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Properties

Label 2352.4.a.bz.1.2
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
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] = [2352,4,Mod(1,2352)] 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("2352.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,6,0,-6,0,0,0,18,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +4.54983 q^{5} +9.00000 q^{9} +40.7492 q^{11} -53.2990 q^{13} +13.6495 q^{15} -4.54983 q^{17} +122.598 q^{19} -131.347 q^{23} -104.299 q^{25} +27.0000 q^{27} -216.598 q^{29} -251.794 q^{31} +122.248 q^{33} +11.8970 q^{37} -159.897 q^{39} +111.752 q^{41} -369.196 q^{43} +40.9485 q^{45} -262.694 q^{47} -13.6495 q^{51} -567.100 q^{53} +185.402 q^{55} +367.794 q^{57} +839.890 q^{59} +485.794 q^{61} -242.502 q^{65} +333.691 q^{67} -394.042 q^{69} -590.248 q^{71} -490.701 q^{73} -312.897 q^{75} -121.691 q^{79} +81.0000 q^{81} +609.608 q^{83} -20.7010 q^{85} -649.794 q^{87} -719.038 q^{89} -755.382 q^{93} +557.801 q^{95} +637.877 q^{97} +366.743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 6 q^{5} + 18 q^{9} + 6 q^{11} - 16 q^{13} - 18 q^{15} + 6 q^{17} + 64 q^{19} - 6 q^{23} - 118 q^{25} + 54 q^{27} - 252 q^{29} + 40 q^{31} + 18 q^{33} - 248 q^{37} - 48 q^{39} + 450 q^{41}+ \cdots + 54 q^{99}+O(q^{100}) \) 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 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 4.54983 0.406950 0.203475 − 0.979080i \(-0.434777\pi\)
0.203475 + 0.979080i \(0.434777\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 40.7492 1.11694 0.558470 − 0.829525i \(-0.311389\pi\)
0.558470 + 0.829525i \(0.311389\pi\)
\(12\) 0 0
\(13\) −53.2990 −1.13711 −0.568557 − 0.822644i \(-0.692498\pi\)
−0.568557 + 0.822644i \(0.692498\pi\)
\(14\) 0 0
\(15\) 13.6495 0.234952
\(16\) 0 0
\(17\) −4.54983 −0.0649116 −0.0324558 − 0.999473i \(-0.510333\pi\)
−0.0324558 + 0.999473i \(0.510333\pi\)
\(18\) 0 0
\(19\) 122.598 1.48031 0.740156 − 0.672436i \(-0.234752\pi\)
0.740156 + 0.672436i \(0.234752\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −131.347 −1.19077 −0.595387 − 0.803439i \(-0.703001\pi\)
−0.595387 + 0.803439i \(0.703001\pi\)
\(24\) 0 0
\(25\) −104.299 −0.834392
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −216.598 −1.38694 −0.693470 − 0.720486i \(-0.743919\pi\)
−0.693470 + 0.720486i \(0.743919\pi\)
\(30\) 0 0
\(31\) −251.794 −1.45882 −0.729412 − 0.684075i \(-0.760206\pi\)
−0.729412 + 0.684075i \(0.760206\pi\)
\(32\) 0 0
\(33\) 122.248 0.644865
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 11.8970 0.0528610 0.0264305 − 0.999651i \(-0.491586\pi\)
0.0264305 + 0.999651i \(0.491586\pi\)
\(38\) 0 0
\(39\) −159.897 −0.656513
\(40\) 0 0
\(41\) 111.752 0.425678 0.212839 − 0.977087i \(-0.431729\pi\)
0.212839 + 0.977087i \(0.431729\pi\)
\(42\) 0 0
\(43\) −369.196 −1.30935 −0.654673 − 0.755912i \(-0.727194\pi\)
−0.654673 + 0.755912i \(0.727194\pi\)
\(44\) 0 0
\(45\) 40.9485 0.135650
\(46\) 0 0
\(47\) −262.694 −0.815275 −0.407637 − 0.913144i \(-0.633647\pi\)
