Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [465,2,Mod(119,465)] 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("465.119"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 119.3
Root \(1.40294 + 1.01575i\) of defining polynomial
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.c.254.3

$q$-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+2.44949 q^{2} +(-0.178197 + 1.72286i) q^{3} +4.00000 q^{4} +(1.93649 - 1.11803i) q^{5} +(-0.436492 + 4.22013i) q^{6} +4.89898 q^{8} +(-2.93649 - 0.614017i) q^{9} +(4.74342 - 2.73861i) q^{10} +(-1.93649 - 3.35410i) q^{11} +(-0.712788 + 6.89144i) q^{12} +(1.58114 + 2.73861i) q^{13} +(1.58114 + 3.53553i) q^{15} +4.00000 q^{16} +(-4.89898 - 2.82843i) q^{17} +(-7.19291 - 1.50403i) q^{18} +(-1.00000 + 1.73205i) q^{19} +(7.74597 - 4.47214i) q^{20} +(-4.74342 - 8.21584i) q^{22} +5.65685i q^{23} +(-0.872983 + 8.44025i) q^{24} +(2.50000 - 4.33013i) q^{25} +(3.87298 + 6.70820i) q^{26} +(1.58114 - 4.94975i) q^{27} -3.87298 q^{29} +(3.87298 + 8.66025i) q^{30} +(-3.50000 + 4.33013i) q^{31} +(6.12372 - 2.73861i) q^{33} +(-12.0000 - 6.92820i) q^{34} +(-11.7460 - 2.45607i) q^{36} +(-3.16228 + 5.47723i) q^{37} +(-2.44949 + 4.24264i) q^{38} +(-5.00000 + 2.23607i) q^{39} +(9.48683 - 5.47723i) q^{40} +(9.68246 - 5.59017i) q^{41} +(1.58114 - 2.73861i) q^{43} +(-7.74597 - 13.4164i) q^{44} +(-6.37298 + 2.09406i) q^{45} +13.8564i q^{46} +9.79796 q^{47} +(-0.712788 + 6.89144i) q^{48} +(-3.50000 - 6.06218i) q^{49} +(6.12372 - 10.6066i) q^{50} +(5.74597 - 7.93624i) q^{51} +(6.32456 + 10.9545i) q^{52} +(4.89898 - 2.82843i) q^{53} +(3.87298 - 12.1244i) q^{54} +(-7.50000 - 4.33013i) q^{55} +(-2.80588 - 2.03151i) q^{57} -9.48683 q^{58} +(1.93649 + 1.11803i) q^{59} +(6.32456 + 14.1421i) q^{60} +1.73205i q^{61} +(-8.57321 + 10.6066i) q^{62} -8.00000 q^{64} +(6.12372 + 3.53553i) q^{65} +(15.0000 - 6.70820i) q^{66} +(-19.5959 - 11.3137i) q^{68} +(-9.74597 - 1.00803i) q^{69} +(-1.93649 + 1.11803i) q^{71} +(-14.3858 - 3.00806i) q^{72} +(1.58114 + 2.73861i) q^{73} +(-7.74597 + 13.4164i) q^{74} +(7.01471 + 5.07877i) q^{75} +(-4.00000 + 6.92820i) q^{76} +(-12.2474 + 5.47723i) q^{78} +(4.50000 + 2.59808i) q^{79} +(7.74597 - 4.47214i) q^{80} +(8.24597 + 3.60611i) q^{81} +(23.7171 - 13.6931i) q^{82} +(8.57321 - 4.94975i) q^{83} -12.6491 q^{85} +(3.87298 - 6.70820i) q^{86} +(0.690154 - 6.67261i) q^{87} +(-9.48683 - 16.4317i) q^{88} -11.6190 q^{89} +(-15.6106 + 5.12938i) q^{90} +22.6274i q^{92} +(-6.83651 - 6.80162i) q^{93} +24.0000 q^{94} +4.47214i q^{95} -10.9545i q^{97} +(-8.57321 - 14.8492i) q^{98} +(3.62702 + 11.0383i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{4} + 12 q^{6} - 8 q^{9} + 32 q^{16} - 8 q^{19} + 24 q^{24} + 20 q^{25} - 28 q^{31} - 96 q^{34} - 32 q^{36} - 40 q^{39} - 20 q^{45} - 28 q^{49} - 16 q^{51} - 60 q^{55} - 64 q^{64} + 120 q^{66}+ \cdots + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.44949 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(3\) −0.178197 + 1.72286i −0.102882 + 0.994694i
