Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,4,Mod(199,225)] 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("225.199"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,14,0,0,0,0,0,0,-104] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.4.b.e.199.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +7.00000 q^{4} +24.0000i q^{7} +15.0000i q^{8} -52.0000 q^{11} +22.0000i q^{13} -24.0000 q^{14} +41.0000 q^{16} -14.0000i q^{17} +20.0000 q^{19} -52.0000i q^{22} +168.000i q^{23} -22.0000 q^{26} +168.000i q^{28} +230.000 q^{29} -288.000 q^{31} +161.000i q^{32} +14.0000 q^{34} +34.0000i q^{37} +20.0000i q^{38} -122.000 q^{41} -188.000i q^{43} -364.000 q^{44} -168.000 q^{46} +256.000i q^{47} -233.000 q^{49} +154.000i q^{52} +338.000i q^{53} -360.000 q^{56} +230.000i q^{58} +100.000 q^{59} +742.000 q^{61} -288.000i q^{62} +167.000 q^{64} +84.0000i q^{67} -98.0000i q^{68} +328.000 q^{71} -38.0000i q^{73} -34.0000 q^{74} +140.000 q^{76} -1248.00i q^{77} +240.000 q^{79} -122.000i q^{82} -1212.00i q^{83} +188.000 q^{86} -780.000i q^{88} +330.000 q^{89} -528.000 q^{91} +1176.00i q^{92} -256.000 q^{94} -866.000i q^{97} -233.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} - 104 q^{11} - 48 q^{14} + 82 q^{16} + 40 q^{19} - 44 q^{26} + 460 q^{29} - 576 q^{31} + 28 q^{34} - 244 q^{41} - 728 q^{44} - 336 q^{46} - 466 q^{49} - 720 q^{56} + 200 q^{59} + 1484 q^{61}+ \cdots - 512 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.353553i 0.984251 + 0.176777i \(0.0565670\pi\)
−0.984251 + 0.176777i \(0.943433\pi\)
\(3\) 0 0
\(4\) 7.00000 0.875000
\(5\) 0 0
\(6\) 0 0
\(7\) 24.0000i 1.29588i 0.761692 + 0.647939i \(0.224369\pi\)
−0.761692 + 0.647939i \(0.775631\pi\)
\(8\) 15.0000i 0.662913i
\(9\) 0 0
\(10\) 0 0
\(11\) −52.0000 −1.42533 −0.712663 − 0.701506i \(-0.752511\pi\)
−0.712663 + 0.701506i \(0.752511\pi\)
\(12\) 0 0
\(13\) 22.0000i 0.469362i 0.972072 + 0.234681i \(0.0754045\pi\)
−0.972072 + 0.234681i \(0.924595\pi\)
\(14\) −24.0000 −0.458162
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) − 14.0000i − 0.199735i −0.995001 − 0.0998676i \(-0.968158\pi\)
0.995001 − 0.0998676i \(-0.0318419\pi\)
\(18\) 0 0
\(19\) 20.0000 0.241490 0.120745 − 0.992684i \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 52.0000i − 0.503929i
\(23\) 168.000i 1.52306i 0.648129 + 0.761531i \(0.275552\pi\)
−0.648129 + 0.761531i \(0.724448\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −22.0000 −0.165944
\(27\) 0 0
\(28\) 168.000i 1.13389i
\(29\) 230.000 1.47276 0.736378 − 0.676570i \(-0.236535\pi\)
0.736378 + 0.676570i \(0.236535\pi\)
\(30\) 0 0
\(31\) −288.000 −1.66859 −0.834296 − 0.551317i \(-0.814125\pi\)
−0.834296 + 0.551317i \(0.814125\pi\)
\(32\) 161.000i 0.889408i
\(33\) 0 0
\(34\) 14.0000 0.0706171
\(35\) 0 0
\(36\) 0 0
\(37\) 34.0000i 0.151069i 0.997143 + 0.0755347i \(0.0240664\pi\)
−0.997143 + 0.0755347i \(0.975934\pi\)
\(38\) 20.0000i 0.0853797i
\(39\) 0 0
\(40\) 0 0
\(41\) −122.000 −0.464712 −0.232356 − 0.972631i \(-0.574643\pi\)
−0.232356 + 0.972631i \(0.574643\pi\)
\(42\) 0 0
\(43\) − 188.000i − 0.666738i −0.942796 − 0.333369i \(-0.891815\pi\)
0.942796 − 0.333369i \(-0.108185\pi\)
\(44\) −364.000 −1.24716
\(45\) 0 0
\(46\) −168.000 −0.538484
