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Properties

Label 252.8.a.b.1.1
Level $252$
Weight $8$
Character 252.1
Self dual yes
Analytic conductor $78.721$
Analytic rank $1$
Dimension $1$
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 = [1,0,0,0,240,0,343] 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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q+240.000 q^{5} +343.000 q^{7} -702.000 q^{11} -3958.00 q^{13} +3408.00 q^{17} -49036.0 q^{19} +11514.0 q^{23} -20525.0 q^{25} -49662.0 q^{29} -113320. q^{31} +82320.0 q^{35} -66886.0 q^{37} +360900. q^{41} -765292. q^{43} +1.34488e6 q^{47} +117649. q^{49} -358962. q^{53} -168480. q^{55} -930528. q^{59} -1.31883e6 q^{61} -949920. q^{65} +1.89346e6 q^{67} -227994. q^{71} +784934. q^{73} -240786. q^{77} -2.10089e6 q^{79} -8.62931e6 q^{83} +817920. q^{85} -5.90310e6 q^{89} -1.35759e6 q^{91} -1.17686e7 q^{95} +773846. q^{97} +O(q^{100})\)

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
\(5\) 240.000 0.858650 0.429325 − 0.903150i \(-0.358751\pi\)
0.429325 + 0.903150i \(0.358751\pi\)
\(6\) 0 0
\(7\) 343.000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −702.000 −0.159024 −0.0795120 − 0.996834i \(-0.525336\pi\)
−0.0795120 + 0.996834i \(0.525336\pi\)
\(12\) 0 0
\(13\) −3958.00 −0.499659 −0.249830 − 0.968290i \(-0.580375\pi\)
−0.249830 + 0.968290i \(0.580375\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3408.00 0.168240 0.0841198 − 0.996456i \(-0.473192\pi\)
0.0841198 + 0.996456i \(0.473192\pi\)
\(18\) 0 0
\(19\) −49036.0 −1.64013 −0.820063 − 0.572273i \(-0.806062\pi\)
−0.820063 + 0.572273i \(0.806062\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 11514.0 0.197323 0.0986617 − 0.995121i \(-0.468544\pi\)
0.0986617 + 0.995121i \(0.468544\pi\)
\(24\) 0 0
\(25\) −20525.0 −0.262720
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −49662.0 −0.378121 −0.189061 − 0.981965i \(-0.560544\pi\)
−0.189061 + 0.981965i \(0.560544\pi\)
\(30\) 0 0
\(31\) −113320. −0.683189 −0.341594 − 0.939847i \(-0.610967\pi\)
−0.341594 + 0.939847i \(0.610967\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 82320.0 0.324539
\(36\) 0 0
\(37\) −66886.0 −0.217085 −0.108542 − 0.994092i \(-0.534618\pi\)
−0.108542 + 0.994092i \(0.534618\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 360900. 0.817793 0.408896 − 0.912581i \(-0.365914\pi\)
0.408896 + 0.912581i \(0.365914\pi\)
\(42\) 0 0
\(43\) −765292. −1.46787 −0.733935 − 0.679220i \(-0.762318\pi\)
−0.733935 + 0.679220i \(0.762318\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.34488e6 1.88947 0.944734 − 0.327837i \(-0.106320\pi\)
0.944734 + 0.327837i \(0.106320\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −358962. −0.331194 −0.165597 − 0.986193i \(-0.552955\pi\)
−0.165597 + 0.986193i \(0.552955\pi\)
\(54\) 0 0
\(55\) −168480. −0.136546
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −930528. −0.589858 −0.294929 − 0.955519i \(-0.595296\pi\)
−0.294929 + 0.955519i \(0.595296\pi\)
\(60\) 0 0
\(61\) −1.31883e6 −0.743936 −0.371968 − 0.928246i \(-0.621317\pi\)
−0.371968 + 0.928246i \(0.621317\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −949920. −0.429033
\(66\) 0 0
\(67\) 1.89346e6 0.769122 0.384561 − 0.923100i \(-0.374353\pi\)
0.384561 + 0.923100i \(0.374353\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −227994. −0.0755995 −0.0377998 − 0.999285i \(-0.512035\pi\)
−0.0377998 + 0.999285i \(0.512035\pi\)
\(72\) 0 0
\(73\) 784934. 0.236158 0.118079 − 0.993004i \(-0.462326\pi\)
0.118079 + 0.993004i \(0.462326\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −240786. −0.0601054
\(78\) 0 0
\(79\) −2.10089e6 −0.479412 −0.239706 − 0.970846i \(-0.577051\pi\)
−0.239706 + 0.970846i \(0.577051\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.62931e6 −1.65654 −0.828271 − 0.560327i \(-0.810675\pi\)
−0.828271 + 0.560327i \(0.810675\pi\)
\(84\) 0 0
\(85\) 817920. 0.144459
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.90310e6 −0.887596 −0.443798 − 0.896127i \(-0.646369\pi\)
−0.443798 + 0.896127i \(0.646369\pi\)
\(90\) 0 0
\(91\) −1.35759e6 −0.188853
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.17686e7 −1.40830
\(96\) 0 0
\(97\) 773846. 0.0860902 0.0430451 − 0.999073i \(-0.486294\pi\)
0.0430451 + 0.999073i \(0.486294\pi\)
\(98\) 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.b.1.1 1
3.2 odd 2 84.8.a.b.1.1 ✓ 1
12.11 even 2 336.8.a.c.1.1 1
21.2 odd 6 588.8.i.d.361.1 2
21.5 even 6 588.8.i.e.361.1 2
21.11 odd 6 588.8.i.d.373.1 2
21.17 even 6 588.8.i.e.373.1 2
21.20 even 2 588.8.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.a.b.1.1 ✓ 1 3.2 odd 2
252.8.a.b.1.1 1 1.1 even 1 trivial
336.8.a.c.1.1 1 12.11 even 2
588.8.a.b.1.1 1 21.20 even 2
588.8.i.d.361.1 2 21.2 odd 6
588.8.i.d.373.1 2 21.11 odd 6
588.8.i.e.361.1 2 21.5 even 6
588.8.i.e.373.1 2 21.17 even 6