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Properties

Label 480.4.k.b.241.2
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 241.2
Root \(1.61974 + 1.17321i\) of defining polynomial
Character \(\chi\) \(=\) 480.241
Dual form 480.4.k.b.241.6

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} +5.00000i q^{5} +3.73490 q^{7} -9.00000 q^{9} -39.1696i q^{11} +50.0873i q^{13} +15.0000 q^{15} +6.84324 q^{17} -91.4369i q^{19} -11.2047i q^{21} +32.1332 q^{23} -25.0000 q^{25} +27.0000i q^{27} -199.984i q^{29} +68.7861 q^{31} -117.509 q^{33} +18.6745i q^{35} +31.0616i q^{37} +150.262 q^{39} +129.004 q^{41} -455.610i q^{43} -45.0000i q^{45} +253.592 q^{47} -329.051 q^{49} -20.5297i q^{51} -269.270i q^{53} +195.848 q^{55} -274.311 q^{57} +282.249i q^{59} -639.547i q^{61} -33.6141 q^{63} -250.437 q^{65} -830.362i q^{67} -96.3997i q^{69} -659.040 q^{71} -486.232 q^{73} +75.0000i q^{75} -146.294i q^{77} -1161.63 q^{79} +81.0000 q^{81} -698.131i q^{83} +34.2162i q^{85} -599.951 q^{87} +1243.30 q^{89} +187.071i q^{91} -206.358i q^{93} +457.184 q^{95} +1167.81 q^{97} +352.526i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 80 q^{7} - 72 q^{9} + 120 q^{15} - 216 q^{17} + 32 q^{23} - 200 q^{25} + 136 q^{31} - 24 q^{33} - 72 q^{39} + 176 q^{41} + 848 q^{47} - 1320 q^{49} + 40 q^{55} + 192 q^{57} - 720 q^{63} + 120 q^{65}+ \cdots - 4256 q^{97}+O(q^{100}) \) 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
\(7\) 3.73490 0.201666 0.100833 − 0.994903i \(-0.467849\pi\)
0.100833 + 0.994903i \(0.467849\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) − 39.1696i − 1.07364i −0.843696 − 0.536821i \(-0.819625\pi\)
0.843696 − 0.536821i \(-0.180375\pi\)
\(12\) 0 0
\(13\) 50.0873i 1.06859i 0.845297 + 0.534297i \(0.179424\pi\)
−0.845297 + 0.534297i \(0.820576\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) 6.84324 0.0976312 0.0488156 − 0.998808i \(-0.484455\pi\)
0.0488156 + 0.998808i \(0.484455\pi\)
\(18\) 0 0
\(19\) − 91.4369i − 1.10406i −0.833826 − 0.552028i \(-0.813854\pi\)
0.833826 − 0.552028i \(-0.186146\pi\)
\(20\) 0 0
\(21\) − 11.2047i − 0.116432i
\(22\) 0 0
\(23\) 32.1332 0.291315 0.145657 − 0.989335i \(-0.453470\pi\)
0.145657 + 0.989335i \(0.453470\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) − 199.984i − 1.28055i −0.768145 − 0.640276i \(-0.778820\pi\)
0.768145 − 0.640276i \(-0.221180\pi\)
\(30\) 0 0
\(31\) 68.7861 0.398527 0.199264 − 0.979946i \(-0.436145\pi\)
0.199264 + 0.979946i \(0.436145\pi\)
\(32\) 0 0
\(33\) −117.509 −0.619868
\(34\) 0 0
\(35\) 18.6745i 0.0901877i
\(36\) 0 0
\(37\) 31.0616i 0.138013i 0.997616 + 0.0690066i \(0.0219830\pi\)
−0.997616 + 0.0690066i \(0.978017\pi\)
\(38\) 0 0
\(39\) 150.262 0.616953
