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Properties

Label 2304.2.d.k.1153.1
Level $2304$
Weight $2$
Character 2304.1153
Analytic conductor $18.398$
Analytic rank $0$
Dimension $2$
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] = [2304,2,Mod(1153,2304)] 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("2304.1153"); 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([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-4,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == 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: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1153.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1153
Dual form 2304.2.d.k.1153.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{5} -4.00000i q^{11} -2.00000i q^{13} -2.00000 q^{17} +4.00000i q^{19} +8.00000 q^{23} +1.00000 q^{25} -6.00000i q^{29} -8.00000 q^{31} -6.00000i q^{37} -6.00000 q^{41} +4.00000i q^{43} -7.00000 q^{49} -2.00000i q^{53} -8.00000 q^{55} -4.00000i q^{59} -2.00000i q^{61} -4.00000 q^{65} +4.00000i q^{67} -8.00000 q^{71} -10.0000 q^{73} +8.00000 q^{79} -4.00000i q^{83} +4.00000i q^{85} -6.00000 q^{89} +8.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{17} + 16 q^{23} + 2 q^{25} - 16 q^{31} - 12 q^{41} - 14 q^{49} - 16 q^{55} - 8 q^{65} - 16 q^{71} - 20 q^{73} + 16 q^{79} - 12 q^{89} + 16 q^{95} + 4 q^{97}+O(q^{100}) \) 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
\(5\) − 2.00000i − 0.894427i −0.894427 − 0.447214i \(-0.852416\pi\)
0.894427 − 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.00000i − 1.20605i −0.797724 − 0.603023i \(-0.793963\pi\)
0.797724 − 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.00000i − 1.11417i −0.830455 − 0.557086i \(-0.811919\pi\)
0.830455 − 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.00000i − 0.986394i −0.869918 − 0.493197i \(-0.835828\pi\)
0.869918 − 0.493197i \(-0.164172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 − 0.137361i \(-0.956138\pi\)
0.990521 − 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 4.00000i − 0.520756i −0.965507 − 0.260378i \(-0.916153\pi\)
0.965507 − 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) − 2.00000i − 0.256074i −0.991769 − 0.128037i \(-0.959132\pi\)
0.991769 − 0.128037i \(-0.0408676\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 4.00000i − 0.439057i −0.975606 − 0.219529i \(-0.929548\pi\)
0.975606 − 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 4.00000i 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 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.d.k.1153.1 2
3.2 odd 2 768.2.d.d.385.2 2
4.3 odd 2 2304.2.d.i.1153.1 2
8.3 odd 2 2304.2.d.i.1153.2 2
8.5 even 2 inner 2304.2.d.k.1153.2 2
12.11 even 2 768.2.d.e.385.1 2
16.3 odd 4 576.2.a.d.1.1 1
16.5 even 4 144.2.a.b.1.1 1
16.11 odd 4 72.2.a.a.1.1 1
16.13 even 4 576.2.a.b.1.1 1
24.5 odd 2 768.2.d.d.385.1 2
24.11 even 2 768.2.d.e.385.2 2
48.5 odd 4 48.2.a.a.1.1 1
48.11 even 4 24.2.a.a.1.1 ✓ 1
48.29 odd 4 192.2.a.b.1.1 1
48.35 even 4 192.2.a.d.1.1 1
80.27 even 4 1800.2.f.c.649.2 2
80.37 odd 4 3600.2.f.r.2449.2 2
80.43 even 4 1800.2.f.c.649.1 2
80.53 odd 4 3600.2.f.r.2449.1 2
80.59 odd 4 1800.2.a.m.1.1 1
80.69 even 4 3600.2.a.v.1.1 1
112.11 odd 12 3528.2.s.j.3313.1 2
112.27 even 4 3528.2.a.d.1.1 1
112.59 even 12 3528.2.s.y.3313.1 2
112.69 odd 4 7056.2.a.q.1.1 1
112.75 even 12 3528.2.s.y.361.1 2
112.107 odd 12 3528.2.s.j.361.1 2
144.5 odd 12 1296.2.i.m.865.1 2
144.11 even 12 648.2.i.g.433.1 2
144.43 odd 12 648.2.i.b.433.1 2
