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Properties

Label 1008.5.f.h.433.3
Level $1008$
Weight $5$
Character 1008.433
Analytic conductor $104.197$
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] = [1008,5,Mod(433,1008)] 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("1008.433"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1008.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,76,0,0,0,360] 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: \(104.196922789\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.1308672.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 72x^{2} + 1278 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 433.3
Root \(-6.34371i\) of defining polynomial
Character \(\chi\) \(=\) 1008.433
Dual form 1008.5.f.h.433.2

$q$-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)
 
\(f(q)\) \(=\) \(q+23.1980i q^{5} +(-6.45584 - 48.5729i) q^{7} +191.823 q^{11} -48.5729i q^{13} +181.977i q^{17} -599.915i q^{19} -469.529 q^{23} +86.8519 q^{25} +338.881 q^{29} -267.556i q^{31} +(1126.79 - 149.763i) q^{35} -668.530 q^{37} -1323.85i q^{41} -1940.23 q^{43} +2936.89i q^{47} +(-2317.64 + 627.158i) q^{49} +1460.94 q^{53} +4449.92i q^{55} -1730.83i q^{59} +246.343i q^{61} +1126.79 q^{65} +1076.59 q^{67} -2276.39 q^{71} -7106.94i q^{73} +(-1238.38 - 9317.41i) q^{77} -7012.38 q^{79} -1448.36i q^{83} -4221.52 q^{85} +2133.73i q^{89} +(-2359.32 + 313.579i) q^{91} +13916.8 q^{95} -5898.76i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 76 q^{7} + 360 q^{11} - 792 q^{23} - 2300 q^{25} - 1224 q^{29} + 4032 q^{35} - 3896 q^{37} - 3688 q^{43} - 1532 q^{49} - 5832 q^{53} + 4032 q^{65} + 1048 q^{67} - 21528 q^{71} - 3528 q^{77} - 12776 q^{79}+ \cdots + 36864 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(5\) 23.1980i 0.927921i 0.885856 + 0.463960i \(0.153572\pi\)
−0.885856 + 0.463960i \(0.846428\pi\)
\(6\) 0 0
\(7\) −6.45584 − 48.5729i −0.131752 − 0.991283i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 191.823 1.58532 0.792659 − 0.609666i \(-0.208696\pi\)
0.792659 + 0.609666i \(0.208696\pi\)
\(12\) 0 0
\(13\) − 48.5729i − 0.287413i −0.989620 − 0.143707i \(-0.954098\pi\)
0.989620 − 0.143707i \(-0.0459022\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 181.977i 0.629680i 0.949145 + 0.314840i \(0.101951\pi\)
−0.949145 + 0.314840i \(0.898049\pi\)
\(18\) 0 0
\(19\) − 599.915i − 1.66182i −0.556410 − 0.830908i \(-0.687822\pi\)
0.556410 − 0.830908i \(-0.312178\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −469.529 −0.887578 −0.443789 − 0.896131i \(-0.646366\pi\)
−0.443789 + 0.896131i \(0.646366\pi\)
\(24\) 0 0
\(25\) 86.8519 0.138963
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 338.881 0.402951 0.201475 − 0.979494i \(-0.435426\pi\)
0.201475 + 0.979494i \(0.435426\pi\)
\(30\) 0 0
\(31\) − 267.556i − 0.278414i −0.990263 − 0.139207i \(-0.955545\pi\)
0.990263 − 0.139207i \(-0.0444554\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1126.79 − 149.763i 0.919832 − 0.122255i
\(36\) 0 0
\(37\) −668.530 −0.488334 −0.244167 − 0.969733i \(-0.578515\pi\)
−0.244167 + 0.969733i \(0.578515\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 1323.85i − 0.787534i −0.919210 − 0.393767i \(-0.871172\pi\)
0.919210 − 0.393767i \(-0.128828\pi\)
\(42\) 0 0
\(43\) −1940.23 −1.04934 −0.524671 − 0.851305i \(-0.675812\pi\)
