↑

Properties

Label 9280.2.a.bg.1.1
Level $9280$
Weight $2$
Character 9280.1
Self dual yes
Analytic conductor $74.101$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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,-3,0,3,0,7,0,2,0,2,0,9,0,-3,0,1,0,6,0,-20] 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: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4640)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.11491\) 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)
 
\(f(q)\) \(=\) \(q-3.11491 q^{3} +1.00000 q^{5} +4.75698 q^{7} +6.70265 q^{9} -2.94567 q^{11} +0.885092 q^{13} -3.11491 q^{15} -1.47283 q^{17} -2.22982 q^{19} -14.8176 q^{21} -3.11491 q^{23} +1.00000 q^{25} -11.5334 q^{27} -1.00000 q^{29} -4.18869 q^{31} +9.17548 q^{33} +4.75698 q^{35} -7.51396 q^{37} -2.75698 q^{39} -0.945668 q^{41} -3.70265 q^{43} +6.70265 q^{45} -3.17548 q^{47} +15.6289 q^{49} +4.58774 q^{51} +13.9325 q^{53} -2.94567 q^{55} +6.94567 q^{57} +13.2361 q^{59} -4.62887 q^{61} +31.8844 q^{63} +0.885092 q^{65} -0.824517 q^{67} +9.70265 q^{69} -9.89134 q^{71} -2.90454 q^{73} -3.11491 q^{75} -14.0125 q^{77} +6.54661 q^{79} +15.8176 q^{81} -4.94567 q^{83} -1.47283 q^{85} +3.11491 q^{87} +13.6615 q^{89} +4.21037 q^{91} +13.0474 q^{93} -2.22982 q^{95} +11.5745 q^{97} -19.7438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} + 7 q^{7} + 2 q^{9} + 2 q^{11} + 9 q^{13} - 3 q^{15} + q^{17} + 6 q^{19} - 20 q^{21} - 3 q^{23} + 3 q^{25} - 12 q^{27} - 3 q^{29} - 9 q^{31} + 4 q^{33} + 7 q^{35} - 8 q^{37} - q^{39}+ \cdots - 32 q^{99}+O(q^{100}) \) 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
\(3\) −3.11491 −1.79839 −0.899196 − 0.437545i \(-0.855848\pi\)
−0.899196 + 0.437545i \(0.855848\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.75698 1.79797 0.898985 − 0.437980i \(-0.144306\pi\)
0.898985 + 0.437980i \(0.144306\pi\)
\(8\) 0 0
\(9\) 6.70265 2.23422
\(10\) 0 0
\(11\) −2.94567 −0.888152 −0.444076 − 0.895989i \(-0.646468\pi\)
−0.444076 + 0.895989i \(0.646468\pi\)
\(12\) 0 0
\(13\) 0.885092 0.245480 0.122740 − 0.992439i \(-0.460832\pi\)
0.122740 + 0.992439i \(0.460832\pi\)
\(14\) 0 0
\(15\) −3.11491 −0.804266
\(16\) 0 0
\(17\) −1.47283 −0.357215 −0.178607 − 0.983920i \(-0.557159\pi\)
−0.178607 + 0.983920i \(0.557159\pi\)
\(18\) 0 0
\(19\) −2.22982 −0.511555 −0.255777 − 0.966736i \(-0.582331\pi\)
−0.255777 + 0.966736i \(0.582331\pi\)
\(20\) 0 0
\(21\) −14.8176 −3.23346
\(22\) 0 0
\(23\) −3.11491 −0.649503 −0.324752 − 0.945799i \(-0.605281\pi\)
−0.324752 + 0.945799i \(0.605281\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −11.5334 −2.21961
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −4.18869 −0.752310 −0.376155 − 0.926557i \(-0.622754\pi\)
