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Properties

Label 1.76.a.a.1.5
Level $1$
Weight $76$
Character 1.1
Self dual yes
Analytic conductor $35.623$
Analytic rank $0$
Dimension $6$
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] = [1,76,Mod(1,1)] 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("1.1"); S:= CuspForms(chi, 76); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 76, names="a")
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 76 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.6228392822\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3 x^{5} + \cdots - 67\!\cdots\!50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{58}\cdot 3^{26}\cdot 5^{7}\cdot 7^{3}\cdot 11\cdot 13\cdot 19 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(3.19848e9\) of defining polynomial
Character \(\chi\) \(=\) 1.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.20777e11 q^{2} +1.08881e18 q^{3} +1.09637e22 q^{4} -2.30711e25 q^{5} +2.40385e29 q^{6} +1.96027e31 q^{7} -5.92020e33 q^{8} +5.77242e35 q^{9} -5.09358e36 q^{10} +9.40509e38 q^{11} +1.19374e40 q^{12} +1.14147e42 q^{13} +4.32784e42 q^{14} -2.51201e43 q^{15} -1.72124e45 q^{16} +1.79040e46 q^{17} +1.27442e47 q^{18} +1.43901e48 q^{19} -2.52945e47 q^{20} +2.13436e49 q^{21} +2.07643e50 q^{22} -1.23136e51 q^{23} -6.44597e51 q^{24} -2.59375e52 q^{25} +2.52010e53 q^{26} -3.37806e52 q^{27} +2.14918e53 q^{28} -2.48097e54 q^{29} -5.54594e54 q^{30} +1.95055e55 q^{31} -1.56352e56 q^{32} +1.02404e57 q^{33} +3.95280e57 q^{34} -4.52257e56 q^{35} +6.32870e57 q^{36} -2.65296e58 q^{37} +3.17702e59 q^{38} +1.24284e60 q^{39} +1.36586e59 q^{40} +3.61594e59 q^{41} +4.71219e60 q^{42} +8.98859e59 q^{43} +1.03115e61 q^{44} -1.33176e61 q^{45} -2.71857e62 q^{46} +2.48355e62 q^{47} -1.87411e63 q^{48} -2.02760e63 q^{49} -5.72641e63 q^{50} +1.94941e64 q^{51} +1.25147e64 q^{52} +3.59994e64 q^{53} -7.45799e63 q^{54} -2.16986e64 q^{55} -1.16052e65 q^{56} +1.56681e66 q^{57} -5.47742e65 q^{58} -4.00811e66 q^{59} -2.75409e65 q^{60} -1.59203e67 q^{61} +4.30637e66 q^{62} +1.13155e67 q^{63} +3.05076e67 q^{64} -2.63349e67 q^{65} +2.26084e68 q^{66} +1.61324e68 q^{67} +1.96294e68 q^{68} -1.34072e69 q^{69} -9.98480e67 q^{70} +2.63303e69 q^{71} -3.41738e69 q^{72} -2.52652e69 q^{73} -5.85713e69 q^{74} -2.82410e70 q^{75} +1.57769e70 q^{76} +1.84365e70 q^{77} +2.74391e71 q^{78} -9.29901e70 q^{79} +3.97110e70 q^{80} -3.87898e71 q^{81} +7.98318e70 q^{82} -7.84087e71 q^{83} +2.34005e71 q^{84} -4.13066e71 q^{85} +1.98448e71 q^{86} -2.70130e72 q^{87} -5.56800e72 q^{88} +2.05618e73 q^{89} -2.94023e72 q^{90} +2.23759e73 q^{91} -1.35003e73 q^{92} +2.12378e73 q^{93} +5.48312e73 q^{94} -3.31997e73 q^{95} -1.70238e74 q^{96} -4.10454e74 q^{97} -4.47648e74 q^{98} +5.42901e74 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 57080822040 q^{2} - 78\!