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Properties

Label 2.88.a.a.1.1
Level $2$
Weight $88$
Character 2.1
Self dual yes
Analytic conductor $95.867$
Analytic rank $1$
Dimension $3$
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] = [2,88,Mod(1,2)] 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("2.1"); S:= CuspForms(chi, 88); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 88, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 88 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(95.8667262922\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 11473904362186221800312196301729x - 156905659743614387346100645850205598702591560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{9}\cdot 5^{3}\cdot 7\cdot 11\cdot 29 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.39413e15\) of defining polynomial
Character \(\chi\) \(=\) 2.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.79609e12 q^{2} -8.60735e20 q^{3} +7.73713e25 q^{4} -5.73526e28 q^{5} -7.57111e33 q^{6} +1.36940e36 q^{7} +6.80565e38 q^{8} +4.17607e41 q^{9} -5.04479e41 q^{10} -2.71122e45 q^{11} -6.65962e46 q^{12} +2.87976e47 q^{13} +1.20454e49 q^{14} +4.93654e49 q^{15} +5.98631e51 q^{16} +3.10251e53 q^{17} +3.67331e54 q^{18} -1.86576e55 q^{19} -4.43744e54 q^{20} -1.17869e57 q^{21} -2.38481e58 q^{22} +2.58977e59 q^{23} -5.85786e59 q^{24} -6.45906e60 q^{25} +2.53306e60 q^{26} -8.12096e61 q^{27} +1.05952e62 q^{28} +2.68328e63 q^{29} +4.34223e62 q^{30} +6.88153e64 q^{31} +5.26561e64 q^{32} +2.33364e66 q^{33} +2.72900e66 q^{34} -7.85389e64 q^{35} +3.23108e67 q^{36} +2.93207e68 q^{37} -1.64114e68 q^{38} -2.47871e68 q^{39} -3.90322e67 q^{40} -2.76945e70 q^{41} -1.03679e70 q^{42} -2.01972e70 q^{43} -2.09770e71 q^{44} -2.39509e70 q^{45} +2.27798e72 q^{46} +5.35314e72 q^{47} -5.15263e72 q^{48} -3.15081e73 q^{49} -5.68145e73 q^{50} -2.67044e74 q^{51} +2.22811e73 q^{52} -1.25785e75 q^{53} -7.14327e74 q^{54} +1.55495e74 q^{55} +9.31968e74 q^{56} +1.60593e76 q^{57} +2.36024e76 q^{58} -4.72394e76 q^{59} +3.81946e75 q^{60} -2.91100e77 q^{61} +6.05306e77 q^{62} +5.71872e77 q^{63} +4.63168e77 q^{64} -1.65162e76 q^{65} +2.05269e79 q^{66} +1.58265e78 q^{67} +2.40045e79 q^{68} -2.22910e80 q^{69} -6.90835e77 q^{70} -4.85042e80 q^{71} +2.84209e80 q^{72} -8.17817e80 q^{73} +2.57908e81 q^{74} +5.55954e81 q^{75} -1.44357e81 q^{76} -3.71275e81 q^{77} -2.18030e81 q^{78} +1.04086e82 q^{79} -3.43331e80 q^{80} -6.50948e82 q^{81} -2.43603e83 q^{82} +4.94033e83 q^{83} -9.11970e82 q^{84} -1.77937e82 q^{85} -1.77656e83 q^{86} -2.30960e84 q^{87} -1.84516e84 q^{88} -1.60988e84 q^{89} -2.10674e83 q^{90} +3.94355e83 q^{91} +2.00374e85 q^{92} -5.92317e85 q^{93} +4.70867e85 q^{94} +1.07006e84 q^{95} -4.53230e85 q^{96} -3.52888e86 q^{97} -2.77148e86 q^{98} -1.13222e87 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 26388279066624 q^{2} - 32\!