Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,18,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, 18); 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, 18, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,338] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(137.416565508\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 580318x^{4} + 45393344x^{3} + 72695152416x^{2} - 6623241804288x - 149217035286528 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{5}\cdot 5^{7} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-410.696\) 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)
 
\(f(q)\) \(=\) \(q-354.696 q^{2} -6561.00 q^{3} -5262.54 q^{4} +2.32716e6 q^{6} -1.41469e7 q^{7} +4.83574e7 q^{8} +4.30467e7 q^{9} +5.09867e8 q^{11} +3.45275e7 q^{12} +4.37542e9 q^{13} +5.01784e9 q^{14} -1.64624e10 q^{16} +1.49113e10 q^{17} -1.52685e10 q^{18} +1.24854e11 q^{19} +9.28176e10 q^{21} -1.80848e11 q^{22} +8.35954e10 q^{23} -3.17273e11 q^{24} -1.55194e12 q^{26} -2.82430e11 q^{27} +7.44485e10 q^{28} +5.15565e12 q^{29} +3.28299e12 q^{31} -4.99142e11 q^{32} -3.34523e12 q^{33} -5.28897e12 q^{34} -2.26535e11 q^{36} +1.60563e13 q^{37} -4.42852e13 q^{38} -2.87071e13 q^{39} +4.59655e13 q^{41} -3.29221e13 q^{42} +7.93712e13 q^{43} -2.68319e12 q^{44} -2.96510e13 q^{46} -1.62631e13 q^{47} +1.08010e14 q^{48} -3.24966e13 q^{49} -9.78329e13 q^{51} -2.30258e13 q^{52} -2.27707e14 q^{53} +1.00177e14 q^{54} -6.84105e14 q^{56} -8.19166e14 q^{57} -1.82869e15 q^{58} -7.24114e14 q^{59} +2.08878e15 q^{61} -1.16446e15 q^{62} -6.08976e14 q^{63} +2.33480e15 q^{64} +1.18654e15 q^{66} -1.17513e15 q^{67} -7.84712e13 q^{68} -5.48469e14 q^{69} -5.82360e15 q^{71} +2.08163e15 q^{72} -4.08549e15 q^{73} -5.69510e15 q^{74} -6.57048e14 q^{76} -7.21302e15 q^{77} +1.01823e16 q^{78} +2.19279e16 q^{79} +1.85302e15 q^{81} -1.63038e16 q^{82} -2.80193e16 q^{83} -4.88456e14 q^{84} -2.81527e16 q^{86} -3.38262e16 q^{87} +2.46558e16 q^{88} +3.90478e16 q^{89} -6.18985e16 q^{91} -4.39924e14 q^{92} -2.15397e16 q^{93} +5.76847e15 q^{94} +3.27487e15 q^{96} -1.01770e17 q^{97} +1.15264e16 q^{98} +2.19481e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 338 q^{2} - 39366 q^{3} + 393248 q^{4} - 2217618 q^{6} - 20999794 q^{7} - 25242756 q^{8} + 258280326 q^{9} + 85907324 q^{11} - 2580100128 q^{12} - 344649098 q^{13} + 4962466602 q^{14} + 13788291320 q^{16}+ \cdots + 36\!\cdots\!04 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −354.696 −0.979719 −0.489860 − 0.871801i \(-0.662952\pi\)
−0.489860 + 0.871801i \(0.662952\pi\)
\(3\) −6561.00 −0.577350
\(4\) −5262.54 −0.0401500
\(5\) 0 0
\(6\) 2.32716e6 0.565641
\(7\) −1.41469e7 −0.927528 −0.463764 − 0.885959i \(-0.653501\pi\)
−0.463764 + 0.885959i \(0.653501\pi\)
\(8\) 4.83574e7 1.01906
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) 5.09867e8 0.717164 0.358582 − 0.933498i \(-0.383260\pi\)
0.358582 + 0.933498i \(0.383260\pi\)
\(12\) 3.45275e7 0.0231806
\(13\) 4.37542e9 1.48765 0.743826 − 0.668374i \(-0.233009\pi\)
0.743826 + 0.668374i \(0.233009\pi\)
\(14\) 5.01784e9 0.908717
\(15\) 0 0
\(16\) −1.64624e10 −0.958238
\(17\) 1.49113e10 0.518440 0.259220 − 0.965818i \(-0.416534\pi\)
0.259220 + 0.965818i \(0.416534\pi\)
\(18\) −1.52685e10 −0.326573
\(19\) 1.24854e11 1.68654 0.843269 − 0.537492i \(-0.180628\pi\)
0.843269 + 0.537492i \(0.180628\pi\)
\(20\) 0 0
\(21\) 9.28176e10 0.535508
\(22\) −1.80848e11 −0.702620
\(23\) 8.35954e10 0.222585 0.111292 − 0.993788i \(-0.464501\pi\)
