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Properties

Label 5625.2.a.o.1.5
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 1.5
Root \(-2.02791\) 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)
 
\(f(q)\) \(=\) \(q+2.02791 q^{2} +2.11242 q^{4} +0.505614 q^{7} +0.227977 q^{8} -0.687513 q^{11} -5.78627 q^{13} +1.02534 q^{14} -3.76252 q^{16} +4.74333 q^{17} +4.23709 q^{19} -1.39422 q^{22} -8.36239 q^{23} -11.7340 q^{26} +1.06807 q^{28} -4.88758 q^{29} +2.68853 q^{31} -8.08601 q^{32} +9.61905 q^{34} -11.3806 q^{37} +8.59244 q^{38} -0.144247 q^{41} +5.94442 q^{43} -1.45232 q^{44} -16.9582 q^{46} +6.10650 q^{47} -6.74435 q^{49} -12.2230 q^{52} -10.8398 q^{53} +0.115269 q^{56} -9.91157 q^{58} +6.96817 q^{59} -3.98042 q^{61} +5.45211 q^{62} -8.87266 q^{64} -1.31351 q^{67} +10.0199 q^{68} +3.79321 q^{71} -10.9252 q^{73} -23.0787 q^{74} +8.95051 q^{76} -0.347616 q^{77} -1.89869 q^{79} -0.292519 q^{82} -2.51849 q^{83} +12.0548 q^{86} -0.156737 q^{88} -15.0809 q^{89} -2.92562 q^{91} -17.6649 q^{92} +12.3834 q^{94} +1.17394 q^{97} -13.6769 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8} + 4 q^{14} + 17 q^{16} - 2 q^{17} - 2 q^{19} + 9 q^{22} + q^{23} - 37 q^{26} - 44 q^{28} - 31 q^{29} - 2 q^{31} - 33 q^{32} + 37 q^{34} - 22 q^{37} - 27 q^{38}+ \cdots - 15 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.02791 1.43395 0.716975 − 0.697099i \(-0.245526\pi\)
0.716975 + 0.697099i \(0.245526\pi\)
\(3\) 0 0
\(4\) 2.11242 1.05621
\(5\) 0 0
\(6\) 0 0
\(7\) 0.505614 0.191104 0.0955521 − 0.995424i \(-0.469538\pi\)
0.0955521 + 0.995424i \(0.469538\pi\)
\(8\) 0.227977 0.0806021
\(9\) 0 0
\(10\) 0 0
\(11\) −0.687513 −0.207293 −0.103647 − 0.994614i \(-0.533051\pi\)
−0.103647 + 0.994614i \(0.533051\pi\)
\(12\) 0 0
\(13\) −5.78627 −1.60482 −0.802412 − 0.596771i \(-0.796450\pi\)
−0.802412 + 0.596771i \(0.796450\pi\)
\(14\) 1.02534 0.274034
\(15\) 0 0
\(16\) −3.76252 −0.940631
\(17\) 4.74333 1.15043 0.575214 − 0.818003i \(-0.304919\pi\)
0.575214 + 0.818003i \(0.304919\pi\)
\(18\) 0 0
\(19\) 4.23709 0.972055 0.486027 − 0.873944i \(-0.338446\pi\)
0.486027 + 0.873944i \(0.338446\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.39422 −0.297248
\(23\) −8.36239 −1.74368 −0.871839 − 0.489792i \(-0.837073\pi\)
−0.871839 + 0.489792i \(0.837073\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −11.7340 −2.30124
\(27\) 0 0
\(28\) 1.06807 0.201846
\(29\) −4.88758 −0.907601 −0.453800 − 0.891103i \(-0.649932\pi\)
−0.453800 + 0.891103i \(0.649932\pi\)
\(30\) 0 0
\(31\) 2.68853 0.482875 0.241437 − 0.970416i \(-0.422381\pi\)
0.241437 + 0.970416i \(0.422381\pi\)
\(32\) −8.08601 −1.42942
\(33\) 0 0
\(34\) 9.61905 1.64965
\(35\) 0 0
\(36\) 0 0
