Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8464 = 2^{4} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8464.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,-2,0,7,0,-8,0,-1,0,-13,0,4] 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: yes
Analytic conductor: \(67.5853802708\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) 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 23)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.68251\) of defining polynomial
Character \(\chi\) \(=\) 8464.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.478891 q^{3} -1.51334 q^{5} -2.54620 q^{7} -2.77066 q^{9} -3.37279 q^{11} +3.29177 q^{13} +0.724724 q^{15} +6.27686 q^{17} -4.23092 q^{19} +1.21935 q^{21} -2.70981 q^{25} +2.76352 q^{27} +6.28621 q^{29} +4.74204 q^{31} +1.61520 q^{33} +3.85326 q^{35} -0.0379723 q^{37} -1.57640 q^{39} +3.24666 q^{41} +1.28887 q^{43} +4.19295 q^{45} -3.41741 q^{47} -0.516864 q^{49} -3.00593 q^{51} +6.72578 q^{53} +5.10416 q^{55} +2.02615 q^{57} +10.4186 q^{59} -3.59980 q^{61} +7.05466 q^{63} -4.98156 q^{65} -6.81939 q^{67} -4.08974 q^{71} +4.36149 q^{73} +1.29771 q^{75} +8.58779 q^{77} +2.40198 q^{79} +6.98856 q^{81} +2.95715 q^{83} -9.49900 q^{85} -3.01041 q^{87} -12.3607 q^{89} -8.38151 q^{91} -2.27092 q^{93} +6.40281 q^{95} +10.6165 q^{97} +9.34485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{3} + 7 q^{5} - 8 q^{7} - q^{9} - 13 q^{11} + 4 q^{13} - 5 q^{15} + 16 q^{17} - 10 q^{19} + q^{21} - 2 q^{25} + 13 q^{27} + 7 q^{29} - 5 q^{31} + 3 q^{33} - 9 q^{35} - 7 q^{37} + 5 q^{39} + 9 q^{41}+ \cdots + 18 q^{99}+O(q^{100}) \) 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
\(3\) −0.478891 −0.276488 −0.138244 − 0.990398i \(-0.544146\pi\)
−0.138244 + 0.990398i \(0.544146\pi\)
\(4\) 0 0
\(5\) −1.51334 −0.676785 −0.338392 − 0.941005i \(-0.609883\pi\)
−0.338392 + 0.941005i \(0.609883\pi\)
\(6\) 0 0
\(7\) −2.54620 −0.962373 −0.481187 − 0.876618i \(-0.659794\pi\)
−0.481187 + 0.876618i \(0.659794\pi\)
\(8\) 0 0
\(9\) −2.77066 −0.923554
\(10\) 0 0
\(11\) −3.37279 −1.01693 −0.508467 − 0.861082i \(-0.669787\pi\)
−0.508467 + 0.861082i \(0.669787\pi\)
\(12\) 0 0
\(13\) 3.29177 0.912973 0.456487 − 0.889730i \(-0.349108\pi\)
0.456487 + 0.889730i \(0.349108\pi\)
\(14\) 0 0
\(15\) 0.724724 0.187123
\(16\) 0 0
\(17\) 6.27686 1.52236 0.761181 − 0.648540i \(-0.224620\pi\)
0.761181 + 0.648540i \(0.224620\pi\)
\(18\) 0 0
\(19\) −4.23092 −0.970639 −0.485320 − 0.874337i \(-0.661297\pi\)
−0.485320 + 0.874337i \(0.661297\pi\)
\(20\) 0 0
