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Properties

Label 1008.2.cs.o.271.1
Level $1008$
Weight $2$
Character 1008.271
Analytic conductor $8.049$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [1008,2,Mod(271,1008)] 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("1008.271"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cs (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 271.1
Root \(0.809017 + 1.40126i\) of defining polynomial
Character \(\chi\) \(=\) 1008.271
Dual form 1008.2.cs.o.703.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.35410 - 1.93649i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(3.35410 - 1.93649i) q^{11} +3.46410i q^{13} +(-2.00000 + 3.46410i) q^{19} +(-6.70820 - 3.87298i) q^{23} +(5.00000 + 8.66025i) q^{25} +6.70820 q^{29} +(0.500000 + 0.866025i) q^{31} +(6.70820 + 7.74597i) q^{35} +(-2.00000 + 3.46410i) q^{37} +7.74597i q^{41} +6.92820i q^{43} +(-6.70820 + 11.6190i) q^{47} +(5.50000 + 4.33013i) q^{49} +(3.35410 + 5.80948i) q^{53} -15.0000 q^{55} +(-3.35410 - 5.80948i) q^{59} +(9.00000 + 5.19615i) q^{61} +(6.70820 - 11.6190i) q^{65} +(6.00000 - 3.46410i) q^{67} -7.74597i q^{71} +(-6.00000 + 3.46410i) q^{73} +(-10.0623 + 1.93649i) q^{77} +(-10.5000 - 6.06218i) q^{79} +6.70820 q^{83} +(-6.70820 - 3.87298i) q^{89} +(3.00000 - 8.66025i) q^{91} +(13.4164 - 7.74597i) q^{95} -5.19615i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{7} - 8 q^{19} + 20 q^{25} + 2 q^{31} - 8 q^{37} + 22 q^{49} - 60 q^{55} + 36 q^{61} + 24 q^{67} - 24 q^{73} - 42 q^{79} + 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) −3.35410 − 1.93649i −1.50000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −2.50000 − 0.866025i −0.944911 − 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.35410 − 1.93649i 1.01130 − 0.583874i 0.0997278 − 0.995015i \(-0.468203\pi\)
0.911572 + 0.411141i \(0.134869\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) −2.00000 + 3.46410i −0.458831 + 0.794719i −0.998899 − 0.0469020i \(-0.985065\pi\)
0.540068 + 0.841621i \(0.318398\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.70820 − 3.87298i −1.39876 − 0.807573i −0.404495 − 0.914540i \(-0.632553\pi\)
−0.994263 + 0.106967i \(0.965886\pi\)
\(24\) 0 0
\(25\) 5.00000 + 8.66025i 1.00000 + 1.73205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.70820 1.24568 0.622841 − 0.782348i \(-0.285978\pi\)
0.622841 + 0.782348i \(0.285978\pi\)
\(30\) 0 0
\(31\) 0.500000 + 0.866025i 0.0898027 + 0.155543i 0.907428 − 0.420208i \(-0.138043\pi\)
−0.817625 + 0.575751i \(0.804710\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.70820 + 7.74597i 1.13389 + 1.30931i
\(36\) 0 0
\(37\) −2.00000 + 3.46410i −0.328798 + 0.569495i −0.982274 − 0.187453i \(-0.939977\pi\)
0.653476 + 0.756948i \(0.273310\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.74597i 1.20972i 0.796333 + 0.604858i \(0.206770\pi\)
−0.796333 + 0.604858i \(0.793230\pi\)
\(42\) 0 0
\(43\) 6.92820i 1.05654i 0.849076 + 0.528271i \(0.177159\pi\)
−0.849076 + 0.528271i \(0.822841\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.70820 + 11.6190i −0.978492 + 1.69480i −0.310599 + 0.950541i \(0.600530\pi\)
