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Properties

Label 31.8.c.a.5.15
Level $31$
Weight $8$
Character 31.5
Analytic conductor $9.684$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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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] = [31,8,Mod(5,31)] 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("31.5"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 31.c (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(9.68393579001\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 5.15
Character \(\chi\) \(=\) 31.5
Dual form 31.8.c.a.25.15

$q$-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+14.6433 q^{2} +(-11.3125 + 19.5939i) q^{3} +86.4251 q^{4} +(170.496 + 295.308i) q^{5} +(-165.652 + 286.918i) q^{6} +(-28.8903 + 50.0395i) q^{7} -608.792 q^{8} +(837.554 + 1450.69i) q^{9} +(2496.62 + 4324.27i) q^{10} +(-170.386 - 295.118i) q^{11} +(-977.685 + 1693.40i) q^{12} +(6001.30 + 10394.6i) q^{13} +(-423.048 + 732.741i) q^{14} -7714.97 q^{15} -19977.1 q^{16} +(12648.4 - 21907.8i) q^{17} +(12264.5 + 21242.8i) q^{18} +(14377.5 - 24902.6i) q^{19} +(14735.2 + 25522.0i) q^{20} +(-653.644 - 1132.14i) q^{21} +(-2495.01 - 4321.49i) q^{22} -35005.6 q^{23} +(6886.97 - 11928.6i) q^{24} +(-19075.5 + 33039.7i) q^{25} +(87878.7 + 152210. i) q^{26} -87380.3 q^{27} +(-2496.85 + 4324.67i) q^{28} +151108. q^{29} -112972. q^{30} +(-24080.0 - 164112. i) q^{31} -214605. q^{32} +7710.00 q^{33} +(185215. - 320801. i) q^{34} -19702.8 q^{35} +(72385.7 + 125376. i) q^{36} +(98671.5 - 170904. i) q^{37} +(210534. - 364656. i) q^{38} -271559. q^{39} +(-103797. - 179781. i) q^{40} +(-284089. - 492057. i) q^{41} +(-9571.48 - 16578.3i) q^{42} +(-40504.7 + 70156.2i) q^{43} +(-14725.7 - 25505.6i) q^{44} +(-285600. + 494673. i) q^{45} -512596. q^{46} +640277. q^{47} +(225991. - 391429. i) q^{48} +(410102. + 710318. i) q^{49} +(-279327. + 483808. i) q^{50} +(286172. + 495664. i) q^{51} +(518663. + 898351. i) q^{52} +(569792. + 986908. i) q^{53} -1.27953e6 q^{54} +(58100.5 - 100633. i) q^{55} +(17588.2 - 30463.6i) q^{56} +(325292. + 563423. i) q^{57} +2.21272e6 q^{58} +(-917072. + 1.58842e6i) q^{59} -666767. q^{60} +1.64989e6 q^{61} +(-352610. - 2.40314e6i) q^{62} -96788.7 q^{63} -585442. q^{64} +(-2.04640e6 + 3.54447e6i) q^{65} +112899. q^{66} +(-707061. - 1.22467e6i) q^{67} +(1.09314e6 - 1.89338e6i) q^{68} +(396001. - 685894. i) q^{69} -288513. q^{70} +(-114289. - 197955. i) q^{71} +(-509896. - 883166. i) q^{72} +(-105590. - 182887. i) q^{73} +(1.44487e6 - 2.50259e6i) q^{74} +(-431583. - 747523. i) q^{75} +(1.24258e6 - 2.15221e6i) q^{76} +19690.1 q^{77} -3.97652e6 q^{78} +(978531. - 1.69487e6i) q^{79} +(-3.40602e6 - 5.89941e6i) q^{80} +(-843239. + 1.46053e6i) q^{81} +(-4.15999e6 - 7.20532e6i) q^{82} +(3.72286e6 + 6.44818e6i) q^{83} +(-56491.2 - 97845.7i) q^{84} +8.62605e6 q^{85} +(-593121. + 1.02732e6i) q^{86} +(-1.70941e6 + 2.96079e6i) q^{87} +(103730. + 179665. i) q^{88} -3.29748e6 q^{89} +(-4.18211e6 + 7.24363e6i) q^{90} -693518. q^{91} -3.02536e6 q^{92} +(3.48799e6 + 1.38470e6i) q^{93} +9.37575e6 q^{94} +9.80527e6 q^{95} +(2.42772e6 - 4.20493e6i) q^{96} -6.94981e6 q^{97} +(6.00523e6 + 1.04014e7i) q^{98} +(285416. - 494354. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 4 q^{2} - 2 q^{3} + 2352 q^{4} + 70 q^{5} - 528 q^{6} - 2158 q^{7} + 3084 q^{8} - 14552 q^{9} + 5942 q^{10} - 4666 q^{11} - 11134 q^{12} - 10502 q^{13} - 19440 q^{14} + 99588 q^{15} + 172360 q^{16}+ \cdots + 23529296 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.6433 1.29429 0.647147 − 0.762365i \(-0.275962\pi\)
