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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [450,4,Mod(199,450)] 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("450.199"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-8,0,0,0,0,0,0,-72,0,0,44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.5508595026\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 450.199
Dual form 450.4.c.b.199.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -4.00000 q^{4} +11.0000i q^{7} +8.00000i q^{8} -36.0000 q^{11} -17.0000i q^{13} +22.0000 q^{14} +16.0000 q^{16} -12.0000i q^{17} +91.0000 q^{19} +72.0000i q^{22} -60.0000i q^{23} -34.0000 q^{26} -44.0000i q^{28} +276.000 q^{29} +191.000 q^{31} -32.0000i q^{32} -24.0000 q^{34} +254.000i q^{37} -182.000i q^{38} -60.0000 q^{41} +49.0000i q^{43} +144.000 q^{44} -120.000 q^{46} -600.000i q^{47} +222.000 q^{49} +68.0000i q^{52} -612.000i q^{53} -88.0000 q^{56} -552.000i q^{58} +744.000 q^{59} +167.000 q^{61} -382.000i q^{62} -64.0000 q^{64} -457.000i q^{67} +48.0000i q^{68} -588.000 q^{71} +970.000i q^{73} +508.000 q^{74} -364.000 q^{76} -396.000i q^{77} -164.000 q^{79} +120.000i q^{82} -696.000i q^{83} +98.0000 q^{86} -288.000i q^{88} +1248.00 q^{89} +187.000 q^{91} +240.000i q^{92} -1200.00 q^{94} -1099.00i q^{97} -444.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 72 q^{11} + 44 q^{14} + 32 q^{16} + 182 q^{19} - 68 q^{26} + 552 q^{29} + 382 q^{31} - 48 q^{34} - 120 q^{41} + 288 q^{44} - 240 q^{46} + 444 q^{49} - 176 q^{56} + 1488 q^{59} + 334 q^{61}+ \cdots - 2400 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) 0 0
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 11.0000i 0.593944i 0.954886 + 0.296972i \(0.0959768\pi\)
−0.954886 + 0.296972i \(0.904023\pi\)
\(8\) 8.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −36.0000 −0.986764 −0.493382 − 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) 0 0
\(13\) − 17.0000i − 0.362689i −0.983420 − 0.181344i \(-0.941955\pi\)
0.983420 − 0.181344i \(-0.0580448\pi\)
\(14\) 22.0000 0.419982
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 12.0000i − 0.171202i −0.996330 − 0.0856008i \(-0.972719\pi\)
0.996330 − 0.0856008i \(-0.0272810\pi\)
\(18\) 0 0
\(19\) 91.0000 1.09878 0.549390 − 0.835566i \(-0.314860\pi\)
0.549390 + 0.835566i \(0.314860\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 72.0000i 0.697748i
\(23\) − 60.0000i − 0.543951i −0.962304 − 0.271975i \(-0.912323\pi\)
0.962304 − 0.271975i \(-0.0876769\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −34.0000 −0.256460
\(27\) 0 0
\(28\) − 44.0000i − 0.296972i
\(29\) 276.000 1.76731 0.883654 − 0.468141i \(-0.155076\pi\)
0.883654 + 0.468141i \(0.155076\pi\)
\(30\) 0 0
\(31\) 191.000 1.10660 0.553300 − 0.832982i \(-0.313368\pi\)
0.553300 + 0.832982i \(0.313368\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 0 0
\(34\) −24.0000 −0.121058
\(35\) 0 0
\(36\) 0 0
