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Properties

Label 480.4.k.b.241.5
Level $480$
Weight $4$
Character 480.241
Analytic conductor $28.321$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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] = [480,4,Mod(241,480)] 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("480.241"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 480.k (of order \(2\), degree \(1\), not 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: \(28.3209168028\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.55839580416.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 7x^{6} - 6x^{5} + 18x^{4} - 24x^{3} + 112x^{2} - 128x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 241.5
Root \(0.694547 + 1.87553i\) of defining polynomial
Character \(\chi\) \(=\) 480.241
Dual form 480.4.k.b.241.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} -5.00000i q^{5} -1.05849 q^{7} -9.00000 q^{9} +12.0304i q^{11} +1.58299i q^{13} +15.0000 q^{15} -5.63366 q^{17} -26.2836i q^{19} -3.17546i q^{21} -104.197 q^{23} -25.0000 q^{25} -27.0000i q^{27} +32.7778i q^{29} -250.045 q^{31} -36.0912 q^{33} +5.29243i q^{35} +235.185i q^{37} -4.74897 q^{39} +138.027 q^{41} -355.886i q^{43} +45.0000i q^{45} -462.599 q^{47} -341.880 q^{49} -16.9010i q^{51} +5.63429i q^{53} +60.1520 q^{55} +78.8508 q^{57} -46.7356i q^{59} -340.563i q^{61} +9.52638 q^{63} +7.91495 q^{65} -790.372i q^{67} -312.590i q^{69} -971.879 q^{71} -1026.48 q^{73} -75.0000i q^{75} -12.7340i q^{77} +693.490 q^{79} +81.0000 q^{81} -117.875i q^{83} +28.1683i q^{85} -98.3335 q^{87} +106.132 q^{89} -1.67557i q^{91} -750.136i q^{93} -131.418 q^{95} -1266.82 q^{97} -108.274i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 80 q^{7} - 72 q^{9} + 120 q^{15} - 216 q^{17} + 32 q^{23} - 200 q^{25} + 136 q^{31} - 24 q^{33} - 72 q^{39} + 176 q^{41} + 848 q^{47} - 1320 q^{49} + 40 q^{55} + 192 q^{57} - 720 q^{63} + 120 q^{65}+ \cdots - 4256 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(421\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) − 5.00000i − 0.447214i
\(6\) 0 0
\(7\) −1.05849 −0.0571529 −0.0285764 − 0.999592i \(-0.509097\pi\)
−0.0285764 + 0.999592i \(0.509097\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 12.0304i 0.329755i 0.986314 + 0.164877i \(0.0527228\pi\)
−0.986314 + 0.164877i \(0.947277\pi\)
\(12\) 0 0
\(13\) 1.58299i 0.0337725i 0.999857 + 0.0168863i \(0.00537532\pi\)
−0.999857 + 0.0168863i \(0.994625\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) −5.63366 −0.0803744 −0.0401872 − 0.999192i \(-0.512795\pi\)
−0.0401872 + 0.999192i \(0.512795\pi\)
\(18\) 0 0
\(19\) − 26.2836i − 0.317361i −0.987330 − 0.158681i \(-0.949276\pi\)
0.987330 − 0.158681i \(-0.0507241\pi\)
\(20\) 0 0
\(21\) − 3.17546i − 0.0329972i
\(22\) 0 0
\(23\) −104.197 −0.944630 −0.472315 − 0.881430i \(-0.656581\pi\)
−0.472315 + 0.881430i \(0.656581\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) 32.7778i 0.209886i 0.994478 + 0.104943i \(0.0334660\pi\)
−0.994478 + 0.104943i \(0.966534\pi\)
\(30\) 0 0
