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Properties

Label 6125.2.a.w.1.5
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 1.5
Root \(2.88263\) 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)
 
\(f(q)\) \(=\) \(q+0.499926 q^{2} -0.163530 q^{3} -1.75007 q^{4} -0.0817529 q^{6} -1.87476 q^{8} -2.97326 q^{9} +2.39553 q^{11} +0.286190 q^{12} -5.78206 q^{13} +2.56291 q^{16} +0.521660 q^{17} -1.48641 q^{18} -6.06954 q^{19} +1.19759 q^{22} -0.376644 q^{23} +0.306580 q^{24} -2.89060 q^{26} +0.976807 q^{27} +2.08573 q^{29} -5.79035 q^{31} +5.03078 q^{32} -0.391742 q^{33} +0.260791 q^{34} +5.20342 q^{36} +5.24477 q^{37} -3.03432 q^{38} +0.945541 q^{39} -6.88221 q^{41} +0.390724 q^{43} -4.19236 q^{44} -0.188294 q^{46} +4.25634 q^{47} -0.419112 q^{48} -0.0853070 q^{51} +10.1190 q^{52} -11.8303 q^{53} +0.488331 q^{54} +0.992552 q^{57} +1.04271 q^{58} -6.50760 q^{59} +0.862565 q^{61} -2.89475 q^{62} -2.61079 q^{64} -0.195842 q^{66} -14.8145 q^{67} -0.912943 q^{68} +0.0615927 q^{69} +7.68498 q^{71} +5.57414 q^{72} +1.45903 q^{73} +2.62200 q^{74} +10.6221 q^{76} +0.472701 q^{78} +10.9321 q^{79} +8.76004 q^{81} -3.44059 q^{82} +4.45165 q^{83} +0.195333 q^{86} -0.341080 q^{87} -4.49105 q^{88} +9.83519 q^{89} +0.659156 q^{92} +0.946896 q^{93} +2.12785 q^{94} -0.822684 q^{96} -16.2904 q^{97} -7.12254 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 8 q^{3} + 13 q^{4} - 2 q^{6} + 12 q^{8} + 18 q^{9} - 5 q^{11} + 20 q^{12} + 6 q^{13} + 35 q^{16} - 13 q^{17} + 3 q^{18} - 13 q^{19} + 22 q^{22} - 5 q^{23} + 3 q^{24} + 5 q^{26} + 29 q^{27}+ \cdots - 31 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.499926 0.353501 0.176751 − 0.984256i \(-0.443441\pi\)
0.176751 + 0.984256i \(0.443441\pi\)
\(3\) −0.163530 −0.0944141 −0.0472071 − 0.998885i \(-0.515032\pi\)
−0.0472071 + 0.998885i \(0.515032\pi\)
\(4\) −1.75007 −0.875037
\(5\) 0 0
\(6\) −0.0817529 −0.0333755
\(7\) 0 0
\(8\) −1.87476 −0.662828
\(9\) −2.97326 −0.991086
\(10\) 0 0
\(11\) 2.39553 0.722280 0.361140 − 0.932512i \(-0.382388\pi\)
0.361140 + 0.932512i \(0.382388\pi\)
\(12\) 0.286190 0.0826159
\(13\) −5.78206 −1.60366 −0.801828 − 0.597555i \(-0.796139\pi\)
−0.801828 + 0.597555i \(0.796139\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.56291 0.640727
\(17\) 0.521660 0.126521 0.0632605 − 0.997997i \(-0.479850\pi\)
0.0632605 + 0.997997i \(0.479850\pi\)
\(18\) −1.48641 −0.350350
\(19\) −6.06954 −1.39245 −0.696224 − 0.717825i \(-0.745138\pi\)
−0.696224 + 0.717825i \(0.745138\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.19759 0.255327
\(23\) −0.376644 −0.0785358 −0.0392679 − 0.999229i \(-0.512503\pi\)
−0.0392679 + 0.999229i \(0.512503\pi\)
\(24\) 0.306580 0.0625803
\(25\) 0 0
\(26\) −2.89060 −0.566894
\(27\) 0.976807 0.187987
\(28\) 0 0
\(29\) 2.08573 0.387310 0.193655 − 0.981070i \(-0.437966\pi\)
0.193655 + 0.981070i \(0.437966\pi\)
\(30\) 0 0
\(31\) −5.79035 −1.03998 −0.519988 − 0.854173i \(-0.674064\pi\)
−0.519988 + 0.854173i \(0.674064\pi\)
\(32\) 5.03078 0.889325
\(33\) −0.391742 −0.0681935
\(34\) 0.260791 0.0447253
