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Properties

Label 168.2.j.a.155.1
Level $168$
Weight $2$
Character 168.155
Analytic conductor $1.341$
Analytic rank $0$
Dimension $4$
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] = [168,2,Mod(155,168)] 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("168.155"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.j (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4] 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: no
Analytic conductor: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 155.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 168.155
Dual form 168.2.j.a.155.3

$q$-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+(-1.00000 - 1.00000i) q^{2} +(-1.61803 - 0.618034i) q^{3} +2.00000i q^{4} +3.23607 q^{5} +(1.00000 + 2.23607i) q^{6} +1.00000i q^{7} +(2.00000 - 2.00000i) q^{8} +(2.23607 + 2.00000i) q^{9} +(-3.23607 - 3.23607i) q^{10} -4.47214i q^{11} +(1.23607 - 3.23607i) q^{12} -1.23607i q^{13} +(1.00000 - 1.00000i) q^{14} +(-5.23607 - 2.00000i) q^{15} -4.00000 q^{16} -2.00000i q^{17} +(-0.236068 - 4.23607i) q^{18} +7.23607 q^{19} +6.47214i q^{20} +(0.618034 - 1.61803i) q^{21} +(-4.47214 + 4.47214i) q^{22} -0.472136 q^{23} +(-4.47214 + 2.00000i) q^{24} +5.47214 q^{25} +(-1.23607 + 1.23607i) q^{26} +(-2.38197 - 4.61803i) q^{27} -2.00000 q^{28} +(3.23607 + 7.23607i) q^{30} +8.94427i q^{31} +(4.00000 + 4.00000i) q^{32} +(-2.76393 + 7.23607i) q^{33} +(-2.00000 + 2.00000i) q^{34} +3.23607i q^{35} +(-4.00000 + 4.47214i) q^{36} -6.47214i q^{37} +(-7.23607 - 7.23607i) q^{38} +(-0.763932 + 2.00000i) q^{39} +(6.47214 - 6.47214i) q^{40} +4.47214i q^{41} +(-2.23607 + 1.00000i) q^{42} -0.472136 q^{43} +8.94427 q^{44} +(7.23607 + 6.47214i) q^{45} +(0.472136 + 0.472136i) q^{46} +2.47214 q^{47} +(6.47214 + 2.47214i) q^{48} -1.00000 q^{49} +(-5.47214 - 5.47214i) q^{50} +(-1.23607 + 3.23607i) q^{51} +2.47214 q^{52} -10.4721 q^{53} +(-2.23607 + 7.00000i) q^{54} -14.4721i q^{55} +(2.00000 + 2.00000i) q^{56} +(-11.7082 - 4.47214i) q^{57} +1.23607i q^{59} +(4.00000 - 10.4721i) q^{60} +11.7082i q^{61} +(8.94427 - 8.94427i) q^{62} +(-2.00000 + 2.23607i) q^{63} -8.00000i q^{64} -4.00000i q^{65} +(10.0000 - 4.47214i) q^{66} -10.9443 q^{67} +4.00000 q^{68} +(0.763932 + 0.291796i) q^{69} +(3.23607 - 3.23607i) q^{70} -6.94427 q^{71} +(8.47214 - 0.472136i) q^{72} -0.472136 q^{73} +(-6.47214 + 6.47214i) q^{74} +(-8.85410 - 3.38197i) q^{75} +14.4721i q^{76} +4.47214 q^{77} +(2.76393 - 1.23607i) q^{78} -4.94427i q^{79} -12.9443 q^{80} +(1.00000 + 8.94427i) q^{81} +(4.47214 - 4.47214i) q^{82} +13.2361i q^{83} +(3.23607 + 1.23607i) q^{84} -6.47214i q^{85} +(0.472136 + 0.472136i) q^{86} +(-8.94427 - 8.94427i) q^{88} -6.00000i q^{89} +(-0.763932 - 13.7082i) q^{90} +1.23607 q^{91} -0.944272i q^{92} +(5.52786 - 14.4721i) q^{93} +(-2.47214 - 2.47214i) q^{94} +23.4164 q^{95} +(-4.00000 - 8.94427i) q^{96} +3.52786 q^{97} +(1.00000 + 1.00000i) q^{98} +(8.94427 - 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{5} + 4 q^{6} + 8 q^{8} - 4 q^{10} - 4 q^{12} + 4 q^{14} - 12 q^{15} - 16 q^{16} + 8 q^{18} + 20 q^{19} - 2 q^{21} + 16 q^{23} + 4 q^{25} + 4 q^{26} - 14 q^{27} - 8 q^{28}+ \cdots + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 − 1.00000i −0.707107 − 0.707107i
