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Properties

Label 9248.2.a.f.1.1
Level $9248$
Weight $2$
Character 9248.1
Self dual yes
Analytic conductor $73.846$
Analytic rank $0$
Dimension $1$
CM discriminant -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] = [9248,2,Mod(1,9248)] 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("9248.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9248 = 2^{5} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9248.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,2,0,0,0,-3,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,10, 0,0,0,0,0,0,0,2,0,0,0,-10,0,0,0,-6,0,0,0,-7,0,0,0,14,0,0,0,0,0,0,0,10, 0,0,0,12,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,10,0,0,0,0,0,0, 0,-18,0,0,0,-2,0,0,0,0,0,0,0,-6,0,0,0,14,0,0,0,-18,0,0,0,-11,0,0,0,-12, 0,0,0,0,0,0,0,0,0,0,0,-22,0,0,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(145)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(73.8456517893\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9248.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{5} -3.00000 q^{9} +6.00000 q^{13} -1.00000 q^{25} +10.0000 q^{29} +2.00000 q^{37} -10.0000 q^{41} -6.00000 q^{45} -7.00000 q^{49} +14.0000 q^{53} +10.0000 q^{61} +12.0000 q^{65} +6.00000 q^{73} +9.00000 q^{81} +10.0000 q^{89} -18.0000 q^{97} +O(q^{100})\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −6.00000 −0.894427
\(46\) 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
\(53\) 14.0000 1.92305 0.961524 − 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 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 9248.2.a.f.1.1 1
4.3 odd 2 CM 9248.2.a.f.1.1 1
17.16 even 2 32.2.a.a.1.1 ✓ 1
51.50 odd 2 288.2.a.d.1.1 1
68.67 odd 2 32.2.a.a.1.1 ✓ 1
85.33 odd 4 800.2.c.e.449.2 2
85.67 odd 4 800.2.c.e.449.1 2
85.84 even 2 800.2.a.d.1.1 1
119.16 even 6 1568.2.i.g.1537.1 2
119.33 odd 6 1568.2.i.f.1537.1 2
119.67 even 6 1568.2.i.g.961.1 2
119.101 odd 6 1568.2.i.f.961.1 2
119.118 odd 2 1568.2.a.e.1.1 1
136.67 odd 2 64.2.a.a.1.1 1
136.101 even 2 64.2.a.a.1.1 1
153.16 even 6 2592.2.i.t.1729.1 2
153.50 odd 6 2592.2.i.e.865.1 2
153.67 even 6 2592.2.i.t.865.1 2
153.101 odd 6 2592.2.i.e.1729.1 2
187.186 odd 2 3872.2.a.f.1.1 1
204.203 even 2 288.2.a.d.1.1 1
221.220 even 2 5408.2.a.g.1.1 1
255.152 even 4 7200.2.f.m.6049.1 2
255.203 even 4 7200.2.f.m.6049.2 2
255.254 odd 2 7200.2.a.v.1.1 1
272.67 odd 4 256.2.b.b.129.2 2
272.101 even 4 256.2.b.b.129.1 2
272.203 odd 4 256.2.b.b.129.1 2
272.237 even 4 256.2.b.b.129.2 2
340.67 even 4 800.2.c.e.449.1 2
340.203 even 4 800.2.c.e.449.2 2
340.339 odd 2 800.2.a.d.1.1 1
408.101 odd 2 576.2.a.c.1.1 1
408.203 even 2 576.2.a.c.1.1 1
476.67 odd 6 1568.2.i.g.961.1 2
476.135 odd 6 1568.2.i.g.1537.1 2
476.271 even 6 1568.2.i.f.1537.1 2
476.339 even 6 1568.2.i.f.961.1 2
476.475 even 2 1568.2.a.e.1.1 1
544.67 odd 8 1024.2.e.j.257.2 4
544.101 even 8 1024.2.e.j.769.1 4
544.203 odd 8 1024.2.e.j.769.2 4
544.237 even 8 1024.2.e.j.257.1 4
544.339 odd 8 1024.2.e.j.257.1 4
544.373 even 8 1024.2.e.j.769.2 4
544.475 odd 8 1024.2.e.j.769.1 4
