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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,22,Mod(49,75)] 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("75.49"); 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, 1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-1777664] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(209.608008215\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.22.b.b.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1728.00i q^{2} -59049.0i q^{3} -888832. q^{4} -1.02037e8 q^{6} -5.38430e8i q^{7} -2.08798e9i q^{8} -3.48678e9 q^{9} -6.41130e10 q^{11} +5.24846e10i q^{12} -1.30980e11i q^{13} -9.30407e11 q^{14} -5.47204e12 q^{16} -8.24203e12i q^{17} +6.02516e12i q^{18} -1.34921e13 q^{19} -3.17937e13 q^{21} +1.10787e14i q^{22} -2.33185e14i q^{23} -1.23293e14 q^{24} -2.26334e14 q^{26} +2.05891e14i q^{27} +4.78574e14i q^{28} +2.02456e15 q^{29} -6.86919e15 q^{31} +5.07688e15i q^{32} +3.78581e15i q^{33} -1.42422e16 q^{34} +3.09917e15 q^{36} -3.44400e15i q^{37} +2.33144e16i q^{38} -7.73424e15 q^{39} -2.18424e16 q^{41} +5.49396e16i q^{42} -7.17928e16i q^{43} +5.69857e16 q^{44} -4.02943e17 q^{46} -2.83545e17i q^{47} +3.23118e17i q^{48} +2.68639e17 q^{49} -4.86684e17 q^{51} +1.16419e17i q^{52} -2.17229e18i q^{53} +3.55780e17 q^{54} -1.12423e18 q^{56} +7.96695e17i q^{57} -3.49844e18i q^{58} -1.53483e18 q^{59} +4.31159e18 q^{61} +1.18700e19i q^{62} +1.87739e18i q^{63} -2.70285e18 q^{64} +6.54188e18 q^{66} -9.24391e18i q^{67} +7.32578e18i q^{68} -1.37693e19 q^{69} -2.03874e19 q^{71} +7.28033e18i q^{72} +1.66178e19i q^{73} -5.95123e18 q^{74} +1.19922e19 q^{76} +3.45204e19i q^{77} +1.33648e19i q^{78} -6.79403e19 q^{79} +1.21577e19 q^{81} +3.77437e19i q^{82} +3.95037e19i q^{83} +2.82593e19 q^{84} -1.24058e20 q^{86} -1.19548e20i q^{87} +1.33867e20i q^{88} -4.16117e19 q^{89} -7.05236e19 q^{91} +2.07262e20i q^{92} +4.05619e20i q^{93} -4.89965e20 q^{94} +2.99785e20 q^{96} -5.71815e19i q^{97} -4.64209e20i q^{98} +2.23548e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1777664 q^{4} - 204073344 q^{6} - 6973568802 q^{9} - 128226080376 q^{11} - 1860813416448 q^{14} - 10944079986688 q^{16} - 26984203506040 q^{19} - 63587483465184 q^{21} - 246585903022080 q^{24} - 452667253199616 q^{26}+ \cdots + 44\!\cdots\!76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1728.00i − 1.19324i −0.802523 − 0.596621i \(-0.796509\pi\)
0.802523 − 0.596621i \(-0.203491\pi\)
\(3\) − 59049.0i − 0.577350i
\(4\) −888832. −0.423828
\(5\) 0 0
\(6\) −1.02037e8 −0.688919
\(7\) − 5.38430e8i − 0.720443i −0.932867 − 0.360222i \(-0.882701\pi\)
0.932867 − 0.360222i \(-0.117299\pi\)
\(8\) − 2.08798e9i − 0.687513i
\(9\) −3.48678e9 −0.333333
\(10\) 0 0
\(11\) −6.41130e10 −0.745286 −0.372643 − 0.927975i \(-0.621548\pi\)
−0.372643 + 0.927975i \(0.621548\pi\)
\(12\) 5.24846e10i 0.244697i
\(13\) − 1.30980e11i − 0.263512i −0.991282 − 0.131756i \(-0.957938\pi\)
0.991282 − 0.131756i \(-0.0420615\pi\)
\(14\) −9.30407e11 −0.859663
\(15\) 0 0
\(16\) −5.47204e12 −1.24420
\(17\) − 8.24203e12i − 0.991563i −0.868447 − 0.495782i \(-0.834882\pi\)
0.868447 − 0.495782i \(-0.165118\pi\)
\(18\) 6.02516e12i 0.397748i
\(19\) −1.34921e13 −0.504855 −0.252428 − 0.967616i \(-0.581229\pi\)
−0.252428 + 0.967616i \(0.581229\pi\)
\(20\) 0 0
\(21\) −3.17937e13 −0.415948
\(22\) 1.10787e14i 0.889308i
\(23\) − 2.33185e14i − 1.17370i −0.809695 − 0.586851i \(-0.800367\pi\)
0.809695 − 0.586851i \(-0.199633\pi\)
\(24\) −1.23293e14 −0.396936
\(25\) 0 0
\(26\) −2.26334e14 −0.314434
