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Properties

Label 7002.2.a.e.1.3
Level $7002$
Weight $2$
Character 7002.1
Self dual yes
Analytic conductor $55.911$
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] = [7002,2,Mod(1,7002)] 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("7002.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7002, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7002 = 2 \cdot 3^{2} \cdot 389 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7002.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4,0,4,7] 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: \(55.9112514953\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 778)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.95630\) of defining polynomial
Character \(\chi\) \(=\) 7002.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.20906 q^{5} -4.57433 q^{7} -1.00000 q^{8} -2.20906 q^{10} -1.46747 q^{11} -6.29456 q^{13} +4.57433 q^{14} +1.00000 q^{16} -3.73968 q^{17} -1.27977 q^{19} +2.20906 q^{20} +1.46747 q^{22} -0.306784 q^{23} -0.120067 q^{25} +6.29456 q^{26} -4.57433 q^{28} -3.37441 q^{29} -8.86889 q^{31} -1.00000 q^{32} +3.73968 q^{34} -10.1050 q^{35} +2.65418 q^{37} +1.27977 q^{38} -2.20906 q^{40} -2.33070 q^{41} +7.53062 q^{43} -1.46747 q^{44} +0.306784 q^{46} +3.18205 q^{47} +13.9245 q^{49} +0.120067 q^{50} -6.29456 q^{52} +4.49448 q^{53} -3.24171 q^{55} +4.57433 q^{56} +3.37441 q^{58} +3.15622 q^{59} -11.5231 q^{61} +8.86889 q^{62} +1.00000 q^{64} -13.9050 q^{65} +4.30678 q^{67} -3.73968 q^{68} +10.1050 q^{70} -10.2071 q^{71} +3.16535 q^{73} -2.65418 q^{74} -1.27977 q^{76} +6.71267 q^{77} +1.44355 q^{79} +2.20906 q^{80} +2.33070 q^{82} +5.77892 q^{83} -8.26117 q^{85} -7.53062 q^{86} +1.46747 q^{88} +16.4495 q^{89} +28.7934 q^{91} -0.306784 q^{92} -3.18205 q^{94} -2.82709 q^{95} +1.63282 q^{97} -13.9245 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} + 7 q^{5} - 5 q^{7} - 4 q^{8} - 7 q^{10} + q^{11} - 12 q^{13} + 5 q^{14} + 4 q^{16} + 9 q^{17} - 5 q^{19} + 7 q^{20} - q^{22} + 6 q^{23} + q^{25} + 12 q^{26} - 5 q^{28} - q^{29}+ \cdots + 3 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.20906 0.987920 0.493960 − 0.869485i \(-0.335549\pi\)
0.493960 + 0.869485i \(0.335549\pi\)
\(6\) 0 0
\(7\) −4.57433 −1.72893 −0.864467 − 0.502690i \(-0.832344\pi\)
−0.864467 + 0.502690i \(0.832344\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −2.20906 −0.698565
\(11\) −1.46747 −0.442457 −0.221229 − 0.975222i \(-0.571007\pi\)
−0.221229 + 0.975222i \(0.571007\pi\)
\(12\) 0 0
\(13\) −6.29456 −1.74580 −0.872898 − 0.487903i \(-0.837762\pi\)
−0.872898 + 0.487903i \(0.837762\pi\)
\(14\) 4.57433 1.22254
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.73968 −0.907006 −0.453503 − 0.891255i \(-0.649826\pi\)
−0.453503 + 0.891255i \(0.649826\pi\)
\(18\) 0 0
\(19\) −1.27977 −0.293600 −0.146800 − 0.989166i \(-0.546897\pi\)
−0.146800 + 0.989166i \(0.546897\pi\)
\(20\) 2.20906 0.493960
\(21\) 0 0
\(22\) 1.46747 0.312865
\(23\) −0.306784 −0.0639688 −0.0319844 − 0.999488i \(-0.510183\pi\)
