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Properties

Label 2940.2.q.r.961.2
Level $2940$
Weight $2$
Character 2940.961
Analytic conductor $23.476$
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] = [2940,2,Mod(361,2940)] 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("2940.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,2,0,-2,0,0,0,-2,0,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.4760181943\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.2
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 2940.961
Dual form 2940.2.q.r.361.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(1.00000 + 1.73205i) q^{11} +2.82843 q^{13} -1.00000 q^{15} +(-0.585786 - 1.01461i) q^{17} +(2.70711 - 4.68885i) q^{19} +(3.70711 - 6.42090i) q^{23} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +3.65685 q^{29} +(2.12132 + 3.67423i) q^{31} +(-1.00000 + 1.73205i) q^{33} +(5.24264 - 9.08052i) q^{37} +(1.41421 + 2.44949i) q^{39} +2.00000 q^{41} +1.65685 q^{43} +(-0.500000 - 0.866025i) q^{45} +(-5.24264 + 9.08052i) q^{47} +(0.585786 - 1.01461i) q^{51} +(2.29289 + 3.97141i) q^{53} -2.00000 q^{55} +5.41421 q^{57} +(1.41421 + 2.44949i) q^{59} +(-4.94975 + 8.57321i) q^{61} +(-1.41421 + 2.44949i) q^{65} +(3.41421 + 5.91359i) q^{67} +7.41421 q^{69} -10.4853 q^{71} +(2.24264 + 3.88437i) q^{73} +(0.500000 - 0.866025i) q^{75} +(5.00000 - 8.66025i) q^{79} +(-0.500000 - 0.866025i) q^{81} +4.82843 q^{83} +1.17157 q^{85} +(1.82843 + 3.16693i) q^{87} +(-2.41421 + 4.18154i) q^{89} +(-2.12132 + 3.67423i) q^{93} +(2.70711 + 4.68885i) q^{95} -4.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{5} - 2 q^{9} + 4 q^{11} - 4 q^{15} - 8 q^{17} + 8 q^{19} + 12 q^{23} - 2 q^{25} - 4 q^{27} - 8 q^{29} - 4 q^{33} + 4 q^{37} + 8 q^{41} - 16 q^{43} - 2 q^{45} - 4 q^{47} + 8 q^{51} + 12 q^{53}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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 0
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 − 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 − 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −0.585786 − 1.01461i −0.142074 − 0.246080i 0.786203 − 0.617968i \(-0.212044\pi\)
−0.928278 + 0.371888i \(0.878710\pi\)
\(18\) 0 0
\(19\) 2.70711 − 4.68885i 0.621053 − 1.07570i −0.368237 − 0.929732i \(-0.620039\pi\)
0.989290 − 0.145963i \(-0.0466281\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.70711 − 6.42090i 0.772985 − 1.33885i −0.162935 − 0.986637i \(-0.552096\pi\)
0.935920 − 0.352213i \(-0.114571\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.65685 0.679061 0.339530 − 0.940595i \(-0.389732\pi\)
0.339530 + 0.940595i \(0.389732\pi\)
\(30\) 0 0
\(31\) 2.12132 + 3.67423i 0.381000 + 0.659912i 0.991206 − 0.132331i \(-0.0422463\pi\)
−0.610205 + 0.792243i \(0.708913\pi\)
\(32\) 0 0
\(33\) −1.00000 + 1.73205i −0.174078 + 0.301511i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.24264 − 9.08052i 0.861885 − 1.49283i −0.00822261 − 0.999966i \(-0.502617\pi\)
