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Properties

Label 40.2.f.a.29.1
Level $40$
Weight $2$
Character 40.29
Analytic conductor $0.319$
Analytic rank $0$
Dimension $4$
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] = [40,2,Mod(29,40)] 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("40.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 40.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.319401608085\)
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: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 29.1
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 40.29
Dual form 40.2.f.a.29.2

$q$-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} +1.41421 q^{3} +(-1.00000 + 1.73205i) q^{4} +(-1.41421 - 1.73205i) q^{5} +(-1.00000 - 1.73205i) q^{6} +2.44949i q^{7} +2.82843 q^{8} -1.00000 q^{9} +(-1.12132 + 2.95680i) q^{10} +3.46410i q^{11} +(-1.41421 + 2.44949i) q^{12} +(3.00000 - 1.73205i) q^{14} +(-2.00000 - 2.44949i) q^{15} +(-2.00000 - 3.46410i) q^{16} -4.89898i q^{17} +(0.707107 + 1.22474i) q^{18} -3.46410i q^{19} +(4.41421 - 0.717439i) q^{20} +3.46410i q^{21} +(4.24264 - 2.44949i) q^{22} -2.44949i q^{23} +4.00000 q^{24} +(-1.00000 + 4.89898i) q^{25} -5.65685 q^{27} +(-4.24264 - 2.44949i) q^{28} +(-1.58579 + 4.18154i) q^{30} +4.00000 q^{31} +(-2.82843 + 4.89898i) q^{32} +4.89898i q^{33} +(-6.00000 + 3.46410i) q^{34} +(4.24264 - 3.46410i) q^{35} +(1.00000 - 1.73205i) q^{36} +8.48528 q^{37} +(-4.24264 + 2.44949i) q^{38} +(-4.00000 - 4.89898i) q^{40} +(4.24264 - 2.44949i) q^{42} -4.24264 q^{43} +(-6.00000 - 3.46410i) q^{44} +(1.41421 + 1.73205i) q^{45} +(-3.00000 + 1.73205i) q^{46} +7.34847i q^{47} +(-2.82843 - 4.89898i) q^{48} +1.00000 q^{49} +(6.70711 - 2.23936i) q^{50} -6.92820i q^{51} -5.65685 q^{53} +(4.00000 + 6.92820i) q^{54} +(6.00000 - 4.89898i) q^{55} +6.92820i q^{56} -4.89898i q^{57} +10.3923i q^{59} +(6.24264 - 1.01461i) q^{60} -3.46410i q^{61} +(-2.82843 - 4.89898i) q^{62} -2.44949i q^{63} +8.00000 q^{64} +(6.00000 - 3.46410i) q^{66} +4.24264 q^{67} +(8.48528 + 4.89898i) q^{68} -3.46410i q^{69} +(-7.24264 - 2.74666i) q^{70} -12.0000 q^{71} -2.82843 q^{72} -4.89898i q^{73} +(-6.00000 - 10.3923i) q^{74} +(-1.41421 + 6.92820i) q^{75} +(6.00000 + 3.46410i) q^{76} -8.48528 q^{77} -4.00000 q^{79} +(-3.17157 + 8.36308i) q^{80} -5.00000 q^{81} +9.89949 q^{83} +(-6.00000 - 3.46410i) q^{84} +(-8.48528 + 6.92820i) q^{85} +(3.00000 + 5.19615i) q^{86} +9.79796i q^{88} +6.00000 q^{89} +(1.12132 - 2.95680i) q^{90} +(4.24264 + 2.44949i) q^{92} +5.65685 q^{93} +(9.00000 - 5.19615i) q^{94} +(-6.00000 + 4.89898i) q^{95} +(-4.00000 + 6.92820i) q^{96} +4.89898i q^{97} +(-0.707107 - 