↑

Properties

Label 168.2.j.d.155.4
Level $168$
Weight $2$
Character 168.155
Analytic conductor $1.341$
Analytic rank $0$
Dimension $12$
Inner twists $4$

Related objects

Downloads

Learn more

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 155.4
Root \(1.19877 + 0.750295i\) of defining polynomial
Character \(\chi\) \(=\) 168.155
Dual form 168.2.j.d.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+(-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 \(0.0825629\pi\)
−0.966549 + 0.256480i \(0.917437\pi\)
\(14\) −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 \(-0.833295\pi\)
0.865965 − 0.500104i \(-0.166705\pi\)
\(18\) −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 \(-0.658297\pi\)
0.477060 − 0.878871i \(-0.341703\pi\)
\(38\) 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 \(-0.709699\pi\)
0.612161 − 0.790733i \(-0.290301\pi\)
\(42\) −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 \(0.148258\pi\)
−0.893478 + 0.449107i \(0.851742\pi\)
\(60\) −0.588066 − 3.05099i −0.0759190 − 0.393881i
\(61\) 4.54143i 0.581471i 0.956803 + 0.290735i \(0.0939000\pi\)
−0.956803 + 0.290735i \(0.906100\pi\)
\(62\) 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 \(-0.728654\pi\)
0.658134 − 0.752901i \(-0.271346\pi\)
\(80\) −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 \(0.261815\pi\)
−0.680380 + 0.732860i \(0.738185\pi\)
\(84\) 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 \(0.259016\pi\)
−0.686798 + 0.726849i \(0.740984\pi\)
\(90\) −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.155.4 yes 12
3.2 odd 2 inner 168.2.j.d.155.9 yes 12
4.3 odd 2 672.2.j.d.239.2 12
8.3 odd 2 inner 168.2.j.d.155.10 yes 12
8.5 even 2 672.2.j.d.239.1 12
12.11 even 2 672.2.j.d.239.3 12
24.5 odd 2 672.2.j.d.239.4 12
24.11 even 2 inner 168.2.j.d.155.3 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.j.d.155.3 ✓ 12 24.11 even 2 inner
168.2.j.d.155.4 yes 12 1.1 even 1 trivial
168.2.j.d.155.9 yes 12 3.2 odd 2 inner
168.2.j.d.155.10 yes 12 8.3 odd 2 inner
672.2.j.d.239.1 12 8.5 even 2
672.2.j.d.239.2 12 4.3 odd 2
672.2.j.d.239.3 12 12.11 even 2
672.2.j.d.239.4 12 24.5 odd 2