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Properties

Label 2.72.a.b.1.2
Level $2$
Weight $72$
Character 2.1
Self dual yes
Analytic conductor $63.849$
Analytic rank $0$
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] = [2,72,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, 72); 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, 72, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 72 \)
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: \(63.8492321122\)
Analytic rank: \(0\)
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: \( x^{3} - 71437129084791448795855051x - 180952663419752575975880178936282470070 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{6}\cdot 5^{3}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(9.51117e12\) 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)
 
\(f(q)\) \(=\) \(q-3.43597e10 q^{2} +4.72491e16 q^{3} +1.18059e21 q^{4} -7.30618e24 q^{5} -1.62347e27 q^{6} -2.33219e28 q^{7} -4.05648e31 q^{8} -5.27699e33 q^{9} +2.51038e35 q^{10} -1.13918e37 q^{11} +5.57819e37 q^{12} -4.82382e39 q^{13} +8.01335e38 q^{14} -3.45211e41 q^{15} +1.39380e42 q^{16} +5.37011e43 q^{17} +1.81316e44 q^{18} +3.28254e44 q^{19} -8.62561e45 q^{20} -1.10194e45 q^{21} +3.91420e47 q^{22} +1.06232e48 q^{23} -1.91665e48 q^{24} +1.10286e49 q^{25} +1.65745e50 q^{26} -6.04149e50 q^{27} -2.75337e49 q^{28} -1.14189e52 q^{29} +1.18613e52 q^{30} +1.80143e52 q^{31} -4.78905e52 q^{32} -5.38254e53 q^{33} -1.84515e54 q^{34} +1.70394e53 q^{35} -6.22996e54 q^{36} -4.58287e55 q^{37} -1.12787e55 q^{38} -2.27921e56 q^{39} +2.96374e56 q^{40} +2.53761e57 q^{41} +3.78624e55 q^{42} +2.43971e57 q^{43} -1.34491e58 q^{44} +3.85546e58 q^{45} -3.65011e58 q^{46} +5.94323e58 q^{47} +6.58557e58 q^{48} -1.00398e60 q^{49} -3.78939e59 q^{50} +2.53733e60 q^{51} -5.69496e60 q^{52} -6.61258e60 q^{53} +2.07584e61 q^{54} +8.32306e61 q^{55} +9.46049e59 q^{56} +1.55097e61 q^{57} +3.92350e62 q^{58} -1.13910e63 q^{59} -4.07553e62 q^{60} +3.97263e63 q^{61} -6.18965e62 q^{62} +1.23069e62 q^{63} +1.64550e63 q^{64} +3.52437e64 q^{65} +1.84943e64 q^{66} +1.05200e65 q^{67} +6.33990e64 q^{68} +5.01938e64 q^{69} -5.85470e63 q^{70} +6.61153e65 q^{71} +2.14060e65 q^{72} +1.33268e66 q^{73} +1.57466e66 q^{74} +5.21091e65 q^{75} +3.87534e65 q^{76} +2.65679e65 q^{77} +7.83131e66 q^{78} +2.53823e67 q^{79} -1.01833e67 q^{80} +1.10818e67 q^{81} -8.71917e67 q^{82} -2.11949e68 q^{83} -1.30094e66 q^{84} -3.92350e68 q^{85} -8.38278e67 q^{86} -5.39532e68 q^{87} +4.62107e68 q^{88} -4.02156e66 q^{89} -1.32473e69 q^{90} +1.12501e68 q^{91} +1.25417e69 q^{92} +8.51158e68 q^{93} -2.04208e69 q^{94} -2.39828e69 q^{95} -2.26278e69 q^{96} +3.31804e70 q^{97} +3.44965e70 q^{98} +6.01144e70 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 103079215104 q^{2} + 23\!