Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7360,2,Mod(1,7360)] 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("7360.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,-2,0,-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: \(58.7698958877\)
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 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 7360.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.61803 q^{3} -1.00000 q^{5} +0.618034 q^{7} -0.381966 q^{9} -2.85410 q^{11} +7.09017 q^{13} -1.61803 q^{15} +6.09017 q^{17} +1.85410 q^{19} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.47214 q^{27} +9.23607 q^{29} -9.09017 q^{31} -4.61803 q^{33} -0.618034 q^{35} -6.47214 q^{37} +11.4721 q^{39} +3.32624 q^{41} +0.381966 q^{45} +3.70820 q^{47} -6.61803 q^{49} +9.85410 q^{51} -0.472136 q^{53} +2.85410 q^{55} +3.00000 q^{57} +1.70820 q^{59} +9.32624 q^{61} -0.236068 q^{63} -7.09017 q^{65} +14.4721 q^{67} -1.61803 q^{69} +4.09017 q^{71} +3.23607 q^{73} +1.61803 q^{75} -1.76393 q^{77} -1.52786 q^{79} -7.70820 q^{81} -6.94427 q^{83} -6.09017 q^{85} +14.9443 q^{87} -10.4721 q^{89} +4.38197 q^{91} -14.7082 q^{93} -1.85410 q^{95} +12.3820 q^{97} +1.09017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{5} - q^{7} - 3 q^{9} + q^{11} + 3 q^{13} - q^{15} + q^{17} - 3 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} - 2 q^{27} + 14 q^{29} - 7 q^{31} - 7 q^{33} + q^{35} - 4 q^{37} + 14 q^{39}+ \cdots - 9 q^{99}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.618034 0.233595 0.116797 − 0.993156i \(-0.462737\pi\)
0.116797 + 0.993156i \(0.462737\pi\)
\(8\) 0 0
\(9\) −0.381966 −0.127322
\(10\) 0 0
\(11\) −2.85410 −0.860544 −0.430272 − 0.902699i \(-0.641582\pi\)
−0.430272 + 0.902699i \(0.641582\pi\)
\(12\) 0 0
\(13\) 7.09017 1.96646 0.983230 − 0.182372i \(-0.0583774\pi\)
0.983230 + 0.182372i \(0.0583774\pi\)
\(14\) 0 0
\(15\) −1.61803 −0.417775
\(16\) 0 0
\(17\) 6.09017 1.47708 0.738542 − 0.674208i \(-0.235515\pi\)
0.738542 + 0.674208i \(0.235515\pi\)
\(18\) 0 0
\(19\) 1.85410 0.425360 0.212680 − 0.977122i \(-0.431781\pi\)
0.212680 + 0.977122i \(0.431781\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.47214 −1.05311
\(28\) 0 0
\(29\) 9.23607 1.71509 0.857547 − 0.514405i \(-0.171987\pi\)
0.857547 + 0.514405i \(0.171987\pi\)
\(30\) 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
\(35\) −0.618034 −0.104467
\(36\) 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
\(45\) 0.381966 0.0569401
\(46\) 0 0
\(47\) 3.70820 0.540897 0.270449 − 0.962734i \(-0.412828\pi\)
0.270449 + 0.962734i \(0.412828\pi\)
\(48\) 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
\(55\) 2.85410 0.384847
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 0 0
\(59\) 1.70820 0.222389 0.111195 − 0.993799i \(-0.464532\pi\)
0.111195 + 0.993799i \(0.464532\pi\)
\(60\) 0 0
\(61\) 9.32624 1.19410 0.597051 − 0.802203i \(-0.296339\pi\)
0.597051 + 0.802203i \(0.296339\pi\)
\(62\) 0 0
\(63\) −0.236068 −0.0297418
\(64\) 0 0
\(65\) −7.09017 −0.879427
\(66\) 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 \(-0.439353\pi\)
0.189377 + 0.981905i \(0.439353\pi\)
\(74\) 0 0
\(75\) 1.61803 0.186834
\(76\) 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
\(85\) −6.09017 −0.660572
\(86\) 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
\(95\) −1.85410 −0.190227
\(96\) 0 0
\(97\) 12.3820 1.25720 0.628599 − 0.777730i \(-0.283629\pi\)
0.628599 + 0.777730i \(0.283629\pi\)
\(98\) 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 7360.2.a.bn.1.2 2
4.3 odd 2 7360.2.a.bh.1.1 2
8.3 odd 2 230.2.a.c.1.2 ✓ 2
8.5 even 2 1840.2.a.l.1.1 2
24.11 even 2 2070.2.a.u.1.1 2
40.3 even 4 1150.2.b.i.599.2 4
40.19 odd 2 1150.2.a.j.1.1 2
40.27 even 4 1150.2.b.i.599.3 4
40.29 even 2 9200.2.a.bu.1.2 2
184.91 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 8.3 odd 2
1150.2.a.j.1.1 2 40.19 odd 2
1150.2.b.i.599.2 4 40.3 even 4
1150.2.b.i.599.3 4 40.27 even 4
1840.2.a.l.1.1 2 8.5 even 2
2070.2.a.u.1.1 2 24.11 even 2
5290.2.a.o.1.2 2 184.91 even 2
7360.2.a.bh.1.1 2 4.3 odd 2
7360.2.a.bn.1.2 2 1.1 even 1 trivial
9200.2.a.bu.1.2 2 40.29 even 2