−0.524671 + 0.851305i \(0.675812\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2936.89i 1.32951i 0.747062 + 0.664755i \(0.231464\pi\)
−0.747062 + 0.664755i \(0.768536\pi\)
\(48\) 0 0
\(49\) −2317.64 + 627.158i −0.965283 + 0.261207i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1460.94 0.520091 0.260046 − 0.965596i \(-0.416262\pi\)
0.260046 + 0.965596i \(0.416262\pi\)
\(54\) 0 0
\(55\) 4449.92i 1.47105i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 1730.83i − 0.497223i −0.968603 − 0.248612i \(-0.920026\pi\)
0.968603 − 0.248612i \(-0.0799743\pi\)
\(60\) 0 0
\(61\) 246.343i 0.0662034i 0.999452 + 0.0331017i \(0.0105385\pi\)
−0.999452 + 0.0331017i \(0.989461\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1126.79 0.266697
\(66\) 0 0
\(67\) 1076.59 0.239828 0.119914 − 0.992784i \(-0.461738\pi\)
0.119914 + 0.992784i \(0.461738\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2276.39 −0.451574 −0.225787 − 0.974177i \(-0.572495\pi\)
−0.225787 + 0.974177i \(0.572495\pi\)
\(72\) 0 0
\(73\) − 7106.94i − 1.33363i −0.745221 − 0.666817i \(-0.767656\pi\)
0.745221 − 0.666817i \(-0.232344\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1238.38 − 9317.41i −0.208869 − 1.57150i
\(78\) 0 0
\(79\) −7012.38 −1.12360 −0.561799 − 0.827274i \(-0.689891\pi\)
−0.561799 + 0.827274i \(0.689891\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 1448.36i − 0.210243i −0.994459 − 0.105121i \(-0.966477\pi\)
0.994459 − 0.105121i \(-0.0335231\pi\)
\(84\) 0 0
\(85\) −4221.52 −0.584293
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2133.73i 0.269376i 0.990888 + 0.134688i \(0.0430032\pi\)
−0.990888 + 0.134688i \(0.956997\pi\)
\(90\) 0 0
\(91\) −2359.32 + 313.579i −0.284908 + 0.0378673i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 13916.8 1.54203
\(96\) 0 0
\(97\) − 5898.76i − 0.626928i −0.949600 − 0.313464i \(-0.898511\pi\)
0.949600 − 0.313464i \(-0.101489\pi\)
\(98\) 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 1008.5.f.h.433.3 4
3.2 odd 2 112.5.c.c.97.4 4
4.3 odd 2 126.5.c.a.55.4 4
7.6 odd 2 inner 1008.5.f.h.433.2 4
12.11 even 2 14.5.b.a.13.1 ✓ 4
21.20 even 2 112.5.c.c.97.1 4
24.5 odd 2 448.5.c.f.321.1 4
24.11 even 2 448.5.c.e.321.4 4
28.27 even 2 126.5.c.a.55.3 4
60.23 odd 4 350.5.d.a.349.8 8
60.47 odd 4 350.5.d.a.349.1 8
60.59 even 2 350.5.b.a.251.4 4
84.11 even 6 98.5.d.d.19.3 8
84.23 even 6 98.5.d.d.31.4 8
84.47 odd 6 98.5.d.d.31.3 8
84.59 odd 6 98.5.d.d.19.4 8
84.83 odd 2 14.5.b.a.13.2 yes 4
168.83 odd 2 448.5.c.e.321.1 4
168.125 even 2 448.5.c.f.321.4 4
420.83 even 4 350.5.d.a.349.5 8
420.167 even 4 350.5.d.a.349.4 8
420.419 odd 2 350.5.b.a.251.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.5.b.a.13.1 ✓ 4 12.11 even 2
14.5.b.a.13.2 yes 4 84.83 odd 2
98.5.d.d.19.3 8 84.11 even 6
98.5.d.d.19.4 8 84.59 odd 6
98.5.d.d.31.3 8 84.47 odd 6
98.5.d.d.31.4 8 84.23 even 6
112.5.c.c.97.1 4 21.20 even 2
112.5.c.c.97.4 4 3.2 odd 2
126.5.c.a.55.3 4 28.27 even 2
126.5.c.a.55.4 4 4.3 odd 2
350.5.b.a.251.3 4 420.419 odd 2
350.5.b.a.251.4 4 60.59 even 2
350.5.d.a.349.1 8 60.47 odd 4
350.5.d.a.349.4 8 420.167 even 4
350.5.d.a.349.5 8 420.83 even 4
350.5.d.a.349.8 8 60.23 odd 4
448.5.c.e.321.1 4 168.83 odd 2
448.5.c.e.321.4 4 24.11 even 2
448.5.c.f.321.1 4 24.5 odd 2
448.5.c.f.321.4 4 168.125 even 2
1008.5.f.h.433.2 4 7.6 odd 2 inner
1008.5.f.h.433.3 4 1.1 even 1 trivial