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Properties

Label 75.18.a.f.1.3
Level $75$
Weight $18$
Character 75.1
Self dual yes
Analytic conductor $137.417$
Analytic rank $0$
Dimension $4$
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 = [4,33] 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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 481686x^{2} + 26523040x + 36023696000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(359.218\) 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)
 
\(f(q)\) \(=\) \(q+367.218 q^{2} +6561.00 q^{3} +3777.06 q^{4} +2.40932e6 q^{6} +1.91248e7 q^{7} -4.67450e7 q^{8} +4.30467e7 q^{9} -1.08277e9 q^{11} +2.47813e7 q^{12} +4.07421e9 q^{13} +7.02299e9 q^{14} -1.76607e10 q^{16} +1.03327e9 q^{17} +1.58075e10 q^{18} -8.27218e10 q^{19} +1.25478e11 q^{21} -3.97612e11 q^{22} +5.01376e11 q^{23} -3.06694e11 q^{24} +1.49612e12 q^{26} +2.82430e11 q^{27} +7.22357e10 q^{28} +3.86913e12 q^{29} +8.19265e12 q^{31} -3.58356e11 q^{32} -7.10405e12 q^{33} +3.79437e11 q^{34} +1.62590e11 q^{36} -1.23893e13 q^{37} -3.03770e13 q^{38} +2.67309e13 q^{39} -7.08479e13 q^{41} +4.60778e13 q^{42} +3.21849e13 q^{43} -4.08968e12 q^{44} +1.84114e14 q^{46} -1.79604e13 q^{47} -1.15872e14 q^{48} +1.33129e14 q^{49} +6.77931e12 q^{51} +1.53885e13 q^{52} +1.35368e14 q^{53} +1.03713e14 q^{54} -8.93991e14 q^{56} -5.42738e14 q^{57} +1.42081e15 q^{58} -1.26236e15 q^{59} +2.30641e15 q^{61} +3.00849e15 q^{62} +8.23262e14 q^{63} +2.18322e15 q^{64} -2.60873e15 q^{66} +4.62557e15 q^{67} +3.90274e12 q^{68} +3.28953e15 q^{69} +1.04536e16 q^{71} -2.01222e15 q^{72} -6.57154e15 q^{73} -4.54956e15 q^{74} -3.12445e14 q^{76} -2.07078e16 q^{77} +9.81605e15 q^{78} +7.36663e15 q^{79} +1.85302e15 q^{81} -2.60166e16 q^{82} +6.77164e15 q^{83} +4.73938e14 q^{84} +1.18189e16 q^{86} +2.53854e16 q^{87} +5.06140e16 q^{88} +6.56709e15 q^{89} +7.79186e16 q^{91} +1.89373e15 q^{92} +5.37520e16 q^{93} -6.59539e15 q^{94} -2.35117e15 q^{96} -2.22965e16 q^{97} +4.88875e16 q^{98} -4.66097e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 33 q^{2} + 26244 q^{3} + 439357 q^{4} + 216513 q^{6} - 17583104 q^{7} - 63651621 q^{8} + 172186884 q^{9} - 575495184 q^{11} + 2882621277 q^{12} + 5049645832 q^{13} - 6699316032 q^{14} + 7512683905 q^{16}+ \cdots - 24\!\cdots\!64 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 367.218 1.01431 0.507153 − 0.861856i \(-0.330698\pi\)
0.507153 + 0.861856i \(0.330698\pi\)
\(3\) 6561.00 0.577350
\(4\) 3777.06 0.0288167
\(5\) 0 0
\(6\) 2.40932e6 0.585610
\(7\) 1.91248e7 1.25391 0.626953 − 0.779057i \(-0.284302\pi\)
0.626953 + 0.779057i \(0.284302\pi\)
\(8\) −4.67450e7 −0.985077
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) −1.08277e9 −1.52299 −0.761497 − 0.648169i \(-0.775535\pi\)
−0.761497 + 0.648169i \(0.775535\pi\)
\(12\) 2.47813e7 0.0166373
\(13\) 4.07421e9 1.38524 0.692619 − 0.721304i \(-0.256457\pi\)
0.692619 + 0.721304i \(0.256457\pi\)
\(14\) 7.02299e9 1.27184
\(15\) 0 0
\(16\) −1.76607e10 −1.02799
\(17\) 1.03327e9 0.0359252 0.0179626 − 0.999839i \(-0.494282\pi\)
0.0179626 + 0.999839i \(0.494282\pi\)
\(18\) 1.58075e10 0.338102
\(19\) −8.27218e10 −1.11742 −0.558708 − 0.829365i \(-0.688703\pi\)
−0.558708 + 0.829365i \(0.688703\pi\)
\(20\) 0 0
\(21\) 1.25478e11 0.723942
\(22\) −3.97612e11 −1.54478
\(23\) 5.01376e11 1.33499 0.667494 − 0.744615i \(-0.267367\pi\)
0.667494 + 0.744615i \(0.267367\pi\)
\(24\) −3.06694e11 −0.568735
