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,-3,0,-2,0,3,0,5,0,7] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) 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 \(-1.30278\) 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)
 
\(f(q)\) \(=\) \(q+0.302776 q^{3} -1.00000 q^{5} +3.30278 q^{7} -2.90833 q^{9} +1.69722 q^{11} -3.30278 q^{13} -0.302776 q^{15} +6.90833 q^{17} -5.90833 q^{19} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.78890 q^{27} +2.60555 q^{29} -7.90833 q^{31} +0.513878 q^{33} -3.30278 q^{35} -8.00000 q^{37} -1.00000 q^{39} +0.908327 q^{41} +9.21110 q^{43} +2.90833 q^{45} -2.60555 q^{47} +3.90833 q^{49} +2.09167 q^{51} +11.2111 q^{53} -1.69722 q^{55} -1.78890 q^{57} +3.39445 q^{59} -11.5139 q^{61} -9.60555 q^{63} +3.30278 q^{65} +4.00000 q^{67} -0.302776 q^{69} -16.3028 q^{71} -5.81665 q^{73} +0.302776 q^{75} +5.60555 q^{77} -14.4222 q^{79} +8.18335 q^{81} -11.2111 q^{83} -6.90833 q^{85} +0.788897 q^{87} -10.9083 q^{91} -2.39445 q^{93} +5.90833 q^{95} +6.30278 q^{97} -4.93608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 2 q^{5} + 3 q^{7} + 5 q^{9} + 7 q^{11} - 3 q^{13} + 3 q^{15} + 3 q^{17} - q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} - 18 q^{27} - 2 q^{29} - 5 q^{31} - 17 q^{33} - 3 q^{35} - 16 q^{37} - 2 q^{39}+ \cdots + 37 q^{99}+O(q^{100}) \) 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
\(3\) 0.302776 0.174808 0.0874038 − 0.996173i \(-0.472143\pi\)
0.0874038 + 0.996173i \(0.472143\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.30278 1.24833 0.624166 − 0.781292i \(-0.285439\pi\)
0.624166 + 0.781292i \(0.285439\pi\)
\(8\) 0 0
\(9\) −2.90833 −0.969442
\(10\) 0 0
\(11\) 1.69722 0.511732 0.255866 − 0.966712i \(-0.417639\pi\)
0.255866 + 0.966712i \(0.417639\pi\)
\(12\) 0 0
\(13\) −3.30278 −0.916025 −0.458013 − 0.888946i \(-0.651439\pi\)
−0.458013 + 0.888946i \(0.651439\pi\)
\(14\) 0 0
\(15\) −0.302776 −0.0781763
\(16\) 0 0
\(17\) 6.90833 1.67552 0.837758 − 0.546042i \(-0.183866\pi\)
0.837758 + 0.546042i \(0.183866\pi\)
\(18\) 0 0
\(19\) −5.90833 −1.35546 −0.677732 − 0.735309i \(-0.737037\pi\)
−0.677732 + 0.735309i \(0.737037\pi\)
\(20\) 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
\(27\) −1.78890 −0.344273
\(28\) 0 0
\(29\) 2.60555 0.483839 0.241919 − 0.970296i \(-0.422223\pi\)
0.241919 + 0.970296i \(0.422223\pi\)
\(30\) 0 0
\(31\) −7.90833 −1.42038 −0.710189 − 0.704011i \(-0.751390\pi\)
−0.710189 + 0.704011i \(0.751390\pi\)
\(32\) 0 0
\(33\) 0.513878 0.0894547
\(34\) 0 0
\(35\) −3.30278 −0.558271
