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Properties

Label 1776.2.q.m.1009.1
Level $1776$
Weight $2$
Character 1776.1009
Analytic conductor $14.181$
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] = [1776,2,Mod(433,1776)] 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("1776.433"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1776, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1776.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-3,0,3,0,3,0,-3,0,2] 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: no
Analytic conductor: \(14.1814313990\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.50898483.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} - 10x^{3} + 64x^{2} - 40x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 888)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1009.1
Root \(1.21966 + 2.11251i\) of defining polynomial
Character \(\chi\) \(=\) 1776.1009
Dual form 1776.2.q.m.433.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(-0.719656 - 1.24648i) q^{5} +(0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +0.950241 q^{11} +(-0.219656 - 0.380455i) q^{13} +(-0.719656 + 1.24648i) q^{15} +(3.19478 - 5.53352i) q^{17} +(1.91443 + 3.31589i) q^{19} +(0.500000 - 0.866025i) q^{21} -8.21842 q^{23} +(1.46419 - 2.53605i) q^{25} +1.00000 q^{27} +6.48907 q^{29} +3.38955 q^{31} +(-0.475121 - 0.822933i) q^{33} +(0.719656 - 1.24648i) q^{35} +(-6.07340 - 0.337363i) q^{37} +(-0.219656 + 0.380455i) q^{39} +(-3.86467 - 6.69381i) q^{41} -2.43931 q^{43} +1.43931 q^{45} +4.48907 q^{47} +(3.00000 - 5.19615i) q^{49} -6.38955 q^{51} +(1.63409 - 2.83032i) q^{53} +(-0.683847 - 1.18446i) q^{55} +(1.91443 - 3.31589i) q^{57} +(0.964191 - 1.67003i) q^{59} +(-0.0497586 - 0.0861844i) q^{61} -1.00000 q^{63} +(-0.316153 + 0.547593i) q^{65} +(2.46419 + 4.26810i) q^{67} +(4.10921 + 7.11736i) q^{69} +(-7.87862 - 13.6462i) q^{71} -5.29004 q^{73} -2.92838 q^{75} +(0.475121 + 0.822933i) q^{77} +(-6.80399 - 11.7848i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(-6.26818 + 10.8568i) q^{83} -9.19656 q^{85} +(-3.24454 - 5.61970i) q^{87} +(5.82886 - 10.0959i) q^{89} +(0.219656 - 0.380455i) q^{91} +(-1.69478 - 2.93544i) q^{93} +(2.75546 - 4.77261i) q^{95} -7.80701 q^{97} +(-0.475121 + 0.822933i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 3 q^{5} + 3 q^{7} - 3 q^{9} + 2 q^{11} + 6 q^{13} + 3 q^{15} + 10 q^{17} - 5 q^{19} + 3 q^{21} + 2 q^{23} - 4 q^{25} + 6 q^{27} + 28 q^{29} + 2 q^{31} - q^{33} - 3 q^{35} + 2 q^{37} + 6 q^{39}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(593\) \(1297\) \(1333\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) −0.500000 − 0.866025i −0.288675 − 0.500000i
\(4\) 0 0
\(5\) −0.719656 − 1.24648i −0.321840 − 0.557443i 0.659028 − 0.752119i \(-0.270968\pi\)
−0.980868 + 0.194676i \(0.937635\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i 0.944911 − 0.327327i \(-0.106148\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 0.950241 0.286509 0.143254 − 0.989686i \(-0.454243\pi\)
0.143254 + 0.989686i \(0.454243\pi\)
\(12\) 0 0
\(13\) −0.219656 − 0.380455i −0.0609216 − 0.105519i 0.833956 − 0.551831i \(-0.186071\pi\)
−0.894878 + 0.446312i \(0.852737\pi\)
\(14\) 0 0
\(15\) −0.719656 + 1.24648i −0.185814 + 0.321840i
\(16\) 0 0
\(17\) 3.19478 − 5.53352i 0.774847 − 1.34207i −0.160033 − 0.987112i \(-0.551160\pi\)
0.934880 − 0.354963i \(-0.115507\pi\)
\(18\) 0 0
\(19\) 1.91443 + 3.31589i 0.439201 + 0.760718i 0.997628 − 0.0688353i \(-0.0219283\pi\)
−0.558427 + 0.829554i \(0.688595\pi\)
\(20\) 0 0
\(21\) 0.500000 − 0.866025i 0.109109 − 0.188982i
\(22\) 0 0
\(23\) −8.21842 −1.71366 −0.856829 − 0.515600i \(-0.827569\pi\)
