↑

Properties

Label 1.76.a.a.1.6
Level $1$
Weight $76$
Character 1.1
Self dual yes
Analytic conductor $35.623$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.6
Root \(5.23553e9\) 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)
 
\(f(q)\) \(=\) \(q+3.67444e11 q^{2} -8.83049e17 q^{3} +9.72365e22 q^{4} +2.43619e26 q^{5} -3.24471e29 q^{6} +1.21530e31 q^{7} +2.18473e34 q^{8} +1.71508e35 q^{9} +8.95165e37 q^{10} -5.65037e38 q^{11} -8.58645e40 q^{12} +5.30382e40 q^{13} +4.46553e42 q^{14} -2.15128e44 q^{15} +4.35419e45 q^{16} +1.14821e45 q^{17} +6.30198e46 q^{18} +1.24645e48 q^{19} +2.36887e49 q^{20} -1.07316e49 q^{21} -2.07620e50 q^{22} +8.02205e50 q^{23} -1.92923e52 q^{24} +3.28805e52 q^{25} +1.94886e52 q^{26} +3.85679e53 q^{27} +1.18171e54 q^{28} +8.53673e54 q^{29} -7.90474e55 q^{30} -6.18737e55 q^{31} +7.74555e56 q^{32} +4.98956e56 q^{33} +4.21904e56 q^{34} +2.96069e57 q^{35} +1.66769e58 q^{36} -5.98766e58 q^{37} +4.58002e59 q^{38} -4.68354e58 q^{39} +5.32243e60 q^{40} -5.32628e60 q^{41} -3.94328e60 q^{42} +2.59254e61 q^{43} -5.49422e61 q^{44} +4.17827e61 q^{45} +2.94766e62 q^{46} -5.76440e62 q^{47} -3.84496e63 q^{48} -2.26417e63 q^{49} +1.20818e64 q^{50} -1.01393e63 q^{51} +5.15725e63 q^{52} +5.92558e63 q^{53} +1.41716e65 q^{54} -1.37654e65 q^{55} +2.65510e65 q^{56} -1.10068e66 q^{57} +3.13677e66 q^{58} +2.82712e66 q^{59} -2.09183e67 q^{60} +5.43239e66 q^{61} -2.27352e67 q^{62} +2.08433e66 q^{63} +1.20109e68 q^{64} +1.29211e67 q^{65} +1.83338e68 q^{66} -2.61376e68 q^{67} +1.11648e68 q^{68} -7.08386e68 q^{69} +1.08789e69 q^{70} +1.78760e67 q^{71} +3.74700e69 q^{72} -1.09051e70 q^{73} -2.20013e70 q^{74} -2.90351e70 q^{75} +1.21201e71 q^{76} -6.86687e69 q^{77} -1.72094e70 q^{78} -1.49375e71 q^{79} +1.06076e72 q^{80} -4.44896e71 q^{81} -1.95711e72 q^{82} +3.39100e71 q^{83} -1.04351e72 q^{84} +2.79727e71 q^{85} +9.52613e72 q^{86} -7.53835e72 q^{87} -1.23446e73 q^{88} +2.45405e72 q^{89} +1.53528e73 q^{90} +6.44571e71 q^{91} +7.80036e73 q^{92} +5.46375e73 q^{93} -2.11810e74 q^{94} +3.03660e74 q^{95} -6.83970e74 q^{96} +1.36838e74 q^{97} -8.31957e74 q^{98} -9.69086e73 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 57080822040 q^{2} - 78\!\cdots\!40 q^{3} + 17\!\cdots\!28 q^{4} - 38\!\cdots\!40 q^{5} + 31\!\cdots\!92 q^{6} + 19\!\cdots\!00 q^{7} + 44\!\cdots\!20 q^{8} + 21\!\cdots\!82 q^{9} + 13\!\cdots\!60 q^{10}+ \cdots - 18\!\cdots\!36 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.67444e11 1.89046 0.945229 − 0.326409i \(-0.105839\pi\)
