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Properties

Label 320.2.f.b.289.4
Level $320$
Weight $2$
Character 320.289
Analytic conductor $2.555$
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] = [320,2,Mod(289,320)] 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("320.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.55521286468\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 289.4
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 320.289
Dual form 320.2.f.b.289.3

$q$-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)
 
\(f(q)\) \(=\) \(q-2.44949 q^{3} +(1.41421 + 1.73205i) q^{5} -1.41421i q^{7} +3.00000 q^{9} +2.00000i q^{11} -5.65685 q^{13} +(-3.46410 - 4.24264i) q^{15} +4.89898i q^{17} +6.00000i q^{19} +3.46410i q^{21} +7.07107i q^{23} +(-1.00000 + 4.89898i) q^{25} -6.92820i q^{29} -6.92820 q^{31} -4.89898i q^{33} +(2.44949 - 2.00000i) q^{35} +2.82843 q^{37} +13.8564 q^{39} -4.00000 q^{41} -2.44949 q^{43} +(4.24264 + 5.19615i) q^{45} -4.24264i q^{47} +5.00000 q^{49} -12.0000i q^{51} +(-3.46410 + 2.82843i) q^{55} -14.6969i q^{57} -2.00000i q^{59} +3.46410i q^{61} -4.24264i q^{63} +(-8.00000 - 9.79796i) q^{65} +2.44949 q^{67} -17.3205i q^{69} +6.92820 q^{71} +4.89898i q^{73} +(2.44949 - 12.0000i) q^{75} +2.82843 q^{77} +6.92820 q^{79} -9.00000 q^{81} +12.2474 q^{83} +(-8.48528 + 6.92820i) q^{85} +16.9706i q^{87} -2.00000 q^{89} +8.00000i q^{91} +16.9706 q^{93} +(-10.3923 + 8.48528i) q^{95} +14.6969i q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9} - 8 q^{25} - 32 q^{41} + 40 q^{49} - 64 q^{65} - 72 q^{81} - 16 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) −2.44949 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) 1.41421 + 1.73205i 0.632456 + 0.774597i
\(6\) 0 0
\(7\) − 1.41421i − 0.534522i −0.963624 − 0.267261i \(-0.913881\pi\)
0.963624 − 0.267261i \(-0.0861187\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 − 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 0 0
\(15\) −3.46410 − 4.24264i −0.894427 − 1.09545i
\(16\) 0 0
\(17\) 4.89898i 1.18818i 0.804400 + 0.594089i \(0.202487\pi\)
−0.804400 + 0.594089i \(0.797513\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) 3.46410i 0.755929i
\(22\) 0 0
\(23\) 7.07107i 1.47442i 0.675664 + 0.737210i \(0.263857\pi\)
−0.675664 + 0.737210i \(0.736143\pi\)
\(24\) 0 0
\(25\) −1.00000 + 4.89898i −0.200000 + 0.979796i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.92820i − 1.28654i −0.765641 − 0.643268i \(-0.777578\pi\)
0.765641 − 0.643268i \(-0.222422\pi\)
\(30\) 0 0
\(31\) −6.92820 −1.24434 −0.622171 − 0.782881i \(-0.713749\pi\)
−0.622171 + 0.782881i \(0.713749\pi\)
\(32\) 0 0
\(33\) − 4.89898i − 0.852803i
\(34\) 0 0
\(35\) 2.44949 − 2.00000i 0.414039 − 0.338062i
\(36\) 0 0
\(37\) 2.82843 0.464991 0.232495 − 0.972598i \(-0.425311\pi\)
0.232495 + 0.972598i \(0.425311\pi\)
\(38\) 0 0
\(39\) 13.8564 2.21880
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −2.44949 −0.373544 −0.186772 − 0.982403i \(-0.559803\pi\)
−0.186772 + 0.982403i \(0.559803\pi\)
\(44\) 0 0
\(45\) 4.24264 + 5.19615i 0.632456 + 0.774597i
\(46\) 0 0
\(47\) − 4.24264i − 0.618853i −0.950923 − 0.309426i \(-0.899863\pi\)
0.950923 − 0.309426i \(-0.100137\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) − 12.0000i − 1.68034i
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −3.46410 + 2.82843i −0.467099 + 0.381385i
\(56\) 0 0
\(57\) − 14.6969i − 1.94666i
\(58\) 0 0
\(59\) − 2.00000i − 0.260378i −0.991489 − 0.130189i \(-0.958442\pi\)
0.991489 − 0.130189i \(-0.0415584\pi\)
\(60\) 0 0
\(61\) 3.46410i 0.443533i 0.975100 + 0.221766i \(0.0711822\pi\)
−0.975100 + 0.221766i \(0.928818\pi\)
\(62\) 0 0
\(63\) − 4.24264i − 0.534522i
\(64\) 0 0
\(65\) −8.00000 − 9.79796i −0.992278 − 1.21529i
\(66\) 0 0