0.870107 − 0.492862i \(-0.164049\pi\)
\(38\) 0 0
\(39\) 1.41421 + 2.44949i 0.226455 + 0.392232i
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 1.65685 0.252668 0.126334 − 0.991988i \(-0.459679\pi\)
0.126334 + 0.991988i \(0.459679\pi\)
\(44\) 0 0
\(45\) −0.500000 − 0.866025i −0.0745356 − 0.129099i
\(46\) 0 0
\(47\) −5.24264 + 9.08052i −0.764718 + 1.32453i 0.175678 + 0.984448i \(0.443788\pi\)
−0.940396 + 0.340082i \(0.889545\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0.585786 − 1.01461i 0.0820265 − 0.142074i
\(52\) 0 0
\(53\) 2.29289 + 3.97141i 0.314953 + 0.545515i 0.979428 − 0.201796i \(-0.0646779\pi\)
−0.664474 + 0.747311i \(0.731345\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 5.41421 0.717130
\(58\) 0 0
\(59\) 1.41421 + 2.44949i 0.184115 + 0.318896i 0.943278 − 0.332004i \(-0.107725\pi\)
−0.759163 + 0.650901i \(0.774391\pi\)
\(60\) 0 0
\(61\) −4.94975 + 8.57321i −0.633750 + 1.09769i 0.353028 + 0.935613i \(0.385152\pi\)
−0.986778 + 0.162075i \(0.948181\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.41421 + 2.44949i −0.175412 + 0.303822i
\(66\) 0 0
\(67\) 3.41421 + 5.91359i 0.417113 + 0.722460i 0.995648 − 0.0931973i \(-0.0297087\pi\)
−0.578535 + 0.815657i \(0.696375\pi\)
\(68\) 0 0
\(69\) 7.41421 0.892566
\(70\) 0 0
\(71\) −10.4853 −1.24437 −0.622187 − 0.782869i \(-0.713756\pi\)
−0.622187 + 0.782869i \(0.713756\pi\)
\(72\) 0 0
\(73\) 2.24264 + 3.88437i 0.262481 + 0.454631i 0.966901 − 0.255153i \(-0.0821259\pi\)
−0.704419 + 0.709784i \(0.748793\pi\)
\(74\) 0 0
\(75\) 0.500000 − 0.866025i 0.0577350 − 0.100000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.00000 − 8.66025i 0.562544 − 0.974355i −0.434730 − 0.900561i \(-0.643156\pi\)
0.997274 − 0.0737937i \(-0.0235106\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 4.82843 0.529989 0.264994 − 0.964250i \(-0.414630\pi\)
0.264994 + 0.964250i \(0.414630\pi\)
\(84\) 0 0
\(85\) 1.17157 0.127075
\(86\) 0 0
\(87\) 1.82843 + 3.16693i 0.196028 + 0.339530i
\(88\) 0 0
\(89\) −2.41421 + 4.18154i −0.255906 + 0.443242i −0.965141 − 0.261730i \(-0.915707\pi\)
0.709235 + 0.704972i \(0.249040\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.12132 + 3.67423i −0.219971 + 0.381000i
\(94\) 0 0
\(95\) 2.70711 + 4.68885i 0.277743 + 0.481065i
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 − 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
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 2940.2.q.r.961.2 4
7.2 even 3 2940.2.a.o.1.2 ✓ 2
7.3 odd 6 2940.2.q.p.361.1 4
7.4 even 3 inner 2940.2.q.r.361.2 4
7.5 odd 6 2940.2.a.q.1.1 yes 2
7.6 odd 2 2940.2.q.p.961.1 4
21.2 odd 6 8820.2.a.bh.1.2 2
21.5 even 6 8820.2.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2940.2.a.o.1.2 ✓ 2 7.2 even 3
2940.2.a.q.1.1 yes 2 7.5 odd 6
2940.2.q.p.361.1 4 7.3 odd 6
2940.2.q.p.961.1 4 7.6 odd 2
2940.2.q.r.361.2 4 7.4 even 3 inner
2940.2.q.r.961.2 4 1.1 even 1 trivial
8820.2.a.bh.1.2 2 21.2 odd 6
8820.2.a.bm.1.1 2 21.5 even 6