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Properties

Label 75.18.a.i.1.3
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 1.3
Root \(112.575\) 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)
 
\(f(q)\) \(=\) \(q-168.575 q^{2} +6561.00 q^{3} -102655. q^{4} -1.10602e6 q^{6} +2.46799e6 q^{7} +3.94004e7 q^{8} +4.30467e7 q^{9} -9.18646e8 q^{11} -6.73517e8 q^{12} -5.59047e9 q^{13} -4.16040e8 q^{14} +6.81325e9 q^{16} +1.89701e10 q^{17} -7.25658e9 q^{18} -1.07413e11 q^{19} +1.61925e10 q^{21} +1.54860e11 q^{22} +6.17937e11 q^{23} +2.58506e11 q^{24} +9.42411e11 q^{26} +2.82430e11 q^{27} -2.53350e11 q^{28} -3.24945e11 q^{29} -7.66951e12 q^{31} -6.31282e12 q^{32} -6.02724e12 q^{33} -3.19787e12 q^{34} -4.41894e12 q^{36} +1.16830e13 q^{37} +1.81071e13 q^{38} -3.66791e13 q^{39} +1.26785e13 q^{41} -2.72964e12 q^{42} +6.34524e13 q^{43} +9.43033e13 q^{44} -1.04168e14 q^{46} -2.95498e14 q^{47} +4.47017e13 q^{48} -2.26540e14 q^{49} +1.24463e14 q^{51} +5.73888e14 q^{52} +5.71911e14 q^{53} -4.76104e13 q^{54} +9.72397e13 q^{56} -7.04738e14 q^{57} +5.47774e13 q^{58} -1.31940e15 q^{59} +1.00730e14 q^{61} +1.29288e15 q^{62} +1.06239e14 q^{63} +1.71155e14 q^{64} +1.01604e15 q^{66} -1.83567e15 q^{67} -1.94737e15 q^{68} +4.05428e15 q^{69} +5.00202e15 q^{71} +1.69606e15 q^{72} -2.35185e15 q^{73} -1.96946e15 q^{74} +1.10265e16 q^{76} -2.26721e15 q^{77} +6.18316e15 q^{78} -2.06977e16 q^{79} +1.85302e15 q^{81} -2.13727e15 q^{82} +1.09097e16 q^{83} -1.66223e15 q^{84} -1.06965e16 q^{86} -2.13196e15 q^{87} -3.61950e16 q^{88} +4.23635e16 q^{89} -1.37972e16 q^{91} -6.34341e16 q^{92} -5.03197e16 q^{93} +4.98135e16 q^{94} -4.14184e16 q^{96} -9.31195e16 q^{97} +3.81888e16 q^{98} -3.95447e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 338 q^{2} + 39366 q^{3} + 393248 q^{4} - 2217618 q^{6} + 20999794 q^{7} + 25242756 q^{8} + 258280326 q^{9} + 85907324 q^{11} + 2580100128 q^{12} + 344649098 q^{13} + 4962466602 q^{14} + 13788291320 q^{16}+ \cdots + 36\!\cdots\!04 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −168.575 −0.465626 −0.232813 − 0.972522i \(-0.574793\pi\)
−0.232813 + 0.972522i \(0.574793\pi\)
\(3\) 6561.00 0.577350
\(4\) −102655. −0.783193
\(5\) 0 0
\(6\) −1.10602e6 −0.268829
\(7\) 2.46799e6 0.161812 0.0809058 − 0.996722i \(-0.474219\pi\)
0.0809058 + 0.996722i \(0.474219\pi\)
\(8\) 3.94004e7 0.830300
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) −9.18646e8 −1.29214 −0.646071 − 0.763277i \(-0.723589\pi\)
−0.646071 + 0.763277i \(0.723589\pi\)
\(12\) −6.73517e8 −0.452176
\(13\) −5.59047e9 −1.90077 −0.950386 − 0.311074i \(-0.899311\pi\)
−0.950386 + 0.311074i \(0.899311\pi\)
\(14\) −4.16040e8 −0.0753437
\(15\) 0 0
\(16\) 6.81325e9 0.396583
\(17\) 1.89701e10 0.659559 0.329779 − 0.944058i \(-0.393026\pi\)
0.329779 + 0.944058i \(0.393026\pi\)
\(18\) −7.25658e9 −0.155209
\(19\) −1.07413e11 −1.45095 −0.725474 − 0.688250i \(-0.758379\pi\)
−0.725474 + 0.688250i \(0.758379\pi\)
\(20\) 0 0
\(21\) 1.61925e10 0.0934220
\(22\) 1.54860e11 0.601655
\(23\) 6.17937e11 1.64535 0.822673 − 0.568514i \(-0.192482\pi\)
0.822673 + 0.568514i \(0.192482\pi\)
\(24\) 2.58506e11 0.479374
\(25\) 0 0
\(26\) 9.42411e11 0.885048
\(27\) 2.82430e11 0.192450
\(28\) −2.53350e11 −0.126730
\(29\) −3.24945e11 −0.120622 −0.0603110 − 0.998180i \(-0.519209\pi\)
