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Properties

Label 36.3.f.a.31.1
Level $36$
Weight $3$
Character 36.31
Analytic conductor $0.981$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [36,3,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 = [2,-2] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 36.31
Dual form 36.3.f.a.7.1

$q$-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+(-1.00000 + 1.73205i) q^{2} +(1.50000 + 2.59808i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(-2.00000 + 3.46410i) q^{5} -6.00000 q^{6} +(3.00000 - 1.73205i) q^{7} +8.00000 q^{8} +(-4.50000 + 7.79423i) q^{9} +(-4.00000 - 6.92820i) q^{10} +(10.5000 - 6.06218i) q^{11} +(6.00000 - 10.3923i) q^{12} +(11.0000 - 19.0526i) q^{13} +6.92820i q^{14} -12.0000 q^{15} +(-8.00000 + 13.8564i) q^{16} -11.0000 q^{17} +(-9.00000 - 15.5885i) q^{18} +15.5885i q^{19} +16.0000 q^{20} +(9.00000 + 5.19615i) q^{21} +24.2487i q^{22} +(-21.0000 - 12.1244i) q^{23} +(12.0000 + 20.7846i) q^{24} +(4.50000 + 7.79423i) q^{25} +(22.0000 + 38.1051i) q^{26} -27.0000 q^{27} +(-12.0000 - 6.92820i) q^{28} +(-17.0000 - 29.4449i) q^{29} +(12.0000 - 20.7846i) q^{30} +(-6.00000 - 3.46410i) q^{31} +(-16.0000 - 27.7128i) q^{32} +(31.5000 + 18.1865i) q^{33} +(11.0000 - 19.0526i) q^{34} +13.8564i q^{35} +36.0000 q^{36} -16.0000 q^{37} +(-27.0000 - 15.5885i) q^{38} +66.0000 q^{39} +(-16.0000 + 27.7128i) q^{40} +(-6.50000 + 11.2583i) q^{41} +(-18.0000 + 10.3923i) q^{42} +(43.5000 - 25.1147i) q^{43} +(-42.0000 - 24.2487i) q^{44} +(-18.0000 - 31.1769i) q^{45} +(42.0000 - 24.2487i) q^{46} +(-3.00000 + 1.73205i) q^{47} -48.0000 q^{48} +(-18.5000 + 32.0429i) q^{49} -18.0000 q^{50} +(-16.5000 - 28.5788i) q^{51} -88.0000 q^{52} +52.0000 q^{53} +(27.0000 - 46.7654i) q^{54} +48.4974i q^{55} +(24.0000 - 13.8564i) q^{56} +(-40.5000 + 23.3827i) q^{57} +68.0000 q^{58} +(46.5000 + 26.8468i) q^{59} +(24.0000 + 41.5692i) q^{60} +(8.00000 + 13.8564i) q^{61} +(12.0000 - 6.92820i) q^{62} +31.1769i q^{63} +64.0000 q^{64} +(44.0000 + 76.2102i) q^{65} +(-63.0000 + 36.3731i) q^{66} +(-100.500 - 58.0237i) q^{67} +(22.0000 + 38.1051i) q^{68} -72.7461i q^{69} +(-24.0000 - 13.8564i) q^{70} +(-36.0000 + 62.3538i) q^{72} -25.0000 q^{73} +(16.0000 - 27.7128i) q^{74} +(-13.5000 + 23.3827i) q^{75} +(54.0000 - 31.1769i) q^{76} +(21.0000 - 36.3731i) q^{77} +(-66.0000 + 114.315i) q^{78} +(-24.0000 + 13.8564i) q^{79} +(-32.0000 - 55.4256i) q^{80} +(-40.5000 - 70.1481i) q^{81} +(-13.0000 - 22.5167i) q^{82} +(-30.0000 + 17.3205i) q^{83} -41.5692i