\(67\) 2.44949 0.299253 0.149626 − 0.988743i \(-0.452193\pi\)
0.149626 + 0.988743i \(0.452193\pi\)
\(68\) 0 0
\(69\) − 17.3205i − 2.08514i
\(70\) 0 0
\(71\) 6.92820 0.822226 0.411113 − 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 0 0
\(73\) 4.89898i 0.573382i 0.958023 + 0.286691i \(0.0925553\pi\)
−0.958023 + 0.286691i \(0.907445\pi\)
\(74\) 0 0
\(75\) 2.44949 − 12.0000i 0.282843 − 1.38564i
\(76\) 0 0
\(77\) 2.82843 0.322329
\(78\) 0 0
\(79\) 6.92820 0.779484 0.389742 − 0.920924i \(-0.372564\pi\)
0.389742 + 0.920924i \(0.372564\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 12.2474 1.34433 0.672166 − 0.740400i \(-0.265364\pi\)
0.672166 + 0.740400i \(0.265364\pi\)
\(84\) 0 0
\(85\) −8.48528 + 6.92820i −0.920358 + 0.751469i
\(86\) 0 0
\(87\) 16.9706i 1.81944i
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 8.00000i 0.838628i
\(92\) 0 0
\(93\) 16.9706 1.75977
\(94\) 0 0
\(95\) −10.3923 + 8.48528i −1.06623 + 0.870572i
\(96\) 0 0
\(97\) 14.6969i 1.49225i 0.665807 + 0.746124i \(0.268087\pi\)
−0.665807 + 0.746124i \(0.731913\pi\)
\(98\) 0 0
\(99\) 6.00000i 0.603023i
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 320.2.f.b.289.4 yes 8
3.2 odd 2 2880.2.d.g.289.1 8
4.3 odd 2 inner 320.2.f.b.289.8 yes 8
5.2 odd 4 1600.2.d.i.801.8 8
5.3 odd 4 1600.2.d.i.801.2 8
5.4 even 2 inner 320.2.f.b.289.6 yes 8
8.3 odd 2 inner 320.2.f.b.289.1 ✓ 8
8.5 even 2 inner 320.2.f.b.289.5 yes 8
12.11 even 2 2880.2.d.g.289.2 8
15.14 odd 2 2880.2.d.g.289.6 8
16.3 odd 4 1280.2.c.g.769.4 4
16.5 even 4 1280.2.c.g.769.3 4
16.11 odd 4 1280.2.c.h.769.1 4
16.13 even 4 1280.2.c.h.769.2 4
20.3 even 4 1600.2.d.i.801.7 8
20.7 even 4 1600.2.d.i.801.1 8
20.19 odd 2 inner 320.2.f.b.289.2 yes 8
24.5 odd 2 2880.2.d.g.289.7 8
24.11 even 2 2880.2.d.g.289.8 8
40.3 even 4 1600.2.d.i.801.4 8
40.13 odd 4 1600.2.d.i.801.5 8
40.19 odd 2 inner 320.2.f.b.289.7 yes 8
40.27 even 4 1600.2.d.i.801.6 8
40.29 even 2 inner 320.2.f.b.289.3 yes 8
40.37 odd 4 1600.2.d.i.801.3 8
60.59 even 2 2880.2.d.g.289.5 8
80.3 even 4 6400.2.a.ct.1.1 4
80.13 odd 4 6400.2.a.cu.1.4 4
80.19 odd 4 1280.2.c.g.769.2 4
80.27 even 4 6400.2.a.cu.1.2 4
80.29 even 4 1280.2.c.h.769.4 4
80.37 odd 4 6400.2.a.ct.1.3 4
80.43 even 4 6400.2.a.cu.1.3 4
80.53 odd 4 6400.2.a.ct.1.2 4
80.59 odd 4 1280.2.c.h.769.3 4
80.67 even 4 6400.2.a.ct.1.4 4
80.69 even 4 1280.2.c.g.769.1 4
80.77 odd 4 6400.2.a.cu.1.1 4
120.29 odd 2 2880.2.d.g.289.4 8
120.59 even 2 2880.2.d.g.289.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.f.b.289.1 ✓ 8 8.3 odd 2 inner
320.2.f.b.289.2 yes 8 20.19 odd 2 inner
320.2.f.b.289.3 yes 8 40.29 even 2 inner
320.2.f.b.289.4 yes 8 1.1 even 1 trivial
320.2.f.b.289.5 yes 8 8.5 even 2 inner
320.2.f.b.289.6 yes 8 5.4 even 2 inner
320.2.f.b.289.7 yes 8 40.19 odd 2 inner
320.2.f.b.289.8 yes 8 4.3 odd 2 inner
1280.2.c.g.769.1 4 80.69 even 4
1280.2.c.g.769.2 4 80.19 odd 4
1280.2.c.g.769.3 4 16.5 even 4
1280.2.c.g.769.4 4 16.3 odd 4
1280.2.c.h.769.1 4 16.11 odd 4
1280.2.c.h.769.2 4 16.13 even 4
1280.2.c.h.769.3 4 80.59 odd 4
1280.2.c.h.769.4 4 80.29 even 4
1600.2.d.i.801.1 8 20.7 even 4
1600.2.d.i.801.2 8 5.3 odd 4
1600.2.d.i.801.3 8 40.37 odd 4
1600.2.d.i.801.4 8 40.3 even 4
1600.2.d.i.801.5 8 40.13 odd 4
1600.2.d.i.801.6 8 40.27 even 4
1600.2.d.i.801.7 8 20.3 even 4
1600.2.d.i.801.8 8 5.2 odd 4
2880.2.d.g.289.1 8 3.2 odd 2
2880.2.d.g.289.2 8 12.11 even 2
2880.2.d.g.289.3 8 120.59 even 2
2880.2.d.g.289.4 8 120.29 odd 2
2880.2.d.g.289.5 8 60.59 even 2
2880.2.d.g.289.6 8 15.14 odd 2
2880.2.d.g.289.7 8 24.5 odd 2
2880.2.d.g.289.8 8 24.11 even 2
6400.2.a.ct.1.1 4 80.3 even 4
6400.2.a.ct.1.2 4 80.53 odd 4
6400.2.a.ct.1.3 4 80.37 odd 4
6400.2.a.ct.1.4 4 80.67 even 4
6400.2.a.cu.1.1 4 80.77 odd 4
6400.2.a.cu.1.2 4 80.27 even 4
6400.2.a.cu.1.3 4 80.43 even 4
6400.2.a.cu.1.4 4 80.13 odd 4