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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [480,4,Mod(431,480)] 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("480.431"); 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([1, 1, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 480.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,-120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.3209168028\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 431.8
Character \(\chi\) \(=\) 480.431
Dual form 480.4.b.a.431.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.31357 + 4.00253i) q^{3} -5.00000 q^{5} -30.8728i q^{7} +(-5.04045 - 26.5253i) q^{9} -45.8002i q^{11} +76.8279i q^{13} +(16.5679 - 20.0126i) q^{15} +13.8879i q^{17} -45.8869 q^{19} +(123.569 + 102.299i) q^{21} +74.3371 q^{23} +25.0000 q^{25} +(122.870 + 67.7191i) q^{27} -85.8680 q^{29} +227.728i q^{31} +(183.317 + 151.762i) q^{33} +154.364i q^{35} -5.58615i q^{37} +(-307.506 - 254.575i) q^{39} -61.0663i q^{41} -65.1565 q^{43} +(25.2023 + 132.627i) q^{45} -529.167 q^{47} -610.132 q^{49} +(-55.5867 - 46.0186i) q^{51} -22.7454 q^{53} +229.001i q^{55} +(152.050 - 183.664i) q^{57} -384.963i q^{59} +588.386i q^{61} +(-818.912 + 155.613i) q^{63} -384.139i q^{65} -369.398 q^{67} +(-246.321 + 297.536i) q^{69} -102.090 q^{71} +897.637 q^{73} +(-82.8394 + 100.063i) q^{75} -1413.98 q^{77} +1020.54i q^{79} +(-678.188 + 267.400i) q^{81} +588.935i q^{83} -69.4395i q^{85} +(284.530 - 343.689i) q^{87} +1260.99i q^{89} +2371.89 q^{91} +(-911.486 - 754.593i) q^{93} +229.435 q^{95} +936.527 q^{97} +(-1214.87 + 230.854i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 120 q^{5} - 12 q^{19} + 4 q^{21} + 228 q^{23} + 600 q^{25} - 132 q^{27} + 116 q^{33} - 656 q^{39} - 924 q^{47} - 816 q^{49} + 700 q^{51} - 528 q^{53} - 172 q^{57} + 476 q^{63} - 1632 q^{67} - 980 q^{69}+ \cdots + 1328 q^{99}+O(q^{100}) \) 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
\(3\) −3.31357 + 4.00253i −0.637698 + 0.770287i
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) − 30.8728i − 1.66698i −0.552537 − 0.833488i \(-0.686340\pi\)
0.552537 − 0.833488i \(-0.313660\pi\)
\(8\) 0 0
\(9\) −5.04045 − 26.5253i −0.186683 − 0.982420i
\(10\) 0 0
\(11\) − 45.8002i − 1.25539i −0.778459 − 0.627695i \(-0.783999\pi\)
0.778459 − 0.627695i \(-0.216001\pi\)
\(12\) 0 0
\(13\) 76.8279i 1.63909i 0.573012 + 0.819547i \(0.305775\pi\)
−0.573012 + 0.819547i \(0.694225\pi\)
\(14\) 0 0
\(15\) 16.5679 − 20.0126i 0.285187 − 0.344483i
\(16\) 0 0
\(17\) 13.8879i 0.198136i 0.995081 + 0.0990679i \(0.0315861\pi\)
−0.995081 + 0.0990679i \(0.968414\pi\)
\(18\) 0 0
\(19\) −45.8869 −0.554062 −0.277031 − 0.960861i \(-0.589350\pi\)
−0.277031 + 0.960861i \(0.589350\pi\)
\(20\) 0 0
\(21\) 123.569 + 102.299i 1.28405 + 1.06303i
\(22\) 0 0
\(23\) 74.3371 0.673928 0.336964 − 0.941517i \(-0.390600\pi\)
0.336964 + 0.941517i \(0.390600\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 122.870 + 67.7191i 0.875793 + 0.482687i
\(28\) 0 0
\(29\) −85.8680 −0.549838 −0.274919 − 0.961467i \(-0.588651\pi\)
−0.274919 + 0.961467i \(0.588651\pi\)
\(30\) 0 0
\(31\) 227.728i 1.31939i 0.751533 + 0.659695i \(0.229315\pi\)
−0.751533 + 0.659695i \(0.770685\pi\)