q^{84} +(22.0000 - 38.1051i) q^{85} +100.459i q^{86} +(51.0000 - 88.3346i) q^{87} +(84.0000 - 48.4974i) q^{88} -2.00000 q^{89} +72.0000 q^{90} -76.2102i q^{91} +96.9948i q^{92} -20.7846i q^{93} -6.92820i q^{94} +(-54.0000 - 31.1769i) q^{95} +(48.0000 - 83.1384i) q^{96} +(21.5000 + 37.2391i) q^{97} +(-37.0000 - 64.0859i) q^{98} +109.119i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} - 4 q^{4} - 4 q^{5} - 12 q^{6} + 6 q^{7} + 16 q^{8} - 9 q^{9} - 8 q^{10} + 21 q^{11} + 12 q^{12} + 22 q^{13} - 24 q^{15} - 16 q^{16} - 22 q^{17} - 18 q^{18} + 32 q^{20} + 18 q^{21}+ \cdots - 74 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 + 1.73205i −0.500000 + 0.866025i
\(3\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(4\) −2.00000 − 3.46410i −0.500000 − 0.866025i
\(5\) −2.00000 + 3.46410i −0.400000 + 0.692820i −0.993725 − 0.111847i \(-0.964323\pi\)
0.593725 + 0.804668i \(0.297657\pi\)
\(6\) −6.00000 −1.00000
\(7\) 3.00000 − 1.73205i 0.428571 − 0.247436i −0.270166 − 0.962814i \(-0.587079\pi\)
0.698738 + 0.715378i \(0.253745\pi\)
\(8\) 8.00000 1.00000
\(9\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(10\) −4.00000 − 6.92820i −0.400000 − 0.692820i
\(11\) 10.5000 − 6.06218i 0.954545 − 0.551107i 0.0600555 − 0.998195i \(-0.480872\pi\)
0.894490 + 0.447088i \(0.147539\pi\)
\(12\) 6.00000 − 10.3923i 0.500000 − 0.866025i
\(13\) 11.0000 − 19.0526i 0.846154 − 1.46558i −0.0384615 − 0.999260i \(-0.512246\pi\)
0.884615 − 0.466321i \(-0.154421\pi\)
\(14\) 6.92820i 0.494872i
\(15\) −12.0000 −0.800000
\(16\) −8.00000 + 13.8564i −0.500000 + 0.866025i
\(17\) −11.0000 −0.647059 −0.323529 − 0.946218i \(-0.604869\pi\)
−0.323529 + 0.946218i \(0.604869\pi\)
\(18\) −9.00000 − 15.5885i −0.500000 − 0.866025i
\(19\) 15.5885i 0.820445i 0.911985 + 0.410223i \(0.134549\pi\)
−0.911985 + 0.410223i \(0.865451\pi\)
\(20\) 16.0000 0.800000
\(21\) 9.00000 + 5.19615i 0.428571 + 0.247436i
\(22\) 24.2487i 1.10221i
\(23\) −21.0000 − 12.1244i −0.913043 − 0.527146i −0.0316343 − 0.999500i \(-0.510071\pi\)
−0.881409 + 0.472354i \(0.843405\pi\)
\(24\) 12.0000 + 20.7846i 0.500000 + 0.866025i
\(25\) 4.50000 + 7.79423i 0.180000 + 0.311769i
\(26\) 22.0000 + 38.1051i 0.846154 + 1.46558i
\(27\) −27.0000 −1.00000
\(28\) −12.0000 − 6.92820i −0.428571 − 0.247436i
\(29\) −17.0000 − 29.4449i −0.586207 − 1.01534i −0.994724 − 0.102589i \(-0.967287\pi\)
0.408517 − 0.912751i \(-0.366046\pi\)
\(30\) 12.0000 − 20.7846i 0.400000 − 0.692820i
\(31\) −6.00000 − 3.46410i −0.193548 − 0.111745i 0.400094 − 0.916474i \(-0.368977\pi\)
