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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [168,2,Mod(125,168)] 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("168.125"); 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([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == 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: \(\Q(\sqrt{2}, \sqrt{-3})\)
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: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 125.3
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 168.125
Dual form 168.2.i.b.125.4

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +3.46410i q^{5} -2.44949i q^{6} +(1.00000 - 2.44949i) q^{7} +2.82843 q^{8} -3.00000 q^{9} +4.89898i q^{10} -5.65685 q^{11} -3.46410i q^{12} +(1.41421 - 3.46410i) q^{14} +6.00000 q^{15} +4.00000 q^{16} -4.24264 q^{18} +6.92820i q^{20} +(-4.24264 - 1.73205i) q^{21} -8.00000 q^{22} -4.89898i q^{24} -7.00000 q^{25} +5.19615i q^{27} +(2.00000 - 4.89898i) q^{28} -2.82843 q^{29} +8.48528 q^{30} +4.89898i q^{31} +5.65685 q^{32} +9.79796i q^{33} +(8.48528 + 3.46410i) q^{35} -6.00000 q^{36} +9.79796i q^{40} +(-6.00000 - 2.44949i) q^{42} -11.3137 q^{44} -10.3923i q^{45} -6.92820i q^{48} +(-5.00000 - 4.89898i) q^{49} -9.89949 q^{50} +14.1421 q^{53} +7.34847i q^{54} -19.5959i q^{55} +(2.82843 - 6.92820i) q^{56} -4.00000 q^{58} -10.3923i q^{59} +12.0000 q^{60} +6.92820i q^{62} +(-3.00000 + 7.34847i) q^{63} +8.00000 q^{64} +13.8564i q^{66} +(12.0000 + 4.89898i) q^{70} -8.48528 q^{72} -9.79796i q^{73} +12.1244i q^{75} +(-5.65685 + 13.8564i) q^{77} +10.0000 q^{79} +13.8564i q^{80} +9.00000 q^{81} -17.3205i q^{83} +(-8.48528 - 3.46410i) q^{84} +4.89898i q^{87} -16.0000 q^{88} -14.6969i q^{90} +8.48528 q^{93} -9.79796i q^{96} +19.5959i q^{97} +(-7.07107 - 6.92820i) q^{98} +16.9706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + 4 q^{7} - 12 q^{9} + 24 q^{15} + 16 q^{16} - 32 q^{22} - 28 q^{25} + 8 q^{28} - 24 q^{36} - 24 q^{42} - 20 q^{49} - 16 q^{58} + 48 q^{60} - 12 q^{63} + 32 q^{64} + 48 q^{70} + 40 q^{79}+ \cdots - 64 q^{88}+O(q^{100}) \) 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
\(3\) − 1.73205i − 1.00000i
\(4\) 2.00000 1.00000
\(5\) 3.46410i 1.54919i 0.632456 + 0.774597i \(0.282047\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) − 2.44949i − 1.00000i
\(7\) 1.00000 − 2.44949i 0.377964 − 0.925820i
\(8\) 2.82843 1.00000
\(9\) −3.00000 −1.00000
\(10\) 4.89898i 1.54919i
\(11\) −5.65685 −1.70561 −0.852803 − 0.522233i \(-0.825099\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) − 3.46410i − 1.00000i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 1.41421 − 3.46410i 0.377964 − 0.925820i
\(15\) 6.00000 1.54919
\(16\) 4.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −4.24264 −1.00000
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 6.92820i 1.54919i
\(21\) −4.24264 − 1.73205i −0.925820 − 0.377964i
\(22\) −8.00000 −1.70561
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) − 4.89898i − 1.00000i
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 2.00000 − 4.89898i 0.377964 − 0.925820i
\(29\) −2.82843 −0.525226 −0.262613 − 0.964901i \(-0.584584\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 8.48528 1.54919
\(31\) 4.89898i 0.879883i 0.898027 + 0.439941i \(0.145001\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 5.65685 1.00000