\(32\) 0 0
\(33\) 183.317 + 151.762i 0.967010 + 0.800559i
\(34\) 0 0
\(35\) 154.364i 0.745494i
\(36\) 0 0
\(37\) − 5.58615i − 0.0248205i −0.999923 − 0.0124102i \(-0.996050\pi\)
0.999923 − 0.0124102i \(-0.00395041\pi\)
\(38\) 0 0
\(39\) −307.506 − 254.575i −1.26257 − 1.04525i
\(40\) 0 0
\(41\) − 61.0663i − 0.232609i −0.993214 − 0.116304i \(-0.962895\pi\)
0.993214 − 0.116304i \(-0.0371048\pi\)
\(42\) 0 0
\(43\) −65.1565 −0.231076 −0.115538 − 0.993303i \(-0.536859\pi\)
−0.115538 + 0.993303i \(0.536859\pi\)
\(44\) 0 0
\(45\) 25.2023 + 132.627i 0.0834874 + 0.439352i
\(46\) 0 0
\(47\) −529.167 −1.64228 −0.821138 − 0.570730i \(-0.806660\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(48\) 0 0
\(49\) −610.132 −1.77881
\(50\) 0 0
\(51\) −55.5867 − 46.0186i −0.152621 − 0.126351i
\(52\) 0 0
\(53\) −22.7454 −0.0589496 −0.0294748 − 0.999566i \(-0.509383\pi\)
−0.0294748 + 0.999566i \(0.509383\pi\)
\(54\) 0 0
\(55\) 229.001i 0.561427i
\(56\) 0 0
\(57\) 152.050 − 183.664i 0.353324 − 0.426787i
\(58\) 0 0
\(59\) − 384.963i − 0.849456i −0.905321 − 0.424728i \(-0.860370\pi\)
0.905321 − 0.424728i \(-0.139630\pi\)
\(60\) 0 0
\(61\) 588.386i 1.23500i 0.786571 + 0.617500i \(0.211855\pi\)
−0.786571 + 0.617500i \(0.788145\pi\)
\(62\) 0 0
\(63\) −818.912 + 155.613i −1.63767 + 0.311197i
\(64\) 0 0
\(65\) − 384.139i − 0.733025i
\(66\) 0 0
\(67\) −369.398 −0.673570 −0.336785 − 0.941582i \(-0.609340\pi\)
−0.336785 + 0.941582i \(0.609340\pi\)
\(68\) 0 0
\(69\) −246.321 + 297.536i −0.429762 + 0.519118i
\(70\) 0 0
\(71\) −102.090 −0.170646 −0.0853229 − 0.996353i \(-0.527192\pi\)
−0.0853229 + 0.996353i \(0.527192\pi\)
\(72\) 0 0
\(73\) 897.637 1.43918 0.719592 − 0.694397i \(-0.244329\pi\)
0.719592 + 0.694397i \(0.244329\pi\)
\(74\) 0 0
\(75\) −82.8394 + 100.063i −0.127540 + 0.154057i
\(76\) 0 0
\(77\) −1413.98 −2.09271
\(78\) 0 0
\(79\) 1020.54i 1.45341i 0.686951 + 0.726704i \(0.258949\pi\)
−0.686951 + 0.726704i \(0.741051\pi\)
\(80\) 0 0
\(81\) −678.188 + 267.400i −0.930299 + 0.366803i
\(82\) 0 0
\(83\) 588.935i 0.778844i 0.921059 + 0.389422i \(0.127325\pi\)
−0.921059 + 0.389422i \(0.872675\pi\)
\(84\) 0 0
\(85\) − 69.4395i − 0.0886091i
\(86\) 0 0
\(87\) 284.530 − 343.689i 0.350630 − 0.423533i
\(88\) 0 0
\(89\) 1260.99i 1.50185i 0.660387 + 0.750925i \(0.270392\pi\)
−0.660387 + 0.750925i \(0.729608\pi\)
\(90\) 0 0
\(91\) 2371.89 2.73233
\(92\) 0 0
\(93\) −911.486 − 754.593i −1.01631 − 0.841372i
\(94\) 0 0
\(95\) 229.435 0.247784
\(96\) 0 0
\(97\) 936.527 0.980308 0.490154 − 0.871636i \(-0.336941\pi\)
0.490154 + 0.871636i \(0.336941\pi\)
\(98\) 0 0
\(99\) −1214.87 + 230.854i −1.23332 + 0.234361i
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.b.a.431.8 24
3.2 odd 2 480.4.b.b.431.7 24
4.3 odd 2 120.4.b.b.11.12 yes 24
8.3 odd 2 480.4.b.b.431.8 24
8.5 even 2 120.4.b.a.11.14 yes 24
12.11 even 2 120.4.b.a.11.13 ✓ 24
24.5 odd 2 120.4.b.b.11.11 yes 24
24.11 even 2 inner 480.4.b.a.431.7 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.b.a.11.13 ✓ 24 12.11 even 2
120.4.b.a.11.14 yes 24 8.5 even 2
120.4.b.b.11.11 yes 24 24.5 odd 2
120.4.b.b.11.12 yes 24 4.3 odd 2
480.4.b.a.431.7 24 24.11 even 2 inner
480.4.b.a.431.8 24 1.1 even 1 trivial
480.4.b.b.431.7 24 3.2 odd 2
480.4.b.b.431.8 24 8.3 odd 2