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Properties

Label 2.72.a.b.1.1
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 1.1
Root \(-6.65040e12\) 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)
 
\(f(q)\) \(=\) \(q-3.43597e10 q^{2} -1.08417e17 q^{3} +1.18059e21 q^{4} +7.90573e22 q^{5} +3.72520e27 q^{6} +6.08855e29 q^{7} -4.05648e31 q^{8} +4.24488e33 q^{9} -2.71639e33 q^{10} +1.54925e37 q^{11} -1.27997e38 q^{12} +2.44465e39 q^{13} -2.09201e40 q^{14} -8.57119e39 q^{15} +1.39380e42 q^{16} +4.66619e43 q^{17} -1.45853e44 q^{18} -2.14466e44 q^{19} +9.33344e43 q^{20} -6.60105e46 q^{21} -5.32320e47 q^{22} -3.03623e48 q^{23} +4.39793e48 q^{24} -4.23454e49 q^{25} -8.39977e49 q^{26} +3.53938e50 q^{27} +7.18809e50 q^{28} +5.23008e51 q^{29} +2.94504e50 q^{30} -2.94169e52 q^{31} -4.78905e52 q^{32} -1.67966e54 q^{33} -1.60329e54 q^{34} +4.81344e52 q^{35} +5.01147e54 q^{36} -4.30443e54 q^{37} +7.36899e54 q^{38} -2.65043e56 q^{39} -3.20694e54 q^{40} -5.89629e56 q^{41} +2.26810e57 q^{42} +6.09960e57 q^{43} +1.82904e58 q^{44} +3.35589e56 q^{45} +1.04324e59 q^{46} -3.56597e59 q^{47} -1.51112e59 q^{48} -6.33821e59 q^{49} +1.45498e60 q^{50} -5.05896e60 q^{51} +2.88614e60 q^{52} +3.09537e61 q^{53} -1.21612e61 q^{54} +1.22480e60 q^{55} -2.46981e61 q^{56} +2.32518e61 q^{57} -1.79704e62 q^{58} +1.36513e63 q^{59} -1.01191e61 q^{60} -1.19961e63 q^{61} +1.01076e63 q^{62} +2.58452e63 q^{63} +1.64550e63 q^{64} +1.93268e62 q^{65} +5.77128e64 q^{66} +1.19801e65 q^{67} +5.50886e64 q^{68} +3.29181e65 q^{69} -1.65389e63 q^{70} -3.22364e65 q^{71} -1.72193e65 q^{72} +2.31415e66 q^{73} +1.47899e65 q^{74} +4.59098e66 q^{75} -2.53196e65 q^{76} +9.43271e66 q^{77} +9.10682e66 q^{78} -9.14450e66 q^{79} +1.10190e65 q^{80} -7.02499e67 q^{81} +2.02595e67 q^{82} -1.67738e68 q^{83} -7.79315e67 q^{84} +3.68896e66 q^{85} -2.09581e68 q^{86} -5.67032e68 q^{87} -6.28452e68 q^{88} -7.15572e68 q^{89} -1.15307e67 q^{90} +1.48844e69 q^{91} -3.58455e69 q^{92} +3.18930e69 q^{93} +1.22526e70 q^{94} -1.69551e67 q^{95} +5.19216e69 q^{96} -2.08258e70 q^{97} +2.17779e70 q^{98} +6.57640e70 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 103079215104 q^{2} + 23\!\cdots\!36 q^{3} + 35\!\cdots\!72 q^{4} + 47\!\cdots\!50 q^{5} - 81\!\cdots\!48 q^{6} - 68\!\cdots\!72 q^{7} - 12\!\cdots\!96 q^{8} - 45\!\cdots\!09 q^{9} - 16\!\cdots\!00 q^{10}+ \cdots + 12\!\cdots\!32 q^{99}+O(q^{100}) \) 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
\(3\) −1.08417e17 −1.25111 −0.625554 − 0.780181i \(-0.715127\pi\)
−0.625554 + 0.780181i \(0.715127\pi\)
\(4\) 1.18059e21 0.500000
\(5\) 7.90573e22 0.0121481 0.00607403 − 0.999982i \(-0.498067\pi\)
0.00607403 + 0.999982i \(0.498067\pi\)
\(6\) 3.72520e27 0.884667
\(7\) 6.08855e29 0.607482 0.303741 − 0.952755i \(-0.401764\pi\)
0.303741 + 0.952755i \(0.401764\pi\)
\(8\) −4.05648e31 −0.353553
\(9\) 4.24488e33 0.565270
\(10\) −2.71639e33 −0.00858997
\(11\) 1.54925e37 1.66220 0.831098 − 0.556126i \(-0.187713\pi\)
0.831098 + 0.556126i \(0.187713\pi\)
\(12\) −1.27997e38 −0.625554
\(13\) 2.44465e39 0.696993 0.348497 − 0.937310i \(-0.386692\pi\)
0.348497 + 0.937310i \(0.386692\pi\)
\(14\) −2.09201e40 −0.429555
\(15\) −8.57119e39 −0.0151985
\(16\) 1.39380e42 0.250000
\(17\) 4.66619e43 0.972804 0.486402 − 0.873735i \(-0.338309\pi\)
0.486402 + 0.873735i \(0.338309\pi\)
\(18\) −1.45853e44 −0.399707
\(19\) −2.14466e44 −0.0862194 −0.0431097 − 0.999070i \(-0.513727\pi\)
−0.0431097 + 0.999070i \(0.513727\pi\)
\(20\) 9.33344e43 0.00607403
\(21\) −6.60105e46 −0.760025
\(22\) −5.32320e47 −1.17535
\(23\) −3.03623e48 −1.38356 −0.691778 − 0.722110i \(-0.743172\pi\)
