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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1140,2,Mod(49,1140)] 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("1140.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1140, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1140.bg (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.10294583043\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1140.49
Dual form 1140.2.bg.b.349.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{3} +(-1.23205 - 1.86603i) q^{5} +(0.500000 + 0.866025i) q^{9} +5.00000 q^{11} +(1.73205 - 1.00000i) q^{13} +(-0.133975 - 2.23205i) q^{15} +(-5.19615 - 3.00000i) q^{17} +(-3.50000 - 2.59808i) q^{19} +(5.19615 - 3.00000i) q^{23} +(-1.96410 + 4.59808i) q^{25} +1.00000i q^{27} +(0.500000 + 0.866025i) q^{29} +3.00000 q^{31} +(4.33013 + 2.50000i) q^{33} -2.00000i q^{37} +2.00000 q^{39} +(3.00000 - 5.19615i) q^{41} +(-5.19615 - 3.00000i) q^{43} +(1.00000 - 2.00000i) q^{45} +(5.19615 - 3.00000i) q^{47} +7.00000 q^{49} +(-3.00000 - 5.19615i) q^{51} +(3.46410 - 2.00000i) q^{53} +(-6.16025 - 9.33013i) q^{55} +(-1.73205 - 4.00000i) q^{57} +(5.50000 - 9.52628i) q^{59} +(1.50000 + 2.59808i) q^{61} +(-4.00000 - 2.00000i) q^{65} +(-1.73205 + 1.00000i) q^{67} +6.00000 q^{69} +(4.50000 - 7.79423i) q^{71} +(-1.73205 - 1.00000i) q^{73} +(-4.00000 + 3.00000i) q^{75} +(-2.50000 + 4.33013i) q^{79} +(-0.500000 + 0.866025i) q^{81} +6.00000i q^{83} +(0.803848 + 13.3923i) q^{85} +1.00000i q^{87} +(0.500000 + 0.866025i) q^{89} +(2.59808 + 1.50000i) q^{93} +(-0.535898 + 9.73205i) q^{95} +(-8.66025 - 5.00000i) q^{97} +(2.50000 + 4.33013i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 2 q^{9} + 20 q^{11} - 4 q^{15} - 14 q^{19} + 6 q^{25} + 2 q^{29} + 12 q^{31} + 8 q^{39} + 12 q^{41} + 4 q^{45} + 28 q^{49} - 12 q^{51} + 10 q^{55} + 22 q^{59} + 6 q^{61} - 16 q^{65} + 24 q^{69}+ \cdots + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1140\mathbb{Z}\right)^\times\).

\(n\) \(457\) \(571\) \(761\) \(781\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0.866025 + 0.500000i 0.500000 + 0.288675i
\(4\) 0 0
\(5\) −1.23205 − 1.86603i −0.550990 − 0.834512i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 1.73205 − 1.00000i 0.480384 − 0.277350i −0.240192 − 0.970725i \(-0.577210\pi\)
0.720577 + 0.693375i \(0.243877\pi\)
\(14\) 0 0
\(15\) −0.133975 − 2.23205i −0.0345921 − 0.576313i
\(16\) 0 0
\(17\) −5.19615 − 3.00000i −1.26025 − 0.727607i −0.287129 − 0.957892i \(-0.592701\pi\)
−0.973123 + 0.230285i \(0.926034\pi\)
\(18\) 0 0
\(19\) −3.50000 − 2.59808i −0.802955 − 0.596040i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.19615 − 3.00000i 1.08347 − 0.625543i 0.151642 − 0.988436i \(-0.451544\pi\)
0.931831 + 0.362892i \(0.118211\pi\)
\(24\) 0 0
\(25\) −1.96410 + 4.59808i −0.392820 + 0.919615i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 0.500000 + 0.866025i 0.0928477 + 0.160817i 0.908708 − 0.417432i \(-0.137070\pi\)
−0.815861 + 0.578249i \(0.803736\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 4.33013 + 2.50000i 0.753778 + 0.435194i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 3.00000 − 5.19615i 0.468521 − 0.811503i −0.530831 − 0.847477i \(-0.678120\pi\)