\cdots\!36 q^{3} + 35\!\cdots\!72 q^{4} + 47\!\cdots\!50 q^{5} - 81\!\cdots\!48 q^{6} - 68\!\cdots\!72 q^{7} - 12\!\cdots\!96 q^{8} - 45\!\cdots\!09 q^{9} - 16\!\cdots\!00 q^{10}+ \cdots + 12\!\cdots\!32 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.43597e10 −0.707107
\(3\) 4.72491e16 0.545242 0.272621 − 0.962121i \(-0.412109\pi\)
0.272621 + 0.962121i \(0.412109\pi\)
\(4\) 1.18059e21 0.500000
\(5\) −7.30618e24 −1.12268 −0.561339 − 0.827586i \(-0.689714\pi\)
−0.561339 + 0.827586i \(0.689714\pi\)
\(6\) −1.62347e27 −0.385544
\(7\) −2.33219e28 −0.0232693 −0.0116347 − 0.999932i \(-0.503704\pi\)
−0.0116347 + 0.999932i \(0.503704\pi\)
\(8\) −4.05648e31 −0.353553
\(9\) −5.27699e33 −0.702711
\(10\) 2.51038e35 0.793853
\(11\) −1.13918e37 −1.22223 −0.611115 − 0.791542i \(-0.709279\pi\)
−0.611115 + 0.791542i \(0.709279\pi\)
\(12\) 5.57819e37 0.272621
\(13\) −4.82382e39 −1.37531 −0.687657 − 0.726036i \(-0.741361\pi\)
−0.687657 + 0.726036i \(0.741361\pi\)
\(14\) 8.01335e38 0.0164539
\(15\) −3.45211e41 −0.612131
\(16\) 1.39380e42 0.250000
\(17\) 5.37011e43 1.11956 0.559779 − 0.828642i \(-0.310886\pi\)
0.559779 + 0.828642i \(0.310886\pi\)
\(18\) 1.81316e44 0.496892
\(19\) 3.28254e44 0.131964 0.0659822 − 0.997821i \(-0.478982\pi\)
0.0659822 + 0.997821i \(0.478982\pi\)
\(20\) −8.62561e45 −0.561339
\(21\) −1.10194e45 −0.0126874
\(22\) 3.91420e47 0.864247
\(23\) 1.06232e48 0.484081 0.242040 − 0.970266i \(-0.422183\pi\)
0.242040 + 0.970266i \(0.422183\pi\)
\(24\) −1.91665e48 −0.192772
\(25\) 1.10286e49 0.260405
\(26\) 1.65745e50 0.972494
\(27\) −6.04149e50 −0.928390
\(28\) −2.75337e49 −0.0116347
\(29\) −1.14189e52 −1.38834 −0.694168 − 0.719813i \(-0.744227\pi\)
−0.694168 + 0.719813i \(0.744227\pi\)
\(30\) 1.18613e52 0.432842
\(31\) 1.80143e52 0.205247 0.102623 − 0.994720i \(-0.467276\pi\)
0.102623 + 0.994720i \(0.467276\pi\)
\(32\) −4.78905e52 −0.176777
\(33\) −5.38254e53 −0.666411
\(34\) −1.84515e54 −0.791646
\(35\) 1.70394e53 0.0261240
\(36\) −6.22996e54 −0.351356
\(37\) −4.58287e55 −0.977185 −0.488592 − 0.872512i \(-0.662489\pi\)
−0.488592 + 0.872512i \(0.662489\pi\)
\(38\) −1.12787e55 −0.0933130
\(39\) −2.27921e56 −0.749879
\(40\) 2.96374e56 0.396926
\(41\) 2.53761e57 1.41449 0.707243 − 0.706971i \(-0.249939\pi\)
0.707243 + 0.706971i \(0.249939\pi\)
\(42\) 3.78624e55 0.00897136
\(43\) 2.43971e57 0.250731 0.125365 − 0.992111i \(-0.459990\pi\)
