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Properties

Label 875.2.a.j.1.8
Level $875$
Weight $2$
Character 875.1
Self dual yes
Analytic conductor $6.987$
Analytic rank $0$
Dimension $8$
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] = [875,2,Mod(1,875)] 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("875.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(875, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 875 = 5^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 875.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,1,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.98691017686\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 10x^{6} + 30x^{5} + 29x^{4} - 79x^{3} - 43x^{2} + 62x + 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.453202\) of defining polynomial
Character \(\chi\) \(=\) 875.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.78847 q^{2} +1.35133 q^{3} +5.77559 q^{4} +3.76815 q^{6} -1.00000 q^{7} +10.5281 q^{8} -1.17391 q^{9} -1.44463 q^{11} +7.80473 q^{12} -5.95512 q^{13} -2.78847 q^{14} +17.8062 q^{16} +3.35316 q^{17} -3.27341 q^{18} -1.49663 q^{19} -1.35133 q^{21} -4.02832 q^{22} -4.58430 q^{23} +14.2270 q^{24} -16.6057 q^{26} -5.64033 q^{27} -5.77559 q^{28} +3.52127 q^{29} -4.81340 q^{31} +28.5960 q^{32} -1.95218 q^{33} +9.35020 q^{34} -6.78000 q^{36} +9.38481 q^{37} -4.17330 q^{38} -8.04734 q^{39} +1.47252 q^{41} -3.76815 q^{42} -0.743696 q^{43} -8.34360 q^{44} -12.7832 q^{46} +4.16186 q^{47} +24.0621 q^{48} +1.00000 q^{49} +4.53123 q^{51} -34.3943 q^{52} -6.90991 q^{53} -15.7279 q^{54} -10.5281 q^{56} -2.02243 q^{57} +9.81896 q^{58} -5.52874 q^{59} +0.0619354 q^{61} -13.4221 q^{62} +1.17391 q^{63} +44.1267 q^{64} -5.44359 q^{66} +3.66299 q^{67} +19.3665 q^{68} -6.19490 q^{69} +14.2391 q^{71} -12.3590 q^{72} -11.3078 q^{73} +26.1693 q^{74} -8.64389 q^{76} +1.44463 q^{77} -22.4398 q^{78} +8.62641 q^{79} -4.10023 q^{81} +4.10607 q^{82} +5.45554 q^{83} -7.80473 q^{84} -2.07378 q^{86} +4.75839 q^{87} -15.2093 q^{88} +11.7133 q^{89} +5.95512 q^{91} -26.4770 q^{92} -6.50450 q^{93} +11.6052 q^{94} +38.6426 q^{96} -11.8475 q^{97} +2.78847 q^{98} +1.69586 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 8 q^{3} + 13 q^{4} + 2 q^{6} - 8 q^{7} + 12 q^{8} + 18 q^{9} - 5 q^{11} - 20 q^{12} - 6 q^{13} - q^{14} + 35 q^{16} + 13 q^{17} + 3 q^{18} + 13 q^{19} + 8 q^{21} + 22 q^{22} - 5 q^{23} - 3 q^{24}+ \cdots - 31 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.78847 1.97175 0.985875 − 0.167486i \(-0.0535648\pi\)
0.985875 + 0.167486i \(0.0535648\pi\)
\(3\) 1.35133 0.780191 0.390095 − 0.920774i \(-0.372442\pi\)
0.390095 + 0.920774i \(0.372442\pi\)
\(4\) 5.77559 2.88779
\(5\) 0 0
\(6\) 3.76815 1.53834
\(7\) −1.00000 −0.377964
\(8\) 10.5281 3.72226
\(9\) −1.17391 −0.391302
\(10\) 0 0
\(11\) −1.44463 −0.435573 −0.217787 − 0.975996i \(-0.569884\pi\)
−0.217787 + 0.975996i \(0.569884\pi\)
\(12\) 7.80473 2.25303
\(13\) −5.95512 −1.65165 −0.825827 − 0.563923i \(-0.809291\pi\)
−0.825827 + 0.563923i \(0.809291\pi\)
\(14\) −2.78847 −0.745251
\(15\) 0 0
\(16\) 17.8062 4.45156
\(17\) 3.35316 0.813261 0.406630 − 0.913593i \(-0.366704\pi\)
0.406630 + 0.913593i \(0.366704\pi\)
\(18\) −3.27341 −0.771550
\(19\) −1.49663 −0.343349 −0.171675 − 0.985154i \(-0.554918\pi\)
−0.171675 + 0.985154i \(0.554918\pi\)
\(20\) 0 0
\(21\) −1.35133 −0.294884
\(22\) −4.02832 −0.858841
\(23\) −4.58430 −0.955892 −0.477946 − 0.878389i \(-0.658619\pi\)
−0.477946 + 0.878389i \(0.658619\pi\)
