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Properties

Label 4840.2.a.bg.1.2
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $8$
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] = [4840,2,Mod(1,4840)] 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("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,1,0,8,0,-6,0,19,0,0,0,12] 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: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 \(-2.77190\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.77190 q^{3} +1.00000 q^{5} -4.55506 q^{7} +4.68342 q^{9} +2.90026 q^{13} -2.77190 q^{15} -7.08802 q^{17} -7.14347 q^{19} +12.6262 q^{21} -1.86699 q^{23} +1.00000 q^{25} -4.66626 q^{27} -1.01070 q^{29} +4.41034 q^{31} -4.55506 q^{35} -5.39094 q^{37} -8.03922 q^{39} -4.48790 q^{41} -6.73002 q^{43} +4.68342 q^{45} +7.85017 q^{47} +13.7486 q^{49} +19.6473 q^{51} +0.714458 q^{53} +19.8010 q^{57} -3.80330 q^{59} +2.11595 q^{61} -21.3332 q^{63} +2.90026 q^{65} -12.8384 q^{67} +5.17511 q^{69} +4.21095 q^{71} +1.36576 q^{73} -2.77190 q^{75} -12.2375 q^{79} -1.11585 q^{81} -16.3351 q^{83} -7.08802 q^{85} +2.80156 q^{87} -11.8165 q^{89} -13.2108 q^{91} -12.2250 q^{93} -7.14347 q^{95} -10.8544 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9} + 12 q^{13} + q^{15} + 2 q^{17} - 6 q^{19} + 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} + 8 q^{29} + 19 q^{31} - 6 q^{35} + 12 q^{37} - 21 q^{39} - 3 q^{41}+ \cdots + 21 q^{97}+O(q^{100}) \) 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
\(3\) −2.77190 −1.60036 −0.800178 − 0.599763i \(-0.795262\pi\)
−0.800178 + 0.599763i \(0.795262\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.55506 −1.72165 −0.860825 − 0.508901i \(-0.830052\pi\)
−0.860825 + 0.508901i \(0.830052\pi\)
\(8\) 0 0
\(9\) 4.68342 1.56114
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 2.90026 0.804387 0.402193 − 0.915555i \(-0.368248\pi\)
0.402193 + 0.915555i \(0.368248\pi\)
\(14\) 0 0
\(15\) −2.77190 −0.715701
\(16\) 0 0
\(17\) −7.08802 −1.71910 −0.859548 − 0.511054i \(-0.829255\pi\)
−0.859548 + 0.511054i \(0.829255\pi\)
\(18\) 0 0
\(19\) −7.14347 −1.63882 −0.819412 − 0.573205i \(-0.805700\pi\)
−0.819412 + 0.573205i \(0.805700\pi\)
\(20\) 0 0
\(21\) 12.6262 2.75525
\(22\) 0 0
\(23\) −1.86699 −0.389295 −0.194647 − 0.980873i \(-0.562356\pi\)
−0.194647 + 0.980873i \(0.562356\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.66626 −0.898023
\(28\) 0 0
\(29\) −1.01070 −0.187682 −0.0938412 − 0.995587i \(-0.529915\pi\)
−0.0938412 + 0.995587i \(0.529915\pi\)
\(30\) 0 0
\(31\) 4.41034 0.792121 0.396061 − 0.918224i \(-0.370377\pi\)
0.396061 + 0.918224i \(0.370377\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.55506 −0.769945
\(36\) 0 0
