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Properties

Label 1280.4.a.bb.1.2
Level $1280$
Weight $4$
Character 1280.1
Self dual yes
Analytic conductor $75.522$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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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] = [1280,4,Mod(1,1280)] 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("1280.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-6,0,30,0,-14,0,54,0,-44,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(75.5224448073\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 21x^{4} + 38x^{3} + 93x^{2} - 108x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.46129\) of defining polynomial
Character \(\chi\) \(=\) 1280.1

$q$-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)
 
\(f(q)\) \(=\) \(q-6.25785 q^{3} +5.00000 q^{5} +34.6280 q^{7} +12.1606 q^{9} +7.91595 q^{11} -11.0346 q^{13} -31.2892 q^{15} +57.6152 q^{17} -141.133 q^{19} -216.696 q^{21} -129.328 q^{23} +25.0000 q^{25} +92.8625 q^{27} -89.1664 q^{29} -78.3307 q^{31} -49.5368 q^{33} +173.140 q^{35} -249.332 q^{37} +69.0530 q^{39} -91.8705 q^{41} -194.184 q^{43} +60.8031 q^{45} -72.6149 q^{47} +856.096 q^{49} -360.547 q^{51} +456.782 q^{53} +39.5797 q^{55} +883.187 q^{57} -341.098 q^{59} -217.067 q^{61} +421.098 q^{63} -55.1731 q^{65} +529.237 q^{67} +809.314 q^{69} -381.540 q^{71} +876.902 q^{73} -156.446 q^{75} +274.113 q^{77} -203.950 q^{79} -909.456 q^{81} -996.654 q^{83} +288.076 q^{85} +557.989 q^{87} -172.456 q^{89} -382.107 q^{91} +490.182 q^{93} -705.664 q^{95} +1058.49 q^{97} +96.2629 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 30 q^{5} - 14 q^{7} + 54 q^{9} - 44 q^{11} - 30 q^{15} - 152 q^{19} + 4 q^{21} - 302 q^{23} + 150 q^{25} - 216 q^{27} - 132 q^{31} - 116 q^{33} - 70 q^{35} - 68 q^{37} - 300 q^{39} - 20 q^{41}+ \cdots - 5516 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −6.25785 −1.20432 −0.602161 − 0.798374i \(-0.705694\pi\)
−0.602161 + 0.798374i \(0.705694\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 34.6280 1.86973 0.934867 − 0.354998i \(-0.115518\pi\)
0.934867 + 0.354998i \(0.115518\pi\)
\(8\) 0 0
\(9\) 12.1606 0.450394
\(10\) 0 0
\(11\) 7.91595 0.216977 0.108489 − 0.994098i \(-0.465399\pi\)
0.108489 + 0.994098i \(0.465399\pi\)
\(12\) 0 0
\(13\) −11.0346 −0.235420 −0.117710 − 0.993048i \(-0.537555\pi\)
−0.117710 + 0.993048i \(0.537555\pi\)
\(14\) 0 0
\(15\) −31.2892 −0.538590
\(16\) 0 0
\(17\) 57.6152 0.821985 0.410992 − 0.911639i \(-0.365182\pi\)
0.410992 + 0.911639i \(0.365182\pi\)
\(18\) 0 0
\(19\) −141.133 −1.70411 −0.852055 − 0.523453i \(-0.824644\pi\)
−0.852055 + 0.523453i \(0.824644\pi\)
\(20\) 0 0
\(21\) −216.696 −2.25176
\(22\) 0 0
\(23\) −129.328 −1.17247 −0.586233 − 0.810143i \(-0.699390\pi\)
−0.586233 + 0.810143i \(0.699390\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 92.8625 0.661903
\(28\) 0 0
\(29\) −89.1664 −0.570958 −0.285479 − 0.958385i \(-0.592153\pi\)
−0.285479 + 0.958385i \(0.592153\pi\)
\(30\) 0 0
\(31\) −78.3307 −0.453826 −0.226913 − 0.973915i \(-0.572863\pi\)
−0.226913 + 0.973915i \(0.572863\pi\)
\(32\) 0 0
\(33\) −49.5368 −0.261310
\(34\) 0 0
\(35\) 173.140 0.836171
\(36\) 0 0
\(37\) −249.332 −1.10783 −0.553917 − 0.832572i \(-0.686868\pi\)
−0.553917 + 0.832572i \(0.686868\pi\)
\(38\) 0 0
\(39\) 69.0530 0.283521
\(40\) 0 0
\(41\) −91.8705 −0.349946 −0.174973 − 0.984573i \(-0.555984\pi\)
−0.174973 + 0.984573i \(0.555984\pi\)
\(42\) 0 0
