Properties

Label 600.4.a.s.1.1
Level $600$
Weight $4$
Character 600.1
Self dual yes
Analytic conductor $35.401$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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] = [600,4,Mod(1,600)] 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("600.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-6,0,0,0,2,0,18,0,-16] 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: \(35.4011460034\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{109}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.72015\) of defining polynomial
Character \(\chi\) \(=\) 600.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -19.8806 q^{7} +9.00000 q^{9} -70.6418 q^{11} -0.761226 q^{13} +108.403 q^{17} -125.881 q^{19} +59.6418 q^{21} +40.8806 q^{23} -27.0000 q^{27} -140.881 q^{29} +296.926 q^{31} +211.926 q^{33} -26.8061 q^{37} +2.28368 q^{39} -20.6418 q^{41} +32.5969 q^{43} +11.4326 q^{47} +52.2388 q^{49} -325.209 q^{51} +159.358 q^{53} +377.642 q^{57} -374.955 q^{59} +303.478 q^{61} -178.926 q^{63} +877.583 q^{67} -122.642 q^{69} +1159.54 q^{71} +120.716 q^{73} +1404.40 q^{77} -1248.66 q^{79} +81.0000 q^{81} +654.180 q^{83} +422.642 q^{87} +795.522 q^{89} +15.1336 q^{91} -890.777 q^{93} +850.910 q^{97} -635.777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 2 q^{7} + 18 q^{9} - 16 q^{11} + 82 q^{13} + 8 q^{17} - 210 q^{19} - 6 q^{21} + 40 q^{23} - 54 q^{27} - 240 q^{29} + 218 q^{31} + 48 q^{33} + 364 q^{37} - 246 q^{39} + 84 q^{41} + 274 q^{43}+ \cdots - 144 q^{99}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −19.8806 −1.07345 −0.536726 − 0.843757i \(-0.680339\pi\)
−0.536726 + 0.843757i \(0.680339\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −70.6418 −1.93630 −0.968151 − 0.250368i \(-0.919448\pi\)
−0.968151 + 0.250368i \(0.919448\pi\)
\(12\) 0 0
\(13\) −0.761226 −0.0162405 −0.00812024 − 0.999967i \(-0.502585\pi\)
−0.00812024 + 0.999967i \(0.502585\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 108.403 1.54657 0.773283 − 0.634062i \(-0.218613\pi\)
0.773283 + 0.634062i \(0.218613\pi\)
\(18\) 0 0
\(19\) −125.881 −1.51995 −0.759974 − 0.649954i \(-0.774788\pi\)
−0.759974 + 0.649954i \(0.774788\pi\)
\(20\) 0 0
\(21\) 59.6418 0.619758
\(22\) 0 0
\(23\) 40.8806 0.370617 0.185309 − 0.982680i \(-0.440672\pi\)
0.185309 + 0.982680i \(0.440672\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −140.881 −0.902099 −0.451050 − 0.892499i \(-0.648950\pi\)
−0.451050 + 0.892499i \(0.648950\pi\)
\(30\) 0 0
\(31\) 296.926 1.72030 0.860152 − 0.510039i \(-0.170369\pi\)
0.860152 + 0.510039i \(0.170369\pi\)
\(32\) 0 0
\(33\) 211.926 1.11792
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −26.8061 −0.119105 −0.0595527 − 0.998225i \(-0.518967\pi\)
−0.0595527 + 0.998225i \(0.518967\pi\)
\(38\) 0 0
\(39\) 2.28368 0.00937644
\(40\) 0 0
