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Properties

Label 760.2.a.i.1.2
Level $760$
Weight $2$
Character 760.1
Self dual yes
Analytic conductor $6.069$
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] = [760,2,Mod(1,760)] 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("760.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(760, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-1,0,-3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.06863055362\)
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: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 760.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.470683 q^{3} -1.00000 q^{5} +2.71982 q^{7} -2.77846 q^{9} -5.55691 q^{11} +2.02760 q^{13} +0.470683 q^{15} -3.77846 q^{17} +1.00000 q^{19} -1.28018 q^{21} -5.77846 q^{23} +1.00000 q^{25} +2.71982 q^{27} -5.66119 q^{29} +7.55691 q^{31} +2.61555 q^{33} -2.71982 q^{35} -3.75086 q^{37} -0.954357 q^{39} -12.6155 q^{41} -9.43965 q^{43} +2.77846 q^{45} +11.1138 q^{47} +0.397442 q^{49} +1.77846 q^{51} -8.85170 q^{53} +5.55691 q^{55} -0.470683 q^{57} +11.4526 q^{59} -10.6155 q^{61} -7.55691 q^{63} -2.02760 q^{65} -11.5845 q^{67} +2.71982 q^{69} -9.45264 q^{73} -0.470683 q^{75} -15.1138 q^{77} +8.94137 q^{79} +7.05520 q^{81} -4.94137 q^{83} +3.77846 q^{85} +2.66463 q^{87} +15.4948 q^{89} +5.51471 q^{91} -3.55691 q^{93} -1.00000 q^{95} +10.8647 q^{97} +15.4396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 3 q^{5} - q^{7} - 11 q^{13} + q^{15} - 3 q^{17} + 3 q^{19} - 13 q^{21} - 9 q^{23} + 3 q^{25} - q^{27} - 7 q^{29} + 6 q^{31} - 8 q^{33} + q^{35} - 20 q^{37} + 3 q^{39} - 22 q^{41} - 10 q^{43}+ \cdots + 28 q^{99}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 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 \(-0.816121\pi\)
−0.837736 + 0.546075i \(0.816121\pi\)
\(12\) 0 0
\(13\) 2.02760 0.562354 0.281177 − 0.959656i \(-0.409275\pi\)
0.281177 + 0.959656i \(0.409275\pi\)
\(14\) 0 0
\(15\) 0.470683 0.121530
\(16\) 0 0
\(17\) −3.77846 −0.916410 −0.458205 − 0.888846i \(-0.651508\pi\)
−0.458205 + 0.888846i \(0.651508\pi\)
\(18\) 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
\(25\) 1.00000 0.200000
\(26\) 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 \(-0.262574\pi\)
0.678631 + 0.734479i \(0.262574\pi\)
\(32\) 0 0
\(33\) 2.61555 0.455308
\(34\) 0 0
\(35\) −2.71982 −0.459734
\(36\) 0 0
\(37\) −3.75086 −0.616637 −0.308319 − 0.951283i \(-0.599766\pi\)
−0.308319 + 0.951283i \(0.599766\pi\)
\(38\) 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
\(45\) 2.77846 0.414188
\(46\) 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 \(-0.708003\pi\)
−0.607937 + 0.793985i \(0.708003\pi\)
\(54\) 0 0
\(55\) 5.55691 0.749294
\(56\) 0 0
\(57\) −0.470683 −0.0623435
\(58\) 0 0
\(59\) 11.4526 1.49101 0.745503 − 0.666502i \(-0.232209\pi\)
0.745503 + 0.666502i \(0.232209\pi\)
\(60\) 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
\(65\) −2.02760 −0.251493
\(66\) 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 \(-0.686583\pi\)
−0.553174 + 0.833066i \(0.686583\pi\)
\(74\) 0 0
\(75\) −0.470683 −0.0543498
\(76\) 0 0
\(77\) −15.1138 −1.72238
\(78\) 0 0
\(79\) 8.94137 1.00598 0.502991 − 0.864292i \(-0.332233\pi\)
0.502991 + 0.864292i \(0.332233\pi\)
\(80\) 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
\(85\) 3.77846 0.409831
\(86\) 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
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 10.8647 1.10314 0.551571 − 0.834128i \(-0.314029\pi\)
0.551571 + 0.834128i \(0.314029\pi\)
\(98\) 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 760.2.a.i.1.2 ✓ 3
3.2 odd 2 6840.2.a.bm.1.3 3
4.3 odd 2 1520.2.a.q.1.2 3
5.2 odd 4 3800.2.d.n.3649.4 6
5.3 odd 4 3800.2.d.n.3649.3 6
5.4 even 2 3800.2.a.w.1.2 3
8.3 odd 2 6080.2.a.br.1.2 3
8.5 even 2 6080.2.a.bx.1.2 3
20.19 odd 2 7600.2.a.bp.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.2 ✓ 3 1.1 even 1 trivial
1520.2.a.q.1.2 3 4.3 odd 2
3800.2.a.w.1.2 3 5.4 even 2
3800.2.d.n.3649.3 6 5.3 odd 4
3800.2.d.n.3649.4 6 5.2 odd 4
6080.2.a.br.1.2 3 8.3 odd 2
6080.2.a.bx.1.2 3 8.5 even 2
6840.2.a.bm.1.3 3 3.2 odd 2
7600.2.a.bp.1.2 3 20.19 odd 2