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Properties

Label 720.2.h.a.431.3
Level $720$
Weight $2$
Character 720.431
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(431,720)] 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("720.431"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 431.3
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 720.431
Dual form 720.2.h.a.431.5

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{5} +2.44949i q^{7} -5.91359 q^{11} -6.24264 q^{13} +4.89898i q^{19} +1.43488 q^{23} -1.00000 q^{25} -6.00000i q^{29} +1.43488i q^{31} +2.44949 q^{35} -2.24264 q^{37} -4.24264i q^{41} +11.8272i q^{43} -11.8272 q^{47} +1.00000 q^{49} +8.48528i q^{53} +5.91359i q^{55} -5.91359 q^{59} -6.48528 q^{61} +6.24264i q^{65} -6.92820i q^{67} -2.86976 q^{71} +6.48528 q^{73} -14.4853i q^{77} -8.36308i q^{79} +14.6969 q^{83} +7.75736i q^{89} -15.2913i q^{91} +4.89898 q^{95} +6.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{13} - 8 q^{25} + 16 q^{37} + 8 q^{49} + 16 q^{61} - 16 q^{73} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 0
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) 2.44949i 0.925820i 0.886405 + 0.462910i \(0.153195\pi\)
−0.886405 + 0.462910i \(0.846805\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.91359 −1.78301 −0.891507 − 0.453006i \(-0.850352\pi\)
−0.891507 + 0.453006i \(0.850352\pi\)
\(12\) 0 0
\(13\) −6.24264 −1.73140 −0.865699 − 0.500566i \(-0.833125\pi\)
−0.865699 + 0.500566i \(0.833125\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 4.89898i 1.12390i 0.827170 + 0.561951i \(0.189949\pi\)
−0.827170 + 0.561951i \(0.810051\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.43488 0.299193 0.149596 − 0.988747i \(-0.452203\pi\)
0.149596 + 0.988747i \(0.452203\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.00000i − 1.11417i −0.830455 − 0.557086i \(-0.811919\pi\)
0.830455 − 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) 1.43488i 0.257712i 0.991663 + 0.128856i \(0.0411304\pi\)
−0.991663 + 0.128856i \(0.958870\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.44949 0.414039
\(36\) 0 0
\(37\) −2.24264 −0.368688 −0.184344 − 0.982862i \(-0.559016\pi\)
−0.184344 + 0.982862i \(0.559016\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 4.24264i − 0.662589i −0.943527 − 0.331295i \(-0.892515\pi\)
0.943527 − 0.331295i \(-0.107485\pi\)
\(42\) 0 0
\(43\) 11.8272i 1.80363i 0.432124 + 0.901814i \(0.357764\pi\)
−0.432124 + 0.901814i \(0.642236\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.8272 −1.72517 −0.862586 − 0.505911i \(-0.831157\pi\)
−0.862586 + 0.505911i \(0.831157\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.48528i 1.16554i 0.812636 + 0.582772i \(0.198032\pi\)
−0.812636 + 0.582772i \(0.801968\pi\)
\(54\) 0 0
\(55\) 5.91359i 0.797388i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.91359 −0.769884 −0.384942 − 0.922941i \(-0.625779\pi\)
−0.384942 + 0.922941i \(0.625779\pi\)
\(60\) 0 0
\(61\) −6.48528 −0.830355 −0.415178 − 0.909740i \(-0.636281\pi\)
−0.415178 + 0.909740i \(0.636281\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.24264i 0.774304i
\(66\) 0 0
\(67\) − 6.92820i − 0.846415i −0.906033 − 0.423207i \(-0.860904\pi\)
0.906033 − 0.423207i \(-0.139096\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.86976 −0.340577 −0.170289 − 0.985394i \(-0.554470\pi\)
−0.170289 + 0.985394i \(0.554470\pi\)
\(72\) 0 0
\(73\) 6.48528 0.759045 0.379522 − 0.925183i \(-0.376088\pi\)
0.379522 + 0.925183i \(0.376088\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 14.4853i − 1.65075i
\(78\) 0 0
\(79\) − 8.36308i − 0.940920i −0.882421 − 0.470460i \(-0.844088\pi\)
0.882421 − 0.470460i \(-0.155912\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.6969 1.61320 0.806599 − 0.591099i \(-0.201306\pi\)
0.806599 + 0.591099i \(0.201306\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.75736i 0.822278i 0.911573 + 0.411139i \(0.134869\pi\)
−0.911573 + 0.411139i \(0.865131\pi\)
\(90\) 0 0
\(91\) − 15.2913i − 1.60296i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.89898 0.502625
\(96\) 0 0
\(97\) 6.48528 0.658481 0.329240 − 0.944246i \(-0.393207\pi\)
0.329240 + 0.944246i \(0.393207\pi\)
\(98\) 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.h.a.431.3 yes 8
3.2 odd 2 inner 720.2.h.a.431.8 yes 8
4.3 odd 2 inner 720.2.h.a.431.2 ✓ 8
5.2 odd 4 3600.2.o.d.3599.2 8
5.3 odd 4 3600.2.o.c.3599.5 8
5.4 even 2 3600.2.h.j.1151.1 8
8.3 odd 2 2880.2.h.f.1151.5 8
8.5 even 2 2880.2.h.f.1151.8 8
12.11 even 2 inner 720.2.h.a.431.5 yes 8
15.2 even 4 3600.2.o.c.3599.4 8
15.8 even 4 3600.2.o.d.3599.7 8
15.14 odd 2 3600.2.h.j.1151.4 8
20.3 even 4 3600.2.o.c.3599.3 8
20.7 even 4 3600.2.o.d.3599.8 8
20.19 odd 2 3600.2.h.j.1151.8 8
24.5 odd 2 2880.2.h.f.1151.3 8
24.11 even 2 2880.2.h.f.1151.2 8
60.23 odd 4 3600.2.o.d.3599.1 8
60.47 odd 4 3600.2.o.c.3599.6 8
60.59 even 2 3600.2.h.j.1151.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
720.2.h.a.431.2 ✓ 8 4.3 odd 2 inner
720.2.h.a.431.3 yes 8 1.1 even 1 trivial
720.2.h.a.431.5 yes 8 12.11 even 2 inner
720.2.h.a.431.8 yes 8 3.2 odd 2 inner
2880.2.h.f.1151.2 8 24.11 even 2
2880.2.h.f.1151.3 8 24.5 odd 2
2880.2.h.f.1151.5 8 8.3 odd 2
2880.2.h.f.1151.8 8 8.5 even 2
3600.2.h.j.1151.1 8 5.4 even 2
3600.2.h.j.1151.4 8 15.14 odd 2
3600.2.h.j.1151.5 8 60.59 even 2
3600.2.h.j.1151.8 8 20.19 odd 2
3600.2.o.c.3599.3 8 20.3 even 4
3600.2.o.c.3599.4 8 15.2 even 4
3600.2.o.c.3599.5 8 5.3 odd 4
3600.2.o.c.3599.6 8 60.47 odd 4
3600.2.o.d.3599.1 8 60.23 odd 4
3600.2.o.d.3599.2 8 5.2 odd 4
3600.2.o.d.3599.7 8 15.8 even 4
3600.2.o.d.3599.8 8 20.7 even 4