−0.376155 + 0.926557i \(0.622754\pi\)
\(32\) 0 0
\(33\) 9.17548 1.59725
\(34\) 0 0
\(35\) 4.75698 0.804077
\(36\) 0 0
\(37\) −7.51396 −1.23529 −0.617644 − 0.786458i \(-0.711913\pi\)
−0.617644 + 0.786458i \(0.711913\pi\)
\(38\) 0 0
\(39\) −2.75698 −0.441470
\(40\) 0 0
\(41\) −0.945668 −0.147689 −0.0738443 − 0.997270i \(-0.523527\pi\)
−0.0738443 + 0.997270i \(0.523527\pi\)
\(42\) 0 0
\(43\) −3.70265 −0.564649 −0.282324 − 0.959319i \(-0.591105\pi\)
−0.282324 + 0.959319i \(0.591105\pi\)
\(44\) 0 0
\(45\) 6.70265 0.999172
\(46\) 0 0
\(47\) −3.17548 −0.463192 −0.231596 − 0.972812i \(-0.574395\pi\)
−0.231596 + 0.972812i \(0.574395\pi\)
\(48\) 0 0
\(49\) 15.6289 2.23270
\(50\) 0 0
\(51\) 4.58774 0.642412
\(52\) 0 0
\(53\) 13.9325 1.91377 0.956886 − 0.290465i \(-0.0938100\pi\)
0.956886 + 0.290465i \(0.0938100\pi\)
\(54\) 0 0
\(55\) −2.94567 −0.397194
\(56\) 0 0
\(57\) 6.94567 0.919976
\(58\) 0 0
\(59\) 13.2361 1.72319 0.861594 − 0.507598i \(-0.169466\pi\)
0.861594 + 0.507598i \(0.169466\pi\)
\(60\) 0 0
\(61\) −4.62887 −0.592666 −0.296333 − 0.955085i \(-0.595764\pi\)
−0.296333 + 0.955085i \(0.595764\pi\)
\(62\) 0 0
\(63\) 31.8844 4.01705
\(64\) 0 0
\(65\) 0.885092 0.109782
\(66\) 0 0
\(67\) −0.824517 −0.100731 −0.0503654 − 0.998731i \(-0.516039\pi\)
−0.0503654 + 0.998731i \(0.516039\pi\)
\(68\) 0 0
\(69\) 9.70265 1.16806
\(70\) 0 0
\(71\) −9.89134 −1.17389 −0.586943 − 0.809628i \(-0.699669\pi\)
−0.586943 + 0.809628i \(0.699669\pi\)
\(72\) 0 0
\(73\) −2.90454 −0.339951 −0.169975 − 0.985448i \(-0.554369\pi\)
−0.169975 + 0.985448i \(0.554369\pi\)
\(74\) 0 0
\(75\) −3.11491 −0.359679
\(76\) 0 0
\(77\) −14.0125 −1.59687
\(78\) 0 0
\(79\) 6.54661 0.736552 0.368276 − 0.929717i \(-0.379948\pi\)
0.368276 + 0.929717i \(0.379948\pi\)
\(80\) 0 0
\(81\) 15.8176 1.75751
\(82\) 0 0
\(83\) −4.94567 −0.542858 −0.271429 − 0.962459i \(-0.587496\pi\)
−0.271429 + 0.962459i \(0.587496\pi\)
\(84\) 0 0
\(85\) −1.47283 −0.159751
\(86\) 0 0
\(87\) 3.11491 0.333953
\(88\) 0 0
\(89\) 13.6615 1.44812 0.724059 − 0.689738i \(-0.242274\pi\)
0.724059 + 0.689738i \(0.242274\pi\)
\(90\) 0 0
\(91\) 4.21037 0.441367
\(92\) 0 0
\(93\) 13.0474 1.35295
\(94\) 0 0
\(95\) −2.22982 −0.228774
\(96\) 0 0
\(97\) 11.5745 1.17522 0.587608 − 0.809146i \(-0.300070\pi\)
0.587608 + 0.809146i \(0.300070\pi\)
\(98\) 0 0
\(99\) −19.7438 −1.98432
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.bg.1.1 3
4.3 odd 2 9280.2.a.bx.1.3 3
8.3 odd 2 4640.2.a.j.1.1 ✓ 3
8.5 even 2 4640.2.a.k.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4640.2.a.j.1.1 ✓ 3 8.3 odd 2
4640.2.a.k.1.3 yes 3 8.5 even 2
9280.2.a.bg.1.1 3 1.1 even 1 trivial
9280.2.a.bx.1.3 3 4.3 odd 2