\(37\) −11.3806 −1.87095 −0.935476 − 0.353390i \(-0.885029\pi\)
−0.935476 + 0.353390i \(0.885029\pi\)
\(38\) 8.59244 1.39388
\(39\) 0 0
\(40\) 0 0
\(41\) −0.144247 −0.0225275 −0.0112638 − 0.999937i \(-0.503585\pi\)
−0.0112638 + 0.999937i \(0.503585\pi\)
\(42\) 0 0
\(43\) 5.94442 0.906516 0.453258 − 0.891379i \(-0.350262\pi\)
0.453258 + 0.891379i \(0.350262\pi\)
\(44\) −1.45232 −0.218945
\(45\) 0 0
\(46\) −16.9582 −2.50035
\(47\) 6.10650 0.890725 0.445362 − 0.895350i \(-0.353075\pi\)
0.445362 + 0.895350i \(0.353075\pi\)
\(48\) 0 0
\(49\) −6.74435 −0.963479
\(50\) 0 0
\(51\) 0 0
\(52\) −12.2230 −1.69503
\(53\) −10.8398 −1.48896 −0.744481 − 0.667643i \(-0.767303\pi\)
−0.744481 + 0.667643i \(0.767303\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.115269 0.0154034
\(57\) 0 0
\(58\) −9.91157 −1.30145
\(59\) 6.96817 0.907179 0.453589 − 0.891211i \(-0.350143\pi\)
0.453589 + 0.891211i \(0.350143\pi\)
\(60\) 0 0
\(61\) −3.98042 −0.509641 −0.254820 − 0.966988i \(-0.582016\pi\)
−0.254820 + 0.966988i \(0.582016\pi\)
\(62\) 5.45211 0.692418
\(63\) 0 0
\(64\) −8.87266 −1.10908
\(65\) 0 0
\(66\) 0 0
\(67\) −1.31351 −0.160470 −0.0802352 − 0.996776i \(-0.525567\pi\)
−0.0802352 + 0.996776i \(0.525567\pi\)
\(68\) 10.0199 1.21509
\(69\) 0 0
\(70\) 0 0
\(71\) 3.79321 0.450172 0.225086 − 0.974339i \(-0.427734\pi\)
0.225086 + 0.974339i \(0.427734\pi\)
\(72\) 0 0
\(73\) −10.9252 −1.27870 −0.639351 − 0.768915i \(-0.720797\pi\)
−0.639351 + 0.768915i \(0.720797\pi\)
\(74\) −23.0787 −2.68285
\(75\) 0 0
\(76\) 8.95051 1.02669
\(77\) −0.347616 −0.0396146
\(78\) 0 0
\(79\) −1.89869 −0.213620 −0.106810 − 0.994279i \(-0.534064\pi\)
−0.106810 + 0.994279i \(0.534064\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.292519 −0.0323033
\(83\) −2.51849 −0.276441 −0.138220 − 0.990402i \(-0.544138\pi\)
−0.138220 + 0.990402i \(0.544138\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0548 1.29990
\(87\) 0 0
\(88\) −0.156737 −0.0167083
\(89\) −15.0809 −1.59858 −0.799289 − 0.600947i \(-0.794790\pi\)
−0.799289 + 0.600947i \(0.794790\pi\)
\(90\) 0 0
\(91\) −2.92562 −0.306689
\(92\) −17.6649 −1.84169
\(93\) 0 0
\(94\) 12.3834 1.27725
\(95\) 0 0
\(96\) 0 0
\(97\) 1.17394 0.119195 0.0595977 − 0.998222i \(-0.481018\pi\)
0.0595977 + 0.998222i \(0.481018\pi\)
\(98\) −13.6769 −1.38158
\(99\) 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.1.5 6
3.2 odd 2 1875.2.a.l.1.2 yes 6
5.4 even 2 5625.2.a.r.1.2 6
15.2 even 4 1875.2.b.e.1249.4 12
15.8 even 4 1875.2.b.e.1249.9 12
15.14 odd 2 1875.2.a.i.1.5 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.i.1.5 ✓ 6 15.14 odd 2
1875.2.a.l.1.2 yes 6 3.2 odd 2
1875.2.b.e.1249.4 12 15.2 even 4
1875.2.b.e.1249.9 12 15.8 even 4
5625.2.a.o.1.5 6 1.1 even 1 trivial
5625.2.a.r.1.2 6 5.4 even 2