0.999353 + 0.0359748i \(0.0114536\pi\)
\(42\) 0 0
\(43\) −5.19615 − 3.00000i −0.792406 − 0.457496i 0.0484030 − 0.998828i \(-0.484587\pi\)
−0.840809 + 0.541332i \(0.817920\pi\)
\(44\) 0 0
\(45\) 1.00000 − 2.00000i 0.149071 − 0.298142i
\(46\) 0 0
\(47\) 5.19615 − 3.00000i 0.757937 − 0.437595i −0.0706177 − 0.997503i \(-0.522497\pi\)
0.828554 + 0.559908i \(0.189164\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) −3.00000 − 5.19615i −0.420084 − 0.727607i
\(52\) 0 0
\(53\) 3.46410 − 2.00000i 0.475831 − 0.274721i −0.242846 − 0.970065i \(-0.578081\pi\)
0.718677 + 0.695344i \(0.244748\pi\)
\(54\) 0 0
\(55\) −6.16025 − 9.33013i −0.830648 − 1.25807i
\(56\) 0 0
\(57\) −1.73205 − 4.00000i −0.229416 − 0.529813i
\(58\) 0 0
\(59\) 5.50000 − 9.52628i 0.716039 − 1.24022i −0.246518 − 0.969138i \(-0.579287\pi\)
0.962557 − 0.271078i \(-0.0873801\pi\)
\(60\) 0 0
\(61\) 1.50000 + 2.59808i 0.192055 + 0.332650i 0.945931 − 0.324367i \(-0.105151\pi\)
−0.753876 + 0.657017i \(0.771818\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.00000 − 2.00000i −0.496139 − 0.248069i
\(66\) 0 0
\(67\) −1.73205 + 1.00000i −0.211604 + 0.122169i −0.602056 − 0.798454i \(-0.705652\pi\)
0.390453 + 0.920623i \(0.372318\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 4.50000 − 7.79423i 0.534052 − 0.925005i −0.465157 − 0.885228i \(-0.654002\pi\)
0.999209 − 0.0397765i \(-0.0126646\pi\)
\(72\) 0 0
\(73\) −1.73205 − 1.00000i −0.202721 − 0.117041i 0.395203 − 0.918594i \(-0.370674\pi\)
−0.597924 + 0.801553i \(0.704008\pi\)
\(74\) 0 0
\(75\) −4.00000 + 3.00000i −0.461880 + 0.346410i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.50000 + 4.33013i −0.281272 + 0.487177i −0.971698 − 0.236225i \(-0.924090\pi\)
0.690426 + 0.723403i \(0.257423\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0.803848 + 13.3923i 0.0871895 + 1.45260i
\(86\) 0 0
\(87\) 1.00000i 0.107211i
\(88\) 0 0
\(89\) 0.500000 + 0.866025i 0.0529999 + 0.0917985i 0.891308 − 0.453398i \(-0.149788\pi\)
−0.838308 + 0.545197i \(0.816455\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.59808 + 1.50000i 0.269408 + 0.155543i
\(94\) 0 0
\(95\) −0.535898 + 9.73205i −0.0549820 + 0.998487i
\(96\) 0 0
\(97\) −8.66025 − 5.00000i −0.879316 − 0.507673i −0.00888289 − 0.999961i \(-0.502828\pi\)
−0.870433 + 0.492287i \(0.836161\pi\)
\(98\) 0 0
\(99\) 2.50000 + 4.33013i 0.251259 + 0.435194i
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 1140.2.bg.b.49.2 yes 4
3.2 odd 2 3420.2.bj.a.1189.2 4
5.4 even 2 inner 1140.2.bg.b.49.1 ✓ 4
15.14 odd 2 3420.2.bj.a.1189.1 4
19.7 even 3 inner 1140.2.bg.b.349.1 yes 4
57.26 odd 6 3420.2.bj.a.2629.1 4
95.64 even 6 inner 1140.2.bg.b.349.2 yes 4
285.254 odd 6 3420.2.bj.a.2629.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.bg.b.49.1 ✓ 4 5.4 even 2 inner
1140.2.bg.b.49.2 yes 4 1.1 even 1 trivial
1140.2.bg.b.349.1 yes 4 19.7 even 3 inner
1140.2.bg.b.349.2 yes 4 95.64 even 6 inner
3420.2.bj.a.1189.1 4 15.14 odd 2
3420.2.bj.a.1189.2 4 3.2 odd 2
3420.2.bj.a.2629.1 4 57.26 odd 6
3420.2.bj.a.2629.2 4 285.254 odd 6