\(47\) 256.000i 0.794499i 0.917711 + 0.397249i \(0.130035\pi\)
−0.917711 + 0.397249i \(0.869965\pi\)
\(48\) 0 0
\(49\) −233.000 −0.679300
\(50\) 0 0
\(51\) 0 0
\(52\) 154.000i 0.410691i
\(53\) 338.000i 0.875998i 0.898976 + 0.437999i \(0.144313\pi\)
−0.898976 + 0.437999i \(0.855687\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −360.000 −0.859054
\(57\) 0 0
\(58\) 230.000i 0.520698i
\(59\) 100.000 0.220659 0.110330 − 0.993895i \(-0.464809\pi\)
0.110330 + 0.993895i \(0.464809\pi\)
\(60\) 0 0
\(61\) 742.000 1.55743 0.778716 − 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(62\) − 288.000i − 0.589936i
\(63\) 0 0
\(64\) 167.000 0.326172
\(65\) 0 0
\(66\) 0 0
\(67\) 84.0000i 0.153168i 0.997063 + 0.0765838i \(0.0244013\pi\)
−0.997063 + 0.0765838i \(0.975599\pi\)
\(68\) − 98.0000i − 0.174768i
\(69\) 0 0
\(70\) 0 0
\(71\) 328.000 0.548260 0.274130 − 0.961693i \(-0.411610\pi\)
0.274130 + 0.961693i \(0.411610\pi\)
\(72\) 0 0
\(73\) − 38.0000i − 0.0609255i −0.999536 − 0.0304628i \(-0.990302\pi\)
0.999536 − 0.0304628i \(-0.00969810\pi\)
\(74\) −34.0000 −0.0534111
\(75\) 0 0
\(76\) 140.000 0.211304
\(77\) − 1248.00i − 1.84705i
\(78\) 0 0
\(79\) 240.000 0.341799 0.170899 − 0.985288i \(-0.445333\pi\)
0.170899 + 0.985288i \(0.445333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 122.000i − 0.164301i
\(83\) − 1212.00i − 1.60282i −0.598114 − 0.801411i \(-0.704083\pi\)
0.598114 − 0.801411i \(-0.295917\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 188.000 0.235727
\(87\) 0 0
\(88\) − 780.000i − 0.944867i
\(89\) 330.000 0.393033 0.196516 − 0.980501i \(-0.437037\pi\)
0.196516 + 0.980501i \(0.437037\pi\)
\(90\) 0 0
\(91\) −528.000 −0.608236
\(92\) 1176.00i 1.33268i
\(93\) 0 0
\(94\) −256.000 −0.280898
\(95\) 0 0
\(96\) 0 0
\(97\) − 866.000i − 0.906484i −0.891387 − 0.453242i \(-0.850267\pi\)
0.891387 − 0.453242i \(-0.149733\pi\)
\(98\) − 233.000i − 0.240169i
\(99\) 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 225.4.b.e.199.2 2
3.2 odd 2 75.4.b.b.49.1 2
5.2 odd 4 45.4.a.c.1.1 1
5.3 odd 4 225.4.a.f.1.1 1
5.4 even 2 inner 225.4.b.e.199.1 2
12.11 even 2 1200.4.f.b.49.1 2
15.2 even 4 15.4.a.a.1.1 ✓ 1
15.8 even 4 75.4.a.b.1.1 1
15.14 odd 2 75.4.b.b.49.2 2
20.7 even 4 720.4.a.n.1.1 1
35.27 even 4 2205.4.a.l.1.1 1
45.2 even 12 405.4.e.g.271.1 2
45.7 odd 12 405.4.e.i.271.1 2
45.22 odd 12 405.4.e.i.136.1 2
45.32 even 12 405.4.e.g.136.1 2
60.23 odd 4 1200.4.a.t.1.1 1
60.47 odd 4 240.4.a.e.1.1 1
60.59 even 2 1200.4.f.b.49.2 2
105.62 odd 4 735.4.a.e.1.1 1
120.77 even 4 960.4.a.b.1.1 1
120.107 odd 4 960.4.a.ba.1.1 1
165.32 odd 4 1815.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 ✓ 1 15.2 even 4
45.4.a.c.1.1 1 5.2 odd 4
75.4.a.b.1.1 1 15.8 even 4
75.4.b.b.49.1 2 3.2 odd 2
75.4.b.b.49.2 2 15.14 odd 2
225.4.a.f.1.1 1 5.3 odd 4
225.4.b.e.199.1 2 5.4 even 2 inner
225.4.b.e.199.2 2 1.1 even 1 trivial
240.4.a.e.1.1 1 60.47 odd 4
405.4.e.g.136.1 2 45.32 even 12
405.4.e.g.271.1 2 45.2 even 12
405.4.e.i.136.1 2 45.22 odd 12
405.4.e.i.271.1 2 45.7 odd 12
720.4.a.n.1.1 1 20.7 even 4
735.4.a.e.1.1 1 105.62 odd 4
960.4.a.b.1.1 1 120.77 even 4
960.4.a.ba.1.1 1 120.107 odd 4
1200.4.a.t.1.1 1 60.23 odd 4
1200.4.f.b.49.1 2 12.11 even 2
1200.4.f.b.49.2 2 60.59 even 2
1815.4.a.e.1.1 1 165.32 odd 4
2205.4.a.l.1.1 1 35.27 even 4