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Properties

Label 875.2.a.j.1.7
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 1.7
Root \(-2.52916\) 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)
 
\(f(q)\) \(=\) \(q+2.66234 q^{2} -3.18114 q^{3} +5.08806 q^{4} -8.46928 q^{6} -1.00000 q^{7} +8.22148 q^{8} +7.11965 q^{9} -0.271966 q^{11} -16.1858 q^{12} +1.86091 q^{13} -2.66234 q^{14} +11.7123 q^{16} +3.94864 q^{17} +18.9549 q^{18} +3.11983 q^{19} +3.18114 q^{21} -0.724065 q^{22} -2.16261 q^{23} -26.1537 q^{24} +4.95439 q^{26} -13.1052 q^{27} -5.08806 q^{28} +8.32888 q^{29} +8.98849 q^{31} +14.7391 q^{32} +0.865161 q^{33} +10.5126 q^{34} +36.2252 q^{36} -0.317877 q^{37} +8.30604 q^{38} -5.91983 q^{39} -9.46696 q^{41} +8.46928 q^{42} -6.82386 q^{43} -1.38378 q^{44} -5.75761 q^{46} -5.48266 q^{47} -37.2583 q^{48} +1.00000 q^{49} -12.5612 q^{51} +9.46845 q^{52} +5.31422 q^{53} -34.8904 q^{54} -8.22148 q^{56} -9.92460 q^{57} +22.1743 q^{58} +3.74847 q^{59} -2.29764 q^{61} +23.9304 q^{62} -7.11965 q^{63} +15.8159 q^{64} +2.30335 q^{66} +7.37418 q^{67} +20.0909 q^{68} +6.87957 q^{69} -15.2000 q^{71} +58.5340 q^{72} +0.554591 q^{73} -0.846298 q^{74} +15.8739 q^{76} +0.271966 q^{77} -15.7606 q^{78} -4.33995 q^{79} +20.3304 q^{81} -25.2043 q^{82} +2.82289 q^{83} +16.1858 q^{84} -18.1674 q^{86} -26.4953 q^{87} -2.23596 q^{88} +2.80062 q^{89} -1.86091 q^{91} -11.0035 q^{92} -28.5936 q^{93} -14.5967 q^{94} -46.8870 q^{96} +0.822734 q^{97} +2.66234 q^{98} -1.93630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 8 q^{3} + 13 q^{4} + 2 q^{6} - 8 q^{7} + 12 q^{8} + 18 q^{9} - 5 q^{11} - 20 q^{12} - 6 q^{13} - q^{14} + 35 q^{16} + 13 q^{17} + 3 q^{18} + 13 q^{19} + 8 q^{21} + 22 q^{22} - 5 q^{23} - 3 q^{24}+ \cdots - 31 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.66234 1.88256 0.941280 − 0.337627i \(-0.109624\pi\)
0.941280 + 0.337627i \(0.109624\pi\)
\(3\) −3.18114 −1.83663 −0.918316 − 0.395849i \(-0.870450\pi\)
−0.918316 + 0.395849i \(0.870450\pi\)
\(4\) 5.08806 2.54403
\(5\) 0 0
\(6\) −8.46928 −3.45757
\(7\) −1.00000 −0.377964
\(8\) 8.22148 2.90673
\(9\) 7.11965 2.37322
\(10\) 0 0
\(11\) −0.271966 −0.0820007 −0.0410004 − 0.999159i \(-0.513054\pi\)
−0.0410004 + 0.999159i \(0.513054\pi\)
\(12\) −16.1858 −4.67245
\(13\) 1.86091 0.516125 0.258062 − 0.966128i \(-0.416916\pi\)
0.258062 + 0.966128i \(0.416916\pi\)
\(14\) −2.66234 −0.711541
\(15\) 0 0
\(16\) 11.7123 2.92806
\(17\) 3.94864 0.957685 0.478842 − 0.877901i \(-0.341057\pi\)
0.478842 + 0.877901i \(0.341057\pi\)
\(18\) 18.9549 4.46772
\(19\) 3.11983 0.715737 0.357868 − 0.933772i \(-0.383504\pi\)
0.357868 + 0.933772i \(0.383504\pi\)
\(20\) 0 0
\(21\) 3.18114 0.694181
\(22\) −0.724065 −0.154371
\(23\) −2.16261 −0.450936 −0.225468 − 0.974251i \(-0.572391\pi\)
−0.225468 + 0.974251i \(0.572391\pi\)
\(24\) −26.1537 −5.33859
\(25\) 0 0
\(26\) 4.95439 0.971636
\(27\) −13.1052 −2.52209
\(28\) −5.08806 −0.961553
\(29\) 8.32888 1.54663 0.773317 − 0.634019i \(-0.218596\pi\)
0.773317 + 0.634019i \(0.218596\pi\)
\(30\) 0 0
\(31\) 8.98849 1.61438 0.807190 − 0.590292i \(-0.200987\pi\)
0.807190 + 0.590292i \(0.200987\pi\)
