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Properties

Label 1.96.a.a.1.2
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 1.2
Root \(-1.46162e13\) 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)
 
\(f(q)\) \(=\) \(q-3.51519e14 q^{2} +6.56090e22 q^{3} +8.39514e28 q^{4} +2.79946e33 q^{5} -2.30628e37 q^{6} +5.53054e39 q^{7} -1.55854e43 q^{8} +2.18364e45 q^{9} -9.84062e47 q^{10} +3.60788e48 q^{11} +5.50796e51 q^{12} +1.06916e53 q^{13} -1.94409e54 q^{14} +1.83670e56 q^{15} +2.15290e57 q^{16} +3.05031e58 q^{17} -7.67592e59 q^{18} -5.36133e58 q^{19} +2.35018e62 q^{20} +3.62853e62 q^{21} -1.26824e63 q^{22} +3.27765e64 q^{23} -1.02254e66 q^{24} +5.31261e66 q^{25} -3.75830e67 q^{26} +4.11692e66 q^{27} +4.64296e68 q^{28} -3.24861e69 q^{29} -6.45633e70 q^{30} -4.63359e70 q^{31} -1.39384e71 q^{32} +2.36709e71 q^{33} -1.07224e73 q^{34} +1.54825e73 q^{35} +1.83320e74 q^{36} +2.22715e74 q^{37} +1.88461e73 q^{38} +7.01465e75 q^{39} -4.36306e76 q^{40} -8.69382e75 q^{41} -1.27550e77 q^{42} +4.86197e77 q^{43} +3.02887e77 q^{44} +6.11302e78 q^{45} -1.15216e79 q^{46} -1.28855e79 q^{47} +1.41250e80 q^{48} -1.61861e80 q^{49} -1.86748e81 q^{50} +2.00128e81 q^{51} +8.97574e81 q^{52} +4.42467e81 q^{53} -1.44717e81 q^{54} +1.01001e82 q^{55} -8.61956e82 q^{56} -3.51751e81 q^{57} +1.14195e84 q^{58} -1.94954e84 q^{59} +1.54193e85 q^{60} -5.98141e84 q^{61} +1.62879e85 q^{62} +1.20767e85 q^{63} -3.62891e85 q^{64} +2.99307e86 q^{65} -8.32078e85 q^{66} -6.60161e86 q^{67} +2.56078e87 q^{68} +2.15044e87 q^{69} -5.44240e87 q^{70} -7.94968e87 q^{71} -3.40329e88 q^{72} -4.40521e88 q^{73} -7.82883e88 q^{74} +3.48555e89 q^{75} -4.50091e87 q^{76} +1.99535e88 q^{77} -2.46578e90 q^{78} +1.51977e90 q^{79} +6.02695e90 q^{80} -4.36117e90 q^{81} +3.05604e90 q^{82} +1.48872e91 q^{83} +3.04620e91 q^{84} +8.53923e91 q^{85} -1.70907e92 q^{86} -2.13138e92 q^{87} -5.62302e91 q^{88} +5.31083e92 q^{89} -2.14884e93 q^{90} +5.91303e92 q^{91} +2.75163e93 q^{92} -3.04005e93 q^{93} +4.52949e93 q^{94} -1.50088e92 q^{95} -9.14485e93 q^{96} -2.46425e94 q^{97} +5.68973e94 q^{98} +7.87833e93 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5835659138280 q^{2} - 95\!\cdots\!80 q^{3} + 20\!\cdots\!84 q^{4} + 19\!\cdots\!60 q^{5} + 10\!\cdots\!76 q^{6} + 31\!\cdots\!00 q^{7} - 14\!\cdots\!60 q^{8} + 92\!\cdots\!36 q^{9} - 35\!\cdots\!40 q^{10}+ \cdots + 30\!\cdots\!72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.51519e14 −1.76613 −0.883067 − 0.469247i \(-0.844526\pi\)
