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Properties

Label 280.10.a.f.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 = [7,0,11,0,-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: \( x^{7} - 3 x^{6} - 75886 x^{5} + 1263838 x^{4} + 1492027269 x^{3} - 43705600687 x^{2} + \cdots + 112929763661700 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{21}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-51.7160\) 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)
 
\(f(q)\) \(=\) \(q+53.7160 q^{3} -625.000 q^{5} -2401.00 q^{7} -16797.6 q^{9} +20897.5 q^{11} +9502.88 q^{13} -33572.5 q^{15} +466462. q^{17} -634122. q^{19} -128972. q^{21} +1.17287e6 q^{23} +390625. q^{25} -1.95959e6 q^{27} +4.14686e6 q^{29} +6.45599e6 q^{31} +1.12253e6 q^{33} +1.50062e6 q^{35} +811236. q^{37} +510457. q^{39} -2.39034e7 q^{41} -1.42410e7 q^{43} +1.04985e7 q^{45} +4.09361e7 q^{47} +5.76480e6 q^{49} +2.50565e7 q^{51} -1.01033e8 q^{53} -1.30609e7 q^{55} -3.40625e7 q^{57} -3.15180e7 q^{59} +5.03726e7 q^{61} +4.03310e7 q^{63} -5.93930e6 q^{65} -2.39612e8 q^{67} +6.30017e7 q^{69} +3.86742e8 q^{71} -1.62503e8 q^{73} +2.09828e7 q^{75} -5.01749e7 q^{77} -5.89381e8 q^{79} +2.25365e8 q^{81} -5.35738e8 q^{83} -2.91539e8 q^{85} +2.22753e8 q^{87} +7.23809e8 q^{89} -2.28164e7 q^{91} +3.46790e8 q^{93} +3.96326e8 q^{95} -4.11436e8 q^{97} -3.51028e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 11 q^{3} - 4375 q^{5} - 16807 q^{7} + 14016 q^{9} + 11047 q^{11} + 56289 q^{13} - 6875 q^{15} + 114003 q^{17} + 35578 q^{19} - 26411 q^{21} + 264730 q^{23} + 2734375 q^{25} + 3586193 q^{27} + 3733791 q^{29}+ \cdots + 590955218 q^{99}+O(q^{100}) \) 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
\(3\) 53.7160 0.382876 0.191438 − 0.981505i \(-0.438685\pi\)
0.191438 + 0.981505i \(0.438685\pi\)
\(4\) 0 0
\(5\) −625.000 −0.447214
\(6\) 0 0
\(7\) −2401.00 −0.377964
\(8\) 0 0
\(9\) −16797.6 −0.853406
\(10\) 0 0
\(11\) 20897.5 0.430355 0.215178 − 0.976575i \(-0.430967\pi\)
0.215178 + 0.976575i \(0.430967\pi\)
\(12\) 0 0
\(13\) 9502.88 0.0922805 0.0461403 − 0.998935i \(-0.485308\pi\)
0.0461403 + 0.998935i \(0.485308\pi\)
\(14\) 0 0
\(15\) −33572.5 −0.171227
\(16\) 0 0
\(17\) 466462. 1.35455 0.677276 − 0.735729i \(-0.263160\pi\)
0.677276 + 0.735729i \(0.263160\pi\)
\(18\) 0 0
\(19\) −634122. −1.11630 −0.558151 − 0.829739i \(-0.688489\pi\)
−0.558151 + 0.829739i \(0.688489\pi\)
\(20\) 0 0
\(21\) −128972. −0.144714
\(22\) 0 0
\(23\) 1.17287e6 0.873923 0.436961 − 0.899480i \(-0.356055\pi\)
0.436961 + 0.899480i \(0.356055\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 0 0
\(27\) −1.95959e6 −0.709625
\(28\) 0 0
\(29\) 4.14686e6 1.08875 0.544376 − 0.838841i \(-0.316767\pi\)
