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(5,168)] 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("168.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 101.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 168.101
Dual form 168.2.ba.b.5.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.707107 + 1.22474i) q^{2} +(1.50000 + 0.866025i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(0.621320 - 0.358719i) q^{5} +2.44949i q^{6} +(-2.62132 - 0.358719i) q^{7} -2.82843 q^{8} +(1.50000 + 2.59808i) q^{9} +(0.878680 + 0.507306i) q^{10} +(2.91421 - 5.04757i) q^{11} +(-3.00000 + 1.73205i) q^{12} +(-1.41421 - 3.46410i) q^{14} +1.24264 q^{15} +(-2.00000 - 3.46410i) q^{16} +(-2.12132 + 3.67423i) q^{18} +1.43488i q^{20} +(-3.62132 - 2.80821i) q^{21} +8.24264 q^{22} +(-4.24264 - 2.44949i) q^{24} +(-2.24264 + 3.88437i) q^{25} +5.19615i q^{27} +(3.24264 - 4.18154i) q^{28} +7.58579 q^{29} +(0.878680 + 1.52192i) q^{30} +(-9.62132 - 5.55487i) q^{31} +(2.82843 - 4.89898i) q^{32} +(8.74264 - 5.04757i) q^{33} +(-1.75736 + 0.717439i) q^{35} -6.00000 q^{36} +(-1.75736 + 1.01461i) q^{40} +(0.878680 - 6.42090i) q^{42} +(5.82843 + 10.0951i) q^{44} +(1.86396 + 1.07616i) q^{45} -6.92820i q^{48} +(6.74264 + 1.88064i) q^{49} -6.34315 q^{50} +(-2.03553 + 3.52565i) q^{53} +(-6.36396 + 3.67423i) q^{54} -4.18154i q^{55} +(7.41421 + 1.01461i) q^{56} +(5.36396 + 9.29065i) q^{58} +(-12.9853 - 7.49706i) q^{59} +(-1.24264 + 2.15232i) q^{60} -15.7116i q^{62} +(-3.00000 - 7.34847i) q^{63} +8.00000 q^{64} +(12.3640 + 7.13834i) q^{66} +(-2.12132 - 1.64501i) q^{70} +(-4.24264 - 7.34847i) q^{72} +(-8.48528 - 4.89898i) q^{73} +(-6.72792 + 3.88437i) q^{75} +(-9.44975 + 12.1859i) q^{77} +(8.86396 + 15.3528i) q^{79} +(-2.48528 - 1.43488i) q^{80} +(-4.50000 + 7.79423i) q^{81} +13.5592i q^{83} +(8.48528 - 3.46410i) q^{84} +(11.3787 + 6.56948i) q^{87} +(-8.24264 + 14.2767i) q^{88} +3.04384i q^{90} +(-9.62132 - 16.6646i) q^{93} +(8.48528 - 4.89898i) q^{96} +8.06591i q^{97} +(2.46447 + 9.58783i) q^{98} +17.4853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 4 q^{4} - 6 q^{5} - 2 q^{7} + 6 q^{9} + 12 q^{10} + 6 q^{11} - 12 q^{12} - 12 q^{15} - 8 q^{16} - 6 q^{21} + 16 q^{22} + 8 q^{25} - 4 q^{28} + 36 q^{29} + 12 q^{30} - 30 q^{31} + 18 q^{33}+ \cdots + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 + 1.22474i 0.500000 + 0.866025i
\(3\) 1.50000 + 0.866025i 0.866025 + 0.500000i
\(4\) −1.00000 + 1.73205i −0.500000 + 0.866025i
\(5\) 0.621320 − 0.358719i 0.277863 − 0.160424i −0.354593 − 0.935021i \(-0.615380\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) 2.44949i 1.00000i
\(7\) −2.62132 − 0.358719i −0.990766 − 0.135583i
\(8\) −2.82843 −1.00000
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0.878680 + 0.507306i 0.277863 + 0.160424i
\(11\) 2.91421 − 5.04757i 0.878668 − 1.52190i 0.0258656 − 0.999665i \(-0.491766\pi\)
0.852803 − 0.522233i \(-0.174901\pi\)
\(12\) −3.00000 + 1.73205i −0.866025 + 0.500000i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −1.41421 − 3.46410i −0.377964 − 0.925820i
\(15\) 1.24264 0.320848
\(16\) −2.00000 − 3.46410i −0.500000 − 0.866025i
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) −2.12132 + 3.67423i −0.500000 + 0.866025i
\(19\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 1.43488i 0.320848i
\(21\) −3.62132 − 2.80821i −0.790237 − 0.612801i
\(22\) 8.24264 1.75734
\(23\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) −4.24264 − 2.44949i −0.866025 − 0.500000i
\(25\) −2.24264 + 3.88437i −0.448528 + 0.776874i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 3.24264 − 4.18154i 0.612801 − 0.790237i
\(29\) 7.58579 1.40865 0.704323 − 0.709880i \(-0.251251\pi\)
0.704323 + 0.709880i \(0.251251\pi\)
\(30\) 0.878680 + 1.52192i 0.160424 + 0.277863i
\(31\) −9.62132 − 5.55487i −1.72804 − 0.997684i −0.898027 − 0.439941i \(-0.854999\pi\)
