Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9300,2,Mod(1,9300)] 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("9300.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9300.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,0,0,0,0,3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(74.2608738798\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.7636.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 16x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1860)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.22881\) of defining polynomial
Character \(\chi\) \(=\) 9300.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.22881 q^{7} +1.00000 q^{9} +2.00000 q^{11} +3.22881 q^{13} -6.88279 q^{17} +4.00000 q^{19} +3.22881 q^{21} +6.88279 q^{23} -1.00000 q^{27} -1.65399 q^{29} -1.00000 q^{31} -2.00000 q^{33} +5.22881 q^{37} -3.22881 q^{39} +4.00000 q^{41} -2.00000 q^{43} -1.57482 q^{47} +3.42518 q^{49} +6.88279 q^{51} -0.425182 q^{53} -4.00000 q^{57} +7.65399 q^{59} -11.7656 q^{61} -3.22881 q^{63} -14.9944 q^{67} -6.88279 q^{69} -4.80362 q^{71} +7.68642 q^{73} -6.45761 q^{77} -11.3404 q^{79} +1.00000 q^{81} -2.88279 q^{83} +1.65399 q^{87} +1.88840 q^{89} -10.4252 q^{91} +1.00000 q^{93} +9.76558 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{9} + 6 q^{11} - 2 q^{17} + 12 q^{19} + 2 q^{23} - 3 q^{27} + 4 q^{29} - 3 q^{31} - 6 q^{33} + 6 q^{37} + 12 q^{41} - 6 q^{43} - 4 q^{47} + 11 q^{49} + 2 q^{51} - 2 q^{53} - 12 q^{57}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.22881 −1.22037 −0.610187 − 0.792258i \(-0.708906\pi\)
−0.610187 + 0.792258i \(0.708906\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 3.22881 0.895509 0.447755 − 0.894156i \(-0.352224\pi\)
0.447755 + 0.894156i \(0.352224\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.88279 −1.66932 −0.834661 − 0.550764i \(-0.814337\pi\)
−0.834661 + 0.550764i \(0.814337\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 3.22881 0.704583
\(22\) 0 0
\(23\) 6.88279 1.43516 0.717581 − 0.696475i \(-0.245249\pi\)
0.717581 + 0.696475i \(0.245249\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.65399 −0.307138 −0.153569 − 0.988138i \(-0.549077\pi\)
−0.153569 + 0.988138i \(0.549077\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.22881 0.859610 0.429805 − 0.902922i \(-0.358582\pi\)
0.429805 + 0.902922i \(0.358582\pi\)
\(38\) 0 0
\(39\) −3.22881 −0.517023
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.57482 −0.229711 −0.114855 − 0.993382i \(-0.536640\pi\)
−0.114855 + 0.993382i \(0.536640\pi\)
\(48\) 0 0
\(49\) 3.42518 0.489312
\(50\) 0 0
\(51\) 6.88279 0.963784
\(52\) 0 0
\(53\) −0.425182 −0.0584033 −0.0292016 − 0.999574i \(-0.509296\pi\)
−0.0292016 + 0.999574i \(0.509296\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 7.65399 0.996464 0.498232 − 0.867044i \(-0.333983\pi\)
0.498232 + 0.867044i \(0.333983\pi\)
\(60\) 0 0
\(61\) −11.7656 −1.50643 −0.753214 − 0.657775i \(-0.771498\pi\)
−0.753214 + 0.657775i \(0.771498\pi\)
\(62\) 0 0
\(63\) −3.22881 −0.406791
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −14.9944 −1.83186 −0.915928 − 0.401342i \(-0.868544\pi\)
−0.915928 + 0.401342i \(0.868544\pi\)
\(68\) 0 0
\(69\) −6.88279 −0.828591
\(70\) 0 0
\(71\) −4.80362 −0.570085 −0.285043 − 0.958515i \(-0.592008\pi\)
−0.285043 + 0.958515i \(0.592008\pi\)
\(72\) 0 0
\(73\) 7.68642 0.899627 0.449813 − 0.893123i \(-0.351490\pi\)
0.449813 + 0.893123i \(0.351490\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.45761 −0.735913
\(78\) 0 0
\(79\) −11.3404 −1.27589 −0.637947 − 0.770080i \(-0.720216\pi\)
−0.637947 + 0.770080i \(0.720216\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.88279 −0.316428 −0.158214 − 0.987405i \(-0.550574\pi\)
−0.158214 + 0.987405i \(0.550574\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.65399 0.177326
\(88\) 0 0
\(89\) 1.88840 0.200170 0.100085 − 0.994979i \(-0.468088\pi\)
0.100085 + 0.994979i \(0.468088\pi\)
\(90\) 0 0
\(91\) −10.4252 −1.09286
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.76558 0.991545 0.495772 − 0.868452i \(-0.334885\pi\)
0.495772 + 0.868452i \(0.334885\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
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 9300.2.a.t.1.1 3
5.2 odd 4 9300.2.g.r.3349.4 6
5.3 odd 4 9300.2.g.r.3349.3 6
5.4 even 2 1860.2.a.h.1.3 ✓ 3
15.14 odd 2 5580.2.a.i.1.3 3
20.19 odd 2 7440.2.a.bq.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.a.h.1.3 ✓ 3 5.4 even 2
5580.2.a.i.1.3 3 15.14 odd 2
7440.2.a.bq.1.1 3 20.19 odd 2
9300.2.a.t.1.1 3 1.1 even 1 trivial
9300.2.g.r.3349.3 6 5.3 odd 4
9300.2.g.r.3349.4 6 5.2 odd 4