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Properties

Label 2254.2.a.x.1.1
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 1.1
Root \(2.06150\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.57641 q^{3} +1.00000 q^{4} -1.32664 q^{5} -2.57641 q^{6} +1.00000 q^{8} +3.63791 q^{9} -1.32664 q^{10} +2.09133 q^{11} -2.57641 q^{12} -3.11142 q^{13} +3.41797 q^{15} +1.00000 q^{16} +6.94919 q^{17} +3.63791 q^{18} -7.76653 q^{19} -1.32664 q^{20} +2.09133 q^{22} -1.00000 q^{23} -2.57641 q^{24} -3.24003 q^{25} -3.11142 q^{26} -1.64353 q^{27} +1.15845 q^{29} +3.41797 q^{30} +8.71388 q^{31} +1.00000 q^{32} -5.38814 q^{33} +6.94919 q^{34} +3.63791 q^{36} -3.30944 q^{37} -7.76653 q^{38} +8.01631 q^{39} -1.32664 q^{40} +4.26425 q^{41} -8.36716 q^{43} +2.09133 q^{44} -4.82619 q^{45} -1.00000 q^{46} +3.48130 q^{47} -2.57641 q^{48} -3.24003 q^{50} -17.9040 q^{51} -3.11142 q^{52} +4.24003 q^{53} -1.64353 q^{54} -2.77444 q^{55} +20.0098 q^{57} +1.15845 q^{58} -9.18450 q^{59} +3.41797 q^{60} -12.0387 q^{61} +8.71388 q^{62} +1.00000 q^{64} +4.12772 q^{65} -5.38814 q^{66} -7.26319 q^{67} +6.94919 q^{68} +2.57641 q^{69} -7.41708 q^{71} +3.63791 q^{72} +4.36805 q^{73} -3.30944 q^{74} +8.34767 q^{75} -7.76653 q^{76} +8.01631 q^{78} -0.797250 q^{79} -1.32664 q^{80} -6.67932 q^{81} +4.26425 q^{82} -10.3746 q^{83} -9.21906 q^{85} -8.36716 q^{86} -2.98464 q^{87} +2.09133 q^{88} -16.6426 q^{89} -4.82619 q^{90} -1.00000 q^{92} -22.4506 q^{93} +3.48130 q^{94} +10.3034 q^{95} -2.57641 q^{96} -6.65800 q^{97} +7.60808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 3 q^{3} + 4 q^{4} - 7 q^{5} - 3 q^{6} + 4 q^{8} - q^{9} - 7 q^{10} + 2 q^{11} - 3 q^{12} - q^{13} + 9 q^{15} + 4 q^{16} - 5 q^{17} - q^{18} - 11 q^{19} - 7 q^{20} + 2 q^{22} - 4 q^{23}+ \cdots + 13 q^{99}+O(q^{100}) \) 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
\(3\) −2.57641 −1.48749 −0.743747 − 0.668461i \(-0.766953\pi\)
−0.743747 + 0.668461i \(0.766953\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.32664 −0.593290 −0.296645 − 0.954988i \(-0.595868\pi\)
−0.296645 + 0.954988i \(0.595868\pi\)
\(6\) −2.57641 −1.05182
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 3.63791 1.21264
\(10\) −1.32664 −0.419520
\(11\) 2.09133 0.630560 0.315280 − 0.948999i \(-0.397902\pi\)
0.315280 + 0.948999i \(0.397902\pi\)
\(12\) −2.57641 −0.743747
\(13\) −3.11142 −0.862952 −0.431476 − 0.902124i \(-0.642007\pi\)
−0.431476 + 0.902124i \(0.642007\pi\)
\(14\) 0 0
\(15\) 3.41797 0.882516
\(16\) 1.00000 0.250000
\(17\) 6.94919 1.68543 0.842713 − 0.538363i \(-0.180957\pi\)
0.842713 + 0.538363i \(0.180957\pi\)
\(18\) 3.63791 0.857464
\(19\) −7.76653 −1.78176 −0.890882 − 0.454235i \(-0.849913\pi\)
−0.890882 + 0.454235i \(0.849913\pi\)
\(20\) −1.32664 −0.296645
\(21\) 0 0
\(22\) 2.09133 0.445873
\(23\) −1.00000 −0.208514
\(24\) −2.57641 −0.525908
\(25\) −3.24003 −0.648007
\(26\) −3.11142 −0.610199
\(27\) −1.64353 −0.316298
\(28\) 0 0
\(29\) 1.15845 0.215118 0.107559 − 0.994199i \(-0.465697\pi\)
0.107559 + 0.994199i \(0.465697\pi\)
\(30\) 3.41797 0.624033
