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Properties

Label 6125.2.a.w.1.7
Level $6125$
Weight $2$
Character 6125.1
Self dual yes
Analytic conductor $48.908$
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] = [6125,2,Mod(1,6125)] 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("6125.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6125, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6125 = 5^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6125.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,1,8,13,0,-2,0,12,18,0,-5,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(12)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.9083712380\)
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: no (minimal twist has level 875)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-2.52916\) of defining polynomial
Character \(\chi\) \(=\) 6125.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.66234 q^{2} +3.18114 q^{3} +5.08806 q^{4} +8.46928 q^{6} +8.22148 q^{8} +7.11965 q^{9} -0.271966 q^{11} +16.1858 q^{12} -1.86091 q^{13} +11.7123 q^{16} -3.94864 q^{17} +18.9549 q^{18} -3.11983 q^{19} -0.724065 q^{22} -2.16261 q^{23} +26.1537 q^{24} -4.95439 q^{26} +13.1052 q^{27} +8.32888 q^{29} -8.98849 q^{31} +14.7391 q^{32} -0.865161 q^{33} -10.5126 q^{34} +36.2252 q^{36} -0.317877 q^{37} -8.30604 q^{38} -5.91983 q^{39} +9.46696 q^{41} -6.82386 q^{43} -1.38378 q^{44} -5.75761 q^{46} +5.48266 q^{47} +37.2583 q^{48} -12.5612 q^{51} -9.46845 q^{52} +5.31422 q^{53} +34.8904 q^{54} -9.92460 q^{57} +22.1743 q^{58} -3.74847 q^{59} +2.29764 q^{61} -23.9304 q^{62} +15.8159 q^{64} -2.30335 q^{66} +7.37418 q^{67} -20.0909 q^{68} -6.87957 q^{69} -15.2000 q^{71} +58.5340 q^{72} -0.554591 q^{73} -0.846298 q^{74} -15.8739 q^{76} -15.7606 q^{78} -4.33995 q^{79} +20.3304 q^{81} +25.2043 q^{82} -2.82289 q^{83} -18.1674 q^{86} +26.4953 q^{87} -2.23596 q^{88} -2.80062 q^{89} -11.0035 q^{92} -28.5936 q^{93} +14.5967 q^{94} +46.8870 q^{96} -0.822734 q^{97} -1.93630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 8 q^{3} + 13 q^{4} - 2 q^{6} + 12 q^{8} + 18 q^{9} - 5 q^{11} + 20 q^{12} + 6 q^{13} + 35 q^{16} - 13 q^{17} + 3 q^{18} - 13 q^{19} + 22 q^{22} - 5 q^{23} + 3 q^{24} + 5 q^{26} + 29 q^{27}+ \cdots - 31 q^{99}+O(q^{100}) \) 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 \(-0.129550\pi\)
0.918316 + 0.395849i \(0.129550\pi\)
\(4\) 5.08806 2.54403
\(5\) 0 0
\(6\) 8.46928 3.45757
\(7\) 0 0
\(8\) 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 \(-0.583084\pi\)
−0.258062 + 0.966128i \(0.583084\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 11.7123 2.92806
\(17\) −3.94864 −0.957685 −0.478842 − 0.877901i \(-0.658943\pi\)
−0.478842 + 0.877901i \(0.658943\pi\)
\(18\) 18.9549 4.46772
\(19\) −3.11983 −0.715737 −0.357868 − 0.933772i \(-0.616496\pi\)
−0.357868 + 0.933772i \(0.616496\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −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
\(28\) 0 0
\(29\) 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 \(-0.799013\pi\)
−0.807190 + 0.590292i \(0.799013\pi\)
\(32\) 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 \(-0.235182\pi\)
0.739245 + 0.673436i \(0.235182\pi\)
\(42\) 0 0
\(43\) −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 \(-0.369057\pi\)
0.399864 + 0.916575i \(0.369057\pi\)
\(48\) 37.2583 5.37777
\(49\) 0 0
\(50\) 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
\(56\) 0 0
\(57\) −9.92460 −1.31455
\(58\) 22.1743 2.91163
\(59\) −3.74847 −0.488009 −0.244005 − 0.969774i \(-0.578461\pi\)
−0.244005 + 0.969774i \(0.578461\pi\)
\(60\) 0 0
\(61\) 2.29764 0.294183 0.147091 − 0.989123i \(-0.453009\pi\)
0.147091 + 0.989123i \(0.453009\pi\)
\(62\) −23.9304 −3.03917
\(63\) 0 0
\(64\) 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 \(-0.510333\pi\)
−0.0324550 + 0.999473i \(0.510333\pi\)
\(74\) −0.846298 −0.0983802
\(75\) 0 0
\(76\) −15.8739 −1.82086
\(77\) 0 0
\(78\) −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 \(-0.549514\pi\)
−0.154926 + 0.987926i \(0.549514\pi\)
\(84\) 0 0
\(85\) 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 \(-0.547423\pi\)
−0.148433 + 0.988923i \(0.547423\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −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 \(-0.513299\pi\)
−0.0417680 + 0.999127i \(0.513299\pi\)
\(98\) 0 0
\(99\) −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 6125.2.a.w.1.7 8
5.4 even 2 6125.2.a.v.1.2 8
7.6 odd 2 875.2.a.j.1.7 yes 8
21.20 even 2 7875.2.a.w.1.2 8
35.13 even 4 875.2.b.e.624.2 16
35.27 even 4 875.2.b.e.624.15 16
35.34 odd 2 875.2.a.i.1.2 ✓ 8
105.104 even 2 7875.2.a.bb.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
875.2.a.i.1.2 ✓ 8 35.34 odd 2
875.2.a.j.1.7 yes 8 7.6 odd 2
875.2.b.e.624.2 16 35.13 even 4
875.2.b.e.624.15 16 35.27 even 4
6125.2.a.v.1.2 8 5.4 even 2
6125.2.a.w.1.7 8 1.1 even 1 trivial
7875.2.a.w.1.2 8 21.20 even 2
7875.2.a.bb.1.7 8 105.104 even 2