−0.830014 − 0.557743i \(-0.811667\pi\)
\(32\) 2.82843 − 4.89898i 0.500000 − 0.866025i
\(33\) 8.74264 − 5.04757i 1.52190 − 0.878668i
\(34\) 0 0
\(35\) −1.75736 + 0.717439i −0.297048 + 0.121269i
\(36\) −6.00000 −1.00000
\(37\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.75736 + 1.01461i −0.277863 + 0.160424i
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0.878680 − 6.42090i 0.135583 − 0.990766i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 5.82843 + 10.0951i 0.878668 + 1.52190i
\(45\) 1.86396 + 1.07616i 0.277863 + 0.160424i
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) − 6.92820i − 1.00000i
\(49\) 6.74264 + 1.88064i 0.963234 + 0.268662i
\(50\) −6.34315 −0.897056
\(51\) 0 0
\(52\) 0 0
\(53\) −2.03553 + 3.52565i −0.279602 + 0.484285i −0.971286 − 0.237915i \(-0.923536\pi\)
0.691684 + 0.722200i \(0.256869\pi\)
\(54\) −6.36396 + 3.67423i −0.866025 + 0.500000i
\(55\) − 4.18154i − 0.563839i
\(56\) 7.41421 + 1.01461i 0.990766 + 0.135583i
\(57\) 0 0
\(58\) 5.36396 + 9.29065i 0.704323 + 1.21992i
\(59\) −12.9853 − 7.49706i −1.69054 − 0.976034i −0.954080 − 0.299552i \(-0.903163\pi\)
−0.736460 − 0.676481i \(-0.763504\pi\)
\(60\) −1.24264 + 2.15232i −0.160424 + 0.277863i
\(61\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) − 15.7116i − 1.99537i
\(63\) −3.00000 − 7.34847i −0.377964 − 0.925820i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 12.3640 + 7.13834i 1.52190 + 0.878668i
\(67\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −2.12132 − 1.64501i −0.253546 − 0.196616i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −4.24264 − 7.34847i −0.500000 − 0.866025i
\(73\) −8.48528 − 4.89898i −0.993127 − 0.573382i −0.0869195 − 0.996215i \(-0.527702\pi\)
−0.906208 + 0.422833i \(0.861036\pi\)
\(74\) 0 0
\(75\) −6.72792 + 3.88437i −0.776874 + 0.448528i
\(76\) 0 0
\(77\) −9.44975 + 12.1859i −1.07690 + 1.38871i
\(78\) 0 0
\(79\) 8.86396 + 15.3528i 0.997274 + 1.72733i 0.562544 + 0.826767i \(0.309823\pi\)
0.434730 + 0.900561i \(0.356844\pi\)
\(80\) −2.48528 − 1.43488i −0.277863 − 0.160424i
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 13.5592i 1.48832i 0.668002 + 0.744160i \(0.267150\pi\)
−0.668002 + 0.744160i \(0.732850\pi\)
\(84\) 8.48528 − 3.46410i 0.925820 − 0.377964i
\(85\) 0 0
\(86\) 0 0
\(87\) 11.3787 + 6.56948i 1.21992 + 0.704323i
\(88\) −8.24264 + 14.2767i −0.878668 + 1.52190i
\(89\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 3.04384i 0.320848i
\(91\) 0 0
\(92\) 0 0
\(93\) −9.62132 − 16.6646i −0.997684 − 1.72804i
\(94\) 0 0
\(95\) 0 0
\(96\) 8.48528 − 4.89898i 0.866025 − 0.500000i
\(97\) 8.06591i 0.818969i 0.912317 + 0.409484i \(0.134291\pi\)
−0.912317 + 0.409484i \(0.865709\pi\)
\(98\) 2.46447 + 9.58783i 0.248949 + 0.968517i
\(99\) 17.4853 1.75734
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.ba.b.101.2 yes 4
3.2 odd 2 168.2.ba.a.101.1 yes 4
4.3 odd 2 672.2.bi.a.17.2 4
7.5 odd 6 inner 168.2.ba.b.5.2 yes 4
8.3 odd 2 672.2.bi.b.17.1 4
8.5 even 2 168.2.ba.a.101.1 yes 4
12.11 even 2 672.2.bi.b.17.1 4
21.5 even 6 168.2.ba.a.5.1 ✓ 4
24.5 odd 2 CM 168.2.ba.b.101.2 yes 4
24.11 even 2 672.2.bi.a.17.2 4
28.19 even 6 672.2.bi.a.593.2 4
56.5 odd 6 168.2.ba.a.5.1 ✓ 4
56.19 even 6 672.2.bi.b.593.1 4
84.47 odd 6 672.2.bi.b.593.1 4
168.5 even 6 inner 168.2.ba.b.5.2 yes 4
168.131 odd 6 672.2.bi.a.593.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.ba.a.5.1 ✓ 4 21.5 even 6
168.2.ba.a.5.1 ✓ 4 56.5 odd 6
168.2.ba.a.101.1 yes 4 3.2 odd 2
168.2.ba.a.101.1 yes 4 8.5 even 2
168.2.ba.b.5.2 yes 4 7.5 odd 6 inner
168.2.ba.b.5.2 yes 4 168.5 even 6 inner
168.2.ba.b.101.2 yes 4 1.1 even 1 trivial
168.2.ba.b.101.2 yes 4 24.5 odd 2 CM
672.2.bi.a.17.2 4 4.3 odd 2
672.2.bi.a.17.2 4 24.11 even 2
672.2.bi.a.593.2 4 28.19 even 6
672.2.bi.a.593.2 4 168.131 odd 6
672.2.bi.b.17.1 4 8.3 odd 2
672.2.bi.b.17.1 4 12.11 even 2
672.2.bi.b.593.1 4 56.19 even 6
672.2.bi.b.593.1 4 84.47 odd 6