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Properties

Label 75.18.a.d.1.2
Level $75$
Weight $18$
Character 75.1
Self dual yes
Analytic conductor $137.417$
Analytic rank $1$
Dimension $3$
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 = [3,253] 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: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 182396x + 3921120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(21.5502\) 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)
 
\(f(q)\) \(=\) \(q+105.550 q^{2} -6561.00 q^{3} -119931. q^{4} -692515. q^{6} +2.39385e7 q^{7} -2.64934e7 q^{8} +4.30467e7 q^{9} -5.09412e8 q^{11} +7.86868e8 q^{12} -4.71682e9 q^{13} +2.52671e9 q^{14} +1.29232e10 q^{16} +4.44993e10 q^{17} +4.54359e9 q^{18} -4.25101e10 q^{19} -1.57060e11 q^{21} -5.37686e10 q^{22} -4.70013e11 q^{23} +1.73823e11 q^{24} -4.97862e11 q^{26} -2.82430e11 q^{27} -2.87097e12 q^{28} +3.15724e12 q^{29} +3.46790e12 q^{31} +4.83660e12 q^{32} +3.34226e12 q^{33} +4.69691e12 q^{34} -5.16264e12 q^{36} -3.18740e13 q^{37} -4.48695e12 q^{38} +3.09471e13 q^{39} +8.13537e13 q^{41} -1.65778e13 q^{42} +1.24496e14 q^{43} +6.10944e13 q^{44} -4.96099e13 q^{46} +7.05130e13 q^{47} -8.47893e13 q^{48} +3.40421e14 q^{49} -2.91960e14 q^{51} +5.65694e14 q^{52} -2.51778e14 q^{53} -2.98105e13 q^{54} -6.34213e14 q^{56} +2.78909e14 q^{57} +3.33247e14 q^{58} -6.39615e14 q^{59} -4.96148e13 q^{61} +3.66037e14 q^{62} +1.03047e15 q^{63} -1.18337e15 q^{64} +3.52776e14 q^{66} +2.75976e15 q^{67} -5.33686e15 q^{68} +3.08375e15 q^{69} +2.50612e15 q^{71} -1.14046e15 q^{72} -8.85474e14 q^{73} -3.36430e15 q^{74} +5.09829e15 q^{76} -1.21946e16 q^{77} +3.26647e15 q^{78} -6.85544e13 q^{79} +1.85302e15 q^{81} +8.58690e15 q^{82} -3.38359e16 q^{83} +1.88364e16 q^{84} +1.31405e16 q^{86} -2.07146e16 q^{87} +1.34961e16 q^{88} -3.37871e16 q^{89} -1.12914e17 q^{91} +5.63692e16 q^{92} -2.27529e16 q^{93} +7.44266e15 q^{94} -3.17329e16 q^{96} -6.48330e16 q^{97} +3.59315e16 q^{98} -2.19285e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 253 q^{2} - 19683 q^{3} - 7087 q^{4} - 1659933 q^{6} + 4332484 q^{7} + 16188513 q^{8} + 129140163 q^{9} + 943563680 q^{11} + 46497807 q^{12} - 4257013150 q^{13} + 1847483988 q^{14} - 21943871359 q^{16}+ \cdots + 40\!\cdots\!80 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 105.550 0.291544 0.145772 − 0.989318i \(-0.453433\pi\)
0.145772 + 0.989318i \(0.453433\pi\)
\(3\) −6561.00 −0.577350
\(4\) −119931. −0.915002
\(5\) 0 0
\(6\) −692515. −0.168323
\(7\) 2.39385e7 1.56951 0.784754 − 0.619807i \(-0.212789\pi\)
0.784754 + 0.619807i \(0.212789\pi\)
\(8\) −2.64934e7 −0.558307
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) −5.09412e8 −0.716526 −0.358263 − 0.933621i \(-0.616631\pi\)
−0.358263 + 0.933621i \(0.616631\pi\)
\(12\) 7.86868e8 0.528277
\(13\) −4.71682e9 −1.60373 −0.801865 − 0.597506i \(-0.796159\pi\)
−0.801865 + 0.597506i \(0.796159\pi\)
\(14\) 2.52671e9 0.457580
\(15\) 0 0
\(16\) 1.29232e10 0.752231
\(17\) 4.44993e10 1.54717 0.773584 − 0.633693i \(-0.218462\pi\)
0.773584 + 0.633693i \(0.218462\pi\)
\(18\) 4.54359e9 0.0971813
\(19\) −4.25101e10 −0.574231 −0.287116 − 0.957896i \(-0.592696\pi\)
−0.287116 + 0.957896i \(0.592696\pi\)
\(20\) 0 0
\(21\) −1.57060e11 −0.906156
\(22\) −5.37686e10 −0.208899
\(23\) −4.70013e11 −1.25148 −0.625739 − 0.780033i \(-0.715202\pi\)
−0.625739 + 0.780033i \(0.715202\pi\)
\(24\) 1.73823e11 0.322339
\(25\) 0 0
