Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [405,2,Mod(1,405)] 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("405.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-1,0,5,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.51414\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.51414 q^{2} +4.32088 q^{4} -1.00000 q^{5} -0.514137 q^{7} -5.83502 q^{8} +2.51414 q^{10} +3.32088 q^{11} -1.32088 q^{13} +1.29261 q^{14} +6.02827 q^{16} +3.32088 q^{17} -1.32088 q^{19} -4.32088 q^{20} -8.34916 q^{22} -4.12763 q^{23} +1.00000 q^{25} +3.32088 q^{26} -2.22153 q^{28} +1.38650 q^{29} +8.73566 q^{31} -3.48586 q^{32} -8.34916 q^{34} +0.514137 q^{35} +0.292611 q^{37} +3.32088 q^{38} +5.83502 q^{40} +11.3492 q^{41} +10.3492 q^{43} +14.3492 q^{44} +10.3774 q^{46} +4.86330 q^{47} -6.73566 q^{49} -2.51414 q^{50} -5.70739 q^{52} +5.02827 q^{53} -3.32088 q^{55} +3.00000 q^{56} -3.48586 q^{58} +5.02827 q^{59} +7.34916 q^{61} -21.9627 q^{62} -3.29261 q^{64} +1.32088 q^{65} +9.44852 q^{67} +14.3492 q^{68} -1.29261 q^{70} -8.99093 q^{71} +6.05655 q^{73} -0.735663 q^{74} -5.70739 q^{76} -1.70739 q^{77} -8.05655 q^{79} -6.02827 q^{80} -28.5333 q^{82} -1.54241 q^{83} -3.32088 q^{85} -26.0192 q^{86} -19.3774 q^{88} +3.00000 q^{89} +0.679116 q^{91} -17.8350 q^{92} -12.2270 q^{94} +1.32088 q^{95} -12.2553 q^{97} +16.9344 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8} + q^{10} + 2 q^{11} + 4 q^{13} + 9 q^{14} + 5 q^{16} + 2 q^{17} + 4 q^{19} - 5 q^{20} - 4 q^{22} - 3 q^{23} + 3 q^{25} + 2 q^{26} + 5 q^{28} + 7 q^{29}+ \cdots + 40 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.51414 −1.77776 −0.888882 − 0.458137i \(-0.848517\pi\)
−0.888882 + 0.458137i \(0.848517\pi\)
\(3\) 0 0
\(4\) 4.32088 2.16044
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.514137 −0.194325 −0.0971627 − 0.995269i \(-0.530977\pi\)
−0.0971627 + 0.995269i \(0.530977\pi\)
\(8\) −5.83502 −2.06299
\(9\) 0 0
\(10\) 2.51414 0.795040
\(11\) 3.32088 1.00128 0.500642 − 0.865654i \(-0.333097\pi\)
0.500642 + 0.865654i \(0.333097\pi\)
\(12\) 0 0
\(13\) −1.32088 −0.366347 −0.183174 − 0.983081i \(-0.558637\pi\)
−0.183174 + 0.983081i \(0.558637\pi\)
\(14\) 1.29261 0.345465
\(15\) 0 0
\(16\) 6.02827 1.50707
\(17\) 3.32088 0.805433 0.402716 − 0.915325i \(-0.368066\pi\)
0.402716 + 0.915325i \(0.368066\pi\)
\(18\) 0 0
\(19\) −1.32088 −0.303032 −0.151516 − 0.988455i \(-0.548415\pi\)
