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Properties

Label 252.4.a.e.1.2
Level $252$
Weight $4$
Character 252.1
Self dual yes
Analytic conductor $14.868$
Analytic rank $1$
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] = [252,4,Mod(1,252)] 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("252.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-14,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.8684813214\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 252.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.29150 q^{5} -7.00000 q^{7} -68.7895 q^{11} -30.0000 q^{13} +121.705 q^{17} -100.000 q^{19} +89.9555 q^{23} -97.0000 q^{25} -232.826 q^{29} -180.000 q^{31} -37.0405 q^{35} -118.000 q^{37} -15.8745 q^{41} -412.000 q^{43} +285.741 q^{47} +49.0000 q^{49} +285.741 q^{53} -364.000 q^{55} -836.057 q^{59} -378.000 q^{61} -158.745 q^{65} +244.000 q^{67} +439.195 q^{71} +670.000 q^{73} +481.527 q^{77} +216.000 q^{79} +804.308 q^{83} +644.000 q^{85} -968.345 q^{89} +210.000 q^{91} -529.150 q^{95} +574.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{7} - 60 q^{13} - 200 q^{19} - 194 q^{25} - 360 q^{31} - 236 q^{37} - 824 q^{43} + 98 q^{49} - 728 q^{55} - 756 q^{61} + 488 q^{67} + 1340 q^{73} + 432 q^{79} + 1288 q^{85} + 420 q^{91} + 1148 q^{97}+O(q^{100}) \) 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 0
\(4\) 0 0
\(5\) 5.29150 0.473286 0.236643 − 0.971597i \(-0.423953\pi\)
0.236643 + 0.971597i \(0.423953\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −68.7895 −1.88553 −0.942765 − 0.333459i \(-0.891784\pi\)
−0.942765 + 0.333459i \(0.891784\pi\)
\(12\) 0 0
\(13\) −30.0000 −0.640039 −0.320019 − 0.947411i \(-0.603689\pi\)
−0.320019 + 0.947411i \(0.603689\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 121.705 1.73633 0.868167 − 0.496271i \(-0.165298\pi\)
0.868167 + 0.496271i \(0.165298\pi\)
\(18\) 0 0
\(19\) −100.000 −1.20745 −0.603726 − 0.797192i \(-0.706318\pi\)
−0.603726 + 0.797192i \(0.706318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 89.9555 0.815523 0.407761 − 0.913088i \(-0.366309\pi\)
0.407761 + 0.913088i \(0.366309\pi\)
\(24\) 0 0
\(25\) −97.0000 −0.776000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −232.826 −1.49085 −0.745426 − 0.666588i \(-0.767754\pi\)
−0.745426 + 0.666588i \(0.767754\pi\)
\(30\) 0 0
\(31\) −180.000 −1.04287 −0.521435 − 0.853291i \(-0.674603\pi\)
−0.521435 + 0.853291i \(0.674603\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −37.0405 −0.178885
\(36\) 0 0
\(37\) −118.000 −0.524299 −0.262150 − 0.965027i \(-0.584431\pi\)
−0.262150 + 0.965027i \(0.584431\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −15.8745 −0.0604678 −0.0302339 − 0.999543i \(-0.509625\pi\)
−0.0302339 + 0.999543i \(0.509625\pi\)
\(42\) 0 0
\(43\) −412.000 −1.46115 −0.730575 − 0.682833i \(-0.760748\pi\)
−0.730575 + 0.682833i \(0.760748\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 285.741 0.886801 0.443400 − 0.896324i \(-0.353772\pi\)
0.443400 + 0.896324i \(0.353772\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 285.741 0.740558 0.370279 − 0.928921i \(-0.379262\pi\)
0.370279 + 0.928921i \(0.379262\pi\)
\(54\) 0 0
\(55\) −364.000 −0.892395
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −836.057 −1.84484 −0.922419 − 0.386191i \(-0.873790\pi\)
−0.922419 + 0.386191i \(0.873790\pi\)
\(60\) 0 0
\(61\) −378.000 −0.793409 −0.396704 − 0.917946i \(-0.629846\pi\)
−0.396704 + 0.917946i \(0.629846\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −158.745 −0.302922
\(66\) 0 0
\(67\) 244.000 0.444916 0.222458 − 0.974942i \(-0.428592\pi\)
0.222458 + 0.974942i \(0.428592\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 439.195 0.734124 0.367062 − 0.930196i \(-0.380364\pi\)
0.367062 + 0.930196i \(0.380364\pi\)
\(72\) 0 0
\(73\) 670.000 1.07421 0.537107 − 0.843514i \(-0.319517\pi\)
0.537107 + 0.843514i \(0.319517\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 481.527 0.712663
\(78\) 0 0
\(79\) 216.000 0.307619 0.153809 − 0.988101i \(-0.450846\pi\)
0.153809 + 0.988101i \(0.450846\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 804.308 1.06367 0.531833 − 0.846849i \(-0.321503\pi\)
0.531833 + 0.846849i \(0.321503\pi\)
\(84\) 0 0
\(85\) 644.000 0.821784
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −968.345 −1.15331 −0.576654 − 0.816989i \(-0.695642\pi\)
−0.576654 + 0.816989i \(0.695642\pi\)
\(90\) 0 0
\(91\) 210.000 0.241912
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −529.150 −0.571470
\(96\) 0 0
\(97\) 574.000 0.600834 0.300417 − 0.953808i \(-0.402874\pi\)
0.300417 + 0.953808i \(0.402874\pi\)
\(98\) 0 0
\(99\) 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 252.4.a.e.1.2 yes 2
3.2 odd 2 inner 252.4.a.e.1.1 ✓ 2
4.3 odd 2 1008.4.a.bd.1.2 2
7.2 even 3 1764.4.k.t.361.1 4
7.3 odd 6 1764.4.k.w.1549.2 4
7.4 even 3 1764.4.k.t.1549.1 4
7.5 odd 6 1764.4.k.w.361.2 4
7.6 odd 2 1764.4.a.t.1.1 2
12.11 even 2 1008.4.a.bd.1.1 2
21.2 odd 6 1764.4.k.t.361.2 4
21.5 even 6 1764.4.k.w.361.1 4
21.11 odd 6 1764.4.k.t.1549.2 4
21.17 even 6 1764.4.k.w.1549.1 4
21.20 even 2 1764.4.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.4.a.e.1.1 ✓ 2 3.2 odd 2 inner
252.4.a.e.1.2 yes 2 1.1 even 1 trivial
1008.4.a.bd.1.1 2 12.11 even 2
1008.4.a.bd.1.2 2 4.3 odd 2
1764.4.a.t.1.1 2 7.6 odd 2
1764.4.a.t.1.2 2 21.20 even 2
1764.4.k.t.361.1 4 7.2 even 3
1764.4.k.t.361.2 4 21.2 odd 6
1764.4.k.t.1549.1 4 7.4 even 3
1764.4.k.t.1549.2 4 21.11 odd 6
1764.4.k.w.361.1 4 21.5 even 6
1764.4.k.w.361.2 4 7.5 odd 6
1764.4.k.w.1549.1 4 21.17 even 6
1764.4.k.w.1549.2 4 7.3 odd 6