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Properties

Label 75.18.a.i.1.5
Level $75$
Weight $18$
Character 75.1
Self dual yes
Analytic conductor $137.417$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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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 1.5
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 \(-0.337048\pi\)
0.489860 + 0.871801i \(0.337048\pi\)
\(3\) 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 \(-0.346499\pi\)
0.463764 + 0.885959i \(0.346499\pi\)
\(8\) −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 \(-0.766991\pi\)
−0.743826 + 0.668374i \(0.766991\pi\)
\(14\) 5.01784e9 0.908717
\(15\) 0 0
\(16\) −1.64624e10 −0.958238
\(17\) −1.49113e10 −0.518440 −0.259220 − 0.965818i \(-0.583466\pi\)
−0.259220 + 0.965818i \(0.583466\pi\)
\(18\) 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 \(-0.535499\pi\)
−0.111292 + 0.993788i \(0.535499\pi\)
\(24\) −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 \(-0.622615\pi\)
−0.375751 + 0.926721i \(0.622615\pi\)
\(38\) 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 \(-0.673244\pi\)
−0.517787 + 0.855510i \(0.673244\pi\)
\(44\) −2.68319e12 −0.0287941
\(45\) 0 0
\(46\) −2.96510e13 −0.218071
\(47\) 1.62631e13 0.0996259 0.0498129 − 0.998759i \(-0.484137\pi\)
0.0498129 + 0.998759i \(0.484137\pi\)
\(48\) −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 \(-0.419178\pi\)
0.251190 + 0.967938i \(0.419178\pi\)
\(54\) 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 \(-0.443434\pi\)
0.176774 + 0.984251i \(0.443434\pi\)
\(68\) 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 \(-0.404193\pi\)
0.296463 + 0.955044i \(0.404193\pi\)
\(74\) −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 \(-0.260783\pi\)
0.682752 + 0.730650i \(0.260783\pi\)
\(84\) −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 \(-0.270886\pi\)
0.659220 + 0.751950i \(0.270886\pi\)
\(98\) −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.i.1.5 ✓ 6
5.2 odd 4 75.18.b.h.49.9 12
5.3 odd 4 75.18.b.h.49.4 12
5.4 even 2 75.18.a.j.1.2 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.18.a.i.1.5 ✓ 6 1.1 even 1 trivial
75.18.a.j.1.2 yes 6 5.4 even 2
75.18.b.h.49.4 12 5.3 odd 4
75.18.b.h.49.9 12 5.2 odd 4