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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [180,8,Mod(109,180)] 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("180.109"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 180.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-500] 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: \(56.2293045871\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 109.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 180.109
Dual form 180.8.d.a.109.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-250.000 - 125.000i) q^{5} +722.000i q^{7} +3994.00 q^{11} +3030.00i q^{13} -20582.0i q^{17} +25320.0 q^{19} -66652.0i q^{23} +(46875.0 + 62500.0i) q^{25} -152664. q^{29} -123776. q^{31} +(90250.0 - 180500. i) q^{35} +337886. i q^{37} -396530. q^{41} +442852. i q^{43} +170432. i q^{47} +302259. q^{49} +1.23943e6i q^{53} +(-998500. - 499250. i) q^{55} -302354. q^{59} -2.83020e6 q^{61} +(378750. - 757500. i) q^{65} -3.74127e6i q^{67} +1.00758e6 q^{71} +2.40464e6i q^{73} +2.88367e6i q^{77} -7.51783e6 q^{79} +5.29963e6i q^{83} +(-2.57275e6 + 5.14550e6i) q^{85} -7.65025e6 q^{89} -2.18766e6 q^{91} +(-6.33000e6 - 3.16500e6i) q^{95} +1.00559e7i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 500 q^{5} + 7988 q^{11} + 50640 q^{19} + 93750 q^{25} - 305328 q^{29} - 247552 q^{31} + 180500 q^{35} - 793060 q^{41} + 604518 q^{49} - 1997000 q^{55} - 604708 q^{59} - 5660396 q^{61} + 757500 q^{65}+ \cdots - 12660000 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 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
\(9\) 0 0
\(10\) 0 0
\(11\) 3994.00 0.904761 0.452380 − 0.891825i \(-0.350575\pi\)
0.452380 + 0.891825i \(0.350575\pi\)
\(12\) 0 0
\(13\) 3030.00i 0.382508i 0.981541 + 0.191254i \(0.0612555\pi\)
−0.981541 + 0.191254i \(0.938745\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 20582.0i − 1.01605i −0.861341 − 0.508026i \(-0.830375\pi\)
0.861341 − 0.508026i \(-0.169625\pi\)
\(18\) 0 0
\(19\) 25320.0 0.846888 0.423444 − 0.905922i \(-0.360821\pi\)
0.423444 + 0.905922i \(0.360821\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 66652.0i − 1.14226i −0.820859 − 0.571131i \(-0.806505\pi\)
0.820859 − 0.571131i \(-0.193495\pi\)
\(24\) 0 0
\(25\) 46875.0 + 62500.0i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −152664. −1.16237 −0.581184 − 0.813772i \(-0.697410\pi\)
−0.581184 + 0.813772i \(0.697410\pi\)
\(30\) 0 0
\(31\) −123776. −0.746226 −0.373113 − 0.927786i \(-0.621710\pi\)
−0.373113 + 0.927786i \(0.621710\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 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
\(39\) 0 0
\(40\) 0 0
\(41\) −396530. −0.898530 −0.449265 − 0.893399i \(-0.648314\pi\)
−0.449265 + 0.893399i \(0.648314\pi\)
\(42\) 0 0
\(43\) 442852.i 0.849413i 0.905331 + 0.424707i \(0.139623\pi\)
−0.905331 + 0.424707i \(0.860377\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 170432.i 0.239447i 0.992807 + 0.119723i \(0.0382007\pi\)
−0.992807 + 0.119723i \(0.961799\pi\)
\(48\) 0 0
\(49\) 302259. 0.367023
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.23943e6i 1.14355i 0.820411 + 0.571775i \(0.193745\pi\)
−0.820411 + 0.571775i \(0.806255\pi\)
\(54\) 0 0
\(55\) −998500. − 499250.i −0.809242 − 0.404621i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −302354. −0.191661 −0.0958305 − 0.995398i \(-0.530551\pi\)
−0.0958305 + 0.995398i \(0.530551\pi\)
\(60\) 0 0
\(61\) −2.83020e6 −1.59648 −0.798238 − 0.602342i \(-0.794234\pi\)
−0.798238 + 0.602342i \(0.794234\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 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
\(69\) 0 0
\(70\) 0 0
\(71\) 1.00758e6 0.334099 0.167050 − 0.985949i \(-0.446576\pi\)
0.167050 + 0.985949i \(0.446576\pi\)
\(72\) 0 0
\(73\) 2.40464e6i 0.723468i 0.932281 + 0.361734i \(0.117815\pi\)
−0.932281 + 0.361734i \(0.882185\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 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
\(81\) 0 0
\(82\) 0 0
\(83\) 5.29963e6i 1.01735i 0.860957 + 0.508677i \(0.169865\pi\)
−0.860957 + 0.508677i \(0.830135\pi\)
\(84\) 0 0
\(85\) −2.57275e6 + 5.14550e6i −0.454393 + 0.908785i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.65025e6 −1.15030 −0.575149 − 0.818048i \(-0.695056\pi\)
−0.575149 + 0.818048i \(0.695056\pi\)
\(90\) 0 0
\(91\) −2.18766e6 −0.304323
\(92\) 0 0
\(93\) 0 0
\(94\) 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
\(99\) 0 0
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 180.8.d.a.109.1 2
3.2 odd 2 60.8.d.a.49.1 ✓ 2
5.4 even 2 inner 180.8.d.a.109.2 2
12.11 even 2 240.8.f.b.49.2 2
15.2 even 4 300.8.a.b.1.1 1
15.8 even 4 300.8.a.f.1.1 1
15.14 odd 2 60.8.d.a.49.2 yes 2
60.59 even 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 3.2 odd 2
60.8.d.a.49.2 yes 2 15.14 odd 2
180.8.d.a.109.1 2 1.1 even 1 trivial
180.8.d.a.109.2 2 5.4 even 2 inner
240.8.f.b.49.1 2 60.59 even 2
240.8.f.b.49.2 2 12.11 even 2
300.8.a.b.1.1 1 15.2 even 4
300.8.a.f.1.1 1 15.8 even 4