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Properties

Label 800.4.a.p.1.2
Level $800$
Weight $4$
Character 800.1
Self dual yes
Analytic conductor $47.202$
Analytic rank $0$
Dimension $2$
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] = [800,4,Mod(1,800)] 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("800.1"); 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, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,26,0,0,0,-76] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.16228\) of defining polynomial
Character \(\chi\) \(=\) 800.1

$q$-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)
 
\(f(q)\) \(=\) \(q+6.32456 q^{3} +18.9737 q^{7} +13.0000 q^{9} +12.6491 q^{11} -38.0000 q^{13} -34.0000 q^{17} +101.193 q^{19} +120.000 q^{21} +82.2192 q^{23} -88.5438 q^{27} +270.000 q^{29} +341.526 q^{31} +80.0000 q^{33} -206.000 q^{37} -240.333 q^{39} -270.000 q^{41} +537.587 q^{43} -132.816 q^{47} +17.0000 q^{49} -215.035 q^{51} +258.000 q^{53} +640.000 q^{57} +75.8947 q^{59} -250.000 q^{61} +246.658 q^{63} -815.868 q^{67} +520.000 q^{69} +645.105 q^{71} +1078.00 q^{73} +240.000 q^{77} +278.280 q^{79} -911.000 q^{81} -1106.80 q^{83} +1707.63 q^{87} +890.000 q^{89} -720.999 q^{91} +2160.00 q^{93} +254.000 q^{97} +164.438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 26 q^{9} - 76 q^{13} - 68 q^{17} + 240 q^{21} + 540 q^{29} + 160 q^{33} - 412 q^{37} - 540 q^{41} + 34 q^{49} + 516 q^{53} + 1280 q^{57} - 500 q^{61} + 1040 q^{69} + 2156 q^{73} + 480 q^{77} - 1822 q^{81}+ \cdots + 508 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 6.32456 1.21716 0.608581 − 0.793492i \(-0.291739\pi\)
0.608581 + 0.793492i \(0.291739\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 18.9737 1.02448 0.512241 − 0.858842i \(-0.328816\pi\)
0.512241 + 0.858842i \(0.328816\pi\)
\(8\) 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
\(13\) −38.0000 −0.810716 −0.405358 − 0.914158i \(-0.632853\pi\)
−0.405358 + 0.914158i \(0.632853\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −34.0000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 101.193 1.22185 0.610927 − 0.791687i \(-0.290797\pi\)
0.610927 + 0.791687i \(0.290797\pi\)
\(20\) 0 0
\(21\) 120.000 1.24696
\(22\) 0 0
\(23\) 82.2192 0.745387 0.372693 − 0.927955i \(-0.378434\pi\)
0.372693 + 0.927955i \(0.378434\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −88.5438 −0.631121
\(28\) 0 0
\(29\) 270.000 1.72889 0.864444 − 0.502729i \(-0.167671\pi\)
0.864444 + 0.502729i \(0.167671\pi\)
\(30\) 0 0
\(31\) 341.526 1.97871 0.989353 − 0.145537i \(-0.0464908\pi\)
0.989353 + 0.145537i \(0.0464908\pi\)
\(32\) 0 0
\(33\) 80.0000 0.422006
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −206.000 −0.915302 −0.457651 − 0.889132i \(-0.651309\pi\)
−0.457651 + 0.889132i \(0.651309\pi\)
\(38\) 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
\(43\) 537.587 1.90654 0.953271 − 0.302117i \(-0.0976935\pi\)
0.953271 + 0.302117i \(0.0976935\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −132.816 −0.412195 −0.206097 − 0.978531i \(-0.566076\pi\)
−0.206097 + 0.978531i \(0.566076\pi\)
\(48\) 0 0
\(49\) 17.0000 0.0495627
\(50\) 0 0
\(51\) −215.035 −0.590410
\(52\) 0 0
\(53\) 258.000 0.668661 0.334330 − 0.942456i \(-0.391490\pi\)
0.334330 + 0.942456i \(0.391490\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 640.000 1.48719
\(58\) 0 0
\(59\) 75.8947 0.167469 0.0837343 − 0.996488i \(-0.473315\pi\)
0.0837343 + 0.996488i \(0.473315\pi\)
\(60\) 0 0
\(61\) −250.000 −0.524741 −0.262371 − 0.964967i \(-0.584504\pi\)
−0.262371 + 0.964967i \(0.584504\pi\)
\(62\) 0 0
\(63\) 246.658 0.493269
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −815.868 −1.48767 −0.743837 − 0.668362i \(-0.766996\pi\)
−0.743837 + 0.668362i \(0.766996\pi\)
\(68\) 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
\(73\) 1078.00 1.72836 0.864181 − 0.503182i \(-0.167837\pi\)
0.864181 + 0.503182i \(0.167837\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 240.000 0.355202
\(78\) 0 0
\(79\) 278.280 0.396316 0.198158 − 0.980170i \(-0.436504\pi\)
0.198158 + 0.980170i \(0.436504\pi\)
\(80\) 0 0
\(81\) −911.000 −1.24966
\(82\) 0 0
\(83\) −1106.80 −1.46370 −0.731848 − 0.681468i \(-0.761342\pi\)
−0.731848 + 0.681468i \(0.761342\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1707.63 2.10434
\(88\) 0 0
\(89\) 890.000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −720.999 −0.830563
\(92\) 0 0
\(93\) 2160.00 2.40840
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 254.000 0.265874 0.132937 − 0.991124i \(-0.457559\pi\)
0.132937 + 0.991124i \(0.457559\pi\)
\(98\) 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.a.p.1.2 2
4.3 odd 2 inner 800.4.a.p.1.1 2
5.2 odd 4 800.4.c.j.449.2 4
5.3 odd 4 800.4.c.j.449.4 4
5.4 even 2 160.4.a.f.1.1 ✓ 2
8.3 odd 2 1600.4.a.ch.1.2 2
8.5 even 2 1600.4.a.ch.1.1 2
15.14 odd 2 1440.4.a.v.1.1 2
20.3 even 4 800.4.c.j.449.1 4
20.7 even 4 800.4.c.j.449.3 4
20.19 odd 2 160.4.a.f.1.2 yes 2
40.19 odd 2 320.4.a.p.1.1 2
40.29 even 2 320.4.a.p.1.2 2
60.59 even 2 1440.4.a.v.1.2 2
80.19 odd 4 1280.4.d.u.641.3 4
80.29 even 4 1280.4.d.u.641.1 4
80.59 odd 4 1280.4.d.u.641.2 4
80.69 even 4 1280.4.d.u.641.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.f.1.1 ✓ 2 5.4 even 2
160.4.a.f.1.2 yes 2 20.19 odd 2
320.4.a.p.1.1 2 40.19 odd 2
320.4.a.p.1.2 2 40.29 even 2
800.4.a.p.1.1 2 4.3 odd 2 inner
800.4.a.p.1.2 2 1.1 even 1 trivial
800.4.c.j.449.1 4 20.3 even 4
800.4.c.j.449.2 4 5.2 odd 4
800.4.c.j.449.3 4 20.7 even 4
800.4.c.j.449.4 4 5.3 odd 4
1280.4.d.u.641.1 4 80.29 even 4
1280.4.d.u.641.2 4 80.59 odd 4
1280.4.d.u.641.3 4 80.19 odd 4
1280.4.d.u.641.4 4 80.69 even 4
1440.4.a.v.1.1 2 15.14 odd 2
1440.4.a.v.1.2 2 60.59 even 2
1600.4.a.ch.1.1 2 8.5 even 2
1600.4.a.ch.1.2 2 8.3 odd 2