Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [45,6,Mod(19,45)] 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("45.19"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 45.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.21727189158\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 19.1
Root \(0.500000 + 1.65831i\) of defining polynomial
Character \(\chi\) \(=\) 45.19
Dual form 45.6.b.b.19.2

$q$-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)
 
\(f(q)\) \(=\) \(q-6.63325i q^{2} -12.0000 q^{4} +(45.0000 - 33.1662i) q^{5} -59.6992i q^{7} -132.665i q^{8} +(-220.000 - 298.496i) q^{10} -252.000 q^{11} -119.398i q^{13} -396.000 q^{14} -1264.00 q^{16} +689.858i q^{17} -220.000 q^{19} +(-540.000 + 397.995i) q^{20} +1671.58i q^{22} -2434.40i q^{23} +(925.000 - 2984.96i) q^{25} -792.000 q^{26} +716.391i q^{28} +6930.00 q^{29} +6752.00 q^{31} +4139.15i q^{32} +4576.00 q^{34} +(-1980.00 - 2686.47i) q^{35} +13969.6i q^{37} +1459.31i q^{38} +(-4400.00 - 5969.92i) q^{40} +198.000 q^{41} -417.895i q^{43} +3024.00 q^{44} -16148.0 q^{46} +10540.2i q^{47} +13243.0 q^{49} +(-19800.0 - 6135.76i) q^{50} +1432.78i q^{52} +5823.99i q^{53} +(-11340.0 + 8357.89i) q^{55} -7920.00 q^{56} -45968.4i q^{58} +24660.0 q^{59} -5698.00 q^{61} -44787.7i q^{62} -12992.0 q^{64} +(-3960.00 - 5372.93i) q^{65} -43640.1i q^{67} -8278.30i q^{68} +(-17820.0 + 13133.8i) q^{70} -53352.0 q^{71} +70922.7i q^{73} +92664.0 q^{74} +2640.00 q^{76} +15044.2i q^{77} +51920.0 q^{79} +(-56880.0 + 41922.1i) q^{80} -1313.38i q^{82} +61841.8i q^{83} +(22880.0 + 31043.6i) q^{85} -2772.00 q^{86} +33431.6i q^{88} +9990.00 q^{89} -7128.00 q^{91} +29212.8i q^{92} +69916.0 q^{94} +(-9900.00 + 7296.57i) q^{95} -101250. i q^{97} -87844.1i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{4} + 90 q^{5} - 440 q^{10} - 504 q^{11} - 792 q^{14} - 2528 q^{16} - 440 q^{19} - 1080 q^{20} + 1850 q^{25} - 1584 q^{26} + 13860 q^{29} + 13504 q^{31} + 9152 q^{34} - 3960 q^{35} - 8800 q^{40}+ \cdots - 19800 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 6.63325i − 1.17260i −0.810093 − 0.586302i \(-0.800583\pi\)
0.810093 − 0.586302i \(-0.199417\pi\)
\(3\) 0 0
\(4\) −12.0000 −0.375000
\(5\) 45.0000 − 33.1662i 0.804984 − 0.593296i
\(6\) 0 0
\(7\) − 59.6992i − 0.460494i −0.973132 − 0.230247i \(-0.926047\pi\)
0.973132 − 0.230247i \(-0.0739534\pi\)
\(8\) − 132.665i − 0.732877i
\(9\) 0 0
\(10\) −220.000 − 298.496i −0.695701 − 0.943928i
\(11\) −252.000 −0.627941 −0.313970 − 0.949433i \(-0.601659\pi\)
