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Properties

Label 888.2.q.c.433.1
Level $888$
Weight $2$
Character 888.433
Analytic conductor $7.091$
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] = [888,2,Mod(121,888)] 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("888.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.q (of order \(3\), degree \(2\), 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: \(7.09071569949\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 433.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 888.433
Dual form 888.2.q.c.121.1

$q$-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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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 −0.0599153 + 0.998203i \(0.519083\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i −0.944911 − 0.327327i \(-0.893852\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −1.50000 + 2.59808i −0.416025 + 0.720577i −0.995535 − 0.0943882i \(-0.969911\pi\)
0.579510 + 0.814965i \(0.303244\pi\)
\(14\) 0 0
\(15\) 2.00000 + 3.46410i 0.516398 + 0.894427i
\(16\) 0 0
\(17\) −4.00000 − 6.92820i −0.970143 − 1.68034i −0.695113 − 0.718900i \(-0.744646\pi\)
−0.275029 − 0.961436i \(-0.588688\pi\)
\(18\) 0 0
\(19\) 4.00000 − 6.92820i 0.917663 − 1.58944i 0.114708 − 0.993399i \(-0.463407\pi\)
0.802955 − 0.596040i \(-0.203260\pi\)
\(20\) 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 \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 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 −0.777312 − 0.629115i \(-0.783417\pi\)
0.933486 + 0.358614i \(0.116751\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 − 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 4.00000 0.596285
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 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 0.583036 − 0.812447i \(-0.301865\pi\)
−0.995117 + 0.0987002i \(0.968532\pi\)
\(54\) 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 0.931282 + 0.364299i \(0.118692\pi\)
−0.150148 + 0.988663i \(0.547975\pi\)
\(60\) 0 0
\(61\) 1.00000 − 1.73205i 0.128037 − 0.221766i −0.794879 − 0.606768i \(-0.792466\pi\)
0.922916 + 0.385002i \(0.125799\pi\)
\(62\) 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 0.448528 + 0.893769i \(0.351948\pi\)
−0.998290 + 0.0584478i \(0.981385\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.00000 + 8.66025i −0.593391 + 1.02778i 0.400381 + 0.916349i \(0.368878\pi\)
−0.993772 + 0.111434i \(0.964456\pi\)
\(72\) 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 −0.493684 − 0.869641i \(-0.664350\pi\)
0.999974 − 0.00727784i \(-0.00231663\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 6.00000 + 10.3923i 0.658586 + 1.14070i 0.980982 + 0.194099i \(0.0621783\pi\)
−0.322396 + 0.946605i \(0.604488\pi\)
\(84\) 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 0.980071 − 0.198650i \(-0.0636557\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(90\) 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 888.2.q.c.433.1 yes 2
3.2 odd 2 2664.2.r.g.433.1 2
4.3 odd 2 1776.2.q.a.433.1 2
37.10 even 3 inner 888.2.q.c.121.1 ✓ 2
111.47 odd 6 2664.2.r.g.1009.1 2
148.47 odd 6 1776.2.q.a.1009.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.q.c.121.1 ✓ 2 37.10 even 3 inner
888.2.q.c.433.1 yes 2 1.1 even 1 trivial
1776.2.q.a.433.1 2 4.3 odd 2
1776.2.q.a.1009.1 2 148.47 odd 6
2664.2.r.g.433.1 2 3.2 odd 2
2664.2.r.g.1009.1 2 111.47 odd 6