−0.883067 + 0.469247i \(0.844526\pi\)
\(3\) 6.56090e22 1.42464 0.712318 − 0.701857i \(-0.247646\pi\)
0.712318 + 0.701857i \(0.247646\pi\)
\(4\) 8.39514e28 2.11923
\(5\) 2.79946e33 1.76197 0.880986 − 0.473143i \(-0.156881\pi\)
0.880986 + 0.473143i \(0.156881\pi\)
\(6\) −2.30628e37 −2.51610
\(7\) 5.53054e39 0.398667 0.199334 − 0.979932i \(-0.436122\pi\)
0.199334 + 0.979932i \(0.436122\pi\)
\(8\) −1.55854e43 −1.97671
\(9\) 2.18364e45 1.02959
\(10\) −9.84062e47 −3.11188
\(11\) 3.60788e48 0.123339 0.0616694 − 0.998097i \(-0.480358\pi\)
0.0616694 + 0.998097i \(0.480358\pi\)
\(12\) 5.50796e51 3.01913
\(13\) 1.06916e53 1.30838 0.654189 − 0.756331i \(-0.273010\pi\)
0.654189 + 0.756331i \(0.273010\pi\)
\(14\) −1.94409e54 −0.704100
\(15\) 1.83670e56 2.51017
\(16\) 2.15290e57 1.37191
\(17\) 3.05031e58 1.09149 0.545746 − 0.837950i \(-0.316246\pi\)
0.545746 + 0.837950i \(0.316246\pi\)
\(18\) −7.67592e59 −1.81839
\(19\) −5.36133e58 −0.00973815 −0.00486907 − 0.999988i \(-0.501550\pi\)
−0.00486907 + 0.999988i \(0.501550\pi\)
\(20\) 2.35018e62 3.73402
\(21\) 3.62853e62 0.567956
\(22\) −1.26824e63 −0.217833
\(23\) 3.27765e64 0.681539 0.340770 − 0.940147i \(-0.389312\pi\)
0.340770 + 0.940147i \(0.389312\pi\)
\(24\) −1.02254e66 −2.81609
\(25\) 5.31261e66 2.10454
\(26\) −3.75830e67 −2.31077
\(27\) 4.11692e66 0.0421496
\(28\) 4.64296e68 0.844868
\(29\) −3.24861e69 −1.11633 −0.558165 − 0.829730i \(-0.688495\pi\)
−0.558165 + 0.829730i \(0.688495\pi\)
\(30\) −6.45633e70 −4.43329
\(31\) −4.63359e70 −0.670251 −0.335125 − 0.942174i \(-0.608779\pi\)
−0.335125 + 0.942174i \(0.608779\pi\)
\(32\) −1.39384e71 −0.446261
\(33\) 2.36709e71 0.175713
\(34\) −1.07224e73 −1.92772
\(35\) 1.54825e73 0.702440
\(36\) 1.83320e74 2.18193
\(37\) 2.22715e74 0.721384 0.360692 − 0.932685i \(-0.382541\pi\)
0.360692 + 0.932685i \(0.382541\pi\)
\(38\) 1.88461e73 0.0171989
\(39\) 7.01465e75 1.86396
\(40\) −4.36306e76 −3.48291
\(41\) −8.69382e75 −0.214772 −0.107386 − 0.994217i \(-0.534248\pi\)
−0.107386 + 0.994217i \(0.534248\pi\)
\(42\) −1.27550e77 −1.00309
\(43\) 4.86197e77 1.25043 0.625217 − 0.780451i \(-0.285010\pi\)
0.625217 + 0.780451i \(0.285010\pi\)
\(44\) 3.02887e77 0.261383
\(45\) 6.11302e78 1.81410
\(46\) −1.15216e79 −1.20369
\(47\) −1.28855e79 −0.484677 −0.242339 − 0.970192i \(-0.577915\pi\)
−0.242339 + 0.970192i \(0.577915\pi\)
