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Properties

Label 1200.2.v.l.593.3
Level $1200$
Weight $2$
Character 1200.593
Analytic conductor $9.582$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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] = [1200,2,Mod(257,1200)] 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("1200.257"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.v (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 593.3
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1200.593
Dual form 1200.2.v.l.257.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.448288 - 1.67303i) q^{3} +(2.44949 - 2.44949i) q^{7} +(-2.59808 - 1.50000i) q^{9} -5.19615i q^{11} +(2.12132 + 2.12132i) q^{17} -1.00000i q^{19} +(-3.00000 - 5.19615i) q^{21} +(-4.24264 + 4.24264i) q^{23} +(-3.67423 + 3.67423i) q^{27} +2.00000 q^{31} +(-8.69333 - 2.32937i) q^{33} +(2.44949 - 2.44949i) q^{37} -5.19615i q^{41} +(-2.44949 - 2.44949i) q^{43} -5.00000i q^{49} +(4.50000 - 2.59808i) q^{51} +(-4.24264 + 4.24264i) q^{53} +(-1.67303 - 0.448288i) q^{57} -10.3923 q^{59} +14.0000 q^{61} +(-10.0382 + 2.68973i) q^{63} +(3.67423 - 3.67423i) q^{67} +(5.19615 + 9.00000i) q^{69} +(6.12372 + 6.12372i) q^{73} +(-12.7279 - 12.7279i) q^{77} -14.0000i q^{79} +(4.50000 + 7.79423i) q^{81} +(2.12132 - 2.12132i) q^{83} -15.5885 q^{89} +(0.896575 - 3.34607i) q^{93} +(-4.89898 + 4.89898i) q^{97} +(-7.79423 + 13.5000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{21} + 16 q^{31} + 36 q^{51} + 112 q^{61} + 36 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

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
\(3\) 0.448288 − 1.67303i 0.258819 − 0.965926i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.44949 − 2.44949i 0.925820 − 0.925820i −0.0716124 − 0.997433i \(-0.522814\pi\)
0.997433 + 0.0716124i \(0.0228145\pi\)
\(8\) 0 0
\(9\) −2.59808 − 1.50000i −0.866025 − 0.500000i
\(10\) 0 0
\(11\) − 5.19615i − 1.56670i −0.621582 − 0.783349i \(-0.713510\pi\)
0.621582 − 0.783349i \(-0.286490\pi\)
\(12\) 0 0
\(13\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.12132 + 2.12132i 0.514496 + 0.514496i 0.915901 − 0.401405i \(-0.131478\pi\)
−0.401405 + 0.915901i \(0.631478\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i −0.993399 − 0.114708i \(-0.963407\pi\)
0.993399 − 0.114708i \(-0.0365932\pi\)
\(20\) 0 0
\(21\) −3.00000 − 5.19615i −0.654654 − 1.13389i
\(22\) 0 0
\(23\) −4.24264 + 4.24264i −0.884652 + 0.884652i −0.994003 − 0.109351i \(-0.965123\pi\)
0.109351 + 0.994003i \(0.465123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.67423 + 3.67423i −0.707107 + 0.707107i
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) −8.69333 − 2.32937i −1.51331 − 0.405492i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.44949 − 2.44949i 0.402694 − 0.402694i −0.476488 − 0.879181i \(-0.658090\pi\)
0.879181 + 0.476488i \(0.158090\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 5.19615i − 0.811503i −0.913984 − 0.405751i \(-0.867010\pi\)
0.913984 − 0.405751i \(-0.132990\pi\)
\(42\) 0 0
