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Properties

Label 575.2.a.a.1.1
Level $575$
Weight $2$
Character 575.1
Self dual yes
Analytic conductor $4.591$
Analytic rank $1$
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] = [575,2,Mod(1,575)] 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("575.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.59139811622\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 575.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -2.00000 q^{3} +2.00000 q^{4} +4.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} +5.00000 q^{17} -2.00000 q^{18} +8.00000 q^{19} +2.00000 q^{21} +1.00000 q^{23} +4.00000 q^{26} +4.00000 q^{27} -2.00000 q^{28} -5.00000 q^{29} -5.00000 q^{31} +8.00000 q^{32} -10.0000 q^{34} +2.00000 q^{36} -7.00000 q^{37} -16.0000 q^{38} +4.00000 q^{39} -7.00000 q^{41} -4.00000 q^{42} -4.00000 q^{43} -2.00000 q^{46} +2.00000 q^{47} +8.00000 q^{48} -6.00000 q^{49} -10.0000 q^{51} -4.00000 q^{52} +1.00000 q^{53} -8.00000 q^{54} -16.0000 q^{57} +10.0000 q^{58} +3.00000 q^{59} -6.00000 q^{61} +10.0000 q^{62} -1.00000 q^{63} -8.00000 q^{64} -13.0000 q^{67} +10.0000 q^{68} -2.00000 q^{69} +13.0000 q^{71} -8.00000 q^{73} +14.0000 q^{74} +16.0000 q^{76} -8.00000 q^{78} -14.0000 q^{79} -11.0000 q^{81} +14.0000 q^{82} +3.00000 q^{83} +4.00000 q^{84} +8.00000 q^{86} +10.0000 q^{87} -14.0000 q^{89} +2.00000 q^{91} +2.00000 q^{92} +10.0000 q^{93} -4.00000 q^{94} -16.0000 q^{96} -14.0000 q^{97} +12.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 4.00000 1.63299
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −4.00000 −1.15470
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) −2.00000 −0.471405
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 4.00000 0.769800
\(28\) −2.00000 −0.377964
\(29\) −5.00000 −0.928477 −0.464238 − 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 − 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) −10.0000 −1.71499
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −16.0000 −2.59554
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 − 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) −4.00000 −0.617213
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 8.00000 1.15470
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −10.0000 −1.40028
\(52\) −4.00000 −0.554700
\(53\) 1.00000 0.137361 0.0686803 − 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) −8.00000 −1.08866
\(55\) 0 0
\(56\) 0 0
\(57\) −16.0000 −2.11925
\(58\) 10.0000 1.31306
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 10.0000 1.27000
\(63\) −1.00000 −0.125988
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 − 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 10.0000 1.21268
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) 13.0000 1.54282 0.771408 − 0.636341i \(-0.219553\pi\)
0.771408 + 0.636341i \(0.219553\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 − 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 14.0000 1.62747
\(75\) 0 0
\(76\) 16.0000 1.83533
\(77\) 0 0
\(78\) −8.00000 −0.905822
\(79\) −14.0000 −1.57512 −0.787562 − 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 14.0000 1.54604
\(83\) 3.00000 0.329293 0.164646 − 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 2.00000 0.208514
\(93\) 10.0000 1.03695
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) −16.0000 −1.63299
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 12.0000 1.21218
\(99\) 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 575.2.a.a.1.1 1
3.2 odd 2 5175.2.a.z.1.1 1
4.3 odd 2 9200.2.a.bg.1.1 1
5.2 odd 4 115.2.b.a.24.1 ✓ 2
5.3 odd 4 115.2.b.a.24.2 yes 2
5.4 even 2 575.2.a.e.1.1 1
15.2 even 4 1035.2.b.a.829.2 2
15.8 even 4 1035.2.b.a.829.1 2
15.14 odd 2 5175.2.a.a.1.1 1
20.3 even 4 1840.2.e.b.369.2 2
20.7 even 4 1840.2.e.b.369.1 2
20.19 odd 2 9200.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.b.a.24.1 ✓ 2 5.2 odd 4
115.2.b.a.24.2 yes 2 5.3 odd 4
575.2.a.a.1.1 1 1.1 even 1 trivial
575.2.a.e.1.1 1 5.4 even 2
1035.2.b.a.829.1 2 15.8 even 4
1035.2.b.a.829.2 2 15.2 even 4
1840.2.e.b.369.1 2 20.7 even 4
1840.2.e.b.369.2 2 20.3 even 4
5175.2.a.a.1.1 1 15.14 odd 2
5175.2.a.z.1.1 1 3.2 odd 2
9200.2.a.g.1.1 1 20.19 odd 2
9200.2.a.bg.1.1 1 4.3 odd 2