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Properties

Label 25.16.b.a.24.2
Level $25$
Weight $16$
Character 25.24
Analytic conductor $35.673$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [25,16,Mod(24,25)] 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("25.24"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(35.6733762750\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.16.b.a.24.1

$q$-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)
 
\(f(q)\) \(=\) \(q+216.000i q^{2} +3348.00i q^{3} -13888.0 q^{4} -723168. q^{6} +2.82246e6i q^{7} +4.07808e6i q^{8} +3.13980e6 q^{9} +2.05869e7 q^{11} -4.64970e7i q^{12} +1.90073e8i q^{13} -6.09650e8 q^{14} -1.33595e9 q^{16} +1.64653e9i q^{17} +6.78197e8i q^{18} -1.56326e9 q^{19} -9.44958e9 q^{21} +4.44676e9i q^{22} -9.45112e9i q^{23} -1.36534e10 q^{24} -4.10558e10 q^{26} +5.85522e10i q^{27} -3.91983e10i q^{28} +3.69026e10 q^{29} +7.15885e10 q^{31} -1.54934e11i q^{32} +6.89248e10i q^{33} -3.55650e11 q^{34} -4.36056e10 q^{36} -1.03365e12i q^{37} -3.37664e11i q^{38} -6.36366e11 q^{39} +1.64197e12 q^{41} -2.04111e12i q^{42} +4.92403e11i q^{43} -2.85910e11 q^{44} +2.04144e12 q^{46} -3.41068e12i q^{47} -4.47275e12i q^{48} -3.21870e12 q^{49} -5.51258e12 q^{51} -2.63974e12i q^{52} -6.79715e12i q^{53} -1.26473e13 q^{54} -1.15102e13 q^{56} -5.23379e12i q^{57} +7.97095e12i q^{58} -9.85886e12 q^{59} +4.93184e12 q^{61} +1.54631e13i q^{62} +8.86196e12i q^{63} -1.03106e13 q^{64} -1.48878e13 q^{66} -2.88378e13i q^{67} -2.28670e13i q^{68} +3.16423e13 q^{69} +1.25050e14 q^{71} +1.28044e13i q^{72} +8.21715e13i q^{73} +2.23269e14 q^{74} +2.17105e13 q^{76} +5.81055e13i q^{77} -1.37455e14i q^{78} +2.54131e13 q^{79} -1.50980e14 q^{81} +3.54666e14i q^{82} +2.81737e14i q^{83} +1.31236e14 q^{84} -1.06359e14 q^{86} +1.23550e14i q^{87} +8.39548e13i q^{88} -7.15619e14 q^{89} -5.36474e14 q^{91} +1.31257e14i q^{92} +2.39678e14i q^{93} +7.36708e14 q^{94} +5.18719e14 q^{96} +6.12786e14i q^{97} -6.95238e14i q^{98} +6.46387e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 27776 q^{4} - 1446336 q^{6} + 6279606 q^{9} + 41173704 q^{11} - 1219300992 q^{14} - 2671894528 q^{16} - 3126514360 q^{19} - 18899165376 q^{21} - 27306823680 q^{24} - 82111682016 q^{26} + 73805136660 q^{29}+ \cdots + 129277319340312 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 216.000i 1.19324i 0.802523 + 0.596621i \(0.203491\pi\)
−0.802523 + 0.596621i \(0.796509\pi\)
\(3\) 3348.00i 0.883845i 0.897053 + 0.441922i \(0.145703\pi\)
−0.897053 + 0.441922i \(0.854297\pi\)
