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Properties

Label 5625.2.a.o.1.6
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $1$
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] = [5625,2,Mod(1,5625)] 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("5625.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5625, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-1,0,11,0,0,-2,-6,0,0,0,0,0,4,0,17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.9158511370\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.46840000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1875)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.38719\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.38719 q^{2} +3.69868 q^{4} -3.31671 q^{7} +4.05506 q^{8} +4.36655 q^{11} -5.85502 q^{13} -7.91762 q^{14} +2.28286 q^{16} +0.407830 q^{17} -6.64447 q^{19} +10.4238 q^{22} +4.61860 q^{23} -13.9771 q^{26} -12.2674 q^{28} -3.30132 q^{29} -8.77495 q^{31} -2.66052 q^{32} +0.973567 q^{34} -3.09911 q^{37} -15.8616 q^{38} -2.89349 q^{41} -1.33270 q^{43} +16.1505 q^{44} +11.0255 q^{46} -11.0609 q^{47} +4.00057 q^{49} -21.6558 q^{52} +2.70021 q^{53} -13.4495 q^{56} -7.88089 q^{58} +5.80518 q^{59} +11.2366 q^{61} -20.9475 q^{62} -10.9169 q^{64} +0.0418484 q^{67} +1.50843 q^{68} +16.2062 q^{71} -5.35799 q^{73} -7.39817 q^{74} -24.5757 q^{76} -14.4826 q^{77} -3.55370 q^{79} -6.90732 q^{82} +4.98952 q^{83} -3.18140 q^{86} +17.7066 q^{88} +1.94025 q^{89} +19.4194 q^{91} +17.0827 q^{92} -26.4044 q^{94} -5.99258 q^{97} +9.55012 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8} + 4 q^{14} + 17 q^{16} - 2 q^{17} - 2 q^{19} + 9 q^{22} + q^{23} - 37 q^{26} - 44 q^{28} - 31 q^{29} - 2 q^{31} - 33 q^{32} + 37 q^{34} - 22 q^{37} - 27 q^{38}+ \cdots - 15 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.38719 1.68800 0.843999 − 0.536345i \(-0.180195\pi\)
0.843999 + 0.536345i \(0.180195\pi\)
\(3\) 0 0
\(4\) 3.69868 1.84934
\(5\) 0 0
\(6\) 0 0
\(7\) −3.31671 −1.25360 −0.626799 − 0.779181i \(-0.715635\pi\)
−0.626799 + 0.779181i \(0.715635\pi\)
\(8\) 4.05506 1.43368
\(9\) 0 0
\(10\) 0 0
\(11\) 4.36655 1.31656 0.658282 − 0.752771i \(-0.271283\pi\)
0.658282 + 0.752771i \(0.271283\pi\)
\(12\) 0 0
\(13\) −5.85502 −1.62389 −0.811946 − 0.583733i \(-0.801591\pi\)
−0.811946 + 0.583733i \(0.801591\pi\)
\(14\) −7.91762 −2.11607
\(15\) 0 0
\(16\) 2.28286 0.570714
\(17\) 0.407830 0.0989132 0.0494566 − 0.998776i \(-0.484251\pi\)
0.0494566 + 0.998776i \(0.484251\pi\)
\(18\) 0 0
\(19\) −6.64447 −1.52435 −0.762173 − 0.647374i \(-0.775867\pi\)
−0.762173 + 0.647374i \(0.775867\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 10.4238 2.22236
\(23\) 4.61860 0.963045 0.481523 − 0.876434i \(-0.340084\pi\)
0.481523 + 0.876434i \(0.340084\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −13.9771 −2.74113
\(27\) 0 0
\(28\) −12.2674 −2.31833
\(29\) −3.30132 −0.613040 −0.306520 − 0.951864i \(-0.599165\pi\)
−0.306520 + 0.951864i \(0.599165\pi\)
