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Properties

Label 3520.2.a.bl.1.1
Level $3520$
Weight $2$
Character 3520.1
Self dual yes
Analytic conductor $28.107$
Analytic rank $0$
Dimension $2$
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] = [3520,2,Mod(1,3520)] 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("3520.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3520.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,2,0,3,0,3,0,-2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(28.1073415115\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 3520.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.56155 q^{3} +1.00000 q^{5} -0.561553 q^{7} +3.56155 q^{9} -1.00000 q^{11} -5.12311 q^{13} -2.56155 q^{15} +1.43845 q^{17} -6.56155 q^{19} +1.43845 q^{21} -1.12311 q^{23} +1.00000 q^{25} -1.43845 q^{27} +4.56155 q^{29} -3.68466 q^{31} +2.56155 q^{33} -0.561553 q^{35} +10.8078 q^{37} +13.1231 q^{39} -10.0000 q^{41} +3.12311 q^{43} +3.56155 q^{45} +1.12311 q^{47} -6.68466 q^{49} -3.68466 q^{51} +8.56155 q^{53} -1.00000 q^{55} +16.8078 q^{57} -11.3693 q^{59} -0.561553 q^{61} -2.00000 q^{63} -5.12311 q^{65} +2.87689 q^{69} -6.56155 q^{71} +13.1231 q^{73} -2.56155 q^{75} +0.561553 q^{77} +9.12311 q^{79} -7.00000 q^{81} -10.0000 q^{83} +1.43845 q^{85} -11.6847 q^{87} -10.8078 q^{89} +2.87689 q^{91} +9.43845 q^{93} -6.56155 q^{95} +8.87689 q^{97} -3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} + 3 q^{7} + 3 q^{9} - 2 q^{11} - 2 q^{13} - q^{15} + 7 q^{17} - 9 q^{19} + 7 q^{21} + 6 q^{23} + 2 q^{25} - 7 q^{27} + 5 q^{29} + 5 q^{31} + q^{33} + 3 q^{35} + q^{37} + 18 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) \) 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 0
\(3\) −2.56155 −1.47891 −0.739457 − 0.673204i \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.561553 −0.212247 −0.106124 − 0.994353i \(-0.533844\pi\)
−0.106124 + 0.994353i \(0.533844\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −5.12311 −1.42089 −0.710447 − 0.703751i \(-0.751507\pi\)
−0.710447 + 0.703751i \(0.751507\pi\)
\(14\) 0 0
\(15\) −2.56155 −0.661390
\(16\) 0 0
\(17\) 1.43845 0.348875 0.174437 − 0.984668i \(-0.444189\pi\)
0.174437 + 0.984668i \(0.444189\pi\)
\(18\) 0 0
\(19\) −6.56155 −1.50532 −0.752662 − 0.658407i \(-0.771230\pi\)
−0.752662 + 0.658407i \(0.771230\pi\)
\(20\) 0 0
\(21\) 1.43845 0.313895
\(22\) 0 0
\(23\) −1.12311 −0.234184 −0.117092 − 0.993121i \(-0.537357\pi\)
−0.117092 + 0.993121i \(0.537357\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) 4.56155 0.847059 0.423530 − 0.905882i \(-0.360791\pi\)
0.423530 + 0.905882i \(0.360791\pi\)
\(30\) 0 0
\(31\) −3.68466 −0.661784 −0.330892 − 0.943669i \(-0.607350\pi\)
−0.330892 + 0.943669i \(0.607350\pi\)
\(32\) 0 0
\(33\) 2.56155 0.445909
\(34\) 0 0
\(35\) −0.561553 −0.0949197
\(36\) 0 0
\(37\) 10.8078 1.77679 0.888393 − 0.459084i \(-0.151822\pi\)
