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 = [4,0] 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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 155.3
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 168.155
Dual form 168.2.j.b.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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +(1.00000 - 1.41421i) q^{3} +2.00000 q^{4} -2.82843 q^{5} +(1.41421 - 2.00000i) q^{6} -1.00000i q^{7} +2.82843 q^{8} +(-1.00000 - 2.82843i) q^{9} -4.00000 q^{10} +5.65685i q^{11} +(2.00000 - 2.82843i) q^{12} +4.00000i q^{13} -1.41421i q^{14} +(-2.82843 + 4.00000i) q^{15} +4.00000 q^{16} -2.82843i q^{17} +(-1.41421 - 4.00000i) q^{18} -2.00000 q^{19} -5.65685 q^{20} +(-1.41421 - 1.00000i) q^{21} +8.00000i q^{22} -2.82843 q^{23} +(2.82843 - 4.00000i) q^{24} +3.00000 q^{25} +5.65685i q^{26} +(-5.00000 - 1.41421i) q^{27} -2.00000i q^{28} +5.65685 q^{29} +(-4.00000 + 5.65685i) q^{30} +5.65685 q^{32} +(8.00000 + 5.65685i) q^{33} -4.00000i q^{34} +2.82843i q^{35} +(-2.00000 - 5.65685i) q^{36} -8.00000i q^{37} -2.82843 q^{38} +(5.65685 + 4.00000i) q^{39} -8.00000 q^{40} -2.82843i q^{41} +(-2.00000 - 1.41421i) q^{42} -10.0000 q^{43} +11.3137i q^{44} +(2.82843 + 8.00000i) q^{45} -4.00000 q^{46} +(4.00000 - 5.65685i) q^{48} -1.00000 q^{49} +4.24264 q^{50} +(-4.00000 - 2.82843i) q^{51} +8.00000i q^{52} -5.65685 q^{53} +(-7.07107 - 2.00000i) q^{54} -16.0000i q^{55} -2.82843i q^{56} +(-2.00000 + 2.82843i) q^{57} +8.00000 q^{58} +8.48528i q^{59} +(-5.65685 + 8.00000i) q^{60} -12.0000i q^{61} +(-2.82843 + 1.00000i) q^{63} +8.00000 q^{64} -11.3137i q^{65} +(11.3137 + 8.00000i) q^{66} +6.00000 q^{67} -5.65685i q^{68} +(-2.82843 + 4.00000i) q^{69} +4.00000i q^{70} +8.48528 q^{71} +(-2.82843 - 8.00000i) q^{72} +10.0000 q^{73} -11.3137i q^{74} +(3.00000 - 4.24264i) q^{75} -4.00000 q^{76} +5.65685 q^{77} +(8.00000 + 5.65685i) q^{78} +8.00000i q^{79} -11.3137 q^{80} +(-7.00000 + 5.65685i) q^{81} -4.00000i q^{82} -2.82843i q^{83} +(-2.82843 - 2.00000i) q^{84} +8.00000i q^{85} -14.1421 q^{86} +(5.65685 - 8.00000i) q^{87} +16.0000i q^{88} +2.82843i q^{89} +(4.00000 + 11.3137i) q^{90} +4.00000 q^{91} -5.65685 q^{92} +5.65685 q^{95} +(5.65685 - 8.00000i) q^{96} -2.00000 q^{97} -1.41421 q^{98} +(16.0000 - 5.65685i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 8 q^{4} - 4 q^{9} - 16 q^{10} + 8 q^{12} + 16 q^{16} - 8 q^{19} + 12 q^{25} - 20 q^{27} - 16 q^{30} + 32 q^{33} - 8 q^{36} - 32 q^{40} - 8 q^{42} - 40 q^{43} - 16 q^{46} + 16 q^{48} - 4 q^{49}+ \cdots + 64 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 1.00000
\(3\) 1.00000 − 1.41421i 0.577350 − 0.816497i
\(4\) 2.00000 1.00000
\(5\) −2.82843 −1.26491 −0.632456 − 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 1.41421 − 2.00000i 0.577350 − 0.816497i
\(7\) − 1.00000i − 0.377964i
\(8\) 2.82843 1.00000
\(9\) −1.00000 − 2.82843i −0.333333 − 0.942809i
\(10\) −4.00000 −1.26491
\(11\) 5.65685i 1.70561i 0.522233 + 0.852803i \(0.325099\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 2.00000 − 2.82843i 0.577350 − 0.816497i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) − 1.41421i − 0.377964i
\(15\) −2.82843 + 4.00000i −0.730297 + 1.03280i
\(16\) 4.00000 1.00000
\(17\) − 2.82843i − 0.685994i −0.939336 − 0.342997i \(-0.888558\pi\)
0.939336 − 0.342997i \(-0.111442\pi\)
\(18\) −1.41421 − 4.00000i −0.333333 − 0.942809i
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −5.65685 −1.26491
\(21\) −1.41421 − 1.00000i −0.308607 − 0.218218i
\(22\) 8.00000i 1.70561i
\(23\) −2.82843 −0.589768 −0.294884 − 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 2.82843 − 4.00000i 0.577350 − 0.816497i
\(25\) 3.00000 0.600000
\(26\) 5.65685i 1.10940i
\(27\) −5.00000 − 1.41421i −0.962250 − 0.272166i
\(28\) − 2.00000i − 0.377964i