\(30\) 0 0
\(31\) −8.77495 −1.57603 −0.788014 − 0.615658i \(-0.788890\pi\)
−0.788014 + 0.615658i \(0.788890\pi\)
\(32\) −2.66052 −0.470318
\(33\) 0 0
\(34\) 0.973567 0.166965
\(35\) 0 0
\(36\) 0 0
\(37\) −3.09911 −0.509491 −0.254745 − 0.967008i \(-0.581992\pi\)
−0.254745 + 0.967008i \(0.581992\pi\)
\(38\) −15.8616 −2.57309
\(39\) 0 0
\(40\) 0 0
\(41\) −2.89349 −0.451888 −0.225944 − 0.974140i \(-0.572547\pi\)
−0.225944 + 0.974140i \(0.572547\pi\)
\(42\) 0 0
\(43\) −1.33270 −0.203234 −0.101617 − 0.994824i \(-0.532402\pi\)
−0.101617 + 0.994824i \(0.532402\pi\)
\(44\) 16.1505 2.43477
\(45\) 0 0
\(46\) 11.0255 1.62562
\(47\) −11.0609 −1.61339 −0.806696 − 0.590967i \(-0.798746\pi\)
−0.806696 + 0.590967i \(0.798746\pi\)
\(48\) 0 0
\(49\) 4.00057 0.571510
\(50\) 0 0
\(51\) 0 0
\(52\) −21.6558 −3.00312
\(53\) 2.70021 0.370903 0.185451 − 0.982653i \(-0.440625\pi\)
0.185451 + 0.982653i \(0.440625\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −13.4495 −1.79726
\(57\) 0 0
\(58\) −7.88089 −1.03481
\(59\) 5.80518 0.755770 0.377885 − 0.925852i \(-0.376651\pi\)
0.377885 + 0.925852i \(0.376651\pi\)
\(60\) 0 0
\(61\) 11.2366 1.43870 0.719352 − 0.694646i \(-0.244439\pi\)
0.719352 + 0.694646i \(0.244439\pi\)
\(62\) −20.9475 −2.66033
\(63\) 0 0
\(64\) −10.9169 −1.36461
\(65\) 0 0
\(66\) 0 0
\(67\) 0.0418484 0.00511260 0.00255630 − 0.999997i \(-0.499186\pi\)
0.00255630 + 0.999997i \(0.499186\pi\)
\(68\) 1.50843 0.182924
\(69\) 0 0
\(70\) 0 0
\(71\) 16.2062 1.92332 0.961659 − 0.274250i \(-0.0884295\pi\)
0.961659 + 0.274250i \(0.0884295\pi\)
\(72\) 0 0
\(73\) −5.35799 −0.627105 −0.313553 − 0.949571i \(-0.601519\pi\)
−0.313553 + 0.949571i \(0.601519\pi\)
\(74\) −7.39817 −0.860019
\(75\) 0 0
\(76\) −24.5757 −2.81903
\(77\) −14.4826 −1.65044
\(78\) 0 0
\(79\) −3.55370 −0.399822 −0.199911 − 0.979814i \(-0.564065\pi\)
−0.199911 + 0.979814i \(0.564065\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.90732 −0.762786
\(83\) 4.98952 0.547671 0.273836 − 0.961776i \(-0.411708\pi\)
0.273836 + 0.961776i \(0.411708\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −3.18140 −0.343059
\(87\) 0 0
\(88\) 17.7066 1.88753
\(89\) 1.94025 0.205666 0.102833 − 0.994699i \(-0.467209\pi\)
0.102833 + 0.994699i \(0.467209\pi\)
\(90\) 0 0
\(91\) 19.4194 2.03571
\(92\) 17.0827 1.78100
\(93\) 0 0
\(94\) −26.4044 −2.72340
\(95\) 0 0
\(96\) 0 0
\(97\) −5.99258 −0.608454 −0.304227 − 0.952600i \(-0.598398\pi\)
−0.304227 + 0.952600i \(0.598398\pi\)
\(98\) 9.55012 0.964708
\(99\) 0 0
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 5625.2.a.o.1.6 6
3.2 odd 2 1875.2.a.l.1.1 yes 6
5.4 even 2 5625.2.a.r.1.1 6
15.2 even 4 1875.2.b.e.1249.2 12
15.8 even 4 1875.2.b.e.1249.11 12
15.14 odd 2 1875.2.a.i.1.6 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.i.1.6 ✓ 6 15.14 odd 2
1875.2.a.l.1.1 yes 6 3.2 odd 2
1875.2.b.e.1249.2 12 15.2 even 4
1875.2.b.e.1249.11 12 15.8 even 4
5625.2.a.o.1.6 6 1.1 even 1 trivial
5625.2.a.r.1.1 6 5.4 even 2