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 = [2,0,1,0,0,0,3,0,-3,0,9] 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: \(0\)
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)
 
\(f(q)\) \(=\) \(q-0.618034 q^{3} +4.85410 q^{7} -2.61803 q^{9} +3.38197 q^{11} -0.381966 q^{13} +5.85410 q^{17} +6.85410 q^{19} -3.00000 q^{21} +1.00000 q^{23} +3.47214 q^{27} +3.70820 q^{29} +8.85410 q^{31} -2.09017 q^{33} -3.70820 q^{37} +0.236068 q^{39} -3.38197 q^{41} +6.76393 q^{43} -11.7082 q^{47} +16.5623 q^{49} -3.61803 q^{51} +2.00000 q^{53} -4.23607 q^{57} +6.00000 q^{59} -3.85410 q^{61} -12.7082 q^{63} +0.763932 q^{67} -0.618034 q^{69} -2.61803 q^{71} +7.52786 q^{73} +16.4164 q^{77} -5.70820 q^{79} +5.70820 q^{81} -5.70820 q^{83} -2.29180 q^{87} -9.70820 q^{89} -1.85410 q^{91} -5.47214 q^{93} +16.0344 q^{97} -8.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 3 q^{7} - 3 q^{9} + 9 q^{11} - 3 q^{13} + 5 q^{17} + 7 q^{19} - 6 q^{21} + 2 q^{23} - 2 q^{27} - 6 q^{29} + 11 q^{31} + 7 q^{33} + 6 q^{37} - 4 q^{39} - 9 q^{41} + 18 q^{43} - 10 q^{47} + 13 q^{49}+ \cdots - 11 q^{99}+O(q^{100}) \) 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
\(7\) 4.85410 1.83468 0.917339 − 0.398107i \(-0.130333\pi\)
0.917339 + 0.398107i \(0.130333\pi\)
\(8\) 0 0
\(9\) −2.61803 −0.872678
\(10\) 0 0
\(11\) 3.38197 1.01970 0.509851 − 0.860263i \(-0.329701\pi\)
0.509851 + 0.860263i \(0.329701\pi\)
\(12\) 0 0
\(13\) −0.381966 −0.105938 −0.0529692 − 0.998596i \(-0.516869\pi\)
−0.0529692 + 0.998596i \(0.516869\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.85410 1.41983 0.709914 − 0.704288i \(-0.248734\pi\)
0.709914 + 0.704288i \(0.248734\pi\)
\(18\) 0 0
\(19\) 6.85410 1.57244 0.786219 − 0.617947i \(-0.212036\pi\)
0.786219 + 0.617947i \(0.212036\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.47214 0.668213
\(28\) 0 0
\(29\) 3.70820 0.688596 0.344298 − 0.938860i \(-0.388117\pi\)
0.344298 + 0.938860i \(0.388117\pi\)
\(30\) 0 0
\(31\) 8.85410 1.59024 0.795122 − 0.606450i \(-0.207407\pi\)
0.795122 + 0.606450i \(0.207407\pi\)
\(32\) 0 0
\(33\) −2.09017 −0.363852
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.70820 −0.609625 −0.304812 − 0.952412i \(-0.598594\pi\)
−0.304812 + 0.952412i \(0.598594\pi\)
\(38\) 0 0
\(39\) 0.236068 0.0378011
\(40\) 0 0
\(41\) −3.38197 −0.528174 −0.264087 − 0.964499i \(-0.585071\pi\)
−0.264087 + 0.964499i \(0.585071\pi\)
\(42\) 0 0
\(43\) 6.76393 1.03149 0.515745 − 0.856742i \(-0.327515\pi\)
0.515745 + 0.856742i \(0.327515\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.7082 −1.70782 −0.853909 − 0.520423i \(-0.825774\pi\)
−0.853909 + 0.520423i \(0.825774\pi\)
\(48\) 0 0
\(49\) 16.5623 2.36604
\(50\) 0 0
\(51\) −3.61803 −0.506626
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.23607 −0.561081
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −3.85410 −0.493467 −0.246734 − 0.969083i \(-0.579357\pi\)
−0.246734 + 0.969083i \(0.579357\pi\)
\(62\) 0 0
\(63\) −12.7082 −1.60108
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.763932 0.0933292 0.0466646 − 0.998911i \(-0.485141\pi\)
0.0466646 + 0.998911i \(0.485141\pi\)
\(68\) 0 0
\(69\) −0.618034 −0.0744025
\(70\) 0 0
\(71\) −2.61803 −0.310703 −0.155352 − 0.987859i \(-0.549651\pi\)
−0.155352 + 0.987859i \(0.549651\pi\)
\(72\) 0 0
\(73\) 7.52786 0.881070 0.440535 − 0.897735i \(-0.354789\pi\)
0.440535 + 0.897735i \(0.354789\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.4164 1.87082
\(78\) 0 0
\(79\) −5.70820 −0.642223 −0.321112 − 0.947041i \(-0.604056\pi\)
−0.321112 + 0.947041i \(0.604056\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) 0 0
\(83\) −5.70820 −0.626557 −0.313278 − 0.949661i \(-0.601427\pi\)
−0.313278 + 0.949661i \(0.601427\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.29180 −0.245706
\(88\) 0 0
\(89\) −9.70820 −1.02907 −0.514534 − 0.857470i \(-0.672035\pi\)
−0.514534 + 0.857470i \(0.672035\pi\)
\(90\) 0 0
\(91\) −1.85410 −0.194363
\(92\) 0 0
\(93\) −5.47214 −0.567434
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.0344 1.62805 0.814025 − 0.580829i \(-0.197272\pi\)
0.814025 + 0.580829i \(0.197272\pi\)
\(98\) 0 0
\(99\) −8.85410 −0.889871
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.by.1.1 2
4.3 odd 2 1150.2.a.n.1.2 2
5.2 odd 4 1840.2.e.c.369.3 4
5.3 odd 4 1840.2.e.c.369.2 4
5.4 even 2 9200.2.a.bo.1.2 2
20.3 even 4 230.2.b.a.139.2 ✓ 4
20.7 even 4 230.2.b.a.139.3 yes 4
20.19 odd 2 1150.2.a.l.1.1 2
60.23 odd 4 2070.2.d.c.829.3 4
60.47 odd 4 2070.2.d.c.829.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.b.a.139.2 ✓ 4 20.3 even 4
230.2.b.a.139.3 yes 4 20.7 even 4
1150.2.a.l.1.1 2 20.19 odd 2
1150.2.a.n.1.2 2 4.3 odd 2
1840.2.e.c.369.2 4 5.3 odd 4
1840.2.e.c.369.3 4 5.2 odd 4
2070.2.d.c.829.1 4 60.47 odd 4
2070.2.d.c.829.3 4 60.23 odd 4
9200.2.a.bo.1.2 2 5.4 even 2
9200.2.a.by.1.1 2 1.1 even 1 trivial