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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [36,16,Mod(35,36)] 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("36.35"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 36.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28] 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: \(51.3696618360\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 35.23
Character \(\chi\) \(=\) 36.35
Dual form 36.16.b.b.35.24

$q$-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)
 
\(f(q)\) \(=\) \(q+(138.866 - 116.121i) q^{2} +(5799.65 - 32250.7i) q^{4} -170244. i q^{5} -1.33330e6i q^{7} +(-2.93962e6 - 5.15199e6i) q^{8} +(-1.97689e7 - 2.36411e7i) q^{10} -4.94923e7 q^{11} -2.82138e7 q^{13} +(-1.54825e8 - 1.85151e8i) q^{14} +(-1.00647e9 - 3.74085e8i) q^{16} -7.58870e8i q^{17} -3.85845e8i q^{19} +(-5.49047e9 - 9.87353e8i) q^{20} +(-6.87281e9 + 5.74712e9i) q^{22} +1.37125e10 q^{23} +1.53471e9 q^{25} +(-3.91794e9 + 3.27622e9i) q^{26} +(-4.29999e10 - 7.73269e9i) q^{28} +1.81751e11i q^{29} -2.17010e11i q^{31} +(-1.83204e11 + 6.49249e10i) q^{32} +(-8.81211e10 - 1.05381e11i) q^{34} -2.26986e11 q^{35} -7.64630e11 q^{37} +(-4.48049e10 - 5.35808e10i) q^{38} +(-8.77094e11 + 5.00451e11i) q^{40} -4.19418e11i q^{41} +1.71738e12i q^{43} +(-2.87038e11 + 1.59616e12i) q^{44} +(1.90421e12 - 1.59232e12i) q^{46} -4.19351e12 q^{47} +2.96987e12 q^{49} +(2.13119e11 - 1.78212e11i) q^{50} +(-1.63630e11 + 9.09913e11i) q^{52} -8.08314e11i q^{53} +8.42575e12i q^{55} +(-6.86916e12 + 3.91940e12i) q^{56} +(2.11052e13 + 2.52391e13i) q^{58} -9.64431e12 q^{59} +2.02603e13 q^{61} +(-2.51995e13 - 3.01353e13i) q^{62} +(-1.79017e13 + 3.02898e13i) q^{64} +4.80321e12i q^{65} +4.31350e13i q^{67} +(-2.44741e13 - 4.40118e12i) q^{68} +(-3.15207e13 + 2.63579e13i) q^{70} -1.15937e14 q^{71} +5.08663e12 q^{73} +(-1.06181e14 + 8.87898e13i) q^{74} +(-1.24438e13 - 2.23777e12i) q^{76} +6.59882e13i q^{77} -1.38810e13i q^{79} +(-6.36856e13 + 1.71345e14i) q^{80} +(-4.87034e13 - 5.82430e13i) q^{82} -3.65172e14 q^{83} -1.29193e14 q^{85} +(1.99425e14 + 2.38486e14i) q^{86} +(1.45488e14 + 2.54984e14i) q^{88} -1.78389e14i q^{89} +3.76175e13i q^{91} +(7.95280e13 - 4.42239e14i) q^{92} +(-5.82337e14 + 4.86957e14i) q^{94} -6.56876e13 q^{95} +6.64914e14 q^{97} +(4.12414e14 - 3.44865e14i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 11596 q^{4} - 37877668 q^{10} - 763573936 q^{13} + 1758880264 q^{16} + 24400430688 q^{22} + 46317352308 q^{25} - 39902121696 q^{28} - 852887075116 q^{34} + 1561242268376 q^{37} + 4554896912024 q^{40}+ \cdots + 19\!\cdots\!88 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 138.866 − 116.121i 0.767135 − 0.641486i
