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Properties

Label 2.72.a.b.1.3
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 1.3
Root \(-2.86077e12\) 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)
 
\(f(q)\) \(=\) \(q-3.43597e10 q^{2} +6.35309e16 q^{3} +1.18059e21 q^{4} +1.19804e25 q^{5} -2.18290e27 q^{6} -1.27132e30 q^{7} -4.05648e31 q^{8} -3.47329e33 q^{9} -4.11644e35 q^{10} -1.31209e35 q^{11} +7.50040e37 q^{12} +4.90880e39 q^{13} +4.36824e40 q^{14} +7.61127e41 q^{15} +1.39380e42 q^{16} -6.48706e43 q^{17} +1.19341e44 q^{18} +2.72740e45 q^{19} +1.41440e46 q^{20} -8.07683e46 q^{21} +4.50829e45 q^{22} +2.34677e48 q^{23} -2.57712e48 q^{24} +1.01179e50 q^{25} -1.68665e50 q^{26} -6.97744e50 q^{27} -1.50091e51 q^{28} +9.02709e51 q^{29} -2.61521e52 q^{30} -4.95831e52 q^{31} -4.78905e52 q^{32} -8.33579e51 q^{33} +2.22894e54 q^{34} -1.52310e55 q^{35} -4.10054e54 q^{36} -1.80430e55 q^{37} -9.37127e55 q^{38} +3.11861e56 q^{39} -4.85984e56 q^{40} +6.04763e56 q^{41} +2.77518e57 q^{42} +1.03698e58 q^{43} -1.54904e56 q^{44} -4.16115e58 q^{45} -8.06344e58 q^{46} +2.53196e59 q^{47} +8.85491e58 q^{48} +6.11739e59 q^{49} -3.47649e60 q^{50} -4.12129e60 q^{51} +5.79529e60 q^{52} +6.53033e60 q^{53} +2.39743e61 q^{54} -1.57193e60 q^{55} +5.15710e61 q^{56} +1.73274e62 q^{57} -3.10168e62 q^{58} +2.34221e62 q^{59} +8.98581e62 q^{60} -2.75624e63 q^{61} +1.70366e63 q^{62} +4.41568e63 q^{63} +1.64550e63 q^{64} +5.88096e64 q^{65} +2.86416e62 q^{66} +6.97669e64 q^{67} -7.65857e64 q^{68} +1.49092e65 q^{69} +5.23333e65 q^{70} +5.81114e65 q^{71} +1.40893e65 q^{72} -8.45661e65 q^{73} +6.19952e65 q^{74} +6.42799e66 q^{75} +3.21994e66 q^{76} +1.66809e65 q^{77} -1.07155e67 q^{78} +1.96236e67 q^{79} +1.66983e67 q^{80} -1.82458e67 q^{81} -2.07795e67 q^{82} -1.11774e68 q^{83} -9.53544e67 q^{84} -7.77178e68 q^{85} -3.56304e68 q^{86} +5.73499e68 q^{87} +5.32245e66 q^{88} +5.42522e68 q^{89} +1.42976e69 q^{90} -6.24068e69 q^{91} +2.77058e69 q^{92} -3.15006e69 q^{93} -8.69974e69 q^{94} +3.26754e70 q^{95} -3.04253e69 q^{96} +2.87774e70 q^{97} -2.10192e70 q^{98} +4.55726e68 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 103079215104 q^{2} + 23\!\cdots\!36 q^{3} + 35\!\cdots\!72 q^{4} + 47\!\cdots\!50 q^{5} - 81\!\cdots\!48 q^{6} - 68\!\cdots\!72 q^{7} - 12\!\cdots\!96 q^{8} - 45\!\cdots\!09 q^{9} - 16\!\cdots\!00 q^{10}+ \cdots + 12\!\cdots\!32 q^{99}+O(q^{100}) \) 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
\(3\) 6.35309e16 0.733129 0.366565 − 0.930393i \(-0.380534\pi\)
0.366565 + 0.930393i \(0.380534\pi\)
\(4\) 1.18059e21 0.500000
\(5\) 1.19804e25 1.84093 0.920465 − 0.390825i \(-0.127810\pi\)
0.920465 + 0.390825i \(0.127810\pi\)
\(6\) −2.18290e27 −0.518400
\(7\) −1.27132e30 −1.26846 −0.634229 − 0.773146i \(-0.718682\pi\)
−0.634229 + 0.773146i \(0.718682\pi\)
\(8\) −4.05648e31 −0.353553
\(9\) −3.47329e33 −0.462522
\(10\) −4.11644e35 −1.30173
\(11\) −1.31209e35 −0.0140774 −0.00703869 − 0.999975i \(-0.502241\pi\)
−0.00703869 + 0.999975i \(0.502241\pi\)
\(12\) 7.50040e37 0.366565
\(13\) 4.90880e39 1.39954 0.699772 − 0.714366i \(-0.253285\pi\)
0.699772 + 0.714366i \(0.253285\pi\)
\(14\) 4.36824e40 0.896935
\(15\) 7.61127e41 1.34964
\(16\) 1.39380e42 0.250000
\(17\) −6.48706e43 −1.35242 −0.676209 − 0.736710i \(-0.736379\pi\)
−0.676209 + 0.736710i \(0.736379\pi\)
\(18\) 1.19341e44 0.327052
\(19\) 2.72740e45 1.09647 0.548234 − 0.836325i \(-0.315300\pi\)
0.548234 + 0.836325i \(0.315300\pi\)
\(20\) 1.41440e46 0.920465
\(21\) −8.07683e46 −0.929943
\(22\) 4.50829e45 0.00995421