−0.0603110 + 0.998180i \(0.519209\pi\)
\(30\) 0 0
\(31\) −7.66951e12 −1.61508 −0.807539 − 0.589815i \(-0.799201\pi\)
−0.807539 + 0.589815i \(0.799201\pi\)
\(32\) −6.31282e12 −1.01496
\(33\) −6.02724e12 −0.746019
\(34\) −3.19787e12 −0.307107
\(35\) 0 0
\(36\) −4.41894e12 −0.261064
\(37\) 1.16830e13 0.546816 0.273408 − 0.961898i \(-0.411849\pi\)
0.273408 + 0.961898i \(0.411849\pi\)
\(38\) 1.81071e13 0.675599
\(39\) −3.66791e13 −1.09741
\(40\) 0 0
\(41\) 1.26785e13 0.247974 0.123987 − 0.992284i \(-0.460432\pi\)
0.123987 + 0.992284i \(0.460432\pi\)
\(42\) −2.72964e12 −0.0434997
\(43\) 6.34524e13 0.827878 0.413939 − 0.910305i \(-0.364153\pi\)
0.413939 + 0.910305i \(0.364153\pi\)
\(44\) 9.43033e13 1.01200
\(45\) 0 0
\(46\) −1.04168e14 −0.766116
\(47\) −2.95498e14 −1.81019 −0.905093 − 0.425214i \(-0.860199\pi\)
−0.905093 + 0.425214i \(0.860199\pi\)
\(48\) 4.47017e13 0.228967
\(49\) −2.26540e14 −0.973817
\(50\) 0 0
\(51\) 1.24463e14 0.380796
\(52\) 5.73888e14 1.48867
\(53\) 5.71911e14 1.26178 0.630890 − 0.775872i \(-0.282690\pi\)
0.630890 + 0.775872i \(0.282690\pi\)
\(54\) −4.76104e13 −0.0896097
\(55\) 0 0
\(56\) 9.72397e13 0.134352
\(57\) −7.04738e14 −0.837705
\(58\) 5.47774e13 0.0561647
\(59\) −1.31940e15 −1.16986 −0.584932 − 0.811082i \(-0.698879\pi\)
−0.584932 + 0.811082i \(0.698879\pi\)
\(60\) 0 0
\(61\) 1.00730e14 0.0672755 0.0336378 − 0.999434i \(-0.489291\pi\)
0.0336378 + 0.999434i \(0.489291\pi\)
\(62\) 1.29288e15 0.752022
\(63\) 1.06239e14 0.0539372
\(64\) 1.71155e14 0.0760082
\(65\) 0 0
\(66\) 1.01604e15 0.347366
\(67\) −1.83567e15 −0.552278 −0.276139 − 0.961118i \(-0.589055\pi\)
−0.276139 + 0.961118i \(0.589055\pi\)
\(68\) −1.94737e15 −0.516561
\(69\) 4.05428e15 0.949942
\(70\) 0 0
\(71\) 5.00202e15 0.919284 0.459642 − 0.888104i \(-0.347978\pi\)
0.459642 + 0.888104i \(0.347978\pi\)
\(72\) 1.69606e15 0.276767
\(73\) −2.35185e15 −0.341322 −0.170661 − 0.985330i \(-0.554590\pi\)
−0.170661 + 0.985330i \(0.554590\pi\)
\(74\) −1.96946e15 −0.254611
\(75\) 0 0
\(76\) 1.10265e16 1.13637
\(77\) −2.26721e15 −0.209084
\(78\) 6.18316e15 0.510983
\(79\) −2.06977e16 −1.53494 −0.767469 − 0.641086i \(-0.778484\pi\)
−0.767469 + 0.641086i \(0.778484\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) −2.13727e15 −0.115463
\(83\) 1.09097e16 0.531677 0.265839 − 0.964018i \(-0.414351\pi\)
0.265839 + 0.964018i \(0.414351\pi\)
\(84\) −1.66223e15 −0.0731674
\(85\) 0 0
\(86\) −1.06965e16 −0.385481
\(87\) −2.13196e15 −0.0696411
\(88\) −3.61950e16 −1.07287
\(89\) 4.23635e16 1.14071 0.570356 − 0.821397i \(-0.306805\pi\)
0.570356 + 0.821397i \(0.306805\pi\)
\(90\) 0 0
\(91\) −1.37972e16 −0.307567
\(92\) −6.34341e16 −1.28862
\(93\) −5.03197e16 −0.932465
\(94\) 4.98135e16 0.842869
\(95\) 0 0
\(96\) −4.14184e16 −0.585987
\(97\) −9.31195e16 −1.20637 −0.603186 − 0.797601i \(-0.706102\pi\)
−0.603186 + 0.797601i \(0.706102\pi\)
\(98\) 3.81888e16 0.453434
\(99\) −3.95447e16 −0.430714
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.1.3 ✓ 6
5.2 odd 4 75.18.b.h.49.5 12
5.3 odd 4 75.18.b.h.49.8 12
5.4 even 2 75.18.a.j.1.4 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.18.a.i.1.3 ✓ 6 1.1 even 1 trivial
75.18.a.j.1.4 yes 6 5.4 even 2
75.18.b.h.49.5 12 5.2 odd 4
75.18.b.h.49.8 12 5.3 odd 4