Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [58,6,Mod(7,58)] 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("58.7"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([6])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 58.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36] 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: no
Analytic conductor: \(9.30226154915\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 7.3
Character \(\chi\) \(=\) 58.7
Dual form 58.6.d.a.25.3

$q$-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+(-0.890084 - 3.89971i) q^{2} +(-4.43755 - 2.13701i) q^{3} +(-14.4155 + 6.94214i) q^{4} +(4.41541 + 19.3452i) q^{5} +(-4.38393 + 19.2073i) q^{6} +(196.591 + 94.6733i) q^{7} +(39.9033 + 50.0372i) q^{8} +(-136.383 - 171.019i) q^{9} +(71.5105 - 34.4377i) q^{10} +(354.673 - 444.746i) q^{11} +78.8049 q^{12} +(-84.5352 + 106.004i) q^{13} +(194.216 - 850.915i) q^{14} +(21.7472 - 95.2809i) q^{15} +(159.613 - 200.149i) q^{16} -113.873 q^{17} +(-545.532 + 684.076i) q^{18} +(1356.62 - 653.316i) q^{19} +(-197.947 - 248.218i) q^{20} +(-670.064 - 840.234i) q^{21} +(-2050.07 - 987.262i) q^{22} +(810.214 - 3549.78i) q^{23} +(-70.1429 - 307.316i) q^{24} +(2460.79 - 1185.05i) q^{25} +(488.628 + 235.311i) q^{26} +(506.061 + 2217.20i) q^{27} -3491.19 q^{28} +(3578.70 + 2775.62i) q^{29} -390.925 q^{30} +(454.302 + 1990.43i) q^{31} +(-922.592 - 444.297i) q^{32} +(-2524.30 + 1215.64i) q^{33} +(101.356 + 444.072i) q^{34} +(-963.441 + 4221.11i) q^{35} +(3153.27 + 1518.53i) q^{36} +(-645.086 - 808.912i) q^{37} +(-3755.25 - 4708.94i) q^{38} +(601.660 - 289.744i) q^{39} +(-791.789 + 992.872i) q^{40} +13113.3 q^{41} +(-2680.26 + 3360.94i) q^{42} +(-1817.68 + 7963.79i) q^{43} +(-2025.30 + 8873.43i) q^{44} +(2706.20 - 3393.47i) q^{45} -14564.3 q^{46} +(-18388.1 + 23057.9i) q^{47} +(-1136.01 + 547.075i) q^{48} +(19206.0 + 24083.6i) q^{49} +(-6811.67 - 8541.57i) q^{50} +(505.316 + 243.348i) q^{51} +(482.724 - 2114.95i) q^{52} +(-8793.68 - 38527.6i) q^{53} +(8196.00 - 3946.98i) q^{54} +(10169.7 + 4897.48i) q^{55} +(3107.45 + 13614.6i) q^{56} -7416.22 q^{57} +(7638.79 - 16426.4i) q^{58} -25881.7 q^{59} +(347.956 + 1524.49i) q^{60} +(-10997.5 - 5296.10i) q^{61} +(7357.72 - 3543.29i) q^{62} +(-10620.8 - 46532.6i) q^{63} +(-911.446 + 3993.30i) q^{64} +(-2423.92 - 1167.30i) q^{65} +(6987.49 + 8762.04i) q^{66} +(21751.1 + 27275.0i) q^{67} +(1641.54 - 790.522i) q^{68} +(-11181.3 + 14020.9i) q^{69} +17318.7 q^{70} +(25669.1 - 32188.0i) q^{71} +(3115.17 - 