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Properties

Label 720.2.x.f.127.4
Level $720$
Weight $2$
Character 720.127
Analytic conductor $5.749$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [720,2,Mod(127,720)] 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("720.127"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,-8] 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: \(5.74922894553\)
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: \( 2^{6} \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 127.4
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 720.127
Dual form 720.2.x.f.703.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.224745 - 2.22474i) q^{5} +(3.14626 - 3.14626i) q^{7} -1.41421i q^{11} +(-2.44949 + 2.44949i) q^{13} +(-4.44949 - 4.44949i) q^{17} +0.635674 q^{19} +(2.04989 + 2.04989i) q^{23} +(-4.89898 - 1.00000i) q^{25} +9.34847i q^{29} -8.48528i q^{31} +(-6.29253 - 7.70674i) q^{35} +(-1.55051 - 1.55051i) q^{37} +1.10102 q^{41} +(0.635674 + 0.635674i) q^{43} +(5.51399 - 5.51399i) q^{47} -12.7980i q^{49} +(2.00000 - 2.00000i) q^{53} +(-3.14626 - 0.317837i) q^{55} +9.89949 q^{59} +2.89898 q^{61} +(4.89898 + 6.00000i) q^{65} +(6.29253 - 6.29253i) q^{67} +12.5851i q^{71} +(3.00000 - 3.00000i) q^{73} +(-4.44949 - 4.44949i) q^{77} -9.75663 q^{79} +(-1.41421 - 1.41421i) q^{83} +(-10.8990 + 8.89898i) q^{85} -2.00000i q^{89} +15.4135i q^{91} +(0.142865 - 1.41421i) q^{95} +(3.00000 + 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5} - 16 q^{17} - 32 q^{37} + 48 q^{41} + 16 q^{53} - 16 q^{61} + 24 q^{73} - 16 q^{77} - 48 q^{85} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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 0
\(4\) 0 0
\(5\) 0.224745 − 2.22474i 0.100509 − 0.994936i
\(6\) 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
\(9\) 0 0
\(10\) 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 −0.959857 − 0.280491i \(-0.909503\pi\)
0.280491 + 0.959857i \(0.409503\pi\)
\(14\) 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 \(-0.476769\pi\)
0.0729169 + 0.997338i \(0.476769\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.04989 + 2.04989i 0.427431 + 0.427431i 0.887752 − 0.460321i \(-0.152266\pi\)
−0.460321 + 0.887752i \(0.652266\pi\)
\(24\) 0 0
\(25\) −4.89898 − 1.00000i −0.979796 − 0.200000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.34847i 1.73597i 0.496593 + 0.867984i \(0.334584\pi\)
−0.496593 + 0.867984i \(0.665416\pi\)
\(30\) 0 0
\(31\) − 8.48528i − 1.52400i −0.647576 − 0.762001i \(-0.724217\pi\)
0.647576 − 0.762001i \(-0.275783\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.29253 − 7.70674i −1.06363 − 1.30268i
\(36\) 0 0
\(37\) −1.55051 − 1.55051i −0.254902 − 0.254902i 0.568075 − 0.822977i \(-0.307688\pi\)
−0.822977 + 0.568075i \(0.807688\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.10102 0.171951 0.0859753 − 0.996297i \(-0.472599\pi\)
0.0859753 + 0.996297i \(0.472599\pi\)
\(42\) 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 −0.179466 − 0.983764i \(-0.557437\pi\)
0.983764 + 0.179466i \(0.0574370\pi\)
\(48\) 0 0
\(49\) − 12.7980i − 1.82828i
\(50\) 0 0
\(51\) 0 0
\(52\) 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
\(55\) −3.14626 − 0.317837i −0.424242 − 0.0428572i
\(56\) 0 0
\(57\) 0 0
\(58\) 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
\(63\) 0 0
\(64\) 0 0
\(65\) 4.89898 + 6.00000i 0.607644 + 0.744208i
\(66\) 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
\(69\) 0 0
\(70\) 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 −0.509404 − 0.860527i \(-0.670134\pi\)
0.860527 + 0.509404i \(0.170134\pi\)
\(74\) 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 \(-0.684935\pi\)
−0.548853 + 0.835919i \(0.684935\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.41421 − 1.41421i −0.155230 − 0.155230i 0.625219 − 0.780449i \(-0.285010\pi\)
−0.780449 + 0.625219i \(0.785010\pi\)
\(84\) 0 0
\(85\) −10.8990 + 8.89898i −1.18216 + 0.965230i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 2.00000i − 0.212000i −0.994366 − 0.106000i \(-0.966196\pi\)
0.994366 − 0.106000i \(-0.0338043\pi\)
\(90\) 0 0
\(91\) 15.4135i 1.61577i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.142865 − 1.41421i 0.0146576 − 0.145095i
\(96\) 0 0
\(97\) 3.00000 + 3.00000i 0.304604 + 0.304604i 0.842812 − 0.538208i \(-0.180899\pi\)
−0.538208 + 0.842812i \(0.680899\pi\)
\(98\) 0 0
\(99\) 0 0
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 720.2.x.f.127.4 8
3.2 odd 2 240.2.w.b.127.3 yes 8
4.3 odd 2 inner 720.2.x.f.127.3 8
5.2 odd 4 3600.2.x.n.2143.4 8
5.3 odd 4 inner 720.2.x.f.703.3 8
5.4 even 2 3600.2.x.n.3007.1 8
12.11 even 2 240.2.w.b.127.1 ✓ 8
15.2 even 4 1200.2.w.b.943.4 8
15.8 even 4 240.2.w.b.223.1 yes 8
15.14 odd 2 1200.2.w.b.607.1 8
20.3 even 4 inner 720.2.x.f.703.4 8
20.7 even 4 3600.2.x.n.2143.1 8
20.19 odd 2 3600.2.x.n.3007.4 8
24.5 odd 2 960.2.w.d.127.2 8
24.11 even 2 960.2.w.d.127.4 8
60.23 odd 4 240.2.w.b.223.3 yes 8
60.47 odd 4 1200.2.w.b.943.1 8
60.59 even 2 1200.2.w.b.607.4 8
120.53 even 4 960.2.w.d.703.4 8
120.83 odd 4 960.2.w.d.703.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
240.2.w.b.127.1 ✓ 8 12.11 even 2
240.2.w.b.127.3 yes 8 3.2 odd 2
240.2.w.b.223.1 yes 8 15.8 even 4
240.2.w.b.223.3 yes 8 60.23 odd 4
720.2.x.f.127.3 8 4.3 odd 2 inner
720.2.x.f.127.4 8 1.1 even 1 trivial
720.2.x.f.703.3 8 5.3 odd 4 inner
720.2.x.f.703.4 8 20.3 even 4 inner
960.2.w.d.127.2 8 24.5 odd 2
960.2.w.d.127.4 8 24.11 even 2
960.2.w.d.703.2 8 120.83 odd 4
960.2.w.d.703.4 8 120.53 even 4
1200.2.w.b.607.1 8 15.14 odd 2
1200.2.w.b.607.4 8 60.59 even 2
1200.2.w.b.943.1 8 60.47 odd 4
1200.2.w.b.943.4 8 15.2 even 4
3600.2.x.n.2143.1 8 20.7 even 4
3600.2.x.n.2143.4 8 5.2 odd 4
3600.2.x.n.3007.1 8 5.4 even 2
3600.2.x.n.3007.4 8 20.19 odd 2