Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,12,Mod(1,75)] 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("75.1"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,46,972] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(57.6257385420\)
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: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-62.6246\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-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)
 
\(f(q)\) \(=\) \(q-51.6246 q^{2} +243.000 q^{3} +617.104 q^{4} -12544.8 q^{6} +74399.4 q^{7} +73869.5 q^{8} +59049.0 q^{9} +715447. q^{11} +149956. q^{12} +1.87736e6 q^{13} -3.84084e6 q^{14} -5.07732e6 q^{16} +890725. q^{17} -3.04838e6 q^{18} +1.87502e7 q^{19} +1.80791e7 q^{21} -3.69347e7 q^{22} -563809. q^{23} +1.79503e7 q^{24} -9.69180e7 q^{26} +1.43489e7 q^{27} +4.59121e7 q^{28} +1.17569e8 q^{29} -1.96393e7 q^{31} +1.10830e8 q^{32} +1.73854e8 q^{33} -4.59834e7 q^{34} +3.64394e7 q^{36} -6.18818e8 q^{37} -9.67970e8 q^{38} +4.56198e8 q^{39} -1.30299e9 q^{41} -9.33325e8 q^{42} +4.05437e7 q^{43} +4.41505e8 q^{44} +2.91064e7 q^{46} -1.59171e9 q^{47} -1.23379e9 q^{48} +3.55794e9 q^{49} +2.16446e8 q^{51} +1.15853e9 q^{52} -1.39586e9 q^{53} -7.40757e8 q^{54} +5.49585e9 q^{56} +4.55629e9 q^{57} -6.06944e9 q^{58} +6.33376e8 q^{59} +2.84972e9 q^{61} +1.01387e9 q^{62} +4.39321e9 q^{63} +4.67679e9 q^{64} -8.97513e9 q^{66} +4.48583e9 q^{67} +5.49670e8 q^{68} -1.37006e8 q^{69} +1.58898e10 q^{71} +4.36192e9 q^{72} +4.54481e9 q^{73} +3.19462e10 q^{74} +1.15708e10 q^{76} +5.32288e10 q^{77} -2.35511e10 q^{78} +1.60636e10 q^{79} +3.48678e9 q^{81} +6.72664e10 q^{82} -3.01315e10 q^{83} +1.11566e10 q^{84} -2.09305e9 q^{86} +2.85692e10 q^{87} +5.28497e10 q^{88} -2.70045e10 q^{89} +1.39674e11 q^{91} -3.47929e8 q^{92} -4.77236e9 q^{93} +8.21717e10 q^{94} +2.69316e10 q^{96} -1.16395e11 q^{97} -1.83677e11 q^{98} +4.22464e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 46 q^{2} + 972 q^{3} + 4648 q^{4} + 11178 q^{6} + 68372 q^{7} + 386172 q^{8} + 236196 q^{9} - 99944 q^{11} + 1129464 q^{12} + 2306276 q^{13} + 3107022 q^{14} + 7622200 q^{16} - 3443816 q^{17} + 2716254 q^{18}+ \cdots - 5901593256 q^{99}+O(q^{100}) \) 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 \(-0.693203\pi\)
−0.570377 + 0.821383i \(0.693203\pi\)
\(3\) 243.000 0.577350
\(4\) 617.104 0.301320
\(5\) 0 0
\(6\) −12544.8 −0.658615
\(7\) 74399.4 1.67313 0.836566 − 0.547866i \(-0.184560\pi\)
0.836566 + 0.547866i \(0.184560\pi\)
\(8\) 73869.5 0.797022
\(9\) 59049.0 0.333333
\(10\) 0 0
\(11\) 715447. 1.33942 0.669711 − 0.742622i \(-0.266418\pi\)
0.669711 + 0.742622i \(0.266418\pi\)
\(12\) 149956. 0.173967
\(13\) 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 \(-0.475761\pi\)
0.0760754 + 0.997102i \(0.475761\pi\)
\(18\) −3.04838e6 −0.380251
\(19\) 1.87502e7 1.73724 0.868620 − 0.495478i \(-0.165007\pi\)
0.868620 + 0.495478i \(0.165007\pi\)
\(20\) 0 0
\(21\) 1.80791e7 0.965983
\(22\) −3.69347e7 −1.52795