\(4\) 4.00000 2.00000
\(5\) 1.93649 − 1.11803i 0.866025 − 0.500000i
\(6\) −0.436492 + 4.22013i −0.178197 + 1.72286i
\(7\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 4.89898 1.73205
\(9\) −2.93649 − 0.614017i −0.978831 − 0.204672i
\(10\) 4.74342 − 2.73861i 1.50000 − 0.866025i
\(11\) −1.93649 − 3.35410i −0.583874 − 1.01130i −0.995015 − 0.0997278i \(-0.968203\pi\)
0.411141 − 0.911572i \(-0.365131\pi\)
\(12\) −0.712788 + 6.89144i −0.205764 + 1.98939i
\(13\) 1.58114 + 2.73861i 0.438529 + 0.759555i 0.997576 − 0.0695813i \(-0.0221663\pi\)
−0.559047 + 0.829136i \(0.688833\pi\)
\(14\) 0 0
\(15\) 1.58114 + 3.53553i 0.408248 + 0.912871i
\(16\) 4.00000 1.00000
\(17\) −4.89898 − 2.82843i −1.18818 − 0.685994i −0.230285 − 0.973123i \(-0.573966\pi\)
−0.957892 + 0.287129i \(0.907299\pi\)
\(18\) −7.19291 − 1.50403i −1.69538 − 0.354503i
\(19\) −1.00000 + 1.73205i −0.229416 + 0.397360i −0.957635 − 0.287984i \(-0.907015\pi\)
0.728219 + 0.685344i \(0.240348\pi\)
\(20\) 7.74597 − 4.47214i 1.73205 − 1.00000i
\(21\) 0 0
\(22\) −4.74342 − 8.21584i −1.01130 − 1.75162i
\(23\) 5.65685i 1.17954i 0.807573 + 0.589768i \(0.200781\pi\)
−0.807573 + 0.589768i \(0.799219\pi\)
\(24\) −0.872983 + 8.44025i −0.178197 + 1.72286i
\(25\) 2.50000 − 4.33013i 0.500000 − 0.866025i
\(26\) 3.87298 + 6.70820i 0.759555 + 1.31559i
\(27\) 1.58114 − 4.94975i 0.304290 − 0.952579i
\(28\) 0 0
\(29\) −3.87298 −0.719195 −0.359597 − 0.933108i \(-0.617086\pi\)
−0.359597 + 0.933108i \(0.617086\pi\)
\(30\) 3.87298 + 8.66025i 0.707107 + 1.58114i
\(31\) −3.50000 + 4.33013i −0.628619 + 0.777714i
\(32\) 0 0
\(33\) 6.12372 − 2.73861i 1.06600 − 0.476731i
\(34\) −12.0000 − 6.92820i −2.05798 − 1.18818i
\(35\) 0 0
\(36\) −11.7460 − 2.45607i −1.95766 − 0.409345i
\(37\) −3.16228 + 5.47723i −0.519875 + 0.900450i 0.479858 + 0.877346i \(0.340688\pi\)
−0.999733 + 0.0231041i \(0.992645\pi\)
\(38\) −2.44949 + 4.24264i −0.397360 + 0.688247i
\(39\) −5.00000 + 2.23607i −0.800641 + 0.358057i
\(40\) 9.48683 − 5.47723i 1.50000 − 0.866025i
\(41\) 9.68246 − 5.59017i 1.51215 − 0.873038i 0.512247 − 0.858838i \(-0.328813\pi\)
0.999899 − 0.0141996i \(-0.00452001\pi\)
\(42\) 0 0
\(43\) 1.58114 − 2.73861i 0.241121 − 0.417635i −0.719913 − 0.694065i \(-0.755818\pi\)
0.961034 + 0.276430i \(0.0891515\pi\)
\(44\) −7.74597 − 13.4164i −1.16775 − 2.02260i
\(45\) −6.37298 + 2.09406i −0.950028 + 0.312164i
\(46\) 13.8564i 2.04302i
\(47\) 9.79796 1.42918 0.714590 − 0.699544i \(-0.246613\pi\)
0.714590 + 0.699544i \(0.246613\pi\)
\(48\) −0.712788 + 6.89144i −0.102882 + 0.994694i
\(49\) −3.50000 − 6.06218i −0.500000 − 0.866025i
\(50\) 6.12372 − 10.6066i 0.866025 − 1.50000i
\(51\) 5.74597 − 7.93624i 0.804596 − 1.11130i
\(52\) 6.32456 + 10.9545i 0.877058 + 1.51911i
\(53\) 4.89898 − 2.82843i 0.672927 − 0.388514i −0.124258 − 0.992250i \(-0.539655\pi\)
0.797185 + 0.603736i \(0.206322\pi\)
\(54\) 3.87298 − 12.1244i 0.527046 − 1.64992i
\(55\) −7.50000 − 4.33013i −1.01130 − 0.583874i
\(56\) 0 0
