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Properties

Label 720.2.h.a.431.7
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 431.7
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 720.431
Dual form 720.2.h.a.431.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{5} +2.44949i q^{7} -1.01461 q^{11} +2.24264 q^{13} +4.89898i q^{19} -8.36308 q^{23} -1.00000 q^{25} +6.00000i q^{29} +8.36308i q^{31} -2.44949 q^{35} +6.24264 q^{37} -4.24264i q^{41} -2.02922i q^{43} -2.02922 q^{47} +1.00000 q^{49} +8.48528i q^{53} -1.01461i q^{55} -1.01461 q^{59} +10.4853 q^{61} +2.24264i q^{65} +6.92820i q^{67} +16.7262 q^{71} -10.4853 q^{73} -2.48528i q^{77} -1.43488i q^{79} -14.6969 q^{83} -16.2426i q^{89} +5.49333i q^{91} -4.89898 q^{95} -10.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{13} - 8 q^{25} + 16 q^{37} + 8 q^{49} + 16 q^{61} - 16 q^{73} - 16 q^{97}+O(q^{100}) \) 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
\(11\) −1.01461 −0.305917 −0.152958 − 0.988233i \(-0.548880\pi\)
−0.152958 + 0.988233i \(0.548880\pi\)
\(12\) 0 0
\(13\) 2.24264 0.621997 0.310998 − 0.950410i \(-0.399337\pi\)
0.310998 + 0.950410i \(0.399337\pi\)
\(14\) 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
\(23\) −8.36308 −1.74382 −0.871911 − 0.489664i \(-0.837120\pi\)
−0.871911 + 0.489664i \(0.837120\pi\)
\(24\) 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 \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 8.36308i 1.50205i 0.660272 + 0.751027i \(0.270441\pi\)
−0.660272 + 0.751027i \(0.729559\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.44949 −0.414039
\(36\) 0 0
\(37\) 6.24264 1.02628 0.513142 − 0.858304i \(-0.328481\pi\)
0.513142 + 0.858304i \(0.328481\pi\)
\(38\) 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
\(43\) − 2.02922i − 0.309454i −0.987957 − 0.154727i \(-0.950550\pi\)
0.987957 − 0.154727i \(-0.0494498\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.02922 −0.295993 −0.147996 − 0.988988i \(-0.547282\pi\)
−0.147996 + 0.988988i \(0.547282\pi\)
\(48\) 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
\(55\) − 1.01461i − 0.136810i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.01461 −0.132091 −0.0660456 − 0.997817i \(-0.521038\pi\)
−0.0660456 + 0.997817i \(0.521038\pi\)
\(60\) 0 0
\(61\) 10.4853 1.34250 0.671251 − 0.741230i \(-0.265757\pi\)
0.671251 + 0.741230i \(0.265757\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.24264i 0.278165i
\(66\) 0 0
\(67\) 6.92820i 0.846415i 0.906033 + 0.423207i \(0.139096\pi\)
−0.906033 + 0.423207i \(0.860904\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.7262 1.98503 0.992515 − 0.122122i \(-0.0389698\pi\)
0.992515 + 0.122122i \(0.0389698\pi\)
\(72\) 0 0
\(73\) −10.4853 −1.22721 −0.613605 − 0.789613i \(-0.710281\pi\)
−0.613605 + 0.789613i \(0.710281\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.48528i − 0.283224i
\(78\) 0 0
\(79\) − 1.43488i − 0.161436i −0.996737 − 0.0807182i \(-0.974279\pi\)
0.996737 − 0.0807182i \(-0.0257214\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −14.6969 −1.61320 −0.806599 − 0.591099i \(-0.798694\pi\)
−0.806599 + 0.591099i \(0.798694\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 16.2426i − 1.72172i −0.508845 − 0.860858i \(-0.669927\pi\)
0.508845 − 0.860858i \(-0.330073\pi\)
\(90\) 0 0
\(91\) 5.49333i 0.575857i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.89898 −0.502625
\(96\) 0 0
\(97\) −10.4853 −1.06462 −0.532310 − 0.846550i \(-0.678676\pi\)
−0.532310 + 0.846550i \(0.678676\pi\)
\(98\) 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.431.7 yes 8
3.2 odd 2 inner 720.2.h.a.431.4 yes 8
4.3 odd 2 inner 720.2.h.a.431.6 yes 8
5.2 odd 4 3600.2.o.c.3599.1 8
5.3 odd 4 3600.2.o.d.3599.6 8
5.4 even 2 3600.2.h.j.1151.2 8
8.3 odd 2 2880.2.h.f.1151.1 8
8.5 even 2 2880.2.h.f.1151.4 8
12.11 even 2 inner 720.2.h.a.431.1 ✓ 8
15.2 even 4 3600.2.o.d.3599.3 8
15.8 even 4 3600.2.o.c.3599.8 8
15.14 odd 2 3600.2.h.j.1151.3 8
20.3 even 4 3600.2.o.d.3599.4 8
20.7 even 4 3600.2.o.c.3599.7 8
20.19 odd 2 3600.2.h.j.1151.7 8
24.5 odd 2 2880.2.h.f.1151.7 8
24.11 even 2 2880.2.h.f.1151.6 8
60.23 odd 4 3600.2.o.c.3599.2 8
60.47 odd 4 3600.2.o.d.3599.5 8
60.59 even 2 3600.2.h.j.1151.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
720.2.h.a.431.1 ✓ 8 12.11 even 2 inner
720.2.h.a.431.4 yes 8 3.2 odd 2 inner
720.2.h.a.431.6 yes 8 4.3 odd 2 inner
720.2.h.a.431.7 yes 8 1.1 even 1 trivial
2880.2.h.f.1151.1 8 8.3 odd 2
2880.2.h.f.1151.4 8 8.5 even 2
2880.2.h.f.1151.6 8 24.11 even 2
2880.2.h.f.1151.7 8 24.5 odd 2
3600.2.h.j.1151.2 8 5.4 even 2
3600.2.h.j.1151.3 8 15.14 odd 2
3600.2.h.j.1151.6 8 60.59 even 2
3600.2.h.j.1151.7 8 20.19 odd 2
3600.2.o.c.3599.1 8 5.2 odd 4
3600.2.o.c.3599.2 8 60.23 odd 4
3600.2.o.c.3599.7 8 20.7 even 4
3600.2.o.c.3599.8 8 15.8 even 4
3600.2.o.d.3599.3 8 15.2 even 4
3600.2.o.d.3599.4 8 20.3 even 4
3600.2.o.d.3599.5 8 60.47 odd 4
3600.2.o.d.3599.6 8 5.3 odd 4