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Properties

Label 2254.2.a.x.1.3
Level $2254$
Weight $2$
Character 2254.1
Self dual yes
Analytic conductor $17.998$
Analytic rank $1$
Dimension $4$
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] = [2254,2,Mod(1,2254)] 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("2254.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2254, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2254.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,-3,4,-7,-3,0,4,-1,-7,2] 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: \(17.9982806156\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.1957.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 322)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.76401\) of defining polynomial
Character \(\chi\) \(=\) 2254.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.197126 q^{3} +1.00000 q^{4} +0.308875 q^{5} +0.197126 q^{6} +1.00000 q^{8} -2.96114 q^{9} +0.308875 q^{10} +0.369762 q^{11} +0.197126 q^{12} -6.29639 q^{13} +0.0608874 q^{15} +1.00000 q^{16} -4.61341 q^{17} -2.96114 q^{18} +0.352932 q^{19} +0.308875 q^{20} +0.369762 q^{22} -1.00000 q^{23} +0.197126 q^{24} -4.90460 q^{25} -6.29639 q^{26} -1.17510 q^{27} +1.74199 q^{29} +0.0608874 q^{30} -2.93477 q^{31} +1.00000 q^{32} +0.0728899 q^{33} -4.61341 q^{34} -2.96114 q^{36} -6.37930 q^{37} +0.352932 q^{38} -1.24118 q^{39} +0.308875 q^{40} +1.90213 q^{41} +6.55252 q^{43} +0.369762 q^{44} -0.914622 q^{45} -1.00000 q^{46} -11.7347 q^{47} +0.197126 q^{48} -4.90460 q^{50} -0.909424 q^{51} -6.29639 q^{52} +5.90460 q^{53} -1.17510 q^{54} +0.114210 q^{55} +0.0695722 q^{57} +1.74199 q^{58} +2.29204 q^{59} +0.0608874 q^{60} -6.78792 q^{61} -2.93477 q^{62} +1.00000 q^{64} -1.94480 q^{65} +0.0728899 q^{66} +12.7442 q^{67} -4.61341 q^{68} -0.197126 q^{69} +0.492118 q^{71} -2.96114 q^{72} -5.99951 q^{73} -6.37930 q^{74} -0.966825 q^{75} +0.352932 q^{76} -1.24118 q^{78} +0.666150 q^{79} +0.308875 q^{80} +8.65178 q^{81} +1.90213 q^{82} +6.44785 q^{83} -1.42497 q^{85} +6.55252 q^{86} +0.343391 q^{87} +0.369762 q^{88} -11.6221 q^{89} -0.914622 q^{90} -1.00000 q^{92} -0.578520 q^{93} -11.7347 q^{94} +0.109012 q^{95} +0.197126 q^{96} -4.96548 q^{97} -1.09492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 3 q^{3} + 4 q^{4} - 7 q^{5} - 3 q^{6} + 4 q^{8} - q^{9} - 7 q^{10} + 2 q^{11} - 3 q^{12} - q^{13} + 9 q^{15} + 4 q^{16} - 5 q^{17} - q^{18} - 11 q^{19} - 7 q^{20} + 2 q^{22} - 4 q^{23}+ \cdots + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0.197126 0.113811 0.0569055 − 0.998380i \(-0.481877\pi\)
0.0569055 + 0.998380i \(0.481877\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.308875 0.138133 0.0690665 − 0.997612i \(-0.477998\pi\)
0.0690665 + 0.997612i \(0.477998\pi\)
\(6\) 0.197126 0.0804765
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −2.96114 −0.987047
\(10\) 0.308875 0.0976748
\(11\) 0.369762 0.111488 0.0557438 − 0.998445i \(-0.482247\pi\)
0.0557438 + 0.998445i \(0.482247\pi\)
\(12\) 0.197126 0.0569055
\(13\) −6.29639 −1.74630 −0.873152 − 0.487448i \(-0.837928\pi\)
−0.873152 + 0.487448i \(0.837928\pi\)
\(14\) 0 0
\(15\) 0.0608874 0.0157211
\(16\) 1.00000 0.250000
\(17\) −4.61341 −1.11892 −0.559458 − 0.828859i \(-0.688991\pi\)
−0.559458 + 0.828859i \(0.688991\pi\)
\(18\) −2.96114 −0.697948
\(19\) 0.352932 0.0809682 0.0404841 − 0.999180i \(-0.487110\pi\)
0.0404841 + 0.999180i \(0.487110\pi\)
\(20\) 0.308875 0.0690665
