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Properties

Label 1200.2.w.c.943.3
Level $1200$
Weight $2$
Character 1200.943
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(607,1200)] 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("1200.607"); 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([2, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.w (of order \(4\), degree \(2\), 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,-16,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == 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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 943.3
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1200.943
Dual form 1200.2.w.c.607.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.707107 - 0.707107i) q^{3} +(-1.41421 - 1.41421i) q^{7} -1.00000i q^{9} -3.46410i q^{11} +(-2.44949 - 2.44949i) q^{13} +(-4.89898 + 4.89898i) q^{17} -3.46410 q^{19} -2.00000 q^{21} +(-0.707107 - 0.707107i) q^{27} +3.46410i q^{31} +(-2.44949 - 2.44949i) q^{33} +(-7.34847 + 7.34847i) q^{37} -3.46410 q^{39} +6.00000 q^{41} +(5.65685 - 5.65685i) q^{43} +(-8.48528 - 8.48528i) q^{47} -3.00000i q^{49} +6.92820i q^{51} +(-4.89898 - 4.89898i) q^{53} +(-2.44949 + 2.44949i) q^{57} +10.3923 q^{59} -10.0000 q^{61} +(-1.41421 + 1.41421i) q^{63} +(-2.82843 - 2.82843i) q^{67} -13.8564i q^{71} +(-4.89898 - 4.89898i) q^{73} +(-4.89898 + 4.89898i) q^{77} +3.46410 q^{79} -1.00000 q^{81} +(8.48528 - 8.48528i) q^{83} +18.0000i q^{89} +6.92820i q^{91} +(2.44949 + 2.44949i) q^{93} +(-4.89898 + 4.89898i) q^{97} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{21} + 48 q^{41} - 80 q^{61} - 8 q^{81}+O(q^{100}) \) 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
\(3\) 0.707107 − 0.707107i 0.408248 − 0.408248i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.41421 − 1.41421i −0.534522 − 0.534522i 0.387392 − 0.921915i \(-0.373376\pi\)
−0.921915 + 0.387392i \(0.873376\pi\)
\(8\) 0 0
\(9\) − 1.00000i − 0.333333i
\(10\) 0 0
\(11\) − 3.46410i − 1.04447i −0.852803 − 0.522233i \(-0.825099\pi\)
0.852803 − 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) −2.44949 − 2.44949i −0.679366 − 0.679366i 0.280491 − 0.959857i \(-0.409503\pi\)
−0.959857 + 0.280491i \(0.909503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.89898 + 4.89898i −1.18818 + 1.18818i −0.210606 + 0.977571i \(0.567544\pi\)
−0.977571 + 0.210606i \(0.932456\pi\)
\(18\) 0 0
\(19\) −3.46410 −0.794719 −0.397360 − 0.917663i \(-0.630073\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 − 0.707107i −0.136083 − 0.136083i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 0 0
\(33\) −2.44949 − 2.44949i −0.426401 − 0.426401i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.34847 + 7.34847i −1.20808 + 1.20808i −0.236433 + 0.971648i \(0.575978\pi\)
−0.971648 + 0.236433i \(0.924022\pi\)
\(38\) 0 0
\(39\) −3.46410 −0.554700
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 5.65685 − 5.65685i 0.862662 − 0.862662i −0.128984 − 0.991647i \(-0.541172\pi\)
