Properties

Label 75.22.a.a.1.1
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [75,22,Mod(1,75)] 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("75.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1728] 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: yes
Analytic conductor: \(209.608008215\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 75.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1728.00 q^{2} +59049.0 q^{3} +888832. q^{4} -1.02037e8 q^{6} -5.38430e8 q^{7} +2.08798e9 q^{8} +3.48678e9 q^{9} -6.41130e10 q^{11} +5.24846e10 q^{12} +1.30980e11 q^{13} +9.30407e11 q^{14} -5.47204e12 q^{16} -8.24203e12 q^{17} -6.02516e12 q^{18} +1.34921e13 q^{19} -3.17937e13 q^{21} +1.10787e14 q^{22} +2.33185e14 q^{23} +1.23293e14 q^{24} -2.26334e14 q^{26} +2.05891e14 q^{27} -4.78574e14 q^{28} -2.02456e15 q^{29} -6.86919e15 q^{31} +5.07688e15 q^{32} -3.78581e15 q^{33} +1.42422e16 q^{34} +3.09917e15 q^{36} -3.44400e15 q^{37} -2.33144e16 q^{38} +7.73424e15 q^{39} -2.18424e16 q^{41} +5.49396e16 q^{42} +7.17928e16 q^{43} -5.69857e16 q^{44} -4.02943e17 q^{46} -2.83545e17 q^{47} -3.23118e17 q^{48} -2.68639e17 q^{49} -4.86684e17 q^{51} +1.16419e17 q^{52} +2.17229e18 q^{53} -3.55780e17 q^{54} -1.12423e18 q^{56} +7.96695e17 q^{57} +3.49844e18 q^{58} +1.53483e18 q^{59} +4.31159e18 q^{61} +1.18700e19 q^{62} -1.87739e18 q^{63} +2.70285e18 q^{64} +6.54188e18 q^{66} -9.24391e18 q^{67} -7.32578e18 q^{68} +1.37693e19 q^{69} -2.03874e19 q^{71} +7.28033e18 q^{72} -1.66178e19 q^{73} +5.95123e18 q^{74} +1.19922e19 q^{76} +3.45204e19 q^{77} -1.33648e19 q^{78} +6.79403e19 q^{79} +1.21577e19 q^{81} +3.77437e19 q^{82} -3.95037e19 q^{83} -2.82593e19 q^{84} -1.24058e20 q^{86} -1.19548e20 q^{87} -1.33867e20 q^{88} +4.16117e19 q^{89} -7.05236e19 q^{91} +2.07262e20 q^{92} -4.05619e20 q^{93} +4.89965e20 q^{94} +2.99785e20 q^{96} -5.71815e19 q^{97} +4.64209e20 q^{98} -2.23548e20 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1728.00 −1.19324 −0.596621 − 0.802523i \(-0.703491\pi\)
−0.596621 + 0.802523i \(0.703491\pi\)
\(3\) 59049.0 0.577350
\(4\) 888832. 0.423828
\(5\) 0 0
\(6\) −1.02037e8 −0.688919
\(7\) −5.38430e8 −0.720443 −0.360222 − 0.932867i \(-0.617299\pi\)
−0.360222 + 0.932867i \(0.617299\pi\)
\(8\) 2.08798e9 0.687513
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) −6.41130e10 −0.745286 −0.372643 − 0.927975i \(-0.621548\pi\)
−0.372643 + 0.927975i \(0.621548\pi\)
\(12\) 5.24846e10 0.244697
\(13\) 1.30980e11 0.263512 0.131756 − 0.991282i \(-0.457938\pi\)
0.131756 + 0.991282i \(0.457938\pi\)
\(14\) 9.30407e11 0.859663
\(15\) 0 0
\(16\) −5.47204e12 −1.24420
\(17\) −8.24203e12 −0.991563 −0.495782 − 0.868447i \(-0.665118\pi\)
−0.495782 + 0.868447i \(0.665118\pi\)
\(18\) −6.02516e12 −0.397748
\(19\) 1.34921e13 0.504855 0.252428 − 0.967616i \(-0.418771\pi\)
0.252428 + 0.967616i \(0.418771\pi\)
\(20\) 0 0
\(21\) −3.17937e13 −0.415948
\(22\) 1.10787e14 0.889308
\(23\) 2.33185e14 1.17370 0.586851 − 0.809695i \(-0.300367\pi\)
0.586851 + 0.809695i \(0.300367\pi\)
\(24\) 1.23293e14 0.396936
\(25\) 0 0
\(26\) −2.26334e14 −0.314434
\(27\) 2.05891e14 0.192450
\(28\) −4.78574e14 −0.305344
\(29\) −2.02456e15 −0.893618 −0.446809 − 0.894629i \(-0.647440\pi\)
−0.446809 + 0.894629i \(0.647440\pi\)
\(30\) 0 0
