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Properties

Label 3520.2.a.bl.1.2
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 1.2
Root \(-1.56155\) 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)
 
\(f(q)\) \(=\) \(q+1.56155 q^{3} +1.00000 q^{5} +3.56155 q^{7} -0.561553 q^{9} -1.00000 q^{11} +3.12311 q^{13} +1.56155 q^{15} +5.56155 q^{17} -2.43845 q^{19} +5.56155 q^{21} +7.12311 q^{23} +1.00000 q^{25} -5.56155 q^{27} +0.438447 q^{29} +8.68466 q^{31} -1.56155 q^{33} +3.56155 q^{35} -9.80776 q^{37} +4.87689 q^{39} -10.0000 q^{41} -5.12311 q^{43} -0.561553 q^{45} -7.12311 q^{47} +5.68466 q^{49} +8.68466 q^{51} +4.43845 q^{53} -1.00000 q^{55} -3.80776 q^{57} +13.3693 q^{59} +3.56155 q^{61} -2.00000 q^{63} +3.12311 q^{65} +11.1231 q^{69} -2.43845 q^{71} +4.87689 q^{73} +1.56155 q^{75} -3.56155 q^{77} +0.876894 q^{79} -7.00000 q^{81} -10.0000 q^{83} +5.56155 q^{85} +0.684658 q^{87} +9.80776 q^{89} +11.1231 q^{91} +13.5616 q^{93} -2.43845 q^{95} +17.1231 q^{97} +0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} + 3 q^{7} + 3 q^{9} - 2 q^{11} - 2 q^{13} - q^{15} + 7 q^{17} - 9 q^{19} + 7 q^{21} + 6 q^{23} + 2 q^{25} - 7 q^{27} + 5 q^{29} + 5 q^{31} + q^{33} + 3 q^{35} + q^{37} + 18 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) \) 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
\(3\) 1.56155 0.901563 0.450781 − 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.56155 1.34614 0.673070 − 0.739579i \(-0.264975\pi\)
0.673070 + 0.739579i \(0.264975\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 3.12311 0.866194 0.433097 − 0.901347i \(-0.357421\pi\)
0.433097 + 0.901347i \(0.357421\pi\)
\(14\) 0 0
\(15\) 1.56155 0.403191
\(16\) 0 0
\(17\) 5.56155 1.34887 0.674437 − 0.738332i \(-0.264386\pi\)
0.674437 + 0.738332i \(0.264386\pi\)
\(18\) 0 0
\(19\) −2.43845 −0.559418 −0.279709 − 0.960085i \(-0.590238\pi\)
−0.279709 + 0.960085i \(0.590238\pi\)
\(20\) 0 0
\(21\) 5.56155 1.21363
\(22\) 0 0
\(23\) 7.12311 1.48527 0.742635 − 0.669696i \(-0.233576\pi\)
0.742635 + 0.669696i \(0.233576\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) 0.438447 0.0814176 0.0407088 − 0.999171i \(-0.487038\pi\)
0.0407088 + 0.999171i \(0.487038\pi\)
\(30\) 0 0
\(31\) 8.68466 1.55981 0.779905 − 0.625897i \(-0.215267\pi\)
0.779905 + 0.625897i \(0.215267\pi\)
\(32\) 0 0
\(33\) −1.56155 −0.271831
\(34\) 0 0
\(35\) 3.56155 0.602012
\(36\) 0 0
\(37\) −9.80776 −1.61239 −0.806193 − 0.591652i \(-0.798476\pi\)
−0.806193 + 0.591652i \(0.798476\pi\)
\(38\) 0 0
\(39\) 4.87689 0.780928
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −5.12311 −0.781266 −0.390633 − 0.920546i \(-0.627744\pi\)