\(3\) −1.61803 − 0.618034i −0.934172 − 0.356822i
\(4\) 2.00000i 1.00000i
\(5\) 3.23607 1.44721 0.723607 − 0.690212i \(-0.242483\pi\)
0.723607 + 0.690212i \(0.242483\pi\)
\(6\) 1.00000 + 2.23607i 0.408248 + 0.912871i
\(7\) 1.00000i 0.377964i
\(8\) 2.00000 − 2.00000i 0.707107 − 0.707107i
\(9\) 2.23607 + 2.00000i 0.745356 + 0.666667i
\(10\) −3.23607 − 3.23607i −1.02333 − 1.02333i
\(11\) − 4.47214i − 1.34840i −0.738549 − 0.674200i \(-0.764489\pi\)
0.738549 − 0.674200i \(-0.235511\pi\)
\(12\) 1.23607 − 3.23607i 0.356822 − 0.934172i
\(13\) − 1.23607i − 0.342824i −0.985199 − 0.171412i \(-0.945167\pi\)
0.985199 − 0.171412i \(-0.0548329\pi\)
\(14\) 1.00000 − 1.00000i 0.267261 − 0.267261i
\(15\) −5.23607 − 2.00000i −1.35195 − 0.516398i
\(16\) −4.00000 −1.00000
\(17\) − 2.00000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) −0.236068 − 4.23607i −0.0556418 − 0.998451i
\(19\) 7.23607 1.66007 0.830034 − 0.557713i \(-0.188321\pi\)
0.830034 + 0.557713i \(0.188321\pi\)
\(20\) 6.47214i 1.44721i
\(21\) 0.618034 − 1.61803i 0.134866 − 0.353084i
\(22\) −4.47214 + 4.47214i −0.953463 + 0.953463i
\(23\) −0.472136 −0.0984472 −0.0492236 − 0.998788i \(-0.515675\pi\)
−0.0492236 + 0.998788i \(0.515675\pi\)
\(24\) −4.47214 + 2.00000i −0.912871 + 0.408248i
\(25\) 5.47214 1.09443
\(26\) −1.23607 + 1.23607i −0.242413 + 0.242413i
\(27\) −2.38197 − 4.61803i −0.458410 − 0.888741i
\(28\) −2.00000 −0.377964
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 3.23607 + 7.23607i 0.590822 + 1.32112i
\(31\) 8.94427i 1.60644i 0.595683 + 0.803219i \(0.296881\pi\)
−0.595683 + 0.803219i \(0.703119\pi\)
\(32\) 4.00000 + 4.00000i 0.707107 + 0.707107i
\(33\) −2.76393 + 7.23607i −0.481139 + 1.25964i
\(34\) −2.00000 + 2.00000i −0.342997 + 0.342997i
\(35\) 3.23607i 0.546995i
\(36\) −4.00000 + 4.47214i −0.666667 + 0.745356i
\(37\) − 6.47214i − 1.06401i −0.846740 − 0.532006i \(-0.821438\pi\)
0.846740 − 0.532006i \(-0.178562\pi\)
\(38\) −7.23607 − 7.23607i −1.17385 − 1.17385i
\(39\) −0.763932 + 2.00000i −0.122327 + 0.320256i
\(40\) 6.47214 − 6.47214i 1.02333 − 1.02333i
\(41\) 4.47214i 0.698430i 0.937043 + 0.349215i \(0.113552\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) −2.23607 + 1.00000i −0.345033 + 0.154303i
\(43\) −0.472136 −0.0720001 −0.0360000 − 0.999352i \(-0.511462\pi\)
−0.0360000 + 0.999352i \(0.511462\pi\)
\(44\) 8.94427 1.34840
\(45\) 7.23607 + 6.47214i 1.07869 + 0.964809i
\(46\) 0.472136 + 0.472136i 0.0696126 + 0.0696126i
\(47\) 2.47214 0.360598 0.180299 − 0.983612i \(-0.442293\pi\)
0.180299 + 0.983612i \(0.442293\pi\)
\(48\) 6.47214 + 2.47214i 0.934172 + 0.356822i
\(49\) −1.00000 −0.142857
\(50\) −5.47214 − 5.47214i −0.773877 − 0.773877i
\(51\) −1.23607 + 3.23607i −0.173084 + 0.453140i
\(52\) 2.47214 0.342824
\(53\) −10.4721 −1.43846 −0.719229 − 0.694773i \(-0.755505\pi\)
