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 1.2
Root \(1.61803\) 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)
 
\(f(q)\) \(=\) \(q+1.61803 q^{3} -0.618034 q^{7} -0.381966 q^{9} +2.85410 q^{11} +7.09017 q^{13} -6.09017 q^{17} -1.85410 q^{19} -1.00000 q^{21} +1.00000 q^{23} -5.47214 q^{27} -9.23607 q^{29} -9.09017 q^{31} +4.61803 q^{33} -6.47214 q^{37} +11.4721 q^{39} +3.32624 q^{41} -3.70820 q^{47} -6.61803 q^{49} -9.85410 q^{51} -0.472136 q^{53} -3.00000 q^{57} -1.70820 q^{59} -9.32624 q^{61} +0.236068 q^{63} +14.4721 q^{67} +1.61803 q^{69} +4.09017 q^{71} -3.23607 q^{73} -1.76393 q^{77} -1.52786 q^{79} -7.70820 q^{81} -6.94427 q^{83} -14.9443 q^{87} -10.4721 q^{89} -4.38197 q^{91} -14.7082 q^{93} -12.3820 q^{97} -1.09017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{7} - 3 q^{9} - q^{11} + 3 q^{13} - q^{17} + 3 q^{19} - 2 q^{21} + 2 q^{23} - 2 q^{27} - 14 q^{29} - 7 q^{31} + 7 q^{33} - 4 q^{37} + 14 q^{39} - 9 q^{41} + 6 q^{47} - 11 q^{49} - 13 q^{51}+ \cdots + 9 q^{99}+O(q^{100}) \) 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
\(3\) 1.61803 0.934172 0.467086 − 0.884212i \(-0.345304\pi\)
0.467086 + 0.884212i \(0.345304\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.618034 −0.233595 −0.116797 − 0.993156i \(-0.537263\pi\)
−0.116797 + 0.993156i \(0.537263\pi\)
\(8\) 0 0
\(9\) −0.381966 −0.127322
\(10\) 0 0
\(11\) 2.85410 0.860544 0.430272 − 0.902699i \(-0.358418\pi\)
0.430272 + 0.902699i \(0.358418\pi\)
\(12\) 0 0
\(13\) 7.09017 1.96646 0.983230 − 0.182372i \(-0.0583774\pi\)
0.983230 + 0.182372i \(0.0583774\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.09017 −1.47708 −0.738542 − 0.674208i \(-0.764485\pi\)
−0.738542 + 0.674208i \(0.764485\pi\)
\(18\) 0 0
\(19\) −1.85410 −0.425360 −0.212680 − 0.977122i \(-0.568219\pi\)
−0.212680 + 0.977122i \(0.568219\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.47214 −1.05311
\(28\) 0 0
\(29\) −9.23607 −1.71509 −0.857547 − 0.514405i \(-0.828013\pi\)
−0.857547 + 0.514405i \(0.828013\pi\)
\(30\) 0 0
\(31\) −9.09017 −1.63264 −0.816321 − 0.577598i \(-0.803990\pi\)
−0.816321 + 0.577598i \(0.803990\pi\)
\(32\) 0 0
\(33\) 4.61803 0.803897
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.47214 −1.06401 −0.532006 − 0.846740i \(-0.678562\pi\)
−0.532006 + 0.846740i \(0.678562\pi\)
\(38\) 0 0
\(39\) 11.4721 1.83701
\(40\) 0 0
\(41\) 3.32624 0.519471 0.259736 − 0.965680i \(-0.416365\pi\)
0.259736 + 0.965680i \(0.416365\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.70820 −0.540897 −0.270449 − 0.962734i \(-0.587172\pi\)
−0.270449 + 0.962734i \(0.587172\pi\)
\(48\) 0 0
\(49\) −6.61803 −0.945433
\(50\) 0 0
\(51\) −9.85410 −1.37985
\(52\) 0 0
\(53\) −0.472136 −0.0648529 −0.0324264 − 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) 0 0
\(59\) −1.70820 −0.222389 −0.111195 − 0.993799i \(-0.535468\pi\)
−0.111195 + 0.993799i \(0.535468\pi\)
\(60\) 0 0
\(61\) −9.32624 −1.19410 −0.597051 − 0.802203i \(-0.703661\pi\)
−0.597051 + 0.802203i \(0.703661\pi\)
\(62\) 0 0
\(63\) 0.236068 0.0297418
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 14.4721 1.76805 0.884026 − 0.467437i \(-0.154823\pi\)
0.884026 + 0.467437i \(0.154823\pi\)
\(68\) 0 0
\(69\) 1.61803 0.194788
\(70\) 0 0
\(71\) 4.09017 0.485414 0.242707 − 0.970100i \(-0.421965\pi\)
0.242707 + 0.970100i \(0.421965\pi\)
\(72\) 0 0
\(73\) −3.23607 −0.378753 −0.189377 − 0.981905i \(-0.560647\pi\)
−0.189377 + 0.981905i \(0.560647\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.76393 −0.201019
\(78\) 0 0
\(79\) −1.52786 −0.171898 −0.0859491 − 0.996300i \(-0.527392\pi\)
−0.0859491 + 0.996300i \(0.527392\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) 0 0
\(83\) −6.94427 −0.762233 −0.381116 − 0.924527i \(-0.624460\pi\)
−0.381116 + 0.924527i \(0.624460\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −14.9443 −1.60219
\(88\) 0 0
\(89\) −10.4721 −1.11004 −0.555022 − 0.831836i \(-0.687290\pi\)
−0.555022 + 0.831836i \(0.687290\pi\)
\(90\) 0 0
\(91\) −4.38197 −0.459355
\(92\) 0 0
\(93\) −14.7082 −1.52517
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.3820 −1.25720 −0.628599 − 0.777730i \(-0.716371\pi\)
−0.628599 + 0.777730i \(0.716371\pi\)
\(98\) 0 0
\(99\) −1.09017 −0.109566
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.1.2 2
4.3 odd 2 1150.2.a.j.1.1 2
5.4 even 2 1840.2.a.l.1.1 2
20.3 even 4 1150.2.b.i.599.3 4
20.7 even 4 1150.2.b.i.599.2 4
20.19 odd 2 230.2.a.c.1.2 ✓ 2
40.19 odd 2 7360.2.a.bh.1.1 2
40.29 even 2 7360.2.a.bn.1.2 2
60.59 even 2 2070.2.a.u.1.1 2
460.459 even 2 5290.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.2 ✓ 2 20.19 odd 2
1150.2.a.j.1.1 2 4.3 odd 2
1150.2.b.i.599.2 4 20.7 even 4
1150.2.b.i.599.3 4 20.3 even 4
1840.2.a.l.1.1 2 5.4 even 2
2070.2.a.u.1.1 2 60.59 even 2
5290.2.a.o.1.2 2 460.459 even 2
7360.2.a.bh.1.1 2 40.19 odd 2
7360.2.a.bn.1.2 2 40.29 even 2
9200.2.a.bu.1.2 2 1.1 even 1 trivial