Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,0,-2,0,0,-8,0,0,0,4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{14 +2 \sqrt{5}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 555)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.54336\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.618034 q^{2} -1.61803 q^{4} +0.497212 q^{7} -2.23607 q^{8} -0.543362 q^{11} -0.335816 q^{13} +0.307294 q^{14} +1.85410 q^{16} +3.70476 q^{17} -5.04057 q^{19} -0.335816 q^{22} +2.75836 q^{23} -0.207546 q^{26} -0.804506 q^{28} -2.68713 q^{29} +1.31074 q^{31} +5.61803 q^{32} +2.28967 q^{34} -1.00000 q^{37} -3.11525 q^{38} +6.33582 q^{41} -0.971478 q^{43} +0.879178 q^{44} +1.70476 q^{46} -0.0176277 q^{47} -6.75278 q^{49} +0.543362 q^{52} -8.08115 q^{53} -1.11180 q^{56} -1.66074 q^{58} -8.27107 q^{59} -6.39746 q^{61} +0.810082 q^{62} -0.236068 q^{64} +2.63009 q^{67} -5.99442 q^{68} +15.2020 q^{71} +6.27107 q^{73} -0.618034 q^{74} +8.15582 q^{76} -0.270166 q^{77} +4.25212 q^{79} +3.91575 q^{82} +9.43804 q^{83} -0.600406 q^{86} +1.21499 q^{88} +2.98050 q^{89} -0.166972 q^{91} -4.46311 q^{92} -0.0108945 q^{94} +8.13632 q^{97} -4.17345 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} - 8 q^{7} + 4 q^{11} - 2 q^{13} + 4 q^{14} - 6 q^{16} - 2 q^{17} - 4 q^{19} - 2 q^{22} + 6 q^{26} + 4 q^{28} + 12 q^{29} - 2 q^{31} + 18 q^{32} + 6 q^{34} - 4 q^{37} + 2 q^{38} + 26 q^{41}+ \cdots + 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 − 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) 0 0
\(7\) 0.497212 0.187928 0.0939642 − 0.995576i \(-0.470046\pi\)
0.0939642 + 0.995576i \(0.470046\pi\)
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) 0 0
\(11\) −0.543362 −0.163830 −0.0819149 − 0.996639i \(-0.526104\pi\)
−0.0819149 + 0.996639i \(0.526104\pi\)
\(12\) 0 0
\(13\) −0.335816 −0.0931386 −0.0465693 − 0.998915i \(-0.514829\pi\)
−0.0465693 + 0.998915i \(0.514829\pi\)
\(14\) 0.307294 0.0821278
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 3.70476 0.898536 0.449268 − 0.893397i \(-0.351685\pi\)
0.449268 + 0.893397i \(0.351685\pi\)
\(18\) 0 0
\(19\) −5.04057 −1.15639 −0.578193 − 0.815900i \(-0.696242\pi\)
−0.578193 + 0.815900i \(0.696242\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.335816 −0.0715962
\(23\) 2.75836 0.575157 0.287579 − 0.957757i \(-0.407150\pi\)
0.287579 + 0.957757i \(0.407150\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.207546 −0.0407031
\(27\) 0 0
\(28\) −0.804506 −0.152037
\(29\) −2.68713 −0.498988 −0.249494 − 0.968376i \(-0.580264\pi\)
−0.249494 + 0.968376i \(0.580264\pi\)
\(30\) 0 0
\(31\) 1.31074 0.235416 0.117708 − 0.993048i \(-0.462445\pi\)
0.117708 + 0.993048i \(0.462445\pi\)
\(32\) 5.61803 0.993137
\(33\) 0 0
\(34\) 2.28967 0.392675
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −3.11525 −0.505360