\(41\) −20.6418 −0.0786272 −0.0393136 − 0.999227i \(-0.512517\pi\)
−0.0393136 + 0.999227i \(0.512517\pi\)
\(42\) 0 0
\(43\) 32.5969 0.115604 0.0578022 − 0.998328i \(-0.481591\pi\)
0.0578022 + 0.998328i \(0.481591\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.4326 0.0354813 0.0177407 − 0.999843i \(-0.494353\pi\)
0.0177407 + 0.999843i \(0.494353\pi\)
\(48\) 0 0
\(49\) 52.2388 0.152300
\(50\) 0 0
\(51\) −325.209 −0.892910
\(52\) 0 0
\(53\) 159.358 0.413010 0.206505 − 0.978446i \(-0.433791\pi\)
0.206505 + 0.978446i \(0.433791\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 377.642 0.877542
\(58\) 0 0
\(59\) −374.955 −0.827373 −0.413686 − 0.910419i \(-0.635759\pi\)
−0.413686 + 0.910419i \(0.635759\pi\)
\(60\) 0 0
\(61\) 303.478 0.636989 0.318494 − 0.947925i \(-0.396823\pi\)
0.318494 + 0.947925i \(0.396823\pi\)
\(62\) 0 0
\(63\) −178.926 −0.357817
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 877.583 1.60021 0.800103 − 0.599863i \(-0.204778\pi\)
0.800103 + 0.599863i \(0.204778\pi\)
\(68\) 0 0
\(69\) −122.642 −0.213976
\(70\) 0 0
\(71\) 1159.54 1.93819 0.969097 − 0.246679i \(-0.0793393\pi\)
0.969097 + 0.246679i \(0.0793393\pi\)
\(72\) 0 0
\(73\) 120.716 0.193545 0.0967724 − 0.995307i \(-0.469148\pi\)
0.0967724 + 0.995307i \(0.469148\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1404.40 2.07853
\(78\) 0 0
\(79\) −1248.66 −1.77829 −0.889145 − 0.457626i \(-0.848700\pi\)
−0.889145 + 0.457626i \(0.848700\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 654.180 0.865127 0.432564 − 0.901603i \(-0.357609\pi\)
0.432564 + 0.901603i \(0.357609\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 422.642 0.520827
\(88\) 0 0
\(89\) 795.522 0.947474 0.473737 − 0.880666i \(-0.342905\pi\)
0.473737 + 0.880666i \(0.342905\pi\)
\(90\) 0 0
\(91\) 15.1336 0.0174334
\(92\) 0 0
\(93\) −890.777 −0.993217
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 850.910 0.890689 0.445345 − 0.895359i \(-0.353081\pi\)
0.445345 + 0.895359i \(0.353081\pi\)
\(98\) 0 0
\(99\) −635.777 −0.645434
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 600.4.a.s.1.1 ✓ 2
3.2 odd 2 1800.4.a.bo.1.1 2
4.3 odd 2 1200.4.a.br.1.2 2
5.2 odd 4 600.4.f.j.49.3 4
5.3 odd 4 600.4.f.j.49.2 4
5.4 even 2 600.4.a.u.1.2 yes 2
15.2 even 4 1800.4.f.z.649.2 4
15.8 even 4 1800.4.f.z.649.3 4
15.14 odd 2 1800.4.a.bm.1.2 2
20.3 even 4 1200.4.f.x.49.3 4
20.7 even 4 1200.4.f.x.49.2 4
20.19 odd 2 1200.4.a.bp.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.4.a.s.1.1 ✓ 2 1.1 even 1 trivial
600.4.a.u.1.2 yes 2 5.4 even 2
600.4.f.j.49.2 4 5.3 odd 4
600.4.f.j.49.3 4 5.2 odd 4
1200.4.a.bp.1.1 2 20.19 odd 2
1200.4.a.br.1.2 2 4.3 odd 2
1200.4.f.x.49.2 4 20.7 even 4
1200.4.f.x.49.3 4 20.3 even 4
1800.4.a.bm.1.2 2 15.14 odd 2
1800.4.a.bo.1.1 2 3.2 odd 2
1800.4.f.z.649.2 4 15.2 even 4
1800.4.f.z.649.3 4 15.8 even 4