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Properties

Label 81.8.a.c.1.4
Level $81$
Weight $8$
Character 81.1
Self dual yes
Analytic conductor $25.303$
Analytic rank $1$
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] = [81,8,Mod(1,81)] 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("81.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 81.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.3031870642\)
Analytic rank: \(1\)
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: \( x^{6} - 401x^{4} - 1212x^{3} + 17752x^{2} + 15108x - 22632 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{9} \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-9.27695\) of defining polynomial
Character \(\chi\) \(=\) 81.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.07224 q^{2} -126.850 q^{4} +95.9733 q^{5} +378.001 q^{7} -273.261 q^{8} +102.906 q^{10} +6873.26 q^{11} -9653.28 q^{13} +405.308 q^{14} +15943.8 q^{16} -21431.3 q^{17} +5518.94 q^{19} -12174.2 q^{20} +7369.79 q^{22} -62972.9 q^{23} -68914.1 q^{25} -10350.6 q^{26} -47949.5 q^{28} -222226. q^{29} +115458. q^{31} +52073.0 q^{32} -22979.5 q^{34} +36277.9 q^{35} +81737.7 q^{37} +5917.64 q^{38} -26225.7 q^{40} -597547. q^{41} +67748.4 q^{43} -871875. q^{44} -67522.1 q^{46} -303481. q^{47} -680659. q^{49} -73892.5 q^{50} +1.22452e6 q^{52} -846755. q^{53} +659649. q^{55} -103293. q^{56} -238280. q^{58} -1.58624e6 q^{59} -2.25412e6 q^{61} +123799. q^{62} -1.98498e6 q^{64} -926457. q^{65} +3.02345e6 q^{67} +2.71856e6 q^{68} +38898.7 q^{70} +4.41675e6 q^{71} +2.21484e6 q^{73} +87642.5 q^{74} -700080. q^{76} +2.59809e6 q^{77} +307642. q^{79} +1.53018e6 q^{80} -640714. q^{82} +3.15469e6 q^{83} -2.05683e6 q^{85} +72642.6 q^{86} -1.87819e6 q^{88} -1.93441e6 q^{89} -3.64895e6 q^{91} +7.98813e6 q^{92} -325404. q^{94} +529671. q^{95} +9.89056e6 q^{97} -729830. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{2} + 321 q^{4} - 180 q^{5} + 84 q^{7} - 2961 q^{8} + 126 q^{10} - 8460 q^{11} + 1848 q^{13} - 16272 q^{14} + 12417 q^{16} - 15282 q^{17} + 12216 q^{19} - 40788 q^{20} + 35001 q^{22} - 51588 q^{23}+ \cdots + 47916657 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.07224 0.0947736 0.0473868 − 0.998877i \(-0.484911\pi\)
0.0473868 + 0.998877i \(0.484911\pi\)
\(3\) 0 0
\(4\) −126.850 −0.991018
\(5\) 95.9733 0.343364 0.171682 − 0.985152i \(-0.445080\pi\)
0.171682 + 0.985152i \(0.445080\pi\)
\(6\) 0 0
\(7\) 378.001 0.416533 0.208266 − 0.978072i \(-0.433218\pi\)
0.208266 + 0.978072i \(0.433218\pi\)
\(8\) −273.261 −0.188696
\(9\) 0 0
\(10\) 102.906 0.0325419
\(11\) 6873.26 1.55700 0.778499 − 0.627646i \(-0.215981\pi\)
0.778499 + 0.627646i \(0.215981\pi\)
\(12\) 0 0
\(13\) −9653.28 −1.21863 −0.609317 − 0.792927i \(-0.708556\pi\)
−0.609317 + 0.792927i \(0.708556\pi\)
\(14\) 405.308 0.0394763
\(15\) 0 0
\(16\) 15943.8 0.973135
\(17\) −21431.3 −1.05798 −0.528989 − 0.848629i \(-0.677429\pi\)
−0.528989 + 0.848629i \(0.677429\pi\)
\(18\) 0 0
\(19\) 5518.94 0.184594 0.0922972 − 0.995732i \(-0.470579\pi\)
0.0922972 + 0.995732i \(0.470579\pi\)
\(20\) −12174.2 −0.340280
\(21\) 0 0
\(22\) 7369.79 0.147562
\(23\) −62972.9 −1.07921 −0.539605 − 0.841918i \(-0.681426\pi\)
−0.539605 + 0.841918i \(0.681426\pi\)
\(24\) 0 0
\(25\) −68914.1 −0.882101
\(26\) −10350.6 −0.115494
\(27\) 0 0
\(28\) −47949.5 −0.412792
