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Properties

Label 1152.4.d.i.577.2
Level $1152$
Weight $4$
Character 1152.577
Analytic conductor $67.970$
Analytic rank $0$
Dimension $4$
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] = [1152,4,Mod(577,1152)] 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("1152.577"); S:= CuspForms(chi, 4); 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, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-32,0,0,0,0,0,0,0,0,0,72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(67.9702003266\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.2
Root \(-1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1152.577
Dual form 1152.4.d.i.577.3

$q$-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)
 
\(f(q)\) \(=\) \(q-10.4222i q^{5} +6.42221 q^{7} -61.6888i q^{11} -64.8444i q^{13} +75.6888 q^{17} +10.3112i q^{19} +156.844 q^{23} +16.3776 q^{25} -53.7998i q^{29} -227.489 q^{31} -66.9335i q^{35} +10.3112i q^{37} +70.4441 q^{41} +298.311i q^{43} -89.9109 q^{47} -301.755 q^{49} -388.333i q^{53} -642.934 q^{55} -324.000i q^{59} -324.000i q^{61} -675.822 q^{65} +920.266i q^{67} +995.156 q^{71} +362.266 q^{73} -396.178i q^{77} +1098.91 q^{79} -791.822i q^{83} -788.844i q^{85} +150.622 q^{89} -416.444i q^{91} +107.465 q^{95} -1879.15 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{7} + 72 q^{17} + 512 q^{23} - 396 q^{25} - 160 q^{31} - 872 q^{41} + 448 q^{47} - 284 q^{49} - 3264 q^{55} - 1088 q^{65} + 4096 q^{71} - 1320 q^{73} + 992 q^{79} + 1064 q^{89} - 4416 q^{95}+ \cdots - 2440 q^{97}+O(q^{100}) \) 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
\(5\) − 10.4222i − 0.932190i −0.884735 − 0.466095i \(-0.845660\pi\)
0.884735 − 0.466095i \(-0.154340\pi\)
\(6\) 0 0
\(7\) 6.42221 0.346766 0.173383 − 0.984854i \(-0.444530\pi\)
0.173383 + 0.984854i \(0.444530\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 61.6888i − 1.69090i −0.534056 − 0.845449i \(-0.679333\pi\)
0.534056 − 0.845449i \(-0.320667\pi\)
\(12\) 0 0
\(13\) − 64.8444i − 1.38343i −0.722170 − 0.691716i \(-0.756855\pi\)
0.722170 − 0.691716i \(-0.243145\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 75.6888 1.07984 0.539919 − 0.841717i \(-0.318455\pi\)
0.539919 + 0.841717i \(0.318455\pi\)
\(18\) 0 0
\(19\) 10.3112i 0.124502i 0.998061 + 0.0622512i \(0.0198280\pi\)
−0.998061 + 0.0622512i \(0.980172\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 156.844 1.42193 0.710963 − 0.703229i \(-0.248259\pi\)
0.710963 + 0.703229i \(0.248259\pi\)
\(24\) 0 0
\(25\) 16.3776 0.131021
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 53.7998i − 0.344496i −0.985054 − 0.172248i \(-0.944897\pi\)
0.985054 − 0.172248i \(-0.0551030\pi\)
\(30\) 0 0
\(31\) −227.489 −1.31801 −0.659003 − 0.752141i \(-0.729021\pi\)
−0.659003 + 0.752141i \(0.729021\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 66.9335i − 0.323252i
\(36\) 0 0
\(37\) 10.3112i 0.0458148i 0.999738 + 0.0229074i \(0.00729229\pi\)
−0.999738 + 0.0229074i \(0.992708\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 70.4441 0.268330 0.134165 − 0.990959i \(-0.457165\pi\)