\(3\) 0 0
\(4\) 5799.65 − 32250.7i 0.176991 − 0.984212i
\(5\) − 170244.i − 0.974531i −0.873254 − 0.487266i \(-0.837994\pi\)
0.873254 − 0.487266i \(-0.162006\pi\)
\(6\) 0 0
\(7\) − 1.33330e6i − 0.611918i −0.952045 − 0.305959i \(-0.901023\pi\)
0.952045 − 0.305959i \(-0.0989771\pi\)
\(8\) −2.93962e6 − 5.15199e6i −0.495582 − 0.868561i
\(9\) 0 0
\(10\) −1.97689e7 − 2.36411e7i −0.625148 − 0.747597i
\(11\) −4.94923e7 −0.765760 −0.382880 − 0.923798i \(-0.625068\pi\)
−0.382880 + 0.923798i \(0.625068\pi\)
\(12\) 0 0
\(13\) −2.82138e7 −0.124706 −0.0623528 − 0.998054i \(-0.519860\pi\)
−0.0623528 + 0.998054i \(0.519860\pi\)
\(14\) −1.54825e8 − 1.85151e8i −0.392537 − 0.469424i
\(15\) 0 0
\(16\) −1.00647e9 − 3.74085e8i −0.937348 − 0.348394i
\(17\) − 7.58870e8i − 0.448539i −0.974527 − 0.224270i \(-0.928000\pi\)
0.974527 − 0.224270i \(-0.0719996\pi\)
\(18\) 0 0
\(19\) − 3.85845e8i − 0.0990287i −0.998773 − 0.0495143i \(-0.984233\pi\)
0.998773 − 0.0495143i \(-0.0157674\pi\)
\(20\) −5.49047e9 − 9.87353e8i −0.959146 − 0.172483i
\(21\) 0 0
\(22\) −6.87281e9 + 5.74712e9i −0.587441 + 0.491224i
\(23\) 1.37125e10 0.839768 0.419884 − 0.907578i \(-0.362071\pi\)
0.419884 + 0.907578i \(0.362071\pi\)
\(24\) 0 0
\(25\) 1.53471e9 0.0502892
\(26\) −3.91794e9 + 3.27622e9i −0.0956660 + 0.0799969i
\(27\) 0 0
\(28\) −4.29999e10 − 7.73269e9i −0.602257 − 0.108304i
\(29\) 1.81751e11i 1.95655i 0.207304 + 0.978277i \(0.433531\pi\)
−0.207304 + 0.978277i \(0.566469\pi\)
\(30\) 0 0
\(31\) − 2.17010e11i − 1.41666i −0.705880 − 0.708332i \(-0.749448\pi\)
0.705880 − 0.708332i \(-0.250552\pi\)
\(32\) −1.83204e11 + 6.49249e10i −0.942562 + 0.334031i
\(33\) 0 0
\(34\) −8.81211e10 − 1.05381e11i −0.287732 − 0.344090i
\(35\) −2.26986e11 −0.596333
\(36\) 0 0
\(37\) −7.64630e11 −1.32415 −0.662076 − 0.749436i \(-0.730325\pi\)
−0.662076 + 0.749436i \(0.730325\pi\)
\(38\) −4.48049e10 − 5.35808e10i −0.0635255 − 0.0759683i
\(39\) 0 0
\(40\) −8.77094e11 + 5.00451e11i −0.846440 + 0.482960i
\(41\) − 4.19418e11i − 0.336332i −0.985759 − 0.168166i \(-0.946215\pi\)
0.985759 − 0.168166i \(-0.0537845\pi\)
\(42\) 0 0
\(43\) 1.71738e12i 0.963504i 0.876308 + 0.481752i \(0.159999\pi\)
−0.876308 + 0.481752i \(0.840001\pi\)
\(44\) −2.87038e11 + 1.59616e12i −0.135533 + 0.753670i
\(45\) 0 0
\(46\) 1.90421e12 − 1.59232e12i 0.644215 − 0.538700i
\(47\) −4.19351e12 −1.20738 −0.603690 − 0.797219i \(-0.706304\pi\)
−0.603690 + 0.797219i \(0.706304\pi\)
\(48\) 0 0
\(49\) 2.96987e12 0.625556
\(50\) 2.13119e11 − 1.78212e11i 0.0385786 − 0.0322598i
\(51\) 0 0
\(52\) −1.63630e11 + 9.09913e11i −0.0220718 + 0.122737i
