↑

Properties

Label 5800.2.a.r.1.2
Level $5800$
Weight $2$
Character 5800.1
Self dual yes
Analytic conductor $46.313$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5800,2,Mod(1,5800)] 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("5800.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5800 = 2^{3} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,2,0,0,0,-2,0,3,0,-4] 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: yes
Analytic conductor: \(46.3132331723\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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 1160)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 5800.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.806063 q^{3} -4.15633 q^{7} -2.35026 q^{9} -1.19394 q^{11} +5.35026 q^{13} -3.76845 q^{17} -6.15633 q^{19} -3.35026 q^{21} +0.806063 q^{23} -4.31265 q^{27} +1.00000 q^{29} -4.54420 q^{31} -0.962389 q^{33} -2.15633 q^{37} +4.31265 q^{39} +1.03761 q^{41} +6.41819 q^{43} +6.73084 q^{47} +10.2750 q^{49} -3.03761 q^{51} +12.8872 q^{53} -4.96239 q^{57} -6.57452 q^{59} +7.27504 q^{61} +9.76845 q^{63} +15.5066 q^{67} +0.649738 q^{69} +2.57452 q^{71} +6.80606 q^{73} +4.96239 q^{77} -15.4314 q^{79} +3.57452 q^{81} -12.7816 q^{83} +0.806063 q^{87} +7.61213 q^{89} -22.2374 q^{91} -3.66291 q^{93} +13.0435 q^{97} +2.80606 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 2 q^{7} + 3 q^{9} - 4 q^{11} + 6 q^{13} - 8 q^{19} + 2 q^{23} + 8 q^{27} + 3 q^{29} - 4 q^{31} + 8 q^{33} + 4 q^{37} - 8 q^{39} + 14 q^{41} + 18 q^{43} - 2 q^{47} - q^{49} - 20 q^{51} + 6 q^{53}+ \cdots + 8 q^{99}+O(q^{100}) \) 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
\(3\) 0.806063 0.465381 0.232690 − 0.972551i \(-0.425247\pi\)
0.232690 + 0.972551i \(0.425247\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.15633 −1.57094 −0.785472 − 0.618898i \(-0.787580\pi\)
−0.785472 + 0.618898i \(0.787580\pi\)
\(8\) 0 0
\(9\) −2.35026 −0.783421
\(10\) 0 0
\(11\) −1.19394 −0.359985 −0.179993 − 0.983668i \(-0.557607\pi\)
−0.179993 + 0.983668i \(0.557607\pi\)
\(12\) 0 0
\(13\) 5.35026 1.48390 0.741948 − 0.670458i \(-0.233902\pi\)
0.741948 + 0.670458i \(0.233902\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.76845 −0.913984 −0.456992 − 0.889471i \(-0.651073\pi\)
−0.456992 + 0.889471i \(0.651073\pi\)
\(18\) 0 0
\(19\) −6.15633 −1.41236 −0.706179 − 0.708033i \(-0.749583\pi\)
−0.706179 + 0.708033i \(0.749583\pi\)
\(20\) 0 0
\(21\) −3.35026 −0.731087
\(22\) 0 0
\(23\) 0.806063 0.168076 0.0840379 − 0.996463i \(-0.473218\pi\)
0.0840379 + 0.996463i \(0.473218\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.31265 −0.829970
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −4.54420 −0.816162 −0.408081 − 0.912946i \(-0.633802\pi\)
−0.408081 + 0.912946i \(0.633802\pi\)
\(32\) 0 0
\(33\) −0.962389 −0.167530
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.15633 −0.354498 −0.177249 − 0.984166i \(-0.556720\pi\)
−0.177249 + 0.984166i \(0.556720\pi\)
\(38\) 0 0
\(39\) 4.31265 0.690577
\(40\) 0 0
\(41\) 1.03761 0.162048 0.0810238 − 0.996712i \(-0.474181\pi\)
0.0810238 + 0.996712i \(0.474181\pi\)
\(42\) 0 0
\(43\) 6.41819 0.978765 0.489382 − 0.872069i \(-0.337222\pi\)
0.489382 + 0.872069i \(0.337222\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.73084 0.981794 0.490897 − 0.871218i \(-0.336669\pi\)
0.490897 + 0.871218i \(0.336669\pi\)
\(48\) 0 0
\(49\) 10.2750 1.46786
\(50\) 0 0
\(51\) −3.03761 −0.425351
\(52\) 0 0
\(53\) 12.8872 1.77019 0.885094 − 0.465412i \(-0.154094\pi\)
0.885094 + 0.465412i \(0.154094\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.96239 −0.657284
\(58\) 0 0
\(59\) −6.57452 −0.855929 −0.427965 − 0.903796i \(-0.640769\pi\)
−0.427965 + 0.903796i \(0.640769\pi\)
\(60\) 0 0
\(61\) 7.27504 0.931473 0.465737 − 0.884923i \(-0.345789\pi\)
0.465737 + 0.884923i \(0.345789\pi\)
\(62\) 0 0
\(63\) 9.76845 1.23071
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 15.5066 1.89443 0.947216 − 0.320597i \(-0.103884\pi\)
0.947216 + 0.320597i \(0.103884\pi\)
\(68\) 0 0
\(69\) 0.649738 0.0782193
\(70\) 0 0
\(71\) 2.57452 0.305539 0.152769 − 0.988262i \(-0.451181\pi\)
0.152769 + 0.988262i \(0.451181\pi\)
\(72\) 0 0
\(73\) 6.80606 0.796589 0.398295 − 0.917257i \(-0.369602\pi\)
0.398295 + 0.917257i \(0.369602\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.96239 0.565517
\(78\) 0 0
\(79\) −15.4314 −1.73616 −0.868082 − 0.496421i \(-0.834647\pi\)
−0.868082 + 0.496421i \(0.834647\pi\)
\(80\) 0 0
\(81\) 3.57452 0.397168
\(82\) 0 0
\(83\) −12.7816 −1.40297 −0.701483 − 0.712686i \(-0.747478\pi\)
−0.701483 + 0.712686i \(0.747478\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.806063 0.0864191
\(88\) 0 0
\(89\) 7.61213 0.806884 0.403442 − 0.915005i \(-0.367814\pi\)
0.403442 + 0.915005i \(0.367814\pi\)
\(90\) 0 0
\(91\) −22.2374 −2.33112
\(92\) 0 0
\(93\) −3.66291 −0.379826
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.0435 1.32437 0.662183 − 0.749342i \(-0.269630\pi\)
0.662183 + 0.749342i \(0.269630\pi\)
\(98\) 0 0
\(99\) 2.80606 0.282020
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 5800.2.a.r.1.2 3
5.4 even 2 1160.2.a.e.1.2 ✓ 3
20.19 odd 2 2320.2.a.r.1.2 3
40.19 odd 2 9280.2.a.bl.1.2 3
40.29 even 2 9280.2.a.bv.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1160.2.a.e.1.2 ✓ 3 5.4 even 2
2320.2.a.r.1.2 3 20.19 odd 2
5800.2.a.r.1.2 3 1.1 even 1 trivial
9280.2.a.bl.1.2 3 40.19 odd 2
9280.2.a.bv.1.2 3 40.29 even 2