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(1,225)] 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("225.1"); 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, 0])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-46,0,4648,0,0,68372] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(172.877215626\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 6154x^{2} - 41770x + 5647125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-62.6246\) of defining polynomial
Character \(\chi\) \(=\) 225.1

$q$-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)
 
\(f(q)\) \(=\) \(q+51.6246 q^{2} +617.104 q^{4} +74399.4 q^{7} -73869.5 q^{8} -715447. q^{11} +1.87736e6 q^{13} +3.84084e6 q^{14} -5.07732e6 q^{16} -890725. q^{17} +1.87502e7 q^{19} -3.69347e7 q^{22} +563809. q^{23} +9.69180e7 q^{26} +4.59121e7 q^{28} -1.17569e8 q^{29} -1.96393e7 q^{31} -1.10830e8 q^{32} -4.59834e7 q^{34} -6.18818e8 q^{37} +9.67970e8 q^{38} +1.30299e9 q^{41} +4.05437e7 q^{43} -4.41505e8 q^{44} +2.91064e7 q^{46} +1.59171e9 q^{47} +3.55794e9 q^{49} +1.15853e9 q^{52} +1.39586e9 q^{53} -5.49585e9 q^{56} -6.06944e9 q^{58} -6.33376e8 q^{59} +2.84972e9 q^{61} -1.01387e9 q^{62} +4.67679e9 q^{64} +4.48583e9 q^{67} -5.49670e8 q^{68} -1.58898e10 q^{71} +4.54481e9 q^{73} -3.19462e10 q^{74} +1.15708e10 q^{76} -5.32288e10 q^{77} +1.60636e10 q^{79} +6.72664e10 q^{82} +3.01315e10 q^{83} +2.09305e9 q^{86} +5.28497e10 q^{88} +2.70045e10 q^{89} +1.39674e11 q^{91} +3.47929e8 q^{92} +8.21717e10 q^{94} -1.16395e11 q^{97} +1.83677e11 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 46 q^{2} + 4648 q^{4} + 68372 q^{7} - 386172 q^{8} + 99944 q^{11} + 2306276 q^{13} - 3107022 q^{14} + 7622200 q^{16} + 3443816 q^{17} - 4214548 q^{19} - 34106532 q^{22} - 52691304 q^{23} + 8766722 q^{26}+ \cdots + 139912937716 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 51.6246 1.14075 0.570377 − 0.821383i \(-0.306797\pi\)
0.570377 + 0.821383i \(0.306797\pi\)
\(3\) 0 0
\(4\) 617.104 0.301320
\(5\) 0 0
\(6\) 0 0
\(7\) 74399.4 1.67313 0.836566 − 0.547866i \(-0.184560\pi\)
0.836566 + 0.547866i \(0.184560\pi\)
\(8\) −73869.5 −0.797022
\(9\) 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
\(13\) 1.87736e6 1.40236 0.701179 − 0.712985i \(-0.252657\pi\)
0.701179 + 0.712985i \(0.252657\pi\)
\(14\) 3.84084e6 1.90863
\(15\) 0 0
\(16\) −5.07732e6 −1.21053
\(17\) −890725. −0.152151 −0.0760754 − 0.997102i \(-0.524239\pi\)
−0.0760754 + 0.997102i \(0.524239\pi\)
\(18\) 0 0
\(19\) 1.87502e7 1.73724 0.868620 − 0.495478i \(-0.165007\pi\)
0.868620 + 0.495478i \(0.165007\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.69347e7 −1.52795
\(23\) 563809. 0.0182654 0.00913270 − 0.999958i \(-0.497093\pi\)
0.00913270 + 0.999958i \(0.497093\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 9.69180e7 1.59975
\(27\) 0 0
\(28\) 4.59121e7 0.504148
\(29\) −1.17569e8 −1.06439 −0.532197 − 0.846621i \(-0.678633\pi\)
−0.532197 + 0.846621i \(0.678633\pi\)
