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Properties

Label 280.10.a.e.1.5
Level $280$
Weight $10$
Character 280.1
Self dual yes
Analytic conductor $144.210$
Analytic rank $1$
Dimension $7$
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] = [280,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 280.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,0,-151,0,4375] 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: \(144.210034126\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 68982 x^{5} - 1651764 x^{4} + 1100124770 x^{3} - 1398985356 x^{2} + \cdots + 198660733545904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{3}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(62.6530\) 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)
 
\(f(q)\) \(=\) \(q+113.026 q^{3} +625.000 q^{5} +2401.00 q^{7} -6908.15 q^{9} -26940.9 q^{11} -136556. q^{13} +70641.2 q^{15} +38170.8 q^{17} +1.05090e6 q^{19} +271375. q^{21} +2.18648e6 q^{23} +390625. q^{25} -3.00549e6 q^{27} -6.05248e6 q^{29} -2.24311e6 q^{31} -3.04502e6 q^{33} +1.50062e6 q^{35} -1.77025e7 q^{37} -1.54343e7 q^{39} +2.10270e7 q^{41} +970296. q^{43} -4.31760e6 q^{45} +5.10285e6 q^{47} +5.76480e6 q^{49} +4.31429e6 q^{51} -7.59495e7 q^{53} -1.68381e7 q^{55} +1.18779e8 q^{57} -8.43051e7 q^{59} -9.58227e7 q^{61} -1.65865e7 q^{63} -8.53474e7 q^{65} -1.61663e7 q^{67} +2.47129e8 q^{69} +1.51801e8 q^{71} -4.53805e8 q^{73} +4.41507e7 q^{75} -6.46851e7 q^{77} +8.05450e7 q^{79} -2.03725e8 q^{81} -3.35952e8 q^{83} +2.38568e7 q^{85} -6.84087e8 q^{87} +2.88479e8 q^{89} -3.27871e8 q^{91} -2.53530e8 q^{93} +6.56812e8 q^{95} +1.40961e9 q^{97} +1.86112e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 151 q^{3} + 4375 q^{5} + 16807 q^{7} + 55400 q^{9} - 40697 q^{11} - 58501 q^{13} - 94375 q^{15} - 220319 q^{17} - 922730 q^{19} - 362551 q^{21} - 3289058 q^{23} + 2734375 q^{25} - 9979909 q^{27} - 7316353 q^{29}+ \cdots - 823065222 q^{99}+O(q^{100}) \) 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
\(3\) 113.026 0.805624 0.402812 − 0.915283i \(-0.368033\pi\)
0.402812 + 0.915283i \(0.368033\pi\)
\(4\) 0 0
\(5\) 625.000 0.447214
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 0 0
\(9\) −6908.15 −0.350971
\(10\) 0 0
\(11\) −26940.9 −0.554811 −0.277405 − 0.960753i \(-0.589474\pi\)
−0.277405 + 0.960753i \(0.589474\pi\)
\(12\) 0 0
\(13\) −136556. −1.32607 −0.663033 − 0.748590i \(-0.730731\pi\)
−0.663033 + 0.748590i \(0.730731\pi\)
\(14\) 0 0
\(15\) 70641.2 0.360286
\(16\) 0 0
\(17\) 38170.8 0.110844 0.0554219 − 0.998463i \(-0.482350\pi\)
0.0554219 + 0.998463i \(0.482350\pi\)
\(18\) 0 0
\(19\) 1.05090e6 1.84999 0.924996 − 0.379977i \(-0.124068\pi\)
0.924996 + 0.379977i \(0.124068\pi\)
\(20\) 0 0
\(21\) 271375. 0.304497
\(22\) 0 0
\(23\) 2.18648e6 1.62919 0.814594 − 0.580032i \(-0.196960\pi\)