−0.151516 + 0.988455i \(0.548415\pi\)
\(20\) −4.32088 −0.966179
\(21\) 0 0
\(22\) −8.34916 −1.78005
\(23\) −4.12763 −0.860671 −0.430335 − 0.902669i \(-0.641605\pi\)
−0.430335 + 0.902669i \(0.641605\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 3.32088 0.651279
\(27\) 0 0
\(28\) −2.22153 −0.419829
\(29\) 1.38650 0.257467 0.128734 − 0.991679i \(-0.458909\pi\)
0.128734 + 0.991679i \(0.458909\pi\)
\(30\) 0 0
\(31\) 8.73566 1.56897 0.784486 − 0.620147i \(-0.212927\pi\)
0.784486 + 0.620147i \(0.212927\pi\)
\(32\) −3.48586 −0.616219
\(33\) 0 0
\(34\) −8.34916 −1.43187
\(35\) 0.514137 0.0869050
\(36\) 0 0
\(37\) 0.292611 0.0481049 0.0240524 − 0.999711i \(-0.492343\pi\)
0.0240524 + 0.999711i \(0.492343\pi\)
\(38\) 3.32088 0.538719
\(39\) 0 0
\(40\) 5.83502 0.922598
\(41\) 11.3492 1.77244 0.886220 − 0.463264i \(-0.153322\pi\)
0.886220 + 0.463264i \(0.153322\pi\)
\(42\) 0 0
\(43\) 10.3492 1.57823 0.789116 − 0.614244i \(-0.210539\pi\)
0.789116 + 0.614244i \(0.210539\pi\)
\(44\) 14.3492 2.16322
\(45\) 0 0
\(46\) 10.3774 1.53007
\(47\) 4.86330 0.709385 0.354692 − 0.934983i \(-0.384586\pi\)
0.354692 + 0.934983i \(0.384586\pi\)
\(48\) 0 0
\(49\) −6.73566 −0.962238
\(50\) −2.51414 −0.355553
\(51\) 0 0
\(52\) −5.70739 −0.791472
\(53\) 5.02827 0.690687 0.345343 − 0.938476i \(-0.387762\pi\)
0.345343 + 0.938476i \(0.387762\pi\)
\(54\) 0 0
\(55\) −3.32088 −0.447788
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −3.48586 −0.457716
\(59\) 5.02827 0.654625 0.327313 − 0.944916i \(-0.393857\pi\)
0.327313 + 0.944916i \(0.393857\pi\)
\(60\) 0 0
\(61\) 7.34916 0.940963 0.470482 − 0.882410i \(-0.344080\pi\)
0.470482 + 0.882410i \(0.344080\pi\)
\(62\) −21.9627 −2.78926
\(63\) 0 0
\(64\) −3.29261 −0.411576
\(65\) 1.32088 0.163836
\(66\) 0 0
\(67\) 9.44852 1.15432 0.577160 − 0.816631i \(-0.304161\pi\)
0.577160 + 0.816631i \(0.304161\pi\)
\(68\) 14.3492 1.74009
\(69\) 0 0
\(70\) −1.29261 −0.154497
\(71\) −8.99093 −1.06703 −0.533513 − 0.845792i \(-0.679129\pi\)
−0.533513 + 0.845792i \(0.679129\pi\)
\(72\) 0 0
\(73\) 6.05655 0.708865 0.354433 − 0.935082i \(-0.384674\pi\)
0.354433 + 0.935082i \(0.384674\pi\)
\(74\) −0.735663 −0.0855191
\(75\) 0 0
\(76\) −5.70739 −0.654682
\(77\) −1.70739 −0.194575
\(78\) 0 0
\(79\) −8.05655 −0.906432 −0.453216 − 0.891401i \(-0.649723\pi\)
−0.453216 + 0.891401i \(0.649723\pi\)
\(80\) −6.02827 −0.673982
\(81\) 0 0
\(82\) −28.5333 −3.15098
