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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,7,Mod(53,72)] 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("72.53"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 72.h (of order \(2\), degree \(1\), 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: \(16.5638940206\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 53.2
Character \(\chi\) \(=\) 72.53
Dual form 72.7.h.a.53.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-7.66719 + 2.28345i) q^{2} +(53.5717 - 35.0153i) q^{4} +149.398 q^{5} +498.755 q^{7} +(-330.790 + 390.797i) q^{8} +(-1145.46 + 341.142i) q^{10} -394.875 q^{11} -3962.14i q^{13} +(-3824.05 + 1138.88i) q^{14} +(1643.86 - 3751.66i) q^{16} +6929.82i q^{17} +1077.01i q^{19} +(8003.51 - 5231.21i) q^{20} +(3027.59 - 901.677i) q^{22} -10605.7i q^{23} +6694.74 q^{25} +(9047.34 + 30378.5i) q^{26} +(26719.2 - 17464.0i) q^{28} +13037.3 q^{29} +53775.2 q^{31} +(-4037.12 + 32518.4i) q^{32} +(-15823.9 - 53132.3i) q^{34} +74512.9 q^{35} +40996.0i q^{37} +(-2459.29 - 8257.62i) q^{38} +(-49419.3 + 58384.3i) q^{40} -71003.1i q^{41} -70391.7i q^{43} +(-21154.2 + 13826.7i) q^{44} +(24217.5 + 81315.7i) q^{46} -13021.7i q^{47} +131107. q^{49} +(-51329.8 + 15287.1i) q^{50} +(-138735. - 212259. i) q^{52} +27017.7 q^{53} -58993.6 q^{55} +(-164983. + 194912. i) q^{56} +(-99959.2 + 29769.9i) q^{58} -40464.9 q^{59} +277921. i q^{61} +(-412305. + 122793. i) q^{62} +(-43300.5 - 258543. i) q^{64} -591936. i q^{65} +339817. i q^{67} +(242650. + 371243. i) q^{68} +(-571305. + 170146. i) q^{70} -456044. i q^{71} -303469. q^{73} +(-93612.2 - 314325. i) q^{74} +(37711.7 + 57697.1i) q^{76} -196946. q^{77} +471240. q^{79} +(245590. - 560490. i) q^{80} +(162132. + 544394. i) q^{82} +1.09854e6 q^{83} +1.03530e6i q^{85} +(160736. + 539707. i) q^{86} +(130621. - 154316. i) q^{88} +853707. i q^{89} -1.97614e6i q^{91} +(-371360. - 568164. i) q^{92} +(29734.3 + 99839.6i) q^{94} +160903. i q^{95} -35614.1 q^{97} +(-1.00522e6 + 299376. i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 132 q^{4} + 588 q^{10} + 4536 q^{16} + 11784 q^{22} + 75000 q^{25} + 25896 q^{28} + 122496 q^{31} + 74724 q^{34} - 56880 q^{40} - 74088 q^{46} + 197448 q^{49} - 149064 q^{52} - 232704 q^{55} - 196140 q^{58}+ \cdots + 2960832 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.66719 + 2.28345i −0.958399 + 0.285431i
\(3\) 0 0
\(4\) 53.5717 − 35.0153i 0.837059 − 0.547113i
\(5\) 149.398 1.19518 0.597592 − 0.801801i \(-0.296124\pi\)
