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Properties

Label 2254.2.a.z.1.1
Level $2254$
Weight $2$
Character 2254.1
Self dual yes
Analytic conductor $17.998$
Analytic rank $0$
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: \(0\)
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 1.1
Root \(0.396339\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.12676 q^{3} +1.00000 q^{4} +1.71616 q^{5} -1.12676 q^{6} +1.00000 q^{8} -1.73042 q^{9} +1.71616 q^{10} -3.64985 q^{11} -1.12676 q^{12} -3.79833 q^{13} -1.93369 q^{15} +1.00000 q^{16} +4.17700 q^{17} -1.73042 q^{18} +8.12271 q^{19} +1.71616 q^{20} -3.64985 q^{22} -1.00000 q^{23} -1.12676 q^{24} -2.05480 q^{25} -3.79833 q^{26} +5.33003 q^{27} +2.80694 q^{29} -1.93369 q^{30} +8.54301 q^{31} +1.00000 q^{32} +4.11250 q^{33} +4.17700 q^{34} -1.73042 q^{36} +8.93530 q^{37} +8.12271 q^{38} +4.27979 q^{39} +1.71616 q^{40} +10.0518 q^{41} +8.11069 q^{43} -3.64985 q^{44} -2.96967 q^{45} -1.00000 q^{46} -2.95137 q^{47} -1.12676 q^{48} -2.05480 q^{50} -4.70646 q^{51} -3.79833 q^{52} +3.05480 q^{53} +5.33003 q^{54} -6.26373 q^{55} -9.15232 q^{57} +2.80694 q^{58} +4.18902 q^{59} -1.93369 q^{60} -11.3156 q^{61} +8.54301 q^{62} +1.00000 q^{64} -6.51854 q^{65} +4.11250 q^{66} -13.2368 q^{67} +4.17700 q^{68} +1.12676 q^{69} +14.3054 q^{71} -1.73042 q^{72} -4.26097 q^{73} +8.93530 q^{74} +2.31526 q^{75} +8.12271 q^{76} +4.27979 q^{78} -13.4482 q^{79} +1.71616 q^{80} -0.814396 q^{81} +10.0518 q^{82} +9.43848 q^{83} +7.16839 q^{85} +8.11069 q^{86} -3.16274 q^{87} -3.64985 q^{88} +3.95837 q^{89} -2.96967 q^{90} -1.00000 q^{92} -9.62589 q^{93} -2.95137 q^{94} +13.9399 q^{95} -1.12676 q^{96} +0.121105 q^{97} +6.31577 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 3 q^{3} + 4 q^{4} + 7 q^{5} + 3 q^{6} + 4 q^{8} - q^{9} + 7 q^{10} + 2 q^{11} + 3 q^{12} + q^{13} + 9 q^{15} + 4 q^{16} + 5 q^{17} - q^{18} + 11 q^{19} + 7 q^{20} + 2 q^{22} - 4 q^{23}+ \cdots + 13 q^{99}+O(q^{100}) \) 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
\(3\) −1.12676 −0.650533 −0.325267 − 0.945622i \(-0.605454\pi\)
−0.325267 + 0.945622i \(0.605454\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.71616 0.767490 0.383745 − 0.923439i \(-0.374634\pi\)
0.383745 + 0.923439i \(0.374634\pi\)
\(6\) −1.12676 −0.459997
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −1.73042 −0.576806
\(10\) 1.71616 0.542697
\(11\) −3.64985 −1.10047 −0.550236 − 0.835009i \(-0.685462\pi\)
−0.550236 + 0.835009i \(0.685462\pi\)
\(12\) −1.12676 −0.325267
\(13\) −3.79833 −1.05347 −0.526733 − 0.850031i \(-0.676583\pi\)
−0.526733 + 0.850031i \(0.676583\pi\)
\(14\) 0 0
\(15\) −1.93369 −0.499278
\(16\) 1.00000 0.250000
\(17\) 4.17700 1.01307 0.506535 − 0.862219i \(-0.330926\pi\)
0.506535 + 0.862219i \(0.330926\pi\)
\(18\) −1.73042 −0.407864
\(19\) 8.12271 1.86348 0.931739 − 0.363129i \(-0.118292\pi\)
0.931739 + 0.363129i \(0.118292\pi\)
\(20\) 1.71616 0.383745
\(21\) 0 0
\(22\) −3.64985 −0.778151
\(23\) −1.00000 −0.208514
\(24\) −1.12676 −0.229998
\(25\) −2.05480 −0.410960
\(26\) −3.79833 −0.744914
\(27\) 5.33003 1.02577
\(28\) 0 0
\(29\) 2.80694 0.521235 0.260618 − 0.965442i \(-0.416074\pi\)
