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Properties

Label 2.74.a.b.1.2
Level $2$
Weight $74$
Character 2.1
Self dual yes
Analytic conductor $67.497$
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] = [2,74,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, 74); 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, 74, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 74 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] 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: \(67.4967947474\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2 x^{3} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{14}\cdot 5^{5}\cdot 7^{2}\cdot 11 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.25645e13\) 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)
 
\(f(q)\) \(=\) \(q+6.87195e10 q^{2} -2.46573e16 q^{3} +4.72237e21 q^{4} -4.21515e24 q^{5} -1.69444e27 q^{6} -1.19188e31 q^{7} +3.24519e32 q^{8} -6.69772e34 q^{9} -2.89663e35 q^{10} -1.68299e38 q^{11} -1.16441e38 q^{12} +6.97112e40 q^{13} -8.19053e41 q^{14} +1.03934e41 q^{15} +2.23007e43 q^{16} +7.07443e44 q^{17} -4.60264e45 q^{18} +5.30898e46 q^{19} -1.99055e46 q^{20} +2.93885e47 q^{21} -1.15654e49 q^{22} +3.48043e49 q^{23} -8.00176e48 q^{24} -1.04102e51 q^{25} +4.79052e51 q^{26} +3.31795e51 q^{27} -5.62849e52 q^{28} +8.51468e52 q^{29} +7.14232e51 q^{30} +1.30244e54 q^{31} +1.53250e54 q^{32} +4.14981e54 q^{33} +4.86151e55 q^{34} +5.02395e55 q^{35} -3.16291e56 q^{36} +2.88717e57 q^{37} +3.64831e57 q^{38} -1.71889e57 q^{39} -1.36790e57 q^{40} +2.55851e58 q^{41} +2.01957e58 q^{42} -1.38982e59 q^{43} -7.94771e59 q^{44} +2.82319e59 q^{45} +2.39174e60 q^{46} -2.03016e61 q^{47} -5.49877e59 q^{48} +9.28358e61 q^{49} -7.15386e61 q^{50} -1.74436e61 q^{51} +3.29202e62 q^{52} +8.52520e61 q^{53} +2.28008e62 q^{54} +7.09407e62 q^{55} -3.86787e63 q^{56} -1.30905e63 q^{57} +5.85124e63 q^{58} -2.99429e64 q^{59} +4.90816e62 q^{60} +3.92014e64 q^{61} +8.95032e64 q^{62} +7.98287e65 q^{63} +1.05312e65 q^{64} -2.93843e65 q^{65} +2.85173e65 q^{66} +4.85230e66 q^{67} +3.34080e66 q^{68} -8.58182e65 q^{69} +3.45243e66 q^{70} +4.78682e67 q^{71} -2.17353e67 q^{72} +1.55085e68 q^{73} +1.98405e68 q^{74} +2.56689e67 q^{75} +2.50710e68 q^{76} +2.00592e69 q^{77} -1.18121e68 q^{78} -1.59580e69 q^{79} -9.40011e67 q^{80} +4.44486e69 q^{81} +1.75819e69 q^{82} -7.89354e69 q^{83} +1.38783e69 q^{84} -2.98198e69 q^{85} -9.55078e69 q^{86} -2.09949e69 q^{87} -5.46162e70 q^{88} +4.19168e69 q^{89} +1.94008e70 q^{90} -8.30873e71 q^{91} +1.64359e71 q^{92} -3.21148e70 q^{93} -1.39512e72 q^{94} -2.23782e71 q^{95} -3.77872e70 q^{96} -1.72372e72 q^{97} +6.37963e72 q^{98} +1.12722e73 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 274877906944 q^{2} + 30\!