\(48\) 1.41250e80 1.95447
\(49\) −1.61861e80 −0.841064
\(50\) −1.86748e81 −3.71690
\(51\) 2.00128e81 1.55498
\(52\) 8.97574e81 2.77275
\(53\) 4.42467e81 0.553066 0.276533 − 0.961004i \(-0.410815\pi\)
0.276533 + 0.961004i \(0.410815\pi\)
\(54\) −1.44717e81 −0.0744418
\(55\) 1.01001e82 0.217319
\(56\) −8.61956e82 −0.788050
\(57\) −3.51751e81 −0.0138733
\(58\) 1.14195e84 1.97159
\(59\) −1.94954e84 −1.49438 −0.747190 − 0.664610i \(-0.768598\pi\)
−0.747190 + 0.664610i \(0.768598\pi\)
\(60\) 1.54193e85 5.31962
\(61\) −5.98141e84 −0.941102 −0.470551 − 0.882373i \(-0.655945\pi\)
−0.470551 + 0.882373i \(0.655945\pi\)
\(62\) 1.62879e85 1.18375
\(63\) 1.20767e85 0.410462
\(64\) −3.62891e85 −0.583750
\(65\) 2.99307e86 2.30532
\(66\) −8.32078e85 −0.310332
\(67\) −6.60161e86 −1.20530 −0.602652 − 0.798004i \(-0.705889\pi\)
−0.602652 + 0.798004i \(0.705889\pi\)
\(68\) 2.56078e87 2.31312
\(69\) 2.15044e87 0.970945
\(70\) −5.44240e87 −1.24060
\(71\) −7.94968e87 −0.923801 −0.461900 − 0.886932i \(-0.652832\pi\)
−0.461900 + 0.886932i \(0.652832\pi\)
\(72\) −3.40329e88 −2.03519
\(73\) −4.40521e88 −1.36814 −0.684070 − 0.729417i \(-0.739792\pi\)
−0.684070 + 0.729417i \(0.739792\pi\)
\(74\) −7.82883e88 −1.27406
\(75\) 3.48555e89 2.99821
\(76\) −4.50091e87 −0.0206374
\(77\) 1.99535e88 0.0491712
\(78\) −2.46578e90 −3.29201
\(79\) 1.51977e90 1.10787 0.553937 − 0.832558i \(-0.313125\pi\)
0.553937 + 0.832558i \(0.313125\pi\)
\(80\) 6.02695e90 2.41726
\(81\) −4.36117e90 −0.969538
\(82\) 3.05604e90 0.379316
\(83\) 1.48872e91 1.03897 0.519485 − 0.854479i \(-0.326124\pi\)
0.519485 + 0.854479i \(0.326124\pi\)
\(84\) 3.04620e91 1.20363
\(85\) 8.53923e91 1.92318
\(86\) −1.70907e92 −2.20843
\(87\) −2.13138e92 −1.59036
\(88\) −5.62302e91 −0.243805
\(89\) 5.31083e92 1.34629 0.673143 − 0.739512i \(-0.264944\pi\)
0.673143 + 0.739512i \(0.264944\pi\)
\(90\) −2.14884e93 −3.20395
\(91\) 5.91303e92 0.521608
\(92\) 2.75163e93 1.44434
\(93\) −3.04005e93 −0.954863
\(94\) 4.52949e93 0.856005
\(95\) −1.50088e92 −0.0171583
\(96\) −9.14485e93 −0.635759
\(97\) −2.46425e94 −1.04719 −0.523597 − 0.851966i \(-0.675410\pi\)
−0.523597 + 0.851966i \(0.675410\pi\)
\(98\) 5.68973e94 1.48543
\(99\) 7.87833e93 0.126988
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.1.2 ✓ 8
3.2 odd 2 9.96.a.c.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.96.a.a.1.2 ✓ 8 1.1 even 1 trivial
9.96.a.c.1.7 8 3.2 odd 2