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Properties

Label 75.18.a.i.1.4
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 1.4
Root \(-18.7333\) 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)
 
\(f(q)\) \(=\) \(q-37.2667 q^{2} +6561.00 q^{3} -129683. q^{4} -244507. q^{6} -4.45476e6 q^{7} +9.71748e6 q^{8} +4.30467e7 q^{9} +8.94121e8 q^{11} -8.50851e8 q^{12} +4.68045e9 q^{13} +1.66014e8 q^{14} +1.66357e10 q^{16} -7.14036e9 q^{17} -1.60421e9 q^{18} +1.60299e10 q^{19} -2.92277e10 q^{21} -3.33209e10 q^{22} -2.00833e11 q^{23} +6.37564e10 q^{24} -1.74425e11 q^{26} +2.82430e11 q^{27} +5.77707e11 q^{28} -3.38903e12 q^{29} -7.72370e11 q^{31} -1.89365e12 q^{32} +5.86633e12 q^{33} +2.66097e11 q^{34} -5.58244e12 q^{36} -1.98749e13 q^{37} -5.97382e11 q^{38} +3.07084e13 q^{39} -6.17738e13 q^{41} +1.08922e12 q^{42} +6.28002e13 q^{43} -1.15952e14 q^{44} +7.48437e12 q^{46} +1.59956e14 q^{47} +1.09147e14 q^{48} -2.12786e14 q^{49} -4.68479e13 q^{51} -6.06976e14 q^{52} +4.53728e14 q^{53} -1.05252e13 q^{54} -4.32890e13 q^{56} +1.05172e14 q^{57} +1.26298e14 q^{58} +7.87172e14 q^{59} +8.05401e14 q^{61} +2.87837e13 q^{62} -1.91763e14 q^{63} -2.10990e15 q^{64} -2.18619e14 q^{66} +4.65610e15 q^{67} +9.25985e14 q^{68} -1.31766e15 q^{69} +9.90027e15 q^{71} +4.18306e14 q^{72} -1.59824e15 q^{73} +7.40671e14 q^{74} -2.07881e15 q^{76} -3.98309e15 q^{77} -1.14440e15 q^{78} +1.37101e16 q^{79} +1.85302e15 q^{81} +2.30210e15 q^{82} -7.48072e15 q^{83} +3.79034e15 q^{84} -2.34035e15 q^{86} -2.22354e16 q^{87} +8.68860e15 q^{88} -2.79037e16 q^{89} -2.08503e16 q^{91} +2.60446e16 q^{92} -5.06752e15 q^{93} -5.96102e15 q^{94} -1.24242e16 q^{96} -1.10172e17 q^{97} +7.92981e15 q^{98} +3.84890e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 338 q^{2} + 39366 q^{3} + 393248 q^{4} - 2217618 q^{6} + 20999794 q^{7} + 25242756 q^{8} + 258280326 q^{9} + 85907324 q^{11} + 2580100128 q^{12} + 344649098 q^{13} + 4962466602 q^{14} + 13788291320 q^{16}+ \cdots + 36\!\cdots\!04 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −37.2667 −0.102936 −0.0514678 − 0.998675i \(-0.516390\pi\)
−0.0514678 + 0.998675i \(0.516390\pi\)
\(3\) 6561.00 0.577350
\(4\) −129683. −0.989404
\(5\) 0 0
\(6\) −244507. −0.0594299
\(7\) −4.45476e6 −0.292073 −0.146036 − 0.989279i \(-0.546652\pi\)
−0.146036 + 0.989279i \(0.546652\pi\)
\(8\) 9.71748e6 0.204781
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) 8.94121e8 1.25765 0.628823 − 0.777548i \(-0.283537\pi\)
0.628823 + 0.777548i \(0.283537\pi\)
\(12\) −8.50851e8 −0.571233
\(13\) 4.68045e9 1.59136 0.795682 − 0.605715i \(-0.207113\pi\)
0.795682 + 0.605715i \(0.207113\pi\)
\(14\) 1.66014e8 0.0300647
\(15\) 0 0
\(16\) 1.66357e10 0.968325
\(17\) −7.14036e9 −0.248259 −0.124129 − 0.992266i \(-0.539614\pi\)
−0.124129 + 0.992266i \(0.539614\pi\)
\(18\) −1.60421e9 −0.0343119
\(19\) 1.60299e10 0.216534 0.108267 − 0.994122i \(-0.465470\pi\)
0.108267 + 0.994122i \(0.465470\pi\)
\(20\) 0 0
\(21\) −2.92277e10 −0.168628
\(22\) −3.33209e10 −0.129457
\(23\) −2.00833e11 −0.534747 −0.267373 − 0.963593i \(-0.586156\pi\)
−0.267373 + 0.963593i \(0.586156\pi\)
\(24\) 6.37564e10 0.118230
\(25\) 0 0
\(26\) −1.74425e11 −0.163808
\(27\) 2.82430e11 0.192450
\(28\) 5.77707e11 0.288978
\(29\) −3.38903e12 −1.25803 −0.629017 − 0.777392i \(-0.716542\pi\)
