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Properties

Label 280.10.a.a.1.3
Level $280$
Weight $10$
Character 280.1
Self dual yes
Analytic conductor $144.210$
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] = [280,10,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 = [6,0,-84,0,-3750] 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: \(6\)
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: \( x^{6} - 60766x^{4} - 506116x^{3} + 731940897x^{2} - 5328001988x - 1003443514620 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15}\cdot 3\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(46.3193\) 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)
 
\(f(q)\) \(=\) \(q-60.3193 q^{3} -625.000 q^{5} +2401.00 q^{7} -16044.6 q^{9} +41307.9 q^{11} +66523.5 q^{13} +37699.6 q^{15} -71396.0 q^{17} -906592. q^{19} -144827. q^{21} -365492. q^{23} +390625. q^{25} +2.15506e6 q^{27} -2.44561e6 q^{29} +7.98192e6 q^{31} -2.49166e6 q^{33} -1.50062e6 q^{35} +1.90741e7 q^{37} -4.01265e6 q^{39} +2.25656e7 q^{41} -2.52786e7 q^{43} +1.00279e7 q^{45} -5.02935e7 q^{47} +5.76480e6 q^{49} +4.30656e6 q^{51} -2.68351e7 q^{53} -2.58174e7 q^{55} +5.46850e7 q^{57} +1.17216e8 q^{59} +2.60225e7 q^{61} -3.85230e7 q^{63} -4.15772e7 q^{65} +2.15456e8 q^{67} +2.20463e7 q^{69} -9.10840e7 q^{71} +1.04930e8 q^{73} -2.35622e7 q^{75} +9.91802e7 q^{77} +1.10864e8 q^{79} +1.85813e8 q^{81} -5.58947e8 q^{83} +4.46225e7 q^{85} +1.47518e8 q^{87} -1.98606e8 q^{89} +1.59723e8 q^{91} -4.81464e8 q^{93} +5.66620e8 q^{95} -9.55552e8 q^{97} -6.62767e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 84 q^{3} - 3750 q^{5} + 14406 q^{7} + 4610 q^{9} - 29892 q^{11} + 16288 q^{13} + 52500 q^{15} - 373400 q^{17} + 426816 q^{19} - 201684 q^{21} - 695088 q^{23} + 2343750 q^{25} - 3332412 q^{27} + 180920 q^{29}+ \cdots - 1678978192 q^{99}+O(q^{100}) \) 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
\(3\) −60.3193 −0.429943 −0.214971 − 0.976620i \(-0.568966\pi\)
−0.214971 + 0.976620i \(0.568966\pi\)
\(4\) 0 0
\(5\) −625.000 −0.447214
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 0 0
\(9\) −16044.6 −0.815149
\(10\) 0 0
\(11\) 41307.9 0.850679 0.425339 − 0.905034i \(-0.360155\pi\)
0.425339 + 0.905034i \(0.360155\pi\)
\(12\) 0 0
\(13\) 66523.5 0.645996 0.322998 − 0.946400i \(-0.395309\pi\)
0.322998 + 0.946400i \(0.395309\pi\)
\(14\) 0 0
\(15\) 37699.6 0.192276
\(16\) 0 0
\(17\) −71396.0 −0.207326 −0.103663 − 0.994612i \(-0.533056\pi\)
−0.103663 + 0.994612i \(0.533056\pi\)
\(18\) 0 0
\(19\) −906592. −1.59596 −0.797978 − 0.602687i \(-0.794097\pi\)
−0.797978 + 0.602687i \(0.794097\pi\)
\(20\) 0 0
\(21\) −144827. −0.162503
\(22\) 0 0
\(23\) −365492. −0.272335 −0.136167 − 0.990686i \(-0.543478\pi\)
−0.136167 + 0.990686i \(0.543478\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 0 0