0.647147 + 0.762365i \(0.275962\pi\)
\(3\) −11.3125 + 19.5939i −0.241899 + 0.418982i −0.961255 − 0.275660i \(-0.911104\pi\)
0.719356 + 0.694642i \(0.244437\pi\)
\(4\) 86.4251 0.675196
\(5\) 170.496 + 295.308i 0.609986 + 1.05653i 0.991242 + 0.132057i \(0.0421583\pi\)
−0.381256 + 0.924469i \(0.624508\pi\)
\(6\) −165.652 + 286.918i −0.313089 + 0.542286i
\(7\) −28.8903 + 50.0395i −0.0318353 + 0.0551404i −0.881504 − 0.472176i \(-0.843469\pi\)
0.849669 + 0.527317i \(0.176802\pi\)
\(8\) −608.792 −0.420392
\(9\) 837.554 + 1450.69i 0.382969 + 0.663322i
\(10\) 2496.62 + 4324.27i 0.789501 + 1.36746i
\(11\) −170.386 − 295.118i −0.0385976 − 0.0668530i 0.846081 − 0.533054i \(-0.178956\pi\)
−0.884679 + 0.466201i \(0.845622\pi\)
\(12\) −977.685 + 1693.40i −0.163330 + 0.282895i
\(13\) 6001.30 + 10394.6i 0.757607 + 1.31221i 0.944068 + 0.329751i \(0.106965\pi\)
−0.186461 + 0.982462i \(0.559702\pi\)
\(14\) −423.048 + 732.741i −0.0412042 + 0.0713678i
\(15\) −7714.97 −0.590221
\(16\) −19977.1 −1.21931
\(17\) 12648.4 − 21907.8i 0.624404 − 1.08150i −0.364251 − 0.931301i \(-0.618675\pi\)
0.988656 − 0.150200i \(-0.0479916\pi\)
\(18\) 12264.5 + 21242.8i 0.495675 + 0.858534i
\(19\) 14377.5 − 24902.6i 0.480891 − 0.832928i −0.518868 − 0.854854i \(-0.673646\pi\)
0.999760 + 0.0219259i \(0.00697980\pi\)
\(20\) 14735.2 + 25522.0i 0.411860 + 0.713363i
\(21\) −653.644 − 1132.14i −0.0154019 − 0.0266768i
\(22\) −2495.01 − 4321.49i −0.0499566 − 0.0865275i
\(23\) −35005.6 −0.599915 −0.299958 − 0.953953i \(-0.596973\pi\)
−0.299958 + 0.953953i \(0.596973\pi\)
\(24\) 6886.97 − 11928.6i 0.101693 − 0.176137i
\(25\) −19075.5 + 33039.7i −0.244166 + 0.422908i
\(26\) 87878.7 + 152210.i 0.980566 + 1.69839i
\(27\) −87380.3 −0.854359
\(28\) −2496.85 + 4324.67i −0.0214951 + 0.0372305i
\(29\) 151108. 1.15052 0.575261 − 0.817970i \(-0.304900\pi\)
0.575261 + 0.817970i \(0.304900\pi\)
\(30\) −112972. −0.763919
\(31\) −24080.0 − 164112.i −0.145175 − 0.989406i
\(32\) −214605. −1.15775
\(33\) 7710.00 0.0373470
\(34\) 185215. − 320801.i 0.808163 − 1.39978i
\(35\) −19702.8 −0.0776763
\(36\) 72385.7 + 125376.i 0.258579 + 0.447873i
\(37\) 98671.5 − 170904.i 0.320247 − 0.554685i −0.660292 − 0.751009i \(-0.729567\pi\)
0.980539 + 0.196325i \(0.0629006\pi\)
\(38\) 210534. − 364656.i 0.622415 − 1.07805i
\(39\) −271559. −0.733059
\(40\) −103797. − 179781.i −0.256433 − 0.444155i
\(41\) −284089. − 492057.i −0.643741 − 1.11499i −0.984591 − 0.174875i \(-0.944048\pi\)
0.340849 − 0.940118i \(-0.389285\pi\)
\(42\) −9571.48 − 16578.3i −0.0199346 − 0.0345277i
\(43\) −40504.7 + 70156.2i −0.0776901 + 0.134563i −0.902253 − 0.431207i \(-0.858088\pi\)
0.824563 + 0.565770i \(0.191421\pi\)
\(44\) −14725.7 − 25505.6i −0.0260610 − 0.0451389i
\(45\) −285600. + 494673.i −0.467212 + 0.809235i
\(46\) −512596. −0.776466
\(47\) 640277. 0.899551 0.449775 − 0.893142i \(-0.351504\pi\)
