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Properties

Label 31.6.a.b.1.4
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 1.4
Root \(2.21681\) 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)
 
\(f(q)\) \(=\) \(q-1.21681 q^{2} -24.0001 q^{3} -30.5194 q^{4} +80.9514 q^{5} +29.2035 q^{6} +175.330 q^{7} +76.0739 q^{8} +333.006 q^{9} -98.5022 q^{10} -506.435 q^{11} +732.469 q^{12} +75.1097 q^{13} -213.342 q^{14} -1942.85 q^{15} +884.053 q^{16} +2340.12 q^{17} -405.204 q^{18} +1946.64 q^{19} -2470.59 q^{20} -4207.94 q^{21} +616.233 q^{22} -829.073 q^{23} -1825.78 q^{24} +3428.14 q^{25} -91.3939 q^{26} -2160.17 q^{27} -5350.95 q^{28} -619.340 q^{29} +2364.07 q^{30} +961.000 q^{31} -3510.09 q^{32} +12154.5 q^{33} -2847.47 q^{34} +14193.2 q^{35} -10163.2 q^{36} -3487.14 q^{37} -2368.68 q^{38} -1802.64 q^{39} +6158.29 q^{40} +13137.9 q^{41} +5120.24 q^{42} -14336.7 q^{43} +15456.1 q^{44} +26957.4 q^{45} +1008.82 q^{46} -2946.60 q^{47} -21217.4 q^{48} +13933.5 q^{49} -4171.38 q^{50} -56163.1 q^{51} -2292.30 q^{52} +21954.4 q^{53} +2628.50 q^{54} -40996.7 q^{55} +13338.0 q^{56} -46719.6 q^{57} +753.616 q^{58} +28172.2 q^{59} +59294.5 q^{60} -12952.3 q^{61} -1169.35 q^{62} +58385.9 q^{63} -24018.6 q^{64} +6080.24 q^{65} -14789.7 q^{66} -34804.0 q^{67} -71419.0 q^{68} +19897.9 q^{69} -17270.4 q^{70} +19907.8 q^{71} +25333.1 q^{72} +45077.6 q^{73} +4243.18 q^{74} -82275.7 q^{75} -59410.3 q^{76} -88793.1 q^{77} +2193.47 q^{78} +63368.0 q^{79} +71565.4 q^{80} -29076.3 q^{81} -15986.2 q^{82} -24250.2 q^{83} +128424. q^{84} +189436. q^{85} +17445.0 q^{86} +14864.2 q^{87} -38526.5 q^{88} +7020.86 q^{89} -32801.9 q^{90} +13169.0 q^{91} +25302.8 q^{92} -23064.1 q^{93} +3585.44 q^{94} +157583. q^{95} +84242.5 q^{96} +23289.5 q^{97} -16954.3 q^{98} -168646. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 7 q^{2} - 2 q^{3} + 149 q^{4} + 128 q^{5} + 72 q^{6} + 88 q^{7} + 924 q^{8} + 1512 q^{9} + 1581 q^{10} + 574 q^{11} - 46 q^{12} - 122 q^{13} - 309 q^{14} - 524 q^{15} + 833 q^{16} + 1932 q^{17} - 6845 q^{18}+ \cdots - 180150 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.21681 −0.215103 −0.107551 − 0.994200i \(-0.534301\pi\)
−0.107551 + 0.994200i \(0.534301\pi\)
\(3\) −24.0001 −1.53961 −0.769805 − 0.638280i \(-0.779646\pi\)
−0.769805 + 0.638280i \(0.779646\pi\)
\(4\) −30.5194 −0.953731
\(5\) 80.9514 1.44810 0.724052 − 0.689746i \(-0.242278\pi\)
0.724052 + 0.689746i \(0.242278\pi\)
\(6\) 29.2035 0.331174
\(7\) 175.330 1.35242 0.676208 − 0.736711i \(-0.263622\pi\)
0.676208 + 0.736711i \(0.263622\pi\)
\(8\) 76.0739 0.420253
\(9\) 333.006 1.37040
\(10\) −98.5022 −0.311491
\(11\) −506.435 −1.26195 −0.630975 − 0.775803i \(-0.717345\pi\)
−0.630975 + 0.775803i \(0.717345\pi\)
\(12\) 732.469 1.46837
\(13\) 75.1097 0.123264 0.0616322 − 0.998099i \(-0.480369\pi\)
0.0616322 + 0.998099i \(0.480369\pi\)
\(14\) −213.342 −0.290909
\(15\) −1942.85 −2.22951
\(16\) 884.053 0.863333
\(17\) 2340.12 1.96388 0.981941 − 0.189187i \(-0.0605853\pi\)
0.981941 + 0.189187i \(0.0605853\pi\)
\(18\) −405.204 −0.294776
\(19\) 1946.64 1.23709 0.618546 − 0.785749i \(-0.287722\pi\)
0.618546 + 0.785749i \(0.287722\pi\)
\(20\) −2470.59 −1.38110
\(21\) −4207.94 −2.08219
\(22\) 616.233 0.271449
\(23\) −829.073 −0.326793 −0.163397 − 0.986560i \(-0.552245\pi\)
−0.163397 + 0.986560i \(0.552245\pi\)
\(24\) −1825.78 −0.647026