−0.719229 + 0.694773i \(0.755505\pi\)
\(54\) −2.23607 + 7.00000i −0.304290 + 0.952579i
\(55\) − 14.4721i − 1.95142i
\(56\) 2.00000 + 2.00000i 0.267261 + 0.267261i
\(57\) −11.7082 − 4.47214i −1.55079 − 0.592349i
\(58\) 0 0
\(59\) 1.23607i 0.160922i 0.996758 + 0.0804612i \(0.0256393\pi\)
−0.996758 + 0.0804612i \(0.974361\pi\)
\(60\) 4.00000 − 10.4721i 0.516398 − 1.35195i
\(61\) 11.7082i 1.49908i 0.661958 + 0.749541i \(0.269726\pi\)
−0.661958 + 0.749541i \(0.730274\pi\)
\(62\) 8.94427 − 8.94427i 1.13592 − 1.13592i
\(63\) −2.00000 + 2.23607i −0.251976 + 0.281718i
\(64\) − 8.00000i − 1.00000i
\(65\) − 4.00000i − 0.496139i
\(66\) 10.0000 − 4.47214i 1.23091 − 0.550482i
\(67\) −10.9443 −1.33706 −0.668528 − 0.743687i \(-0.733075\pi\)
−0.668528 + 0.743687i \(0.733075\pi\)
\(68\) 4.00000 0.485071
\(69\) 0.763932 + 0.291796i 0.0919666 + 0.0351281i
\(70\) 3.23607 − 3.23607i 0.386784 − 0.386784i
\(71\) −6.94427 −0.824133 −0.412067 − 0.911154i \(-0.635193\pi\)
−0.412067 + 0.911154i \(0.635193\pi\)
\(72\) 8.47214 − 0.472136i 0.998451 − 0.0556418i
\(73\) −0.472136 −0.0552593 −0.0276297 − 0.999618i \(-0.508796\pi\)
−0.0276297 + 0.999618i \(0.508796\pi\)
\(74\) −6.47214 + 6.47214i −0.752371 + 0.752371i
\(75\) −8.85410 − 3.38197i −1.02238 − 0.390516i
\(76\) 14.4721i 1.66007i
\(77\) 4.47214 0.509647
\(78\) 2.76393 − 1.23607i 0.312954 − 0.139957i
\(79\) − 4.94427i − 0.556274i −0.960541 − 0.278137i \(-0.910283\pi\)
0.960541 − 0.278137i \(-0.0897169\pi\)
\(80\) −12.9443 −1.44721
\(81\) 1.00000 + 8.94427i 0.111111 + 0.993808i
\(82\) 4.47214 − 4.47214i 0.493865 − 0.493865i
\(83\) 13.2361i 1.45285i 0.687247 + 0.726424i \(0.258819\pi\)
−0.687247 + 0.726424i \(0.741181\pi\)
\(84\) 3.23607 + 1.23607i 0.353084 + 0.134866i
\(85\) − 6.47214i − 0.702002i
\(86\) 0.472136 + 0.472136i 0.0509117 + 0.0509117i
\(87\) 0 0
\(88\) −8.94427 − 8.94427i −0.953463 − 0.953463i
\(89\) − 6.00000i − 0.635999i −0.948091 − 0.317999i \(-0.896989\pi\)
0.948091 − 0.317999i \(-0.103011\pi\)
\(90\) −0.763932 − 13.7082i −0.0805255 − 1.44497i
\(91\) 1.23607 0.129575
\(92\) − 0.944272i − 0.0984472i
\(93\) 5.52786 − 14.4721i 0.573213 − 1.50069i
\(94\) −2.47214 − 2.47214i −0.254981 − 0.254981i
\(95\) 23.4164 2.40247
\(96\) −4.00000 − 8.94427i −0.408248 − 0.912871i
\(97\) 3.52786 0.358200 0.179100 − 0.983831i \(-0.442681\pi\)
0.179100 + 0.983831i \(0.442681\pi\)
\(98\) 1.00000 + 1.00000i 0.101015 + 0.101015i
\(99\) 8.94427 − 10.0000i 0.898933 − 1.00504i
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 168.2.j.a.155.1 ✓ 4
3.2 odd 2 168.2.j.c.155.3 yes 4
4.3 odd 2 672.2.j.c.239.4 4
8.3 odd 2 168.2.j.c.155.1 yes 4
8.5 even 2 672.2.j.b.239.4 4
12.11 even 2 672.2.j.b.239.3 4
24.5 odd 2 672.2.j.c.239.3 4
24.11 even 2 inner 168.2.j.a.155.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.j.a.155.1 ✓ 4 1.1 even 1 trivial
168.2.j.a.155.3 yes 4 24.11 even 2 inner
168.2.j.c.155.1 yes 4 8.3 odd 2
168.2.j.c.155.3 yes 4 3.2 odd 2
672.2.j.b.239.3 4 12.11 even 2
672.2.j.b.239.4 4 8.5 even 2
672.2.j.c.239.3 4 24.5 odd 2
672.2.j.c.239.4 4 4.3 odd 2