\(31\) 8.71388 1.56506 0.782530 − 0.622613i \(-0.213929\pi\)
0.782530 + 0.622613i \(0.213929\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.38814 −0.937954
\(34\) 6.94919 1.19178
\(35\) 0 0
\(36\) 3.63791 0.606319
\(37\) −3.30944 −0.544069 −0.272034 − 0.962288i \(-0.587696\pi\)
−0.272034 + 0.962288i \(0.587696\pi\)
\(38\) −7.76653 −1.25990
\(39\) 8.01631 1.28364
\(40\) −1.32664 −0.209760
\(41\) 4.26425 0.665964 0.332982 − 0.942933i \(-0.391945\pi\)
0.332982 + 0.942933i \(0.391945\pi\)
\(42\) 0 0
\(43\) −8.36716 −1.27598 −0.637990 − 0.770045i \(-0.720234\pi\)
−0.637990 + 0.770045i \(0.720234\pi\)
\(44\) 2.09133 0.315280
\(45\) −4.82619 −0.719446
\(46\) −1.00000 −0.147442
\(47\) 3.48130 0.507800 0.253900 − 0.967230i \(-0.418287\pi\)
0.253900 + 0.967230i \(0.418287\pi\)
\(48\) −2.57641 −0.371873
\(49\) 0 0
\(50\) −3.24003 −0.458210
\(51\) −17.9040 −2.50706
\(52\) −3.11142 −0.431476
\(53\) 4.24003 0.582413 0.291207 − 0.956660i \(-0.405943\pi\)
0.291207 + 0.956660i \(0.405943\pi\)
\(54\) −1.64353 −0.223656
\(55\) −2.77444 −0.374105
\(56\) 0 0
\(57\) 20.0098 2.65036
\(58\) 1.15845 0.152111
\(59\) −9.18450 −1.19572 −0.597860 − 0.801601i \(-0.703982\pi\)
−0.597860 + 0.801601i \(0.703982\pi\)
\(60\) 3.41797 0.441258
\(61\) −12.0387 −1.54140 −0.770698 − 0.637201i \(-0.780092\pi\)
−0.770698 + 0.637201i \(0.780092\pi\)
\(62\) 8.71388 1.10666
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.12772 0.511981
\(66\) −5.38814 −0.663234
\(67\) −7.26319 −0.887340 −0.443670 − 0.896190i \(-0.646324\pi\)
−0.443670 + 0.896190i \(0.646324\pi\)
\(68\) 6.94919 0.842713
\(69\) 2.57641 0.310164
\(70\) 0 0
\(71\) −7.41708 −0.880245 −0.440123 − 0.897938i \(-0.645065\pi\)
−0.440123 + 0.897938i \(0.645065\pi\)
\(72\) 3.63791 0.428732
\(73\) 4.36805 0.511241 0.255621 − 0.966777i \(-0.417720\pi\)
0.255621 + 0.966777i \(0.417720\pi\)
\(74\) −3.30944 −0.384715
\(75\) 8.34767 0.963906
\(76\) −7.76653 −0.890882
\(77\) 0 0
\(78\) 8.01631 0.907668
\(79\) −0.797250 −0.0896977 −0.0448488 − 0.998994i \(-0.514281\pi\)
−0.0448488 + 0.998994i \(0.514281\pi\)
\(80\) −1.32664 −0.148323
\(81\) −6.67932 −0.742147
\(82\) 4.26425 0.470907
\(83\) −10.3746 −1.13876 −0.569381 − 0.822074i \(-0.692817\pi\)
−0.569381 + 0.822074i \(0.692817\pi\)
\(84\) 0 0
\(85\) −9.21906 −0.999947
\(86\) −8.36716 −0.902254
\(87\) −2.98464 −0.319987
\(88\) 2.09133 0.222937
\(89\) −16.6426 −1.76412 −0.882058 − 0.471140i \(-0.843842\pi\)
−0.882058 + 0.471140i \(0.843842\pi\)
\(90\) −4.82619 −0.508725
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −22.4506 −2.32802
\(94\) 3.48130 0.359069
\(95\) 10.3034 1.05710
\(96\) −2.57641 −0.262954
\(97\) −6.65800 −0.676018 −0.338009 − 0.941143i \(-0.609753\pi\)
−0.338009 + 0.941143i \(0.609753\pi\)
\(98\) 0 0
\(99\) 7.60808 0.764641
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.1.1 4
7.3 odd 6 322.2.e.a.93.1 ✓ 8
7.5 odd 6 322.2.e.a.277.1 yes 8
7.6 odd 2 2254.2.a.z.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.e.a.93.1 ✓ 8 7.3 odd 6
322.2.e.a.277.1 yes 8 7.5 odd 6
2254.2.a.x.1.1 4 1.1 even 1 trivial
2254.2.a.z.1.4 4 7.6 odd 2