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Properties

Label 2304.2.c.h.2303.2
Level $2304$
Weight $2$
Character 2304.2303
Analytic conductor $18.398$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2304,2,Mod(2303,2304)] 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("2304.2303"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,8,0,4,0,0,0,0,0,0,0,0,0,8,0,6,0,0,0,0,0, 0,0,0,0,-8,0,16,0,0,0,0,0,0,0,0,0,-24,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
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 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2303.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2303
Dual form 2304.2.c.h.2303.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{5} +2.82843i q^{7} +4.00000 q^{11} +2.00000 q^{13} -1.41421i q^{17} -5.65685i q^{19} +4.00000 q^{23} +3.00000 q^{25} +7.07107i q^{29} -8.48528i q^{31} -4.00000 q^{35} +8.00000 q^{37} -4.24264i q^{41} +11.3137i q^{43} -12.0000 q^{47} -1.00000 q^{49} +12.7279i q^{53} +5.65685i q^{55} +8.00000 q^{61} +2.82843i q^{65} -5.65685i q^{67} +4.00000 q^{71} -8.00000 q^{73} +11.3137i q^{77} -2.82843i q^{79} -12.0000 q^{83} +2.00000 q^{85} +15.5563i q^{89} +5.65685i q^{91} +8.00000 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{11} + 4 q^{13} + 8 q^{23} + 6 q^{25} - 8 q^{35} + 16 q^{37} - 24 q^{47} - 2 q^{49} + 16 q^{61} + 8 q^{71} - 16 q^{73} - 24 q^{83} + 4 q^{85} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 0
\(4\) 0 0
\(5\) 1.41421i 0.632456i 0.948683 + 0.316228i \(0.102416\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 0 0
\(7\) 2.82843i 1.06904i 0.845154 + 0.534522i \(0.179509\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.41421i − 0.342997i −0.985184 − 0.171499i \(-0.945139\pi\)
0.985184 − 0.171499i \(-0.0548609\pi\)
\(18\) 0 0
\(19\) − 5.65685i − 1.29777i −0.760886 − 0.648886i \(-0.775235\pi\)
0.760886 − 0.648886i \(-0.224765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.07107i 1.31306i 0.754298 + 0.656532i \(0.227977\pi\)
−0.754298 + 0.656532i \(0.772023\pi\)
\(30\) 0 0
\(31\) − 8.48528i − 1.52400i −0.647576 − 0.762001i \(-0.724217\pi\)
0.647576 − 0.762001i \(-0.275783\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 4.24264i − 0.662589i −0.943527 − 0.331295i \(-0.892515\pi\)
0.943527 − 0.331295i \(-0.107485\pi\)
\(42\) 0 0
\(43\) 11.3137i 1.72532i 0.505781 + 0.862662i \(0.331205\pi\)
−0.505781 + 0.862662i \(0.668795\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.7279i 1.74831i 0.485643 + 0.874157i \(0.338586\pi\)
−0.485643 + 0.874157i \(0.661414\pi\)
\(54\) 0 0
\(55\) 5.65685i 0.762770i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.82843i 0.350823i
\(66\) 0 0
\(67\) − 5.65685i − 0.691095i −0.938401 − 0.345547i \(-0.887693\pi\)
0.938401 − 0.345547i \(-0.112307\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 − 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.3137i 1.28932i
\(78\) 0 0
\(79\) − 2.82843i − 0.318223i −0.987261 − 0.159111i \(-0.949137\pi\)
0.987261 − 0.159111i \(-0.0508629\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.5563i 1.64897i 0.565884 + 0.824485i \(0.308535\pi\)
−0.565884 + 0.824485i \(0.691465\pi\)
\(90\) 0 0
\(91\) 5.65685i 0.592999i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 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 2304.2.c.h.2303.2 2
3.2 odd 2 2304.2.c.b.2303.1 2
4.3 odd 2 2304.2.c.b.2303.2 2
8.3 odd 2 2304.2.c.g.2303.1 2
8.5 even 2 2304.2.c.a.2303.1 2
12.11 even 2 inner 2304.2.c.h.2303.1 2
16.3 odd 4 1152.2.f.d.575.4 yes 4
16.5 even 4 1152.2.f.a.575.1 ✓ 4
16.11 odd 4 1152.2.f.d.575.2 yes 4
16.13 even 4 1152.2.f.a.575.3 yes 4
24.5 odd 2 2304.2.c.g.2303.2 2
24.11 even 2 2304.2.c.a.2303.2 2
48.5 odd 4 1152.2.f.d.575.3 yes 4
48.11 even 4 1152.2.f.a.575.4 yes 4
48.29 odd 4 1152.2.f.d.575.1 yes 4
48.35 even 4 1152.2.f.a.575.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.f.a.575.1 ✓ 4 16.5 even 4
1152.2.f.a.575.2 yes 4 48.35 even 4
1152.2.f.a.575.3 yes 4 16.13 even 4
1152.2.f.a.575.4 yes 4 48.11 even 4
1152.2.f.d.575.1 yes 4 48.29 odd 4
1152.2.f.d.575.2 yes 4 16.11 odd 4
1152.2.f.d.575.3 yes 4 48.5 odd 4
1152.2.f.d.575.4 yes 4 16.3 odd 4
2304.2.c.a.2303.1 2 8.5 even 2
2304.2.c.a.2303.2 2 24.11 even 2
2304.2.c.b.2303.1 2 3.2 odd 2
2304.2.c.b.2303.2 2 4.3 odd 2
2304.2.c.g.2303.1 2 8.3 odd 2
2304.2.c.g.2303.2 2 24.5 odd 2
2304.2.c.h.2303.1 2 12.11 even 2 inner
2304.2.c.h.2303.2 2 1.1 even 1 trivial