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Properties

Label 158.6.a.c.1.3
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 1.3
Root \(2.42477\) 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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -11.0374 q^{3} +16.0000 q^{4} +98.5110 q^{5} +44.1494 q^{6} -177.207 q^{7} -64.0000 q^{8} -121.177 q^{9} -394.044 q^{10} +481.923 q^{11} -176.598 q^{12} -149.648 q^{13} +708.828 q^{14} -1087.30 q^{15} +256.000 q^{16} -1941.67 q^{17} +484.707 q^{18} +2489.43 q^{19} +1576.18 q^{20} +1955.90 q^{21} -1927.69 q^{22} -1050.35 q^{23} +706.391 q^{24} +6579.41 q^{25} +598.593 q^{26} +4019.55 q^{27} -2835.31 q^{28} +708.738 q^{29} +4349.20 q^{30} -2863.85 q^{31} -1024.00 q^{32} -5319.16 q^{33} +7766.68 q^{34} -17456.8 q^{35} -1938.83 q^{36} -7501.67 q^{37} -9957.73 q^{38} +1651.72 q^{39} -6304.70 q^{40} -18714.5 q^{41} -7823.59 q^{42} -9180.79 q^{43} +7710.77 q^{44} -11937.2 q^{45} +4201.41 q^{46} -1429.85 q^{47} -2825.56 q^{48} +14595.3 q^{49} -26317.6 q^{50} +21430.9 q^{51} -2394.37 q^{52} -30437.1 q^{53} -16078.2 q^{54} +47474.7 q^{55} +11341.3 q^{56} -27476.7 q^{57} -2834.95 q^{58} -25971.4 q^{59} -17396.8 q^{60} -19177.0 q^{61} +11455.4 q^{62} +21473.4 q^{63} +4096.00 q^{64} -14742.0 q^{65} +21276.6 q^{66} +453.188 q^{67} -31066.7 q^{68} +11593.1 q^{69} +69827.4 q^{70} +78429.5 q^{71} +7755.31 q^{72} -41282.9 q^{73} +30006.7 q^{74} -72619.3 q^{75} +39830.9 q^{76} -85400.2 q^{77} -6606.89 q^{78} -6241.00 q^{79} +25218.8 q^{80} -14919.2 q^{81} +74858.1 q^{82} -66264.3 q^{83} +31294.4 q^{84} -191276. q^{85} +36723.2 q^{86} -7822.60 q^{87} -30843.1 q^{88} -69113.1 q^{89} +47748.9 q^{90} +26518.7 q^{91} -16805.6 q^{92} +31609.4 q^{93} +5719.41 q^{94} +245236. q^{95} +11302.3 q^{96} +44049.2 q^{97} -58381.4 q^{98} -58397.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 28 q^{2} - 9 q^{3} + 112 q^{4} + 31 q^{5} + 36 q^{6} - 235 q^{7} - 448 q^{8} + 514 q^{9} - 124 q^{10} + 52 q^{11} - 144 q^{12} - 1047 q^{13} + 940 q^{14} - 469 q^{15} + 1792 q^{16} - 2056 q^{17} - 2056 q^{18}+ \cdots + 260836 q^{99}+O(q^{100}) \) 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
\(3\) −11.0374 −0.708047 −0.354023 − 0.935237i \(-0.615187\pi\)
−0.354023 + 0.935237i \(0.615187\pi\)
\(4\) 16.0000 0.500000
\(5\) 98.5110 1.76222 0.881109 − 0.472914i \(-0.156798\pi\)
0.881109 + 0.472914i \(0.156798\pi\)
\(6\) 44.1494 0.500665
\(7\) −177.207 −1.36690 −0.683449 − 0.729999i \(-0.739521\pi\)
−0.683449 + 0.729999i \(0.739521\pi\)
\(8\) −64.0000 −0.353553
\(9\) −121.177 −0.498670
\(10\) −394.044 −1.24608
\(11\) 481.923 1.20087 0.600435 − 0.799673i \(-0.294994\pi\)
0.600435 + 0.799673i \(0.294994\pi\)
\(12\) −176.598 −0.354023
\(13\) −149.648 −0.245592 −0.122796 − 0.992432i \(-0.539186\pi\)
−0.122796 + 0.992432i \(0.539186\pi\)
\(14\) 708.828 0.966542
\(15\) −1087.30 −1.24773
\(16\) 256.000 0.250000
\(17\) −1941.67 −1.62950 −0.814748 − 0.579815i \(-0.803125\pi\)
−0.814748 + 0.579815i \(0.803125\pi\)
\(18\) 484.707 0.352613
\(19\) 2489.43 1.58204 0.791018 − 0.611793i \(-0.209552\pi\)
0.791018 + 0.611793i \(0.209552\pi\)
\(20\) 1576.18 0.881109
\(21\) 1955.90 0.967827
\(22\) −1927.69 −0.849143
\(23\) −1050.35 −0.414015 −0.207007 − 0.978339i \(-0.566372\pi\)