0.544376 + 0.838841i \(0.316767\pi\)
\(30\) 0 0
\(31\) 6.45599e6 1.25555 0.627777 − 0.778393i \(-0.283965\pi\)
0.627777 + 0.778393i \(0.283965\pi\)
\(32\) 0 0
\(33\) 1.12253e6 0.164773
\(34\) 0 0
\(35\) 1.50062e6 0.169031
\(36\) 0 0
\(37\) 811236. 0.0711606 0.0355803 − 0.999367i \(-0.488672\pi\)
0.0355803 + 0.999367i \(0.488672\pi\)
\(38\) 0 0
\(39\) 510457. 0.0353320
\(40\) 0 0
\(41\) −2.39034e7 −1.32109 −0.660546 − 0.750786i \(-0.729675\pi\)
−0.660546 + 0.750786i \(0.729675\pi\)
\(42\) 0 0
\(43\) −1.42410e7 −0.635232 −0.317616 − 0.948219i \(-0.602882\pi\)
−0.317616 + 0.948219i \(0.602882\pi\)
\(44\) 0 0
\(45\) 1.04985e7 0.381655
\(46\) 0 0
\(47\) 4.09361e7 1.22368 0.611838 − 0.790983i \(-0.290430\pi\)
0.611838 + 0.790983i \(0.290430\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) 2.50565e7 0.518626
\(52\) 0 0
\(53\) −1.01033e8 −1.75883 −0.879413 − 0.476060i \(-0.842064\pi\)
−0.879413 + 0.476060i \(0.842064\pi\)
\(54\) 0 0
\(55\) −1.30609e7 −0.192461
\(56\) 0 0
\(57\) −3.40625e7 −0.427406
\(58\) 0 0
\(59\) −3.15180e7 −0.338629 −0.169315 − 0.985562i \(-0.554155\pi\)
−0.169315 + 0.985562i \(0.554155\pi\)
\(60\) 0 0
\(61\) 5.03726e7 0.465812 0.232906 − 0.972499i \(-0.425177\pi\)
0.232906 + 0.972499i \(0.425177\pi\)
\(62\) 0 0
\(63\) 4.03310e7 0.322557
\(64\) 0 0
\(65\) −5.93930e6 −0.0412691
\(66\) 0 0
\(67\) −2.39612e8 −1.45269 −0.726343 − 0.687333i \(-0.758782\pi\)
−0.726343 + 0.687333i \(0.758782\pi\)
\(68\) 0 0
\(69\) 6.30017e7 0.334604
\(70\) 0 0
\(71\) 3.86742e8 1.80617 0.903086 − 0.429459i \(-0.141296\pi\)
0.903086 + 0.429459i \(0.141296\pi\)
\(72\) 0 0
\(73\) −1.62503e8 −0.669742 −0.334871 − 0.942264i \(-0.608693\pi\)
−0.334871 + 0.942264i \(0.608693\pi\)
\(74\) 0 0
\(75\) 2.09828e7 0.0765752
\(76\) 0 0
\(77\) −5.01749e7 −0.162659
\(78\) 0 0
\(79\) −5.89381e8 −1.70245 −0.851225 − 0.524801i \(-0.824140\pi\)
−0.851225 + 0.524801i \(0.824140\pi\)
\(80\) 0 0
\(81\) 2.25365e8 0.581707
\(82\) 0 0
\(83\) −5.35738e8 −1.23908 −0.619542 − 0.784963i \(-0.712682\pi\)
−0.619542 + 0.784963i \(0.712682\pi\)
\(84\) 0 0
\(85\) −2.91539e8 −0.605774
\(86\) 0 0
\(87\) 2.22753e8 0.416857
\(88\) 0 0
\(89\) 7.23809e8 1.22284 0.611419 − 0.791307i \(-0.290599\pi\)
0.611419 + 0.791307i \(0.290599\pi\)
\(90\) 0 0
\(91\) −2.28164e7 −0.0348788
\(92\) 0 0
\(93\) 3.46790e8 0.480721
\(94\) 0 0
\(95\) 3.96326e8 0.499226
\(96\) 0 0
\(97\) −4.11436e8 −0.471877 −0.235939 − 0.971768i \(-0.575816\pi\)
−0.235939 + 0.971768i \(0.575816\pi\)
\(98\) 0 0
\(99\) −3.51028e8 −0.367268
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.f.1.5 ✓ 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.10.a.f.1.5 ✓ 7 1.1 even 1 trivial