9.5 odd 6
324.3.d.g.163.1 8 36.11 even 6
324.3.d.g.163.2 8 9.2 odd 6
324.3.d.i.163.7 8 9.7 even 3
324.3.d.i.163.8 8 36.7 odd 6
576.3.o.g.319.2 16 72.13 even 6
576.3.o.g.319.7 16 72.67 odd 6
576.3.o.g.511.2 16 8.3 odd 2
576.3.o.g.511.7 16 8.5 even 2
1728.3.o.g.127.7 16 24.11 even 2
1728.3.o.g.127.8 16 24.5 odd 2
1728.3.o.g.1279.7 16 72.5 odd 6
1728.3.o.g.1279.8 16 72.59 even 6