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Properties

Label 280.10.a.f.1.6
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 1.6
Root \(-178.506\) 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)
 
\(f(q)\) \(=\) \(q+180.506 q^{3} -625.000 q^{5} -2401.00 q^{7} +12899.5 q^{9} +25672.0 q^{11} +163180. q^{13} -112816. q^{15} -640683. q^{17} -326711. q^{19} -433396. q^{21} +356439. q^{23} +390625. q^{25} -1.22446e6 q^{27} -1.78271e6 q^{29} -3.63203e6 q^{31} +4.63396e6 q^{33} +1.50062e6 q^{35} +1.28465e7 q^{37} +2.94550e7 q^{39} +1.99829e7 q^{41} -3.20874e7 q^{43} -8.06220e6 q^{45} +7.80318e6 q^{47} +5.76480e6 q^{49} -1.15647e8 q^{51} -5.49807e7 q^{53} -1.60450e7 q^{55} -5.89734e7 q^{57} +5.07137e7 q^{59} -6.52337e7 q^{61} -3.09717e7 q^{63} -1.01987e8 q^{65} +1.97776e8 q^{67} +6.43395e7 q^{69} -2.29453e8 q^{71} -2.57344e8 q^{73} +7.05103e7 q^{75} -6.16385e7 q^{77} -1.89873e8 q^{79} -4.74924e8 q^{81} +1.19157e8 q^{83} +4.00427e8 q^{85} -3.21791e8 q^{87} -7.17169e8 q^{89} -3.91795e8 q^{91} -6.55604e8 q^{93} +2.04195e8 q^{95} +2.20430e8 q^{97} +3.31157e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 11 q^{3} - 4375 q^{5} - 16807 q^{7} + 14016 q^{9} + 11047 q^{11} + 56289 q^{13} - 6875 q^{15} + 114003 q^{17} + 35578 q^{19} - 26411 q^{21} + 264730 q^{23} + 2734375 q^{25} + 3586193 q^{27} + 3733791 q^{29}+ \cdots + 590955218 q^{99}+O(q^{100}) \) 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
\(3\) 180.506 1.28661 0.643305 − 0.765610i \(-0.277563\pi\)
0.643305 + 0.765610i \(0.277563\pi\)
\(4\) 0 0
\(5\) −625.000 −0.447214
\(6\) 0 0
\(7\) −2401.00 −0.377964
\(8\) 0 0
\(9\) 12899.5 0.655363
\(10\) 0 0
\(11\) 25672.0 0.528680 0.264340 − 0.964430i \(-0.414846\pi\)
0.264340 + 0.964430i \(0.414846\pi\)
\(12\) 0 0
\(13\) 163180. 1.58461 0.792303 − 0.610128i \(-0.208882\pi\)
0.792303 + 0.610128i \(0.208882\pi\)
\(14\) 0 0
\(15\) −112816. −0.575389
\(16\) 0 0
\(17\) −640683. −1.86047 −0.930235 − 0.366964i \(-0.880397\pi\)
−0.930235 + 0.366964i \(0.880397\pi\)
\(18\) 0 0
\(19\) −326711. −0.575139 −0.287570 − 0.957760i \(-0.592847\pi\)
−0.287570 + 0.957760i \(0.592847\pi\)
\(20\) 0 0
\(21\) −433396. −0.486293
\(22\) 0 0
\(23\) 356439. 0.265589 0.132794 − 0.991144i \(-0.457605\pi\)
0.132794 + 0.991144i \(0.457605\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 0 0
\(27\) −1.22446e6 −0.443413
\(28\) 0 0
\(29\) −1.78271e6 −0.468049 −0.234024 − 0.972231i \(-0.575190\pi\)
−0.234024 + 0.972231i \(0.575190\pi\)
\(30\) 0 0
\(31\) −3.63203e6 −0.706353 −0.353176 − 0.935557i \(-0.614898\pi\)
