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Properties

Label 1.96.a.a.1.4
Level $1$
Weight $96$
Character 1.1
Self dual yes
Analytic conductor $57.154$
Analytic rank $0$
Dimension $8$
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] = [1,96,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, 96); 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, 96, names="a")
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 96 \)
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: \(57.1535908815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{104}\cdot 3^{38}\cdot 5^{12}\cdot 7^{7}\cdot 11\cdot 13\cdot 19^{3} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.67307e12\) 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)
 
\(f(q)\) \(=\) \(q-4.08831e13 q^{2} -7.46060e22 q^{3} -3.79427e28 q^{4} +2.52173e33 q^{5} +3.05013e36 q^{6} +2.11876e40 q^{7} +3.17076e42 q^{8} +3.44515e45 q^{9} -1.03096e47 q^{10} +2.27998e49 q^{11} +2.83075e51 q^{12} +2.29728e52 q^{13} -8.66214e53 q^{14} -1.88136e56 q^{15} +1.37343e57 q^{16} -1.90158e58 q^{17} -1.40849e59 q^{18} -1.95604e60 q^{19} -9.56809e61 q^{20} -1.58072e63 q^{21} -9.32129e62 q^{22} +6.96784e64 q^{23} -2.36558e65 q^{24} +3.83474e66 q^{25} -9.39201e65 q^{26} -9.87975e67 q^{27} -8.03912e68 q^{28} +3.29733e69 q^{29} +7.69158e69 q^{30} -1.05470e70 q^{31} -1.81757e71 q^{32} -1.70100e72 q^{33} +7.77425e71 q^{34} +5.34292e73 q^{35} -1.30718e74 q^{36} +6.15205e73 q^{37} +7.99690e73 q^{38} -1.71391e75 q^{39} +7.99579e75 q^{40} -5.28549e75 q^{41} +6.46247e76 q^{42} -5.20413e77 q^{43} -8.65086e77 q^{44} +8.68773e78 q^{45} -2.84867e78 q^{46} -1.26681e79 q^{47} -1.02466e80 q^{48} +2.56465e80 q^{49} -1.56776e80 q^{50} +1.41869e81 q^{51} -8.71650e80 q^{52} -3.16779e81 q^{53} +4.03915e81 q^{54} +5.74949e82 q^{55} +6.71807e82 q^{56} +1.45932e83 q^{57} -1.34805e83 q^{58} +4.60990e83 q^{59} +7.13837e84 q^{60} +4.35502e84 q^{61} +4.31195e83 q^{62} +7.29944e85 q^{63} -4.69765e85 q^{64} +5.79311e85 q^{65} +6.95424e85 q^{66} +1.06266e87 q^{67} +7.21510e86 q^{68} -5.19842e87 q^{69} -2.18435e87 q^{70} -2.68493e87 q^{71} +1.09238e88 q^{72} -4.60158e88 q^{73} -2.51515e87 q^{74} -2.86095e89 q^{75} +7.42173e88 q^{76} +4.83073e89 q^{77} +7.00700e88 q^{78} -9.17980e89 q^{79} +3.46342e90 q^{80} +6.40740e88 q^{81} +2.16088e89 q^{82} +9.48142e90 q^{83} +5.99766e91 q^{84} -4.79526e91 q^{85} +2.12761e91 q^{86} -2.46001e92 q^{87} +7.22929e91 q^{88} -2.01039e92 q^{89} -3.55182e92 q^{90} +4.86738e92 q^{91} -2.64378e93 q^{92} +7.86870e92 q^{93} +5.17914e92 q^{94} -4.93259e93 q^{95} +1.35602e94 q^{96} +4.06342e94 q^{97} -1.04851e94 q^{98} +7.85489e94 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5835659138280 q^{2} - 95\!