\(37\) −5.39094 −0.886265 −0.443132 − 0.896456i \(-0.646133\pi\)
−0.443132 + 0.896456i \(0.646133\pi\)
\(38\) 0 0
\(39\) −8.03922 −1.28730
\(40\) 0 0
\(41\) −4.48790 −0.700892 −0.350446 − 0.936583i \(-0.613970\pi\)
−0.350446 + 0.936583i \(0.613970\pi\)
\(42\) 0 0
\(43\) −6.73002 −1.02632 −0.513159 − 0.858293i \(-0.671525\pi\)
−0.513159 + 0.858293i \(0.671525\pi\)
\(44\) 0 0
\(45\) 4.68342 0.698163
\(46\) 0 0
\(47\) 7.85017 1.14507 0.572533 − 0.819882i \(-0.305961\pi\)
0.572533 + 0.819882i \(0.305961\pi\)
\(48\) 0 0
\(49\) 13.7486 1.96408
\(50\) 0 0
\(51\) 19.6473 2.75117
\(52\) 0 0
\(53\) 0.714458 0.0981383 0.0490692 − 0.998795i \(-0.484375\pi\)
0.0490692 + 0.998795i \(0.484375\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 19.8010 2.62270
\(58\) 0 0
\(59\) −3.80330 −0.495147 −0.247574 − 0.968869i \(-0.579633\pi\)
−0.247574 + 0.968869i \(0.579633\pi\)
\(60\) 0 0
\(61\) 2.11595 0.270920 0.135460 − 0.990783i \(-0.456749\pi\)
0.135460 + 0.990783i \(0.456749\pi\)
\(62\) 0 0
\(63\) −21.3332 −2.68774
\(64\) 0 0
\(65\) 2.90026 0.359733
\(66\) 0 0
\(67\) −12.8384 −1.56847 −0.784233 − 0.620467i \(-0.786943\pi\)
−0.784233 + 0.620467i \(0.786943\pi\)
\(68\) 0 0
\(69\) 5.17511 0.623010
\(70\) 0 0
\(71\) 4.21095 0.499747 0.249874 − 0.968278i \(-0.419611\pi\)
0.249874 + 0.968278i \(0.419611\pi\)
\(72\) 0 0
\(73\) 1.36576 0.159850 0.0799249 − 0.996801i \(-0.474532\pi\)
0.0799249 + 0.996801i \(0.474532\pi\)
\(74\) 0 0
\(75\) −2.77190 −0.320071
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.2375 −1.37682 −0.688411 − 0.725320i \(-0.741692\pi\)
−0.688411 + 0.725320i \(0.741692\pi\)
\(80\) 0 0
\(81\) −1.11585 −0.123983
\(82\) 0 0
\(83\) −16.3351 −1.79301 −0.896503 − 0.443037i \(-0.853901\pi\)
−0.896503 + 0.443037i \(0.853901\pi\)
\(84\) 0 0
\(85\) −7.08802 −0.768803
\(86\) 0 0
\(87\) 2.80156 0.300359
\(88\) 0 0
\(89\) −11.8165 −1.25255 −0.626273 − 0.779604i \(-0.715420\pi\)
−0.626273 + 0.779604i \(0.715420\pi\)
\(90\) 0 0
\(91\) −13.2108 −1.38487
\(92\) 0 0
\(93\) −12.2250 −1.26768
\(94\) 0 0
\(95\) −7.14347 −0.732904
\(96\) 0 0
\(97\) −10.8544 −1.10210 −0.551048 − 0.834473i \(-0.685772\pi\)
−0.551048 + 0.834473i \(0.685772\pi\)
\(98\) 0 0
\(99\) 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 4840.2.a.bg.1.2 8
4.3 odd 2 9680.2.a.df.1.7 8
11.5 even 5 440.2.y.d.201.1 yes 16
11.9 even 5 440.2.y.d.81.1 ✓ 16
11.10 odd 2 4840.2.a.bh.1.2 8
44.27 odd 10 880.2.bo.k.641.4 16
44.31 odd 10 880.2.bo.k.81.4 16
44.43 even 2 9680.2.a.de.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.d.81.1 ✓ 16 11.9 even 5
440.2.y.d.201.1 yes 16 11.5 even 5
880.2.bo.k.81.4 16 44.31 odd 10
880.2.bo.k.641.4 16 44.27 odd 10
4840.2.a.bg.1.2 8 1.1 even 1 trivial
4840.2.a.bh.1.2 8 11.10 odd 2
9680.2.a.de.1.7 8 44.43 even 2
9680.2.a.df.1.7 8 4.3 odd 2