\(31\) −6.86919e15 −1.50525 −0.752624 − 0.658451i \(-0.771212\pi\)
−0.752624 + 0.658451i \(0.771212\pi\)
\(32\) 5.07688e15 0.797117
\(33\) −3.78581e15 −0.430291
\(34\) 1.42422e16 1.18318
\(35\) 0 0
\(36\) 3.09917e15 0.141276
\(37\) −3.44400e15 −0.117746 −0.0588728 − 0.998265i \(-0.518751\pi\)
−0.0588728 + 0.998265i \(0.518751\pi\)
\(38\) −2.33144e16 −0.602415
\(39\) 7.73424e15 0.152139
\(40\) 0 0
\(41\) −2.18424e16 −0.254138 −0.127069 − 0.991894i \(-0.540557\pi\)
−0.127069 + 0.991894i \(0.540557\pi\)
\(42\) 5.49396e16 0.496327
\(43\) 7.17928e16 0.506597 0.253298 − 0.967388i \(-0.418485\pi\)
0.253298 + 0.967388i \(0.418485\pi\)
\(44\) −5.69857e16 −0.315873
\(45\) 0 0
\(46\) −4.02943e17 −1.40051
\(47\) −2.83545e17 −0.786310 −0.393155 − 0.919472i \(-0.628617\pi\)
−0.393155 + 0.919472i \(0.628617\pi\)
\(48\) −3.23118e17 −0.718338
\(49\) −2.68639e17 −0.480962
\(50\) 0 0
\(51\) −4.86684e17 −0.572479
\(52\) 1.16419e17 0.111684
\(53\) 2.17229e18 1.70616 0.853081 − 0.521779i \(-0.174731\pi\)
0.853081 + 0.521779i \(0.174731\pi\)
\(54\) −3.55780e17 −0.229640
\(55\) 0 0
\(56\) −1.12423e18 −0.495314
\(57\) 7.96695e17 0.291478
\(58\) 3.49844e18 1.06630
\(59\) 1.53483e18 0.390944 0.195472 − 0.980709i \(-0.437376\pi\)
0.195472 + 0.980709i \(0.437376\pi\)
\(60\) 0 0
\(61\) 4.31159e18 0.773881 0.386940 − 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(62\) 1.18700e19 1.79613
\(63\) −1.87739e18 −0.240148
\(64\) 2.70285e18 0.293044
\(65\) 0 0
\(66\) 6.54188e18 0.513442
\(67\) −9.24391e18 −0.619541 −0.309771 − 0.950811i \(-0.600252\pi\)
−0.309771 + 0.950811i \(0.600252\pi\)
\(68\) −7.32578e18 −0.420252
\(69\) 1.37693e19 0.677637
\(70\) 0 0
\(71\) −2.03874e19 −0.743273 −0.371636 − 0.928378i \(-0.621203\pi\)
−0.371636 + 0.928378i \(0.621203\pi\)
\(72\) 7.28033e18 0.229171
\(73\) −1.66178e19 −0.452566 −0.226283 − 0.974062i \(-0.572657\pi\)
−0.226283 + 0.974062i \(0.572657\pi\)
\(74\) 5.95123e18 0.140499
\(75\) 0 0
\(76\) 1.19922e19 0.213972
\(77\) 3.45204e19 0.536936
\(78\) −1.33648e19 −0.181538
\(79\) 6.79403e19 0.807315 0.403658 − 0.914910i \(-0.367739\pi\)
0.403658 + 0.914910i \(0.367739\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 3.77437e19 0.303249
\(83\) −3.95037e19 −0.279459 −0.139730 − 0.990190i \(-0.544623\pi\)
−0.139730 + 0.990190i \(0.544623\pi\)
\(84\) −2.82593e19 −0.176290
\(85\) 0 0
\(86\) −1.24058e20 −0.604493
\(87\) −1.19548e20 −0.515931
\(88\) −1.33867e20 −0.512394
\(89\) 4.16117e19 0.141456 0.0707278 − 0.997496i \(-0.477468\pi\)
0.0707278 + 0.997496i \(0.477468\pi\)
\(90\) 0 0
\(91\) −7.05236e19 −0.189845
\(92\) 2.07262e20 0.497448
\(93\) −4.05619e20 −0.869055
\(94\) 4.89965e20 0.938259
\(95\) 0 0
\(96\) 2.99785e20 0.460216
\(97\) −5.71815e19 −0.0787322 −0.0393661 − 0.999225i \(-0.512534\pi\)
−0.0393661 + 0.999225i \(0.512534\pi\)
\(98\) 4.64209e20 0.573904
\(99\) −2.23548e20 −0.248429
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 75.22.a.a.1.1 1
5.2 odd 4 75.22.b.b.49.1 2
5.3 odd 4 75.22.b.b.49.2 2
5.4 even 2 3.22.a.b.1.1 ✓ 1
15.14 odd 2 9.22.a.a.1.1 1
20.19 odd 2 48.22.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.b.1.1 ✓ 1 5.4 even 2
9.22.a.a.1.1 1 15.14 odd 2
48.22.a.d.1.1 1 20.19 odd 2
75.22.a.a.1.1 1 1.1 even 1 trivial
75.22.b.b.49.1 2 5.2 odd 4
75.22.b.b.49.2 2 5.3 odd 4