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 0.908327 0.141857 0.0709284 − 0.997481i \(-0.477404\pi\)
0.0709284 + 0.997481i \(0.477404\pi\)
\(42\) 0 0
\(43\) 9.21110 1.40468 0.702340 − 0.711842i \(-0.252139\pi\)
0.702340 + 0.711842i \(0.252139\pi\)
\(44\) 0 0
\(45\) 2.90833 0.433548
\(46\) 0 0
\(47\) −2.60555 −0.380059 −0.190029 − 0.981778i \(-0.560858\pi\)
−0.190029 + 0.981778i \(0.560858\pi\)
\(48\) 0 0
\(49\) 3.90833 0.558332
\(50\) 0 0
\(51\) 2.09167 0.292893
\(52\) 0 0
\(53\) 11.2111 1.53996 0.769982 − 0.638066i \(-0.220265\pi\)
0.769982 + 0.638066i \(0.220265\pi\)
\(54\) 0 0
\(55\) −1.69722 −0.228854
\(56\) 0 0
\(57\) −1.78890 −0.236945
\(58\) 0 0
\(59\) 3.39445 0.441920 0.220960 − 0.975283i \(-0.429081\pi\)
0.220960 + 0.975283i \(0.429081\pi\)
\(60\) 0 0
\(61\) −11.5139 −1.47420 −0.737101 − 0.675783i \(-0.763806\pi\)
−0.737101 + 0.675783i \(0.763806\pi\)
\(62\) 0 0
\(63\) −9.60555 −1.21019
\(64\) 0 0
\(65\) 3.30278 0.409659
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −0.302776 −0.0364499
\(70\) 0 0
\(71\) −16.3028 −1.93478 −0.967392 − 0.253285i \(-0.918489\pi\)
−0.967392 + 0.253285i \(0.918489\pi\)
\(72\) 0 0
\(73\) −5.81665 −0.680788 −0.340394 − 0.940283i \(-0.610560\pi\)
−0.340394 + 0.940283i \(0.610560\pi\)
\(74\) 0 0
\(75\) 0.302776 0.0349615
\(76\) 0 0
\(77\) 5.60555 0.638812
\(78\) 0 0
\(79\) −14.4222 −1.62262 −0.811312 − 0.584613i \(-0.801246\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) 0 0
\(81\) 8.18335 0.909261
\(82\) 0 0
\(83\) −11.2111 −1.23058 −0.615289 − 0.788301i \(-0.710961\pi\)
−0.615289 + 0.788301i \(0.710961\pi\)
\(84\) 0 0
\(85\) −6.90833 −0.749313
\(86\) 0 0
\(87\) 0.788897 0.0845787
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −10.9083 −1.14350
\(92\) 0 0
\(93\) −2.39445 −0.248293
\(94\) 0 0
\(95\) 5.90833 0.606182
\(96\) 0 0
\(97\) 6.30278 0.639950 0.319975 − 0.947426i \(-0.396325\pi\)
0.319975 + 0.947426i \(0.396325\pi\)
\(98\) 0 0
\(99\) −4.93608 −0.496095
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.bc.1.2 2
4.3 odd 2 7360.2.a.bu.1.1 2
8.3 odd 2 1840.2.a.j.1.2 2
8.5 even 2 230.2.a.b.1.1 ✓ 2
24.5 odd 2 2070.2.a.w.1.2 2
40.13 odd 4 1150.2.b.f.599.3 4
40.19 odd 2 9200.2.a.ca.1.1 2
40.29 even 2 1150.2.a.m.1.2 2
40.37 odd 4 1150.2.b.f.599.2 4
184.45 odd 2 5290.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.1 ✓ 2 8.5 even 2
1150.2.a.m.1.2 2 40.29 even 2
1150.2.b.f.599.2 4 40.37 odd 4
1150.2.b.f.599.3 4 40.13 odd 4
1840.2.a.j.1.2 2 8.3 odd 2
2070.2.a.w.1.2 2 24.5 odd 2
5290.2.a.j.1.1 2 184.45 odd 2
7360.2.a.bc.1.2 2 1.1 even 1 trivial
7360.2.a.bu.1.1 2 4.3 odd 2
9200.2.a.ca.1.1 2 40.19 odd 2