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Properties

Label 9680.2.a.cu.1.2
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
Dimension $4$
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] = [9680,2,Mod(1,9680)] 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("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,2,0,4,0,7,0,-4,0,0,0,3,0,2,0,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{22 +2 \sqrt{5}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) 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 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.477260\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.294963 q^{3} +1.00000 q^{5} +4.39026 q^{7} -2.91300 q^{9} +5.96281 q^{13} -0.294963 q^{15} -1.66785 q^{17} -7.69351 q^{19} -1.29496 q^{21} +0.904706 q^{23} +1.00000 q^{25} +1.74411 q^{27} -4.73899 q^{29} +5.12608 q^{31} +4.39026 q^{35} -0.184976 q^{37} -1.75881 q^{39} -2.62237 q^{41} -3.59822 q^{43} -2.91300 q^{45} +0.776180 q^{47} +12.2744 q^{49} +0.491953 q^{51} -9.59554 q^{53} +2.26930 q^{57} +11.2538 q^{59} +13.1898 q^{61} -12.7888 q^{63} +5.96281 q^{65} +7.79954 q^{67} -0.266855 q^{69} +6.97072 q^{71} +12.7541 q^{73} -0.294963 q^{75} +10.0313 q^{79} +8.22454 q^{81} -4.09529 q^{83} -1.66785 q^{85} +1.39783 q^{87} -0.466291 q^{89} +26.1783 q^{91} -1.51200 q^{93} -7.69351 q^{95} +7.09963 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{5} + 7 q^{7} - 4 q^{9} + 3 q^{13} + 2 q^{15} + 11 q^{17} - 2 q^{19} - 2 q^{21} + 11 q^{23} + 4 q^{25} - q^{27} + 4 q^{29} + 17 q^{31} + 7 q^{35} - 3 q^{37} - q^{39} + 13 q^{41} + 7 q^{43}+ \cdots + 2 q^{97}+O(q^{100}) \) 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
\(3\) −0.294963 −0.170297 −0.0851485 − 0.996368i \(-0.527136\pi\)
−0.0851485 + 0.996368i \(0.527136\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.39026 1.65936 0.829681 − 0.558238i \(-0.188523\pi\)
0.829681 + 0.558238i \(0.188523\pi\)
\(8\) 0 0
\(9\) −2.91300 −0.970999
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 5.96281 1.65379 0.826893 − 0.562359i \(-0.190106\pi\)
0.826893 + 0.562359i \(0.190106\pi\)
\(14\) 0 0
\(15\) −0.294963 −0.0761591
\(16\) 0 0
\(17\) −1.66785 −0.404513 −0.202256 − 0.979333i \(-0.564827\pi\)
−0.202256 + 0.979333i \(0.564827\pi\)
\(18\) 0 0
\(19\) −7.69351 −1.76501 −0.882506 − 0.470301i \(-0.844145\pi\)
−0.882506 + 0.470301i \(0.844145\pi\)
\(20\) 0 0
\(21\) −1.29496 −0.282584
\(22\) 0 0
\(23\) 0.904706 0.188644 0.0943221 − 0.995542i \(-0.469932\pi\)
0.0943221 + 0.995542i \(0.469932\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.74411 0.335655
\(28\) 0 0
\(29\) −4.73899 −0.880008 −0.440004 − 0.897996i \(-0.645023\pi\)
−0.440004 + 0.897996i \(0.645023\pi\)
\(30\) 0 0
\(31\) 5.12608 0.920671 0.460336 − 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.39026 0.742089
