Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9200,2,Mod(1,9200)] 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("9200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,0,0,1,0,-3,0,-1] 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: \(73.4623698596\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 9200.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.618034 q^{3} +1.61803 q^{7} -2.61803 q^{9} -3.85410 q^{11} -4.09017 q^{13} +5.09017 q^{17} +4.85410 q^{19} -1.00000 q^{21} +1.00000 q^{23} +3.47214 q^{27} -4.76393 q^{29} +2.09017 q^{31} +2.38197 q^{33} +2.47214 q^{37} +2.52786 q^{39} -12.3262 q^{41} +9.70820 q^{47} -4.38197 q^{49} -3.14590 q^{51} +8.47214 q^{53} -3.00000 q^{57} +11.7082 q^{59} +6.32624 q^{61} -4.23607 q^{63} +5.52786 q^{67} -0.618034 q^{69} -7.09017 q^{71} +1.23607 q^{73} -6.23607 q^{77} -10.4721 q^{79} +5.70820 q^{81} +10.9443 q^{83} +2.94427 q^{87} -1.52786 q^{89} -6.61803 q^{91} -1.29180 q^{93} -14.6180 q^{97} +10.0902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{7} - 3 q^{9} - q^{11} + 3 q^{13} - q^{17} + 3 q^{19} - 2 q^{21} + 2 q^{23} - 2 q^{27} - 14 q^{29} - 7 q^{31} + 7 q^{33} - 4 q^{37} + 14 q^{39} - 9 q^{41} + 6 q^{47} - 11 q^{49} - 13 q^{51}+ \cdots + 9 q^{99}+O(q^{100}) \) 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
\(3\) −0.618034 −0.356822 −0.178411 − 0.983956i \(-0.557096\pi\)
−0.178411 + 0.983956i \(0.557096\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.61803 0.611559 0.305780 − 0.952102i \(-0.401083\pi\)
0.305780 + 0.952102i \(0.401083\pi\)
\(8\) 0 0
\(9\) −2.61803 −0.872678
\(10\) 0 0
\(11\) −3.85410 −1.16206 −0.581028 − 0.813884i \(-0.697349\pi\)
−0.581028 + 0.813884i \(0.697349\pi\)
\(12\) 0 0
\(13\) −4.09017 −1.13441 −0.567205 − 0.823577i \(-0.691975\pi\)
−0.567205 + 0.823577i \(0.691975\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.09017 1.23455 0.617274 − 0.786748i \(-0.288237\pi\)
0.617274 + 0.786748i \(0.288237\pi\)
\(18\) 0 0
\(19\) 4.85410 1.11361 0.556804 − 0.830644i \(-0.312028\pi\)
0.556804 + 0.830644i \(0.312028\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.47214 0.668213
\(28\) 0 0
\(29\) −4.76393 −0.884640 −0.442320 − 0.896857i \(-0.645844\pi\)
−0.442320 + 0.896857i \(0.645844\pi\)
\(30\) 0 0
\(31\) 2.09017 0.375406 0.187703 − 0.982226i \(-0.439896\pi\)
0.187703 + 0.982226i \(0.439896\pi\)
\(32\) 0 0
\(33\) 2.38197 0.414647
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.47214 0.406417 0.203208 − 0.979136i \(-0.434863\pi\)
0.203208 + 0.979136i \(0.434863\pi\)
\(38\) 0 0
\(39\) 2.52786 0.404782
\(40\) 0 0
\(41\) −12.3262 −1.92503 −0.962517 − 0.271220i \(-0.912573\pi\)
−0.962517 + 0.271220i \(0.912573\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.70820 1.41609 0.708044 − 0.706169i \(-0.249578\pi\)
0.708044 + 0.706169i \(0.249578\pi\)
\(48\) 0 0
\(49\) −4.38197 −0.625995
\(50\) 0 0
\(51\) −3.14590 −0.440514
\(52\) 0 0
\(53\) 8.47214 1.16374 0.581869 − 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) 0 0
\(59\) 11.7082 1.52428 0.762139 − 0.647413i \(-0.224149\pi\)
0.762139 + 0.647413i \(0.224149\pi\)
\(60\) 0 0
\(61\) 6.32624 0.809992 0.404996 − 0.914319i \(-0.367273\pi\)
0.404996 + 0.914319i \(0.367273\pi\)
\(62\) 0 0
\(63\) −4.23607 −0.533694
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.52786 0.675336 0.337668 − 0.941265i \(-0.390362\pi\)
0.337668 + 0.941265i \(0.390362\pi\)
\(68\) 0 0
\(69\) −0.618034 −0.0744025
\(70\) 0 0
\(71\) −7.09017 −0.841448 −0.420724 − 0.907189i \(-0.638224\pi\)
−0.420724 + 0.907189i \(0.638224\pi\)
\(72\) 0 0
\(73\) 1.23607 0.144671 0.0723354 − 0.997380i \(-0.476955\pi\)
0.0723354 + 0.997380i \(0.476955\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.23607 −0.710666
\(78\) 0 0
\(79\) −10.4721 −1.17821 −0.589104 − 0.808057i \(-0.700519\pi\)
−0.589104 + 0.808057i \(0.700519\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) 0 0
\(83\) 10.9443 1.20129 0.600645 − 0.799516i \(-0.294911\pi\)
0.600645 + 0.799516i \(0.294911\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.94427 0.315659
\(88\) 0 0
\(89\) −1.52786 −0.161953 −0.0809766 − 0.996716i \(-0.525804\pi\)
−0.0809766 + 0.996716i \(0.525804\pi\)
\(90\) 0 0
\(91\) −6.61803 −0.693758
\(92\) 0 0
\(93\) −1.29180 −0.133953
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.6180 −1.48424 −0.742118 − 0.670269i \(-0.766179\pi\)
−0.742118 + 0.670269i \(0.766179\pi\)
\(98\) 0 0
\(99\) 10.0902 1.01410
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 9200.2.a.bu.1.1 2
4.3 odd 2 1150.2.a.j.1.2 2
5.4 even 2 1840.2.a.l.1.2 2
20.3 even 4 1150.2.b.i.599.4 4
20.7 even 4 1150.2.b.i.599.1 4
20.19 odd 2 230.2.a.c.1.1 ✓ 2
40.19 odd 2 7360.2.a.bh.1.2 2
40.29 even 2 7360.2.a.bn.1.1 2
60.59 even 2 2070.2.a.u.1.2 2
460.459 even 2 5290.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.1 ✓ 2 20.19 odd 2
1150.2.a.j.1.2 2 4.3 odd 2
1150.2.b.i.599.1 4 20.7 even 4
1150.2.b.i.599.4 4 20.3 even 4
1840.2.a.l.1.2 2 5.4 even 2
2070.2.a.u.1.2 2 60.59 even 2
5290.2.a.o.1.1 2 460.459 even 2
7360.2.a.bh.1.2 2 40.19 odd 2
7360.2.a.bn.1.1 2 40.29 even 2
9200.2.a.bu.1.1 2 1.1 even 1 trivial