Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1300,2,Mod(649,1300)] 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("1300.649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,-6,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9144576.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(-2.26180i\) of defining polynomial
Character \(\chi\) \(=\) 1300.649
Dual form 1300.2.d.c.649.5

$q$-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)
 
\(f(q)\) \(=\) \(q-2.26180i q^{3} -1.11575 q^{7} -2.11575 q^{9} -5.37755i q^{11} +(1.26180 - 3.37755i) q^{13} +7.90116i q^{19} +2.52360i q^{21} -6.49330i q^{23} -2.00000i q^{27} -3.63935 q^{29} +3.14605i q^{31} -12.1630 q^{33} -7.40786 q^{37} +(-7.63935 - 2.85395i) q^{39} +4.75510i q^{41} -4.78541i q^{43} +6.16296 q^{47} -5.75510 q^{49} +0.292106i q^{53} +17.8709 q^{57} -11.3776i q^{59} -5.34725 q^{61} +2.36065 q^{63} -11.8709 q^{67} -14.6866 q^{69} -3.43816i q^{71} +4.59214 q^{73} +6.00000i q^{77} +10.8157 q^{79} -10.8709 q^{81} -7.40786 q^{83} +8.23150i q^{87} +14.8157i q^{89} +(-1.40786 + 3.76850i) q^{91} +7.11575 q^{93} +3.76850 q^{97} +11.3776i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9} - 6 q^{13} + 12 q^{29} - 12 q^{33} - 24 q^{37} - 12 q^{39} - 24 q^{47} + 6 q^{49} + 60 q^{57} - 12 q^{61} + 48 q^{63} - 24 q^{67} + 48 q^{73} + 24 q^{79} - 18 q^{81} - 24 q^{83} + 12 q^{91}+ \cdots + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) − 2.26180i − 1.30585i −0.757422 − 0.652926i \(-0.773541\pi\)
0.757422 − 0.652926i \(-0.226459\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.11575 −0.421714 −0.210857 − 0.977517i \(-0.567625\pi\)
−0.210857 + 0.977517i \(0.567625\pi\)
\(8\) 0 0
\(9\) −2.11575 −0.705250
\(10\) 0 0
\(11\) − 5.37755i − 1.62139i −0.585467 − 0.810696i \(-0.699089\pi\)
0.585467 − 0.810696i \(-0.300911\pi\)
\(12\) 0 0
\(13\) 1.26180 − 3.37755i 0.349961 − 0.936764i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 7.90116i 1.81265i 0.422582 + 0.906325i \(0.361124\pi\)
−0.422582 + 0.906325i \(0.638876\pi\)
\(20\) 0 0
\(21\) 2.52360i 0.550696i
\(22\) 0 0
\(23\) − 6.49330i − 1.35395i −0.736007 − 0.676973i \(-0.763291\pi\)
0.736007 − 0.676973i \(-0.236709\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 2.00000i − 0.384900i
\(28\) 0 0
\(29\) −3.63935 −0.675811 −0.337906 − 0.941180i \(-0.609718\pi\)
−0.337906 + 0.941180i \(0.609718\pi\)
\(30\) 0 0
\(31\) 3.14605i 0.565048i 0.959260 + 0.282524i \(0.0911716\pi\)
−0.959260 + 0.282524i \(0.908828\pi\)
\(32\) 0 0
\(33\) −12.1630 −2.11730
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.40786 −1.21784 −0.608922 − 0.793230i \(-0.708398\pi\)
−0.608922 + 0.793230i \(0.708398\pi\)
\(38\) 0 0
\(39\) −7.63935 − 2.85395i −1.22328 − 0.456997i
\(40\) 0 0
\(41\) 4.75510i 0.742622i 0.928508 + 0.371311i \(0.121092\pi\)
−0.928508 + 0.371311i \(0.878908\pi\)
\(42\) 0 0
