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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [60,8,Mod(49,60)] 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("60.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(60, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 60.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] 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: \(18.7431015290\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 60.49
Dual form 60.8.d.a.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q-27.0000i q^{3} +(250.000 + 125.000i) q^{5} +722.000i q^{7} -729.000 q^{9} -3994.00 q^{11} +3030.00i q^{13} +(3375.00 - 6750.00i) q^{15} +20582.0i q^{17} +25320.0 q^{19} +19494.0 q^{21} +66652.0i q^{23} +(46875.0 + 62500.0i) q^{25} +19683.0i q^{27} +152664. q^{29} -123776. q^{31} +107838. i q^{33} +(-90250.0 + 180500. i) q^{35} +337886. i q^{37} +81810.0 q^{39} +396530. q^{41} +442852. i q^{43} +(-182250. - 91125.0i) q^{45} -170432. i q^{47} +302259. q^{49} +555714. q^{51} -1.23943e6i q^{53} +(-998500. - 499250. i) q^{55} -683640. i q^{57} +302354. q^{59} -2.83020e6 q^{61} -526338. i q^{63} +(-378750. + 757500. i) q^{65} -3.74127e6i q^{67} +1.79960e6 q^{69} -1.00758e6 q^{71} +2.40464e6i q^{73} +(1.68750e6 - 1.26562e6i) q^{75} -2.88367e6i q^{77} -7.51783e6 q^{79} +531441. q^{81} -5.29963e6i q^{83} +(-2.57275e6 + 5.14550e6i) q^{85} -4.12193e6i q^{87} +7.65025e6 q^{89} -2.18766e6 q^{91} +3.34195e6i q^{93} +(6.33000e6 + 3.16500e6i) q^{95} +1.00559e7i q^{97} +2.91163e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 500 q^{5} - 1458 q^{9} - 7988 q^{11} + 6750 q^{15} + 50640 q^{19} + 38988 q^{21} + 93750 q^{25} + 305328 q^{29} - 247552 q^{31} - 180500 q^{35} + 163620 q^{39} + 793060 q^{41} - 364500 q^{45} + 604518 q^{49}+ \cdots + 5823252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 27.0000i − 0.577350i
\(4\) 0 0
\(5\) 250.000 + 125.000i 0.894427 + 0.447214i
\(6\) 0 0
\(7\) 722.000i 0.795599i 0.917472 + 0.397799i \(0.130226\pi\)
−0.917472 + 0.397799i \(0.869774\pi\)
\(8\) 0 0
\(9\) −729.000 −0.333333
\(10\) 0 0
\(11\) −3994.00 −0.904761 −0.452380 − 0.891825i \(-0.649425\pi\)
−0.452380 + 0.891825i \(0.649425\pi\)
\(12\) 0 0
\(13\) 3030.00i 0.382508i 0.981541 + 0.191254i \(0.0612555\pi\)
−0.981541 + 0.191254i \(0.938745\pi\)
\(14\) 0 0
\(15\) 3375.00 − 6750.00i 0.258199 − 0.516398i
\(16\) 0 0
\(17\) 20582.0i 1.01605i 0.861341 + 0.508026i \(0.169625\pi\)
−0.861341 + 0.508026i \(0.830375\pi\)
\(18\) 0 0
\(19\) 25320.0 0.846888 0.423444 − 0.905922i \(-0.360821\pi\)
0.423444 + 0.905922i \(0.360821\pi\)
\(20\) 0 0
\(21\) 19494.0 0.459339
\(22\) 0 0
\(23\) 66652.0i 1.14226i 0.820859 + 0.571131i \(0.193495\pi\)
−0.820859 + 0.571131i \(0.806505\pi\)
\(24\) 0 0
\(25\) 46875.0 + 62500.0i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 19683.0i 0.192450i
\(28\) 0 0
\(29\) 152664. 1.16237 0.581184 − 0.813772i \(-0.302590\pi\)
0.581184 + 0.813772i \(0.302590\pi\)
\(30\) 0 0
\(31\) −123776. −0.746226 −0.373113 − 0.927786i \(-0.621710\pi\)
−0.373113 + 0.927786i \(0.621710\pi\)
\(32\) 0 0
