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Properties

Label 800.4.c.j.449.2
Level $800$
Weight $4$
Character 800.449
Analytic conductor $47.202$
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] = [800,4,Mod(449,800)] 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("800.449"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-52,0,0] 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: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.58114 + 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 800.449
Dual form 800.4.c.j.449.4

$q$-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)
 
\(f(q)\) \(=\) \(q-6.32456i q^{3} +18.9737i q^{7} -13.0000 q^{9} +12.6491 q^{11} +38.0000i q^{13} -34.0000i q^{17} -101.193 q^{19} +120.000 q^{21} -82.2192i q^{23} -88.5438i q^{27} -270.000 q^{29} +341.526 q^{31} -80.0000i q^{33} -206.000i q^{37} +240.333 q^{39} -270.000 q^{41} -537.587i q^{43} -132.816i q^{47} -17.0000 q^{49} -215.035 q^{51} -258.000i q^{53} +640.000i q^{57} -75.8947 q^{59} -250.000 q^{61} -246.658i q^{63} -815.868i q^{67} -520.000 q^{69} +645.105 q^{71} -1078.00i q^{73} +240.000i q^{77} -278.280 q^{79} -911.000 q^{81} +1106.80i q^{83} +1707.63i q^{87} -890.000 q^{89} -720.999 q^{91} -2160.00i q^{93} +254.000i q^{97} -164.438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 52 q^{9} + 480 q^{21} - 1080 q^{29} - 1080 q^{41} - 68 q^{49} - 1000 q^{61} - 2080 q^{69} - 3644 q^{81} - 3560 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 6.32456i − 1.21716i −0.793492 − 0.608581i \(-0.791739\pi\)
0.793492 − 0.608581i \(-0.208261\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 18.9737i 1.02448i 0.858842 + 0.512241i \(0.171184\pi\)
−0.858842 + 0.512241i \(0.828816\pi\)
\(8\) 0 0
\(9\) −13.0000 −0.481481
\(10\) 0 0
\(11\) 12.6491 0.346714 0.173357 − 0.984859i \(-0.444539\pi\)
0.173357 + 0.984859i \(0.444539\pi\)
\(12\) 0 0
\(13\) 38.0000i 0.810716i 0.914158 + 0.405358i \(0.132853\pi\)
−0.914158 + 0.405358i \(0.867147\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 34.0000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) −101.193 −1.22185 −0.610927 − 0.791687i \(-0.709203\pi\)
−0.610927 + 0.791687i \(0.709203\pi\)
\(20\) 0 0
\(21\) 120.000 1.24696
\(22\) 0 0
\(23\) − 82.2192i − 0.745387i −0.927955 − 0.372693i \(-0.878434\pi\)
0.927955 − 0.372693i \(-0.121566\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 88.5438i − 0.631121i
\(28\) 0 0
\(29\) −270.000 −1.72889 −0.864444 − 0.502729i \(-0.832329\pi\)
−0.864444 + 0.502729i \(0.832329\pi\)
\(30\) 0 0
\(31\) 341.526 1.97871 0.989353 − 0.145537i \(-0.0464908\pi\)
0.989353 + 0.145537i \(0.0464908\pi\)
\(32\) 0 0
\(33\) − 80.0000i − 0.422006i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 206.000i − 0.915302i −0.889132 − 0.457651i \(-0.848691\pi\)
0.889132 − 0.457651i \(-0.151309\pi\)
\(38\) 0 0
\(39\) 240.333 0.986772
\(40\) 0 0
\(41\) −270.000 −1.02846 −0.514231 − 0.857652i \(-0.671922\pi\)
−0.514231 + 0.857652i \(0.671922\pi\)
\(42\) 0 0
\(43\) − 537.587i − 1.90654i −0.302117 − 0.953271i \(-0.597693\pi\)