−0.593643 + 0.804729i \(0.702311\pi\)
\(32\) −16.0000 − 27.7128i −0.500000 − 0.866025i
\(33\) 31.5000 + 18.1865i 0.954545 + 0.551107i
\(34\) 11.0000 − 19.0526i 0.323529 − 0.560369i
\(35\) 13.8564i 0.395897i
\(36\) 36.0000 1.00000
\(37\) −16.0000 −0.432432 −0.216216 − 0.976346i \(-0.569372\pi\)
−0.216216 + 0.976346i \(0.569372\pi\)
\(38\) −27.0000 − 15.5885i −0.710526 − 0.410223i
\(39\) 66.0000 1.69231
\(40\) −16.0000 + 27.7128i −0.400000 + 0.692820i
\(41\) −6.50000 + 11.2583i −0.158537 + 0.274593i −0.934341 − 0.356380i \(-0.884011\pi\)
0.775805 + 0.630973i \(0.217344\pi\)
\(42\) −18.0000 + 10.3923i −0.428571 + 0.247436i
\(43\) 43.5000 − 25.1147i 1.01163 − 0.584064i 0.0999600 − 0.994991i \(-0.468129\pi\)
0.911668 + 0.410928i \(0.134795\pi\)
\(44\) −42.0000 − 24.2487i −0.954545 − 0.551107i
\(45\) −18.0000 − 31.1769i −0.400000 − 0.692820i
\(46\) 42.0000 − 24.2487i 0.913043 − 0.527146i
\(47\) −3.00000 + 1.73205i −0.0638298 + 0.0368521i −0.531575 − 0.847011i \(-0.678400\pi\)
0.467745 + 0.883863i \(0.345066\pi\)
\(48\) −48.0000 −1.00000
\(49\) −18.5000 + 32.0429i −0.377551 + 0.653938i
\(50\) −18.0000 −0.360000
\(51\) −16.5000 − 28.5788i −0.323529 − 0.560369i
\(52\) −88.0000 −1.69231
\(53\) 52.0000 0.981132 0.490566 − 0.871404i \(-0.336790\pi\)
0.490566 + 0.871404i \(0.336790\pi\)
\(54\) 27.0000 − 46.7654i 0.500000 − 0.866025i
\(55\) 48.4974i 0.881771i
\(56\) 24.0000 − 13.8564i 0.428571 − 0.247436i
\(57\) −40.5000 + 23.3827i −0.710526 + 0.410223i
\(58\) 68.0000 1.17241
\(59\) 46.5000 + 26.8468i 0.788136 + 0.455030i 0.839306 − 0.543660i \(-0.182962\pi\)
−0.0511702 + 0.998690i \(0.516295\pi\)
\(60\) 24.0000 + 41.5692i 0.400000 + 0.692820i
\(61\) 8.00000 + 13.8564i 0.131148 + 0.227154i 0.924119 − 0.382104i \(-0.124801\pi\)
−0.792972 + 0.609259i \(0.791467\pi\)
\(62\) 12.0000 − 6.92820i 0.193548 − 0.111745i
\(63\) 31.1769i 0.494872i
\(64\) 64.0000 1.00000
\(65\) 44.0000 + 76.2102i 0.676923 + 1.17247i
\(66\) −63.0000 + 36.3731i −0.954545 + 0.551107i
\(67\) −100.500 − 58.0237i −1.50000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(68\) 22.0000 + 38.1051i 0.323529 + 0.560369i
\(69\) − 72.7461i − 1.05429i
\(70\) −24.0000 − 13.8564i −0.342857 − 0.197949i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −36.0000 + 62.3538i −0.500000 + 0.866025i
\(73\) −25.0000 −0.342466 −0.171233 − 0.985231i \(-0.554775\pi\)
−0.171233 + 0.985231i \(0.554775\pi\)
\(74\) 16.0000 − 27.7128i 0.216216 − 0.374497i
\(75\) −13.5000 + 23.3827i −0.180000 + 0.311769i
\(76\) 54.0000 − 31.1769i 0.710526 − 0.410223i