−0.0319844 + 0.999488i \(0.510183\pi\)
\(24\) 0 0
\(25\) −0.120067 −0.0240135
\(26\) 6.29456 1.23446
\(27\) 0 0
\(28\) −4.57433 −0.864467
\(29\) −3.37441 −0.626612 −0.313306 − 0.949652i \(-0.601437\pi\)
−0.313306 + 0.949652i \(0.601437\pi\)
\(30\) 0 0
\(31\) −8.86889 −1.59290 −0.796449 − 0.604705i \(-0.793291\pi\)
−0.796449 + 0.604705i \(0.793291\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.73968 0.641350
\(35\) −10.1050 −1.70805
\(36\) 0 0
\(37\) 2.65418 0.436345 0.218172 − 0.975910i \(-0.429990\pi\)
0.218172 + 0.975910i \(0.429990\pi\)
\(38\) 1.27977 0.207607
\(39\) 0 0
\(40\) −2.20906 −0.349283
\(41\) −2.33070 −0.363995 −0.181997 − 0.983299i \(-0.558256\pi\)
−0.181997 + 0.983299i \(0.558256\pi\)
\(42\) 0 0
\(43\) 7.53062 1.14841 0.574205 − 0.818712i \(-0.305311\pi\)
0.574205 + 0.818712i \(0.305311\pi\)
\(44\) −1.46747 −0.221229
\(45\) 0 0
\(46\) 0.306784 0.0452328
\(47\) 3.18205 0.464149 0.232075 − 0.972698i \(-0.425449\pi\)
0.232075 + 0.972698i \(0.425449\pi\)
\(48\) 0 0
\(49\) 13.9245 1.98921
\(50\) 0.120067 0.0169801
\(51\) 0 0
\(52\) −6.29456 −0.872898
\(53\) 4.49448 0.617364 0.308682 − 0.951165i \(-0.400112\pi\)
0.308682 + 0.951165i \(0.400112\pi\)
\(54\) 0 0
\(55\) −3.24171 −0.437113
\(56\) 4.57433 0.611270
\(57\) 0 0
\(58\) 3.37441 0.443082
\(59\) 3.15622 0.410904 0.205452 − 0.978667i \(-0.434134\pi\)
0.205452 + 0.978667i \(0.434134\pi\)
\(60\) 0 0
\(61\) −11.5231 −1.47538 −0.737689 − 0.675141i \(-0.764083\pi\)
−0.737689 + 0.675141i \(0.764083\pi\)
\(62\) 8.86889 1.12635
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −13.9050 −1.72471
\(66\) 0 0
\(67\) 4.30678 0.526157 0.263079 − 0.964774i \(-0.415262\pi\)
0.263079 + 0.964774i \(0.415262\pi\)
\(68\) −3.73968 −0.453503
\(69\) 0 0
\(70\) 10.1050 1.20777
\(71\) −10.2071 −1.21137 −0.605683 − 0.795706i \(-0.707100\pi\)
−0.605683 + 0.795706i \(0.707100\pi\)
\(72\) 0 0
\(73\) 3.16535 0.370476 0.185238 − 0.982694i \(-0.440694\pi\)
0.185238 + 0.982694i \(0.440694\pi\)
\(74\) −2.65418 −0.308542
\(75\) 0 0
\(76\) −1.27977 −0.146800
\(77\) 6.71267 0.764980
\(78\) 0 0
\(79\) 1.44355 0.162412 0.0812058 − 0.996697i \(-0.474123\pi\)
0.0812058 + 0.996697i \(0.474123\pi\)
\(80\) 2.20906 0.246980
\(81\) 0 0
\(82\) 2.33070 0.257383
\(83\) 5.77892 0.634319 0.317159 − 0.948372i \(-0.397271\pi\)
0.317159 + 0.948372i \(0.397271\pi\)
\(84\) 0 0
\(85\) −8.26117 −0.896050
\(86\) −7.53062 −0.812048
\(87\) 0 0
\(88\) 1.46747 0.156432
\(89\) 16.4495 1.74364 0.871820 − 0.489827i \(-0.162940\pi\)
0.871820 + 0.489827i \(0.162940\pi\)
\(90\) 0 0
\(91\) 28.7934 3.01837
\(92\) −0.306784 −0.0319844
\(93\) 0 0
\(94\) −3.18205 −0.328203
\(95\) −2.82709 −0.290053
\(96\) 0 0
\(97\) 1.63282 0.165788 0.0828938 − 0.996558i \(-0.473584\pi\)
0.0828938 + 0.996558i \(0.473584\pi\)
\(98\) −13.9245 −1.40659
\(99\) 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 7002.2.a.e.1.3 4
3.2 odd 2 778.2.a.a.1.4 ✓ 4
12.11 even 2 6224.2.a.h.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
778.2.a.a.1.4 ✓ 4 3.2 odd 2
6224.2.a.h.1.1 4 12.11 even 2
7002.2.a.e.1.3 4 1.1 even 1 trivial