\(4\) −13888.0 −0.423828
\(5\) 0 0
\(6\) −723168. −1.05464
\(7\) 2.82246e6i 1.29536i 0.761911 + 0.647682i \(0.224261\pi\)
−0.761911 + 0.647682i \(0.775739\pi\)
\(8\) 4.07808e6i 0.687513i
\(9\) 3.13980e6 0.218818
\(10\) 0 0
\(11\) 2.05869e7 0.318526 0.159263 − 0.987236i \(-0.449088\pi\)
0.159263 + 0.987236i \(0.449088\pi\)
\(12\) − 4.64970e7i − 0.374598i
\(13\) 1.90073e8i 0.840129i 0.907494 + 0.420065i \(0.137993\pi\)
−0.907494 + 0.420065i \(0.862007\pi\)
\(14\) −6.09650e8 −1.54568
\(15\) 0 0
\(16\) −1.33595e9 −1.24420
\(17\) 1.64653e9i 0.973200i 0.873625 + 0.486600i \(0.161763\pi\)
−0.873625 + 0.486600i \(0.838237\pi\)
\(18\) 6.78197e8i 0.261103i
\(19\) −1.56326e9 −0.401216 −0.200608 − 0.979672i \(-0.564292\pi\)
−0.200608 + 0.979672i \(0.564292\pi\)
\(20\) 0 0
\(21\) −9.44958e9 −1.14490
\(22\) 4.44676e9i 0.380079i
\(23\) − 9.45112e9i − 0.578794i −0.957209 − 0.289397i \(-0.906545\pi\)
0.957209 − 0.289397i \(-0.0934548\pi\)
\(24\) −1.36534e10 −0.607655
\(25\) 0 0
\(26\) −4.10558e10 −1.00248
\(27\) 5.85522e10i 1.07725i
\(28\) − 3.91983e10i − 0.549012i
\(29\) 3.69026e10 0.397257 0.198629 − 0.980075i \(-0.436351\pi\)
0.198629 + 0.980075i \(0.436351\pi\)
\(30\) 0 0
\(31\) 7.15885e10 0.467337 0.233669 − 0.972316i \(-0.424927\pi\)
0.233669 + 0.972316i \(0.424927\pi\)
\(32\) − 1.54934e11i − 0.797117i
\(33\) 6.89248e10i 0.281528i
\(34\) −3.55650e11 −1.16126
\(35\) 0 0
\(36\) −4.36056e10 −0.0927413
\(37\) − 1.03365e12i − 1.79003i −0.446031 − 0.895017i \(-0.647163\pi\)
0.446031 − 0.895017i \(-0.352837\pi\)
\(38\) − 3.37664e11i − 0.478748i
\(39\) −6.36366e11 −0.742544
\(40\) 0 0
\(41\) 1.64197e12 1.31670 0.658351 − 0.752711i \(-0.271254\pi\)
0.658351 + 0.752711i \(0.271254\pi\)
\(42\) − 2.04111e12i − 1.36614i
\(43\) 4.92403e11i 0.276253i 0.990415 + 0.138127i \(0.0441081\pi\)
−0.990415 + 0.138127i \(0.955892\pi\)
\(44\) −2.85910e11 −0.135000
\(45\) 0 0
\(46\) 2.04144e12 0.690642
\(47\) − 3.41068e12i − 0.981991i −0.871162 − 0.490996i \(-0.836633\pi\)
0.871162 − 0.490996i \(-0.163367\pi\)
\(48\) − 4.47275e12i − 1.09968i
\(49\) −3.21870e12 −0.677968
\(50\) 0 0
\(51\) −5.51258e12 −0.860158
\(52\) − 2.63974e12i − 0.356070i
\(53\) − 6.79715e12i − 0.794800i −0.917645 − 0.397400i \(-0.869913\pi\)
0.917645 − 0.397400i \(-0.130087\pi\)
\(54\) −1.26473e13 −1.28542
\(55\) 0 0
\(56\) −1.15102e13 −0.890580
\(57\) − 5.23379e12i − 0.354613i
\(58\) 7.97095e12i 0.474024i
\(59\) −9.85886e12 −0.515747 −0.257873 − 0.966179i \(-0.583022\pi\)
−0.257873 + 0.966179i \(0.583022\pi\)
\(60\) 0 0
\(61\) 4.93184e12 0.200926 0.100463 − 0.994941i \(-0.467968\pi\)
