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Properties

Label 1776.2.a.r.1.3
Level $1776$
Weight $2$
Character 1776.1
Self dual yes
Analytic conductor $14.181$
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] = [1776,2,Mod(1,1776)] 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("1776.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1776, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1776.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,0,0,1,0,3,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.1814313990\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 888)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 1776.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.64002 q^{5} +0.484862 q^{7} +1.00000 q^{9} -2.48486 q^{11} +0.484862 q^{13} -2.64002 q^{15} -7.37466 q^{17} -7.76491 q^{19} -0.484862 q^{21} -7.12489 q^{23} +1.96972 q^{25} -1.00000 q^{27} -2.64002 q^{29} +6.24977 q^{31} +2.48486 q^{33} +1.28005 q^{35} -1.00000 q^{37} -0.484862 q^{39} +2.00000 q^{41} -7.21949 q^{43} +2.64002 q^{45} +7.52982 q^{47} -6.76491 q^{49} +7.37466 q^{51} +12.9844 q^{53} -6.56009 q^{55} +7.76491 q^{57} -1.11021 q^{59} -6.00000 q^{61} +0.484862 q^{63} +1.28005 q^{65} +5.21949 q^{67} +7.12489 q^{69} -10.5601 q^{71} -9.70436 q^{73} -1.96972 q^{75} -1.20482 q^{77} +11.5298 q^{79} +1.00000 q^{81} -13.0450 q^{83} -19.4693 q^{85} +2.64002 q^{87} -9.43521 q^{89} +0.235091 q^{91} -6.24977 q^{93} -20.4995 q^{95} +15.7796 q^{97} -2.48486 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + q^{7} + 3 q^{9} - 7 q^{11} + q^{13} + 3 q^{17} - 7 q^{19} - q^{21} - 13 q^{23} + 5 q^{25} - 3 q^{27} + 2 q^{31} + 7 q^{33} - 12 q^{35} - 3 q^{37} - q^{39} + 6 q^{41} - 4 q^{43} - 10 q^{47}+ \cdots - 7 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.64002 1.18065 0.590327 − 0.807164i \(-0.298999\pi\)
0.590327 + 0.807164i \(0.298999\pi\)
\(6\) 0 0
\(7\) 0.484862 0.183261 0.0916303 − 0.995793i \(-0.470792\pi\)
0.0916303 + 0.995793i \(0.470792\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.48486 −0.749214 −0.374607 − 0.927184i \(-0.622222\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(12\) 0 0
\(13\) 0.484862 0.134477 0.0672383 − 0.997737i \(-0.478581\pi\)
0.0672383 + 0.997737i \(0.478581\pi\)
\(14\) 0 0
\(15\) −2.64002 −0.681651
\(16\) 0 0
\(17\) −7.37466 −1.78862 −0.894308 − 0.447451i \(-0.852332\pi\)
−0.894308 + 0.447451i \(0.852332\pi\)
\(18\) 0 0
\(19\) −7.76491 −1.78139 −0.890696 − 0.454599i \(-0.849783\pi\)
−0.890696 + 0.454599i \(0.849783\pi\)
\(20\) 0 0
\(21\) −0.484862 −0.105806
\(22\) 0 0
\(23\) −7.12489 −1.48564 −0.742821 − 0.669490i \(-0.766512\pi\)
−0.742821 + 0.669490i \(0.766512\pi\)
\(24\) 0 0
\(25\) 1.96972 0.393945
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.64002 −0.490240 −0.245120 − 0.969493i \(-0.578827\pi\)
−0.245120 + 0.969493i \(0.578827\pi\)
\(30\) 0 0
\(31\) 6.24977 1.12249 0.561246 − 0.827649i \(-0.310322\pi\)
0.561246 + 0.827649i \(0.310322\pi\)
\(32\) 0 0
\(33\) 2.48486 0.432559
\(34\) 0 0
\(35\) 1.28005 0.216367
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) −0.484862 −0.0776400
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −7.21949 −1.10096 −0.550481 − 0.834847i \(-0.685556\pi\)
−0.550481 + 0.834847i \(0.685556\pi\)
\(44\) 0 0
\(45\) 2.64002 0.393551
\(46\) 0 0
\(47\) 7.52982 1.09834 0.549168 − 0.835712i \(-0.314945\pi\)
0.549168 + 0.835712i \(0.314945\pi\)
\(48\) 0 0
\(49\) −6.76491 −0.966416
\(50\) 0 0
\(51\) 7.37466 1.03266
\(52\) 0 0
\(53\) 12.9844 1.78355 0.891773 − 0.452484i \(-0.149462\pi\)
0.891773 + 0.452484i \(0.149462\pi\)
\(54\) 0 0
\(55\) −6.56009 −0.884563
\(56\) 0 0
\(57\) 7.76491 1.02849
\(58\) 0 0
\(59\) −1.11021 −0.144536 −0.0722682 − 0.997385i \(-0.523024\pi\)
−0.0722682 + 0.997385i \(0.523024\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 0.484862 0.0610869
\(64\) 0 0
\(65\) 1.28005 0.158770
\(66\) 0 0
\(67\) 5.21949 0.637663 0.318831 − 0.947811i \(-0.396710\pi\)
0.318831 + 0.947811i \(0.396710\pi\)
\(68\) 0 0
\(69\) 7.12489 0.857735
\(70\) 0 0
\(71\) −10.5601 −1.25325 −0.626626 − 0.779320i \(-0.715565\pi\)
−0.626626 + 0.779320i \(0.715565\pi\)
\(72\) 0 0
\(73\) −9.70436 −1.13581 −0.567905 − 0.823094i \(-0.692246\pi\)
−0.567905 + 0.823094i \(0.692246\pi\)
\(74\) 0 0
\(75\) −1.96972 −0.227444
\(76\) 0 0
\(77\) −1.20482 −0.137301
\(78\) 0 0
\(79\) 11.5298 1.29721 0.648603 − 0.761127i \(-0.275354\pi\)
0.648603 + 0.761127i \(0.275354\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.0450 −1.43187 −0.715935 − 0.698167i \(-0.753999\pi\)
−0.715935 + 0.698167i \(0.753999\pi\)
\(84\) 0 0
\(85\) −19.4693 −2.11174
\(86\) 0 0
\(87\) 2.64002 0.283040
\(88\) 0 0
\(89\) −9.43521 −1.00013 −0.500065 − 0.865988i \(-0.666691\pi\)
−0.500065 + 0.865988i \(0.666691\pi\)
\(90\) 0 0
\(91\) 0.235091 0.0246442
\(92\) 0 0
\(93\) −6.24977 −0.648071
\(94\) 0 0
\(95\) −20.4995 −2.10321
\(96\) 0 0
\(97\) 15.7796 1.60217 0.801087 − 0.598548i \(-0.204255\pi\)
0.801087 + 0.598548i \(0.204255\pi\)
\(98\) 0 0
\(99\) −2.48486 −0.249738
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 1776.2.a.r.1.3 3
3.2 odd 2 5328.2.a.bm.1.1 3
4.3 odd 2 888.2.a.j.1.3 ✓ 3
8.3 odd 2 7104.2.a.br.1.1 3
8.5 even 2 7104.2.a.bx.1.1 3
12.11 even 2 2664.2.a.o.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.a.j.1.3 ✓ 3 4.3 odd 2
1776.2.a.r.1.3 3 1.1 even 1 trivial
2664.2.a.o.1.1 3 12.11 even 2
5328.2.a.bm.1.1 3 3.2 odd 2
7104.2.a.br.1.1 3 8.3 odd 2
7104.2.a.bx.1.1 3 8.5 even 2