\(35\) 0 0
\(36\) 5.20342 0.867237
\(37\) 5.24477 0.862235 0.431117 − 0.902296i \(-0.358119\pi\)
0.431117 + 0.902296i \(0.358119\pi\)
\(38\) −3.03432 −0.492232
\(39\) 0.945541 0.151408
\(40\) 0 0
\(41\) −6.88221 −1.07482 −0.537410 − 0.843321i \(-0.680597\pi\)
−0.537410 + 0.843321i \(0.680597\pi\)
\(42\) 0 0
\(43\) 0.390724 0.0595849 0.0297924 − 0.999556i \(-0.490515\pi\)
0.0297924 + 0.999556i \(0.490515\pi\)
\(44\) −4.19236 −0.632022
\(45\) 0 0
\(46\) −0.188294 −0.0277625
\(47\) 4.25634 0.620851 0.310425 − 0.950598i \(-0.399528\pi\)
0.310425 + 0.950598i \(0.399528\pi\)
\(48\) −0.419112 −0.0604936
\(49\) 0 0
\(50\) 0 0
\(51\) −0.0853070 −0.0119454
\(52\) 10.1190 1.40326
\(53\) −11.8303 −1.62502 −0.812512 − 0.582945i \(-0.801900\pi\)
−0.812512 + 0.582945i \(0.801900\pi\)
\(54\) 0.488331 0.0664535
\(55\) 0 0
\(56\) 0 0
\(57\) 0.992552 0.131467
\(58\) 1.04271 0.136915
\(59\) −6.50760 −0.847217 −0.423608 − 0.905845i \(-0.639237\pi\)
−0.423608 + 0.905845i \(0.639237\pi\)
\(60\) 0 0
\(61\) 0.862565 0.110440 0.0552200 − 0.998474i \(-0.482414\pi\)
0.0552200 + 0.998474i \(0.482414\pi\)
\(62\) −2.89475 −0.367633
\(63\) 0 0
\(64\) −2.61079 −0.326349
\(65\) 0 0
\(66\) −0.195842 −0.0241065
\(67\) −14.8145 −1.80987 −0.904937 − 0.425545i \(-0.860082\pi\)
−0.904937 + 0.425545i \(0.860082\pi\)
\(68\) −0.912943 −0.110711
\(69\) 0.0615927 0.00741489
\(70\) 0 0
\(71\) 7.68498 0.912039 0.456019 − 0.889970i \(-0.349275\pi\)
0.456019 + 0.889970i \(0.349275\pi\)
\(72\) 5.57414 0.656919
\(73\) 1.45903 0.170766 0.0853830 − 0.996348i \(-0.472789\pi\)
0.0853830 + 0.996348i \(0.472789\pi\)
\(74\) 2.62200 0.304801
\(75\) 0 0
\(76\) 10.6221 1.21844
\(77\) 0 0
\(78\) 0.472701 0.0535228
\(79\) 10.9321 1.22996 0.614978 − 0.788545i \(-0.289165\pi\)
0.614978 + 0.788545i \(0.289165\pi\)
\(80\) 0 0
\(81\) 8.76004 0.973337
\(82\) −3.44059 −0.379950
\(83\) 4.45165 0.488633 0.244316 − 0.969696i \(-0.421436\pi\)
0.244316 + 0.969696i \(0.421436\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.195333 0.0210633
\(87\) −0.341080 −0.0365676
\(88\) −4.49105 −0.478747
\(89\) 9.83519 1.04253 0.521264 − 0.853395i \(-0.325461\pi\)
0.521264 + 0.853395i \(0.325461\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.659156 0.0687217
\(93\) 0.946896 0.0981885
\(94\) 2.12785 0.219471
\(95\) 0 0
\(96\) −0.822684 −0.0839649
\(97\) −16.2904 −1.65404 −0.827018 − 0.562175i \(-0.809965\pi\)
−0.827018 + 0.562175i \(0.809965\pi\)
\(98\) 0 0
\(99\) −7.12254 −0.715842
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.1.5 8
5.4 even 2 6125.2.a.v.1.4 8
7.6 odd 2 875.2.a.j.1.5 yes 8
21.20 even 2 7875.2.a.w.1.4 8
35.13 even 4 875.2.b.e.624.8 16
35.27 even 4 875.2.b.e.624.9 16
35.34 odd 2 875.2.a.i.1.4 ✓ 8
105.104 even 2 7875.2.a.bb.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
875.2.a.i.1.4 ✓ 8 35.34 odd 2
875.2.a.j.1.5 yes 8 7.6 odd 2
875.2.b.e.624.8 16 35.13 even 4
875.2.b.e.624.9 16 35.27 even 4
6125.2.a.v.1.4 8 5.4 even 2
6125.2.a.w.1.5 8 1.1 even 1 trivial
7875.2.a.w.1.4 8 21.20 even 2
7875.2.a.bb.1.5 8 105.104 even 2