\(24\) 14.2270 2.90407
\(25\) 0 0
\(26\) −16.6057 −3.25665
\(27\) −5.64033 −1.08548
\(28\) −5.77559 −1.09148
\(29\) 3.52127 0.653883 0.326941 − 0.945045i \(-0.393982\pi\)
0.326941 + 0.945045i \(0.393982\pi\)
\(30\) 0 0
\(31\) −4.81340 −0.864513 −0.432256 − 0.901751i \(-0.642282\pi\)
−0.432256 + 0.901751i \(0.642282\pi\)
\(32\) 28.5960 5.05510
\(33\) −1.95218 −0.339830
\(34\) 9.35020 1.60355
\(35\) 0 0
\(36\) −6.78000 −1.13000
\(37\) 9.38481 1.54285 0.771426 − 0.636318i \(-0.219544\pi\)
0.771426 + 0.636318i \(0.219544\pi\)
\(38\) −4.17330 −0.676999
\(39\) −8.04734 −1.28861
\(40\) 0 0
\(41\) 1.47252 0.229968 0.114984 − 0.993367i \(-0.463318\pi\)
0.114984 + 0.993367i \(0.463318\pi\)
\(42\) −3.76815 −0.581438
\(43\) −0.743696 −0.113413 −0.0567063 − 0.998391i \(-0.518060\pi\)
−0.0567063 + 0.998391i \(0.518060\pi\)
\(44\) −8.34360 −1.25785
\(45\) 0 0
\(46\) −12.7832 −1.88478
\(47\) 4.16186 0.607069 0.303535 − 0.952820i \(-0.401833\pi\)
0.303535 + 0.952820i \(0.401833\pi\)
\(48\) 24.0621 3.47307
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.53123 0.634499
\(52\) −34.3943 −4.76964
\(53\) −6.90991 −0.949150 −0.474575 − 0.880215i \(-0.657398\pi\)
−0.474575 + 0.880215i \(0.657398\pi\)
\(54\) −15.7279 −2.14030
\(55\) 0 0
\(56\) −10.5281 −1.40688
\(57\) −2.02243 −0.267878
\(58\) 9.81896 1.28929
\(59\) −5.52874 −0.719781 −0.359890 − 0.932995i \(-0.617186\pi\)
−0.359890 + 0.932995i \(0.617186\pi\)
\(60\) 0 0
\(61\) 0.0619354 0.00793002 0.00396501 − 0.999992i \(-0.498738\pi\)
0.00396501 + 0.999992i \(0.498738\pi\)
\(62\) −13.4221 −1.70460
\(63\) 1.17391 0.147898
\(64\) 44.1267 5.51584
\(65\) 0 0
\(66\) −5.44359 −0.670060
\(67\) 3.66299 0.447506 0.223753 − 0.974646i \(-0.428169\pi\)
0.223753 + 0.974646i \(0.428169\pi\)
\(68\) 19.3665 2.34853
\(69\) −6.19490 −0.745778
\(70\) 0 0
\(71\) 14.2391 1.68987 0.844936 − 0.534867i \(-0.179638\pi\)
0.844936 + 0.534867i \(0.179638\pi\)
\(72\) −12.3590 −1.45653
\(73\) −11.3078 −1.32348 −0.661739 − 0.749734i \(-0.730181\pi\)
−0.661739 + 0.749734i \(0.730181\pi\)
\(74\) 26.1693 3.04212
\(75\) 0 0
\(76\) −8.64389 −0.991522
\(77\) 1.44463 0.164631
\(78\) −22.4398 −2.54081
\(79\) 8.62641 0.970547 0.485273 − 0.874362i \(-0.338720\pi\)
0.485273 + 0.874362i \(0.338720\pi\)
\(80\) 0 0
\(81\) −4.10023 −0.455581
\(82\) 4.10607 0.453440
\(83\) 5.45554 0.598823 0.299412 − 0.954124i \(-0.403210\pi\)
0.299412 + 0.954124i \(0.403210\pi\)
\(84\) −7.80473 −0.851566
\(85\) 0 0
\(86\) −2.07378 −0.223621
\(87\) 4.75839 0.510153
\(88\) −15.2093 −1.62131
\(89\) 11.7133 1.24160 0.620801 − 0.783968i \(-0.286807\pi\)
0.620801 + 0.783968i \(0.286807\pi\)
\(90\) 0 0
\(91\) 5.95512 0.624267
\(92\) −26.4770 −2.76042
\(93\) −6.50450 −0.674485
\(94\) 11.6052 1.19699
\(95\) 0 0
\(96\) 38.6426 3.94395
\(97\) −11.8475 −1.20293 −0.601465 − 0.798899i \(-0.705416\pi\)
−0.601465 + 0.798899i \(0.705416\pi\)
\(98\) 2.78847 0.281678
\(99\) 1.69586 0.170441
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 875.2.a.j.1.8 yes 8
3.2 odd 2 7875.2.a.w.1.1 8
5.2 odd 4 875.2.b.e.624.16 16
5.3 odd 4 875.2.b.e.624.1 16
5.4 even 2 875.2.a.i.1.1 ✓ 8
7.6 odd 2 6125.2.a.w.1.8 8
15.14 odd 2 7875.2.a.bb.1.8 8
35.34 odd 2 6125.2.a.v.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
875.2.a.i.1.1 ✓ 8 5.4 even 2
875.2.a.j.1.8 yes 8 1.1 even 1 trivial
875.2.b.e.624.1 16 5.3 odd 4
875.2.b.e.624.16 16 5.2 odd 4
6125.2.a.v.1.1 8 35.34 odd 2
6125.2.a.w.1.8 8 7.6 odd 2
7875.2.a.w.1.1 8 3.2 odd 2
7875.2.a.bb.1.8 8 15.14 odd 2