1.22474i) q^{98} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 4 q^{10} + 12 q^{14} - 8 q^{15} - 8 q^{16} + 12 q^{20} + 16 q^{24} - 4 q^{25} - 12 q^{30} + 16 q^{31} - 24 q^{34} + 4 q^{36} - 16 q^{40} - 24 q^{44} - 12 q^{46} + 4 q^{49}+ \cdots - 16 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 1.41421 0.816497 0.408248 − 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) −1.00000 + 1.73205i −0.500000 + 0.866025i
\(5\) −1.41421 − 1.73205i −0.632456 − 0.774597i
\(6\) −1.00000 − 1.73205i −0.408248 − 0.707107i
\(7\) 2.44949i 0.925820i 0.886405 + 0.462910i \(0.153195\pi\)
−0.886405 + 0.462910i \(0.846805\pi\)
\(8\) 2.82843 1.00000
\(9\) −1.00000 −0.333333
\(10\) −1.12132 + 2.95680i −0.354593 + 0.935021i
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) −1.41421 + 2.44949i −0.408248 + 0.707107i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 3.00000 − 1.73205i 0.801784 − 0.462910i
\(15\) −2.00000 − 2.44949i −0.516398 − 0.632456i
\(16\) −2.00000 − 3.46410i −0.500000 − 0.866025i
\(17\) − 4.89898i − 1.18818i −0.804400 − 0.594089i \(-0.797513\pi\)
0.804400 − 0.594089i \(-0.202487\pi\)
\(18\) 0.707107 + 1.22474i 0.166667 + 0.288675i
\(19\) − 3.46410i − 0.794719i −0.917663 − 0.397360i \(-0.869927\pi\)
0.917663 − 0.397360i \(-0.130073\pi\)
\(20\) 4.41421 − 0.717439i 0.987048 − 0.160424i
\(21\) 3.46410i 0.755929i
\(22\) 4.24264 − 2.44949i 0.904534 − 0.522233i
\(23\) − 2.44949i − 0.510754i −0.966842 − 0.255377i \(-0.917800\pi\)
0.966842 − 0.255377i \(-0.0821996\pi\)
\(24\) 4.00000 0.816497
\(25\) −1.00000 + 4.89898i −0.200000 + 0.979796i
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) −4.24264 − 2.44949i −0.801784 − 0.462910i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −1.58579 + 4.18154i −0.289524 + 0.763441i
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −2.82843 + 4.89898i −0.500000 + 0.866025i
\(33\) 4.89898i 0.852803i
\(34\) −6.00000 + 3.46410i −1.02899 + 0.594089i
\(35\) 4.24264 − 3.46410i 0.717137 − 0.585540i
\(36\) 1.00000 − 1.73205i 0.166667 − 0.288675i
\(37\) 8.48528 1.39497 0.697486 − 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) −4.24264 + 2.44949i −0.688247 + 0.397360i
\(39\) 0 0
\(40\) −4.00000 − 4.89898i −0.632456 − 0.774597i
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 4.24264 − 2.44949i 0.654654 − 0.377964i
\(43\) −4.24264 −0.646997 −0.323498 − 0.946229i \(-0.604859\pi\)
−0.323498 + 0.946229i \(0.604859\pi\)
\(44\) −6.00000 − 3.46410i −0.904534 − 0.522233i
\(45\) 1.41421 + 1.73205i 0.210819 + 0.258199i
\(46\) −3.00000 + 1.73205i −0.442326 + 0.255377i
\(47\) 7.34847i 1.07188i 0.844255 + 0.535942i \(0.180044\pi\)
−0.844255 + 0.535942i \(0.819956\pi\)
\(48\) −2.82843 − 4.89898i −0.408248 − 0.707107i