0.260618 + 0.965442i \(0.416074\pi\)
\(30\) −1.93369 −0.353043
\(31\) 8.54301 1.53437 0.767185 − 0.641426i \(-0.221657\pi\)
0.767185 + 0.641426i \(0.221657\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.11250 0.715894
\(34\) 4.17700 0.716349
\(35\) 0 0
\(36\) −1.73042 −0.288403
\(37\) 8.93530 1.46895 0.734477 − 0.678634i \(-0.237428\pi\)
0.734477 + 0.678634i \(0.237428\pi\)
\(38\) 8.12271 1.31768
\(39\) 4.27979 0.685315
\(40\) 1.71616 0.271349
\(41\) 10.0518 1.56983 0.784917 − 0.619601i \(-0.212706\pi\)
0.784917 + 0.619601i \(0.212706\pi\)
\(42\) 0 0
\(43\) 8.11069 1.23687 0.618434 − 0.785837i \(-0.287767\pi\)
0.618434 + 0.785837i \(0.287767\pi\)
\(44\) −3.64985 −0.550236
\(45\) −2.96967 −0.442693
\(46\) −1.00000 −0.147442
\(47\) −2.95137 −0.430501 −0.215250 − 0.976559i \(-0.569057\pi\)
−0.215250 + 0.976559i \(0.569057\pi\)
\(48\) −1.12676 −0.162633
\(49\) 0 0
\(50\) −2.05480 −0.290593
\(51\) −4.70646 −0.659036
\(52\) −3.79833 −0.526733
\(53\) 3.05480 0.419609 0.209804 − 0.977743i \(-0.432717\pi\)
0.209804 + 0.977743i \(0.432717\pi\)
\(54\) 5.33003 0.725326
\(55\) −6.26373 −0.844601
\(56\) 0 0
\(57\) −9.15232 −1.21225
\(58\) 2.80694 0.368569
\(59\) 4.18902 0.545363 0.272682 − 0.962104i \(-0.412089\pi\)
0.272682 + 0.962104i \(0.412089\pi\)
\(60\) −1.93369 −0.249639
\(61\) −11.3156 −1.44881 −0.724405 − 0.689375i \(-0.757885\pi\)
−0.724405 + 0.689375i \(0.757885\pi\)
\(62\) 8.54301 1.08496
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.51854 −0.808525
\(66\) 4.11250 0.506213
\(67\) −13.2368 −1.61713 −0.808567 − 0.588404i \(-0.799756\pi\)
−0.808567 + 0.588404i \(0.799756\pi\)
\(68\) 4.17700 0.506535
\(69\) 1.12676 0.135646
\(70\) 0 0
\(71\) 14.3054 1.69773 0.848867 − 0.528607i \(-0.177285\pi\)
0.848867 + 0.528607i \(0.177285\pi\)
\(72\) −1.73042 −0.203932
\(73\) −4.26097 −0.498709 −0.249355 − 0.968412i \(-0.580218\pi\)
−0.249355 + 0.968412i \(0.580218\pi\)
\(74\) 8.93530 1.03871
\(75\) 2.31526 0.267343
\(76\) 8.12271 0.931739
\(77\) 0 0
\(78\) 4.27979 0.484591
\(79\) −13.4482 −1.51304 −0.756519 − 0.653971i \(-0.773102\pi\)
−0.756519 + 0.653971i \(0.773102\pi\)
\(80\) 1.71616 0.191872
\(81\) −0.814396 −0.0904885
\(82\) 10.0518 1.11004
\(83\) 9.43848 1.03601 0.518004 − 0.855378i \(-0.326675\pi\)
0.518004 + 0.855378i \(0.326675\pi\)
\(84\) 0 0
\(85\) 7.16839 0.777521
\(86\) 8.11069 0.874598
\(87\) −3.16274 −0.339081
\(88\) −3.64985 −0.389076
\(89\) 3.95837 0.419586 0.209793 − 0.977746i \(-0.432721\pi\)
0.209793 + 0.977746i \(0.432721\pi\)
\(90\) −2.96967 −0.313031
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −9.62589 −0.998159
\(94\) −2.95137 −0.304410
\(95\) 13.9399 1.43020
\(96\) −1.12676 −0.114999
\(97\) 0.121105 0.0122964 0.00614819 − 0.999981i \(-0.498043\pi\)
0.00614819 + 0.999981i \(0.498043\pi\)
\(98\) 0 0
\(99\) 6.31577 0.634759
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.z.1.1 4
7.2 even 3 322.2.e.a.277.4 yes 8
7.4 even 3 322.2.e.a.93.4 ✓ 8
7.6 odd 2 2254.2.a.x.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.e.a.93.4 ✓ 8 7.4 even 3
322.2.e.a.277.4 yes 8 7.2 even 3
2254.2.a.x.1.4 4 7.6 odd 2
2254.2.a.z.1.1 4 1.1 even 1 trivial