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Properties

Label 3800.2.d.n.3649.4
Level $3800$
Weight $2$
Character 3800.3649
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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] = [3800,2,Mod(3649,3800)] 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("3800.3649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.4
Root \(0.264658 - 1.38923i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.n.3649.3

$q$-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)
 
\(f(q)\) \(=\) \(q+0.470683i q^{3} +2.71982i q^{7} +2.77846 q^{9} -5.55691 q^{11} -2.02760i q^{13} -3.77846i q^{17} -1.00000 q^{19} -1.28018 q^{21} +5.77846i q^{23} +2.71982i q^{27} +5.66119 q^{29} +7.55691 q^{31} -2.61555i q^{33} -3.75086i q^{37} +0.954357 q^{39} -12.6155 q^{41} +9.43965i q^{43} +11.1138i q^{47} -0.397442 q^{49} +1.77846 q^{51} +8.85170i q^{53} -0.470683i q^{57} -11.4526 q^{59} -10.6155 q^{61} +7.55691i q^{63} -11.5845i q^{67} -2.71982 q^{69} +9.45264i q^{73} -15.1138i q^{77} -8.94137 q^{79} +7.05520 q^{81} +4.94137i q^{83} +2.66463i q^{87} -15.4948 q^{89} +5.51471 q^{91} +3.55691i q^{93} +10.8647i q^{97} -15.4396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{19} - 26 q^{21} + 14 q^{29} + 12 q^{31} - 6 q^{39} - 44 q^{41} - 24 q^{49} - 6 q^{51} - 22 q^{59} - 32 q^{61} + 2 q^{69} - 52 q^{79} - 26 q^{81} + 12 q^{89} + 58 q^{91} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 0.470683i 0.271749i 0.990726 + 0.135875i \(0.0433844\pi\)
−0.990726 + 0.135875i \(0.956616\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.71982i 1.02800i 0.857791 + 0.513998i \(0.171836\pi\)
−0.857791 + 0.513998i \(0.828164\pi\)
\(8\) 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
\(13\) − 2.02760i − 0.562354i −0.959656 − 0.281177i \(-0.909275\pi\)
0.959656 − 0.281177i \(-0.0907249\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.77846i − 0.916410i −0.888846 − 0.458205i \(-0.848492\pi\)
0.888846 − 0.458205i \(-0.151508\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.28018 −0.279357
\(22\) 0 0
\(23\) 5.77846i 1.20489i 0.798160 + 0.602446i \(0.205807\pi\)
−0.798160 + 0.602446i \(0.794193\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.71982i 0.523430i
\(28\) 0 0
\(29\) 5.66119 1.05126 0.525628 − 0.850714i \(-0.323830\pi\)
0.525628 + 0.850714i \(0.323830\pi\)
\(30\) 0 0
\(31\) 7.55691 1.35726 0.678631 − 0.734479i \(-0.262574\pi\)
0.678631 + 0.734479i \(0.262574\pi\)
\(32\) 0 0
\(33\) − 2.61555i − 0.455308i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 3.75086i − 0.616637i −0.951283 − 0.308319i \(-0.900234\pi\)
0.951283 − 0.308319i \(-0.0997663\pi\)
\(38\) 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
\(43\) 9.43965i 1.43953i 0.694216 + 0.719766i \(0.255751\pi\)
−0.694216 + 0.719766i \(0.744249\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.1138i 1.62112i 0.585657 + 0.810559i \(0.300837\pi\)
−0.585657 + 0.810559i \(0.699163\pi\)
\(48\) 0 0
\(49\) −0.397442 −0.0567775
\(50\) 0 0
\(51\) 1.77846 0.249034
\(52\) 0 0
\(53\) 8.85170i 1.21587i 0.793985 + 0.607937i \(0.208003\pi\)
−0.793985 + 0.607937i \(0.791997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 0.470683i − 0.0623435i
\(58\) 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
\(63\) 7.55691i 0.952082i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 11.5845i − 1.41527i −0.706576 − 0.707637i \(-0.749761\pi\)
0.706576 − 0.707637i \(-0.250239\pi\)
\(68\) 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
\(73\) 9.45264i 1.10635i 0.833066 + 0.553174i \(0.186583\pi\)
−0.833066 + 0.553174i \(0.813417\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 15.1138i − 1.72238i
\(78\) 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
\(83\) 4.94137i 0.542385i 0.962525 + 0.271193i \(0.0874181\pi\)
−0.962525 + 0.271193i \(0.912582\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.66463i 0.285678i
\(88\) 0 0
\(89\) −15.4948 −1.64245 −0.821225 − 0.570604i \(-0.806709\pi\)
−0.821225 + 0.570604i \(0.806709\pi\)
\(90\) 0 0
\(91\) 5.51471 0.578099
\(92\) 0 0
\(93\) 3.55691i 0.368835i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.8647i 1.10314i 0.834128 + 0.551571i \(0.185971\pi\)
−0.834128 + 0.551571i \(0.814029\pi\)
\(98\) 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 3800.2.d.n.3649.4 6
5.2 odd 4 3800.2.a.w.1.2 3
5.3 odd 4 760.2.a.i.1.2 ✓ 3
5.4 even 2 inner 3800.2.d.n.3649.3 6
15.8 even 4 6840.2.a.bm.1.3 3
20.3 even 4 1520.2.a.q.1.2 3
20.7 even 4 7600.2.a.bp.1.2 3
40.3 even 4 6080.2.a.br.1.2 3
40.13 odd 4 6080.2.a.bx.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.2 ✓ 3 5.3 odd 4
1520.2.a.q.1.2 3 20.3 even 4
3800.2.a.w.1.2 3 5.2 odd 4
3800.2.d.n.3649.3 6 5.4 even 2 inner
3800.2.d.n.3649.4 6 1.1 even 1 trivial
6080.2.a.br.1.2 3 40.3 even 4
6080.2.a.bx.1.2 3 40.13 odd 4
6840.2.a.bm.1.3 3 15.8 even 4
7600.2.a.bp.1.2 3 20.7 even 4