\(26\) −4.97862e11 −0.467558
\(27\) −2.82430e11 −0.192450
\(28\) −2.87097e12 −1.43610
\(29\) 3.15724e12 1.17199 0.585996 − 0.810314i \(-0.300704\pi\)
0.585996 + 0.810314i \(0.300704\pi\)
\(30\) 0 0
\(31\) 3.46790e12 0.730285 0.365142 − 0.930952i \(-0.381020\pi\)
0.365142 + 0.930952i \(0.381020\pi\)
\(32\) 4.83660e12 0.777616
\(33\) 3.34226e12 0.413686
\(34\) 4.69691e12 0.451068
\(35\) 0 0
\(36\) −5.16264e12 −0.305001
\(37\) −3.18740e13 −1.49184 −0.745919 − 0.666037i \(-0.767989\pi\)
−0.745919 + 0.666037i \(0.767989\pi\)
\(38\) −4.48695e12 −0.167414
\(39\) 3.09471e13 0.925914
\(40\) 0 0
\(41\) 8.13537e13 1.59116 0.795581 − 0.605847i \(-0.207166\pi\)
0.795581 + 0.605847i \(0.207166\pi\)
\(42\) −1.65778e13 −0.264184
\(43\) 1.24496e14 1.62432 0.812161 − 0.583433i \(-0.198291\pi\)
0.812161 + 0.583433i \(0.198291\pi\)
\(44\) 6.10944e13 0.655623
\(45\) 0 0
\(46\) −4.96099e13 −0.364861
\(47\) 7.05130e13 0.431954 0.215977 − 0.976398i \(-0.430706\pi\)
0.215977 + 0.976398i \(0.430706\pi\)
\(48\) −8.47893e13 −0.434301
\(49\) 3.40421e14 1.46336
\(50\) 0 0
\(51\) −2.91960e14 −0.893258
\(52\) 5.65694e14 1.46742
\(53\) −2.51778e14 −0.555485 −0.277743 − 0.960656i \(-0.589586\pi\)
−0.277743 + 0.960656i \(0.589586\pi\)
\(54\) −2.98105e13 −0.0561076
\(55\) 0 0
\(56\) −6.34213e14 −0.876268
\(57\) 2.78909e14 0.331533
\(58\) 3.33247e14 0.341687
\(59\) −6.39615e14 −0.567122 −0.283561 − 0.958954i \(-0.591516\pi\)
−0.283561 + 0.958954i \(0.591516\pi\)
\(60\) 0 0
\(61\) −4.96148e13 −0.0331366 −0.0165683 − 0.999863i \(-0.505274\pi\)
−0.0165683 + 0.999863i \(0.505274\pi\)
\(62\) 3.66037e14 0.212910
\(63\) 1.03047e15 0.523169
\(64\) −1.18337e15 −0.525522
\(65\) 0 0
\(66\) 3.52776e14 0.120608
\(67\) 2.75976e15 0.830300 0.415150 − 0.909753i \(-0.363729\pi\)
0.415150 + 0.909753i \(0.363729\pi\)
\(68\) −5.33686e15 −1.41566
\(69\) 3.08375e15 0.722541
\(70\) 0 0
\(71\) 2.50612e15 0.460582 0.230291 − 0.973122i \(-0.426032\pi\)
0.230291 + 0.973122i \(0.426032\pi\)
\(72\) −1.14046e15 −0.186102
\(73\) −8.85474e14 −0.128508 −0.0642542 − 0.997934i \(-0.520467\pi\)
−0.0642542 + 0.997934i \(0.520467\pi\)
\(74\) −3.36430e15 −0.434936
\(75\) 0 0
\(76\) 5.09829e15 0.525423
\(77\) −1.21946e16 −1.12459
\(78\) 3.26647e15 0.269944
\(79\) −6.85544e13 −0.00508400 −0.00254200 − 0.999997i \(-0.500809\pi\)
−0.00254200 + 0.999997i \(0.500809\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 8.58690e15 0.463894
\(83\) −3.38359e16 −1.64897 −0.824487 − 0.565881i \(-0.808536\pi\)
−0.824487 + 0.565881i \(0.808536\pi\)
\(84\) 1.88364e16 0.829135
\(85\) 0 0
\(86\) 1.31405e16 0.473561
\(87\) −2.07146e16 −0.676650
\(88\) 1.34961e16 0.400041
\(89\) −3.37871e16 −0.909778 −0.454889 − 0.890548i \(-0.650321\pi\)
−0.454889 + 0.890548i \(0.650321\pi\)
\(90\) 0 0
\(91\) −1.12914e17 −2.51707
\(92\) 5.63692e16 1.14510
\(93\) −2.27529e16 −0.421630
\(94\) 7.44266e15 0.125934
\(95\) 0 0
\(96\) −3.17329e16 −0.448957
\(97\) −6.48330e16 −0.839917 −0.419958 − 0.907543i \(-0.637955\pi\)
−0.419958 + 0.907543i \(0.637955\pi\)
\(98\) 3.59315e16 0.426632
\(99\) −2.19285e16 −0.238842
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.d.1.2 3
5.2 odd 4 75.18.b.e.49.4 6
5.3 odd 4 75.18.b.e.49.3 6
5.4 even 2 15.18.a.c.1.2 ✓ 3
15.14 odd 2 45.18.a.d.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.c.1.2 ✓ 3 5.4 even 2
45.18.a.d.1.2 3 15.14 odd 2
75.18.a.d.1.2 3 1.1 even 1 trivial
75.18.b.e.49.3 6 5.3 odd 4
75.18.b.e.49.4 6 5.2 odd 4