0.945229 + 0.326409i \(0.105839\pi\)
\(3\) −8.83049e17 −1.13224 −0.566119 − 0.824324i \(-0.691556\pi\)
−0.566119 + 0.824324i \(0.691556\pi\)
\(4\) 9.72365e22 2.57383
\(5\) 2.43619e26 1.49739 0.748697 − 0.662912i \(-0.230680\pi\)
0.748697 + 0.662912i \(0.230680\pi\)
\(6\) −3.24471e29 −2.14045
\(7\) 1.21530e31 0.247460 0.123730 − 0.992316i \(-0.460514\pi\)
0.123730 + 0.992316i \(0.460514\pi\)
\(8\) 2.18473e34 2.97525
\(9\) 1.71508e35 0.281962
\(10\) 8.95165e37 2.83076
\(11\) −5.65037e38 −0.501016 −0.250508 − 0.968115i \(-0.580598\pi\)
−0.250508 + 0.968115i \(0.580598\pi\)
\(12\) −8.58645e40 −2.91419
\(13\) 5.30382e40 0.0894774 0.0447387 − 0.998999i \(-0.485754\pi\)
0.0447387 + 0.998999i \(0.485754\pi\)
\(14\) 4.46553e42 0.467813
\(15\) −2.15128e44 −1.69541
\(16\) 4.35419e45 3.05076
\(17\) 1.14821e45 0.0828300 0.0414150 − 0.999142i \(-0.486813\pi\)
0.0414150 + 0.999142i \(0.486813\pi\)
\(18\) 6.30198e46 0.533038
\(19\) 1.24645e48 1.38808 0.694042 − 0.719935i \(-0.255828\pi\)
0.694042 + 0.719935i \(0.255828\pi\)
\(20\) 2.36887e49 3.85404
\(21\) −1.07316e49 −0.280184
\(22\) −2.07620e50 −0.947149
\(23\) 8.02205e50 0.691022 0.345511 − 0.938415i \(-0.387706\pi\)
0.345511 + 0.938415i \(0.387706\pi\)
\(24\) −1.92923e52 −3.36869
\(25\) 3.28805e52 1.24219
\(26\) 1.94886e52 0.169153
\(27\) 3.85679e53 0.812989
\(28\) 1.18171e54 0.636920
\(29\) 8.53673e54 1.23415 0.617073 − 0.786906i \(-0.288318\pi\)
0.617073 + 0.786906i \(0.288318\pi\)
\(30\) −7.90474e55 −3.20509
\(31\) −6.18737e55 −0.733572 −0.366786 − 0.930305i \(-0.619542\pi\)
−0.366786 + 0.930305i \(0.619542\pi\)
\(32\) 7.74555e56 2.79208
\(33\) 4.98956e56 0.567269
\(34\) 4.21904e56 0.156587
\(35\) 2.96069e57 0.370546
\(36\) 1.66769e58 0.725723
\(37\) −5.98766e58 −0.932594 −0.466297 − 0.884628i \(-0.654412\pi\)
−0.466297 + 0.884628i \(0.654412\pi\)
\(38\) 4.58002e59 2.62411
\(39\) −4.68354e58 −0.101310
\(40\) 5.32243e60 4.45513
\(41\) −5.32628e60 −1.76616 −0.883079 − 0.469224i \(-0.844534\pi\)
−0.883079 + 0.469224i \(0.844534\pi\)
\(42\) −3.94328e60 −0.529675
\(43\) 2.59254e61 1.44098 0.720489 − 0.693466i \(-0.243917\pi\)
0.720489 + 0.693466i \(0.243917\pi\)
\(44\) −5.49422e61 −1.28953
\(45\) 4.17827e61 0.422209
\(46\) 2.94766e62 1.30635
\(47\) −5.76440e62 −1.14047 −0.570236 − 0.821481i \(-0.693148\pi\)
−0.570236 + 0.821481i \(0.693148\pi\)
