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Properties

Label 75.18.a.e.1.1
Level $75$
Weight $18$
Character 75.1
Self dual yes
Analytic conductor $137.417$
Analytic rank $0$
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 = [3,442] 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: \(3\)
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: \( x^{3} - 37234x - 350700 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-9.44141\) 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)
 
\(f(q)\) \(=\) \(q-442.567 q^{2} +6561.00 q^{3} +64793.8 q^{4} -2.90368e6 q^{6} -2.47993e7 q^{7} +2.93326e7 q^{8} +4.30467e7 q^{9} -7.66684e6 q^{11} +4.25112e8 q^{12} +3.40086e9 q^{13} +1.09754e10 q^{14} -2.14743e10 q^{16} +5.35306e10 q^{17} -1.90511e10 q^{18} -1.29760e11 q^{19} -1.62708e11 q^{21} +3.39309e9 q^{22} -2.11422e11 q^{23} +1.92451e11 q^{24} -1.50511e12 q^{26} +2.82430e11 q^{27} -1.60684e12 q^{28} +4.13975e12 q^{29} -5.31377e12 q^{31} +5.65914e12 q^{32} -5.03022e10 q^{33} -2.36909e13 q^{34} +2.78916e12 q^{36} -1.85634e13 q^{37} +5.74274e13 q^{38} +2.23130e13 q^{39} -5.57404e13 q^{41} +7.20094e13 q^{42} +7.31648e13 q^{43} -4.96764e11 q^{44} +9.35684e13 q^{46} +1.22507e14 q^{47} -1.40893e14 q^{48} +3.82376e14 q^{49} +3.51215e14 q^{51} +2.20355e14 q^{52} -2.74251e14 q^{53} -1.24994e14 q^{54} -7.27428e14 q^{56} -8.51354e14 q^{57} -1.83212e15 q^{58} +6.31977e14 q^{59} -1.14361e15 q^{61} +2.35170e15 q^{62} -1.06753e15 q^{63} +3.10129e14 q^{64} +2.22621e13 q^{66} -6.08292e14 q^{67} +3.46845e15 q^{68} -1.38714e15 q^{69} +1.25518e15 q^{71} +1.26267e15 q^{72} -7.71376e15 q^{73} +8.21553e15 q^{74} -8.40763e15 q^{76} +1.90133e14 q^{77} -9.87502e15 q^{78} -2.63076e16 q^{79} +1.85302e15 q^{81} +2.46689e16 q^{82} -1.45466e16 q^{83} -1.05425e16 q^{84} -3.23803e16 q^{86} +2.71609e16 q^{87} -2.24888e14 q^{88} +2.61450e16 q^{89} -8.43391e16 q^{91} -1.36988e16 q^{92} -3.48637e16 q^{93} -5.42175e16 q^{94} +3.71296e16 q^{96} +9.64972e15 q^{97} -1.69227e17 q^{98} -3.30032e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 442 q^{2} + 19683 q^{3} + 298148 q^{4} + 2899962 q^{6} - 4962644 q^{7} + 108831912 q^{8} + 129140163 q^{9} + 1049849720 q^{11} + 1956149028 q^{12} + 3091742090 q^{13} + 27586028328 q^{14} + 22392797456 q^{16}+ \cdots + 45\!\cdots\!20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −442.567 −1.22243 −0.611215 − 0.791464i \(-0.709319\pi\)
−0.611215 + 0.791464i \(0.709319\pi\)
\(3\) 6561.00 0.577350
\(4\) 64793.8 0.494337
\(5\) 0 0
\(6\) −2.90368e6 −0.705771
\(7\) −2.47993e7 −1.62595 −0.812974 − 0.582300i \(-0.802153\pi\)
−0.812974 + 0.582300i \(0.802153\pi\)
\(8\) 2.93326e7 0.618138
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) −7.66684e6 −0.0107840 −0.00539199 − 0.999985i \(-0.501716\pi\)
−0.00539199 + 0.999985i \(0.501716\pi\)
\(12\) 4.25112e8 0.285406
\(13\) 3.40086e9 1.15630 0.578150 − 0.815931i \(-0.303775\pi\)
0.578150 + 0.815931i \(0.303775\pi\)
\(14\) 1.09754e10 1.98761
\(15\) 0 0
\(16\) −2.14743e10 −1.24997
\(17\) 5.35306e10 1.86117 0.930586 − 0.366073i \(-0.119298\pi\)
0.930586 + 0.366073i \(0.119298\pi\)
\(18\) −1.90511e10 −0.407477
\(19\) −1.29760e11 −1.75281 −0.876404 − 0.481576i \(-0.840065\pi\)
−0.876404 + 0.481576i \(0.840065\pi\)
\(20\) 0 0
\(21\) −1.62708e11 −0.938742
\(22\) 3.39309e9 0.0131827
\(23\) −2.11422e11 −0.562941 −0.281471 − 0.959570i \(-0.590822\pi\)
−0.281471 + 0.959570i \(0.590822\pi\)
\(24\) 1.92451e11 0.356882