\(33\) 107838.i 0.522364i
\(34\) 0 0
\(35\) −90250.0 + 180500.i −0.355803 + 0.711605i
\(36\) 0 0
\(37\) 337886.i 1.09664i 0.836269 + 0.548320i \(0.184732\pi\)
−0.836269 + 0.548320i \(0.815268\pi\)
\(38\) 0 0
\(39\) 81810.0 0.220841
\(40\) 0 0
\(41\) 396530. 0.898530 0.449265 − 0.893399i \(-0.351686\pi\)
0.449265 + 0.893399i \(0.351686\pi\)
\(42\) 0 0
\(43\) 442852.i 0.849413i 0.905331 + 0.424707i \(0.139623\pi\)
−0.905331 + 0.424707i \(0.860377\pi\)
\(44\) 0 0
\(45\) −182250. − 91125.0i −0.298142 − 0.149071i
\(46\) 0 0
\(47\) − 170432.i − 0.239447i −0.992807 − 0.119723i \(-0.961799\pi\)
0.992807 − 0.119723i \(-0.0382007\pi\)
\(48\) 0 0
\(49\) 302259. 0.367023
\(50\) 0 0
\(51\) 555714. 0.586618
\(52\) 0 0
\(53\) − 1.23943e6i − 1.14355i −0.820411 − 0.571775i \(-0.806255\pi\)
0.820411 − 0.571775i \(-0.193745\pi\)
\(54\) 0 0
\(55\) −998500. − 499250.i −0.809242 − 0.404621i
\(56\) 0 0
\(57\) − 683640.i − 0.488951i
\(58\) 0 0
\(59\) 302354. 0.191661 0.0958305 − 0.995398i \(-0.469449\pi\)
0.0958305 + 0.995398i \(0.469449\pi\)
\(60\) 0 0
\(61\) −2.83020e6 −1.59648 −0.798238 − 0.602342i \(-0.794234\pi\)
−0.798238 + 0.602342i \(0.794234\pi\)
\(62\) 0 0
\(63\) − 526338.i − 0.265200i
\(64\) 0 0
\(65\) −378750. + 757500.i −0.171063 + 0.342126i
\(66\) 0 0
\(67\) − 3.74127e6i − 1.51970i −0.650099 − 0.759849i \(-0.725273\pi\)
0.650099 − 0.759849i \(-0.274727\pi\)
\(68\) 0 0
\(69\) 1.79960e6 0.659485
\(70\) 0 0
\(71\) −1.00758e6 −0.334099 −0.167050 − 0.985949i \(-0.553424\pi\)
−0.167050 + 0.985949i \(0.553424\pi\)
\(72\) 0 0
\(73\) 2.40464e6i 0.723468i 0.932281 + 0.361734i \(0.117815\pi\)
−0.932281 + 0.361734i \(0.882185\pi\)
\(74\) 0 0
\(75\) 1.68750e6 − 1.26562e6i 0.461880 − 0.346410i
\(76\) 0 0
\(77\) − 2.88367e6i − 0.719826i
\(78\) 0 0
\(79\) −7.51783e6 −1.71553 −0.857764 − 0.514044i \(-0.828147\pi\)
−0.857764 + 0.514044i \(0.828147\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) − 5.29963e6i − 1.01735i −0.860957 − 0.508677i \(-0.830135\pi\)
0.860957 − 0.508677i \(-0.169865\pi\)
\(84\) 0 0
\(85\) −2.57275e6 + 5.14550e6i −0.454393 + 0.908785i
\(86\) 0 0
\(87\) − 4.12193e6i − 0.671093i
\(88\) 0 0
\(89\) 7.65025e6 1.15030 0.575149 − 0.818048i \(-0.304944\pi\)
0.575149 + 0.818048i \(0.304944\pi\)
\(90\) 0 0
\(91\) −2.18766e6 −0.304323
\(92\) 0 0
\(93\) 3.34195e6i 0.430834i
\(94\) 0 0
\(95\) 6.33000e6 + 3.16500e6i 0.757480 + 0.378740i
\(96\) 0 0
\(97\) 1.00559e7i 1.11872i 0.828925 + 0.559360i \(0.188953\pi\)
−0.828925 + 0.559360i \(0.811047\pi\)
\(98\) 0 0
\(99\) 2.91163e6 0.301587
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 60.8.d.a.49.1 ✓ 2
3.2 odd 2 180.8.d.a.109.1 2
4.3 odd 2 240.8.f.b.49.2 2
5.2 odd 4 300.8.a.b.1.1 1
5.3 odd 4 300.8.a.f.1.1 1
5.4 even 2 inner 60.8.d.a.49.2 yes 2
15.14 odd 2 180.8.d.a.109.2 2
20.19 odd 2 240.8.f.b.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.8.d.a.49.1 ✓ 2 1.1 even 1 trivial
60.8.d.a.49.2 yes 2 5.4 even 2 inner
180.8.d.a.109.1 2 3.2 odd 2
180.8.d.a.109.2 2 15.14 odd 2
240.8.f.b.49.1 2 20.19 odd 2
240.8.f.b.49.2 2 4.3 odd 2
300.8.a.b.1.1 1 5.2 odd 4
300.8.a.f.1.1 1 5.3 odd 4