Properties

Label 5041.2.a.t.1.32
Level $5041$
Weight $2$
Character 5041.1
Self dual yes
Analytic conductor $40.253$
Analytic rank $0$
Dimension $60$
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] = [5041,2,Mod(1,5041)] 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("5041.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5041, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5041 = 71^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5041.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60,-2,-2,46,10,4,25] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(40.2525876589\)
Analytic rank: \(0\)
Dimension: \(60\)
Twist minimal: no (minimal twist has level 71)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.32
Character \(\chi\) \(=\) 5041.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.108509 q^{2} +1.83798 q^{3} -1.98823 q^{4} +2.24405 q^{5} -0.199438 q^{6} +4.53451 q^{7} +0.432759 q^{8} +0.378188 q^{9} -0.243500 q^{10} +2.81754 q^{11} -3.65433 q^{12} -2.63928 q^{13} -0.492036 q^{14} +4.12453 q^{15} +3.92949 q^{16} -6.56042 q^{17} -0.0410369 q^{18} +3.33774 q^{19} -4.46168 q^{20} +8.33436 q^{21} -0.305729 q^{22} +0.737950 q^{23} +0.795405 q^{24} +0.0357678 q^{25} +0.286386 q^{26} -4.81885 q^{27} -9.01563 q^{28} -1.77304 q^{29} -0.447550 q^{30} +6.58599 q^{31} -1.29191 q^{32} +5.17859 q^{33} +0.711866 q^{34} +10.1757 q^{35} -0.751922 q^{36} -2.17921 q^{37} -0.362176 q^{38} -4.85095 q^{39} +0.971134 q^{40} -0.927931 q^{41} -0.904355 q^{42} +10.0472 q^{43} -5.60190 q^{44} +0.848673 q^{45} -0.0800744 q^{46} +12.6873 q^{47} +7.22235 q^{48} +13.5618 q^{49} -0.00388114 q^{50} -12.0579 q^{51} +5.24748 q^{52} +11.8870 q^{53} +0.522890 q^{54} +6.32270 q^{55} +1.96235 q^{56} +6.13472 q^{57} +0.192391 q^{58} +3.17391 q^{59} -8.20050 q^{60} +2.64823 q^{61} -0.714641 q^{62} +1.71490 q^{63} -7.71880 q^{64} -5.92268 q^{65} -0.561925 q^{66} +0.539324 q^{67} +13.0436 q^{68} +1.35634 q^{69} -1.10416 q^{70} +0.163664 q^{72} +5.37137 q^{73} +0.236464 q^{74} +0.0657407 q^{75} -6.63618 q^{76} +12.7762 q^{77} +0.526373 q^{78} -9.27507 q^{79} +8.81799 q^{80} -9.99154 q^{81} +0.100689 q^{82} +1.31034 q^{83} -16.5706 q^{84} -14.7219 q^{85} -1.09021 q^{86} -3.25881 q^{87} +1.21932 q^{88} +1.65799 q^{89} -0.0920888 q^{90} -11.9678 q^{91} -1.46721 q^{92} +12.1049 q^{93} -1.37669 q^{94} +7.49006 q^{95} -2.37450 q^{96} -9.57675 q^{97} -1.47158 q^{98} +1.06556 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 2 q^{2} - 2 q^{3} + 46 q^{4} + 10 q^{5} + 4 q^{6} + 25 q^{7} + 9 q^{8} + 32 q^{9} + 18 q^{10} + 53 q^{11} + 44 q^{12} + 13 q^{13} + 49 q^{14} - 20 q^{15} + 26 q^{16} + 33 q^{17} - 48 q^{18} + 3 q^{19}+ \cdots + 135 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.108509 −0.0767276 −0.0383638 − 0.999264i \(-0.512215\pi\)
−0.0383638 + 0.999264i \(0.512215\pi\)
\(3\) 1.83798 1.06116 0.530580 − 0.847635i \(-0.321974\pi\)
0.530580 + 0.847635i \(0.321974\pi\)
\(4\) −1.98823 −0.994113
\(5\) 2.24405 1.00357 0.501785 − 0.864992i \(-0.332677\pi\)
0.501785 + 0.864992i \(0.332677\pi\)
\(6\) −0.199438 −0.0814204
\(7\) 4.53451 1.71388 0.856942 − 0.515413i \(-0.172361\pi\)
0.856942 + 0.515413i \(0.172361\pi\)
\(8\) 0.432759 0.153004
\(9\) 0.378188 0.126063
\(10\) −0.243500 −0.0770016
\(11\) 2.81754 0.849519 0.424760 − 0.905306i \(-0.360359\pi\)
0.424760 + 0.905306i \(0.360359\pi\)
\(12\) −3.65433 −1.05491
\(13\) −2.63928 −0.732004 −0.366002 − 0.930614i \(-0.619274\pi\)
−0.366002 + 0.930614i \(0.619274\pi\)
\(14\) −0.492036 −0.131502
\(15\) 4.12453 1.06495
\(16\) 3.92949 0.982373
\(17\) −6.56042 −1.59114 −0.795568 − 0.605865i \(-0.792827\pi\)
−0.795568 + 0.605865i \(0.792827\pi\)
\(18\) −0.0410369 −0.00967248
\(19\) 3.33774 0.765730 0.382865 − 0.923804i \(-0.374937\pi\)
