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Properties

Label 2254.2.a.y.1.3
Level $2254$
Weight $2$
Character 2254.1
Self dual yes
Analytic conductor $17.998$
Analytic rank $0$
Dimension $4$
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] = [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 = [4,4,0,4,0,0,0,4,8,0,8] 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: \(\Q(\sqrt{10 +2 \sqrt{17}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.13578\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.936426 q^{3} +1.00000 q^{4} +3.33513 q^{5} +0.936426 q^{6} +1.00000 q^{8} -2.12311 q^{9} +3.33513 q^{10} +2.00000 q^{11} +0.936426 q^{12} +1.87285 q^{13} +3.12311 q^{15} +1.00000 q^{16} -0.936426 q^{17} -2.12311 q^{18} +2.39871 q^{19} +3.33513 q^{20} +2.00000 q^{22} +1.00000 q^{23} +0.936426 q^{24} +6.12311 q^{25} +1.87285 q^{26} -4.79741 q^{27} -1.12311 q^{29} +3.12311 q^{30} -0.936426 q^{31} +1.00000 q^{32} +1.87285 q^{33} -0.936426 q^{34} -2.12311 q^{36} +3.12311 q^{37} +2.39871 q^{38} +1.75379 q^{39} +3.33513 q^{40} -4.79741 q^{41} -1.12311 q^{43} +2.00000 q^{44} -7.08084 q^{45} +1.00000 q^{46} +0.936426 q^{47} +0.936426 q^{48} +6.12311 q^{50} -0.876894 q^{51} +1.87285 q^{52} +13.3693 q^{53} -4.79741 q^{54} +6.67026 q^{55} +2.24621 q^{57} -1.12311 q^{58} +3.86098 q^{59} +3.12311 q^{60} -11.8782 q^{61} -0.936426 q^{62} +1.00000 q^{64} +6.24621 q^{65} +1.87285 q^{66} +10.0000 q^{67} -0.936426 q^{68} +0.936426 q^{69} -2.12311 q^{72} -10.4160 q^{73} +3.12311 q^{74} +5.73384 q^{75} +2.39871 q^{76} +1.75379 q^{78} -13.3693 q^{79} +3.33513 q^{80} +1.87689 q^{81} -4.79741 q^{82} -4.27156 q^{83} -3.12311 q^{85} -1.12311 q^{86} -1.05171 q^{87} +2.00000 q^{88} -0.936426 q^{89} -7.08084 q^{90} +1.00000 q^{92} -0.876894 q^{93} +0.936426 q^{94} +8.00000 q^{95} +0.936426 q^{96} +12.4041 q^{97} -4.24621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 8 q^{9} + 8 q^{11} - 4 q^{15} + 4 q^{16} + 8 q^{18} + 8 q^{22} + 4 q^{23} + 8 q^{25} + 12 q^{29} - 4 q^{30} + 4 q^{32} + 8 q^{36} - 4 q^{37} + 40 q^{39} + 12 q^{43} + 8 q^{44}+ \cdots + 16 q^{99}+O(q^{100}) \) 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
\(3\) 0.936426 0.540646 0.270323 − 0.962770i \(-0.412870\pi\)
0.270323 + 0.962770i \(0.412870\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.33513 1.49152 0.745758 − 0.666217i \(-0.232087\pi\)
0.745758 + 0.666217i \(0.232087\pi\)
\(6\) 0.936426 0.382294
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −2.12311 −0.707702
\(10\) 3.33513 1.05466
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0.936426 0.270323
\(13\) 1.87285 0.519436 0.259718 − 0.965685i \(-0.416370\pi\)
0.259718 + 0.965685i \(0.416370\pi\)
\(14\) 0 0
\(15\) 3.12311 0.806382
\(16\) 1.00000 0.250000
\(17\) −0.936426 −0.227117 −0.113558 − 0.993531i \(-0.536225\pi\)
−0.113558 + 0.993531i \(0.536225\pi\)
\(18\) −2.12311 −0.500421
\(19\) 2.39871 0.550301 0.275150 − 0.961401i \(-0.411272\pi\)
0.275150 + 0.961401i \(0.411272\pi\)
\(20\) 3.33513 0.745758
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) 0.936426 0.191147