−0.313970 + 0.949433i \(0.601659\pi\)
\(12\) 0 0
\(13\) − 119.398i − 0.195948i −0.995189 − 0.0979739i \(-0.968764\pi\)
0.995189 − 0.0979739i \(-0.0312362\pi\)
\(14\) −396.000 −0.539977
\(15\) 0 0
\(16\) −1264.00 −1.23438
\(17\) 689.858i 0.578945i 0.957186 + 0.289473i \(0.0934799\pi\)
−0.957186 + 0.289473i \(0.906520\pi\)
\(18\) 0 0
\(19\) −220.000 −0.139810 −0.0699051 − 0.997554i \(-0.522270\pi\)
−0.0699051 + 0.997554i \(0.522270\pi\)
\(20\) −540.000 + 397.995i −0.301869 + 0.222486i
\(21\) 0 0
\(22\) 1671.58i 0.736326i
\(23\) − 2434.40i − 0.959561i −0.877388 − 0.479781i \(-0.840716\pi\)
0.877388 − 0.479781i \(-0.159284\pi\)
\(24\) 0 0
\(25\) 925.000 − 2984.96i 0.296000 − 0.955188i
\(26\) −792.000 −0.229769
\(27\) 0 0
\(28\) 716.391i 0.172685i
\(29\) 6930.00 1.53016 0.765082 − 0.643932i \(-0.222698\pi\)
0.765082 + 0.643932i \(0.222698\pi\)
\(30\) 0 0
\(31\) 6752.00 1.26191 0.630955 − 0.775820i \(-0.282663\pi\)
0.630955 + 0.775820i \(0.282663\pi\)
\(32\) 4139.15i 0.714556i
\(33\) 0 0
\(34\) 4576.00 0.678873
\(35\) −1980.00 − 2686.47i −0.273209 − 0.370690i
\(36\) 0 0
\(37\) 13969.6i 1.67757i 0.544464 + 0.838785i \(0.316733\pi\)
−0.544464 + 0.838785i \(0.683267\pi\)
\(38\) 1459.31i 0.163942i
\(39\) 0 0
\(40\) −4400.00 − 5969.92i −0.434813 − 0.589955i
\(41\) 198.000 0.0183952 0.00919762 − 0.999958i \(-0.497072\pi\)
0.00919762 + 0.999958i \(0.497072\pi\)
\(42\) 0 0
\(43\) − 417.895i − 0.0344664i −0.999851 − 0.0172332i \(-0.994514\pi\)
0.999851 − 0.0172332i \(-0.00548577\pi\)
\(44\) 3024.00 0.235478
\(45\) 0 0
\(46\) −16148.0 −1.12519
\(47\) 10540.2i 0.695994i 0.937496 + 0.347997i \(0.113138\pi\)
−0.937496 + 0.347997i \(0.886862\pi\)
\(48\) 0 0
\(49\) 13243.0 0.787945
\(50\) −19800.0 − 6135.76i −1.12006 − 0.347091i
\(51\) 0 0
\(52\) 1432.78i 0.0734804i
\(53\) 5823.99i 0.284794i 0.989810 + 0.142397i \(0.0454810\pi\)
−0.989810 + 0.142397i \(0.954519\pi\)
\(54\) 0 0
\(55\) −11340.0 + 8357.89i −0.505483 + 0.372555i
\(56\) −7920.00 −0.337485
\(57\) 0 0
\(58\) − 45968.4i − 1.79428i
\(59\) 24660.0 0.922281 0.461140 − 0.887327i \(-0.347440\pi\)
0.461140 + 0.887327i \(0.347440\pi\)
\(60\) 0 0
\(61\) −5698.00 −0.196064 −0.0980320 − 0.995183i \(-0.531255\pi\)
−0.0980320 + 0.995183i \(0.531255\pi\)
\(62\) − 44787.7i − 1.47972i
\(63\) 0 0
\(64\) −12992.0 −0.396484
\(65\) −3960.00 − 5372.93i −0.116255 − 0.157735i
\(66\) 0 0
\(67\) − 43640.1i − 1.18768i −0.804583 − 0.593840i \(-0.797611\pi\)
0.804583 − 0.593840i \(-0.202389\pi\)
