↑

Properties

Label 1200.2.w.b.607.4
Level $1200$
Weight $2$
Character 1200.607
Analytic conductor $9.582$
Analytic rank $0$
Dimension $8$
Inner twists $4$

Related objects

Downloads

Learn more

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,-16,0,0,0,16,0,0,0,0,0,0,0,0, 0,0,0,8,0,0,0,32,0,0,0,-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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 607.4
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1200.607
Dual form 1200.2.w.b.943.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.707107 + 0.707107i) q^{3} +(3.14626 - 3.14626i) q^{7} +1.00000i q^{9} -1.41421i q^{11} +(2.44949 - 2.44949i) q^{13} +(-4.44949 - 4.44949i) q^{17} -0.635674 q^{19} +4.44949 q^{21} +(-2.04989 - 2.04989i) q^{23} +(-0.707107 + 0.707107i) q^{27} -9.34847i q^{29} +8.48528i q^{31} +(1.00000 - 1.00000i) q^{33} +(1.55051 + 1.55051i) q^{37} +3.46410 q^{39} -1.10102 q^{41} +(0.635674 + 0.635674i) q^{43} +(-5.51399 + 5.51399i) q^{47} -12.7980i q^{49} -6.29253i q^{51} +(2.00000 - 2.00000i) q^{53} +(-0.449490 - 0.449490i) q^{57} +9.89949 q^{59} +2.89898 q^{61} +(3.14626 + 3.14626i) q^{63} +(6.29253 - 6.29253i) q^{67} -2.89898i q^{69} +12.5851i q^{71} +(-3.00000 + 3.00000i) q^{73} +(-4.44949 - 4.44949i) q^{77} +9.75663 q^{79} -1.00000 q^{81} +(1.41421 + 1.41421i) q^{83} +(6.61037 - 6.61037i) q^{87} +2.00000i q^{89} -15.4135i q^{91} +(-6.00000 + 6.00000i) q^{93} +(-3.00000 - 3.00000i) q^{97} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{17} + 16 q^{21} + 8 q^{33} + 32 q^{37} - 48 q^{41} + 16 q^{53} + 16 q^{57} - 16 q^{61} - 24 q^{73} - 16 q^{77} - 8 q^{81} - 48 q^{93} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(7\) 3.14626 − 3.14626i 1.18918 − 1.18918i 0.211881 − 0.977296i \(-0.432041\pi\)
0.977296 − 0.211881i \(-0.0679588\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) − 1.41421i − 0.426401i −0.977008 − 0.213201i \(-0.931611\pi\)
0.977008 − 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) 2.44949 − 2.44949i 0.679366 − 0.679366i −0.280491 − 0.959857i \(-0.590497\pi\)
0.959857 + 0.280491i \(0.0904971\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.44949 − 4.44949i −1.07916 − 1.07916i −0.996585 − 0.0825749i \(-0.973686\pi\)
−0.0825749 − 0.996585i \(-0.526314\pi\)
\(18\) 0 0
\(19\) −0.635674 −0.145834 −0.0729169 − 0.997338i \(-0.523231\pi\)
−0.0729169 + 0.997338i \(0.523231\pi\)
\(20\) 0 0
\(21\) 4.44949 0.970958
\(22\) 0 0
\(23\) −2.04989 − 2.04989i −0.427431 − 0.427431i 0.460321 − 0.887752i \(-0.347734\pi\)
−0.887752 + 0.460321i \(0.847734\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 + 0.707107i −0.136083 + 0.136083i
\(28\) 0 0
\(29\) − 9.34847i − 1.73597i −0.496593 − 0.867984i \(-0.665416\pi\)
0.496593 − 0.867984i \(-0.334584\pi\)
\(30\) 0 0
\(31\) 8.48528i 1.52400i 0.647576 + 0.762001i \(0.275783\pi\)
−0.647576 + 0.762001i \(0.724217\pi\)
\(32\) 0 0
\(33\) 1.00000 − 1.00000i 0.174078 − 0.174078i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.55051 + 1.55051i 0.254902 + 0.254902i 0.822977 − 0.568075i \(-0.192312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(38\) 0 0
\(39\) 3.46410 0.554700
\(40\) 0 0
\(41\) −1.10102 −0.171951 −0.0859753 − 0.996297i \(-0.527401\pi\)
−0.0859753 + 0.996297i \(0.527401\pi\)
\(42\) 0 0
\(43\) 0.635674 + 0.635674i 0.0969395 + 0.0969395i 0.753913 − 0.656974i \(-0.228164\pi\)