\(43\) −194.184 −0.688668 −0.344334 − 0.938847i \(-0.611895\pi\)
−0.344334 + 0.938847i \(0.611895\pi\)
\(44\) 0 0
\(45\) 60.8031 0.201422
\(46\) 0 0
\(47\) −72.6149 −0.225361 −0.112680 − 0.993631i \(-0.535944\pi\)
−0.112680 + 0.993631i \(0.535944\pi\)
\(48\) 0 0
\(49\) 856.096 2.49591
\(50\) 0 0
\(51\) −360.547 −0.989935
\(52\) 0 0
\(53\) 456.782 1.18385 0.591923 − 0.805995i \(-0.298369\pi\)
0.591923 + 0.805995i \(0.298369\pi\)
\(54\) 0 0
\(55\) 39.5797 0.0970351
\(56\) 0 0
\(57\) 883.187 2.05230
\(58\) 0 0
\(59\) −341.098 −0.752664 −0.376332 − 0.926485i \(-0.622815\pi\)
−0.376332 + 0.926485i \(0.622815\pi\)
\(60\) 0 0
\(61\) −217.067 −0.455616 −0.227808 − 0.973706i \(-0.573156\pi\)
−0.227808 + 0.973706i \(0.573156\pi\)
\(62\) 0 0
\(63\) 421.098 0.842117
\(64\) 0 0
\(65\) −55.1731 −0.105283
\(66\) 0 0
\(67\) 529.237 0.965024 0.482512 − 0.875889i \(-0.339724\pi\)
0.482512 + 0.875889i \(0.339724\pi\)
\(68\) 0 0
\(69\) 809.314 1.41203
\(70\) 0 0
\(71\) −381.540 −0.637754 −0.318877 − 0.947796i \(-0.603306\pi\)
−0.318877 + 0.947796i \(0.603306\pi\)
\(72\) 0 0
\(73\) 876.902 1.40594 0.702970 − 0.711220i \(-0.251857\pi\)
0.702970 + 0.711220i \(0.251857\pi\)
\(74\) 0 0
\(75\) −156.446 −0.240865
\(76\) 0 0
\(77\) 274.113 0.405690
\(78\) 0 0
\(79\) −203.950 −0.290458 −0.145229 − 0.989398i \(-0.546392\pi\)
−0.145229 + 0.989398i \(0.546392\pi\)
\(80\) 0 0
\(81\) −909.456 −1.24754
\(82\) 0 0
\(83\) −996.654 −1.31804 −0.659018 − 0.752127i \(-0.729028\pi\)
−0.659018 + 0.752127i \(0.729028\pi\)
\(84\) 0 0
\(85\) 288.076 0.367603
\(86\) 0 0
\(87\) 557.989 0.687618
\(88\) 0 0
\(89\) −172.456 −0.205396 −0.102698 − 0.994713i \(-0.532748\pi\)
−0.102698 + 0.994713i \(0.532748\pi\)
\(90\) 0 0
\(91\) −382.107 −0.440172
\(92\) 0 0
\(93\) 490.182 0.546553
\(94\) 0 0
\(95\) −705.664 −0.762101
\(96\) 0 0
\(97\) 1058.49 1.10797 0.553984 − 0.832527i \(-0.313107\pi\)
0.553984 + 0.832527i \(0.313107\pi\)
\(98\) 0 0
\(99\) 96.2629 0.0977251
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 1280.4.a.bb.1.2 6
4.3 odd 2 1280.4.a.bd.1.5 6
8.3 odd 2 1280.4.a.ba.1.2 6
8.5 even 2 1280.4.a.bc.1.5 6
16.3 odd 4 160.4.d.a.81.10 12
16.5 even 4 40.4.d.a.21.2 yes 12
16.11 odd 4 160.4.d.a.81.3 12
16.13 even 4 40.4.d.a.21.1 ✓ 12
48.5 odd 4 360.4.k.c.181.11 12
48.11 even 4 1440.4.k.c.721.6 12
48.29 odd 4 360.4.k.c.181.12 12
48.35 even 4 1440.4.k.c.721.12 12
80.3 even 4 800.4.f.b.49.3 12
80.13 odd 4 200.4.f.c.149.4 12
80.19 odd 4 800.4.d.d.401.3 12
80.27 even 4 800.4.f.b.49.4 12
80.29 even 4 200.4.d.b.101.12 12
80.37 odd 4 200.4.f.c.149.3 12
80.43 even 4 800.4.f.c.49.9 12
80.53 odd 4 200.4.f.b.149.10 12
80.59 odd 4 800.4.d.d.401.10 12
80.67 even 4 800.4.f.c.49.10 12
80.69 even 4 200.4.d.b.101.11 12
80.77 odd 4 200.4.f.b.149.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.d.a.21.1 ✓ 12 16.13 even 4
40.4.d.a.21.2 yes 12 16.5 even 4
160.4.d.a.81.3 12 16.11 odd 4
160.4.d.a.81.10 12 16.3 odd 4
200.4.d.b.101.11 12 80.69 even 4
200.4.d.b.101.12 12 80.29 even 4
200.4.f.b.149.9 12 80.77 odd 4
200.4.f.b.149.10 12 80.53 odd 4
200.4.f.c.149.3 12 80.37 odd 4
200.4.f.c.149.4 12 80.13 odd 4
360.4.k.c.181.11 12 48.5 odd 4
360.4.k.c.181.12 12 48.29 odd 4
800.4.d.d.401.3 12 80.19 odd 4
800.4.d.d.401.10 12 80.59 odd 4
800.4.f.b.49.3 12 80.3 even 4
800.4.f.b.49.4 12 80.27 even 4
800.4.f.c.49.9 12 80.43 even 4
800.4.f.c.49.10 12 80.67 even 4
1280.4.a.ba.1.2 6 8.3 odd 2
1280.4.a.bb.1.2 6 1.1 even 1 trivial
1280.4.a.bc.1.5 6 8.5 even 2
1280.4.a.bd.1.5 6 4.3 odd 2
1440.4.k.c.721.6 12 48.11 even 4
1440.4.k.c.721.12 12 48.35 even 4