\(49\) 1.00000 0.142857
\(50\) 6.70711 − 2.23936i 0.948528 − 0.316693i
\(51\) − 6.92820i − 0.970143i
\(52\) 0 0
\(53\) −5.65685 −0.777029 −0.388514 − 0.921443i \(-0.627012\pi\)
−0.388514 + 0.921443i \(0.627012\pi\)
\(54\) 4.00000 + 6.92820i 0.544331 + 0.942809i
\(55\) 6.00000 − 4.89898i 0.809040 − 0.660578i
\(56\) 6.92820i 0.925820i
\(57\) − 4.89898i − 0.648886i
\(58\) 0 0
\(59\) 10.3923i 1.35296i 0.736460 + 0.676481i \(0.236496\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 6.24264 − 1.01461i 0.805921 − 0.130986i
\(61\) − 3.46410i − 0.443533i −0.975100 − 0.221766i \(-0.928818\pi\)
0.975100 − 0.221766i \(-0.0711822\pi\)
\(62\) −2.82843 − 4.89898i −0.359211 − 0.622171i
\(63\) − 2.44949i − 0.308607i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 6.00000 − 3.46410i 0.738549 − 0.426401i
\(67\) 4.24264 0.518321 0.259161 − 0.965834i \(-0.416554\pi\)
0.259161 + 0.965834i \(0.416554\pi\)
\(68\) 8.48528 + 4.89898i 1.02899 + 0.594089i
\(69\) − 3.46410i − 0.417029i
\(70\) −7.24264 − 2.74666i −0.865661 − 0.328289i
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −2.82843 −0.333333
\(73\) − 4.89898i − 0.573382i −0.958023 − 0.286691i \(-0.907445\pi\)
0.958023 − 0.286691i \(-0.0925553\pi\)
\(74\) −6.00000 − 10.3923i −0.697486 − 1.20808i
\(75\) −1.41421 + 6.92820i −0.163299 + 0.800000i
\(76\) 6.00000 + 3.46410i 0.688247 + 0.397360i
\(77\) −8.48528 −0.966988
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −3.17157 + 8.36308i −0.354593 + 0.935021i
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 9.89949 1.08661 0.543305 − 0.839535i \(-0.317173\pi\)
0.543305 + 0.839535i \(0.317173\pi\)
\(84\) −6.00000 − 3.46410i −0.654654 − 0.377964i
\(85\) −8.48528 + 6.92820i −0.920358 + 0.751469i
\(86\) 3.00000 + 5.19615i 0.323498 + 0.560316i
\(87\) 0 0
\(88\) 9.79796i 1.04447i
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 1.12132 − 2.95680i 0.118198 − 0.311674i
\(91\) 0 0
\(92\) 4.24264 + 2.44949i 0.442326 + 0.255377i
\(93\) 5.65685 0.586588
\(94\) 9.00000 − 5.19615i 0.928279 − 0.535942i
\(95\) −6.00000 + 4.89898i −0.615587 + 0.502625i
\(96\) −4.00000 + 6.92820i −0.408248 + 0.707107i
\(97\) 4.89898i 0.497416i 0.968579 + 0.248708i \(0.0800060\pi\)
−0.968579 + 0.248708i \(0.919994\pi\)
\(98\) −0.707107 − 1.22474i −0.0714286 − 0.123718i
\(99\) − 3.46410i − 0.348155i
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 40.2.f.a.29.1 ✓ 4
3.2 odd 2 360.2.d.b.109.4 4
4.3 odd 2 160.2.f.a.49.1 4
5.2 odd 4 200.2.d.e.101.3 4
5.3 odd 4 200.2.d.e.101.2 4
5.4 even 2 inner 40.2.f.a.29.4 yes 4
8.3 odd 2 160.2.f.a.49.4 4
8.5 even 2 inner 40.2.f.a.29.3 yes 4
12.11 even 2 1440.2.d.c.1009.4 4
15.2 even 4 1800.2.k.m.901.2 4
15.8 even 4 1800.2.k.m.901.3 4