\(37\) 254.000i 1.12858i 0.825578 + 0.564288i \(0.190849\pi\)
−0.825578 + 0.564288i \(0.809151\pi\)
\(38\) − 182.000i − 0.776955i
\(39\) 0 0
\(40\) 0 0
\(41\) −60.0000 −0.228547 −0.114273 − 0.993449i \(-0.536454\pi\)
−0.114273 + 0.993449i \(0.536454\pi\)
\(42\) 0 0
\(43\) 49.0000i 0.173777i 0.996218 + 0.0868887i \(0.0276925\pi\)
−0.996218 + 0.0868887i \(0.972308\pi\)
\(44\) 144.000 0.493382
\(45\) 0 0
\(46\) −120.000 −0.384631
\(47\) − 600.000i − 1.86211i −0.364884 − 0.931053i \(-0.618891\pi\)
0.364884 − 0.931053i \(-0.381109\pi\)
\(48\) 0 0
\(49\) 222.000 0.647230
\(50\) 0 0
\(51\) 0 0
\(52\) 68.0000i 0.181344i
\(53\) − 612.000i − 1.58613i −0.609140 − 0.793063i \(-0.708485\pi\)
0.609140 − 0.793063i \(-0.291515\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −88.0000 −0.209991
\(57\) 0 0
\(58\) − 552.000i − 1.24968i
\(59\) 744.000 1.64170 0.820852 − 0.571141i \(-0.193499\pi\)
0.820852 + 0.571141i \(0.193499\pi\)
\(60\) 0 0
\(61\) 167.000 0.350527 0.175264 − 0.984522i \(-0.443922\pi\)
0.175264 + 0.984522i \(0.443922\pi\)
\(62\) − 382.000i − 0.782485i
\(63\) 0 0
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 457.000i − 0.833305i −0.909066 − 0.416653i \(-0.863203\pi\)
0.909066 − 0.416653i \(-0.136797\pi\)
\(68\) 48.0000i 0.0856008i
\(69\) 0 0
\(70\) 0 0
\(71\) −588.000 −0.982856 −0.491428 − 0.870918i \(-0.663525\pi\)
−0.491428 + 0.870918i \(0.663525\pi\)
\(72\) 0 0
\(73\) 970.000i 1.55520i 0.628757 + 0.777602i \(0.283564\pi\)
−0.628757 + 0.777602i \(0.716436\pi\)
\(74\) 508.000 0.798024
\(75\) 0 0
\(76\) −364.000 −0.549390
\(77\) − 396.000i − 0.586083i
\(78\) 0 0
\(79\) −164.000 −0.233563 −0.116781 − 0.993158i \(-0.537258\pi\)
−0.116781 + 0.993158i \(0.537258\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 120.000i 0.161607i
\(83\) − 696.000i − 0.920433i −0.887807 − 0.460216i \(-0.847772\pi\)
0.887807 − 0.460216i \(-0.152228\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 98.0000 0.122879
\(87\) 0 0
\(88\) − 288.000i − 0.348874i
\(89\) 1248.00 1.48638 0.743190 − 0.669081i \(-0.233312\pi\)
0.743190 + 0.669081i \(0.233312\pi\)
\(90\) 0 0
\(91\) 187.000 0.215417
\(92\) 240.000i 0.271975i
\(93\) 0 0
\(94\) −1200.00 −1.31671
\(95\) 0 0
\(96\) 0 0
\(97\) − 1099.00i − 1.15038i −0.818021 − 0.575188i \(-0.804929\pi\)
0.818021 − 0.575188i \(-0.195071\pi\)
\(98\) − 444.000i − 0.457661i
\(99\) 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 450.4.c.b.199.1 2
3.2 odd 2 450.4.c.i.199.2 2
5.2 odd 4 450.4.a.n.1.1 yes 1
5.3 odd 4 450.4.a.g.1.1 yes 1
5.4 even 2 inner 450.4.c.b.199.2 2
15.2 even 4 450.4.a.d.1.1 ✓ 1
15.8 even 4 450.4.a.s.1.1 yes 1
15.14 odd 2 450.4.c.i.199.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.4.a.d.1.1 ✓ 1 15.2 even 4
450.4.a.g.1.1 yes 1 5.3 odd 4
450.4.a.n.1.1 yes 1 5.2 odd 4
450.4.a.s.1.1 yes 1 15.8 even 4
450.4.c.b.199.1 2 1.1 even 1 trivial
450.4.c.b.199.2 2 5.4 even 2 inner
450.4.c.i.199.1 2 15.14 odd 2
450.4.c.i.199.2 2 3.2 odd 2