Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,12,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, 12); 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, 12, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-9296,0,0,0,0,0,0,199888] 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: \(172.877215626\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} + 2 x^{6} - 20886 x^{5} + 6668329 x^{4} - 80402480 x^{3} + 365580800 x^{2} + \cdots + 1959440040000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(31.8123 + 31.8123i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.12.b.o.199.7

$q$-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)
 
\(f(q)\) \(=\) \(q-51.6246i q^{2} -617.104 q^{4} -74399.4i q^{7} -73869.5i q^{8} -715447. q^{11} +1.87736e6i q^{13} -3.84084e6 q^{14} -5.07732e6 q^{16} +890725. i q^{17} -1.87502e7 q^{19} +3.69347e7i q^{22} +563809. i q^{23} +9.69180e7 q^{26} +4.59121e7i q^{28} +1.17569e8 q^{29} -1.96393e7 q^{31} +1.10830e8i q^{32} +4.59834e7 q^{34} +6.18818e8i q^{37} +9.67970e8i q^{38} +1.30299e9 q^{41} +4.05437e7i q^{43} +4.41505e8 q^{44} +2.91064e7 q^{46} -1.59171e9i q^{47} -3.55794e9 q^{49} -1.15853e9i q^{52} +1.39586e9i q^{53} -5.49585e9 q^{56} -6.06944e9i q^{58} +6.33376e8 q^{59} +2.84972e9 q^{61} +1.01387e9i q^{62} -4.67679e9 q^{64} -4.48583e9i q^{67} -5.49670e8i q^{68} -1.58898e10 q^{71} +4.54481e9i q^{73} +3.19462e10 q^{74} +1.15708e10 q^{76} +5.32288e10i q^{77} -1.60636e10 q^{79} -6.72664e10i q^{82} +3.01315e10i q^{83} +2.09305e9 q^{86} +5.28497e10i q^{88} -2.70045e10 q^{89} +1.39674e11 q^{91} -3.47929e8i q^{92} -8.21717e10 q^{94} +1.16395e11i q^{97} +1.83677e11i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 9296 q^{4} + 199888 q^{11} + 6214044 q^{14} + 15244400 q^{16} + 8429096 q^{19} + 17533444 q^{26} + 434786608 q^{29} - 652634072 q^{31} + 1129672488 q^{34} + 44138912 q^{41} - 1984948352 q^{44} + 5395707048 q^{46}+ \cdots - 386741434776 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 51.6246i − 1.14075i −0.821383 − 0.570377i \(-0.806797\pi\)
0.821383 − 0.570377i \(-0.193203\pi\)
\(3\) 0 0
\(4\) −617.104 −0.301320
\(5\) 0 0
\(6\) 0 0
\(7\) − 74399.4i − 1.67313i −0.547866 − 0.836566i \(-0.684560\pi\)
0.547866 − 0.836566i \(-0.315440\pi\)
\(8\) − 73869.5i − 0.797022i
\(9\) 0 0
\(10\) 0 0
\(11\) −715447. −1.33942 −0.669711 − 0.742622i \(-0.733582\pi\)
−0.669711 + 0.742622i \(0.733582\pi\)
\(12\) 0 0
\(13\) 1.87736e6i 1.40236i 0.712985 + 0.701179i \(0.247343\pi\)
−0.712985 + 0.701179i \(0.752657\pi\)
\(14\) −3.84084e6 −1.90863
\(15\) 0 0
\(16\) −5.07732e6 −1.21053
\(17\) 890725.i 0.152151i 0.997102 + 0.0760754i \(0.0242390\pi\)
−0.997102 + 0.0760754i \(0.975761\pi\)
\(18\) 0 0
\(19\) −1.87502e7 −1.73724 −0.868620 − 0.495478i \(-0.834993\pi\)
−0.868620 + 0.495478i \(0.834993\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.69347e7i 1.52795i
\(23\) 563809.i 0.0182654i 0.999958 + 0.00913270i \(0.00290707\pi\)
−0.999958 + 0.00913270i \(0.997093\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 9.69180e7 1.59975
\(27\) 0 0
\(28\) 4.59121e7i 0.504148i
\(29\) 1.17569e8 1.06439 0.532197 − 0.846621i \(-0.321367\pi\)
0.532197 + 0.846621i \(0.321367\pi\)
\(30\) 0 0