0.134165 + 0.990959i \(0.457165\pi\)
\(42\) 0 0
\(43\) 298.311i 1.05795i 0.848636 + 0.528977i \(0.177424\pi\)
−0.848636 + 0.528977i \(0.822576\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −89.9109 −0.279039 −0.139520 − 0.990219i \(-0.544556\pi\)
−0.139520 + 0.990219i \(0.544556\pi\)
\(48\) 0 0
\(49\) −301.755 −0.879753
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 388.333i − 1.00645i −0.864157 − 0.503223i \(-0.832147\pi\)
0.864157 − 0.503223i \(-0.167853\pi\)
\(54\) 0 0
\(55\) −642.934 −1.57624
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 324.000i − 0.714936i −0.933925 − 0.357468i \(-0.883640\pi\)
0.933925 − 0.357468i \(-0.116360\pi\)
\(60\) 0 0
\(61\) − 324.000i − 0.680065i −0.940414 − 0.340032i \(-0.889562\pi\)
0.940414 − 0.340032i \(-0.110438\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −675.822 −1.28962
\(66\) 0 0
\(67\) 920.266i 1.67804i 0.544104 + 0.839018i \(0.316870\pi\)
−0.544104 + 0.839018i \(0.683130\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 995.156 1.66343 0.831713 − 0.555206i \(-0.187361\pi\)
0.831713 + 0.555206i \(0.187361\pi\)
\(72\) 0 0
\(73\) 362.266 0.580822 0.290411 − 0.956902i \(-0.406208\pi\)
0.290411 + 0.956902i \(0.406208\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 396.178i − 0.586347i
\(78\) 0 0
\(79\) 1098.91 1.56503 0.782513 − 0.622634i \(-0.213938\pi\)
0.782513 + 0.622634i \(0.213938\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 791.822i − 1.04715i −0.851979 − 0.523577i \(-0.824597\pi\)
0.851979 − 0.523577i \(-0.175403\pi\)
\(84\) 0 0
\(85\) − 788.844i − 1.00661i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 150.622 0.179393 0.0896963 − 0.995969i \(-0.471410\pi\)
0.0896963 + 0.995969i \(0.471410\pi\)
\(90\) 0 0
\(91\) − 416.444i − 0.479728i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 107.465 0.116060
\(96\) 0 0
\(97\) −1879.15 −1.96700 −0.983501 − 0.180903i \(-0.942098\pi\)
−0.983501 + 0.180903i \(0.942098\pi\)
\(98\) 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 1152.4.d.i.577.2 4
3.2 odd 2 384.4.d.c.193.2 ✓ 4
4.3 odd 2 1152.4.d.o.577.2 4
8.3 odd 2 1152.4.d.o.577.3 4
8.5 even 2 inner 1152.4.d.i.577.3 4
12.11 even 2 384.4.d.e.193.4 yes 4
16.3 odd 4 2304.4.a.bp.1.1 2
16.5 even 4 2304.4.a.t.1.2 2
16.11 odd 4 2304.4.a.s.1.2 2
16.13 even 4 2304.4.a.bq.1.1 2
24.5 odd 2 384.4.d.c.193.3 yes 4
24.11 even 2 384.4.d.e.193.1 yes 4
48.5 odd 4 768.4.a.j.1.1 2
48.11 even 4 768.4.a.p.1.1 2
48.29 odd 4 768.4.a.k.1.2 2
48.35 even 4 768.4.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.c.193.2 ✓ 4 3.2 odd 2
384.4.d.c.193.3 yes 4 24.5 odd 2
384.4.d.e.193.1 yes 4 24.11 even 2
384.4.d.e.193.4 yes 4 12.11 even 2
768.4.a.e.1.2 2 48.35 even 4
768.4.a.j.1.1 2 48.5 odd 4
768.4.a.k.1.2 2 48.29 odd 4
768.4.a.p.1.1 2 48.11 even 4
1152.4.d.i.577.2 4 1.1 even 1 trivial
1152.4.d.i.577.3 4 8.5 even 2 inner
1152.4.d.o.577.2 4 4.3 odd 2
1152.4.d.o.577.3 4 8.3 odd 2
2304.4.a.s.1.2 2 16.11 odd 4
2304.4.a.t.1.2 2 16.5 even 4
2304.4.a.bp.1.1 2 16.3 odd 4
2304.4.a.bq.1.1 2 16.13 even 4