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Properties

Label 5184.2.d.p.2593.2
Level $5184$
Weight $2$
Character 5184.2593
Analytic conductor $41.394$
Analytic rank $0$
Dimension $8$
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] = [5184,2,Mod(2593,5184)] 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("5184.2593"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5184, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,-32,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2593.2
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 5184.2593
Dual form 5184.2.d.p.2593.8

$q$-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)
 
\(f(q)\) \(=\) \(q-4.18154i q^{5} +2.44949 q^{7} -4.24264i q^{11} +0.717439i q^{13} +3.00000 q^{17} -6.24264i q^{19} -1.01461 q^{23} -12.4853 q^{25} -4.18154i q^{29} -8.36308 q^{31} -10.2426i q^{35} -7.64564i q^{37} +8.48528 q^{41} +12.2426i q^{43} -8.36308 q^{47} -1.00000 q^{49} +2.02922i q^{53} -17.7408 q^{55} +5.61642i q^{61} +3.00000 q^{65} -2.24264i q^{67} +11.4069 q^{71} +7.48528 q^{73} -10.3923i q^{77} -5.91359 q^{79} -6.00000i q^{83} -12.5446i q^{85} +17.4853 q^{89} +1.75736i q^{91} -26.1039 q^{95} +0.485281 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{17} - 32 q^{25} - 8 q^{49} + 24 q^{65} - 8 q^{73} + 72 q^{89} - 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) − 4.18154i − 1.87004i −0.354593 − 0.935021i \(-0.615380\pi\)
0.354593 − 0.935021i \(-0.384620\pi\)
\(6\) 0 0
\(7\) 2.44949 0.925820 0.462910 − 0.886405i \(-0.346805\pi\)
0.462910 + 0.886405i \(0.346805\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.24264i − 1.27920i −0.768706 − 0.639602i \(-0.779099\pi\)
0.768706 − 0.639602i \(-0.220901\pi\)
\(12\) 0 0
\(13\) 0.717439i 0.198982i 0.995038 + 0.0994909i \(0.0317214\pi\)
−0.995038 + 0.0994909i \(0.968279\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) − 6.24264i − 1.43216i −0.698018 − 0.716080i \(-0.745935\pi\)
0.698018 − 0.716080i \(-0.254065\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.01461 −0.211561 −0.105781 − 0.994389i \(-0.533734\pi\)
−0.105781 + 0.994389i \(0.533734\pi\)
\(24\) 0 0
\(25\) −12.4853 −2.49706
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.18154i − 0.776493i −0.921556 − 0.388246i \(-0.873081\pi\)
0.921556 − 0.388246i \(-0.126919\pi\)
\(30\) 0 0
\(31\) −8.36308 −1.50205 −0.751027 − 0.660272i \(-0.770441\pi\)
−0.751027 + 0.660272i \(0.770441\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 10.2426i − 1.73132i
\(36\) 0 0
\(37\) − 7.64564i − 1.25694i −0.777836 − 0.628468i \(-0.783682\pi\)
0.777836 − 0.628468i \(-0.216318\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.48528 1.32518 0.662589 − 0.748983i \(-0.269458\pi\)
0.662589 + 0.748983i \(0.269458\pi\)
\(42\) 0 0
\(43\) 12.2426i 1.86699i 0.358597 + 0.933493i \(0.383255\pi\)
−0.358597 + 0.933493i \(0.616745\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.36308 −1.21988 −0.609940 − 0.792447i \(-0.708807\pi\)
−0.609940 + 0.792447i \(0.708807\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.02922i 0.278735i 0.990241 + 0.139368i \(0.0445070\pi\)
−0.990241 + 0.139368i \(0.955493\pi\)
\(54\) 0 0
\(55\) −17.7408 −2.39217
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 5.61642i 0.719109i 0.933124 + 0.359554i \(0.117071\pi\)
−0.933124 + 0.359554i \(0.882929\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) − 2.24264i − 0.273982i −0.990572 − 0.136991i \(-0.956257\pi\)
0.990572 − 0.136991i \(-0.0437432\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.4069 1.35375 0.676876 − 0.736097i \(-0.263333\pi\)
0.676876 + 0.736097i \(0.263333\pi\)
\(72\) 0 0
\(73\) 7.48528 0.876086 0.438043 − 0.898954i \(-0.355672\pi\)
0.438043 + 0.898954i \(0.355672\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 10.3923i − 1.18431i
\(78\) 0 0
\(79\) −5.91359 −0.665331 −0.332666 − 0.943045i \(-0.607948\pi\)
−0.332666 + 0.943045i \(0.607948\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 − 0.329293i \(-0.893190\pi\)
0.944228 − 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) − 12.5446i − 1.36066i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 17.4853 1.85344 0.926718 − 0.375757i \(-0.122617\pi\)
0.926718 + 0.375757i \(0.122617\pi\)
\(90\) 0 0
\(91\) 1.75736i 0.184221i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −26.1039 −2.67820
\(96\) 0 0
\(97\) 0.485281 0.0492729 0.0246364 − 0.999696i \(-0.492157\pi\)
0.0246364 + 0.999696i \(0.492157\pi\)
\(98\) 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 5184.2.d.p.2593.2 yes 8
3.2 odd 2 5184.2.d.o.2593.8 yes 8
4.3 odd 2 inner 5184.2.d.p.2593.1 yes 8
8.3 odd 2 inner 5184.2.d.p.2593.7 yes 8
8.5 even 2 inner 5184.2.d.p.2593.8 yes 8
12.11 even 2 5184.2.d.o.2593.7 yes 8
24.5 odd 2 5184.2.d.o.2593.2 yes 8
24.11 even 2 5184.2.d.o.2593.1 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5184.2.d.o.2593.1 ✓ 8 24.11 even 2
5184.2.d.o.2593.2 yes 8 24.5 odd 2
5184.2.d.o.2593.7 yes 8 12.11 even 2
5184.2.d.o.2593.8 yes 8 3.2 odd 2
5184.2.d.p.2593.1 yes 8 4.3 odd 2 inner
5184.2.d.p.2593.2 yes 8 1.1 even 1 trivial
5184.2.d.p.2593.7 yes 8 8.3 odd 2 inner
5184.2.d.p.2593.8 yes 8 8.5 even 2 inner