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Properties

Label 2312.2.b.h.577.3
Level $2312$
Weight $2$
Character 2312.577
Analytic conductor $18.461$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [2312,2,Mod(577,2312)] 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("2312.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2312.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-10,0,0,0,10,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.4614129473\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.3
Root \(1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 2312.577
Dual form 2312.2.b.h.577.2

$q$-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)
 
\(f(q)\) \(=\) \(q+0.302776i q^{3} -1.30278i q^{5} +1.30278i q^{7} +2.90833 q^{9} -1.60555i q^{11} +4.30278 q^{13} +0.394449 q^{15} -2.30278 q^{19} -0.394449 q^{21} +0.697224i q^{23} +3.30278 q^{25} +1.78890i q^{27} +4.30278i q^{29} -10.2111i q^{31} +0.486122 q^{33} +1.69722 q^{35} -1.39445i q^{37} +1.30278i q^{39} +6.00000i q^{41} -10.2111 q^{43} -3.78890i q^{45} +4.39445 q^{47} +5.30278 q^{49} +0.302776 q^{53} -2.09167 q^{55} -0.697224i q^{57} +11.2111 q^{59} +5.60555i q^{61} +3.78890i q^{63} -5.60555i q^{65} -6.60555 q^{67} -0.211103 q^{69} -16.4222i q^{71} -9.60555i q^{73} +1.00000i q^{75} +2.09167 q^{77} +10.0000i q^{79} +8.18335 q^{81} +12.3028 q^{83} -1.30278 q^{87} -7.21110 q^{89} +5.60555i q^{91} +3.09167 q^{93} +3.00000i q^{95} +9.11943i q^{97} -4.66947i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{9} + 10 q^{13} + 16 q^{15} - 2 q^{19} - 16 q^{21} + 6 q^{25} + 38 q^{33} + 14 q^{35} - 12 q^{43} + 32 q^{47} + 14 q^{49} - 6 q^{53} - 30 q^{55} + 16 q^{59} - 12 q^{67} + 28 q^{69} + 30 q^{77}+ \cdots + 34 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1735\) \(1737\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0.302776i 0.174808i 0.996173 + 0.0874038i \(0.0278570\pi\)
−0.996173 + 0.0874038i \(0.972143\pi\)
\(4\) 0 0
\(5\) − 1.30278i − 0.582619i −0.956629 − 0.291309i \(-0.905909\pi\)
0.956629 − 0.291309i \(-0.0940909\pi\)
\(6\) 0 0
\(7\) 1.30278i 0.492403i 0.969219 + 0.246201i \(0.0791825\pi\)
−0.969219 + 0.246201i \(0.920818\pi\)
\(8\) 0 0
\(9\) 2.90833 0.969442
\(10\) 0 0
\(11\) − 1.60555i − 0.484092i −0.970265 − 0.242046i \(-0.922182\pi\)
0.970265 − 0.242046i \(-0.0778185\pi\)
\(12\) 0 0
\(13\) 4.30278 1.19338 0.596688 − 0.802474i \(-0.296483\pi\)
0.596688 + 0.802474i \(0.296483\pi\)
\(14\) 0 0
\(15\) 0.394449 0.101846
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) −2.30278 −0.528293 −0.264146 − 0.964483i \(-0.585090\pi\)
−0.264146 + 0.964483i \(0.585090\pi\)
\(20\) 0 0
\(21\) −0.394449 −0.0860758
\(22\) 0 0
\(23\) 0.697224i 0.145381i 0.997355 + 0.0726907i \(0.0231586\pi\)
−0.997355 + 0.0726907i \(0.976841\pi\)
\(24\) 0 0
\(25\) 3.30278 0.660555
\(26\) 0 0
\(27\) 1.78890i 0.344273i
\(28\) 0 0
\(29\) 4.30278i 0.799005i 0.916732 + 0.399503i \(0.130817\pi\)