0.125365 + 0.992111i \(0.459990\pi\)
\(44\) −1.34491e58 −0.611115
\(45\) 3.85546e58 0.788918
\(46\) −3.65011e58 −0.342297
\(47\) 5.94323e58 0.259746 0.129873 − 0.991531i \(-0.458543\pi\)
0.129873 + 0.991531i \(0.458543\pi\)
\(48\) 6.58557e58 0.136311
\(49\) −1.00398e60 −0.999459
\(50\) −3.78939e59 −0.184134
\(51\) 2.53733e60 0.610430
\(52\) −5.69496e60 −0.687657
\(53\) −6.61258e60 −0.406048 −0.203024 − 0.979174i \(-0.565077\pi\)
−0.203024 + 0.979174i \(0.565077\pi\)
\(54\) 2.07584e61 0.656471
\(55\) 8.32306e61 1.37217
\(56\) 9.46049e59 0.00822695
\(57\) 1.55097e61 0.0719526
\(58\) 3.92350e62 0.981701
\(59\) −1.13910e63 −1.55352 −0.776759 − 0.629798i \(-0.783138\pi\)
−0.776759 + 0.629798i \(0.783138\pi\)
\(60\) −4.07553e62 −0.306066
\(61\) 3.97263e63 1.65909 0.829543 − 0.558443i \(-0.188601\pi\)
0.829543 + 0.558443i \(0.188601\pi\)
\(62\) −6.18965e62 −0.145131
\(63\) 1.23069e62 0.0163516
\(64\) 1.64550e63 0.125000
\(65\) 3.52437e64 1.54403
\(66\) 1.84943e64 0.471224
\(67\) 1.05200e65 1.57166 0.785830 − 0.618442i \(-0.212236\pi\)
0.785830 + 0.618442i \(0.212236\pi\)
\(68\) 6.33990e64 0.559779
\(69\) 5.01938e64 0.263941
\(70\) −5.85470e63 −0.0184724
\(71\) 6.61153e65 1.26075 0.630377 − 0.776289i \(-0.282900\pi\)
0.630377 + 0.776289i \(0.282900\pi\)
\(72\) 2.14060e65 0.248446
\(73\) 1.33268e66 0.947904 0.473952 − 0.880551i \(-0.342827\pi\)
0.473952 + 0.880551i \(0.342827\pi\)
\(74\) 1.57466e66 0.690974
\(75\) 5.21091e65 0.141984
\(76\) 3.87534e65 0.0659822
\(77\) 2.65679e65 0.0284405
\(78\) 7.83131e66 0.530245
\(79\) 2.53823e67 1.09338 0.546688 − 0.837336i \(-0.315888\pi\)
0.546688 + 0.837336i \(0.315888\pi\)
\(80\) −1.01833e67 −0.280669
\(81\) 1.10818e67 0.196514
\(82\) −8.71917e67 −1.00019
\(83\) −2.11949e68 −1.58110 −0.790549 − 0.612398i \(-0.790205\pi\)
−0.790549 + 0.612398i \(0.790205\pi\)
\(84\) −1.30094e66 −0.00634371
\(85\) −3.92350e68 −1.25690
\(86\) −8.38278e67 −0.177294
\(87\) −5.39532e68 −0.756979
\(88\) 4.62107e68 0.432123
\(89\) −4.02156e66 −0.00251796 −0.00125898 − 0.999999i \(-0.500401\pi\)
−0.00125898 + 0.999999i \(0.500401\pi\)
\(90\) −1.32473e69 −0.557849
\(91\) 1.12501e68 0.0320026
\(92\) 1.25417e69 0.242040
\(93\) 8.51158e68 0.111909
\(94\) −2.04208e69 −0.183668
\(95\) −2.39828e69 −0.148154
\(96\) −2.26278e69 −0.0963861
\(97\) 3.31804e70 0.978329 0.489165 − 0.872192i \(-0.337302\pi\)
0.489165 + 0.872192i \(0.337302\pi\)
\(98\) 3.44965e70 0.706724
\(99\) 6.01144e70 0.858874
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 2.72.a.b.1.2 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.72.a.b.1.2 ✓ 3 1.1 even 1 trivial