0.597592 + 0.801801i \(0.296124\pi\)
\(6\) 0 0
\(7\) 498.755 1.45409 0.727047 − 0.686587i \(-0.240892\pi\)
0.727047 + 0.686587i \(0.240892\pi\)
\(8\) −330.790 + 390.797i −0.646073 + 0.763275i
\(9\) 0 0
\(10\) −1145.46 + 341.142i −1.14546 + 0.341142i
\(11\) −394.875 −0.296676 −0.148338 − 0.988937i \(-0.547392\pi\)
−0.148338 + 0.988937i \(0.547392\pi\)
\(12\) 0 0
\(13\) − 3962.14i − 1.80343i −0.432327 − 0.901717i \(-0.642308\pi\)
0.432327 − 0.901717i \(-0.357692\pi\)
\(14\) −3824.05 + 1138.88i −1.39360 + 0.415043i
\(15\) 0 0
\(16\) 1643.86 − 3751.66i 0.401334 − 0.915932i
\(17\) 6929.82i 1.41051i 0.708955 + 0.705254i \(0.249167\pi\)
−0.708955 + 0.705254i \(0.750833\pi\)
\(18\) 0 0
\(19\) 1077.01i 0.157021i 0.996913 + 0.0785105i \(0.0250164\pi\)
−0.996913 + 0.0785105i \(0.974984\pi\)
\(20\) 8003.51 − 5231.21i 1.00044 − 0.653901i
\(21\) 0 0
\(22\) 3027.59 − 901.677i 0.284334 − 0.0846804i
\(23\) − 10605.7i − 0.871675i −0.900025 − 0.435837i \(-0.856452\pi\)
0.900025 − 0.435837i \(-0.143548\pi\)
\(24\) 0 0
\(25\) 6694.74 0.428463
\(26\) 9047.34 + 30378.5i 0.514755 + 1.72841i
\(27\) 0 0
\(28\) 26719.2 − 17464.0i 1.21716 − 0.795555i
\(29\) 13037.3 0.534555 0.267277 − 0.963620i \(-0.413876\pi\)
0.267277 + 0.963620i \(0.413876\pi\)
\(30\) 0 0
\(31\) 53775.2 1.80508 0.902541 − 0.430604i \(-0.141699\pi\)
0.902541 + 0.430604i \(0.141699\pi\)
\(32\) −4037.12 + 32518.4i −0.123203 + 0.992381i
\(33\) 0 0
\(34\) −15823.9 − 53132.3i −0.402602 − 1.35183i
\(35\) 74512.9 1.73791
\(36\) 0 0
\(37\) 40996.0i 0.809351i 0.914460 + 0.404675i \(0.132615\pi\)
−0.914460 + 0.404675i \(0.867385\pi\)
\(38\) −2459.29 − 8257.62i −0.0448186 − 0.150489i
\(39\) 0 0
\(40\) −49419.3 + 58384.3i −0.772176 + 0.912254i
\(41\) − 71003.1i − 1.03021i −0.857127 − 0.515105i \(-0.827753\pi\)
0.857127 − 0.515105i \(-0.172247\pi\)
\(42\) 0 0
\(43\) − 70391.7i − 0.885352i −0.896682 − 0.442676i \(-0.854029\pi\)
0.896682 − 0.442676i \(-0.145971\pi\)
\(44\) −21154.2 + 13826.7i −0.248335 + 0.162315i
\(45\) 0 0
\(46\) 24217.5 + 81315.7i 0.248803 + 0.835412i
\(47\) − 13021.7i − 0.125422i −0.998032 − 0.0627109i \(-0.980025\pi\)
0.998032 − 0.0627109i \(-0.0199746\pi\)
\(48\) 0 0
\(49\) 131107. 1.11439
\(50\) −51329.8 + 15287.1i −0.410639 + 0.122297i
\(51\) 0 0
\(52\) −138735. − 212259.i −0.986682 − 1.50958i
\(53\) 27017.7 0.181477 0.0907383 − 0.995875i \(-0.471077\pi\)
0.0907383 + 0.995875i \(0.471077\pi\)
\(54\) 0 0
\(55\) −58993.6 −0.354582
\(56\) −164983. + 194912.i −0.939452 + 1.10987i
\(57\) 0 0
