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,-1,0,2,0,5,0,3,0,2,0,-6] 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: \(1\)
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)
 
\(f(q)\) \(=\) \(q-2.56155 q^{3} +1.00000 q^{5} +4.56155 q^{7} +3.56155 q^{9} +1.00000 q^{11} +1.12311 q^{13} -2.56155 q^{15} -7.68466 q^{17} -1.43845 q^{19} -11.6847 q^{21} +1.12311 q^{23} +1.00000 q^{25} -1.43845 q^{27} -8.56155 q^{29} -1.43845 q^{31} -2.56155 q^{33} +4.56155 q^{35} -7.43845 q^{37} -2.87689 q^{39} -12.2462 q^{41} +3.12311 q^{43} +3.56155 q^{45} -11.3693 q^{47} +13.8078 q^{49} +19.6847 q^{51} -9.68466 q^{53} +1.00000 q^{55} +3.68466 q^{57} -1.12311 q^{59} +12.5616 q^{61} +16.2462 q^{63} +1.12311 q^{65} -2.87689 q^{69} +3.68466 q^{71} +1.12311 q^{73} -2.56155 q^{75} +4.56155 q^{77} -11.3693 q^{79} -7.00000 q^{81} +6.00000 q^{83} -7.68466 q^{85} +21.9309 q^{87} +9.68466 q^{89} +5.12311 q^{91} +3.68466 q^{93} -1.43845 q^{95} -4.87689 q^{97} +3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} + 5 q^{7} + 3 q^{9} + 2 q^{11} - 6 q^{13} - q^{15} - 3 q^{17} - 7 q^{19} - 11 q^{21} - 6 q^{23} + 2 q^{25} - 7 q^{27} - 13 q^{29} - 7 q^{31} - q^{33} + 5 q^{35} - 19 q^{37} - 14 q^{39}+ \cdots + 3 q^{99}+O(q^{100}) \) 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
\(7\) 4.56155 1.72410 0.862052 − 0.506819i \(-0.169179\pi\)
0.862052 + 0.506819i \(0.169179\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.12311 0.311493 0.155747 − 0.987797i \(-0.450222\pi\)
0.155747 + 0.987797i \(0.450222\pi\)
\(14\) 0 0
\(15\) −2.56155 −0.661390
\(16\) 0 0
\(17\) −7.68466 −1.86380 −0.931902 − 0.362711i \(-0.881851\pi\)
−0.931902 + 0.362711i \(0.881851\pi\)
\(18\) 0 0
\(19\) −1.43845 −0.330002 −0.165001 − 0.986293i \(-0.552763\pi\)
−0.165001 + 0.986293i \(0.552763\pi\)
\(20\) 0 0
\(21\) −11.6847 −2.54980
\(22\) 0 0
\(23\) 1.12311 0.234184 0.117092 − 0.993121i \(-0.462643\pi\)
0.117092 + 0.993121i \(0.462643\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) −8.56155 −1.58984 −0.794920 − 0.606714i \(-0.792487\pi\)
−0.794920 + 0.606714i \(0.792487\pi\)
\(30\) 0 0
\(31\) −1.43845 −0.258353 −0.129176 − 0.991622i \(-0.541233\pi\)
−0.129176 + 0.991622i \(0.541233\pi\)
\(32\) 0 0
\(33\) −2.56155 −0.445909
\(34\) 0 0
\(35\) 4.56155 0.771043
\(36\) 0 0
\(37\) −7.43845 −1.22287 −0.611437 − 0.791293i \(-0.709408\pi\)
−0.611437 + 0.791293i \(0.709408\pi\)
\(38\) 0 0
\(39\) −2.87689 −0.460672
\(40\) 0 0
\(41\) −12.2462 −1.91254 −0.956268 − 0.292490i \(-0.905516\pi\)
−0.956268 + 0.292490i \(0.905516\pi\)