\(43\) −2.44949 − 2.44949i −0.373544 − 0.373544i 0.495222 − 0.868766i \(-0.335087\pi\)
−0.868766 + 0.495222i \(0.835087\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) − 5.00000i − 0.714286i
\(50\) 0 0
\(51\) 4.50000 − 2.59808i 0.630126 − 0.363803i
\(52\) 0 0
\(53\) −4.24264 + 4.24264i −0.582772 + 0.582772i −0.935664 − 0.352892i \(-0.885198\pi\)
0.352892 + 0.935664i \(0.385198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.67303 − 0.448288i −0.221599 − 0.0593772i
\(58\) 0 0
\(59\) −10.3923 −1.35296 −0.676481 − 0.736460i \(-0.736496\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) −10.0382 + 2.68973i −1.26469 + 0.338874i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.67423 − 3.67423i 0.448879 − 0.448879i −0.446103 − 0.894982i \(-0.647188\pi\)
0.894982 + 0.446103i \(0.147188\pi\)
\(68\) 0 0
\(69\) 5.19615 + 9.00000i 0.625543 + 1.08347i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 6.12372 + 6.12372i 0.716728 + 0.716728i 0.967934 − 0.251206i \(-0.0808271\pi\)
−0.251206 + 0.967934i \(0.580827\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.7279 − 12.7279i −1.45048 − 1.45048i
\(78\) 0 0
\(79\) − 14.0000i − 1.57512i −0.616236 − 0.787562i \(-0.711343\pi\)
0.616236 − 0.787562i \(-0.288657\pi\)
\(80\) 0 0
\(81\) 4.50000 + 7.79423i 0.500000 + 0.866025i
\(82\) 0 0
\(83\) 2.12132 − 2.12132i 0.232845 − 0.232845i −0.581034 − 0.813879i \(-0.697352\pi\)
0.813879 + 0.581034i \(0.197352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −15.5885 −1.65237 −0.826187 − 0.563397i \(-0.809494\pi\)
−0.826187 + 0.563397i \(0.809494\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0.896575 − 3.34607i 0.0929705 − 0.346971i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.89898 + 4.89898i −0.497416 + 0.497416i −0.910633 − 0.413217i \(-0.864405\pi\)
0.413217 + 0.910633i \(0.364405\pi\)
\(98\) 0 0
\(99\) −7.79423 + 13.5000i −0.783349 + 1.35680i
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 1200.2.v.l.593.3 8
3.2 odd 2 inner 1200.2.v.l.593.1 8
4.3 odd 2 150.2.e.b.143.2 yes 8
5.2 odd 4 inner 1200.2.v.l.257.1 8
5.3 odd 4 inner 1200.2.v.l.257.4 8
5.4 even 2 inner 1200.2.v.l.593.2 8
12.11 even 2 150.2.e.b.143.4 yes 8
15.2 even 4 inner 1200.2.v.l.257.3 8
15.8 even 4 inner 1200.2.v.l.257.2 8
15.14 odd 2 inner 1200.2.v.l.593.4 8
20.3 even 4 150.2.e.b.107.1 ✓ 8
20.7 even 4 150.2.e.b.107.4 yes 8
20.19 odd 2 150.2.e.b.143.3 yes 8
60.23 odd 4 150.2.e.b.107.3 yes 8
60.47 odd 4 150.2.e.b.107.2 yes 8
60.59 even 2 150.2.e.b.143.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.2.e.b.107.1 ✓ 8 20.3 even 4
150.2.e.b.107.2 yes 8 60.47 odd 4
150.2.e.b.107.3 yes 8 60.23 odd 4
150.2.e.b.107.4 yes 8 20.7 even 4
150.2.e.b.143.1 yes 8 60.59 even 2
150.2.e.b.143.2 yes 8 4.3 odd 2
150.2.e.b.143.3 yes 8 20.19 odd 2
150.2.e.b.143.4 yes 8 12.11 even 2
1200.2.v.l.257.1 8 5.2 odd 4 inner
1200.2.v.l.257.2 8 15.8 even 4 inner
1200.2.v.l.257.3 8 15.2 even 4 inner
1200.2.v.l.257.4 8 5.3 odd 4 inner
1200.2.v.l.593.1 8 3.2 odd 2 inner
1200.2.v.l.593.2 8 5.4 even 2 inner
1200.2.v.l.593.3 8 1.1 even 1 trivial
1200.2.v.l.593.4 8 15.14 odd 2 inner