\(48\) −3.84496e63 −3.45419
\(49\) −2.26417e63 −0.938763
\(50\) 1.20818e64 2.34831
\(51\) −1.01393e63 −0.0937833
\(52\) 5.15725e63 0.230299
\(53\) 5.92558e63 0.129535 0.0647673 − 0.997900i \(-0.479370\pi\)
0.0647673 + 0.997900i \(0.479370\pi\)
\(54\) 1.41716e65 1.53692
\(55\) −1.37654e65 −0.750219
\(56\) 2.65510e65 0.736257
\(57\) −1.10068e66 −1.57164
\(58\) 3.13677e66 2.33310
\(59\) 2.82712e66 1.10763 0.553814 − 0.832640i \(-0.313172\pi\)
0.553814 + 0.832640i \(0.313172\pi\)
\(60\) −2.09183e67 −4.36369
\(61\) 5.43239e66 0.609708 0.304854 − 0.952399i \(-0.401392\pi\)
0.304854 + 0.952399i \(0.401392\pi\)
\(62\) −2.27352e67 −1.38679
\(63\) 2.08433e66 0.0697745
\(64\) 1.20109e68 2.22755
\(65\) 1.29211e67 0.133983
\(66\) 1.83338e68 1.07240
\(67\) −2.61376e68 −0.869883 −0.434941 − 0.900459i \(-0.643231\pi\)
−0.434941 + 0.900459i \(0.643231\pi\)
\(68\) 1.11648e68 0.213190
\(69\) −7.08386e68 −0.782401
\(70\) 1.08789e69 0.700501
\(71\) 1.78760e67 0.00676213 0.00338106 − 0.999994i \(-0.498924\pi\)
0.00338106 + 0.999994i \(0.498924\pi\)
\(72\) 3.74700e69 0.838910
\(73\) −1.09051e70 −1.45554 −0.727770 − 0.685821i \(-0.759443\pi\)
−0.727770 + 0.685821i \(0.759443\pi\)
\(74\) −2.20013e70 −1.76303
\(75\) −2.90351e70 −1.40646
\(76\) 1.21201e71 3.57269
\(77\) −6.86687e69 −0.123981
\(78\) −1.72094e70 −0.191522
\(79\) −1.49375e71 −1.03101 −0.515504 − 0.856887i \(-0.672395\pi\)
−0.515504 + 0.856887i \(0.672395\pi\)
\(80\) 1.06076e72 4.56820
\(81\) −4.44896e71 −1.20246
\(82\) −1.95711e72 −3.33885
\(83\) 3.39100e71 0.367197 0.183599 − 0.983001i \(-0.441225\pi\)
0.183599 + 0.983001i \(0.441225\pi\)
\(84\) −1.04351e72 −0.721145
\(85\) 2.79727e71 0.124029
\(86\) 9.52613e72 2.72411
\(87\) −7.53835e72 −1.39735
\(88\) −1.23446e73 −1.49065
\(89\) 2.45405e72 0.193981 0.0969905 − 0.995285i \(-0.469078\pi\)
0.0969905 + 0.995285i \(0.469078\pi\)
\(90\) 1.53528e73 0.798168
\(91\) 6.44571e71 0.0221421
\(92\) 7.80036e73 1.77857
\(93\) 5.46375e73 0.830578
\(94\) −2.11810e74 −2.15601
\(95\) 3.03660e74 2.07851
\(96\) −6.83970e74 −3.16130
\(97\) 1.36838e74 0.428811 0.214405 − 0.976745i \(-0.431219\pi\)
0.214405 + 0.976745i \(0.431219\pi\)
\(98\) −8.31957e74 −1.77469
\(99\) −9.69086e73 −0.141268
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.1.6 ✓ 6
3.2 odd 2 9.76.a.c.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.76.a.a.1.6 ✓ 6 1.1 even 1 trivial
9.76.a.c.1.1 6 3.2 odd 2