144.59 even 12 648.2.i.g.217.1 2
144.85 even 12 1296.2.i.e.865.1 2
144.101 odd 12 1296.2.i.m.433.1 2
144.133 even 12 1296.2.i.e.433.1 2
144.139 odd 12 648.2.i.b.217.1 2
176.43 even 4 8712.2.a.u.1.1 1
240.29 odd 4 4800.2.a.cc.1.1 1
240.53 even 4 1200.2.f.b.49.2 2
240.59 even 4 600.2.a.h.1.1 1
240.77 even 4 4800.2.f.bg.3649.2 2
240.83 odd 4 4800.2.f.d.3649.2 2
240.107 odd 4 600.2.f.e.49.2 2
240.149 odd 4 1200.2.a.d.1.1 1
240.173 even 4 4800.2.f.bg.3649.1 2
240.179 even 4 4800.2.a.q.1.1 1
240.197 even 4 1200.2.f.b.49.1 2
240.203 odd 4 600.2.f.e.49.1 2
240.227 odd 4 4800.2.f.d.3649.1 2
336.5 even 12 2352.2.q.r.1537.1 2
336.11 even 12 1176.2.q.i.961.1 2
336.53 odd 12 2352.2.q.l.961.1 2
336.59 odd 12 1176.2.q.a.961.1 2
336.83 odd 4 9408.2.a.h.1.1 1
336.101 even 12 2352.2.q.r.961.1 2
336.107 even 12 1176.2.q.i.361.1 2
336.125 even 4 9408.2.a.cc.1.1 1
336.149 odd 12 2352.2.q.l.1537.1 2
336.251 odd 4 1176.2.a.i.1.1 1
336.293 even 4 2352.2.a.i.1.1 1
336.299 odd 12 1176.2.q.a.361.1 2
528.197 even 4 5808.2.a.s.1.1 1
528.395 odd 4 2904.2.a.c.1.1 1
624.155 even 4 4056.2.a.i.1.1 1
624.203 odd 4 4056.2.c.e.337.1 2
624.389 odd 4 8112.2.a.be.1.1 1
624.395 odd 4 4056.2.c.e.337.2 2
816.203 even 4 6936.2.a.p.1.1 1
912.683 odd 4 8664.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.a.a.1.1 ✓ 1 48.11 even 4
48.2.a.a.1.1 1 48.5 odd 4
72.2.a.a.1.1 1 16.11 odd 4
144.2.a.b.1.1 1 16.5 even 4
192.2.a.b.1.1 1 48.29 odd 4
192.2.a.d.1.1 1 48.35 even 4
576.2.a.b.1.1 1 16.13 even 4
576.2.a.d.1.1 1 16.3 odd 4
600.2.a.h.1.1 1 240.59 even 4
600.2.f.e.49.1 2 240.203 odd 4
600.2.f.e.49.2 2 240.107 odd 4
648.2.i.b.217.1 2 144.139 odd 12
648.2.i.b.433.1 2 144.43 odd 12
648.2.i.g.217.1 2 144.59 even 12
648.2.i.g.433.1 2 144.11 even 12
768.2.d.d.385.1 2 24.5 odd 2
768.2.d.d.385.2 2 3.2 odd 2
768.2.d.e.385.1 2 12.11 even 2
768.2.d.e.385.2 2 24.11 even 2
1176.2.a.i.1.1 1 336.251 odd 4
1176.2.q.a.361.1 2 336.299 odd 12
1176.2.q.a.961.1 2 336.59 odd 12
1176.2.q.i.361.1 2 336.107 even 12
1176.2.q.i.961.1 2 336.11 even 12
1200.2.a.d.1.1 1 240.149 odd 4
1200.2.f.b.49.1 2 240.197 even 4
1200.2.f.b.49.2 2 240.53 even 4
1296.2.i.e.433.1 2 144.133 even 12
1296.2.i.e.865.1 2 144.85 even 12
1296.2.i.m.433.1 2 144.101 odd 12
1296.2.i.m.865.1 2 144.5 odd 12
1800.2.a.m.1.1 1 80.59 odd 4
1800.2.f.c.649.1 2 80.43 even 4
1800.2.f.c.649.2 2 80.27 even 4
2304.2.d.i.1153.1 2 4.3 odd 2
2304.2.d.i.1153.2 2 8.3 odd 2
2304.2.d.k.1153.1 2 1.1 even 1 trivial
2304.2.d.k.1153.2 2 8.5 even 2 inner
2352.2.a.i.1.1 1 336.293 even 4
2352.2.q.l.961.1 2 336.53 odd 12
2352.2.q.l.1537.1 2 336.149 odd 12
2352.2.q.r.961.1 2 336.101 even 12
2352.2.q.r.1537.1 2 336.5 even 12
2904.2.a.c.1.1 1 528.395 odd 4
3528.2.a.d.1.1 1 112.27 even 4
3528.2.s.j.361.1 2 112.107 odd 12
3528.2.s.j.3313.1 2 112.11 odd 12
3528.2.s.y.361.1 2 112.75 even 12
3528.2.s.y.3313.1 2 112.59 even 12
3600.2.a.v.1.1 1 80.69 even 4
3600.2.f.r.2449.1 2 80.53 odd 4
3600.2.f.r.2449.2 2 80.37 odd 4
4056.2.a.i.1.1 1 624.155 even 4
4056.2.c.e.337.1 2 624.203 odd 4
4056.2.c.e.337.2 2 624.395 odd 4
4800.2.a.q.1.1 1 240.179 even 4
4800.2.a.cc.1.1 1 240.29 odd 4
4800.2.f.d.3649.1 2 240.227 odd 4
4800.2.f.d.3649.2 2 240.83 odd 4
4800.2.f.bg.3649.1 2 240.173 even 4
4800.2.f.bg.3649.2 2 240.77 even 4
5808.2.a.s.1.1 1 528.197 even 4
6936.2.a.p.1.1 1 816.203 even 4
7056.2.a.q.1.1 1 112.69 odd 4
8112.2.a.be.1.1 1 624.389 odd 4
8664.2.a.j.1.1 1 912.683 odd 4
8712.2.a.u.1.1 1 176.43 even 4
9408.2.a.h.1.1 1 336.83 odd 4
9408.2.a.cc.1.1 1 336.125 even 4