−0.353176 + 0.935557i \(0.614898\pi\)
\(32\) 0 0
\(33\) 4.63396e6 0.680204
\(34\) 0 0
\(35\) 1.50062e6 0.169031
\(36\) 0 0
\(37\) 1.28465e7 1.12688 0.563438 − 0.826158i \(-0.309478\pi\)
0.563438 + 0.826158i \(0.309478\pi\)
\(38\) 0 0
\(39\) 2.94550e7 2.03877
\(40\) 0 0
\(41\) 1.99829e7 1.10441 0.552207 − 0.833707i \(-0.313786\pi\)
0.552207 + 0.833707i \(0.313786\pi\)
\(42\) 0 0
\(43\) −3.20874e7 −1.43129 −0.715644 − 0.698465i \(-0.753867\pi\)
−0.715644 + 0.698465i \(0.753867\pi\)
\(44\) 0 0
\(45\) −8.06220e6 −0.293087
\(46\) 0 0
\(47\) 7.80318e6 0.233255 0.116628 − 0.993176i \(-0.462792\pi\)
0.116628 + 0.993176i \(0.462792\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) −1.15647e8 −2.39370
\(52\) 0 0
\(53\) −5.49807e7 −0.957125 −0.478563 − 0.878053i \(-0.658842\pi\)
−0.478563 + 0.878053i \(0.658842\pi\)
\(54\) 0 0
\(55\) −1.60450e7 −0.236433
\(56\) 0 0
\(57\) −5.89734e7 −0.739979
\(58\) 0 0
\(59\) 5.07137e7 0.544868 0.272434 − 0.962175i \(-0.412171\pi\)
0.272434 + 0.962175i \(0.412171\pi\)
\(60\) 0 0
\(61\) −6.52337e7 −0.603236 −0.301618 − 0.953429i \(-0.597527\pi\)
−0.301618 + 0.953429i \(0.597527\pi\)
\(62\) 0 0
\(63\) −3.09717e7 −0.247704
\(64\) 0 0
\(65\) −1.01987e8 −0.708657
\(66\) 0 0
\(67\) 1.97776e8 1.19905 0.599523 − 0.800357i \(-0.295357\pi\)
0.599523 + 0.800357i \(0.295357\pi\)
\(68\) 0 0
\(69\) 6.43395e7 0.341709
\(70\) 0 0
\(71\) −2.29453e8 −1.07160 −0.535799 − 0.844346i \(-0.679990\pi\)
−0.535799 + 0.844346i \(0.679990\pi\)
\(72\) 0 0
\(73\) −2.57344e8 −1.06062 −0.530312 − 0.847803i \(-0.677925\pi\)
−0.530312 + 0.847803i \(0.677925\pi\)
\(74\) 0 0
\(75\) 7.05103e7 0.257322
\(76\) 0 0
\(77\) −6.16385e7 −0.199822
\(78\) 0 0
\(79\) −1.89873e8 −0.548456 −0.274228 − 0.961665i \(-0.588422\pi\)
−0.274228 + 0.961665i \(0.588422\pi\)
\(80\) 0 0
\(81\) −4.74924e8 −1.22586
\(82\) 0 0
\(83\) 1.19157e8 0.275594 0.137797 − 0.990461i \(-0.455998\pi\)
0.137797 + 0.990461i \(0.455998\pi\)
\(84\) 0 0
\(85\) 4.00427e8 0.832028
\(86\) 0 0
\(87\) −3.21791e8 −0.602196
\(88\) 0 0
\(89\) −7.17169e8 −1.21162 −0.605810 − 0.795609i \(-0.707151\pi\)
−0.605810 + 0.795609i \(0.707151\pi\)
\(90\) 0 0
\(91\) −3.91795e8 −0.598925
\(92\) 0 0
\(93\) −6.55604e8 −0.908800
\(94\) 0 0
\(95\) 2.04195e8 0.257210
\(96\) 0 0
\(97\) 2.20430e8 0.252813 0.126406 − 0.991979i \(-0.459656\pi\)
0.126406 + 0.991979i \(0.459656\pi\)
\(98\) 0 0
\(99\) 3.31157e8 0.346477
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.1.6 ✓ 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.10.a.f.1.6 ✓ 7 1.1 even 1 trivial