\(27\) 2.05891e14i 0.192450i
\(28\) 4.78574e14i 0.305344i
\(29\) 2.02456e15 0.893618 0.446809 − 0.894629i \(-0.352560\pi\)
0.446809 + 0.894629i \(0.352560\pi\)
\(30\) 0 0
\(31\) −6.86919e15 −1.50525 −0.752624 − 0.658451i \(-0.771212\pi\)
−0.752624 + 0.658451i \(0.771212\pi\)
\(32\) 5.07688e15i 0.797117i
\(33\) 3.78581e15i 0.430291i
\(34\) −1.42422e16 −1.18318
\(35\) 0 0
\(36\) 3.09917e15 0.141276
\(37\) − 3.44400e15i − 0.117746i −0.998265 − 0.0588728i \(-0.981249\pi\)
0.998265 − 0.0588728i \(-0.0187506\pi\)
\(38\) 2.33144e16i 0.602415i
\(39\) −7.73424e15 −0.152139
\(40\) 0 0
\(41\) −2.18424e16 −0.254138 −0.127069 − 0.991894i \(-0.540557\pi\)
−0.127069 + 0.991894i \(0.540557\pi\)
\(42\) 5.49396e16i 0.496327i
\(43\) − 7.17928e16i − 0.506597i −0.967388 − 0.253298i \(-0.918485\pi\)
0.967388 − 0.253298i \(-0.0815154\pi\)
\(44\) 5.69857e16 0.315873
\(45\) 0 0
\(46\) −4.02943e17 −1.40051
\(47\) − 2.83545e17i − 0.786310i −0.919472 − 0.393155i \(-0.871383\pi\)
0.919472 − 0.393155i \(-0.128617\pi\)
\(48\) 3.23118e17i 0.718338i
\(49\) 2.68639e17 0.480962
\(50\) 0 0
\(51\) −4.86684e17 −0.572479
\(52\) 1.16419e17i 0.111684i
\(53\) − 2.17229e18i − 1.70616i −0.521779 − 0.853081i \(-0.674731\pi\)
0.521779 − 0.853081i \(-0.325269\pi\)
\(54\) 3.55780e17 0.229640
\(55\) 0 0
\(56\) −1.12423e18 −0.495314
\(57\) 7.96695e17i 0.291478i
\(58\) − 3.49844e18i − 1.06630i
\(59\) −1.53483e18 −0.390944 −0.195472 − 0.980709i \(-0.562624\pi\)
−0.195472 + 0.980709i \(0.562624\pi\)
\(60\) 0 0
\(61\) 4.31159e18 0.773881 0.386940 − 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(62\) 1.18700e19i 1.79613i
\(63\) 1.87739e18i 0.240148i
\(64\) −2.70285e18 −0.293044
\(65\) 0 0
\(66\) 6.54188e18 0.513442
\(67\) − 9.24391e18i − 0.619541i −0.950811 − 0.309771i \(-0.899748\pi\)
0.950811 − 0.309771i \(-0.100252\pi\)
\(68\) 7.32578e18i 0.420252i
\(69\) −1.37693e19 −0.677637
\(70\) 0 0
\(71\) −2.03874e19 −0.743273 −0.371636 − 0.928378i \(-0.621203\pi\)
−0.371636 + 0.928378i \(0.621203\pi\)
\(72\) 7.28033e18i 0.229171i
\(73\) 1.66178e19i 0.452566i 0.974062 + 0.226283i \(0.0726575\pi\)
−0.974062 + 0.226283i \(0.927343\pi\)
\(74\) −5.95123e18 −0.140499
\(75\) 0 0
\(76\) 1.19922e19 0.213972
\(77\) 3.45204e19i 0.536936i
\(78\) 1.33648e19i 0.181538i
\(79\) −6.79403e19 −0.807315 −0.403658 − 0.914910i \(-0.632261\pi\)
−0.403658 + 0.914910i \(0.632261\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 3.77437e19i 0.303249i
\(83\) 3.95037e19i 0.279459i 0.990190 + 0.139730i \(0.0446233\pi\)
−0.990190 + 0.139730i \(0.955377\pi\)
\(84\) 2.82593e19 0.176290
\(85\) 0 0
\(86\) −1.24058e20 −0.604493
\(87\) − 1.19548e20i − 0.515931i
\(88\) 1.33867e20i 0.512394i
\(89\) −4.16117e19 −0.141456 −0.0707278 − 0.997496i \(-0.522532\pi\)
−0.0707278 + 0.997496i \(0.522532\pi\)
\(90\) 0 0
\(91\) −7.05236e19 −0.189845
\(92\) 2.07262e20i 0.497448i
\(93\) 4.05619e20i 0.869055i
\(94\) −4.89965e20 −0.938259
\(95\) 0 0
\(96\) 2.99785e20 0.460216
\(97\) − 5.71815e19i − 0.0787322i −0.999225 − 0.0393661i \(-0.987466\pi\)
0.999225 − 0.0393661i \(-0.0125339\pi\)
\(98\) − 4.64209e20i − 0.573904i
\(99\) 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.b.b.49.1 2
5.2 odd 4 3.22.a.b.1.1 ✓ 1
5.3 odd 4 75.22.a.a.1.1 1
5.4 even 2 inner 75.22.b.b.49.2 2
15.2 even 4 9.22.a.a.1.1 1
20.7 even 4 48.22.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.b.1.1 ✓ 1 5.2 odd 4
9.22.a.a.1.1 1 15.2 even 4
48.22.a.d.1.1 1 20.7 even 4
75.22.a.a.1.1 1 5.3 odd 4
75.22.b.b.49.1 2 1.1 even 1 trivial
75.22.b.b.49.2 2 5.4 even 2 inner