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Properties

Label 875.2.a.j.1.4
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 1.4
Root \(-1.22981\) 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)
 
\(f(q)\) \(=\) \(q-0.582954 q^{2} +2.60791 q^{3} -1.66016 q^{4} -1.52029 q^{6} -1.00000 q^{7} +2.13371 q^{8} +3.80121 q^{9} -4.09198 q^{11} -4.32956 q^{12} +2.06359 q^{13} +0.582954 q^{14} +2.07647 q^{16} +5.05725 q^{17} -2.21593 q^{18} +6.14252 q^{19} -2.60791 q^{21} +2.38544 q^{22} +6.05624 q^{23} +5.56452 q^{24} -1.20298 q^{26} +2.08948 q^{27} +1.66016 q^{28} +3.73419 q^{29} +8.44329 q^{31} -5.47791 q^{32} -10.6715 q^{33} -2.94815 q^{34} -6.31063 q^{36} -8.10158 q^{37} -3.58081 q^{38} +5.38167 q^{39} +5.43385 q^{41} +1.52029 q^{42} -0.577054 q^{43} +6.79335 q^{44} -3.53051 q^{46} -1.65702 q^{47} +5.41526 q^{48} +1.00000 q^{49} +13.1889 q^{51} -3.42590 q^{52} -0.334974 q^{53} -1.21807 q^{54} -2.13371 q^{56} +16.0192 q^{57} -2.17686 q^{58} -8.69884 q^{59} +13.3810 q^{61} -4.92205 q^{62} -3.80121 q^{63} -0.959580 q^{64} +6.22101 q^{66} -1.36127 q^{67} -8.39587 q^{68} +15.7941 q^{69} -13.0966 q^{71} +8.11067 q^{72} -9.75665 q^{73} +4.72285 q^{74} -10.1976 q^{76} +4.09198 q^{77} -3.13727 q^{78} -8.88698 q^{79} -5.95444 q^{81} -3.16769 q^{82} -8.92647 q^{83} +4.32956 q^{84} +0.336396 q^{86} +9.73844 q^{87} -8.73108 q^{88} +9.10123 q^{89} -2.06359 q^{91} -10.0544 q^{92} +22.0194 q^{93} +0.965968 q^{94} -14.2859 q^{96} +5.40173 q^{97} -0.582954 q^{98} -15.5545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 8 q^{3} + 13 q^{4} + 2 q^{6} - 8 q^{7} + 12 q^{8} + 18 q^{9} - 5 q^{11} - 20 q^{12} - 6 q^{13} - q^{14} + 35 q^{16} + 13 q^{17} + 3 q^{18} + 13 q^{19} + 8 q^{21} + 22 q^{22} - 5 q^{23} - 3 q^{24}+ \cdots - 31 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.582954 −0.412211 −0.206105 − 0.978530i \(-0.566079\pi\)
−0.206105 + 0.978530i \(0.566079\pi\)
\(3\) 2.60791 1.50568 0.752840 − 0.658204i \(-0.228684\pi\)
0.752840 + 0.658204i \(0.228684\pi\)
\(4\) −1.66016 −0.830082
\(5\) 0 0
\(6\) −1.52029 −0.620657
\(7\) −1.00000 −0.377964
\(8\) 2.13371 0.754380
\(9\) 3.80121 1.26707
\(10\) 0 0
\(11\) −4.09198 −1.23378 −0.616889 − 0.787050i \(-0.711607\pi\)
−0.616889 + 0.787050i \(0.711607\pi\)
\(12\) −4.32956 −1.24984
\(13\) 2.06359 0.572337 0.286169 − 0.958179i \(-0.407618\pi\)
0.286169 + 0.958179i \(0.407618\pi\)
\(14\) 0.582954 0.155801
\(15\) 0 0
\(16\) 2.07647 0.519119
\(17\) 5.05725 1.22656 0.613282 − 0.789864i \(-0.289849\pi\)
0.613282 + 0.789864i \(0.289849\pi\)
\(18\) −2.21593 −0.522300
\(19\) 6.14252 1.40919 0.704596 − 0.709609i \(-0.251128\pi\)
0.704596 + 0.709609i \(0.251128\pi\)
\(20\) 0 0
\(21\) −2.60791 −0.569093
\(22\) 2.38544 0.508576
\(23\) 6.05624 1.26281 0.631407 − 0.775452i \(-0.282478\pi\)
0.631407 + 0.775452i \(0.282478\pi\)
\(24\) 5.56452 1.13585
\(25\) 0 0
\(26\) −1.20298 −0.235924
\(27\) 2.08948 0.402121
\(28\) 1.66016 0.313742
\(29\) 3.73419 0.693422 0.346711 − 0.937972i \(-0.387299\pi\)
0.346711 + 0.937972i \(0.387299\pi\)
\(30\) 0 0
\(31\) 8.44329 1.51646 0.758230 − 0.651987i \(-0.226064\pi\)
0.758230 + 0.651987i \(0.226064\pi\)
\(32\) −5.47791 −0.968366