\(29\) 5.65685 1.05045 0.525226 − 0.850963i \(-0.323981\pi\)
0.525226 + 0.850963i \(0.323981\pi\)
\(30\) −4.00000 + 5.65685i −0.730297 + 1.03280i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 5.65685 1.00000
\(33\) 8.00000 + 5.65685i 1.39262 + 0.984732i
\(34\) − 4.00000i − 0.685994i
\(35\) 2.82843i 0.478091i
\(36\) −2.00000 − 5.65685i −0.333333 − 0.942809i
\(37\) − 8.00000i − 1.31519i −0.753371 − 0.657596i \(-0.771573\pi\)
0.753371 − 0.657596i \(-0.228427\pi\)
\(38\) −2.82843 −0.458831
\(39\) 5.65685 + 4.00000i 0.905822 + 0.640513i
\(40\) −8.00000 −1.26491
\(41\) − 2.82843i − 0.441726i −0.975305 − 0.220863i \(-0.929113\pi\)
0.975305 − 0.220863i \(-0.0708874\pi\)
\(42\) −2.00000 − 1.41421i −0.308607 − 0.218218i
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 11.3137i 1.70561i
\(45\) 2.82843 + 8.00000i 0.421637 + 1.19257i
\(46\) −4.00000 −0.589768
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 4.00000 − 5.65685i 0.577350 − 0.816497i
\(49\) −1.00000 −0.142857
\(50\) 4.24264 0.600000
\(51\) −4.00000 − 2.82843i −0.560112 − 0.396059i
\(52\) 8.00000i 1.10940i
\(53\) −5.65685 −0.777029 −0.388514 − 0.921443i \(-0.627012\pi\)
−0.388514 + 0.921443i \(0.627012\pi\)
\(54\) −7.07107 − 2.00000i −0.962250 − 0.272166i
\(55\) − 16.0000i − 2.15744i
\(56\) − 2.82843i − 0.377964i
\(57\) −2.00000 + 2.82843i −0.264906 + 0.374634i
\(58\) 8.00000 1.05045
\(59\) 8.48528i 1.10469i 0.833616 + 0.552345i \(0.186267\pi\)
−0.833616 + 0.552345i \(0.813733\pi\)
\(60\) −5.65685 + 8.00000i −0.730297 + 1.03280i
\(61\) − 12.0000i − 1.53644i −0.640184 − 0.768221i \(-0.721142\pi\)
0.640184 − 0.768221i \(-0.278858\pi\)
\(62\) 0 0
\(63\) −2.82843 + 1.00000i −0.356348 + 0.125988i
\(64\) 8.00000 1.00000
\(65\) − 11.3137i − 1.40329i
\(66\) 11.3137 + 8.00000i 1.39262 + 0.984732i
\(67\) 6.00000 0.733017 0.366508 − 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) − 5.65685i − 0.685994i
\(69\) −2.82843 + 4.00000i −0.340503 + 0.481543i
\(70\) 4.00000i 0.478091i
\(71\) 8.48528 1.00702 0.503509 − 0.863990i \(-0.332042\pi\)
0.503509 + 0.863990i \(0.332042\pi\)
\(72\) −2.82843 − 8.00000i −0.333333 − 0.942809i
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) − 11.3137i − 1.31519i
\(75\) 3.00000 − 4.24264i 0.346410 − 0.489898i
\(76\) −4.00000 −0.458831
\(77\) 5.65685 0.644658
\(78\) 8.00000 + 5.65685i 0.905822 + 0.640513i
\(79\) 8.00000i 0.900070i 0.893011 + 0.450035i \(0.148589\pi\)
−0.893011 + 0.450035i \(0.851411\pi\)
\(80\) −11.3137 −1.26491
\(81\) −7.00000 + 5.65685i −0.777778 + 0.628539i
\(82\) − 4.00000i − 0.441726i
\(83\) − 2.82843i − 0.310460i −0.987878 − 0.155230i \(-0.950388\pi\)
0.987878 − 0.155230i \(-0.0496119\pi\)
\(84\) −2.82843 − 2.00000i −0.308607 − 0.218218i
\(85\) 8.00000i 0.867722i
\(86\) −14.1421 −1.52499
\(87\) 5.65685 − 8.00000i 0.606478 − 0.857690i
\(88\) 16.0000i 1.70561i
\(89\) 2.82843i 0.299813i 0.988700 + 0.149906i \(0.0478972\pi\)
−0.988700 + 0.149906i \(0.952103\pi\)
\(90\) 4.00000 + 11.3137i 0.421637 + 1.19257i
\(91\) 4.00000 0.419314
\(92\) −5.65685 −0.589768
\(93\) 0 0
\(94\) 0 0
\(95\) 5.65685 0.580381
\(96\) 5.65685 − 8.00000i 0.577350 − 0.816497i
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.41421 −0.142857
\(99\) 16.0000 − 5.65685i 1.60806 − 0.568535i
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.b.155.3 yes 4
3.2 odd 2 inner 168.2.j.b.155.2 yes 4
4.3 odd 2 672.2.j.a.239.3 4
8.3 odd 2 inner 168.2.j.b.155.1 ✓ 4
8.5 even 2 672.2.j.a.239.4 4
12.11 even 2 672.2.j.a.239.2 4
24.5 odd 2 672.2.j.a.239.1 4
24.11 even 2 inner 168.2.j.b.155.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.j.b.155.1 ✓ 4 8.3 odd 2 inner
168.2.j.b.155.2 yes 4 3.2 odd 2 inner
168.2.j.b.155.3 yes 4 1.1 even 1 trivial
168.2.j.b.155.4 yes 4 24.11 even 2 inner
672.2.j.a.239.1 4 24.5 odd 2
672.2.j.a.239.2 4 12.11 even 2
672.2.j.a.239.3 4 4.3 odd 2
672.2.j.a.239.4 4 8.5 even 2