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Properties

Label 5625.2.a.o.1.4
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 1.4
Root \(-0.364088\) 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)
 
\(f(q)\) \(=\) \(q+0.364088 q^{2} -1.86744 q^{4} +2.24941 q^{7} -1.40809 q^{8} -4.63962 q^{11} +3.75730 q^{13} +0.818981 q^{14} +3.22221 q^{16} +5.36779 q^{17} -5.66369 q^{19} -1.68923 q^{22} -1.32214 q^{23} +1.36799 q^{26} -4.20063 q^{28} -8.86744 q^{29} +1.21200 q^{31} +3.98934 q^{32} +1.95435 q^{34} -2.15975 q^{37} -2.06208 q^{38} -3.49965 q^{41} +1.16800 q^{43} +8.66420 q^{44} -0.481374 q^{46} +2.36614 q^{47} -1.94017 q^{49} -7.01653 q^{52} +14.1656 q^{53} -3.16736 q^{56} -3.22853 q^{58} -0.367089 q^{59} +5.29590 q^{61} +0.441273 q^{62} -4.99195 q^{64} +8.06723 q^{67} -10.0240 q^{68} -11.4185 q^{71} -13.7580 q^{73} -0.786340 q^{74} +10.5766 q^{76} -10.4364 q^{77} +11.6247 q^{79} -1.27418 q^{82} -11.1027 q^{83} +0.425253 q^{86} +6.53299 q^{88} -16.6526 q^{89} +8.45169 q^{91} +2.46901 q^{92} +0.861482 q^{94} +14.4790 q^{97} -0.706393 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8} + 4 q^{14} + 17 q^{16} - 2 q^{17} - 2 q^{19} + 9 q^{22} + q^{23} - 37 q^{26} - 44 q^{28} - 31 q^{29} - 2 q^{31} - 33 q^{32} + 37 q^{34} - 22 q^{37} - 27 q^{38}+ \cdots - 15 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.364088 0.257449 0.128724 − 0.991680i \(-0.458912\pi\)
0.128724 + 0.991680i \(0.458912\pi\)
\(3\) 0 0
\(4\) −1.86744 −0.933720
\(5\) 0 0
\(6\) 0 0
\(7\) 2.24941 0.850196 0.425098 − 0.905147i \(-0.360240\pi\)
0.425098 + 0.905147i \(0.360240\pi\)
\(8\) −1.40809 −0.497834
\(9\) 0 0
\(10\) 0 0
\(11\) −4.63962 −1.39890 −0.699448 − 0.714683i \(-0.746571\pi\)
−0.699448 + 0.714683i \(0.746571\pi\)
\(12\) 0 0
\(13\) 3.75730 1.04209 0.521044 − 0.853530i \(-0.325543\pi\)
0.521044 + 0.853530i \(0.325543\pi\)
\(14\) 0.818981 0.218882
\(15\) 0 0
\(16\) 3.22221 0.805553
\(17\) 5.36779 1.30188 0.650940 − 0.759129i \(-0.274375\pi\)
0.650940 + 0.759129i \(0.274375\pi\)
\(18\) 0 0
\(19\) −5.66369 −1.29934 −0.649669 − 0.760217i \(-0.725093\pi\)
−0.649669 + 0.760217i \(0.725093\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.68923 −0.360144
\(23\) −1.32214 −0.275685 −0.137842 − 0.990454i \(-0.544017\pi\)
−0.137842 + 0.990454i \(0.544017\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.36799 0.268284
\(27\) 0 0
\(28\) −4.20063 −0.793845
\(29\) −8.86744 −1.64664 −0.823321 − 0.567576i \(-0.807881\pi\)
−0.823321 + 0.567576i \(0.807881\pi\)
\(30\) 0 0
\(31\) 1.21200 0.217681 0.108841 − 0.994059i \(-0.465286\pi\)
0.108841 + 0.994059i \(0.465286\pi\)
\(32\) 3.98934 0.705223
\(33\) 0 0
\(34\) 1.95435 0.335168
\(35\) 0 0
\(36\) 0 0