\(23\) −563809. −0.0182654 −0.00913270 − 0.999958i \(-0.502907\pi\)
−0.00913270 + 0.999958i \(0.502907\pi\)
\(24\) 1.79503e7 0.460161
\(25\) 0 0
\(26\) −9.69180e7 −1.59975
\(27\) 1.43489e7 0.192450
\(28\) 4.59121e7 0.504148
\(29\) 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 \(0.519622\pi\)
\(32\) 1.10830e8 0.583891
\(33\) 1.73854e8 0.773316
\(34\) −4.59834e7 −0.173567
\(35\) 0 0
\(36\) 3.64394e7 0.100440
\(37\) −6.18818e8 −1.46708 −0.733539 − 0.679647i \(-0.762133\pi\)
−0.733539 + 0.679647i \(0.762133\pi\)
\(38\) −9.67970e8 −1.98176
\(39\) 4.56198e8 0.809652
\(40\) 0 0
\(41\) −1.30299e9 −1.75643 −0.878213 − 0.478269i \(-0.841264\pi\)
−0.878213 + 0.478269i \(0.841264\pi\)
\(42\) −9.33325e8 −1.10195
\(43\) 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 \(-0.668939\pi\)
−0.506170 + 0.862433i \(0.668939\pi\)
\(48\) −1.23379e9 −0.698898
\(49\) 3.55794e9 1.79937
\(50\) 0 0
\(51\) 2.16446e8 0.0878444
\(52\) 1.15853e9 0.422559
\(53\) −1.39586e9 −0.458484 −0.229242 − 0.973369i \(-0.573625\pi\)
−0.229242 + 0.973369i \(0.573625\pi\)
\(54\) −7.40757e8 −0.219538
\(55\) 0 0
\(56\) 5.49585e9 1.33352
\(57\) 4.55629e9 1.00300
\(58\) −6.06944e9 −1.21421
\(59\) 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 \(0.430698\pi\)
\(62\) 1.01387e9 0.140550
\(63\) 4.39321e9 0.557711
\(64\) 4.67679e9 0.544450
\(65\) 0 0
\(66\) −8.97513e9 −0.882164
\(67\) 4.48583e9 0.405912 0.202956 − 0.979188i \(-0.434945\pi\)
0.202956 + 0.979188i \(0.434945\pi\)
\(68\) 5.49670e8 0.0458461
\(69\) −1.37006e8 −0.0105455
\(70\) 0 0
\(71\) 1.58898e10 1.04519 0.522596 − 0.852580i \(-0.324964\pi\)
0.522596 + 0.852580i \(0.324964\pi\)
\(72\) 4.36192e9 0.265674
\(73\) 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
\(78\) −2.35511e10 −0.923614
\(79\) 1.60636e10 0.587344 0.293672 − 0.955906i \(-0.405123\pi\)
0.293672 + 0.955906i \(0.405123\pi\)
\(80\) 0 0
\(81\) 3.48678e9 0.111111
\(82\) 6.72664e10 2.00365
\(83\) −3.01315e10 −0.839636 −0.419818 − 0.907608i \(-0.637906\pi\)
−0.419818 + 0.907608i \(0.637906\pi\)
\(84\) 1.11566e10 0.291070
\(85\) 0 0
\(86\) −2.09305e9 −0.0479776
\(87\) 2.85692e10 0.614528
\(88\) 5.28497e10 1.06755
\(89\) −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
\(92\) −3.47929e8 −0.00550373
\(93\) −4.77236e9 −0.0711340
\(94\) 8.21717e10 1.15483
\(95\) 0 0
\(96\) 2.69316e10 0.337110
\(97\) −1.16395e11 −1.37623 −0.688114 − 0.725603i \(-0.741561\pi\)
−0.688114 + 0.725603i \(0.741561\pi\)
\(98\) −1.83677e11 −2.05264
\(99\) 4.22464e10 0.446474
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 75.12.a.i.1.1 yes 4
3.2 odd 2 225.12.a.q.1.4 4
5.2 odd 4 75.12.b.g.49.2 8
5.3 odd 4 75.12.b.g.49.7 8
5.4 even 2 75.12.a.h.1.4 ✓ 4
15.2 even 4 225.12.b.o.199.7 8
15.8 even 4 225.12.b.o.199.2 8
15.14 odd 2 225.12.a.s.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.12.a.h.1.4 ✓ 4 5.4 even 2
75.12.a.i.1.1 yes 4 1.1 even 1 trivial
75.12.b.g.49.2 8 5.2 odd 4
75.12.b.g.49.7 8 5.3 odd 4
225.12.a.q.1.4 4 3.2 odd 2
225.12.a.s.1.1 4 15.14 odd 2
225.12.b.o.199.2 8 15.8 even 4
225.12.b.o.199.7 8 15.2 even 4