\cdots\!80 q^{3} + 20\!\cdots\!84 q^{4} + 19\!\cdots\!60 q^{5} + 10\!\cdots\!76 q^{6} + 31\!\cdots\!00 q^{7} - 14\!\cdots\!60 q^{8} + 92\!\cdots\!36 q^{9} - 35\!\cdots\!40 q^{10}+ \cdots + 30\!\cdots\!72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.08831e13 −0.205409 −0.102704 − 0.994712i \(-0.532750\pi\)
−0.102704 + 0.994712i \(0.532750\pi\)
\(3\) −7.46060e22 −1.62000 −0.809998 − 0.586433i \(-0.800532\pi\)
−0.809998 + 0.586433i \(0.800532\pi\)
\(4\) −3.79427e28 −0.957807
\(5\) 2.52173e33 1.58717 0.793583 − 0.608461i \(-0.208213\pi\)
0.793583 + 0.608461i \(0.208213\pi\)
\(6\) 3.05013e36 0.332762
\(7\) 2.11876e40 1.52730 0.763650 − 0.645631i \(-0.223406\pi\)
0.763650 + 0.645631i \(0.223406\pi\)
\(8\) 3.17076e42 0.402151
\(9\) 3.44515e45 1.62439
\(10\) −1.03096e47 −0.326018
\(11\) 2.27998e49 0.779434 0.389717 − 0.920935i \(-0.372573\pi\)
0.389717 + 0.920935i \(0.372573\pi\)
\(12\) 2.83075e51 1.55164
\(13\) 2.29728e52 0.281129 0.140564 − 0.990072i \(-0.455108\pi\)
0.140564 + 0.990072i \(0.455108\pi\)
\(14\) −8.66214e53 −0.313721
\(15\) −1.88136e56 −2.57120
\(16\) 1.37343e57 0.875202
\(17\) −1.90158e58 −0.680441 −0.340221 − 0.940346i \(-0.610502\pi\)
−0.340221 + 0.940346i \(0.610502\pi\)
\(18\) −1.40849e59 −0.333664
\(19\) −1.95604e60 −0.355289 −0.177644 − 0.984095i \(-0.556848\pi\)
−0.177644 + 0.984095i \(0.556848\pi\)
\(20\) −9.56809e61 −1.52020
\(21\) −1.58072e63 −2.47422
\(22\) −9.32129e62 −0.160103
\(23\) 6.96784e64 1.44886 0.724429 − 0.689350i \(-0.242104\pi\)
0.724429 + 0.689350i \(0.242104\pi\)
\(24\) −2.36558e65 −0.651483
\(25\) 3.83474e66 1.51910
\(26\) −9.39201e65 −0.0577463
\(27\) −9.87975e67 −1.01150
\(28\) −8.03912e68 −1.46286
\(29\) 3.29733e69 1.13307 0.566536 − 0.824037i \(-0.308283\pi\)
0.566536 + 0.824037i \(0.308283\pi\)
\(30\) 7.69158e69 0.528148
\(31\) −1.05470e70 −0.152563 −0.0762815 − 0.997086i \(-0.524305\pi\)
−0.0762815 + 0.997086i \(0.524305\pi\)
\(32\) −1.81757e71 −0.581925
\(33\) −1.70100e72 −1.26268
\(34\) 7.77425e71 0.139769
\(35\) 5.34292e73 2.42408
\(36\) −1.30718e74 −1.55585
\(37\) 6.15205e73 0.199268 0.0996341 − 0.995024i \(-0.468233\pi\)
0.0996341 + 0.995024i \(0.468233\pi\)
\(38\) 7.99690e73 0.0729795
\(39\) −1.71391e75 −0.455427
\(40\) 7.99579e75 0.638281
\(41\) −5.28549e75 −0.130573 −0.0652864 − 0.997867i \(-0.520796\pi\)
−0.0652864 + 0.997867i \(0.520796\pi\)
\(42\) 6.46247e76 0.508227
\(43\) −5.20413e77 −1.33843 −0.669217 − 0.743067i \(-0.733370\pi\)
−0.669217 + 0.743067i \(0.733370\pi\)