\(21\) 1.21935 0.266085
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −2.70981 −0.541962
\(26\) 0 0
\(27\) 2.76352 0.531840
\(28\) 0 0
\(29\) 6.28621 1.16732 0.583660 − 0.811998i \(-0.301620\pi\)
0.583660 + 0.811998i \(0.301620\pi\)
\(30\) 0 0
\(31\) 4.74204 0.851696 0.425848 − 0.904795i \(-0.359976\pi\)
0.425848 + 0.904795i \(0.359976\pi\)
\(32\) 0 0
\(33\) 1.61520 0.281170
\(34\) 0 0
\(35\) 3.85326 0.651320
\(36\) 0 0
\(37\) −0.0379723 −0.00624261 −0.00312131 − 0.999995i \(-0.500994\pi\)
−0.00312131 + 0.999995i \(0.500994\pi\)
\(38\) 0 0
\(39\) −1.57640 −0.252426
\(40\) 0 0
\(41\) 3.24666 0.507043 0.253521 − 0.967330i \(-0.418411\pi\)
0.253521 + 0.967330i \(0.418411\pi\)
\(42\) 0 0
\(43\) 1.28887 0.196552 0.0982758 − 0.995159i \(-0.468667\pi\)
0.0982758 + 0.995159i \(0.468667\pi\)
\(44\) 0 0
\(45\) 4.19295 0.625048
\(46\) 0 0
\(47\) −3.41741 −0.498480 −0.249240 − 0.968442i \(-0.580181\pi\)
−0.249240 + 0.968442i \(0.580181\pi\)
\(48\) 0 0
\(49\) −0.516864 −0.0738377
\(50\) 0 0
\(51\) −3.00593 −0.420915
\(52\) 0 0
\(53\) 6.72578 0.923857 0.461929 − 0.886917i \(-0.347158\pi\)
0.461929 + 0.886917i \(0.347158\pi\)
\(54\) 0 0
\(55\) 5.10416 0.688245
\(56\) 0 0
\(57\) 2.02615 0.268370
\(58\) 0 0
\(59\) 10.4186 1.35639 0.678194 − 0.734883i \(-0.262763\pi\)
0.678194 + 0.734883i \(0.262763\pi\)
\(60\) 0 0
\(61\) −3.59980 −0.460908 −0.230454 − 0.973083i \(-0.574021\pi\)
−0.230454 + 0.973083i \(0.574021\pi\)
\(62\) 0 0
\(63\) 7.05466 0.888804
\(64\) 0 0
\(65\) −4.98156 −0.617886
\(66\) 0 0
\(67\) −6.81939 −0.833121 −0.416561 − 0.909108i \(-0.636765\pi\)
−0.416561 + 0.909108i \(0.636765\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.08974 −0.485363 −0.242681 − 0.970106i \(-0.578027\pi\)
−0.242681 + 0.970106i \(0.578027\pi\)
\(72\) 0 0
\(73\) 4.36149 0.510473 0.255237 − 0.966879i \(-0.417847\pi\)
0.255237 + 0.966879i \(0.417847\pi\)
\(74\) 0 0
\(75\) 1.29771 0.149846
\(76\) 0 0
\(77\) 8.58779 0.978669
\(78\) 0 0
\(79\) 2.40198 0.270244 0.135122 − 0.990829i \(-0.456857\pi\)
0.135122 + 0.990829i \(0.456857\pi\)
\(80\) 0 0
\(81\) 6.98856 0.776507
\(82\) 0 0
\(83\) 2.95715 0.324590 0.162295 − 0.986742i \(-0.448110\pi\)
0.162295 + 0.986742i \(0.448110\pi\)
\(84\) 0 0
\(85\) −9.49900 −1.03031
\(86\) 0 0
\(87\) −3.01041 −0.322750
\(88\) 0 0
\(89\) −12.3607 −1.31023 −0.655117 − 0.755528i \(-0.727381\pi\)
−0.655117 + 0.755528i \(0.727381\pi\)
\(90\) 0 0
\(91\) −8.38151 −0.878621
\(92\) 0 0
\(93\) −2.27092 −0.235484
\(94\) 0 0
\(95\) 6.40281 0.656914
\(96\) 0 0
