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Properties

Label 7600.2.a.bp.1.2
Level $7600$
Weight $2$
Character 7600.1
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
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] = [7600,2,Mod(1,7600)] 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("7600.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-1,0,0,0,-1,0,0,0,0,0,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(60.6863055362\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) 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 760)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 7600.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.470683 q^{3} +2.71982 q^{7} -2.77846 q^{9} +5.55691 q^{11} -2.02760 q^{13} +3.77846 q^{17} -1.00000 q^{19} -1.28018 q^{21} -5.77846 q^{23} +2.71982 q^{27} -5.66119 q^{29} -7.55691 q^{31} -2.61555 q^{33} +3.75086 q^{37} +0.954357 q^{39} -12.6155 q^{41} -9.43965 q^{43} +11.1138 q^{47} +0.397442 q^{49} -1.77846 q^{51} +8.85170 q^{53} +0.470683 q^{57} -11.4526 q^{59} -10.6155 q^{61} -7.55691 q^{63} -11.5845 q^{67} +2.71982 q^{69} +9.45264 q^{73} +15.1138 q^{77} -8.94137 q^{79} +7.05520 q^{81} -4.94137 q^{83} +2.66463 q^{87} +15.4948 q^{89} -5.51471 q^{91} +3.55691 q^{93} -10.8647 q^{97} -15.4396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - q^{7} + 11 q^{13} + 3 q^{17} - 3 q^{19} - 13 q^{21} - 9 q^{23} - q^{27} - 7 q^{29} - 6 q^{31} + 8 q^{33} + 20 q^{37} - 3 q^{39} - 22 q^{41} - 10 q^{43} + 12 q^{49} + 3 q^{51} + 7 q^{53}+ \cdots - 28 q^{99}+O(q^{100}) \) 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
\(3\) −0.470683 −0.271749 −0.135875 − 0.990726i \(-0.543384\pi\)
−0.135875 + 0.990726i \(0.543384\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.71982 1.02800 0.513998 − 0.857791i \(-0.328164\pi\)
0.513998 + 0.857791i \(0.328164\pi\)
\(8\) 0 0
\(9\) −2.77846 −0.926152
\(10\) 0 0
\(11\) 5.55691 1.67547 0.837736 − 0.546075i \(-0.183879\pi\)
0.837736 + 0.546075i \(0.183879\pi\)
\(12\) 0 0
\(13\) −2.02760 −0.562354 −0.281177 − 0.959656i \(-0.590725\pi\)
−0.281177 + 0.959656i \(0.590725\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.77846 0.916410 0.458205 − 0.888846i \(-0.348492\pi\)
0.458205 + 0.888846i \(0.348492\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.28018 −0.279357
\(22\) 0 0
\(23\) −5.77846 −1.20489 −0.602446 − 0.798160i \(-0.705807\pi\)
−0.602446 + 0.798160i \(0.705807\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.71982 0.523430
\(28\) 0 0
\(29\) −5.66119 −1.05126 −0.525628 − 0.850714i \(-0.676170\pi\)
−0.525628 + 0.850714i \(0.676170\pi\)
\(30\) 0 0
\(31\) −7.55691 −1.35726 −0.678631 − 0.734479i \(-0.737426\pi\)
−0.678631 + 0.734479i \(0.737426\pi\)
\(32\) 0 0
\(33\) −2.61555 −0.455308
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.75086 0.616637 0.308319 − 0.951283i \(-0.400234\pi\)
0.308319 + 0.951283i \(0.400234\pi\)
\(38\) 0 0
\(39\) 0.954357 0.152819
\(40\) 0 0
\(41\) −12.6155 −1.97022 −0.985109 − 0.171932i \(-0.944999\pi\)
−0.985109 + 0.171932i \(0.944999\pi\)
\(42\) 0 0
\(43\) −9.43965 −1.43953 −0.719766 − 0.694216i \(-0.755751\pi\)
−0.719766 + 0.694216i \(0.755751\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.1138 1.62112 0.810559 − 0.585657i \(-0.199163\pi\)
0.810559 + 0.585657i \(0.199163\pi\)
\(48\) 0 0
\(49\) 0.397442 0.0567775
\(50\) 0 0
\(51\) −1.77846 −0.249034
\(52\) 0 0
\(53\) 8.85170 1.21587 0.607937 − 0.793985i \(-0.291997\pi\)
0.607937 + 0.793985i \(0.291997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.470683 0.0623435
\(58\) 0 0
\(59\) −11.4526 −1.49101 −0.745503 − 0.666502i \(-0.767791\pi\)
−0.745503 + 0.666502i \(0.767791\pi\)
\(60\) 0 0
\(61\) −10.6155 −1.35918 −0.679591 − 0.733591i \(-0.737843\pi\)
−0.679591 + 0.733591i \(0.737843\pi\)
\(62\) 0 0
\(63\) −7.55691 −0.952082
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −11.5845 −1.41527 −0.707637 − 0.706576i \(-0.750239\pi\)
−0.707637 + 0.706576i \(0.750239\pi\)
\(68\) 0 0
\(69\) 2.71982 0.327428
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 9.45264 1.10635 0.553174 − 0.833066i \(-0.313417\pi\)
0.553174 + 0.833066i \(0.313417\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 15.1138 1.72238
\(78\) 0 0
\(79\) −8.94137 −1.00598 −0.502991 − 0.864292i \(-0.667767\pi\)
−0.502991 + 0.864292i \(0.667767\pi\)
\(80\) 0 0
\(81\) 7.05520 0.783911
\(82\) 0 0
\(83\) −4.94137 −0.542385 −0.271193 − 0.962525i \(-0.587418\pi\)
−0.271193 + 0.962525i \(0.587418\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.66463 0.285678
\(88\) 0 0
\(89\) 15.4948 1.64245 0.821225 − 0.570604i \(-0.193291\pi\)
0.821225 + 0.570604i \(0.193291\pi\)
\(90\) 0 0
\(91\) −5.51471 −0.578099
\(92\) 0 0
\(93\) 3.55691 0.368835
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.8647 −1.10314 −0.551571 − 0.834128i \(-0.685971\pi\)
−0.551571 + 0.834128i \(0.685971\pi\)
\(98\) 0 0
\(99\) −15.4396 −1.55174
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 7600.2.a.bp.1.2 3
4.3 odd 2 3800.2.a.w.1.2 3
5.4 even 2 1520.2.a.q.1.2 3
20.3 even 4 3800.2.d.n.3649.4 6
20.7 even 4 3800.2.d.n.3649.3 6
20.19 odd 2 760.2.a.i.1.2 ✓ 3
40.19 odd 2 6080.2.a.bx.1.2 3
40.29 even 2 6080.2.a.br.1.2 3
60.59 even 2 6840.2.a.bm.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.2 ✓ 3 20.19 odd 2
1520.2.a.q.1.2 3 5.4 even 2
3800.2.a.w.1.2 3 4.3 odd 2
3800.2.d.n.3649.3 6 20.7 even 4
3800.2.d.n.3649.4 6 20.3 even 4
6080.2.a.br.1.2 3 40.29 even 2
6080.2.a.bx.1.2 3 40.19 odd 2
6840.2.a.bm.1.3 3 60.59 even 2
7600.2.a.bp.1.2 3 1.1 even 1 trivial