−0.629017 + 0.777392i \(0.716542\pi\)
\(30\) 0 0
\(31\) −7.72370e11 −0.162649 −0.0813244 − 0.996688i \(-0.525915\pi\)
−0.0813244 + 0.996688i \(0.525915\pi\)
\(32\) −1.89365e12 −0.304456
\(33\) 5.86633e12 0.726103
\(34\) 2.66097e11 0.0255546
\(35\) 0 0
\(36\) −5.58244e12 −0.329801
\(37\) −1.98749e13 −0.930230 −0.465115 − 0.885250i \(-0.653987\pi\)
−0.465115 + 0.885250i \(0.653987\pi\)
\(38\) −5.97382e11 −0.0222890
\(39\) 3.07084e13 0.918774
\(40\) 0 0
\(41\) −6.17738e13 −1.20821 −0.604103 − 0.796906i \(-0.706469\pi\)
−0.604103 + 0.796906i \(0.706469\pi\)
\(42\) 1.08922e12 0.0173578
\(43\) 6.28002e13 0.819368 0.409684 − 0.912227i \(-0.365639\pi\)
0.409684 + 0.912227i \(0.365639\pi\)
\(44\) −1.15952e14 −1.24432
\(45\) 0 0
\(46\) 7.48437e12 0.0550445
\(47\) 1.59956e14 0.979870 0.489935 − 0.871759i \(-0.337020\pi\)
0.489935 + 0.871759i \(0.337020\pi\)
\(48\) 1.09147e14 0.559063
\(49\) −2.12786e14 −0.914694
\(50\) 0 0
\(51\) −4.68479e13 −0.143332
\(52\) −6.06976e14 −1.57450
\(53\) 4.53728e14 1.00104 0.500519 − 0.865725i \(-0.333142\pi\)
0.500519 + 0.865725i \(0.333142\pi\)
\(54\) −1.05252e13 −0.0198100
\(55\) 0 0
\(56\) −4.32890e13 −0.0598108
\(57\) 1.05172e14 0.125016
\(58\) 1.26298e14 0.129496
\(59\) 7.87172e14 0.697955 0.348978 − 0.937131i \(-0.386529\pi\)
0.348978 + 0.937131i \(0.386529\pi\)
\(60\) 0 0
\(61\) 8.05401e14 0.537909 0.268954 − 0.963153i \(-0.413322\pi\)
0.268954 + 0.963153i \(0.413322\pi\)
\(62\) 2.87837e13 0.0167424
\(63\) −1.91763e14 −0.0973575
\(64\) −2.10990e15 −0.936986
\(65\) 0 0
\(66\) −2.18619e14 −0.0747418
\(67\) 4.65610e15 1.40083 0.700416 − 0.713735i \(-0.252998\pi\)
0.700416 + 0.713735i \(0.252998\pi\)
\(68\) 9.25985e14 0.245628
\(69\) −1.31766e15 −0.308736
\(70\) 0 0
\(71\) 9.90027e15 1.81950 0.909748 − 0.415161i \(-0.136275\pi\)
0.909748 + 0.415161i \(0.136275\pi\)
\(72\) 4.18306e14 0.0682602
\(73\) −1.59824e15 −0.231952 −0.115976 − 0.993252i \(-0.537000\pi\)
−0.115976 + 0.993252i \(0.537000\pi\)
\(74\) 7.40671e14 0.0957538
\(75\) 0 0
\(76\) −2.07881e15 −0.214239
\(77\) −3.98309e15 −0.367324
\(78\) −1.14440e15 −0.0945745
\(79\) 1.37101e16 1.01674 0.508371 − 0.861138i \(-0.330248\pi\)
0.508371 + 0.861138i \(0.330248\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 2.30210e15 0.124367
\(83\) −7.48072e15 −0.364569 −0.182284 − 0.983246i \(-0.558349\pi\)
−0.182284 + 0.983246i \(0.558349\pi\)
\(84\) 3.79034e15 0.166841
\(85\) 0 0
\(86\) −2.34035e15 −0.0843422
\(87\) −2.22354e16 −0.726326
\(88\) 8.68860e15 0.257541
\(89\) −2.79037e16 −0.751357 −0.375678 − 0.926750i \(-0.622590\pi\)
−0.375678 + 0.926750i \(0.622590\pi\)
\(90\) 0 0
\(91\) −2.08503e16 −0.464794
\(92\) 2.60446e16 0.529081
\(93\) −5.06752e15 −0.0939054
\(94\) −5.96102e15 −0.100864
\(95\) 0 0
\(96\) −1.24242e16 −0.175778
\(97\) −1.10172e17 −1.42729 −0.713647 − 0.700505i \(-0.752958\pi\)
−0.713647 + 0.700505i \(0.752958\pi\)
\(98\) 7.92981e15 0.0941545
\(99\) 3.84890e16 0.419215
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.1.4 ✓ 6
5.2 odd 4 75.18.b.h.49.6 12
5.3 odd 4 75.18.b.h.49.7 12
5.4 even 2 75.18.a.j.1.3 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.18.a.i.1.4 ✓ 6 1.1 even 1 trivial
75.18.a.j.1.3 yes 6 5.4 even 2
75.18.b.h.49.6 12 5.2 odd 4
75.18.b.h.49.7 12 5.3 odd 4