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Properties

Label 1152.5.h.a.449.1
Level $1152$
Weight $5$
Character 1152.449
Analytic conductor $119.082$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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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] = [1152,5,Mod(449,1152)] 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("1152.449"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1152.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-188] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(119.082197473\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1152.449
Dual form 1152.5.h.a.449.2

$q$-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)
 
\(f(q)\) \(=\) \(q-24.0416 q^{5} -238.000i q^{13} +567.100i q^{17} -47.0000 q^{25} +1245.92 q^{29} +1680.00i q^{37} -1129.96i q^{41} -2401.00 q^{49} +1808.78 q^{53} +2640.00i q^{61} +5721.91i q^{65} -10560.0 q^{73} -13634.0i q^{85} -1924.74i q^{89} +18720.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 188 q^{25} - 9604 q^{49} - 42240 q^{73} + 74880 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) −24.0416 −0.961665 −0.480833 − 0.876812i \(-0.659666\pi\)
−0.480833 + 0.876812i \(0.659666\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 238.000i − 1.40828i −0.710059 − 0.704142i \(-0.751332\pi\)
0.710059 − 0.704142i \(-0.248668\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 567.100i 1.96228i 0.193292 + 0.981141i \(0.438083\pi\)
−0.193292 + 0.981141i \(0.561917\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −47.0000 −0.0752000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1245.92 1.48148 0.740738 − 0.671793i \(-0.234476\pi\)
0.740738 + 0.671793i \(0.234476\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1680.00i 1.22717i 0.789627 + 0.613587i \(0.210274\pi\)
−0.789627 + 0.613587i \(0.789726\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 1129.96i − 0.672193i −0.941828 − 0.336097i \(-0.890893\pi\)
0.941828 − 0.336097i \(-0.109107\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −2401.00 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1808.78 0.643923 0.321961 − 0.946753i \(-0.395658\pi\)
0.321961 + 0.946753i \(0.395658\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2640.00i 0.709487i 0.934964 + 0.354743i \(0.115432\pi\)
−0.934964 + 0.354743i \(0.884568\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5721.91i 1.35430i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −10560.0 −1.98161 −0.990805 − 0.135297i \(-0.956801\pi\)
−0.990805 + 0.135297i \(0.956801\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) − 13634.0i − 1.88706i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1924.74i − 0.242993i −0.992592 − 0.121496i \(-0.961231\pi\)
0.992592 − 0.121496i \(-0.0387693\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18720.0 1.98958 0.994792 − 0.101924i \(-0.0324998\pi\)
0.994792 + 0.101924i \(0.0324998\pi\)
\(98\) 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 1152.5.h.a.449.1 ✓ 4
3.2 odd 2 inner 1152.5.h.a.449.3 yes 4
4.3 odd 2 CM 1152.5.h.a.449.1 ✓ 4
8.3 odd 2 inner 1152.5.h.a.449.4 yes 4
8.5 even 2 inner 1152.5.h.a.449.4 yes 4
12.11 even 2 inner 1152.5.h.a.449.3 yes 4
24.5 odd 2 inner 1152.5.h.a.449.2 yes 4
24.11 even 2 inner 1152.5.h.a.449.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.5.h.a.449.1 ✓ 4 1.1 even 1 trivial
1152.5.h.a.449.1 ✓ 4 4.3 odd 2 CM
1152.5.h.a.449.2 yes 4 24.5 odd 2 inner
1152.5.h.a.449.2 yes 4 24.11 even 2 inner
1152.5.h.a.449.3 yes 4 3.2 odd 2 inner
1152.5.h.a.449.3 yes 4 12.11 even 2 inner
1152.5.h.a.449.4 yes 4 8.3 odd 2 inner
1152.5.h.a.449.4 yes 4 8.5 even 2 inner