\(29\) −222226. −1.69201 −0.846004 − 0.533177i \(-0.820998\pi\)
−0.846004 + 0.533177i \(0.820998\pi\)
\(30\) 0 0
\(31\) 115458. 0.696079 0.348040 − 0.937480i \(-0.386847\pi\)
0.348040 + 0.937480i \(0.386847\pi\)
\(32\) 52073.0 0.280923
\(33\) 0 0
\(34\) −22979.5 −0.100268
\(35\) 36277.9 0.143023
\(36\) 0 0
\(37\) 81737.7 0.265287 0.132644 − 0.991164i \(-0.457653\pi\)
0.132644 + 0.991164i \(0.457653\pi\)
\(38\) 5917.64 0.0174947
\(39\) 0 0
\(40\) −26225.7 −0.0647915
\(41\) −597547. −1.35403 −0.677015 − 0.735969i \(-0.736727\pi\)
−0.677015 + 0.735969i \(0.736727\pi\)
\(42\) 0 0
\(43\) 67748.4 0.129945 0.0649725 − 0.997887i \(-0.479304\pi\)
0.0649725 + 0.997887i \(0.479304\pi\)
\(44\) −871875. −1.54301
\(45\) 0 0
\(46\) −67522.1 −0.102281
\(47\) −303481. −0.426372 −0.213186 − 0.977012i \(-0.568384\pi\)
−0.213186 + 0.977012i \(0.568384\pi\)
\(48\) 0 0
\(49\) −680659. −0.826500
\(50\) −73892.5 −0.0835999
\(51\) 0 0
\(52\) 1.22452e6 1.20769
\(53\) −846755. −0.781254 −0.390627 − 0.920549i \(-0.627742\pi\)
−0.390627 + 0.920549i \(0.627742\pi\)
\(54\) 0 0
\(55\) 659649. 0.534618
\(56\) −103293. −0.0785981
\(57\) 0 0
\(58\) −238280. −0.160358
\(59\) −1.58624e6 −1.00551 −0.502755 − 0.864429i \(-0.667680\pi\)
−0.502755 + 0.864429i \(0.667680\pi\)
\(60\) 0 0
\(61\) −2.25412e6 −1.27152 −0.635760 − 0.771886i \(-0.719313\pi\)
−0.635760 + 0.771886i \(0.719313\pi\)
\(62\) 123799. 0.0659699
\(63\) 0 0
\(64\) −1.98498e6 −0.946510
\(65\) −926457. −0.418435
\(66\) 0 0
\(67\) 3.02345e6 1.22812 0.614060 − 0.789259i \(-0.289535\pi\)
0.614060 + 0.789259i \(0.289535\pi\)
\(68\) 2.71856e6 1.04848
\(69\) 0 0
\(70\) 38898.7 0.0135548
\(71\) 4.41675e6 1.46453 0.732266 − 0.681018i \(-0.238463\pi\)
0.732266 + 0.681018i \(0.238463\pi\)
\(72\) 0 0
\(73\) 2.21484e6 0.666366 0.333183 − 0.942862i \(-0.391877\pi\)
0.333183 + 0.942862i \(0.391877\pi\)
\(74\) 87642.5 0.0251422
\(75\) 0 0
\(76\) −700080. −0.182936
\(77\) 2.59809e6 0.648541
\(78\) 0 0
\(79\) 307642. 0.0702022 0.0351011 − 0.999384i \(-0.488825\pi\)
0.0351011 + 0.999384i \(0.488825\pi\)
\(80\) 1.53018e6 0.334140
\(81\) 0 0
\(82\) −640714. −0.128326
\(83\) 3.15469e6 0.605597 0.302798 − 0.953055i \(-0.402079\pi\)
0.302798 + 0.953055i \(0.402079\pi\)
\(84\) 0 0
\(85\) −2.05683e6 −0.363272
\(86\) 72642.6 0.0123154
\(87\) 0 0
\(88\) −1.87819e6 −0.293799
\(89\) −1.93441e6 −0.290859 −0.145430 − 0.989369i \(-0.546456\pi\)
−0.145430 + 0.989369i \(0.546456\pi\)
\(90\) 0 0
\(91\) −3.64895e6 −0.507601
\(92\) 7.98813e6 1.06952
\(93\) 0 0
\(94\) −325404. −0.0404088
\(95\) 529671. 0.0633831
\(96\) 0 0
\(97\) 9.89056e6 1.10032 0.550161 − 0.835059i \(-0.314566\pi\)
0.550161 + 0.835059i \(0.314566\pi\)
\(98\) −729830. −0.0783304
\(99\) 0 0
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 81.8.a.c.1.4 6
3.2 odd 2 81.8.a.e.1.3 6
9.2 odd 6 9.8.c.a.4.4 ✓ 12
9.4 even 3 27.8.c.a.19.3 12
9.5 odd 6 9.8.c.a.7.4 yes 12
9.7 even 3 27.8.c.a.10.3 12
36.7 odd 6 432.8.i.c.145.3 12
36.11 even 6 144.8.i.c.49.4 12
36.23 even 6 144.8.i.c.97.4 12
36.31 odd 6 432.8.i.c.289.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.4 ✓ 12 9.2 odd 6
9.8.c.a.7.4 yes 12 9.5 odd 6
27.8.c.a.10.3 12 9.7 even 3
27.8.c.a.19.3 12 9.4 even 3
81.8.a.c.1.4 6 1.1 even 1 trivial
81.8.a.e.1.3 6 3.2 odd 2
144.8.i.c.49.4 12 36.11 even 6
144.8.i.c.97.4 12 36.23 even 6
432.8.i.c.145.3 12 36.7 odd 6
432.8.i.c.289.3 12 36.31 odd 6