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Properties

Label 7600.2.a.cl.1.3
Level $7600$
Weight $2$
Character 7600.1
Self dual yes
Analytic conductor $60.686$
Analytic rank $0$
Dimension $6$
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] = [7600,2,Mod(1,7600)] 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("7600.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,2,0,0,0,2,0,6,0,-3,0,1] 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: \(60.6863055362\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3800)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.185519\) of defining polynomial
Character \(\chi\) \(=\) 7600.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.185519 q^{3} -4.45651 q^{7} -2.96558 q^{9} -2.64623 q^{11} -1.30142 q^{13} +3.51716 q^{17} +1.00000 q^{19} +0.826767 q^{21} -6.52882 q^{23} +1.10673 q^{27} -5.20946 q^{29} -10.8219 q^{31} +0.490925 q^{33} -2.04607 q^{37} +0.241439 q^{39} -3.80044 q^{41} +4.77089 q^{43} +1.49093 q^{47} +12.8605 q^{49} -0.652501 q^{51} -0.225583 q^{53} -0.185519 q^{57} +2.86056 q^{59} -6.31449 q^{61} +13.2161 q^{63} -13.1831 q^{67} +1.21122 q^{69} -12.3310 q^{71} +5.42276 q^{73} +11.7929 q^{77} -14.9688 q^{79} +8.69143 q^{81} +3.84470 q^{83} +0.966455 q^{87} +1.67666 q^{89} +5.79981 q^{91} +2.00768 q^{93} -9.48523 q^{97} +7.84760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + 2 q^{7} + 6 q^{9} - 3 q^{11} + q^{13} + 14 q^{17} + 6 q^{19} + 15 q^{21} + 12 q^{23} + 8 q^{27} + 9 q^{29} - 5 q^{31} - 2 q^{33} + 8 q^{37} - 12 q^{39} + 3 q^{41} + 15 q^{43} + 4 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(3\) −0.185519 −0.107109 −0.0535547 − 0.998565i \(-0.517055\pi\)
−0.0535547 + 0.998565i \(0.517055\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.45651 −1.68440 −0.842201 − 0.539164i \(-0.818740\pi\)
−0.842201 + 0.539164i \(0.818740\pi\)
\(8\) 0 0
\(9\) −2.96558 −0.988528
\(10\) 0 0
\(11\) −2.64623 −0.797867 −0.398934 − 0.916980i \(-0.630620\pi\)
−0.398934 + 0.916980i \(0.630620\pi\)
\(12\) 0 0
\(13\) −1.30142 −0.360950 −0.180475 − 0.983580i \(-0.557764\pi\)
−0.180475 + 0.983580i \(0.557764\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.51716 0.853037 0.426519 − 0.904479i \(-0.359740\pi\)
0.426519 + 0.904479i \(0.359740\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.826767 0.180415
\(22\) 0 0
\(23\) −6.52882 −1.36135 −0.680677 − 0.732584i \(-0.738314\pi\)
−0.680677 + 0.732584i \(0.738314\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.10673 0.212990
\(28\) 0 0
\(29\) −5.20946 −0.967373 −0.483686 − 0.875241i \(-0.660702\pi\)
−0.483686 + 0.875241i \(0.660702\pi\)
\(30\) 0 0
\(31\) −10.8219 −1.94368 −0.971839 − 0.235646i \(-0.924279\pi\)
−0.971839 + 0.235646i \(0.924279\pi\)
\(32\) 0 0
\(33\) 0.490925 0.0854591
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.04607 −0.336373 −0.168186 − 0.985755i \(-0.553791\pi\)
−0.168186 + 0.985755i \(0.553791\pi\)
\(38\) 0 0
\(39\) 0.241439 0.0386612
\(40\) 0 0
\(41\) −3.80044 −0.593529 −0.296764 − 0.954951i \(-0.595908\pi\)
−0.296764 + 0.954951i \(0.595908\pi\)
\(42\) 0 0
\(43\) 4.77089 0.727554 0.363777 − 0.931486i \(-0.381487\pi\)
0.363777 + 0.931486i \(0.381487\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.49093 0.217474 0.108737 − 0.994071i \(-0.465319\pi\)
0.108737 + 0.994071i \(0.465319\pi\)
\(48\) 0 0
\(49\) 12.8605 1.83721
\(50\) 0 0
\(51\) −0.652501 −0.0913684
\(52\) 0 0
\(53\) −0.225583 −0.0309862 −0.0154931 − 0.999880i \(-0.504932\pi\)
−0.0154931 + 0.999880i \(0.504932\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.185519 −0.0245726
\(58\) 0 0
\(59\) 2.86056 0.372413 0.186206 − 0.982511i \(-0.440381\pi\)
0.186206 + 0.982511i \(0.440381\pi\)
\(60\) 0 0
\(61\) −6.31449 −0.808488 −0.404244 − 0.914651i \(-0.632465\pi\)
−0.404244 + 0.914651i \(0.632465\pi\)
\(62\) 0 0
\(63\) 13.2161 1.66508
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −13.1831 −1.61058 −0.805288 − 0.592884i \(-0.797989\pi\)
−0.805288 + 0.592884i \(0.797989\pi\)
\(68\) 0 0
\(69\) 1.21122 0.145814
\(70\) 0 0
\(71\) −12.3310 −1.46342 −0.731711 − 0.681615i \(-0.761278\pi\)
−0.731711 + 0.681615i \(0.761278\pi\)
\(72\) 0 0
\(73\) 5.42276 0.634686 0.317343 − 0.948311i \(-0.397209\pi\)
0.317343 + 0.948311i \(0.397209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.7929 1.34393
\(78\) 0 0
\(79\) −14.9688 −1.68413 −0.842063 − 0.539379i \(-0.818659\pi\)
−0.842063 + 0.539379i \(0.818659\pi\)
\(80\) 0 0
\(81\) 8.69143 0.965714
\(82\) 0 0
\(83\) 3.84470 0.422011 0.211005 − 0.977485i \(-0.432326\pi\)
0.211005 + 0.977485i \(0.432326\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.966455 0.103615
\(88\) 0 0
\(89\) 1.67666 0.177726 0.0888629 − 0.996044i \(-0.471677\pi\)
0.0888629 + 0.996044i \(0.471677\pi\)
\(90\) 0 0
\(91\) 5.79981 0.607985
\(92\) 0 0
\(93\) 2.00768 0.208186
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.48523 −0.963079 −0.481539 − 0.876424i \(-0.659922\pi\)
−0.481539 + 0.876424i \(0.659922\pi\)
\(98\) 0 0
\(99\) 7.84760 0.788714
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 7600.2.a.cl.1.3 6
4.3 odd 2 3800.2.a.ba.1.4 ✓ 6
5.4 even 2 7600.2.a.ch.1.4 6
20.3 even 4 3800.2.d.q.3649.7 12
20.7 even 4 3800.2.d.q.3649.6 12
20.19 odd 2 3800.2.a.bc.1.3 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.4 ✓ 6 4.3 odd 2
3800.2.a.bc.1.3 yes 6 20.19 odd 2
3800.2.d.q.3649.6 12 20.7 even 4
3800.2.d.q.3649.7 12 20.3 even 4
7600.2.a.ch.1.4 6 5.4 even 2
7600.2.a.cl.1.3 6 1.1 even 1 trivial