0.302117 − 0.953271i \(-0.402307\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 132.816i − 0.412195i −0.978531 − 0.206097i \(-0.933924\pi\)
0.978531 − 0.206097i \(-0.0660764\pi\)
\(48\) 0 0
\(49\) −17.0000 −0.0495627
\(50\) 0 0
\(51\) −215.035 −0.590410
\(52\) 0 0
\(53\) − 258.000i − 0.668661i −0.942456 − 0.334330i \(-0.891490\pi\)
0.942456 − 0.334330i \(-0.108510\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 640.000i 1.48719i
\(58\) 0 0
\(59\) −75.8947 −0.167469 −0.0837343 − 0.996488i \(-0.526685\pi\)
−0.0837343 + 0.996488i \(0.526685\pi\)
\(60\) 0 0
\(61\) −250.000 −0.524741 −0.262371 − 0.964967i \(-0.584504\pi\)
−0.262371 + 0.964967i \(0.584504\pi\)
\(62\) 0 0
\(63\) − 246.658i − 0.493269i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 815.868i − 1.48767i −0.668362 − 0.743837i \(-0.733004\pi\)
0.668362 − 0.743837i \(-0.266996\pi\)
\(68\) 0 0
\(69\) −520.000 −0.907256
\(70\) 0 0
\(71\) 645.105 1.07831 0.539154 − 0.842207i \(-0.318744\pi\)
0.539154 + 0.842207i \(0.318744\pi\)
\(72\) 0 0
\(73\) − 1078.00i − 1.72836i −0.503182 − 0.864181i \(-0.667837\pi\)
0.503182 − 0.864181i \(-0.332163\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 240.000i 0.355202i
\(78\) 0 0
\(79\) −278.280 −0.396316 −0.198158 − 0.980170i \(-0.563496\pi\)
−0.198158 + 0.980170i \(0.563496\pi\)
\(80\) 0 0
\(81\) −911.000 −1.24966
\(82\) 0 0
\(83\) 1106.80i 1.46370i 0.681468 + 0.731848i \(0.261342\pi\)
−0.681468 + 0.731848i \(0.738658\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1707.63i 2.10434i
\(88\) 0 0
\(89\) −890.000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −720.999 −0.830563
\(92\) 0 0
\(93\) − 2160.00i − 2.40840i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 254.000i 0.265874i 0.991124 + 0.132937i \(0.0424408\pi\)
−0.991124 + 0.132937i \(0.957559\pi\)
\(98\) 0 0
\(99\) −164.438 −0.166936
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 800.4.c.j.449.2 4
4.3 odd 2 inner 800.4.c.j.449.3 4
5.2 odd 4 160.4.a.f.1.1 ✓ 2
5.3 odd 4 800.4.a.p.1.2 2
5.4 even 2 inner 800.4.c.j.449.4 4
15.2 even 4 1440.4.a.v.1.1 2
20.3 even 4 800.4.a.p.1.1 2
20.7 even 4 160.4.a.f.1.2 yes 2
20.19 odd 2 inner 800.4.c.j.449.1 4
40.3 even 4 1600.4.a.ch.1.2 2
40.13 odd 4 1600.4.a.ch.1.1 2
40.27 even 4 320.4.a.p.1.1 2
40.37 odd 4 320.4.a.p.1.2 2
60.47 odd 4 1440.4.a.v.1.2 2
80.27 even 4 1280.4.d.u.641.2 4
80.37 odd 4 1280.4.d.u.641.4 4
80.67 even 4 1280.4.d.u.641.3 4
80.77 odd 4 1280.4.d.u.641.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.f.1.1 ✓ 2 5.2 odd 4
160.4.a.f.1.2 yes 2 20.7 even 4
320.4.a.p.1.1 2 40.27 even 4
320.4.a.p.1.2 2 40.37 odd 4
800.4.a.p.1.1 2 20.3 even 4
800.4.a.p.1.2 2 5.3 odd 4
800.4.c.j.449.1 4 20.19 odd 2 inner
800.4.c.j.449.2 4 1.1 even 1 trivial
800.4.c.j.449.3 4 4.3 odd 2 inner
800.4.c.j.449.4 4 5.4 even 2 inner
1280.4.d.u.641.1 4 80.77 odd 4
1280.4.d.u.641.2 4 80.27 even 4
1280.4.d.u.641.3 4 80.67 even 4
1280.4.d.u.641.4 4 80.37 odd 4
1440.4.a.v.1.1 2 15.2 even 4
1440.4.a.v.1.2 2 60.47 odd 4
1600.4.a.ch.1.1 2 40.13 odd 4
1600.4.a.ch.1.2 2 40.3 even 4