\(33\) 9.79796i 1.70561i
\(34\) 0 0
\(35\) 8.48528 + 3.46410i 1.43427 + 0.585540i
\(36\) −6.00000 −1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 9.79796i 1.54919i
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −6.00000 − 2.44949i −0.925820 − 0.377964i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −11.3137 −1.70561
\(45\) − 10.3923i − 1.54919i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) − 6.92820i − 1.00000i
\(49\) −5.00000 − 4.89898i −0.714286 − 0.699854i
\(50\) −9.89949 −1.40000
\(51\) 0 0
\(52\) 0 0
\(53\) 14.1421 1.94257 0.971286 − 0.237915i \(-0.0764641\pi\)
0.971286 + 0.237915i \(0.0764641\pi\)
\(54\) 7.34847i 1.00000i
\(55\) − 19.5959i − 2.64231i
\(56\) 2.82843 − 6.92820i 0.377964 − 0.925820i
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) − 10.3923i − 1.35296i −0.736460 − 0.676481i \(-0.763504\pi\)
0.736460 − 0.676481i \(-0.236496\pi\)
\(60\) 12.0000 1.54919
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 6.92820i 0.879883i
\(63\) −3.00000 + 7.34847i −0.377964 + 0.925820i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 13.8564i 1.70561i
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 12.0000 + 4.89898i 1.43427 + 0.585540i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −8.48528 −1.00000
\(73\) − 9.79796i − 1.14676i −0.819288 − 0.573382i \(-0.805631\pi\)
0.819288 − 0.573382i \(-0.194369\pi\)
\(74\) 0 0
\(75\) 12.1244i 1.40000i
\(76\) 0 0
\(77\) −5.65685 + 13.8564i −0.644658 + 1.57908i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 13.8564i 1.54919i
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 17.3205i − 1.90117i −0.310460 − 0.950586i \(-0.600483\pi\)
0.310460 − 0.950586i \(-0.399517\pi\)
\(84\) −8.48528 − 3.46410i −0.925820 − 0.377964i
\(85\) 0 0
\(86\) 0 0
\(87\) 4.89898i 0.525226i
\(88\) −16.0000 −1.70561
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) − 14.6969i − 1.54919i
\(91\) 0 0
\(92\) 0 0
\(93\) 8.48528 0.879883
\(94\) 0 0
\(95\) 0 0
\(96\) − 9.79796i − 1.00000i
\(97\) 19.5959i 1.98966i 0.101535 + 0.994832i \(0.467625\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −7.07107 − 6.92820i −0.714286 − 0.699854i
\(99\) 16.9706 1.70561
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.i.b.125.3 yes 4
3.2 odd 2 inner 168.2.i.b.125.2 yes 4
4.3 odd 2 672.2.i.b.209.4 4
7.6 odd 2 inner 168.2.i.b.125.4 yes 4
8.3 odd 2 672.2.i.b.209.2 4
8.5 even 2 inner 168.2.i.b.125.2 yes 4
12.11 even 2 672.2.i.b.209.2 4
21.20 even 2 inner 168.2.i.b.125.1 ✓ 4
24.5 odd 2 CM 168.2.i.b.125.3 yes 4
24.11 even 2 672.2.i.b.209.4 4
28.27 even 2 672.2.i.b.209.1 4
56.13 odd 2 inner 168.2.i.b.125.1 ✓ 4
56.27 even 2 672.2.i.b.209.3 4
84.83 odd 2 672.2.i.b.209.3 4
168.83 odd 2 672.2.i.b.209.1 4
168.125 even 2 inner 168.2.i.b.125.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.i.b.125.1 ✓ 4 21.20 even 2 inner
168.2.i.b.125.1 ✓ 4 56.13 odd 2 inner
168.2.i.b.125.2 yes 4 3.2 odd 2 inner
168.2.i.b.125.2 yes 4 8.5 even 2 inner
168.2.i.b.125.3 yes 4 1.1 even 1 trivial
168.2.i.b.125.3 yes 4 24.5 odd 2 CM
168.2.i.b.125.4 yes 4 7.6 odd 2 inner
168.2.i.b.125.4 yes 4 168.125 even 2 inner
672.2.i.b.209.1 4 28.27 even 2
672.2.i.b.209.1 4 168.83 odd 2
672.2.i.b.209.2 4 8.3 odd 2
672.2.i.b.209.2 4 12.11 even 2
672.2.i.b.209.3 4 56.27 even 2
672.2.i.b.209.3 4 84.83 odd 2
672.2.i.b.209.4 4 4.3 odd 2
672.2.i.b.209.4 4 24.11 even 2