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Properties

Label 31.6.a.b.1.7
Level $31$
Weight $6$
Character 31.1
Self dual yes
Analytic conductor $4.972$
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] = [31,6,Mod(1,31)] 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("31.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 31.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.97189841420\)
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: \( x^{8} - x^{7} - 199x^{6} + 256x^{5} + 12633x^{4} - 18583x^{3} - 260319x^{2} + 410640x + 275908 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 5\cdot 13 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-7.89102\) of defining polynomial
Character \(\chi\) \(=\) 31.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.89102 q^{2} +18.9313 q^{3} +47.0502 q^{4} -33.8630 q^{5} +168.319 q^{6} -134.850 q^{7} +133.811 q^{8} +115.395 q^{9} -301.077 q^{10} -191.415 q^{11} +890.722 q^{12} +766.219 q^{13} -1198.95 q^{14} -641.071 q^{15} -315.888 q^{16} +1357.39 q^{17} +1025.98 q^{18} -164.832 q^{19} -1593.26 q^{20} -2552.89 q^{21} -1701.88 q^{22} +3313.68 q^{23} +2533.22 q^{24} -1978.30 q^{25} +6812.46 q^{26} -2415.73 q^{27} -6344.71 q^{28} -6675.50 q^{29} -5699.78 q^{30} +961.000 q^{31} -7090.52 q^{32} -3623.74 q^{33} +12068.6 q^{34} +4566.42 q^{35} +5429.35 q^{36} +10334.6 q^{37} -1465.53 q^{38} +14505.5 q^{39} -4531.25 q^{40} +13434.9 q^{41} -22697.7 q^{42} +16907.4 q^{43} -9006.12 q^{44} -3907.62 q^{45} +29462.0 q^{46} -18714.2 q^{47} -5980.17 q^{48} +1377.48 q^{49} -17589.1 q^{50} +25697.2 q^{51} +36050.7 q^{52} +11410.8 q^{53} -21478.3 q^{54} +6481.90 q^{55} -18044.4 q^{56} -3120.50 q^{57} -59351.9 q^{58} +7475.61 q^{59} -30162.5 q^{60} -55634.9 q^{61} +8544.27 q^{62} -15561.0 q^{63} -52933.5 q^{64} -25946.5 q^{65} -32218.8 q^{66} +26388.2 q^{67} +63865.5 q^{68} +62732.4 q^{69} +40600.1 q^{70} -14258.2 q^{71} +15441.1 q^{72} +6721.72 q^{73} +91885.2 q^{74} -37451.8 q^{75} -7755.39 q^{76} +25812.3 q^{77} +128969. q^{78} -29059.9 q^{79} +10696.9 q^{80} -73774.0 q^{81} +119450. q^{82} -37568.9 q^{83} -120114. q^{84} -45965.3 q^{85} +150324. q^{86} -126376. q^{87} -25613.5 q^{88} +91193.1 q^{89} -34742.7 q^{90} -103325. q^{91} +155909. q^{92} +18193.0 q^{93} -166388. q^{94} +5581.72 q^{95} -134233. q^{96} +95596.4 q^{97} +12247.2 q^{98} -22088.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 7 q^{2} - 2 q^{3} + 149 q^{4} + 128 q^{5} + 72 q^{6} + 88 q^{7} + 924 q^{8} + 1512 q^{9} + 1581 q^{10} + 574 q^{11} - 46 q^{12} - 122 q^{13} - 309 q^{14} - 524 q^{15} + 833 q^{16} + 1932 q^{17} - 6845 q^{18}+ \cdots - 180150 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.89102 1.57172 0.785862 − 0.618402i \(-0.212220\pi\)
0.785862 + 0.618402i \(0.212220\pi\)
\(3\) 18.9313 1.21444 0.607222 − 0.794532i \(-0.292284\pi\)
0.607222 + 0.794532i \(0.292284\pi\)
\(4\) 47.0502 1.47032
\(5\) −33.8630 −0.605760 −0.302880 − 0.953029i \(-0.597948\pi\)
−0.302880 + 0.953029i \(0.597948\pi\)
\(6\) 168.319 1.90877
\(7\) −134.850 −1.04017 −0.520086 − 0.854114i \(-0.674100\pi\)
−0.520086 + 0.854114i \(0.674100\pi\)
\(8\) 133.811 0.739209
\(9\) 115.395 0.474876
\(10\) −301.077 −0.952088
\(11\) −191.415 −0.476974 −0.238487 − 0.971146i \(-0.576651\pi\)
−0.238487 + 0.971146i \(0.576651\pi\)
\(12\) 890.722 1.78562
\(13\) 766.219 1.25746 0.628730 − 0.777623i \(-0.283575\pi\)
0.628730 + 0.777623i \(0.283575\pi\)
\(14\) −1198.95 −1.63486
\(15\) −641.071 −0.735662
\(16\) −315.888 −0.308484
\(17\) 1357.39 1.13915 0.569577 − 0.821938i \(-0.307107\pi\)
0.569577 + 0.821938i \(0.307107\pi\)
\(18\) 1025.98 0.746375
