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Properties

Label 252.8.a.f.1.1
Level $252$
Weight $8$
Character 252.1
Self dual yes
Analytic conductor $78.721$
Analytic rank $0$
Dimension $2$
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] = [252,8,Mod(1,252)] 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("252.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,294,0,-686] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.7210264220\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1009}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 252 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(16.3824\) of defining polynomial
Character \(\chi\) \(=\) 252.1

$q$-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)
 
\(f(q)\) \(=\) \(q-202.412 q^{5} -343.000 q^{7} +7527.19 q^{11} -1700.28 q^{13} +2481.86 q^{17} -39816.0 q^{19} -25560.1 q^{23} -37154.2 q^{25} -148830. q^{29} -33538.4 q^{31} +69427.4 q^{35} +400557. q^{37} +362750. q^{41} -324652. q^{43} +708899. q^{47} +117649. q^{49} +185924. q^{53} -1.52360e6 q^{55} +1.19372e6 q^{59} +2.51204e6 q^{61} +344158. q^{65} -2.85983e6 q^{67} -3.22083e6 q^{71} +5.01287e6 q^{73} -2.58182e6 q^{77} +5.94208e6 q^{79} +1.02006e7 q^{83} -502359. q^{85} -1.85373e6 q^{89} +583197. q^{91} +8.05925e6 q^{95} -1.52142e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 294 q^{5} - 686 q^{7} + 3492 q^{11} - 16170 q^{13} + 29232 q^{17} - 3206 q^{19} + 9360 q^{23} + 131146 q^{25} - 184704 q^{29} + 165060 q^{31} - 100842 q^{35} + 286144 q^{37} + 116760 q^{41} - 294428 q^{43}+ \cdots - 26702368 q^{97}+O(q^{100}) \) 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 0
\(3\) 0 0
\(4\) 0 0
\(5\) −202.412 −0.724172 −0.362086 − 0.932145i \(-0.617935\pi\)
−0.362086 + 0.932145i \(0.617935\pi\)
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 7527.19 1.70513 0.852567 − 0.522619i \(-0.175045\pi\)
0.852567 + 0.522619i \(0.175045\pi\)
\(12\) 0 0
\(13\) −1700.28 −0.214644 −0.107322 − 0.994224i \(-0.534228\pi\)
−0.107322 + 0.994224i \(0.534228\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2481.86 0.122520 0.0612599 − 0.998122i \(-0.480488\pi\)
0.0612599 + 0.998122i \(0.480488\pi\)
\(18\) 0 0
\(19\) −39816.0 −1.33174 −0.665871 − 0.746067i \(-0.731940\pi\)
−0.665871 + 0.746067i \(0.731940\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −25560.1 −0.438041 −0.219020 − 0.975720i \(-0.570286\pi\)
−0.219020 + 0.975720i \(0.570286\pi\)
\(24\) 0 0
\(25\) −37154.2 −0.475574
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −148830. −1.13317 −0.566587 − 0.824002i \(-0.691737\pi\)
−0.566587 + 0.824002i \(0.691737\pi\)
\(30\) 0 0
\(31\) −33538.4 −0.202198 −0.101099 − 0.994876i \(-0.532236\pi\)
−0.101099 + 0.994876i \(0.532236\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 69427.4 0.273711
\(36\) 0 0
\(37\) 400557. 1.30005 0.650023 − 0.759915i \(-0.274759\pi\)
0.650023 + 0.759915i \(0.274759\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 362750. 0.821985 0.410992 − 0.911639i \(-0.365182\pi\)
0.410992 + 0.911639i \(0.365182\pi\)
\(42\) 0 0
\(43\) −324652. −0.622699 −0.311350 − 0.950295i \(-0.600781\pi\)
−0.311350 + 0.950295i \(0.600781\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 708899. 0.995960 0.497980 − 0.867188i \(-0.334075\pi\)
0.497980 + 0.867188i \(0.334075\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 185924. 0.171542 0.0857709 − 0.996315i \(-0.472665\pi\)
0.0857709 + 0.996315i \(0.472665\pi\)
\(54\) 0 0
\(55\) −1.52360e6 −1.23481
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.19372e6 0.756692 0.378346 − 0.925664i \(-0.376493\pi\)
0.378346 + 0.925664i \(0.376493\pi\)
\(60\) 0 0
\(61\) 2.51204e6 1.41701 0.708504 − 0.705707i \(-0.249370\pi\)
0.708504 + 0.705707i \(0.249370\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 344158. 0.155440
\(66\) 0 0
\(67\) −2.85983e6 −1.16166 −0.580829 − 0.814025i \(-0.697272\pi\)
−0.580829 + 0.814025i \(0.697272\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.22083e6 −1.06798 −0.533990 − 0.845491i \(-0.679308\pi\)
−0.533990 + 0.845491i \(0.679308\pi\)
\(72\) 0 0
\(73\) 5.01287e6 1.50819 0.754095 − 0.656765i \(-0.228076\pi\)
0.754095 + 0.656765i \(0.228076\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.58182e6 −0.644480
\(78\) 0 0
\(79\) 5.94208e6 1.35595 0.677975 − 0.735085i \(-0.262858\pi\)
0.677975 + 0.735085i \(0.262858\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.02006e7 1.95819 0.979094 − 0.203407i \(-0.0652014\pi\)
0.979094 + 0.203407i \(0.0652014\pi\)
\(84\) 0 0
\(85\) −502359. −0.0887255
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.85373e6 −0.278729 −0.139365 − 0.990241i \(-0.544506\pi\)
−0.139365 + 0.990241i \(0.544506\pi\)
\(90\) 0 0
\(91\) 583197. 0.0811280
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.05925e6 0.964411
\(96\) 0 0
\(97\) −1.52142e7 −1.69257 −0.846287 − 0.532727i \(-0.821167\pi\)
−0.846287 + 0.532727i \(0.821167\pi\)
\(98\) 0 0
\(99\) 0 0
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 252.8.a.f.1.1 2
3.2 odd 2 28.8.a.b.1.1 ✓ 2
12.11 even 2 112.8.a.h.1.2 2
21.2 odd 6 196.8.e.b.165.2 4
21.5 even 6 196.8.e.c.165.1 4
21.11 odd 6 196.8.e.b.177.2 4
21.17 even 6 196.8.e.c.177.1 4
21.20 even 2 196.8.a.a.1.2 2
24.5 odd 2 448.8.a.o.1.2 2
24.11 even 2 448.8.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.8.a.b.1.1 ✓ 2 3.2 odd 2
112.8.a.h.1.2 2 12.11 even 2
196.8.a.a.1.2 2 21.20 even 2
196.8.e.b.165.2 4 21.2 odd 6
196.8.e.b.177.2 4 21.11 odd 6
196.8.e.c.165.1 4 21.5 even 6
196.8.e.c.177.1 4 21.17 even 6
252.8.a.f.1.1 2 1.1 even 1 trivial
448.8.a.o.1.2 2 24.5 odd 2
448.8.a.q.1.1 2 24.11 even 2