0.814594 + 0.580032i \(0.196960\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 0 0
\(27\) −3.00549e6 −1.08837
\(28\) 0 0
\(29\) −6.05248e6 −1.58907 −0.794534 − 0.607220i \(-0.792285\pi\)
−0.794534 + 0.607220i \(0.792285\pi\)
\(30\) 0 0
\(31\) −2.24311e6 −0.436238 −0.218119 − 0.975922i \(-0.569992\pi\)
−0.218119 + 0.975922i \(0.569992\pi\)
\(32\) 0 0
\(33\) −3.04502e6 −0.446969
\(34\) 0 0
\(35\) 1.50062e6 0.169031
\(36\) 0 0
\(37\) −1.77025e7 −1.55284 −0.776421 − 0.630215i \(-0.782967\pi\)
−0.776421 + 0.630215i \(0.782967\pi\)
\(38\) 0 0
\(39\) −1.54343e7 −1.06831
\(40\) 0 0
\(41\) 2.10270e7 1.16212 0.581059 − 0.813861i \(-0.302638\pi\)
0.581059 + 0.813861i \(0.302638\pi\)
\(42\) 0 0
\(43\) 970296. 0.0432809 0.0216404 − 0.999766i \(-0.493111\pi\)
0.0216404 + 0.999766i \(0.493111\pi\)
\(44\) 0 0
\(45\) −4.31760e6 −0.156959
\(46\) 0 0
\(47\) 5.10285e6 0.152536 0.0762680 − 0.997087i \(-0.475700\pi\)
0.0762680 + 0.997087i \(0.475700\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) 4.31429e6 0.0892984
\(52\) 0 0
\(53\) −7.59495e7 −1.32216 −0.661079 − 0.750316i \(-0.729901\pi\)
−0.661079 + 0.750316i \(0.729901\pi\)
\(54\) 0 0
\(55\) −1.68381e7 −0.248119
\(56\) 0 0
\(57\) 1.18779e8 1.49040
\(58\) 0 0
\(59\) −8.43051e7 −0.905773 −0.452887 − 0.891568i \(-0.649606\pi\)
−0.452887 + 0.891568i \(0.649606\pi\)
\(60\) 0 0
\(61\) −9.58227e7 −0.886103 −0.443052 − 0.896496i \(-0.646104\pi\)
−0.443052 + 0.896496i \(0.646104\pi\)
\(62\) 0 0
\(63\) −1.65865e7 −0.132654
\(64\) 0 0
\(65\) −8.53474e7 −0.593035
\(66\) 0 0
\(67\) −1.61663e7 −0.0980111 −0.0490055 − 0.998799i \(-0.515605\pi\)
−0.0490055 + 0.998799i \(0.515605\pi\)
\(68\) 0 0
\(69\) 2.47129e8 1.31251
\(70\) 0 0
\(71\) 1.51801e8 0.708945 0.354473 − 0.935066i \(-0.384660\pi\)
0.354473 + 0.935066i \(0.384660\pi\)
\(72\) 0 0
\(73\) −4.53805e8 −1.87032 −0.935160 − 0.354225i \(-0.884745\pi\)
−0.935160 + 0.354225i \(0.884745\pi\)
\(74\) 0 0
\(75\) 4.41507e7 0.161125
\(76\) 0 0
\(77\) −6.46851e7 −0.209699
\(78\) 0 0
\(79\) 8.05450e7 0.232657 0.116329 − 0.993211i \(-0.462887\pi\)
0.116329 + 0.993211i \(0.462887\pi\)
\(80\) 0 0
\(81\) −2.03725e8 −0.525849
\(82\) 0 0
\(83\) −3.35952e8 −0.777010 −0.388505 − 0.921447i \(-0.627008\pi\)
−0.388505 + 0.921447i \(0.627008\pi\)
\(84\) 0 0
\(85\) 2.38568e7 0.0495709
\(86\) 0 0
\(87\) −6.84087e8 −1.28019
\(88\) 0 0
\(89\) 2.88479e8 0.487370 0.243685 − 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(90\) 0 0
\(91\) −3.27871e8 −0.501206
\(92\) 0 0
\(93\) −2.53530e8 −0.351444
\(94\) 0 0
\(95\) 6.56812e8 0.827342
\(96\) 0 0
\(97\) 1.40961e9 1.61669 0.808346 − 0.588708i \(-0.200363\pi\)
0.808346 + 0.588708i \(0.200363\pi\)
\(98\) 0 0
\(99\) 1.86112e8 0.194722
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 280.10.a.e.1.5 ✓ 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.10.a.e.1.5 ✓ 7 1.1 even 1 trivial