0.991647 + 0.128984i \(0.0411717\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.48528 − 8.48528i −1.23771 − 1.23771i −0.960936 − 0.276769i \(-0.910736\pi\)
−0.276769 − 0.960936i \(-0.589264\pi\)
\(48\) 0 0
\(49\) − 3.00000i − 0.428571i
\(50\) 0 0
\(51\) 6.92820i 0.970143i
\(52\) 0 0
\(53\) −4.89898 − 4.89898i −0.672927 − 0.672927i 0.285463 − 0.958390i \(-0.407853\pi\)
−0.958390 + 0.285463i \(0.907853\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.44949 + 2.44949i −0.324443 + 0.324443i
\(58\) 0 0
\(59\) 10.3923 1.35296 0.676481 − 0.736460i \(-0.263504\pi\)
0.676481 + 0.736460i \(0.263504\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −1.41421 + 1.41421i −0.178174 + 0.178174i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.82843 − 2.82843i −0.345547 − 0.345547i 0.512901 − 0.858448i \(-0.328571\pi\)
−0.858448 + 0.512901i \(0.828571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 13.8564i − 1.64445i −0.569160 − 0.822226i \(-0.692732\pi\)
0.569160 − 0.822226i \(-0.307268\pi\)
\(72\) 0 0
\(73\) −4.89898 − 4.89898i −0.573382 − 0.573382i 0.359690 − 0.933072i \(-0.382883\pi\)
−0.933072 + 0.359690i \(0.882883\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.89898 + 4.89898i −0.558291 + 0.558291i
\(78\) 0 0
\(79\) 3.46410 0.389742 0.194871 − 0.980829i \(-0.437571\pi\)
0.194871 + 0.980829i \(0.437571\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 8.48528 − 8.48528i 0.931381 − 0.931381i −0.0664117 − 0.997792i \(-0.521155\pi\)
0.997792 + 0.0664117i \(0.0211551\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0000i 1.90800i 0.299813 + 0.953998i \(0.403076\pi\)
−0.299813 + 0.953998i \(0.596924\pi\)
\(90\) 0 0
\(91\) 6.92820i 0.726273i
\(92\) 0 0
\(93\) 2.44949 + 2.44949i 0.254000 + 0.254000i
\(94\) 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
\(99\) −3.46410 −0.348155
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.w.c.943.3 yes 8
3.2 odd 2 3600.2.x.h.2143.2 8
4.3 odd 2 inner 1200.2.w.c.943.2 yes 8
5.2 odd 4 inner 1200.2.w.c.607.1 ✓ 8
5.3 odd 4 inner 1200.2.w.c.607.3 yes 8
5.4 even 2 inner 1200.2.w.c.943.1 yes 8
12.11 even 2 3600.2.x.h.2143.3 8
15.2 even 4 3600.2.x.h.3007.4 8
15.8 even 4 3600.2.x.h.3007.2 8
15.14 odd 2 3600.2.x.h.2143.4 8
20.3 even 4 inner 1200.2.w.c.607.2 yes 8
20.7 even 4 inner 1200.2.w.c.607.4 yes 8
20.19 odd 2 inner 1200.2.w.c.943.4 yes 8
60.23 odd 4 3600.2.x.h.3007.3 8
60.47 odd 4 3600.2.x.h.3007.1 8
60.59 even 2 3600.2.x.h.2143.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1200.2.w.c.607.1 ✓ 8 5.2 odd 4 inner
1200.2.w.c.607.2 yes 8 20.3 even 4 inner
1200.2.w.c.607.3 yes 8 5.3 odd 4 inner
1200.2.w.c.607.4 yes 8 20.7 even 4 inner
1200.2.w.c.943.1 yes 8 5.4 even 2 inner
1200.2.w.c.943.2 yes 8 4.3 odd 2 inner
1200.2.w.c.943.3 yes 8 1.1 even 1 trivial
1200.2.w.c.943.4 yes 8 20.19 odd 2 inner
3600.2.x.h.2143.1 8 60.59 even 2
3600.2.x.h.2143.2 8 3.2 odd 2
3600.2.x.h.2143.3 8 12.11 even 2
3600.2.x.h.2143.4 8 15.14 odd 2
3600.2.x.h.3007.1 8 60.47 odd 4
3600.2.x.h.3007.2 8 15.8 even 4
3600.2.x.h.3007.3 8 60.23 odd 4
3600.2.x.h.3007.4 8 15.2 even 4