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Properties

Label 4080.2.m.o.2449.3
Level $4080$
Weight $2$
Character 4080.2449
Analytic conductor $32.579$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [4080,2,Mod(2449,4080)] 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("4080.2449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4080.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,4,0,0,0,-4,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.5789640247\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 510)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2449.3
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 4080.2449
Dual form 4080.2.m.o.2449.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +(-0.224745 + 2.22474i) q^{5} +2.44949i q^{7} -1.00000 q^{9} +4.89898 q^{11} -6.00000i q^{13} +(-2.22474 - 0.224745i) q^{15} +1.00000i q^{17} +6.89898 q^{19} -2.44949 q^{21} +6.44949i q^{23} +(-4.89898 - 1.00000i) q^{25} -1.00000i q^{27} +9.34847 q^{29} +6.44949 q^{31} +4.89898i q^{33} +(-5.44949 - 0.550510i) q^{35} -0.449490i q^{37} +6.00000 q^{39} -1.10102 q^{41} -2.89898i q^{43} +(0.224745 - 2.22474i) q^{45} -4.89898i q^{47} +1.00000 q^{49} -1.00000 q^{51} +1.10102i q^{53} +(-1.10102 + 10.8990i) q^{55} +6.89898i q^{57} +5.79796 q^{59} +13.3485 q^{61} -2.44949i q^{63} +(13.3485 + 1.34847i) q^{65} -4.00000i q^{67} -6.44949 q^{69} -2.44949 q^{71} -14.8990i q^{73} +(1.00000 - 4.89898i) q^{75} +12.0000i q^{77} -1.55051 q^{79} +1.00000 q^{81} +2.89898i q^{83} +(-2.22474 - 0.224745i) q^{85} +9.34847i q^{87} +1.79796 q^{89} +14.6969 q^{91} +6.44949i q^{93} +(-1.55051 + 15.3485i) q^{95} -3.79796i q^{97} -4.89898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 4 q^{9} - 4 q^{15} + 8 q^{19} + 8 q^{29} + 16 q^{31} - 12 q^{35} + 24 q^{39} - 24 q^{41} - 4 q^{45} + 4 q^{49} - 4 q^{51} - 24 q^{55} - 16 q^{59} + 24 q^{61} + 24 q^{65} - 16 q^{69} + 4 q^{75}+ \cdots - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4080\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(817\) \(1361\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −0.224745 + 2.22474i −0.100509 + 0.994936i
\(6\) 0 0
\(7\) 2.44949i 0.925820i 0.886405 + 0.462910i \(0.153195\pi\)
−0.886405 + 0.462910i \(0.846805\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.89898 1.47710 0.738549 − 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 − 0.832050i \(-0.687167\pi\)
0.554700 − 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) −2.22474 − 0.224745i −0.574427 − 0.0580289i
\(16\) 0 0
\(17\) 1.00000i 0.242536i
\(18\) 0 0
\(19\) 6.89898 1.58273 0.791367 − 0.611341i \(-0.209370\pi\)
0.791367 + 0.611341i \(0.209370\pi\)
\(20\) 0 0
\(21\) −2.44949 −0.534522
\(22\) 0 0
\(23\) 6.44949i 1.34481i 0.740183 + 0.672406i \(0.234739\pi\)
−0.740183 + 0.672406i \(0.765261\pi\)
\(24\) 0 0
\(25\) −4.89898 − 1.00000i −0.979796 − 0.200000i
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 9.34847 1.73597 0.867984 − 0.496593i \(-0.165416\pi\)
0.867984 + 0.496593i \(0.165416\pi\)
\(30\) 0 0
\(31\) 6.44949 1.15836 0.579181 − 0.815199i \(-0.303372\pi\)
0.579181 + 0.815199i \(0.303372\pi\)
