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Properties

Label 2940.2.q.p.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.p.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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(1.00000 + 1.73205i) q^{11} +2.82843 q^{13} -1.00000 q^{15} +(3.41421 + 5.91359i) q^{17} +(-1.29289 + 2.23936i) q^{19} +(2.29289 - 3.97141i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} -7.65685 q^{29} +(2.12132 + 3.67423i) q^{31} +(1.00000 - 1.73205i) q^{33} +(-3.24264 + 5.61642i) q^{37} +(-1.41421 - 2.44949i) q^{39} -2.00000 q^{41} -9.65685 q^{43} +(0.500000 + 0.866025i) q^{45} +(-3.24264 + 5.61642i) q^{47} +(3.41421 - 5.91359i) q^{51} +(3.70711 + 6.42090i) q^{53} +2.00000 q^{55} +2.58579 q^{57} +(1.41421 + 2.44949i) q^{59} +(-4.94975 + 8.57321i) q^{61} +(1.41421 - 2.44949i) q^{65} +(0.585786 + 1.01461i) q^{67} -4.58579 q^{69} +6.48528 q^{71} +(6.24264 + 10.8126i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(5.00000 - 8.66025i) q^{79} +(-0.500000 - 0.866025i) q^{81} +0.828427 q^{83} +6.82843 q^{85} +(3.82843 + 6.63103i) q^{87} +(-0.414214 + 0.717439i) q^{89} +(2.12132 - 3.67423i) q^{93} +(1.29289 + 2.23936i) q^{95} +4.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{5} - 2 q^{9} + 4 q^{11} - 4 q^{15} + 8 q^{17} - 8 q^{19} + 12 q^{23} - 2 q^{25} + 4 q^{27} - 8 q^{29} + 4 q^{33} + 4 q^{37} - 8 q^{41} - 16 q^{43} + 2 q^{45} + 4 q^{47} + 8 q^{51} + 12 q^{53}+ \cdots - 8 q^{99}+O(q^{100}) \) 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
\(17\) 3.41421 + 5.91359i 0.828068 + 1.43426i 0.899551 + 0.436815i \(0.143893\pi\)
−0.0714831 + 0.997442i \(0.522773\pi\)
\(18\) 0 0
\(19\) −1.29289 + 2.23936i −0.296610 + 0.513744i −0.975358 − 0.220628i \(-0.929189\pi\)
0.678748 + 0.734371i \(0.262523\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.29289 − 3.97141i 0.478101 − 0.828096i −0.521584 − 0.853200i \(-0.674659\pi\)
0.999685 + 0.0251045i \(0.00799185\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.65685 −1.42184 −0.710921 − 0.703272i \(-0.751722\pi\)
−0.710921 + 0.703272i \(0.751722\pi\)
\(30\) 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
\(37\) −3.24264 + 5.61642i −0.533087 + 0.923334i 0.466166 + 0.884697i \(0.345635\pi\)
−0.999253 + 0.0386365i \(0.987699\pi\)
\(38\) 0 0
\(39\) −1.41421 − 2.44949i −0.226455 − 0.392232i
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −9.65685 −1.47266 −0.736328 − 0.676625i \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) 0 0
\(45\) 0.500000 + 0.866025i 0.0745356 + 0.129099i
\(46\) 0 0
\(47\) −3.24264 + 5.61642i −0.472988 + 0.819239i −0.999522 − 0.0309151i \(-0.990158\pi\)
0.526534 + 0.850154i \(0.323491\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 3.41421 − 5.91359i 0.478086 − 0.828068i
\(52\) 0 0
\(53\) 3.70711 + 6.42090i 0.509210 + 0.881978i 0.999943 + 0.0106680i \(0.00339578\pi\)
−0.490733 + 0.871310i \(0.663271\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 2.58579 0.342496
\(58\) 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
\(67\) 0.585786 + 1.01461i 0.0715652 + 0.123955i 0.899587 − 0.436741i \(-0.143867\pi\)
−0.828022 + 0.560695i \(0.810534\pi\)
\(68\) 0 0
\(69\) −4.58579 −0.552064
\(70\) 0 0
\(71\) 6.48528 0.769661 0.384831 − 0.922987i \(-0.374260\pi\)
0.384831 + 0.922987i \(0.374260\pi\)
\(72\) 0 0
\(73\) 6.24264 + 10.8126i 0.730646 + 1.26552i 0.956608 + 0.291380i \(0.0941142\pi\)
−0.225962 + 0.974136i \(0.572552\pi\)
\(74\) 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
\(83\) 0.828427 0.0909317 0.0454658 − 0.998966i \(-0.485523\pi\)
0.0454658 + 0.998966i \(0.485523\pi\)
\(84\) 0 0
\(85\) 6.82843 0.740647
\(86\) 0 0
\(87\) 3.82843 + 6.63103i 0.410450 + 0.710921i
\(88\) 0 0
\(89\) −0.414214 + 0.717439i −0.0439065 + 0.0760484i −0.887144 − 0.461494i \(-0.847314\pi\)
0.843237 + 0.537542i \(0.180647\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.12132 − 3.67423i 0.219971 − 0.381000i
\(94\) 0 0
\(95\) 1.29289 + 2.23936i 0.132648 + 0.229753i
\(96\) 0 0
\(97\) 4.00000 0.406138 0.203069 − 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 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.p.961.2 4
7.2 even 3 2940.2.a.q.1.2 yes 2
7.3 odd 6 2940.2.q.r.361.1 4
7.4 even 3 inner 2940.2.q.p.361.2 4
7.5 odd 6 2940.2.a.o.1.1 ✓ 2
7.6 odd 2 2940.2.q.r.961.1 4
21.2 odd 6 8820.2.a.bm.1.2 2
21.5 even 6 8820.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2940.2.a.o.1.1 ✓ 2 7.5 odd 6
2940.2.a.q.1.2 yes 2 7.2 even 3
2940.2.q.p.361.2 4 7.4 even 3 inner
2940.2.q.p.961.2 4 1.1 even 1 trivial
2940.2.q.r.361.1 4 7.3 odd 6
2940.2.q.r.961.1 4 7.6 odd 2
8820.2.a.bh.1.1 2 21.5 even 6
8820.2.a.bm.1.2 2 21.2 odd 6