\(42\) 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
\(47\) −11.3693 −1.65839 −0.829193 − 0.558963i \(-0.811199\pi\)
−0.829193 + 0.558963i \(0.811199\pi\)
\(48\) 0 0
\(49\) 13.8078 1.97254
\(50\) 0 0
\(51\) 19.6847 2.75640
\(52\) 0 0
\(53\) −9.68466 −1.33029 −0.665145 − 0.746714i \(-0.731630\pi\)
−0.665145 + 0.746714i \(0.731630\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 3.68466 0.488045
\(58\) 0 0
\(59\) −1.12311 −0.146216 −0.0731079 − 0.997324i \(-0.523292\pi\)
−0.0731079 + 0.997324i \(0.523292\pi\)
\(60\) 0 0
\(61\) 12.5616 1.60834 0.804171 − 0.594398i \(-0.202610\pi\)
0.804171 + 0.594398i \(0.202610\pi\)
\(62\) 0 0
\(63\) 16.2462 2.04683
\(64\) 0 0
\(65\) 1.12311 0.139304
\(66\) 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
\(71\) 3.68466 0.437289 0.218644 − 0.975805i \(-0.429837\pi\)
0.218644 + 0.975805i \(0.429837\pi\)
\(72\) 0 0
\(73\) 1.12311 0.131450 0.0657248 − 0.997838i \(-0.479064\pi\)
0.0657248 + 0.997838i \(0.479064\pi\)
\(74\) 0 0
\(75\) −2.56155 −0.295783
\(76\) 0 0
\(77\) 4.56155 0.519837
\(78\) 0 0
\(79\) −11.3693 −1.27915 −0.639574 − 0.768729i \(-0.720889\pi\)
−0.639574 + 0.768729i \(0.720889\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −7.68466 −0.833518
\(86\) 0 0
\(87\) 21.9309 2.35124
\(88\) 0 0
\(89\) 9.68466 1.02657 0.513286 − 0.858218i \(-0.328428\pi\)
0.513286 + 0.858218i \(0.328428\pi\)
\(90\) 0 0
\(91\) 5.12311 0.537047
\(92\) 0 0
\(93\) 3.68466 0.382081
\(94\) 0 0
\(95\) −1.43845 −0.147582
\(96\) 0 0
\(97\) −4.87689 −0.495174 −0.247587 − 0.968866i \(-0.579638\pi\)
−0.247587 + 0.968866i \(0.579638\pi\)
\(98\) 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.bm.1.1 2
4.3 odd 2 3520.2.a.br.1.2 2
8.3 odd 2 880.2.a.k.1.1 2
8.5 even 2 440.2.a.g.1.2 ✓ 2
24.5 odd 2 3960.2.a.bf.1.2 2
24.11 even 2 7920.2.a.by.1.1 2
40.3 even 4 4400.2.b.w.4049.1 4
40.13 odd 4 2200.2.b.f.1849.4 4
40.19 odd 2 4400.2.a.bt.1.2 2
40.27 even 4 4400.2.b.w.4049.4 4
40.29 even 2 2200.2.a.l.1.1 2
40.37 odd 4 2200.2.b.f.1849.1 4
88.21 odd 2 4840.2.a.m.1.2 2
88.43 even 2 9680.2.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.g.1.2 ✓ 2 8.5 even 2
880.2.a.k.1.1 2 8.3 odd 2
2200.2.a.l.1.1 2 40.29 even 2
2200.2.b.f.1849.1 4 40.37 odd 4
2200.2.b.f.1849.4 4 40.13 odd 4
3520.2.a.bm.1.1 2 1.1 even 1 trivial
3520.2.a.br.1.2 2 4.3 odd 2
3960.2.a.bf.1.2 2 24.5 odd 2
4400.2.a.bt.1.2 2 40.19 odd 2
4400.2.b.w.4049.1 4 40.3 even 4
4400.2.b.w.4049.4 4 40.27 even 4
4840.2.a.m.1.2 2 88.21 odd 2
7920.2.a.by.1.1 2 24.11 even 2
9680.2.a.bm.1.1 2 88.43 even 2