\(77\) 21.0000 − 36.3731i 0.272727 − 0.472377i
\(78\) −66.0000 + 114.315i −0.846154 + 1.46558i
\(79\) −24.0000 + 13.8564i −0.303797 + 0.175398i −0.644148 − 0.764901i \(-0.722788\pi\)
0.340350 + 0.940299i \(0.389454\pi\)
\(80\) −32.0000 − 55.4256i −0.400000 − 0.692820i
\(81\) −40.5000 − 70.1481i −0.500000 − 0.866025i
\(82\) −13.0000 − 22.5167i −0.158537 − 0.274593i
\(83\) −30.0000 + 17.3205i −0.361446 + 0.208681i −0.669715 − 0.742618i \(-0.733584\pi\)
0.308269 + 0.951299i \(0.400250\pi\)
\(84\) − 41.5692i − 0.494872i
\(85\) 22.0000 − 38.1051i 0.258824 − 0.448296i
\(86\) 100.459i 1.16813i
\(87\) 51.0000 − 88.3346i 0.586207 − 1.01534i
\(88\) 84.0000 − 48.4974i 0.954545 − 0.551107i
\(89\) −2.00000 −0.0224719 −0.0112360 − 0.999937i \(-0.503577\pi\)
−0.0112360 + 0.999937i \(0.503577\pi\)
\(90\) 72.0000 0.800000
\(91\) − 76.2102i − 0.837475i
\(92\) 96.9948i 1.05429i
\(93\) − 20.7846i − 0.223490i
\(94\) − 6.92820i − 0.0737043i
\(95\) −54.0000 − 31.1769i −0.568421 − 0.328178i
\(96\) 48.0000 − 83.1384i 0.500000 − 0.866025i
\(97\) 21.5000 + 37.2391i 0.221649 + 0.383908i 0.955309 − 0.295609i \(-0.0955226\pi\)
−0.733659 + 0.679517i \(0.762189\pi\)
\(98\) −37.0000 − 64.0859i −0.377551 − 0.653938i
\(99\) 109.119i 1.10221i
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.a.31.1 yes 2
3.2 odd 2 108.3.f.b.91.1 2
4.3 odd 2 36.3.f.b.31.1 yes 2
8.3 odd 2 576.3.o.b.319.1 2
8.5 even 2 576.3.o.a.319.1 2
9.2 odd 6 108.3.f.a.19.1 2
9.4 even 3 324.3.d.b.163.1 2
9.5 odd 6 324.3.d.c.163.2 2
9.7 even 3 36.3.f.b.7.1 yes 2
12.11 even 2 108.3.f.a.91.1 2
24.5 odd 2 1728.3.o.b.1279.1 2
24.11 even 2 1728.3.o.a.1279.1 2
36.7 odd 6 inner 36.3.f.a.7.1 ✓ 2
36.11 even 6 108.3.f.b.19.1 2
36.23 even 6 324.3.d.c.163.1 2
36.31 odd 6 324.3.d.b.163.2 2
72.11 even 6 1728.3.o.b.127.1 2
72.29 odd 6 1728.3.o.a.127.1 2
72.43 odd 6 576.3.o.a.511.1 2
72.61 even 6 576.3.o.b.511.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.f.a.7.1 ✓ 2 36.7 odd 6 inner
36.3.f.a.31.1 yes 2 1.1 even 1 trivial
36.3.f.b.7.1 yes 2 9.7 even 3
36.3.f.b.31.1 yes 2 4.3 odd 2
108.3.f.a.19.1 2 9.2 odd 6
108.3.f.a.91.1 2 12.11 even 2
108.3.f.b.19.1 2 36.11 even 6
108.3.f.b.91.1 2 3.2 odd 2
324.3.d.b.163.1 2 9.4 even 3
324.3.d.b.163.2 2 36.31 odd 6
324.3.d.c.163.1 2 36.23 even 6
324.3.d.c.163.2 2 9.5 odd 6
576.3.o.a.319.1 2 8.5 even 2
576.3.o.a.511.1 2 72.43 odd 6
576.3.o.b.319.1 2 8.3 odd 2
576.3.o.b.511.1 2 72.61 even 6
1728.3.o.a.127.1 2 72.29 odd 6
1728.3.o.a.1279.1 2 24.11 even 2
1728.3.o.b.127.1 2 72.11 even 6
1728.3.o.b.1279.1 2 24.5 odd 2