\(97\) 10.6165 1.07794 0.538971 − 0.842324i \(-0.318813\pi\)
0.538971 + 0.842324i \(0.318813\pi\)
\(98\) 0 0
\(99\) 9.34485 0.939193
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 8464.2.a.bt.1.3 5
4.3 odd 2 529.2.a.j.1.2 5
12.11 even 2 4761.2.a.bn.1.4 5
23.15 odd 22 368.2.m.c.225.1 10
23.20 odd 22 368.2.m.c.193.1 10
23.22 odd 2 8464.2.a.bs.1.3 5
92.3 odd 22 529.2.c.a.170.1 10
92.7 even 22 529.2.c.d.118.1 10
92.11 even 22 529.2.c.i.466.1 10
92.15 even 22 23.2.c.a.18.1 yes 10
92.19 even 22 529.2.c.g.177.1 10
92.27 odd 22 529.2.c.f.177.1 10
92.31 odd 22 529.2.c.a.501.1 10
92.35 odd 22 529.2.c.h.466.1 10
92.39 odd 22 529.2.c.e.118.1 10
92.43 even 22 23.2.c.a.9.1 ✓ 10
92.51 even 22 529.2.c.b.255.1 10
92.55 odd 22 529.2.c.c.334.1 10
92.59 odd 22 529.2.c.e.399.1 10
92.63 even 22 529.2.c.g.266.1 10
92.67 even 22 529.2.c.i.487.1 10
92.71 odd 22 529.2.c.h.487.1 10
92.75 odd 22 529.2.c.f.266.1 10
92.79 even 22 529.2.c.d.399.1 10
92.83 even 22 529.2.c.b.334.1 10
92.87 odd 22 529.2.c.c.255.1 10
92.91 even 2 529.2.a.i.1.2 5
276.107 odd 22 207.2.i.c.64.1 10
276.227 odd 22 207.2.i.c.55.1 10
276.275 odd 2 4761.2.a.bo.1.4 5
460.43 odd 44 575.2.p.b.124.2 20
460.107 odd 44 575.2.p.b.524.2 20
460.199 even 22 575.2.k.b.501.1 10
460.227 odd 44 575.2.p.b.124.1 20
460.319 even 22 575.2.k.b.101.1 10
460.383 odd 44 575.2.p.b.524.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.2.c.a.9.1 ✓ 10 92.43 even 22
23.2.c.a.18.1 yes 10 92.15 even 22
207.2.i.c.55.1 10 276.227 odd 22
207.2.i.c.64.1 10 276.107 odd 22
368.2.m.c.193.1 10 23.20 odd 22
368.2.m.c.225.1 10 23.15 odd 22
529.2.a.i.1.2 5 92.91 even 2
529.2.a.j.1.2 5 4.3 odd 2
529.2.c.a.170.1 10 92.3 odd 22
529.2.c.a.501.1 10 92.31 odd 22
529.2.c.b.255.1 10 92.51 even 22
529.2.c.b.334.1 10 92.83 even 22
529.2.c.c.255.1 10 92.87 odd 22
529.2.c.c.334.1 10 92.55 odd 22
529.2.c.d.118.1 10 92.7 even 22
529.2.c.d.399.1 10 92.79 even 22
529.2.c.e.118.1 10 92.39 odd 22
529.2.c.e.399.1 10 92.59 odd 22
529.2.c.f.177.1 10 92.27 odd 22
529.2.c.f.266.1 10 92.75 odd 22
529.2.c.g.177.1 10 92.19 even 22
529.2.c.g.266.1 10 92.63 even 22
529.2.c.h.466.1 10 92.35 odd 22
529.2.c.h.487.1 10 92.71 odd 22
529.2.c.i.466.1 10 92.11 even 22
529.2.c.i.487.1 10 92.67 even 22
575.2.k.b.101.1 10 460.319 even 22
575.2.k.b.501.1 10 460.199 even 22
575.2.p.b.124.1 20 460.227 odd 44
575.2.p.b.124.2 20 460.43 odd 44
575.2.p.b.524.1 20 460.383 odd 44
575.2.p.b.524.2 20 460.107 odd 44
4761.2.a.bn.1.4 5 12.11 even 2
4761.2.a.bo.1.4 5 276.275 odd 2
8464.2.a.bs.1.3 5 23.22 odd 2
8464.2.a.bt.1.3 5 1.1 even 1 trivial