\(32\) 14.7391 2.60552
\(33\) 0.865161 0.150605
\(34\) 10.5126 1.80290
\(35\) 0 0
\(36\) 36.2252 6.03753
\(37\) −0.317877 −0.0522587 −0.0261294 − 0.999659i \(-0.508318\pi\)
−0.0261294 + 0.999659i \(0.508318\pi\)
\(38\) 8.30604 1.34742
\(39\) −5.91983 −0.947931
\(40\) 0 0
\(41\) −9.46696 −1.47849 −0.739245 − 0.673436i \(-0.764818\pi\)
−0.739245 + 0.673436i \(0.764818\pi\)
\(42\) 8.46928 1.30684
\(43\) −6.82386 −1.04063 −0.520315 − 0.853975i \(-0.674185\pi\)
−0.520315 + 0.853975i \(0.674185\pi\)
\(44\) −1.38378 −0.208612
\(45\) 0 0
\(46\) −5.75761 −0.848913
\(47\) −5.48266 −0.799728 −0.399864 − 0.916575i \(-0.630943\pi\)
−0.399864 + 0.916575i \(0.630943\pi\)
\(48\) −37.2583 −5.37777
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −12.5612 −1.75891
\(52\) 9.46845 1.31304
\(53\) 5.31422 0.729964 0.364982 − 0.931015i \(-0.381075\pi\)
0.364982 + 0.931015i \(0.381075\pi\)
\(54\) −34.8904 −4.74799
\(55\) 0 0
\(56\) −8.22148 −1.09864
\(57\) −9.92460 −1.31455
\(58\) 22.1743 2.91163
\(59\) 3.74847 0.488009 0.244005 − 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(60\) 0 0
\(61\) −2.29764 −0.294183 −0.147091 − 0.989123i \(-0.546991\pi\)
−0.147091 + 0.989123i \(0.546991\pi\)
\(62\) 23.9304 3.03917
\(63\) −7.11965 −0.896991
\(64\) 15.8159 1.97699
\(65\) 0 0
\(66\) 2.30335 0.283523
\(67\) 7.37418 0.900899 0.450450 − 0.892802i \(-0.351264\pi\)
0.450450 + 0.892802i \(0.351264\pi\)
\(68\) 20.0909 2.43638
\(69\) 6.87957 0.828202
\(70\) 0 0
\(71\) −15.2000 −1.80391 −0.901954 − 0.431833i \(-0.857867\pi\)
−0.901954 + 0.431833i \(0.857867\pi\)
\(72\) 58.5340 6.89830
\(73\) 0.554591 0.0649100 0.0324550 − 0.999473i \(-0.489667\pi\)
0.0324550 + 0.999473i \(0.489667\pi\)
\(74\) −0.846298 −0.0983802
\(75\) 0 0
\(76\) 15.8739 1.82086
\(77\) 0.271966 0.0309934
\(78\) −15.7606 −1.78454
\(79\) −4.33995 −0.488283 −0.244141 − 0.969740i \(-0.578506\pi\)
−0.244141 + 0.969740i \(0.578506\pi\)
\(80\) 0 0
\(81\) 20.3304 2.25894
\(82\) −25.2043 −2.78335
\(83\) 2.82289 0.309853 0.154926 − 0.987926i \(-0.450486\pi\)
0.154926 + 0.987926i \(0.450486\pi\)
\(84\) 16.1858 1.76602
\(85\) 0 0
\(86\) −18.1674 −1.95905
\(87\) −26.4953 −2.84060
\(88\) −2.23596 −0.238354
\(89\) 2.80062 0.296865 0.148433 − 0.988923i \(-0.452577\pi\)
0.148433 + 0.988923i \(0.452577\pi\)
\(90\) 0 0
\(91\) −1.86091 −0.195077
\(92\) −11.0035 −1.14719
\(93\) −28.5936 −2.96502
\(94\) −14.5967 −1.50553
\(95\) 0 0
\(96\) −46.8870 −4.78539
\(97\) 0.822734 0.0835360 0.0417680 − 0.999127i \(-0.486701\pi\)
0.0417680 + 0.999127i \(0.486701\pi\)
\(98\) 2.66234 0.268937
\(99\) −1.93630 −0.194605
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.1.7 yes 8
3.2 odd 2 7875.2.a.w.1.2 8
5.2 odd 4 875.2.b.e.624.15 16
5.3 odd 4 875.2.b.e.624.2 16
5.4 even 2 875.2.a.i.1.2 ✓ 8
7.6 odd 2 6125.2.a.w.1.7 8
15.14 odd 2 7875.2.a.bb.1.7 8
35.34 odd 2 6125.2.a.v.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
875.2.a.i.1.2 ✓ 8 5.4 even 2
875.2.a.j.1.7 yes 8 1.1 even 1 trivial
875.2.b.e.624.2 16 5.3 odd 4
875.2.b.e.624.15 16 5.2 odd 4
6125.2.a.v.1.2 8 35.34 odd 2
6125.2.a.w.1.7 8 7.6 odd 2
7875.2.a.w.1.2 8 3.2 odd 2
7875.2.a.bb.1.7 8 15.14 odd 2