\(25\) 3428.14 1.09700
\(26\) −91.3939 −0.0265145
\(27\) −2160.17 −0.570267
\(28\) −5350.95 −1.28984
\(29\) −619.340 −0.136752 −0.0683760 − 0.997660i \(-0.521782\pi\)
−0.0683760 + 0.997660i \(0.521782\pi\)
\(30\) 2364.07 0.479575
\(31\) 961.000 0.179605
\(32\) −3510.09 −0.605958
\(33\) 12154.5 1.94291
\(34\) −2847.47 −0.422437
\(35\) 14193.2 1.95844
\(36\) −10163.2 −1.30699
\(37\) −3487.14 −0.418760 −0.209380 − 0.977834i \(-0.567145\pi\)
−0.209380 + 0.977834i \(0.567145\pi\)
\(38\) −2368.68 −0.266102
\(39\) −1802.64 −0.189779
\(40\) 6158.29 0.608570
\(41\) 13137.9 1.22058 0.610288 − 0.792179i \(-0.291054\pi\)
0.610288 + 0.792179i \(0.291054\pi\)
\(42\) 5120.24 0.447885
\(43\) −14336.7 −1.18244 −0.591219 − 0.806511i \(-0.701353\pi\)
−0.591219 + 0.806511i \(0.701353\pi\)
\(44\) 15456.1 1.20356
\(45\) 26957.4 1.98448
\(46\) 1008.82 0.0702942
\(47\) −2946.60 −0.194571 −0.0972853 − 0.995257i \(-0.531016\pi\)
−0.0972853 + 0.995257i \(0.531016\pi\)
\(48\) −21217.4 −1.32920
\(49\) 13933.5 0.829029
\(50\) −4171.38 −0.235969
\(51\) −56163.1 −3.02361
\(52\) −2292.30 −0.117561
\(53\) 21954.4 1.07357 0.536786 − 0.843718i \(-0.319638\pi\)
0.536786 + 0.843718i \(0.319638\pi\)
\(54\) 2628.50 0.122666
\(55\) −40996.7 −1.82744
\(56\) 13338.0 0.568357
\(57\) −46719.6 −1.90464
\(58\) 753.616 0.0294158
\(59\) 28172.2 1.05364 0.526818 − 0.849978i \(-0.323385\pi\)
0.526818 + 0.849978i \(0.323385\pi\)
\(60\) 59294.5 2.12636
\(61\) −12952.3 −0.445679 −0.222839 − 0.974855i \(-0.571533\pi\)
−0.222839 + 0.974855i \(0.571533\pi\)
\(62\) −1169.35 −0.0386336
\(63\) 58385.9 1.85335
\(64\) −24018.6 −0.732990
\(65\) 6080.24 0.178500
\(66\) −14789.7 −0.417926
\(67\) −34804.0 −0.947201 −0.473601 − 0.880740i \(-0.657046\pi\)
−0.473601 + 0.880740i \(0.657046\pi\)
\(68\) −71419.0 −1.87301
\(69\) 19897.9 0.503134
\(70\) −17270.4 −0.421266
\(71\) 19907.8 0.468682 0.234341 − 0.972155i \(-0.424707\pi\)
0.234341 + 0.972155i \(0.424707\pi\)
\(72\) 25333.1 0.575914
\(73\) 45077.6 0.990042 0.495021 − 0.868881i \(-0.335160\pi\)
0.495021 + 0.868881i \(0.335160\pi\)
\(74\) 4243.18 0.0900765
\(75\) −82275.7 −1.68896
\(76\) −59410.3 −1.17985
\(77\) −88793.1 −1.70668
\(78\) 2193.47 0.0408220
\(79\) 63368.0 1.14236 0.571179 − 0.820825i \(-0.306486\pi\)
0.571179 + 0.820825i \(0.306486\pi\)
\(80\) 71565.4 1.25020
\(81\) −29076.3 −0.492409
\(82\) −15986.2 −0.262550
\(83\) −24250.2 −0.386385 −0.193193 − 0.981161i \(-0.561884\pi\)
−0.193193 + 0.981161i \(0.561884\pi\)
\(84\) 128424. 1.98585
\(85\) 189436. 2.84390
\(86\) 17445.0 0.254346
\(87\) 14864.2 0.210545
\(88\) −38526.5 −0.530339
\(89\) 7020.86 0.0939540 0.0469770 − 0.998896i \(-0.485041\pi\)
0.0469770 + 0.998896i \(0.485041\pi\)
\(90\) −32801.9 −0.426867
\(91\) 13169.0 0.166705
\(92\) 25302.8 0.311673
\(93\) −23064.1 −0.276522
\(94\) 3585.44 0.0418527
\(95\) 157583. 1.79144
\(96\) 84242.5 0.932939
\(97\) 23289.5 0.251322 0.125661 − 0.992073i \(-0.459895\pi\)
0.125661 + 0.992073i \(0.459895\pi\)
\(98\) −16954.3 −0.178326
\(99\) −168646. −1.72937
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.1.4 ✓ 8
3.2 odd 2 279.6.a.f.1.5 8
4.3 odd 2 496.6.a.h.1.7 8
5.4 even 2 775.6.a.b.1.5 8
31.30 odd 2 961.6.a.c.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.6.a.b.1.4 ✓ 8 1.1 even 1 trivial
279.6.a.f.1.5 8 3.2 odd 2
496.6.a.h.1.7 8 4.3 odd 2
775.6.a.b.1.5 8 5.4 even 2
961.6.a.c.1.4 8 31.30 odd 2