\cdots\!40 q^{3} + 17\!\cdots\!28 q^{4} - 38\!\cdots\!40 q^{5} + 31\!\cdots\!92 q^{6} + 19\!\cdots\!00 q^{7} + 44\!\cdots\!20 q^{8} + 21\!\cdots\!82 q^{9} + 13\!\cdots\!60 q^{10}+ \cdots - 18\!\cdots\!36 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.20777e11 1.13587 0.567936 − 0.823072i \(-0.307742\pi\)
0.567936 + 0.823072i \(0.307742\pi\)
\(3\) 1.08881e18 1.39606 0.698032 − 0.716067i \(-0.254059\pi\)
0.698032 + 0.716067i \(0.254059\pi\)
\(4\) 1.09637e22 0.290207
\(5\) −2.30711e25 −0.141806 −0.0709028 − 0.997483i \(-0.522588\pi\)
−0.0709028 + 0.997483i \(0.522588\pi\)
\(6\) 2.40385e29 1.58575
\(7\) 1.96027e31 0.399153 0.199577 − 0.979882i \(-0.436043\pi\)
0.199577 + 0.979882i \(0.436043\pi\)
\(8\) −5.92020e33 −0.806235
\(9\) 5.77242e35 0.948994
\(10\) −5.09358e36 −0.161073
\(11\) 9.40509e38 0.833945 0.416972 − 0.908919i \(-0.363091\pi\)
0.416972 + 0.908919i \(0.363091\pi\)
\(12\) 1.19374e40 0.405147
\(13\) 1.14147e42 1.92570 0.962848 − 0.270042i \(-0.0870377\pi\)
0.962848 + 0.270042i \(0.0870377\pi\)
\(14\) 4.32784e42 0.453387
\(15\) −2.51201e43 −0.197970
\(16\) −1.72124e45 −1.20599
\(17\) 1.79040e46 1.29156 0.645782 − 0.763522i \(-0.276531\pi\)
0.645782 + 0.763522i \(0.276531\pi\)
\(18\) 1.27442e47 1.07794
\(19\) 1.43901e48 1.60253 0.801263 − 0.598312i \(-0.204162\pi\)
0.801263 + 0.598312i \(0.204162\pi\)
\(20\) −2.52945e47 −0.0411529
\(21\) 2.13436e49 0.557244
\(22\) 2.07643e50 0.947255
\(23\) −1.23136e51 −1.06070 −0.530349 − 0.847779i \(-0.677939\pi\)
−0.530349 + 0.847779i \(0.677939\pi\)
\(24\) −6.44597e51 −1.12556
\(25\) −2.59375e52 −0.979891
\(26\) 2.52010e53 2.18735
\(27\) −3.37806e52 −0.0712075
\(28\) 2.14918e53 0.115837
\(29\) −2.48097e54 −0.358671 −0.179336 − 0.983788i \(-0.557395\pi\)
−0.179336 + 0.983788i \(0.557395\pi\)
\(30\) −5.54594e54 −0.224868
\(31\) 1.95055e55 0.231256 0.115628 − 0.993293i \(-0.463112\pi\)
0.115628 + 0.993293i \(0.463112\pi\)
\(32\) −1.56352e56 −0.563613
\(33\) 1.02404e57 1.16424
\(34\) 3.95280e57 1.46705
\(35\) −4.52257e56 −0.0566022
\(36\) 6.32870e57 0.275404
\(37\) −2.65296e58 −0.413205 −0.206603 − 0.978425i \(-0.566241\pi\)
−0.206603 + 0.978425i \(0.566241\pi\)
\(38\) 3.17702e59 1.82027
\(39\) 1.24284e60 2.68840
\(40\) 1.36586e59 0.114329
\(41\) 3.61594e59 0.119902 0.0599511 − 0.998201i \(-0.480906\pi\)
0.0599511 + 0.998201i \(0.480906\pi\)
\(42\) 4.71219e60 0.632958
\(43\) 8.98859e59 0.0499603 0.0249801 − 0.999688i \(-0.492048\pi\)