0.888393 + 0.459084i \(0.151822\pi\)
\(38\) 0 0
\(39\) 13.1231 2.10138
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 3.12311 0.476269 0.238135 − 0.971232i \(-0.423464\pi\)
0.238135 + 0.971232i \(0.423464\pi\)
\(44\) 0 0
\(45\) 3.56155 0.530925
\(46\) 0 0
\(47\) 1.12311 0.163822 0.0819109 − 0.996640i \(-0.473898\pi\)
0.0819109 + 0.996640i \(0.473898\pi\)
\(48\) 0 0
\(49\) −6.68466 −0.954951
\(50\) 0 0
\(51\) −3.68466 −0.515955
\(52\) 0 0
\(53\) 8.56155 1.17602 0.588010 − 0.808854i \(-0.299912\pi\)
0.588010 + 0.808854i \(0.299912\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 16.8078 2.22624
\(58\) 0 0
\(59\) −11.3693 −1.48016 −0.740079 − 0.672519i \(-0.765212\pi\)
−0.740079 + 0.672519i \(0.765212\pi\)
\(60\) 0 0
\(61\) −0.561553 −0.0718995 −0.0359497 − 0.999354i \(-0.511446\pi\)
−0.0359497 + 0.999354i \(0.511446\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) −5.12311 −0.635443
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 2.87689 0.346337
\(70\) 0 0
\(71\) −6.56155 −0.778713 −0.389357 − 0.921087i \(-0.627303\pi\)
−0.389357 + 0.921087i \(0.627303\pi\)
\(72\) 0 0
\(73\) 13.1231 1.53594 0.767972 − 0.640484i \(-0.221266\pi\)
0.767972 + 0.640484i \(0.221266\pi\)
\(74\) 0 0
\(75\) −2.56155 −0.295783
\(76\) 0 0
\(77\) 0.561553 0.0639949
\(78\) 0 0
\(79\) 9.12311 1.02643 0.513215 − 0.858260i \(-0.328454\pi\)
0.513215 + 0.858260i \(0.328454\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −10.0000 −1.09764 −0.548821 − 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) 1.43845 0.156022
\(86\) 0 0
\(87\) −11.6847 −1.25273
\(88\) 0 0
\(89\) −10.8078 −1.14562 −0.572810 − 0.819688i \(-0.694147\pi\)
−0.572810 + 0.819688i \(0.694147\pi\)
\(90\) 0 0
\(91\) 2.87689 0.301580
\(92\) 0 0
\(93\) 9.43845 0.978721
\(94\) 0 0
\(95\) −6.56155 −0.673201
\(96\) 0 0
\(97\) 8.87689 0.901312 0.450656 − 0.892698i \(-0.351190\pi\)
0.450656 + 0.892698i \(0.351190\pi\)
\(98\) 0 0
\(99\) −3.56155 −0.357950
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 3520.2.a.bl.1.1 2
4.3 odd 2 3520.2.a.bs.1.2 2
8.3 odd 2 880.2.a.l.1.1 2
8.5 even 2 440.2.a.f.1.2 ✓ 2
24.5 odd 2 3960.2.a.be.1.1 2
24.11 even 2 7920.2.a.ca.1.2 2
40.3 even 4 4400.2.b.u.4049.1 4
40.13 odd 4 2200.2.b.h.1849.4 4
40.19 odd 2 4400.2.a.br.1.2 2
40.27 even 4 4400.2.b.u.4049.4 4
40.29 even 2 2200.2.a.m.1.1 2
40.37 odd 4 2200.2.b.h.1849.1 4
88.21 odd 2 4840.2.a.n.1.2 2
88.43 even 2 9680.2.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.f.1.2 ✓ 2 8.5 even 2
880.2.a.l.1.1 2 8.3 odd 2
2200.2.a.m.1.1 2 40.29 even 2
2200.2.b.h.1849.1 4 40.37 odd 4
2200.2.b.h.1849.4 4 40.13 odd 4
3520.2.a.bl.1.1 2 1.1 even 1 trivial
3520.2.a.bs.1.2 2 4.3 odd 2
3960.2.a.be.1.1 2 24.5 odd 2
4400.2.a.br.1.2 2 40.19 odd 2
4400.2.b.u.4049.1 4 40.3 even 4
4400.2.b.u.4049.4 4 40.27 even 4
4840.2.a.n.1.2 2 88.21 odd 2
7920.2.a.ca.1.2 2 24.11 even 2
9680.2.a.bl.1.1 2 88.43 even 2