−0.916732 + 0.399503i \(0.869183\pi\)
\(30\) 0 0
\(31\) − 10.2111i − 1.83397i −0.398924 − 0.916984i \(-0.630616\pi\)
0.398924 − 0.916984i \(-0.369384\pi\)
\(32\) 0 0
\(33\) 0.486122 0.0846229
\(34\) 0 0
\(35\) 1.69722 0.286883
\(36\) 0 0
\(37\) − 1.39445i − 0.229246i −0.993409 − 0.114623i \(-0.963434\pi\)
0.993409 − 0.114623i \(-0.0365660\pi\)
\(38\) 0 0
\(39\) 1.30278i 0.208611i
\(40\) 0 0
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) 0 0
\(43\) −10.2111 −1.55718 −0.778589 − 0.627534i \(-0.784064\pi\)
−0.778589 + 0.627534i \(0.784064\pi\)
\(44\) 0 0
\(45\) − 3.78890i − 0.564815i
\(46\) 0 0
\(47\) 4.39445 0.640996 0.320498 − 0.947249i \(-0.396150\pi\)
0.320498 + 0.947249i \(0.396150\pi\)
\(48\) 0 0
\(49\) 5.30278 0.757539
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.302776 0.0415894 0.0207947 − 0.999784i \(-0.493380\pi\)
0.0207947 + 0.999784i \(0.493380\pi\)
\(54\) 0 0
\(55\) −2.09167 −0.282041
\(56\) 0 0
\(57\) − 0.697224i − 0.0923496i
\(58\) 0 0
\(59\) 11.2111 1.45956 0.729781 − 0.683681i \(-0.239622\pi\)
0.729781 + 0.683681i \(0.239622\pi\)
\(60\) 0 0
\(61\) 5.60555i 0.717717i 0.933392 + 0.358859i \(0.116834\pi\)
−0.933392 + 0.358859i \(0.883166\pi\)
\(62\) 0 0
\(63\) 3.78890i 0.477356i
\(64\) 0 0
\(65\) − 5.60555i − 0.695283i
\(66\) 0 0
\(67\) −6.60555 −0.806997 −0.403498 − 0.914980i \(-0.632206\pi\)
−0.403498 + 0.914980i \(0.632206\pi\)
\(68\) 0 0
\(69\) −0.211103 −0.0254138
\(70\) 0 0
\(71\) − 16.4222i − 1.94896i −0.224480 − 0.974479i \(-0.572068\pi\)
0.224480 − 0.974479i \(-0.427932\pi\)
\(72\) 0 0
\(73\) − 9.60555i − 1.12424i −0.827054 − 0.562122i \(-0.809985\pi\)
0.827054 − 0.562122i \(-0.190015\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 2.09167 0.238368
\(78\) 0 0
\(79\) 10.0000i 1.12509i 0.826767 + 0.562544i \(0.190177\pi\)
−0.826767 + 0.562544i \(0.809823\pi\)
\(80\) 0 0
\(81\) 8.18335 0.909261
\(82\) 0 0
\(83\) 12.3028 1.35041 0.675203 − 0.737632i \(-0.264056\pi\)
0.675203 + 0.737632i \(0.264056\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.30278 −0.139672
\(88\) 0 0
\(89\) −7.21110 −0.764375 −0.382188 − 0.924085i \(-0.624829\pi\)
−0.382188 + 0.924085i \(0.624829\pi\)
\(90\) 0 0
\(91\) 5.60555i 0.587621i
\(92\) 0 0
\(93\) 3.09167 0.320592
\(94\) 0 0
\(95\) 3.00000i 0.307794i
\(96\) 0 0
\(97\) 9.11943i 0.925938i 0.886375 + 0.462969i \(0.153216\pi\)
−0.886375 + 0.462969i \(0.846784\pi\)
\(98\) 0 0
\(99\) − 4.66947i − 0.469299i
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 2312.2.b.h.577.3 4
17.4 even 4 2312.2.a.e.1.2 ✓ 2
17.13 even 4 2312.2.a.n.1.1 yes 2
17.16 even 2 inner 2312.2.b.h.577.2 4
68.47 odd 4 4624.2.a.g.1.2 2
68.55 odd 4 4624.2.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2312.2.a.e.1.2 ✓ 2 17.4 even 4
2312.2.a.n.1.1 yes 2 17.13 even 4
2312.2.b.h.577.2 4 17.16 even 2 inner
2312.2.b.h.577.3 4 1.1 even 1 trivial
4624.2.a.g.1.2 2 68.47 odd 4
4624.2.a.y.1.1 2 68.55 odd 4