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Properties

Label 168.2.j.d.155.3
Level $168$
Weight $2$
Character 168.155
Analytic conductor $1.341$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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 = [12,0] 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: \(12\)
Coefficient field: 12.0.2593100598870016.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{10} + x^{8} + 4x^{6} + 4x^{4} - 32x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 155.3
Root \(1.19877 - 0.750295i\) of defining polynomial
Character \(\chi\) \(=\) 168.155
Dual form 168.2.j.d.155.4

$q$-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+(-0.750295 - 1.19877i) q^{2} +(1.67298 - 0.448478i) q^{3} +(-0.874114 + 1.79887i) q^{4} +0.896956 q^{5} +(-1.79285 - 1.66903i) q^{6} -1.00000i q^{7} +(2.81228 - 0.301817i) q^{8} +(2.59774 - 1.50059i) q^{9} +(-0.672982 - 1.07525i) q^{10} +(-0.655624 + 3.40149i) q^{12} -1.84951i q^{13} +(-1.19877 + 0.750295i) q^{14} +(1.50059 - 0.402265i) q^{15} +(-2.47185 - 3.14483i) q^{16} +4.12397i q^{17} +(-3.74794 - 1.98821i) q^{18} -0.654037 q^{19} +(-0.784042 + 1.61350i) q^{20} +(-0.448478 - 1.67298i) q^{21} -7.12515 q^{23} +(4.56953 - 1.76618i) q^{24} -4.19547 q^{25} +(-2.21714 + 1.38768i) q^{26} +(3.67298 - 3.67549i) q^{27} +(1.79887 + 0.874114i) q^{28} +7.79627 q^{29} +(-1.60811 - 1.49705i) q^{30} +(-1.91532 + 5.32274i) q^{32} +(4.94370 - 3.09419i) q^{34} -0.896956i q^{35} +(0.428647 + 5.98467i) q^{36} +10.6919i q^{37} +(0.490721 + 0.784042i) q^{38} +(-0.829463 - 3.09419i) q^{39} +(2.52249 - 0.270717i) q^{40} +10.1263i q^{41} +(-1.66903 + 1.79285i) q^{42} -1.19547 q^{43} +(2.33005 - 1.34596i) q^{45} +(5.34596 + 8.54143i) q^{46} -9.59019 q^{47} +(-5.54575 - 4.15267i) q^{48} -1.00000 q^{49} +(3.14784 + 5.02942i) q^{50} +(1.84951 + 6.89932i) q^{51} +(3.32702 + 1.61668i) q^{52} +8.24793 q^{53} +(-7.16190 - 1.64537i) q^{54} +(-0.301817 - 2.81228i) q^{56} +(-1.09419 + 0.293321i) q^{57} +(-5.84951 - 9.34596i) q^{58} -6.89932i q^{59} +(-0.588066 + 3.05099i) q^{60} -4.54143i q^{61} +(-1.50059 - 2.59774i) q^{63} +(7.81781 - 1.69759i) q^{64} -1.65893i q^{65} -5.49646 q^{67} +(-7.41847 - 3.60482i) q^{68} +(-11.9202 + 3.19547i) q^{69} +(-1.07525 + 0.672982i) q^{70} +4.12397 q^{71} +(6.85265 - 5.00412i) q^{72} -6.00000 q^{73} +(12.8172 - 8.02210i) q^{74} +(-7.01894 + 1.88158i) q^{75} +(0.571703 - 1.17653i) q^{76} +(-3.08689 + 3.31590i) q^{78} +13.3839i q^{79} +(-2.21714 - 2.82077i) q^{80} +(4.49646 - 7.79627i) q^{81} +(12.1392 - 7.59774i) q^{82} -13.3533i q^{83} +(3.40149 + 0.655624i) q^{84} +3.69901i q^{85} +(0.896956 + 1.43310i) q^{86} +(13.0430 - 3.49646i) q^{87} -13.7142i q^{89} +(-3.36173 - 1.78334i) q^{90} -1.84951 q^{91} +(6.22819 - 12.8172i) q^{92} +(7.19547 + 11.4965i) q^{94} -0.586642 q^{95} +(-0.817168 + 9.76382i) q^{96} -8.99291 q^{97} +(0.750295 + 1.19877i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{4} - 10 q^{6} + 4 q^{9} + 12 q^{10} - 14 q^{12} + 4 q^{16} - 8 q^{18} - 48 q^{19} + 22 q^{24} + 4 q^{25} + 24 q^{27} + 8 q^{28} - 20 q^{30} - 8 q^{34} + 20 q^{36} - 4 q^{40} - 6 q^{42} + 40 q^{43}+ \cdots - 56 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.750295 − 1.19877i −0.530539 − 0.847661i