−0.407637 + 0.913144i \(0.633647\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −13.6495 −0.0374767
\(52\) 0 0
\(53\) −567.100 −1.46976 −0.734879 − 0.678199i \(-0.762761\pi\)
−0.734879 + 0.678199i \(0.762761\pi\)
\(54\) 0 0
\(55\) 185.402 0.454538
\(56\) 0 0
\(57\) 367.794 0.854658
\(58\) 0 0
\(59\) 839.890 1.85330 0.926648 − 0.375931i \(-0.122677\pi\)
0.926648 + 0.375931i \(0.122677\pi\)
\(60\) 0 0
\(61\) 485.794 1.01966 0.509832 − 0.860274i \(-0.329707\pi\)
0.509832 + 0.860274i \(0.329707\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −242.502 −0.462748
\(66\) 0 0
\(67\) 333.691 0.608460 0.304230 − 0.952599i \(-0.401601\pi\)
0.304230 + 0.952599i \(0.401601\pi\)
\(68\) 0 0
\(69\) −394.042 −0.687493
\(70\) 0 0
\(71\) −590.248 −0.986613 −0.493306 − 0.869856i \(-0.664212\pi\)
−0.493306 + 0.869856i \(0.664212\pi\)
\(72\) 0 0
\(73\) −490.701 −0.786743 −0.393371 − 0.919380i \(-0.628691\pi\)
−0.393371 + 0.919380i \(0.628691\pi\)
\(74\) 0 0
\(75\) −312.897 −0.481736
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −121.691 −0.173308 −0.0866539 − 0.996238i \(-0.527617\pi\)
−0.0866539 + 0.996238i \(0.527617\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 609.608 0.806183 0.403091 − 0.915160i \(-0.367936\pi\)
0.403091 + 0.915160i \(0.367936\pi\)
\(84\) 0 0
\(85\) −20.7010 −0.0264157
\(86\) 0 0
\(87\) −649.794 −0.800750
\(88\) 0 0
\(89\) −719.038 −0.856381 −0.428190 − 0.903689i \(-0.640849\pi\)
−0.428190 + 0.903689i \(0.640849\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −755.382 −0.842252
\(94\) 0 0
\(95\) 557.801 0.602412
\(96\) 0 0
\(97\) 637.877 0.667697 0.333849 − 0.942627i \(-0.391653\pi\)
0.333849 + 0.942627i \(0.391653\pi\)
\(98\) 0 0
\(99\) 366.743 0.372313
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 2352.4.a.bz.1.2 2
4.3 odd 2 147.4.a.i.1.2 2
7.6 odd 2 336.4.a.m.1.1 2
12.11 even 2 441.4.a.r.1.1 2
21.20 even 2 1008.4.a.ba.1.2 2
28.3 even 6 147.4.e.l.79.1 4
28.11 odd 6 147.4.e.m.79.1 4
28.19 even 6 147.4.e.l.67.1 4
28.23 odd 6 147.4.e.m.67.1 4
28.27 even 2 21.4.a.c.1.2 ✓ 2
56.13 odd 2 1344.4.a.bo.1.2 2
56.27 even 2 1344.4.a.bg.1.2 2
84.11 even 6 441.4.e.p.226.2 4
84.23 even 6 441.4.e.p.361.2 4
84.47 odd 6 441.4.e.q.361.2 4
84.59 odd 6 441.4.e.q.226.2 4
84.83 odd 2 63.4.a.e.1.1 2
140.27 odd 4 525.4.d.g.274.3 4
140.83 odd 4 525.4.d.g.274.2 4
140.139 even 2 525.4.a.n.1.1 2
420.419 odd 2 1575.4.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.c.1.2 ✓ 2 28.27 even 2
63.4.a.e.1.1 2 84.83 odd 2
147.4.a.i.1.2 2 4.3 odd 2
147.4.e.l.67.1 4 28.19 even 6
147.4.e.l.79.1 4 28.3 even 6
147.4.e.m.67.1 4 28.23 odd 6
147.4.e.m.79.1 4 28.11 odd 6
336.4.a.m.1.1 2 7.6 odd 2
441.4.a.r.1.1 2 12.11 even 2
441.4.e.p.226.2 4 84.11 even 6
441.4.e.p.361.2 4 84.23 even 6
441.4.e.q.226.2 4 84.59 odd 6
441.4.e.q.361.2 4 84.47 odd 6
525.4.a.n.1.1 2 140.139 even 2
525.4.d.g.274.2 4 140.83 odd 4
525.4.d.g.274.3 4 140.27 odd 4
1008.4.a.ba.1.2 2 21.20 even 2
1344.4.a.bg.1.2 2 56.27 even 2
1344.4.a.bo.1.2 2 56.13 odd 2
1575.4.a.p.1.2 2 420.419 odd 2
2352.4.a.bz.1.2 2 1.1 even 1 trivial