\(32\) 0 0
\(33\) 4.89898i 0.852803i
\(34\) 0 0
\(35\) −5.44949 − 0.550510i −0.921132 − 0.0930532i
\(36\) 0 0
\(37\) − 0.449490i − 0.0738957i −0.999317 − 0.0369478i \(-0.988236\pi\)
0.999317 − 0.0369478i \(-0.0117635\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −1.10102 −0.171951 −0.0859753 − 0.996297i \(-0.527401\pi\)
−0.0859753 + 0.996297i \(0.527401\pi\)
\(42\) 0 0
\(43\) − 2.89898i − 0.442090i −0.975264 − 0.221045i \(-0.929053\pi\)
0.975264 − 0.221045i \(-0.0709468\pi\)
\(44\) 0 0
\(45\) 0.224745 − 2.22474i 0.0335030 − 0.331645i
\(46\) 0 0
\(47\) − 4.89898i − 0.714590i −0.933992 − 0.357295i \(-0.883699\pi\)
0.933992 − 0.357295i \(-0.116301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) 1.10102i 0.151237i 0.997137 + 0.0756184i \(0.0240931\pi\)
−0.997137 + 0.0756184i \(0.975907\pi\)
\(54\) 0 0
\(55\) −1.10102 + 10.8990i −0.148462 + 1.46962i
\(56\) 0 0
\(57\) 6.89898i 0.913792i
\(58\) 0 0
\(59\) 5.79796 0.754830 0.377415 − 0.926044i \(-0.376813\pi\)
0.377415 + 0.926044i \(0.376813\pi\)
\(60\) 0 0
\(61\) 13.3485 1.70910 0.854548 − 0.519372i \(-0.173834\pi\)
0.854548 + 0.519372i \(0.173834\pi\)
\(62\) 0 0
\(63\) − 2.44949i − 0.308607i
\(64\) 0 0
\(65\) 13.3485 + 1.34847i 1.65567 + 0.167257i
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) −6.44949 −0.776427
\(70\) 0 0
\(71\) −2.44949 −0.290701 −0.145350 − 0.989380i \(-0.546431\pi\)
−0.145350 + 0.989380i \(0.546431\pi\)
\(72\) 0 0
\(73\) − 14.8990i − 1.74379i −0.489690 − 0.871897i \(-0.662890\pi\)
0.489690 − 0.871897i \(-0.337110\pi\)
\(74\) 0 0
\(75\) 1.00000 − 4.89898i 0.115470 − 0.565685i
\(76\) 0 0
\(77\) 12.0000i 1.36753i
\(78\) 0 0
\(79\) −1.55051 −0.174446 −0.0872230 − 0.996189i \(-0.527799\pi\)
−0.0872230 + 0.996189i \(0.527799\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.89898i 0.318204i 0.987262 + 0.159102i \(0.0508599\pi\)
−0.987262 + 0.159102i \(0.949140\pi\)
\(84\) 0 0
\(85\) −2.22474 − 0.224745i −0.241307 − 0.0243770i
\(86\) 0 0
\(87\) 9.34847i 1.00226i
\(88\) 0 0
\(89\) 1.79796 0.190583 0.0952916 − 0.995449i \(-0.469622\pi\)
0.0952916 + 0.995449i \(0.469622\pi\)
\(90\) 0 0
\(91\) 14.6969 1.54066
\(92\) 0 0
\(93\) 6.44949i 0.668781i
\(94\) 0 0
\(95\) −1.55051 + 15.3485i −0.159079 + 1.57472i
\(96\) 0 0
\(97\) − 3.79796i − 0.385624i −0.981236 − 0.192812i \(-0.938239\pi\)
0.981236 − 0.192812i \(-0.0617608\pi\)
\(98\) 0 0
\(99\) −4.89898 −0.492366
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 4080.2.m.o.2449.3 4
4.3 odd 2 510.2.d.c.409.1 ✓ 4
5.4 even 2 inner 4080.2.m.o.2449.1 4
12.11 even 2 1530.2.d.e.919.4 4
20.3 even 4 2550.2.a.bi.1.1 2
20.7 even 4 2550.2.a.bj.1.2 2
20.19 odd 2 510.2.d.c.409.3 yes 4
60.23 odd 4 7650.2.a.dg.1.1 2
60.47 odd 4 7650.2.a.ct.1.2 2
60.59 even 2 1530.2.d.e.919.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.d.c.409.1 ✓ 4 4.3 odd 2
510.2.d.c.409.3 yes 4 20.19 odd 2
1530.2.d.e.919.2 4 60.59 even 2
1530.2.d.e.919.4 4 12.11 even 2
2550.2.a.bi.1.1 2 20.3 even 4
2550.2.a.bj.1.2 2 20.7 even 4
4080.2.m.o.2449.1 4 5.4 even 2 inner
4080.2.m.o.2449.3 4 1.1 even 1 trivial
7650.2.a.ct.1.2 2 60.47 odd 4
7650.2.a.dg.1.1 2 60.23 odd 4