\(23\) 2.34677e48 1.06938 0.534690 − 0.845048i \(-0.320428\pi\)
0.534690 + 0.845048i \(0.320428\pi\)
\(24\) −2.57712e48 −0.259200
\(25\) 1.01179e50 2.38902
\(26\) −1.68665e50 −0.989627
\(27\) −6.97744e50 −1.07222
\(28\) −1.50091e51 −0.634229
\(29\) 9.02709e51 1.09754 0.548768 − 0.835975i \(-0.315097\pi\)
0.548768 + 0.835975i \(0.315097\pi\)
\(30\) −2.61521e52 −0.954339
\(31\) −4.95831e52 −0.564929 −0.282465 − 0.959278i \(-0.591152\pi\)
−0.282465 + 0.959278i \(0.591152\pi\)
\(32\) −4.78905e52 −0.176777
\(33\) −8.33579e51 −0.0103205
\(34\) 2.22894e54 0.956304
\(35\) −1.52310e55 −2.33514
\(36\) −4.10054e54 −0.231261
\(37\) −1.80430e55 −0.384722 −0.192361 − 0.981324i \(-0.561614\pi\)
−0.192361 + 0.981324i \(0.561614\pi\)
\(38\) −9.37127e55 −0.775320
\(39\) 3.11861e56 1.02605
\(40\) −4.85984e56 −0.650867
\(41\) 6.04763e56 0.337099 0.168550 − 0.985693i \(-0.446092\pi\)
0.168550 + 0.985693i \(0.446092\pi\)
\(42\) 2.77518e57 0.657569
\(43\) 1.03698e58 1.06571 0.532857 − 0.846206i \(-0.321119\pi\)
0.532857 + 0.846206i \(0.321119\pi\)
\(44\) −1.54904e56 −0.00703869
\(45\) −4.16115e58 −0.851470
\(46\) −8.06344e58 −0.756166
\(47\) 2.53196e59 1.10658 0.553289 − 0.832989i \(-0.313372\pi\)
0.553289 + 0.832989i \(0.313372\pi\)
\(48\) 8.85491e58 0.183282
\(49\) 6.11739e59 0.608983
\(50\) −3.47649e60 −1.68929
\(51\) −4.12129e60 −0.991497
\(52\) 5.79529e60 0.699772
\(53\) 6.53033e60 0.400997 0.200499 − 0.979694i \(-0.435744\pi\)
0.200499 + 0.979694i \(0.435744\pi\)
\(54\) 2.39743e61 0.758172
\(55\) −1.57193e60 −0.0259155
\(56\) 5.15710e61 0.448467
\(57\) 1.73274e62 0.803852
\(58\) −3.10168e62 −0.776075
\(59\) 2.34221e62 0.319433 0.159716 − 0.987163i \(-0.448942\pi\)
0.159716 + 0.987163i \(0.448942\pi\)
\(60\) 8.98581e62 0.674820
\(61\) −2.75624e63 −1.15109 −0.575543 − 0.817771i \(-0.695209\pi\)
−0.575543 + 0.817771i \(0.695209\pi\)
\(62\) 1.70366e63 0.399465
\(63\) 4.41568e63 0.586689
\(64\) 1.64550e63 0.125000
\(65\) 5.88096e64 2.57646
\(66\) 2.86416e62 0.00729772
\(67\) 6.97669e64 1.04230 0.521151 − 0.853464i \(-0.325503\pi\)
0.521151 + 0.853464i \(0.325503\pi\)
\(68\) −7.65857e64 −0.676209
\(69\) 1.49092e65 0.783994
\(70\) 5.23333e65 1.65119
\(71\) 5.81114e65 1.10813 0.554064 − 0.832474i \(-0.313076\pi\)
0.554064 + 0.832474i \(0.313076\pi\)
\(72\) 1.40893e65 0.163526
\(73\) −8.45661e65 −0.601499 −0.300749 − 0.953703i \(-0.597237\pi\)
−0.300749 + 0.953703i \(0.597237\pi\)
\(74\) 6.19952e65 0.272040
\(75\) 6.42799e66 1.75146
\(76\) 3.21994e66 0.548234
\(77\) 1.66809e65 0.0178565
\(78\) −1.07155e67 −0.725525
\(79\) 1.96236e67 0.845312 0.422656 − 0.906290i \(-0.361098\pi\)
0.422656 + 0.906290i \(0.361098\pi\)
\(80\) 1.66983e67 0.460232
\(81\) −1.82458e67 −0.323552
\(82\) −2.07795e67 −0.238365
\(83\) −1.11774e68 −0.833816 −0.416908 − 0.908949i \(-0.636886\pi\)
−0.416908 + 0.908949i \(0.636886\pi\)
\(84\) −9.53544e67 −0.464971
\(85\) −7.77178e68 −2.48971
\(86\) −3.56304e68 −0.753573
\(87\) 5.73499e68 0.804635
\(88\) 5.32245e66 0.00497710
\(89\) 5.42522e68 0.339682 0.169841 − 0.985472i \(-0.445675\pi\)
0.169841 + 0.985472i \(0.445675\pi\)
\(90\) 1.42976e69 0.602080
\(91\) −6.24068e69 −1.77526
\(92\) 2.77058e69 0.534690
\(93\) −3.15006e69 −0.414166
\(94\) −8.69974e69 −0.782469
\(95\) 3.26754e70 2.01852
\(96\) −3.04253e69 −0.129600
\(97\) 2.87774e70 0.848505 0.424252 − 0.905544i \(-0.360537\pi\)
0.424252 + 0.905544i \(0.360537\pi\)
\(98\) −2.10192e70 −0.430616
\(99\) 4.55726e68 0.00651109
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.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.72.a.b.1.3 ✓ 3 1.1 even 1 trivial