\(53\) − 8.08314e11i − 0.0945172i −0.998883 − 0.0472586i \(-0.984952\pi\)
0.998883 − 0.0472586i \(-0.0150485\pi\)
\(54\) 0 0
\(55\) 8.42575e12i 0.746257i
\(56\) −6.86916e12 + 3.91940e12i −0.531488 + 0.303256i
\(57\) 0 0
\(58\) 2.11052e13 + 2.52391e13i 1.25510 + 1.50094i
\(59\) −9.64431e12 −0.504523 −0.252262 − 0.967659i \(-0.581174\pi\)
−0.252262 + 0.967659i \(0.581174\pi\)
\(60\) 0 0
\(61\) 2.02603e13 0.825413 0.412707 − 0.910864i \(-0.364583\pi\)
0.412707 + 0.910864i \(0.364583\pi\)
\(62\) −2.51995e13 − 3.01353e13i −0.908770 − 1.08677i
\(63\) 0 0
\(64\) −1.79017e13 + 3.02898e13i −0.508796 + 0.860887i
\(65\) 4.80321e12i 0.121529i
\(66\) 0 0
\(67\) 4.31350e13i 0.869498i 0.900552 + 0.434749i \(0.143163\pi\)
−0.900552 + 0.434749i \(0.856837\pi\)
\(68\) −2.44741e13 − 4.40118e12i −0.441458 − 0.0793876i
\(69\) 0 0
\(70\) −3.15207e13 + 2.63579e13i −0.457468 + 0.382539i
\(71\) −1.15937e14 −1.51281 −0.756404 − 0.654105i \(-0.773045\pi\)
−0.756404 + 0.654105i \(0.773045\pi\)
\(72\) 0 0
\(73\) 5.08663e12 0.0538901 0.0269450 − 0.999637i \(-0.491422\pi\)
0.0269450 + 0.999637i \(0.491422\pi\)
\(74\) −1.06181e14 + 8.87898e13i −1.01580 + 0.849426i
\(75\) 0 0
\(76\) −1.24438e13 − 2.23777e12i −0.0974653 − 0.0175272i
\(77\) 6.59882e13i 0.468582i
\(78\) 0 0
\(79\) − 1.38810e13i − 0.0813236i −0.999173 − 0.0406618i \(-0.987053\pi\)
0.999173 − 0.0406618i \(-0.0129466\pi\)
\(80\) −6.36856e13 + 1.71345e14i −0.339521 + 0.913475i
\(81\) 0 0
\(82\) −4.87034e13 − 5.82430e13i −0.215752 − 0.258012i
\(83\) −3.65172e14 −1.47711 −0.738553 − 0.674195i \(-0.764491\pi\)
−0.738553 + 0.674195i \(0.764491\pi\)
\(84\) 0 0
\(85\) −1.29193e14 −0.437116
\(86\) 1.99425e14 + 2.38486e14i 0.618074 + 0.739137i
\(87\) 0 0
\(88\) 1.45488e14 + 2.54984e14i 0.379497 + 0.665109i
\(89\) − 1.78389e14i − 0.427507i −0.976888 − 0.213753i \(-0.931431\pi\)
0.976888 − 0.213753i \(-0.0685689\pi\)
\(90\) 0 0
\(91\) 3.76175e13i 0.0763096i
\(92\) 7.95280e13 − 4.42239e14i 0.148632 − 0.826510i
\(93\) 0 0
\(94\) −5.82337e14 + 4.86957e14i −0.926223 + 0.774518i
\(95\) −6.56876e13 −0.0965065
\(96\) 0 0
\(97\) 6.64914e14 0.835560 0.417780 − 0.908548i \(-0.362808\pi\)
0.417780 + 0.908548i \(0.362808\pi\)
\(98\) 4.12414e14 − 3.44865e14i 0.479886 − 0.401286i
\(99\) 0 0
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 36.16.b.b.35.23 yes 28
3.2 odd 2 inner 36.16.b.b.35.6 yes 28
4.3 odd 2 inner 36.16.b.b.35.5 ✓ 28
12.11 even 2 inner 36.16.b.b.35.24 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.16.b.b.35.5 ✓ 28 4.3 odd 2 inner
36.16.b.b.35.6 yes 28 3.2 odd 2 inner
36.16.b.b.35.23 yes 28 1.1 even 1 trivial
36.16.b.b.35.24 yes 28 12.11 even 2 inner