0.0249801 + 0.999688i \(0.492048\pi\)
\(44\) 1.03115e61 0.242016
\(45\) −1.33176e61 −0.134573
\(46\) −2.71857e62 −1.20482
\(47\) 2.48355e62 0.491364 0.245682 − 0.969350i \(-0.420988\pi\)
0.245682 + 0.969350i \(0.420988\pi\)
\(48\) −1.87411e63 −1.68363
\(49\) −2.02760e63 −0.840677
\(50\) −5.72641e63 −1.11303
\(51\) 1.94941e64 1.80311
\(52\) 1.25147e64 0.558850
\(53\) 3.59994e64 0.786955 0.393477 − 0.919334i \(-0.371272\pi\)
0.393477 + 0.919334i \(0.371272\pi\)
\(54\) −7.45799e63 −0.0808827
\(55\) −2.16986e64 −0.118258
\(56\) −1.16052e65 −0.321811
\(57\) 1.56681e66 2.23723
\(58\) −5.47742e65 −0.407405
\(59\) −4.00811e66 −1.57032 −0.785162 − 0.619291i \(-0.787420\pi\)
−0.785162 + 0.619291i \(0.787420\pi\)
\(60\) −2.75409e65 −0.0574521
\(61\) −1.59203e67 −1.78683 −0.893413 − 0.449237i \(-0.851696\pi\)
−0.893413 + 0.449237i \(0.851696\pi\)
\(62\) 4.30637e66 0.262678
\(63\) 1.13155e67 0.378794
\(64\) 3.05076e67 0.565795
\(65\) −2.63349e67 −0.273075
\(66\) 2.26084e68 1.32243
\(67\) 1.61324e68 0.536900 0.268450 − 0.963294i \(-0.413489\pi\)
0.268450 + 0.963294i \(0.413489\pi\)
\(68\) 1.96294e68 0.374821
\(69\) −1.34072e69 −1.48080
\(70\) −9.98480e67 −0.0642929
\(71\) 2.63303e69 0.996020 0.498010 − 0.867171i \(-0.334064\pi\)
0.498010 + 0.867171i \(0.334064\pi\)
\(72\) −3.41738e69 −0.765112
\(73\) −2.52652e69 −0.337221 −0.168611 − 0.985683i \(-0.553928\pi\)
−0.168611 + 0.985683i \(0.553928\pi\)
\(74\) −5.85713e69 −0.469348
\(75\) −2.82410e70 −1.36799
\(76\) 1.57769e70 0.465064
\(77\) 1.84365e70 0.332872
\(78\) 2.74391e71 3.05368
\(79\) −9.29901e70 −0.641831 −0.320916 − 0.947108i \(-0.603991\pi\)
−0.320916 + 0.947108i \(0.603991\pi\)
\(80\) 3.97110e70 0.171016
\(81\) −3.87898e71 −1.04840
\(82\) 7.98318e70 0.136194
\(83\) −7.84087e71 −0.849055 −0.424528 − 0.905415i \(-0.639560\pi\)
−0.424528 + 0.905415i \(0.639560\pi\)
\(84\) 2.34005e71 0.161716
\(85\) −4.13066e71 −0.183151
\(86\) 1.98448e71 0.0567485
\(87\) −2.70130e72 −0.500728
\(88\) −5.56800e72 −0.672355
\(89\) 2.05618e73 1.62531 0.812656 − 0.582744i \(-0.198021\pi\)
0.812656 + 0.582744i \(0.198021\pi\)
\(90\) −2.94023e72 −0.152857
\(91\) 2.23759e73 0.768648
\(92\) −1.35003e73 −0.307822
\(93\) 2.12378e73 0.322848
\(94\) 5.48312e73 0.558127
\(95\) −3.31997e73 −0.227247
\(96\) −1.70238e74 −0.786839
\(97\) −4.10454e74 −1.28625 −0.643123 − 0.765763i \(-0.722362\pi\)
−0.643123 + 0.765763i \(0.722362\pi\)
\(98\) −4.47648e74 −0.954902
\(99\) 5.42901e74 0.791409
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 1.76.a.a.1.5 ✓ 6
3.2 odd 2 9.76.a.c.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.76.a.a.1.5 ✓ 6 1.1 even 1 trivial
9.76.a.c.1.2 6 3.2 odd 2