\(25\) 6.12311 1.22462
\(26\) 1.87285 0.367297
\(27\) −4.79741 −0.923262
\(28\) 0 0
\(29\) −1.12311 −0.208555 −0.104278 − 0.994548i \(-0.533253\pi\)
−0.104278 + 0.994548i \(0.533253\pi\)
\(30\) 3.12311 0.570198
\(31\) −0.936426 −0.168187 −0.0840936 − 0.996458i \(-0.526799\pi\)
−0.0840936 + 0.996458i \(0.526799\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.87285 0.326022
\(34\) −0.936426 −0.160596
\(35\) 0 0
\(36\) −2.12311 −0.353851
\(37\) 3.12311 0.513435 0.256718 − 0.966486i \(-0.417359\pi\)
0.256718 + 0.966486i \(0.417359\pi\)
\(38\) 2.39871 0.389121
\(39\) 1.75379 0.280831
\(40\) 3.33513 0.527331
\(41\) −4.79741 −0.749230 −0.374615 − 0.927181i \(-0.622225\pi\)
−0.374615 + 0.927181i \(0.622225\pi\)
\(42\) 0 0
\(43\) −1.12311 −0.171272 −0.0856360 − 0.996326i \(-0.527292\pi\)
−0.0856360 + 0.996326i \(0.527292\pi\)
\(44\) 2.00000 0.301511
\(45\) −7.08084 −1.05555
\(46\) 1.00000 0.147442
\(47\) 0.936426 0.136592 0.0682959 − 0.997665i \(-0.478244\pi\)
0.0682959 + 0.997665i \(0.478244\pi\)
\(48\) 0.936426 0.135162
\(49\) 0 0
\(50\) 6.12311 0.865938
\(51\) −0.876894 −0.122790
\(52\) 1.87285 0.259718
\(53\) 13.3693 1.83642 0.918208 − 0.396098i \(-0.129636\pi\)
0.918208 + 0.396098i \(0.129636\pi\)
\(54\) −4.79741 −0.652845
\(55\) 6.67026 0.899418
\(56\) 0 0
\(57\) 2.24621 0.297518
\(58\) −1.12311 −0.147471
\(59\) 3.86098 0.502657 0.251329 − 0.967902i \(-0.419132\pi\)
0.251329 + 0.967902i \(0.419132\pi\)
\(60\) 3.12311 0.403191
\(61\) −11.8782 −1.52085 −0.760427 − 0.649423i \(-0.775010\pi\)
−0.760427 + 0.649423i \(0.775010\pi\)
\(62\) −0.936426 −0.118926
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.24621 0.774747
\(66\) 1.87285 0.230532
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) −0.936426 −0.113558
\(69\) 0.936426 0.112732
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −2.12311 −0.250210
\(73\) −10.4160 −1.21910 −0.609549 − 0.792749i \(-0.708649\pi\)
−0.609549 + 0.792749i \(0.708649\pi\)
\(74\) 3.12311 0.363054
\(75\) 5.73384 0.662087
\(76\) 2.39871 0.275150
\(77\) 0 0
\(78\) 1.75379 0.198577
\(79\) −13.3693 −1.50417 −0.752083 − 0.659069i \(-0.770951\pi\)
−0.752083 + 0.659069i \(0.770951\pi\)
\(80\) 3.33513 0.372879
\(81\) 1.87689 0.208544
\(82\) −4.79741 −0.529785
\(83\) −4.27156 −0.468864 −0.234432 − 0.972132i \(-0.575323\pi\)
−0.234432 + 0.972132i \(0.575323\pi\)
\(84\) 0 0
\(85\) −3.12311 −0.338748
\(86\) −1.12311 −0.121108
\(87\) −1.05171 −0.112755
\(88\) 2.00000 0.213201
\(89\) −0.936426 −0.0992610 −0.0496305 − 0.998768i \(-0.515804\pi\)
−0.0496305 + 0.998768i \(0.515804\pi\)
\(90\) −7.08084 −0.746386
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −0.876894 −0.0909297
\(94\) 0.936426 0.0965850
\(95\) 8.00000 0.820783
\(96\) 0.936426 0.0955736
\(97\) 12.4041 1.25945 0.629723 − 0.776820i \(-0.283168\pi\)
0.629723 + 0.776820i \(0.283168\pi\)
\(98\) 0 0
\(99\) −4.24621 −0.426760
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.y.1.3 yes 4
7.6 odd 2 inner 2254.2.a.y.1.2 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2254.2.a.y.1.2 ✓ 4 7.6 odd 2 inner
2254.2.a.y.1.3 yes 4 1.1 even 1 trivial