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Properties

Label 6728.2.a.bf.1.11
Level $6728$
Weight $2$
Character 6728.1
Self dual yes
Analytic conductor $53.723$
Analytic rank $0$
Dimension $24$
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] = [6728,2,Mod(1,6728)] 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("6728.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6728 = 2^{3} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6728.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,4,0,8,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(53.7233504799\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 232)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Character \(\chi\) \(=\) 6728.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.271416 q^{3} -1.42299 q^{5} +2.04647 q^{7} -2.92633 q^{9} -5.48669 q^{11} +5.80145 q^{13} +0.386221 q^{15} +0.840415 q^{17} +7.61426 q^{19} -0.555443 q^{21} -5.83450 q^{23} -2.97511 q^{25} +1.60850 q^{27} +0.324611 q^{31} +1.48917 q^{33} -2.91209 q^{35} +9.78461 q^{37} -1.57460 q^{39} +3.13993 q^{41} -8.93930 q^{43} +4.16414 q^{45} +1.21823 q^{47} -2.81198 q^{49} -0.228102 q^{51} -9.20976 q^{53} +7.80749 q^{55} -2.06663 q^{57} -7.46966 q^{59} -3.56407 q^{61} -5.98864 q^{63} -8.25539 q^{65} +6.06340 q^{67} +1.58358 q^{69} +3.86484 q^{71} +5.78442 q^{73} +0.807491 q^{75} -11.2283 q^{77} +1.63589 q^{79} +8.34243 q^{81} +9.73068 q^{83} -1.19590 q^{85} -9.52886 q^{89} +11.8725 q^{91} -0.0881046 q^{93} -10.8350 q^{95} +14.0115 q^{97} +16.0559 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 4 q^{3} + 8 q^{5} - 2 q^{7} + 32 q^{9} + 6 q^{11} + 2 q^{13} + 32 q^{15} + 20 q^{17} + 44 q^{19} + 4 q^{21} + 2 q^{23} + 32 q^{25} + 34 q^{27} + 10 q^{31} - 12 q^{33} - 32 q^{35} + 40 q^{37} + 18 q^{39}+ \cdots + 60 q^{99}+O(q^{100}) \) 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
\(3\) −0.271416 −0.156702 −0.0783510 − 0.996926i \(-0.524965\pi\)
−0.0783510 + 0.996926i \(0.524965\pi\)
\(4\) 0 0
\(5\) −1.42299 −0.636379 −0.318190 − 0.948027i \(-0.603075\pi\)
−0.318190 + 0.948027i \(0.603075\pi\)
\(6\) 0 0
\(7\) 2.04647 0.773491 0.386746 − 0.922186i \(-0.373599\pi\)
0.386746 + 0.922186i \(0.373599\pi\)
\(8\) 0 0
\(9\) −2.92633 −0.975444
\(10\) 0 0
\(11\) −5.48669 −1.65430 −0.827149 − 0.561982i \(-0.810039\pi\)
−0.827149 + 0.561982i \(0.810039\pi\)
\(12\) 0 0
\(13\) 5.80145 1.60903 0.804516 − 0.593931i \(-0.202425\pi\)
0.804516 + 0.593931i \(0.202425\pi\)
\(14\) 0 0
\(15\) 0.386221 0.0997219
\(16\) 0 0
\(17\) 0.840415 0.203831 0.101915 − 0.994793i \(-0.467503\pi\)
0.101915 + 0.994793i \(0.467503\pi\)
\(18\) 0 0
\(19\) 7.61426 1.74683 0.873416 − 0.486976i \(-0.161900\pi\)
0.873416 + 0.486976i \(0.161900\pi\)
\(20\) 0 0
\(21\) −0.555443 −0.121208
\(22\) 0 0
\(23\) −5.83450 −1.21658 −0.608289 − 0.793716i \(-0.708144\pi\)
−0.608289 + 0.793716i \(0.708144\pi\)
\(24\) 0 0
\(25\) −2.97511 −0.595021
\(26\) 0 0