\cdots\!76 q^{3} + 18\!\cdots\!84 q^{4} - 78\!\cdots\!60 q^{5} + 20\!\cdots\!36 q^{6} + 36\!\cdots\!12 q^{7} + 12\!\cdots\!24 q^{8} + 12\!\cdots\!52 q^{9} - 53\!\cdots\!60 q^{10}+ \cdots + 28\!\cdots\!24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.87195e10 0.707107
\(3\) −2.46573e16 −0.0948462 −0.0474231 − 0.998875i \(-0.515101\pi\)
−0.0474231 + 0.998875i \(0.515101\pi\)
\(4\) 4.72237e21 0.500000
\(5\) −4.21515e24 −0.129541 −0.0647707 − 0.997900i \(-0.520632\pi\)
−0.0647707 + 0.997900i \(0.520632\pi\)
\(6\) −1.69444e27 −0.0670664
\(7\) −1.19188e31 −1.69884 −0.849422 − 0.527713i \(-0.823049\pi\)
−0.849422 + 0.527713i \(0.823049\pi\)
\(8\) 3.24519e32 0.353553
\(9\) −6.69772e34 −0.991004
\(10\) −2.89663e35 −0.0915995
\(11\) −1.68299e38 −1.64153 −0.820766 − 0.571265i \(-0.806453\pi\)
−0.820766 + 0.571265i \(0.806453\pi\)
\(12\) −1.16441e38 −0.0474231
\(13\) 6.97112e40 1.52887 0.764434 − 0.644702i \(-0.223018\pi\)
0.764434 + 0.644702i \(0.223018\pi\)
\(14\) −8.19053e41 −1.20126
\(15\) 1.03934e41 0.0122865
\(16\) 2.23007e43 0.250000
\(17\) 7.07443e44 0.867572 0.433786 − 0.901016i \(-0.357177\pi\)
0.433786 + 0.901016i \(0.357177\pi\)
\(18\) −4.60264e45 −0.700746
\(19\) 5.30898e46 1.12332 0.561662 − 0.827367i \(-0.310162\pi\)
0.561662 + 0.827367i \(0.310162\pi\)
\(20\) −1.99055e46 −0.0647707
\(21\) 2.93885e47 0.161129
\(22\) −1.15654e49 −1.16074
\(23\) 3.48043e49 0.689553 0.344776 − 0.938685i \(-0.387955\pi\)
0.344776 + 0.938685i \(0.387955\pi\)
\(24\) −8.00176e48 −0.0335332
\(25\) −1.04102e51 −0.983219
\(26\) 4.79052e51 1.08107
\(27\) 3.31795e51 0.188839
\(28\) −5.62849e52 −0.849422
\(29\) 8.51468e52 0.356978 0.178489 − 0.983942i \(-0.442879\pi\)
0.178489 + 0.983942i \(0.442879\pi\)
\(30\) 7.14232e51 0.00868787
\(31\) 1.30244e54 0.478693 0.239346 − 0.970934i \(-0.423067\pi\)
0.239346 + 0.970934i \(0.423067\pi\)
\(32\) 1.53250e54 0.176777
\(33\) 4.14981e54 0.155693
\(34\) 4.86151e55 0.613466
\(35\) 5.02395e55 0.220071
\(36\) −3.16291e56 −0.495502
\(37\) 2.88717e57 1.66383 0.831917 − 0.554900i \(-0.187244\pi\)
0.831917 + 0.554900i \(0.187244\pi\)
\(38\) 3.64831e57 0.794310
\(39\) −1.71889e57 −0.145007
\(40\) −1.36790e57 −0.0457998
\(41\) 2.55851e58 0.347837 0.173919 − 0.984760i \(-0.444357\pi\)
0.173919 + 0.984760i \(0.444357\pi\)
\(42\) 2.01957e58 0.113935
\(43\) −1.38982e59 −0.332170 −0.166085 − 0.986111i \(-0.553113\pi\)
−0.166085 + 0.986111i \(0.553113\pi\)
