↑

Properties

Label 45.18.a.d.1.1
Level $45$
Weight $18$
Character 45.1
Self dual yes
Analytic conductor $82.450$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [45,18,Mod(1,45)] 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("45.1"); S:= CuspForms(chi, 18); 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, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,253] 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: yes
Analytic conductor: \(82.4499393051\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 182396x + 3921120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-436.958\) of defining polynomial
Character \(\chi\) \(=\) 45.1

$q$-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)
 
\(f(q)\) \(=\) \(q-352.958 q^{2} -6492.31 q^{4} +390625. q^{5} +1.07009e7 q^{7} +4.85545e7 q^{8} -1.37874e8 q^{10} -1.00429e9 q^{11} +3.43757e9 q^{13} -3.77697e9 q^{14} -1.62868e10 q^{16} -4.81590e10 q^{17} +8.08158e9 q^{19} -2.53606e9 q^{20} +3.54473e11 q^{22} +3.08039e11 q^{23} +1.52588e11 q^{25} -1.21332e12 q^{26} -6.94735e10 q^{28} -7.97487e11 q^{29} -3.17371e12 q^{31} -6.15585e11 q^{32} +1.69981e13 q^{34} +4.18003e12 q^{35} +1.61437e13 q^{37} -2.85246e12 q^{38} +1.89666e13 q^{40} -7.63188e13 q^{41} +1.29020e14 q^{43} +6.52018e12 q^{44} -1.08725e14 q^{46} +2.44935e14 q^{47} -1.18121e14 q^{49} -5.38572e13 q^{50} -2.23178e13 q^{52} +8.60208e14 q^{53} -3.92301e14 q^{55} +5.19576e14 q^{56} +2.81480e14 q^{58} -1.21582e14 q^{59} -4.27987e14 q^{61} +1.12019e15 q^{62} +2.35201e15 q^{64} +1.34280e15 q^{65} -1.16734e15 q^{67} +3.12664e14 q^{68} -1.47538e15 q^{70} -3.01633e15 q^{71} -7.32396e15 q^{73} -5.69806e15 q^{74} -5.24682e13 q^{76} -1.07468e16 q^{77} -7.07645e15 q^{79} -6.36201e15 q^{80} +2.69374e16 q^{82} +1.18761e16 q^{83} -1.88121e16 q^{85} -4.55388e16 q^{86} -4.87629e16 q^{88} -5.69376e16 q^{89} +3.67851e16 q^{91} -1.99989e15 q^{92} -8.64519e16 q^{94} +3.15687e15 q^{95} -8.11777e16 q^{97} +4.16920e16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 253 q^{2} - 7087 q^{4} + 1171875 q^{5} - 4332484 q^{7} + 16188513 q^{8} + 98828125 q^{10} - 943563680 q^{11} + 4257013150 q^{13} - 1847483988 q^{14} - 21943871359 q^{16} - 3063169514 q^{17} - 78122492996 q^{19}+ \cdots + 896231490630245 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −352.958 −0.974919 −0.487460 − 0.873146i \(-0.662076\pi\)
−0.487460 + 0.873146i \(0.662076\pi\)
\(3\) 0 0
\(4\) −6492.31 −0.0495324
\(5\) 390625. 0.447214
\(6\) 0 0
\(7\) 1.07009e7 0.701595 0.350798 − 0.936451i \(-0.385911\pi\)
0.350798 + 0.936451i \(0.385911\pi\)
\(8\) 4.85545e7 1.02321
\(9\) 0 0
\(10\) −1.37874e8 −0.435997
\(11\) −1.00429e9 −1.41261 −0.706305 − 0.707908i \(-0.749639\pi\)
−0.706305 + 0.707908i \(0.749639\pi\)
\(12\) 0 0
\(13\) 3.43757e9 1.16878 0.584391 − 0.811473i \(-0.301334\pi\)
0.584391 + 0.811473i \(0.301334\pi\)
\(14\) −3.77697e9 −0.683999
\(15\) 0 0
\(16\) −1.62868e10 −0.948014
\(17\) −4.81590e10 −1.67441 −0.837205 − 0.546889i \(-0.815812\pi\)
−0.837205 + 0.546889i \(0.815812\pi\)
\(18\) 0 0
\(19\) 8.08158e9 0.109167 0.0545834 − 0.998509i \(-0.482617\pi\)
0.0545834 + 0.998509i \(0.482617\pi\)
\(20\) −2.53606e9 −0.0221516
\(21\) 0 0
\(22\) 3.54473e11 1.37718
\(23\) 3.08039e11 0.820199 0.410099 − 0.912041i \(-0.365494\pi\)
0.410099 + 0.912041i \(0.365494\pi\)