\(33\) −10.6715 −1.85767
\(34\) −2.94815 −0.505603
\(35\) 0 0
\(36\) −6.31063 −1.05177
\(37\) −8.10158 −1.33189 −0.665946 − 0.746000i \(-0.731972\pi\)
−0.665946 + 0.746000i \(0.731972\pi\)
\(38\) −3.58081 −0.580884
\(39\) 5.38167 0.861756
\(40\) 0 0
\(41\) 5.43385 0.848625 0.424312 − 0.905516i \(-0.360516\pi\)
0.424312 + 0.905516i \(0.360516\pi\)
\(42\) 1.52029 0.234586
\(43\) −0.577054 −0.0879999 −0.0439999 − 0.999032i \(-0.514010\pi\)
−0.0439999 + 0.999032i \(0.514010\pi\)
\(44\) 6.79335 1.02414
\(45\) 0 0
\(46\) −3.53051 −0.520545
\(47\) −1.65702 −0.241702 −0.120851 − 0.992671i \(-0.538562\pi\)
−0.120851 + 0.992671i \(0.538562\pi\)
\(48\) 5.41526 0.781626
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 13.1889 1.84681
\(52\) −3.42590 −0.475087
\(53\) −0.334974 −0.0460123 −0.0230061 − 0.999735i \(-0.507324\pi\)
−0.0230061 + 0.999735i \(0.507324\pi\)
\(54\) −1.21807 −0.165759
\(55\) 0 0
\(56\) −2.13371 −0.285129
\(57\) 16.0192 2.12179
\(58\) −2.17686 −0.285836
\(59\) −8.69884 −1.13249 −0.566246 − 0.824236i \(-0.691605\pi\)
−0.566246 + 0.824236i \(0.691605\pi\)
\(60\) 0 0
\(61\) 13.3810 1.71326 0.856631 − 0.515929i \(-0.172553\pi\)
0.856631 + 0.515929i \(0.172553\pi\)
\(62\) −4.92205 −0.625101
\(63\) −3.80121 −0.478907
\(64\) −0.959580 −0.119947
\(65\) 0 0
\(66\) 6.22101 0.765753
\(67\) −1.36127 −0.166306 −0.0831530 − 0.996537i \(-0.526499\pi\)
−0.0831530 + 0.996537i \(0.526499\pi\)
\(68\) −8.39587 −1.01815
\(69\) 15.7941 1.90139
\(70\) 0 0
\(71\) −13.0966 −1.55428 −0.777138 − 0.629330i \(-0.783329\pi\)
−0.777138 + 0.629330i \(0.783329\pi\)
\(72\) 8.11067 0.955852
\(73\) −9.75665 −1.14193 −0.570965 − 0.820975i \(-0.693431\pi\)
−0.570965 + 0.820975i \(0.693431\pi\)
\(74\) 4.72285 0.549020
\(75\) 0 0
\(76\) −10.1976 −1.16974
\(77\) 4.09198 0.466324
\(78\) −3.13727 −0.355225
\(79\) −8.88698 −0.999864 −0.499932 − 0.866065i \(-0.666642\pi\)
−0.499932 + 0.866065i \(0.666642\pi\)
\(80\) 0 0
\(81\) −5.95444 −0.661605
\(82\) −3.16769 −0.349812
\(83\) −8.92647 −0.979807 −0.489903 − 0.871777i \(-0.662968\pi\)
−0.489903 + 0.871777i \(0.662968\pi\)
\(84\) 4.32956 0.472394
\(85\) 0 0
\(86\) 0.336396 0.0362745
\(87\) 9.73844 1.04407
\(88\) −8.73108 −0.930737
\(89\) 9.10123 0.964728 0.482364 − 0.875971i \(-0.339778\pi\)
0.482364 + 0.875971i \(0.339778\pi\)
\(90\) 0 0
\(91\) −2.06359 −0.216323
\(92\) −10.0544 −1.04824
\(93\) 22.0194 2.28330
\(94\) 0.965968 0.0996320
\(95\) 0 0
\(96\) −14.2859 −1.45805
\(97\) 5.40173 0.548462 0.274231 − 0.961664i \(-0.411577\pi\)
0.274231 + 0.961664i \(0.411577\pi\)
\(98\) −0.582954 −0.0588873
\(99\) −15.5545 −1.56328
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.1.4 yes 8
3.2 odd 2 7875.2.a.w.1.5 8
5.2 odd 4 875.2.b.e.624.7 16
5.3 odd 4 875.2.b.e.624.10 16
5.4 even 2 875.2.a.i.1.5 ✓ 8
7.6 odd 2 6125.2.a.w.1.4 8
15.14 odd 2 7875.2.a.bb.1.4 8
35.34 odd 2 6125.2.a.v.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
875.2.a.i.1.5 ✓ 8 5.4 even 2
875.2.a.j.1.4 yes 8 1.1 even 1 trivial
875.2.b.e.624.7 16 5.2 odd 4
875.2.b.e.624.10 16 5.3 odd 4
6125.2.a.v.1.5 8 35.34 odd 2
6125.2.a.w.1.4 8 7.6 odd 2
7875.2.a.w.1.5 8 3.2 odd 2
7875.2.a.bb.1.4 8 15.14 odd 2