−0.667893 + 0.744257i \(0.732804\pi\)
\(48\) 0 0
\(49\) 5.50000 + 4.33013i 0.785714 + 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.35410 + 5.80948i 0.460721 + 0.797993i 0.998997 − 0.0447760i \(-0.0142574\pi\)
−0.538276 + 0.842769i \(0.680924\pi\)
\(54\) 0 0
\(55\) −15.0000 −2.02260
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.35410 − 5.80948i −0.436667 − 0.756329i 0.560763 − 0.827976i \(-0.310508\pi\)
−0.997430 + 0.0716470i \(0.977174\pi\)
\(60\) 0 0
\(61\) 9.00000 + 5.19615i 1.15233 + 0.665299i 0.949454 − 0.313905i \(-0.101637\pi\)
0.202878 + 0.979204i \(0.434971\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.70820 − 11.6190i 0.832050 − 1.44115i
\(66\) 0 0
\(67\) 6.00000 − 3.46410i 0.733017 − 0.423207i −0.0865081 − 0.996251i \(-0.527571\pi\)
0.819525 + 0.573044i \(0.194238\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 7.74597i − 0.919277i −0.888106 − 0.459639i \(-0.847979\pi\)
0.888106 − 0.459639i \(-0.152021\pi\)
\(72\) 0 0
\(73\) −6.00000 + 3.46410i −0.702247 + 0.405442i −0.808184 − 0.588930i \(-0.799549\pi\)
0.105937 + 0.994373i \(0.466216\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.0623 + 1.93649i −1.14671 + 0.220684i
\(78\) 0 0
\(79\) −10.5000 − 6.06218i −1.18134 − 0.682048i −0.225018 − 0.974355i \(-0.572244\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.70820 0.736321 0.368161 − 0.929762i \(-0.379988\pi\)
0.368161 + 0.929762i \(0.379988\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.70820 − 3.87298i −0.711068 − 0.410535i 0.100388 − 0.994948i \(-0.467992\pi\)
−0.811456 + 0.584413i \(0.801325\pi\)
\(90\) 0 0
\(91\) 3.00000 − 8.66025i 0.314485 − 0.907841i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 13.4164 − 7.74597i 1.37649 − 0.794719i
\(96\) 0 0
\(97\) − 5.19615i − 0.527589i −0.964579 − 0.263795i \(-0.915026\pi\)
0.964579 − 0.263795i \(-0.0849741\pi\)
\(98\) 0 0
\(99\) 0 0
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 1008.2.cs.o.271.1 ✓ 4
3.2 odd 2 inner 1008.2.cs.o.271.2 yes 4
4.3 odd 2 1008.2.cs.p.271.1 yes 4
7.2 even 3 7056.2.b.v.1567.4 4
7.3 odd 6 1008.2.cs.p.703.1 yes 4
7.5 odd 6 7056.2.b.o.1567.1 4
12.11 even 2 1008.2.cs.p.271.2 yes 4
21.2 odd 6 7056.2.b.v.1567.2 4
21.5 even 6 7056.2.b.o.1567.3 4
21.17 even 6 1008.2.cs.p.703.2 yes 4
28.3 even 6 inner 1008.2.cs.o.703.1 yes 4
28.19 even 6 7056.2.b.v.1567.1 4
28.23 odd 6 7056.2.b.o.1567.4 4
84.23 even 6 7056.2.b.o.1567.2 4
84.47 odd 6 7056.2.b.v.1567.3 4
84.59 odd 6 inner 1008.2.cs.o.703.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.cs.o.271.1 ✓ 4 1.1 even 1 trivial
1008.2.cs.o.271.2 yes 4 3.2 odd 2 inner
1008.2.cs.o.703.1 yes 4 28.3 even 6 inner
1008.2.cs.o.703.2 yes 4 84.59 odd 6 inner
1008.2.cs.p.271.1 yes 4 4.3 odd 2
1008.2.cs.p.271.2 yes 4 12.11 even 2
1008.2.cs.p.703.1 yes 4 7.3 odd 6
1008.2.cs.p.703.2 yes 4 21.17 even 6
7056.2.b.o.1567.1 4 7.5 odd 6
7056.2.b.o.1567.2 4 84.23 even 6
7056.2.b.o.1567.3 4 21.5 even 6
7056.2.b.o.1567.4 4 28.23 odd 6
7056.2.b.v.1567.1 4 28.19 even 6
7056.2.b.v.1567.2 4 21.2 odd 6
7056.2.b.v.1567.3 4 84.47 odd 6
7056.2.b.v.1567.4 4 7.2 even 3