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Properties

Label 158.6.a.c.1.4
Level $158$
Weight $6$
Character 158.1
Self dual yes
Analytic conductor $25.341$
Analytic rank $1$
Dimension $7$
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] = [158,6,Mod(1,158)] 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("158.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(158, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 158 = 2 \cdot 79 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 158.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,-28] 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: \(25.3406435305\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 424x^{5} - 1337x^{4} + 29651x^{3} + 148738x^{2} - 123584x - 916096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-15.4801\) of defining polynomial
Character \(\chi\) \(=\) 158.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -3.80292 q^{3} +16.0000 q^{4} +19.7521 q^{5} +15.2117 q^{6} +78.5412 q^{7} -64.0000 q^{8} -228.538 q^{9} -79.0086 q^{10} -328.656 q^{11} -60.8467 q^{12} +335.314 q^{13} -314.165 q^{14} -75.1158 q^{15} +256.000 q^{16} +541.150 q^{17} +914.151 q^{18} +113.414 q^{19} +316.034 q^{20} -298.686 q^{21} +1314.63 q^{22} +2437.66 q^{23} +243.387 q^{24} -2734.85 q^{25} -1341.25 q^{26} +1793.22 q^{27} +1256.66 q^{28} +1835.55 q^{29} +300.463 q^{30} -10409.1 q^{31} -1024.00 q^{32} +1249.85 q^{33} -2164.60 q^{34} +1551.36 q^{35} -3656.60 q^{36} -3483.79 q^{37} -453.656 q^{38} -1275.17 q^{39} -1264.14 q^{40} -3859.71 q^{41} +1194.74 q^{42} -8652.81 q^{43} -5258.50 q^{44} -4514.11 q^{45} -9750.66 q^{46} +11470.9 q^{47} -973.547 q^{48} -10638.3 q^{49} +10939.4 q^{50} -2057.95 q^{51} +5365.02 q^{52} -28053.8 q^{53} -7172.88 q^{54} -6491.67 q^{55} -5026.64 q^{56} -431.305 q^{57} -7342.19 q^{58} -47707.3 q^{59} -1201.85 q^{60} -15032.0 q^{61} +41636.5 q^{62} -17949.6 q^{63} +4096.00 q^{64} +6623.16 q^{65} -4999.41 q^{66} -40584.4 q^{67} +8658.39 q^{68} -9270.24 q^{69} -6205.43 q^{70} -38935.7 q^{71} +14626.4 q^{72} +39872.9 q^{73} +13935.2 q^{74} +10400.4 q^{75} +1814.62 q^{76} -25813.1 q^{77} +5100.68 q^{78} -6241.00 q^{79} +5056.55 q^{80} +48715.2 q^{81} +15438.8 q^{82} +31050.2 q^{83} -4778.97 q^{84} +10688.9 q^{85} +34611.2 q^{86} -6980.44 q^{87} +21034.0 q^{88} -36913.0 q^{89} +18056.4 q^{90} +26335.9 q^{91} +39002.6 q^{92} +39585.1 q^{93} -45883.8 q^{94} +2240.17 q^{95} +3894.19 q^{96} -127865. q^{97} +42553.1 q^{98} +75110.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 28 q^{2} - 9 q^{3} + 112 q^{4} + 31 q^{5} + 36 q^{6} - 235 q^{7} - 448 q^{8} + 514 q^{9} - 124 q^{10} + 52 q^{11} - 144 q^{12} - 1047 q^{13} + 940 q^{14} - 469 q^{15} + 1792 q^{16} - 2056 q^{17} - 2056 q^{18}+ \cdots + 260836 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) −3.80292 −0.243957 −0.121979 − 0.992533i \(-0.538924\pi\)
−0.121979 + 0.992533i \(0.538924\pi\)
\(4\) 16.0000 0.500000
\(5\) 19.7521 0.353337 0.176669 − 0.984270i \(-0.443468\pi\)
0.176669 + 0.984270i \(0.443468\pi\)
\(6\) 15.2117 0.172504
\(7\) 78.5412 0.605832 0.302916 − 0.953017i \(-0.402040\pi\)
0.302916 + 0.953017i \(0.402040\pi\)
\(8\) −64.0000 −0.353553
\(9\) −228.538 −0.940485
\(10\) −79.0086 −0.249847
\(11\) −328.656 −0.818955 −0.409478 − 0.912320i \(-0.634289\pi\)
−0.409478 + 0.912320i \(0.634289\pi\)
\(12\) −60.8467 −0.121979
\(13\) 335.314 0.550291 0.275146 − 0.961403i \(-0.411274\pi\)
0.275146 + 0.961403i \(0.411274\pi\)
\(14\) −314.165 −0.428388
\(15\) −75.1158 −0.0861992
\(16\) 256.000 0.250000
\(17\) 541.150 0.454146 0.227073 − 0.973878i \(-0.427084\pi\)