\(83\) −1.54241 −0.169302 −0.0846508 − 0.996411i \(-0.526977\pi\)
−0.0846508 + 0.996411i \(0.526977\pi\)
\(84\) 0 0
\(85\) −3.32088 −0.360200
\(86\) −26.0192 −2.80572
\(87\) 0 0
\(88\) −19.3774 −2.06564
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 0.679116 0.0711906
\(92\) −17.8350 −1.85943
\(93\) 0 0
\(94\) −12.2270 −1.26112
\(95\) 1.32088 0.135520
\(96\) 0 0
\(97\) −12.2553 −1.24433 −0.622167 − 0.782885i \(-0.713747\pi\)
−0.622167 + 0.782885i \(0.713747\pi\)
\(98\) 16.9344 1.71063
\(99\) 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 405.2.a.i.1.1 3
3.2 odd 2 405.2.a.j.1.3 3
4.3 odd 2 6480.2.a.bs.1.3 3
5.2 odd 4 2025.2.b.m.649.1 6
5.3 odd 4 2025.2.b.m.649.6 6
5.4 even 2 2025.2.a.o.1.3 3
9.2 odd 6 45.2.e.b.31.1 yes 6
9.4 even 3 135.2.e.b.46.3 6
9.5 odd 6 45.2.e.b.16.1 ✓ 6
9.7 even 3 135.2.e.b.91.3 6
12.11 even 2 6480.2.a.bv.1.3 3
15.2 even 4 2025.2.b.l.649.6 6
15.8 even 4 2025.2.b.l.649.1 6
15.14 odd 2 2025.2.a.n.1.1 3
36.7 odd 6 2160.2.q.k.1441.1 6
36.11 even 6 720.2.q.i.481.1 6
36.23 even 6 720.2.q.i.241.1 6
36.31 odd 6 2160.2.q.k.721.1 6
45.2 even 12 225.2.k.b.49.6 12
45.4 even 6 675.2.e.b.451.1 6
45.7 odd 12 675.2.k.b.199.1 12
45.13 odd 12 675.2.k.b.424.1 12
45.14 odd 6 225.2.e.b.151.3 6
45.22 odd 12 675.2.k.b.424.6 12
45.23 even 12 225.2.k.b.124.6 12
45.29 odd 6 225.2.e.b.76.3 6
45.32 even 12 225.2.k.b.124.1 12
45.34 even 6 675.2.e.b.226.1 6
45.38 even 12 225.2.k.b.49.1 12
45.43 odd 12 675.2.k.b.199.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.1 ✓ 6 9.5 odd 6
45.2.e.b.31.1 yes 6 9.2 odd 6
135.2.e.b.46.3 6 9.4 even 3
135.2.e.b.91.3 6 9.7 even 3
225.2.e.b.76.3 6 45.29 odd 6
225.2.e.b.151.3 6 45.14 odd 6
225.2.k.b.49.1 12 45.38 even 12
225.2.k.b.49.6 12 45.2 even 12
225.2.k.b.124.1 12 45.32 even 12
225.2.k.b.124.6 12 45.23 even 12
405.2.a.i.1.1 3 1.1 even 1 trivial
405.2.a.j.1.3 3 3.2 odd 2
675.2.e.b.226.1 6 45.34 even 6
675.2.e.b.451.1 6 45.4 even 6
675.2.k.b.199.1 12 45.7 odd 12
675.2.k.b.199.6 12 45.43 odd 12
675.2.k.b.424.1 12 45.13 odd 12
675.2.k.b.424.6 12 45.22 odd 12
720.2.q.i.241.1 6 36.23 even 6
720.2.q.i.481.1 6 36.11 even 6
2025.2.a.n.1.1 3 15.14 odd 2
2025.2.a.o.1.3 3 5.4 even 2
2025.2.b.l.649.1 6 15.8 even 4
2025.2.b.l.649.6 6 15.2 even 4
2025.2.b.m.649.1 6 5.2 odd 4
2025.2.b.m.649.6 6 5.3 odd 4
2160.2.q.k.721.1 6 36.31 odd 6
2160.2.q.k.1441.1 6 36.7 odd 6
6480.2.a.bs.1.3 3 4.3 odd 2
6480.2.a.bv.1.3 3 12.11 even 2