↑

Properties

Label 1776.2.a.r.1.2
Level $1776$
Weight $2$
Character 1776.1
Self dual yes
Analytic conductor $14.181$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.2
Root \(-1.76156\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.864641 q^{5} -2.62620 q^{7} +1.00000 q^{9} +0.626198 q^{11} -2.62620 q^{13} -0.864641 q^{15} +7.28467 q^{17} -1.10308 q^{19} +2.62620 q^{21} -2.23844 q^{23} -4.25240 q^{25} -1.00000 q^{27} -0.864641 q^{29} -3.52311 q^{31} -0.626198 q^{33} -2.27072 q^{35} -1.00000 q^{37} +2.62620 q^{39} +2.00000 q^{41} +8.77551 q^{43} +0.864641 q^{45} -5.79383 q^{47} -0.103084 q^{49} -7.28467 q^{51} -9.67243 q^{53} +0.541436 q^{55} +1.10308 q^{57} -12.6585 q^{59} -6.00000 q^{61} -2.62620 q^{63} -2.27072 q^{65} -10.7755 q^{67} +2.23844 q^{69} -3.45856 q^{71} +9.40171 q^{73} +4.25240 q^{75} -1.64452 q^{77} -1.79383 q^{79} +1.00000 q^{81} -2.83237 q^{83} +6.29862 q^{85} +0.864641 q^{87} -7.22012 q^{89} +6.89692 q^{91} +3.52311 q^{93} -0.953771 q^{95} -7.31695 q^{97} +0.626198 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + q^{7} + 3 q^{9} - 7 q^{11} + q^{13} + 3 q^{17} - 7 q^{19} - q^{21} - 13 q^{23} + 5 q^{25} - 3 q^{27} + 2 q^{31} + 7 q^{33} - 12 q^{35} - 3 q^{37} - q^{39} + 6 q^{41} - 4 q^{43} - 10 q^{47}+ \cdots - 7 q^{99}+O(q^{100}) \) 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
\(5\) 0.864641 0.386679 0.193340 − 0.981132i \(-0.438068\pi\)
0.193340 + 0.981132i \(0.438068\pi\)
\(6\) 0 0
\(7\) −2.62620 −0.992610 −0.496305 − 0.868148i \(-0.665310\pi\)
−0.496305 + 0.868148i \(0.665310\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.626198 0.188806 0.0944029 − 0.995534i \(-0.469906\pi\)
0.0944029 + 0.995534i \(0.469906\pi\)
\(12\) 0 0
\(13\) −2.62620 −0.728376 −0.364188 − 0.931325i \(-0.618653\pi\)
−0.364188 + 0.931325i \(0.618653\pi\)
\(14\) 0 0
\(15\) −0.864641 −0.223249
\(16\) 0 0
\(17\) 7.28467 1.76679 0.883396 − 0.468627i \(-0.155251\pi\)
0.883396 + 0.468627i \(0.155251\pi\)
\(18\) 0 0
\(19\) −1.10308 −0.253065 −0.126532 − 0.991962i \(-0.540385\pi\)
−0.126532 + 0.991962i \(0.540385\pi\)
\(20\) 0 0
\(21\) 2.62620 0.573083
\(22\) 0 0
\(23\) −2.23844 −0.466748 −0.233374 − 0.972387i \(-0.574977\pi\)
−0.233374 + 0.972387i \(0.574977\pi\)
\(24\) 0 0
\(25\) −4.25240 −0.850479
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.864641 −0.160560 −0.0802799 − 0.996772i \(-0.525581\pi\)
−0.0802799 + 0.996772i \(0.525581\pi\)
\(30\) 0 0
\(31\) −3.52311 −0.632770 −0.316385 − 0.948631i \(-0.602469\pi\)
−0.316385 + 0.948631i \(0.602469\pi\)