\(43\) − 4.78541i − 0.729768i −0.931053 − 0.364884i \(-0.881109\pi\)
0.931053 − 0.364884i \(-0.118891\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.16296 0.898960 0.449480 − 0.893290i \(-0.351609\pi\)
0.449480 + 0.893290i \(0.351609\pi\)
\(48\) 0 0
\(49\) −5.75510 −0.822158
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.292106i 0.0401238i 0.999799 + 0.0200619i \(0.00638633\pi\)
−0.999799 + 0.0200619i \(0.993614\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 17.8709 2.36705
\(58\) 0 0
\(59\) − 11.3776i − 1.48123i −0.671929 − 0.740616i \(-0.734534\pi\)
0.671929 − 0.740616i \(-0.265466\pi\)
\(60\) 0 0
\(61\) −5.34725 −0.684645 −0.342322 − 0.939583i \(-0.611214\pi\)
−0.342322 + 0.939583i \(0.611214\pi\)
\(62\) 0 0
\(63\) 2.36065 0.297413
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −11.8709 −1.45026 −0.725128 − 0.688614i \(-0.758219\pi\)
−0.725128 + 0.688614i \(0.758219\pi\)
\(68\) 0 0
\(69\) −14.6866 −1.76805
\(70\) 0 0
\(71\) − 3.43816i − 0.408034i −0.978967 − 0.204017i \(-0.934600\pi\)
0.978967 − 0.204017i \(-0.0653998\pi\)
\(72\) 0 0
\(73\) 4.59214 0.537470 0.268735 − 0.963214i \(-0.413394\pi\)
0.268735 + 0.963214i \(0.413394\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) 10.8157 1.21686 0.608431 − 0.793607i \(-0.291799\pi\)
0.608431 + 0.793607i \(0.291799\pi\)
\(80\) 0 0
\(81\) −10.8709 −1.20787
\(82\) 0 0
\(83\) −7.40786 −0.813118 −0.406559 − 0.913625i \(-0.633271\pi\)
−0.406559 + 0.913625i \(0.633271\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.23150i 0.882509i
\(88\) 0 0
\(89\) 14.8157i 1.57046i 0.619203 + 0.785231i \(0.287456\pi\)
−0.619203 + 0.785231i \(0.712544\pi\)
\(90\) 0 0
\(91\) −1.40786 + 3.76850i −0.147583 + 0.395046i
\(92\) 0 0
\(93\) 7.11575 0.737869
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.76850 0.382633 0.191317 − 0.981528i \(-0.438724\pi\)
0.191317 + 0.981528i \(0.438724\pi\)
\(98\) 0 0
\(99\) 11.3776i 1.14349i
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 1300.2.d.c.649.2 6
5.2 odd 4 260.2.f.a.181.2 yes 6
5.3 odd 4 1300.2.f.e.701.6 6
5.4 even 2 1300.2.d.d.649.5 6
13.12 even 2 1300.2.d.d.649.2 6
15.2 even 4 2340.2.c.d.181.2 6
20.7 even 4 1040.2.k.c.961.6 6
65.12 odd 4 260.2.f.a.181.1 ✓ 6
65.38 odd 4 1300.2.f.e.701.5 6
65.47 even 4 3380.2.a.m.1.1 3
65.57 even 4 3380.2.a.n.1.1 3
65.64 even 2 inner 1300.2.d.c.649.5 6
195.77 even 4 2340.2.c.d.181.5 6
260.207 even 4 1040.2.k.c.961.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.f.a.181.1 ✓ 6 65.12 odd 4
260.2.f.a.181.2 yes 6 5.2 odd 4
1040.2.k.c.961.5 6 260.207 even 4
1040.2.k.c.961.6 6 20.7 even 4
1300.2.d.c.649.2 6 1.1 even 1 trivial
1300.2.d.c.649.5 6 65.64 even 2 inner
1300.2.d.d.649.2 6 13.12 even 2
1300.2.d.d.649.5 6 5.4 even 2
1300.2.f.e.701.5 6 65.38 odd 4
1300.2.f.e.701.6 6 5.3 odd 4
2340.2.c.d.181.2 6 15.2 even 4
2340.2.c.d.181.5 6 195.77 even 4
3380.2.a.m.1.1 3 65.47 even 4
3380.2.a.n.1.1 3 65.57 even 4