\(58\) −99959.2 + 29769.9i −0.512317 + 0.152578i
\(59\) −40464.9 −0.197026 −0.0985128 − 0.995136i \(-0.531409\pi\)
−0.0985128 + 0.995136i \(0.531409\pi\)
\(60\) 0 0
\(61\) 277921.i 1.22442i 0.790694 + 0.612212i \(0.209720\pi\)
−0.790694 + 0.612212i \(0.790280\pi\)
\(62\) −412305. + 122793.i −1.72999 + 0.515226i
\(63\) 0 0
\(64\) −43300.5 − 258543.i −0.165178 − 0.986264i
\(65\) − 591936.i − 2.15543i
\(66\) 0 0
\(67\) 339817.i 1.12985i 0.825143 + 0.564924i \(0.191095\pi\)
−0.825143 + 0.564924i \(0.808905\pi\)
\(68\) 242650. + 371243.i 0.771707 + 1.18068i
\(69\) 0 0
\(70\) −571305. + 170146.i −1.66561 + 0.496053i
\(71\) − 456044.i − 1.27418i −0.770788 − 0.637091i \(-0.780137\pi\)
0.770788 − 0.637091i \(-0.219863\pi\)
\(72\) 0 0
\(73\) −303469. −0.780091 −0.390045 − 0.920796i \(-0.627541\pi\)
−0.390045 + 0.920796i \(0.627541\pi\)
\(74\) −93612.2 − 314325.i −0.231014 − 0.775681i
\(75\) 0 0
\(76\) 37711.7 + 57697.1i 0.0859083 + 0.131436i
\(77\) −196946. −0.431395
\(78\) 0 0
\(79\) 471240. 0.955786 0.477893 − 0.878418i \(-0.341401\pi\)
0.477893 + 0.878418i \(0.341401\pi\)
\(80\) 245590. − 560490.i 0.479668 − 1.09471i
\(81\) 0 0
\(82\) 162132. + 544394.i 0.294053 + 0.987352i
\(83\) 1.09854e6 1.92124 0.960620 − 0.277866i \(-0.0896272\pi\)
0.960620 + 0.277866i \(0.0896272\pi\)
\(84\) 0 0
\(85\) 1.03530e6i 1.68581i
\(86\) 160736. + 539707.i 0.252707 + 0.848521i
\(87\) 0 0
\(88\) 130621. − 154316.i 0.191674 − 0.226445i
\(89\) 853707.i 1.21099i 0.795851 + 0.605493i \(0.207024\pi\)
−0.795851 + 0.605493i \(0.792976\pi\)
\(90\) 0 0
\(91\) − 1.97614e6i − 2.62236i
\(92\) −371360. − 568164.i −0.476905 − 0.729643i
\(93\) 0 0
\(94\) 29734.3 + 99839.6i 0.0357992 + 0.120204i
\(95\) 160903.i 0.187669i
\(96\) 0 0
\(97\) −35614.1 −0.0390218 −0.0195109 − 0.999810i \(-0.506211\pi\)
−0.0195109 + 0.999810i \(0.506211\pi\)
\(98\) −1.00522e6 + 299376.i −1.06803 + 0.318082i
\(99\) 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 72.7.h.a.53.2 yes 24
3.2 odd 2 inner 72.7.h.a.53.23 yes 24
4.3 odd 2 288.7.h.a.17.20 24
8.3 odd 2 288.7.h.a.17.5 24
8.5 even 2 inner 72.7.h.a.53.24 yes 24
12.11 even 2 288.7.h.a.17.6 24
24.5 odd 2 inner 72.7.h.a.53.1 ✓ 24
24.11 even 2 288.7.h.a.17.19 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.7.h.a.53.1 ✓ 24 24.5 odd 2 inner
72.7.h.a.53.2 yes 24 1.1 even 1 trivial
72.7.h.a.53.23 yes 24 3.2 odd 2 inner
72.7.h.a.53.24 yes 24 8.5 even 2 inner
288.7.h.a.17.5 24 8.3 odd 2
288.7.h.a.17.6 24 12.11 even 2
288.7.h.a.17.19 24 24.11 even 2
288.7.h.a.17.20 24 4.3 odd 2