\(36\) 0 0
\(37\) −0.184976 −0.0304098 −0.0152049 − 0.999884i \(-0.504840\pi\)
−0.0152049 + 0.999884i \(0.504840\pi\)
\(38\) 0 0
\(39\) −1.75881 −0.281635
\(40\) 0 0
\(41\) −2.62237 −0.409545 −0.204773 − 0.978810i \(-0.565645\pi\)
−0.204773 + 0.978810i \(0.565645\pi\)
\(42\) 0 0
\(43\) −3.59822 −0.548723 −0.274361 − 0.961627i \(-0.588466\pi\)
−0.274361 + 0.961627i \(0.588466\pi\)
\(44\) 0 0
\(45\) −2.91300 −0.434244
\(46\) 0 0
\(47\) 0.776180 0.113217 0.0566087 − 0.998396i \(-0.481971\pi\)
0.0566087 + 0.998396i \(0.481971\pi\)
\(48\) 0 0
\(49\) 12.2744 1.75348
\(50\) 0 0
\(51\) 0.491953 0.0688872
\(52\) 0 0
\(53\) −9.59554 −1.31805 −0.659024 − 0.752122i \(-0.729031\pi\)
−0.659024 + 0.752122i \(0.729031\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.26930 0.300576
\(58\) 0 0
\(59\) 11.2538 1.46512 0.732561 − 0.680701i \(-0.238325\pi\)
0.732561 + 0.680701i \(0.238325\pi\)
\(60\) 0 0
\(61\) 13.1898 1.68878 0.844390 − 0.535729i \(-0.179963\pi\)
0.844390 + 0.535729i \(0.179963\pi\)
\(62\) 0 0
\(63\) −12.7888 −1.61124
\(64\) 0 0
\(65\) 5.96281 0.739596
\(66\) 0 0
\(67\) 7.79954 0.952866 0.476433 − 0.879211i \(-0.341930\pi\)
0.476433 + 0.879211i \(0.341930\pi\)
\(68\) 0 0
\(69\) −0.266855 −0.0321255
\(70\) 0 0
\(71\) 6.97072 0.827273 0.413636 − 0.910442i \(-0.364258\pi\)
0.413636 + 0.910442i \(0.364258\pi\)
\(72\) 0 0
\(73\) 12.7541 1.49275 0.746375 − 0.665526i \(-0.231793\pi\)
0.746375 + 0.665526i \(0.231793\pi\)
\(74\) 0 0
\(75\) −0.294963 −0.0340594
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0313 1.12861 0.564303 − 0.825568i \(-0.309145\pi\)
0.564303 + 0.825568i \(0.309145\pi\)
\(80\) 0 0
\(81\) 8.22454 0.913838
\(82\) 0 0
\(83\) −4.09529 −0.449517 −0.224758 − 0.974415i \(-0.572159\pi\)
−0.224758 + 0.974415i \(0.572159\pi\)
\(84\) 0 0
\(85\) −1.66785 −0.180904
\(86\) 0 0
\(87\) 1.39783 0.149863
\(88\) 0 0
\(89\) −0.466291 −0.0494267 −0.0247134 − 0.999695i \(-0.507867\pi\)
−0.0247134 + 0.999695i \(0.507867\pi\)
\(90\) 0 0
\(91\) 26.1783 2.74423
\(92\) 0 0
\(93\) −1.51200 −0.156787
\(94\) 0 0
\(95\) −7.69351 −0.789338
\(96\) 0 0
\(97\) 7.09963 0.720858 0.360429 − 0.932787i \(-0.382630\pi\)
0.360429 + 0.932787i \(0.382630\pi\)
\(98\) 0 0
\(99\) 0 0
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 9680.2.a.cu.1.2 4
4.3 odd 2 4840.2.a.y.1.3 4
11.5 even 5 880.2.bo.d.641.1 8
11.9 even 5 880.2.bo.d.81.1 8
11.10 odd 2 9680.2.a.ct.1.2 4
44.27 odd 10 440.2.y.a.201.2 yes 8
44.31 odd 10 440.2.y.a.81.2 ✓ 8
44.43 even 2 4840.2.a.z.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.a.81.2 ✓ 8 44.31 odd 10
440.2.y.a.201.2 yes 8 44.27 odd 10
880.2.bo.d.81.1 8 11.9 even 5
880.2.bo.d.641.1 8 11.5 even 5
4840.2.a.y.1.3 4 4.3 odd 2
4840.2.a.z.1.3 4 44.43 even 2
9680.2.a.ct.1.2 4 11.10 odd 2
9680.2.a.cu.1.2 4 1.1 even 1 trivial