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Properties

Label 2448.2.c.d.577.1
Level $2448$
Weight $2$
Character 2448.577
Analytic conductor $19.547$
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] = [2448,2,Mod(577,2448)] 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("2448.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2448.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,-8,0,0,0,-6,0,8,0,0,0,0,0,-6,0,0,0,0, 0,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(35)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.5473784148\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 2448.577
Dual form 2448.2.c.d.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)
 
\(f(q)\) \(=\) \(q-2.82843i q^{5} +4.24264i q^{7} -1.41421i q^{11} -4.00000 q^{13} +(-3.00000 + 2.82843i) q^{17} +4.00000 q^{19} -1.41421i q^{23} -3.00000 q^{25} -2.82843i q^{29} -4.24264i q^{31} +12.0000 q^{35} -8.48528i q^{37} -11.3137i q^{41} -8.00000 q^{43} -12.0000 q^{47} -11.0000 q^{49} +6.00000 q^{53} -4.00000 q^{55} -8.48528i q^{61} +11.3137i q^{65} +4.00000 q^{67} +7.07107i q^{71} +6.00000 q^{77} -4.24264i q^{79} +(8.00000 + 8.48528i) q^{85} -12.0000 q^{89} -16.9706i q^{91} -11.3137i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{13} - 6 q^{17} + 8 q^{19} - 6 q^{25} + 24 q^{35} - 16 q^{43} - 24 q^{47} - 22 q^{49} + 12 q^{53} - 8 q^{55} + 8 q^{67} + 12 q^{77} + 16 q^{85} - 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times\).

