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Properties

Label 75.18.a.e.1.2
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 1.2
Root \(197.509\) 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)
 
\(f(q)\) \(=\) \(q+213.954 q^{2} +6561.00 q^{3} -85295.6 q^{4} +1.40375e6 q^{6} -7.24373e6 q^{7} -4.62928e7 q^{8} +4.30467e7 q^{9} +4.10790e7 q^{11} -5.59624e8 q^{12} -1.35746e9 q^{13} -1.54983e9 q^{14} +1.27533e9 q^{16} -1.84008e10 q^{17} +9.21003e9 q^{18} +1.42200e9 q^{19} -4.75261e10 q^{21} +8.78904e9 q^{22} +1.96071e10 q^{23} -3.03727e11 q^{24} -2.90434e11 q^{26} +2.82430e11 q^{27} +6.17858e11 q^{28} -3.22761e12 q^{29} -1.93640e12 q^{31} +6.34055e12 q^{32} +2.69520e11 q^{33} -3.93694e12 q^{34} -3.67169e12 q^{36} +2.47063e13 q^{37} +3.04244e11 q^{38} -8.90629e12 q^{39} +1.91037e13 q^{41} -1.01684e13 q^{42} +9.80229e12 q^{43} -3.50386e12 q^{44} +4.19503e12 q^{46} -6.08880e13 q^{47} +8.36742e12 q^{48} -1.80159e14 q^{49} -1.20728e14 q^{51} +1.15785e14 q^{52} -3.72760e14 q^{53} +6.04270e13 q^{54} +3.35332e14 q^{56} +9.32977e12 q^{57} -6.90561e14 q^{58} +1.78795e15 q^{59} +1.98485e15 q^{61} -4.14302e14 q^{62} -3.11819e14 q^{63} +1.18943e15 q^{64} +5.76649e13 q^{66} +4.22235e15 q^{67} +1.56951e15 q^{68} +1.28642e14 q^{69} -1.65823e15 q^{71} -1.99275e15 q^{72} +1.26706e16 q^{73} +5.28602e15 q^{74} -1.21291e14 q^{76} -2.97565e14 q^{77} -1.90554e15 q^{78} -1.63691e16 q^{79} +1.85302e15 q^{81} +4.08732e15 q^{82} +2.45355e16 q^{83} +4.05377e15 q^{84} +2.09724e15 q^{86} -2.11763e16 q^{87} -1.90166e15 q^{88} -4.52816e15 q^{89} +9.83307e15 q^{91} -1.67240e15 q^{92} -1.27047e16 q^{93} -1.30272e16 q^{94} +4.16003e16 q^{96} -3.59936e16 q^{97} -3.85458e16 q^{98} +1.76832e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 442 q^{2} + 19683 q^{3} + 298148 q^{4} + 2899962 q^{6} - 4962644 q^{7} + 108831912 q^{8} + 129140163 q^{9} + 1049849720 q^{11} + 1956149028 q^{12} + 3091742090 q^{13} + 27586028328 q^{14} + 22392797456 q^{16}+ \cdots + 45\!\cdots\!20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 213.954 0.590971 0.295485 − 0.955347i \(-0.404519\pi\)
0.295485 + 0.955347i \(0.404519\pi\)
\(3\) 6561.00 0.577350
\(4\) −85295.6 −0.650754
\(5\) 0 0
\(6\) 1.40375e6 0.341197
\(7\) −7.24373e6 −0.474929 −0.237465 − 0.971396i \(-0.576316\pi\)
−0.237465 + 0.971396i \(0.576316\pi\)
\(8\) −4.62928e7 −0.975547
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) 4.10790e7 0.0577807 0.0288903 − 0.999583i \(-0.490803\pi\)
0.0288903 + 0.999583i \(0.490803\pi\)
\(12\) −5.59624e8 −0.375713
\(13\) −1.35746e9 −0.461539 −0.230769 − 0.973008i \(-0.574124\pi\)
−0.230769 + 0.973008i \(0.574124\pi\)
\(14\) −1.54983e9 −0.280669
\(15\) 0 0
\(16\) 1.27533e9 0.0742338
\(17\) −1.84008e10 −0.639766 −0.319883 − 0.947457i \(-0.603644\pi\)
−0.319883 + 0.947457i \(0.603644\pi\)
\(18\) 9.21003e9 0.196990
\(19\) 1.42200e9 0.0192086 0.00960429 − 0.999954i \(-0.496943\pi\)
0.00960429 + 0.999954i \(0.496943\pi\)
\(20\) 0 0
\(21\) −4.75261e10 −0.274201
\(22\) 8.78904e9 0.0341467
\(23\) 1.96071e10 0.0522068 0.0261034 − 0.999659i \(-0.491690\pi\)
0.0261034 + 0.999659i \(0.491690\pi\)
\(24\) −3.03727e11 −0.563232
\(25\) 0 0
\(26\) −2.90434e11 −0.272756
\(27\) 2.82430e11 0.192450
\(28\) 6.17858e11 0.309062
\(29\) −3.22761e12 −1.19811 −0.599057 − 0.800707i \(-0.704458\pi\)