\(24\) 0 0
\(25\) 1.52588e11 0.200000
\(26\) −1.21332e12 −1.13947
\(27\) 0 0
\(28\) −6.94735e10 −0.0347517
\(29\) −7.97487e11 −0.296033 −0.148017 − 0.988985i \(-0.547289\pi\)
−0.148017 + 0.988985i \(0.547289\pi\)
\(30\) 0 0
\(31\) −3.17371e12 −0.668334 −0.334167 − 0.942514i \(-0.608455\pi\)
−0.334167 + 0.942514i \(0.608455\pi\)
\(32\) −6.15585e11 −0.0989722
\(33\) 0 0
\(34\) 1.69981e13 1.63241
\(35\) 4.18003e12 0.313763
\(36\) 0 0
\(37\) 1.61437e13 0.755595 0.377797 − 0.925888i \(-0.376682\pi\)
0.377797 + 0.925888i \(0.376682\pi\)
\(38\) −2.85246e12 −0.106429
\(39\) 0 0
\(40\) 1.89666e13 0.457593
\(41\) −7.63188e13 −1.49269 −0.746344 − 0.665560i \(-0.768193\pi\)
−0.746344 + 0.665560i \(0.768193\pi\)
\(42\) 0 0
\(43\) 1.29020e14 1.68336 0.841679 − 0.539978i \(-0.181568\pi\)
0.841679 + 0.539978i \(0.181568\pi\)
\(44\) 6.52018e12 0.0699700
\(45\) 0 0
\(46\) −1.08725e14 −0.799628
\(47\) 2.44935e14 1.50044 0.750221 − 0.661187i \(-0.229947\pi\)
0.750221 + 0.661187i \(0.229947\pi\)
\(48\) 0 0
\(49\) −1.18121e14 −0.507764
\(50\) −5.38572e13 −0.194984
\(51\) 0 0
\(52\) −2.23178e13 −0.0578926
\(53\) 8.60208e14 1.89784 0.948918 − 0.315523i \(-0.102180\pi\)
0.948918 + 0.315523i \(0.102180\pi\)
\(54\) 0 0
\(55\) −3.92301e14 −0.631738
\(56\) 5.19576e14 0.717879
\(57\) 0 0
\(58\) 2.81480e14 0.288609
\(59\) −1.21582e14 −0.107802 −0.0539012 − 0.998546i \(-0.517166\pi\)
−0.0539012 + 0.998546i \(0.517166\pi\)
\(60\) 0 0
\(61\) −4.27987e14 −0.285842 −0.142921 − 0.989734i \(-0.545650\pi\)
−0.142921 + 0.989734i \(0.545650\pi\)
\(62\) 1.12019e15 0.651572
\(63\) 0 0
\(64\) 2.35201e15 1.04450
\(65\) 1.34280e15 0.522695
\(66\) 0 0
\(67\) −1.16734e15 −0.351206 −0.175603 − 0.984461i \(-0.556187\pi\)
−0.175603 + 0.984461i \(0.556187\pi\)
\(68\) 3.12664e14 0.0829376
\(69\) 0 0
\(70\) −1.47538e15 −0.305893
\(71\) −3.01633e15 −0.554348 −0.277174 − 0.960820i \(-0.589398\pi\)
−0.277174 + 0.960820i \(0.589398\pi\)
\(72\) 0 0
\(73\) −7.32396e15 −1.06292 −0.531461 − 0.847082i \(-0.678357\pi\)
−0.531461 + 0.847082i \(0.678357\pi\)
\(74\) −5.69806e15 −0.736644
\(75\) 0 0
\(76\) −5.24682e13 −0.00540730
\(77\) −1.07468e16 −0.991080
\(78\) 0 0
\(79\) −7.07645e15 −0.524789 −0.262395 − 0.964961i \(-0.584512\pi\)
−0.262395 + 0.964961i \(0.584512\pi\)
\(80\) −6.36201e15 −0.423965
\(81\) 0 0
\(82\) 2.69374e16 1.45525
\(83\) 1.18761e16 0.578773 0.289387 − 0.957212i \(-0.406549\pi\)
0.289387 + 0.957212i \(0.406549\pi\)
\(84\) 0 0
\(85\) −1.88121e16 −0.748819
\(86\) −4.55388e16 −1.64114
\(87\) 0 0
\(88\) −4.87629e16 −1.44540
\(89\) −5.69376e16 −1.53315 −0.766574 − 0.642156i \(-0.778040\pi\)
−0.766574 + 0.642156i \(0.778040\pi\)
\(90\) 0 0
\(91\) 3.67851e16 0.820011
\(92\) −1.99989e15 −0.0406264
\(93\) 0 0
\(94\) −8.64519e16 −1.46281
\(95\) 3.15687e15 0.0488209
\(96\) 0 0
\(97\) −8.11777e16 −1.05166 −0.525832 − 0.850588i \(-0.676246\pi\)
−0.525832 + 0.850588i \(0.676246\pi\)
\(98\) 4.16920e16 0.495029
\(99\) 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 45.18.a.d.1.1 3
3.2 odd 2 15.18.a.c.1.3 ✓ 3
15.2 even 4 75.18.b.e.49.5 6
15.8 even 4 75.18.b.e.49.2 6
15.14 odd 2 75.18.a.d.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.c.1.3 ✓ 3 3.2 odd 2
45.18.a.d.1.1 3 1.1 even 1 trivial
75.18.a.d.1.1 3 15.14 odd 2
75.18.b.e.49.2 6 15.8 even 4
75.18.b.e.49.5 6 15.2 even 4