\(25\) 0 0
\(26\) −1.50511e12 −1.41350
\(27\) 2.82430e11 0.192450
\(28\) −1.60684e12 −0.803767
\(29\) 4.13975e12 1.53671 0.768354 − 0.640026i \(-0.221076\pi\)
0.768354 + 0.640026i \(0.221076\pi\)
\(30\) 0 0
\(31\) −5.31377e12 −1.11900 −0.559498 − 0.828832i \(-0.689006\pi\)
−0.559498 + 0.828832i \(0.689006\pi\)
\(32\) 5.65914e12 0.909862
\(33\) −5.03022e10 −0.00622613
\(34\) −2.36909e13 −2.27515
\(35\) 0 0
\(36\) 2.78916e12 0.164779
\(37\) −1.85634e13 −0.868844 −0.434422 − 0.900709i \(-0.643047\pi\)
−0.434422 + 0.900709i \(0.643047\pi\)
\(38\) 5.74274e13 2.14269
\(39\) 2.23130e13 0.667590
\(40\) 0 0
\(41\) −5.57404e13 −1.09020 −0.545101 − 0.838370i \(-0.683509\pi\)
−0.545101 + 0.838370i \(0.683509\pi\)
\(42\) 7.20094e13 1.14755
\(43\) 7.31648e13 0.954597 0.477299 − 0.878741i \(-0.341616\pi\)
0.477299 + 0.878741i \(0.341616\pi\)
\(44\) −4.96764e11 −0.00533092
\(45\) 0 0
\(46\) 9.35684e13 0.688157
\(47\) 1.22507e14 0.750461 0.375231 − 0.926931i \(-0.377564\pi\)
0.375231 + 0.926931i \(0.377564\pi\)
\(48\) −1.40893e14 −0.721669
\(49\) 3.82376e14 1.64371
\(50\) 0 0
\(51\) 3.51215e14 1.07455
\(52\) 2.20355e14 0.571602
\(53\) −2.74251e14 −0.605068 −0.302534 − 0.953139i \(-0.597832\pi\)
−0.302534 + 0.953139i \(0.597832\pi\)
\(54\) −1.24994e14 −0.235257
\(55\) 0 0
\(56\) −7.27428e14 −1.00506
\(57\) −8.51354e14 −1.01198
\(58\) −1.83212e15 −1.87852
\(59\) 6.31977e14 0.560350 0.280175 − 0.959949i \(-0.409608\pi\)
0.280175 + 0.959949i \(0.409608\pi\)
\(60\) 0 0
\(61\) −1.14361e15 −0.763791 −0.381895 − 0.924206i \(-0.624729\pi\)
−0.381895 + 0.924206i \(0.624729\pi\)
\(62\) 2.35170e15 1.36790
\(63\) −1.06753e15 −0.541983
\(64\) 3.10129e14 0.137725
\(65\) 0 0
\(66\) 2.22621e13 0.00761101
\(67\) −6.08292e14 −0.183011 −0.0915053 − 0.995805i \(-0.529168\pi\)
−0.0915053 + 0.995805i \(0.529168\pi\)
\(68\) 3.46845e15 0.920047
\(69\) −1.38714e15 −0.325014
\(70\) 0 0
\(71\) 1.25518e15 0.230681 0.115340 − 0.993326i \(-0.463204\pi\)
0.115340 + 0.993326i \(0.463204\pi\)
\(72\) 1.26267e15 0.206046
\(73\) −7.71376e15 −1.11949 −0.559747 − 0.828664i \(-0.689102\pi\)
−0.559747 + 0.828664i \(0.689102\pi\)
\(74\) 8.21553e15 1.06210
\(75\) 0 0
\(76\) −8.40763e15 −0.866479
\(77\) 1.90133e14 0.0175342
\(78\) −9.87502e15 −0.816082
\(79\) −2.63076e16 −1.95097 −0.975486 − 0.220061i \(-0.929374\pi\)
−0.975486 + 0.220061i \(0.929374\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 2.46689e16 1.33270
\(83\) −1.45466e16 −0.708922 −0.354461 − 0.935071i \(-0.615336\pi\)
−0.354461 + 0.935071i \(0.615336\pi\)
\(84\) −1.05425e16 −0.464055
\(85\) 0 0
\(86\) −3.23803e16 −1.16693
\(87\) 2.71609e16 0.887218
\(88\) −2.24888e14 −0.00666598
\(89\) 2.61450e16 0.704001 0.352000 − 0.936000i \(-0.385502\pi\)
0.352000 + 0.936000i \(0.385502\pi\)
\(90\) 0 0
\(91\) −8.43391e16 −1.88008
\(92\) −1.36988e16 −0.278283
\(93\) −3.48637e16 −0.646053
\(94\) −5.42175e16 −0.917387
\(95\) 0 0
\(96\) 3.71296e16 0.525309
\(97\) 9.64972e15 0.125013 0.0625064 − 0.998045i \(-0.480091\pi\)
0.0625064 + 0.998045i \(0.480091\pi\)
\(98\) −1.69227e17 −2.00932
\(99\) −3.30032e14 −0.00359466
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.e.1.1 3
5.2 odd 4 75.18.b.d.49.2 6
5.3 odd 4 75.18.b.d.49.5 6
5.4 even 2 15.18.a.b.1.3 ✓ 3
15.14 odd 2 45.18.a.e.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.b.1.3 ✓ 3 5.4 even 2
45.18.a.e.1.1 3 15.14 odd 2
75.18.a.e.1.1 3 1.1 even 1 trivial
75.18.b.d.49.2 6 5.2 odd 4
75.18.b.d.49.5 6 5.3 odd 4