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Properties

Label 280.8.a.h.1.6
Level $280$
Weight $8$
Character 280.1
Self dual yes
Analytic conductor $87.468$
Analytic rank $1$
Dimension $6$
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] = [280,8,Mod(1,280)] 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("280.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 280.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-12,0,750] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(87.4678071356\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 9435x^{4} - 52400x^{3} + 18225435x^{2} + 213666960x - 1334442969 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(84.8268\) of defining polynomial
Character \(\chi\) \(=\) 280.1

$q$-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)
 
\(f(q)\) \(=\) \(q+80.2180 q^{3} +125.000 q^{5} -343.000 q^{7} +4247.92 q^{9} -2903.87 q^{11} -9271.14 q^{13} +10027.2 q^{15} -18677.6 q^{17} -47626.2 q^{19} -27514.8 q^{21} -68366.4 q^{23} +15625.0 q^{25} +165323. q^{27} +213686. q^{29} -320828. q^{31} -232943. q^{33} -42875.0 q^{35} -431472. q^{37} -743712. q^{39} +678495. q^{41} -249771. q^{43} +530990. q^{45} +393067. q^{47} +117649. q^{49} -1.49828e6 q^{51} -1.80338e6 q^{53} -362984. q^{55} -3.82047e6 q^{57} +376762. q^{59} -3715.81 q^{61} -1.45704e6 q^{63} -1.15889e6 q^{65} +980616. q^{67} -5.48422e6 q^{69} +5.97800e6 q^{71} +409082. q^{73} +1.25341e6 q^{75} +996029. q^{77} +6.70172e6 q^{79} +3.97168e6 q^{81} -7.95045e6 q^{83} -2.33471e6 q^{85} +1.71415e7 q^{87} +1.94968e6 q^{89} +3.18000e6 q^{91} -2.57362e7 q^{93} -5.95327e6 q^{95} +1.18589e7 q^{97} -1.23354e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{3} + 750 q^{5} - 2058 q^{7} + 6450 q^{9} - 4844 q^{11} - 5200 q^{13} - 1500 q^{15} - 2976 q^{17} - 68872 q^{19} + 4116 q^{21} - 35776 q^{23} + 93750 q^{25} - 160044 q^{27} + 66696 q^{29} - 184584 q^{31}+ \cdots - 44604192 q^{99}+O(q^{100}) \) 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
\(3\) 80.2180 1.71533 0.857664 − 0.514210i \(-0.171915\pi\)
0.857664 + 0.514210i \(0.171915\pi\)
\(4\) 0 0
\(5\) 125.000 0.447214
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 0 0
\(9\) 4247.92 1.94235
\(10\) 0 0
\(11\) −2903.87 −0.657814 −0.328907 − 0.944362i \(-0.606680\pi\)
−0.328907 + 0.944362i \(0.606680\pi\)
\(12\) 0 0
\(13\) −9271.14 −1.17039 −0.585196 − 0.810892i \(-0.698982\pi\)
−0.585196 + 0.810892i \(0.698982\pi\)
\(14\) 0 0
\(15\) 10027.2 0.767118
\(16\) 0 0
\(17\) −18677.6 −0.922042 −0.461021 − 0.887389i \(-0.652517\pi\)
−0.461021 + 0.887389i \(0.652517\pi\)
\(18\) 0 0
\(19\) −47626.2 −1.59297 −0.796486 − 0.604657i \(-0.793310\pi\)
−0.796486 + 0.604657i \(0.793310\pi\)
\(20\) 0 0
\(21\) −27514.8 −0.648333
\(22\) 0 0
\(23\) −68366.4 −1.17164 −0.585822 − 0.810440i \(-0.699228\pi\)