\(30\) 0 0
\(31\) −1.96393e7 −0.123208 −0.0616038 − 0.998101i \(-0.519622\pi\)
−0.0616038 + 0.998101i \(0.519622\pi\)
\(32\) −1.10830e8 −0.583891
\(33\) 0 0
\(34\) −4.59834e7 −0.173567
\(35\) 0 0
\(36\) 0 0
\(37\) −6.18818e8 −1.46708 −0.733539 − 0.679647i \(-0.762133\pi\)
−0.733539 + 0.679647i \(0.762133\pi\)
\(38\) 9.67970e8 1.98176
\(39\) 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
\(43\) 4.05437e7 0.0420578 0.0210289 − 0.999779i \(-0.493306\pi\)
0.0210289 + 0.999779i \(0.493306\pi\)
\(44\) −4.41505e8 −0.403595
\(45\) 0 0
\(46\) 2.91064e7 0.0208363
\(47\) 1.59171e9 1.01234 0.506170 − 0.862433i \(-0.331061\pi\)
0.506170 + 0.862433i \(0.331061\pi\)
\(48\) 0 0
\(49\) 3.55794e9 1.79937
\(50\) 0 0
\(51\) 0 0
\(52\) 1.15853e9 0.422559
\(53\) 1.39586e9 0.458484 0.229242 − 0.973369i \(-0.426375\pi\)
0.229242 + 0.973369i \(0.426375\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −5.49585e9 −1.33352
\(57\) 0 0
\(58\) −6.06944e9 −1.21421
\(59\) −6.33376e8 −0.115339 −0.0576694 − 0.998336i \(-0.518367\pi\)
−0.0576694 + 0.998336i \(0.518367\pi\)
\(60\) 0 0
\(61\) 2.84972e9 0.432004 0.216002 − 0.976393i \(-0.430698\pi\)
0.216002 + 0.976393i \(0.430698\pi\)
\(62\) −1.01387e9 −0.140550
\(63\) 0 0
\(64\) 4.67679e9 0.544450
\(65\) 0 0
\(66\) 0 0
\(67\) 4.48583e9 0.405912 0.202956 − 0.979188i \(-0.434945\pi\)
0.202956 + 0.979188i \(0.434945\pi\)
\(68\) −5.49670e8 −0.0458461
\(69\) 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
\(73\) 4.54481e9 0.256590 0.128295 − 0.991736i \(-0.459050\pi\)
0.128295 + 0.991736i \(0.459050\pi\)
\(74\) −3.19462e10 −1.67358
\(75\) 0 0
\(76\) 1.15708e10 0.523466
\(77\) −5.32288e10 −2.24103
\(78\) 0 0
\(79\) 1.60636e10 0.587344 0.293672 − 0.955906i \(-0.405123\pi\)
0.293672 + 0.955906i \(0.405123\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.72664e10 2.00365
\(83\) 3.01315e10 0.839636 0.419818 − 0.907608i \(-0.362094\pi\)
0.419818 + 0.907608i \(0.362094\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.09305e9 0.0479776
\(87\) 0 0
\(88\) 5.28497e10 1.06755
\(89\) 2.70045e10 0.512615 0.256308 − 0.966595i \(-0.417494\pi\)
0.256308 + 0.966595i \(0.417494\pi\)
\(90\) 0 0
\(91\) 1.39674e11 2.34633
\(92\) 3.47929e8 0.00550373
\(93\) 0 0
\(94\) 8.21717e10 1.15483
\(95\) 0 0
\(96\) 0 0
\(97\) −1.16395e11 −1.37623 −0.688114 − 0.725603i \(-0.741561\pi\)
−0.688114 + 0.725603i \(0.741561\pi\)
\(98\) 1.83677e11 2.05264
\(99\) 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.a.q.1.4 4
3.2 odd 2 75.12.a.i.1.1 yes 4
5.2 odd 4 225.12.b.o.199.7 8
5.3 odd 4 225.12.b.o.199.2 8
5.4 even 2 225.12.a.s.1.1 4
15.2 even 4 75.12.b.g.49.2 8
15.8 even 4 75.12.b.g.49.7 8
15.14 odd 2 75.12.a.h.1.4 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.12.a.h.1.4 ✓ 4 15.14 odd 2
75.12.a.i.1.1 yes 4 3.2 odd 2
75.12.b.g.49.2 8 15.2 even 4
75.12.b.g.49.7 8 15.8 even 4
225.12.a.q.1.4 4 1.1 even 1 trivial
225.12.a.s.1.1 4 5.4 even 2
225.12.b.o.199.2 8 5.3 odd 4
225.12.b.o.199.7 8 5.2 odd 4