0.111292 + 0.993788i \(0.464501\pi\)
\(24\) −3.17273e11 −0.588352
\(25\) 0 0
\(26\) −1.55194e12 −1.45748
\(27\) −2.82430e11 −0.192450
\(28\) 7.44485e10 0.0372402
\(29\) 5.15565e12 1.91382 0.956909 − 0.290389i \(-0.0937846\pi\)
0.956909 + 0.290389i \(0.0937846\pi\)
\(30\) 0 0
\(31\) 3.28299e12 0.691345 0.345673 − 0.938355i \(-0.387651\pi\)
0.345673 + 0.938355i \(0.387651\pi\)
\(32\) −4.99142e11 −0.0802508
\(33\) −3.34523e12 −0.414055
\(34\) −5.28897e12 −0.507926
\(35\) 0 0
\(36\) −2.26535e11 −0.0133833
\(37\) 1.60563e13 0.751501 0.375751 − 0.926721i \(-0.377385\pi\)
0.375751 + 0.926721i \(0.377385\pi\)
\(38\) −4.42852e13 −1.65233
\(39\) −2.87071e13 −0.858896
\(40\) 0 0
\(41\) 4.59655e13 0.899020 0.449510 − 0.893275i \(-0.351599\pi\)
0.449510 + 0.893275i \(0.351599\pi\)
\(42\) −3.29221e13 −0.524648
\(43\) 7.93712e13 1.03557 0.517787 − 0.855510i \(-0.326756\pi\)
0.517787 + 0.855510i \(0.326756\pi\)
\(44\) −2.68319e12 −0.0287941
\(45\) 0 0
\(46\) −2.96510e13 −0.218071
\(47\) −1.62631e13 −0.0996259 −0.0498129 − 0.998759i \(-0.515863\pi\)
−0.0498129 + 0.998759i \(0.515863\pi\)
\(48\) 1.08010e14 0.553239
\(49\) −3.24966e13 −0.139692
\(50\) 0 0
\(51\) −9.78329e13 −0.299322
\(52\) −2.30258e13 −0.0597292
\(53\) −2.27707e14 −0.502380 −0.251190 − 0.967938i \(-0.580822\pi\)
−0.251190 + 0.967938i \(0.580822\pi\)
\(54\) 1.00177e14 0.188547
\(55\) 0 0
\(56\) −6.84105e14 −0.945202
\(57\) −8.19166e14 −0.973723
\(58\) −1.82869e15 −1.87500
\(59\) −7.24114e14 −0.642044 −0.321022 − 0.947072i \(-0.604026\pi\)
−0.321022 + 0.947072i \(0.604026\pi\)
\(60\) 0 0
\(61\) 2.08878e15 1.39505 0.697524 − 0.716562i \(-0.254285\pi\)
0.697524 + 0.716562i \(0.254285\pi\)
\(62\) −1.16446e15 −0.677324
\(63\) −6.08976e14 −0.309176
\(64\) 2.33480e15 1.03686
\(65\) 0 0
\(66\) 1.18654e15 0.405658
\(67\) −1.17513e15 −0.353549 −0.176774 − 0.984251i \(-0.556566\pi\)
−0.176774 + 0.984251i \(0.556566\pi\)
\(68\) −7.84712e13 −0.0208154
\(69\) −5.48469e14 −0.128509
\(70\) 0 0
\(71\) −5.82360e15 −1.07028 −0.535138 − 0.844765i \(-0.679740\pi\)
−0.535138 + 0.844765i \(0.679740\pi\)
\(72\) 2.08163e15 0.339685
\(73\) −4.08549e15 −0.592926 −0.296463 − 0.955044i \(-0.595807\pi\)
−0.296463 + 0.955044i \(0.595807\pi\)
\(74\) −5.69510e15 −0.736260
\(75\) 0 0
\(76\) −6.57048e14 −0.0677145
\(77\) −7.21302e15 −0.665190
\(78\) 1.01823e16 0.841477
\(79\) 2.19279e16 1.62617 0.813087 − 0.582142i \(-0.197785\pi\)
0.813087 + 0.582142i \(0.197785\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) −1.63038e16 −0.880788
\(83\) −2.80193e16 −1.36550 −0.682752 − 0.730650i \(-0.739217\pi\)
−0.682752 + 0.730650i \(0.739217\pi\)
\(84\) −4.88456e14 −0.0215007
\(85\) 0 0
\(86\) −2.81527e16 −1.01457
\(87\) −3.38262e16 −1.10494
\(88\) 2.46558e16 0.730830
\(89\) 3.90478e16 1.05143 0.525717 − 0.850660i \(-0.323797\pi\)
0.525717 + 0.850660i \(0.323797\pi\)
\(90\) 0 0
\(91\) −6.18985e16 −1.37984
\(92\) −4.39924e14 −0.00893678
\(93\) −2.15397e16 −0.399148
\(94\) 5.76847e15 0.0976054
\(95\) 0 0
\(96\) 3.27487e15 0.0463328
\(97\) −1.01770e17 −1.31844 −0.659220 − 0.751950i \(-0.729114\pi\)
−0.659220 + 0.751950i \(0.729114\pi\)
\(98\) 1.15264e16 0.136859
\(99\) 2.19481e16 0.239055
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 75.18.a.j.1.2 yes 6
5.2 odd 4 75.18.b.h.49.4 12
5.3 odd 4 75.18.b.h.49.9 12
5.4 even 2 75.18.a.i.1.5 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.18.a.i.1.5 ✓ 6 5.4 even 2
75.18.a.j.1.2 yes 6 1.1 even 1 trivial
75.18.b.h.49.4 12 5.2 odd 4
75.18.b.h.49.9 12 5.3 odd 4