−0.585822 + 0.810440i \(0.699228\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 0 0
\(27\) 165323. 1.61644
\(28\) 0 0
\(29\) 213686. 1.62699 0.813493 − 0.581575i \(-0.197563\pi\)
0.813493 + 0.581575i \(0.197563\pi\)
\(30\) 0 0
\(31\) −320828. −1.93422 −0.967111 − 0.254356i \(-0.918136\pi\)
−0.967111 + 0.254356i \(0.918136\pi\)
\(32\) 0 0
\(33\) −232943. −1.12837
\(34\) 0 0
\(35\) −42875.0 −0.169031
\(36\) 0 0
\(37\) −431472. −1.40038 −0.700192 − 0.713955i \(-0.746902\pi\)
−0.700192 + 0.713955i \(0.746902\pi\)
\(38\) 0 0
\(39\) −743712. −2.00761
\(40\) 0 0
\(41\) 678495. 1.53746 0.768729 − 0.639574i \(-0.220889\pi\)
0.768729 + 0.639574i \(0.220889\pi\)
\(42\) 0 0
\(43\) −249771. −0.479075 −0.239537 − 0.970887i \(-0.576996\pi\)
−0.239537 + 0.970887i \(0.576996\pi\)
\(44\) 0 0
\(45\) 530990. 0.868646
\(46\) 0 0
\(47\) 393067. 0.552235 0.276117 − 0.961124i \(-0.410952\pi\)
0.276117 + 0.961124i \(0.410952\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) −1.49828e6 −1.58161
\(52\) 0 0
\(53\) −1.80338e6 −1.66388 −0.831938 − 0.554868i \(-0.812769\pi\)
−0.831938 + 0.554868i \(0.812769\pi\)
\(54\) 0 0
\(55\) −362984. −0.294183
\(56\) 0 0
\(57\) −3.82047e6 −2.73247
\(58\) 0 0
\(59\) 376762. 0.238828 0.119414 − 0.992845i \(-0.461898\pi\)
0.119414 + 0.992845i \(0.461898\pi\)
\(60\) 0 0
\(61\) −3715.81 −0.00209604 −0.00104802 − 0.999999i \(-0.500334\pi\)
−0.00104802 + 0.999999i \(0.500334\pi\)
\(62\) 0 0
\(63\) −1.45704e6 −0.734140
\(64\) 0 0
\(65\) −1.15889e6 −0.523415
\(66\) 0 0
\(67\) 980616. 0.398325 0.199162 − 0.979967i \(-0.436178\pi\)
0.199162 + 0.979967i \(0.436178\pi\)
\(68\) 0 0
\(69\) −5.48422e6 −2.00975
\(70\) 0 0
\(71\) 5.97800e6 1.98222 0.991109 − 0.133050i \(-0.0424771\pi\)
0.991109 + 0.133050i \(0.0424771\pi\)
\(72\) 0 0
\(73\) 409082. 0.123078 0.0615390 − 0.998105i \(-0.480399\pi\)
0.0615390 + 0.998105i \(0.480399\pi\)
\(74\) 0 0
\(75\) 1.25341e6 0.343066
\(76\) 0 0
\(77\) 996029. 0.248630
\(78\) 0 0
\(79\) 6.70172e6 1.52929 0.764647 − 0.644449i \(-0.222913\pi\)
0.764647 + 0.644449i \(0.222913\pi\)
\(80\) 0 0
\(81\) 3.97168e6 0.830379
\(82\) 0 0
\(83\) −7.95045e6 −1.52622 −0.763112 − 0.646266i \(-0.776330\pi\)
−0.763112 + 0.646266i \(0.776330\pi\)
\(84\) 0 0
\(85\) −2.33471e6 −0.412350
\(86\) 0 0
\(87\) 1.71415e7 2.79082
\(88\) 0 0
\(89\) 1.94968e6 0.293156 0.146578 − 0.989199i \(-0.453174\pi\)
0.146578 + 0.989199i \(0.453174\pi\)
\(90\) 0 0
\(91\) 3.18000e6 0.442367
\(92\) 0 0
\(93\) −2.57362e7 −3.31782
\(94\) 0 0
\(95\) −5.95327e6 −0.712398
\(96\) 0 0
\(97\) 1.18589e7 1.31930 0.659649 − 0.751573i \(-0.270705\pi\)
0.659649 + 0.751573i \(0.270705\pi\)
\(98\) 0 0
\(99\) −1.23354e7 −1.27771
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 280.8.a.h.1.6 ✓ 6
4.3 odd 2 560.8.a.v.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.8.a.h.1.6 ✓ 6 1.1 even 1 trivial
560.8.a.v.1.1 6 4.3 odd 2