\cdots\!16 q^{3} + 23\!\cdots\!92 q^{4} + 11\!\cdots\!50 q^{5} - 28\!\cdots\!28 q^{6} - 28\!\cdots\!88 q^{7} + 20\!\cdots\!36 q^{8} + 58\!\cdots\!91 q^{9} + 98\!\cdots\!00 q^{10}+ \cdots - 21\!\cdots\!68 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.79609e12 0.707107
\(3\) −8.60735e20 −1.51389 −0.756946 − 0.653477i \(-0.773309\pi\)
−0.756946 + 0.653477i \(0.773309\pi\)
\(4\) 7.73713e25 0.500000
\(5\) −5.73526e28 −0.0225610 −0.0112805 − 0.999936i \(-0.503591\pi\)
−0.0112805 + 0.999936i \(0.503591\pi\)
\(6\) −7.57111e33 −1.07048
\(7\) 1.36940e36 0.237010 0.118505 − 0.992953i \(-0.462190\pi\)
0.118505 + 0.992953i \(0.462190\pi\)
\(8\) 6.80565e38 0.353553
\(9\) 4.17607e41 1.29187
\(10\) −5.04479e41 −0.0159530
\(11\) −2.71122e45 −1.35701 −0.678504 − 0.734596i \(-0.737372\pi\)
−0.678504 + 0.734596i \(0.737372\pi\)
\(12\) −6.65962e46 −0.756946
\(13\) 2.87976e47 0.100652 0.0503258 − 0.998733i \(-0.483974\pi\)
0.0503258 + 0.998733i \(0.483974\pi\)
\(14\) 1.20454e49 0.167591
\(15\) 4.93654e49 0.0341549
\(16\) 5.98631e51 0.250000
\(17\) 3.10251e53 0.927225 0.463612 − 0.886038i \(-0.346553\pi\)
0.463612 + 0.886038i \(0.346553\pi\)
\(18\) 3.67331e54 0.913490
\(19\) −1.86576e55 −0.441647 −0.220823 − 0.975314i \(-0.570874\pi\)
−0.220823 + 0.975314i \(0.570874\pi\)
\(20\) −4.43744e54 −0.0112805
\(21\) −1.17869e57 −0.358808
\(22\) −2.38481e58 −0.959550
\(23\) 2.58977e59 1.50695 0.753477 − 0.657474i \(-0.228375\pi\)
0.753477 + 0.657474i \(0.228375\pi\)
\(24\) −5.85786e59 −0.535242
\(25\) −6.45906e60 −0.999491
\(26\) 2.53306e60 0.0711714
\(27\) −8.12096e61 −0.441859
\(28\) 1.05952e62 0.118505
\(29\) 2.68328e63 0.652159 0.326080 − 0.945342i \(-0.394272\pi\)
0.326080 + 0.945342i \(0.394272\pi\)
\(30\) 4.34223e62 0.0241512
\(31\) 6.88153e64 0.919287 0.459644 − 0.888103i \(-0.347977\pi\)
0.459644 + 0.888103i \(0.347977\pi\)
\(32\) 5.26561e64 0.176777
\(33\) 2.33364e66 2.05437
\(34\) 2.72900e66 0.655647
\(35\) −7.85389e64 −0.00534718
\(36\) 3.23108e67 0.645935
\(37\) 2.93207e68 1.77992 0.889959 − 0.456041i \(-0.150733\pi\)
0.889959 + 0.456041i \(0.150733\pi\)
\(38\) −1.64114e68 −0.312291
\(39\) −2.47871e68 −0.152376
\(40\) −3.90322e67 −0.00797651
\(41\) −2.76945e70 −1.93328 −0.966642 − 0.256132i \(-0.917552\pi\)
−0.966642 + 0.256132i \(0.917552\pi\)
\(42\) −1.03679e70 −0.253715
\(43\) −2.01972e70 −0.177588 −0.0887938 − 0.996050i \(-0.528301\pi\)