−0.856829 + 0.515600i \(0.827569\pi\)
\(24\) 0 0
\(25\) 1.46419 − 2.53605i 0.292838 − 0.507211i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.48907 1.20499 0.602495 − 0.798123i \(-0.294173\pi\)
0.602495 + 0.798123i \(0.294173\pi\)
\(30\) 0 0
\(31\) 3.38955 0.608782 0.304391 − 0.952547i \(-0.401547\pi\)
0.304391 + 0.952547i \(0.401547\pi\)
\(32\) 0 0
\(33\) −0.475121 − 0.822933i −0.0827079 − 0.143254i
\(34\) 0 0
\(35\) 0.719656 − 1.24648i 0.121644 − 0.210694i
\(36\) 0 0
\(37\) −6.07340 − 0.337363i −0.998461 − 0.0554621i
\(38\) 0 0
\(39\) −0.219656 + 0.380455i −0.0351731 + 0.0609216i
\(40\) 0 0
\(41\) −3.86467 − 6.69381i −0.603561 − 1.04540i −0.992277 − 0.124040i \(-0.960415\pi\)
0.388717 − 0.921357i \(-0.372919\pi\)
\(42\) 0 0
\(43\) −2.43931 −0.371992 −0.185996 − 0.982551i \(-0.559551\pi\)
−0.185996 + 0.982551i \(0.559551\pi\)
\(44\) 0 0
\(45\) 1.43931 0.214560
\(46\) 0 0
\(47\) 4.48907 0.654798 0.327399 − 0.944886i \(-0.393828\pi\)
0.327399 + 0.944886i \(0.393828\pi\)
\(48\) 0 0
\(49\) 3.00000 − 5.19615i 0.428571 − 0.742307i
\(50\) 0 0
\(51\) −6.38955 −0.894716
\(52\) 0 0
\(53\) 1.63409 − 2.83032i 0.224459 − 0.388775i −0.731698 − 0.681629i \(-0.761272\pi\)
0.956157 + 0.292854i \(0.0946051\pi\)
\(54\) 0 0
\(55\) −0.683847 − 1.18446i −0.0922099 − 0.159712i
\(56\) 0 0
\(57\) 1.91443 − 3.31589i 0.253573 − 0.439201i
\(58\) 0 0
\(59\) 0.964191 − 1.67003i 0.125527 − 0.217419i −0.796412 − 0.604755i \(-0.793271\pi\)
0.921939 + 0.387336i \(0.126605\pi\)
\(60\) 0 0
\(61\) −0.0497586 − 0.0861844i −0.00637093 − 0.0110348i 0.862822 − 0.505507i \(-0.168695\pi\)
−0.869193 + 0.494472i \(0.835361\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −0.316153 + 0.547593i −0.0392140 + 0.0679206i
\(66\) 0 0
\(67\) 2.46419 + 4.26810i 0.301049 + 0.521432i 0.976374 − 0.216088i \(-0.0693299\pi\)
−0.675325 + 0.737520i \(0.735997\pi\)
\(68\) 0 0
\(69\) 4.10921 + 7.11736i 0.494691 + 0.856829i
\(70\) 0 0
\(71\) −7.87862 − 13.6462i −0.935021 − 1.61950i −0.774597 − 0.632455i \(-0.782047\pi\)
−0.160423 − 0.987048i \(-0.551286\pi\)
\(72\) 0 0
\(73\) −5.29004 −0.619152 −0.309576 − 0.950875i \(-0.600187\pi\)
−0.309576 + 0.950875i \(0.600187\pi\)
\(74\) 0 0
\(75\) −2.92838 −0.338140
\(76\) 0 0
\(77\) 0.475121 + 0.822933i 0.0541450 + 0.0937819i
\(78\) 0 0
\(79\) −6.80399 − 11.7848i −0.765508 − 1.32590i −0.939977 − 0.341236i \(-0.889154\pi\)
0.174469 − 0.984663i \(-0.444179\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −6.26818 + 10.8568i −0.688022 + 1.19169i 0.284455 + 0.958689i \(0.408187\pi\)
−0.972477 + 0.232999i \(0.925146\pi\)
\(84\) 0 0
\(85\) −9.19656 −0.997507
\(86\) 0 0
\(87\) −3.24454 − 5.61970i −0.347851 − 0.602495i
\(88\) 0 0
\(89\) 5.82886 − 10.0959i 0.617858 − 1.07016i −0.372017 − 0.928226i \(-0.621334\pi\)
0.989876 − 0.141936i \(-0.0453329\pi\)
\(90\) 0 0
\(91\) 0.219656 − 0.380455i 0.0230262 − 0.0398825i
\(92\) 0 0
\(93\) −1.69478 − 2.93544i −0.175740 − 0.304391i
\(94\) 0 0
\(95\) 2.75546 − 4.77261i 0.282705 − 0.489659i
\(96\) 0 0
\(97\) −7.80701 −0.792681 −0.396341 − 0.918104i \(-0.629720\pi\)
−0.396341 + 0.918104i \(0.629720\pi\)
\(98\) 0 0
\(99\) −0.475121 + 0.822933i −0.0477514 + 0.0827079i
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.q.m.1009.1 6
4.3 odd 2 888.2.q.h.121.1 ✓ 6
12.11 even 2 2664.2.r.h.1009.3 6
37.26 even 3 inner 1776.2.q.m.433.1 6
148.63 odd 6 888.2.q.h.433.1 yes 6
444.359 even 6 2664.2.r.h.433.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.h.121.1 ✓ 6 4.3 odd 2
888.2.q.h.433.1 yes 6 148.63 odd 6
1776.2.q.m.433.1 6 37.26 even 3 inner
1776.2.q.m.1009.1 6 1.1 even 1 trivial
2664.2.r.h.433.3 6 444.359 even 6
2664.2.r.h.1009.3 6 12.11 even 2