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Properties

Label 380.2.a.c.1.2
Level $380$
Weight $2$
Character 380.1
Self dual yes
Analytic conductor $3.034$
Analytic rank $1$
Dimension $2$
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] = [380,2,Mod(1,380)] 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("380.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.03431527681\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 380.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.585786 q^{3} +1.00000 q^{5} -4.82843 q^{7} -2.65685 q^{9} -2.00000 q^{11} +2.24264 q^{13} -0.585786 q^{15} -4.82843 q^{17} -1.00000 q^{19} +2.82843 q^{21} -6.00000 q^{23} +1.00000 q^{25} +3.31371 q^{27} +10.4853 q^{29} -1.17157 q^{31} +1.17157 q^{33} -4.82843 q^{35} -10.2426 q^{37} -1.31371 q^{39} -7.65685 q^{41} -0.828427 q^{43} -2.65685 q^{45} +0.828427 q^{47} +16.3137 q^{49} +2.82843 q^{51} -5.07107 q^{53} -2.00000 q^{55} +0.585786 q^{57} +2.34315 q^{59} -2.34315 q^{61} +12.8284 q^{63} +2.24264 q^{65} -0.585786 q^{67} +3.51472 q^{69} +10.8284 q^{71} +14.4853 q^{73} -0.585786 q^{75} +9.65685 q^{77} -9.65685 q^{79} +6.02944 q^{81} +9.31371 q^{83} -4.82843 q^{85} -6.14214 q^{87} +10.4853 q^{89} -10.8284 q^{91} +0.686292 q^{93} -1.00000 q^{95} -1.75736 q^{97} +5.31371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9} - 4 q^{11} - 4 q^{13} - 4 q^{15} - 4 q^{17} - 2 q^{19} - 12 q^{23} + 2 q^{25} - 16 q^{27} + 4 q^{29} - 8 q^{31} + 8 q^{33} - 4 q^{35} - 12 q^{37} + 20 q^{39}+ \cdots - 12 q^{99}+O(q^{100}) \) 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
\(3\) −0.585786 −0.338204 −0.169102 − 0.985599i \(-0.554087\pi\)
−0.169102 + 0.985599i \(0.554087\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.82843 −1.82497 −0.912487 − 0.409106i \(-0.865841\pi\)
−0.912487 + 0.409106i \(0.865841\pi\)
\(8\) 0 0
\(9\) −2.65685 −0.885618
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 2.24264 0.621997 0.310998 − 0.950410i \(-0.399337\pi\)
0.310998 + 0.950410i \(0.399337\pi\)
\(14\) 0 0
\(15\) −0.585786 −0.151249
\(16\) 0 0
\(17\) −4.82843 −1.17107 −0.585533 − 0.810649i \(-0.699115\pi\)
−0.585533 + 0.810649i \(0.699115\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 2.82843 0.617213
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.31371 0.637723
\(28\) 0 0
\(29\) 10.4853 1.94707 0.973534 − 0.228543i \(-0.0733960\pi\)
0.973534 + 0.228543i \(0.0733960\pi\)
\(30\) 0 0
\(31\) −1.17157 −0.210421 −0.105210 − 0.994450i \(-0.533552\pi\)
−0.105210 + 0.994450i \(0.533552\pi\)
\(32\) 0 0
\(33\) 1.17157 0.203945
\(34\) 0 0
\(35\) −4.82843 −0.816153
\(36\) 0 0
\(37\) −10.2426 −1.68388 −0.841940 − 0.539571i \(-0.818586\pi\)
−0.841940 + 0.539571i \(0.818586\pi\)
\(38\) 0 0
\(39\) −1.31371 −0.210362