\(21\) 0 0
\(22\) 0.369762 0.0788336
\(23\) −1.00000 −0.208514
\(24\) 0.197126 0.0402382
\(25\) −4.90460 −0.980919
\(26\) −6.29639 −1.23482
\(27\) −1.17510 −0.226148
\(28\) 0 0
\(29\) 1.74199 0.323479 0.161739 − 0.986834i \(-0.448290\pi\)
0.161739 + 0.986834i \(0.448290\pi\)
\(30\) 0.0608874 0.0111165
\(31\) −2.93477 −0.527100 −0.263550 − 0.964646i \(-0.584893\pi\)
−0.263550 + 0.964646i \(0.584893\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.0728899 0.0126885
\(34\) −4.61341 −0.791193
\(35\) 0 0
\(36\) −2.96114 −0.493524
\(37\) −6.37930 −1.04875 −0.524375 − 0.851487i \(-0.675701\pi\)
−0.524375 + 0.851487i \(0.675701\pi\)
\(38\) 0.352932 0.0572532
\(39\) −1.24118 −0.198748
\(40\) 0.308875 0.0488374
\(41\) 1.90213 0.297064 0.148532 − 0.988908i \(-0.452545\pi\)
0.148532 + 0.988908i \(0.452545\pi\)
\(42\) 0 0
\(43\) 6.55252 0.999250 0.499625 − 0.866242i \(-0.333471\pi\)
0.499625 + 0.866242i \(0.333471\pi\)
\(44\) 0.369762 0.0557438
\(45\) −0.914622 −0.136344
\(46\) −1.00000 −0.147442
\(47\) −11.7347 −1.71168 −0.855841 − 0.517239i \(-0.826960\pi\)
−0.855841 + 0.517239i \(0.826960\pi\)
\(48\) 0.197126 0.0284527
\(49\) 0 0
\(50\) −4.90460 −0.693615
\(51\) −0.909424 −0.127345
\(52\) −6.29639 −0.873152
\(53\) 5.90460 0.811059 0.405529 − 0.914082i \(-0.367087\pi\)
0.405529 + 0.914082i \(0.367087\pi\)
\(54\) −1.17510 −0.159911
\(55\) 0.114210 0.0154001
\(56\) 0 0
\(57\) 0.0695722 0.00921506
\(58\) 1.74199 0.228734
\(59\) 2.29204 0.298399 0.149199 − 0.988807i \(-0.452330\pi\)
0.149199 + 0.988807i \(0.452330\pi\)
\(60\) 0.0608874 0.00786053
\(61\) −6.78792 −0.869105 −0.434552 − 0.900647i \(-0.643093\pi\)
−0.434552 + 0.900647i \(0.643093\pi\)
\(62\) −2.93477 −0.372716
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.94480 −0.241222
\(66\) 0.0728899 0.00897212
\(67\) 12.7442 1.55696 0.778478 − 0.627672i \(-0.215992\pi\)
0.778478 + 0.627672i \(0.215992\pi\)
\(68\) −4.61341 −0.559458
\(69\) −0.197126 −0.0237312
\(70\) 0 0
\(71\) 0.492118 0.0584037 0.0292018 − 0.999574i \(-0.490703\pi\)
0.0292018 + 0.999574i \(0.490703\pi\)
\(72\) −2.96114 −0.348974
\(73\) −5.99951 −0.702190 −0.351095 − 0.936340i \(-0.614191\pi\)
−0.351095 + 0.936340i \(0.614191\pi\)
\(74\) −6.37930 −0.741579
\(75\) −0.966825 −0.111639
\(76\) 0.352932 0.0404841
\(77\) 0 0
\(78\) −1.24118 −0.140536
\(79\) 0.666150 0.0749477 0.0374738 − 0.999298i \(-0.488069\pi\)
0.0374738 + 0.999298i \(0.488069\pi\)
\(80\) 0.308875 0.0345333
\(81\) 8.65178 0.961309
\(82\) 1.90213 0.210056
\(83\) 6.44785 0.707744 0.353872 − 0.935294i \(-0.384865\pi\)
0.353872 + 0.935294i \(0.384865\pi\)
\(84\) 0 0
\(85\) −1.42497 −0.154559
\(86\) 6.55252 0.706576
\(87\) 0.343391 0.0368154
\(88\) 0.369762 0.0394168
\(89\) −11.6221 −1.23194 −0.615970 − 0.787770i \(-0.711236\pi\)
−0.615970 + 0.787770i \(0.711236\pi\)
\(90\) −0.914622 −0.0964097
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −0.578520 −0.0599898
\(94\) −11.7347 −1.21034
\(95\) 0.109012 0.0111844
\(96\) 0.197126 0.0201191
\(97\) −4.96548 −0.504168 −0.252084 − 0.967705i \(-0.581116\pi\)
−0.252084 + 0.967705i \(0.581116\pi\)
\(98\) 0 0
\(99\) −1.09492 −0.110043
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 2254.2.a.x.1.3 4
7.3 odd 6 322.2.e.a.93.3 ✓ 8
7.5 odd 6 322.2.e.a.277.3 yes 8
7.6 odd 2 2254.2.a.z.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.e.a.93.3 ✓ 8 7.3 odd 6
322.2.e.a.277.3 yes 8 7.5 odd 6
2254.2.a.x.1.3 4 1.1 even 1 trivial
2254.2.a.z.1.2 4 7.6 odd 2