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Properties

Label 1776.2.q.a.1009.1
Level $1776$
Weight $2$
Character 1776.1009
Analytic conductor $14.181$
Analytic rank $1$
Dimension $2$
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 = [2,0,-1,0,-4,0,1,0,-1,0,4] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1776.1009
Dual form 1776.2.q.a.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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +2.00000 q^{11} +(-1.50000 - 2.59808i) q^{13} +(-2.00000 + 3.46410i) q^{15} +(-4.00000 + 6.92820i) q^{17} +(-4.00000 - 6.92820i) q^{19} +(0.500000 - 0.866025i) q^{21} +(-5.50000 + 9.52628i) q^{25} +1.00000 q^{27} -10.0000 q^{29} +7.00000 q^{31} +(-1.00000 - 1.73205i) q^{33} +(2.00000 - 3.46410i) q^{35} +(5.00000 - 3.46410i) q^{37} +(-1.50000 + 2.59808i) q^{39} +(1.00000 + 1.73205i) q^{41} -5.00000 q^{43} +4.00000 q^{45} +6.00000 q^{47} +(3.00000 - 5.19615i) q^{49} +8.00000 q^{51} +(-3.00000 + 5.19615i) q^{53} +(-4.00000 - 6.92820i) q^{55} +(-4.00000 + 6.92820i) q^{57} +(-6.00000 + 10.3923i) q^{59} +(1.00000 + 1.73205i) q^{61} -1.00000 q^{63} +(-6.00000 + 10.3923i) q^{65} +(4.50000 + 7.79423i) q^{67} +(5.00000 + 8.66025i) q^{71} -11.0000 q^{73} +11.0000 q^{75} +(1.00000 + 1.73205i) q^{77} +(-4.50000 - 7.79423i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(-6.00000 + 10.3923i) q^{83} +32.0000 q^{85} +(5.00000 + 8.66025i) q^{87} +(3.00000 - 5.19615i) q^{89} +(1.50000 - 2.59808i) q^{91} +(-3.50000 - 6.06218i) q^{93} +(-16.0000 + 27.7128i) q^{95} +1.00000 q^{97} +(-1.00000 + 1.73205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 4 q^{5} + q^{7} - q^{9} + 4 q^{11} - 3 q^{13} - 4 q^{15} - 8 q^{17} - 8 q^{19} + q^{21} - 11 q^{25} + 2 q^{27} - 20 q^{29} + 14 q^{31} - 2 q^{33} + 4 q^{35} + 10 q^{37} - 3 q^{39} + 2 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(5\) −2.00000 − 3.46410i −0.894427 − 1.54919i −0.834512 − 0.550990i \(-0.814250\pi\)
−0.0599153 − 0.998203i \(-0.519083\pi\)
\(6\) 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
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −1.50000 − 2.59808i −0.416025 − 0.720577i 0.579510 − 0.814965i \(-0.303244\pi\)
−0.995535 + 0.0943882i \(0.969911\pi\)
\(14\) 0 0
\(15\) −2.00000 + 3.46410i −0.516398 + 0.894427i
\(16\) 0 0
\(17\) −4.00000 + 6.92820i −0.970143 + 1.68034i −0.275029 + 0.961436i \(0.588688\pi\)
−0.695113 + 0.718900i \(0.744646\pi\)
\(18\) 0 0
\(19\) −4.00000 − 6.92820i −0.917663 − 1.58944i −0.802955 − 0.596040i \(-0.796740\pi\)
−0.114708 − 0.993399i \(-0.536593\pi\)
\(20\) 0 0
\(21\) 0.500000 − 0.866025i 0.109109 − 0.188982i
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.50000 + 9.52628i −1.10000 + 1.90526i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 − 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) −1.00000 − 1.73205i −0.174078 − 0.301511i
\(34\) 0 0
\(35\) 2.00000 − 3.46410i 0.338062 − 0.585540i
\(36\) 0 0
\(37\) 5.00000 − 3.46410i 0.821995 − 0.569495i
\(38\) 0 0
\(39\) −1.50000 + 2.59808i −0.240192 + 0.416025i
\(40\) 0 0
\(41\) 1.00000 + 1.73205i 0.156174 + 0.270501i 0.933486 − 0.358614i \(-0.116751\pi\)
−0.777312 + 0.629115i \(0.783417\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 − 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 4.00000 0.596285
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 3.00000 − 5.19615i 0.428571 − 0.742307i
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 − 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) −4.00000 − 6.92820i −0.539360 − 0.934199i
\(56\) 0 0
\(57\) −4.00000 + 6.92820i −0.529813 + 0.917663i
\(58\) 0 0
\(59\) −6.00000 + 10.3923i −0.781133 + 1.35296i 0.150148 + 0.988663i \(0.452025\pi\)
−0.931282 + 0.364299i \(0.881308\pi\)
\(60\) 0 0
\(61\) 1.00000 + 1.73205i 0.128037 + 0.221766i 0.922916 − 0.385002i \(-0.125799\pi\)
−0.794879 + 0.606768i \(0.792466\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −6.00000 + 10.3923i −0.744208 + 1.28901i
\(66\) 0 0
\(67\) 4.50000 + 7.79423i 0.549762 + 0.952217i 0.998290 + 0.0584478i \(0.0186151\pi\)
−0.448528 + 0.893769i \(0.648052\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.00000 + 8.66025i 0.593391 + 1.02778i 0.993772 + 0.111434i \(0.0355445\pi\)
−0.400381 + 0.916349i \(0.631122\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 − 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 11.0000 1.27017
\(76\) 0 0
\(77\) 1.00000 + 1.73205i 0.113961 + 0.197386i
\(78\) 0 0
\(79\) −4.50000 − 7.79423i −0.506290 − 0.876919i −0.999974 − 0.00727784i \(-0.997683\pi\)
0.493684 − 0.869641i \(-0.335650\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −6.00000 + 10.3923i −0.658586 + 1.14070i 0.322396 + 0.946605i \(0.395512\pi\)
−0.980982 + 0.194099i \(0.937822\pi\)
\(84\) 0 0
\(85\) 32.0000 3.47089
\(86\) 0 0
\(87\) 5.00000 + 8.66025i 0.536056 + 0.928477i
\(88\) 0 0
\(89\) 3.00000 − 5.19615i 0.317999 − 0.550791i −0.662071 − 0.749441i \(-0.730322\pi\)
0.980071 + 0.198650i \(0.0636557\pi\)
\(90\) 0 0
\(91\) 1.50000 − 2.59808i 0.157243 − 0.272352i
\(92\) 0 0
\(93\) −3.50000 − 6.06218i −0.362933 − 0.628619i
\(94\) 0 0
\(95\) −16.0000 + 27.7128i −1.64157 + 2.84327i
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) −1.00000 + 1.73205i −0.100504 + 0.174078i
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.a.1009.1 2
4.3 odd 2 888.2.q.c.121.1 ✓ 2
12.11 even 2 2664.2.r.g.1009.1 2
37.26 even 3 inner 1776.2.q.a.433.1 2
148.63 odd 6 888.2.q.c.433.1 yes 2
444.359 even 6 2664.2.r.g.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.c.121.1 ✓ 2 4.3 odd 2
888.2.q.c.433.1 yes 2 148.63 odd 6
1776.2.q.a.433.1 2 37.26 even 3 inner
1776.2.q.a.1009.1 2 1.1 even 1 trivial
2664.2.r.g.433.1 2 444.359 even 6
2664.2.r.g.1009.1 2 12.11 even 2