\(40\) 0 0
\(41\) −7.65685 −1.19580 −0.597900 − 0.801571i \(-0.703998\pi\)
−0.597900 + 0.801571i \(0.703998\pi\)
\(42\) 0 0
\(43\) −0.828427 −0.126334 −0.0631670 − 0.998003i \(-0.520120\pi\)
−0.0631670 + 0.998003i \(0.520120\pi\)
\(44\) 0 0
\(45\) −2.65685 −0.396060
\(46\) 0 0
\(47\) 0.828427 0.120839 0.0604193 − 0.998173i \(-0.480756\pi\)
0.0604193 + 0.998173i \(0.480756\pi\)
\(48\) 0 0
\(49\) 16.3137 2.33053
\(50\) 0 0
\(51\) 2.82843 0.396059
\(52\) 0 0
\(53\) −5.07107 −0.696565 −0.348282 − 0.937390i \(-0.613235\pi\)
−0.348282 + 0.937390i \(0.613235\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0.585786 0.0775893
\(58\) 0 0
\(59\) 2.34315 0.305052 0.152526 − 0.988299i \(-0.451259\pi\)
0.152526 + 0.988299i \(0.451259\pi\)
\(60\) 0 0
\(61\) −2.34315 −0.300009 −0.150005 − 0.988685i \(-0.547929\pi\)
−0.150005 + 0.988685i \(0.547929\pi\)
\(62\) 0 0
\(63\) 12.8284 1.61623
\(64\) 0 0
\(65\) 2.24264 0.278165
\(66\) 0 0
\(67\) −0.585786 −0.0715652 −0.0357826 − 0.999360i \(-0.511392\pi\)
−0.0357826 + 0.999360i \(0.511392\pi\)
\(68\) 0 0
\(69\) 3.51472 0.423122
\(70\) 0 0
\(71\) 10.8284 1.28510 0.642549 − 0.766245i \(-0.277877\pi\)
0.642549 + 0.766245i \(0.277877\pi\)
\(72\) 0 0
\(73\) 14.4853 1.69537 0.847687 − 0.530497i \(-0.177995\pi\)
0.847687 + 0.530497i \(0.177995\pi\)
\(74\) 0 0
\(75\) −0.585786 −0.0676408
\(76\) 0 0
\(77\) 9.65685 1.10050
\(78\) 0 0
\(79\) −9.65685 −1.08648 −0.543240 − 0.839577i \(-0.682803\pi\)
−0.543240 + 0.839577i \(0.682803\pi\)
\(80\) 0 0
\(81\) 6.02944 0.669937
\(82\) 0 0
\(83\) 9.31371 1.02231 0.511156 − 0.859488i \(-0.329217\pi\)
0.511156 + 0.859488i \(0.329217\pi\)
\(84\) 0 0
\(85\) −4.82843 −0.523716
\(86\) 0 0
\(87\) −6.14214 −0.658506
\(88\) 0 0
\(89\) 10.4853 1.11144 0.555719 − 0.831370i \(-0.312443\pi\)
0.555719 + 0.831370i \(0.312443\pi\)
\(90\) 0 0
\(91\) −10.8284 −1.13513
\(92\) 0 0
\(93\) 0.686292 0.0711651
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −1.75736 −0.178433 −0.0892164 − 0.996012i \(-0.528436\pi\)
−0.0892164 + 0.996012i \(0.528436\pi\)
\(98\) 0 0
\(99\) 5.31371 0.534048
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 380.2.a.c.1.2 ✓ 2
3.2 odd 2 3420.2.a.g.1.1 2
4.3 odd 2 1520.2.a.o.1.1 2
5.2 odd 4 1900.2.c.d.1749.3 4
5.3 odd 4 1900.2.c.d.1749.2 4
5.4 even 2 1900.2.a.e.1.1 2
8.3 odd 2 6080.2.a.y.1.2 2
8.5 even 2 6080.2.a.bl.1.1 2
19.18 odd 2 7220.2.a.m.1.1 2
20.19 odd 2 7600.2.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.c.1.2 ✓ 2 1.1 even 1 trivial
1520.2.a.o.1.1 2 4.3 odd 2
1900.2.a.e.1.1 2 5.4 even 2
1900.2.c.d.1749.2 4 5.3 odd 4
1900.2.c.d.1749.3 4 5.2 odd 4
3420.2.a.g.1.1 2 3.2 odd 2
6080.2.a.y.1.2 2 8.3 odd 2
6080.2.a.bl.1.1 2 8.5 even 2
7220.2.a.m.1.1 2 19.18 odd 2
7600.2.a.u.1.2 2 20.19 odd 2