\(25\) 0 0
\(26\) 1.49612e12 1.40506
\(27\) 2.82430e11 0.192450
\(28\) 7.22357e10 0.0361334
\(29\) 3.86913e12 1.43625 0.718126 − 0.695914i \(-0.245000\pi\)
0.718126 + 0.695914i \(0.245000\pi\)
\(30\) 0 0
\(31\) 8.19265e12 1.72524 0.862621 − 0.505851i \(-0.168821\pi\)
0.862621 + 0.505851i \(0.168821\pi\)
\(32\) −3.58356e11 −0.0576156
\(33\) −7.10405e12 −0.879301
\(34\) 3.79437e11 0.0364392
\(35\) 0 0
\(36\) 1.62590e11 0.00960556
\(37\) −1.23893e13 −0.579870 −0.289935 − 0.957046i \(-0.593634\pi\)
−0.289935 + 0.957046i \(0.593634\pi\)
\(38\) −3.03770e13 −1.13340
\(39\) 2.67309e13 0.799768
\(40\) 0 0
\(41\) −7.08479e13 −1.38568 −0.692842 − 0.721089i \(-0.743642\pi\)
−0.692842 + 0.721089i \(0.743642\pi\)
\(42\) 4.60778e13 0.734299
\(43\) 3.21849e13 0.419923 0.209962 − 0.977710i \(-0.432666\pi\)
0.209962 + 0.977710i \(0.432666\pi\)
\(44\) −4.08968e12 −0.0438876
\(45\) 0 0
\(46\) 1.84114e14 1.35409
\(47\) −1.79604e13 −0.110023 −0.0550117 − 0.998486i \(-0.517520\pi\)
−0.0550117 + 0.998486i \(0.517520\pi\)
\(48\) −1.15872e14 −0.593508
\(49\) 1.33129e14 0.572278
\(50\) 0 0
\(51\) 6.77931e12 0.0207414
\(52\) 1.53885e13 0.0399180
\(53\) 1.35368e14 0.298657 0.149328 − 0.988788i \(-0.452289\pi\)
0.149328 + 0.988788i \(0.452289\pi\)
\(54\) 1.03713e14 0.195203
\(55\) 0 0
\(56\) −8.93991e14 −1.23519
\(57\) −5.42738e14 −0.645140
\(58\) 1.42081e15 1.45680
\(59\) −1.26236e15 −1.11928 −0.559642 − 0.828734i \(-0.689061\pi\)
−0.559642 + 0.828734i \(0.689061\pi\)
\(60\) 0 0
\(61\) 2.30641e15 1.54040 0.770200 − 0.637803i \(-0.220157\pi\)
0.770200 + 0.637803i \(0.220157\pi\)
\(62\) 3.00849e15 1.74992
\(63\) 8.23262e14 0.417968
\(64\) 2.18322e15 0.969546
\(65\) 0 0
\(66\) −2.60873e15 −0.891880
\(67\) 4.62557e15 1.39165 0.695825 − 0.718212i \(-0.255039\pi\)
0.695825 + 0.718212i \(0.255039\pi\)
\(68\) 3.90274e12 0.00103525
\(69\) 3.28953e15 0.770755
\(70\) 0 0
\(71\) 1.04536e16 1.92119 0.960597 − 0.277946i \(-0.0896536\pi\)
0.960597 + 0.277946i \(0.0896536\pi\)
\(72\) −2.01222e15 −0.328359
\(73\) −6.57154e15 −0.953724 −0.476862 − 0.878978i \(-0.658226\pi\)
−0.476862 + 0.878978i \(0.658226\pi\)
\(74\) −4.54956e15 −0.588165
\(75\) 0 0
\(76\) −3.12445e14 −0.0322002
\(77\) −2.07078e16 −1.90969
\(78\) 9.81605e15 0.811209
\(79\) 7.36663e15 0.546310 0.273155 − 0.961970i \(-0.411933\pi\)
0.273155 + 0.961970i \(0.411933\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) −2.60166e16 −1.40551
\(83\) 6.77164e15 0.330012 0.165006 − 0.986293i \(-0.447236\pi\)
0.165006 + 0.986293i \(0.447236\pi\)
\(84\) 4.73938e14 0.0208616
\(85\) 0 0
\(86\) 1.18189e16 0.425931
\(87\) 2.53854e16 0.829220
\(88\) 5.06140e16 1.50027
\(89\) 6.56709e15 0.176831 0.0884153 − 0.996084i \(-0.471820\pi\)
0.0884153 + 0.996084i \(0.471820\pi\)
\(90\) 0 0
\(91\) 7.79186e16 1.73696
\(92\) 1.89373e15 0.0384699
\(93\) 5.37520e16 0.996069
\(94\) −6.59539e15 −0.111597
\(95\) 0 0
\(96\) −2.35117e15 −0.0332644
\(97\) −2.22965e16 −0.288853 −0.144426 − 0.989516i \(-0.546134\pi\)
−0.144426 + 0.989516i \(0.546134\pi\)
\(98\) 4.88875e16 0.580465
\(99\) −4.66097e16 −0.507665
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.f.1.3 4
5.2 odd 4 75.18.b.f.49.6 8
5.3 odd 4 75.18.b.f.49.3 8
5.4 even 2 15.18.a.d.1.2 ✓ 4
15.14 odd 2 45.18.a.f.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.d.1.2 ✓ 4 5.4 even 2
45.18.a.f.1.3 4 15.14 odd 2
75.18.a.f.1.3 4 1.1 even 1 trivial
75.18.b.f.49.3 8 5.3 odd 4
75.18.b.f.49.6 8 5.2 odd 4