\(44\) −7.94771e59 −0.820766
\(45\) 2.82319e59 0.128376
\(46\) 2.39174e60 0.487587
\(47\) −2.03016e61 −1.88781 −0.943906 − 0.330214i \(-0.892879\pi\)
−0.943906 + 0.330214i \(0.892879\pi\)
\(48\) −5.49877e59 −0.0237116
\(49\) 9.28358e61 1.88607
\(50\) −7.15386e61 −0.695241
\(51\) −1.74436e61 −0.0822859
\(52\) 3.29202e62 0.764434
\(53\) 8.52520e61 0.0987724 0.0493862 − 0.998780i \(-0.484273\pi\)
0.0493862 + 0.998780i \(0.484273\pi\)
\(54\) 2.28008e62 0.133530
\(55\) 7.09407e62 0.212646
\(56\) −3.86787e63 −0.600632
\(57\) −1.30905e63 −0.106543
\(58\) 5.85124e63 0.252422
\(59\) −2.99429e64 −0.692143 −0.346071 − 0.938208i \(-0.612484\pi\)
−0.346071 + 0.938208i \(0.612484\pi\)
\(60\) 4.90816e62 0.00614325
\(61\) 3.92014e64 0.268388 0.134194 − 0.990955i \(-0.457156\pi\)
0.134194 + 0.990955i \(0.457156\pi\)
\(62\) 8.95032e64 0.338487
\(63\) 7.98287e65 1.68356
\(64\) 1.05312e65 0.125000
\(65\) −2.93843e65 −0.198052
\(66\) 2.85173e65 0.110092
\(67\) 4.85230e66 1.08198 0.540988 − 0.841030i \(-0.318050\pi\)
0.540988 + 0.841030i \(0.318050\pi\)
\(68\) 3.34080e66 0.433786
\(69\) −8.58182e65 −0.0654015
\(70\) 3.45243e66 0.155613
\(71\) 4.78682e67 1.28563 0.642817 − 0.766020i \(-0.277766\pi\)
0.642817 + 0.766020i \(0.277766\pi\)
\(72\) −2.17353e67 −0.350373
\(73\) 1.55085e68 1.51108 0.755538 − 0.655105i \(-0.227376\pi\)
0.755538 + 0.655105i \(0.227376\pi\)
\(74\) 1.98405e68 1.17651
\(75\) 2.56689e67 0.0932546
\(76\) 2.50710e68 0.561662
\(77\) 2.00592e69 2.78871
\(78\) −1.18121e68 −0.102536
\(79\) −1.59580e69 −0.870139 −0.435070 − 0.900397i \(-0.643276\pi\)
−0.435070 + 0.900397i \(0.643276\pi\)
\(80\) −9.40011e67 −0.0323853
\(81\) 4.44486e69 0.973093
\(82\) 1.75819e69 0.245958
\(83\) −7.89354e69 −0.709449 −0.354725 − 0.934971i \(-0.615425\pi\)
−0.354725 + 0.934971i \(0.615425\pi\)
\(84\) 1.38783e69 0.0805645
\(85\) −2.98198e69 −0.112386
\(86\) −9.55078e69 −0.234880
\(87\) −2.09949e69 −0.0338580
\(88\) −5.46162e70 −0.580369
\(89\) 4.19168e69 0.0294885 0.0147443 − 0.999891i \(-0.495307\pi\)
0.0147443 + 0.999891i \(0.495307\pi\)
\(90\) 1.94008e70 0.0907755
\(91\) −8.30873e71 −2.59731
\(92\) 1.64359e71 0.344776
\(93\) −3.21148e70 −0.0454022
\(94\) −1.39512e72 −1.33488
\(95\) −2.23782e71 −0.145517
\(96\) −3.77872e70 −0.0167666
\(97\) −1.72372e72 −0.523961 −0.261980 − 0.965073i \(-0.584376\pi\)
−0.261980 + 0.965073i \(0.584376\pi\)
\(98\) 6.37963e72 1.33366
\(99\) 1.12722e73 1.62676
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 2.74.a.b.1.2 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.74.a.b.1.2 ✓ 4 1.1 even 1 trivial