−0.656974 + 0.753913i \(0.728164\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.51399 + 5.51399i −0.804298 + 0.804298i −0.983764 − 0.179466i \(-0.942563\pi\)
0.179466 + 0.983764i \(0.442563\pi\)
\(48\) 0 0
\(49\) − 12.7980i − 1.82828i
\(50\) 0 0
\(51\) − 6.29253i − 0.881130i
\(52\) 0 0
\(53\) 2.00000 − 2.00000i 0.274721 − 0.274721i −0.556276 − 0.830997i \(-0.687770\pi\)
0.830997 + 0.556276i \(0.187770\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.449490 − 0.449490i −0.0595364 − 0.0595364i
\(58\) 0 0
\(59\) 9.89949 1.28880 0.644402 − 0.764687i \(-0.277106\pi\)
0.644402 + 0.764687i \(0.277106\pi\)
\(60\) 0 0
\(61\) 2.89898 0.371176 0.185588 − 0.982628i \(-0.440581\pi\)
0.185588 + 0.982628i \(0.440581\pi\)
\(62\) 0 0
\(63\) 3.14626 + 3.14626i 0.396392 + 0.396392i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.29253 − 6.29253i 0.768755 − 0.768755i −0.209133 − 0.977887i \(-0.567064\pi\)
0.977887 + 0.209133i \(0.0670640\pi\)
\(68\) 0 0
\(69\) − 2.89898i − 0.348996i
\(70\) 0 0
\(71\) 12.5851i 1.49357i 0.665065 + 0.746786i \(0.268404\pi\)
−0.665065 + 0.746786i \(0.731596\pi\)
\(72\) 0 0
\(73\) −3.00000 + 3.00000i −0.351123 + 0.351123i −0.860527 − 0.509404i \(-0.829866\pi\)
0.509404 + 0.860527i \(0.329866\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.44949 − 4.44949i −0.507066 − 0.507066i
\(78\) 0 0
\(79\) 9.75663 1.09771 0.548853 − 0.835919i \(-0.315065\pi\)
0.548853 + 0.835919i \(0.315065\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 1.41421 + 1.41421i 0.155230 + 0.155230i 0.780449 − 0.625219i \(-0.214990\pi\)
−0.625219 + 0.780449i \(0.714990\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.61037 − 6.61037i 0.708706 − 0.708706i
\(88\) 0 0
\(89\) 2.00000i 0.212000i 0.994366 + 0.106000i \(0.0338043\pi\)
−0.994366 + 0.106000i \(0.966196\pi\)
\(90\) 0 0
\(91\) − 15.4135i − 1.61577i
\(92\) 0 0
\(93\) −6.00000 + 6.00000i −0.622171 + 0.622171i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.00000 − 3.00000i −0.304604 − 0.304604i 0.538208 − 0.842812i \(-0.319101\pi\)
−0.842812 + 0.538208i \(0.819101\pi\)
\(98\) 0 0
\(99\) 1.41421 0.142134
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.b.607.4 8
3.2 odd 2 3600.2.x.n.3007.4 8
4.3 odd 2 inner 1200.2.w.b.607.1 8
5.2 odd 4 240.2.w.b.223.3 yes 8
5.3 odd 4 inner 1200.2.w.b.943.1 8
5.4 even 2 240.2.w.b.127.1 ✓ 8
12.11 even 2 3600.2.x.n.3007.1 8
15.2 even 4 720.2.x.f.703.4 8
15.8 even 4 3600.2.x.n.2143.1 8
15.14 odd 2 720.2.x.f.127.3 8
20.3 even 4 inner 1200.2.w.b.943.4 8
20.7 even 4 240.2.w.b.223.1 yes 8
20.19 odd 2 240.2.w.b.127.3 yes 8
40.19 odd 2 960.2.w.d.127.2 8
40.27 even 4 960.2.w.d.703.4 8
40.29 even 2 960.2.w.d.127.4 8
40.37 odd 4 960.2.w.d.703.2 8
60.23 odd 4 3600.2.x.n.2143.4 8
60.47 odd 4 720.2.x.f.703.3 8
60.59 even 2 720.2.x.f.127.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
240.2.w.b.127.1 ✓ 8 5.4 even 2
240.2.w.b.127.3 yes 8 20.19 odd 2
240.2.w.b.223.1 yes 8 20.7 even 4
240.2.w.b.223.3 yes 8 5.2 odd 4
720.2.x.f.127.3 8 15.14 odd 2
720.2.x.f.127.4 8 60.59 even 2
720.2.x.f.703.3 8 60.47 odd 4
720.2.x.f.703.4 8 15.2 even 4
960.2.w.d.127.2 8 40.19 odd 2
960.2.w.d.127.4 8 40.29 even 2
960.2.w.d.703.2 8 40.37 odd 4
960.2.w.d.703.4 8 40.27 even 4
1200.2.w.b.607.1 8 4.3 odd 2 inner
1200.2.w.b.607.4 8 1.1 even 1 trivial
1200.2.w.b.943.1 8 5.3 odd 4 inner
1200.2.w.b.943.4 8 20.3 even 4 inner
3600.2.x.n.2143.1 8 15.8 even 4
3600.2.x.n.2143.4 8 60.23 odd 4
3600.2.x.n.3007.1 8 12.11 even 2
3600.2.x.n.3007.4 8 3.2 odd 2