\(57\) −2.80588 − 2.03151i −0.371648 − 0.269080i
\(58\) −9.48683 −1.24568
\(59\) 1.93649 + 1.11803i 0.252110 + 0.145556i 0.620730 − 0.784024i \(-0.286836\pi\)
−0.368620 + 0.929580i \(0.620170\pi\)
\(60\) 6.32456 + 14.1421i 0.816497 + 1.82574i
\(61\) 1.73205i 0.221766i 0.993833 + 0.110883i \(0.0353679\pi\)
−0.993833 + 0.110883i \(0.964632\pi\)
\(62\) −8.57321 + 10.6066i −1.08880 + 1.34704i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 6.12372 + 3.53553i 0.759555 + 0.438529i
\(66\) 15.0000 − 6.70820i 1.84637 − 0.825723i
\(67\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) −19.5959 − 11.3137i −2.37635 − 1.37199i
\(69\) −9.74597 − 1.00803i −1.17328 − 0.121353i
\(70\) 0 0
\(71\) −1.93649 + 1.11803i −0.229819 + 0.132686i −0.610489 − 0.792025i \(-0.709027\pi\)
0.380669 + 0.924711i \(0.375694\pi\)
\(72\) −14.3858 − 3.00806i −1.69538 − 0.354503i
\(73\) 1.58114 + 2.73861i 0.185058 + 0.320530i 0.943596 − 0.331099i \(-0.107419\pi\)
−0.758538 + 0.651629i \(0.774086\pi\)
\(74\) −7.74597 + 13.4164i −0.900450 + 1.55963i
\(75\) 7.01471 + 5.07877i 0.809989 + 0.586445i
\(76\) −4.00000 + 6.92820i −0.458831 + 0.794719i
\(77\) 0 0
\(78\) −12.2474 + 5.47723i −1.38675 + 0.620174i
\(79\) 4.50000 + 2.59808i 0.506290 + 0.292306i 0.731307 − 0.682048i \(-0.238911\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 7.74597 − 4.47214i 0.866025 − 0.500000i
\(81\) 8.24597 + 3.60611i 0.916219 + 0.400679i
\(82\) 23.7171 − 13.6931i 2.61911 − 1.51215i
\(83\) 8.57321 − 4.94975i 0.941033 − 0.543305i 0.0507487 − 0.998711i \(-0.483839\pi\)
0.890284 + 0.455406i \(0.150506\pi\)
\(84\) 0 0
\(85\) −12.6491 −1.37199
\(86\) 3.87298 − 6.70820i 0.417635 − 0.723364i
\(87\) 0.690154 − 6.67261i 0.0739923 − 0.715379i
\(88\) −9.48683 − 16.4317i −1.01130 − 1.75162i
\(89\) −11.6190 −1.23161 −0.615803 − 0.787900i \(-0.711168\pi\)
−0.615803 + 0.787900i \(0.711168\pi\)
\(90\) −15.6106 + 5.12938i −1.64550 + 0.540684i
\(91\) 0 0
\(92\) 22.6274i 2.35907i
\(93\) −6.83651 − 6.80162i −0.708913 − 0.705296i
\(94\) 24.0000 2.47541
\(95\) 4.47214i 0.458831i
\(96\) 0 0
\(97\) − 10.9545i − 1.11226i −0.831097 − 0.556128i \(-0.812286\pi\)
0.831097 − 0.556128i \(-0.187714\pi\)
\(98\) −8.57321 − 14.8492i −0.866025 − 1.50000i
\(99\) 3.62702 + 11.0383i 0.364529 + 1.10939i
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 465.2.t.c.119.3 yes 8
3.2 odd 2 inner 465.2.t.c.119.1 ✓ 8
5.4 even 2 inner 465.2.t.c.119.2 yes 8
15.14 odd 2 inner 465.2.t.c.119.4 yes 8
31.6 odd 6 inner 465.2.t.c.254.4 yes 8
93.68 even 6 inner 465.2.t.c.254.2 yes 8
155.99 odd 6 inner 465.2.t.c.254.1 yes 8
465.254 even 6 inner 465.2.t.c.254.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.c.119.1 ✓ 8 3.2 odd 2 inner
465.2.t.c.119.2 yes 8 5.4 even 2 inner
465.2.t.c.119.3 yes 8 1.1 even 1 trivial
465.2.t.c.119.4 yes 8 15.14 odd 2 inner
465.2.t.c.254.1 yes 8 155.99 odd 6 inner
465.2.t.c.254.2 yes 8 93.68 even 6 inner
465.2.t.c.254.3 yes 8 465.254 even 6 inner
465.2.t.c.254.4 yes 8 31.6 odd 6 inner