\(19\) −164.832 −0.104751 −0.0523755 − 0.998627i \(-0.516679\pi\)
−0.0523755 + 0.998627i \(0.516679\pi\)
\(20\) −1593.26 −0.890659
\(21\) −2552.89 −1.26323
\(22\) −1701.88 −0.749672
\(23\) 3313.68 1.30615 0.653073 − 0.757295i \(-0.273480\pi\)
0.653073 + 0.757295i \(0.273480\pi\)
\(24\) 2533.22 0.897729
\(25\) −1978.30 −0.633055
\(26\) 6812.46 1.97638
\(27\) −2415.73 −0.637734
\(28\) −6344.71 −1.52938
\(29\) −6675.50 −1.47397 −0.736985 − 0.675909i \(-0.763751\pi\)
−0.736985 + 0.675909i \(0.763751\pi\)
\(30\) −5699.78 −1.15626
\(31\) 961.000 0.179605
\(32\) −7090.52 −1.22406
\(33\) −3623.74 −0.579259
\(34\) 12068.6 1.79044
\(35\) 4566.42 0.630095
\(36\) 5429.35 0.698219
\(37\) 10334.6 1.24105 0.620526 − 0.784186i \(-0.286919\pi\)
0.620526 + 0.784186i \(0.286919\pi\)
\(38\) −1465.53 −0.164640
\(39\) 14505.5 1.52712
\(40\) −4531.25 −0.447783
\(41\) 13434.9 1.24817 0.624085 − 0.781356i \(-0.285472\pi\)
0.624085 + 0.781356i \(0.285472\pi\)
\(42\) −22697.7 −1.98545
\(43\) 16907.4 1.39446 0.697228 − 0.716850i \(-0.254417\pi\)
0.697228 + 0.716850i \(0.254417\pi\)
\(44\) −9006.12 −0.701303
\(45\) −3907.62 −0.287661
\(46\) 29462.0 2.05290
\(47\) −18714.2 −1.23574 −0.617869 − 0.786281i \(-0.712004\pi\)
−0.617869 + 0.786281i \(0.712004\pi\)
\(48\) −5980.17 −0.374637
\(49\) 1377.48 0.0819589
\(50\) −17589.1 −0.994988
\(51\) 25697.2 1.38344
\(52\) 36050.7 1.84887
\(53\) 11410.8 0.557990 0.278995 − 0.960293i \(-0.409999\pi\)
0.278995 + 0.960293i \(0.409999\pi\)
\(54\) −21478.3 −1.00234
\(55\) 6481.90 0.288932
\(56\) −18044.4 −0.768905
\(57\) −3120.50 −0.127214
\(58\) −59351.9 −2.31667
\(59\) 7475.61 0.279587 0.139793 − 0.990181i \(-0.455356\pi\)
0.139793 + 0.990181i \(0.455356\pi\)
\(60\) −30162.5 −1.08166
\(61\) −55634.9 −1.91436 −0.957179 − 0.289498i \(-0.906512\pi\)
−0.957179 + 0.289498i \(0.906512\pi\)
\(62\) 8544.27 0.282290
\(63\) −15561.0 −0.493953
\(64\) −52933.5 −1.61540
\(65\) −25946.5 −0.761719
\(66\) −32218.8 −0.910435
\(67\) 26388.2 0.718161 0.359081 − 0.933307i \(-0.383090\pi\)
0.359081 + 0.933307i \(0.383090\pi\)
\(68\) 63865.5 1.67492
\(69\) 62732.4 1.58624
\(70\) 40600.1 0.990335
\(71\) −14258.2 −0.335674 −0.167837 − 0.985815i \(-0.553678\pi\)
−0.167837 + 0.985815i \(0.553678\pi\)
\(72\) 15441.1 0.351033
\(73\) 6721.72 0.147630 0.0738148 − 0.997272i \(-0.476483\pi\)
0.0738148 + 0.997272i \(0.476483\pi\)
\(74\) 91885.2 1.95059
\(75\) −37451.8 −0.768810
\(76\) −7755.39 −0.154017
\(77\) 25812.3 0.496135
\(78\) 128969. 2.40021
\(79\) −29059.9 −0.523874 −0.261937 − 0.965085i \(-0.584361\pi\)
−0.261937 + 0.965085i \(0.584361\pi\)
\(80\) 10696.9 0.186867
\(81\) −73774.0 −1.24937
\(82\) 119450. 1.96178
\(83\) −37568.9 −0.598595 −0.299298 − 0.954160i \(-0.596752\pi\)
−0.299298 + 0.954160i \(0.596752\pi\)
\(84\) −120114. −1.85735
\(85\) −45965.3 −0.690054
\(86\) 150324. 2.19170
\(87\) −126376. −1.79005
\(88\) −25613.5 −0.352584
\(89\) 91193.1 1.22036 0.610179 − 0.792264i \(-0.291098\pi\)
0.610179 + 0.792264i \(0.291098\pi\)
\(90\) −34742.7 −0.452124
\(91\) −103325. −1.30798
\(92\) 155909. 1.92045
\(93\) 18193.0 0.218121
\(94\) −166388. −1.94224
\(95\) 5581.72 0.0634540
\(96\) −134233. −1.48655
\(97\) 95596.4 1.03160 0.515801 − 0.856708i \(-0.327494\pi\)
0.515801 + 0.856708i \(0.327494\pi\)
\(98\) 12247.2 0.128817
\(99\) −22088.4 −0.226504
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 31.6.a.b.1.7 ✓ 8
3.2 odd 2 279.6.a.f.1.2 8
4.3 odd 2 496.6.a.h.1.3 8
5.4 even 2 775.6.a.b.1.2 8
31.30 odd 2 961.6.a.c.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.6.a.b.1.7 ✓ 8 1.1 even 1 trivial
279.6.a.f.1.2 8 3.2 odd 2
496.6.a.h.1.3 8 4.3 odd 2
775.6.a.b.1.2 8 5.4 even 2
961.6.a.c.1.7 8 31.30 odd 2