\(3\) 1.67298 − 0.448478i 0.965896 − 0.258929i
\(4\) −0.874114 + 1.79887i −0.437057 + 0.899434i
\(5\) 0.896956 0.401131 0.200565 − 0.979680i \(-0.435722\pi\)
0.200565 + 0.979680i \(0.435722\pi\)
\(6\) −1.79285 − 1.66903i −0.731929 − 0.681381i
\(7\) − 1.00000i − 0.377964i
\(8\) 2.81228 − 0.301817i 0.994290 − 0.106709i
\(9\) 2.59774 − 1.50059i 0.865912 − 0.500197i
\(10\) −0.672982 − 1.07525i −0.212815 − 0.340023i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −0.655624 + 3.40149i −0.189262 + 0.981927i
\(13\) − 1.84951i − 0.512961i −0.966549 − 0.256480i \(-0.917437\pi\)
0.966549 − 0.256480i \(-0.0825629\pi\)
\(14\) −1.19877 + 0.750295i −0.320386 + 0.200525i
\(15\) 1.50059 − 0.402265i 0.387451 − 0.103864i
\(16\) −2.47185 − 3.14483i −0.617962 − 0.786208i
\(17\) 4.12397i 1.00021i 0.865965 + 0.500104i \(0.166705\pi\)
−0.865965 + 0.500104i \(0.833295\pi\)
\(18\) −3.74794 − 1.98821i −0.883397 − 0.468625i
\(19\) −0.654037 −0.150046 −0.0750232 − 0.997182i \(-0.523903\pi\)
−0.0750232 + 0.997182i \(0.523903\pi\)
\(20\) −0.784042 + 1.61350i −0.175317 + 0.360791i
\(21\) −0.448478 − 1.67298i −0.0978659 − 0.365075i
\(22\) 0 0
\(23\) −7.12515 −1.48570 −0.742848 − 0.669460i \(-0.766525\pi\)
−0.742848 + 0.669460i \(0.766525\pi\)
\(24\) 4.56953 − 1.76618i 0.932752 − 0.360520i
\(25\) −4.19547 −0.839094
\(26\) −2.21714 + 1.38768i −0.434817 + 0.272146i
\(27\) 3.67298 − 3.67549i 0.706866 − 0.707348i
\(28\) 1.79887 + 0.874114i 0.339954 + 0.165192i
\(29\) 7.79627 1.44773 0.723866 − 0.689941i \(-0.242363\pi\)
0.723866 + 0.689941i \(0.242363\pi\)
\(30\) −1.60811 − 1.49705i −0.293599 − 0.273323i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.91532 + 5.32274i −0.338584 + 0.940936i
\(33\) 0 0
\(34\) 4.94370 − 3.09419i 0.847837 − 0.530649i
\(35\) − 0.896956i − 0.151613i
\(36\) 0.428647 + 5.98467i 0.0714411 + 0.997445i
\(37\) 10.6919i 1.75774i 0.477060 + 0.878871i \(0.341703\pi\)
−0.477060 + 0.878871i \(0.658297\pi\)
\(38\) 0.490721 + 0.784042i 0.0796054 + 0.127188i
\(39\) −0.829463 − 3.09419i −0.132820 − 0.495467i
\(40\) 2.52249 − 0.270717i 0.398840 − 0.0428041i
\(41\) 10.1263i 1.58147i 0.612161 + 0.790733i \(0.290301\pi\)
−0.612161 + 0.790733i \(0.709699\pi\)
\(42\) −1.66903 + 1.79285i −0.257538 + 0.276643i
\(43\) −1.19547 −0.182308 −0.0911538 − 0.995837i \(-0.529055\pi\)
−0.0911538 + 0.995837i \(0.529055\pi\)
\(44\) 0 0
\(45\) 2.33005 − 1.34596i 0.347344 − 0.200644i
\(46\) 5.34596 + 8.54143i 0.788219 + 1.25937i
\(47\) −9.59019 −1.39887 −0.699436 − 0.714695i \(-0.746565\pi\)
−0.699436 + 0.714695i \(0.746565\pi\)
\(48\) −5.54575 − 4.15267i −0.800459 − 0.599387i
\(49\) −1.00000 −0.142857
\(50\) 3.14784 + 5.02942i 0.445172 + 0.711267i
\(51\) 1.84951 + 6.89932i 0.258983 + 0.966098i
\(52\) 3.32702 + 1.61668i 0.461374 + 0.224193i
\(53\) 8.24793 1.13294 0.566470 − 0.824082i \(-0.308309\pi\)