\(31\) −1.96393e7 −0.123208 −0.0616038 − 0.998101i \(-0.519622\pi\)
−0.0616038 + 0.998101i \(0.519622\pi\)
\(32\) 1.10830e8i 0.583891i
\(33\) 0 0
\(34\) 4.59834e7 0.173567
\(35\) 0 0
\(36\) 0 0
\(37\) 6.18818e8i 1.46708i 0.679647 + 0.733539i \(0.262133\pi\)
−0.679647 + 0.733539i \(0.737867\pi\)
\(38\) 9.67970e8i 1.98176i
\(39\) 0 0
\(40\) 0 0
\(41\) 1.30299e9 1.75643 0.878213 − 0.478269i \(-0.158736\pi\)
0.878213 + 0.478269i \(0.158736\pi\)
\(42\) 0 0
\(43\) 4.05437e7i 0.0420578i 0.999779 + 0.0210289i \(0.00669420\pi\)
−0.999779 + 0.0210289i \(0.993306\pi\)
\(44\) 4.41505e8 0.403595
\(45\) 0 0
\(46\) 2.91064e7 0.0208363
\(47\) − 1.59171e9i − 1.01234i −0.862433 − 0.506170i \(-0.831061\pi\)
0.862433 − 0.506170i \(-0.168939\pi\)
\(48\) 0 0
\(49\) −3.55794e9 −1.79937
\(50\) 0 0
\(51\) 0 0
\(52\) − 1.15853e9i − 0.422559i
\(53\) 1.39586e9i 0.458484i 0.973369 + 0.229242i \(0.0736247\pi\)
−0.973369 + 0.229242i \(0.926375\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −5.49585e9 −1.33352
\(57\) 0 0
\(58\) − 6.06944e9i − 1.21421i
\(59\) 6.33376e8 0.115339 0.0576694 − 0.998336i \(-0.481633\pi\)
0.0576694 + 0.998336i \(0.481633\pi\)
\(60\) 0 0
\(61\) 2.84972e9 0.432004 0.216002 − 0.976393i \(-0.430698\pi\)
0.216002 + 0.976393i \(0.430698\pi\)
\(62\) 1.01387e9i 0.140550i
\(63\) 0 0
\(64\) −4.67679e9 −0.544450
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.48583e9i − 0.405912i −0.979188 − 0.202956i \(-0.934945\pi\)
0.979188 − 0.202956i \(-0.0650548\pi\)
\(68\) − 5.49670e8i − 0.0458461i
\(69\) 0 0
\(70\) 0 0
\(71\) −1.58898e10 −1.04519 −0.522596 − 0.852580i \(-0.675036\pi\)
−0.522596 + 0.852580i \(0.675036\pi\)
\(72\) 0 0
\(73\) 4.54481e9i 0.256590i 0.991736 + 0.128295i \(0.0409505\pi\)
−0.991736 + 0.128295i \(0.959050\pi\)
\(74\) 3.19462e10 1.67358
\(75\) 0 0
\(76\) 1.15708e10 0.523466
\(77\) 5.32288e10i 2.24103i
\(78\) 0 0
\(79\) −1.60636e10 −0.587344 −0.293672 − 0.955906i \(-0.594877\pi\)
−0.293672 + 0.955906i \(0.594877\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 6.72664e10i − 2.00365i
\(83\) 3.01315e10i 0.839636i 0.907608 + 0.419818i \(0.137906\pi\)
−0.907608 + 0.419818i \(0.862094\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.09305e9 0.0479776
\(87\) 0 0
\(88\) 5.28497e10i 1.06755i
\(89\) −2.70045e10 −0.512615 −0.256308 − 0.966595i \(-0.582506\pi\)
−0.256308 + 0.966595i \(0.582506\pi\)
\(90\) 0 0
\(91\) 1.39674e11 2.34633
\(92\) − 3.47929e8i − 0.00550373i
\(93\) 0 0
\(94\) −8.21717e10 −1.15483
\(95\) 0 0
\(96\) 0 0
\(97\) 1.16395e11i 1.37623i 0.725603 + 0.688114i \(0.241561\pi\)
−0.725603 + 0.688114i \(0.758439\pi\)
\(98\) 1.83677e11i 2.05264i
\(99\) 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 225.12.b.o.199.2 8
3.2 odd 2 75.12.b.g.49.7 8
5.2 odd 4 225.12.a.q.1.4 4
5.3 odd 4 225.12.a.s.1.1 4
5.4 even 2 inner 225.12.b.o.199.7 8
15.2 even 4 75.12.a.i.1.1 yes 4
15.8 even 4 75.12.a.h.1.4 ✓ 4
15.14 odd 2 75.12.b.g.49.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.12.a.h.1.4 ✓ 4 15.8 even 4
75.12.a.i.1.1 yes 4 15.2 even 4
75.12.b.g.49.2 8 15.14 odd 2
75.12.b.g.49.7 8 3.2 odd 2
225.12.a.q.1.4 4 5.2 odd 4
225.12.a.s.1.1 4 5.3 odd 4
225.12.b.o.199.2 8 1.1 even 1 trivial
225.12.b.o.199.7 8 5.4 even 2 inner