\(27\) 2.15506e6 0.780411
\(28\) 0 0
\(29\) −2.44561e6 −0.642092 −0.321046 − 0.947064i \(-0.604034\pi\)
−0.321046 + 0.947064i \(0.604034\pi\)
\(30\) 0 0
\(31\) 7.98192e6 1.55231 0.776157 − 0.630539i \(-0.217166\pi\)
0.776157 + 0.630539i \(0.217166\pi\)
\(32\) 0 0
\(33\) −2.49166e6 −0.365743
\(34\) 0 0
\(35\) −1.50062e6 −0.169031
\(36\) 0 0
\(37\) 1.90741e7 1.67316 0.836578 − 0.547847i \(-0.184552\pi\)
0.836578 + 0.547847i \(0.184552\pi\)
\(38\) 0 0
\(39\) −4.01265e6 −0.277742
\(40\) 0 0
\(41\) 2.25656e7 1.24715 0.623577 − 0.781762i \(-0.285679\pi\)
0.623577 + 0.781762i \(0.285679\pi\)
\(42\) 0 0
\(43\) −2.52786e7 −1.12757 −0.563787 − 0.825920i \(-0.690656\pi\)
−0.563787 + 0.825920i \(0.690656\pi\)
\(44\) 0 0
\(45\) 1.00279e7 0.364546
\(46\) 0 0
\(47\) −5.02935e7 −1.50339 −0.751695 − 0.659511i \(-0.770763\pi\)
−0.751695 + 0.659511i \(0.770763\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) 4.30656e6 0.0891383
\(52\) 0 0
\(53\) −2.68351e7 −0.467157 −0.233578 − 0.972338i \(-0.575044\pi\)
−0.233578 + 0.972338i \(0.575044\pi\)
\(54\) 0 0
\(55\) −2.58174e7 −0.380435
\(56\) 0 0
\(57\) 5.46850e7 0.686170
\(58\) 0 0
\(59\) 1.17216e8 1.25937 0.629686 − 0.776850i \(-0.283183\pi\)
0.629686 + 0.776850i \(0.283183\pi\)
\(60\) 0 0
\(61\) 2.60225e7 0.240638 0.120319 − 0.992735i \(-0.461608\pi\)
0.120319 + 0.992735i \(0.461608\pi\)
\(62\) 0 0
\(63\) −3.85230e7 −0.308097
\(64\) 0 0
\(65\) −4.15772e7 −0.288898
\(66\) 0 0
\(67\) 2.15456e8 1.30624 0.653120 − 0.757255i \(-0.273460\pi\)
0.653120 + 0.757255i \(0.273460\pi\)
\(68\) 0 0
\(69\) 2.20463e7 0.117088
\(70\) 0 0
\(71\) −9.10840e7 −0.425383 −0.212691 − 0.977119i \(-0.568223\pi\)
−0.212691 + 0.977119i \(0.568223\pi\)
\(72\) 0 0
\(73\) 1.04930e8 0.432460 0.216230 − 0.976342i \(-0.430624\pi\)
0.216230 + 0.976342i \(0.430624\pi\)
\(74\) 0 0
\(75\) −2.35622e7 −0.0859886
\(76\) 0 0
\(77\) 9.91802e7 0.321526
\(78\) 0 0
\(79\) 1.10864e8 0.320235 0.160118 − 0.987098i \(-0.448813\pi\)
0.160118 + 0.987098i \(0.448813\pi\)
\(80\) 0 0
\(81\) 1.85813e8 0.479617
\(82\) 0 0
\(83\) −5.58947e8 −1.29276 −0.646382 − 0.763014i \(-0.723719\pi\)
−0.646382 + 0.763014i \(0.723719\pi\)
\(84\) 0 0
\(85\) 4.46225e7 0.0927190
\(86\) 0 0
\(87\) 1.47518e8 0.276063
\(88\) 0 0
\(89\) −1.98606e8 −0.335534 −0.167767 − 0.985827i \(-0.553656\pi\)
−0.167767 + 0.985827i \(0.553656\pi\)
\(90\) 0 0
\(91\) 1.59723e8 0.244164
\(92\) 0 0
\(93\) −4.81464e8 −0.667407
\(94\) 0 0
\(95\) 5.66620e8 0.713733
\(96\) 0 0
\(97\) −9.55552e8 −1.09593 −0.547964 − 0.836502i \(-0.684597\pi\)
−0.547964 + 0.836502i \(0.684597\pi\)
\(98\) 0 0
\(99\) −6.62767e8 −0.693430
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.a.1.3 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.10.a.a.1.3 ✓ 6 1.1 even 1 trivial