\(31\) −250.045 −1.44869 −0.724346 − 0.689436i \(-0.757858\pi\)
−0.724346 + 0.689436i \(0.757858\pi\)
\(32\) 0 0
\(33\) −36.0912 −0.190384
\(34\) 0 0
\(35\) 5.29243i 0.0255595i
\(36\) 0 0
\(37\) 235.185i 1.04498i 0.852646 + 0.522488i \(0.174996\pi\)
−0.852646 + 0.522488i \(0.825004\pi\)
\(38\) 0 0
\(39\) −4.74897 −0.0194986
\(40\) 0 0
\(41\) 138.027 0.525763 0.262881 − 0.964828i \(-0.415327\pi\)
0.262881 + 0.964828i \(0.415327\pi\)
\(42\) 0 0
\(43\) − 355.886i − 1.26214i −0.775726 − 0.631070i \(-0.782616\pi\)
0.775726 − 0.631070i \(-0.217384\pi\)
\(44\) 0 0
\(45\) 45.0000i 0.149071i
\(46\) 0 0
\(47\) −462.599 −1.43568 −0.717840 − 0.696208i \(-0.754869\pi\)
−0.717840 + 0.696208i \(0.754869\pi\)
\(48\) 0 0
\(49\) −341.880 −0.996734
\(50\) 0 0
\(51\) − 16.9010i − 0.0464042i
\(52\) 0 0
\(53\) 5.63429i 0.0146024i 0.999973 + 0.00730122i \(0.00232407\pi\)
−0.999973 + 0.00730122i \(0.997676\pi\)
\(54\) 0 0
\(55\) 60.1520 0.147471
\(56\) 0 0
\(57\) 78.8508 0.183229
\(58\) 0 0
\(59\) − 46.7356i − 0.103126i −0.998670 − 0.0515632i \(-0.983580\pi\)
0.998670 − 0.0515632i \(-0.0164203\pi\)
\(60\) 0 0
\(61\) − 340.563i − 0.714830i −0.933946 − 0.357415i \(-0.883658\pi\)
0.933946 − 0.357415i \(-0.116342\pi\)
\(62\) 0 0
\(63\) 9.52638 0.0190510
\(64\) 0 0
\(65\) 7.91495 0.0151035
\(66\) 0 0
\(67\) − 790.372i − 1.44118i −0.693360 − 0.720592i \(-0.743870\pi\)
0.693360 − 0.720592i \(-0.256130\pi\)
\(68\) 0 0
\(69\) − 312.590i − 0.545382i
\(70\) 0 0
\(71\) −971.879 −1.62452 −0.812259 − 0.583296i \(-0.801763\pi\)
−0.812259 + 0.583296i \(0.801763\pi\)
\(72\) 0 0
\(73\) −1026.48 −1.64575 −0.822876 − 0.568220i \(-0.807632\pi\)
−0.822876 + 0.568220i \(0.807632\pi\)
\(74\) 0 0
\(75\) − 75.0000i − 0.115470i
\(76\) 0 0
\(77\) − 12.7340i − 0.0188464i
\(78\) 0 0
\(79\) 693.490 0.987642 0.493821 − 0.869563i \(-0.335600\pi\)
0.493821 + 0.869563i \(0.335600\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 117.875i − 0.155885i −0.996958 − 0.0779423i \(-0.975165\pi\)
0.996958 − 0.0779423i \(-0.0248350\pi\)
\(84\) 0 0
\(85\) 28.1683i 0.0359445i
\(86\) 0 0
\(87\) −98.3335 −0.121178
\(88\) 0 0
\(89\) 106.132 0.126404 0.0632021 − 0.998001i \(-0.479869\pi\)
0.0632021 + 0.998001i \(0.479869\pi\)
\(90\) 0 0
\(91\) − 1.67557i − 0.00193020i
\(92\) 0 0
\(93\) − 750.136i − 0.836403i
\(94\) 0 0
\(95\) −131.418 −0.141928
\(96\) 0 0
\(97\) −1266.82 −1.32604 −0.663021 − 0.748601i \(-0.730726\pi\)
−0.663021 + 0.748601i \(0.730726\pi\)
\(98\) 0 0
\(99\) − 108.274i − 0.109918i
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 480.4.k.b.241.5 8
3.2 odd 2 1440.4.k.b.721.5 8
4.3 odd 2 120.4.k.b.61.7 ✓ 8
8.3 odd 2 120.4.k.b.61.8 yes 8
8.5 even 2 inner 480.4.k.b.241.1 8
12.11 even 2 360.4.k.b.181.2 8
24.5 odd 2 1440.4.k.b.721.1 8
24.11 even 2 360.4.k.b.181.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.k.b.61.7 ✓ 8 4.3 odd 2
120.4.k.b.61.8 yes 8 8.3 odd 2
360.4.k.b.181.1 8 24.11 even 2
360.4.k.b.181.2 8 12.11 even 2
480.4.k.b.241.1 8 8.5 even 2 inner
480.4.k.b.241.5 8 1.1 even 1 trivial
1440.4.k.b.721.1 8 24.5 odd 2
1440.4.k.b.721.5 8 3.2 odd 2