544.509 even 8 1024.2.e.j.257.2 4
612.67 odd 6 2592.2.i.t.865.1 2
612.203 even 6 2592.2.i.e.865.1 2
612.407 even 6 2592.2.i.e.1729.1 2
612.475 odd 6 2592.2.i.t.1729.1 2
680.67 even 4 1600.2.c.l.449.2 2
680.203 even 4 1600.2.c.l.449.1 2
680.237 odd 4 1600.2.c.l.449.2 2
680.339 odd 2 1600.2.a.n.1.1 1
680.373 odd 4 1600.2.c.l.449.1 2
680.509 even 2 1600.2.a.n.1.1 1
748.747 even 2 3872.2.a.f.1.1 1
816.101 odd 4 2304.2.d.j.1153.2 2
816.203 even 4 2304.2.d.j.1153.2 2
816.509 odd 4 2304.2.d.j.1153.1 2
816.611 even 4 2304.2.d.j.1153.1 2
884.883 odd 2 5408.2.a.g.1.1 1
952.237 odd 2 3136.2.a.m.1.1 1
952.475 even 2 3136.2.a.m.1.1 1
1020.203 odd 4 7200.2.f.m.6049.2 2
1020.407 odd 4 7200.2.f.m.6049.1 2
1020.1019 even 2 7200.2.a.v.1.1 1
1496.373 odd 2 7744.2.a.v.1.1 1
1496.747 even 2 7744.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.2.a.a.1.1 ✓ 1 17.16 even 2
32.2.a.a.1.1 ✓ 1 68.67 odd 2
64.2.a.a.1.1 1 136.67 odd 2
64.2.a.a.1.1 1 136.101 even 2
256.2.b.b.129.1 2 272.101 even 4
256.2.b.b.129.1 2 272.203 odd 4
256.2.b.b.129.2 2 272.67 odd 4
256.2.b.b.129.2 2 272.237 even 4
288.2.a.d.1.1 1 51.50 odd 2
288.2.a.d.1.1 1 204.203 even 2
576.2.a.c.1.1 1 408.101 odd 2
576.2.a.c.1.1 1 408.203 even 2
800.2.a.d.1.1 1 85.84 even 2
800.2.a.d.1.1 1 340.339 odd 2
800.2.c.e.449.1 2 85.67 odd 4
800.2.c.e.449.1 2 340.67 even 4
800.2.c.e.449.2 2 85.33 odd 4
800.2.c.e.449.2 2 340.203 even 4
1024.2.e.j.257.1 4 544.237 even 8
1024.2.e.j.257.1 4 544.339 odd 8
1024.2.e.j.257.2 4 544.67 odd 8
1024.2.e.j.257.2 4 544.509 even 8
1024.2.e.j.769.1 4 544.101 even 8
1024.2.e.j.769.1 4 544.475 odd 8
1024.2.e.j.769.2 4 544.203 odd 8
1024.2.e.j.769.2 4 544.373 even 8
1568.2.a.e.1.1 1 119.118 odd 2
1568.2.a.e.1.1 1 476.475 even 2
1568.2.i.f.961.1 2 119.101 odd 6
1568.2.i.f.961.1 2 476.339 even 6
1568.2.i.f.1537.1 2 119.33 odd 6
1568.2.i.f.1537.1 2 476.271 even 6
1568.2.i.g.961.1 2 119.67 even 6
1568.2.i.g.961.1 2 476.67 odd 6
1568.2.i.g.1537.1 2 119.16 even 6
1568.2.i.g.1537.1 2 476.135 odd 6
1600.2.a.n.1.1 1 680.339 odd 2
1600.2.a.n.1.1 1 680.509 even 2
1600.2.c.l.449.1 2 680.203 even 4
1600.2.c.l.449.1 2 680.373 odd 4
1600.2.c.l.449.2 2 680.67 even 4
1600.2.c.l.449.2 2 680.237 odd 4
2304.2.d.j.1153.1 2 816.509 odd 4
2304.2.d.j.1153.1 2 816.611 even 4
2304.2.d.j.1153.2 2 816.101 odd 4
2304.2.d.j.1153.2 2 816.203 even 4
2592.2.i.e.865.1 2 153.50 odd 6
2592.2.i.e.865.1 2 612.203 even 6
2592.2.i.e.1729.1 2 153.101 odd 6
2592.2.i.e.1729.1 2 612.407 even 6
2592.2.i.t.865.1 2 153.67 even 6
2592.2.i.t.865.1 2 612.67 odd 6
2592.2.i.t.1729.1 2 153.16 even 6
2592.2.i.t.1729.1 2 612.475 odd 6
3136.2.a.m.1.1 1 952.237 odd 2
3136.2.a.m.1.1 1 952.475 even 2
3872.2.a.f.1.1 1 187.186 odd 2
3872.2.a.f.1.1 1 748.747 even 2
5408.2.a.g.1.1 1 221.220 even 2
5408.2.a.g.1.1 1 884.883 odd 2
7200.2.a.v.1.1 1 255.254 odd 2
7200.2.a.v.1.1 1 1020.1019 even 2
7200.2.f.m.6049.1 2 255.152 even 4
7200.2.f.m.6049.1 2 1020.407 odd 4
7200.2.f.m.6049.2 2 255.203 even 4
7200.2.f.m.6049.2 2 1020.203 odd 4
7744.2.a.v.1.1 1 1496.373 odd 2
7744.2.a.v.1.1 1 1496.747 even 2
9248.2.a.f.1.1 1 1.1 even 1 trivial
9248.2.a.f.1.1 1 4.3 odd 2 CM