13648.5i) q^{72} +(-4094.37 + 17938.6i) q^{73} +(-2580.34 + 3235.65i) q^{74} -13452.3 q^{75} +(-15021.0 + 18835.7i) q^{76} +(111831. - 53855.0i) q^{77} +(-1665.45 - 2088.41i) q^{78} +(-52328.4 - 65617.7i) q^{79} +(4576.67 + 2204.01i) q^{80} +(-9335.41 + 40901.1i) q^{81} +(-11671.9 - 51137.9i) q^{82} +(30667.0 - 14768.5i) q^{83} +(15492.3 + 7460.71i) q^{84} +(-502.796 - 2202.89i) q^{85} +32674.4 q^{86} +(-9949.10 - 19964.7i) q^{87} +36406.5 q^{88} +(581.131 + 2546.10i) q^{89} +(-15642.3 - 7532.94i) q^{90} +(-26654.6 + 12836.2i) q^{91} +(12963.4 + 56796.5i) q^{92} +(2237.57 - 9803.46i) q^{93} +(106286. + 51184.7i) q^{94} +(18628.6 + 23359.5i) q^{95} +(3144.58 + 3943.18i) q^{96} +(-121987. + 58746.0i) q^{97} +(76824.1 - 96334.3i) q^{98} -124431. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 24 q^{2} + q^{3} - 96 q^{4} - 228 q^{5} - 136 q^{6} - 9 q^{7} - 384 q^{8} - 435 q^{9} - 44 q^{10} + 1002 q^{11} + 1024 q^{12} + 1169 q^{13} - 428 q^{14} + 1423 q^{15} - 1536 q^{16} + 3076 q^{17}+ \cdots + 6916 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{7}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.890084 − 3.89971i −0.157346 − 0.689378i
\(3\) −4.43755 − 2.13701i −0.284669 − 0.137089i 0.286101 − 0.958200i \(-0.407641\pi\)
−0.570770 + 0.821110i \(0.693355\pi\)
\(4\) −14.4155 + 6.94214i −0.450484 + 0.216942i
\(5\) 4.41541 + 19.3452i 0.0789853 + 0.346057i 0.998943 − 0.0459598i \(-0.0146346\pi\)
−0.919958 + 0.392017i \(0.871777\pi\)
\(6\) −4.38393 + 19.2073i −0.0497148 + 0.217815i
\(7\) 196.591 + 94.6733i 1.51642 + 0.730268i 0.992585 − 0.121554i \(-0.0387877\pi\)
0.523832 + 0.851822i \(0.324502\pi\)
\(8\) 39.9033 + 50.0372i 0.220437 + 0.276419i
\(9\) −136.383 − 171.019i −0.561247 − 0.703781i
\(10\) 71.5105 − 34.4377i 0.226136 − 0.108901i
\(11\) 354.673 − 444.746i 0.883785 − 1.10823i −0.109666 − 0.993969i \(-0.534978\pi\)
0.993451 − 0.114263i \(-0.0364505\pi\)
\(12\) 78.8049 0.157979
\(13\) −84.5352 + 106.004i −0.138733 + 0.173966i −0.846344 − 0.532636i \(-0.821201\pi\)
0.707611 + 0.706602i \(0.249773\pi\)
\(14\) 194.216 − 850.915i 0.264828 − 1.16029i
\(15\) 21.7472 − 95.2809i 0.0249561 − 0.109340i
\(16\) 159.613 − 200.149i 0.155872 − 0.195458i
\(17\) −113.873 −0.0955648 −0.0477824 − 0.998858i \(-0.515215\pi\)
−0.0477824 + 0.998858i \(0.515215\pi\)
\(18\) −545.532 + 684.076i −0.396862 + 0.497649i
\(19\) 1356.62 − 653.316i 0.862135 − 0.415182i 0.0500676 − 0.998746i \(-0.484056\pi\)
0.812068 + 0.583563i \(0.198342\pi\)
\(20\) −197.947 − 248.218i −0.110656 − 0.138758i