\(27\) 1.60850 0.309556
\(28\) 0 0
\(29\) 0 0
\(30\) 0 0
\(31\) 0.324611 0.0583019 0.0291509 − 0.999575i \(-0.490720\pi\)
0.0291509 + 0.999575i \(0.490720\pi\)
\(32\) 0 0
\(33\) 1.48917 0.259232
\(34\) 0 0
\(35\) −2.91209 −0.492234
\(36\) 0 0
\(37\) 9.78461 1.60858 0.804290 − 0.594237i \(-0.202546\pi\)
0.804290 + 0.594237i \(0.202546\pi\)
\(38\) 0 0
\(39\) −1.57460 −0.252139
\(40\) 0 0
\(41\) 3.13993 0.490375 0.245188 − 0.969476i \(-0.421150\pi\)
0.245188 + 0.969476i \(0.421150\pi\)
\(42\) 0 0
\(43\) −8.93930 −1.36323 −0.681615 − 0.731711i \(-0.738722\pi\)
−0.681615 + 0.731711i \(0.738722\pi\)
\(44\) 0 0
\(45\) 4.16414 0.620753
\(46\) 0 0
\(47\) 1.21823 0.177697 0.0888483 − 0.996045i \(-0.471681\pi\)
0.0888483 + 0.996045i \(0.471681\pi\)
\(48\) 0 0
\(49\) −2.81198 −0.401711
\(50\) 0 0
\(51\) −0.228102 −0.0319407
\(52\) 0 0
\(53\) −9.20976 −1.26506 −0.632529 − 0.774536i \(-0.717983\pi\)
−0.632529 + 0.774536i \(0.717983\pi\)
\(54\) 0 0
\(55\) 7.80749 1.05276
\(56\) 0 0
\(57\) −2.06663 −0.273732
\(58\) 0 0
\(59\) −7.46966 −0.972467 −0.486234 − 0.873829i \(-0.661630\pi\)
−0.486234 + 0.873829i \(0.661630\pi\)
\(60\) 0 0
\(61\) −3.56407 −0.456333 −0.228166 − 0.973622i \(-0.573273\pi\)
−0.228166 + 0.973622i \(0.573273\pi\)
\(62\) 0 0
\(63\) −5.98864 −0.754498
\(64\) 0 0
\(65\) −8.25539 −1.02395
\(66\) 0 0
\(67\) 6.06340 0.740763 0.370381 − 0.928880i \(-0.379227\pi\)
0.370381 + 0.928880i \(0.379227\pi\)
\(68\) 0 0
\(69\) 1.58358 0.190640
\(70\) 0 0
\(71\) 3.86484 0.458672 0.229336 − 0.973347i \(-0.426345\pi\)
0.229336 + 0.973347i \(0.426345\pi\)
\(72\) 0 0
\(73\) 5.78442 0.677015 0.338508 − 0.940964i \(-0.390078\pi\)
0.338508 + 0.940964i \(0.390078\pi\)
\(74\) 0 0
\(75\) 0.807491 0.0932411
\(76\) 0 0
\(77\) −11.2283 −1.27959
\(78\) 0 0
\(79\) 1.63589 0.184052 0.0920258 − 0.995757i \(-0.470666\pi\)
0.0920258 + 0.995757i \(0.470666\pi\)
\(80\) 0 0
\(81\) 8.34243 0.926936
\(82\) 0 0
\(83\) 9.73068 1.06808 0.534040 − 0.845459i \(-0.320673\pi\)
0.534040 + 0.845459i \(0.320673\pi\)
\(84\) 0 0
\(85\) −1.19590 −0.129714
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.52886 −1.01006 −0.505028 − 0.863103i \(-0.668518\pi\)
−0.505028 + 0.863103i \(0.668518\pi\)
\(90\) 0 0
\(91\) 11.8725 1.24457
\(92\) 0 0
\(93\) −0.0881046 −0.00913602
\(94\) 0 0
\(95\) −10.8350 −1.11165
\(96\) 0 0
\(97\) 14.0115 1.42265 0.711324 − 0.702864i \(-0.248096\pi\)
0.711324 + 0.702864i \(0.248096\pi\)
\(98\) 0 0
\(99\) 16.0559 1.61368
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 6728.2.a.bf.1.11 24
29.3 odd 28 232.2.q.a.9.4 ✓ 48
29.10 odd 28 232.2.q.a.129.4 yes 48
29.28 even 2 6728.2.a.be.1.14 24
116.3 even 28 464.2.y.e.241.5 48
116.39 even 28 464.2.y.e.129.5 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.2.q.a.9.4 ✓ 48 29.3 odd 28
232.2.q.a.129.4 yes 48 29.10 odd 28
464.2.y.e.129.5 48 116.39 even 28
464.2.y.e.241.5 48 116.3 even 28
6728.2.a.be.1.14 24 29.28 even 2
6728.2.a.bf.1.11 24 1.1 even 1 trivial