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Properties

Label 9280.2.a.bj.1.2
Level $9280$
Weight $2$
Character 9280.1
Self dual yes
Analytic conductor $74.101$
Analytic rank $0$
Dimension $3$
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] = [9280,2,Mod(1,9280)] 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("9280.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9280.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-2,0,-3,0,4,0,3,0,-2,0,2,0,2,0,-4,0,10,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(74.1011730757\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 9280.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.806063 q^{3} -1.00000 q^{5} +1.19394 q^{7} -2.35026 q^{9} -4.15633 q^{11} -2.96239 q^{13} +0.806063 q^{15} +5.50659 q^{17} +3.19394 q^{19} -0.962389 q^{21} +1.84367 q^{23} +1.00000 q^{25} +4.31265 q^{27} +1.00000 q^{29} -4.80606 q^{31} +3.35026 q^{33} -1.19394 q^{35} +9.50659 q^{37} +2.38787 q^{39} -11.2750 q^{41} +0.0303172 q^{43} +2.35026 q^{45} +4.80606 q^{47} -5.57452 q^{49} -4.43866 q^{51} +1.35026 q^{53} +4.15633 q^{55} -2.57452 q^{57} -13.2750 q^{59} -8.88717 q^{61} -2.80606 q^{63} +2.96239 q^{65} -5.84367 q^{67} -1.48612 q^{69} -1.27504 q^{71} -15.2447 q^{73} -0.806063 q^{75} -4.96239 q^{77} -4.93207 q^{79} +3.57452 q^{81} -4.41819 q^{83} -5.50659 q^{85} -0.806063 q^{87} -3.61213 q^{89} -3.53690 q^{91} +3.87399 q^{93} -3.19394 q^{95} -1.38058 q^{97} +9.76845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} - 4 q^{17} + 10 q^{19} + 8 q^{21} + 16 q^{23} + 3 q^{25} - 8 q^{27} + 3 q^{29} - 14 q^{31} - 4 q^{35} + 8 q^{37} + 8 q^{39} - 2 q^{41}+ \cdots + 18 q^{99}+O(q^{100}) \) 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
\(3\) −0.806063 −0.465381 −0.232690 − 0.972551i \(-0.574753\pi\)
−0.232690 + 0.972551i \(0.574753\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.19394 0.451266 0.225633 − 0.974212i \(-0.427555\pi\)
0.225633 + 0.974212i \(0.427555\pi\)
\(8\) 0 0
\(9\) −2.35026 −0.783421
\(10\) 0 0
\(11\) −4.15633 −1.25318 −0.626590 − 0.779349i \(-0.715550\pi\)
−0.626590 + 0.779349i \(0.715550\pi\)
\(12\) 0 0
\(13\) −2.96239 −0.821619 −0.410809 − 0.911721i \(-0.634754\pi\)
−0.410809 + 0.911721i \(0.634754\pi\)
\(14\) 0 0
\(15\) 0.806063 0.208125
\(16\) 0 0
\(17\) 5.50659 1.33554 0.667772 − 0.744366i \(-0.267248\pi\)
0.667772 + 0.744366i \(0.267248\pi\)
\(18\) 0 0
\(19\) 3.19394 0.732739 0.366370 − 0.930469i \(-0.380601\pi\)
0.366370 + 0.930469i \(0.380601\pi\)
\(20\) 0 0
\(21\) −0.962389 −0.210010
\(22\) 0 0
\(23\) 1.84367 0.384433 0.192216 − 0.981353i \(-0.438432\pi\)
0.192216 + 0.981353i \(0.438432\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.31265 0.829970
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −4.80606 −0.863194 −0.431597 − 0.902066i \(-0.642050\pi\)
−0.431597 + 0.902066i \(0.642050\pi\)
\(32\) 0 0
\(33\) 3.35026 0.583206
\(34\) 0 0
\(35\) −1.19394 −0.201812
\(36\) 0 0