\(37\) −2.15975 −0.355061 −0.177531 − 0.984115i \(-0.556811\pi\)
−0.177531 + 0.984115i \(0.556811\pi\)
\(38\) −2.06208 −0.334513
\(39\) 0 0
\(40\) 0 0
\(41\) −3.49965 −0.546553 −0.273277 − 0.961935i \(-0.588107\pi\)
−0.273277 + 0.961935i \(0.588107\pi\)
\(42\) 0 0
\(43\) 1.16800 0.178118 0.0890589 − 0.996026i \(-0.471614\pi\)
0.0890589 + 0.996026i \(0.471614\pi\)
\(44\) 8.66420 1.30618
\(45\) 0 0
\(46\) −0.481374 −0.0709748
\(47\) 2.36614 0.345137 0.172568 − 0.984998i \(-0.444793\pi\)
0.172568 + 0.984998i \(0.444793\pi\)
\(48\) 0 0
\(49\) −1.94017 −0.277167
\(50\) 0 0
\(51\) 0 0
\(52\) −7.01653 −0.973018
\(53\) 14.1656 1.94580 0.972901 − 0.231223i \(-0.0742728\pi\)
0.972901 + 0.231223i \(0.0742728\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.16736 −0.423256
\(57\) 0 0
\(58\) −3.22853 −0.423926
\(59\) −0.367089 −0.0477909 −0.0238955 − 0.999714i \(-0.507607\pi\)
−0.0238955 + 0.999714i \(0.507607\pi\)
\(60\) 0 0
\(61\) 5.29590 0.678070 0.339035 − 0.940774i \(-0.389899\pi\)
0.339035 + 0.940774i \(0.389899\pi\)
\(62\) 0.441273 0.0560417
\(63\) 0 0
\(64\) −4.99195 −0.623994
\(65\) 0 0
\(66\) 0 0
\(67\) 8.06723 0.985570 0.492785 − 0.870151i \(-0.335979\pi\)
0.492785 + 0.870151i \(0.335979\pi\)
\(68\) −10.0240 −1.21559
\(69\) 0 0
\(70\) 0 0
\(71\) −11.4185 −1.35513 −0.677563 − 0.735465i \(-0.736964\pi\)
−0.677563 + 0.735465i \(0.736964\pi\)
\(72\) 0 0
\(73\) −13.7580 −1.61025 −0.805126 − 0.593104i \(-0.797902\pi\)
−0.805126 + 0.593104i \(0.797902\pi\)
\(74\) −0.786340 −0.0914101
\(75\) 0 0
\(76\) 10.5766 1.21322
\(77\) −10.4364 −1.18934
\(78\) 0 0
\(79\) 11.6247 1.30789 0.653943 − 0.756544i \(-0.273114\pi\)
0.653943 + 0.756544i \(0.273114\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.27418 −0.140710
\(83\) −11.1027 −1.21868 −0.609338 − 0.792910i \(-0.708565\pi\)
−0.609338 + 0.792910i \(0.708565\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.425253 0.0458562
\(87\) 0 0
\(88\) 6.53299 0.696419
\(89\) −16.6526 −1.76518 −0.882588 − 0.470148i \(-0.844201\pi\)
−0.882588 + 0.470148i \(0.844201\pi\)
\(90\) 0 0
\(91\) 8.45169 0.885978
\(92\) 2.46901 0.257412
\(93\) 0 0
\(94\) 0.861482 0.0888551
\(95\) 0 0
\(96\) 0 0
\(97\) 14.4790 1.47012 0.735061 − 0.678001i \(-0.237153\pi\)
0.735061 + 0.678001i \(0.237153\pi\)
\(98\) −0.706393 −0.0713565
\(99\) 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.1.4 6
3.2 odd 2 1875.2.a.l.1.3 yes 6
5.4 even 2 5625.2.a.r.1.3 6
15.2 even 4 1875.2.b.e.1249.6 12
15.8 even 4 1875.2.b.e.1249.7 12
15.14 odd 2 1875.2.a.i.1.4 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.i.1.4 ✓ 6 15.14 odd 2
1875.2.a.l.1.3 yes 6 3.2 odd 2
1875.2.b.e.1249.6 12 15.2 even 4
1875.2.b.e.1249.7 12 15.8 even 4
5625.2.a.o.1.4 6 1.1 even 1 trivial
5625.2.a.r.1.3 6 5.4 even 2