−0.691778 + 0.722110i \(0.743172\pi\)
\(24\) 4.39793e48 0.442333
\(25\) −4.23454e49 −0.999852
\(26\) −8.39977e49 −0.492849
\(27\) 3.53938e50 0.543894
\(28\) 7.18809e50 0.303741
\(29\) 5.23008e51 0.635886 0.317943 − 0.948110i \(-0.397008\pi\)
0.317943 + 0.948110i \(0.397008\pi\)
\(30\) 2.94504e50 0.0107470
\(31\) −2.94169e52 −0.335163 −0.167582 − 0.985858i \(-0.553596\pi\)
−0.167582 + 0.985858i \(0.553596\pi\)
\(32\) −4.78905e52 −0.176777
\(33\) −1.67966e54 −2.07959
\(34\) −1.60329e54 −0.687876
\(35\) 4.81344e52 0.00737973
\(36\) 5.01147e54 0.282635
\(37\) −4.30443e54 −0.0917814 −0.0458907 − 0.998946i \(-0.514613\pi\)
−0.0458907 + 0.998946i \(0.514613\pi\)
\(38\) 7.36899e54 0.0609663
\(39\) −2.65043e56 −0.872013
\(40\) −3.20694e54 −0.00429499
\(41\) −5.89629e56 −0.328664 −0.164332 − 0.986405i \(-0.552547\pi\)
−0.164332 + 0.986405i \(0.552547\pi\)
\(42\) 2.26810e57 0.537419
\(43\) 6.09960e57 0.626861 0.313430 − 0.949611i \(-0.398522\pi\)
0.313430 + 0.949611i \(0.398522\pi\)
\(44\) 1.82904e58 0.831098
\(45\) 3.35589e56 0.00686694
\(46\) 1.04324e59 0.978322
\(47\) −3.56597e59 −1.55849 −0.779244 − 0.626721i \(-0.784397\pi\)
−0.779244 + 0.626721i \(0.784397\pi\)
\(48\) −1.51112e59 −0.312777
\(49\) −6.33821e59 −0.630966
\(50\) 1.45498e60 0.707002
\(51\) −5.05896e60 −1.21708
\(52\) 2.88614e60 0.348497
\(53\) 3.09537e61 1.90072 0.950362 − 0.311146i \(-0.100713\pi\)
0.950362 + 0.311146i \(0.100713\pi\)
\(54\) −1.21612e61 −0.384591
\(55\) 1.22480e60 0.0201925
\(56\) −2.46981e61 −0.214777
\(57\) 2.32518e61 0.107870
\(58\) −1.79704e62 −0.449640
\(59\) 1.36513e63 1.86177 0.930887 − 0.365307i \(-0.119036\pi\)
0.930887 + 0.365307i \(0.119036\pi\)
\(60\) −1.01191e61 −0.00759926
\(61\) −1.19961e63 −0.500994 −0.250497 − 0.968117i \(-0.580594\pi\)
−0.250497 + 0.968117i \(0.580594\pi\)
\(62\) 1.01076e63 0.236996
\(63\) 2.58452e63 0.343392
\(64\) 1.64550e63 0.125000
\(65\) 1.93268e62 0.00846711
\(66\) 5.77128e64 1.47049
\(67\) 1.19801e65 1.78981 0.894904 − 0.446260i \(-0.147244\pi\)
0.894904 + 0.446260i \(0.147244\pi\)
\(68\) 5.50886e64 0.486402
\(69\) 3.29181e65 1.73098
\(70\) −1.65389e63 −0.00521825
\(71\) −3.22364e65 −0.614717 −0.307358 − 0.951594i \(-0.599445\pi\)
−0.307358 + 0.951594i \(0.599445\pi\)
\(72\) −1.72193e65 −0.199853
\(73\) 2.31415e66 1.64600 0.823002 − 0.568039i \(-0.192298\pi\)
0.823002 + 0.568039i \(0.192298\pi\)
\(74\) 1.47899e65 0.0648993
\(75\) 4.59098e66 1.25092
\(76\) −2.53196e65 −0.0431097
\(77\) 9.43271e66 1.00975
\(78\) 9.10682e66 0.616607
\(79\) −9.14450e66 −0.393911 −0.196956 − 0.980412i \(-0.563105\pi\)
−0.196956 + 0.980412i \(0.563105\pi\)
\(80\) 1.10190e65 0.00303701
\(81\) −7.02499e67 −1.24574
\(82\) 2.02595e67 0.232400
\(83\) −1.67738e68 −1.25129 −0.625646 − 0.780107i \(-0.715165\pi\)
−0.625646 + 0.780107i \(0.715165\pi\)
\(84\) −7.79315e67 −0.380013
\(85\) 3.68896e66 0.0118177
\(86\) −2.09581e68 −0.443257
\(87\) −5.67032e68 −0.795562
\(88\) −6.28452e68 −0.587675
\(89\) −7.15572e68 −0.448031 −0.224016 − 0.974586i \(-0.571917\pi\)
−0.224016 + 0.974586i \(0.571917\pi\)
\(90\) −1.15307e67 −0.00485566
\(91\) 1.48844e69 0.423411
\(92\) −3.58455e69 −0.691778
\(93\) 3.18930e69 0.419326
\(94\) 1.22526e70 1.10202
\(95\) −1.69551e67 −0.00104740
\(96\) 5.19216e69 0.221167
\(97\) −2.08258e70 −0.614053 −0.307027 − 0.951701i \(-0.599334\pi\)
−0.307027 + 0.951701i \(0.599334\pi\)
\(98\) 2.17779e70 0.446160
\(99\) 6.57640e70 0.939591
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.1.1 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.72.a.b.1.1 ✓ 3 1.1 even 1 trivial