\(40\) 0 0
\(41\) 129.004 0.491390 0.245695 − 0.969347i \(-0.420984\pi\)
0.245695 + 0.969347i \(0.420984\pi\)
\(42\) 0 0
\(43\) − 455.610i − 1.61581i −0.589312 − 0.807905i \(-0.700601\pi\)
0.589312 − 0.807905i \(-0.299399\pi\)
\(44\) 0 0
\(45\) − 45.0000i − 0.149071i
\(46\) 0 0
\(47\) 253.592 0.787026 0.393513 − 0.919319i \(-0.371260\pi\)
0.393513 + 0.919319i \(0.371260\pi\)
\(48\) 0 0
\(49\) −329.051 −0.959331
\(50\) 0 0
\(51\) − 20.5297i − 0.0563674i
\(52\) 0 0
\(53\) − 269.270i − 0.697870i −0.937147 − 0.348935i \(-0.886543\pi\)
0.937147 − 0.348935i \(-0.113457\pi\)
\(54\) 0 0
\(55\) 195.848 0.480148
\(56\) 0 0
\(57\) −274.311 −0.637427
\(58\) 0 0
\(59\) 282.249i 0.622807i 0.950278 + 0.311404i \(0.100799\pi\)
−0.950278 + 0.311404i \(0.899201\pi\)
\(60\) 0 0
\(61\) − 639.547i − 1.34239i −0.741282 − 0.671193i \(-0.765782\pi\)
0.741282 − 0.671193i \(-0.234218\pi\)
\(62\) 0 0
\(63\) −33.6141 −0.0672219
\(64\) 0 0
\(65\) −250.437 −0.477890
\(66\) 0 0
\(67\) − 830.362i − 1.51410i −0.653356 − 0.757051i \(-0.726639\pi\)
0.653356 − 0.757051i \(-0.273361\pi\)
\(68\) 0 0
\(69\) − 96.3997i − 0.168191i
\(70\) 0 0
\(71\) −659.040 −1.10160 −0.550801 − 0.834637i \(-0.685678\pi\)
−0.550801 + 0.834637i \(0.685678\pi\)
\(72\) 0 0
\(73\) −486.232 −0.779577 −0.389788 − 0.920904i \(-0.627452\pi\)
−0.389788 + 0.920904i \(0.627452\pi\)
\(74\) 0 0
\(75\) 75.0000i 0.115470i
\(76\) 0 0
\(77\) − 146.294i − 0.216517i
\(78\) 0 0
\(79\) −1161.63 −1.65435 −0.827174 − 0.561946i \(-0.810053\pi\)
−0.827174 + 0.561946i \(0.810053\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 698.131i − 0.923251i −0.887075 − 0.461626i \(-0.847266\pi\)
0.887075 − 0.461626i \(-0.152734\pi\)
\(84\) 0 0
\(85\) 34.2162i 0.0436620i
\(86\) 0 0
\(87\) −599.951 −0.739327
\(88\) 0 0
\(89\) 1243.30 1.48078 0.740390 − 0.672177i \(-0.234641\pi\)
0.740390 + 0.672177i \(0.234641\pi\)
\(90\) 0 0
\(91\) 187.071i 0.215499i
\(92\) 0 0
\(93\) − 206.358i − 0.230090i
\(94\) 0 0
\(95\) 457.184 0.493749
\(96\) 0 0
\(97\) 1167.81 1.22240 0.611201 − 0.791475i \(-0.290687\pi\)
0.611201 + 0.791475i \(0.290687\pi\)
\(98\) 0 0
\(99\) 352.526i 0.357881i
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.241.2 8
3.2 odd 2 1440.4.k.b.721.2 8
4.3 odd 2 120.4.k.b.61.5 ✓ 8
8.3 odd 2 120.4.k.b.61.6 yes 8
8.5 even 2 inner 480.4.k.b.241.6 8
12.11 even 2 360.4.k.b.181.4 8
24.5 odd 2 1440.4.k.b.721.6 8
24.11 even 2 360.4.k.b.181.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.k.b.61.5 ✓ 8 4.3 odd 2
120.4.k.b.61.6 yes 8 8.3 odd 2
360.4.k.b.181.3 8 24.11 even 2
360.4.k.b.181.4 8 12.11 even 2
480.4.k.b.241.2 8 1.1 even 1 trivial
480.4.k.b.241.6 8 8.5 even 2 inner
1440.4.k.b.721.2 8 3.2 odd 2
1440.4.k.b.721.6 8 24.5 odd 2