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Properties

Label 4752.2.a.ba.1.1
Level $4752$
Weight $2$
Character 4752.1
Self dual yes
Analytic conductor $37.945$
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] = [4752,2,Mod(1,4752)] 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("4752.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4752, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4752 = 2^{4} \cdot 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4752.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,2,0,0,0,-2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(37.9449110405\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2376)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4752.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421 q^{5} +2.41421 q^{7} -1.00000 q^{11} +0.414214 q^{13} +2.24264 q^{17} -3.24264 q^{19} -5.65685 q^{23} -3.00000 q^{25} -3.41421 q^{29} +7.65685 q^{31} -3.41421 q^{35} +1.82843 q^{37} +2.82843 q^{41} +0.343146 q^{43} +0.242641 q^{47} -1.17157 q^{49} -13.8995 q^{53} +1.41421 q^{55} -5.41421 q^{59} -10.4142 q^{61} -0.585786 q^{65} +14.3137 q^{67} -4.00000 q^{71} -8.89949 q^{73} -2.41421 q^{77} +14.0711 q^{79} +1.65685 q^{83} -3.17157 q^{85} -8.24264 q^{89} +1.00000 q^{91} +4.58579 q^{95} -3.34315 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} - 2 q^{11} - 2 q^{13} - 4 q^{17} + 2 q^{19} - 6 q^{25} - 4 q^{29} + 4 q^{31} - 4 q^{35} - 2 q^{37} + 12 q^{43} - 8 q^{47} - 8 q^{49} - 8 q^{53} - 8 q^{59} - 18 q^{61} - 4 q^{65} + 6 q^{67}+ \cdots - 18 q^{97}+O(q^{100}) \) 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
\(5\) −1.41421 −0.632456 −0.316228 − 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 2.41421 0.912487 0.456243 − 0.889855i \(-0.349195\pi\)
0.456243 + 0.889855i \(0.349195\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.414214 0.114882 0.0574411 − 0.998349i \(-0.481706\pi\)
0.0574411 + 0.998349i \(0.481706\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.24264 0.543920 0.271960 − 0.962309i \(-0.412328\pi\)
0.271960 + 0.962309i \(0.412328\pi\)
\(18\) 0 0
\(19\) −3.24264 −0.743913 −0.371956 − 0.928250i \(-0.621313\pi\)
−0.371956 + 0.928250i \(0.621313\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.65685 −1.17954 −0.589768 − 0.807573i \(-0.700781\pi\)
−0.589768 + 0.807573i \(0.700781\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.41421 −0.634004 −0.317002 − 0.948425i \(-0.602676\pi\)
−0.317002 + 0.948425i \(0.602676\pi\)
\(30\) 0 0
\(31\) 7.65685 1.37521 0.687606 − 0.726084i \(-0.258662\pi\)
0.687606 + 0.726084i \(0.258662\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.41421 −0.577107
\(36\) 0 0
\(37\) 1.82843 0.300592 0.150296 − 0.988641i \(-0.451977\pi\)
0.150296 + 0.988641i \(0.451977\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.82843 0.441726 0.220863 − 0.975305i \(-0.429113\pi\)
0.220863 + 0.975305i \(0.429113\pi\)
\(42\) 0 0
\(43\) 0.343146 0.0523292 0.0261646 − 0.999658i \(-0.491671\pi\)
0.0261646 + 0.999658i \(0.491671\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.242641 0.0353928 0.0176964 − 0.999843i \(-0.494367\pi\)
0.0176964 + 0.999843i \(0.494367\pi\)
\(48\) 0 0
\(49\) −1.17157 −0.167368
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13.8995 −1.90924 −0.954621 − 0.297823i \(-0.903740\pi\)
−0.954621 + 0.297823i \(0.903740\pi\)
\(54\) 0 0
\(55\) 1.41421 0.190693
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.41421 −0.704871 −0.352435 − 0.935836i \(-0.614646\pi\)
−0.352435 + 0.935836i \(0.614646\pi\)
\(60\) 0 0
\(61\) −10.4142 −1.33340 −0.666702 − 0.745325i \(-0.732294\pi\)
−0.666702 + 0.745325i \(0.732294\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.585786 −0.0726579
\(66\) 0 0
\(67\) 14.3137 1.74870 0.874349 − 0.485298i \(-0.161289\pi\)
0.874349 + 0.485298i \(0.161289\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −8.89949 −1.04161 −0.520804 − 0.853677i \(-0.674368\pi\)
−0.520804 + 0.853677i \(0.674368\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.41421 −0.275125
\(78\) 0 0
\(79\) 14.0711 1.58312 0.791559 − 0.611092i \(-0.209270\pi\)
0.791559 + 0.611092i \(0.209270\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.65685 0.181863 0.0909317 − 0.995857i \(-0.471016\pi\)
0.0909317 + 0.995857i \(0.471016\pi\)
\(84\) 0 0
\(85\) −3.17157 −0.344005
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.24264 −0.873718 −0.436859 − 0.899530i \(-0.643909\pi\)
−0.436859 + 0.899530i \(0.643909\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.58579 0.470492
\(96\) 0 0
\(97\) −3.34315 −0.339445 −0.169723 − 0.985492i \(-0.554287\pi\)
−0.169723 + 0.985492i \(0.554287\pi\)
\(98\) 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 4752.2.a.ba.1.1 2
3.2 odd 2 4752.2.a.bc.1.2 2
4.3 odd 2 2376.2.a.h.1.1 yes 2
12.11 even 2 2376.2.a.f.1.2 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2376.2.a.f.1.2 ✓ 2 12.11 even 2
2376.2.a.h.1.1 yes 2 4.3 odd 2
4752.2.a.ba.1.1 2 1.1 even 1 trivial
4752.2.a.bc.1.2 2 3.2 odd 2