\(n\) \(613\) \(1361\) \(1873\) \(2143\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(5\) − 2.82843i − 1.26491i −0.774597 − 0.632456i \(-0.782047\pi\)
0.774597 − 0.632456i \(-0.217953\pi\)
\(6\) 0 0
\(7\) 4.24264i 1.60357i 0.597614 + 0.801784i \(0.296115\pi\)
−0.597614 + 0.801784i \(0.703885\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.41421i − 0.426401i −0.977008 − 0.213201i \(-0.931611\pi\)
0.977008 − 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 + 2.82843i −0.727607 + 0.685994i
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 1.41421i − 0.294884i −0.989071 − 0.147442i \(-0.952896\pi\)
0.989071 − 0.147442i \(-0.0471040\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 2.82843i − 0.525226i −0.964901 − 0.262613i \(-0.915416\pi\)
0.964901 − 0.262613i \(-0.0845842\pi\)
\(30\) 0 0
\(31\) − 4.24264i − 0.762001i −0.924575 − 0.381000i \(-0.875580\pi\)
0.924575 − 0.381000i \(-0.124420\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) 0 0
\(37\) − 8.48528i − 1.39497i −0.716599 − 0.697486i \(-0.754302\pi\)
0.716599 − 0.697486i \(-0.245698\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 11.3137i − 1.76690i −0.468521 − 0.883452i \(-0.655213\pi\)
0.468521 − 0.883452i \(-0.344787\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 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
\(49\) −11.0000 −1.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) − 8.48528i − 1.08643i −0.839594 − 0.543214i \(-0.817207\pi\)
0.839594 − 0.543214i \(-0.182793\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 11.3137i 1.40329i
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.07107i 0.839181i 0.907713 + 0.419591i \(0.137826\pi\)
−0.907713 + 0.419591i \(0.862174\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) − 4.24264i − 0.477334i −0.971101 − 0.238667i \(-0.923290\pi\)
0.971101 − 0.238667i \(-0.0767105\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 8.00000 + 8.48528i 0.867722 + 0.920358i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) − 16.9706i − 1.77900i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 11.3137i − 1.16076i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 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 2448.2.c.d.577.1 2
3.2 odd 2 272.2.b.c.33.1 2
4.3 odd 2 612.2.b.a.577.1 2
12.11 even 2 68.2.b.a.33.2 yes 2
17.16 even 2 inner 2448.2.c.d.577.2 2
24.5 odd 2 1088.2.b.f.577.2 2
24.11 even 2 1088.2.b.e.577.1 2
51.38 odd 4 4624.2.a.n.1.1 2
51.47 odd 4 4624.2.a.n.1.2 2
51.50 odd 2 272.2.b.c.33.2 2
60.23 odd 4 1700.2.g.a.849.1 4
60.47 odd 4 1700.2.g.a.849.3 4
60.59 even 2 1700.2.c.a.101.1 2
68.67 odd 2 612.2.b.a.577.2 2
84.83 odd 2 3332.2.b.a.2549.1 2
204.11 odd 16 1156.2.h.d.1001.1 8
204.23 odd 16 1156.2.h.d.1001.2 8
204.47 even 4 1156.2.a.c.1.1 2
204.59 even 8 1156.2.e.b.905.1 2
204.71 odd 16 1156.2.h.d.977.1 8
204.83 even 8 1156.2.e.a.829.1 2
204.95 odd 16 1156.2.h.d.733.1 8
204.107 odd 16 1156.2.h.d.757.1 8
204.131 odd 16 1156.2.h.d.757.2 8
204.143 odd 16 1156.2.h.d.733.2 8
204.155 even 8 1156.2.e.b.829.1 2
204.167 odd 16 1156.2.h.d.977.2 8
204.179 even 8 1156.2.e.a.905.1 2
204.191 even 4 1156.2.a.c.1.2 2
204.203 even 2 68.2.b.a.33.1 ✓ 2
408.101 odd 2 1088.2.b.f.577.1 2
408.203 even 2 1088.2.b.e.577.2 2
1020.203 odd 4 1700.2.g.a.849.4 4
1020.407 odd 4 1700.2.g.a.849.2 4
1020.1019 even 2 1700.2.c.a.101.2 2
1428.1427 odd 2 3332.2.b.a.2549.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
68.2.b.a.33.1 ✓ 2 204.203 even 2
68.2.b.a.33.2 yes 2 12.11 even 2
272.2.b.c.33.1 2 3.2 odd 2
272.2.b.c.33.2 2 51.50 odd 2
612.2.b.a.577.1 2 4.3 odd 2
612.2.b.a.577.2 2 68.67 odd 2
1088.2.b.e.577.1 2 24.11 even 2
1088.2.b.e.577.2 2 408.203 even 2
1088.2.b.f.577.1 2 408.101 odd 2
1088.2.b.f.577.2 2 24.5 odd 2
1156.2.a.c.1.1 2 204.47 even 4
1156.2.a.c.1.2 2 204.191 even 4
1156.2.e.a.829.1 2 204.83 even 8
1156.2.e.a.905.1 2 204.179 even 8
1156.2.e.b.829.1 2 204.155 even 8
1156.2.e.b.905.1 2 204.59 even 8
1156.2.h.d.733.1 8 204.95 odd 16
1156.2.h.d.733.2 8 204.143 odd 16
1156.2.h.d.757.1 8 204.107 odd 16
1156.2.h.d.757.2 8 204.131 odd 16
1156.2.h.d.977.1 8 204.71 odd 16
1156.2.h.d.977.2 8 204.167 odd 16
1156.2.h.d.1001.1 8 204.11 odd 16
1156.2.h.d.1001.2 8 204.23 odd 16
1700.2.c.a.101.1 2 60.59 even 2
1700.2.c.a.101.2 2 1020.1019 even 2
1700.2.g.a.849.1 4 60.23 odd 4
1700.2.g.a.849.2 4 1020.407 odd 4
1700.2.g.a.849.3 4 60.47 odd 4
1700.2.g.a.849.4 4 1020.203 odd 4
2448.2.c.d.577.1 2 1.1 even 1 trivial
2448.2.c.d.577.2 2 17.16 even 2 inner
3332.2.b.a.2549.1 2 84.83 odd 2
3332.2.b.a.2549.2 2 1428.1427 odd 2
4624.2.a.n.1.1 2 51.38 odd 4
4624.2.a.n.1.2 2 51.47 odd 4