−0.207007 + 0.978339i \(0.566372\pi\)
\(24\) 706.391 0.250332
\(25\) 6579.41 2.10541
\(26\) 598.593 0.173659
\(27\) 4019.55 1.06113
\(28\) −2835.31 −0.683449
\(29\) 708.738 0.156492 0.0782458 − 0.996934i \(-0.475068\pi\)
0.0782458 + 0.996934i \(0.475068\pi\)
\(30\) 4349.20 0.882280
\(31\) −2863.85 −0.535238 −0.267619 − 0.963525i \(-0.586237\pi\)
−0.267619 + 0.963525i \(0.586237\pi\)
\(32\) −1024.00 −0.176777
\(33\) −5319.16 −0.850272
\(34\) 7766.68 1.15223
\(35\) −17456.8 −2.40877
\(36\) −1938.83 −0.249335
\(37\) −7501.67 −0.900852 −0.450426 − 0.892814i \(-0.648728\pi\)
−0.450426 + 0.892814i \(0.648728\pi\)
\(38\) −9957.73 −1.11867
\(39\) 1651.72 0.173890
\(40\) −6304.70 −0.623038
\(41\) −18714.5 −1.73868 −0.869339 − 0.494217i \(-0.835455\pi\)
−0.869339 + 0.494217i \(0.835455\pi\)
\(42\) −7823.59 −0.684357
\(43\) −9180.79 −0.757197 −0.378598 − 0.925561i \(-0.623594\pi\)
−0.378598 + 0.925561i \(0.623594\pi\)
\(44\) 7710.77 0.600435
\(45\) −11937.2 −0.878765
\(46\) 4201.41 0.292752
\(47\) −1429.85 −0.0944162 −0.0472081 − 0.998885i \(-0.515032\pi\)
−0.0472081 + 0.998885i \(0.515032\pi\)
\(48\) −2825.56 −0.177012
\(49\) 14595.3 0.868409
\(50\) −26317.6 −1.48875
\(51\) 21430.9 1.15376
\(52\) −2394.37 −0.122796
\(53\) −30437.1 −1.48838 −0.744191 − 0.667967i \(-0.767165\pi\)
−0.744191 + 0.667967i \(0.767165\pi\)
\(54\) −16078.2 −0.750331
\(55\) 47474.7 2.11619
\(56\) 11341.3 0.483271
\(57\) −27476.7 −1.12016
\(58\) −2834.95 −0.110656
\(59\) −25971.4 −0.971328 −0.485664 − 0.874145i \(-0.661422\pi\)
−0.485664 + 0.874145i \(0.661422\pi\)
\(60\) −17396.8 −0.623866
\(61\) −19177.0 −0.659868 −0.329934 − 0.944004i \(-0.607027\pi\)
−0.329934 + 0.944004i \(0.607027\pi\)
\(62\) 11455.4 0.378470
\(63\) 21473.4 0.681630
\(64\) 4096.00 0.125000
\(65\) −14742.0 −0.432786
\(66\) 21276.6 0.601233
\(67\) 453.188 0.0123337 0.00616683 − 0.999981i \(-0.498037\pi\)
0.00616683 + 0.999981i \(0.498037\pi\)
\(68\) −31066.7 −0.814748
\(69\) 11593.1 0.293142
\(70\) 69827.4 1.70326
\(71\) 78429.5 1.84643 0.923216 − 0.384281i \(-0.125551\pi\)
0.923216 + 0.384281i \(0.125551\pi\)
\(72\) 7755.31 0.176306
\(73\) −41282.9 −0.906699 −0.453350 − 0.891333i \(-0.649771\pi\)
−0.453350 + 0.891333i \(0.649771\pi\)
\(74\) 30006.7 0.636999
\(75\) −72619.3 −1.49073
\(76\) 39830.9 0.791018
\(77\) −85400.2 −1.64147
\(78\) −6606.89 −0.122959
\(79\) −6241.00 −0.112509
\(80\) 25218.8 0.440554
\(81\) −14919.2 −0.252659
\(82\) 74858.1 1.22943
\(83\) −66264.3 −1.05581 −0.527903 − 0.849304i \(-0.677022\pi\)
−0.527903 + 0.849304i \(0.677022\pi\)
\(84\) 31294.4 0.483914
\(85\) −191276. −2.87153
\(86\) 36723.2 0.535419
\(87\) −7822.60 −0.110803
\(88\) −30843.1 −0.424572
\(89\) −69113.1 −0.924880 −0.462440 − 0.886651i \(-0.653026\pi\)
−0.462440 + 0.886651i \(0.653026\pi\)
\(90\) 47748.9 0.621380
\(91\) 26518.7 0.335698
\(92\) −16805.6 −0.207007
\(93\) 31609.4 0.378973
\(94\) 5719.41 0.0667623
\(95\) 245236. 2.78789
\(96\) 11302.3 0.125166
\(97\) 44049.2 0.475344 0.237672 − 0.971345i \(-0.423616\pi\)
0.237672 + 0.971345i \(0.423616\pi\)
\(98\) −58381.4 −0.614058
\(99\) −58397.9 −0.598838
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.1.3 ✓ 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
158.6.a.c.1.3 ✓ 7 1.1 even 1 trivial