\(44\) −8.65086e77 −0.746547
\(45\) 8.68773e78 2.57817
\(46\) −2.84867e78 −0.297608
\(47\) −1.26681e79 −0.476502 −0.238251 − 0.971204i \(-0.576574\pi\)
−0.238251 + 0.971204i \(0.576574\pi\)
\(48\) −1.02466e80 −1.41782
\(49\) 2.56465e80 1.33264
\(50\) −1.56776e80 −0.312037
\(51\) 1.41869e81 1.10231
\(52\) −8.71650e80 −0.269267
\(53\) −3.16779e81 −0.395961 −0.197981 − 0.980206i \(-0.563438\pi\)
−0.197981 + 0.980206i \(0.563438\pi\)
\(54\) 4.03915e81 0.207772
\(55\) 5.74949e82 1.23709
\(56\) 6.71807e82 0.614205
\(57\) 1.45932e83 0.575566
\(58\) −1.34805e83 −0.232743
\(59\) 4.60990e83 0.353363 0.176682 − 0.984268i \(-0.443464\pi\)
0.176682 + 0.984268i \(0.443464\pi\)
\(60\) 7.13837e84 2.46272
\(61\) 4.35502e84 0.685209 0.342605 − 0.939480i \(-0.388691\pi\)
0.342605 + 0.939480i \(0.388691\pi\)
\(62\) 4.31195e83 0.0313378
\(63\) 7.29944e85 2.48092
\(64\) −4.69765e85 −0.755669
\(65\) 5.79311e85 0.446198
\(66\) 6.95424e85 0.259366
\(67\) 1.06266e87 1.94017 0.970085 − 0.242764i \(-0.0780539\pi\)
0.970085 + 0.242764i \(0.0780539\pi\)
\(68\) 7.21510e86 0.651732
\(69\) −5.19842e87 −2.34714
\(70\) −2.18435e87 −0.497928
\(71\) −2.68493e87 −0.312005 −0.156003 − 0.987757i \(-0.549861\pi\)
−0.156003 + 0.987757i \(0.549861\pi\)
\(72\) 1.09238e88 0.653249
\(73\) −4.60158e88 −1.42913 −0.714564 − 0.699570i \(-0.753375\pi\)
−0.714564 + 0.699570i \(0.753375\pi\)
\(74\) −2.51515e87 −0.0409315
\(75\) −2.86095e89 −2.46093
\(76\) 7.42173e88 0.340298
\(77\) 4.83073e89 1.19043
\(78\) 7.00700e88 0.0935488
\(79\) −9.17980e89 −0.669185 −0.334593 − 0.942363i \(-0.608599\pi\)
−0.334593 + 0.942363i \(0.608599\pi\)
\(80\) 3.46342e90 1.38909
\(81\) 6.40740e88 0.0142444
\(82\) 2.16088e89 0.0268208
\(83\) 9.48142e90 0.661704 0.330852 − 0.943683i \(-0.392664\pi\)
0.330852 + 0.943683i \(0.392664\pi\)
\(84\) 5.99766e91 2.36982
\(85\) −4.79526e91 −1.07997
\(86\) 2.12761e91 0.274926
\(87\) −2.46001e92 −1.83557
\(88\) 7.22929e91 0.313450
\(89\) −2.01039e92 −0.509630 −0.254815 − 0.966990i \(-0.582015\pi\)
−0.254815 + 0.966990i \(0.582015\pi\)
\(90\) −3.55182e92 −0.529580
\(91\) 4.86738e92 0.429367
\(92\) −2.64378e93 −1.38773
\(93\) 7.86870e92 0.247151
\(94\) 5.17914e92 0.0978778
\(95\) −4.93259e93 −0.563903
\(96\) 1.35602e94 0.942717
\(97\) 4.06342e94 1.72677 0.863384 − 0.504548i \(-0.168341\pi\)
0.863384 + 0.504548i \(0.168341\pi\)
\(98\) −1.04851e94 −0.273737
\(99\) 7.85489e94 1.26610
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 1.96.a.a.1.4 ✓ 8
3.2 odd 2 9.96.a.c.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.96.a.a.1.4 ✓ 8 1.1 even 1 trivial
9.96.a.c.1.5 8 3.2 odd 2