0.449775 + 0.893142i \(0.351504\pi\)
\(48\) 225991. − 391429.i 0.294950 − 0.510868i
\(49\) 410102. + 710318.i 0.497973 + 0.862515i
\(50\) −279327. + 483808.i −0.316022 + 0.547367i
\(51\) 286172. + 495664.i 0.302086 + 0.523229i
\(52\) 518663. + 898351.i 0.511533 + 0.886001i
\(53\) 569792. + 986908.i 0.525715 + 0.910565i 0.999551 + 0.0299524i \(0.00953556\pi\)
−0.473836 + 0.880613i \(0.657131\pi\)
\(54\) −1.27953e6 −1.10579
\(55\) 58100.5 − 100633.i 0.0470880 − 0.0815588i
\(56\) 17588.2 − 30463.6i 0.0133833 − 0.0231806i
\(57\) 325292. + 563423.i 0.232655 + 0.402970i
\(58\) 2.21272e6 1.48911
\(59\) −917072. + 1.58842e6i −0.581328 + 1.00689i 0.413994 + 0.910280i \(0.364134\pi\)
−0.995322 + 0.0966107i \(0.969200\pi\)
\(60\) −666767. −0.398515
\(61\) 1.64989e6 0.930678 0.465339 − 0.885133i \(-0.345932\pi\)
0.465339 + 0.885133i \(0.345932\pi\)
\(62\) −352610. − 2.40314e6i −0.187899 − 1.28058i
\(63\) −96788.7 −0.0487678
\(64\) −585442. −0.279160
\(65\) −2.04640e6 + 3.54447e6i −0.924259 + 1.60086i
\(66\) 112899. 0.0483379
\(67\) −707061. − 1.22467e6i −0.287207 − 0.497457i 0.685935 − 0.727663i \(-0.259393\pi\)
−0.973142 + 0.230206i \(0.926060\pi\)
\(68\) 1.09314e6 − 1.89338e6i 0.421595 − 0.730225i
\(69\) 396001. − 685894.i 0.145119 − 0.251354i
\(70\) −288513. −0.100536
\(71\) −114289. − 197955.i −0.0378966 − 0.0656389i 0.846455 − 0.532460i \(-0.178732\pi\)
−0.884352 + 0.466821i \(0.845399\pi\)
\(72\) −509896. − 883166.i −0.160997 − 0.278855i
\(73\) −105590. − 182887.i −0.0317682 − 0.0550241i 0.849704 − 0.527260i \(-0.176780\pi\)
−0.881472 + 0.472236i \(0.843447\pi\)
\(74\) 1.44487e6 − 2.50259e6i 0.414494 − 0.717925i
\(75\) −431583. − 747523.i −0.118127 − 0.204602i
\(76\) 1.24258e6 − 2.15221e6i 0.324696 − 0.562390i
\(77\) 19690.1 0.00491507
\(78\) −3.97652e6 −0.948793
\(79\) 978531. − 1.69487e6i 0.223295 − 0.386759i −0.732511 − 0.680755i \(-0.761652\pi\)
0.955807 + 0.293996i \(0.0949852\pi\)
\(80\) −3.40602e6 − 5.89941e6i −0.743760 − 1.28823i
\(81\) −843239. + 1.46053e6i −0.176300 + 0.305361i
\(82\) −4.15999e6 − 7.20532e6i −0.833190 − 1.44313i
\(83\) 3.72286e6 + 6.44818e6i 0.714666 + 1.23784i 0.963088 + 0.269186i \(0.0867545\pi\)
−0.248423 + 0.968652i \(0.579912\pi\)
\(84\) −56491.2 − 97845.7i −0.0103993 − 0.0180121i
\(85\) 8.62605e6 1.52351
\(86\) −593121. + 1.02732e6i −0.100554 + 0.174164i
\(87\) −1.70941e6 + 2.96079e6i −0.278311 + 0.482048i
\(88\) 103730. + 179665.i 0.0162261 + 0.0281045i
\(89\) −3.29748e6 −0.495811 −0.247906 − 0.968784i \(-0.579742\pi\)
−0.247906 + 0.968784i \(0.579742\pi\)
\(90\) −4.18211e6 + 7.24363e6i −0.604709 + 1.04739i
\(91\) −693518. −0.0964746
\(92\) −3.02536e6 −0.405060
\(93\) 3.48799e6 + 1.38470e6i 0.449661 + 0.178511i
\(94\) 9.37575e6 1.16428
\(95\) 9.80527e6 1.17335
\(96\) 2.42772e6 − 4.20493e6i 0.280059 − 0.485076i
\(97\) −6.94981e6 −0.773165 −0.386582 − 0.922255i \(-0.626344\pi\)
−0.386582 + 0.922255i \(0.626344\pi\)
\(98\) 6.00523e6 + 1.04014e7i 0.644523 + 1.11635i
\(99\) 285416. − 494354.i 0.0295634 − 0.0512053i
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 31.8.c.a.5.15 ✓ 36
31.25 even 3 inner 31.8.c.a.25.15 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.8.c.a.5.15 ✓ 36 1.1 even 1 trivial
31.8.c.a.25.15 yes 36 31.25 even 3 inner