0.566470 + 0.824082i \(0.308309\pi\)
\(54\) −7.16190 − 1.64537i −0.974611 − 0.223907i
\(55\) 0 0
\(56\) −0.301817 − 2.81228i −0.0403320 − 0.375806i
\(57\) −1.09419 + 0.293321i −0.144929 + 0.0388513i
\(58\) −5.84951 − 9.34596i −0.768078 − 1.22719i
\(59\) − 6.89932i − 0.898215i −0.893478 − 0.449107i \(-0.851742\pi\)
0.893478 − 0.449107i \(-0.148258\pi\)
\(60\) −0.588066 + 3.05099i −0.0759190 + 0.393881i
\(61\) − 4.54143i − 0.581471i −0.956803 − 0.290735i \(-0.906100\pi\)
0.956803 − 0.290735i \(-0.0939000\pi\)
\(62\) 0 0
\(63\) −1.50059 − 2.59774i −0.189057 − 0.327284i
\(64\) 7.81781 − 1.69759i 0.977227 − 0.212199i
\(65\) − 1.65893i − 0.205764i
\(66\) 0 0
\(67\) −5.49646 −0.671499 −0.335750 − 0.941951i \(-0.608990\pi\)
−0.335750 + 0.941951i \(0.608990\pi\)
\(68\) −7.41847 − 3.60482i −0.899621 − 0.437148i
\(69\) −11.9202 + 3.19547i −1.43503 + 0.384689i
\(70\) −1.07525 + 0.672982i −0.128517 + 0.0804367i
\(71\) 4.12397 0.489425 0.244712 − 0.969596i \(-0.421306\pi\)
0.244712 + 0.969596i \(0.421306\pi\)
\(72\) 6.85265 − 5.00412i 0.807592 − 0.589741i
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 12.8172 − 8.02210i 1.48997 − 0.932550i
\(75\) −7.01894 + 1.88158i −0.810478 + 0.217266i
\(76\) 0.571703 − 1.17653i 0.0655788 − 0.134957i
\(77\) 0 0
\(78\) −3.08689 + 3.31590i −0.349522 + 0.375451i
\(79\) 13.3839i 1.50580i 0.658134 + 0.752901i \(0.271346\pi\)
−0.658134 + 0.752901i \(0.728654\pi\)
\(80\) −2.21714 − 2.82077i −0.247884 − 0.315372i
\(81\) 4.49646 − 7.79627i 0.499606 − 0.866253i
\(82\) 12.1392 − 7.59774i 1.34055 − 0.839029i
\(83\) − 13.3533i − 1.46572i −0.680380 − 0.732860i \(-0.738185\pi\)
0.680380 − 0.732860i \(-0.261815\pi\)
\(84\) 3.40149 + 0.655624i 0.371133 + 0.0715345i
\(85\) 3.69901i 0.401214i
\(86\) 0.896956 + 1.43310i 0.0967212 + 0.154535i
\(87\) 13.0430 − 3.49646i 1.39836 − 0.374859i
\(88\) 0 0
\(89\) − 13.7142i − 1.45370i −0.686798 − 0.726849i \(-0.740984\pi\)
0.686798 − 0.726849i \(-0.259016\pi\)
\(90\) −3.36173 − 1.78334i −0.354358 − 0.187980i
\(91\) −1.84951 −0.193881
\(92\) 6.22819 − 12.8172i 0.649334 − 1.33628i
\(93\) 0 0
\(94\) 7.19547 + 11.4965i 0.742156 + 1.18577i
\(95\) −0.586642 −0.0601882
\(96\) −0.817168 + 9.76382i −0.0834019 + 0.996516i
\(97\) −8.99291 −0.913092 −0.456546 − 0.889700i \(-0.650914\pi\)
−0.456546 + 0.889700i \(0.650914\pi\)
\(98\) 0.750295 + 1.19877i 0.0757913 + 0.121094i
\(99\) 0 0
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.d.155.3 ✓ 12
3.2 odd 2 inner 168.2.j.d.155.10 yes 12
4.3 odd 2 672.2.j.d.239.4 12
8.3 odd 2 inner 168.2.j.d.155.9 yes 12
8.5 even 2 672.2.j.d.239.3 12
12.11 even 2 672.2.j.d.239.1 12
24.5 odd 2 672.2.j.d.239.2 12
24.11 even 2 inner 168.2.j.d.155.4 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.j.d.155.3 ✓ 12 1.1 even 1 trivial
168.2.j.d.155.4 yes 12 24.11 even 2 inner
168.2.j.d.155.9 yes 12 8.3 odd 2 inner
168.2.j.d.155.10 yes 12 3.2 odd 2 inner
672.2.j.d.239.1 12 12.11 even 2
672.2.j.d.239.2 12 24.5 odd 2
672.2.j.d.239.3 12 8.5 even 2
672.2.j.d.239.4 12 4.3 odd 2