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Properties

Label 280.10.a.e.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,-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 1.6
Root \(80.7365\) 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)
 
\(f(q)\) \(=\) \(q+127.408 q^{3} +625.000 q^{5} +2401.00 q^{7} -3450.12 q^{9} +33976.1 q^{11} +83397.8 q^{13} +79630.2 q^{15} -544342. q^{17} -748975. q^{19} +305907. q^{21} -1.19433e6 q^{23} +390625. q^{25} -2.94735e6 q^{27} -2.27588e6 q^{29} +8.90332e6 q^{31} +4.32884e6 q^{33} +1.50062e6 q^{35} -2.14270e7 q^{37} +1.06256e7 q^{39} -5.43659e6 q^{41} +9.15359e6 q^{43} -2.15633e6 q^{45} -4.17313e7 q^{47} +5.76480e6 q^{49} -6.93537e7 q^{51} +4.73171e6 q^{53} +2.12351e7 q^{55} -9.54256e7 q^{57} -9.99246e7 q^{59} -1.09394e7 q^{61} -8.28375e6 q^{63} +5.21236e7 q^{65} +3.32994e7 q^{67} -1.52167e8 q^{69} +2.53957e8 q^{71} +2.10851e8 q^{73} +4.97689e7 q^{75} +8.15766e7 q^{77} +1.72596e8 q^{79} -3.07608e8 q^{81} -7.22430e8 q^{83} -3.40214e8 q^{85} -2.89966e8 q^{87} -9.98575e8 q^{89} +2.00238e8 q^{91} +1.13436e9 q^{93} -4.68109e8 q^{95} -1.53254e9 q^{97} -1.17222e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 151 q^{3} + 4375 q^{5} + 16807 q^{7} + 55400 q^{9} - 40697 q^{11} - 58501 q^{13} - 94375 q^{15} - 220319 q^{17} - 922730 q^{19} - 362551 q^{21} - 3289058 q^{23} + 2734375 q^{25} - 9979909 q^{27} - 7316353 q^{29}+ \cdots - 823065222 q^{99}+O(q^{100}) \) 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
\(3\) 127.408 0.908139 0.454069 − 0.890966i \(-0.349972\pi\)
0.454069 + 0.890966i \(0.349972\pi\)
\(4\) 0 0
\(5\) 625.000 0.447214
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 0 0
\(9\) −3450.12 −0.175284
\(10\) 0 0
\(11\) 33976.1 0.699691 0.349846 − 0.936807i \(-0.386234\pi\)
0.349846 + 0.936807i \(0.386234\pi\)
\(12\) 0 0
\(13\) 83397.8 0.809859 0.404929 − 0.914348i \(-0.367296\pi\)
0.404929 + 0.914348i \(0.367296\pi\)
\(14\) 0 0
\(15\) 79630.2 0.406132
\(16\) 0 0
\(17\) −544342. −1.58071 −0.790354 − 0.612650i \(-0.790103\pi\)
−0.790354 + 0.612650i \(0.790103\pi\)
\(18\) 0 0
\(19\) −748975. −1.31849 −0.659244 − 0.751929i \(-0.729123\pi\)
−0.659244 + 0.751929i \(0.729123\pi\)
\(20\) 0 0
\(21\) 305907. 0.343244
\(22\) 0 0
\(23\) −1.19433e6 −0.889914 −0.444957 − 0.895552i \(-0.646781\pi\)
−0.444957 + 0.895552i \(0.646781\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 0 0
\(27\) −2.94735e6 −1.06732
\(28\) 0 0
\(29\) −2.27588e6 −0.597528 −0.298764 − 0.954327i \(-0.596574\pi\)
−0.298764 + 0.954327i \(0.596574\pi\)
\(30\) 0 0