\(39\) 0 0
\(40\) 0 0
\(41\) 6.33582 0.989488 0.494744 − 0.869039i \(-0.335262\pi\)
0.494744 + 0.869039i \(0.335262\pi\)
\(42\) 0 0
\(43\) −0.971478 −0.148149 −0.0740745 − 0.997253i \(-0.523600\pi\)
−0.0740745 + 0.997253i \(0.523600\pi\)
\(44\) 0.879178 0.132541
\(45\) 0 0
\(46\) 1.70476 0.251353
\(47\) −0.0176277 −0.00257126 −0.00128563 − 0.999999i \(-0.500409\pi\)
−0.00128563 + 0.999999i \(0.500409\pi\)
\(48\) 0 0
\(49\) −6.75278 −0.964683
\(50\) 0 0
\(51\) 0 0
\(52\) 0.543362 0.0753507
\(53\) −8.08115 −1.11003 −0.555016 − 0.831840i \(-0.687288\pi\)
−0.555016 + 0.831840i \(0.687288\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.11180 −0.148571
\(57\) 0 0
\(58\) −1.66074 −0.218066
\(59\) −8.27107 −1.07680 −0.538401 − 0.842689i \(-0.680971\pi\)
−0.538401 + 0.842689i \(0.680971\pi\)
\(60\) 0 0
\(61\) −6.39746 −0.819111 −0.409556 − 0.912285i \(-0.634316\pi\)
−0.409556 + 0.912285i \(0.634316\pi\)
\(62\) 0.810082 0.102881
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) 0 0
\(67\) 2.63009 0.321316 0.160658 − 0.987010i \(-0.448638\pi\)
0.160658 + 0.987010i \(0.448638\pi\)
\(68\) −5.99442 −0.726931
\(69\) 0 0
\(70\) 0 0
\(71\) 15.2020 1.80414 0.902071 − 0.431588i \(-0.142046\pi\)
0.902071 + 0.431588i \(0.142046\pi\)
\(72\) 0 0
\(73\) 6.27107 0.733973 0.366986 − 0.930226i \(-0.380390\pi\)
0.366986 + 0.930226i \(0.380390\pi\)
\(74\) −0.618034 −0.0718450
\(75\) 0 0
\(76\) 8.15582 0.935537
\(77\) −0.270166 −0.0307883
\(78\) 0 0
\(79\) 4.25212 0.478401 0.239201 − 0.970970i \(-0.423115\pi\)
0.239201 + 0.970970i \(0.423115\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.91575 0.432422
\(83\) 9.43804 1.03596 0.517980 − 0.855393i \(-0.326684\pi\)
0.517980 + 0.855393i \(0.326684\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.600406 −0.0647435
\(87\) 0 0
\(88\) 1.21499 0.129519
\(89\) 2.98050 0.315932 0.157966 − 0.987445i \(-0.449506\pi\)
0.157966 + 0.987445i \(0.449506\pi\)
\(90\) 0 0
\(91\) −0.166972 −0.0175034
\(92\) −4.46311 −0.465312
\(93\) 0 0
\(94\) −0.0108945 −0.00112368
\(95\) 0 0
\(96\) 0 0
\(97\) 8.13632 0.826118 0.413059 − 0.910704i \(-0.364460\pi\)
0.413059 + 0.910704i \(0.364460\pi\)
\(98\) −4.17345 −0.421582
\(99\) 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 8325.2.a.bt.1.4 4
3.2 odd 2 2775.2.a.y.1.2 4
5.2 odd 4 1665.2.c.d.334.5 8
5.3 odd 4 1665.2.c.d.334.3 8
5.4 even 2 8325.2.a.bw.1.1 4
15.2 even 4 555.2.c.b.334.4 ✓ 8
15.8 even 4 555.2.c.b.334.6 yes 8
15.14 odd 2 2775.2.a.w.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.c.b.334.4 ✓ 8 15.2 even 4
555.2.c.b.334.6 yes 8 15.8 even 4
1665.2.c.d.334.3 8 5.3 odd 4
1665.2.c.d.334.5 8 5.2 odd 4
2775.2.a.w.1.3 4 15.14 odd 2
2775.2.a.y.1.2 4 3.2 odd 2
8325.2.a.bt.1.4 4 1.1 even 1 trivial
8325.2.a.bw.1.1 4 5.4 even 2