\(32\) 0 0
\(33\) −0.626198 −0.109007
\(34\) 0 0
\(35\) −2.27072 −0.383821
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 2.62620 0.420528
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 8.77551 1.33825 0.669126 − 0.743149i \(-0.266668\pi\)
0.669126 + 0.743149i \(0.266668\pi\)
\(44\) 0 0
\(45\) 0.864641 0.128893
\(46\) 0 0
\(47\) −5.79383 −0.845117 −0.422559 − 0.906336i \(-0.638868\pi\)
−0.422559 + 0.906336i \(0.638868\pi\)
\(48\) 0 0
\(49\) −0.103084 −0.0147262
\(50\) 0 0
\(51\) −7.28467 −1.02006
\(52\) 0 0
\(53\) −9.67243 −1.32861 −0.664305 − 0.747462i \(-0.731272\pi\)
−0.664305 + 0.747462i \(0.731272\pi\)
\(54\) 0 0
\(55\) 0.541436 0.0730073
\(56\) 0 0
\(57\) 1.10308 0.146107
\(58\) 0 0
\(59\) −12.6585 −1.64799 −0.823996 − 0.566595i \(-0.808260\pi\)
−0.823996 + 0.566595i \(0.808260\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −2.62620 −0.330870
\(64\) 0 0
\(65\) −2.27072 −0.281648
\(66\) 0 0
\(67\) −10.7755 −1.31644 −0.658219 − 0.752826i \(-0.728690\pi\)
−0.658219 + 0.752826i \(0.728690\pi\)
\(68\) 0 0
\(69\) 2.23844 0.269477
\(70\) 0 0
\(71\) −3.45856 −0.410456 −0.205228 − 0.978714i \(-0.565794\pi\)
−0.205228 + 0.978714i \(0.565794\pi\)
\(72\) 0 0
\(73\) 9.40171 1.10039 0.550193 − 0.835037i \(-0.314554\pi\)
0.550193 + 0.835037i \(0.314554\pi\)
\(74\) 0 0
\(75\) 4.25240 0.491024
\(76\) 0 0
\(77\) −1.64452 −0.187410
\(78\) 0 0
\(79\) −1.79383 −0.201822 −0.100911 − 0.994895i \(-0.532176\pi\)
−0.100911 + 0.994895i \(0.532176\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.83237 −0.310893 −0.155446 − 0.987844i \(-0.549682\pi\)
−0.155446 + 0.987844i \(0.549682\pi\)
\(84\) 0 0
\(85\) 6.29862 0.683182
\(86\) 0 0
\(87\) 0.864641 0.0926992
\(88\) 0 0
\(89\) −7.22012 −0.765331 −0.382666 − 0.923887i \(-0.624994\pi\)
−0.382666 + 0.923887i \(0.624994\pi\)
\(90\) 0 0
\(91\) 6.89692 0.722993
\(92\) 0 0
\(93\) 3.52311 0.365330
\(94\) 0 0
\(95\) −0.953771 −0.0978549
\(96\) 0 0
\(97\) −7.31695 −0.742923 −0.371462 − 0.928448i \(-0.621143\pi\)
−0.371462 + 0.928448i \(0.621143\pi\)
\(98\) 0 0
\(99\) 0.626198 0.0629353
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.1.2 3
3.2 odd 2 5328.2.a.bm.1.2 3
4.3 odd 2 888.2.a.j.1.2 ✓ 3
8.3 odd 2 7104.2.a.br.1.2 3
8.5 even 2 7104.2.a.bx.1.2 3
12.11 even 2 2664.2.a.o.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.a.j.1.2 ✓ 3 4.3 odd 2
1776.2.a.r.1.2 3 1.1 even 1 trivial
2664.2.a.o.1.2 3 12.11 even 2
5328.2.a.bm.1.2 3 3.2 odd 2
7104.2.a.br.1.2 3 8.3 odd 2
7104.2.a.bx.1.2 3 8.5 even 2