0.382865 + 0.923804i \(0.374937\pi\)
\(20\) −4.46168 −0.997662
\(21\) 8.33436 1.81871
\(22\) −0.305729 −0.0651816
\(23\) 0.737950 0.153873 0.0769366 − 0.997036i \(-0.475486\pi\)
0.0769366 + 0.997036i \(0.475486\pi\)
\(24\) 0.795405 0.162361
\(25\) 0.0357678 0.00715357
\(26\) 0.286386 0.0561649
\(27\) −4.81885 −0.927388
\(28\) −9.01563 −1.70379
\(29\) −1.77304 −0.329244 −0.164622 − 0.986357i \(-0.552640\pi\)
−0.164622 + 0.986357i \(0.552640\pi\)
\(30\) −0.447550 −0.0817111
\(31\) 6.58599 1.18288 0.591439 − 0.806349i \(-0.298560\pi\)
0.591439 + 0.806349i \(0.298560\pi\)
\(32\) −1.29191 −0.228379
\(33\) 5.17859 0.901477
\(34\) 0.711866 0.122084
\(35\) 10.1757 1.72000
\(36\) −0.751922 −0.125320
\(37\) −2.17921 −0.358260 −0.179130 − 0.983825i \(-0.557328\pi\)
−0.179130 + 0.983825i \(0.557328\pi\)
\(38\) −0.362176 −0.0587527
\(39\) −4.85095 −0.776774
\(40\) 0.971134 0.153550
\(41\) −0.927931 −0.144919 −0.0724593 − 0.997371i \(-0.523085\pi\)
−0.0724593 + 0.997371i \(0.523085\pi\)
\(42\) −0.904355 −0.139545
\(43\) 10.0472 1.53218 0.766090 − 0.642734i \(-0.222200\pi\)
0.766090 + 0.642734i \(0.222200\pi\)
\(44\) −5.60190 −0.844518
\(45\) 0.848673 0.126513
\(46\) −0.0800744 −0.0118063
\(47\) 12.6873 1.85063 0.925316 − 0.379196i \(-0.123800\pi\)
0.925316 + 0.379196i \(0.123800\pi\)
\(48\) 7.22235 1.04246
\(49\) 13.5618 1.93740
\(50\) −0.00388114 −0.000548876 0
\(51\) −12.0579 −1.68845
\(52\) 5.24748 0.727695
\(53\) 11.8870 1.63280 0.816399 − 0.577488i \(-0.195967\pi\)
0.816399 + 0.577488i \(0.195967\pi\)
\(54\) 0.522890 0.0711563
\(55\) 6.32270 0.852553
\(56\) 1.96235 0.262230
\(57\) 6.13472 0.812563
\(58\) 0.192391 0.0252621
\(59\) 3.17391 0.413208 0.206604 − 0.978425i \(-0.433759\pi\)
0.206604 + 0.978425i \(0.433759\pi\)
\(60\) −8.20050 −1.05868
\(61\) 2.64823 0.339071 0.169535 − 0.985524i \(-0.445773\pi\)
0.169535 + 0.985524i \(0.445773\pi\)
\(62\) −0.714641 −0.0907595
\(63\) 1.71490 0.216057
\(64\) −7.71880 −0.964850
\(65\) −5.92268 −0.734618
\(66\) −0.561925 −0.0691682
\(67\) 0.539324 0.0658890 0.0329445 − 0.999457i \(-0.489512\pi\)
0.0329445 + 0.999457i \(0.489512\pi\)
\(68\) 13.0436 1.58177
\(69\) 1.35634 0.163284
\(70\) −1.10416 −0.131972
\(71\) 0 0
\(72\) 0.163664 0.0192880
\(73\) 5.37137 0.628672 0.314336 − 0.949312i \(-0.398218\pi\)
0.314336 + 0.949312i \(0.398218\pi\)
\(74\) 0.236464 0.0274884
\(75\) 0.0657407 0.00759108
\(76\) −6.63618 −0.761222
\(77\) 12.7762 1.45598
\(78\) 0.526373 0.0596000
\(79\) −9.27507 −1.04353 −0.521763 − 0.853090i \(-0.674726\pi\)
−0.521763 + 0.853090i \(0.674726\pi\)
\(80\) 8.81799 0.985881
\(81\) −9.99154 −1.11017
\(82\) 0.100689 0.0111193
\(83\) 1.31034 0.143828 0.0719142 − 0.997411i \(-0.477089\pi\)
0.0719142 + 0.997411i \(0.477089\pi\)
\(84\) −16.5706 −1.80800
\(85\) −14.7219 −1.59682
\(86\) −1.09021 −0.117560
\(87\) −3.25881 −0.349381
\(88\) 1.21932 0.129979
\(89\) 1.65799 0.175747 0.0878734 − 0.996132i \(-0.471993\pi\)
0.0878734 + 0.996132i \(0.471993\pi\)
\(90\) −0.0920888 −0.00970702
\(91\) −11.9678 −1.25457
\(92\) −1.46721 −0.152967
\(93\) 12.1049 1.25522
\(94\) −1.37669 −0.141995
\(95\) 7.49006 0.768464
\(96\) −2.37450 −0.242347
\(97\) −9.57675 −0.972371 −0.486186 − 0.873856i \(-0.661612\pi\)
−0.486186 + 0.873856i \(0.661612\pi\)
\(98\) −1.47158 −0.148652
\(99\) 1.06556 0.107093
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 5041.2.a.t.1.32 60
71.52 odd 70 71.2.g.a.6.4 ✓ 120
71.56 odd 70 71.2.g.a.12.4 yes 120
71.70 odd 2 5041.2.a.s.1.32 60
213.56 even 70 639.2.v.a.154.2 120
213.194 even 70 639.2.v.a.361.2 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
71.2.g.a.6.4 ✓ 120 71.52 odd 70
71.2.g.a.12.4 yes 120 71.56 odd 70
639.2.v.a.154.2 120 213.56 even 70
639.2.v.a.361.2 120 213.194 even 70
5041.2.a.s.1.32 60 71.70 odd 2
5041.2.a.t.1.32 60 1.1 even 1 trivial