0.100463 + 0.994941i \(0.467968\pi\)
\(62\) 1.54631e13i 0.557647i
\(63\) 8.86196e12i 0.283449i
\(64\) −1.03106e13 −0.293044
\(65\) 0 0
\(66\) −1.48878e13 −0.335931
\(67\) − 2.88378e13i − 0.581302i −0.956829 − 0.290651i \(-0.906128\pi\)
0.956829 − 0.290651i \(-0.0938718\pi\)
\(68\) − 2.28670e13i − 0.412470i
\(69\) 3.16423e13 0.511564
\(70\) 0 0
\(71\) 1.25050e14 1.63172 0.815862 − 0.578247i \(-0.196263\pi\)
0.815862 + 0.578247i \(0.196263\pi\)
\(72\) 1.28044e13i 0.150440i
\(73\) 8.21715e13i 0.870562i 0.900295 + 0.435281i \(0.143351\pi\)
−0.900295 + 0.435281i \(0.856649\pi\)
\(74\) 2.23269e14 2.13595
\(75\) 0 0
\(76\) 2.17105e13 0.170047
\(77\) 5.81055e13i 0.412607i
\(78\) − 1.37455e14i − 0.886035i
\(79\) 2.54131e13 0.148886 0.0744430 − 0.997225i \(-0.476282\pi\)
0.0744430 + 0.997225i \(0.476282\pi\)
\(80\) 0 0
\(81\) −1.50980e14 −0.733300
\(82\) 3.54666e14i 1.57114i
\(83\) 2.81737e14i 1.13961i 0.821779 + 0.569807i \(0.192982\pi\)
−0.821779 + 0.569807i \(0.807018\pi\)
\(84\) 1.31236e14 0.485241
\(85\) 0 0
\(86\) −1.06359e14 −0.329637
\(87\) 1.23550e14i 0.351114i
\(88\) 8.39548e13i 0.218991i
\(89\) −7.15619e14 −1.71497 −0.857485 − 0.514509i \(-0.827974\pi\)
−0.857485 + 0.514509i \(0.827974\pi\)
\(90\) 0 0
\(91\) −5.36474e14 −1.08827
\(92\) 1.31257e14i 0.245309i
\(93\) 2.39678e14i 0.413054i
\(94\) 7.36708e14 1.17175
\(95\) 0 0
\(96\) 5.18719e14 0.704528
\(97\) 6.12786e14i 0.770054i 0.922905 + 0.385027i \(0.125808\pi\)
−0.922905 + 0.385027i \(0.874192\pi\)
\(98\) − 6.95238e14i − 0.808981i
\(99\) 6.46387e13 0.0696993
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 25.16.b.a.24.2 2
5.2 odd 4 25.16.a.a.1.1 1
5.3 odd 4 1.16.a.a.1.1 ✓ 1
5.4 even 2 inner 25.16.b.a.24.1 2
15.8 even 4 9.16.a.a.1.1 1
20.3 even 4 16.16.a.d.1.1 1
35.3 even 12 49.16.c.b.30.1 2
35.13 even 4 49.16.a.a.1.1 1
35.18 odd 12 49.16.c.c.30.1 2
35.23 odd 12 49.16.c.c.18.1 2
35.33 even 12 49.16.c.b.18.1 2
40.3 even 4 64.16.a.c.1.1 1
40.13 odd 4 64.16.a.i.1.1 1
55.43 even 4 121.16.a.a.1.1 1
60.23 odd 4 144.16.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.16.a.a.1.1 ✓ 1 5.3 odd 4
9.16.a.a.1.1 1 15.8 even 4
16.16.a.d.1.1 1 20.3 even 4
25.16.a.a.1.1 1 5.2 odd 4
25.16.b.a.24.1 2 5.4 even 2 inner
25.16.b.a.24.2 2 1.1 even 1 trivial
49.16.a.a.1.1 1 35.13 even 4
49.16.c.b.18.1 2 35.33 even 12
49.16.c.b.30.1 2 35.3 even 12
49.16.c.c.18.1 2 35.23 odd 12
49.16.c.c.30.1 2 35.18 odd 12
64.16.a.c.1.1 1 40.3 even 4
64.16.a.i.1.1 1 40.13 odd 4
121.16.a.a.1.1 1 55.43 even 4
144.16.a.f.1.1 1 60.23 odd 4