\(68\) − 8278.30i − 0.217104i
\(69\) 0 0
\(70\) −17820.0 + 13133.8i −0.434673 + 0.320366i
\(71\) −53352.0 −1.25604 −0.628022 − 0.778196i \(-0.716135\pi\)
−0.628022 + 0.778196i \(0.716135\pi\)
\(72\) 0 0
\(73\) 70922.7i 1.55768i 0.627223 + 0.778840i \(0.284192\pi\)
−0.627223 + 0.778840i \(0.715808\pi\)
\(74\) 92664.0 1.96712
\(75\) 0 0
\(76\) 2640.00 0.0524288
\(77\) 15044.2i 0.289163i
\(78\) 0 0
\(79\) 51920.0 0.935981 0.467990 − 0.883734i \(-0.344978\pi\)
0.467990 + 0.883734i \(0.344978\pi\)
\(80\) −56880.0 + 41922.1i −0.993653 + 0.732350i
\(81\) 0 0
\(82\) − 1313.38i − 0.0215703i
\(83\) 61841.8i 0.985342i 0.870216 + 0.492671i \(0.163979\pi\)
−0.870216 + 0.492671i \(0.836021\pi\)
\(84\) 0 0
\(85\) 22880.0 + 31043.6i 0.343486 + 0.466042i
\(86\) −2772.00 −0.0404154
\(87\) 0 0
\(88\) 33431.6i 0.460204i
\(89\) 9990.00 0.133687 0.0668437 − 0.997763i \(-0.478707\pi\)
0.0668437 + 0.997763i \(0.478707\pi\)
\(90\) 0 0
\(91\) −7128.00 −0.0902328
\(92\) 29212.8i 0.359836i
\(93\) 0 0
\(94\) 69916.0 0.816125
\(95\) −9900.00 + 7296.57i −0.112545 + 0.0829488i
\(96\) 0 0
\(97\) − 101250.i − 1.09261i −0.837586 − 0.546305i \(-0.816034\pi\)
0.837586 − 0.546305i \(-0.183966\pi\)
\(98\) − 87844.1i − 0.923948i
\(99\) 0 0
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 45.6.b.b.19.1 2
3.2 odd 2 5.6.b.a.4.2 yes 2
4.3 odd 2 720.6.f.f.289.1 2
5.2 odd 4 225.6.a.n.1.2 2
5.3 odd 4 225.6.a.n.1.1 2
5.4 even 2 inner 45.6.b.b.19.2 2
12.11 even 2 80.6.c.a.49.2 2
15.2 even 4 25.6.a.c.1.1 2
15.8 even 4 25.6.a.c.1.2 2
15.14 odd 2 5.6.b.a.4.1 ✓ 2
20.19 odd 2 720.6.f.f.289.2 2
21.20 even 2 245.6.b.a.99.2 2
24.5 odd 2 320.6.c.f.129.2 2
24.11 even 2 320.6.c.g.129.1 2
60.23 odd 4 400.6.a.t.1.1 2
60.47 odd 4 400.6.a.t.1.2 2
60.59 even 2 80.6.c.a.49.1 2
105.104 even 2 245.6.b.a.99.1 2
120.29 odd 2 320.6.c.f.129.1 2
120.59 even 2 320.6.c.g.129.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.b.a.4.1 ✓ 2 15.14 odd 2
5.6.b.a.4.2 yes 2 3.2 odd 2
25.6.a.c.1.1 2 15.2 even 4
25.6.a.c.1.2 2 15.8 even 4
45.6.b.b.19.1 2 1.1 even 1 trivial
45.6.b.b.19.2 2 5.4 even 2 inner
80.6.c.a.49.1 2 60.59 even 2
80.6.c.a.49.2 2 12.11 even 2
225.6.a.n.1.1 2 5.3 odd 4
225.6.a.n.1.2 2 5.2 odd 4
245.6.b.a.99.1 2 105.104 even 2
245.6.b.a.99.2 2 21.20 even 2
320.6.c.f.129.1 2 120.29 odd 2
320.6.c.f.129.2 2 24.5 odd 2
320.6.c.g.129.1 2 24.11 even 2
320.6.c.g.129.2 2 120.59 even 2
400.6.a.t.1.1 2 60.23 odd 4
400.6.a.t.1.2 2 60.47 odd 4
720.6.f.f.289.1 2 4.3 odd 2
720.6.f.f.289.2 2 20.19 odd 2