−0.0887938 + 0.996050i \(0.528301\pi\)
\(44\) −2.09770e71 −0.678504
\(45\) −2.39509e70 −0.0291458
\(46\) 2.27798e72 1.06558
\(47\) 5.35314e72 0.982543 0.491272 − 0.871006i \(-0.336532\pi\)
0.491272 + 0.871006i \(0.336532\pi\)
\(48\) −5.15263e72 −0.378473
\(49\) −3.15081e73 −0.943826
\(50\) −5.68145e73 −0.706747
\(51\) −2.67044e74 −1.40372
\(52\) 2.22811e73 0.0503258
\(53\) −1.25785e75 −1.24059 −0.620295 − 0.784369i \(-0.712987\pi\)
−0.620295 + 0.784369i \(0.712987\pi\)
\(54\) −7.14327e74 −0.312441
\(55\) 1.55495e74 0.0306155
\(56\) 9.31968e74 0.0837957
\(57\) 1.60593e76 0.668606
\(58\) 2.36024e76 0.461146
\(59\) −4.72394e76 −0.438775 −0.219387 − 0.975638i \(-0.570406\pi\)
−0.219387 + 0.975638i \(0.570406\pi\)
\(60\) 3.81946e75 0.0170774
\(61\) −2.91100e77 −0.634153 −0.317077 − 0.948400i \(-0.602701\pi\)
−0.317077 + 0.948400i \(0.602701\pi\)
\(62\) 6.05306e77 0.650034
\(63\) 5.71872e77 0.306186
\(64\) 4.63168e77 0.125000
\(65\) −1.65162e76 −0.00227080
\(66\) 2.05269e79 1.45266
\(67\) 1.58265e78 0.0582277 0.0291138 − 0.999576i \(-0.490731\pi\)
0.0291138 + 0.999576i \(0.490731\pi\)
\(68\) 2.40045e79 0.463612
\(69\) −2.22910e80 −2.28137
\(70\) −6.90835e77 −0.00378103
\(71\) −4.85042e80 −1.43232 −0.716162 − 0.697934i \(-0.754103\pi\)
−0.716162 + 0.697934i \(0.754103\pi\)
\(72\) 2.84209e80 0.456745
\(73\) −8.17817e80 −0.721293 −0.360647 − 0.932703i \(-0.617444\pi\)
−0.360647 + 0.932703i \(0.617444\pi\)
\(74\) 2.57908e81 1.25859
\(75\) 5.55954e81 1.51312
\(76\) −1.44357e81 −0.220823
\(77\) −3.71275e81 −0.321625
\(78\) −2.18030e81 −0.107746
\(79\) 1.04086e82 0.295537 0.147769 − 0.989022i \(-0.452791\pi\)
0.147769 + 0.989022i \(0.452791\pi\)
\(80\) −3.43331e80 −0.00564025
\(81\) −6.50948e82 −0.622943
\(82\) −2.43603e83 −1.36704
\(83\) 4.94033e83 1.63628 0.818142 − 0.575016i \(-0.195004\pi\)
0.818142 + 0.575016i \(0.195004\pi\)
\(84\) −9.11970e82 −0.179404
\(85\) −1.77937e82 −0.0209191
\(86\) −1.77656e83 −0.125573
\(87\) −2.30960e84 −0.987299
\(88\) −1.84516e84 −0.479775
\(89\) −1.60988e84 −0.256051 −0.128026 − 0.991771i \(-0.540864\pi\)
−0.128026 + 0.991771i \(0.540864\pi\)
\(90\) −2.10674e83 −0.0206092
\(91\) 3.94355e83 0.0238554
\(92\) 2.00374e85 0.753477
\(93\) −5.92317e85 −1.39170
\(94\) 4.70867e85 0.694763
\(95\) 1.07006e84 0.00996399
\(96\) −4.53230e85 −0.267621
\(97\) −3.52888e86 −1.32760 −0.663799 − 0.747911i \(-0.731057\pi\)
−0.663799 + 0.747911i \(0.731057\pi\)
\(98\) −2.77148e86 −0.667386
\(99\) −1.13222e87 −1.75308
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 2.88.a.a.1.1 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.88.a.a.1.1 ✓ 3 1.1 even 1 trivial