0.227073 + 0.973878i \(0.427084\pi\)
\(18\) 914.151 0.665023
\(19\) 113.414 0.0720747 0.0360374 − 0.999350i \(-0.488526\pi\)
0.0360374 + 0.999350i \(0.488526\pi\)
\(20\) 316.034 0.176669
\(21\) −298.686 −0.147797
\(22\) 1314.63 0.579089
\(23\) 2437.66 0.960847 0.480423 − 0.877037i \(-0.340483\pi\)
0.480423 + 0.877037i \(0.340483\pi\)
\(24\) 243.387 0.0862520
\(25\) −2734.85 −0.875153
\(26\) −1341.25 −0.389115
\(27\) 1793.22 0.473396
\(28\) 1256.66 0.302916
\(29\) 1835.55 0.405294 0.202647 − 0.979252i \(-0.435046\pi\)
0.202647 + 0.979252i \(0.435046\pi\)
\(30\) 300.463 0.0609520
\(31\) −10409.1 −1.94541 −0.972703 − 0.232055i \(-0.925455\pi\)
−0.972703 + 0.232055i \(0.925455\pi\)
\(32\) −1024.00 −0.176777
\(33\) 1249.85 0.199790
\(34\) −2164.60 −0.321129
\(35\) 1551.36 0.214063
\(36\) −3656.60 −0.470242
\(37\) −3483.79 −0.418358 −0.209179 − 0.977877i \(-0.567079\pi\)
−0.209179 + 0.977877i \(0.567079\pi\)
\(38\) −453.656 −0.0509645
\(39\) −1275.17 −0.134248
\(40\) −1264.14 −0.124924
\(41\) −3859.71 −0.358587 −0.179294 − 0.983796i \(-0.557381\pi\)
−0.179294 + 0.983796i \(0.557381\pi\)
\(42\) 1194.74 0.104508
\(43\) −8652.81 −0.713651 −0.356825 − 0.934171i \(-0.616141\pi\)
−0.356825 + 0.934171i \(0.616141\pi\)
\(44\) −5258.50 −0.409478
\(45\) −4514.11 −0.332308
\(46\) −9750.66 −0.679421
\(47\) 11470.9 0.757451 0.378725 − 0.925509i \(-0.376363\pi\)
0.378725 + 0.925509i \(0.376363\pi\)
\(48\) −973.547 −0.0609894
\(49\) −10638.3 −0.632967
\(50\) 10939.4 0.618827
\(51\) −2057.95 −0.110792
\(52\) 5365.02 0.275146
\(53\) −28053.8 −1.37184 −0.685919 − 0.727678i \(-0.740599\pi\)
−0.685919 + 0.727678i \(0.740599\pi\)
\(54\) −7172.88 −0.334741
\(55\) −6491.67 −0.289367
\(56\) −5026.64 −0.214194
\(57\) −431.305 −0.0175832
\(58\) −7342.19 −0.286586
\(59\) −47707.3 −1.78425 −0.892123 − 0.451793i \(-0.850785\pi\)
−0.892123 + 0.451793i \(0.850785\pi\)
\(60\) −1201.85 −0.0430996
\(61\) −15032.0 −0.517241 −0.258620 − 0.965979i \(-0.583268\pi\)
−0.258620 + 0.965979i \(0.583268\pi\)
\(62\) 41636.5 1.37561
\(63\) −17949.6 −0.569776
\(64\) 4096.00 0.125000
\(65\) 6623.16 0.194438
\(66\) −4999.41 −0.141273
\(67\) −40584.4 −1.10452 −0.552258 − 0.833673i \(-0.686234\pi\)
−0.552258 + 0.833673i \(0.686234\pi\)
\(68\) 8658.39 0.227073
\(69\) −9270.24 −0.234406
\(70\) −6205.43 −0.151365
\(71\) −38935.7 −0.916647 −0.458324 − 0.888785i \(-0.651550\pi\)
−0.458324 + 0.888785i \(0.651550\pi\)
\(72\) 14626.4 0.332512
\(73\) 39872.9 0.875730 0.437865 − 0.899041i \(-0.355735\pi\)
0.437865 + 0.899041i \(0.355735\pi\)
\(74\) 13935.2 0.295824
\(75\) 10400.4 0.213500
\(76\) 1814.62 0.0360374
\(77\) −25813.1 −0.496149
\(78\) 5100.68 0.0949274
\(79\) −6241.00 −0.112509
\(80\) 5056.55 0.0883343
\(81\) 48715.2 0.824996
\(82\) 15438.8 0.253559
\(83\) 31050.2 0.494730 0.247365 − 0.968922i \(-0.420435\pi\)
0.247365 + 0.968922i \(0.420435\pi\)
\(84\) −4778.97 −0.0738986
\(85\) 10688.9 0.160466
\(86\) 34611.2 0.504627
\(87\) −6980.44 −0.0988746
\(88\) 21034.0 0.289544
\(89\) −36913.0 −0.493974 −0.246987 − 0.969019i \(-0.579440\pi\)
−0.246987 + 0.969019i \(0.579440\pi\)
\(90\) 18056.4 0.234977
\(91\) 26335.9 0.333384
\(92\) 39002.6 0.480423
\(93\) 39585.1 0.474596
\(94\) −45883.8 −0.535599
\(95\) 2240.17 0.0254667
\(96\) 3894.19 0.0431260
\(97\) −127865. −1.37982 −0.689910 − 0.723895i \(-0.742350\pi\)
−0.689910 + 0.723895i \(0.742350\pi\)
\(98\) 42553.1 0.447576
\(99\) 75110.4 0.770215
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 158.6.a.c.1.4 ✓ 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
158.6.a.c.1.4 ✓ 7 1.1 even 1 trivial