−0.390633 + 0.920546i \(0.627744\pi\)
\(44\) 0 0
\(45\) −0.561553 −0.0837114
\(46\) 0 0
\(47\) −7.12311 −1.03901 −0.519506 − 0.854467i \(-0.673884\pi\)
−0.519506 + 0.854467i \(0.673884\pi\)
\(48\) 0 0
\(49\) 5.68466 0.812094
\(50\) 0 0
\(51\) 8.68466 1.21610
\(52\) 0 0
\(53\) 4.43845 0.609668 0.304834 − 0.952406i \(-0.401399\pi\)
0.304834 + 0.952406i \(0.401399\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −3.80776 −0.504351
\(58\) 0 0
\(59\) 13.3693 1.74054 0.870268 − 0.492578i \(-0.163945\pi\)
0.870268 + 0.492578i \(0.163945\pi\)
\(60\) 0 0
\(61\) 3.56155 0.456010 0.228005 − 0.973660i \(-0.426780\pi\)
0.228005 + 0.973660i \(0.426780\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 3.12311 0.387374
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 11.1231 1.33906
\(70\) 0 0
\(71\) −2.43845 −0.289390 −0.144695 − 0.989476i \(-0.546220\pi\)
−0.144695 + 0.989476i \(0.546220\pi\)
\(72\) 0 0
\(73\) 4.87689 0.570797 0.285399 − 0.958409i \(-0.407874\pi\)
0.285399 + 0.958409i \(0.407874\pi\)
\(74\) 0 0
\(75\) 1.56155 0.180313
\(76\) 0 0
\(77\) −3.56155 −0.405877
\(78\) 0 0
\(79\) 0.876894 0.0986583 0.0493292 − 0.998783i \(-0.484292\pi\)
0.0493292 + 0.998783i \(0.484292\pi\)
\(80\) 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
\(85\) 5.56155 0.603235
\(86\) 0 0
\(87\) 0.684658 0.0734031
\(88\) 0 0
\(89\) 9.80776 1.03962 0.519810 − 0.854282i \(-0.326003\pi\)
0.519810 + 0.854282i \(0.326003\pi\)
\(90\) 0 0
\(91\) 11.1231 1.16602
\(92\) 0 0
\(93\) 13.5616 1.40627
\(94\) 0 0
\(95\) −2.43845 −0.250179
\(96\) 0 0
\(97\) 17.1231 1.73859 0.869294 − 0.494295i \(-0.164574\pi\)
0.869294 + 0.494295i \(0.164574\pi\)
\(98\) 0 0
\(99\) 0.561553 0.0564382
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.1.2 2
4.3 odd 2 3520.2.a.bs.1.1 2
8.3 odd 2 880.2.a.l.1.2 2
8.5 even 2 440.2.a.f.1.1 ✓ 2
24.5 odd 2 3960.2.a.be.1.2 2
24.11 even 2 7920.2.a.ca.1.1 2
40.3 even 4 4400.2.b.u.4049.3 4
40.13 odd 4 2200.2.b.h.1849.2 4
40.19 odd 2 4400.2.a.br.1.1 2
40.27 even 4 4400.2.b.u.4049.2 4
40.29 even 2 2200.2.a.m.1.2 2
40.37 odd 4 2200.2.b.h.1849.3 4
88.21 odd 2 4840.2.a.n.1.1 2
88.43 even 2 9680.2.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.f.1.1 ✓ 2 8.5 even 2
880.2.a.l.1.2 2 8.3 odd 2
2200.2.a.m.1.2 2 40.29 even 2
2200.2.b.h.1849.2 4 40.13 odd 4
2200.2.b.h.1849.3 4 40.37 odd 4
3520.2.a.bl.1.2 2 1.1 even 1 trivial
3520.2.a.bs.1.1 2 4.3 odd 2
3960.2.a.be.1.2 2 24.5 odd 2
4400.2.a.br.1.1 2 40.19 odd 2
4400.2.b.u.4049.2 4 40.27 even 4
4400.2.b.u.4049.3 4 40.3 even 4
4840.2.a.n.1.1 2 88.21 odd 2
7920.2.a.ca.1.1 2 24.11 even 2
9680.2.a.bl.1.2 2 88.43 even 2