\(37\) 9.50659 1.56287 0.781437 − 0.623985i \(-0.214487\pi\)
0.781437 + 0.623985i \(0.214487\pi\)
\(38\) 0 0
\(39\) 2.38787 0.382366
\(40\) 0 0
\(41\) −11.2750 −1.76087 −0.880433 − 0.474171i \(-0.842748\pi\)
−0.880433 + 0.474171i \(0.842748\pi\)
\(42\) 0 0
\(43\) 0.0303172 0.00462332 0.00231166 − 0.999997i \(-0.499264\pi\)
0.00231166 + 0.999997i \(0.499264\pi\)
\(44\) 0 0
\(45\) 2.35026 0.350356
\(46\) 0 0
\(47\) 4.80606 0.701036 0.350518 − 0.936556i \(-0.386005\pi\)
0.350518 + 0.936556i \(0.386005\pi\)
\(48\) 0 0
\(49\) −5.57452 −0.796359
\(50\) 0 0
\(51\) −4.43866 −0.621536
\(52\) 0 0
\(53\) 1.35026 0.185473 0.0927364 − 0.995691i \(-0.470439\pi\)
0.0927364 + 0.995691i \(0.470439\pi\)
\(54\) 0 0
\(55\) 4.15633 0.560439
\(56\) 0 0
\(57\) −2.57452 −0.341003
\(58\) 0 0
\(59\) −13.2750 −1.72826 −0.864131 − 0.503266i \(-0.832132\pi\)
−0.864131 + 0.503266i \(0.832132\pi\)
\(60\) 0 0
\(61\) −8.88717 −1.13788 −0.568942 − 0.822377i \(-0.692647\pi\)
−0.568942 + 0.822377i \(0.692647\pi\)
\(62\) 0 0
\(63\) −2.80606 −0.353531
\(64\) 0 0
\(65\) 2.96239 0.367439
\(66\) 0 0
\(67\) −5.84367 −0.713919 −0.356959 − 0.934120i \(-0.616187\pi\)
−0.356959 + 0.934120i \(0.616187\pi\)
\(68\) 0 0
\(69\) −1.48612 −0.178908
\(70\) 0 0
\(71\) −1.27504 −0.151319 −0.0756596 − 0.997134i \(-0.524106\pi\)
−0.0756596 + 0.997134i \(0.524106\pi\)
\(72\) 0 0
\(73\) −15.2447 −1.78426 −0.892130 − 0.451779i \(-0.850790\pi\)
−0.892130 + 0.451779i \(0.850790\pi\)
\(74\) 0 0
\(75\) −0.806063 −0.0930762
\(76\) 0 0
\(77\) −4.96239 −0.565517
\(78\) 0 0
\(79\) −4.93207 −0.554901 −0.277451 − 0.960740i \(-0.589490\pi\)
−0.277451 + 0.960740i \(0.589490\pi\)
\(80\) 0 0
\(81\) 3.57452 0.397168
\(82\) 0 0
\(83\) −4.41819 −0.484959 −0.242480 − 0.970156i \(-0.577961\pi\)
−0.242480 + 0.970156i \(0.577961\pi\)
\(84\) 0 0
\(85\) −5.50659 −0.597273
\(86\) 0 0
\(87\) −0.806063 −0.0864191
\(88\) 0 0
\(89\) −3.61213 −0.382885 −0.191442 − 0.981504i \(-0.561316\pi\)
−0.191442 + 0.981504i \(0.561316\pi\)
\(90\) 0 0
\(91\) −3.53690 −0.370768
\(92\) 0 0
\(93\) 3.87399 0.401714
\(94\) 0 0
\(95\) −3.19394 −0.327691
\(96\) 0 0
\(97\) −1.38058 −0.140177 −0.0700883 − 0.997541i \(-0.522328\pi\)
−0.0700883 + 0.997541i \(0.522328\pi\)
\(98\) 0 0
\(99\) 9.76845 0.981766
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 9280.2.a.bj.1.2 3
4.3 odd 2 9280.2.a.br.1.2 3
8.3 odd 2 2320.2.a.n.1.2 3
8.5 even 2 145.2.a.c.1.1 ✓ 3
24.5 odd 2 1305.2.a.p.1.3 3
40.13 odd 4 725.2.b.e.349.5 6
40.29 even 2 725.2.a.e.1.3 3
40.37 odd 4 725.2.b.e.349.2 6
56.13 odd 2 7105.2.a.o.1.1 3
120.29 odd 2 6525.2.a.be.1.1 3
232.173 even 2 4205.2.a.f.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.1 ✓ 3 8.5 even 2
725.2.a.e.1.3 3 40.29 even 2
725.2.b.e.349.2 6 40.37 odd 4
725.2.b.e.349.5 6 40.13 odd 4
1305.2.a.p.1.3 3 24.5 odd 2
2320.2.a.n.1.2 3 8.3 odd 2
4205.2.a.f.1.3 3 232.173 even 2
6525.2.a.be.1.1 3 120.29 odd 2
7105.2.a.o.1.1 3 56.13 odd 2
9280.2.a.bj.1.2 3 1.1 even 1 trivial
9280.2.a.br.1.2 3 4.3 odd 2