\(31\) 8.90332e6 1.73151 0.865754 − 0.500470i \(-0.166840\pi\)
0.865754 + 0.500470i \(0.166840\pi\)
\(32\) 0 0
\(33\) 4.32884e6 0.635416
\(34\) 0 0
\(35\) 1.50062e6 0.169031
\(36\) 0 0
\(37\) −2.14270e7 −1.87955 −0.939775 − 0.341795i \(-0.888965\pi\)
−0.939775 + 0.341795i \(0.888965\pi\)
\(38\) 0 0
\(39\) 1.06256e7 0.735464
\(40\) 0 0
\(41\) −5.43659e6 −0.300469 −0.150234 − 0.988650i \(-0.548003\pi\)
−0.150234 + 0.988650i \(0.548003\pi\)
\(42\) 0 0
\(43\) 9.15359e6 0.408304 0.204152 − 0.978939i \(-0.434556\pi\)
0.204152 + 0.978939i \(0.434556\pi\)
\(44\) 0 0
\(45\) −2.15633e6 −0.0783896
\(46\) 0 0
\(47\) −4.17313e7 −1.24745 −0.623723 − 0.781645i \(-0.714381\pi\)
−0.623723 + 0.781645i \(0.714381\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) −6.93537e7 −1.43550
\(52\) 0 0
\(53\) 4.73171e6 0.0823715 0.0411857 − 0.999152i \(-0.486886\pi\)
0.0411857 + 0.999152i \(0.486886\pi\)
\(54\) 0 0
\(55\) 2.12351e7 0.312911
\(56\) 0 0
\(57\) −9.54256e7 −1.19737
\(58\) 0 0
\(59\) −9.99246e7 −1.07359 −0.536795 − 0.843713i \(-0.680365\pi\)
−0.536795 + 0.843713i \(0.680365\pi\)
\(60\) 0 0
\(61\) −1.09394e7 −0.101160 −0.0505801 − 0.998720i \(-0.516107\pi\)
−0.0505801 + 0.998720i \(0.516107\pi\)
\(62\) 0 0
\(63\) −8.28375e6 −0.0662513
\(64\) 0 0
\(65\) 5.21236e7 0.362180
\(66\) 0 0
\(67\) 3.32994e7 0.201883 0.100941 − 0.994892i \(-0.467815\pi\)
0.100941 + 0.994892i \(0.467815\pi\)
\(68\) 0 0
\(69\) −1.52167e8 −0.808165
\(70\) 0 0
\(71\) 2.53957e8 1.18604 0.593018 − 0.805189i \(-0.297936\pi\)
0.593018 + 0.805189i \(0.297936\pi\)
\(72\) 0 0
\(73\) 2.10851e8 0.869006 0.434503 − 0.900670i \(-0.356924\pi\)
0.434503 + 0.900670i \(0.356924\pi\)
\(74\) 0 0
\(75\) 4.97689e7 0.181628
\(76\) 0 0
\(77\) 8.15766e7 0.264458
\(78\) 0 0
\(79\) 1.72596e8 0.498551 0.249276 − 0.968433i \(-0.419808\pi\)
0.249276 + 0.968433i \(0.419808\pi\)
\(80\) 0 0
\(81\) −3.07608e8 −0.793991
\(82\) 0 0
\(83\) −7.22430e8 −1.67088 −0.835438 − 0.549585i \(-0.814786\pi\)
−0.835438 + 0.549585i \(0.814786\pi\)
\(84\) 0 0
\(85\) −3.40214e8 −0.706914
\(86\) 0 0
\(87\) −2.89966e8 −0.542639
\(88\) 0 0
\(89\) −9.98575e8 −1.68704 −0.843520 − 0.537097i \(-0.819521\pi\)
−0.843520 + 0.537097i \(0.819521\pi\)
\(90\) 0 0
\(91\) 2.00238e8 0.306098
\(92\) 0 0
\(93\) 1.13436e9 1.57245
\(94\) 0 0
\(95\) −4.68109e8 −0.589646
\(96\) 0 0
\(97\) −1.53254e9 −1.75768 −0.878838 − 0.477120i \(-0.841681\pi\)
−0.878838 + 0.477120i \(0.841681\pi\)
\(98\) 0 0
\(99\) −1.17222e8 −0.122645
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.1.6 ✓ 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.10.a.e.1.6 ✓ 7 1.1 even 1 trivial