\(21\) −670.064 − 840.234i −0.331565 − 0.415769i
\(22\) −2050.07 − 987.262i −0.903050 − 0.434886i
\(23\) 810.214 − 3549.78i 0.319360 − 1.39921i −0.519319 − 0.854580i \(-0.673814\pi\)
0.838679 − 0.544626i \(-0.183328\pi\)
\(24\) −70.1429 − 307.316i −0.0248574 − 0.108907i
\(25\) 2460.79 − 1185.05i 0.787452 − 0.379217i
\(26\) 488.628 + 235.311i 0.141757 + 0.0682666i
\(27\) 506.061 + 2217.20i 0.133596 + 0.585322i
\(28\) −3491.19 −0.841548
\(29\) 3578.70 + 2775.62i 0.790187 + 0.612866i
\(30\) −390.925 −0.0793031
\(31\) 454.302 + 1990.43i 0.0849064 + 0.371999i 0.999474 − 0.0324342i \(-0.0103259\pi\)
−0.914568 + 0.404433i \(0.867469\pi\)
\(32\) −922.592 − 444.297i −0.159270 − 0.0767005i
\(33\) −2524.30 + 1215.64i −0.403512 + 0.194321i
\(34\) 101.356 + 444.072i 0.0150368 + 0.0658803i
\(35\) −963.441 + 4221.11i −0.132940 + 0.582447i
\(36\) 3153.27 + 1518.53i 0.405513 + 0.195285i
\(37\) −645.086 − 808.912i −0.0774664 − 0.0971398i 0.741588 − 0.670855i \(-0.234073\pi\)
−0.819055 + 0.573716i \(0.805501\pi\)
\(38\) −3755.25 − 4708.94i −0.421871 − 0.529010i
\(39\) 601.660 − 289.744i 0.0633417 − 0.0305038i
\(40\) −791.789 + 992.872i −0.0782455 + 0.0981168i
\(41\) 13113.3 1.21829 0.609145 − 0.793058i \(-0.291513\pi\)
0.609145 + 0.793058i \(0.291513\pi\)
\(42\) −2680.26 + 3360.94i −0.234452 + 0.293993i
\(43\) −1817.68 + 7963.79i −0.149916 + 0.656824i 0.842991 + 0.537928i \(0.180793\pi\)
−0.992907 + 0.118896i \(0.962064\pi\)
\(44\) −2025.30 + 8873.43i −0.157710 + 0.690971i
\(45\) 2706.20 − 3393.47i 0.199218 − 0.249812i
\(46\) −14564.3 −1.01483
\(47\) −18388.1 + 23057.9i −1.21420 + 1.52256i −0.429023 + 0.903293i \(0.641142\pi\)
−0.785179 + 0.619268i \(0.787429\pi\)
\(48\) −1136.01 + 547.075i −0.0711672 + 0.0342723i
\(49\) 19206.0 + 24083.6i 1.14274 + 1.43295i
\(50\) −6811.67 − 8541.57i −0.385326 − 0.483184i
\(51\) 505.316 + 243.348i 0.0272043 + 0.0131009i
\(52\) 482.724 − 2114.95i 0.0247566 − 0.108466i
\(53\) −8793.68 − 38527.6i −0.430013 − 1.88401i −0.466229 − 0.884664i \(-0.654388\pi\)
0.0362167 − 0.999344i \(-0.488469\pi\)
\(54\) 8196.00 − 3946.98i 0.382488 − 0.184196i
\(55\) 10169.7 + 4897.48i 0.453317 + 0.218306i
\(56\) 3107.45 + 13614.6i 0.132414 + 0.580145i
\(57\) −7416.22 −0.302340
\(58\) 7638.79 − 16426.4i 0.298163 − 0.641170i
\(59\) −25881.7 −0.967973 −0.483987 − 0.875075i \(-0.660812\pi\)
−0.483987 + 0.875075i \(0.660812\pi\)
\(60\) 347.956 + 1524.49i 0.0124780 + 0.0546698i
\(61\) −10997.5 − 5296.10i −0.378415 − 0.182235i 0.235001 − 0.971995i \(-0.424491\pi\)