−0.599057 + 0.800707i \(0.704458\pi\)
\(30\) 0 0
\(31\) −1.93640e12 −0.407776 −0.203888 − 0.978994i \(-0.565358\pi\)
−0.203888 + 0.978994i \(0.565358\pi\)
\(32\) 6.34055e12 1.01942
\(33\) 2.69520e11 0.0333597
\(34\) −3.93694e12 −0.378083
\(35\) 0 0
\(36\) −3.67169e12 −0.216918
\(37\) 2.47063e13 1.15636 0.578180 − 0.815909i \(-0.303763\pi\)
0.578180 + 0.815909i \(0.303763\pi\)
\(38\) 3.04244e11 0.0113517
\(39\) −8.90629e12 −0.266470
\(40\) 0 0
\(41\) 1.91037e13 0.373641 0.186821 − 0.982394i \(-0.440182\pi\)
0.186821 + 0.982394i \(0.440182\pi\)
\(42\) −1.01684e13 −0.162044
\(43\) 9.80229e12 0.127893 0.0639463 − 0.997953i \(-0.479631\pi\)
0.0639463 + 0.997953i \(0.479631\pi\)
\(44\) −3.50386e12 −0.0376010
\(45\) 0 0
\(46\) 4.19503e12 0.0308527
\(47\) −6.08880e13 −0.372992 −0.186496 − 0.982456i \(-0.559713\pi\)
−0.186496 + 0.982456i \(0.559713\pi\)
\(48\) 8.36742e12 0.0428589
\(49\) −1.80159e14 −0.774442
\(50\) 0 0
\(51\) −1.20728e14 −0.369369
\(52\) 1.15785e14 0.300348
\(53\) −3.72760e14 −0.822402 −0.411201 − 0.911545i \(-0.634891\pi\)
−0.411201 + 0.911545i \(0.634891\pi\)
\(54\) 6.04270e13 0.113732
\(55\) 0 0
\(56\) 3.35332e14 0.463316
\(57\) 9.32977e12 0.0110901
\(58\) −6.90561e14 −0.708050
\(59\) 1.78795e15 1.58530 0.792652 − 0.609675i \(-0.208700\pi\)
0.792652 + 0.609675i \(0.208700\pi\)
\(60\) 0 0
\(61\) 1.98485e15 1.32563 0.662817 − 0.748781i \(-0.269361\pi\)
0.662817 + 0.748781i \(0.269361\pi\)
\(62\) −4.14302e14 −0.240983
\(63\) −3.11819e14 −0.158310
\(64\) 1.18943e15 0.528212
\(65\) 0 0
\(66\) 5.76649e13 0.0197146
\(67\) 4.22235e15 1.27033 0.635167 − 0.772375i \(-0.280931\pi\)
0.635167 + 0.772375i \(0.280931\pi\)
\(68\) 1.56951e15 0.416330
\(69\) 1.28642e14 0.0301416
\(70\) 0 0
\(71\) −1.65823e15 −0.304754 −0.152377 − 0.988322i \(-0.548693\pi\)
−0.152377 + 0.988322i \(0.548693\pi\)
\(72\) −1.99275e15 −0.325182
\(73\) 1.26706e16 1.83888 0.919439 − 0.393233i \(-0.128643\pi\)
0.919439 + 0.393233i \(0.128643\pi\)
\(74\) 5.28602e15 0.683375
\(75\) 0 0
\(76\) −1.21291e14 −0.0125001
\(77\) −2.97565e14 −0.0274417
\(78\) −1.90554e15 −0.157476
\(79\) −1.63691e16 −1.21393 −0.606965 − 0.794728i \(-0.707613\pi\)
−0.606965 + 0.794728i \(0.707613\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 4.08732e15 0.220811
\(83\) 2.45355e16 1.19573 0.597863 − 0.801598i \(-0.296017\pi\)
0.597863 + 0.801598i \(0.296017\pi\)
\(84\) 4.05377e15 0.178437
\(85\) 0 0
\(86\) 2.09724e15 0.0755808
\(87\) −2.11763e16 −0.691731
\(88\) −1.90166e15 −0.0563678
\(89\) −4.52816e15 −0.121929 −0.0609645 − 0.998140i \(-0.519418\pi\)
−0.0609645 + 0.998140i \(0.519418\pi\)
\(90\) 0 0
\(91\) 9.83307e15 0.219198
\(92\) −1.67240e15 −0.0339738
\(93\) −1.27047e16 −0.235429
\(94\) −1.30272e16 −0.220427
\(95\) 0 0
\(96\) 4.16003e16 0.588561
\(97\) −3.59936e16 −0.466301 −0.233150 − 0.972441i \(-0.574903\pi\)
−0.233150 + 0.972441i \(0.574903\pi\)
\(98\) −3.85458e16 −0.457673
\(99\) 1.76832e15 0.0192602
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.1.2 3
5.2 odd 4 75.18.b.d.49.4 6
5.3 odd 4 75.18.b.d.49.3 6
5.4 even 2 15.18.a.b.1.2 ✓ 3
15.14 odd 2 45.18.a.e.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.b.1.2 ✓ 3 5.4 even 2
45.18.a.e.1.2 3 15.14 odd 2
75.18.a.e.1.2 3 1.1 even 1 trivial
75.18.b.d.49.3 6 5.3 odd 4
75.18.b.d.49.4 6 5.2 odd 4