15.14 odd 2 360.2.d.b.109.1 4
16.3 odd 4 1280.2.c.k.769.1 4
16.5 even 4 1280.2.c.i.769.2 4
16.11 odd 4 1280.2.c.k.769.4 4
16.13 even 4 1280.2.c.i.769.3 4
20.3 even 4 800.2.d.f.401.1 4
20.7 even 4 800.2.d.f.401.4 4
20.19 odd 2 160.2.f.a.49.3 4
24.5 odd 2 360.2.d.b.109.2 4
24.11 even 2 1440.2.d.c.1009.1 4
40.3 even 4 800.2.d.f.401.3 4
40.13 odd 4 200.2.d.e.101.1 4
40.19 odd 2 160.2.f.a.49.2 4
40.27 even 4 800.2.d.f.401.2 4
40.29 even 2 inner 40.2.f.a.29.2 yes 4
40.37 odd 4 200.2.d.e.101.4 4
60.23 odd 4 7200.2.k.l.3601.2 4
60.47 odd 4 7200.2.k.l.3601.4 4
60.59 even 2 1440.2.d.c.1009.2 4
80.3 even 4 6400.2.a.cm.1.4 4
80.13 odd 4 6400.2.a.co.1.1 4
80.19 odd 4 1280.2.c.k.769.3 4
80.27 even 4 6400.2.a.cm.1.3 4
80.29 even 4 1280.2.c.i.769.1 4
80.37 odd 4 6400.2.a.co.1.2 4
80.43 even 4 6400.2.a.cm.1.2 4
80.53 odd 4 6400.2.a.co.1.3 4
80.59 odd 4 1280.2.c.k.769.2 4
80.67 even 4 6400.2.a.cm.1.1 4
80.69 even 4 1280.2.c.i.769.4 4
80.77 odd 4 6400.2.a.co.1.4 4
120.29 odd 2 360.2.d.b.109.3 4
120.53 even 4 1800.2.k.m.901.4 4
120.59 even 2 1440.2.d.c.1009.3 4
120.77 even 4 1800.2.k.m.901.1 4
120.83 odd 4 7200.2.k.l.3601.1 4
120.107 odd 4 7200.2.k.l.3601.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.f.a.29.1 ✓ 4 1.1 even 1 trivial
40.2.f.a.29.2 yes 4 40.29 even 2 inner
40.2.f.a.29.3 yes 4 8.5 even 2 inner
40.2.f.a.29.4 yes 4 5.4 even 2 inner
160.2.f.a.49.1 4 4.3 odd 2
160.2.f.a.49.2 4 40.19 odd 2
160.2.f.a.49.3 4 20.19 odd 2
160.2.f.a.49.4 4 8.3 odd 2
200.2.d.e.101.1 4 40.13 odd 4
200.2.d.e.101.2 4 5.3 odd 4
200.2.d.e.101.3 4 5.2 odd 4
200.2.d.e.101.4 4 40.37 odd 4
360.2.d.b.109.1 4 15.14 odd 2
360.2.d.b.109.2 4 24.5 odd 2
360.2.d.b.109.3 4 120.29 odd 2
360.2.d.b.109.4 4 3.2 odd 2
800.2.d.f.401.1 4 20.3 even 4
800.2.d.f.401.2 4 40.27 even 4
800.2.d.f.401.3 4 40.3 even 4
800.2.d.f.401.4 4 20.7 even 4
1280.2.c.i.769.1 4 80.29 even 4
1280.2.c.i.769.2 4 16.5 even 4
1280.2.c.i.769.3 4 16.13 even 4
1280.2.c.i.769.4 4 80.69 even 4
1280.2.c.k.769.1 4 16.3 odd 4
1280.2.c.k.769.2 4 80.59 odd 4
1280.2.c.k.769.3 4 80.19 odd 4
1280.2.c.k.769.4 4 16.11 odd 4
1440.2.d.c.1009.1 4 24.11 even 2
1440.2.d.c.1009.2 4 60.59 even 2
1440.2.d.c.1009.3 4 120.59 even 2
1440.2.d.c.1009.4 4 12.11 even 2
1800.2.k.m.901.1 4 120.77 even 4
1800.2.k.m.901.2 4 15.2 even 4
1800.2.k.m.901.3 4 15.8 even 4
1800.2.k.m.901.4 4 120.53 even 4
6400.2.a.cm.1.1 4 80.67 even 4
6400.2.a.cm.1.2 4 80.43 even 4
6400.2.a.cm.1.3 4 80.27 even 4
6400.2.a.cm.1.4 4 80.3 even 4
6400.2.a.co.1.1 4 80.13 odd 4
6400.2.a.co.1.2 4 80.37 odd 4
6400.2.a.co.1.3 4 80.53 odd 4
6400.2.a.co.1.4 4 80.77 odd 4
7200.2.k.l.3601.1 4 120.83 odd 4
7200.2.k.l.3601.2 4 60.23 odd 4
7200.2.k.l.3601.3 4 120.107 odd 4
7200.2.k.l.3601.4 4 60.47 odd 4