−0.613416 + 0.789760i \(0.710205\pi\)
\(62\) 7357.72 − 3543.29i 0.243088 − 0.117065i
\(63\) −10620.8 − 46532.6i −0.337135 − 1.47709i
\(64\) −911.446 + 3993.30i −0.0278151 + 0.121866i
\(65\) −2423.92 − 1167.30i −0.0711598 − 0.0342688i
\(66\) 6987.49 + 8762.04i 0.197452 + 0.247597i
\(67\) 21751.1 + 27275.0i 0.591961 + 0.742296i 0.984101 − 0.177610i \(-0.0568365\pi\)
−0.392140 + 0.919906i \(0.628265\pi\)
\(68\) 1641.54 − 790.522i 0.0430505 − 0.0207320i
\(69\) −11181.3 + 14020.9i −0.282728 + 0.354530i
\(70\) 17318.7 0.422444
\(71\) 25669.1 − 32188.0i 0.604317 − 0.757790i −0.381727 − 0.924275i \(-0.624670\pi\)
0.986044 + 0.166486i \(0.0532419\pi\)
\(72\) 3115.17 − 13648.5i 0.0708191 − 0.310279i
\(73\) −4094.37 + 17938.6i −0.0899249 + 0.393987i −0.999781 − 0.0209376i \(-0.993335\pi\)
0.909856 + 0.414924i \(0.136192\pi\)
\(74\) −2580.34 + 3235.65i −0.0547770 + 0.0686882i
\(75\) −13452.3 −0.276150
\(76\) −15021.0 + 18835.7i −0.298308 + 0.374066i
\(77\) 111831. − 53855.0i 2.14949 − 1.03514i
\(78\) −1665.45 − 2088.41i −0.0309952 − 0.0388668i
\(79\) −52328.4 − 65617.7i −0.943343 − 1.18291i −0.982982 − 0.183700i \(-0.941192\pi\)
0.0396395 − 0.999214i \(-0.487379\pi\)
\(80\) 4576.67 + 2204.01i 0.0799512 + 0.0385025i
\(81\) −9335.41 + 40901.1i −0.158096 + 0.692664i
\(82\) −11671.9 − 51137.9i −0.191693 − 0.839863i
\(83\) 30667.0 − 14768.5i 0.488626 − 0.235310i −0.173314 − 0.984867i \(-0.555448\pi\)
0.661940 + 0.749557i \(0.269733\pi\)
\(84\) 15492.3 + 7460.71i 0.239562 + 0.115367i
\(85\) −502.796 − 2202.89i −0.00754821 − 0.0330709i
\(86\) 32674.4 0.476389
\(87\) −9949.10 − 19964.7i −0.140924 − 0.282790i
\(88\) 36406.5 0.501155
\(89\) 581.131 + 2546.10i 0.00777677 + 0.0340722i 0.978667 − 0.205451i \(-0.0658661\pi\)
−0.970891 + 0.239523i \(0.923009\pi\)
\(90\) −15642.3 − 7532.94i −0.203561 − 0.0980298i
\(91\) −26654.6 + 12836.2i −0.337418 + 0.162492i
\(92\) 12963.4 + 56796.5i 0.159680 + 0.699603i
\(93\) 2237.57 − 9803.46i 0.0268269 − 0.117536i
\(94\) 106286. + 51184.7i 1.24067 + 0.597476i
\(95\) 18628.6 + 23359.5i 0.211773 + 0.265555i
\(96\) 3144.58 + 3943.18i 0.0348245 + 0.0436685i
\(97\) −121987. + 58746.0i −1.31639 + 0.633941i −0.954481 − 0.298272i \(-0.903590\pi\)
−0.361911 + 0.932213i \(0.617875\pi\)
\(98\) 76824.1 − 96334.3i 0.808039 − 1.01325i
\(99\) −124431. −1.27597
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 58.6.d.a.7.3 ✓ 36
29.25 even 7 inner 58.6.d.a.25.3 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.6.d.a.7.3 ✓ 36 1.1 even 1 trivial
58.6.d.a.25.3 yes 36 29.25 even 7 inner